DT2H2X---1.9.5.THM-Ref.s

%------------------------------------------------------------------------------
% File     : DT2H2X---1.9.5
% Problem  : ITP256^1 : TPTP v9.2.1. Bugfixed v9.2.1.
% Transfm  : none
% Format   : tptp:raw
% Command  : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300

% Computer : n019.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Sat Feb 28 09:12:28 AM UTC 2026

% Result   : Theorem 21.59s 7.92s
% Output   : Refutation 21.59s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :   13
%            Number of leaves      :   11
% Syntax   : Number of formulae    :   49 (  33 unt;   0 typ;   0 def)
%            Number of atoms       :  168 (  64 equ;   0 cnn)
%            Maximal formula atoms :    5 (   3 avg)
%            Number of connectives :  394 (  26   ~;   9   |;  10   &; 339   @)
%                                         (   4 <=>;   6  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   3 avg)
%            Number of types       :  174 ( 173 usr)
%            Number of type conns  :    0 (   0   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1399 (1396 usr;  78 con; 0-6 aty)
%            Number of variables   :   43 (  12   ^;  31   !;   0   ?;  43   :)

% Comments : 
%------------------------------------------------------------------------------
thf(type_def_5,type,
    rat: $tType ).

thf(type_def_6,type,
    int: $tType ).

thf(type_def_8,type,
    real: $tType ).

thf(type_def_9,type,
    set_Pr1261947904930325089at_nat: $tType ).

thf(type_def_10,type,
    nat: $tType ).

thf(type_def_11,type,
    product_prod_int_int: $tType ).

thf(type_def_12,type,
    product_prod_nat_nat: $tType ).

thf(type_def_13,type,
    complex: $tType ).

thf(type_def_14,type,
    num: $tType ).

thf(type_def_15,type,
    option_num: $tType ).

thf(type_def_16,type,
    product_prod_num_num: $tType ).

thf(type_def_17,type,
    code_integer: $tType ).

thf(type_def_18,type,
    produc6271795597528267376eger_o: $tType ).

thf(type_def_19,type,
    produc8923325533196201883nteger: $tType ).

thf(type_def_20,type,
    set_fi4554929511873752355omplex: $tType ).

thf(type_def_21,type,
    filter6041513312241820739omplex: $tType ).

thf(type_def_22,type,
    set_fi7789364187291644575l_real: $tType ).

thf(type_def_23,type,
    filter2146258269922977983l_real: $tType ).

thf(type_def_24,type,
    set_nat: $tType ).

thf(type_def_25,type,
    set_set_nat: $tType ).

thf(type_def_26,type,
    filter_real: $tType ).

thf(type_def_27,type,
    filter_nat: $tType ).

thf(type_def_28,type,
    filter1242075044329608583at_nat: $tType ).

thf(type_def_29,type,
    set_Pr5085853215250843933omplex: $tType ).

thf(type_def_30,type,
    set_Pr6218003697084177305l_real: $tType ).

thf(type_def_31,type,
    set_complex: $tType ).

thf(type_def_32,type,
    set_list_nat: $tType ).

thf(type_def_33,type,
    set_Product_unit: $tType ).

thf(type_def_34,type,
    set_o: $tType ).

thf(type_def_35,type,
    set_int: $tType ).

thf(type_def_36,type,
    set_list_o: $tType ).

thf(type_def_37,type,
    set_list_complex: $tType ).

thf(type_def_38,type,
    set_list_int: $tType ).

thf(type_def_39,type,
    set_li5450038453877631591at_nat: $tType ).

thf(type_def_40,type,
    set_list_VEBT_VEBT: $tType ).

thf(type_def_41,type,
    set_num: $tType ).

thf(type_def_42,type,
    set_Pr4329608150637261639at_nat: $tType ).

thf(type_def_43,type,
    set_rat: $tType ).

thf(type_def_44,type,
    set_real: $tType ).

thf(type_def_45,type,
    set_set_complex: $tType ).

thf(type_def_46,type,
    set_set_int: $tType ).

thf(type_def_47,type,
    set_se7855581050983116737at_nat: $tType ).

thf(type_def_48,type,
    set_VEBT_VEBT: $tType ).

thf(type_def_49,type,
    set_Pr8693737435421807431at_nat: $tType ).

thf(type_def_50,type,
    produc1319942482725812455at_nat: $tType ).

thf(type_def_51,type,
    list_nat: $tType ).

thf(type_def_52,type,
    extended_enat: $tType ).

thf(type_def_53,type,
    produc3843707927480180839at_nat: $tType ).

thf(type_def_54,type,
    vEBT_VEBT: $tType ).

thf(type_def_55,type,
    list_o: $tType ).

thf(type_def_56,type,
    produc8763457246119570046nteger: $tType ).

thf(type_def_57,type,
    produc7773217078559923341nt_int: $tType ).

thf(type_def_58,type,
    produc1908205239877642774nteger: $tType ).

thf(type_def_59,type,
    produc2285326912895808259nt_int: $tType ).

thf(type_def_60,type,
    list_int: $tType ).

thf(type_def_61,type,
    option_nat: $tType ).

thf(type_def_62,type,
    set_Pr958786334691620121nt_int: $tType ).

thf(type_def_63,type,
    list_complex: $tType ).

thf(type_def_64,type,
    list_P6011104703257516679at_nat: $tType ).

thf(type_def_65,type,
    list_real: $tType ).

thf(type_def_66,type,
    list_set_nat: $tType ).

thf(type_def_67,type,
    list_VEBT_VEBT: $tType ).

thf(type_def_68,type,
    list_P7333126701944960589_nat_o: $tType ).

thf(type_def_69,type,
    list_P3521021558325789923at_int: $tType ).

thf(type_def_70,type,
    list_P5647936690300460905T_VEBT: $tType ).

thf(type_def_71,type,
    option_o: $tType ).

thf(type_def_72,type,
    option_int: $tType ).

thf(type_def_73,type,
    list_num: $tType ).

thf(type_def_74,type,
    option4927543243414619207at_nat: $tType ).

thf(type_def_75,type,
    option_real: $tType ).

thf(type_def_76,type,
    option_set_nat: $tType ).

thf(type_def_77,type,
    option_VEBT_VEBT: $tType ).

thf(type_def_78,type,
    list_C878401137130745250e_term: $tType ).

thf(type_def_79,type,
    set_Co9149898834107579976e_term: $tType ).

thf(type_def_80,type,
    list_i8448526496819171953e_term: $tType ).

thf(type_def_81,type,
    set_in3461395444621081367e_term: $tType ).

thf(type_def_82,type,
    list_P1316552470764441098e_term: $tType ).

thf(type_def_83,type,
    set_Pr7604974323444597168e_term: $tType ).

thf(type_def_84,type,
    list_P1743416141875011707e_term: $tType ).

thf(type_def_85,type,
    set_Pr3642885161833720865e_term: $tType ).

thf(type_def_86,type,
    list_P5311841565141990158nteger: $tType ).

thf(type_def_87,type,
    set_Pr8056137968301705908nteger: $tType ).

thf(type_def_88,type,
    list_P8915022641806594461nt_int: $tType ).

thf(type_def_89,type,
    set_Pr1872883991513573699nt_int: $tType ).

thf(type_def_90,type,
    list_P7828571989066258726nteger: $tType ).

thf(type_def_91,type,
    set_Pr1281608226676607948nteger: $tType ).

thf(type_def_92,type,
    list_P651320350408439699nt_int: $tType ).

thf(type_def_93,type,
    set_Pr9222295170931077689nt_int: $tType ).

thf(type_def_94,type,
    list_P4002435161011370285od_o_o: $tType ).

thf(type_def_95,type,
    set_Product_prod_o_o: $tType ).

thf(type_def_96,type,
    list_P3795440434834930179_o_int: $tType ).

thf(type_def_97,type,
    set_Pr8834758594704517033_o_int: $tType ).

thf(type_def_98,type,
    list_P6285523579766656935_o_nat: $tType ).

thf(type_def_99,type,
    set_Pr2101469702781467981_o_nat: $tType ).

thf(type_def_100,type,
    list_P5232166724548748803o_real: $tType ).

thf(type_def_101,type,
    set_Pr6573716822653411497o_real: $tType ).

thf(type_def_102,type,
    list_P7495141550334521929T_VEBT: $tType ).

thf(type_def_103,type,
    set_Pr7543698050874017315T_VEBT: $tType ).

thf(type_def_104,type,
    list_P5578671422887162913nteger: $tType ).

thf(type_def_105,type,
    set_Pr4811707699266497531nteger: $tType ).

thf(type_def_106,type,
    list_P5707943133018811711nt_int: $tType ).

thf(type_def_107,type,
    list_P6863124054624500543t_real: $tType ).

thf(type_def_108,type,
    set_Pr3538720872664544793t_real: $tType ).

thf(type_def_109,type,
    list_P7524865323317820941T_VEBT: $tType ).

thf(type_def_110,type,
    set_Pr8044002425091019955T_VEBT: $tType ).

thf(type_def_111,type,
    list_P8469869581646625389at_nat: $tType ).

thf(type_def_112,type,
    list_P3595434254542482545real_o: $tType ).

thf(type_def_113,type,
    set_Pr4936984352647145239real_o: $tType ).

thf(type_def_114,type,
    list_P4344331454722006975al_int: $tType ).

thf(type_def_115,type,
    set_Pr1019928272762051225al_int: $tType ).

thf(type_def_116,type,
    list_P6834414599653733731al_nat: $tType ).

thf(type_def_117,type,
    set_Pr3510011417693777981al_nat: $tType ).

thf(type_def_118,type,
    list_P8689742595348180415l_real: $tType ).

thf(type_def_119,type,
    list_P877281246627933069T_VEBT: $tType ).

thf(type_def_120,type,
    set_Pr6019664923565264691T_VEBT: $tType ).

thf(type_def_121,type,
    list_P6254988961118846195et_nat: $tType ).

thf(type_def_122,type,
    set_Pr5488025237498180813et_nat: $tType ).

thf(type_def_123,type,
    list_P5464809261938338413at_nat: $tType ).

thf(type_def_124,type,
    list_P3126845725202233233VEBT_o: $tType ).

thf(type_def_125,type,
    set_Pr3175402225741728619VEBT_o: $tType ).

thf(type_def_126,type,
    list_P4547456442757143711BT_int: $tType ).

thf(type_def_127,type,
    set_Pr5066593544530342725BT_int: $tType ).

thf(type_def_128,type,
    list_P7037539587688870467BT_nat: $tType ).

thf(type_def_129,type,
    set_Pr7556676689462069481BT_nat: $tType ).

thf(type_def_130,type,
    list_P2623026923184700063T_real: $tType ).

thf(type_def_131,type,
    set_Pr7765410600122031685T_real: $tType ).

thf(type_def_132,type,
    list_P7413028617227757229T_VEBT: $tType ).

thf(type_def_133,type,
    set_Pr6192946355708809607T_VEBT: $tType ).

thf(type_def_134,type,
    option6357759511663192854e_term: $tType ).

thf(type_def_135,type,
    produc6241069584506657477e_term: $tType ).

thf(type_def_136,type,
    produc8551481072490612790e_term: $tType ).

thf(type_def_137,type,
    produc8243902056947475879T_VEBT: $tType ).

thf(type_def_138,type,
    set_Pr6227168374412355847list_o: $tType ).

thf(type_def_139,type,
    set_Pr765067013931698361st_int: $tType ).

thf(type_def_140,type,
    set_Pr3451248702717554689st_nat: $tType ).

thf(type_def_141,type,
    set_Pr1916528119006554503T_VEBT: $tType ).

thf(type_def_142,type,
    product_prod_o_o: $tType ).

thf(type_def_143,type,
    product_prod_o_int: $tType ).

thf(type_def_144,type,
    product_prod_o_nat: $tType ).

thf(type_def_145,type,
    produc2504756804600209347T_VEBT: $tType ).

thf(type_def_146,type,
    product_prod_nat_o: $tType ).

thf(type_def_147,type,
    product_prod_nat_int: $tType ).

thf(type_def_148,type,
    produc8025551001238799321T_VEBT: $tType ).

thf(type_def_149,type,
    produc334124729049499915VEBT_o: $tType ).

thf(type_def_150,type,
    produc4894624898956917775BT_int: $tType ).

thf(type_def_151,type,
    produc9072475918466114483BT_nat: $tType ).

thf(type_def_152,type,
    set_Code_integer: $tType ).

thf(type_def_153,type,
    produc4953844613479565601on_nat: $tType ).

thf(type_def_154,type,
    produc2233624965454879586on_nat: $tType ).

thf(type_def_155,type,
    produc8306885398267862888on_nat: $tType ).

thf(type_def_156,type,
    produc3447558737645232053on_num: $tType ).

thf(type_def_157,type,
    produc7036089656553540234on_num: $tType ).

thf(type_def_158,type,
    produc1193250871479095198on_num: $tType ).

thf(type_def_159,type,
    produc6121120109295599847at_nat: $tType ).

thf(type_def_160,type,
    produc5491161045314408544at_nat: $tType ).

thf(type_def_161,type,
    produc5542196010084753463at_nat: $tType ).

thf(type_def_162,type,
    product_prod_o_real: $tType ).

thf(type_def_163,type,
    produc679980390762269497t_real: $tType ).

thf(type_def_164,type,
    produc1531783533982839933T_VEBT: $tType ).

thf(type_def_165,type,
    produc7102631898165422375list_o: $tType ).

thf(type_def_166,type,
    produc1186641810826059865st_int: $tType ).

thf(type_def_167,type,
    produc1828647624359046049st_nat: $tType ).

thf(type_def_168,type,
    produc9211091688327510695T_VEBT: $tType ).

thf(type_def_169,type,
    product_prod_nat_num: $tType ).

thf(type_def_170,type,
    produc859450856879609959at_nat: $tType ).

thf(type_def_171,type,
    product_prod_real_o: $tType ).

thf(type_def_172,type,
    produc8786904178792722361al_int: $tType ).

thf(type_def_173,type,
    produc3741383161447143261al_nat: $tType ).

thf(type_def_174,type,
    produc2422161461964618553l_real: $tType ).

thf(type_def_175,type,
    produc3757001726724277373T_VEBT: $tType ).

thf(type_def_176,type,
    produc7819656566062154093et_nat: $tType ).

thf(type_def_177,type,
    produc5170161368751668367T_real: $tType ).

thf(type_def_178,type,
    produc4411394909380815293omplex: $tType ).

thf(func_def_0,type,
    produc1319942482725812455at_nat: $tType ).

thf(func_def_1,type,
    produc5542196010084753463at_nat: $tType ).

thf(func_def_2,type,
    list_P7828571989066258726nteger: $tType ).

thf(func_def_3,type,
    set_Pr1281608226676607948nteger: $tType ).

thf(func_def_4,type,
    produc1908205239877642774nteger: $tType ).

thf(func_def_5,type,
    produc5491161045314408544at_nat: $tType ).

thf(func_def_6,type,
    list_P651320350408439699nt_int: $tType ).

thf(func_def_7,type,
    set_Pr9222295170931077689nt_int: $tType ).

thf(func_def_8,type,
    produc2285326912895808259nt_int: $tType ).

thf(func_def_9,type,
    list_P5311841565141990158nteger: $tType ).

thf(func_def_10,type,
    set_Pr8056137968301705908nteger: $tType ).

thf(func_def_11,type,
    produc8763457246119570046nteger: $tType ).

thf(func_def_12,type,
    list_P1316552470764441098e_term: $tType ).

thf(func_def_13,type,
    set_Pr7604974323444597168e_term: $tType ).

thf(func_def_14,type,
    list_P1743416141875011707e_term: $tType ).

thf(func_def_15,type,
    set_Pr3642885161833720865e_term: $tType ).

thf(func_def_16,type,
    list_P8915022641806594461nt_int: $tType ).

thf(func_def_17,type,
    set_Pr1872883991513573699nt_int: $tType ).

thf(func_def_18,type,
    produc7773217078559923341nt_int: $tType ).

thf(func_def_19,type,
    list_P5464809261938338413at_nat: $tType ).

thf(func_def_20,type,
    set_Pr4329608150637261639at_nat: $tType ).

thf(func_def_21,type,
    produc1193250871479095198on_num: $tType ).

thf(func_def_22,type,
    produc8306885398267862888on_nat: $tType ).

thf(func_def_23,type,
    produc6121120109295599847at_nat: $tType ).

thf(func_def_24,type,
    produc7036089656553540234on_num: $tType ).

thf(func_def_25,type,
    produc2233624965454879586on_nat: $tType ).

thf(func_def_26,type,
    produc3843707927480180839at_nat: $tType ).

thf(func_def_27,type,
    list_P8469869581646625389at_nat: $tType ).

thf(func_def_28,type,
    list_C878401137130745250e_term: $tType ).

thf(func_def_29,type,
    set_Pr8693737435421807431at_nat: $tType ).

thf(func_def_30,type,
    set_Co9149898834107579976e_term: $tType ).

thf(func_def_31,type,
    list_i8448526496819171953e_term: $tType ).

thf(func_def_32,type,
    set_Pr1916528119006554503T_VEBT: $tType ).

thf(func_def_33,type,
    produc859450856879609959at_nat: $tType ).

thf(func_def_34,type,
    set_in3461395444621081367e_term: $tType ).

thf(func_def_35,type,
    produc9211091688327510695T_VEBT: $tType ).

thf(func_def_36,type,
    produc6241069584506657477e_term: $tType ).

thf(func_def_37,type,
    set_fi4554929511873752355omplex: $tType ).

thf(func_def_38,type,
    set_Pr3451248702717554689st_nat: $tType ).

thf(func_def_39,type,
    set_Pr765067013931698361st_int: $tType ).

thf(func_def_40,type,
    list_P7413028617227757229T_VEBT: $tType ).

thf(func_def_41,type,
    produc8551481072490612790e_term: $tType ).

thf(func_def_42,type,
    set_Pr6192946355708809607T_VEBT: $tType ).

thf(func_def_43,type,
    list_P6254988961118846195et_nat: $tType ).

thf(func_def_44,type,
    list_P5578671422887162913nteger: $tType ).

thf(func_def_45,type,
    set_Pr5488025237498180813et_nat: $tType ).

thf(func_def_46,type,
    set_Pr4811707699266497531nteger: $tType ).

thf(func_def_47,type,
    option6357759511663192854e_term: $tType ).

thf(func_def_48,type,
    produc3447558737645232053on_num: $tType ).

thf(func_def_49,type,
    produc4953844613479565601on_nat: $tType ).

thf(func_def_50,type,
    set_fi7789364187291644575l_real: $tType ).

thf(func_def_51,type,
    filter6041513312241820739omplex: $tType ).

thf(func_def_52,type,
    list_P2623026923184700063T_real: $tType ).

thf(func_def_53,type,
    list_P877281246627933069T_VEBT: $tType ).

thf(func_def_54,type,
    produc1828647624359046049st_nat: $tType ).

thf(func_def_55,type,
    produc1186641810826059865st_int: $tType ).

thf(func_def_56,type,
    set_Pr7765410600122031685T_real: $tType ).

thf(func_def_57,type,
    set_Pr6019664923565264691T_VEBT: $tType ).

thf(func_def_58,type,
    list_P7037539587688870467BT_nat: $tType ).

thf(func_def_59,type,
    list_P4547456442757143711BT_int: $tType ).

thf(func_def_60,type,
    list_P5647936690300460905T_VEBT: $tType ).

thf(func_def_61,type,
    list_P7524865323317820941T_VEBT: $tType ).

thf(func_def_62,type,
    set_Pr6227168374412355847list_o: $tType ).

thf(func_def_63,type,
    set_li5450038453877631591at_nat: $tType ).

thf(func_def_64,type,
    produc8243902056947475879T_VEBT: $tType ).

thf(func_def_65,type,
    set_Pr7556676689462069481BT_nat: $tType ).

thf(func_def_66,type,
    set_Pr5066593544530342725BT_int: $tType ).

thf(func_def_67,type,
    set_Pr8044002425091019955T_VEBT: $tType ).

thf(func_def_68,type,
    set_se7855581050983116737at_nat: $tType ).

thf(func_def_69,type,
    set_Pr5085853215250843933omplex: $tType ).

thf(func_def_70,type,
    produc7819656566062154093et_nat: $tType ).

thf(func_def_71,type,
    produc8923325533196201883nteger: $tType ).

thf(func_def_72,type,
    list_P3126845725202233233VEBT_o: $tType ).

thf(func_def_73,type,
    list_P7495141550334521929T_VEBT: $tType ).

thf(func_def_74,type,
    filter2146258269922977983l_real: $tType ).

thf(func_def_75,type,
    set_Pr3175402225741728619VEBT_o: $tType ).

thf(func_def_76,type,
    set_Pr7543698050874017315T_VEBT: $tType ).

thf(func_def_77,type,
    option4927543243414619207at_nat: $tType ).

thf(func_def_78,type,
    list_P8689742595348180415l_real: $tType ).

thf(func_def_79,type,
    filter1242075044329608583at_nat: $tType ).

thf(func_def_80,type,
    set_Pr6218003697084177305l_real: $tType ).

thf(func_def_81,type,
    list_P6834414599653733731al_nat: $tType ).

thf(func_def_82,type,
    list_P4344331454722006975al_int: $tType ).

thf(func_def_83,type,
    list_P6863124054624500543t_real: $tType ).

thf(func_def_84,type,
    produc5170161368751668367T_real: $tType ).

thf(func_def_85,type,
    produc3757001726724277373T_VEBT: $tType ).

thf(func_def_86,type,
    set_Pr3510011417693777981al_nat: $tType ).

thf(func_def_87,type,
    set_Pr1019928272762051225al_int: $tType ).

thf(func_def_88,type,
    set_Pr3538720872664544793t_real: $tType ).

thf(func_def_89,type,
    produc7102631898165422375list_o: $tType ).

thf(func_def_90,type,
    list_P6011104703257516679at_nat: $tType ).

thf(func_def_91,type,
    list_P3521021558325789923at_int: $tType ).

thf(func_def_92,type,
    list_P5707943133018811711nt_int: $tType ).

thf(func_def_93,type,
    produc9072475918466114483BT_nat: $tType ).

thf(func_def_94,type,
    produc4894624898956917775BT_int: $tType ).

thf(func_def_95,type,
    produc8025551001238799321T_VEBT: $tType ).

thf(func_def_96,type,
    produc1531783533982839933T_VEBT: $tType ).

thf(func_def_97,type,
    set_Pr1261947904930325089at_nat: $tType ).

thf(func_def_98,type,
    set_Pr958786334691620121nt_int: $tType ).

thf(func_def_99,type,
    produc4411394909380815293omplex: $tType ).

thf(func_def_100,type,
    list_P3595434254542482545real_o: $tType ).

thf(func_def_101,type,
    list_P5232166724548748803o_real: $tType ).

thf(func_def_102,type,
    set_Pr4936984352647145239real_o: $tType ).

thf(func_def_103,type,
    set_Pr6573716822653411497o_real: $tType ).

thf(func_def_104,type,
    list_P7333126701944960589_nat_o: $tType ).

thf(func_def_105,type,
    list_P6285523579766656935_o_nat: $tType ).

thf(func_def_106,type,
    list_P3795440434834930179_o_int: $tType ).

thf(func_def_107,type,
    set_list_VEBT_VEBT: $tType ).

thf(func_def_108,type,
    produc334124729049499915VEBT_o: $tType ).

thf(func_def_109,type,
    produc2504756804600209347T_VEBT: $tType ).

thf(func_def_110,type,
    set_Pr2101469702781467981_o_nat: $tType ).

thf(func_def_111,type,
    set_Pr8834758594704517033_o_int: $tType ).

thf(func_def_112,type,
    produc6271795597528267376eger_o: $tType ).

thf(func_def_113,type,
    produc2422161461964618553l_real: $tType ).

thf(func_def_114,type,
    produc3741383161447143261al_nat: $tType ).

thf(func_def_115,type,
    produc8786904178792722361al_int: $tType ).

thf(func_def_116,type,
    produc679980390762269497t_real: $tType ).

thf(func_def_117,type,
    product_prod_num_num: $tType ).

thf(func_def_118,type,
    product_prod_nat_num: $tType ).

thf(func_def_119,type,
    product_prod_nat_nat: $tType ).

thf(func_def_120,type,
    product_prod_nat_int: $tType ).

thf(func_def_121,type,
    product_prod_int_int: $tType ).

thf(func_def_122,type,
    list_P4002435161011370285od_o_o: $tType ).

thf(func_def_123,type,
    set_list_complex: $tType ).

thf(func_def_124,type,
    set_Product_prod_o_o: $tType ).

thf(func_def_125,type,
    set_set_complex: $tType ).

thf(func_def_126,type,
    option_VEBT_VEBT: $tType ).

thf(func_def_127,type,
    option_set_nat: $tType ).

thf(func_def_128,type,
    product_prod_real_o: $tType ).

thf(func_def_129,type,
    product_prod_o_real: $tType ).

thf(func_def_130,type,
    list_VEBT_VEBT: $tType ).

thf(func_def_131,type,
    set_list_nat: $tType ).

thf(func_def_132,type,
    set_list_int: $tType ).

thf(func_def_133,type,
    product_prod_nat_o: $tType ).

thf(func_def_134,type,
    product_prod_o_nat: $tType ).

thf(func_def_135,type,
    product_prod_o_int: $tType ).

thf(func_def_136,type,
    list_set_nat: $tType ).

thf(func_def_137,type,
    set_VEBT_VEBT: $tType ).

thf(func_def_138,type,
    set_set_nat: $tType ).

thf(func_def_139,type,
    set_set_int: $tType ).

thf(func_def_140,type,
    set_Code_integer: $tType ).

thf(func_def_141,type,
    set_Product_unit: $tType ).

thf(func_def_142,type,
    list_complex: $tType ).

thf(func_def_143,type,
    set_list_o: $tType ).

thf(func_def_144,type,
    product_prod_o_o: $tType ).

thf(func_def_145,type,
    set_complex: $tType ).

thf(func_def_146,type,
    option_real: $tType ).

thf(func_def_147,type,
    filter_real: $tType ).

thf(func_def_148,type,
    option_num: $tType ).

thf(func_def_149,type,
    option_nat: $tType ).

thf(func_def_150,type,
    option_int: $tType ).

thf(func_def_151,type,
    filter_nat: $tType ).

thf(func_def_152,type,
    list_real: $tType ).

thf(func_def_153,type,
    set_real: $tType ).

thf(func_def_154,type,
    list_num: $tType ).

thf(func_def_155,type,
    list_nat: $tType ).

thf(func_def_156,type,
    list_int: $tType ).

thf(func_def_157,type,
    vEBT_VEBT: $tType ).

thf(func_def_158,type,
    set_rat: $tType ).

thf(func_def_159,type,
    set_num: $tType ).

thf(func_def_160,type,
    set_nat: $tType ).

thf(func_def_161,type,
    set_int: $tType ).

thf(func_def_162,type,
    code_integer: $tType ).

thf(func_def_163,type,
    option_o: $tType ).

thf(func_def_164,type,
    extended_enat: $tType ).

thf(func_def_165,type,
    list_o: $tType ).

thf(func_def_166,type,
    complex: $tType ).

thf(func_def_167,type,
    set_o: $tType ).

thf(func_def_168,type,
    real: $tType ).

thf(func_def_169,type,
    rat: $tType ).

thf(func_def_170,type,
    num: $tType ).

thf(func_def_171,type,
    nat: $tType ).

thf(func_def_172,type,
    int: $tType ).

thf(func_def_173,type,
    archim2889992004027027881ng_rat: rat > int ).

thf(func_def_174,type,
    archim7802044766580827645g_real: real > int ).

thf(func_def_175,type,
    archim3151403230148437115or_rat: rat > int ).

thf(func_def_176,type,
    archim6058952711729229775r_real: real > int ).

thf(func_def_177,type,
    archimedean_frac_rat: rat > rat ).

thf(func_def_178,type,
    archim2898591450579166408c_real: real > real ).

thf(func_def_179,type,
    archim7778729529865785530nd_rat: rat > int ).

thf(func_def_180,type,
    archim8280529875227126926d_real: real > int ).

thf(func_def_181,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

thf(func_def_182,type,
    bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).

thf(func_def_183,type,
    bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).

thf(func_def_184,type,
    bNF_re4297313714947099218al_o_o: ( ( nat > rat ) > real > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( real > $o ) > $o ).

thf(func_def_185,type,
    bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).

thf(func_def_186,type,
    bNF_re157797125943740599nt_int: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > product_prod_int_int ) > $o ).

thf(func_def_187,type,
    bNF_re3461391660133120880nt_rat: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > rat ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > rat ) > $o ).

thf(func_def_188,type,
    bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).

thf(func_def_189,type,
    bNF_re6250860962936578807nt_int: ( int > int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ).

thf(func_def_190,type,
    bNF_re2214769303045360666nt_rat: ( int > int > $o ) > ( product_prod_int_int > rat > $o ) > ( int > product_prod_int_int ) > ( int > rat ) > $o ).

thf(func_def_191,type,
    bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(func_def_192,type,
    bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).

thf(func_def_193,type,
    bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $o ).

thf(func_def_194,type,
    bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).

thf(func_def_195,type,
    bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).

thf(func_def_196,type,
    bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).

thf(func_def_197,type,
    bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).

thf(func_def_198,type,
    bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).

thf(func_def_199,type,
    bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).

thf(func_def_200,type,
    bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).

thf(func_def_201,type,
    bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).

thf(func_def_202,type,
    bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).

thf(func_def_203,type,
    bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).

thf(func_def_204,type,
    bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).

thf(func_def_205,type,
    bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).

thf(func_def_206,type,
    bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

thf(func_def_207,type,
    bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).

thf(func_def_208,type,
    bNF_We3818239936649020644el_nat: set_Pr1261947904930325089at_nat > $o ).

thf(func_def_209,type,
    binomial: nat > nat > nat ).

thf(func_def_210,type,
    gbinomial_complex: complex > nat > complex ).

thf(func_def_211,type,
    gbinomial_int: int > nat > int ).

thf(func_def_212,type,
    gbinomial_nat: nat > nat > nat ).

thf(func_def_213,type,
    gbinomial_rat: rat > nat > rat ).

thf(func_def_214,type,
    gbinomial_real: real > nat > real ).

thf(func_def_215,type,
    bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).

thf(func_def_216,type,
    bit_and_not_num: num > num > option_num ).

thf(func_def_217,type,
    bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(func_def_218,type,
    bit_concat_bit: nat > int > int > int ).

thf(func_def_219,type,
    bit_ri7919022796975470100ot_int: int > int ).

thf(func_def_220,type,
    bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).

thf(func_def_221,type,
    bit_ri631733984087533419it_int: nat > int > int ).

thf(func_def_222,type,
    bit_se3949692690581998587nteger: code_integer > code_integer > code_integer ).

thf(func_def_223,type,
    bit_se725231765392027082nd_int: int > int > int ).

thf(func_def_224,type,
    bit_se727722235901077358nd_nat: nat > nat > nat ).

thf(func_def_225,type,
    bit_se8568078237143864401it_int: nat > int > int ).

thf(func_def_226,type,
    bit_se8570568707652914677it_nat: nat > nat > nat ).

thf(func_def_227,type,
    bit_se1345352211410354436nteger: nat > code_integer > code_integer ).

thf(func_def_228,type,
    bit_se2159334234014336723it_int: nat > int > int ).

thf(func_def_229,type,
    bit_se2161824704523386999it_nat: nat > nat > nat ).

thf(func_def_230,type,
    bit_se2000444600071755411sk_int: nat > int ).

thf(func_def_231,type,
    bit_se2002935070580805687sk_nat: nat > nat ).

thf(func_def_232,type,
    bit_se1080825931792720795nteger: code_integer > code_integer > code_integer ).

thf(func_def_233,type,
    bit_se1409905431419307370or_int: int > int > int ).

thf(func_def_234,type,
    bit_se1412395901928357646or_nat: nat > nat > nat ).

thf(func_def_235,type,
    bit_se545348938243370406it_int: nat > int > int ).

thf(func_def_236,type,
    bit_se547839408752420682it_nat: nat > nat > nat ).

thf(func_def_237,type,
    bit_se2793503036327961859nteger: nat > code_integer > code_integer ).

thf(func_def_238,type,
    bit_se7879613467334960850it_int: nat > int > int ).

thf(func_def_239,type,
    bit_se7882103937844011126it_nat: nat > nat > nat ).

thf(func_def_240,type,
    bit_se1745604003318907178nteger: nat > code_integer > code_integer ).

thf(func_def_241,type,
    bit_se2923211474154528505it_int: nat > int > int ).

thf(func_def_242,type,
    bit_se2925701944663578781it_nat: nat > nat > nat ).

thf(func_def_243,type,
    bit_se8260200283734997820nteger: nat > code_integer > code_integer ).

thf(func_def_244,type,
    bit_se4203085406695923979it_int: nat > int > int ).

thf(func_def_245,type,
    bit_se4205575877204974255it_nat: nat > nat > nat ).

thf(func_def_246,type,
    bit_se6526347334894502574or_int: int > int > int ).

thf(func_def_247,type,
    bit_se6528837805403552850or_nat: nat > nat > nat ).

thf(func_def_248,type,
    bit_se1146084159140164899it_int: int > nat > $o ).

thf(func_def_249,type,
    bit_se1148574629649215175it_nat: nat > nat > $o ).

thf(func_def_250,type,
    bit_take_bit_num: nat > num > option_num ).

thf(func_def_251,type,
    bit_un7362597486090784418nd_num: num > num > option_num ).

thf(func_def_252,type,
    bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(func_def_253,type,
    bit_un2480387367778600638or_num: num > num > option_num ).

thf(func_def_254,type,
    bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(func_def_255,type,
    code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).

thf(func_def_256,type,
    code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).

thf(func_def_257,type,
    code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).

thf(func_def_258,type,
    code_int_of_integer: code_integer > int ).

thf(func_def_259,type,
    code_integer_of_int: int > code_integer ).

thf(func_def_260,type,
    code_integer_of_nat: nat > code_integer ).

thf(func_def_261,type,
    code_nat_of_integer: code_integer > nat ).

thf(func_def_262,type,
    comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).

thf(func_def_263,type,
    comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).

thf(func_def_264,type,
    complete_Inf_Inf_nat: set_nat > nat ).

thf(func_def_265,type,
    comple7806235888213564991et_nat: set_set_nat > set_nat ).

thf(func_def_266,type,
    complete_Sup_Sup_nat: set_nat > nat ).

thf(func_def_267,type,
    arg: complex > real ).

thf(func_def_268,type,
    cis: real > complex ).

thf(func_def_269,type,
    complex2: real > real > complex ).

thf(func_def_270,type,
    condit2214826472909112428ve_nat: set_nat > $o ).

thf(func_def_271,type,
    differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).

thf(func_def_272,type,
    has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).

thf(func_def_273,type,
    has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).

thf(func_def_274,type,
    adjust_div: product_prod_int_int > int ).

thf(func_def_275,type,
    divmod_nat: nat > nat > product_prod_nat_nat ).

thf(func_def_276,type,
    eucl_rel_int: int > int > product_prod_int_int > $o ).

thf(func_def_277,type,
    unique6319869463603278526ux_int: product_prod_int_int > $o ).

thf(func_def_278,type,
    unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).

thf(func_def_279,type,
    unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).

thf(func_def_280,type,
    unique5052692396658037445od_int: num > num > product_prod_int_int ).

thf(func_def_281,type,
    unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).

thf(func_def_282,type,
    unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_283,type,
    unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).

thf(func_def_284,type,
    unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).

thf(func_def_285,type,
    comm_s8582702949713902594nteger: code_integer > nat > code_integer ).

thf(func_def_286,type,
    comm_s2602460028002588243omplex: complex > nat > complex ).

thf(func_def_287,type,
    comm_s4660882817536571857er_int: int > nat > int ).

thf(func_def_288,type,
    comm_s4663373288045622133er_nat: nat > nat > nat ).

thf(func_def_289,type,
    comm_s4028243227959126397er_rat: rat > nat > rat ).

thf(func_def_290,type,
    comm_s7457072308508201937r_real: real > nat > real ).

thf(func_def_291,type,
    semiri3624122377584611663nteger: nat > code_integer ).

thf(func_def_292,type,
    semiri5044797733671781792omplex: nat > complex ).

thf(func_def_293,type,
    semiri1406184849735516958ct_int: nat > int ).

thf(func_def_294,type,
    semiri1408675320244567234ct_nat: nat > nat ).

thf(func_def_295,type,
    semiri773545260158071498ct_rat: nat > rat ).

thf(func_def_296,type,
    semiri2265585572941072030t_real: nat > real ).

thf(func_def_297,type,
    invers8013647133539491842omplex: complex > complex ).

thf(func_def_298,type,
    inverse_inverse_rat: rat > rat ).

thf(func_def_299,type,
    inverse_inverse_real: real > real ).

thf(func_def_300,type,
    at_bot_real: filter_real ).

thf(func_def_301,type,
    at_top_nat: filter_nat ).

thf(func_def_302,type,
    at_top_real: filter_real ).

thf(func_def_303,type,
    eventually_nat: ( nat > $o ) > filter_nat > $o ).

thf(func_def_304,type,
    eventu1038000079068216329at_nat: ( product_prod_nat_nat > $o ) > filter1242075044329608583at_nat > $o ).

thf(func_def_305,type,
    eventually_real: ( real > $o ) > filter_real > $o ).

thf(func_def_306,type,
    filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).

thf(func_def_307,type,
    filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).

thf(func_def_308,type,
    filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).

thf(func_def_309,type,
    princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex ).

thf(func_def_310,type,
    princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).

thf(func_def_311,type,
    prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat ).

thf(func_def_312,type,
    finite_card_complex: set_complex > nat ).

thf(func_def_313,type,
    finite_card_list_nat: set_list_nat > nat ).

thf(func_def_314,type,
    finite_card_nat: set_nat > nat ).

thf(func_def_315,type,
    finite410649719033368117t_unit: set_Product_unit > nat ).

thf(func_def_316,type,
    finite_card_set_nat: set_set_nat > nat ).

thf(func_def_317,type,
    finite_finite_o: set_o > $o ).

thf(func_def_318,type,
    finite3207457112153483333omplex: set_complex > $o ).

thf(func_def_319,type,
    finite_finite_int: set_int > $o ).

thf(func_def_320,type,
    finite_finite_list_o: set_list_o > $o ).

thf(func_def_321,type,
    finite8712137658972009173omplex: set_list_complex > $o ).

thf(func_def_322,type,
    finite3922522038869484883st_int: set_list_int > $o ).

thf(func_def_323,type,
    finite8100373058378681591st_nat: set_list_nat > $o ).

thf(func_def_324,type,
    finite500796754983035824at_nat: set_li5450038453877631591at_nat > $o ).

thf(func_def_325,type,
    finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o ).

thf(func_def_326,type,
    finite_finite_nat: set_nat > $o ).

thf(func_def_327,type,
    finite_finite_num: set_num > $o ).

thf(func_def_328,type,
    finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o ).

thf(func_def_329,type,
    finite4343798906461161616at_nat: set_Pr4329608150637261639at_nat > $o ).

thf(func_def_330,type,
    finite_finite_rat: set_rat > $o ).

thf(func_def_331,type,
    finite_finite_real: set_real > $o ).

thf(func_def_332,type,
    finite6551019134538273531omplex: set_set_complex > $o ).

thf(func_def_333,type,
    finite6197958912794628473et_int: set_set_int > $o ).

thf(func_def_334,type,
    finite1152437895449049373et_nat: set_set_nat > $o ).

thf(func_def_335,type,
    finite9047747110432174090at_nat: set_se7855581050983116737at_nat > $o ).

thf(func_def_336,type,
    finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o ).

thf(func_def_337,type,
    bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).

thf(func_def_338,type,
    bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).

thf(func_def_339,type,
    bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).

thf(func_def_340,type,
    comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_341,type,
    comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_342,type,
    comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).

thf(func_def_343,type,
    comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).

thf(func_def_344,type,
    inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).

thf(func_def_345,type,
    inj_on_real_real: ( real > real ) > set_real > $o ).

thf(func_def_346,type,
    inj_on_set_nat_nat: ( set_nat > nat ) > set_set_nat > $o ).

thf(func_def_347,type,
    map_fu4960017516451851995nt_int: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > int > int > int ).

thf(func_def_348,type,
    map_fu3667384564859982768at_int: ( int > product_prod_nat_nat ) > ( product_prod_nat_nat > int ) > ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ).

thf(func_def_349,type,
    map_fu4333342158222067775at_rat: ( rat > product_prod_int_int ) > ( ( product_prod_int_int > product_prod_int_int ) > rat > rat ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > rat > rat > rat ).

thf(func_def_350,type,
    map_fu5673905371560938248nt_rat: ( rat > product_prod_int_int ) > ( product_prod_int_int > rat ) > ( product_prod_int_int > product_prod_int_int ) > rat > rat ).

thf(func_def_351,type,
    strict1292158309912662752at_nat: ( nat > nat ) > set_nat > $o ).

thf(func_def_352,type,
    the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).

thf(func_def_353,type,
    fun_max_strict: set_Pr4329608150637261639at_nat ).

thf(func_def_354,type,
    fun_max_weak: set_Pr4329608150637261639at_nat ).

thf(func_def_355,type,
    fun_min_strict: set_Pr4329608150637261639at_nat ).

thf(func_def_356,type,
    fun_min_weak: set_Pr4329608150637261639at_nat ).

thf(func_def_357,type,
    fun_pair_leq: set_Pr8693737435421807431at_nat ).

thf(func_def_358,type,
    fun_pair_less: set_Pr8693737435421807431at_nat ).

thf(func_def_359,type,
    fun_re2478310338295953701at_nat: produc1319942482725812455at_nat > $o ).

thf(func_def_360,type,
    gcd_Gcd_nat: set_nat > nat ).

thf(func_def_361,type,
    bezw: nat > nat > product_prod_int_int ).

thf(func_def_362,type,
    bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(func_def_363,type,
    gcd_gcd_int: int > int > int ).

thf(func_def_364,type,
    gcd_gcd_nat: nat > nat > nat ).

thf(func_def_365,type,
    gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(func_def_366,type,
    abs_abs_Code_integer: code_integer > code_integer ).

thf(func_def_367,type,
    abs_abs_complex: complex > complex ).

thf(func_def_368,type,
    abs_abs_int: int > int ).

thf(func_def_369,type,
    abs_abs_rat: rat > rat ).

thf(func_def_370,type,
    abs_abs_real: real > real ).

thf(func_def_371,type,
    minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).

thf(func_def_372,type,
    minus_1139252259498527702_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).

thf(func_def_373,type,
    minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).

thf(func_def_374,type,
    minus_2270307095948843157_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(func_def_375,type,
    minus_minus_real_o: ( real > $o ) > ( real > $o ) > real > $o ).

thf(func_def_376,type,
    minus_6910147592129066416_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).

thf(func_def_377,type,
    minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).

thf(func_def_378,type,
    minus_minus_complex: complex > complex > complex ).

thf(func_def_379,type,
    minus_minus_int: int > int > int ).

thf(func_def_380,type,
    minus_minus_nat: nat > nat > nat ).

thf(func_def_381,type,
    minus_minus_rat: rat > rat > rat ).

thf(func_def_382,type,
    minus_minus_real: real > real > real ).

thf(func_def_383,type,
    minus_811609699411566653omplex: set_complex > set_complex > set_complex ).

thf(func_def_384,type,
    minus_minus_set_int: set_int > set_int > set_int ).

thf(func_def_385,type,
    minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).

thf(func_def_386,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(func_def_387,type,
    minus_minus_set_num: set_num > set_num > set_num ).

thf(func_def_388,type,
    minus_1356011639430497352at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(func_def_389,type,
    minus_3314409938677909166at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

thf(func_def_390,type,
    minus_minus_set_rat: set_rat > set_rat > set_rat ).

thf(func_def_391,type,
    minus_minus_set_real: set_real > set_real > set_real ).

thf(func_def_392,type,
    minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).

thf(func_def_393,type,
    one_one_Code_integer: code_integer ).

thf(func_def_394,type,
    one_one_complex: complex ).

thf(func_def_395,type,
    one_on7984719198319812577d_enat: extended_enat ).

thf(func_def_396,type,
    one_one_int: int ).

thf(func_def_397,type,
    one_one_nat: nat ).

thf(func_def_398,type,
    one_one_rat: rat ).

thf(func_def_399,type,
    one_one_real: real ).

thf(func_def_400,type,
    plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).

thf(func_def_401,type,
    plus_plus_complex: complex > complex > complex ).

thf(func_def_402,type,
    plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).

thf(func_def_403,type,
    plus_plus_int: int > int > int ).

thf(func_def_404,type,
    plus_plus_nat: nat > nat > nat ).

thf(func_def_405,type,
    plus_plus_num: num > num > num ).

thf(func_def_406,type,
    plus_plus_rat: rat > rat > rat ).

thf(func_def_407,type,
    plus_plus_real: real > real > real ).

thf(func_def_408,type,
    sgn_sgn_Code_integer: code_integer > code_integer ).

thf(func_def_409,type,
    sgn_sgn_complex: complex > complex ).

thf(func_def_410,type,
    sgn_sgn_int: int > int ).

thf(func_def_411,type,
    sgn_sgn_rat: rat > rat ).

thf(func_def_412,type,
    sgn_sgn_real: real > real ).

thf(func_def_413,type,
    times_3573771949741848930nteger: code_integer > code_integer > code_integer ).

thf(func_def_414,type,
    times_times_complex: complex > complex > complex ).

thf(func_def_415,type,
    times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).

thf(func_def_416,type,
    times_times_int: int > int > int ).

thf(func_def_417,type,
    times_times_nat: nat > nat > nat ).

thf(func_def_418,type,
    times_times_rat: rat > rat > rat ).

thf(func_def_419,type,
    times_times_real: real > real > real ).

thf(func_def_420,type,
    uminus_uminus_int_o: ( int > $o ) > int > $o ).

thf(func_def_421,type,
    uminus5770388063884162150_nat_o: ( list_nat > $o ) > list_nat > $o ).

thf(func_def_422,type,
    uminus_uminus_nat_o: ( nat > $o ) > nat > $o ).

thf(func_def_423,type,
    uminus8676089048583255045_nat_o: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(func_def_424,type,
    uminus_uminus_real_o: ( real > $o ) > real > $o ).

thf(func_def_425,type,
    uminus6401447641752708672_nat_o: ( set_nat > $o ) > set_nat > $o ).

thf(func_def_426,type,
    uminus1351360451143612070nteger: code_integer > code_integer ).

thf(func_def_427,type,
    uminus1482373934393186551omplex: complex > complex ).

thf(func_def_428,type,
    uminus_uminus_int: int > int ).

thf(func_def_429,type,
    uminus_uminus_rat: rat > rat ).

thf(func_def_430,type,
    uminus_uminus_real: real > real ).

thf(func_def_431,type,
    uminus1532241313380277803et_int: set_int > set_int ).

thf(func_def_432,type,
    uminus3195874150345416415st_nat: set_list_nat > set_list_nat ).

thf(func_def_433,type,
    uminus5710092332889474511et_nat: set_nat > set_nat ).

thf(func_def_434,type,
    uminus6524753893492686040at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(func_def_435,type,
    uminus935396558254630718at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

thf(func_def_436,type,
    uminus612125837232591019t_real: set_real > set_real ).

thf(func_def_437,type,
    uminus613421341184616069et_nat: set_set_nat > set_set_nat ).

thf(func_def_438,type,
    zero_z3403309356797280102nteger: code_integer ).

thf(func_def_439,type,
    zero_zero_complex: complex ).

thf(func_def_440,type,
    zero_z5237406670263579293d_enat: extended_enat ).

thf(func_def_441,type,
    zero_zero_int: int ).

thf(func_def_442,type,
    zero_zero_nat: nat ).

thf(func_def_443,type,
    zero_zero_rat: rat ).

thf(func_def_444,type,
    zero_zero_real: real ).

thf(func_def_445,type,
    groups8507830703676809646_o_nat: ( $o > nat ) > set_o > nat ).

thf(func_def_446,type,
    groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).

thf(func_def_447,type,
    groups5690904116761175830ex_int: ( complex > int ) > set_complex > int ).

thf(func_def_448,type,
    groups5693394587270226106ex_nat: ( complex > nat ) > set_complex > nat ).

thf(func_def_449,type,
    groups5058264527183730370ex_rat: ( complex > rat ) > set_complex > rat ).

thf(func_def_450,type,
    groups5808333547571424918x_real: ( complex > real ) > set_complex > real ).

thf(func_def_451,type,
    groups7873554091576472773nteger: ( int > code_integer ) > set_int > code_integer ).

thf(func_def_452,type,
    groups3049146728041665814omplex: ( int > complex ) > set_int > complex ).

thf(func_def_453,type,
    groups4538972089207619220nt_int: ( int > int ) > set_int > int ).

thf(func_def_454,type,
    groups4541462559716669496nt_nat: ( int > nat ) > set_int > nat ).

thf(func_def_455,type,
    groups3906332499630173760nt_rat: ( int > rat ) > set_int > rat ).

thf(func_def_456,type,
    groups8778361861064173332t_real: ( int > real ) > set_int > real ).

thf(func_def_457,type,
    groups7501900531339628137nteger: ( nat > code_integer ) > set_nat > code_integer ).

thf(func_def_458,type,
    groups2073611262835488442omplex: ( nat > complex ) > set_nat > complex ).

thf(func_def_459,type,
    groups7108830773950497114d_enat: ( nat > extended_enat ) > set_nat > extended_enat ).

thf(func_def_460,type,
    groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).

thf(func_def_461,type,
    groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).

thf(func_def_462,type,
    groups2906978787729119204at_rat: ( nat > rat ) > set_nat > rat ).

thf(func_def_463,type,
    groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).

thf(func_def_464,type,
    groups977919841031483927at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > nat ).

thf(func_def_465,type,
    groups4567486121110086003t_real: ( product_prod_nat_nat > real ) > set_Pr1261947904930325089at_nat > real ).

thf(func_def_466,type,
    groups3860910324918113789at_nat: ( produc3843707927480180839at_nat > nat ) > set_Pr4329608150637261639at_nat > nat ).

thf(func_def_467,type,
    groups7713935264441627589nteger: ( real > code_integer ) > set_real > code_integer ).

thf(func_def_468,type,
    groups5754745047067104278omplex: ( real > complex ) > set_real > complex ).

thf(func_def_469,type,
    groups1932886352136224148al_int: ( real > int ) > set_real > int ).

thf(func_def_470,type,
    groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).

thf(func_def_471,type,
    groups1300246762558778688al_rat: ( real > rat ) > set_real > rat ).

thf(func_def_472,type,
    groups8097168146408367636l_real: ( real > real ) > set_real > real ).

thf(func_def_473,type,
    groups8294997508430121362at_nat: ( set_nat > nat ) > set_set_nat > nat ).

thf(func_def_474,type,
    groups771621172384141258BT_nat: ( vEBT_VEBT > nat ) > set_VEBT_VEBT > nat ).

thf(func_def_475,type,
    groups808145749697022017omplex: ( complex > complex ) > set_complex > complex ).

thf(func_def_476,type,
    groups2909182065852811199ex_int: ( complex > int ) > set_complex > int ).

thf(func_def_477,type,
    groups2911672536361861475ex_nat: ( complex > nat ) > set_complex > nat ).

thf(func_def_478,type,
    groups2276542476275365739ex_rat: ( complex > rat ) > set_complex > rat ).

thf(func_def_479,type,
    groups5737402329758386879x_real: ( complex > real ) > set_complex > real ).

thf(func_def_480,type,
    groups267424677133301183omplex: ( int > complex ) > set_int > complex ).

thf(func_def_481,type,
    groups2983280209131991357nt_int: ( int > int ) > set_int > int ).

thf(func_def_482,type,
    groups2985770679641041633nt_nat: ( int > nat ) > set_int > nat ).

thf(func_def_483,type,
    groups2350640619554545897nt_rat: ( int > rat ) > set_int > rat ).

thf(func_def_484,type,
    groups1523912220035142973t_real: ( int > real ) > set_int > real ).

thf(func_def_485,type,
    groups8515261248781899619omplex: ( nat > complex ) > set_nat > complex ).

thf(func_def_486,type,
    groups1983926497230936801at_int: ( nat > int ) > set_nat > int ).

thf(func_def_487,type,
    groups1986416967739987077at_nat: ( nat > nat ) > set_nat > nat ).

thf(func_def_488,type,
    groups1351286907653491341at_rat: ( nat > rat ) > set_nat > rat ).

thf(func_def_489,type,
    groups8560362682196896993t_real: ( nat > real ) > set_nat > real ).

thf(func_def_490,type,
    groups5683813829254066239omplex: ( real > complex ) > set_real > complex ).

thf(func_def_491,type,
    groups3901808747961969597al_int: ( real > int ) > set_real > int ).

thf(func_def_492,type,
    groups3904299218471019873al_nat: ( real > nat ) > set_real > nat ).

thf(func_def_493,type,
    groups3269169158384524137al_rat: ( real > rat ) > set_real > rat ).

thf(func_def_494,type,
    groups97945582718554045l_real: ( real > real ) > set_real > real ).

thf(func_def_495,type,
    groups8682486955453173170nteger: ( complex > code_integer ) > set_complex > code_integer ).

thf(func_def_496,type,
    groups3708469109370488835omplex: ( complex > complex ) > set_complex > complex ).

thf(func_def_497,type,
    groups858564598930262913ex_int: ( complex > int ) > set_complex > int ).

thf(func_def_498,type,
    groups861055069439313189ex_nat: ( complex > nat ) > set_complex > nat ).

thf(func_def_499,type,
    groups225925009352817453ex_rat: ( complex > rat ) > set_complex > rat ).

thf(func_def_500,type,
    groups766887009212190081x_real: ( complex > real ) > set_complex > real ).

thf(func_def_501,type,
    groups3827104343326376752nteger: ( int > code_integer ) > set_int > code_integer ).

thf(func_def_502,type,
    groups7440179247065528705omplex: ( int > complex ) > set_int > complex ).

thf(func_def_503,type,
    groups1705073143266064639nt_int: ( int > int ) > set_int > int ).

thf(func_def_504,type,
    groups1707563613775114915nt_nat: ( int > nat ) > set_int > nat ).

thf(func_def_505,type,
    groups1072433553688619179nt_rat: ( int > rat ) > set_int > rat ).

thf(func_def_506,type,
    groups2316167850115554303t_real: ( int > real ) > set_int > real ).

thf(func_def_507,type,
    groups2907647131375434839at_nat: ( list_nat > nat ) > set_list_nat > nat ).

thf(func_def_508,type,
    groups3455450783089532116nteger: ( nat > code_integer ) > set_nat > code_integer ).

thf(func_def_509,type,
    groups6464643781859351333omplex: ( nat > complex ) > set_nat > complex ).

thf(func_def_510,type,
    groups705719431365010083at_int: ( nat > int ) > set_nat > int ).

thf(func_def_511,type,
    groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).

thf(func_def_512,type,
    groups73079841787564623at_rat: ( nat > rat ) > set_nat > rat ).

thf(func_def_513,type,
    groups129246275422532515t_real: ( nat > real ) > set_nat > real ).

thf(func_def_514,type,
    groups4077766827762148844at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > nat ).

thf(func_def_515,type,
    groups6225526099057966256nteger: ( real > code_integer ) > set_real > code_integer ).

thf(func_def_516,type,
    groups713298508707869441omplex: ( real > complex ) > set_real > complex ).

thf(func_def_517,type,
    groups4694064378042380927al_int: ( real > int ) > set_real > int ).

thf(func_def_518,type,
    groups4696554848551431203al_nat: ( real > nat ) > set_real > nat ).

thf(func_def_519,type,
    groups4061424788464935467al_rat: ( real > rat ) > set_real > rat ).

thf(func_def_520,type,
    groups1681761925125756287l_real: ( real > real ) > set_real > real ).

thf(func_def_521,type,
    groups4248547760180025341at_nat: ( set_nat > nat ) > set_set_nat > nat ).

thf(func_def_522,type,
    groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).

thf(func_def_523,type,
    groups4561878855575611511st_nat: list_nat > nat ).

thf(func_def_524,type,
    the_int: ( int > $o ) > int ).

thf(func_def_525,type,
    the_nat: ( nat > $o ) > nat ).

thf(func_def_526,type,
    the_Pr8210177043389155639nteger: ( produc8763457246119570046nteger > $o ) > produc8763457246119570046nteger ).

thf(func_def_527,type,
    the_Pr6882841213465913158nt_int: ( produc7773217078559923341nt_int > $o ) > produc7773217078559923341nt_int ).

thf(func_def_528,type,
    the_Pr6653488032121699663nteger: ( produc1908205239877642774nteger > $o ) > produc1908205239877642774nteger ).

thf(func_def_529,type,
    the_Pr5445864913131713084nt_int: ( produc2285326912895808259nt_int > $o ) > produc2285326912895808259nt_int ).

thf(func_def_530,type,
    the_Pr4378521158711661632nt_int: ( product_prod_int_int > $o ) > product_prod_int_int ).

thf(func_def_531,type,
    the_Pr7557018466319803784at_nat: ( product_prod_nat_nat > $o ) > product_prod_nat_nat ).

thf(func_def_532,type,
    the_Pr4604535484834969198at_nat: ( produc3843707927480180839at_nat > $o ) > produc3843707927480180839at_nat ).

thf(func_def_533,type,
    the_real: ( real > $o ) > real ).

thf(func_def_534,type,
    if_int_int: $o > ( int > int ) > ( int > int ) > int > int ).

thf(func_def_535,type,
    if_Code_integer: $o > code_integer > code_integer > code_integer ).

thf(func_def_536,type,
    if_complex: $o > complex > complex > complex ).

thf(func_def_537,type,
    if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).

thf(func_def_538,type,
    if_int: $o > int > int > int ).

thf(func_def_539,type,
    if_list_int: $o > list_int > list_int > list_int ).

thf(func_def_540,type,
    if_list_nat: $o > list_nat > list_nat > list_nat ).

thf(func_def_541,type,
    if_nat: $o > nat > nat > nat ).

thf(func_def_542,type,
    if_num: $o > num > num > num ).

thf(func_def_543,type,
    if_option_nat: $o > option_nat > option_nat > option_nat ).

thf(func_def_544,type,
    if_option_num: $o > option_num > option_num > option_num ).

thf(func_def_545,type,
    if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).

thf(func_def_546,type,
    if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_547,type,
    if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).

thf(func_def_548,type,
    if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).

thf(func_def_549,type,
    if_rat: $o > rat > rat > rat ).

thf(func_def_550,type,
    if_real: $o > real > real > real ).

thf(func_def_551,type,
    if_set_int: $o > set_int > set_int > set_int ).

thf(func_def_552,type,
    if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).

thf(func_def_553,type,
    infini8530281810654367211te_nat: set_nat > nat > nat ).

thf(func_def_554,type,
    abs_Integ: product_prod_nat_nat > int ).

thf(func_def_555,type,
    rep_Integ: int > product_prod_nat_nat ).

thf(func_def_556,type,
    int_ge_less_than: int > set_Pr958786334691620121nt_int ).

thf(func_def_557,type,
    int_ge_less_than2: int > set_Pr958786334691620121nt_int ).

thf(func_def_558,type,
    intrel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(func_def_559,type,
    nat2: int > nat ).

thf(func_def_560,type,
    pcr_int: product_prod_nat_nat > int > $o ).

thf(func_def_561,type,
    power_int_real: real > int > real ).

thf(func_def_562,type,
    ring_18347121197199848620nteger: int > code_integer ).

thf(func_def_563,type,
    ring_17405671764205052669omplex: int > complex ).

thf(func_def_564,type,
    ring_1_of_int_int: int > int ).

thf(func_def_565,type,
    ring_1_of_int_rat: int > rat ).

thf(func_def_566,type,
    ring_1_of_int_real: int > real ).

thf(func_def_567,type,
    inf_inf_nat: nat > nat > nat ).

thf(func_def_568,type,
    inf_in2572325071724192079at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(func_def_569,type,
    semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(func_def_570,type,
    sup_sup_nat: nat > nat > nat ).

thf(func_def_571,type,
    sup_sup_set_nat: set_nat > set_nat > set_nat ).

thf(func_def_572,type,
    sup_su5525570899277871387at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

thf(func_def_573,type,
    lattic8265883725875713057ax_nat: set_nat > nat ).

thf(func_def_574,type,
    lattic5364784637807008409ex_nat: ( complex > nat ) > set_complex > complex ).

thf(func_def_575,type,
    lattic1922116423962787043ex_num: ( complex > num ) > set_complex > complex ).

thf(func_def_576,type,
    lattic4729654577720512673ex_rat: ( complex > rat ) > set_complex > complex ).

thf(func_def_577,type,
    lattic8794016678065449205x_real: ( complex > real ) > set_complex > complex ).

thf(func_def_578,type,
    lattic5003618458639192673nt_num: ( int > num ) > set_int > int ).

thf(func_def_579,type,
    lattic7811156612396918303nt_rat: ( int > rat ) > set_int > int ).

thf(func_def_580,type,
    lattic2675449441010098035t_real: ( int > real ) > set_int > int ).

thf(func_def_581,type,
    lattic4004264746738138117at_num: ( nat > num ) > set_nat > nat ).

thf(func_def_582,type,
    lattic6811802900495863747at_rat: ( nat > rat ) > set_nat > nat ).

thf(func_def_583,type,
    lattic488527866317076247t_real: ( nat > real ) > set_nat > nat ).

thf(func_def_584,type,
    lattic5055836439445974935al_nat: ( real > nat ) > set_real > real ).

thf(func_def_585,type,
    lattic1613168225601753569al_num: ( real > num ) > set_real > real ).

thf(func_def_586,type,
    lattic4420706379359479199al_rat: ( real > rat ) > set_real > real ).

thf(func_def_587,type,
    lattic8440615504127631091l_real: ( real > real ) > set_real > real ).

thf(func_def_588,type,
    append_int: list_int > list_int > list_int ).

thf(func_def_589,type,
    append_nat: list_nat > list_nat > list_nat ).

thf(func_def_590,type,
    count_list_o: list_o > $o > nat ).

thf(func_def_591,type,
    count_list_complex: list_complex > complex > nat ).

thf(func_def_592,type,
    count_list_int: list_int > int > nat ).

thf(func_def_593,type,
    count_list_nat: list_nat > nat > nat ).

thf(func_def_594,type,
    count_4203492906077236349at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat > nat ).

thf(func_def_595,type,
    count_list_real: list_real > real > nat ).

thf(func_def_596,type,
    count_list_set_nat: list_set_nat > set_nat > nat ).

thf(func_def_597,type,
    count_list_VEBT_VEBT: list_VEBT_VEBT > vEBT_VEBT > nat ).

thf(func_def_598,type,
    distinct_int: list_int > $o ).

thf(func_def_599,type,
    distinct_nat: list_nat > $o ).

thf(func_def_600,type,
    drop_nat: nat > list_nat > list_nat ).

thf(func_def_601,type,
    enumerate_o: nat > list_o > list_P7333126701944960589_nat_o ).

thf(func_def_602,type,
    enumerate_int: nat > list_int > list_P3521021558325789923at_int ).

thf(func_def_603,type,
    enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat ).

thf(func_def_604,type,
    enumerate_VEBT_VEBT: nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

thf(func_def_605,type,
    find_o: ( $o > $o ) > list_o > option_o ).

thf(func_def_606,type,
    find_int: ( int > $o ) > list_int > option_int ).

thf(func_def_607,type,
    find_nat: ( nat > $o ) > list_nat > option_nat ).

thf(func_def_608,type,
    find_num: ( num > $o ) > list_num > option_num ).

thf(func_def_609,type,
    find_P8199882355184865565at_nat: ( product_prod_nat_nat > $o ) > list_P6011104703257516679at_nat > option4927543243414619207at_nat ).

thf(func_def_610,type,
    find_real: ( real > $o ) > list_real > option_real ).

thf(func_def_611,type,
    find_set_nat: ( set_nat > $o ) > list_set_nat > option_set_nat ).

thf(func_def_612,type,
    find_VEBT_VEBT: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > option_VEBT_VEBT ).

thf(func_def_613,type,
    fold_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).

thf(func_def_614,type,
    last_nat: list_nat > nat ).

thf(func_def_615,type,
    linord1735203802627413978nt_int: ( int > int ) > list_int > list_int ).

thf(func_def_616,type,
    linord738340561235409698at_nat: ( nat > nat ) > list_nat > list_nat ).

thf(func_def_617,type,
    linord2614967742042102400et_nat: set_nat > list_nat ).

thf(func_def_618,type,
    cons_int: int > list_int > list_int ).

thf(func_def_619,type,
    cons_nat: nat > list_nat > list_nat ).

thf(func_def_620,type,
    nil_int: list_int ).

thf(func_def_621,type,
    nil_nat: list_nat ).

thf(func_def_622,type,
    hd_nat: list_nat > nat ).

thf(func_def_623,type,
    map_nat_nat: ( nat > nat ) > list_nat > list_nat ).

thf(func_def_624,type,
    set_Co8062243466402858685e_term: list_C878401137130745250e_term > set_Co9149898834107579976e_term ).

thf(func_def_625,type,
    set_in5217446777445088012e_term: list_i8448526496819171953e_term > set_in3461395444621081367e_term ).

thf(func_def_626,type,
    set_Pr8342322266483756581e_term: list_P1316552470764441098e_term > set_Pr7604974323444597168e_term ).

thf(func_def_627,type,
    set_Pr16608062948090134e_term: list_P1743416141875011707e_term > set_Pr3642885161833720865e_term ).

thf(func_def_628,type,
    set_o2: list_o > set_o ).

thf(func_def_629,type,
    set_complex2: list_complex > set_complex ).

thf(func_def_630,type,
    set_int2: list_int > set_int ).

thf(func_def_631,type,
    set_nat2: list_nat > set_nat ).

thf(func_def_632,type,
    set_num2: list_num > set_num ).

thf(func_def_633,type,
    set_Pr2999063419360598313nteger: list_P5311841565141990158nteger > set_Pr8056137968301705908nteger ).

thf(func_def_634,type,
    set_Pr1633835384712236856nt_int: list_P8915022641806594461nt_int > set_Pr1872883991513573699nt_int ).

thf(func_def_635,type,
    set_Pr2135590979564877377nteger: list_P7828571989066258726nteger > set_Pr1281608226676607948nteger ).

thf(func_def_636,type,
    set_Pr4943052134776177454nt_int: list_P651320350408439699nt_int > set_Pr9222295170931077689nt_int ).

thf(func_def_637,type,
    set_Product_prod_o_o2: list_P4002435161011370285od_o_o > set_Product_prod_o_o ).

thf(func_def_638,type,
    set_Pr2828948584524939422_o_int: list_P3795440434834930179_o_int > set_Pr8834758594704517033_o_int ).

thf(func_def_639,type,
    set_Pr7006799604034136130_o_nat: list_P6285523579766656935_o_nat > set_Pr2101469702781467981_o_nat ).

thf(func_def_640,type,
    set_Pr2600826154070092190o_real: list_P5232166724548748803o_real > set_Pr6573716822653411497o_real ).

thf(func_def_641,type,
    set_Pr655345902815428824T_VEBT: list_P7495141550334521929T_VEBT > set_Pr7543698050874017315T_VEBT ).

thf(func_def_642,type,
    set_Pr920681315882439344nteger: list_P5578671422887162913nteger > set_Pr4811707699266497531nteger ).

thf(func_def_643,type,
    set_Pr2470121279949933262nt_int: list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int ).

thf(func_def_644,type,
    set_Pr112895574167722958t_real: list_P6863124054624500543t_real > set_Pr3538720872664544793t_real ).

thf(func_def_645,type,
    set_Pr8714266321650254504T_VEBT: list_P7524865323317820941T_VEBT > set_Pr8044002425091019955T_VEBT ).

thf(func_def_646,type,
    set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).

thf(func_def_647,type,
    set_Pr5518436109238095868at_nat: list_P8469869581646625389at_nat > set_Pr8693737435421807431at_nat ).

thf(func_def_648,type,
    set_Pr5196769464307566348real_o: list_P3595434254542482545real_o > set_Pr4936984352647145239real_o ).

thf(func_def_649,type,
    set_Pr8219819362198175822al_int: list_P4344331454722006975al_int > set_Pr1019928272762051225al_int ).

thf(func_def_650,type,
    set_Pr3174298344852596722al_nat: list_P6834414599653733731al_nat > set_Pr3510011417693777981al_nat ).

thf(func_def_651,type,
    set_Pr5999470521830281550l_real: list_P8689742595348180415l_real > set_Pr6218003697084177305l_real ).

thf(func_def_652,type,
    set_Pr8897343066327330088T_VEBT: list_P877281246627933069T_VEBT > set_Pr6019664923565264691T_VEBT ).

thf(func_def_653,type,
    set_Pr9040384385603167362et_nat: list_P6254988961118846195et_nat > set_Pr5488025237498180813et_nat ).

thf(func_def_654,type,
    set_Pr3765526544606949372at_nat: list_P5464809261938338413at_nat > set_Pr4329608150637261639at_nat ).

thf(func_def_655,type,
    set_Pr7708085864119495200VEBT_o: list_P3126845725202233233VEBT_o > set_Pr3175402225741728619VEBT_o ).

thf(func_def_656,type,
    set_Pr2853735649769556538BT_int: list_P4547456442757143711BT_int > set_Pr5066593544530342725BT_int ).

thf(func_def_657,type,
    set_Pr7031586669278753246BT_nat: list_P7037539587688870467BT_nat > set_Pr7556676689462069481BT_nat ).

thf(func_def_658,type,
    set_Pr1087130671499945274T_real: list_P2623026923184700063T_real > set_Pr7765410600122031685T_real ).

thf(func_def_659,type,
    set_Pr9182192707038809660T_VEBT: list_P7413028617227757229T_VEBT > set_Pr6192946355708809607T_VEBT ).

thf(func_def_660,type,
    set_real2: list_real > set_real ).

thf(func_def_661,type,
    set_set_nat2: list_set_nat > set_set_nat ).

thf(func_def_662,type,
    set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).

thf(func_def_663,type,
    size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).

thf(func_def_664,type,
    tl_nat: list_nat > list_nat ).

thf(func_def_665,type,
    list_u4743598893156345252e_term: list_C878401137130745250e_term > nat > ( code_integer > option6357759511663192854e_term ) > list_C878401137130745250e_term ).

thf(func_def_666,type,
    list_u8946639151299769843e_term: list_i8448526496819171953e_term > nat > ( int > option6357759511663192854e_term ) > list_i8448526496819171953e_term ).

thf(func_def_667,type,
    list_u877304756163299468e_term: list_P1316552470764441098e_term > nat > ( produc6241069584506657477e_term > option6357759511663192854e_term ) > list_P1316552470764441098e_term ).

thf(func_def_668,type,
    list_u3533491785856317309e_term: list_P1743416141875011707e_term > nat > ( produc8551481072490612790e_term > option6357759511663192854e_term ) > list_P1743416141875011707e_term ).

thf(func_def_669,type,
    list_update_o: list_o > nat > $o > list_o ).

thf(func_def_670,type,
    list_update_int: list_int > nat > int > list_int ).

thf(func_def_671,type,
    list_update_nat: list_nat > nat > nat > list_nat ).

thf(func_def_672,type,
    list_u1133519416628930960nteger: list_P5311841565141990158nteger > nat > produc8763457246119570046nteger > list_P5311841565141990158nteger ).

thf(func_def_673,type,
    list_u4780935413889332127nt_int: list_P8915022641806594461nt_int > nat > produc7773217078559923341nt_int > list_P8915022641806594461nt_int ).

thf(func_def_674,type,
    list_u234853988314817064nteger: list_P7828571989066258726nteger > nat > produc1908205239877642774nteger > list_P7828571989066258726nteger ).

thf(func_def_675,type,
    list_u7736365598306452245nt_int: list_P651320350408439699nt_int > nat > produc2285326912895808259nt_int > list_P651320350408439699nt_int ).

thf(func_def_676,type,
    list_u2254550707601501961nteger: list_P5578671422887162913nteger > nat > produc8923325533196201883nteger > list_P5578671422887162913nteger ).

thf(func_def_677,type,
    list_u3002344382305578791nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int > list_P5707943133018811711nt_int ).

thf(func_def_678,type,
    list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).

thf(func_def_679,type,
    list_u4696772448584712917at_nat: list_P5464809261938338413at_nat > nat > produc3843707927480180839at_nat > list_P5464809261938338413at_nat ).

thf(func_def_680,type,
    list_u6961636818849549845T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT > list_P7413028617227757229T_VEBT ).

thf(func_def_681,type,
    list_update_real: list_real > nat > real > list_real ).

thf(func_def_682,type,
    list_update_set_nat: list_set_nat > nat > set_nat > list_set_nat ).

thf(func_def_683,type,
    list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_684,type,
    listrel1_o: set_Product_prod_o_o > set_Pr6227168374412355847list_o ).

thf(func_def_685,type,
    listrel1_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).

thf(func_def_686,type,
    listrel1_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(func_def_687,type,
    listrel1_VEBT_VEBT: set_Pr6192946355708809607T_VEBT > set_Pr1916528119006554503T_VEBT ).

thf(func_def_688,type,
    listrel1p_int: ( int > int > $o ) > list_int > list_int > $o ).

thf(func_def_689,type,
    listrel1p_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).

thf(func_def_690,type,
    nth_o: list_o > nat > $o ).

thf(func_def_691,type,
    nth_int: list_int > nat > int ).

thf(func_def_692,type,
    nth_nat: list_nat > nat > nat ).

thf(func_def_693,type,
    nth_num: list_num > nat > num ).

thf(func_def_694,type,
    nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).

thf(func_def_695,type,
    nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).

thf(func_def_696,type,
    nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat ).

thf(func_def_697,type,
    nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).

thf(func_def_698,type,
    nth_Pr112076138515278198_nat_o: list_P7333126701944960589_nat_o > nat > product_prod_nat_o ).

thf(func_def_699,type,
    nth_Pr3440142176431000676at_int: list_P3521021558325789923at_int > nat > product_prod_nat_int ).

thf(func_def_700,type,
    nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).

thf(func_def_701,type,
    nth_Pr744662078594809490T_VEBT: list_P5647936690300460905T_VEBT > nat > produc8025551001238799321T_VEBT ).

thf(func_def_702,type,
    nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).

thf(func_def_703,type,
    nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).

thf(func_def_704,type,
    nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).

thf(func_def_705,type,
    nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).

thf(func_def_706,type,
    nth_real: list_real > nat > real ).

thf(func_def_707,type,
    nth_set_nat: list_set_nat > nat > set_nat ).

thf(func_def_708,type,
    nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_709,type,
    product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).

thf(func_def_710,type,
    product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).

thf(func_def_711,type,
    product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).

thf(func_def_712,type,
    product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).

thf(func_def_713,type,
    product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).

thf(func_def_714,type,
    produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

thf(func_def_715,type,
    product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).

thf(func_def_716,type,
    produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).

thf(func_def_717,type,
    produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).

thf(func_def_718,type,
    produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).

thf(func_def_719,type,
    remdups_nat: list_nat > list_nat ).

thf(func_def_720,type,
    replicate_o: nat > $o > list_o ).

thf(func_def_721,type,
    replicate_int: nat > int > list_int ).

thf(func_def_722,type,
    replicate_nat: nat > nat > list_nat ).

thf(func_def_723,type,
    replic4235873036481779905at_nat: nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).

thf(func_def_724,type,
    replic2264142908078655527at_nat: nat > produc3843707927480180839at_nat > list_P5464809261938338413at_nat ).

thf(func_def_725,type,
    replicate_real: nat > real > list_real ).

thf(func_def_726,type,
    replicate_set_nat: nat > set_nat > list_set_nat ).

thf(func_def_727,type,
    replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_728,type,
    sorted_wrt_int: ( int > int > $o ) > list_int > $o ).

thf(func_def_729,type,
    sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).

thf(func_def_730,type,
    take_nat: nat > list_nat > list_nat ).

thf(func_def_731,type,
    upt: nat > nat > list_nat ).

thf(func_def_732,type,
    upto: int > int > list_int ).

thf(func_def_733,type,
    upto_aux: int > int > list_int > list_int ).

thf(func_def_734,type,
    upto_rel: product_prod_int_int > product_prod_int_int > $o ).

thf(func_def_735,type,
    zip_Co8729459035503499408nteger: list_C878401137130745250e_term > list_P5578671422887162913nteger > list_P5311841565141990158nteger ).

thf(func_def_736,type,
    zip_in8766932505889695135nt_int: list_i8448526496819171953e_term > list_P5707943133018811711nt_int > list_P8915022641806594461nt_int ).

thf(func_def_737,type,
    zip_Pr8292346330294042792nteger: list_P1316552470764441098e_term > list_P5578671422887162913nteger > list_P7828571989066258726nteger ).

thf(func_def_738,type,
    zip_Pr4168994715204986005nt_int: list_P1743416141875011707e_term > list_P5707943133018811711nt_int > list_P651320350408439699nt_int ).

thf(func_def_739,type,
    zip_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).

thf(func_def_740,type,
    zip_o_int: list_o > list_int > list_P3795440434834930179_o_int ).

thf(func_def_741,type,
    zip_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).

thf(func_def_742,type,
    zip_o_real: list_o > list_real > list_P5232166724548748803o_real ).

thf(func_def_743,type,
    zip_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).

thf(func_def_744,type,
    zip_int_int: list_int > list_int > list_P5707943133018811711nt_int ).

thf(func_def_745,type,
    zip_int_real: list_int > list_real > list_P6863124054624500543t_real ).

thf(func_def_746,type,
    zip_int_VEBT_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).

thf(func_def_747,type,
    zip_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).

thf(func_def_748,type,
    zip_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).

thf(func_def_749,type,
    zip_nat_VEBT_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

thf(func_def_750,type,
    zip_Pr4664179122662387191at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P8469869581646625389at_nat ).

thf(func_def_751,type,
    zip_real_o: list_real > list_o > list_P3595434254542482545real_o ).

thf(func_def_752,type,
    zip_real_int: list_real > list_int > list_P4344331454722006975al_int ).

thf(func_def_753,type,
    zip_real_nat: list_real > list_nat > list_P6834414599653733731al_nat ).

thf(func_def_754,type,
    zip_real_real: list_real > list_real > list_P8689742595348180415l_real ).

thf(func_def_755,type,
    zip_real_VEBT_VEBT: list_real > list_VEBT_VEBT > list_P877281246627933069T_VEBT ).

thf(func_def_756,type,
    zip_set_nat_set_nat: list_set_nat > list_set_nat > list_P6254988961118846195et_nat ).

thf(func_def_757,type,
    zip_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).

thf(func_def_758,type,
    zip_VEBT_VEBT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).

thf(func_def_759,type,
    zip_VEBT_VEBT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).

thf(func_def_760,type,
    zip_VEBT_VEBT_real: list_VEBT_VEBT > list_real > list_P2623026923184700063T_real ).

thf(func_def_761,type,
    zip_VE537291747668921783T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).

thf(func_def_762,type,
    suc: nat > nat ).

thf(func_def_763,type,
    compow_nat_nat: nat > ( nat > nat ) > nat > nat ).

thf(func_def_764,type,
    case_nat_o: $o > ( nat > $o ) > nat > $o ).

thf(func_def_765,type,
    case_nat_nat: nat > ( nat > nat ) > nat > nat ).

thf(func_def_766,type,
    case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).

thf(func_def_767,type,
    pred: nat > nat ).

thf(func_def_768,type,
    semiri4939895301339042750nteger: nat > code_integer ).

thf(func_def_769,type,
    semiri8010041392384452111omplex: nat > complex ).

thf(func_def_770,type,
    semiri4216267220026989637d_enat: nat > extended_enat ).

thf(func_def_771,type,
    semiri1314217659103216013at_int: nat > int ).

thf(func_def_772,type,
    semiri1316708129612266289at_nat: nat > nat ).

thf(func_def_773,type,
    semiri681578069525770553at_rat: nat > rat ).

thf(func_def_774,type,
    semiri5074537144036343181t_real: nat > real ).

thf(func_def_775,type,
    semiri2816024913162550771omplex: ( complex > complex ) > nat > complex > complex ).

thf(func_def_776,type,
    semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).

thf(func_def_777,type,
    semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).

thf(func_def_778,type,
    semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).

thf(func_def_779,type,
    semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).

thf(func_def_780,type,
    size_size_list_o: list_o > nat ).

thf(func_def_781,type,
    size_s3451745648224563538omplex: list_complex > nat ).

thf(func_def_782,type,
    size_size_list_int: list_int > nat ).

thf(func_def_783,type,
    size_size_list_nat: list_nat > nat ).

thf(func_def_784,type,
    size_size_list_num: list_num > nat ).

thf(func_def_785,type,
    size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).

thf(func_def_786,type,
    size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).

thf(func_def_787,type,
    size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat ).

thf(func_def_788,type,
    size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).

thf(func_def_789,type,
    size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).

thf(func_def_790,type,
    size_s2970893825323803983at_int: list_P3521021558325789923at_int > nat ).

thf(func_def_791,type,
    size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).

thf(func_def_792,type,
    size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat ).

thf(func_def_793,type,
    size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).

thf(func_def_794,type,
    size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).

thf(func_def_795,type,
    size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).

thf(func_def_796,type,
    size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).

thf(func_def_797,type,
    size_size_list_real: list_real > nat ).

thf(func_def_798,type,
    size_s3254054031482475050et_nat: list_set_nat > nat ).

thf(func_def_799,type,
    size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).

thf(func_def_800,type,
    size_size_num: num > nat ).

thf(func_def_801,type,
    size_size_option_nat: option_nat > nat ).

thf(func_def_802,type,
    size_size_option_num: option_num > nat ).

thf(func_def_803,type,
    size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).

thf(func_def_804,type,
    size_size_VEBT_VEBT: vEBT_VEBT > nat ).

thf(func_def_805,type,
    nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).

thf(func_def_806,type,
    nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(func_def_807,type,
    nat_set_decode: nat > set_nat ).

thf(func_def_808,type,
    nat_set_encode: set_nat > nat ).

thf(func_def_809,type,
    root: nat > real > real ).

thf(func_def_810,type,
    sqrt: real > real ).

thf(func_def_811,type,
    bitM: num > num ).

thf(func_def_812,type,
    inc: num > num ).

thf(func_def_813,type,
    neg_nu7757733837767384882nteger: code_integer > code_integer ).

thf(func_def_814,type,
    neg_nu6511756317524482435omplex: complex > complex ).

thf(func_def_815,type,
    neg_nu3811975205180677377ec_int: int > int ).

thf(func_def_816,type,
    neg_nu3179335615603231917ec_rat: rat > rat ).

thf(func_def_817,type,
    neg_nu6075765906172075777c_real: real > real ).

thf(func_def_818,type,
    bit0: num > num ).

thf(func_def_819,type,
    bit1: num > num ).

thf(func_def_820,type,
    one: num ).

thf(func_def_821,type,
    case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).

thf(func_def_822,type,
    size_num: num > nat ).

thf(func_def_823,type,
    num_of_nat: nat > num ).

thf(func_def_824,type,
    numera6620942414471956472nteger: num > code_integer ).

thf(func_def_825,type,
    numera6690914467698888265omplex: num > complex ).

thf(func_def_826,type,
    numera1916890842035813515d_enat: num > extended_enat ).

thf(func_def_827,type,
    numeral_numeral_int: num > int ).

thf(func_def_828,type,
    numeral_numeral_nat: num > nat ).

thf(func_def_829,type,
    numeral_numeral_rat: num > rat ).

thf(func_def_830,type,
    numeral_numeral_real: num > real ).

thf(func_def_831,type,
    pred_numeral: num > nat ).

thf(func_def_832,type,
    none_int: option_int ).

thf(func_def_833,type,
    none_nat: option_nat ).

thf(func_def_834,type,
    none_num: option_num ).

thf(func_def_835,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

thf(func_def_836,type,
    none_real: option_real ).

thf(func_def_837,type,
    none_set_nat: option_set_nat ).

thf(func_def_838,type,
    none_VEBT_VEBT: option_VEBT_VEBT ).

thf(func_def_839,type,
    some_o: $o > option_o ).

thf(func_def_840,type,
    some_int: int > option_int ).

thf(func_def_841,type,
    some_nat: nat > option_nat ).

thf(func_def_842,type,
    some_num: num > option_num ).

thf(func_def_843,type,
    some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).

thf(func_def_844,type,
    some_VEBT_VEBT: vEBT_VEBT > option_VEBT_VEBT ).

thf(func_def_845,type,
    case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).

thf(func_def_846,type,
    case_option_num_num: num > ( num > num ) > option_num > num ).

thf(func_def_847,type,
    case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).

thf(func_def_848,type,
    map_option_num_num: ( num > num ) > option_num > option_num ).

thf(func_def_849,type,
    size_option_nat: ( nat > nat ) > option_nat > nat ).

thf(func_def_850,type,
    size_option_num: ( num > nat ) > option_num > nat ).

thf(func_def_851,type,
    size_o8335143837870341156at_nat: ( product_prod_nat_nat > nat ) > option4927543243414619207at_nat > nat ).

thf(func_def_852,type,
    the_nat2: option_nat > nat ).

thf(func_def_853,type,
    the_num: option_num > num ).

thf(func_def_854,type,
    the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat ).

thf(func_def_855,type,
    order_underS_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).

thf(func_def_856,type,
    bot_bo5358457235160185703eger_o: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ).

thf(func_def_857,type,
    bot_bo1403522918969695512_int_o: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ).

thf(func_def_858,type,
    bot_bo3000040243691356879eger_o: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ).

thf(func_def_859,type,
    bot_bo8662317086119403298_int_o: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ).

thf(func_def_860,type,
    bot_bot_int_int_o: int > int > $o ).

thf(func_def_861,type,
    bot_bot_int_o: int > $o ).

thf(func_def_862,type,
    bot_bot_list_nat_o: list_nat > $o ).

thf(func_def_863,type,
    bot_bot_nat_nat_o: nat > nat > $o ).

thf(func_def_864,type,
    bot_bot_nat_o: nat > $o ).

thf(func_def_865,type,
    bot_bo482883023278783056_nat_o: product_prod_nat_nat > $o ).

thf(func_def_866,type,
    bot_bot_real_o: real > $o ).

thf(func_def_867,type,
    bot_bot_set_nat_o: set_nat > $o ).

thf(func_def_868,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

thf(func_def_869,type,
    bot_bot_filter_nat: filter_nat ).

thf(func_def_870,type,
    bot_bot_nat: nat ).

thf(func_def_871,type,
    bot_bot_set_complex: set_complex ).

thf(func_def_872,type,
    bot_bot_set_int: set_int ).

thf(func_def_873,type,
    bot_bot_set_list_nat: set_list_nat ).

thf(func_def_874,type,
    bot_bot_set_nat: set_nat ).

thf(func_def_875,type,
    bot_bot_set_num: set_num ).

thf(func_def_876,type,
    bot_bo3145834390647256904nteger: set_Pr8056137968301705908nteger ).

thf(func_def_877,type,
    bot_bo4508923176915781079nt_int: set_Pr1872883991513573699nt_int ).

thf(func_def_878,type,
    bot_bo5443222936135328352nteger: set_Pr1281608226676607948nteger ).

thf(func_def_879,type,
    bot_bo572930865798478029nt_int: set_Pr9222295170931077689nt_int ).

thf(func_def_880,type,
    bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int ).

thf(func_def_881,type,
    bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).

thf(func_def_882,type,
    bot_bo228742789529271731at_nat: set_Pr4329608150637261639at_nat ).

thf(func_def_883,type,
    bot_bot_set_rat: set_rat ).

thf(func_def_884,type,
    bot_bot_set_real: set_real ).

thf(func_def_885,type,
    bot_bot_set_set_int: set_set_int ).

thf(func_def_886,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(func_def_887,type,
    bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT ).

thf(func_def_888,type,
    ord_Least_nat: ( nat > $o ) > nat ).

thf(func_def_889,type,
    ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).

thf(func_def_890,type,
    ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

thf(func_def_891,type,
    ord_le549003669493604880_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

thf(func_def_892,type,
    ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).

thf(func_def_893,type,
    ord_less_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).

thf(func_def_894,type,
    ord_le6747313008572928689nteger: code_integer > code_integer > $o ).

thf(func_def_895,type,
    ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).

thf(func_def_896,type,
    ord_less_int: int > int > $o ).

thf(func_def_897,type,
    ord_less_nat: nat > nat > $o ).

thf(func_def_898,type,
    ord_less_num: num > num > $o ).

thf(func_def_899,type,
    ord_less_rat: rat > rat > $o ).

thf(func_def_900,type,
    ord_less_real: real > real > $o ).

thf(func_def_901,type,
    ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o ).

thf(func_def_902,type,
    ord_less_set_complex: set_complex > set_complex > $o ).

thf(func_def_903,type,
    ord_less_set_int: set_int > set_int > $o ).

thf(func_def_904,type,
    ord_less_set_nat: set_nat > set_nat > $o ).

thf(func_def_905,type,
    ord_less_set_num: set_num > set_num > $o ).

thf(func_def_906,type,
    ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(func_def_907,type,
    ord_le2604355607129572851at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ).

thf(func_def_908,type,
    ord_less_set_rat: set_rat > set_rat > $o ).

thf(func_def_909,type,
    ord_less_set_real: set_real > set_real > $o ).

thf(func_def_910,type,
    ord_less_set_set_int: set_set_int > set_set_int > $o ).

thf(func_def_911,type,
    ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).

thf(func_def_912,type,
    ord_le3636971675376928563eger_o: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > $o ).

thf(func_def_913,type,
    ord_le2124322318746777828_int_o: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ) > ( ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ) > $o ).

thf(func_def_914,type,
    ord_le4340812435750786203eger_o: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > $o ).

thf(func_def_915,type,
    ord_le5643404153117327598_int_o: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > $o ).

thf(func_def_916,type,
    ord_le6741204236512500942_int_o: ( int > int > $o ) > ( int > int > $o ) > $o ).

thf(func_def_917,type,
    ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).

thf(func_def_918,type,
    ord_le2646555220125990790_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(func_def_919,type,
    ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

thf(func_def_920,type,
    ord_le704812498762024988_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

thf(func_def_921,type,
    ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).

thf(func_def_922,type,
    ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).

thf(func_def_923,type,
    ord_le3102999989581377725nteger: code_integer > code_integer > $o ).

thf(func_def_924,type,
    ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).

thf(func_def_925,type,
    ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).

thf(func_def_926,type,
    ord_less_eq_int: int > int > $o ).

thf(func_def_927,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(func_def_928,type,
    ord_less_eq_num: num > num > $o ).

thf(func_def_929,type,
    ord_less_eq_rat: rat > rat > $o ).

thf(func_def_930,type,
    ord_less_eq_real: real > real > $o ).

thf(func_def_931,type,
    ord_less_eq_set_o: set_o > set_o > $o ).

thf(func_def_932,type,
    ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o ).

thf(func_def_933,type,
    ord_le211207098394363844omplex: set_complex > set_complex > $o ).

thf(func_def_934,type,
    ord_less_eq_set_int: set_int > set_int > $o ).

thf(func_def_935,type,
    ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).

thf(func_def_936,type,
    ord_less_eq_set_nat: set_nat > set_nat > $o ).

thf(func_def_937,type,
    ord_less_eq_set_num: set_num > set_num > $o ).

thf(func_def_938,type,
    ord_le3216752416896350996nteger: set_Pr8056137968301705908nteger > set_Pr8056137968301705908nteger > $o ).

thf(func_def_939,type,
    ord_le135402666524580259nt_int: set_Pr1872883991513573699nt_int > set_Pr1872883991513573699nt_int > $o ).

thf(func_def_940,type,
    ord_le653643898420964396nteger: set_Pr1281608226676607948nteger > set_Pr1281608226676607948nteger > $o ).

thf(func_def_941,type,
    ord_le8725513860283290265nt_int: set_Pr9222295170931077689nt_int > set_Pr9222295170931077689nt_int > $o ).

thf(func_def_942,type,
    ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).

thf(func_def_943,type,
    ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(func_def_944,type,
    ord_le1268244103169919719at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ).

thf(func_def_945,type,
    ord_less_eq_set_rat: set_rat > set_rat > $o ).

thf(func_def_946,type,
    ord_less_eq_set_real: set_real > set_real > $o ).

thf(func_def_947,type,
    ord_le4403425263959731960et_int: set_set_int > set_set_int > $o ).

thf(func_def_948,type,
    ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).

thf(func_def_949,type,
    ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).

thf(func_def_950,type,
    ord_max_Code_integer: code_integer > code_integer > code_integer ).

thf(func_def_951,type,
    ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat ).

thf(func_def_952,type,
    ord_max_int: int > int > int ).

thf(func_def_953,type,
    ord_max_nat: nat > nat > nat ).

thf(func_def_954,type,
    ord_max_num: num > num > num ).

thf(func_def_955,type,
    ord_max_rat: rat > rat > rat ).

thf(func_def_956,type,
    ord_max_real: real > real > real ).

thf(func_def_957,type,
    ord_max_set_int: set_int > set_int > set_int ).

thf(func_def_958,type,
    ord_max_set_nat: set_nat > set_nat > set_nat ).

thf(func_def_959,type,
    ord_ma7524802468073614006at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(func_def_960,type,
    ord_max_set_real: set_real > set_real > set_real ).

thf(func_def_961,type,
    ord_min_nat: nat > nat > nat ).

thf(func_def_962,type,
    order_Greatest_nat: ( nat > $o ) > nat ).

thf(func_def_963,type,
    order_mono_nat_nat: ( nat > nat ) > $o ).

thf(func_def_964,type,
    order_5726023648592871131at_nat: ( nat > nat ) > $o ).

thf(func_def_965,type,
    order_7092887310737990675l_real: ( real > real ) > $o ).

thf(func_def_966,type,
    ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).

thf(func_def_967,type,
    top_top_set_int: set_int ).

thf(func_def_968,type,
    top_top_set_nat: set_nat ).

thf(func_def_969,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

thf(func_def_970,type,
    top_top_set_real: set_real ).

thf(func_def_971,type,
    power_8256067586552552935nteger: code_integer > nat > code_integer ).

thf(func_def_972,type,
    power_power_complex: complex > nat > complex ).

thf(func_def_973,type,
    power_power_int: int > nat > int ).

thf(func_def_974,type,
    power_power_nat: nat > nat > nat ).

thf(func_def_975,type,
    power_power_rat: rat > nat > rat ).

thf(func_def_976,type,
    power_power_real: real > nat > real ).

thf(func_def_977,type,
    produc6137756002093451184nteger: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc8763457246119570046nteger ).

thf(func_def_978,type,
    produc4305682042979456191nt_int: ( int > option6357759511663192854e_term ) > product_prod_int_int > produc7773217078559923341nt_int ).

thf(func_def_979,type,
    produc4035269172776083154on_nat: ( nat > nat > $o ) > produc4953844613479565601on_nat > produc2233624965454879586on_nat ).

thf(func_def_980,type,
    produc8929957630744042906on_nat: ( nat > nat > nat ) > produc4953844613479565601on_nat > produc8306885398267862888on_nat ).

thf(func_def_981,type,
    produc3576312749637752826on_num: ( num > num > $o ) > produc3447558737645232053on_num > produc7036089656553540234on_num ).

thf(func_def_982,type,
    produc5778274026573060048on_num: ( num > num > num ) > produc3447558737645232053on_num > produc1193250871479095198on_num ).

thf(func_def_983,type,
    produc8603105652947943368nteger: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc1908205239877642774nteger ).

thf(func_def_984,type,
    produc5700946648718959541nt_int: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > produc2285326912895808259nt_int ).

thf(func_def_985,type,
    produc3994169339658061776at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc6121120109295599847at_nat > produc5491161045314408544at_nat ).

thf(func_def_986,type,
    produc2899441246263362727at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > produc6121120109295599847at_nat > produc5542196010084753463at_nat ).

thf(func_def_987,type,
    product_Pair_o_o: $o > $o > product_prod_o_o ).

thf(func_def_988,type,
    product_Pair_o_int: $o > int > product_prod_o_int ).

thf(func_def_989,type,
    product_Pair_o_nat: $o > nat > product_prod_o_nat ).

thf(func_def_990,type,
    product_Pair_o_real: $o > real > product_prod_o_real ).

thf(func_def_991,type,
    produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).

thf(func_def_992,type,
    produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).

thf(func_def_993,type,
    produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).

thf(func_def_994,type,
    product_Pair_int_int: int > int > product_prod_int_int ).

thf(func_def_995,type,
    produc801115645435158769t_real: int > real > produc679980390762269497t_real ).

thf(func_def_996,type,
    produc3329399203697025711T_VEBT: int > vEBT_VEBT > produc1531783533982839933T_VEBT ).

thf(func_def_997,type,
    produc8435520187683070743list_o: list_o > list_o > produc7102631898165422375list_o ).

thf(func_def_998,type,
    produc364263696895485585st_int: list_int > list_int > produc1186641810826059865st_int ).

thf(func_def_999,type,
    produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).

thf(func_def_1000,type,
    produc3897820843166775703T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > produc9211091688327510695T_VEBT ).

thf(func_def_1001,type,
    product_Pair_nat_o: nat > $o > product_prod_nat_o ).

thf(func_def_1002,type,
    product_Pair_nat_int: nat > int > product_prod_nat_int ).

thf(func_def_1003,type,
    product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).

thf(func_def_1004,type,
    product_Pair_nat_num: nat > num > product_prod_nat_num ).

thf(func_def_1005,type,
    produc599794634098209291T_VEBT: nat > vEBT_VEBT > produc8025551001238799321T_VEBT ).

thf(func_def_1006,type,
    product_Pair_num_num: num > num > product_prod_num_num ).

thf(func_def_1007,type,
    produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat ).

thf(func_def_1008,type,
    produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num ).

thf(func_def_1009,type,
    produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat ).

thf(func_def_1010,type,
    produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).

thf(func_def_1011,type,
    product_Pair_real_o: real > $o > product_prod_real_o ).

thf(func_def_1012,type,
    produc3179012173361985393al_int: real > int > produc8786904178792722361al_int ).

thf(func_def_1013,type,
    produc3181502643871035669al_nat: real > nat > produc3741383161447143261al_nat ).

thf(func_def_1014,type,
    produc4511245868158468465l_real: real > real > produc2422161461964618553l_real ).

thf(func_def_1015,type,
    produc6931449550656315951T_VEBT: real > vEBT_VEBT > produc3757001726724277373T_VEBT ).

thf(func_def_1016,type,
    produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat ).

thf(func_def_1017,type,
    produc2922128104949294807at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > produc3843707927480180839at_nat ).

thf(func_def_1018,type,
    produc9060074326276436823at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > produc1319942482725812455at_nat ).

thf(func_def_1019,type,
    produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).

thf(func_def_1020,type,
    produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).

thf(func_def_1021,type,
    produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).

thf(func_def_1022,type,
    produc8117437818029410057T_real: vEBT_VEBT > real > produc5170161368751668367T_real ).

thf(func_def_1023,type,
    produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).

thf(func_def_1024,type,
    produc457027306803732586at_nat: set_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).

thf(func_def_1025,type,
    produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_1026,type,
    produc8678311845419106900er_nat: ( code_integer > nat ) > ( code_integer > nat ) > produc8923325533196201883nteger > product_prod_nat_nat ).

thf(func_def_1027,type,
    produc127349428274296955eger_o: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > produc8763457246119570046nteger > $o ).

thf(func_def_1028,type,
    produc3906647086178084059nteger: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc8763457246119570046nteger ) > produc8763457246119570046nteger > produc8763457246119570046nteger ).

thf(func_def_1029,type,
    produc8604463032469472703et_int: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int ) > produc8763457246119570046nteger > set_int ).

thf(func_def_1030,type,
    produc3558942015123893603et_nat: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat ) > produc8763457246119570046nteger > set_nat ).

thf(func_def_1031,type,
    produc815715089573277247t_real: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real ) > produc8763457246119570046nteger > set_real ).

thf(func_def_1032,type,
    produc2558449545302689196_int_o: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc7773217078559923341nt_int > $o ).

thf(func_def_1033,type,
    produc5122537100556696953nt_int: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > produc7773217078559923341nt_int ) > produc7773217078559923341nt_int > produc7773217078559923341nt_int ).

thf(func_def_1034,type,
    produc8289552606927098482et_nat: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat ) > produc7773217078559923341nt_int > set_nat ).

thf(func_def_1035,type,
    produc8709739885379107790t_real: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real ) > produc7773217078559923341nt_int > set_real ).

thf(func_def_1036,type,
    produc6253627499356882019eger_o: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > produc1908205239877642774nteger > $o ).

thf(func_def_1037,type,
    produc6512950862096126219nteger: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc1908205239877642774nteger ) > produc1908205239877642774nteger > produc1908205239877642774nteger ).

thf(func_def_1038,type,
    produc1573362020775583542_int_o: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc2285326912895808259nt_int > $o ).

thf(func_def_1039,type,
    produc8492565224438309093nt_int: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > produc2285326912895808259nt_int ) > produc2285326912895808259nt_int > produc2285326912895808259nt_int ).

thf(func_def_1040,type,
    produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).

thf(func_def_1041,type,
    produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).

thf(func_def_1042,type,
    produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).

thf(func_def_1043,type,
    produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_1044,type,
    produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).

thf(func_def_1045,type,
    produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).

thf(func_def_1046,type,
    produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).

thf(func_def_1047,type,
    produc73460835934605544et_int: ( int > int > set_int ) > product_prod_int_int > set_int ).

thf(func_def_1048,type,
    produc4251311855443802252et_nat: ( int > int > set_nat ) > product_prod_int_int > set_nat ).

thf(func_def_1049,type,
    produc1656060378719767003at_nat: ( int > int > set_Pr1261947904930325089at_nat ) > product_prod_int_int > set_Pr1261947904930325089at_nat ).

thf(func_def_1050,type,
    produc6452406959799940328t_real: ( int > int > set_real ) > product_prod_int_int > set_real ).

thf(func_def_1051,type,
    produc5233655623923918146et_nat: ( int > int > set_set_nat ) > product_prod_int_int > set_set_nat ).

thf(func_def_1052,type,
    produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(func_def_1053,type,
    produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).

thf(func_def_1054,type,
    produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).

thf(func_def_1055,type,
    produc1917071388513777916omplex: ( nat > nat > complex ) > product_prod_nat_nat > complex ).

thf(func_def_1056,type,
    produc2676513652042109336d_enat: ( nat > nat > extended_enat ) > product_prod_nat_nat > extended_enat ).

thf(func_def_1057,type,
    produc6840382203811409530at_int: ( nat > nat > int ) > product_prod_nat_nat > int ).

thf(func_def_1058,type,
    produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).

thf(func_def_1059,type,
    produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).

thf(func_def_1060,type,
    produc6207742614233964070at_rat: ( nat > nat > rat ) > product_prod_nat_nat > rat ).

thf(func_def_1061,type,
    produc1703576794950452218t_real: ( nat > nat > real ) > product_prod_nat_nat > real ).

thf(func_def_1062,type,
    produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).

thf(func_def_1063,type,
    produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).

thf(func_def_1064,type,
    product_fst_int_int: product_prod_int_int > int ).

thf(func_def_1065,type,
    product_fst_nat_nat: product_prod_nat_nat > nat ).

thf(func_def_1066,type,
    product_snd_int_int: product_prod_int_int > int ).

thf(func_def_1067,type,
    product_snd_nat_nat: product_prod_nat_nat > nat ).

thf(func_def_1068,type,
    abs_Rat: product_prod_int_int > rat ).

thf(func_def_1069,type,
    fract: int > int > rat ).

thf(func_def_1070,type,
    frct: product_prod_int_int > rat ).

thf(func_def_1071,type,
    rep_Rat: rat > product_prod_int_int ).

thf(func_def_1072,type,
    field_5140801741446780682s_real: set_real ).

thf(func_def_1073,type,
    normalize: product_prod_int_int > product_prod_int_int ).

thf(func_def_1074,type,
    of_int: int > rat ).

thf(func_def_1075,type,
    pcr_rat: product_prod_int_int > rat > $o ).

thf(func_def_1076,type,
    positive: rat > $o ).

thf(func_def_1077,type,
    quotient_of: rat > product_prod_int_int ).

thf(func_def_1078,type,
    ratrel: product_prod_int_int > product_prod_int_int > $o ).

thf(func_def_1079,type,
    real2: ( nat > rat ) > real ).

thf(func_def_1080,type,
    cauchy: ( nat > rat ) > $o ).

thf(func_def_1081,type,
    pcr_real: ( nat > rat ) > real > $o ).

thf(func_def_1082,type,
    positive2: real > $o ).

thf(func_def_1083,type,
    realrel: ( nat > rat ) > ( nat > rat ) > $o ).

thf(func_def_1084,type,
    rep_real: real > nat > rat ).

thf(func_def_1085,type,
    vanishes: ( nat > rat ) > $o ).

thf(func_def_1086,type,
    real_V3694042436643373181omplex: complex > complex > real ).

thf(func_def_1087,type,
    real_V975177566351809787t_real: real > real > real ).

thf(func_def_1088,type,
    real_V1022390504157884413omplex: complex > real ).

thf(func_def_1089,type,
    real_V7735802525324610683m_real: real > real ).

thf(func_def_1090,type,
    real_V4546457046886955230omplex: real > complex ).

thf(func_def_1091,type,
    real_V2046097035970521341omplex: real > complex > complex ).

thf(func_def_1092,type,
    real_V1485227260804924795R_real: real > real > real ).

thf(func_def_1093,type,
    field_nat: set_Pr1261947904930325089at_nat > set_nat ).

thf(func_def_1094,type,
    algebr932160517623751201me_int: int > int > $o ).

thf(func_def_1095,type,
    algebr934650988132801477me_nat: nat > nat > $o ).

thf(func_def_1096,type,
    divide6298287555418463151nteger: code_integer > code_integer > code_integer ).

thf(func_def_1097,type,
    divide1717551699836669952omplex: complex > complex > complex ).

thf(func_def_1098,type,
    divide_divide_int: int > int > int ).

thf(func_def_1099,type,
    divide_divide_nat: nat > nat > nat ).

thf(func_def_1100,type,
    divide_divide_rat: rat > rat > rat ).

thf(func_def_1101,type,
    divide_divide_real: real > real > real ).

thf(func_def_1102,type,
    dvd_dvd_Code_integer: code_integer > code_integer > $o ).

thf(func_def_1103,type,
    dvd_dvd_complex: complex > complex > $o ).

thf(func_def_1104,type,
    dvd_dvd_int: int > int > $o ).

thf(func_def_1105,type,
    dvd_dvd_nat: nat > nat > $o ).

thf(func_def_1106,type,
    dvd_dvd_rat: rat > rat > $o ).

thf(func_def_1107,type,
    dvd_dvd_real: real > real > $o ).

thf(func_def_1108,type,
    modulo364778990260209775nteger: code_integer > code_integer > code_integer ).

thf(func_def_1109,type,
    modulo_modulo_int: int > int > int ).

thf(func_def_1110,type,
    modulo_modulo_nat: nat > nat > nat ).

thf(func_def_1111,type,
    zero_n356916108424825756nteger: $o > code_integer ).

thf(func_def_1112,type,
    zero_n1201886186963655149omplex: $o > complex ).

thf(func_def_1113,type,
    zero_n2684676970156552555ol_int: $o > int ).

thf(func_def_1114,type,
    zero_n2687167440665602831ol_nat: $o > nat ).

thf(func_def_1115,type,
    zero_n2052037380579107095ol_rat: $o > rat ).

thf(func_def_1116,type,
    zero_n3304061248610475627l_real: $o > real ).

thf(func_def_1117,type,
    suminf_complex: ( nat > complex ) > complex ).

thf(func_def_1118,type,
    suminf_int: ( nat > int ) > int ).

thf(func_def_1119,type,
    suminf_nat: ( nat > nat ) > nat ).

thf(func_def_1120,type,
    suminf_real: ( nat > real ) > real ).

thf(func_def_1121,type,
    summable_complex: ( nat > complex ) > $o ).

thf(func_def_1122,type,
    summable_int: ( nat > int ) > $o ).

thf(func_def_1123,type,
    summable_nat: ( nat > nat ) > $o ).

thf(func_def_1124,type,
    summable_real: ( nat > real ) > $o ).

thf(func_def_1125,type,
    sums_complex: ( nat > complex ) > complex > $o ).

thf(func_def_1126,type,
    sums_int: ( nat > int ) > int > $o ).

thf(func_def_1127,type,
    sums_nat: ( nat > nat ) > nat > $o ).

thf(func_def_1128,type,
    sums_real: ( nat > real ) > real > $o ).

thf(func_def_1129,type,
    collect_Code_integer: ( code_integer > $o ) > set_Code_integer ).

thf(func_def_1130,type,
    collect_complex: ( complex > $o ) > set_complex ).

thf(func_def_1131,type,
    collect_int: ( int > $o ) > set_int ).

thf(func_def_1132,type,
    collect_list_o: ( list_o > $o ) > set_list_o ).

thf(func_def_1133,type,
    collect_list_complex: ( list_complex > $o ) > set_list_complex ).

thf(func_def_1134,type,
    collect_list_int: ( list_int > $o ) > set_list_int ).

thf(func_def_1135,type,
    collect_list_nat: ( list_nat > $o ) > set_list_nat ).

thf(func_def_1136,type,
    collec3343600615725829874at_nat: ( list_P6011104703257516679at_nat > $o ) > set_li5450038453877631591at_nat ).

thf(func_def_1137,type,
    collec5608196760682091941T_VEBT: ( list_VEBT_VEBT > $o ) > set_list_VEBT_VEBT ).

thf(func_def_1138,type,
    collect_nat: ( nat > $o ) > set_nat ).

thf(func_def_1139,type,
    collect_num: ( num > $o ) > set_num ).

thf(func_def_1140,type,
    collec8663557070575231912omplex: ( produc4411394909380815293omplex > $o ) > set_Pr5085853215250843933omplex ).

thf(func_def_1141,type,
    collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).

thf(func_def_1142,type,
    collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).

thf(func_def_1143,type,
    collec3799799289383736868l_real: ( produc2422161461964618553l_real > $o ) > set_Pr6218003697084177305l_real ).

thf(func_def_1144,type,
    collec6321179662152712658at_nat: ( produc3843707927480180839at_nat > $o ) > set_Pr4329608150637261639at_nat ).

thf(func_def_1145,type,
    collect_rat: ( rat > $o ) > set_rat ).

thf(func_def_1146,type,
    collect_real: ( real > $o ) > set_real ).

thf(func_def_1147,type,
    collect_set_complex: ( set_complex > $o ) > set_set_complex ).

thf(func_def_1148,type,
    collect_set_int: ( set_int > $o ) > set_set_int ).

thf(func_def_1149,type,
    collect_set_nat: ( set_nat > $o ) > set_set_nat ).

thf(func_def_1150,type,
    collec5514110066124741708at_nat: ( set_Pr1261947904930325089at_nat > $o ) > set_se7855581050983116737at_nat ).

thf(func_def_1151,type,
    pow_nat: set_nat > set_set_nat ).

thf(func_def_1152,type,
    image_int_int: ( int > int ) > set_int > set_int ).

thf(func_def_1153,type,
    image_int_nat: ( int > nat ) > set_int > set_nat ).

thf(func_def_1154,type,
    image_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(func_def_1155,type,
    image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).

thf(func_def_1156,type,
    image_5971271580939081552omplex: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).

thf(func_def_1157,type,
    image_2178119161166701260l_real: ( real > filter2146258269922977983l_real ) > set_real > set_fi7789364187291644575l_real ).

thf(func_def_1158,type,
    image_real_real: ( real > real ) > set_real > set_real ).

thf(func_def_1159,type,
    insert_complex: complex > set_complex > set_complex ).

thf(func_def_1160,type,
    insert_int: int > set_int > set_int ).

thf(func_def_1161,type,
    insert_list_nat: list_nat > set_list_nat > set_list_nat ).

thf(func_def_1162,type,
    insert_nat: nat > set_nat > set_nat ).

thf(func_def_1163,type,
    insert_num: num > set_num > set_num ).

thf(func_def_1164,type,
    insert5033312907999012233nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).

thf(func_def_1165,type,
    insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(func_def_1166,type,
    insert9069300056098147895at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

thf(func_def_1167,type,
    insert_rat: rat > set_rat > set_rat ).

thf(func_def_1168,type,
    insert_real: real > set_real > set_real ).

thf(func_def_1169,type,
    insert_set_nat: set_nat > set_set_nat > set_set_nat ).

thf(func_def_1170,type,
    insert_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).

thf(func_def_1171,type,
    is_singleton_int: set_int > $o ).

thf(func_def_1172,type,
    is_singleton_nat: set_nat > $o ).

thf(func_def_1173,type,
    is_sin2850979758926227957at_nat: set_Pr1261947904930325089at_nat > $o ).

thf(func_def_1174,type,
    is_sin2937591304547752795at_nat: set_Pr4329608150637261639at_nat > $o ).

thf(func_def_1175,type,
    is_singleton_real: set_real > $o ).

thf(func_def_1176,type,
    is_singleton_set_nat: set_set_nat > $o ).

thf(func_def_1177,type,
    the_elem_int: set_int > int ).

thf(func_def_1178,type,
    the_elem_nat: set_nat > nat ).

thf(func_def_1179,type,
    the_el8326832613380209454nt_int: set_Pr958786334691620121nt_int > product_prod_int_int ).

thf(func_def_1180,type,
    the_el2281957884133575798at_nat: set_Pr1261947904930325089at_nat > product_prod_nat_nat ).

thf(func_def_1181,type,
    the_el221668144340439132at_nat: set_Pr4329608150637261639at_nat > produc3843707927480180839at_nat ).

thf(func_def_1182,type,
    the_elem_real: set_real > real ).

thf(func_def_1183,type,
    vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(func_def_1184,type,
    set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).

thf(func_def_1185,type,
    set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).

thf(func_def_1186,type,
    set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).

thf(func_def_1187,type,
    set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).

thf(func_def_1188,type,
    set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).

thf(func_def_1189,type,
    set_or1266510415728281911st_int: int > int > set_int ).

thf(func_def_1190,type,
    set_or1269000886237332187st_nat: nat > nat > set_nat ).

thf(func_def_1191,type,
    set_or7049704709247886629st_num: num > num > set_num ).

thf(func_def_1192,type,
    set_or633870826150836451st_rat: rat > rat > set_rat ).

thf(func_def_1193,type,
    set_or1222579329274155063t_real: real > real > set_real ).

thf(func_def_1194,type,
    set_or370866239135849197et_int: set_int > set_int > set_set_int ).

thf(func_def_1195,type,
    set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).

thf(func_def_1196,type,
    set_or4662586982721622107an_int: int > int > set_int ).

thf(func_def_1197,type,
    set_or4665077453230672383an_nat: nat > nat > set_nat ).

thf(func_def_1198,type,
    set_or1222409239386451017an_num: num > num > set_num ).

thf(func_def_1199,type,
    set_or4029947393144176647an_rat: rat > rat > set_rat ).

thf(func_def_1200,type,
    set_or66887138388493659n_real: real > real > set_real ).

thf(func_def_1201,type,
    set_or8585797421378605585et_int: set_int > set_int > set_set_int ).

thf(func_def_1202,type,
    set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).

thf(func_def_1203,type,
    set_ord_atLeast_nat: nat > set_nat ).

thf(func_def_1204,type,
    set_ord_atMost_int: int > set_int ).

thf(func_def_1205,type,
    set_ord_atMost_nat: nat > set_nat ).

thf(func_def_1206,type,
    set_ord_atMost_num: num > set_num ).

thf(func_def_1207,type,
    set_ord_atMost_rat: rat > set_rat ).

thf(func_def_1208,type,
    set_ord_atMost_real: real > set_real ).

thf(func_def_1209,type,
    set_or58775011639299419et_int: set_int > set_set_int ).

thf(func_def_1210,type,
    set_or4236626031148496127et_nat: set_nat > set_set_nat ).

thf(func_def_1211,type,
    set_or6656581121297822940st_int: int > int > set_int ).

thf(func_def_1212,type,
    set_or6659071591806873216st_nat: nat > nat > set_nat ).

thf(func_def_1213,type,
    set_or5832277885323065728an_int: int > int > set_int ).

thf(func_def_1214,type,
    set_or5834768355832116004an_nat: nat > nat > set_nat ).

thf(func_def_1215,type,
    set_or1633881224788618240n_real: real > real > set_real ).

thf(func_def_1216,type,
    set_or1210151606488870762an_nat: nat > set_nat ).

thf(func_def_1217,type,
    set_or5849166863359141190n_real: real > set_real ).

thf(func_def_1218,type,
    set_ord_lessThan_int: int > set_int ).

thf(func_def_1219,type,
    set_ord_lessThan_nat: nat > set_nat ).

thf(func_def_1220,type,
    set_ord_lessThan_num: num > set_num ).

thf(func_def_1221,type,
    set_ord_lessThan_rat: rat > set_rat ).

thf(func_def_1222,type,
    set_or5984915006950818249n_real: real > set_real ).

thf(func_def_1223,type,
    set_or890127255671739683et_nat: set_nat > set_set_nat ).

thf(func_def_1224,type,
    topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).

thf(func_def_1225,type,
    topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).

thf(func_def_1226,type,
    topolo6980174941875973593q_real: ( nat > real ) > $o ).

thf(func_def_1227,type,
    topolo2177554685111907308n_real: real > set_real > filter_real ).

thf(func_def_1228,type,
    topolo2815343760600316023s_real: real > filter_real ).

thf(func_def_1229,type,
    topolo6517432010174082258omplex: ( nat > complex ) > $o ).

thf(func_def_1230,type,
    topolo4055970368930404560y_real: ( nat > real ) > $o ).

thf(func_def_1231,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

thf(func_def_1232,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

thf(func_def_1233,type,
    arccos: real > real ).

thf(func_def_1234,type,
    arcosh_real: real > real ).

thf(func_def_1235,type,
    arcsin: real > real ).

thf(func_def_1236,type,
    arctan: real > real ).

thf(func_def_1237,type,
    arsinh_real: real > real ).

thf(func_def_1238,type,
    artanh_real: real > real ).

thf(func_def_1239,type,
    cos_complex: complex > complex ).

thf(func_def_1240,type,
    cos_real: real > real ).

thf(func_def_1241,type,
    cos_coeff: nat > real ).

thf(func_def_1242,type,
    cosh_complex: complex > complex ).

thf(func_def_1243,type,
    cosh_real: real > real ).

thf(func_def_1244,type,
    cot_complex: complex > complex ).

thf(func_def_1245,type,
    cot_real: real > real ).

thf(func_def_1246,type,
    diffs_complex: ( nat > complex ) > nat > complex ).

thf(func_def_1247,type,
    diffs_real: ( nat > real ) > nat > real ).

thf(func_def_1248,type,
    exp_complex: complex > complex ).

thf(func_def_1249,type,
    exp_real: real > real ).

thf(func_def_1250,type,
    ln_ln_real: real > real ).

thf(func_def_1251,type,
    log: real > real > real ).

thf(func_def_1252,type,
    pi: real ).

thf(func_def_1253,type,
    powr_real: real > real > real ).

thf(func_def_1254,type,
    sin_complex: complex > complex ).

thf(func_def_1255,type,
    sin_real: real > real ).

thf(func_def_1256,type,
    sin_coeff: nat > real ).

thf(func_def_1257,type,
    sinh_complex: complex > complex ).

thf(func_def_1258,type,
    sinh_real: real > real ).

thf(func_def_1259,type,
    tan_complex: complex > complex ).

thf(func_def_1260,type,
    tan_real: real > real ).

thf(func_def_1261,type,
    tanh_complex: complex > complex ).

thf(func_def_1262,type,
    tanh_real: real > real ).

thf(func_def_1263,type,
    transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(func_def_1264,type,
    transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(func_def_1265,type,
    vEBT_Leaf: $o > $o > vEBT_VEBT ).

thf(func_def_1266,type,
    vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1267,type,
    vEBT_size_VEBT: vEBT_VEBT > nat ).

thf(func_def_1268,type,
    vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).

thf(func_def_1269,type,
    vEBT_VEBT_high: nat > nat > nat ).

thf(func_def_1270,type,
    vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).

thf(func_def_1271,type,
    vEBT_VEBT_low: nat > nat > nat ).

thf(func_def_1272,type,
    vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).

thf(func_def_1273,type,
    vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_1274,type,
    vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).

thf(func_def_1275,type,
    vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_1276,type,
    vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).

thf(func_def_1277,type,
    vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_1278,type,
    vEBT_invar_vebt: vEBT_VEBT > nat > $o ).

thf(func_def_1279,type,
    vEBT_set_vebt: vEBT_VEBT > set_nat ).

thf(func_def_1280,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

thf(func_def_1281,type,
    vEBT_v4011308405150292612up_rel: nat > nat > $o ).

thf(func_def_1282,type,
    vEBT_vebt_delete: vEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1283,type,
    vEBT_vebt_delete_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_1284,type,
    vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1285,type,
    vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_1286,type,
    vEBT_VEBT_bit_concat: nat > nat > nat > nat ).

thf(func_def_1287,type,
    vEBT_VEBT_minNull: vEBT_VEBT > $o ).

thf(func_def_1288,type,
    vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(func_def_1289,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).

thf(func_def_1290,type,
    vEBT_vebt_member: vEBT_VEBT > nat > $o ).

thf(func_def_1291,type,
    vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_1292,type,
    vEBT_VEBT_add: option_nat > option_nat > option_nat ).

thf(func_def_1293,type,
    vEBT_VEBT_greater: option_nat > option_nat > $o ).

thf(func_def_1294,type,
    vEBT_VEBT_less: option_nat > option_nat > $o ).

thf(func_def_1295,type,
    vEBT_VEBT_lesseq: option_nat > option_nat > $o ).

thf(func_def_1296,type,
    vEBT_VEBT_max_in_set: set_nat > nat > $o ).

thf(func_def_1297,type,
    vEBT_VEBT_min_in_set: set_nat > nat > $o ).

thf(func_def_1298,type,
    vEBT_VEBT_mul: option_nat > option_nat > option_nat ).

thf(func_def_1299,type,
    vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).

thf(func_def_1300,type,
    vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).

thf(func_def_1301,type,
    vEBT_V1502963449132264192at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat ).

thf(func_def_1302,type,
    vEBT_VEBT_power: option_nat > option_nat > option_nat ).

thf(func_def_1303,type,
    vEBT_vebt_maxt: vEBT_VEBT > option_nat ).

thf(func_def_1304,type,
    vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(func_def_1305,type,
    vEBT_vebt_mint: vEBT_VEBT > option_nat ).

thf(func_def_1306,type,
    vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(func_def_1307,type,
    vEBT_is_pred_in_set: set_nat > nat > nat > $o ).

thf(func_def_1308,type,
    vEBT_vebt_pred: vEBT_VEBT > nat > option_nat ).

thf(func_def_1309,type,
    vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_1310,type,
    vEBT_is_succ_in_set: set_nat > nat > nat > $o ).

thf(func_def_1311,type,
    vEBT_vebt_succ: vEBT_VEBT > nat > option_nat ).

thf(func_def_1312,type,
    vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_1313,type,
    accp_nat: ( nat > nat > $o ) > nat > $o ).

thf(func_def_1314,type,
    accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).

thf(func_def_1315,type,
    accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(func_def_1316,type,
    accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).

thf(func_def_1317,type,
    accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).

thf(func_def_1318,type,
    accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).

thf(func_def_1319,type,
    max_ex8135407076693332796at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).

thf(func_def_1320,type,
    min_ex6901939911449802026at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).

thf(func_def_1321,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

thf(func_def_1322,type,
    wf_nat: set_Pr1261947904930325089at_nat > $o ).

thf(func_def_1323,type,
    fChoice_real: ( real > $o ) > real ).

thf(func_def_1324,type,
    member1535805642427569193e_term: ( code_integer > option6357759511663192854e_term ) > set_Co9149898834107579976e_term > $o ).

thf(func_def_1325,type,
    member8845023287901829240e_term: ( int > option6357759511663192854e_term ) > set_in3461395444621081367e_term > $o ).

thf(func_def_1326,type,
    member4242434998011752849e_term: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > set_Pr7604974323444597168e_term > $o ).

thf(func_def_1327,type,
    member3222579708246209666e_term: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr3642885161833720865e_term > $o ).

thf(func_def_1328,type,
    member_o: $o > set_o > $o ).

thf(func_def_1329,type,
    member_complex: complex > set_complex > $o ).

thf(func_def_1330,type,
    member_int: int > set_int > $o ).

thf(func_def_1331,type,
    member_list_o: list_o > set_list_o > $o ).

thf(func_def_1332,type,
    member_list_int: list_int > set_list_int > $o ).

thf(func_def_1333,type,
    member_list_nat: list_nat > set_list_nat > $o ).

thf(func_def_1334,type,
    member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).

thf(func_def_1335,type,
    member_nat: nat > set_nat > $o ).

thf(func_def_1336,type,
    member_num: num > set_num > $o ).

thf(func_def_1337,type,
    member3068662437193594005nteger: produc8763457246119570046nteger > set_Pr8056137968301705908nteger > $o ).

thf(func_def_1338,type,
    member7034335876925520548nt_int: produc7773217078559923341nt_int > set_Pr1872883991513573699nt_int > $o ).

thf(func_def_1339,type,
    member4164122664394876845nteger: produc1908205239877642774nteger > set_Pr1281608226676607948nteger > $o ).

thf(func_def_1340,type,
    member7618704894036264090nt_int: produc2285326912895808259nt_int > set_Pr9222295170931077689nt_int > $o ).

thf(func_def_1341,type,
    member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o ).

thf(func_def_1342,type,
    member7847949116333733898_o_int: product_prod_o_int > set_Pr8834758594704517033_o_int > $o ).

thf(func_def_1343,type,
    member2802428098988154798_o_nat: product_prod_o_nat > set_Pr2101469702781467981_o_nat > $o ).

thf(func_def_1344,type,
    member7400031367953476362o_real: product_prod_o_real > set_Pr6573716822653411497o_real > $o ).

thf(func_def_1345,type,
    member5477980866518848620T_VEBT: produc2504756804600209347T_VEBT > set_Pr7543698050874017315T_VEBT > $o ).

thf(func_def_1346,type,
    member157494554546826820nteger: produc8923325533196201883nteger > set_Pr4811707699266497531nteger > $o ).

thf(func_def_1347,type,
    member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).

thf(func_def_1348,type,
    member2744130022092475746t_real: produc679980390762269497t_real > set_Pr3538720872664544793t_real > $o ).

thf(func_def_1349,type,
    member2056185340421749780T_VEBT: produc1531783533982839933T_VEBT > set_Pr8044002425091019955T_VEBT > $o ).

thf(func_def_1350,type,
    member4159035015898711888list_o: produc7102631898165422375list_o > set_Pr6227168374412355847list_o > $o ).

thf(func_def_1351,type,
    member6698963635872716290st_int: produc1186641810826059865st_int > set_Pr765067013931698361st_int > $o ).

thf(func_def_1352,type,
    member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).

thf(func_def_1353,type,
    member4439316823752958928T_VEBT: produc9211091688327510695T_VEBT > set_Pr1916528119006554503T_VEBT > $o ).

thf(func_def_1354,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

thf(func_def_1355,type,
    member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(func_def_1356,type,
    member772602641336174712real_o: product_prod_real_o > set_Pr4936984352647145239real_o > $o ).

thf(func_def_1357,type,
    member1627681773268152802al_int: produc8786904178792722361al_int > set_Pr1019928272762051225al_int > $o ).

thf(func_def_1358,type,
    member5805532792777349510al_nat: produc3741383161447143261al_nat > set_Pr3510011417693777981al_nat > $o ).

thf(func_def_1359,type,
    member7849222048561428706l_real: produc2422161461964618553l_real > set_Pr6218003697084177305l_real > $o ).

thf(func_def_1360,type,
    member7262085504369356948T_VEBT: produc3757001726724277373T_VEBT > set_Pr6019664923565264691T_VEBT > $o ).

thf(func_def_1361,type,
    member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).

thf(func_def_1362,type,
    member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).

thf(func_def_1363,type,
    member3307348790968139188VEBT_o: produc334124729049499915VEBT_o > set_Pr3175402225741728619VEBT_o > $o ).

thf(func_def_1364,type,
    member5419026705395827622BT_int: produc4894624898956917775BT_int > set_Pr5066593544530342725BT_int > $o ).

thf(func_def_1365,type,
    member373505688050248522BT_nat: produc9072475918466114483BT_nat > set_Pr7556676689462069481BT_nat > $o ).

thf(func_def_1366,type,
    member8675245146396747942T_real: produc5170161368751668367T_real > set_Pr7765410600122031685T_real > $o ).

thf(func_def_1367,type,
    member568628332442017744T_VEBT: produc8243902056947475879T_VEBT > set_Pr6192946355708809607T_VEBT > $o ).

thf(func_def_1368,type,
    member_rat: rat > set_rat > $o ).

thf(func_def_1369,type,
    member_real: real > set_real > $o ).

thf(func_def_1370,type,
    member_set_int: set_int > set_set_int > $o ).

thf(func_def_1371,type,
    member_set_nat: set_nat > set_set_nat > $o ).

thf(func_def_1372,type,
    member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).

thf(func_def_1373,type,
    deg: nat ).

thf(func_def_1374,type,
    m: nat ).

thf(func_def_1375,type,
    ma: nat ).

thf(func_def_1376,type,
    maxi: nat ).

thf(func_def_1377,type,
    mi: nat ).

thf(func_def_1378,type,
    na: nat ).

thf(func_def_1379,type,
    summary: vEBT_VEBT ).

thf(func_def_1380,type,
    treeList: list_VEBT_VEBT ).

thf(func_def_1381,type,
    xa: nat ).

thf(func_def_1382,type,
    ya: nat ).

thf(func_def_1401,type,
    sP0: list_VEBT_VEBT > $o ).

thf(func_def_1402,type,
    sP1: vEBT_VEBT > list_VEBT_VEBT > nat > $o ).

thf(func_def_1403,type,
    sP2: nat > nat > list_VEBT_VEBT > nat > nat > $o ).

thf(func_def_1404,type,
    sP3: nat > nat > list_VEBT_VEBT > nat > nat > $o ).

thf(func_def_1405,type,
    sP4: nat > nat > list_VEBT_VEBT > nat > nat > $o ).

thf(func_def_1406,type,
    sP5: list_VEBT_VEBT > nat > $o ).

thf(func_def_1407,type,
    sP6: nat > nat > list_VEBT_VEBT > $o ).

thf(func_def_1408,type,
    sP7: nat > vEBT_VEBT > list_VEBT_VEBT > $o ).

thf(func_def_1409,type,
    sP8: nat > list_VEBT_VEBT > nat > nat > $o ).

thf(func_def_1410,type,
    sP9: nat > list_VEBT_VEBT > nat > nat > nat > $o ).

thf(func_def_1411,type,
    sP10: nat > list_VEBT_VEBT > nat > nat > nat > $o ).

thf(func_def_1412,type,
    sP11: $o > nat > $o ).

thf(func_def_1413,type,
    sP12: vEBT_VEBT > nat > $o ).

thf(func_def_1414,type,
    sP13: list_VEBT_VEBT > $o ).

thf(func_def_1415,type,
    sP14: nat > vEBT_VEBT > $o ).

thf(func_def_1416,type,
    sP15: $o > nat > $o ).

thf(func_def_1417,type,
    sP16: $o > nat > $o ).

thf(func_def_1418,type,
    sP17: $o > nat > $o > $o ).

thf(func_def_1419,type,
    sP18: $o > $o > nat > $o > $o ).

thf(func_def_1420,type,
    sP19: list_VEBT_VEBT > nat > nat > $o ).

thf(func_def_1421,type,
    sP20: list_VEBT_VEBT > nat > nat > nat > $o ).

thf(func_def_1422,type,
    sP21: list_VEBT_VEBT > nat > nat > nat > nat > $o ).

thf(func_def_1423,type,
    sP22: list_VEBT_VEBT > nat > nat > nat > nat > $o ).

thf(func_def_1424,type,
    sP23: $o > list_VEBT_VEBT > nat > nat > nat > nat > $o ).

thf(func_def_1425,type,
    sP24: $o > vEBT_VEBT > $o ).

thf(func_def_1426,type,
    sP25: vEBT_VEBT > $o > nat > $o ).

thf(func_def_1427,type,
    sP26: $o > vEBT_VEBT > $o ).

thf(func_def_1428,type,
    sP27: $o > vEBT_VEBT > $o ).

thf(func_def_1429,type,
    sP28: nat > $o ).

thf(func_def_1430,type,
    sP29: list_VEBT_VEBT > nat > nat > nat > nat > $o ).

thf(func_def_1431,type,
    sP30: nat > list_VEBT_VEBT > nat > nat > nat > $o ).

thf(func_def_1432,type,
    sP31: nat > nat > list_VEBT_VEBT > nat > nat > $o ).

thf(func_def_1433,type,
    sP32: vEBT_VEBT > list_VEBT_VEBT > nat > $o ).

thf(func_def_1434,type,
    sP33: list_VEBT_VEBT > $o ).

thf(func_def_1435,type,
    sP34: nat > list_VEBT_VEBT > nat > nat > nat > $o ).

thf(func_def_1436,type,
    sP35: list_VEBT_VEBT > nat > $o ).

thf(func_def_1437,type,
    sP36: nat > vEBT_VEBT > list_VEBT_VEBT > $o ).

thf(func_def_1438,type,
    sP37: list_VEBT_VEBT > $o ).

thf(func_def_1439,type,
    sP38: ( nat > vEBT_VEBT > $o ) > nat > $o ).

thf(func_def_1440,type,
    sP39: nat > ( nat > vEBT_VEBT > $o ) > $o ).

thf(func_def_1441,type,
    sP40: nat > vEBT_VEBT > nat > $o ).

thf(func_def_1442,type,
    sP41: vEBT_VEBT > $o ).

thf(func_def_1443,type,
    sP42: list_VEBT_VEBT > nat > nat > nat > nat > $o ).

thf(func_def_1444,type,
    sP43: nat > list_VEBT_VEBT > nat > nat > nat > $o ).

thf(func_def_1445,type,
    sP44: nat > vEBT_VEBT > list_VEBT_VEBT > $o ).

thf(func_def_1446,type,
    sP45: list_VEBT_VEBT > nat > nat > nat > nat > $o ).

thf(func_def_1447,type,
    sP46: list_VEBT_VEBT > vEBT_VEBT > nat > $o ).

thf(func_def_1448,type,
    sP47: nat > list_VEBT_VEBT > nat > nat > nat > $o ).

thf(func_def_1449,type,
    sP48: nat > vEBT_VEBT > $o ).

thf(func_def_1450,type,
    sP49: vEBT_VEBT > nat > $o ).

thf(func_def_1451,type,
    sP50: nat > vEBT_VEBT > $o ).

thf(func_def_1452,type,
    sP51: vEBT_VEBT > nat > $o ).

thf(func_def_1453,type,
    sP52: nat > nat > list_VEBT_VEBT > $o ).

thf(func_def_1454,type,
    sP53: nat > list_VEBT_VEBT > nat > nat > $o ).

thf(func_def_1455,type,
    sP54: nat > list_VEBT_VEBT > nat > nat > nat > $o ).

thf(func_def_1456,type,
    sP55: $o > nat > $o ).

thf(func_def_1457,type,
    sP56: $o > nat > $o ).

thf(func_def_1458,type,
    sP57: $o > nat > $o > $o ).

thf(func_def_1459,type,
    sP58: nat > vEBT_VEBT > $o ).

thf(func_def_1460,type,
    sP59: list_VEBT_VEBT > nat > nat > $o ).

thf(func_def_1461,type,
    sP60: list_VEBT_VEBT > nat > nat > nat > $o ).

thf(func_def_1462,type,
    sP61: list_VEBT_VEBT > nat > nat > nat > nat > $o ).

thf(func_def_1463,type,
    sP62: nat > nat > list_VEBT_VEBT > nat > nat > $o ).

thf(func_def_1464,type,
    sK63: nat > vEBT_VEBT > list_VEBT_VEBT > nat ).

thf(func_def_1465,type,
    sK64: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1466,type,
    sK65: nat > nat > list_VEBT_VEBT > nat > nat > nat ).

thf(func_def_1467,type,
    sK66: nat > nat > list_VEBT_VEBT > nat > nat > nat ).

thf(func_def_1468,type,
    sK67: list_VEBT_VEBT > nat > nat ).

thf(func_def_1469,type,
    sK68: list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1470,type,
    sK69: list_VEBT_VEBT > nat ).

thf(func_def_1471,type,
    sK70: vEBT_VEBT > nat > $o ).

thf(func_def_1472,type,
    sK71: vEBT_VEBT > nat > $o ).

thf(func_def_1473,type,
    sK72: vEBT_VEBT > nat > nat ).

thf(func_def_1474,type,
    sK73: vEBT_VEBT > nat > nat ).

thf(func_def_1475,type,
    sK74: vEBT_VEBT > nat > list_VEBT_VEBT ).

thf(func_def_1476,type,
    sK75: vEBT_VEBT > nat > nat ).

thf(func_def_1477,type,
    sK76: vEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1478,type,
    sK77: nat ).

thf(func_def_1479,type,
    sK78: ( list_VEBT_VEBT > $o ) > list_VEBT_VEBT ).

thf(func_def_1480,type,
    sK79: list_VEBT_VEBT > ( vEBT_VEBT > $o ) > vEBT_VEBT ).

thf(func_def_1481,type,
    sK80: nat > vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1482,type,
    sK81: nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1483,type,
    sK82: nat > vEBT_VEBT > option4927543243414619207at_nat ).

thf(func_def_1484,type,
    sK83: list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1485,type,
    sK84: list_VEBT_VEBT > nat ).

thf(func_def_1486,type,
    sK85: vEBT_VEBT > nat ).

thf(func_def_1487,type,
    sK86: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1488,type,
    sK87: nat > vEBT_VEBT > nat ).

thf(func_def_1489,type,
    sK88: nat > nat ).

thf(func_def_1490,type,
    sK89: vEBT_VEBT > $o ).

thf(func_def_1491,type,
    sK90: vEBT_VEBT > $o ).

thf(func_def_1492,type,
    sK91: vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1493,type,
    sK92: vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1494,type,
    sK93: vEBT_VEBT > product_prod_nat_nat ).

thf(func_def_1495,type,
    sK94: vEBT_VEBT > product_prod_nat_nat ).

thf(func_def_1496,type,
    sK95: vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1497,type,
    sK96: vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1498,type,
    sK97: vEBT_VEBT > $o > nat > $o ).

thf(func_def_1499,type,
    sK98: vEBT_VEBT > $o > nat > $o ).

thf(func_def_1500,type,
    sK99: vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1501,type,
    sK100: vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1502,type,
    sK101: vEBT_VEBT > nat ).

thf(func_def_1503,type,
    sK102: nat > $o > vEBT_VEBT > nat ).

thf(func_def_1504,type,
    sK103: nat > $o > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1505,type,
    sK104: nat > $o > vEBT_VEBT > nat ).

thf(func_def_1506,type,
    sK105: nat > $o > vEBT_VEBT > nat ).

thf(func_def_1507,type,
    sK106: nat > $o > vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1508,type,
    sK107: vEBT_VEBT > $o ).

thf(func_def_1509,type,
    sK108: vEBT_VEBT > $o ).

thf(func_def_1510,type,
    sK109: nat > list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1511,type,
    sK110: nat > list_VEBT_VEBT ).

thf(func_def_1512,type,
    sK111: list_VEBT_VEBT > ( vEBT_VEBT > $o ) > nat ).

thf(func_def_1513,type,
    sK112: list_VEBT_VEBT > ( vEBT_VEBT > $o ) > vEBT_VEBT ).

thf(func_def_1514,type,
    sK113: list_VEBT_VEBT > list_VEBT_VEBT > nat ).

thf(func_def_1515,type,
    sK114: nat > vEBT_VEBT > list_VEBT_VEBT > nat ).

thf(func_def_1516,type,
    sK115: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1517,type,
    sK116: list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1518,type,
    sK117: list_VEBT_VEBT > nat ).

thf(func_def_1519,type,
    sK118: list_VEBT_VEBT > nat > nat ).

thf(func_def_1520,type,
    sK119: nat > nat > list_VEBT_VEBT > nat > nat > nat ).

thf(func_def_1521,type,
    sK120: list_VEBT_VEBT > nat > nat > nat > nat > nat ).

thf(func_def_1522,type,
    sK121: list_VEBT_VEBT > vEBT_VEBT > nat ).

thf(func_def_1523,type,
    sK122: list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1524,type,
    sK123: list_VEBT_VEBT > nat ).

thf(func_def_1525,type,
    sK124: nat > list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1526,type,
    sK125: vEBT_VEBT > nat ).

thf(func_def_1527,type,
    sK126: nat > ( nat > vEBT_VEBT > $o ) > nat ).

thf(func_def_1528,type,
    sK127: nat > ( nat > vEBT_VEBT > $o ) > vEBT_VEBT ).

thf(func_def_1529,type,
    sK128: list_VEBT_VEBT > ( nat > vEBT_VEBT > $o ) > nat > nat ).

thf(func_def_1530,type,
    sK129: ( nat > vEBT_VEBT > $o ) > nat > list_VEBT_VEBT ).

thf(func_def_1531,type,
    sK130: vEBT_VEBT > nat > nat ).

thf(func_def_1532,type,
    sK131: vEBT_VEBT > nat > nat ).

thf(func_def_1533,type,
    sK132: vEBT_VEBT > nat > nat ).

thf(func_def_1534,type,
    sK133: vEBT_VEBT > nat > list_VEBT_VEBT ).

thf(func_def_1535,type,
    sK134: vEBT_VEBT > nat > nat ).

thf(func_def_1536,type,
    sK135: vEBT_VEBT > nat > nat ).

thf(func_def_1537,type,
    sK136: vEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1538,type,
    sK137: nat > vEBT_VEBT > nat ).

thf(func_def_1539,type,
    sK138: nat > vEBT_VEBT > nat ).

thf(func_def_1540,type,
    sK139: nat > vEBT_VEBT > nat ).

thf(func_def_1541,type,
    sK140: nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1542,type,
    sK141: nat > vEBT_VEBT > nat ).

thf(func_def_1543,type,
    sK142: nat > vEBT_VEBT > nat ).

thf(func_def_1544,type,
    sK143: nat > vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1545,type,
    sK144: vEBT_VEBT > nat > nat ).

thf(func_def_1546,type,
    sK145: vEBT_VEBT > nat > nat ).

thf(func_def_1547,type,
    sK146: vEBT_VEBT > nat > list_VEBT_VEBT ).

thf(func_def_1548,type,
    sK147: vEBT_VEBT > nat > nat ).

thf(func_def_1549,type,
    sK148: vEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1550,type,
    sK149: nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1551,type,
    sK150: nat > vEBT_VEBT > nat ).

thf(func_def_1552,type,
    sK151: nat > vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1553,type,
    sK152: nat > vEBT_VEBT > nat ).

thf(func_def_1554,type,
    sK153: nat > vEBT_VEBT > nat ).

thf(func_def_1555,type,
    sK154: nat > list_VEBT_VEBT > nat ).

thf(func_def_1556,type,
    sK155: nat > list_VEBT_VEBT > nat ).

thf(func_def_1557,type,
    sK156: vEBT_VEBT > $o ).

thf(func_def_1558,type,
    sK157: vEBT_VEBT > $o ).

thf(func_def_1559,type,
    sK158: nat > vEBT_VEBT > nat ).

thf(func_def_1560,type,
    sK159: nat > vEBT_VEBT > nat ).

thf(func_def_1561,type,
    sK160: nat > vEBT_VEBT > nat ).

thf(func_def_1562,type,
    sK161: nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1563,type,
    sK162: nat > vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1564,type,
    sK163: vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1565,type,
    sK164: vEBT_VEBT > product_prod_nat_nat ).

thf(func_def_1566,type,
    sK165: vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1567,type,
    sK166: nat > vEBT_VEBT > $o ).

thf(func_def_1568,type,
    sK167: nat > vEBT_VEBT > $o ).

thf(func_def_1569,type,
    sK168: vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1570,type,
    sK169: vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1571,type,
    sK170: vEBT_VEBT > product_prod_nat_nat ).

thf(func_def_1572,type,
    sK171: vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1573,type,
    sK172: vEBT_VEBT > nat ).

thf(func_def_1574,type,
    sK173: vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1575,type,
    sK174: vEBT_VEBT > $o ).

thf(func_def_1576,type,
    sK175: vEBT_VEBT > $o ).

thf(func_def_1577,type,
    sK176: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > nat ).

thf(func_def_1578,type,
    sK177: vEBT_VEBT > $o ).

thf(func_def_1579,type,
    sK178: vEBT_VEBT > $o ).

thf(func_def_1580,type,
    sK179: vEBT_VEBT > nat ).

thf(func_def_1581,type,
    sK180: vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1582,type,
    sK181: vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1583,type,
    sK182: vEBT_VEBT > option4927543243414619207at_nat ).

thf(func_def_1584,type,
    sK183: nat ).

thf(func_def_1585,type,
    sK184: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1587,type,
    ph186: 
      !>[X0: $tType] : X0 ).

thf(f25655,plain,
    $false,
    inference(subsumption_resolution,[status(thm)],[f25654,f25279]) ).

thf(f25279,plain,
    ( ( ord_less_nat @ ( divide_divide_nat @ ma @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) )
   != $true ),
    inference(trivial_inequality_removal,[status(thm)],[f25278]) ).

thf(f25278,plain,
    ( ( ( ord_less_nat @ ( divide_divide_nat @ ma @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) )
     != $true )
    | ( $true != $true ) ),
    inference(superposition,[status(thm)],[f25264,f24577]) ).

thf(f24577,plain,
    ! [X0: nat,X1: list_VEBT_VEBT] :
      ( ( ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ X1 @ X0 ) @ ( set_VEBT_VEBT2 @ X1 ) )
        = $true )
      | ( $true
       != ( ord_less_nat @ X0 @ ( size_s6755466524823107622T_VEBT @ X1 ) ) ) ),
    inference(cnf_transformation,[status(thm)],[f23916]) ).

thf(f23916,plain,
    ! [X0: nat,X1: list_VEBT_VEBT] :
      ( ( $true
       != ( ord_less_nat @ X0 @ ( size_s6755466524823107622T_VEBT @ X1 ) ) )
      | ( ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ X1 @ X0 ) @ ( set_VEBT_VEBT2 @ X1 ) )
        = $true ) ),
    inference(ennf_transformation,[status(thm)],[f23727]) ).

thf(f23727,plain,
    ! [X0: nat,X1: list_VEBT_VEBT] :
      ( ( $true
        = ( ord_less_nat @ X0 @ ( size_s6755466524823107622T_VEBT @ X1 ) ) )
     => ( ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ X1 @ X0 ) @ ( set_VEBT_VEBT2 @ X1 ) )
        = $true ) ),
    inference(fool_elimination,[status(thm)],[f23726]) ).

thf(f23726,plain,
    ! [X0: nat,X1: list_VEBT_VEBT] :
      ( ( ord_less_nat @ X0 @ ( size_s6755466524823107622T_VEBT @ X1 ) )
     => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ X1 @ X0 ) @ ( set_VEBT_VEBT2 @ X1 ) ) ),
    inference(rectify,[status(thm)],[f138]) ).

thf(f138,axiom,
    ! [X6: nat,X18: list_VEBT_VEBT] :
      ( ( ord_less_nat @ X6 @ ( size_s6755466524823107622T_VEBT @ X18 ) )
     => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ X18 @ X6 ) @ ( set_VEBT_VEBT2 @ X18 ) ) ),
    file('/export/starexec/sandbox/tmp/tmp.JxDoxSuNLY/DTF2THF_14050.p',fact_137_nth__mem) ).

thf(f25264,plain,
    ( ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ treeList @ ( divide_divide_nat @ ma @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) @ ( set_VEBT_VEBT2 @ treeList ) )
   != $true ),
    inference(trivial_inequality_removal,[status(thm)],[f25258]) ).

thf(f25258,plain,
    ( ( ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ treeList @ ( divide_divide_nat @ ma @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) @ ( set_VEBT_VEBT2 @ treeList ) )
     != $true )
    | ( $true != $true ) ),
    inference(superposition,[status(thm)],[f25150,f24696]) ).

thf(f24696,plain,
    ! [X0: vEBT_VEBT] :
      ( ( ( vEBT_invar_vebt @ X0 @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = $true )
      | ( ( member_VEBT_VEBT @ X0 @ ( set_VEBT_VEBT2 @ treeList ) )
       != $true ) ),
    inference(definition_unfolding,[status(thm)],[f24517,f24659]) ).

thf(f24659,plain,
    ( na
    = ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ),
    inference(cnf_transformation,[status(thm)],[f2120]) ).

thf(f2120,axiom,
    ( na
    = ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ),
    file('/export/starexec/sandbox/tmp/tmp.JxDoxSuNLY/DTF2THF_14050.p',fact_2119__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062) ).

thf(f24517,plain,
    ! [X0: vEBT_VEBT] :
      ( ( $true
        = ( vEBT_invar_vebt @ X0 @ na ) )
      | ( ( member_VEBT_VEBT @ X0 @ ( set_VEBT_VEBT2 @ treeList ) )
       != $true ) ),
    inference(cnf_transformation,[status(thm)],[f24042]) ).

thf(f24042,plain,
    ! [X0: vEBT_VEBT] :
      ( ( ( $true
          = ( vEBT_invar_vebt @ X0 @ na ) )
        & ( $true
          = ( sP41 @ X0 ) ) )
      | ( ( member_VEBT_VEBT @ X0 @ ( set_VEBT_VEBT2 @ treeList ) )
       != $true ) ),
    inference(definition_folding,[status(thm)],[f23905,f24041,f24040]) ).

thf(f24040,plain,
    ! [X2: nat,X0: vEBT_VEBT,X1: nat] :
      ( ( $true
        = ( sP40 @ X1 @ X0 @ X2 ) )
    <=> ( ( $true
          = ( vEBT_V8194947554948674370ptions @ X0 @ X2 ) )
        & ( X1 != X2 ) ) ),
    introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).

thf(f24041,plain,
    ! [X0: vEBT_VEBT] :
      ( ! [X1: nat,X2: nat] :
          ( ( $true
            = ( sP40 @ X1 @ X0 @ X2 ) )
        <=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ X0 @ X1 ) @ X2 )
            = $true ) )
      | ( $true
       != ( sP41 @ X0 ) ) ),
    introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction]) ).

thf(f23905,plain,
    ! [X0: vEBT_VEBT] :
      ( ( ( $true
          = ( vEBT_invar_vebt @ X0 @ na ) )
        & ! [X1: nat,X2: nat] :
            ( ( ( $true
                = ( vEBT_V8194947554948674370ptions @ X0 @ X2 ) )
              & ( X1 != X2 ) )
          <=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ X0 @ X1 ) @ X2 )
              = $true ) ) )
      | ( ( member_VEBT_VEBT @ X0 @ ( set_VEBT_VEBT2 @ treeList ) )
       != $true ) ),
    inference(ennf_transformation,[status(thm)],[f18342]) ).

thf(f18342,plain,
    ! [X0: vEBT_VEBT] :
      ( ( ( member_VEBT_VEBT @ X0 @ ( set_VEBT_VEBT2 @ treeList ) )
        = $true )
     => ( ( $true
          = ( vEBT_invar_vebt @ X0 @ na ) )
        & ! [X1: nat,X2: nat] :
            ( ( ( $true
                = ( vEBT_V8194947554948674370ptions @ X0 @ X2 ) )
              & ( X1 != X2 ) )
          <=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ X0 @ X1 ) @ X2 )
              = $true ) ) ) ),
    inference(fool_elimination,[status(thm)],[f18341]) ).

thf(f18341,plain,
    ! [X0: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X0 @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( ! [X1: nat,X2: nat] :
            ( ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ X0 @ X1 ) @ X2 )
              = ( vEBT_V8194947554948674370ptions @ X0 @ X2 ) )
            & ( X1 != X2 ) )
        & ( vEBT_invar_vebt @ X0 @ na ) ) ),
    inference(rectify,[status(thm)],[f24]) ).

thf(f24,axiom,
    ! [X10: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X10 @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( ! [X12: nat,X13: nat] :
            ( ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ X10 @ X12 ) @ X13 )
              = ( vEBT_V8194947554948674370ptions @ X10 @ X13 ) )
            & ( X12 != X13 ) )
        & ( vEBT_invar_vebt @ X10 @ na ) ) ),
    file('/export/starexec/sandbox/tmp/tmp.JxDoxSuNLY/DTF2THF_14050.p',fact_23__C4_OIH_C_I1_J) ).

thf(f25150,plain,
    ( $true
   != ( vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( divide_divide_nat @ ma @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ),
    inference(backward_demodulation,[status(thm)],[f24874,f24672]) ).

thf(f24672,plain,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ),
    inference(definition_unfolding,[status(thm)],[f24375,f24665]) ).

thf(f24665,plain,
    ( m
    = ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ),
    inference(definition_unfolding,[status(thm)],[f24510,f24659]) ).

thf(f24510,plain,
    m = na,
    inference(cnf_transformation,[status(thm)],[f1]) ).

thf(f1,axiom,
    m = na,
    file('/export/starexec/sandbox/tmp/tmp.JxDoxSuNLY/DTF2THF_14050.p',fact_0__C4_Ohyps_C_I3_J) ).

thf(f24375,plain,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ),
    inference(cnf_transformation,[status(thm)],[f1705]) ).

thf(f1705,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ),
    file('/export/starexec/sandbox/tmp/tmp.JxDoxSuNLY/DTF2THF_14050.p',fact_1704__C4_Ohyps_C_I2_J) ).

thf(f24874,plain,
    ( ( vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( divide_divide_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
   != $true ),
    inference(beta_eta_normalization,[status(thm)],[f24673]) ).

thf(f24673,plain,
    ( ( vEBT_invar_vebt
      @ ( nth_VEBT_VEBT @ treeList
        @ ( ^ [Y0: nat,Y1: nat] : ( divide_divide_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) )
          @ ma
          @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
   != $true ),
    inference(definition_unfolding,[status(thm)],[f24376,f24619,f24491,f24659,f24659]) ).

thf(f24491,plain,
    xa = ma,
    inference(cnf_transformation,[status(thm)],[f3]) ).

thf(f3,axiom,
    xa = ma,
    file('/export/starexec/sandbox/tmp/tmp.JxDoxSuNLY/DTF2THF_14050.p',fact_2_True) ).

thf(f24619,plain,
    ( vEBT_VEBT_high
    = ( ^ [Y0: nat,Y1: nat] : ( divide_divide_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) ) ) ),
    inference(cnf_transformation,[status(thm)],[f21131]) ).

thf(f21131,plain,
    ( vEBT_VEBT_high
    = ( ^ [Y0: nat,Y1: nat] : ( divide_divide_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) ) ) ),
    inference(fool_elimination,[status(thm)],[f21130]) ).

thf(f21130,plain,
    ( ( ^ [X0: nat,X1: nat] : ( divide_divide_nat @ X0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X1 ) ) )
    = vEBT_VEBT_high ),
    inference(rectify,[status(thm)],[f2119]) ).

thf(f2119,axiom,
    ( ( ^ [X16: nat,X14: nat] : ( divide_divide_nat @ X16 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X14 ) ) )
    = vEBT_VEBT_high ),
    file('/export/starexec/sandbox/tmp/tmp.JxDoxSuNLY/DTF2THF_14050.p',fact_2118_high__def) ).

thf(f24376,plain,
    ( ( vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ na )
   != $true ),
    inference(cnf_transformation,[status(thm)],[f23864]) ).

thf(f23864,plain,
    ( ( vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ na )
   != $true ),
    inference(flattening,[status(thm)],[f14717]) ).

thf(f14717,plain,
    ( ( vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ na )
   != $true ),
    inference(fool_elimination,[status(thm)],[f14716]) ).

thf(f14716,plain,
    ~ ( vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ na ),
    inference(rectify,[status(thm)],[f9374]) ).

thf(f9374,negated_conjecture,
    ~ ( vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ na ),
    inference(negated_conjecture,[status(cth)],[f9373]) ).

thf(f9373,conjecture,
    vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ na,
    file('/export/starexec/sandbox/tmp/tmp.JxDoxSuNLY/DTF2THF_14050.p',conj_0) ).

thf(f25654,plain,
    ( ( ord_less_nat @ ( divide_divide_nat @ ma @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) )
    = $true ),
    inference(forward_demodulation,[status(thm)],[f24865,f24672]) ).

thf(f24865,plain,
    ( $true
    = ( ord_less_nat @ ( divide_divide_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ),
    inference(beta_eta_normalization,[status(thm)],[f24721]) ).

thf(f24721,plain,
    ( ( ord_less_nat
      @ ( ^ [Y0: nat,Y1: nat] : ( divide_divide_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) )
        @ ma
        @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      @ ( size_s6755466524823107622T_VEBT @ treeList ) )
    = $true ),
    inference(definition_unfolding,[status(thm)],[f24650,f24619,f24491,f24659]) ).

thf(f24650,plain,
    ( ( ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( size_s6755466524823107622T_VEBT @ treeList ) )
    = $true ),
    inference(cnf_transformation,[status(thm)],[f15552]) ).

thf(f15552,plain,
    ( ( ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( size_s6755466524823107622T_VEBT @ treeList ) )
    = $true ),
    inference(fool_elimination,[status(thm)],[f15551]) ).

thf(f15551,plain,
    ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( size_s6755466524823107622T_VEBT @ treeList ),
    inference(rectify,[status(thm)],[f9]) ).

thf(f9,axiom,
    ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( size_s6755466524823107622T_VEBT @ treeList ),
    file('/export/starexec/sandbox/tmp/tmp.JxDoxSuNLY/DTF2THF_14050.p',fact_8__092_060open_062high_Ax_An_A_060_Alength_AtreeList_092_060close_062) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12  % Problem  : ITP256^1 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.11/0.13  % Command  : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34  % Computer : n019.cluster.edu
% 0.16/0.34  % Model    : x86_64 x86_64
% 0.16/0.34  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34  % Memory   : 8042.1875MB
% 0.16/0.34  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34  % CPULimit : 300
% 0.16/0.34  % WCLimit  : 300
% 0.16/0.34  % DateTime : Fri Feb 27 03:17:54 EST 2026
% 0.16/0.34  % CPUTime  : 
% 0.16/0.35  Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.33/0.50  ---- Original DTF file ---
% 0.33/0.50  thf(spec,logic,$$dhol).
% 0.33/0.50  %------------------------------------------------------------------------------
% 0.33/0.50  % File     : ITP256^1 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.33/0.50  % Domain   : Interactive Theorem Proving
% 0.33/0.50  % Problem  : Sledgehammer problem VEBT_Delete 00582_038585
% 0.33/0.50  % Version  : [Des22] axioms.
% 0.33/0.50  % English  :
% 0.33/0.50  
% 0.33/0.50  % Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.33/0.50  %          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% 0.33/0.50  % Source   : [Des22]
% 0.33/0.50  % Names    : 0072_VEBT_Delete_00582_038585 [Des22]
% 0.33/0.50  
% 0.33/0.50  % Status   : Theorem
% 0.33/0.50  % Rating   : ? v9.2.1
% 0.33/0.50  % Syntax   : Number of formulae    : 10756 (4299 unt;1383 typ;   0 def)
% 0.33/0.50  %            Number of atoms       : 29996 (11995 equ;   0 cnn)
% 0.33/0.50  %            Maximal formula atoms :   71 (   3 avg)
% 0.33/0.50  %            Number of connectives : 113577 (3521   ~; 532   |;2341   &;94618   @)
% 0.33/0.50  %                                         (   0 <=>;12565  =>;   0  <=;   0 <~>)
% 0.33/0.50  %            Maximal formula depth :   39 (   6 avg)
% 0.33/0.50  %            Number of types       :  174 ( 173 usr)
% 0.33/0.50  %            Number of type conns  : 5638 (5638   >;   0   *;   0   +;   0  <<)
% 0.33/0.50  %            Number of symbols     : 1213 (1210 usr;  76 con; 0-5 aty)
% 0.33/0.50  %            Number of variables   : 25831 (2400   ^;22571   !; 860   ?;25831   :)
% 0.33/0.50  % SPC      : TH0_THM_EQU_NAR
% 0.33/0.50  
% 0.33/0.50  % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.33/0.50  %            from the van Emde Boas Trees session in the Archive of Formal
% 0.33/0.50  %            proofs - 
% 0.33/0.50  %            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 0.33/0.50  %            2022-02-18 05:03:02.176
% 0.33/0.50  % Bugfixes : Added ()s around negated sides of equations.
% 0.33/0.50  %------------------------------------------------------------------------------
% 0.33/0.50  % Could-be-implicit typings (173)
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
% 0.33/0.50      produc1319942482725812455at_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 0.33/0.50      produc5542196010084753463at_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
% 0.33/0.50      list_P7828571989066258726nteger: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
% 0.33/0.50      set_Pr1281608226676607948nteger: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 0.33/0.50      produc1908205239877642774nteger: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 0.33/0.50      produc5491161045314408544at_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
% 0.33/0.50      list_P651320350408439699nt_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
% 0.33/0.50      set_Pr9222295170931077689nt_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.33/0.50      produc2285326912895808259nt_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
% 0.33/0.50      list_P5311841565141990158nteger: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
% 0.33/0.50      set_Pr8056137968301705908nteger: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 0.33/0.50      produc8763457246119570046nteger: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
% 0.33/0.50      list_P1316552470764441098e_term: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
% 0.33/0.50      set_Pr7604974323444597168e_term: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
% 0.33/0.50      list_P1743416141875011707e_term: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
% 0.33/0.50      set_Pr3642885161833720865e_term: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
% 0.33/0.50      list_P8915022641806594461nt_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
% 0.33/0.50      set_Pr1872883991513573699nt_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.33/0.50      produc7773217078559923341nt_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 0.33/0.50      list_P5464809261938338413at_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 0.33/0.50      set_Pr4329608150637261639at_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_I_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
% 0.33/0.50      produc1193250871479095198on_num: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
% 0.33/0.50      produc8306885398267862888on_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.33/0.50      produc6121120109295599847at_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_I_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
% 0.33/0.50      produc7036089656553540234on_num: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
% 0.33/0.50      produc2233624965454879586on_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.33/0.50      produc3843707927480180839at_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.33/0.50      list_P8469869581646625389at_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
% 0.33/0.50      list_C878401137130745250e_term: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.33/0.50      set_Pr8693737435421807431at_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
% 0.33/0.50      set_Co9149898834107579976e_term: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
% 0.33/0.50      list_i8448526496819171953e_term: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__VEBT____Definitions__OVEBT_J_J_J,type,
% 0.33/0.50      set_Pr1916528119006554503T_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.50      produc859450856879609959at_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
% 0.33/0.50      set_in3461395444621081367e_term: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
% 0.33/0.50      produc9211091688327510695T_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J,type,
% 0.33/0.50      produc6241069584506657477e_term: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J_J,type,
% 0.33/0.50      set_fi4554929511873752355omplex: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
% 0.33/0.50      set_Pr3451248702717554689st_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
% 0.33/0.50      set_Pr765067013931698361st_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.33/0.50      list_P7413028617227757229T_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J,type,
% 0.33/0.50      produc8551481072490612790e_term: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.33/0.50      set_Pr6192946355708809607T_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
% 0.33/0.50      list_P6254988961118846195et_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 0.33/0.50      list_P5578671422887162913nteger: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
% 0.33/0.50      set_Pr5488025237498180813et_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 0.33/0.50      set_Pr4811707699266497531nteger: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J,type,
% 0.33/0.50      option6357759511663192854e_term: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
% 0.33/0.50      produc3447558737645232053on_num: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
% 0.33/0.50      produc4953844613479565601on_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J_J,type,
% 0.33/0.50      set_fi7789364187291644575l_real: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 0.33/0.50      filter6041513312241820739omplex: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_J,type,
% 0.33/0.50      list_P2623026923184700063T_real: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.33/0.50      list_P877281246627933069T_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
% 0.33/0.50      produc1828647624359046049st_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
% 0.33/0.50      produc1186641810826059865st_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_J,type,
% 0.33/0.50      set_Pr7765410600122031685T_real: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.33/0.50      set_Pr6019664923565264691T_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
% 0.33/0.50      list_P7037539587688870467BT_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
% 0.33/0.50      list_P4547456442757143711BT_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.33/0.50      list_P5647936690300460905T_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.33/0.50      list_P7524865323317820941T_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__List__Olist_I_Eo_J_J_J,type,
% 0.33/0.50      set_Pr6227168374412355847list_o: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.33/0.50      set_li5450038453877631591at_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.50      produc8243902056947475879T_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
% 0.33/0.50      set_Pr7556676689462069481BT_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
% 0.33/0.50      set_Pr5066593544530342725BT_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.33/0.50      set_Pr8044002425091019955T_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.33/0.50      set_se7855581050983116737at_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 0.33/0.50      set_Pr5085853215250843933omplex: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
% 0.33/0.50      produc7819656566062154093et_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.33/0.50      produc8923325533196201883nteger: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
% 0.33/0.50      list_P3126845725202233233VEBT_o: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.33/0.50      list_P7495141550334521929T_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 0.33/0.50      filter2146258269922977983l_real: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
% 0.33/0.50      set_Pr3175402225741728619VEBT_o: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.33/0.50      set_Pr7543698050874017315T_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.50      option4927543243414619207at_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 0.33/0.50      list_P8689742595348180415l_real: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.50      filter1242075044329608583at_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 0.33/0.50      set_Pr6218003697084177305l_real: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
% 0.33/0.50      list_P6834414599653733731al_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J_J,type,
% 0.33/0.50      list_P4344331454722006975al_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J_J,type,
% 0.33/0.50      list_P6863124054624500543t_real: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
% 0.33/0.50      produc5170161368751668367T_real: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.50      produc3757001726724277373T_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
% 0.33/0.50      set_Pr3510011417693777981al_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J_J,type,
% 0.33/0.50      set_Pr1019928272762051225al_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J_J,type,
% 0.33/0.50      set_Pr3538720872664544793t_real: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__List__Olist_I_Eo_J_J,type,
% 0.33/0.50      produc7102631898165422375list_o: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.50      list_P6011104703257516679at_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
% 0.33/0.50      list_P3521021558325789923at_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.33/0.50      list_P5707943133018811711nt_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 0.33/0.50      produc9072475918466114483BT_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 0.33/0.50      produc4894624898956917775BT_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.50      produc8025551001238799321T_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.50      produc1531783533982839933T_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.50      set_Pr1261947904930325089at_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.33/0.50      set_Pr958786334691620121nt_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 0.33/0.50      produc4411394909380815293omplex: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_M_Eo_J_J,type,
% 0.33/0.50      list_P3595434254542482545real_o: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J_J,type,
% 0.33/0.50      list_P5232166724548748803o_real: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_M_Eo_J_J,type,
% 0.33/0.50      set_Pr4936984352647145239real_o: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J_J,type,
% 0.33/0.50      set_Pr6573716822653411497o_real: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
% 0.33/0.50      list_P7333126701944960589_nat_o: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
% 0.33/0.50      list_P6285523579766656935_o_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
% 0.33/0.50      list_P3795440434834930179_o_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
% 0.33/0.50      set_list_VEBT_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 0.33/0.50      produc334124729049499915VEBT_o: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.50      produc2504756804600209347T_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
% 0.33/0.50      set_Pr2101469702781467981_o_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
% 0.33/0.50      set_Pr8834758594704517033_o_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 0.33/0.50      produc6271795597528267376eger_o: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.33/0.50      produc2422161461964618553l_real: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
% 0.33/0.50      produc3741383161447143261al_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
% 0.33/0.50      produc8786904178792722361al_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
% 0.33/0.50      produc679980390762269497t_real: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 0.33/0.50      product_prod_num_num: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 0.33/0.50      product_prod_nat_num: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.50      product_prod_nat_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
% 0.33/0.50      product_prod_nat_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.33/0.50      product_prod_int_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 0.33/0.50      list_P4002435161011370285od_o_o: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J,type,
% 0.33/0.50      set_list_complex: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 0.33/0.50      set_Product_prod_o_o: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
% 0.33/0.50      set_set_complex: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Option__Ooption_It__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.50      option_VEBT_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.33/0.50      option_set_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_M_Eo_J,type,
% 0.33/0.50      product_prod_real_o: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J,type,
% 0.33/0.50      product_prod_o_real: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.50      list_VEBT_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 0.33/0.50      set_list_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
% 0.33/0.50      set_list_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
% 0.33/0.50      product_prod_nat_o: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
% 0.33/0.50      product_prod_o_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 0.33/0.50      product_prod_o_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.33/0.50      list_set_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.50      set_VEBT_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.33/0.50      set_set_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
% 0.33/0.50      set_set_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
% 0.33/0.50      set_Code_integer: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
% 0.33/0.50      set_Product_unit: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
% 0.33/0.50      list_complex: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
% 0.33/0.50      set_list_o: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
% 0.33/0.50      product_prod_o_o: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
% 0.33/0.50      set_complex: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Option__Ooption_It__Real__Oreal_J,type,
% 0.33/0.50      option_real: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
% 0.33/0.50      filter_real: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
% 0.33/0.50      option_num: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
% 0.33/0.50      option_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
% 0.33/0.50      option_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
% 0.33/0.50      filter_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
% 0.33/0.50      list_real: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 0.33/0.50      set_real: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Num__Onum_J,type,
% 0.33/0.50      list_num: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
% 0.33/0.50      list_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
% 0.33/0.50      list_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__VEBT____Definitions__OVEBT,type,
% 0.33/0.50      vEBT_VEBT: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
% 0.33/0.50      set_rat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
% 0.33/0.50      set_num: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.50      set_nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
% 0.33/0.50      set_int: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Code____Numeral__Ointeger,type,
% 0.33/0.50      code_integer: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Option__Ooption_I_Eo_J,type,
% 0.33/0.50      option_o: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Extended____Nat__Oenat,type,
% 0.33/0.50      extended_enat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__List__Olist_I_Eo_J,type,
% 0.33/0.50      list_o: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Complex__Ocomplex,type,
% 0.33/0.50      complex: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Set__Oset_I_Eo_J,type,
% 0.33/0.50      set_o: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Real__Oreal,type,
% 0.33/0.50      real: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Rat__Orat,type,
% 0.33/0.50      rat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Num__Onum,type,
% 0.33/0.50      num: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Nat__Onat,type,
% 0.33/0.50      nat: $tType ).
% 0.33/0.50  
% 0.33/0.50  thf(ty_n_t__Int__Oint,type,
% 0.33/0.50      int: $tType ).
% 0.33/0.50  
% 0.33/0.50  % Explicit typings (1210)
% 0.33/0.50  thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
% 0.33/0.50      archim2889992004027027881ng_rat: rat > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
% 0.33/0.50      archim7802044766580827645g_real: real > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
% 0.33/0.50      archim3151403230148437115or_rat: rat > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
% 0.33/0.50      archim6058952711729229775r_real: real > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
% 0.33/0.50      archimedean_frac_rat: rat > rat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
% 0.33/0.50      archim2898591450579166408c_real: real > real ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
% 0.33/0.50      archim7778729529865785530nd_rat: rat > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
% 0.33/0.50      archim8280529875227126926d_real: real > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
% 0.33/0.50      bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
% 0.33/0.50      bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_Eo_001_Eo,type,
% 0.33/0.50      bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_Eo_001_Eo,type,
% 0.33/0.50      bNF_re4297313714947099218al_o_o: ( ( nat > rat ) > real > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( real > $o ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
% 0.33/0.50      bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.33/0.50      bNF_re157797125943740599nt_int: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > product_prod_int_int ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Rat__Orat_J,type,
% 0.33/0.50      bNF_re3461391660133120880nt_rat: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > rat ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > rat ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_Eo_001_Eo,type,
% 0.33/0.50      bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.33/0.50      bNF_re6250860962936578807nt_int: ( int > int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 0.33/0.50      bNF_re2214769303045360666nt_rat: ( int > int > $o ) > ( product_prod_int_int > rat > $o ) > ( int > product_prod_int_int ) > ( int > rat ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.33/0.50      bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
% 0.33/0.50      bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 0.33/0.50      bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.33/0.50      bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
% 0.33/0.50      bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.33/0.50      bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
% 0.33/0.50      bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_Eo_001_Eo,type,
% 0.33/0.50      bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 0.33/0.50      bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
% 0.33/0.50      bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.33/0.50      bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_Eo_001_Eo,type,
% 0.33/0.50      bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 0.33/0.50      bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 0.33/0.50      bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.50      bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
% 0.33/0.50      bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.50      bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
% 0.33/0.50      bNF_We3818239936649020644el_nat: set_Pr1261947904930325089at_nat > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Binomial_Obinomial,type,
% 0.33/0.50      binomial: nat > nat > nat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
% 0.33/0.50      gbinomial_complex: complex > nat > complex ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
% 0.33/0.50      gbinomial_int: int > nat > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
% 0.33/0.50      gbinomial_nat: nat > nat > nat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
% 0.33/0.50      gbinomial_rat: rat > nat > rat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
% 0.33/0.50      gbinomial_real: real > nat > real ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Oand__int__rel,type,
% 0.33/0.50      bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Oand__not__num,type,
% 0.33/0.50      bit_and_not_num: num > num > option_num ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
% 0.33/0.50      bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Oconcat__bit,type,
% 0.33/0.50      bit_concat_bit: nat > int > int > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
% 0.33/0.50      bit_ri7919022796975470100ot_int: int > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
% 0.33/0.50      bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
% 0.33/0.50      bit_ri631733984087533419it_int: nat > int > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
% 0.33/0.50      bit_se3949692690581998587nteger: code_integer > code_integer > code_integer ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
% 0.33/0.50      bit_se725231765392027082nd_int: int > int > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
% 0.33/0.50      bit_se727722235901077358nd_nat: nat > nat > nat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
% 0.33/0.50      bit_se8568078237143864401it_int: nat > int > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
% 0.33/0.50      bit_se8570568707652914677it_nat: nat > nat > nat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
% 0.33/0.50      bit_se1345352211410354436nteger: nat > code_integer > code_integer ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
% 0.33/0.50      bit_se2159334234014336723it_int: nat > int > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
% 0.33/0.50      bit_se2161824704523386999it_nat: nat > nat > nat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
% 0.33/0.50      bit_se2000444600071755411sk_int: nat > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
% 0.33/0.50      bit_se2002935070580805687sk_nat: nat > nat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Code____Numeral__Ointeger,type,
% 0.33/0.50      bit_se1080825931792720795nteger: code_integer > code_integer > code_integer ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
% 0.33/0.50      bit_se1409905431419307370or_int: int > int > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
% 0.33/0.50      bit_se1412395901928357646or_nat: nat > nat > nat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
% 0.33/0.50      bit_se545348938243370406it_int: nat > int > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
% 0.33/0.50      bit_se547839408752420682it_nat: nat > nat > nat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
% 0.33/0.50      bit_se2793503036327961859nteger: nat > code_integer > code_integer ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
% 0.33/0.50      bit_se7879613467334960850it_int: nat > int > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
% 0.33/0.50      bit_se7882103937844011126it_nat: nat > nat > nat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Code____Numeral__Ointeger,type,
% 0.33/0.50      bit_se1745604003318907178nteger: nat > code_integer > code_integer ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
% 0.33/0.50      bit_se2923211474154528505it_int: nat > int > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
% 0.33/0.50      bit_se2925701944663578781it_nat: nat > nat > nat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
% 0.33/0.50      bit_se8260200283734997820nteger: nat > code_integer > code_integer ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
% 0.33/0.50      bit_se4203085406695923979it_int: nat > int > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
% 0.33/0.50      bit_se4205575877204974255it_nat: nat > nat > nat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
% 0.33/0.50      bit_se6526347334894502574or_int: int > int > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
% 0.33/0.50      bit_se6528837805403552850or_nat: nat > nat > nat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
% 0.33/0.50      bit_se1146084159140164899it_int: int > nat > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
% 0.33/0.50      bit_se1148574629649215175it_nat: nat > nat > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Otake__bit__num,type,
% 0.33/0.50      bit_take_bit_num: nat > num > option_num ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
% 0.33/0.50      bit_un7362597486090784418nd_num: num > num > option_num ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
% 0.33/0.50      bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
% 0.33/0.50      bit_un2480387367778600638or_num: num > num > option_num ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
% 0.33/0.50      bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Code__Numeral_Obit__cut__integer,type,
% 0.33/0.50      code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Code__Numeral_Odivmod__abs,type,
% 0.33/0.50      code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Code__Numeral_Odivmod__integer,type,
% 0.33/0.50      code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
% 0.33/0.50      code_int_of_integer: code_integer > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
% 0.33/0.50      code_integer_of_int: int > code_integer ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Code__Numeral_Ointeger__of__nat,type,
% 0.33/0.50      code_integer_of_nat: nat > code_integer ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Code__Numeral_Onat__of__integer,type,
% 0.33/0.50      code_nat_of_integer: code_integer > nat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 0.33/0.50      comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 0.33/0.50      comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
% 0.33/0.50      complete_Inf_Inf_nat: set_nat > nat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.50      comple7806235888213564991et_nat: set_set_nat > set_nat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
% 0.33/0.50      complete_Sup_Sup_nat: set_nat > nat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Complex_OArg,type,
% 0.33/0.50      arg: complex > real ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Complex_Ocis,type,
% 0.33/0.50      cis: real > complex ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Complex_Ocomplex_OComplex,type,
% 0.33/0.50      complex2: real > real > complex ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
% 0.33/0.50      condit2214826472909112428ve_nat: set_nat > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.33/0.50      differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Deriv_Ohas__derivative_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.33/0.50      has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
% 0.33/0.50      has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Divides_Oadjust__div,type,
% 0.33/0.50      adjust_div: product_prod_int_int > int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Divides_Odivmod__nat,type,
% 0.33/0.50      divmod_nat: nat > nat > product_prod_nat_nat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Divides_Oeucl__rel__int,type,
% 0.33/0.50      eucl_rel_int: int > int > product_prod_int_int > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
% 0.33/0.50      unique6319869463603278526ux_int: product_prod_int_int > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
% 0.33/0.50      unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
% 0.33/0.50      unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
% 0.33/0.50      unique5052692396658037445od_int: num > num > product_prod_int_int ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
% 0.33/0.50      unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).
% 0.33/0.50  
% 0.33/0.50  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
% 0.33/0.51      unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
% 0.33/0.51      unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      comm_s8582702949713902594nteger: code_integer > nat > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Complex__Ocomplex,type,
% 0.33/0.51      comm_s2602460028002588243omplex: complex > nat > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Int__Oint,type,
% 0.33/0.51      comm_s4660882817536571857er_int: int > nat > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Nat__Onat,type,
% 0.33/0.51      comm_s4663373288045622133er_nat: nat > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Rat__Orat,type,
% 0.33/0.51      comm_s4028243227959126397er_rat: rat > nat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
% 0.33/0.51      comm_s7457072308508201937r_real: real > nat > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      semiri3624122377584611663nteger: nat > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
% 0.33/0.51      semiri5044797733671781792omplex: nat > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
% 0.33/0.51      semiri1406184849735516958ct_int: nat > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
% 0.33/0.51      semiri1408675320244567234ct_nat: nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Rat__Orat,type,
% 0.33/0.51      semiri773545260158071498ct_rat: nat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
% 0.33/0.51      semiri2265585572941072030t_real: nat > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
% 0.33/0.51      invers8013647133539491842omplex: complex > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
% 0.33/0.51      inverse_inverse_rat: rat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
% 0.33/0.51      inverse_inverse_real: real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
% 0.33/0.51      at_bot_real: filter_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
% 0.33/0.51      at_top_nat: filter_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
% 0.33/0.51      at_top_real: filter_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
% 0.33/0.51      eventually_nat: ( nat > $o ) > filter_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Filter_Oeventually_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.51      eventu1038000079068216329at_nat: ( product_prod_nat_nat > $o ) > filter1242075044329608583at_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
% 0.33/0.51      eventually_real: ( real > $o ) > filter_real > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.51      filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.33/0.51      filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.33/0.51      filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 0.33/0.51      princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.33/0.51      princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Filter_Oprod__filter_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.51      prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
% 0.33/0.51      finite_card_complex: set_complex > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
% 0.33/0.51      finite_card_list_nat: set_list_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
% 0.33/0.51      finite_card_nat: set_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
% 0.33/0.51      finite410649719033368117t_unit: set_Product_unit > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      finite_card_set_nat: set_set_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
% 0.33/0.51      finite_finite_o: set_o > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
% 0.33/0.51      finite3207457112153483333omplex: set_complex > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
% 0.33/0.51      finite_finite_int: set_int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_I_Eo_J,type,
% 0.33/0.51      finite_finite_list_o: set_list_o > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 0.33/0.51      finite8712137658972009173omplex: set_list_complex > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Int__Oint_J,type,
% 0.33/0.51      finite3922522038869484883st_int: set_list_int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
% 0.33/0.51      finite8100373058378681591st_nat: set_list_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.51      finite500796754983035824at_nat: set_li5450038453877631591at_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.51      finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 0.33/0.51      finite_finite_nat: set_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ofinite_001t__Num__Onum,type,
% 0.33/0.51      finite_finite_num: set_num > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.51      finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.33/0.51      finite4343798906461161616at_nat: set_Pr4329608150637261639at_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ofinite_001t__Rat__Orat,type,
% 0.33/0.51      finite_finite_rat: set_rat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
% 0.33/0.51      finite_finite_real: set_real > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 0.33/0.51      finite6551019134538273531omplex: set_set_complex > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
% 0.33/0.51      finite6197958912794628473et_int: set_set_int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      finite1152437895449049373et_nat: set_set_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.51      finite9047747110432174090at_nat: set_se7855581050983116737at_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Finite__Set_Ofinite_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 0.33/0.51      bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 0.33/0.51      bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.51      bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun_Ocomp_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
% 0.33/0.51      comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.51      comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.51      inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.33/0.51      inj_on_real_real: ( real > real ) > set_real > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
% 0.33/0.51      inj_on_set_nat_nat: ( set_nat > nat ) > set_set_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.33/0.51      map_fu4960017516451851995nt_int: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > int > int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 0.33/0.51      map_fu3667384564859982768at_int: ( int > product_prod_nat_nat ) > ( product_prod_nat_nat > int ) > ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
% 0.33/0.51      map_fu4333342158222067775at_rat: ( rat > product_prod_int_int ) > ( ( product_prod_int_int > product_prod_int_int ) > rat > rat ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > rat > rat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 0.33/0.51      map_fu5673905371560938248nt_rat: ( rat > product_prod_int_int ) > ( product_prod_int_int > rat ) > ( product_prod_int_int > product_prod_int_int ) > rat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun_Ostrict__mono__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.51      strict1292158309912662752at_nat: ( nat > nat ) > set_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun_Othe__inv__into_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.33/0.51      the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun__Def_Omax__strict,type,
% 0.33/0.51      fun_max_strict: set_Pr4329608150637261639at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun__Def_Omax__weak,type,
% 0.33/0.51      fun_max_weak: set_Pr4329608150637261639at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun__Def_Omin__strict,type,
% 0.33/0.51      fun_min_strict: set_Pr4329608150637261639at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun__Def_Omin__weak,type,
% 0.33/0.51      fun_min_weak: set_Pr4329608150637261639at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun__Def_Opair__leq,type,
% 0.33/0.51      fun_pair_leq: set_Pr8693737435421807431at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun__Def_Opair__less,type,
% 0.33/0.51      fun_pair_less: set_Pr8693737435421807431at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Fun__Def_Oreduction__pair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.51      fun_re2478310338295953701at_nat: produc1319942482725812455at_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
% 0.33/0.51      gcd_Gcd_nat: set_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_GCD_Obezw,type,
% 0.33/0.51      bezw: nat > nat > product_prod_int_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_GCD_Obezw__rel,type,
% 0.33/0.51      bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Int__Oint,type,
% 0.33/0.51      gcd_gcd_int: int > int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Nat__Onat,type,
% 0.33/0.51      gcd_gcd_nat: nat > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_GCD_Ogcd__nat__rel,type,
% 0.33/0.51      gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Oabs__class_Oabs_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      abs_abs_Code_integer: code_integer > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex,type,
% 0.33/0.51      abs_abs_complex: complex > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
% 0.33/0.51      abs_abs_int: int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
% 0.33/0.51      abs_abs_rat: rat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
% 0.33/0.51      abs_abs_real: real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Int__Oint_M_Eo_J,type,
% 0.33/0.51      minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
% 0.33/0.51      minus_1139252259498527702_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.33/0.51      minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 0.33/0.51      minus_2270307095948843157_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Real__Oreal_M_Eo_J,type,
% 0.33/0.51      minus_minus_real_o: ( real > $o ) > ( real > $o ) > real > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
% 0.33/0.51      minus_6910147592129066416_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
% 0.33/0.51      minus_minus_complex: complex > complex > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
% 0.33/0.51      minus_minus_int: int > int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 0.33/0.51      minus_minus_nat: nat > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
% 0.33/0.51      minus_minus_rat: rat > rat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
% 0.33/0.51      minus_minus_real: real > real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 0.33/0.51      minus_811609699411566653omplex: set_complex > set_complex > set_complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
% 0.33/0.51      minus_minus_set_int: set_int > set_int > set_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 0.33/0.51      minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      minus_minus_set_nat: set_nat > set_nat > set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Num__Onum_J,type,
% 0.33/0.51      minus_minus_set_num: set_num > set_num > set_num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.51      minus_1356011639430497352at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 0.33/0.51      minus_3314409938677909166at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Rat__Orat_J,type,
% 0.33/0.51      minus_minus_set_rat: set_rat > set_rat > set_rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.33/0.51      minus_minus_set_real: set_real > set_real > set_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.33/0.51      minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Oone__class_Oone_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      one_one_Code_integer: code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
% 0.33/0.51      one_one_complex: complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
% 0.33/0.51      one_on7984719198319812577d_enat: extended_enat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
% 0.33/0.51      one_one_int: int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 0.33/0.51      one_one_nat: nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
% 0.33/0.51      one_one_rat: rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
% 0.33/0.51      one_one_real: real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Oplus__class_Oplus_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
% 0.33/0.51      plus_plus_complex: complex > complex > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
% 0.33/0.51      plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
% 0.33/0.51      plus_plus_int: int > int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 0.33/0.51      plus_plus_nat: nat > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
% 0.33/0.51      plus_plus_num: num > num > num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
% 0.33/0.51      plus_plus_rat: rat > rat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
% 0.33/0.51      plus_plus_real: real > real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Osgn__class_Osgn_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      sgn_sgn_Code_integer: code_integer > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Osgn__class_Osgn_001t__Complex__Ocomplex,type,
% 0.33/0.51      sgn_sgn_complex: complex > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
% 0.33/0.51      sgn_sgn_int: int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Osgn__class_Osgn_001t__Rat__Orat,type,
% 0.33/0.51      sgn_sgn_rat: rat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Osgn__class_Osgn_001t__Real__Oreal,type,
% 0.33/0.51      sgn_sgn_real: real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Otimes__class_Otimes_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      times_3573771949741848930nteger: code_integer > code_integer > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
% 0.33/0.51      times_times_complex: complex > complex > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nat__Oenat,type,
% 0.33/0.51      times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
% 0.33/0.51      times_times_int: int > int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
% 0.33/0.51      times_times_nat: nat > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
% 0.33/0.51      times_times_rat: rat > rat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
% 0.33/0.51      times_times_real: real > real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Int__Oint_M_Eo_J,type,
% 0.33/0.51      uminus_uminus_int_o: ( int > $o ) > int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
% 0.33/0.51      uminus5770388063884162150_nat_o: ( list_nat > $o ) > list_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.33/0.51      uminus_uminus_nat_o: ( nat > $o ) > nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 0.33/0.51      uminus8676089048583255045_nat_o: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Real__Oreal_M_Eo_J,type,
% 0.33/0.51      uminus_uminus_real_o: ( real > $o ) > real > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
% 0.33/0.51      uminus6401447641752708672_nat_o: ( set_nat > $o ) > set_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      uminus1351360451143612070nteger: code_integer > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
% 0.33/0.51      uminus1482373934393186551omplex: complex > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
% 0.33/0.51      uminus_uminus_int: int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
% 0.33/0.51      uminus_uminus_rat: rat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
% 0.33/0.51      uminus_uminus_real: real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Int__Oint_J,type,
% 0.33/0.51      uminus1532241313380277803et_int: set_int > set_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 0.33/0.51      uminus3195874150345416415st_nat: set_list_nat > set_list_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      uminus5710092332889474511et_nat: set_nat > set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.51      uminus6524753893492686040at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 0.33/0.51      uminus935396558254630718at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.33/0.51      uminus612125837232591019t_real: set_real > set_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.33/0.51      uminus613421341184616069et_nat: set_set_nat > set_set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ozero__class_Ozero_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      zero_z3403309356797280102nteger: code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
% 0.33/0.51      zero_zero_complex: complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
% 0.33/0.51      zero_z5237406670263579293d_enat: extended_enat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
% 0.33/0.51      zero_zero_int: int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.33/0.51      zero_zero_nat: nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,type,
% 0.33/0.51      zero_zero_rat: rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 0.33/0.51      zero_zero_real: real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_Eo_001t__Nat__Onat,type,
% 0.33/0.51      groups8507830703676809646_o_nat: ( $o > nat ) > set_o > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 0.33/0.51      groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Int__Oint,type,
% 0.33/0.51      groups5690904116761175830ex_int: ( complex > int ) > set_complex > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Nat__Onat,type,
% 0.33/0.51      groups5693394587270226106ex_nat: ( complex > nat ) > set_complex > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Rat__Orat,type,
% 0.33/0.51      groups5058264527183730370ex_rat: ( complex > rat ) > set_complex > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Real__Oreal,type,
% 0.33/0.51      groups5808333547571424918x_real: ( complex > real ) > set_complex > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      groups7873554091576472773nteger: ( int > code_integer ) > set_int > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Complex__Ocomplex,type,
% 0.33/0.51      groups3049146728041665814omplex: ( int > complex ) > set_int > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Int__Oint,type,
% 0.33/0.51      groups4538972089207619220nt_int: ( int > int ) > set_int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Nat__Onat,type,
% 0.33/0.51      groups4541462559716669496nt_nat: ( int > nat ) > set_int > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Rat__Orat,type,
% 0.33/0.51      groups3906332499630173760nt_rat: ( int > rat ) > set_int > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Real__Oreal,type,
% 0.33/0.51      groups8778361861064173332t_real: ( int > real ) > set_int > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      groups7501900531339628137nteger: ( nat > code_integer ) > set_nat > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 0.33/0.51      groups2073611262835488442omplex: ( nat > complex ) > set_nat > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
% 0.33/0.51      groups7108830773950497114d_enat: ( nat > extended_enat ) > set_nat > extended_enat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
% 0.33/0.51      groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.51      groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Rat__Orat,type,
% 0.33/0.51      groups2906978787729119204at_rat: ( nat > rat ) > set_nat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.33/0.51      groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 0.33/0.51      groups977919841031483927at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Real__Oreal,type,
% 0.33/0.51      groups4567486121110086003t_real: ( product_prod_nat_nat > real ) > set_Pr1261947904930325089at_nat > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Nat__Onat,type,
% 0.33/0.51      groups3860910324918113789at_nat: ( produc3843707927480180839at_nat > nat ) > set_Pr4329608150637261639at_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      groups7713935264441627589nteger: ( real > code_integer ) > set_real > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Complex__Ocomplex,type,
% 0.33/0.51      groups5754745047067104278omplex: ( real > complex ) > set_real > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Int__Oint,type,
% 0.33/0.51      groups1932886352136224148al_int: ( real > int ) > set_real > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Nat__Onat,type,
% 0.33/0.51      groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Rat__Orat,type,
% 0.33/0.51      groups1300246762558778688al_rat: ( real > rat ) > set_real > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.33/0.51      groups8097168146408367636l_real: ( real > real ) > set_real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
% 0.33/0.51      groups8294997508430121362at_nat: ( set_nat > nat ) > set_set_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 0.33/0.51      groups771621172384141258BT_nat: ( vEBT_VEBT > nat ) > set_VEBT_VEBT > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 0.33/0.51      groups808145749697022017omplex: ( complex > complex ) > set_complex > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Complex__Ocomplex_001t__Int__Oint,type,
% 0.33/0.51      groups2909182065852811199ex_int: ( complex > int ) > set_complex > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Complex__Ocomplex_001t__Nat__Onat,type,
% 0.33/0.51      groups2911672536361861475ex_nat: ( complex > nat ) > set_complex > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Complex__Ocomplex_001t__Rat__Orat,type,
% 0.33/0.51      groups2276542476275365739ex_rat: ( complex > rat ) > set_complex > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Complex__Ocomplex_001t__Real__Oreal,type,
% 0.33/0.51      groups5737402329758386879x_real: ( complex > real ) > set_complex > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Int__Oint_001t__Complex__Ocomplex,type,
% 0.33/0.51      groups267424677133301183omplex: ( int > complex ) > set_int > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Int__Oint_001t__Int__Oint,type,
% 0.33/0.51      groups2983280209131991357nt_int: ( int > int ) > set_int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Int__Oint_001t__Nat__Onat,type,
% 0.33/0.51      groups2985770679641041633nt_nat: ( int > nat ) > set_int > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Int__Oint_001t__Rat__Orat,type,
% 0.33/0.51      groups2350640619554545897nt_rat: ( int > rat ) > set_int > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Int__Oint_001t__Real__Oreal,type,
% 0.33/0.51      groups1523912220035142973t_real: ( int > real ) > set_int > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 0.33/0.51      groups8515261248781899619omplex: ( nat > complex ) > set_nat > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Nat__Onat_001t__Int__Oint,type,
% 0.33/0.51      groups1983926497230936801at_int: ( nat > int ) > set_nat > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.51      groups1986416967739987077at_nat: ( nat > nat ) > set_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Nat__Onat_001t__Rat__Orat,type,
% 0.33/0.51      groups1351286907653491341at_rat: ( nat > rat ) > set_nat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.33/0.51      groups8560362682196896993t_real: ( nat > real ) > set_nat > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Real__Oreal_001t__Complex__Ocomplex,type,
% 0.33/0.51      groups5683813829254066239omplex: ( real > complex ) > set_real > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Real__Oreal_001t__Int__Oint,type,
% 0.33/0.51      groups3901808747961969597al_int: ( real > int ) > set_real > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Real__Oreal_001t__Nat__Onat,type,
% 0.33/0.51      groups3904299218471019873al_nat: ( real > nat ) > set_real > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Real__Oreal_001t__Rat__Orat,type,
% 0.33/0.51      groups3269169158384524137al_rat: ( real > rat ) > set_real > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.33/0.51      groups97945582718554045l_real: ( real > real ) > set_real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      groups8682486955453173170nteger: ( complex > code_integer ) > set_complex > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 0.33/0.51      groups3708469109370488835omplex: ( complex > complex ) > set_complex > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Int__Oint,type,
% 0.33/0.51      groups858564598930262913ex_int: ( complex > int ) > set_complex > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Nat__Onat,type,
% 0.33/0.51      groups861055069439313189ex_nat: ( complex > nat ) > set_complex > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Rat__Orat,type,
% 0.33/0.51      groups225925009352817453ex_rat: ( complex > rat ) > set_complex > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Real__Oreal,type,
% 0.33/0.51      groups766887009212190081x_real: ( complex > real ) > set_complex > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      groups3827104343326376752nteger: ( int > code_integer ) > set_int > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Complex__Ocomplex,type,
% 0.33/0.51      groups7440179247065528705omplex: ( int > complex ) > set_int > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Int__Oint,type,
% 0.33/0.51      groups1705073143266064639nt_int: ( int > int ) > set_int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Nat__Onat,type,
% 0.33/0.51      groups1707563613775114915nt_nat: ( int > nat ) > set_int > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Rat__Orat,type,
% 0.33/0.51      groups1072433553688619179nt_rat: ( int > rat ) > set_int > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Real__Oreal,type,
% 0.33/0.51      groups2316167850115554303t_real: ( int > real ) > set_int > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
% 0.33/0.51      groups2907647131375434839at_nat: ( list_nat > nat ) > set_list_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      groups3455450783089532116nteger: ( nat > code_integer ) > set_nat > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 0.33/0.51      groups6464643781859351333omplex: ( nat > complex ) > set_nat > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Int__Oint,type,
% 0.33/0.51      groups705719431365010083at_int: ( nat > int ) > set_nat > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.51      groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Rat__Orat,type,
% 0.33/0.51      groups73079841787564623at_rat: ( nat > rat ) > set_nat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.33/0.51      groups129246275422532515t_real: ( nat > real ) > set_nat > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 0.33/0.51      groups4077766827762148844at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      groups6225526099057966256nteger: ( real > code_integer ) > set_real > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Complex__Ocomplex,type,
% 0.33/0.51      groups713298508707869441omplex: ( real > complex ) > set_real > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Int__Oint,type,
% 0.33/0.51      groups4694064378042380927al_int: ( real > int ) > set_real > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Nat__Onat,type,
% 0.33/0.51      groups4696554848551431203al_nat: ( real > nat ) > set_real > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Rat__Orat,type,
% 0.33/0.51      groups4061424788464935467al_rat: ( real > rat ) > set_real > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.33/0.51      groups1681761925125756287l_real: ( real > real ) > set_real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
% 0.33/0.51      groups4248547760180025341at_nat: ( set_nat > nat ) > set_set_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001_Eo_001t__Int__Oint,type,
% 0.33/0.51      groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
% 0.33/0.51      groups4561878855575611511st_nat: list_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_HOL_OThe_001t__Int__Oint,type,
% 0.33/0.51      the_int: ( int > $o ) > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_HOL_OThe_001t__Nat__Onat,type,
% 0.33/0.51      the_nat: ( nat > $o ) > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 0.33/0.51      the_Pr8210177043389155639nteger: ( produc8763457246119570046nteger > $o ) > produc8763457246119570046nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.33/0.51      the_Pr6882841213465913158nt_int: ( produc7773217078559923341nt_int > $o ) > produc7773217078559923341nt_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 0.33/0.51      the_Pr6653488032121699663nteger: ( produc1908205239877642774nteger > $o ) > produc1908205239877642774nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.33/0.51      the_Pr5445864913131713084nt_int: ( produc2285326912895808259nt_int > $o ) > produc2285326912895808259nt_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.33/0.51      the_Pr4378521158711661632nt_int: ( product_prod_int_int > $o ) > product_prod_int_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.51      the_Pr7557018466319803784at_nat: ( product_prod_nat_nat > $o ) > product_prod_nat_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.33/0.51      the_Pr4604535484834969198at_nat: ( produc3843707927480180839at_nat > $o ) > produc3843707927480180839at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_HOL_OThe_001t__Real__Oreal,type,
% 0.33/0.51      the_real: ( real > $o ) > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_If_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.33/0.51      if_int_int: $o > ( int > int ) > ( int > int ) > int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_If_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      if_Code_integer: $o > code_integer > code_integer > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_If_001t__Complex__Ocomplex,type,
% 0.33/0.51      if_complex: $o > complex > complex > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_If_001t__Extended____Nat__Oenat,type,
% 0.33/0.51      if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_If_001t__Int__Oint,type,
% 0.33/0.51      if_int: $o > int > int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
% 0.33/0.51      if_list_int: $o > list_int > list_int > list_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
% 0.33/0.51      if_list_nat: $o > list_nat > list_nat > list_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_If_001t__Nat__Onat,type,
% 0.33/0.51      if_nat: $o > nat > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_If_001t__Num__Onum,type,
% 0.33/0.51      if_num: $o > num > num > num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
% 0.33/0.51      if_option_nat: $o > option_nat > option_nat > option_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_If_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.33/0.51      if_option_num: $o > option_num > option_num > option_num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 0.33/0.51      if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.33/0.51      if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.33/0.51      if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.51      if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_If_001t__Rat__Orat,type,
% 0.33/0.51      if_rat: $o > rat > rat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_If_001t__Real__Oreal,type,
% 0.33/0.51      if_real: $o > real > real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
% 0.33/0.51      if_set_int: $o > set_int > set_int > set_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_If_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
% 0.33/0.51      infini8530281810654367211te_nat: set_nat > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Int_OAbs__Integ,type,
% 0.33/0.51      abs_Integ: product_prod_nat_nat > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Int_ORep__Integ,type,
% 0.33/0.51      rep_Integ: int > product_prod_nat_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Int_Oint__ge__less__than,type,
% 0.33/0.51      int_ge_less_than: int > set_Pr958786334691620121nt_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Int_Oint__ge__less__than2,type,
% 0.33/0.51      int_ge_less_than2: int > set_Pr958786334691620121nt_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Int_Ointrel,type,
% 0.33/0.51      intrel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Int_Onat,type,
% 0.33/0.51      nat2: int > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Int_Opcr__int,type,
% 0.33/0.51      pcr_int: product_prod_nat_nat > int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Int_Opower__int_001t__Real__Oreal,type,
% 0.33/0.51      power_int_real: real > int > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      ring_18347121197199848620nteger: int > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
% 0.33/0.51      ring_17405671764205052669omplex: int > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
% 0.33/0.51      ring_1_of_int_int: int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
% 0.33/0.51      ring_1_of_int_rat: int > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
% 0.33/0.51      ring_1_of_int_real: int > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
% 0.33/0.51      inf_inf_nat: nat > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.51      inf_in2572325071724192079at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
% 0.33/0.51      semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
% 0.33/0.51      sup_sup_nat: nat > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      sup_sup_set_nat: set_nat > set_nat > set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 0.33/0.51      sup_su5525570899277871387at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
% 0.33/0.51      lattic8265883725875713057ax_nat: set_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Nat__Onat,type,
% 0.33/0.51      lattic5364784637807008409ex_nat: ( complex > nat ) > set_complex > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Num__Onum,type,
% 0.33/0.51      lattic1922116423962787043ex_num: ( complex > num ) > set_complex > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Rat__Orat,type,
% 0.33/0.51      lattic4729654577720512673ex_rat: ( complex > rat ) > set_complex > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Real__Oreal,type,
% 0.33/0.51      lattic8794016678065449205x_real: ( complex > real ) > set_complex > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Num__Onum,type,
% 0.33/0.51      lattic5003618458639192673nt_num: ( int > num ) > set_int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Rat__Orat,type,
% 0.33/0.51      lattic7811156612396918303nt_rat: ( int > rat ) > set_int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Real__Oreal,type,
% 0.33/0.51      lattic2675449441010098035t_real: ( int > real ) > set_int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Num__Onum,type,
% 0.33/0.51      lattic4004264746738138117at_num: ( nat > num ) > set_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Rat__Orat,type,
% 0.33/0.51      lattic6811802900495863747at_rat: ( nat > rat ) > set_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.33/0.51      lattic488527866317076247t_real: ( nat > real ) > set_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Nat__Onat,type,
% 0.33/0.51      lattic5055836439445974935al_nat: ( real > nat ) > set_real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Num__Onum,type,
% 0.33/0.51      lattic1613168225601753569al_num: ( real > num ) > set_real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Rat__Orat,type,
% 0.33/0.51      lattic4420706379359479199al_rat: ( real > rat ) > set_real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.33/0.51      lattic8440615504127631091l_real: ( real > real ) > set_real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oappend_001t__Int__Oint,type,
% 0.33/0.51      append_int: list_int > list_int > list_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oappend_001t__Nat__Onat,type,
% 0.33/0.51      append_nat: list_nat > list_nat > list_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ocount__list_001_Eo,type,
% 0.33/0.51      count_list_o: list_o > $o > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ocount__list_001t__Complex__Ocomplex,type,
% 0.33/0.51      count_list_complex: list_complex > complex > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ocount__list_001t__Int__Oint,type,
% 0.33/0.51      count_list_int: list_int > int > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
% 0.33/0.51      count_list_nat: list_nat > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ocount__list_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.51      count_4203492906077236349at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ocount__list_001t__Real__Oreal,type,
% 0.33/0.51      count_list_real: list_real > real > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ocount__list_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      count_list_set_nat: list_set_nat > set_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ocount__list_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      count_list_VEBT_VEBT: list_VEBT_VEBT > vEBT_VEBT > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Odistinct_001t__Int__Oint,type,
% 0.33/0.51      distinct_int: list_int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
% 0.33/0.51      distinct_nat: list_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Odrop_001t__Nat__Onat,type,
% 0.33/0.51      drop_nat: nat > list_nat > list_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oenumerate_001_Eo,type,
% 0.33/0.51      enumerate_o: nat > list_o > list_P7333126701944960589_nat_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oenumerate_001t__Int__Oint,type,
% 0.33/0.51      enumerate_int: nat > list_int > list_P3521021558325789923at_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oenumerate_001t__Nat__Onat,type,
% 0.33/0.51      enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oenumerate_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      enumerate_VEBT_VEBT: nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ofind_001_Eo,type,
% 0.33/0.51      find_o: ( $o > $o ) > list_o > option_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ofind_001t__Int__Oint,type,
% 0.33/0.51      find_int: ( int > $o ) > list_int > option_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ofind_001t__Nat__Onat,type,
% 0.33/0.51      find_nat: ( nat > $o ) > list_nat > option_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ofind_001t__Num__Onum,type,
% 0.33/0.51      find_num: ( num > $o ) > list_num > option_num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ofind_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.51      find_P8199882355184865565at_nat: ( product_prod_nat_nat > $o ) > list_P6011104703257516679at_nat > option4927543243414619207at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ofind_001t__Real__Oreal,type,
% 0.33/0.51      find_real: ( real > $o ) > list_real > option_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ofind_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      find_set_nat: ( set_nat > $o ) > list_set_nat > option_set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ofind_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      find_VEBT_VEBT: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > option_VEBT_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.51      fold_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olast_001t__Nat__Onat,type,
% 0.33/0.51      last_nat: list_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olinorder__class_Osort__key_001t__Int__Oint_001t__Int__Oint,type,
% 0.33/0.51      linord1735203802627413978nt_int: ( int > int ) > list_int > list_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olinorder__class_Osort__key_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.51      linord738340561235409698at_nat: ( nat > nat ) > list_nat > list_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
% 0.33/0.51      linord2614967742042102400et_nat: set_nat > list_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
% 0.33/0.51      cons_int: int > list_int > list_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
% 0.33/0.51      cons_nat: nat > list_nat > list_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
% 0.33/0.51      nil_int: list_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
% 0.33/0.51      nil_nat: list_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
% 0.33/0.51      hd_nat: list_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.51      map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 0.33/0.51      set_Co8062243466402858685e_term: list_C878401137130745250e_term > set_Co9149898834107579976e_term ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 0.33/0.51      set_in5217446777445088012e_term: list_i8448526496819171953e_term > set_in3461395444621081367e_term ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 0.33/0.51      set_Pr8342322266483756581e_term: list_P1316552470764441098e_term > set_Pr7604974323444597168e_term ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 0.33/0.51      set_Pr16608062948090134e_term: list_P1743416141875011707e_term > set_Pr3642885161833720865e_term ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001_Eo,type,
% 0.33/0.51      set_o2: list_o > set_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
% 0.33/0.51      set_complex2: list_complex > set_complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
% 0.33/0.51      set_int2: list_int > set_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
% 0.33/0.51      set_nat2: list_nat > set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Num__Onum,type,
% 0.33/0.51      set_num2: list_num > set_num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 0.33/0.51      set_Pr2999063419360598313nteger: list_P5311841565141990158nteger > set_Pr8056137968301705908nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.33/0.51      set_Pr1633835384712236856nt_int: list_P8915022641806594461nt_int > set_Pr1872883991513573699nt_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 0.33/0.51      set_Pr2135590979564877377nteger: list_P7828571989066258726nteger > set_Pr1281608226676607948nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.33/0.51      set_Pr4943052134776177454nt_int: list_P651320350408439699nt_int > set_Pr9222295170931077689nt_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
% 0.33/0.51      set_Product_prod_o_o2: list_P4002435161011370285od_o_o > set_Product_prod_o_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 0.33/0.51      set_Pr2828948584524939422_o_int: list_P3795440434834930179_o_int > set_Pr8834758594704517033_o_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
% 0.33/0.51      set_Pr7006799604034136130_o_nat: list_P6285523579766656935_o_nat > set_Pr2101469702781467981_o_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J,type,
% 0.33/0.51      set_Pr2600826154070092190o_real: list_P5232166724548748803o_real > set_Pr6573716822653411497o_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.51      set_Pr655345902815428824T_VEBT: list_P7495141550334521929T_VEBT > set_Pr7543698050874017315T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.33/0.51      set_Pr920681315882439344nteger: list_P5578671422887162913nteger > set_Pr4811707699266497531nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.33/0.51      set_Pr2470121279949933262nt_int: list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
% 0.33/0.51      set_Pr112895574167722958t_real: list_P6863124054624500543t_real > set_Pr3538720872664544793t_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.51      set_Pr8714266321650254504T_VEBT: list_P7524865323317820941T_VEBT > set_Pr8044002425091019955T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.51      set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.51      set_Pr5518436109238095868at_nat: list_P8469869581646625389at_nat > set_Pr8693737435421807431at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_M_Eo_J,type,
% 0.33/0.51      set_Pr5196769464307566348real_o: list_P3595434254542482545real_o > set_Pr4936984352647145239real_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
% 0.33/0.51      set_Pr8219819362198175822al_int: list_P4344331454722006975al_int > set_Pr1019928272762051225al_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
% 0.33/0.51      set_Pr3174298344852596722al_nat: list_P6834414599653733731al_nat > set_Pr3510011417693777981al_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.33/0.51      set_Pr5999470521830281550l_real: list_P8689742595348180415l_real > set_Pr6218003697084177305l_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.51      set_Pr8897343066327330088T_VEBT: list_P877281246627933069T_VEBT > set_Pr6019664923565264691T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
% 0.33/0.51      set_Pr9040384385603167362et_nat: list_P6254988961118846195et_nat > set_Pr5488025237498180813et_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.33/0.51      set_Pr3765526544606949372at_nat: list_P5464809261938338413at_nat > set_Pr4329608150637261639at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 0.33/0.51      set_Pr7708085864119495200VEBT_o: list_P3126845725202233233VEBT_o > set_Pr3175402225741728619VEBT_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 0.33/0.51      set_Pr2853735649769556538BT_int: list_P4547456442757143711BT_int > set_Pr5066593544530342725BT_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 0.33/0.51      set_Pr7031586669278753246BT_nat: list_P7037539587688870467BT_nat > set_Pr7556676689462069481BT_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
% 0.33/0.51      set_Pr1087130671499945274T_real: list_P2623026923184700063T_real > set_Pr7765410600122031685T_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.51      set_Pr9182192707038809660T_VEBT: list_P7413028617227757229T_VEBT > set_Pr6192946355708809607T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
% 0.33/0.51      set_real2: list_real > set_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      set_set_nat2: list_set_nat > set_set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
% 0.33/0.51      tl_nat: list_nat > list_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist__update_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 0.33/0.51      list_u4743598893156345252e_term: list_C878401137130745250e_term > nat > ( code_integer > option6357759511663192854e_term ) > list_C878401137130745250e_term ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist__update_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 0.33/0.51      list_u8946639151299769843e_term: list_i8448526496819171953e_term > nat > ( int > option6357759511663192854e_term ) > list_i8448526496819171953e_term ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist__update_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 0.33/0.51      list_u877304756163299468e_term: list_P1316552470764441098e_term > nat > ( produc6241069584506657477e_term > option6357759511663192854e_term ) > list_P1316552470764441098e_term ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist__update_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 0.33/0.51      list_u3533491785856317309e_term: list_P1743416141875011707e_term > nat > ( produc8551481072490612790e_term > option6357759511663192854e_term ) > list_P1743416141875011707e_term ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist__update_001_Eo,type,
% 0.33/0.51      list_update_o: list_o > nat > $o > list_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist__update_001t__Int__Oint,type,
% 0.33/0.51      list_update_int: list_int > nat > int > list_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
% 0.33/0.51      list_update_nat: list_nat > nat > nat > list_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 0.33/0.51      list_u1133519416628930960nteger: list_P5311841565141990158nteger > nat > produc8763457246119570046nteger > list_P5311841565141990158nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.33/0.51      list_u4780935413889332127nt_int: list_P8915022641806594461nt_int > nat > produc7773217078559923341nt_int > list_P8915022641806594461nt_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 0.33/0.51      list_u234853988314817064nteger: list_P7828571989066258726nteger > nat > produc1908205239877642774nteger > list_P7828571989066258726nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.33/0.51      list_u7736365598306452245nt_int: list_P651320350408439699nt_int > nat > produc2285326912895808259nt_int > list_P651320350408439699nt_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.33/0.51      list_u2254550707601501961nteger: list_P5578671422887162913nteger > nat > produc8923325533196201883nteger > list_P5578671422887162913nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.33/0.51      list_u3002344382305578791nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int > list_P5707943133018811711nt_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.51      list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.33/0.51      list_u4696772448584712917at_nat: list_P5464809261938338413at_nat > nat > produc3843707927480180839at_nat > list_P5464809261938338413at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.51      list_u6961636818849549845T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT > list_P7413028617227757229T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
% 0.33/0.51      list_update_real: list_real > nat > real > list_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      list_update_set_nat: list_set_nat > nat > set_nat > list_set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olistrel1_001_Eo,type,
% 0.33/0.51      listrel1_o: set_Product_prod_o_o > set_Pr6227168374412355847list_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olistrel1_001t__Int__Oint,type,
% 0.33/0.51      listrel1_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olistrel1_001t__Nat__Onat,type,
% 0.33/0.51      listrel1_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olistrel1_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      listrel1_VEBT_VEBT: set_Pr6192946355708809607T_VEBT > set_Pr1916528119006554503T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olistrel1p_001t__Int__Oint,type,
% 0.33/0.51      listrel1p_int: ( int > int > $o ) > list_int > list_int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Olistrel1p_001t__Nat__Onat,type,
% 0.33/0.51      listrel1p_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Onth_001_Eo,type,
% 0.33/0.51      nth_o: list_o > nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Onth_001t__Int__Oint,type,
% 0.33/0.51      nth_int: list_int > nat > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 0.33/0.51      nth_nat: list_nat > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Onth_001t__Num__Onum,type,
% 0.33/0.51      nth_num: list_num > nat > num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
% 0.33/0.51      nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 0.33/0.51      nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
% 0.33/0.51      nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.51      nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
% 0.33/0.51      nth_Pr112076138515278198_nat_o: list_P7333126701944960589_nat_o > nat > product_prod_nat_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
% 0.33/0.51      nth_Pr3440142176431000676at_int: list_P3521021558325789923at_int > nat > product_prod_nat_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.51      nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.51      nth_Pr744662078594809490T_VEBT: list_P5647936690300460905T_VEBT > nat > produc8025551001238799321T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 0.33/0.51      nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 0.33/0.51      nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 0.33/0.51      nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.51      nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Onth_001t__Real__Oreal,type,
% 0.33/0.51      nth_real: list_real > nat > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      nth_set_nat: list_set_nat > nat > set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
% 0.33/0.51      product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
% 0.33/0.51      product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
% 0.33/0.51      product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
% 0.33/0.51      product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 0.33/0.51      product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 0.33/0.51      produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 0.33/0.51      produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oremdups_001t__Nat__Onat,type,
% 0.33/0.51      remdups_nat: list_nat > list_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oreplicate_001_Eo,type,
% 0.33/0.51      replicate_o: nat > $o > list_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
% 0.33/0.51      replicate_int: nat > int > list_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
% 0.33/0.51      replicate_nat: nat > nat > list_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.51      replic4235873036481779905at_nat: nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.33/0.51      replic2264142908078655527at_nat: nat > produc3843707927480180839at_nat > list_P5464809261938338413at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
% 0.33/0.51      replicate_real: nat > real > list_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      replicate_set_nat: nat > set_nat > list_set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
% 0.33/0.51      sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
% 0.33/0.51      sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Otake_001t__Nat__Onat,type,
% 0.33/0.51      take_nat: nat > list_nat > list_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oupt,type,
% 0.33/0.51      upt: nat > nat > list_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oupto,type,
% 0.33/0.51      upto: int > int > list_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oupto__aux,type,
% 0.33/0.51      upto_aux: int > int > list_int > list_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Oupto__rel,type,
% 0.33/0.51      upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.33/0.51      zip_Co8729459035503499408nteger: list_C878401137130745250e_term > list_P5578671422887162913nteger > list_P5311841565141990158nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.33/0.51      zip_in8766932505889695135nt_int: list_i8448526496819171953e_term > list_P5707943133018811711nt_int > list_P8915022641806594461nt_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.33/0.51      zip_Pr8292346330294042792nteger: list_P1316552470764441098e_term > list_P5578671422887162913nteger > list_P7828571989066258726nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.33/0.51      zip_Pr4168994715204986005nt_int: list_P1743416141875011707e_term > list_P5707943133018811711nt_int > list_P651320350408439699nt_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001_Eo_001_Eo,type,
% 0.33/0.51      zip_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001_Eo_001t__Int__Oint,type,
% 0.33/0.51      zip_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001_Eo_001t__Nat__Onat,type,
% 0.33/0.51      zip_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001_Eo_001t__Real__Oreal,type,
% 0.33/0.51      zip_o_real: list_o > list_real > list_P5232166724548748803o_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      zip_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001t__Int__Oint_001t__Int__Oint,type,
% 0.33/0.51      zip_int_int: list_int > list_int > list_P5707943133018811711nt_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001t__Int__Oint_001t__Real__Oreal,type,
% 0.33/0.51      zip_int_real: list_int > list_real > list_P6863124054624500543t_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      zip_int_VEBT_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001t__Nat__Onat_001_Eo,type,
% 0.33/0.51      zip_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.51      zip_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      zip_nat_VEBT_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.51      zip_Pr4664179122662387191at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P8469869581646625389at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001t__Real__Oreal_001_Eo,type,
% 0.33/0.51      zip_real_o: list_real > list_o > list_P3595434254542482545real_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Int__Oint,type,
% 0.33/0.51      zip_real_int: list_real > list_int > list_P4344331454722006975al_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Nat__Onat,type,
% 0.33/0.51      zip_real_nat: list_real > list_nat > list_P6834414599653733731al_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.33/0.51      zip_real_real: list_real > list_real > list_P8689742595348180415l_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      zip_real_VEBT_VEBT: list_real > list_VEBT_VEBT > list_P877281246627933069T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      zip_set_nat_set_nat: list_set_nat > list_set_nat > list_P6254988961118846195et_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 0.33/0.51      zip_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 0.33/0.51      zip_VEBT_VEBT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 0.33/0.51      zip_VEBT_VEBT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
% 0.33/0.51      zip_VEBT_VEBT_real: list_VEBT_VEBT > list_real > list_P2623026923184700063T_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      zip_VE537291747668921783T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_OSuc,type,
% 0.33/0.51      suc: nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.51      compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 0.33/0.51      case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 0.33/0.51      case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.33/0.51      case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Onat_Opred,type,
% 0.33/0.51      pred: nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      semiri4939895301339042750nteger: nat > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
% 0.33/0.51      semiri8010041392384452111omplex: nat > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
% 0.33/0.51      semiri4216267220026989637d_enat: nat > extended_enat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 0.33/0.51      semiri1314217659103216013at_int: nat > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 0.33/0.51      semiri1316708129612266289at_nat: nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
% 0.33/0.51      semiri681578069525770553at_rat: nat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 0.33/0.51      semiri5074537144036343181t_real: nat > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
% 0.33/0.51      semiri2816024913162550771omplex: ( complex > complex ) > nat > complex > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
% 0.33/0.51      semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
% 0.33/0.51      semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
% 0.33/0.51      semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
% 0.33/0.51      semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
% 0.33/0.51      size_size_list_o: list_o > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 0.33/0.51      size_s3451745648224563538omplex: list_complex > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
% 0.33/0.51      size_size_list_int: list_int > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 0.33/0.51      size_size_list_nat: list_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
% 0.33/0.51      size_size_list_num: list_num > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 0.33/0.51      size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
% 0.33/0.51      size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
% 0.33/0.51      size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.33/0.51      size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
% 0.33/0.51      size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
% 0.33/0.51      size_s2970893825323803983at_int: list_P3521021558325789923at_int > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.51      size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.33/0.51      size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
% 0.33/0.51      size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
% 0.33/0.51      size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
% 0.33/0.51      size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.33/0.51      size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
% 0.33/0.51      size_size_list_real: list_real > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.33/0.51      size_s3254054031482475050et_nat: list_set_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.51      size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
% 0.33/0.51      size_size_num: num > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
% 0.33/0.51      size_size_option_nat: option_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.33/0.51      size_size_option_num: option_num > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.51      size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      size_size_VEBT_VEBT: vEBT_VEBT > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 0.33/0.51      nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 0.33/0.51      nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat__Bijection_Oset__decode,type,
% 0.33/0.51      nat_set_decode: nat > set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Nat__Bijection_Oset__encode,type,
% 0.33/0.51      nat_set_encode: set_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_NthRoot_Oroot,type,
% 0.33/0.51      root: nat > real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_NthRoot_Osqrt,type,
% 0.33/0.51      sqrt: real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_OBitM,type,
% 0.33/0.51      bitM: num > num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_Oinc,type,
% 0.33/0.51      inc: num > num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      neg_nu7757733837767384882nteger: code_integer > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
% 0.33/0.51      neg_nu6511756317524482435omplex: complex > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
% 0.33/0.51      neg_nu3811975205180677377ec_int: int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
% 0.33/0.51      neg_nu3179335615603231917ec_rat: rat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
% 0.33/0.51      neg_nu6075765906172075777c_real: real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_Onum_OBit0,type,
% 0.33/0.51      bit0: num > num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_Onum_OBit1,type,
% 0.33/0.51      bit1: num > num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_Onum_OOne,type,
% 0.33/0.51      one: num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.33/0.51      case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_Onum_Osize__num,type,
% 0.33/0.51      size_num: num > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_Onum__of__nat,type,
% 0.33/0.51      num_of_nat: nat > num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      numera6620942414471956472nteger: num > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
% 0.33/0.51      numera6690914467698888265omplex: num > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
% 0.33/0.51      numera1916890842035813515d_enat: num > extended_enat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
% 0.33/0.51      numeral_numeral_int: num > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
% 0.33/0.51      numeral_numeral_nat: num > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
% 0.33/0.51      numeral_numeral_rat: num > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
% 0.33/0.51      numeral_numeral_real: num > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Num_Opred__numeral,type,
% 0.33/0.51      pred_numeral: num > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_ONone_001t__Int__Oint,type,
% 0.33/0.51      none_int: option_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
% 0.33/0.51      none_nat: option_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
% 0.33/0.51      none_num: option_num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.51      none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_ONone_001t__Real__Oreal,type,
% 0.33/0.51      none_real: option_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      none_set_nat: option_set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_ONone_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      none_VEBT_VEBT: option_VEBT_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_OSome_001_Eo,type,
% 0.33/0.51      some_o: $o > option_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
% 0.33/0.51      some_int: int > option_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
% 0.33/0.51      some_nat: nat > option_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
% 0.33/0.51      some_num: num > option_num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.51      some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_OSome_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      some_VEBT_VEBT: vEBT_VEBT > option_VEBT_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.51      case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
% 0.33/0.51      case_option_num_num: num > ( num > num ) > option_num > num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
% 0.33/0.51      case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
% 0.33/0.51      map_option_num_num: ( num > num ) > option_num > option_num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_Osize__option_001t__Nat__Onat,type,
% 0.33/0.51      size_option_nat: ( nat > nat ) > option_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
% 0.33/0.51      size_option_num: ( num > nat ) > option_num > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.51      size_o8335143837870341156at_nat: ( product_prod_nat_nat > nat ) > option4927543243414619207at_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
% 0.33/0.51      the_nat2: option_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_Othe_001t__Num__Onum,type,
% 0.33/0.51      the_num: option_num > num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Option_Ooption_Othe_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.51      the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Order__Relation_OunderS_001t__Nat__Onat,type,
% 0.33/0.51      order_underS_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
% 0.33/0.51      bot_bo5358457235160185703eger_o: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
% 0.33/0.51      bot_bo1403522918969695512_int_o: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
% 0.33/0.51      bot_bo3000040243691356879eger_o: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
% 0.33/0.51      bot_bo8662317086119403298_int_o: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
% 0.33/0.51      bot_bot_int_int_o: int > int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_Eo_J,type,
% 0.33/0.51      bot_bot_int_o: int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
% 0.33/0.51      bot_bot_list_nat_o: list_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
% 0.33/0.51      bot_bot_nat_nat_o: nat > nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.33/0.51      bot_bot_nat_o: nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 0.33/0.51      bot_bo482883023278783056_nat_o: product_prod_nat_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Real__Oreal_M_Eo_J,type,
% 0.33/0.51      bot_bot_real_o: real > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
% 0.33/0.51      bot_bot_set_nat_o: set_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
% 0.33/0.51      bot_bo4199563552545308370d_enat: extended_enat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Nat__Onat_J,type,
% 0.33/0.51      bot_bot_filter_nat: filter_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
% 0.33/0.51      bot_bot_nat: nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 0.33/0.51      bot_bot_set_complex: set_complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
% 0.33/0.51      bot_bot_set_int: set_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 0.33/0.51      bot_bot_set_list_nat: set_list_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      bot_bot_set_nat: set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
% 0.33/0.51      bot_bot_set_num: set_num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
% 0.33/0.51      bot_bo3145834390647256904nteger: set_Pr8056137968301705908nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
% 0.33/0.51      bot_bo4508923176915781079nt_int: set_Pr1872883991513573699nt_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
% 0.33/0.51      bot_bo5443222936135328352nteger: set_Pr1281608226676607948nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
% 0.33/0.51      bot_bo572930865798478029nt_int: set_Pr9222295170931077689nt_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.33/0.51      bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.51      bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 0.33/0.51      bot_bo228742789529271731at_nat: set_Pr4329608150637261639at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
% 0.33/0.51      bot_bot_set_rat: set_rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.33/0.51      bot_bot_set_real: set_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
% 0.33/0.51      bot_bot_set_set_int: set_set_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.33/0.51      bot_bot_set_set_nat: set_set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.51      bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
% 0.33/0.51      ord_Least_nat: ( nat > $o ) > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Int__Oint_M_Eo_J,type,
% 0.33/0.51      ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.33/0.51      ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 0.33/0.51      ord_le549003669493604880_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Real__Oreal_M_Eo_J,type,
% 0.33/0.51      ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
% 0.33/0.51      ord_less_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      ord_le6747313008572928689nteger: code_integer > code_integer > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
% 0.33/0.51      ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 0.33/0.51      ord_less_int: int > int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.33/0.51      ord_less_nat: nat > nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
% 0.33/0.51      ord_less_num: num > num > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
% 0.33/0.51      ord_less_rat: rat > rat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 0.33/0.51      ord_less_real: real > real > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
% 0.33/0.51      ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 0.33/0.51      ord_less_set_complex: set_complex > set_complex > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
% 0.33/0.51      ord_less_set_int: set_int > set_int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      ord_less_set_nat: set_nat > set_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
% 0.33/0.51      ord_less_set_num: set_num > set_num > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.51      ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 0.33/0.51      ord_le2604355607129572851at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
% 0.33/0.51      ord_less_set_rat: set_rat > set_rat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.33/0.51      ord_less_set_real: set_real > set_real > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
% 0.33/0.51      ord_less_set_set_int: set_set_int > set_set_int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.33/0.51      ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
% 0.33/0.51      ord_le3636971675376928563eger_o: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
% 0.33/0.51      ord_le2124322318746777828_int_o: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ) > ( ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
% 0.33/0.51      ord_le4340812435750786203eger_o: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
% 0.33/0.51      ord_le5643404153117327598_int_o: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
% 0.33/0.51      ord_le6741204236512500942_int_o: ( int > int > $o ) > ( int > int > $o ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_Eo_J,type,
% 0.33/0.51      ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
% 0.33/0.51      ord_le2646555220125990790_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.33/0.51      ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 0.33/0.51      ord_le704812498762024988_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
% 0.33/0.51      ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
% 0.33/0.51      ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      ord_le3102999989581377725nteger: code_integer > code_integer > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
% 0.33/0.51      ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
% 0.33/0.51      ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 0.33/0.51      ord_less_eq_int: int > int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.33/0.51      ord_less_eq_nat: nat > nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
% 0.33/0.51      ord_less_eq_num: num > num > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
% 0.33/0.51      ord_less_eq_rat: rat > rat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 0.33/0.51      ord_less_eq_real: real > real > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
% 0.33/0.51      ord_less_eq_set_o: set_o > set_o > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
% 0.33/0.51      ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 0.33/0.51      ord_le211207098394363844omplex: set_complex > set_complex > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
% 0.33/0.51      ord_less_eq_set_int: set_int > set_int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 0.33/0.51      ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
% 0.33/0.51      ord_less_eq_set_num: set_num > set_num > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
% 0.33/0.51      ord_le3216752416896350996nteger: set_Pr8056137968301705908nteger > set_Pr8056137968301705908nteger > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
% 0.33/0.51      ord_le135402666524580259nt_int: set_Pr1872883991513573699nt_int > set_Pr1872883991513573699nt_int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
% 0.33/0.51      ord_le653643898420964396nteger: set_Pr1281608226676607948nteger > set_Pr1281608226676607948nteger > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
% 0.33/0.51      ord_le8725513860283290265nt_int: set_Pr9222295170931077689nt_int > set_Pr9222295170931077689nt_int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.33/0.51      ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.51      ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 0.33/0.51      ord_le1268244103169919719at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
% 0.33/0.51      ord_less_eq_set_rat: set_rat > set_rat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.33/0.51      ord_less_eq_set_real: set_real > set_real > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
% 0.33/0.51      ord_le4403425263959731960et_int: set_set_int > set_set_int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.33/0.51      ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.51      ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      ord_max_Code_integer: code_integer > code_integer > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
% 0.33/0.51      ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
% 0.33/0.51      ord_max_int: int > int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
% 0.33/0.51      ord_max_nat: nat > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
% 0.33/0.51      ord_max_num: num > num > num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
% 0.33/0.51      ord_max_rat: rat > rat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
% 0.33/0.51      ord_max_real: real > real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
% 0.33/0.51      ord_max_set_int: set_int > set_int > set_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      ord_max_set_nat: set_nat > set_nat > set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.51      ord_ma7524802468073614006at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.33/0.51      ord_max_set_real: set_real > set_real > set_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
% 0.33/0.51      ord_min_nat: nat > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
% 0.33/0.51      order_Greatest_nat: ( nat > $o ) > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.51      order_mono_nat_nat: ( nat > nat ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.51      order_5726023648592871131at_nat: ( nat > nat ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.33/0.51      order_7092887310737990675l_real: ( real > real ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
% 0.33/0.51      ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
% 0.33/0.51      top_top_set_int: set_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      top_top_set_nat: set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
% 0.33/0.51      top_to1996260823553986621t_unit: set_Product_unit ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.33/0.51      top_top_set_real: set_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      power_8256067586552552935nteger: code_integer > nat > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
% 0.33/0.51      power_power_complex: complex > nat > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
% 0.33/0.51      power_power_int: int > nat > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
% 0.33/0.51      power_power_nat: nat > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
% 0.33/0.51      power_power_rat: rat > nat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
% 0.33/0.51      power_power_real: real > nat > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.33/0.51      produc6137756002093451184nteger: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc8763457246119570046nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.33/0.51      produc4305682042979456191nt_int: ( int > option6357759511663192854e_term ) > product_prod_int_int > produc7773217078559923341nt_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
% 0.33/0.51      produc4035269172776083154on_nat: ( nat > nat > $o ) > produc4953844613479565601on_nat > produc2233624965454879586on_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
% 0.33/0.51      produc8929957630744042906on_nat: ( nat > nat > nat ) > produc4953844613479565601on_nat > produc8306885398267862888on_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
% 0.33/0.51      produc3576312749637752826on_num: ( num > num > $o ) > produc3447558737645232053on_num > produc7036089656553540234on_num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
% 0.33/0.51      produc5778274026573060048on_num: ( num > num > num ) > produc3447558737645232053on_num > produc1193250871479095198on_num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.33/0.51      produc8603105652947943368nteger: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc1908205239877642774nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.33/0.51      produc5700946648718959541nt_int: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > produc2285326912895808259nt_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.33/0.51      produc3994169339658061776at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc6121120109295599847at_nat > produc5491161045314408544at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.33/0.51      produc2899441246263362727at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > produc6121120109295599847at_nat > produc5542196010084753463at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
% 0.33/0.51      product_Pair_o_o: $o > $o > product_prod_o_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
% 0.33/0.51      product_Pair_o_int: $o > int > product_prod_o_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
% 0.33/0.51      product_Pair_o_nat: $o > nat > product_prod_o_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001_Eo_001t__Real__Oreal,type,
% 0.33/0.51      product_Pair_o_real: $o > real > product_prod_o_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
% 0.33/0.51      produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
% 0.33/0.51      product_Pair_int_int: int > int > product_prod_int_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Real__Oreal,type,
% 0.33/0.51      produc801115645435158769t_real: int > real > produc679980390762269497t_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      produc3329399203697025711T_VEBT: int > vEBT_VEBT > produc1531783533982839933T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__List__Olist_I_Eo_J_001t__List__Olist_I_Eo_J,type,
% 0.33/0.51      produc8435520187683070743list_o: list_o > list_o > produc7102631898165422375list_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
% 0.33/0.51      produc364263696895485585st_int: list_int > list_int > produc1186641810826059865st_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
% 0.33/0.51      produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.51      produc3897820843166775703T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > produc9211091688327510695T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001_Eo,type,
% 0.33/0.51      product_Pair_nat_o: nat > $o > product_prod_nat_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
% 0.33/0.51      product_Pair_nat_int: nat > int > product_prod_nat_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.51      product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
% 0.33/0.51      product_Pair_nat_num: nat > num > product_prod_nat_num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      produc599794634098209291T_VEBT: nat > vEBT_VEBT > produc8025551001238799321T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
% 0.33/0.51      product_Pair_num_num: num > num > product_prod_num_num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
% 0.33/0.51      produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.33/0.51      produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.51      produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.51      produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001_Eo,type,
% 0.33/0.51      product_Pair_real_o: real > $o > product_prod_real_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Int__Oint,type,
% 0.33/0.51      produc3179012173361985393al_int: real > int > produc8786904178792722361al_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Nat__Onat,type,
% 0.33/0.51      produc3181502643871035669al_nat: real > nat > produc3741383161447143261al_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.33/0.51      produc4511245868158468465l_real: real > real > produc2422161461964618553l_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      produc6931449550656315951T_VEBT: real > vEBT_VEBT > produc3757001726724277373T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.51      produc2922128104949294807at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > produc3843707927480180839at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 0.33/0.51      produc9060074326276436823at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > produc1319942482725812455at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 0.33/0.51      produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 0.33/0.51      produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 0.33/0.51      produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
% 0.33/0.51      produc8117437818029410057T_real: vEBT_VEBT > real > produc5170161368751668367T_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.51      produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.51      produc457027306803732586at_nat: set_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Omap__prod_001t__Code____Numeral__Ointeger_001t__Nat__Onat_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
% 0.33/0.51      produc8678311845419106900er_nat: ( code_integer > nat ) > ( code_integer > nat ) > produc8923325533196201883nteger > product_prod_nat_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_Eo,type,
% 0.33/0.51      produc127349428274296955eger_o: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > produc8763457246119570046nteger > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 0.33/0.51      produc3906647086178084059nteger: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc8763457246119570046nteger ) > produc8763457246119570046nteger > produc8763457246119570046nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Int__Oint_J,type,
% 0.33/0.51      produc8604463032469472703et_int: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int ) > produc8763457246119570046nteger > set_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      produc3558942015123893603et_nat: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat ) > produc8763457246119570046nteger > set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.33/0.51      produc815715089573277247t_real: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real ) > produc8763457246119570046nteger > set_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo,type,
% 0.33/0.51      produc2558449545302689196_int_o: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc7773217078559923341nt_int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.33/0.51      produc5122537100556696953nt_int: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > produc7773217078559923341nt_int ) > produc7773217078559923341nt_int > produc7773217078559923341nt_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      produc8289552606927098482et_nat: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat ) > produc7773217078559923341nt_int > set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.33/0.51      produc8709739885379107790t_real: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real ) > produc7773217078559923341nt_int > set_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_Eo,type,
% 0.33/0.51      produc6253627499356882019eger_o: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > produc1908205239877642774nteger > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 0.33/0.51      produc6512950862096126219nteger: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc1908205239877642774nteger ) > produc1908205239877642774nteger > produc1908205239877642774nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo,type,
% 0.33/0.51      produc1573362020775583542_int_o: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc2285326912895808259nt_int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.33/0.51      produc8492565224438309093nt_int: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > produc2285326912895808259nt_int ) > produc2285326912895808259nt_int > produc2285326912895808259nt_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
% 0.33/0.51      produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
% 0.33/0.51      produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 0.33/0.51      produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.33/0.51      produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
% 0.33/0.51      produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
% 0.33/0.51      produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.33/0.51      produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
% 0.33/0.51      produc73460835934605544et_int: ( int > int > set_int ) > product_prod_int_int > set_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.51      produc4251311855443802252et_nat: ( int > int > set_nat ) > product_prod_int_int > set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.51      produc1656060378719767003at_nat: ( int > int > set_Pr1261947904930325089at_nat ) > product_prod_int_int > set_Pr1261947904930325089at_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.33/0.51      produc6452406959799940328t_real: ( int > int > set_real ) > product_prod_int_int > set_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.33/0.51      produc5233655623923918146et_nat: ( int > int > set_set_nat ) > product_prod_int_int > set_set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 0.33/0.51      produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.51      produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
% 0.33/0.51      produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 0.33/0.51      produc1917071388513777916omplex: ( nat > nat > complex ) > product_prod_nat_nat > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
% 0.33/0.51      produc2676513652042109336d_enat: ( nat > nat > extended_enat ) > product_prod_nat_nat > extended_enat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
% 0.33/0.51      produc6840382203811409530at_int: ( nat > nat > int ) > product_prod_nat_nat > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.51      produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.51      produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
% 0.33/0.51      produc6207742614233964070at_rat: ( nat > nat > rat ) > product_prod_nat_nat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.33/0.51      produc1703576794950452218t_real: ( nat > nat > real ) > product_prod_nat_nat > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.33/0.51      produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
% 0.33/0.51      produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
% 0.33/0.51      product_fst_int_int: product_prod_int_int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.51      product_fst_nat_nat: product_prod_nat_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
% 0.33/0.51      product_snd_int_int: product_prod_int_int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.51      product_snd_nat_nat: product_prod_nat_nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rat_OAbs__Rat,type,
% 0.33/0.51      abs_Rat: product_prod_int_int > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rat_OFract,type,
% 0.33/0.51      fract: int > int > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rat_OFrct,type,
% 0.33/0.51      frct: product_prod_int_int > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rat_ORep__Rat,type,
% 0.33/0.51      rep_Rat: rat > product_prod_int_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
% 0.33/0.51      field_5140801741446780682s_real: set_real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rat_Onormalize,type,
% 0.33/0.51      normalize: product_prod_int_int > product_prod_int_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rat_Oof__int,type,
% 0.33/0.51      of_int: int > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rat_Opcr__rat,type,
% 0.33/0.51      pcr_rat: product_prod_int_int > rat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rat_Opositive,type,
% 0.33/0.51      positive: rat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rat_Oquotient__of,type,
% 0.33/0.51      quotient_of: rat > product_prod_int_int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rat_Oratrel,type,
% 0.33/0.51      ratrel: product_prod_int_int > product_prod_int_int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Real_OReal,type,
% 0.33/0.51      real2: ( nat > rat ) > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Real_Ocauchy,type,
% 0.33/0.51      cauchy: ( nat > rat ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Real_Opcr__real,type,
% 0.33/0.51      pcr_real: ( nat > rat ) > real > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Real_Opositive,type,
% 0.33/0.51      positive2: real > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Real_Orealrel,type,
% 0.33/0.51      realrel: ( nat > rat ) > ( nat > rat ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Real_Orep__real,type,
% 0.33/0.51      rep_real: real > nat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Real_Ovanishes,type,
% 0.33/0.51      vanishes: ( nat > rat ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
% 0.33/0.51      real_V3694042436643373181omplex: complex > complex > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
% 0.33/0.51      real_V975177566351809787t_real: real > real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
% 0.33/0.51      real_V1022390504157884413omplex: complex > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
% 0.33/0.51      real_V7735802525324610683m_real: real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
% 0.33/0.51      real_V4546457046886955230omplex: real > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
% 0.33/0.51      real_V2046097035970521341omplex: real > complex > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
% 0.33/0.51      real_V1485227260804924795R_real: real > real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Relation_OField_001t__Nat__Onat,type,
% 0.33/0.51      field_nat: set_Pr1261947904930325089at_nat > set_nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
% 0.33/0.51      algebr932160517623751201me_int: int > int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
% 0.33/0.51      algebr934650988132801477me_nat: nat > nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
% 0.33/0.51      divide1717551699836669952omplex: complex > complex > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
% 0.33/0.51      divide_divide_int: int > int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
% 0.33/0.51      divide_divide_nat: nat > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
% 0.33/0.51      divide_divide_rat: rat > rat > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
% 0.33/0.51      divide_divide_real: real > real > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      dvd_dvd_Code_integer: code_integer > code_integer > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
% 0.33/0.51      dvd_dvd_complex: complex > complex > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
% 0.33/0.51      dvd_dvd_int: int > int > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
% 0.33/0.51      dvd_dvd_nat: nat > nat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
% 0.33/0.51      dvd_dvd_rat: rat > rat > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
% 0.33/0.51      dvd_dvd_real: real > real > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      modulo364778990260209775nteger: code_integer > code_integer > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
% 0.33/0.51      modulo_modulo_int: int > int > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
% 0.33/0.51      modulo_modulo_nat: nat > nat > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
% 0.33/0.51      zero_n356916108424825756nteger: $o > code_integer ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
% 0.33/0.51      zero_n1201886186963655149omplex: $o > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
% 0.33/0.51      zero_n2684676970156552555ol_int: $o > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
% 0.33/0.51      zero_n2687167440665602831ol_nat: $o > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
% 0.33/0.51      zero_n2052037380579107095ol_rat: $o > rat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
% 0.33/0.51      zero_n3304061248610475627l_real: $o > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
% 0.33/0.51      suminf_complex: ( nat > complex ) > complex ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Series_Osuminf_001t__Int__Oint,type,
% 0.33/0.51      suminf_int: ( nat > int ) > int ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
% 0.33/0.51      suminf_nat: ( nat > nat ) > nat ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
% 0.33/0.51      suminf_real: ( nat > real ) > real ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
% 0.33/0.51      summable_complex: ( nat > complex ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Series_Osummable_001t__Int__Oint,type,
% 0.33/0.51      summable_int: ( nat > int ) > $o ).
% 0.33/0.51  
% 0.33/0.51  thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
% 0.33/0.51      summable_nat: ( nat > nat ) > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
% 0.33/0.52      summable_real: ( nat > real ) > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
% 0.33/0.52      sums_complex: ( nat > complex ) > complex > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Series_Osums_001t__Int__Oint,type,
% 0.33/0.52      sums_int: ( nat > int ) > int > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Series_Osums_001t__Nat__Onat,type,
% 0.33/0.52      sums_nat: ( nat > nat ) > nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Series_Osums_001t__Real__Oreal,type,
% 0.33/0.52      sums_real: ( nat > real ) > real > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
% 0.33/0.52      collect_Code_integer: ( code_integer > $o ) > set_Code_integer ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
% 0.33/0.52      collect_complex: ( complex > $o ) > set_complex ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__Int__Oint,type,
% 0.33/0.52      collect_int: ( int > $o ) > set_int ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__List__Olist_I_Eo_J,type,
% 0.33/0.52      collect_list_o: ( list_o > $o ) > set_list_o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 0.33/0.52      collect_list_complex: ( list_complex > $o ) > set_list_complex ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
% 0.33/0.52      collect_list_int: ( list_int > $o ) > set_list_int ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
% 0.33/0.52      collect_list_nat: ( list_nat > $o ) > set_list_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.52      collec3343600615725829874at_nat: ( list_P6011104703257516679at_nat > $o ) > set_li5450038453877631591at_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.52      collec5608196760682091941T_VEBT: ( list_VEBT_VEBT > $o ) > set_list_VEBT_VEBT ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 0.33/0.52      collect_nat: ( nat > $o ) > set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__Num__Onum,type,
% 0.33/0.52      collect_num: ( num > $o ) > set_num ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 0.33/0.52      collec8663557070575231912omplex: ( produc4411394909380815293omplex > $o ) > set_Pr5085853215250843933omplex ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.33/0.52      collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.52      collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.33/0.52      collec3799799289383736868l_real: ( produc2422161461964618553l_real > $o ) > set_Pr6218003697084177305l_real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.33/0.52      collec6321179662152712658at_nat: ( produc3843707927480180839at_nat > $o ) > set_Pr4329608150637261639at_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
% 0.33/0.52      collect_rat: ( rat > $o ) > set_rat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 0.33/0.52      collect_real: ( real > $o ) > set_real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 0.33/0.52      collect_set_complex: ( set_complex > $o ) > set_set_complex ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
% 0.33/0.52      collect_set_int: ( set_int > $o ) > set_set_int ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.52      collect_set_nat: ( set_nat > $o ) > set_set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.52      collec5514110066124741708at_nat: ( set_Pr1261947904930325089at_nat > $o ) > set_se7855581050983116737at_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_OPow_001t__Nat__Onat,type,
% 0.33/0.52      pow_nat: set_nat > set_set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
% 0.33/0.52      image_int_int: ( int > int ) > set_int > set_int ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
% 0.33/0.52      image_int_nat: ( int > nat ) > set_int > set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.52      image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.52      image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 0.33/0.52      image_5971271580939081552omplex: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 0.33/0.52      image_2178119161166701260l_real: ( real > filter2146258269922977983l_real ) > set_real > set_fi7789364187291644575l_real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.33/0.52      image_real_real: ( real > real ) > set_real > set_real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
% 0.33/0.52      insert_complex: complex > set_complex > set_complex ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
% 0.33/0.52      insert_int: int > set_int > set_int ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
% 0.33/0.52      insert_list_nat: list_nat > set_list_nat > set_list_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
% 0.33/0.52      insert_nat: nat > set_nat > set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Oinsert_001t__Num__Onum,type,
% 0.33/0.52      insert_num: num > set_num > set_num ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.33/0.52      insert5033312907999012233nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.52      insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.33/0.52      insert9069300056098147895at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Oinsert_001t__Rat__Orat,type,
% 0.33/0.52      insert_rat: rat > set_rat > set_rat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
% 0.33/0.52      insert_real: real > set_real > set_real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.52      insert_set_nat: set_nat > set_set_nat > set_set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Oinsert_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.52      insert_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Ois__singleton_001t__Int__Oint,type,
% 0.33/0.52      is_singleton_int: set_int > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
% 0.33/0.52      is_singleton_nat: set_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.52      is_sin2850979758926227957at_nat: set_Pr1261947904930325089at_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.33/0.52      is_sin2937591304547752795at_nat: set_Pr4329608150637261639at_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Ois__singleton_001t__Real__Oreal,type,
% 0.33/0.52      is_singleton_real: set_real > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.52      is_singleton_set_nat: set_set_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Othe__elem_001t__Int__Oint,type,
% 0.33/0.52      the_elem_int: set_int > int ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
% 0.33/0.52      the_elem_nat: set_nat > nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.33/0.52      the_el8326832613380209454nt_int: set_Pr958786334691620121nt_int > product_prod_int_int ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.52      the_el2281957884133575798at_nat: set_Pr1261947904930325089at_nat > product_prod_nat_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.33/0.52      the_el221668144340439132at_nat: set_Pr4329608150637261639at_nat > produc3843707927480180839at_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Othe__elem_001t__Real__Oreal,type,
% 0.33/0.52      the_elem_real: set_real > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.33/0.52      vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
% 0.33/0.52      set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
% 0.33/0.52      set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
% 0.33/0.52      set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
% 0.33/0.52      set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
% 0.33/0.52      set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
% 0.33/0.52      set_or1266510415728281911st_int: int > int > set_int ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
% 0.33/0.52      set_or1269000886237332187st_nat: nat > nat > set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
% 0.33/0.52      set_or7049704709247886629st_num: num > num > set_num ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
% 0.33/0.52      set_or633870826150836451st_rat: rat > rat > set_rat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
% 0.33/0.52      set_or1222579329274155063t_real: real > real > set_real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Int__Oint_J,type,
% 0.33/0.52      set_or370866239135849197et_int: set_int > set_int > set_set_int ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.52      set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
% 0.33/0.52      set_or4662586982721622107an_int: int > int > set_int ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
% 0.33/0.52      set_or4665077453230672383an_nat: nat > nat > set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Num__Onum,type,
% 0.33/0.52      set_or1222409239386451017an_num: num > num > set_num ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Rat__Orat,type,
% 0.33/0.52      set_or4029947393144176647an_rat: rat > rat > set_rat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal,type,
% 0.33/0.52      set_or66887138388493659n_real: real > real > set_real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Int__Oint_J,type,
% 0.33/0.52      set_or8585797421378605585et_int: set_int > set_int > set_set_int ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.52      set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
% 0.33/0.52      set_ord_atLeast_nat: nat > set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
% 0.33/0.52      set_ord_atMost_int: int > set_int ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
% 0.33/0.52      set_ord_atMost_nat: nat > set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Num__Onum,type,
% 0.33/0.52      set_ord_atMost_num: num > set_num ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Rat__Orat,type,
% 0.33/0.52      set_ord_atMost_rat: rat > set_rat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
% 0.33/0.52      set_ord_atMost_real: real > set_real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Int__Oint_J,type,
% 0.33/0.52      set_or58775011639299419et_int: set_int > set_set_int ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.52      set_or4236626031148496127et_nat: set_nat > set_set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
% 0.33/0.52      set_or6656581121297822940st_int: int > int > set_int ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
% 0.33/0.52      set_or6659071591806873216st_nat: nat > nat > set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
% 0.33/0.52      set_or5832277885323065728an_int: int > int > set_int ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
% 0.33/0.52      set_or5834768355832116004an_nat: nat > nat > set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
% 0.33/0.52      set_or1633881224788618240n_real: real > real > set_real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
% 0.33/0.52      set_or1210151606488870762an_nat: nat > set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
% 0.33/0.52      set_or5849166863359141190n_real: real > set_real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
% 0.33/0.52      set_ord_lessThan_int: int > set_int ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
% 0.33/0.52      set_ord_lessThan_nat: nat > set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
% 0.33/0.52      set_ord_lessThan_num: num > set_num ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
% 0.33/0.52      set_ord_lessThan_rat: rat > set_rat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
% 0.33/0.52      set_or5984915006950818249n_real: real > set_real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.52      set_or890127255671739683et_nat: set_nat > set_set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.33/0.52      topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.33/0.52      topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
% 0.33/0.52      topolo6980174941875973593q_real: ( nat > real ) > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
% 0.33/0.52      topolo2177554685111907308n_real: real > set_real > filter_real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
% 0.33/0.52      topolo2815343760600316023s_real: real > filter_real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex,type,
% 0.33/0.52      topolo6517432010174082258omplex: ( nat > complex ) > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
% 0.33/0.52      topolo4055970368930404560y_real: ( nat > real ) > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
% 0.33/0.52      topolo896644834953643431omplex: filter6041513312241820739omplex ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
% 0.33/0.52      topolo1511823702728130853y_real: filter2146258269922977983l_real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Oarccos,type,
% 0.33/0.52      arccos: real > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
% 0.33/0.52      arcosh_real: real > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Oarcsin,type,
% 0.33/0.52      arcsin: real > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Oarctan,type,
% 0.33/0.52      arctan: real > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
% 0.33/0.52      arsinh_real: real > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
% 0.33/0.52      artanh_real: real > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
% 0.33/0.52      cos_complex: complex > complex ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
% 0.33/0.52      cos_real: real > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Ocos__coeff,type,
% 0.33/0.52      cos_coeff: nat > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Ocosh_001t__Complex__Ocomplex,type,
% 0.33/0.52      cosh_complex: complex > complex ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
% 0.33/0.52      cosh_real: real > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Ocot_001t__Complex__Ocomplex,type,
% 0.33/0.52      cot_complex: complex > complex ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
% 0.33/0.52      cot_real: real > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
% 0.33/0.52      diffs_complex: ( nat > complex ) > nat > complex ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
% 0.33/0.52      diffs_real: ( nat > real ) > nat > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
% 0.33/0.52      exp_complex: complex > complex ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
% 0.33/0.52      exp_real: real > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
% 0.33/0.52      ln_ln_real: real > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Olog,type,
% 0.33/0.52      log: real > real > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Opi,type,
% 0.33/0.52      pi: real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
% 0.33/0.52      powr_real: real > real > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
% 0.33/0.52      sin_complex: complex > complex ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
% 0.33/0.52      sin_real: real > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Osin__coeff,type,
% 0.33/0.52      sin_coeff: nat > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Osinh_001t__Complex__Ocomplex,type,
% 0.33/0.52      sinh_complex: complex > complex ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
% 0.33/0.52      sinh_real: real > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
% 0.33/0.52      tan_complex: complex > complex ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
% 0.33/0.52      tan_real: real > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
% 0.33/0.52      tanh_complex: complex > complex ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
% 0.33/0.52      tanh_real: real > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
% 0.33/0.52      transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
% 0.33/0.52      transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
% 0.33/0.52      vEBT_Leaf: $o > $o > vEBT_VEBT ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
% 0.33/0.52      vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
% 0.33/0.52      vEBT_size_VEBT: vEBT_VEBT > nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
% 0.33/0.52      vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
% 0.33/0.52      vEBT_VEBT_high: nat > nat > nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
% 0.33/0.52      vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
% 0.33/0.52      vEBT_VEBT_low: nat > nat > nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
% 0.33/0.52      vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
% 0.33/0.52      vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
% 0.33/0.52      vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
% 0.33/0.52      vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
% 0.33/0.52      vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
% 0.33/0.52      vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
% 0.33/0.52      vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Definitions_Oset__vebt,type,
% 0.33/0.52      vEBT_set_vebt: vEBT_VEBT > set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
% 0.33/0.52      vEBT_vebt_buildup: nat > vEBT_VEBT ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
% 0.33/0.52      vEBT_v4011308405150292612up_rel: nat > nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Delete_Ovebt__delete,type,
% 0.33/0.52      vEBT_vebt_delete: vEBT_VEBT > nat > vEBT_VEBT ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
% 0.33/0.52      vEBT_vebt_delete_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Insert_Ovebt__insert,type,
% 0.33/0.52      vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
% 0.33/0.52      vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
% 0.33/0.52      vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
% 0.33/0.52      vEBT_VEBT_minNull: vEBT_VEBT > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
% 0.33/0.52      vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
% 0.33/0.52      vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Member_Ovebt__member,type,
% 0.33/0.52      vEBT_vebt_member: vEBT_VEBT > nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
% 0.33/0.52      vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
% 0.33/0.52      vEBT_VEBT_add: option_nat > option_nat > option_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
% 0.33/0.52      vEBT_VEBT_greater: option_nat > option_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
% 0.33/0.52      vEBT_VEBT_less: option_nat > option_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
% 0.33/0.52      vEBT_VEBT_lesseq: option_nat > option_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
% 0.33/0.52      vEBT_VEBT_max_in_set: set_nat > nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
% 0.33/0.52      vEBT_VEBT_min_in_set: set_nat > nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
% 0.33/0.52      vEBT_VEBT_mul: option_nat > option_nat > option_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
% 0.33/0.52      vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
% 0.33/0.52      vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.52      vEBT_V1502963449132264192at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
% 0.33/0.52      vEBT_VEBT_power: option_nat > option_nat > option_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
% 0.33/0.52      vEBT_vebt_maxt: vEBT_VEBT > option_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
% 0.33/0.52      vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
% 0.33/0.52      vEBT_vebt_mint: vEBT_VEBT > option_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
% 0.33/0.52      vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Pred_Ois__pred__in__set,type,
% 0.33/0.52      vEBT_is_pred_in_set: set_nat > nat > nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Pred_Ovebt__pred,type,
% 0.33/0.52      vEBT_vebt_pred: vEBT_VEBT > nat > option_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
% 0.33/0.52      vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Succ_Ois__succ__in__set,type,
% 0.33/0.52      vEBT_is_succ_in_set: set_nat > nat > nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Succ_Ovebt__succ,type,
% 0.33/0.52      vEBT_vebt_succ: vEBT_VEBT > nat > option_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
% 0.33/0.52      vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
% 0.33/0.52      accp_nat: ( nat > nat > $o ) > nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.33/0.52      accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.52      accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 0.33/0.52      accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 0.33/0.52      accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.52      accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Wellfounded_Omax__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.52      max_ex8135407076693332796at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Wellfounded_Omin__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.52      min_ex6901939911449802026at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Wellfounded_Opred__nat,type,
% 0.33/0.52      pred_nat: set_Pr1261947904930325089at_nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
% 0.33/0.52      wf_nat: set_Pr1261947904930325089at_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_fChoice_001t__Real__Oreal,type,
% 0.33/0.52      fChoice_real: ( real > $o ) > real ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 0.33/0.52      member1535805642427569193e_term: ( code_integer > option6357759511663192854e_term ) > set_Co9149898834107579976e_term > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 0.33/0.52      member8845023287901829240e_term: ( int > option6357759511663192854e_term ) > set_in3461395444621081367e_term > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 0.33/0.52      member4242434998011752849e_term: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > set_Pr7604974323444597168e_term > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 0.33/0.52      member3222579708246209666e_term: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr3642885161833720865e_term > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001_Eo,type,
% 0.33/0.52      member_o: $o > set_o > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Complex__Ocomplex,type,
% 0.33/0.52      member_complex: complex > set_complex > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Int__Oint,type,
% 0.33/0.52      member_int: int > set_int > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
% 0.33/0.52      member_list_o: list_o > set_list_o > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
% 0.33/0.52      member_list_int: list_int > set_list_int > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
% 0.33/0.52      member_list_nat: list_nat > set_list_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.52      member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Nat__Onat,type,
% 0.33/0.52      member_nat: nat > set_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Num__Onum,type,
% 0.33/0.52      member_num: num > set_num > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 0.33/0.52      member3068662437193594005nteger: produc8763457246119570046nteger > set_Pr8056137968301705908nteger > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.33/0.52      member7034335876925520548nt_int: produc7773217078559923341nt_int > set_Pr1872883991513573699nt_int > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 0.33/0.52      member4164122664394876845nteger: produc1908205239877642774nteger > set_Pr1281608226676607948nteger > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.33/0.52      member7618704894036264090nt_int: produc2285326912895808259nt_int > set_Pr9222295170931077689nt_int > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
% 0.33/0.52      member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 0.33/0.52      member7847949116333733898_o_int: product_prod_o_int > set_Pr8834758594704517033_o_int > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
% 0.33/0.52      member2802428098988154798_o_nat: product_prod_o_nat > set_Pr2101469702781467981_o_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J,type,
% 0.33/0.52      member7400031367953476362o_real: product_prod_o_real > set_Pr6573716822653411497o_real > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.52      member5477980866518848620T_VEBT: produc2504756804600209347T_VEBT > set_Pr7543698050874017315T_VEBT > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.33/0.52      member157494554546826820nteger: produc8923325533196201883nteger > set_Pr4811707699266497531nteger > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.33/0.52      member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
% 0.33/0.52      member2744130022092475746t_real: produc679980390762269497t_real > set_Pr3538720872664544793t_real > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.52      member2056185340421749780T_VEBT: produc1531783533982839933T_VEBT > set_Pr8044002425091019955T_VEBT > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__List__Olist_I_Eo_J_J,type,
% 0.33/0.52      member4159035015898711888list_o: produc7102631898165422375list_o > set_Pr6227168374412355847list_o > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
% 0.33/0.52      member6698963635872716290st_int: produc1186641810826059865st_int > set_Pr765067013931698361st_int > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
% 0.33/0.52      member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
% 0.33/0.52      member4439316823752958928T_VEBT: produc9211091688327510695T_VEBT > set_Pr1916528119006554503T_VEBT > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.33/0.52      member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.33/0.52      member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_M_Eo_J,type,
% 0.33/0.52      member772602641336174712real_o: product_prod_real_o > set_Pr4936984352647145239real_o > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
% 0.33/0.52      member1627681773268152802al_int: produc8786904178792722361al_int > set_Pr1019928272762051225al_int > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
% 0.33/0.52      member5805532792777349510al_nat: produc3741383161447143261al_nat > set_Pr3510011417693777981al_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.33/0.52      member7849222048561428706l_real: produc2422161461964618553l_real > set_Pr6218003697084177305l_real > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.52      member7262085504369356948T_VEBT: produc3757001726724277373T_VEBT > set_Pr6019664923565264691T_VEBT > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
% 0.33/0.52      member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.33/0.52      member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 0.33/0.52      member3307348790968139188VEBT_o: produc334124729049499915VEBT_o > set_Pr3175402225741728619VEBT_o > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 0.33/0.52      member5419026705395827622BT_int: produc4894624898956917775BT_int > set_Pr5066593544530342725BT_int > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 0.33/0.52      member373505688050248522BT_nat: produc9072475918466114483BT_nat > set_Pr7556676689462069481BT_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
% 0.33/0.52      member8675245146396747942T_real: produc5170161368751668367T_real > set_Pr7765410600122031685T_real > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.33/0.52      member568628332442017744T_VEBT: produc8243902056947475879T_VEBT > set_Pr6192946355708809607T_VEBT > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Rat__Orat,type,
% 0.33/0.52      member_rat: rat > set_rat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Real__Oreal,type,
% 0.33/0.52      member_real: real > set_real > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
% 0.33/0.52      member_set_int: set_int > set_set_int > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.33/0.52      member_set_nat: set_nat > set_set_nat > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
% 0.33/0.52      member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_v_deg____,type,
% 0.33/0.52      deg: nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_v_m____,type,
% 0.33/0.52      m: nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_v_ma____,type,
% 0.33/0.52      ma: nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_v_maxi____,type,
% 0.33/0.52      maxi: nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_v_mi____,type,
% 0.33/0.52      mi: nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_v_na____,type,
% 0.33/0.52      na: nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_v_summary____,type,
% 0.33/0.52      summary: vEBT_VEBT ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_v_treeList____,type,
% 0.33/0.52      treeList: list_VEBT_VEBT ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_v_xa____,type,
% 0.33/0.52      xa: nat ).
% 0.33/0.52  
% 0.33/0.52  thf(sy_v_ya____,type,
% 0.33/0.52      ya: nat ).
% 0.33/0.52  
% 0.33/0.52  % Relevant facts (9332)
% 0.33/0.52  thf(fact_0__C4_Ohyps_C_I3_J,axiom,
% 0.33/0.52      m = na ).
% 0.33/0.52  
% 0.33/0.52  % "4.hyps"(3)
% 0.33/0.52  thf(fact_1_xmi,axiom,
% 0.33/0.52      xa != mi ).
% 0.33/0.52  
% 0.33/0.52  % xmi
% 0.33/0.52  thf(fact_2_True,axiom,
% 0.33/0.52      xa = ma ).
% 0.33/0.52  
% 0.33/0.52  % True
% 0.33/0.52  thf(fact_3_valid__eq,axiom,
% 0.33/0.52      vEBT_VEBT_valid = vEBT_invar_vebt ).
% 0.33/0.52  
% 0.33/0.52  % valid_eq
% 0.33/0.52  thf(fact_4_valid__eq1,axiom,
% 0.33/0.52      ! [T: vEBT_VEBT,D: nat] :
% 0.33/0.52        ( ( vEBT_invar_vebt @ T @ D )
% 0.33/0.52       => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 0.33/0.52  
% 0.33/0.52  % valid_eq1
% 0.33/0.52  thf(fact_5_valid__eq2,axiom,
% 0.33/0.52      ! [T: vEBT_VEBT,D: nat] :
% 0.33/0.52        ( ( vEBT_VEBT_valid @ T @ D )
% 0.33/0.52       => ( vEBT_invar_vebt @ T @ D ) ) ).
% 0.33/0.52  
% 0.33/0.52  % valid_eq2
% 0.33/0.52  thf(fact_6_valid__0__not,axiom,
% 0.33/0.52      ! [T: vEBT_VEBT] :
% 0.33/0.52        ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 0.33/0.52  
% 0.33/0.52  % valid_0_not
% 0.33/0.52  thf(fact_7_valid__tree__deg__neq__0,axiom,
% 0.33/0.52      ! [T: vEBT_VEBT] :
% 0.33/0.52        ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 0.33/0.52  
% 0.33/0.52  % valid_tree_deg_neq_0
% 0.33/0.52  thf(fact_8__092_060open_062high_Ax_An_A_060_Alength_AtreeList_092_060close_062,axiom,
% 0.33/0.52      ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ).
% 0.33/0.52  
% 0.33/0.52  % \<open>high x n < length treeList\<close>
% 0.33/0.52  thf(fact_9__C4_Ohyps_C_I1_J,axiom,
% 0.33/0.52      vEBT_invar_vebt @ summary @ m ).
% 0.33/0.52  
% 0.33/0.52  % "4.hyps"(1)
% 0.33/0.52  thf(fact_10_False,axiom,
% 0.33/0.52      ~ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % False
% 0.33/0.52  thf(fact_11_deg__deg__n,axiom,
% 0.33/0.52      ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.33/0.52        ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
% 0.33/0.52       => ( Deg = N ) ) ).
% 0.33/0.52  
% 0.33/0.52  % deg_deg_n
% 0.33/0.52  thf(fact_12_set__vebt__set__vebt_H__valid,axiom,
% 0.33/0.52      ! [T: vEBT_VEBT,N: nat] :
% 0.33/0.52        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.52       => ( ( vEBT_set_vebt @ T )
% 0.33/0.52          = ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % set_vebt_set_vebt'_valid
% 0.33/0.52  thf(fact_13_notemp,axiom,
% 0.33/0.52      ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) @ X_1 ) ).
% 0.33/0.52  
% 0.33/0.52  % notemp
% 0.33/0.52  thf(fact_14_hprolist,axiom,
% 0.33/0.52      ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( vEBT_VEBT_high @ xa @ na ) )
% 0.33/0.52      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % hprolist
% 0.33/0.52  thf(fact_15_not__min__Null__member,axiom,
% 0.33/0.52      ! [T: vEBT_VEBT] :
% 0.33/0.52        ( ~ ( vEBT_VEBT_minNull @ T )
% 0.33/0.52       => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).
% 0.33/0.52  
% 0.33/0.52  % not_min_Null_member
% 0.33/0.52  thf(fact_16_deg__not__0,axiom,
% 0.33/0.52      ! [T: vEBT_VEBT,N: nat] :
% 0.33/0.52        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.52       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.33/0.52  
% 0.33/0.52  % deg_not_0
% 0.33/0.52  thf(fact_17_bit__split__inv,axiom,
% 0.33/0.52      ! [X: nat,D: nat] :
% 0.33/0.52        ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D ) @ ( vEBT_VEBT_low @ X @ D ) @ D )
% 0.33/0.52        = X ) ).
% 0.33/0.52  
% 0.33/0.52  % bit_split_inv
% 0.33/0.52  thf(fact_18__C4_OIH_C_I2_J,axiom,
% 0.33/0.52      ! [X: nat,Y: nat] :
% 0.33/0.52        ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ summary @ X ) @ Y )
% 0.33/0.52        = ( ( X != Y )
% 0.33/0.52          & ( vEBT_V8194947554948674370ptions @ summary @ Y ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % "4.IH"(2)
% 0.33/0.52  thf(fact_19_mimapr,axiom,
% 0.33/0.52      ord_less_nat @ mi @ ma ).
% 0.33/0.52  
% 0.33/0.52  % mimapr
% 0.33/0.52  thf(fact_20__092_060open_062length_AtreeList_A_061_Alength_A_ItreeList_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_J_092_060close_062,axiom,
% 0.33/0.52      ( ( size_s6755466524823107622T_VEBT @ treeList )
% 0.33/0.52      = ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % \<open>length treeList = length (treeList[high x n := vebt_delete (treeList ! high x n) (low x n)])\<close>
% 0.33/0.52  thf(fact_21__C4_Ohyps_C_I7_J,axiom,
% 0.33/0.52      ord_less_eq_nat @ mi @ ma ).
% 0.33/0.52  
% 0.33/0.52  % "4.hyps"(7)
% 0.33/0.52  thf(fact_22__C4_Ohyps_C_I6_J,axiom,
% 0.33/0.52      ( ( mi = ma )
% 0.33/0.52     => ! [X2: vEBT_VEBT] :
% 0.33/0.52          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
% 0.33/0.52         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % "4.hyps"(6)
% 0.33/0.52  thf(fact_23__C4_OIH_C_I1_J,axiom,
% 0.33/0.52      ! [X2: vEBT_VEBT] :
% 0.33/0.52        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
% 0.33/0.52       => ( ( vEBT_invar_vebt @ X2 @ na )
% 0.33/0.52          & ! [Xa: nat,Xb: nat] :
% 0.33/0.52              ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ X2 @ Xa ) @ Xb )
% 0.33/0.52              = ( ( Xa != Xb )
% 0.33/0.52                & ( vEBT_V8194947554948674370ptions @ X2 @ Xb ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % "4.IH"(1)
% 0.33/0.52  thf(fact_24_in__children__def,axiom,
% 0.33/0.52      ( vEBT_V5917875025757280293ildren
% 0.33/0.52      = ( ^ [N2: nat,TreeList2: list_VEBT_VEBT,X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ N2 ) ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % in_children_def
% 0.33/0.52  thf(fact_25_nth__list__update__eq,axiom,
% 0.33/0.52      ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 0.33/0.52        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.52       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ I )
% 0.33/0.52          = X ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_list_update_eq
% 0.33/0.52  thf(fact_26_nth__list__update__eq,axiom,
% 0.33/0.52      ! [I: nat,Xs: list_o,X: $o] :
% 0.33/0.52        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.33/0.52       => ( ( nth_o @ ( list_update_o @ Xs @ I @ X ) @ I )
% 0.33/0.52          = X ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_list_update_eq
% 0.33/0.52  thf(fact_27_nth__list__update__eq,axiom,
% 0.33/0.52      ! [I: nat,Xs: list_nat,X: nat] :
% 0.33/0.52        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 0.33/0.52       => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ I )
% 0.33/0.52          = X ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_list_update_eq
% 0.33/0.52  thf(fact_28_nth__list__update__eq,axiom,
% 0.33/0.52      ! [I: nat,Xs: list_int,X: int] :
% 0.33/0.52        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 0.33/0.52       => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ I )
% 0.33/0.52          = X ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_list_update_eq
% 0.33/0.52  thf(fact_29_buildup__gives__valid,axiom,
% 0.33/0.52      ! [N: nat] :
% 0.33/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.52       => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% 0.33/0.52  
% 0.33/0.52  % buildup_gives_valid
% 0.33/0.52  thf(fact_30_nth__list__update__neq,axiom,
% 0.33/0.52      ! [I: nat,J: nat,Xs: list_nat,X: nat] :
% 0.33/0.52        ( ( I != J )
% 0.33/0.52       => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
% 0.33/0.52          = ( nth_nat @ Xs @ J ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_list_update_neq
% 0.33/0.52  thf(fact_31_nth__list__update__neq,axiom,
% 0.33/0.52      ! [I: nat,J: nat,Xs: list_int,X: int] :
% 0.33/0.52        ( ( I != J )
% 0.33/0.52       => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
% 0.33/0.52          = ( nth_int @ Xs @ J ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_list_update_neq
% 0.33/0.52  thf(fact_32_nth__list__update__neq,axiom,
% 0.33/0.52      ! [I: nat,J: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 0.33/0.52        ( ( I != J )
% 0.33/0.52       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J )
% 0.33/0.52          = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_list_update_neq
% 0.33/0.52  thf(fact_33_list__update__id,axiom,
% 0.33/0.52      ! [Xs: list_nat,I: nat] :
% 0.33/0.52        ( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
% 0.33/0.52        = Xs ) ).
% 0.33/0.52  
% 0.33/0.52  % list_update_id
% 0.33/0.52  thf(fact_34_list__update__id,axiom,
% 0.33/0.52      ! [Xs: list_int,I: nat] :
% 0.33/0.52        ( ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ I ) )
% 0.33/0.52        = Xs ) ).
% 0.33/0.52  
% 0.33/0.52  % list_update_id
% 0.33/0.52  thf(fact_35_list__update__id,axiom,
% 0.33/0.52      ! [Xs: list_VEBT_VEBT,I: nat] :
% 0.33/0.52        ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ I ) )
% 0.33/0.52        = Xs ) ).
% 0.33/0.52  
% 0.33/0.52  % list_update_id
% 0.33/0.52  thf(fact_36_length__list__update,axiom,
% 0.33/0.52      ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
% 0.33/0.52        ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) )
% 0.33/0.52        = ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% 0.33/0.52  
% 0.33/0.52  % length_list_update
% 0.33/0.52  thf(fact_37_length__list__update,axiom,
% 0.33/0.52      ! [Xs: list_o,I: nat,X: $o] :
% 0.33/0.52        ( ( size_size_list_o @ ( list_update_o @ Xs @ I @ X ) )
% 0.33/0.52        = ( size_size_list_o @ Xs ) ) ).
% 0.33/0.52  
% 0.33/0.52  % length_list_update
% 0.33/0.52  thf(fact_38_length__list__update,axiom,
% 0.33/0.52      ! [Xs: list_nat,I: nat,X: nat] :
% 0.33/0.52        ( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X ) )
% 0.33/0.52        = ( size_size_list_nat @ Xs ) ) ).
% 0.33/0.52  
% 0.33/0.52  % length_list_update
% 0.33/0.52  thf(fact_39_length__list__update,axiom,
% 0.33/0.52      ! [Xs: list_int,I: nat,X: int] :
% 0.33/0.52        ( ( size_size_list_int @ ( list_update_int @ Xs @ I @ X ) )
% 0.33/0.52        = ( size_size_list_int @ Xs ) ) ).
% 0.33/0.52  
% 0.33/0.52  % length_list_update
% 0.33/0.52  thf(fact_40_bot__nat__0_Onot__eq__extremum,axiom,
% 0.33/0.52      ! [A: nat] :
% 0.33/0.52        ( ( A != zero_zero_nat )
% 0.33/0.52        = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.33/0.52  
% 0.33/0.52  % bot_nat_0.not_eq_extremum
% 0.33/0.52  thf(fact_41_neq0__conv,axiom,
% 0.33/0.52      ! [N: nat] :
% 0.33/0.52        ( ( N != zero_zero_nat )
% 0.33/0.52        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.33/0.52  
% 0.33/0.52  % neq0_conv
% 0.33/0.52  thf(fact_42_less__nat__zero__code,axiom,
% 0.33/0.52      ! [N: nat] :
% 0.33/0.52        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.33/0.52  
% 0.33/0.52  % less_nat_zero_code
% 0.33/0.52  thf(fact_43_not__gr__zero,axiom,
% 0.33/0.52      ! [N: nat] :
% 0.33/0.52        ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.33/0.52        = ( N = zero_zero_nat ) ) ).
% 0.33/0.52  
% 0.33/0.52  % not_gr_zero
% 0.33/0.52  thf(fact_44__092_060open_062mi_A_092_060le_062_Ax_A_092_060and_062_Ax_A_092_060le_062_Ama_092_060close_062,axiom,
% 0.33/0.52      ( ( ord_less_eq_nat @ mi @ xa )
% 0.33/0.52      & ( ord_less_eq_nat @ xa @ ma ) ) ).
% 0.33/0.52  
% 0.33/0.52  % \<open>mi \<le> x \<and> x \<le> ma\<close>
% 0.33/0.52  thf(fact_45_nth__list__update,axiom,
% 0.33/0.52      ! [I: nat,Xs: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
% 0.33/0.52        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.52       => ( ( ( I = J )
% 0.33/0.52           => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J )
% 0.33/0.52              = X ) )
% 0.33/0.52          & ( ( I != J )
% 0.33/0.52           => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J )
% 0.33/0.52              = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_list_update
% 0.33/0.52  thf(fact_46_nth__list__update,axiom,
% 0.33/0.52      ! [I: nat,Xs: list_o,X: $o,J: nat] :
% 0.33/0.52        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.33/0.52       => ( ( nth_o @ ( list_update_o @ Xs @ I @ X ) @ J )
% 0.33/0.52          = ( ( ( I = J )
% 0.33/0.52             => X )
% 0.33/0.52            & ( ( I != J )
% 0.33/0.52             => ( nth_o @ Xs @ J ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_list_update
% 0.33/0.52  thf(fact_47_nth__list__update,axiom,
% 0.33/0.52      ! [I: nat,Xs: list_nat,J: nat,X: nat] :
% 0.33/0.52        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 0.33/0.52       => ( ( ( I = J )
% 0.33/0.52           => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
% 0.33/0.52              = X ) )
% 0.33/0.52          & ( ( I != J )
% 0.33/0.52           => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
% 0.33/0.52              = ( nth_nat @ Xs @ J ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_list_update
% 0.33/0.52  thf(fact_48_nth__list__update,axiom,
% 0.33/0.52      ! [I: nat,Xs: list_int,J: nat,X: int] :
% 0.33/0.52        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 0.33/0.52       => ( ( ( I = J )
% 0.33/0.52           => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
% 0.33/0.52              = X ) )
% 0.33/0.52          & ( ( I != J )
% 0.33/0.52           => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
% 0.33/0.52              = ( nth_int @ Xs @ J ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_list_update
% 0.33/0.52  thf(fact_49_min__in__set__def,axiom,
% 0.33/0.52      ( vEBT_VEBT_min_in_set
% 0.33/0.52      = ( ^ [Xs2: set_nat,X3: nat] :
% 0.33/0.52            ( ( member_nat @ X3 @ Xs2 )
% 0.33/0.52            & ! [Y2: nat] :
% 0.33/0.52                ( ( member_nat @ Y2 @ Xs2 )
% 0.33/0.52               => ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % min_in_set_def
% 0.33/0.52  thf(fact_50_max__in__set__def,axiom,
% 0.33/0.52      ( vEBT_VEBT_max_in_set
% 0.33/0.52      = ( ^ [Xs2: set_nat,X3: nat] :
% 0.33/0.52            ( ( member_nat @ X3 @ Xs2 )
% 0.33/0.52            & ! [Y2: nat] :
% 0.33/0.52                ( ( member_nat @ Y2 @ Xs2 )
% 0.33/0.52               => ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % max_in_set_def
% 0.33/0.52  thf(fact_51_inthall,axiom,
% 0.33/0.52      ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,N: nat] :
% 0.33/0.52        ( ! [X4: product_prod_nat_nat] :
% 0.33/0.52            ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.33/0.52           => ( P @ X4 ) )
% 0.33/0.52       => ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
% 0.33/0.52         => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ N ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % inthall
% 0.33/0.52  thf(fact_52_inthall,axiom,
% 0.33/0.52      ! [Xs: list_real,P: real > $o,N: nat] :
% 0.33/0.52        ( ! [X4: real] :
% 0.33/0.52            ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
% 0.33/0.52           => ( P @ X4 ) )
% 0.33/0.52       => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
% 0.33/0.52         => ( P @ ( nth_real @ Xs @ N ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % inthall
% 0.33/0.52  thf(fact_53_inthall,axiom,
% 0.33/0.52      ! [Xs: list_set_nat,P: set_nat > $o,N: nat] :
% 0.33/0.52        ( ! [X4: set_nat] :
% 0.33/0.52            ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
% 0.33/0.52           => ( P @ X4 ) )
% 0.33/0.52       => ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.33/0.52         => ( P @ ( nth_set_nat @ Xs @ N ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % inthall
% 0.33/0.52  thf(fact_54_inthall,axiom,
% 0.33/0.52      ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
% 0.33/0.52        ( ! [X4: vEBT_VEBT] :
% 0.33/0.52            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.52           => ( P @ X4 ) )
% 0.33/0.52       => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.52         => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % inthall
% 0.33/0.52  thf(fact_55_inthall,axiom,
% 0.33/0.52      ! [Xs: list_o,P: $o > $o,N: nat] :
% 0.33/0.52        ( ! [X4: $o] :
% 0.33/0.52            ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 0.33/0.52           => ( P @ X4 ) )
% 0.33/0.52       => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 0.33/0.52         => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % inthall
% 0.33/0.52  thf(fact_56_inthall,axiom,
% 0.33/0.52      ! [Xs: list_nat,P: nat > $o,N: nat] :
% 0.33/0.52        ( ! [X4: nat] :
% 0.33/0.52            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 0.33/0.52           => ( P @ X4 ) )
% 0.33/0.52       => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.33/0.52         => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % inthall
% 0.33/0.52  thf(fact_57_inthall,axiom,
% 0.33/0.52      ! [Xs: list_int,P: int > $o,N: nat] :
% 0.33/0.52        ( ! [X4: int] :
% 0.33/0.52            ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 0.33/0.52           => ( P @ X4 ) )
% 0.33/0.52       => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 0.33/0.52         => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % inthall
% 0.33/0.52  thf(fact_58_list__update__overwrite,axiom,
% 0.33/0.52      ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT,Y: vEBT_VEBT] :
% 0.33/0.52        ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ I @ Y )
% 0.33/0.52        = ( list_u1324408373059187874T_VEBT @ Xs @ I @ Y ) ) ).
% 0.33/0.52  
% 0.33/0.52  % list_update_overwrite
% 0.33/0.52  thf(fact_59_le__zero__eq,axiom,
% 0.33/0.52      ! [N: nat] :
% 0.33/0.52        ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.33/0.52        = ( N = zero_zero_nat ) ) ).
% 0.33/0.52  
% 0.33/0.52  % le_zero_eq
% 0.33/0.52  thf(fact_60_le0,axiom,
% 0.33/0.52      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.33/0.52  
% 0.33/0.52  % le0
% 0.33/0.52  thf(fact_61_bot__nat__0_Oextremum,axiom,
% 0.33/0.52      ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.33/0.52  
% 0.33/0.52  % bot_nat_0.extremum
% 0.33/0.52  thf(fact_62_list__update__beyond,axiom,
% 0.33/0.52      ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
% 0.33/0.52        ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ I )
% 0.33/0.52       => ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X )
% 0.33/0.52          = Xs ) ) ).
% 0.33/0.52  
% 0.33/0.52  % list_update_beyond
% 0.33/0.52  thf(fact_63_list__update__beyond,axiom,
% 0.33/0.52      ! [Xs: list_o,I: nat,X: $o] :
% 0.33/0.52        ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ I )
% 0.33/0.52       => ( ( list_update_o @ Xs @ I @ X )
% 0.33/0.52          = Xs ) ) ).
% 0.33/0.52  
% 0.33/0.52  % list_update_beyond
% 0.33/0.52  thf(fact_64_list__update__beyond,axiom,
% 0.33/0.52      ! [Xs: list_nat,I: nat,X: nat] :
% 0.33/0.52        ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
% 0.33/0.52       => ( ( list_update_nat @ Xs @ I @ X )
% 0.33/0.52          = Xs ) ) ).
% 0.33/0.52  
% 0.33/0.52  % list_update_beyond
% 0.33/0.52  thf(fact_65_list__update__beyond,axiom,
% 0.33/0.52      ! [Xs: list_int,I: nat,X: int] :
% 0.33/0.52        ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ I )
% 0.33/0.52       => ( ( list_update_int @ Xs @ I @ X )
% 0.33/0.52          = Xs ) ) ).
% 0.33/0.52  
% 0.33/0.52  % list_update_beyond
% 0.33/0.52  thf(fact_66_mem__Collect__eq,axiom,
% 0.33/0.52      ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
% 0.33/0.52        ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
% 0.33/0.52        = ( P @ A ) ) ).
% 0.33/0.52  
% 0.33/0.52  % mem_Collect_eq
% 0.33/0.52  thf(fact_67_mem__Collect__eq,axiom,
% 0.33/0.52      ! [A: real,P: real > $o] :
% 0.33/0.52        ( ( member_real @ A @ ( collect_real @ P ) )
% 0.33/0.52        = ( P @ A ) ) ).
% 0.33/0.52  
% 0.33/0.52  % mem_Collect_eq
% 0.33/0.52  thf(fact_68_mem__Collect__eq,axiom,
% 0.33/0.52      ! [A: list_nat,P: list_nat > $o] :
% 0.33/0.52        ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
% 0.33/0.52        = ( P @ A ) ) ).
% 0.33/0.52  
% 0.33/0.52  % mem_Collect_eq
% 0.33/0.52  thf(fact_69_mem__Collect__eq,axiom,
% 0.33/0.52      ! [A: set_nat,P: set_nat > $o] :
% 0.33/0.52        ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
% 0.33/0.52        = ( P @ A ) ) ).
% 0.33/0.52  
% 0.33/0.52  % mem_Collect_eq
% 0.33/0.52  thf(fact_70_mem__Collect__eq,axiom,
% 0.33/0.52      ! [A: nat,P: nat > $o] :
% 0.33/0.52        ( ( member_nat @ A @ ( collect_nat @ P ) )
% 0.33/0.52        = ( P @ A ) ) ).
% 0.33/0.52  
% 0.33/0.52  % mem_Collect_eq
% 0.33/0.52  thf(fact_71_mem__Collect__eq,axiom,
% 0.33/0.52      ! [A: int,P: int > $o] :
% 0.33/0.52        ( ( member_int @ A @ ( collect_int @ P ) )
% 0.33/0.52        = ( P @ A ) ) ).
% 0.33/0.52  
% 0.33/0.52  % mem_Collect_eq
% 0.33/0.52  thf(fact_72_Collect__mem__eq,axiom,
% 0.33/0.52      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.33/0.52        ( ( collec3392354462482085612at_nat
% 0.33/0.52          @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A2 ) )
% 0.33/0.52        = A2 ) ).
% 0.33/0.52  
% 0.33/0.52  % Collect_mem_eq
% 0.33/0.52  thf(fact_73_Collect__mem__eq,axiom,
% 0.33/0.52      ! [A2: set_real] :
% 0.33/0.52        ( ( collect_real
% 0.33/0.52          @ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
% 0.33/0.52        = A2 ) ).
% 0.33/0.52  
% 0.33/0.52  % Collect_mem_eq
% 0.33/0.52  thf(fact_74_Collect__mem__eq,axiom,
% 0.33/0.52      ! [A2: set_list_nat] :
% 0.33/0.52        ( ( collect_list_nat
% 0.33/0.52          @ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A2 ) )
% 0.33/0.52        = A2 ) ).
% 0.33/0.52  
% 0.33/0.52  % Collect_mem_eq
% 0.33/0.52  thf(fact_75_Collect__mem__eq,axiom,
% 0.33/0.52      ! [A2: set_set_nat] :
% 0.33/0.52        ( ( collect_set_nat
% 0.33/0.52          @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
% 0.33/0.52        = A2 ) ).
% 0.33/0.52  
% 0.33/0.52  % Collect_mem_eq
% 0.33/0.52  thf(fact_76_Collect__mem__eq,axiom,
% 0.33/0.52      ! [A2: set_nat] :
% 0.33/0.52        ( ( collect_nat
% 0.33/0.52          @ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
% 0.33/0.52        = A2 ) ).
% 0.33/0.52  
% 0.33/0.52  % Collect_mem_eq
% 0.33/0.52  thf(fact_77_Collect__mem__eq,axiom,
% 0.33/0.52      ! [A2: set_int] :
% 0.33/0.52        ( ( collect_int
% 0.33/0.52          @ ^ [X3: int] : ( member_int @ X3 @ A2 ) )
% 0.33/0.52        = A2 ) ).
% 0.33/0.52  
% 0.33/0.52  % Collect_mem_eq
% 0.33/0.52  thf(fact_78_Collect__cong,axiom,
% 0.33/0.52      ! [P: real > $o,Q: real > $o] :
% 0.33/0.52        ( ! [X4: real] :
% 0.33/0.52            ( ( P @ X4 )
% 0.33/0.52            = ( Q @ X4 ) )
% 0.33/0.52       => ( ( collect_real @ P )
% 0.33/0.52          = ( collect_real @ Q ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % Collect_cong
% 0.33/0.52  thf(fact_79_Collect__cong,axiom,
% 0.33/0.52      ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.33/0.52        ( ! [X4: list_nat] :
% 0.33/0.52            ( ( P @ X4 )
% 0.33/0.52            = ( Q @ X4 ) )
% 0.33/0.52       => ( ( collect_list_nat @ P )
% 0.33/0.52          = ( collect_list_nat @ Q ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % Collect_cong
% 0.33/0.52  thf(fact_80_Collect__cong,axiom,
% 0.33/0.52      ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.33/0.52        ( ! [X4: set_nat] :
% 0.33/0.52            ( ( P @ X4 )
% 0.33/0.52            = ( Q @ X4 ) )
% 0.33/0.52       => ( ( collect_set_nat @ P )
% 0.33/0.52          = ( collect_set_nat @ Q ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % Collect_cong
% 0.33/0.52  thf(fact_81_Collect__cong,axiom,
% 0.33/0.52      ! [P: nat > $o,Q: nat > $o] :
% 0.33/0.52        ( ! [X4: nat] :
% 0.33/0.52            ( ( P @ X4 )
% 0.33/0.52            = ( Q @ X4 ) )
% 0.33/0.52       => ( ( collect_nat @ P )
% 0.33/0.52          = ( collect_nat @ Q ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % Collect_cong
% 0.33/0.52  thf(fact_82_Collect__cong,axiom,
% 0.33/0.52      ! [P: int > $o,Q: int > $o] :
% 0.33/0.52        ( ! [X4: int] :
% 0.33/0.52            ( ( P @ X4 )
% 0.33/0.52            = ( Q @ X4 ) )
% 0.33/0.52       => ( ( collect_int @ P )
% 0.33/0.52          = ( collect_int @ Q ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % Collect_cong
% 0.33/0.52  thf(fact_83_set__swap,axiom,
% 0.33/0.52      ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
% 0.33/0.52        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.52       => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.52         => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs @ I ) ) )
% 0.33/0.52            = ( set_VEBT_VEBT2 @ Xs ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % set_swap
% 0.33/0.52  thf(fact_84_set__swap,axiom,
% 0.33/0.52      ! [I: nat,Xs: list_o,J: nat] :
% 0.33/0.52        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.33/0.52       => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
% 0.33/0.52         => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs @ I @ ( nth_o @ Xs @ J ) ) @ J @ ( nth_o @ Xs @ I ) ) )
% 0.33/0.52            = ( set_o2 @ Xs ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % set_swap
% 0.33/0.52  thf(fact_85_set__swap,axiom,
% 0.33/0.52      ! [I: nat,Xs: list_nat,J: nat] :
% 0.33/0.52        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 0.33/0.52       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
% 0.33/0.52         => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
% 0.33/0.52            = ( set_nat2 @ Xs ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % set_swap
% 0.33/0.52  thf(fact_86_set__swap,axiom,
% 0.33/0.52      ! [I: nat,Xs: list_int,J: nat] :
% 0.33/0.52        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 0.33/0.52       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
% 0.33/0.52         => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ J ) ) @ J @ ( nth_int @ Xs @ I ) ) )
% 0.33/0.52            = ( set_int2 @ Xs ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % set_swap
% 0.33/0.52  thf(fact_87_Nat_Oex__has__greatest__nat,axiom,
% 0.33/0.52      ! [P: nat > $o,K: nat,B: nat] :
% 0.33/0.52        ( ( P @ K )
% 0.33/0.52       => ( ! [Y3: nat] :
% 0.33/0.52              ( ( P @ Y3 )
% 0.33/0.52             => ( ord_less_eq_nat @ Y3 @ B ) )
% 0.33/0.52         => ? [X4: nat] :
% 0.33/0.52              ( ( P @ X4 )
% 0.33/0.52              & ! [Y4: nat] :
% 0.33/0.52                  ( ( P @ Y4 )
% 0.33/0.52                 => ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % Nat.ex_has_greatest_nat
% 0.33/0.52  thf(fact_88_nat__le__linear,axiom,
% 0.33/0.52      ! [M: nat,N: nat] :
% 0.33/0.52        ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.52        | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nat_le_linear
% 0.33/0.52  thf(fact_89_le__antisym,axiom,
% 0.33/0.52      ! [M: nat,N: nat] :
% 0.33/0.52        ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.52       => ( ( ord_less_eq_nat @ N @ M )
% 0.33/0.52         => ( M = N ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % le_antisym
% 0.33/0.52  thf(fact_90_eq__imp__le,axiom,
% 0.33/0.52      ! [M: nat,N: nat] :
% 0.33/0.52        ( ( M = N )
% 0.33/0.52       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.33/0.52  
% 0.33/0.52  % eq_imp_le
% 0.33/0.52  thf(fact_91_le__trans,axiom,
% 0.33/0.52      ! [I: nat,J: nat,K: nat] :
% 0.33/0.52        ( ( ord_less_eq_nat @ I @ J )
% 0.33/0.52       => ( ( ord_less_eq_nat @ J @ K )
% 0.33/0.52         => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % le_trans
% 0.33/0.52  thf(fact_92_le__refl,axiom,
% 0.33/0.52      ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.33/0.52  
% 0.33/0.52  % le_refl
% 0.33/0.52  thf(fact_93_zero__le,axiom,
% 0.33/0.52      ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 0.33/0.52  
% 0.33/0.52  % zero_le
% 0.33/0.52  thf(fact_94_le__0__eq,axiom,
% 0.33/0.52      ! [N: nat] :
% 0.33/0.52        ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.33/0.52        = ( N = zero_zero_nat ) ) ).
% 0.33/0.52  
% 0.33/0.52  % le_0_eq
% 0.33/0.52  thf(fact_95_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.33/0.52      ! [A: nat] :
% 0.33/0.52        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.33/0.52       => ( A = zero_zero_nat ) ) ).
% 0.33/0.52  
% 0.33/0.52  % bot_nat_0.extremum_uniqueI
% 0.33/0.52  thf(fact_96_bot__nat__0_Oextremum__unique,axiom,
% 0.33/0.52      ! [A: nat] :
% 0.33/0.52        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.33/0.52        = ( A = zero_zero_nat ) ) ).
% 0.33/0.52  
% 0.33/0.52  % bot_nat_0.extremum_unique
% 0.33/0.52  thf(fact_97_less__eq__nat_Osimps_I1_J,axiom,
% 0.33/0.52      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.33/0.52  
% 0.33/0.52  % less_eq_nat.simps(1)
% 0.33/0.52  thf(fact_98_less__mono__imp__le__mono,axiom,
% 0.33/0.52      ! [F: nat > nat,I: nat,J: nat] :
% 0.33/0.52        ( ! [I2: nat,J2: nat] :
% 0.33/0.52            ( ( ord_less_nat @ I2 @ J2 )
% 0.33/0.52           => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.33/0.52       => ( ( ord_less_eq_nat @ I @ J )
% 0.33/0.52         => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % less_mono_imp_le_mono
% 0.33/0.52  thf(fact_99_le__neq__implies__less,axiom,
% 0.33/0.52      ! [M: nat,N: nat] :
% 0.33/0.52        ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.52       => ( ( M != N )
% 0.33/0.52         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % le_neq_implies_less
% 0.33/0.52  thf(fact_100_less__or__eq__imp__le,axiom,
% 0.33/0.52      ! [M: nat,N: nat] :
% 0.33/0.52        ( ( ( ord_less_nat @ M @ N )
% 0.33/0.52          | ( M = N ) )
% 0.33/0.52       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.33/0.52  
% 0.33/0.52  % less_or_eq_imp_le
% 0.33/0.52  thf(fact_101_le__eq__less__or__eq,axiom,
% 0.33/0.52      ( ord_less_eq_nat
% 0.33/0.52      = ( ^ [M2: nat,N2: nat] :
% 0.33/0.52            ( ( ord_less_nat @ M2 @ N2 )
% 0.33/0.52            | ( M2 = N2 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % le_eq_less_or_eq
% 0.33/0.52  thf(fact_102_less__imp__le__nat,axiom,
% 0.33/0.52      ! [M: nat,N: nat] :
% 0.33/0.52        ( ( ord_less_nat @ M @ N )
% 0.33/0.52       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.33/0.52  
% 0.33/0.52  % less_imp_le_nat
% 0.33/0.52  thf(fact_103_nat__less__le,axiom,
% 0.33/0.52      ( ord_less_nat
% 0.33/0.52      = ( ^ [M2: nat,N2: nat] :
% 0.33/0.52            ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.33/0.52            & ( M2 != N2 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nat_less_le
% 0.33/0.52  thf(fact_104_ex__least__nat__le,axiom,
% 0.33/0.52      ! [P: nat > $o,N: nat] :
% 0.33/0.52        ( ( P @ N )
% 0.33/0.52       => ( ~ ( P @ zero_zero_nat )
% 0.33/0.52         => ? [K2: nat] :
% 0.33/0.52              ( ( ord_less_eq_nat @ K2 @ N )
% 0.33/0.52              & ! [I3: nat] :
% 0.33/0.52                  ( ( ord_less_nat @ I3 @ K2 )
% 0.33/0.52                 => ~ ( P @ I3 ) )
% 0.33/0.52              & ( P @ K2 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % ex_least_nat_le
% 0.33/0.52  thf(fact_105_length__pos__if__in__set,axiom,
% 0.33/0.52      ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
% 0.33/0.52        ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.33/0.52       => ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % length_pos_if_in_set
% 0.33/0.52  thf(fact_106_length__pos__if__in__set,axiom,
% 0.33/0.52      ! [X: real,Xs: list_real] :
% 0.33/0.52        ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.33/0.52       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % length_pos_if_in_set
% 0.33/0.52  thf(fact_107_length__pos__if__in__set,axiom,
% 0.33/0.52      ! [X: set_nat,Xs: list_set_nat] :
% 0.33/0.52        ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 0.33/0.52       => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % length_pos_if_in_set
% 0.33/0.52  thf(fact_108_length__pos__if__in__set,axiom,
% 0.33/0.52      ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.33/0.52        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.52       => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % length_pos_if_in_set
% 0.33/0.52  thf(fact_109_length__pos__if__in__set,axiom,
% 0.33/0.52      ! [X: $o,Xs: list_o] :
% 0.33/0.52        ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 0.33/0.52       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % length_pos_if_in_set
% 0.33/0.52  thf(fact_110_length__pos__if__in__set,axiom,
% 0.33/0.52      ! [X: nat,Xs: list_nat] :
% 0.33/0.52        ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.33/0.52       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % length_pos_if_in_set
% 0.33/0.52  thf(fact_111_length__pos__if__in__set,axiom,
% 0.33/0.52      ! [X: int,Xs: list_int] :
% 0.33/0.52        ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.33/0.52       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % length_pos_if_in_set
% 0.33/0.52  thf(fact_112_all__set__conv__all__nth,axiom,
% 0.33/0.52      ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.33/0.52        ( ( ! [X3: vEBT_VEBT] :
% 0.33/0.52              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.52             => ( P @ X3 ) ) )
% 0.33/0.52        = ( ! [I4: nat] :
% 0.33/0.52              ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.52             => ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % all_set_conv_all_nth
% 0.33/0.52  thf(fact_113_all__set__conv__all__nth,axiom,
% 0.33/0.52      ! [Xs: list_o,P: $o > $o] :
% 0.33/0.52        ( ( ! [X3: $o] :
% 0.33/0.52              ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
% 0.33/0.52             => ( P @ X3 ) ) )
% 0.33/0.52        = ( ! [I4: nat] :
% 0.33/0.52              ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 0.33/0.52             => ( P @ ( nth_o @ Xs @ I4 ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % all_set_conv_all_nth
% 0.33/0.52  thf(fact_114_all__set__conv__all__nth,axiom,
% 0.33/0.52      ! [Xs: list_nat,P: nat > $o] :
% 0.33/0.52        ( ( ! [X3: nat] :
% 0.33/0.52              ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 0.33/0.52             => ( P @ X3 ) ) )
% 0.33/0.52        = ( ! [I4: nat] :
% 0.33/0.52              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 0.33/0.52             => ( P @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % all_set_conv_all_nth
% 0.33/0.52  thf(fact_115_all__set__conv__all__nth,axiom,
% 0.33/0.52      ! [Xs: list_int,P: int > $o] :
% 0.33/0.52        ( ( ! [X3: int] :
% 0.33/0.52              ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 0.33/0.52             => ( P @ X3 ) ) )
% 0.33/0.52        = ( ! [I4: nat] :
% 0.33/0.52              ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 0.33/0.52             => ( P @ ( nth_int @ Xs @ I4 ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % all_set_conv_all_nth
% 0.33/0.52  thf(fact_116_all__nth__imp__all__set,axiom,
% 0.33/0.52      ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,X: product_prod_nat_nat] :
% 0.33/0.52        ( ! [I2: nat] :
% 0.33/0.52            ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 0.33/0.52           => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) ) )
% 0.33/0.52       => ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.33/0.52         => ( P @ X ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % all_nth_imp_all_set
% 0.33/0.52  thf(fact_117_all__nth__imp__all__set,axiom,
% 0.33/0.52      ! [Xs: list_real,P: real > $o,X: real] :
% 0.33/0.52        ( ! [I2: nat] :
% 0.33/0.52            ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
% 0.33/0.52           => ( P @ ( nth_real @ Xs @ I2 ) ) )
% 0.33/0.52       => ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.33/0.52         => ( P @ X ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % all_nth_imp_all_set
% 0.33/0.52  thf(fact_118_all__nth__imp__all__set,axiom,
% 0.33/0.52      ! [Xs: list_set_nat,P: set_nat > $o,X: set_nat] :
% 0.33/0.52        ( ! [I2: nat] :
% 0.33/0.52            ( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.33/0.52           => ( P @ ( nth_set_nat @ Xs @ I2 ) ) )
% 0.33/0.52       => ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 0.33/0.52         => ( P @ X ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % all_nth_imp_all_set
% 0.33/0.52  thf(fact_119_all__nth__imp__all__set,axiom,
% 0.33/0.52      ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
% 0.33/0.52        ( ! [I2: nat] :
% 0.33/0.52            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.52           => ( P @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) )
% 0.33/0.52       => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.52         => ( P @ X ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % all_nth_imp_all_set
% 0.33/0.52  thf(fact_120_all__nth__imp__all__set,axiom,
% 0.33/0.52      ! [Xs: list_o,P: $o > $o,X: $o] :
% 0.33/0.52        ( ! [I2: nat] :
% 0.33/0.52            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 0.33/0.52           => ( P @ ( nth_o @ Xs @ I2 ) ) )
% 0.33/0.52       => ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 0.33/0.52         => ( P @ X ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % all_nth_imp_all_set
% 0.33/0.52  thf(fact_121_all__nth__imp__all__set,axiom,
% 0.33/0.52      ! [Xs: list_nat,P: nat > $o,X: nat] :
% 0.33/0.52        ( ! [I2: nat] :
% 0.33/0.52            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 0.33/0.52           => ( P @ ( nth_nat @ Xs @ I2 ) ) )
% 0.33/0.52       => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.33/0.52         => ( P @ X ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % all_nth_imp_all_set
% 0.33/0.52  thf(fact_122_all__nth__imp__all__set,axiom,
% 0.33/0.52      ! [Xs: list_int,P: int > $o,X: int] :
% 0.33/0.52        ( ! [I2: nat] :
% 0.33/0.52            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 0.33/0.52           => ( P @ ( nth_int @ Xs @ I2 ) ) )
% 0.33/0.52       => ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.33/0.52         => ( P @ X ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % all_nth_imp_all_set
% 0.33/0.52  thf(fact_123_in__set__conv__nth,axiom,
% 0.33/0.52      ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
% 0.33/0.52        ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.33/0.52        = ( ? [I4: nat] :
% 0.33/0.52              ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 0.33/0.52              & ( ( nth_Pr7617993195940197384at_nat @ Xs @ I4 )
% 0.33/0.52                = X ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % in_set_conv_nth
% 0.33/0.52  thf(fact_124_in__set__conv__nth,axiom,
% 0.33/0.52      ! [X: real,Xs: list_real] :
% 0.33/0.52        ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.33/0.52        = ( ? [I4: nat] :
% 0.33/0.52              ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs ) )
% 0.33/0.52              & ( ( nth_real @ Xs @ I4 )
% 0.33/0.52                = X ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % in_set_conv_nth
% 0.33/0.52  thf(fact_125_in__set__conv__nth,axiom,
% 0.33/0.52      ! [X: set_nat,Xs: list_set_nat] :
% 0.33/0.52        ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 0.33/0.52        = ( ? [I4: nat] :
% 0.33/0.52              ( ( ord_less_nat @ I4 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.33/0.52              & ( ( nth_set_nat @ Xs @ I4 )
% 0.33/0.52                = X ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % in_set_conv_nth
% 0.33/0.52  thf(fact_126_in__set__conv__nth,axiom,
% 0.33/0.52      ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.33/0.52        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.52        = ( ? [I4: nat] :
% 0.33/0.52              ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.52              & ( ( nth_VEBT_VEBT @ Xs @ I4 )
% 0.33/0.52                = X ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % in_set_conv_nth
% 0.33/0.52  thf(fact_127_in__set__conv__nth,axiom,
% 0.33/0.52      ! [X: $o,Xs: list_o] :
% 0.33/0.52        ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 0.33/0.52        = ( ? [I4: nat] :
% 0.33/0.52              ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 0.33/0.52              & ( ( nth_o @ Xs @ I4 )
% 0.33/0.52                = X ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % in_set_conv_nth
% 0.33/0.52  thf(fact_128_in__set__conv__nth,axiom,
% 0.33/0.52      ! [X: nat,Xs: list_nat] :
% 0.33/0.52        ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.33/0.52        = ( ? [I4: nat] :
% 0.33/0.52              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 0.33/0.52              & ( ( nth_nat @ Xs @ I4 )
% 0.33/0.52                = X ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % in_set_conv_nth
% 0.33/0.52  thf(fact_129_in__set__conv__nth,axiom,
% 0.33/0.52      ! [X: int,Xs: list_int] :
% 0.33/0.52        ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.33/0.52        = ( ? [I4: nat] :
% 0.33/0.52              ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 0.33/0.52              & ( ( nth_int @ Xs @ I4 )
% 0.33/0.52                = X ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % in_set_conv_nth
% 0.33/0.52  thf(fact_130_list__ball__nth,axiom,
% 0.33/0.52      ! [N: nat,Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.33/0.52        ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.52       => ( ! [X4: vEBT_VEBT] :
% 0.33/0.52              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.52             => ( P @ X4 ) )
% 0.33/0.52         => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % list_ball_nth
% 0.33/0.52  thf(fact_131_list__ball__nth,axiom,
% 0.33/0.52      ! [N: nat,Xs: list_o,P: $o > $o] :
% 0.33/0.52        ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 0.33/0.52       => ( ! [X4: $o] :
% 0.33/0.52              ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 0.33/0.52             => ( P @ X4 ) )
% 0.33/0.52         => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % list_ball_nth
% 0.33/0.52  thf(fact_132_list__ball__nth,axiom,
% 0.33/0.52      ! [N: nat,Xs: list_nat,P: nat > $o] :
% 0.33/0.52        ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.33/0.52       => ( ! [X4: nat] :
% 0.33/0.52              ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 0.33/0.52             => ( P @ X4 ) )
% 0.33/0.52         => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % list_ball_nth
% 0.33/0.52  thf(fact_133_list__ball__nth,axiom,
% 0.33/0.52      ! [N: nat,Xs: list_int,P: int > $o] :
% 0.33/0.52        ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 0.33/0.52       => ( ! [X4: int] :
% 0.33/0.52              ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 0.33/0.52             => ( P @ X4 ) )
% 0.33/0.52         => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % list_ball_nth
% 0.33/0.52  thf(fact_134_nth__mem,axiom,
% 0.33/0.52      ! [N: nat,Xs: list_P6011104703257516679at_nat] :
% 0.33/0.52        ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
% 0.33/0.52       => ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N ) @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_mem
% 0.33/0.52  thf(fact_135_nth__mem,axiom,
% 0.33/0.52      ! [N: nat,Xs: list_real] :
% 0.33/0.52        ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
% 0.33/0.52       => ( member_real @ ( nth_real @ Xs @ N ) @ ( set_real2 @ Xs ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_mem
% 0.33/0.52  thf(fact_136_nth__mem,axiom,
% 0.33/0.52      ! [N: nat,Xs: list_set_nat] :
% 0.33/0.52        ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.33/0.52       => ( member_set_nat @ ( nth_set_nat @ Xs @ N ) @ ( set_set_nat2 @ Xs ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_mem
% 0.33/0.52  thf(fact_137_nth__mem,axiom,
% 0.33/0.52      ! [N: nat,Xs: list_VEBT_VEBT] :
% 0.33/0.52        ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.52       => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_mem
% 0.33/0.52  thf(fact_138_nth__mem,axiom,
% 0.33/0.52      ! [N: nat,Xs: list_o] :
% 0.33/0.52        ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 0.33/0.52       => ( member_o @ ( nth_o @ Xs @ N ) @ ( set_o2 @ Xs ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_mem
% 0.33/0.52  thf(fact_139_nth__mem,axiom,
% 0.33/0.52      ! [N: nat,Xs: list_nat] :
% 0.33/0.52        ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.33/0.52       => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_mem
% 0.33/0.52  thf(fact_140_nth__mem,axiom,
% 0.33/0.52      ! [N: nat,Xs: list_int] :
% 0.33/0.52        ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 0.33/0.52       => ( member_int @ ( nth_int @ Xs @ N ) @ ( set_int2 @ Xs ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_mem
% 0.33/0.52  thf(fact_141_set__update__memI,axiom,
% 0.33/0.52      ! [N: nat,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
% 0.33/0.52        ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
% 0.33/0.52       => ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ N @ X ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % set_update_memI
% 0.33/0.52  thf(fact_142_set__update__memI,axiom,
% 0.33/0.52      ! [N: nat,Xs: list_real,X: real] :
% 0.33/0.52        ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
% 0.33/0.52       => ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs @ N @ X ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % set_update_memI
% 0.33/0.52  thf(fact_143_set__update__memI,axiom,
% 0.33/0.52      ! [N: nat,Xs: list_set_nat,X: set_nat] :
% 0.33/0.52        ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.33/0.52       => ( member_set_nat @ X @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ N @ X ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % set_update_memI
% 0.33/0.52  thf(fact_144_set__update__memI,axiom,
% 0.33/0.52      ! [N: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 0.33/0.52        ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.52       => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N @ X ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % set_update_memI
% 0.33/0.52  thf(fact_145_set__update__memI,axiom,
% 0.33/0.52      ! [N: nat,Xs: list_o,X: $o] :
% 0.33/0.52        ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 0.33/0.52       => ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs @ N @ X ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % set_update_memI
% 0.33/0.52  thf(fact_146_set__update__memI,axiom,
% 0.33/0.52      ! [N: nat,Xs: list_nat,X: nat] :
% 0.33/0.52        ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.33/0.52       => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % set_update_memI
% 0.33/0.52  thf(fact_147_set__update__memI,axiom,
% 0.33/0.52      ! [N: nat,Xs: list_int,X: int] :
% 0.33/0.52        ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 0.33/0.52       => ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs @ N @ X ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % set_update_memI
% 0.33/0.52  thf(fact_148_zero__reorient,axiom,
% 0.33/0.52      ! [X: complex] :
% 0.33/0.52        ( ( zero_zero_complex = X )
% 0.33/0.52        = ( X = zero_zero_complex ) ) ).
% 0.33/0.52  
% 0.33/0.52  % zero_reorient
% 0.33/0.52  thf(fact_149_zero__reorient,axiom,
% 0.33/0.52      ! [X: real] :
% 0.33/0.52        ( ( zero_zero_real = X )
% 0.33/0.52        = ( X = zero_zero_real ) ) ).
% 0.33/0.52  
% 0.33/0.52  % zero_reorient
% 0.33/0.52  thf(fact_150_zero__reorient,axiom,
% 0.33/0.52      ! [X: rat] :
% 0.33/0.52        ( ( zero_zero_rat = X )
% 0.33/0.52        = ( X = zero_zero_rat ) ) ).
% 0.33/0.52  
% 0.33/0.52  % zero_reorient
% 0.33/0.52  thf(fact_151_zero__reorient,axiom,
% 0.33/0.52      ! [X: nat] :
% 0.33/0.52        ( ( zero_zero_nat = X )
% 0.33/0.52        = ( X = zero_zero_nat ) ) ).
% 0.33/0.52  
% 0.33/0.52  % zero_reorient
% 0.33/0.52  thf(fact_152_zero__reorient,axiom,
% 0.33/0.52      ! [X: int] :
% 0.33/0.52        ( ( zero_zero_int = X )
% 0.33/0.52        = ( X = zero_zero_int ) ) ).
% 0.33/0.52  
% 0.33/0.52  % zero_reorient
% 0.33/0.52  thf(fact_153_linorder__neqE__nat,axiom,
% 0.33/0.52      ! [X: nat,Y: nat] :
% 0.33/0.52        ( ( X != Y )
% 0.33/0.52       => ( ~ ( ord_less_nat @ X @ Y )
% 0.33/0.52         => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % linorder_neqE_nat
% 0.33/0.52  thf(fact_154_infinite__descent,axiom,
% 0.33/0.52      ! [P: nat > $o,N: nat] :
% 0.33/0.52        ( ! [N3: nat] :
% 0.33/0.52            ( ~ ( P @ N3 )
% 0.33/0.52           => ? [M3: nat] :
% 0.33/0.52                ( ( ord_less_nat @ M3 @ N3 )
% 0.33/0.52                & ~ ( P @ M3 ) ) )
% 0.33/0.52       => ( P @ N ) ) ).
% 0.33/0.52  
% 0.33/0.52  % infinite_descent
% 0.33/0.52  thf(fact_155_nat__less__induct,axiom,
% 0.33/0.52      ! [P: nat > $o,N: nat] :
% 0.33/0.52        ( ! [N3: nat] :
% 0.33/0.52            ( ! [M3: nat] :
% 0.33/0.52                ( ( ord_less_nat @ M3 @ N3 )
% 0.33/0.52               => ( P @ M3 ) )
% 0.33/0.52           => ( P @ N3 ) )
% 0.33/0.52       => ( P @ N ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nat_less_induct
% 0.33/0.52  thf(fact_156_less__irrefl__nat,axiom,
% 0.33/0.52      ! [N: nat] :
% 0.33/0.52        ~ ( ord_less_nat @ N @ N ) ).
% 0.33/0.52  
% 0.33/0.52  % less_irrefl_nat
% 0.33/0.52  thf(fact_157_less__not__refl3,axiom,
% 0.33/0.52      ! [S: nat,T: nat] :
% 0.33/0.52        ( ( ord_less_nat @ S @ T )
% 0.33/0.52       => ( S != T ) ) ).
% 0.33/0.52  
% 0.33/0.52  % less_not_refl3
% 0.33/0.52  thf(fact_158_less__not__refl2,axiom,
% 0.33/0.52      ! [N: nat,M: nat] :
% 0.33/0.52        ( ( ord_less_nat @ N @ M )
% 0.33/0.52       => ( M != N ) ) ).
% 0.33/0.52  
% 0.33/0.52  % less_not_refl2
% 0.33/0.52  thf(fact_159_less__not__refl,axiom,
% 0.33/0.52      ! [N: nat] :
% 0.33/0.52        ~ ( ord_less_nat @ N @ N ) ).
% 0.33/0.52  
% 0.33/0.52  % less_not_refl
% 0.33/0.52  thf(fact_160_nat__neq__iff,axiom,
% 0.33/0.52      ! [M: nat,N: nat] :
% 0.33/0.52        ( ( M != N )
% 0.33/0.52        = ( ( ord_less_nat @ M @ N )
% 0.33/0.52          | ( ord_less_nat @ N @ M ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nat_neq_iff
% 0.33/0.52  thf(fact_161_size__neq__size__imp__neq,axiom,
% 0.33/0.52      ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
% 0.33/0.52        ( ( ( size_s6755466524823107622T_VEBT @ X )
% 0.33/0.52         != ( size_s6755466524823107622T_VEBT @ Y ) )
% 0.33/0.52       => ( X != Y ) ) ).
% 0.33/0.52  
% 0.33/0.52  % size_neq_size_imp_neq
% 0.33/0.52  thf(fact_162_size__neq__size__imp__neq,axiom,
% 0.33/0.52      ! [X: list_o,Y: list_o] :
% 0.33/0.52        ( ( ( size_size_list_o @ X )
% 0.33/0.52         != ( size_size_list_o @ Y ) )
% 0.33/0.52       => ( X != Y ) ) ).
% 0.33/0.52  
% 0.33/0.52  % size_neq_size_imp_neq
% 0.33/0.52  thf(fact_163_size__neq__size__imp__neq,axiom,
% 0.33/0.52      ! [X: list_nat,Y: list_nat] :
% 0.33/0.52        ( ( ( size_size_list_nat @ X )
% 0.33/0.52         != ( size_size_list_nat @ Y ) )
% 0.33/0.52       => ( X != Y ) ) ).
% 0.33/0.52  
% 0.33/0.52  % size_neq_size_imp_neq
% 0.33/0.52  thf(fact_164_size__neq__size__imp__neq,axiom,
% 0.33/0.52      ! [X: list_int,Y: list_int] :
% 0.33/0.52        ( ( ( size_size_list_int @ X )
% 0.33/0.52         != ( size_size_list_int @ Y ) )
% 0.33/0.52       => ( X != Y ) ) ).
% 0.33/0.52  
% 0.33/0.52  % size_neq_size_imp_neq
% 0.33/0.52  thf(fact_165_size__neq__size__imp__neq,axiom,
% 0.33/0.52      ! [X: num,Y: num] :
% 0.33/0.52        ( ( ( size_size_num @ X )
% 0.33/0.52         != ( size_size_num @ Y ) )
% 0.33/0.52       => ( X != Y ) ) ).
% 0.33/0.52  
% 0.33/0.52  % size_neq_size_imp_neq
% 0.33/0.52  thf(fact_166_neq__if__length__neq,axiom,
% 0.33/0.52      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.33/0.52        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.33/0.52         != ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.33/0.52       => ( Xs != Ys ) ) ).
% 0.33/0.52  
% 0.33/0.52  % neq_if_length_neq
% 0.33/0.52  thf(fact_167_neq__if__length__neq,axiom,
% 0.33/0.52      ! [Xs: list_o,Ys: list_o] :
% 0.33/0.52        ( ( ( size_size_list_o @ Xs )
% 0.33/0.52         != ( size_size_list_o @ Ys ) )
% 0.33/0.52       => ( Xs != Ys ) ) ).
% 0.33/0.52  
% 0.33/0.52  % neq_if_length_neq
% 0.33/0.52  thf(fact_168_neq__if__length__neq,axiom,
% 0.33/0.52      ! [Xs: list_nat,Ys: list_nat] :
% 0.33/0.52        ( ( ( size_size_list_nat @ Xs )
% 0.33/0.52         != ( size_size_list_nat @ Ys ) )
% 0.33/0.52       => ( Xs != Ys ) ) ).
% 0.33/0.52  
% 0.33/0.52  % neq_if_length_neq
% 0.33/0.52  thf(fact_169_neq__if__length__neq,axiom,
% 0.33/0.52      ! [Xs: list_int,Ys: list_int] :
% 0.33/0.52        ( ( ( size_size_list_int @ Xs )
% 0.33/0.52         != ( size_size_list_int @ Ys ) )
% 0.33/0.52       => ( Xs != Ys ) ) ).
% 0.33/0.52  
% 0.33/0.52  % neq_if_length_neq
% 0.33/0.52  thf(fact_170_Ex__list__of__length,axiom,
% 0.33/0.52      ! [N: nat] :
% 0.33/0.52      ? [Xs3: list_VEBT_VEBT] :
% 0.33/0.52        ( ( size_s6755466524823107622T_VEBT @ Xs3 )
% 0.33/0.52        = N ) ).
% 0.33/0.52  
% 0.33/0.52  % Ex_list_of_length
% 0.33/0.52  thf(fact_171_Ex__list__of__length,axiom,
% 0.33/0.52      ! [N: nat] :
% 0.33/0.52      ? [Xs3: list_o] :
% 0.33/0.52        ( ( size_size_list_o @ Xs3 )
% 0.33/0.52        = N ) ).
% 0.33/0.52  
% 0.33/0.52  % Ex_list_of_length
% 0.33/0.52  thf(fact_172_Ex__list__of__length,axiom,
% 0.33/0.52      ! [N: nat] :
% 0.33/0.52      ? [Xs3: list_nat] :
% 0.33/0.52        ( ( size_size_list_nat @ Xs3 )
% 0.33/0.52        = N ) ).
% 0.33/0.52  
% 0.33/0.52  % Ex_list_of_length
% 0.33/0.52  thf(fact_173_Ex__list__of__length,axiom,
% 0.33/0.52      ! [N: nat] :
% 0.33/0.52      ? [Xs3: list_int] :
% 0.33/0.52        ( ( size_size_list_int @ Xs3 )
% 0.33/0.52        = N ) ).
% 0.33/0.52  
% 0.33/0.52  % Ex_list_of_length
% 0.33/0.52  thf(fact_174_list__update__swap,axiom,
% 0.33/0.52      ! [I: nat,I5: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT,X5: vEBT_VEBT] :
% 0.33/0.52        ( ( I != I5 )
% 0.33/0.52       => ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ I5 @ X5 )
% 0.33/0.52          = ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I5 @ X5 ) @ I @ X ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % list_update_swap
% 0.33/0.52  thf(fact_175_zero__less__iff__neq__zero,axiom,
% 0.33/0.52      ! [N: nat] :
% 0.33/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.52        = ( N != zero_zero_nat ) ) ).
% 0.33/0.52  
% 0.33/0.52  % zero_less_iff_neq_zero
% 0.33/0.52  thf(fact_176_gr__implies__not__zero,axiom,
% 0.33/0.52      ! [M: nat,N: nat] :
% 0.33/0.52        ( ( ord_less_nat @ M @ N )
% 0.33/0.52       => ( N != zero_zero_nat ) ) ).
% 0.33/0.52  
% 0.33/0.52  % gr_implies_not_zero
% 0.33/0.52  thf(fact_177_not__less__zero,axiom,
% 0.33/0.52      ! [N: nat] :
% 0.33/0.52        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.33/0.52  
% 0.33/0.52  % not_less_zero
% 0.33/0.52  thf(fact_178_gr__zeroI,axiom,
% 0.33/0.52      ! [N: nat] :
% 0.33/0.52        ( ( N != zero_zero_nat )
% 0.33/0.52       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.33/0.52  
% 0.33/0.52  % gr_zeroI
% 0.33/0.52  thf(fact_179_infinite__descent0,axiom,
% 0.33/0.52      ! [P: nat > $o,N: nat] :
% 0.33/0.52        ( ( P @ zero_zero_nat )
% 0.33/0.52       => ( ! [N3: nat] :
% 0.33/0.52              ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.33/0.52             => ( ~ ( P @ N3 )
% 0.33/0.52               => ? [M3: nat] :
% 0.33/0.52                    ( ( ord_less_nat @ M3 @ N3 )
% 0.33/0.52                    & ~ ( P @ M3 ) ) ) )
% 0.33/0.52         => ( P @ N ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % infinite_descent0
% 0.33/0.52  thf(fact_180_gr__implies__not0,axiom,
% 0.33/0.52      ! [M: nat,N: nat] :
% 0.33/0.52        ( ( ord_less_nat @ M @ N )
% 0.33/0.52       => ( N != zero_zero_nat ) ) ).
% 0.33/0.52  
% 0.33/0.52  % gr_implies_not0
% 0.33/0.52  thf(fact_181_less__zeroE,axiom,
% 0.33/0.52      ! [N: nat] :
% 0.33/0.52        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.33/0.52  
% 0.33/0.52  % less_zeroE
% 0.33/0.52  thf(fact_182_not__less0,axiom,
% 0.33/0.52      ! [N: nat] :
% 0.33/0.52        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.33/0.52  
% 0.33/0.52  % not_less0
% 0.33/0.52  thf(fact_183_not__gr0,axiom,
% 0.33/0.52      ! [N: nat] :
% 0.33/0.52        ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.33/0.52        = ( N = zero_zero_nat ) ) ).
% 0.33/0.52  
% 0.33/0.52  % not_gr0
% 0.33/0.52  thf(fact_184_gr0I,axiom,
% 0.33/0.52      ! [N: nat] :
% 0.33/0.52        ( ( N != zero_zero_nat )
% 0.33/0.52       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.33/0.52  
% 0.33/0.52  % gr0I
% 0.33/0.52  thf(fact_185_bot__nat__0_Oextremum__strict,axiom,
% 0.33/0.52      ! [A: nat] :
% 0.33/0.52        ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.33/0.52  
% 0.33/0.52  % bot_nat_0.extremum_strict
% 0.33/0.52  thf(fact_186_length__induct,axiom,
% 0.33/0.52      ! [P: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 0.33/0.52        ( ! [Xs3: list_VEBT_VEBT] :
% 0.33/0.52            ( ! [Ys2: list_VEBT_VEBT] :
% 0.33/0.52                ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
% 0.33/0.52               => ( P @ Ys2 ) )
% 0.33/0.52           => ( P @ Xs3 ) )
% 0.33/0.52       => ( P @ Xs ) ) ).
% 0.33/0.52  
% 0.33/0.52  % length_induct
% 0.33/0.52  thf(fact_187_length__induct,axiom,
% 0.33/0.52      ! [P: list_o > $o,Xs: list_o] :
% 0.33/0.52        ( ! [Xs3: list_o] :
% 0.33/0.52            ( ! [Ys2: list_o] :
% 0.33/0.52                ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
% 0.33/0.52               => ( P @ Ys2 ) )
% 0.33/0.52           => ( P @ Xs3 ) )
% 0.33/0.52       => ( P @ Xs ) ) ).
% 0.33/0.52  
% 0.33/0.52  % length_induct
% 0.33/0.52  thf(fact_188_length__induct,axiom,
% 0.33/0.52      ! [P: list_nat > $o,Xs: list_nat] :
% 0.33/0.52        ( ! [Xs3: list_nat] :
% 0.33/0.52            ( ! [Ys2: list_nat] :
% 0.33/0.52                ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
% 0.33/0.52               => ( P @ Ys2 ) )
% 0.33/0.52           => ( P @ Xs3 ) )
% 0.33/0.52       => ( P @ Xs ) ) ).
% 0.33/0.52  
% 0.33/0.52  % length_induct
% 0.33/0.52  thf(fact_189_length__induct,axiom,
% 0.33/0.52      ! [P: list_int > $o,Xs: list_int] :
% 0.33/0.52        ( ! [Xs3: list_int] :
% 0.33/0.52            ( ! [Ys2: list_int] :
% 0.33/0.52                ( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs3 ) )
% 0.33/0.52               => ( P @ Ys2 ) )
% 0.33/0.52           => ( P @ Xs3 ) )
% 0.33/0.52       => ( P @ Xs ) ) ).
% 0.33/0.52  
% 0.33/0.52  % length_induct
% 0.33/0.52  thf(fact_190_list__eq__iff__nth__eq,axiom,
% 0.33/0.52      ( ( ^ [Y5: list_VEBT_VEBT,Z: list_VEBT_VEBT] : ( Y5 = Z ) )
% 0.33/0.52      = ( ^ [Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 0.33/0.52            ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.33/0.52              = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
% 0.33/0.52            & ! [I4: nat] :
% 0.33/0.52                ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.33/0.52               => ( ( nth_VEBT_VEBT @ Xs2 @ I4 )
% 0.33/0.52                  = ( nth_VEBT_VEBT @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % list_eq_iff_nth_eq
% 0.33/0.52  thf(fact_191_list__eq__iff__nth__eq,axiom,
% 0.33/0.52      ( ( ^ [Y5: list_o,Z: list_o] : ( Y5 = Z ) )
% 0.33/0.52      = ( ^ [Xs2: list_o,Ys3: list_o] :
% 0.33/0.52            ( ( ( size_size_list_o @ Xs2 )
% 0.33/0.52              = ( size_size_list_o @ Ys3 ) )
% 0.33/0.52            & ! [I4: nat] :
% 0.33/0.52                ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
% 0.33/0.52               => ( ( nth_o @ Xs2 @ I4 )
% 0.33/0.52                  = ( nth_o @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % list_eq_iff_nth_eq
% 0.33/0.52  thf(fact_192_list__eq__iff__nth__eq,axiom,
% 0.33/0.52      ( ( ^ [Y5: list_nat,Z: list_nat] : ( Y5 = Z ) )
% 0.33/0.52      = ( ^ [Xs2: list_nat,Ys3: list_nat] :
% 0.33/0.52            ( ( ( size_size_list_nat @ Xs2 )
% 0.33/0.52              = ( size_size_list_nat @ Ys3 ) )
% 0.33/0.52            & ! [I4: nat] :
% 0.33/0.52                ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
% 0.33/0.52               => ( ( nth_nat @ Xs2 @ I4 )
% 0.33/0.52                  = ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % list_eq_iff_nth_eq
% 0.33/0.52  thf(fact_193_list__eq__iff__nth__eq,axiom,
% 0.33/0.52      ( ( ^ [Y5: list_int,Z: list_int] : ( Y5 = Z ) )
% 0.33/0.52      = ( ^ [Xs2: list_int,Ys3: list_int] :
% 0.33/0.52            ( ( ( size_size_list_int @ Xs2 )
% 0.33/0.52              = ( size_size_list_int @ Ys3 ) )
% 0.33/0.52            & ! [I4: nat] :
% 0.33/0.52                ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
% 0.33/0.52               => ( ( nth_int @ Xs2 @ I4 )
% 0.33/0.52                  = ( nth_int @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % list_eq_iff_nth_eq
% 0.33/0.52  thf(fact_194_Skolem__list__nth,axiom,
% 0.33/0.52      ! [K: nat,P: nat > vEBT_VEBT > $o] :
% 0.33/0.52        ( ( ! [I4: nat] :
% 0.33/0.52              ( ( ord_less_nat @ I4 @ K )
% 0.33/0.52             => ? [X6: vEBT_VEBT] : ( P @ I4 @ X6 ) ) )
% 0.33/0.52        = ( ? [Xs2: list_VEBT_VEBT] :
% 0.33/0.52              ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.33/0.52                = K )
% 0.33/0.52              & ! [I4: nat] :
% 0.33/0.52                  ( ( ord_less_nat @ I4 @ K )
% 0.33/0.52                 => ( P @ I4 @ ( nth_VEBT_VEBT @ Xs2 @ I4 ) ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % Skolem_list_nth
% 0.33/0.52  thf(fact_195_Skolem__list__nth,axiom,
% 0.33/0.52      ! [K: nat,P: nat > $o > $o] :
% 0.33/0.52        ( ( ! [I4: nat] :
% 0.33/0.52              ( ( ord_less_nat @ I4 @ K )
% 0.33/0.52             => ? [X6: $o] : ( P @ I4 @ X6 ) ) )
% 0.33/0.52        = ( ? [Xs2: list_o] :
% 0.33/0.52              ( ( ( size_size_list_o @ Xs2 )
% 0.33/0.52                = K )
% 0.33/0.52              & ! [I4: nat] :
% 0.33/0.52                  ( ( ord_less_nat @ I4 @ K )
% 0.33/0.52                 => ( P @ I4 @ ( nth_o @ Xs2 @ I4 ) ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % Skolem_list_nth
% 0.33/0.52  thf(fact_196_Skolem__list__nth,axiom,
% 0.33/0.52      ! [K: nat,P: nat > nat > $o] :
% 0.33/0.52        ( ( ! [I4: nat] :
% 0.33/0.52              ( ( ord_less_nat @ I4 @ K )
% 0.33/0.52             => ? [X6: nat] : ( P @ I4 @ X6 ) ) )
% 0.33/0.52        = ( ? [Xs2: list_nat] :
% 0.33/0.52              ( ( ( size_size_list_nat @ Xs2 )
% 0.33/0.52                = K )
% 0.33/0.52              & ! [I4: nat] :
% 0.33/0.52                  ( ( ord_less_nat @ I4 @ K )
% 0.33/0.52                 => ( P @ I4 @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % Skolem_list_nth
% 0.33/0.52  thf(fact_197_Skolem__list__nth,axiom,
% 0.33/0.52      ! [K: nat,P: nat > int > $o] :
% 0.33/0.52        ( ( ! [I4: nat] :
% 0.33/0.52              ( ( ord_less_nat @ I4 @ K )
% 0.33/0.52             => ? [X6: int] : ( P @ I4 @ X6 ) ) )
% 0.33/0.52        = ( ? [Xs2: list_int] :
% 0.33/0.52              ( ( ( size_size_list_int @ Xs2 )
% 0.33/0.52                = K )
% 0.33/0.52              & ! [I4: nat] :
% 0.33/0.52                  ( ( ord_less_nat @ I4 @ K )
% 0.33/0.52                 => ( P @ I4 @ ( nth_int @ Xs2 @ I4 ) ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % Skolem_list_nth
% 0.33/0.52  thf(fact_198_nth__equalityI,axiom,
% 0.33/0.52      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.33/0.52        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.33/0.52          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.33/0.52       => ( ! [I2: nat] :
% 0.33/0.52              ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.52             => ( ( nth_VEBT_VEBT @ Xs @ I2 )
% 0.33/0.52                = ( nth_VEBT_VEBT @ Ys @ I2 ) ) )
% 0.33/0.52         => ( Xs = Ys ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_equalityI
% 0.33/0.52  thf(fact_199_nth__equalityI,axiom,
% 0.33/0.52      ! [Xs: list_o,Ys: list_o] :
% 0.33/0.52        ( ( ( size_size_list_o @ Xs )
% 0.33/0.52          = ( size_size_list_o @ Ys ) )
% 0.33/0.52       => ( ! [I2: nat] :
% 0.33/0.52              ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 0.33/0.52             => ( ( nth_o @ Xs @ I2 )
% 0.33/0.52                = ( nth_o @ Ys @ I2 ) ) )
% 0.33/0.52         => ( Xs = Ys ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_equalityI
% 0.33/0.52  thf(fact_200_nth__equalityI,axiom,
% 0.33/0.52      ! [Xs: list_nat,Ys: list_nat] :
% 0.33/0.52        ( ( ( size_size_list_nat @ Xs )
% 0.33/0.52          = ( size_size_list_nat @ Ys ) )
% 0.33/0.52       => ( ! [I2: nat] :
% 0.33/0.52              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 0.33/0.52             => ( ( nth_nat @ Xs @ I2 )
% 0.33/0.52                = ( nth_nat @ Ys @ I2 ) ) )
% 0.33/0.52         => ( Xs = Ys ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_equalityI
% 0.33/0.52  thf(fact_201_nth__equalityI,axiom,
% 0.33/0.52      ! [Xs: list_int,Ys: list_int] :
% 0.33/0.52        ( ( ( size_size_list_int @ Xs )
% 0.33/0.52          = ( size_size_list_int @ Ys ) )
% 0.33/0.52       => ( ! [I2: nat] :
% 0.33/0.52              ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 0.33/0.52             => ( ( nth_int @ Xs @ I2 )
% 0.33/0.52                = ( nth_int @ Ys @ I2 ) ) )
% 0.33/0.52         => ( Xs = Ys ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nth_equalityI
% 0.33/0.52  thf(fact_202_list__update__same__conv,axiom,
% 0.33/0.52      ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 0.33/0.52        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.52       => ( ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X )
% 0.33/0.52            = Xs )
% 0.33/0.52          = ( ( nth_VEBT_VEBT @ Xs @ I )
% 0.33/0.52            = X ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % list_update_same_conv
% 0.33/0.52  thf(fact_203_list__update__same__conv,axiom,
% 0.33/0.52      ! [I: nat,Xs: list_o,X: $o] :
% 0.33/0.52        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.33/0.52       => ( ( ( list_update_o @ Xs @ I @ X )
% 0.33/0.52            = Xs )
% 0.33/0.52          = ( ( nth_o @ Xs @ I )
% 0.33/0.52            = X ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % list_update_same_conv
% 0.33/0.52  thf(fact_204_list__update__same__conv,axiom,
% 0.33/0.52      ! [I: nat,Xs: list_nat,X: nat] :
% 0.33/0.52        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 0.33/0.52       => ( ( ( list_update_nat @ Xs @ I @ X )
% 0.33/0.52            = Xs )
% 0.33/0.52          = ( ( nth_nat @ Xs @ I )
% 0.33/0.52            = X ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % list_update_same_conv
% 0.33/0.52  thf(fact_205_list__update__same__conv,axiom,
% 0.33/0.52      ! [I: nat,Xs: list_int,X: int] :
% 0.33/0.52        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 0.33/0.52       => ( ( ( list_update_int @ Xs @ I @ X )
% 0.33/0.52            = Xs )
% 0.33/0.52          = ( ( nth_int @ Xs @ I )
% 0.33/0.52            = X ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % list_update_same_conv
% 0.33/0.52  thf(fact_206_buildup__nothing__in__leaf,axiom,
% 0.33/0.52      ! [N: nat,X: nat] :
% 0.33/0.52        ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% 0.33/0.52  
% 0.33/0.52  % buildup_nothing_in_leaf
% 0.33/0.52  thf(fact_207_member__correct,axiom,
% 0.33/0.52      ! [T: vEBT_VEBT,N: nat,X: nat] :
% 0.33/0.52        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.52       => ( ( vEBT_vebt_member @ T @ X )
% 0.33/0.52          = ( member_nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % member_correct
% 0.33/0.52  thf(fact_208_maxidef,axiom,
% 0.33/0.52      ( ( some_nat @ maxi )
% 0.33/0.52      = ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % maxidef
% 0.33/0.52  thf(fact_209_both__member__options__equiv__member,axiom,
% 0.33/0.52      ! [T: vEBT_VEBT,N: nat,X: nat] :
% 0.33/0.52        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.52       => ( ( vEBT_V8194947554948674370ptions @ T @ X )
% 0.33/0.52          = ( vEBT_vebt_member @ T @ X ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % both_member_options_equiv_member
% 0.33/0.52  thf(fact_210_valid__member__both__member__options,axiom,
% 0.33/0.52      ! [T: vEBT_VEBT,N: nat,X: nat] :
% 0.33/0.52        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.52       => ( ( vEBT_V8194947554948674370ptions @ T @ X )
% 0.33/0.52         => ( vEBT_vebt_member @ T @ X ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % valid_member_both_member_options
% 0.33/0.52  thf(fact_211_VEBT_Oinject_I1_J,axiom,
% 0.33/0.52      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
% 0.33/0.52        ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 0.33/0.52          = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
% 0.33/0.52        = ( ( X11 = Y11 )
% 0.33/0.52          & ( X12 = Y12 )
% 0.33/0.52          & ( X13 = Y13 )
% 0.33/0.52          & ( X14 = Y14 ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % VEBT.inject(1)
% 0.33/0.52  thf(fact_212_buildup__gives__empty,axiom,
% 0.33/0.52      ! [N: nat] :
% 0.33/0.52        ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
% 0.33/0.52        = bot_bot_set_nat ) ).
% 0.33/0.52  
% 0.33/0.52  % buildup_gives_empty
% 0.33/0.52  thf(fact_213_dual__order_Orefl,axiom,
% 0.33/0.52      ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 0.33/0.52  
% 0.33/0.52  % dual_order.refl
% 0.33/0.52  thf(fact_214_dual__order_Orefl,axiom,
% 0.33/0.52      ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 0.33/0.52  
% 0.33/0.52  % dual_order.refl
% 0.33/0.52  thf(fact_215_dual__order_Orefl,axiom,
% 0.33/0.52      ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 0.33/0.52  
% 0.33/0.52  % dual_order.refl
% 0.33/0.52  thf(fact_216_dual__order_Orefl,axiom,
% 0.33/0.52      ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.33/0.52  
% 0.33/0.52  % dual_order.refl
% 0.33/0.52  thf(fact_217_dual__order_Orefl,axiom,
% 0.33/0.52      ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.33/0.52  
% 0.33/0.52  % dual_order.refl
% 0.33/0.52  thf(fact_218_order__refl,axiom,
% 0.33/0.52      ! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).
% 0.33/0.52  
% 0.33/0.52  % order_refl
% 0.33/0.52  thf(fact_219_order__refl,axiom,
% 0.33/0.52      ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).
% 0.33/0.52  
% 0.33/0.52  % order_refl
% 0.33/0.52  thf(fact_220_order__refl,axiom,
% 0.33/0.52      ! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% 0.33/0.52  
% 0.33/0.52  % order_refl
% 0.33/0.52  thf(fact_221_order__refl,axiom,
% 0.33/0.52      ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 0.33/0.52  
% 0.33/0.52  % order_refl
% 0.33/0.52  thf(fact_222_order__refl,axiom,
% 0.33/0.52      ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 0.33/0.52  
% 0.33/0.52  % order_refl
% 0.33/0.52  thf(fact_223_buildup__nothing__in__min__max,axiom,
% 0.33/0.52      ! [N: nat,X: nat] :
% 0.33/0.52        ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% 0.33/0.52  
% 0.33/0.52  % buildup_nothing_in_min_max
% 0.33/0.52  thf(fact_224__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062maxi_O_ASome_Amaxi_A_061_Avebt__maxt_A_ItreeList_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Ahigh_Ax_An_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.33/0.52      ~ ! [Maxi: nat] :
% 0.33/0.52          ( ( some_nat @ Maxi )
% 0.33/0.52         != ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % \<open>\<And>thesis. (\<And>maxi. Some maxi = vebt_maxt (treeList[high x n := vebt_delete (treeList ! high x n) (low x n)] ! high x n) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.33/0.52  thf(fact_225_set__vebt__finite,axiom,
% 0.33/0.52      ! [T: vEBT_VEBT,N: nat] :
% 0.33/0.52        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.52       => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % set_vebt_finite
% 0.33/0.52  thf(fact_226_min__Null__member,axiom,
% 0.33/0.52      ! [T: vEBT_VEBT,X: nat] :
% 0.33/0.52        ( ( vEBT_VEBT_minNull @ T )
% 0.33/0.52       => ~ ( vEBT_vebt_member @ T @ X ) ) ).
% 0.33/0.52  
% 0.33/0.52  % min_Null_member
% 0.33/0.52  thf(fact_227_maxbmo,axiom,
% 0.33/0.52      ! [T: vEBT_VEBT,X: nat] :
% 0.33/0.52        ( ( ( vEBT_vebt_maxt @ T )
% 0.33/0.52          = ( some_nat @ X ) )
% 0.33/0.52       => ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).
% 0.33/0.52  
% 0.33/0.52  % maxbmo
% 0.33/0.52  thf(fact_228_both__member__options__def,axiom,
% 0.33/0.52      ( vEBT_V8194947554948674370ptions
% 0.33/0.52      = ( ^ [T2: vEBT_VEBT,X3: nat] :
% 0.33/0.52            ( ( vEBT_V5719532721284313246member @ T2 @ X3 )
% 0.33/0.52            | ( vEBT_VEBT_membermima @ T2 @ X3 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % both_member_options_def
% 0.33/0.52  thf(fact_229_maxt__member,axiom,
% 0.33/0.52      ! [T: vEBT_VEBT,N: nat,Maxi2: nat] :
% 0.33/0.52        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.52       => ( ( ( vEBT_vebt_maxt @ T )
% 0.33/0.52            = ( some_nat @ Maxi2 ) )
% 0.33/0.52         => ( vEBT_vebt_member @ T @ Maxi2 ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % maxt_member
% 0.33/0.52  thf(fact_230_member__valid__both__member__options,axiom,
% 0.33/0.52      ! [Tree: vEBT_VEBT,N: nat,X: nat] :
% 0.33/0.52        ( ( vEBT_invar_vebt @ Tree @ N )
% 0.33/0.52       => ( ( vEBT_vebt_member @ Tree @ X )
% 0.33/0.52         => ( ( vEBT_V5719532721284313246member @ Tree @ X )
% 0.33/0.52            | ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % member_valid_both_member_options
% 0.33/0.52  thf(fact_231_maxt__corr__help,axiom,
% 0.33/0.52      ! [T: vEBT_VEBT,N: nat,Maxi2: nat,X: nat] :
% 0.33/0.52        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.52       => ( ( ( vEBT_vebt_maxt @ T )
% 0.33/0.52            = ( some_nat @ Maxi2 ) )
% 0.33/0.52         => ( ( vEBT_vebt_member @ T @ X )
% 0.33/0.52           => ( ord_less_eq_nat @ X @ Maxi2 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % maxt_corr_help
% 0.33/0.52  thf(fact_232_maxt__sound,axiom,
% 0.33/0.52      ! [T: vEBT_VEBT,N: nat,X: nat] :
% 0.33/0.52        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.52       => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
% 0.33/0.52         => ( ( vEBT_vebt_maxt @ T )
% 0.33/0.52            = ( some_nat @ X ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % maxt_sound
% 0.33/0.52  thf(fact_233_maxt__corr,axiom,
% 0.33/0.52      ! [T: vEBT_VEBT,N: nat,X: nat] :
% 0.33/0.52        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.52       => ( ( ( vEBT_vebt_maxt @ T )
% 0.33/0.52            = ( some_nat @ X ) )
% 0.33/0.52         => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % maxt_corr
% 0.33/0.52  thf(fact_234_List_Ofinite__set,axiom,
% 0.33/0.52      ! [Xs: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) ).
% 0.33/0.52  
% 0.33/0.52  % List.finite_set
% 0.33/0.52  thf(fact_235_List_Ofinite__set,axiom,
% 0.33/0.52      ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% 0.33/0.52  
% 0.33/0.52  % List.finite_set
% 0.33/0.52  thf(fact_236_List_Ofinite__set,axiom,
% 0.33/0.52      ! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).
% 0.33/0.52  
% 0.33/0.52  % List.finite_set
% 0.33/0.52  thf(fact_237_List_Ofinite__set,axiom,
% 0.33/0.52      ! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).
% 0.33/0.52  
% 0.33/0.52  % List.finite_set
% 0.33/0.52  thf(fact_238_List_Ofinite__set,axiom,
% 0.33/0.52      ! [Xs: list_P6011104703257516679at_nat] : ( finite6177210948735845034at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ).
% 0.33/0.52  
% 0.33/0.52  % List.finite_set
% 0.33/0.52  thf(fact_239_pred__none__empty,axiom,
% 0.33/0.52      ! [Xs: set_nat,A: nat] :
% 0.33/0.52        ( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs @ A @ X_1 )
% 0.33/0.52       => ( ( finite_finite_nat @ Xs )
% 0.33/0.52         => ~ ? [X2: nat] :
% 0.33/0.52                ( ( member_nat @ X2 @ Xs )
% 0.33/0.52                & ( ord_less_nat @ X2 @ A ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % pred_none_empty
% 0.33/0.52  thf(fact_240_succ__none__empty,axiom,
% 0.33/0.52      ! [Xs: set_nat,A: nat] :
% 0.33/0.52        ( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs @ A @ X_1 )
% 0.33/0.52       => ( ( finite_finite_nat @ Xs )
% 0.33/0.52         => ~ ? [X2: nat] :
% 0.33/0.52                ( ( member_nat @ X2 @ Xs )
% 0.33/0.52                & ( ord_less_nat @ A @ X2 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % succ_none_empty
% 0.33/0.52  thf(fact_241_obtain__set__pred,axiom,
% 0.33/0.52      ! [Z2: nat,X: nat,A2: set_nat] :
% 0.33/0.52        ( ( ord_less_nat @ Z2 @ X )
% 0.33/0.52       => ( ( vEBT_VEBT_min_in_set @ A2 @ Z2 )
% 0.33/0.52         => ( ( finite_finite_nat @ A2 )
% 0.33/0.52           => ? [X_1: nat] : ( vEBT_is_pred_in_set @ A2 @ X @ X_1 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % obtain_set_pred
% 0.33/0.52  thf(fact_242_obtain__set__succ,axiom,
% 0.33/0.52      ! [X: nat,Z2: nat,A2: set_nat,B2: set_nat] :
% 0.33/0.52        ( ( ord_less_nat @ X @ Z2 )
% 0.33/0.52       => ( ( vEBT_VEBT_max_in_set @ A2 @ Z2 )
% 0.33/0.52         => ( ( finite_finite_nat @ B2 )
% 0.33/0.52           => ( ( A2 = B2 )
% 0.33/0.52             => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A2 @ X @ X_1 ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % obtain_set_succ
% 0.33/0.52  thf(fact_243_lesseq__shift,axiom,
% 0.33/0.52      ( ord_less_eq_nat
% 0.33/0.52      = ( ^ [X3: nat,Y2: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X3 ) @ ( some_nat @ Y2 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % lesseq_shift
% 0.33/0.52  thf(fact_244_pred__member,axiom,
% 0.33/0.52      ! [T: vEBT_VEBT,X: nat,Y: nat] :
% 0.33/0.52        ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
% 0.33/0.52        = ( ( vEBT_vebt_member @ T @ Y )
% 0.33/0.52          & ( ord_less_nat @ Y @ X )
% 0.33/0.52          & ! [Z3: nat] :
% 0.33/0.52              ( ( ( vEBT_vebt_member @ T @ Z3 )
% 0.33/0.52                & ( ord_less_nat @ Z3 @ X ) )
% 0.33/0.52             => ( ord_less_eq_nat @ Z3 @ Y ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % pred_member
% 0.33/0.52  thf(fact_245_succ__member,axiom,
% 0.33/0.52      ! [T: vEBT_VEBT,X: nat,Y: nat] :
% 0.33/0.52        ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
% 0.33/0.52        = ( ( vEBT_vebt_member @ T @ Y )
% 0.33/0.52          & ( ord_less_nat @ X @ Y )
% 0.33/0.52          & ! [Z3: nat] :
% 0.33/0.52              ( ( ( vEBT_vebt_member @ T @ Z3 )
% 0.33/0.52                & ( ord_less_nat @ X @ Z3 ) )
% 0.33/0.52             => ( ord_less_eq_nat @ Y @ Z3 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % succ_member
% 0.33/0.52  thf(fact_246_bot_Oextremum,axiom,
% 0.33/0.52      ! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.extremum
% 0.33/0.52  thf(fact_247_bot_Oextremum,axiom,
% 0.33/0.52      ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.extremum
% 0.33/0.52  thf(fact_248_bot_Oextremum,axiom,
% 0.33/0.52      ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.extremum
% 0.33/0.52  thf(fact_249_bot_Oextremum,axiom,
% 0.33/0.52      ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.extremum
% 0.33/0.52  thf(fact_250_bot_Oextremum,axiom,
% 0.33/0.52      ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.extremum
% 0.33/0.52  thf(fact_251_bot_Oextremum__unique,axiom,
% 0.33/0.52      ! [A: set_Pr1261947904930325089at_nat] :
% 0.33/0.52        ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
% 0.33/0.52        = ( A = bot_bo2099793752762293965at_nat ) ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.extremum_unique
% 0.33/0.52  thf(fact_252_bot_Oextremum__unique,axiom,
% 0.33/0.52      ! [A: set_real] :
% 0.33/0.52        ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 0.33/0.52        = ( A = bot_bot_set_real ) ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.extremum_unique
% 0.33/0.52  thf(fact_253_bot_Oextremum__unique,axiom,
% 0.33/0.52      ! [A: set_nat] :
% 0.33/0.52        ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 0.33/0.52        = ( A = bot_bot_set_nat ) ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.extremum_unique
% 0.33/0.52  thf(fact_254_bot_Oextremum__unique,axiom,
% 0.33/0.52      ! [A: set_int] :
% 0.33/0.52        ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 0.33/0.52        = ( A = bot_bot_set_int ) ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.extremum_unique
% 0.33/0.52  thf(fact_255_bot_Oextremum__unique,axiom,
% 0.33/0.52      ! [A: nat] :
% 0.33/0.52        ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 0.33/0.52        = ( A = bot_bot_nat ) ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.extremum_unique
% 0.33/0.52  thf(fact_256_bot_Oextremum__uniqueI,axiom,
% 0.33/0.52      ! [A: set_Pr1261947904930325089at_nat] :
% 0.33/0.52        ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
% 0.33/0.52       => ( A = bot_bo2099793752762293965at_nat ) ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.extremum_uniqueI
% 0.33/0.52  thf(fact_257_bot_Oextremum__uniqueI,axiom,
% 0.33/0.52      ! [A: set_real] :
% 0.33/0.52        ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 0.33/0.52       => ( A = bot_bot_set_real ) ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.extremum_uniqueI
% 0.33/0.52  thf(fact_258_bot_Oextremum__uniqueI,axiom,
% 0.33/0.52      ! [A: set_nat] :
% 0.33/0.52        ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 0.33/0.52       => ( A = bot_bot_set_nat ) ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.extremum_uniqueI
% 0.33/0.52  thf(fact_259_bot_Oextremum__uniqueI,axiom,
% 0.33/0.52      ! [A: set_int] :
% 0.33/0.52        ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 0.33/0.52       => ( A = bot_bot_set_int ) ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.extremum_uniqueI
% 0.33/0.52  thf(fact_260_bot_Oextremum__uniqueI,axiom,
% 0.33/0.52      ! [A: nat] :
% 0.33/0.52        ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 0.33/0.52       => ( A = bot_bot_nat ) ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.extremum_uniqueI
% 0.33/0.52  thf(fact_261_bot_Oextremum__strict,axiom,
% 0.33/0.52      ! [A: set_Pr1261947904930325089at_nat] :
% 0.33/0.52        ~ ( ord_le7866589430770878221at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.extremum_strict
% 0.33/0.52  thf(fact_262_bot_Oextremum__strict,axiom,
% 0.33/0.52      ! [A: set_real] :
% 0.33/0.52        ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.extremum_strict
% 0.33/0.52  thf(fact_263_bot_Oextremum__strict,axiom,
% 0.33/0.52      ! [A: set_nat] :
% 0.33/0.52        ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.extremum_strict
% 0.33/0.52  thf(fact_264_bot_Oextremum__strict,axiom,
% 0.33/0.52      ! [A: set_int] :
% 0.33/0.52        ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.extremum_strict
% 0.33/0.52  thf(fact_265_bot_Oextremum__strict,axiom,
% 0.33/0.52      ! [A: nat] :
% 0.33/0.52        ~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.extremum_strict
% 0.33/0.52  thf(fact_266_bot_Onot__eq__extremum,axiom,
% 0.33/0.52      ! [A: set_Pr1261947904930325089at_nat] :
% 0.33/0.52        ( ( A != bot_bo2099793752762293965at_nat )
% 0.33/0.52        = ( ord_le7866589430770878221at_nat @ bot_bo2099793752762293965at_nat @ A ) ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.not_eq_extremum
% 0.33/0.52  thf(fact_267_bot_Onot__eq__extremum,axiom,
% 0.33/0.52      ! [A: set_real] :
% 0.33/0.52        ( ( A != bot_bot_set_real )
% 0.33/0.52        = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.not_eq_extremum
% 0.33/0.52  thf(fact_268_bot_Onot__eq__extremum,axiom,
% 0.33/0.52      ! [A: set_nat] :
% 0.33/0.52        ( ( A != bot_bot_set_nat )
% 0.33/0.52        = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.not_eq_extremum
% 0.33/0.52  thf(fact_269_bot_Onot__eq__extremum,axiom,
% 0.33/0.52      ! [A: set_int] :
% 0.33/0.52        ( ( A != bot_bot_set_int )
% 0.33/0.52        = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.not_eq_extremum
% 0.33/0.52  thf(fact_270_bot_Onot__eq__extremum,axiom,
% 0.33/0.52      ! [A: nat] :
% 0.33/0.52        ( ( A != bot_bot_nat )
% 0.33/0.52        = ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% 0.33/0.52  
% 0.33/0.52  % bot.not_eq_extremum
% 0.33/0.52  thf(fact_271_finite__list,axiom,
% 0.33/0.52      ! [A2: set_VEBT_VEBT] :
% 0.33/0.52        ( ( finite5795047828879050333T_VEBT @ A2 )
% 0.33/0.52       => ? [Xs3: list_VEBT_VEBT] :
% 0.33/0.52            ( ( set_VEBT_VEBT2 @ Xs3 )
% 0.33/0.52            = A2 ) ) ).
% 0.33/0.52  
% 0.33/0.52  % finite_list
% 0.33/0.52  thf(fact_272_finite__list,axiom,
% 0.33/0.52      ! [A2: set_nat] :
% 0.33/0.52        ( ( finite_finite_nat @ A2 )
% 0.33/0.52       => ? [Xs3: list_nat] :
% 0.33/0.52            ( ( set_nat2 @ Xs3 )
% 0.33/0.52            = A2 ) ) ).
% 0.33/0.52  
% 0.33/0.52  % finite_list
% 0.33/0.52  thf(fact_273_finite__list,axiom,
% 0.33/0.52      ! [A2: set_int] :
% 0.33/0.52        ( ( finite_finite_int @ A2 )
% 0.33/0.52       => ? [Xs3: list_int] :
% 0.33/0.52            ( ( set_int2 @ Xs3 )
% 0.33/0.52            = A2 ) ) ).
% 0.33/0.52  
% 0.33/0.52  % finite_list
% 0.33/0.52  thf(fact_274_finite__list,axiom,
% 0.33/0.52      ! [A2: set_complex] :
% 0.33/0.52        ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.52       => ? [Xs3: list_complex] :
% 0.33/0.52            ( ( set_complex2 @ Xs3 )
% 0.33/0.52            = A2 ) ) ).
% 0.33/0.52  
% 0.33/0.52  % finite_list
% 0.33/0.52  thf(fact_275_finite__list,axiom,
% 0.33/0.52      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.33/0.52        ( ( finite6177210948735845034at_nat @ A2 )
% 0.33/0.52       => ? [Xs3: list_P6011104703257516679at_nat] :
% 0.33/0.52            ( ( set_Pr5648618587558075414at_nat @ Xs3 )
% 0.33/0.52            = A2 ) ) ).
% 0.33/0.52  
% 0.33/0.52  % finite_list
% 0.33/0.52  thf(fact_276_subset__code_I1_J,axiom,
% 0.33/0.52      ! [Xs: list_P6011104703257516679at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.33/0.52        ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ B2 )
% 0.33/0.52        = ( ! [X3: product_prod_nat_nat] :
% 0.33/0.52              ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.33/0.52             => ( member8440522571783428010at_nat @ X3 @ B2 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % subset_code(1)
% 0.33/0.52  thf(fact_277_subset__code_I1_J,axiom,
% 0.33/0.52      ! [Xs: list_real,B2: set_real] :
% 0.33/0.52        ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B2 )
% 0.33/0.52        = ( ! [X3: real] :
% 0.33/0.52              ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 0.33/0.52             => ( member_real @ X3 @ B2 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % subset_code(1)
% 0.33/0.52  thf(fact_278_subset__code_I1_J,axiom,
% 0.33/0.52      ! [Xs: list_set_nat,B2: set_set_nat] :
% 0.33/0.52        ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B2 )
% 0.33/0.52        = ( ! [X3: set_nat] :
% 0.33/0.52              ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
% 0.33/0.52             => ( member_set_nat @ X3 @ B2 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % subset_code(1)
% 0.33/0.52  thf(fact_279_subset__code_I1_J,axiom,
% 0.33/0.52      ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
% 0.33/0.52        ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B2 )
% 0.33/0.52        = ( ! [X3: vEBT_VEBT] :
% 0.33/0.52              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.52             => ( member_VEBT_VEBT @ X3 @ B2 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % subset_code(1)
% 0.33/0.52  thf(fact_280_subset__code_I1_J,axiom,
% 0.33/0.52      ! [Xs: list_nat,B2: set_nat] :
% 0.33/0.52        ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
% 0.33/0.52        = ( ! [X3: nat] :
% 0.33/0.52              ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 0.33/0.52             => ( member_nat @ X3 @ B2 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % subset_code(1)
% 0.33/0.52  thf(fact_281_subset__code_I1_J,axiom,
% 0.33/0.52      ! [Xs: list_int,B2: set_int] :
% 0.33/0.52        ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B2 )
% 0.33/0.52        = ( ! [X3: int] :
% 0.33/0.52              ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 0.33/0.52             => ( member_int @ X3 @ B2 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % subset_code(1)
% 0.33/0.52  thf(fact_282_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
% 0.33/0.52      ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
% 0.33/0.52        ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% 0.33/0.52  
% 0.33/0.52  % VEBT_internal.naive_member.simps(2)
% 0.33/0.52  thf(fact_283_set__update__subsetI,axiom,
% 0.33/0.52      ! [Xs: list_P6011104703257516679at_nat,A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,I: nat] :
% 0.33/0.52        ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ A2 )
% 0.33/0.52       => ( ( member8440522571783428010at_nat @ X @ A2 )
% 0.33/0.52         => ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % set_update_subsetI
% 0.33/0.52  thf(fact_284_set__update__subsetI,axiom,
% 0.33/0.52      ! [Xs: list_real,A2: set_real,X: real,I: nat] :
% 0.33/0.52        ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A2 )
% 0.33/0.52       => ( ( member_real @ X @ A2 )
% 0.33/0.52         => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % set_update_subsetI
% 0.33/0.52  thf(fact_285_set__update__subsetI,axiom,
% 0.33/0.52      ! [Xs: list_set_nat,A2: set_set_nat,X: set_nat,I: nat] :
% 0.33/0.52        ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A2 )
% 0.33/0.52       => ( ( member_set_nat @ X @ A2 )
% 0.33/0.52         => ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % set_update_subsetI
% 0.33/0.52  thf(fact_286_set__update__subsetI,axiom,
% 0.33/0.52      ! [Xs: list_nat,A2: set_nat,X: nat,I: nat] :
% 0.33/0.52        ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
% 0.33/0.52       => ( ( member_nat @ X @ A2 )
% 0.33/0.52         => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % set_update_subsetI
% 0.33/0.52  thf(fact_287_set__update__subsetI,axiom,
% 0.33/0.52      ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
% 0.33/0.52        ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
% 0.33/0.52       => ( ( member_VEBT_VEBT @ X @ A2 )
% 0.33/0.52         => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % set_update_subsetI
% 0.33/0.52  thf(fact_288_set__update__subsetI,axiom,
% 0.33/0.52      ! [Xs: list_int,A2: set_int,X: int,I: nat] :
% 0.33/0.52        ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
% 0.33/0.52       => ( ( member_int @ X @ A2 )
% 0.33/0.52         => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % set_update_subsetI
% 0.33/0.52  thf(fact_289_nle__le,axiom,
% 0.33/0.52      ! [A: rat,B: rat] :
% 0.33/0.52        ( ( ~ ( ord_less_eq_rat @ A @ B ) )
% 0.33/0.52        = ( ( ord_less_eq_rat @ B @ A )
% 0.33/0.52          & ( B != A ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nle_le
% 0.33/0.52  thf(fact_290_nle__le,axiom,
% 0.33/0.52      ! [A: num,B: num] :
% 0.33/0.52        ( ( ~ ( ord_less_eq_num @ A @ B ) )
% 0.33/0.52        = ( ( ord_less_eq_num @ B @ A )
% 0.33/0.52          & ( B != A ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nle_le
% 0.33/0.52  thf(fact_291_nle__le,axiom,
% 0.33/0.52      ! [A: nat,B: nat] :
% 0.33/0.52        ( ( ~ ( ord_less_eq_nat @ A @ B ) )
% 0.33/0.52        = ( ( ord_less_eq_nat @ B @ A )
% 0.33/0.52          & ( B != A ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nle_le
% 0.33/0.52  thf(fact_292_nle__le,axiom,
% 0.33/0.52      ! [A: int,B: int] :
% 0.33/0.52        ( ( ~ ( ord_less_eq_int @ A @ B ) )
% 0.33/0.52        = ( ( ord_less_eq_int @ B @ A )
% 0.33/0.52          & ( B != A ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % nle_le
% 0.33/0.52  thf(fact_293_le__cases3,axiom,
% 0.33/0.52      ! [X: rat,Y: rat,Z2: rat] :
% 0.33/0.52        ( ( ( ord_less_eq_rat @ X @ Y )
% 0.33/0.52         => ~ ( ord_less_eq_rat @ Y @ Z2 ) )
% 0.33/0.52       => ( ( ( ord_less_eq_rat @ Y @ X )
% 0.33/0.52           => ~ ( ord_less_eq_rat @ X @ Z2 ) )
% 0.33/0.52         => ( ( ( ord_less_eq_rat @ X @ Z2 )
% 0.33/0.52             => ~ ( ord_less_eq_rat @ Z2 @ Y ) )
% 0.33/0.52           => ( ( ( ord_less_eq_rat @ Z2 @ Y )
% 0.33/0.52               => ~ ( ord_less_eq_rat @ Y @ X ) )
% 0.33/0.52             => ( ( ( ord_less_eq_rat @ Y @ Z2 )
% 0.33/0.52                 => ~ ( ord_less_eq_rat @ Z2 @ X ) )
% 0.33/0.52               => ~ ( ( ord_less_eq_rat @ Z2 @ X )
% 0.33/0.52                   => ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % le_cases3
% 0.33/0.52  thf(fact_294_le__cases3,axiom,
% 0.33/0.52      ! [X: num,Y: num,Z2: num] :
% 0.33/0.52        ( ( ( ord_less_eq_num @ X @ Y )
% 0.33/0.52         => ~ ( ord_less_eq_num @ Y @ Z2 ) )
% 0.33/0.52       => ( ( ( ord_less_eq_num @ Y @ X )
% 0.33/0.52           => ~ ( ord_less_eq_num @ X @ Z2 ) )
% 0.33/0.52         => ( ( ( ord_less_eq_num @ X @ Z2 )
% 0.33/0.52             => ~ ( ord_less_eq_num @ Z2 @ Y ) )
% 0.33/0.52           => ( ( ( ord_less_eq_num @ Z2 @ Y )
% 0.33/0.52               => ~ ( ord_less_eq_num @ Y @ X ) )
% 0.33/0.52             => ( ( ( ord_less_eq_num @ Y @ Z2 )
% 0.33/0.52                 => ~ ( ord_less_eq_num @ Z2 @ X ) )
% 0.33/0.52               => ~ ( ( ord_less_eq_num @ Z2 @ X )
% 0.33/0.52                   => ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % le_cases3
% 0.33/0.52  thf(fact_295_le__cases3,axiom,
% 0.33/0.52      ! [X: nat,Y: nat,Z2: nat] :
% 0.33/0.52        ( ( ( ord_less_eq_nat @ X @ Y )
% 0.33/0.52         => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
% 0.33/0.52       => ( ( ( ord_less_eq_nat @ Y @ X )
% 0.33/0.52           => ~ ( ord_less_eq_nat @ X @ Z2 ) )
% 0.33/0.52         => ( ( ( ord_less_eq_nat @ X @ Z2 )
% 0.33/0.52             => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
% 0.33/0.52           => ( ( ( ord_less_eq_nat @ Z2 @ Y )
% 0.33/0.52               => ~ ( ord_less_eq_nat @ Y @ X ) )
% 0.33/0.52             => ( ( ( ord_less_eq_nat @ Y @ Z2 )
% 0.33/0.52                 => ~ ( ord_less_eq_nat @ Z2 @ X ) )
% 0.33/0.52               => ~ ( ( ord_less_eq_nat @ Z2 @ X )
% 0.33/0.52                   => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % le_cases3
% 0.33/0.52  thf(fact_296_le__cases3,axiom,
% 0.33/0.52      ! [X: int,Y: int,Z2: int] :
% 0.33/0.52        ( ( ( ord_less_eq_int @ X @ Y )
% 0.33/0.52         => ~ ( ord_less_eq_int @ Y @ Z2 ) )
% 0.33/0.52       => ( ( ( ord_less_eq_int @ Y @ X )
% 0.33/0.52           => ~ ( ord_less_eq_int @ X @ Z2 ) )
% 0.33/0.52         => ( ( ( ord_less_eq_int @ X @ Z2 )
% 0.33/0.52             => ~ ( ord_less_eq_int @ Z2 @ Y ) )
% 0.33/0.52           => ( ( ( ord_less_eq_int @ Z2 @ Y )
% 0.33/0.52               => ~ ( ord_less_eq_int @ Y @ X ) )
% 0.33/0.52             => ( ( ( ord_less_eq_int @ Y @ Z2 )
% 0.33/0.52                 => ~ ( ord_less_eq_int @ Z2 @ X ) )
% 0.33/0.52               => ~ ( ( ord_less_eq_int @ Z2 @ X )
% 0.33/0.52                   => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % le_cases3
% 0.33/0.52  thf(fact_297_order__class_Oorder__eq__iff,axiom,
% 0.33/0.52      ( ( ^ [Y5: set_int,Z: set_int] : ( Y5 = Z ) )
% 0.33/0.52      = ( ^ [X3: set_int,Y2: set_int] :
% 0.33/0.52            ( ( ord_less_eq_set_int @ X3 @ Y2 )
% 0.33/0.52            & ( ord_less_eq_set_int @ Y2 @ X3 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % order_class.order_eq_iff
% 0.33/0.52  thf(fact_298_order__class_Oorder__eq__iff,axiom,
% 0.33/0.52      ( ( ^ [Y5: rat,Z: rat] : ( Y5 = Z ) )
% 0.33/0.52      = ( ^ [X3: rat,Y2: rat] :
% 0.33/0.52            ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.33/0.52            & ( ord_less_eq_rat @ Y2 @ X3 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % order_class.order_eq_iff
% 0.33/0.52  thf(fact_299_order__class_Oorder__eq__iff,axiom,
% 0.33/0.52      ( ( ^ [Y5: num,Z: num] : ( Y5 = Z ) )
% 0.33/0.52      = ( ^ [X3: num,Y2: num] :
% 0.33/0.52            ( ( ord_less_eq_num @ X3 @ Y2 )
% 0.33/0.52            & ( ord_less_eq_num @ Y2 @ X3 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % order_class.order_eq_iff
% 0.33/0.52  thf(fact_300_order__class_Oorder__eq__iff,axiom,
% 0.33/0.52      ( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
% 0.33/0.52      = ( ^ [X3: nat,Y2: nat] :
% 0.33/0.52            ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.33/0.52            & ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % order_class.order_eq_iff
% 0.33/0.52  thf(fact_301_order__class_Oorder__eq__iff,axiom,
% 0.33/0.52      ( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
% 0.33/0.52      = ( ^ [X3: int,Y2: int] :
% 0.33/0.52            ( ( ord_less_eq_int @ X3 @ Y2 )
% 0.33/0.52            & ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % order_class.order_eq_iff
% 0.33/0.52  thf(fact_302_ord__eq__le__trans,axiom,
% 0.33/0.52      ! [A: set_int,B: set_int,C: set_int] :
% 0.33/0.52        ( ( A = B )
% 0.33/0.52       => ( ( ord_less_eq_set_int @ B @ C )
% 0.33/0.52         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % ord_eq_le_trans
% 0.33/0.52  thf(fact_303_ord__eq__le__trans,axiom,
% 0.33/0.52      ! [A: rat,B: rat,C: rat] :
% 0.33/0.52        ( ( A = B )
% 0.33/0.52       => ( ( ord_less_eq_rat @ B @ C )
% 0.33/0.52         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % ord_eq_le_trans
% 0.33/0.52  thf(fact_304_ord__eq__le__trans,axiom,
% 0.33/0.52      ! [A: num,B: num,C: num] :
% 0.33/0.52        ( ( A = B )
% 0.33/0.52       => ( ( ord_less_eq_num @ B @ C )
% 0.33/0.52         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % ord_eq_le_trans
% 0.33/0.52  thf(fact_305_ord__eq__le__trans,axiom,
% 0.33/0.52      ! [A: nat,B: nat,C: nat] :
% 0.33/0.52        ( ( A = B )
% 0.33/0.52       => ( ( ord_less_eq_nat @ B @ C )
% 0.33/0.52         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % ord_eq_le_trans
% 0.33/0.52  thf(fact_306_ord__eq__le__trans,axiom,
% 0.33/0.52      ! [A: int,B: int,C: int] :
% 0.33/0.52        ( ( A = B )
% 0.33/0.52       => ( ( ord_less_eq_int @ B @ C )
% 0.33/0.52         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % ord_eq_le_trans
% 0.33/0.52  thf(fact_307_ord__le__eq__trans,axiom,
% 0.33/0.52      ! [A: set_int,B: set_int,C: set_int] :
% 0.33/0.52        ( ( ord_less_eq_set_int @ A @ B )
% 0.33/0.52       => ( ( B = C )
% 0.33/0.52         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % ord_le_eq_trans
% 0.33/0.52  thf(fact_308_ord__le__eq__trans,axiom,
% 0.33/0.52      ! [A: rat,B: rat,C: rat] :
% 0.33/0.52        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.52       => ( ( B = C )
% 0.33/0.52         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % ord_le_eq_trans
% 0.33/0.52  thf(fact_309_ord__le__eq__trans,axiom,
% 0.33/0.52      ! [A: num,B: num,C: num] :
% 0.33/0.52        ( ( ord_less_eq_num @ A @ B )
% 0.33/0.52       => ( ( B = C )
% 0.33/0.52         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % ord_le_eq_trans
% 0.33/0.52  thf(fact_310_ord__le__eq__trans,axiom,
% 0.33/0.52      ! [A: nat,B: nat,C: nat] :
% 0.33/0.52        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.52       => ( ( B = C )
% 0.33/0.52         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % ord_le_eq_trans
% 0.33/0.52  thf(fact_311_ord__le__eq__trans,axiom,
% 0.33/0.52      ! [A: int,B: int,C: int] :
% 0.33/0.52        ( ( ord_less_eq_int @ A @ B )
% 0.33/0.52       => ( ( B = C )
% 0.33/0.52         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % ord_le_eq_trans
% 0.33/0.52  thf(fact_312_order__antisym,axiom,
% 0.33/0.52      ! [X: set_int,Y: set_int] :
% 0.33/0.52        ( ( ord_less_eq_set_int @ X @ Y )
% 0.33/0.52       => ( ( ord_less_eq_set_int @ Y @ X )
% 0.33/0.52         => ( X = Y ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % order_antisym
% 0.33/0.52  thf(fact_313_order__antisym,axiom,
% 0.33/0.52      ! [X: rat,Y: rat] :
% 0.33/0.52        ( ( ord_less_eq_rat @ X @ Y )
% 0.33/0.52       => ( ( ord_less_eq_rat @ Y @ X )
% 0.33/0.52         => ( X = Y ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % order_antisym
% 0.33/0.52  thf(fact_314_order__antisym,axiom,
% 0.33/0.52      ! [X: num,Y: num] :
% 0.33/0.52        ( ( ord_less_eq_num @ X @ Y )
% 0.33/0.52       => ( ( ord_less_eq_num @ Y @ X )
% 0.33/0.52         => ( X = Y ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % order_antisym
% 0.33/0.52  thf(fact_315_order__antisym,axiom,
% 0.33/0.52      ! [X: nat,Y: nat] :
% 0.33/0.52        ( ( ord_less_eq_nat @ X @ Y )
% 0.33/0.52       => ( ( ord_less_eq_nat @ Y @ X )
% 0.33/0.52         => ( X = Y ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % order_antisym
% 0.33/0.52  thf(fact_316_order__antisym,axiom,
% 0.33/0.52      ! [X: int,Y: int] :
% 0.33/0.52        ( ( ord_less_eq_int @ X @ Y )
% 0.33/0.52       => ( ( ord_less_eq_int @ Y @ X )
% 0.33/0.52         => ( X = Y ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % order_antisym
% 0.33/0.52  thf(fact_317_order_Otrans,axiom,
% 0.33/0.52      ! [A: set_int,B: set_int,C: set_int] :
% 0.33/0.52        ( ( ord_less_eq_set_int @ A @ B )
% 0.33/0.52       => ( ( ord_less_eq_set_int @ B @ C )
% 0.33/0.52         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % order.trans
% 0.33/0.52  thf(fact_318_order_Otrans,axiom,
% 0.33/0.52      ! [A: rat,B: rat,C: rat] :
% 0.33/0.52        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.52       => ( ( ord_less_eq_rat @ B @ C )
% 0.33/0.52         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % order.trans
% 0.33/0.52  thf(fact_319_order_Otrans,axiom,
% 0.33/0.52      ! [A: num,B: num,C: num] :
% 0.33/0.52        ( ( ord_less_eq_num @ A @ B )
% 0.33/0.52       => ( ( ord_less_eq_num @ B @ C )
% 0.33/0.52         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % order.trans
% 0.33/0.52  thf(fact_320_order_Otrans,axiom,
% 0.33/0.52      ! [A: nat,B: nat,C: nat] :
% 0.33/0.52        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.52       => ( ( ord_less_eq_nat @ B @ C )
% 0.33/0.52         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % order.trans
% 0.33/0.52  thf(fact_321_order_Otrans,axiom,
% 0.33/0.52      ! [A: int,B: int,C: int] :
% 0.33/0.52        ( ( ord_less_eq_int @ A @ B )
% 0.33/0.52       => ( ( ord_less_eq_int @ B @ C )
% 0.33/0.52         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % order.trans
% 0.33/0.52  thf(fact_322_order__trans,axiom,
% 0.33/0.52      ! [X: set_int,Y: set_int,Z2: set_int] :
% 0.33/0.52        ( ( ord_less_eq_set_int @ X @ Y )
% 0.33/0.52       => ( ( ord_less_eq_set_int @ Y @ Z2 )
% 0.33/0.52         => ( ord_less_eq_set_int @ X @ Z2 ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % order_trans
% 0.33/0.52  thf(fact_323_order__trans,axiom,
% 0.33/0.52      ! [X: rat,Y: rat,Z2: rat] :
% 0.33/0.52        ( ( ord_less_eq_rat @ X @ Y )
% 0.33/0.52       => ( ( ord_less_eq_rat @ Y @ Z2 )
% 0.33/0.52         => ( ord_less_eq_rat @ X @ Z2 ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % order_trans
% 0.33/0.52  thf(fact_324_order__trans,axiom,
% 0.33/0.52      ! [X: num,Y: num,Z2: num] :
% 0.33/0.52        ( ( ord_less_eq_num @ X @ Y )
% 0.33/0.52       => ( ( ord_less_eq_num @ Y @ Z2 )
% 0.33/0.52         => ( ord_less_eq_num @ X @ Z2 ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % order_trans
% 0.33/0.52  thf(fact_325_order__trans,axiom,
% 0.33/0.52      ! [X: nat,Y: nat,Z2: nat] :
% 0.33/0.52        ( ( ord_less_eq_nat @ X @ Y )
% 0.33/0.52       => ( ( ord_less_eq_nat @ Y @ Z2 )
% 0.33/0.52         => ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % order_trans
% 0.33/0.52  thf(fact_326_order__trans,axiom,
% 0.33/0.52      ! [X: int,Y: int,Z2: int] :
% 0.33/0.52        ( ( ord_less_eq_int @ X @ Y )
% 0.33/0.52       => ( ( ord_less_eq_int @ Y @ Z2 )
% 0.33/0.52         => ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % order_trans
% 0.33/0.52  thf(fact_327_linorder__wlog,axiom,
% 0.33/0.52      ! [P: rat > rat > $o,A: rat,B: rat] :
% 0.33/0.52        ( ! [A3: rat,B3: rat] :
% 0.33/0.52            ( ( ord_less_eq_rat @ A3 @ B3 )
% 0.33/0.52           => ( P @ A3 @ B3 ) )
% 0.33/0.52       => ( ! [A3: rat,B3: rat] :
% 0.33/0.52              ( ( P @ B3 @ A3 )
% 0.33/0.52             => ( P @ A3 @ B3 ) )
% 0.33/0.52         => ( P @ A @ B ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % linorder_wlog
% 0.33/0.52  thf(fact_328_linorder__wlog,axiom,
% 0.33/0.52      ! [P: num > num > $o,A: num,B: num] :
% 0.33/0.52        ( ! [A3: num,B3: num] :
% 0.33/0.52            ( ( ord_less_eq_num @ A3 @ B3 )
% 0.33/0.52           => ( P @ A3 @ B3 ) )
% 0.33/0.52       => ( ! [A3: num,B3: num] :
% 0.33/0.52              ( ( P @ B3 @ A3 )
% 0.33/0.52             => ( P @ A3 @ B3 ) )
% 0.33/0.52         => ( P @ A @ B ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % linorder_wlog
% 0.33/0.52  thf(fact_329_linorder__wlog,axiom,
% 0.33/0.52      ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.33/0.52        ( ! [A3: nat,B3: nat] :
% 0.33/0.52            ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.33/0.52           => ( P @ A3 @ B3 ) )
% 0.33/0.52       => ( ! [A3: nat,B3: nat] :
% 0.33/0.52              ( ( P @ B3 @ A3 )
% 0.33/0.52             => ( P @ A3 @ B3 ) )
% 0.33/0.52         => ( P @ A @ B ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % linorder_wlog
% 0.33/0.52  thf(fact_330_linorder__wlog,axiom,
% 0.33/0.52      ! [P: int > int > $o,A: int,B: int] :
% 0.33/0.52        ( ! [A3: int,B3: int] :
% 0.33/0.52            ( ( ord_less_eq_int @ A3 @ B3 )
% 0.33/0.52           => ( P @ A3 @ B3 ) )
% 0.33/0.52       => ( ! [A3: int,B3: int] :
% 0.33/0.52              ( ( P @ B3 @ A3 )
% 0.33/0.52             => ( P @ A3 @ B3 ) )
% 0.33/0.52         => ( P @ A @ B ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % linorder_wlog
% 0.33/0.52  thf(fact_331_dual__order_Oeq__iff,axiom,
% 0.33/0.52      ( ( ^ [Y5: set_int,Z: set_int] : ( Y5 = Z ) )
% 0.33/0.52      = ( ^ [A4: set_int,B4: set_int] :
% 0.33/0.52            ( ( ord_less_eq_set_int @ B4 @ A4 )
% 0.33/0.52            & ( ord_less_eq_set_int @ A4 @ B4 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % dual_order.eq_iff
% 0.33/0.52  thf(fact_332_dual__order_Oeq__iff,axiom,
% 0.33/0.52      ( ( ^ [Y5: rat,Z: rat] : ( Y5 = Z ) )
% 0.33/0.52      = ( ^ [A4: rat,B4: rat] :
% 0.33/0.52            ( ( ord_less_eq_rat @ B4 @ A4 )
% 0.33/0.52            & ( ord_less_eq_rat @ A4 @ B4 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % dual_order.eq_iff
% 0.33/0.52  thf(fact_333_dual__order_Oeq__iff,axiom,
% 0.33/0.52      ( ( ^ [Y5: num,Z: num] : ( Y5 = Z ) )
% 0.33/0.52      = ( ^ [A4: num,B4: num] :
% 0.33/0.52            ( ( ord_less_eq_num @ B4 @ A4 )
% 0.33/0.52            & ( ord_less_eq_num @ A4 @ B4 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % dual_order.eq_iff
% 0.33/0.52  thf(fact_334_dual__order_Oeq__iff,axiom,
% 0.33/0.52      ( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
% 0.33/0.52      = ( ^ [A4: nat,B4: nat] :
% 0.33/0.52            ( ( ord_less_eq_nat @ B4 @ A4 )
% 0.33/0.52            & ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % dual_order.eq_iff
% 0.33/0.52  thf(fact_335_dual__order_Oeq__iff,axiom,
% 0.33/0.52      ( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
% 0.33/0.52      = ( ^ [A4: int,B4: int] :
% 0.33/0.52            ( ( ord_less_eq_int @ B4 @ A4 )
% 0.33/0.52            & ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % dual_order.eq_iff
% 0.33/0.52  thf(fact_336_dual__order_Oantisym,axiom,
% 0.33/0.52      ! [B: set_int,A: set_int] :
% 0.33/0.52        ( ( ord_less_eq_set_int @ B @ A )
% 0.33/0.52       => ( ( ord_less_eq_set_int @ A @ B )
% 0.33/0.52         => ( A = B ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % dual_order.antisym
% 0.33/0.52  thf(fact_337_dual__order_Oantisym,axiom,
% 0.33/0.52      ! [B: rat,A: rat] :
% 0.33/0.52        ( ( ord_less_eq_rat @ B @ A )
% 0.33/0.52       => ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.52         => ( A = B ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % dual_order.antisym
% 0.33/0.52  thf(fact_338_dual__order_Oantisym,axiom,
% 0.33/0.52      ! [B: num,A: num] :
% 0.33/0.52        ( ( ord_less_eq_num @ B @ A )
% 0.33/0.52       => ( ( ord_less_eq_num @ A @ B )
% 0.33/0.52         => ( A = B ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % dual_order.antisym
% 0.33/0.52  thf(fact_339_dual__order_Oantisym,axiom,
% 0.33/0.52      ! [B: nat,A: nat] :
% 0.33/0.52        ( ( ord_less_eq_nat @ B @ A )
% 0.33/0.52       => ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.52         => ( A = B ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % dual_order.antisym
% 0.33/0.52  thf(fact_340_dual__order_Oantisym,axiom,
% 0.33/0.52      ! [B: int,A: int] :
% 0.33/0.52        ( ( ord_less_eq_int @ B @ A )
% 0.33/0.52       => ( ( ord_less_eq_int @ A @ B )
% 0.33/0.52         => ( A = B ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % dual_order.antisym
% 0.33/0.52  thf(fact_341_dual__order_Otrans,axiom,
% 0.33/0.52      ! [B: set_int,A: set_int,C: set_int] :
% 0.33/0.52        ( ( ord_less_eq_set_int @ B @ A )
% 0.33/0.52       => ( ( ord_less_eq_set_int @ C @ B )
% 0.33/0.52         => ( ord_less_eq_set_int @ C @ A ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % dual_order.trans
% 0.33/0.52  thf(fact_342_dual__order_Otrans,axiom,
% 0.33/0.52      ! [B: rat,A: rat,C: rat] :
% 0.33/0.52        ( ( ord_less_eq_rat @ B @ A )
% 0.33/0.52       => ( ( ord_less_eq_rat @ C @ B )
% 0.33/0.52         => ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % dual_order.trans
% 0.33/0.52  thf(fact_343_dual__order_Otrans,axiom,
% 0.33/0.52      ! [B: num,A: num,C: num] :
% 0.33/0.52        ( ( ord_less_eq_num @ B @ A )
% 0.33/0.52       => ( ( ord_less_eq_num @ C @ B )
% 0.33/0.52         => ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % dual_order.trans
% 0.33/0.52  thf(fact_344_dual__order_Otrans,axiom,
% 0.33/0.52      ! [B: nat,A: nat,C: nat] :
% 0.33/0.52        ( ( ord_less_eq_nat @ B @ A )
% 0.33/0.52       => ( ( ord_less_eq_nat @ C @ B )
% 0.33/0.52         => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % dual_order.trans
% 0.33/0.52  thf(fact_345_dual__order_Otrans,axiom,
% 0.33/0.52      ! [B: int,A: int,C: int] :
% 0.33/0.52        ( ( ord_less_eq_int @ B @ A )
% 0.33/0.52       => ( ( ord_less_eq_int @ C @ B )
% 0.33/0.52         => ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % dual_order.trans
% 0.33/0.52  thf(fact_346_antisym,axiom,
% 0.33/0.52      ! [A: set_int,B: set_int] :
% 0.33/0.52        ( ( ord_less_eq_set_int @ A @ B )
% 0.33/0.52       => ( ( ord_less_eq_set_int @ B @ A )
% 0.33/0.52         => ( A = B ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % antisym
% 0.33/0.52  thf(fact_347_antisym,axiom,
% 0.33/0.52      ! [A: rat,B: rat] :
% 0.33/0.52        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.52       => ( ( ord_less_eq_rat @ B @ A )
% 0.33/0.52         => ( A = B ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % antisym
% 0.33/0.52  thf(fact_348_antisym,axiom,
% 0.33/0.52      ! [A: num,B: num] :
% 0.33/0.52        ( ( ord_less_eq_num @ A @ B )
% 0.33/0.52       => ( ( ord_less_eq_num @ B @ A )
% 0.33/0.52         => ( A = B ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % antisym
% 0.33/0.52  thf(fact_349_antisym,axiom,
% 0.33/0.52      ! [A: nat,B: nat] :
% 0.33/0.52        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.52       => ( ( ord_less_eq_nat @ B @ A )
% 0.33/0.52         => ( A = B ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % antisym
% 0.33/0.52  thf(fact_350_antisym,axiom,
% 0.33/0.52      ! [A: int,B: int] :
% 0.33/0.52        ( ( ord_less_eq_int @ A @ B )
% 0.33/0.52       => ( ( ord_less_eq_int @ B @ A )
% 0.33/0.52         => ( A = B ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % antisym
% 0.33/0.52  thf(fact_351_Orderings_Oorder__eq__iff,axiom,
% 0.33/0.52      ( ( ^ [Y5: set_int,Z: set_int] : ( Y5 = Z ) )
% 0.33/0.52      = ( ^ [A4: set_int,B4: set_int] :
% 0.33/0.52            ( ( ord_less_eq_set_int @ A4 @ B4 )
% 0.33/0.52            & ( ord_less_eq_set_int @ B4 @ A4 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % Orderings.order_eq_iff
% 0.33/0.52  thf(fact_352_Orderings_Oorder__eq__iff,axiom,
% 0.33/0.52      ( ( ^ [Y5: rat,Z: rat] : ( Y5 = Z ) )
% 0.33/0.52      = ( ^ [A4: rat,B4: rat] :
% 0.33/0.52            ( ( ord_less_eq_rat @ A4 @ B4 )
% 0.33/0.52            & ( ord_less_eq_rat @ B4 @ A4 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % Orderings.order_eq_iff
% 0.33/0.52  thf(fact_353_Orderings_Oorder__eq__iff,axiom,
% 0.33/0.52      ( ( ^ [Y5: num,Z: num] : ( Y5 = Z ) )
% 0.33/0.52      = ( ^ [A4: num,B4: num] :
% 0.33/0.52            ( ( ord_less_eq_num @ A4 @ B4 )
% 0.33/0.52            & ( ord_less_eq_num @ B4 @ A4 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % Orderings.order_eq_iff
% 0.33/0.52  thf(fact_354_Orderings_Oorder__eq__iff,axiom,
% 0.33/0.52      ( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
% 0.33/0.52      = ( ^ [A4: nat,B4: nat] :
% 0.33/0.52            ( ( ord_less_eq_nat @ A4 @ B4 )
% 0.33/0.52            & ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % Orderings.order_eq_iff
% 0.33/0.52  thf(fact_355_Orderings_Oorder__eq__iff,axiom,
% 0.33/0.52      ( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
% 0.33/0.52      = ( ^ [A4: int,B4: int] :
% 0.33/0.52            ( ( ord_less_eq_int @ A4 @ B4 )
% 0.33/0.52            & ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% 0.33/0.52  
% 0.33/0.52  % Orderings.order_eq_iff
% 0.33/0.52  thf(fact_356_order__subst1,axiom,
% 0.33/0.52      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.33/0.52        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_subst1
% 0.33/0.53  thf(fact_357_order__subst1,axiom,
% 0.33/0.53      ! [A: rat,F: num > rat,B: num,C: num] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_num @ B @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_subst1
% 0.33/0.53  thf(fact_358_order__subst1,axiom,
% 0.33/0.53      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_nat @ B @ C )
% 0.33/0.53         => ( ! [X4: nat,Y3: nat] :
% 0.33/0.53                ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_subst1
% 0.33/0.53  thf(fact_359_order__subst1,axiom,
% 0.33/0.53      ! [A: rat,F: int > rat,B: int,C: int] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_int @ B @ C )
% 0.33/0.53         => ( ! [X4: int,Y3: int] :
% 0.33/0.53                ( ( ord_less_eq_int @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_subst1
% 0.33/0.53  thf(fact_360_order__subst1,axiom,
% 0.33/0.53      ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.33/0.53        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_subst1
% 0.33/0.53  thf(fact_361_order__subst1,axiom,
% 0.33/0.53      ! [A: num,F: num > num,B: num,C: num] :
% 0.33/0.53        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_num @ B @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_subst1
% 0.33/0.53  thf(fact_362_order__subst1,axiom,
% 0.33/0.53      ! [A: num,F: nat > num,B: nat,C: nat] :
% 0.33/0.53        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_nat @ B @ C )
% 0.33/0.53         => ( ! [X4: nat,Y3: nat] :
% 0.33/0.53                ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_subst1
% 0.33/0.53  thf(fact_363_order__subst1,axiom,
% 0.33/0.53      ! [A: num,F: int > num,B: int,C: int] :
% 0.33/0.53        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_int @ B @ C )
% 0.33/0.53         => ( ! [X4: int,Y3: int] :
% 0.33/0.53                ( ( ord_less_eq_int @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_subst1
% 0.33/0.53  thf(fact_364_order__subst1,axiom,
% 0.33/0.53      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.33/0.53        ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_subst1
% 0.33/0.53  thf(fact_365_order__subst1,axiom,
% 0.33/0.53      ! [A: nat,F: num > nat,B: num,C: num] :
% 0.33/0.53        ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_num @ B @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_subst1
% 0.33/0.53  thf(fact_366_order__subst2,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.53       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_subst2
% 0.33/0.53  thf(fact_367_order__subst2,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.53       => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_subst2
% 0.33/0.53  thf(fact_368_order__subst2,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.53       => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_subst2
% 0.33/0.53  thf(fact_369_order__subst2,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.53       => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_subst2
% 0.33/0.53  thf(fact_370_order__subst2,axiom,
% 0.33/0.53      ! [A: num,B: num,F: num > rat,C: rat] :
% 0.33/0.53        ( ( ord_less_eq_num @ A @ B )
% 0.33/0.53       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_subst2
% 0.33/0.53  thf(fact_371_order__subst2,axiom,
% 0.33/0.53      ! [A: num,B: num,F: num > num,C: num] :
% 0.33/0.53        ( ( ord_less_eq_num @ A @ B )
% 0.33/0.53       => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_subst2
% 0.33/0.53  thf(fact_372_order__subst2,axiom,
% 0.33/0.53      ! [A: num,B: num,F: num > nat,C: nat] :
% 0.33/0.53        ( ( ord_less_eq_num @ A @ B )
% 0.33/0.53       => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_subst2
% 0.33/0.53  thf(fact_373_order__subst2,axiom,
% 0.33/0.53      ! [A: num,B: num,F: num > int,C: int] :
% 0.33/0.53        ( ( ord_less_eq_num @ A @ B )
% 0.33/0.53       => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_subst2
% 0.33/0.53  thf(fact_374_order__subst2,axiom,
% 0.33/0.53      ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 0.33/0.53        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.53       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: nat,Y3: nat] :
% 0.33/0.53                ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_subst2
% 0.33/0.53  thf(fact_375_order__subst2,axiom,
% 0.33/0.53      ! [A: nat,B: nat,F: nat > num,C: num] :
% 0.33/0.53        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.53       => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: nat,Y3: nat] :
% 0.33/0.53                ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_subst2
% 0.33/0.53  thf(fact_376_order__eq__refl,axiom,
% 0.33/0.53      ! [X: set_int,Y: set_int] :
% 0.33/0.53        ( ( X = Y )
% 0.33/0.53       => ( ord_less_eq_set_int @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_eq_refl
% 0.33/0.53  thf(fact_377_order__eq__refl,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ( X = Y )
% 0.33/0.53       => ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_eq_refl
% 0.33/0.53  thf(fact_378_order__eq__refl,axiom,
% 0.33/0.53      ! [X: num,Y: num] :
% 0.33/0.53        ( ( X = Y )
% 0.33/0.53       => ( ord_less_eq_num @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_eq_refl
% 0.33/0.53  thf(fact_379_order__eq__refl,axiom,
% 0.33/0.53      ! [X: nat,Y: nat] :
% 0.33/0.53        ( ( X = Y )
% 0.33/0.53       => ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_eq_refl
% 0.33/0.53  thf(fact_380_order__eq__refl,axiom,
% 0.33/0.53      ! [X: int,Y: int] :
% 0.33/0.53        ( ( X = Y )
% 0.33/0.53       => ( ord_less_eq_int @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_eq_refl
% 0.33/0.53  thf(fact_381_linorder__linear,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ( ord_less_eq_rat @ X @ Y )
% 0.33/0.53        | ( ord_less_eq_rat @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_linear
% 0.33/0.53  thf(fact_382_linorder__linear,axiom,
% 0.33/0.53      ! [X: num,Y: num] :
% 0.33/0.53        ( ( ord_less_eq_num @ X @ Y )
% 0.33/0.53        | ( ord_less_eq_num @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_linear
% 0.33/0.53  thf(fact_383_linorder__linear,axiom,
% 0.33/0.53      ! [X: nat,Y: nat] :
% 0.33/0.53        ( ( ord_less_eq_nat @ X @ Y )
% 0.33/0.53        | ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_linear
% 0.33/0.53  thf(fact_384_linorder__linear,axiom,
% 0.33/0.53      ! [X: int,Y: int] :
% 0.33/0.53        ( ( ord_less_eq_int @ X @ Y )
% 0.33/0.53        | ( ord_less_eq_int @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_linear
% 0.33/0.53  thf(fact_385_ord__eq__le__subst,axiom,
% 0.33/0.53      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.33/0.53        ( ( A
% 0.33/0.53          = ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_le_subst
% 0.33/0.53  thf(fact_386_ord__eq__le__subst,axiom,
% 0.33/0.53      ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.33/0.53        ( ( A
% 0.33/0.53          = ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_le_subst
% 0.33/0.53  thf(fact_387_ord__eq__le__subst,axiom,
% 0.33/0.53      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.33/0.53        ( ( A
% 0.33/0.53          = ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_le_subst
% 0.33/0.53  thf(fact_388_ord__eq__le__subst,axiom,
% 0.33/0.53      ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.33/0.53        ( ( A
% 0.33/0.53          = ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_le_subst
% 0.33/0.53  thf(fact_389_ord__eq__le__subst,axiom,
% 0.33/0.53      ! [A: rat,F: num > rat,B: num,C: num] :
% 0.33/0.53        ( ( A
% 0.33/0.53          = ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_num @ B @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_le_subst
% 0.33/0.53  thf(fact_390_ord__eq__le__subst,axiom,
% 0.33/0.53      ! [A: num,F: num > num,B: num,C: num] :
% 0.33/0.53        ( ( A
% 0.33/0.53          = ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_num @ B @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_le_subst
% 0.33/0.53  thf(fact_391_ord__eq__le__subst,axiom,
% 0.33/0.53      ! [A: nat,F: num > nat,B: num,C: num] :
% 0.33/0.53        ( ( A
% 0.33/0.53          = ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_num @ B @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_le_subst
% 0.33/0.53  thf(fact_392_ord__eq__le__subst,axiom,
% 0.33/0.53      ! [A: int,F: num > int,B: num,C: num] :
% 0.33/0.53        ( ( A
% 0.33/0.53          = ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_num @ B @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_le_subst
% 0.33/0.53  thf(fact_393_ord__eq__le__subst,axiom,
% 0.33/0.53      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.33/0.53        ( ( A
% 0.33/0.53          = ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_nat @ B @ C )
% 0.33/0.53         => ( ! [X4: nat,Y3: nat] :
% 0.33/0.53                ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_le_subst
% 0.33/0.53  thf(fact_394_ord__eq__le__subst,axiom,
% 0.33/0.53      ! [A: num,F: nat > num,B: nat,C: nat] :
% 0.33/0.53        ( ( A
% 0.33/0.53          = ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_nat @ B @ C )
% 0.33/0.53         => ( ! [X4: nat,Y3: nat] :
% 0.33/0.53                ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_le_subst
% 0.33/0.53  thf(fact_395_ord__le__eq__subst,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.53       => ( ( ( F @ B )
% 0.33/0.53            = C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_le_eq_subst
% 0.33/0.53  thf(fact_396_ord__le__eq__subst,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.53       => ( ( ( F @ B )
% 0.33/0.53            = C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_le_eq_subst
% 0.33/0.53  thf(fact_397_ord__le__eq__subst,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.53       => ( ( ( F @ B )
% 0.33/0.53            = C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_le_eq_subst
% 0.33/0.53  thf(fact_398_ord__le__eq__subst,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.53       => ( ( ( F @ B )
% 0.33/0.53            = C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_le_eq_subst
% 0.33/0.53  thf(fact_399_ord__le__eq__subst,axiom,
% 0.33/0.53      ! [A: num,B: num,F: num > rat,C: rat] :
% 0.33/0.53        ( ( ord_less_eq_num @ A @ B )
% 0.33/0.53       => ( ( ( F @ B )
% 0.33/0.53            = C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_le_eq_subst
% 0.33/0.53  thf(fact_400_ord__le__eq__subst,axiom,
% 0.33/0.53      ! [A: num,B: num,F: num > num,C: num] :
% 0.33/0.53        ( ( ord_less_eq_num @ A @ B )
% 0.33/0.53       => ( ( ( F @ B )
% 0.33/0.53            = C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_le_eq_subst
% 0.33/0.53  thf(fact_401_ord__le__eq__subst,axiom,
% 0.33/0.53      ! [A: num,B: num,F: num > nat,C: nat] :
% 0.33/0.53        ( ( ord_less_eq_num @ A @ B )
% 0.33/0.53       => ( ( ( F @ B )
% 0.33/0.53            = C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_le_eq_subst
% 0.33/0.53  thf(fact_402_ord__le__eq__subst,axiom,
% 0.33/0.53      ! [A: num,B: num,F: num > int,C: int] :
% 0.33/0.53        ( ( ord_less_eq_num @ A @ B )
% 0.33/0.53       => ( ( ( F @ B )
% 0.33/0.53            = C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_le_eq_subst
% 0.33/0.53  thf(fact_403_ord__le__eq__subst,axiom,
% 0.33/0.53      ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 0.33/0.53        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.53       => ( ( ( F @ B )
% 0.33/0.53            = C )
% 0.33/0.53         => ( ! [X4: nat,Y3: nat] :
% 0.33/0.53                ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_le_eq_subst
% 0.33/0.53  thf(fact_404_ord__le__eq__subst,axiom,
% 0.33/0.53      ! [A: nat,B: nat,F: nat > num,C: num] :
% 0.33/0.53        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.53       => ( ( ( F @ B )
% 0.33/0.53            = C )
% 0.33/0.53         => ( ! [X4: nat,Y3: nat] :
% 0.33/0.53                ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_le_eq_subst
% 0.33/0.53  thf(fact_405_linorder__le__cases,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ~ ( ord_less_eq_rat @ X @ Y )
% 0.33/0.53       => ( ord_less_eq_rat @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_le_cases
% 0.33/0.53  thf(fact_406_linorder__le__cases,axiom,
% 0.33/0.53      ! [X: num,Y: num] :
% 0.33/0.53        ( ~ ( ord_less_eq_num @ X @ Y )
% 0.33/0.53       => ( ord_less_eq_num @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_le_cases
% 0.33/0.53  thf(fact_407_linorder__le__cases,axiom,
% 0.33/0.53      ! [X: nat,Y: nat] :
% 0.33/0.53        ( ~ ( ord_less_eq_nat @ X @ Y )
% 0.33/0.53       => ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_le_cases
% 0.33/0.53  thf(fact_408_linorder__le__cases,axiom,
% 0.33/0.53      ! [X: int,Y: int] :
% 0.33/0.53        ( ~ ( ord_less_eq_int @ X @ Y )
% 0.33/0.53       => ( ord_less_eq_int @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_le_cases
% 0.33/0.53  thf(fact_409_order__antisym__conv,axiom,
% 0.33/0.53      ! [Y: set_int,X: set_int] :
% 0.33/0.53        ( ( ord_less_eq_set_int @ Y @ X )
% 0.33/0.53       => ( ( ord_less_eq_set_int @ X @ Y )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_antisym_conv
% 0.33/0.53  thf(fact_410_order__antisym__conv,axiom,
% 0.33/0.53      ! [Y: rat,X: rat] :
% 0.33/0.53        ( ( ord_less_eq_rat @ Y @ X )
% 0.33/0.53       => ( ( ord_less_eq_rat @ X @ Y )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_antisym_conv
% 0.33/0.53  thf(fact_411_order__antisym__conv,axiom,
% 0.33/0.53      ! [Y: num,X: num] :
% 0.33/0.53        ( ( ord_less_eq_num @ Y @ X )
% 0.33/0.53       => ( ( ord_less_eq_num @ X @ Y )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_antisym_conv
% 0.33/0.53  thf(fact_412_order__antisym__conv,axiom,
% 0.33/0.53      ! [Y: nat,X: nat] :
% 0.33/0.53        ( ( ord_less_eq_nat @ Y @ X )
% 0.33/0.53       => ( ( ord_less_eq_nat @ X @ Y )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_antisym_conv
% 0.33/0.53  thf(fact_413_order__antisym__conv,axiom,
% 0.33/0.53      ! [Y: int,X: int] :
% 0.33/0.53        ( ( ord_less_eq_int @ Y @ X )
% 0.33/0.53       => ( ( ord_less_eq_int @ X @ Y )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_antisym_conv
% 0.33/0.53  thf(fact_414_lt__ex,axiom,
% 0.33/0.53      ! [X: real] :
% 0.33/0.53      ? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% 0.33/0.53  
% 0.33/0.53  % lt_ex
% 0.33/0.53  thf(fact_415_lt__ex,axiom,
% 0.33/0.53      ! [X: rat] :
% 0.33/0.53      ? [Y3: rat] : ( ord_less_rat @ Y3 @ X ) ).
% 0.33/0.53  
% 0.33/0.53  % lt_ex
% 0.33/0.53  thf(fact_416_lt__ex,axiom,
% 0.33/0.53      ! [X: int] :
% 0.33/0.53      ? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% 0.33/0.53  
% 0.33/0.53  % lt_ex
% 0.33/0.53  thf(fact_417_gt__ex,axiom,
% 0.33/0.53      ! [X: real] :
% 0.33/0.53      ? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% 0.33/0.53  
% 0.33/0.53  % gt_ex
% 0.33/0.53  thf(fact_418_gt__ex,axiom,
% 0.33/0.53      ! [X: rat] :
% 0.33/0.53      ? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).
% 0.33/0.53  
% 0.33/0.53  % gt_ex
% 0.33/0.53  thf(fact_419_gt__ex,axiom,
% 0.33/0.53      ! [X: nat] :
% 0.33/0.53      ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 0.33/0.53  
% 0.33/0.53  % gt_ex
% 0.33/0.53  thf(fact_420_gt__ex,axiom,
% 0.33/0.53      ! [X: int] :
% 0.33/0.53      ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% 0.33/0.53  
% 0.33/0.53  % gt_ex
% 0.33/0.53  thf(fact_421_dense,axiom,
% 0.33/0.53      ! [X: real,Y: real] :
% 0.33/0.53        ( ( ord_less_real @ X @ Y )
% 0.33/0.53       => ? [Z4: real] :
% 0.33/0.53            ( ( ord_less_real @ X @ Z4 )
% 0.33/0.53            & ( ord_less_real @ Z4 @ Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dense
% 0.33/0.53  thf(fact_422_dense,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ( ord_less_rat @ X @ Y )
% 0.33/0.53       => ? [Z4: rat] :
% 0.33/0.53            ( ( ord_less_rat @ X @ Z4 )
% 0.33/0.53            & ( ord_less_rat @ Z4 @ Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dense
% 0.33/0.53  thf(fact_423_less__imp__neq,axiom,
% 0.33/0.53      ! [X: real,Y: real] :
% 0.33/0.53        ( ( ord_less_real @ X @ Y )
% 0.33/0.53       => ( X != Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % less_imp_neq
% 0.33/0.53  thf(fact_424_less__imp__neq,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ( ord_less_rat @ X @ Y )
% 0.33/0.53       => ( X != Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % less_imp_neq
% 0.33/0.53  thf(fact_425_less__imp__neq,axiom,
% 0.33/0.53      ! [X: num,Y: num] :
% 0.33/0.53        ( ( ord_less_num @ X @ Y )
% 0.33/0.53       => ( X != Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % less_imp_neq
% 0.33/0.53  thf(fact_426_less__imp__neq,axiom,
% 0.33/0.53      ! [X: nat,Y: nat] :
% 0.33/0.53        ( ( ord_less_nat @ X @ Y )
% 0.33/0.53       => ( X != Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % less_imp_neq
% 0.33/0.53  thf(fact_427_less__imp__neq,axiom,
% 0.33/0.53      ! [X: int,Y: int] :
% 0.33/0.53        ( ( ord_less_int @ X @ Y )
% 0.33/0.53       => ( X != Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % less_imp_neq
% 0.33/0.53  thf(fact_428_order_Oasym,axiom,
% 0.33/0.53      ! [A: real,B: real] :
% 0.33/0.53        ( ( ord_less_real @ A @ B )
% 0.33/0.53       => ~ ( ord_less_real @ B @ A ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.asym
% 0.33/0.53  thf(fact_429_order_Oasym,axiom,
% 0.33/0.53      ! [A: rat,B: rat] :
% 0.33/0.53        ( ( ord_less_rat @ A @ B )
% 0.33/0.53       => ~ ( ord_less_rat @ B @ A ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.asym
% 0.33/0.53  thf(fact_430_order_Oasym,axiom,
% 0.33/0.53      ! [A: num,B: num] :
% 0.33/0.53        ( ( ord_less_num @ A @ B )
% 0.33/0.53       => ~ ( ord_less_num @ B @ A ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.asym
% 0.33/0.53  thf(fact_431_order_Oasym,axiom,
% 0.33/0.53      ! [A: nat,B: nat] :
% 0.33/0.53        ( ( ord_less_nat @ A @ B )
% 0.33/0.53       => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.asym
% 0.33/0.53  thf(fact_432_order_Oasym,axiom,
% 0.33/0.53      ! [A: int,B: int] :
% 0.33/0.53        ( ( ord_less_int @ A @ B )
% 0.33/0.53       => ~ ( ord_less_int @ B @ A ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.asym
% 0.33/0.53  thf(fact_433_ord__eq__less__trans,axiom,
% 0.33/0.53      ! [A: real,B: real,C: real] :
% 0.33/0.53        ( ( A = B )
% 0.33/0.53       => ( ( ord_less_real @ B @ C )
% 0.33/0.53         => ( ord_less_real @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_less_trans
% 0.33/0.53  thf(fact_434_ord__eq__less__trans,axiom,
% 0.33/0.53      ! [A: rat,B: rat,C: rat] :
% 0.33/0.53        ( ( A = B )
% 0.33/0.53       => ( ( ord_less_rat @ B @ C )
% 0.33/0.53         => ( ord_less_rat @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_less_trans
% 0.33/0.53  thf(fact_435_ord__eq__less__trans,axiom,
% 0.33/0.53      ! [A: num,B: num,C: num] :
% 0.33/0.53        ( ( A = B )
% 0.33/0.53       => ( ( ord_less_num @ B @ C )
% 0.33/0.53         => ( ord_less_num @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_less_trans
% 0.33/0.53  thf(fact_436_ord__eq__less__trans,axiom,
% 0.33/0.53      ! [A: nat,B: nat,C: nat] :
% 0.33/0.53        ( ( A = B )
% 0.33/0.53       => ( ( ord_less_nat @ B @ C )
% 0.33/0.53         => ( ord_less_nat @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_less_trans
% 0.33/0.53  thf(fact_437_ord__eq__less__trans,axiom,
% 0.33/0.53      ! [A: int,B: int,C: int] :
% 0.33/0.53        ( ( A = B )
% 0.33/0.53       => ( ( ord_less_int @ B @ C )
% 0.33/0.53         => ( ord_less_int @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_less_trans
% 0.33/0.53  thf(fact_438_ord__less__eq__trans,axiom,
% 0.33/0.53      ! [A: real,B: real,C: real] :
% 0.33/0.53        ( ( ord_less_real @ A @ B )
% 0.33/0.53       => ( ( B = C )
% 0.33/0.53         => ( ord_less_real @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_less_eq_trans
% 0.33/0.53  thf(fact_439_ord__less__eq__trans,axiom,
% 0.33/0.53      ! [A: rat,B: rat,C: rat] :
% 0.33/0.53        ( ( ord_less_rat @ A @ B )
% 0.33/0.53       => ( ( B = C )
% 0.33/0.53         => ( ord_less_rat @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_less_eq_trans
% 0.33/0.53  thf(fact_440_ord__less__eq__trans,axiom,
% 0.33/0.53      ! [A: num,B: num,C: num] :
% 0.33/0.53        ( ( ord_less_num @ A @ B )
% 0.33/0.53       => ( ( B = C )
% 0.33/0.53         => ( ord_less_num @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_less_eq_trans
% 0.33/0.53  thf(fact_441_ord__less__eq__trans,axiom,
% 0.33/0.53      ! [A: nat,B: nat,C: nat] :
% 0.33/0.53        ( ( ord_less_nat @ A @ B )
% 0.33/0.53       => ( ( B = C )
% 0.33/0.53         => ( ord_less_nat @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_less_eq_trans
% 0.33/0.53  thf(fact_442_ord__less__eq__trans,axiom,
% 0.33/0.53      ! [A: int,B: int,C: int] :
% 0.33/0.53        ( ( ord_less_int @ A @ B )
% 0.33/0.53       => ( ( B = C )
% 0.33/0.53         => ( ord_less_int @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_less_eq_trans
% 0.33/0.53  thf(fact_443_less__induct,axiom,
% 0.33/0.53      ! [P: nat > $o,A: nat] :
% 0.33/0.53        ( ! [X4: nat] :
% 0.33/0.53            ( ! [Y4: nat] :
% 0.33/0.53                ( ( ord_less_nat @ Y4 @ X4 )
% 0.33/0.53               => ( P @ Y4 ) )
% 0.33/0.53           => ( P @ X4 ) )
% 0.33/0.53       => ( P @ A ) ) ).
% 0.33/0.53  
% 0.33/0.53  % less_induct
% 0.33/0.53  thf(fact_444_antisym__conv3,axiom,
% 0.33/0.53      ! [Y: real,X: real] :
% 0.33/0.53        ( ~ ( ord_less_real @ Y @ X )
% 0.33/0.53       => ( ( ~ ( ord_less_real @ X @ Y ) )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % antisym_conv3
% 0.33/0.53  thf(fact_445_antisym__conv3,axiom,
% 0.33/0.53      ! [Y: rat,X: rat] :
% 0.33/0.53        ( ~ ( ord_less_rat @ Y @ X )
% 0.33/0.53       => ( ( ~ ( ord_less_rat @ X @ Y ) )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % antisym_conv3
% 0.33/0.53  thf(fact_446_antisym__conv3,axiom,
% 0.33/0.53      ! [Y: num,X: num] :
% 0.33/0.53        ( ~ ( ord_less_num @ Y @ X )
% 0.33/0.53       => ( ( ~ ( ord_less_num @ X @ Y ) )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % antisym_conv3
% 0.33/0.53  thf(fact_447_antisym__conv3,axiom,
% 0.33/0.53      ! [Y: nat,X: nat] :
% 0.33/0.53        ( ~ ( ord_less_nat @ Y @ X )
% 0.33/0.53       => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % antisym_conv3
% 0.33/0.53  thf(fact_448_antisym__conv3,axiom,
% 0.33/0.53      ! [Y: int,X: int] :
% 0.33/0.53        ( ~ ( ord_less_int @ Y @ X )
% 0.33/0.53       => ( ( ~ ( ord_less_int @ X @ Y ) )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % antisym_conv3
% 0.33/0.53  thf(fact_449_linorder__cases,axiom,
% 0.33/0.53      ! [X: real,Y: real] :
% 0.33/0.53        ( ~ ( ord_less_real @ X @ Y )
% 0.33/0.53       => ( ( X != Y )
% 0.33/0.53         => ( ord_less_real @ Y @ X ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_cases
% 0.33/0.53  thf(fact_450_linorder__cases,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ~ ( ord_less_rat @ X @ Y )
% 0.33/0.53       => ( ( X != Y )
% 0.33/0.53         => ( ord_less_rat @ Y @ X ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_cases
% 0.33/0.53  thf(fact_451_linorder__cases,axiom,
% 0.33/0.53      ! [X: num,Y: num] :
% 0.33/0.53        ( ~ ( ord_less_num @ X @ Y )
% 0.33/0.53       => ( ( X != Y )
% 0.33/0.53         => ( ord_less_num @ Y @ X ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_cases
% 0.33/0.53  thf(fact_452_linorder__cases,axiom,
% 0.33/0.53      ! [X: nat,Y: nat] :
% 0.33/0.53        ( ~ ( ord_less_nat @ X @ Y )
% 0.33/0.53       => ( ( X != Y )
% 0.33/0.53         => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_cases
% 0.33/0.53  thf(fact_453_linorder__cases,axiom,
% 0.33/0.53      ! [X: int,Y: int] :
% 0.33/0.53        ( ~ ( ord_less_int @ X @ Y )
% 0.33/0.53       => ( ( X != Y )
% 0.33/0.53         => ( ord_less_int @ Y @ X ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_cases
% 0.33/0.53  thf(fact_454_dual__order_Oasym,axiom,
% 0.33/0.53      ! [B: real,A: real] :
% 0.33/0.53        ( ( ord_less_real @ B @ A )
% 0.33/0.53       => ~ ( ord_less_real @ A @ B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.asym
% 0.33/0.53  thf(fact_455_dual__order_Oasym,axiom,
% 0.33/0.53      ! [B: rat,A: rat] :
% 0.33/0.53        ( ( ord_less_rat @ B @ A )
% 0.33/0.53       => ~ ( ord_less_rat @ A @ B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.asym
% 0.33/0.53  thf(fact_456_dual__order_Oasym,axiom,
% 0.33/0.53      ! [B: num,A: num] :
% 0.33/0.53        ( ( ord_less_num @ B @ A )
% 0.33/0.53       => ~ ( ord_less_num @ A @ B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.asym
% 0.33/0.53  thf(fact_457_dual__order_Oasym,axiom,
% 0.33/0.53      ! [B: nat,A: nat] :
% 0.33/0.53        ( ( ord_less_nat @ B @ A )
% 0.33/0.53       => ~ ( ord_less_nat @ A @ B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.asym
% 0.33/0.53  thf(fact_458_dual__order_Oasym,axiom,
% 0.33/0.53      ! [B: int,A: int] :
% 0.33/0.53        ( ( ord_less_int @ B @ A )
% 0.33/0.53       => ~ ( ord_less_int @ A @ B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.asym
% 0.33/0.53  thf(fact_459_dual__order_Oirrefl,axiom,
% 0.33/0.53      ! [A: real] :
% 0.33/0.53        ~ ( ord_less_real @ A @ A ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.irrefl
% 0.33/0.53  thf(fact_460_dual__order_Oirrefl,axiom,
% 0.33/0.53      ! [A: rat] :
% 0.33/0.53        ~ ( ord_less_rat @ A @ A ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.irrefl
% 0.33/0.53  thf(fact_461_dual__order_Oirrefl,axiom,
% 0.33/0.53      ! [A: num] :
% 0.33/0.53        ~ ( ord_less_num @ A @ A ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.irrefl
% 0.33/0.53  thf(fact_462_dual__order_Oirrefl,axiom,
% 0.33/0.53      ! [A: nat] :
% 0.33/0.53        ~ ( ord_less_nat @ A @ A ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.irrefl
% 0.33/0.53  thf(fact_463_dual__order_Oirrefl,axiom,
% 0.33/0.53      ! [A: int] :
% 0.33/0.53        ~ ( ord_less_int @ A @ A ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.irrefl
% 0.33/0.53  thf(fact_464_exists__least__iff,axiom,
% 0.33/0.53      ( ( ^ [P2: nat > $o] :
% 0.33/0.53          ? [X7: nat] : ( P2 @ X7 ) )
% 0.33/0.53      = ( ^ [P3: nat > $o] :
% 0.33/0.53          ? [N2: nat] :
% 0.33/0.53            ( ( P3 @ N2 )
% 0.33/0.53            & ! [M2: nat] :
% 0.33/0.53                ( ( ord_less_nat @ M2 @ N2 )
% 0.33/0.53               => ~ ( P3 @ M2 ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % exists_least_iff
% 0.33/0.53  thf(fact_465_linorder__less__wlog,axiom,
% 0.33/0.53      ! [P: real > real > $o,A: real,B: real] :
% 0.33/0.53        ( ! [A3: real,B3: real] :
% 0.33/0.53            ( ( ord_less_real @ A3 @ B3 )
% 0.33/0.53           => ( P @ A3 @ B3 ) )
% 0.33/0.53       => ( ! [A3: real] : ( P @ A3 @ A3 )
% 0.33/0.53         => ( ! [A3: real,B3: real] :
% 0.33/0.53                ( ( P @ B3 @ A3 )
% 0.33/0.53               => ( P @ A3 @ B3 ) )
% 0.33/0.53           => ( P @ A @ B ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_less_wlog
% 0.33/0.53  thf(fact_466_linorder__less__wlog,axiom,
% 0.33/0.53      ! [P: rat > rat > $o,A: rat,B: rat] :
% 0.33/0.53        ( ! [A3: rat,B3: rat] :
% 0.33/0.53            ( ( ord_less_rat @ A3 @ B3 )
% 0.33/0.53           => ( P @ A3 @ B3 ) )
% 0.33/0.53       => ( ! [A3: rat] : ( P @ A3 @ A3 )
% 0.33/0.53         => ( ! [A3: rat,B3: rat] :
% 0.33/0.53                ( ( P @ B3 @ A3 )
% 0.33/0.53               => ( P @ A3 @ B3 ) )
% 0.33/0.53           => ( P @ A @ B ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_less_wlog
% 0.33/0.53  thf(fact_467_linorder__less__wlog,axiom,
% 0.33/0.53      ! [P: num > num > $o,A: num,B: num] :
% 0.33/0.53        ( ! [A3: num,B3: num] :
% 0.33/0.53            ( ( ord_less_num @ A3 @ B3 )
% 0.33/0.53           => ( P @ A3 @ B3 ) )
% 0.33/0.53       => ( ! [A3: num] : ( P @ A3 @ A3 )
% 0.33/0.53         => ( ! [A3: num,B3: num] :
% 0.33/0.53                ( ( P @ B3 @ A3 )
% 0.33/0.53               => ( P @ A3 @ B3 ) )
% 0.33/0.53           => ( P @ A @ B ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_less_wlog
% 0.33/0.53  thf(fact_468_linorder__less__wlog,axiom,
% 0.33/0.53      ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.33/0.53        ( ! [A3: nat,B3: nat] :
% 0.33/0.53            ( ( ord_less_nat @ A3 @ B3 )
% 0.33/0.53           => ( P @ A3 @ B3 ) )
% 0.33/0.53       => ( ! [A3: nat] : ( P @ A3 @ A3 )
% 0.33/0.53         => ( ! [A3: nat,B3: nat] :
% 0.33/0.53                ( ( P @ B3 @ A3 )
% 0.33/0.53               => ( P @ A3 @ B3 ) )
% 0.33/0.53           => ( P @ A @ B ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_less_wlog
% 0.33/0.53  thf(fact_469_linorder__less__wlog,axiom,
% 0.33/0.53      ! [P: int > int > $o,A: int,B: int] :
% 0.33/0.53        ( ! [A3: int,B3: int] :
% 0.33/0.53            ( ( ord_less_int @ A3 @ B3 )
% 0.33/0.53           => ( P @ A3 @ B3 ) )
% 0.33/0.53       => ( ! [A3: int] : ( P @ A3 @ A3 )
% 0.33/0.53         => ( ! [A3: int,B3: int] :
% 0.33/0.53                ( ( P @ B3 @ A3 )
% 0.33/0.53               => ( P @ A3 @ B3 ) )
% 0.33/0.53           => ( P @ A @ B ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_less_wlog
% 0.33/0.53  thf(fact_470_order_Ostrict__trans,axiom,
% 0.33/0.53      ! [A: real,B: real,C: real] :
% 0.33/0.53        ( ( ord_less_real @ A @ B )
% 0.33/0.53       => ( ( ord_less_real @ B @ C )
% 0.33/0.53         => ( ord_less_real @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_trans
% 0.33/0.53  thf(fact_471_order_Ostrict__trans,axiom,
% 0.33/0.53      ! [A: rat,B: rat,C: rat] :
% 0.33/0.53        ( ( ord_less_rat @ A @ B )
% 0.33/0.53       => ( ( ord_less_rat @ B @ C )
% 0.33/0.53         => ( ord_less_rat @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_trans
% 0.33/0.53  thf(fact_472_order_Ostrict__trans,axiom,
% 0.33/0.53      ! [A: num,B: num,C: num] :
% 0.33/0.53        ( ( ord_less_num @ A @ B )
% 0.33/0.53       => ( ( ord_less_num @ B @ C )
% 0.33/0.53         => ( ord_less_num @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_trans
% 0.33/0.53  thf(fact_473_order_Ostrict__trans,axiom,
% 0.33/0.53      ! [A: nat,B: nat,C: nat] :
% 0.33/0.53        ( ( ord_less_nat @ A @ B )
% 0.33/0.53       => ( ( ord_less_nat @ B @ C )
% 0.33/0.53         => ( ord_less_nat @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_trans
% 0.33/0.53  thf(fact_474_order_Ostrict__trans,axiom,
% 0.33/0.53      ! [A: int,B: int,C: int] :
% 0.33/0.53        ( ( ord_less_int @ A @ B )
% 0.33/0.53       => ( ( ord_less_int @ B @ C )
% 0.33/0.53         => ( ord_less_int @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_trans
% 0.33/0.53  thf(fact_475_not__less__iff__gr__or__eq,axiom,
% 0.33/0.53      ! [X: real,Y: real] :
% 0.33/0.53        ( ( ~ ( ord_less_real @ X @ Y ) )
% 0.33/0.53        = ( ( ord_less_real @ Y @ X )
% 0.33/0.53          | ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % not_less_iff_gr_or_eq
% 0.33/0.53  thf(fact_476_not__less__iff__gr__or__eq,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ( ~ ( ord_less_rat @ X @ Y ) )
% 0.33/0.53        = ( ( ord_less_rat @ Y @ X )
% 0.33/0.53          | ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % not_less_iff_gr_or_eq
% 0.33/0.53  thf(fact_477_not__less__iff__gr__or__eq,axiom,
% 0.33/0.53      ! [X: num,Y: num] :
% 0.33/0.53        ( ( ~ ( ord_less_num @ X @ Y ) )
% 0.33/0.53        = ( ( ord_less_num @ Y @ X )
% 0.33/0.53          | ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % not_less_iff_gr_or_eq
% 0.33/0.53  thf(fact_478_not__less__iff__gr__or__eq,axiom,
% 0.33/0.53      ! [X: nat,Y: nat] :
% 0.33/0.53        ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.33/0.53        = ( ( ord_less_nat @ Y @ X )
% 0.33/0.53          | ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % not_less_iff_gr_or_eq
% 0.33/0.53  thf(fact_479_not__less__iff__gr__or__eq,axiom,
% 0.33/0.53      ! [X: int,Y: int] :
% 0.33/0.53        ( ( ~ ( ord_less_int @ X @ Y ) )
% 0.33/0.53        = ( ( ord_less_int @ Y @ X )
% 0.33/0.53          | ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % not_less_iff_gr_or_eq
% 0.33/0.53  thf(fact_480_dual__order_Ostrict__trans,axiom,
% 0.33/0.53      ! [B: real,A: real,C: real] :
% 0.33/0.53        ( ( ord_less_real @ B @ A )
% 0.33/0.53       => ( ( ord_less_real @ C @ B )
% 0.33/0.53         => ( ord_less_real @ C @ A ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_trans
% 0.33/0.53  thf(fact_481_dual__order_Ostrict__trans,axiom,
% 0.33/0.53      ! [B: rat,A: rat,C: rat] :
% 0.33/0.53        ( ( ord_less_rat @ B @ A )
% 0.33/0.53       => ( ( ord_less_rat @ C @ B )
% 0.33/0.53         => ( ord_less_rat @ C @ A ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_trans
% 0.33/0.53  thf(fact_482_dual__order_Ostrict__trans,axiom,
% 0.33/0.53      ! [B: num,A: num,C: num] :
% 0.33/0.53        ( ( ord_less_num @ B @ A )
% 0.33/0.53       => ( ( ord_less_num @ C @ B )
% 0.33/0.53         => ( ord_less_num @ C @ A ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_trans
% 0.33/0.53  thf(fact_483_dual__order_Ostrict__trans,axiom,
% 0.33/0.53      ! [B: nat,A: nat,C: nat] :
% 0.33/0.53        ( ( ord_less_nat @ B @ A )
% 0.33/0.53       => ( ( ord_less_nat @ C @ B )
% 0.33/0.53         => ( ord_less_nat @ C @ A ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_trans
% 0.33/0.53  thf(fact_484_dual__order_Ostrict__trans,axiom,
% 0.33/0.53      ! [B: int,A: int,C: int] :
% 0.33/0.53        ( ( ord_less_int @ B @ A )
% 0.33/0.53       => ( ( ord_less_int @ C @ B )
% 0.33/0.53         => ( ord_less_int @ C @ A ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_trans
% 0.33/0.53  thf(fact_485_order_Ostrict__implies__not__eq,axiom,
% 0.33/0.53      ! [A: real,B: real] :
% 0.33/0.53        ( ( ord_less_real @ A @ B )
% 0.33/0.53       => ( A != B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_implies_not_eq
% 0.33/0.53  thf(fact_486_order_Ostrict__implies__not__eq,axiom,
% 0.33/0.53      ! [A: rat,B: rat] :
% 0.33/0.53        ( ( ord_less_rat @ A @ B )
% 0.33/0.53       => ( A != B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_implies_not_eq
% 0.33/0.53  thf(fact_487_order_Ostrict__implies__not__eq,axiom,
% 0.33/0.53      ! [A: num,B: num] :
% 0.33/0.53        ( ( ord_less_num @ A @ B )
% 0.33/0.53       => ( A != B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_implies_not_eq
% 0.33/0.53  thf(fact_488_order_Ostrict__implies__not__eq,axiom,
% 0.33/0.53      ! [A: nat,B: nat] :
% 0.33/0.53        ( ( ord_less_nat @ A @ B )
% 0.33/0.53       => ( A != B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_implies_not_eq
% 0.33/0.53  thf(fact_489_order_Ostrict__implies__not__eq,axiom,
% 0.33/0.53      ! [A: int,B: int] :
% 0.33/0.53        ( ( ord_less_int @ A @ B )
% 0.33/0.53       => ( A != B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_implies_not_eq
% 0.33/0.53  thf(fact_490_dual__order_Ostrict__implies__not__eq,axiom,
% 0.33/0.53      ! [B: real,A: real] :
% 0.33/0.53        ( ( ord_less_real @ B @ A )
% 0.33/0.53       => ( A != B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_implies_not_eq
% 0.33/0.53  thf(fact_491_dual__order_Ostrict__implies__not__eq,axiom,
% 0.33/0.53      ! [B: rat,A: rat] :
% 0.33/0.53        ( ( ord_less_rat @ B @ A )
% 0.33/0.53       => ( A != B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_implies_not_eq
% 0.33/0.53  thf(fact_492_dual__order_Ostrict__implies__not__eq,axiom,
% 0.33/0.53      ! [B: num,A: num] :
% 0.33/0.53        ( ( ord_less_num @ B @ A )
% 0.33/0.53       => ( A != B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_implies_not_eq
% 0.33/0.53  thf(fact_493_dual__order_Ostrict__implies__not__eq,axiom,
% 0.33/0.53      ! [B: nat,A: nat] :
% 0.33/0.53        ( ( ord_less_nat @ B @ A )
% 0.33/0.53       => ( A != B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_implies_not_eq
% 0.33/0.53  thf(fact_494_dual__order_Ostrict__implies__not__eq,axiom,
% 0.33/0.53      ! [B: int,A: int] :
% 0.33/0.53        ( ( ord_less_int @ B @ A )
% 0.33/0.53       => ( A != B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_implies_not_eq
% 0.33/0.53  thf(fact_495_linorder__neqE,axiom,
% 0.33/0.53      ! [X: real,Y: real] :
% 0.33/0.53        ( ( X != Y )
% 0.33/0.53       => ( ~ ( ord_less_real @ X @ Y )
% 0.33/0.53         => ( ord_less_real @ Y @ X ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_neqE
% 0.33/0.53  thf(fact_496_linorder__neqE,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ( X != Y )
% 0.33/0.53       => ( ~ ( ord_less_rat @ X @ Y )
% 0.33/0.53         => ( ord_less_rat @ Y @ X ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_neqE
% 0.33/0.53  thf(fact_497_linorder__neqE,axiom,
% 0.33/0.53      ! [X: num,Y: num] :
% 0.33/0.53        ( ( X != Y )
% 0.33/0.53       => ( ~ ( ord_less_num @ X @ Y )
% 0.33/0.53         => ( ord_less_num @ Y @ X ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_neqE
% 0.33/0.53  thf(fact_498_linorder__neqE,axiom,
% 0.33/0.53      ! [X: nat,Y: nat] :
% 0.33/0.53        ( ( X != Y )
% 0.33/0.53       => ( ~ ( ord_less_nat @ X @ Y )
% 0.33/0.53         => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_neqE
% 0.33/0.53  thf(fact_499_linorder__neqE,axiom,
% 0.33/0.53      ! [X: int,Y: int] :
% 0.33/0.53        ( ( X != Y )
% 0.33/0.53       => ( ~ ( ord_less_int @ X @ Y )
% 0.33/0.53         => ( ord_less_int @ Y @ X ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_neqE
% 0.33/0.53  thf(fact_500_order__less__asym,axiom,
% 0.33/0.53      ! [X: real,Y: real] :
% 0.33/0.53        ( ( ord_less_real @ X @ Y )
% 0.33/0.53       => ~ ( ord_less_real @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_asym
% 0.33/0.53  thf(fact_501_order__less__asym,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ( ord_less_rat @ X @ Y )
% 0.33/0.53       => ~ ( ord_less_rat @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_asym
% 0.33/0.53  thf(fact_502_order__less__asym,axiom,
% 0.33/0.53      ! [X: num,Y: num] :
% 0.33/0.53        ( ( ord_less_num @ X @ Y )
% 0.33/0.53       => ~ ( ord_less_num @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_asym
% 0.33/0.53  thf(fact_503_order__less__asym,axiom,
% 0.33/0.53      ! [X: nat,Y: nat] :
% 0.33/0.53        ( ( ord_less_nat @ X @ Y )
% 0.33/0.53       => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_asym
% 0.33/0.53  thf(fact_504_order__less__asym,axiom,
% 0.33/0.53      ! [X: int,Y: int] :
% 0.33/0.53        ( ( ord_less_int @ X @ Y )
% 0.33/0.53       => ~ ( ord_less_int @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_asym
% 0.33/0.53  thf(fact_505_linorder__neq__iff,axiom,
% 0.33/0.53      ! [X: real,Y: real] :
% 0.33/0.53        ( ( X != Y )
% 0.33/0.53        = ( ( ord_less_real @ X @ Y )
% 0.33/0.53          | ( ord_less_real @ Y @ X ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_neq_iff
% 0.33/0.53  thf(fact_506_linorder__neq__iff,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ( X != Y )
% 0.33/0.53        = ( ( ord_less_rat @ X @ Y )
% 0.33/0.53          | ( ord_less_rat @ Y @ X ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_neq_iff
% 0.33/0.53  thf(fact_507_linorder__neq__iff,axiom,
% 0.33/0.53      ! [X: num,Y: num] :
% 0.33/0.53        ( ( X != Y )
% 0.33/0.53        = ( ( ord_less_num @ X @ Y )
% 0.33/0.53          | ( ord_less_num @ Y @ X ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_neq_iff
% 0.33/0.53  thf(fact_508_linorder__neq__iff,axiom,
% 0.33/0.53      ! [X: nat,Y: nat] :
% 0.33/0.53        ( ( X != Y )
% 0.33/0.53        = ( ( ord_less_nat @ X @ Y )
% 0.33/0.53          | ( ord_less_nat @ Y @ X ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_neq_iff
% 0.33/0.53  thf(fact_509_linorder__neq__iff,axiom,
% 0.33/0.53      ! [X: int,Y: int] :
% 0.33/0.53        ( ( X != Y )
% 0.33/0.53        = ( ( ord_less_int @ X @ Y )
% 0.33/0.53          | ( ord_less_int @ Y @ X ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_neq_iff
% 0.33/0.53  thf(fact_510_order__less__asym_H,axiom,
% 0.33/0.53      ! [A: real,B: real] :
% 0.33/0.53        ( ( ord_less_real @ A @ B )
% 0.33/0.53       => ~ ( ord_less_real @ B @ A ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_asym'
% 0.33/0.53  thf(fact_511_order__less__asym_H,axiom,
% 0.33/0.53      ! [A: rat,B: rat] :
% 0.33/0.53        ( ( ord_less_rat @ A @ B )
% 0.33/0.53       => ~ ( ord_less_rat @ B @ A ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_asym'
% 0.33/0.53  thf(fact_512_order__less__asym_H,axiom,
% 0.33/0.53      ! [A: num,B: num] :
% 0.33/0.53        ( ( ord_less_num @ A @ B )
% 0.33/0.53       => ~ ( ord_less_num @ B @ A ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_asym'
% 0.33/0.53  thf(fact_513_order__less__asym_H,axiom,
% 0.33/0.53      ! [A: nat,B: nat] :
% 0.33/0.53        ( ( ord_less_nat @ A @ B )
% 0.33/0.53       => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_asym'
% 0.33/0.53  thf(fact_514_order__less__asym_H,axiom,
% 0.33/0.53      ! [A: int,B: int] :
% 0.33/0.53        ( ( ord_less_int @ A @ B )
% 0.33/0.53       => ~ ( ord_less_int @ B @ A ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_asym'
% 0.33/0.53  thf(fact_515_order__less__trans,axiom,
% 0.33/0.53      ! [X: real,Y: real,Z2: real] :
% 0.33/0.53        ( ( ord_less_real @ X @ Y )
% 0.33/0.53       => ( ( ord_less_real @ Y @ Z2 )
% 0.33/0.53         => ( ord_less_real @ X @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_trans
% 0.33/0.53  thf(fact_516_order__less__trans,axiom,
% 0.33/0.53      ! [X: rat,Y: rat,Z2: rat] :
% 0.33/0.53        ( ( ord_less_rat @ X @ Y )
% 0.33/0.53       => ( ( ord_less_rat @ Y @ Z2 )
% 0.33/0.53         => ( ord_less_rat @ X @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_trans
% 0.33/0.53  thf(fact_517_order__less__trans,axiom,
% 0.33/0.53      ! [X: num,Y: num,Z2: num] :
% 0.33/0.53        ( ( ord_less_num @ X @ Y )
% 0.33/0.53       => ( ( ord_less_num @ Y @ Z2 )
% 0.33/0.53         => ( ord_less_num @ X @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_trans
% 0.33/0.53  thf(fact_518_order__less__trans,axiom,
% 0.33/0.53      ! [X: nat,Y: nat,Z2: nat] :
% 0.33/0.53        ( ( ord_less_nat @ X @ Y )
% 0.33/0.53       => ( ( ord_less_nat @ Y @ Z2 )
% 0.33/0.53         => ( ord_less_nat @ X @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_trans
% 0.33/0.53  thf(fact_519_order__less__trans,axiom,
% 0.33/0.53      ! [X: int,Y: int,Z2: int] :
% 0.33/0.53        ( ( ord_less_int @ X @ Y )
% 0.33/0.53       => ( ( ord_less_int @ Y @ Z2 )
% 0.33/0.53         => ( ord_less_int @ X @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_trans
% 0.33/0.53  thf(fact_520_ord__eq__less__subst,axiom,
% 0.33/0.53      ! [A: real,F: real > real,B: real,C: real] :
% 0.33/0.53        ( ( A
% 0.33/0.53          = ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_real @ B @ C )
% 0.33/0.53         => ( ! [X4: real,Y3: real] :
% 0.33/0.53                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_less_subst
% 0.33/0.53  thf(fact_521_ord__eq__less__subst,axiom,
% 0.33/0.53      ! [A: rat,F: real > rat,B: real,C: real] :
% 0.33/0.53        ( ( A
% 0.33/0.53          = ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_real @ B @ C )
% 0.33/0.53         => ( ! [X4: real,Y3: real] :
% 0.33/0.53                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_less_subst
% 0.33/0.53  thf(fact_522_ord__eq__less__subst,axiom,
% 0.33/0.53      ! [A: num,F: real > num,B: real,C: real] :
% 0.33/0.53        ( ( A
% 0.33/0.53          = ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_real @ B @ C )
% 0.33/0.53         => ( ! [X4: real,Y3: real] :
% 0.33/0.53                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_less_subst
% 0.33/0.53  thf(fact_523_ord__eq__less__subst,axiom,
% 0.33/0.53      ! [A: nat,F: real > nat,B: real,C: real] :
% 0.33/0.53        ( ( A
% 0.33/0.53          = ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_real @ B @ C )
% 0.33/0.53         => ( ! [X4: real,Y3: real] :
% 0.33/0.53                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_less_subst
% 0.33/0.53  thf(fact_524_ord__eq__less__subst,axiom,
% 0.33/0.53      ! [A: int,F: real > int,B: real,C: real] :
% 0.33/0.53        ( ( A
% 0.33/0.53          = ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_real @ B @ C )
% 0.33/0.53         => ( ! [X4: real,Y3: real] :
% 0.33/0.53                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_less_subst
% 0.33/0.53  thf(fact_525_ord__eq__less__subst,axiom,
% 0.33/0.53      ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.33/0.53        ( ( A
% 0.33/0.53          = ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_less_subst
% 0.33/0.53  thf(fact_526_ord__eq__less__subst,axiom,
% 0.33/0.53      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.33/0.53        ( ( A
% 0.33/0.53          = ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_less_subst
% 0.33/0.53  thf(fact_527_ord__eq__less__subst,axiom,
% 0.33/0.53      ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.33/0.53        ( ( A
% 0.33/0.53          = ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_less_subst
% 0.33/0.53  thf(fact_528_ord__eq__less__subst,axiom,
% 0.33/0.53      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.33/0.53        ( ( A
% 0.33/0.53          = ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_less_subst
% 0.33/0.53  thf(fact_529_ord__eq__less__subst,axiom,
% 0.33/0.53      ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.33/0.53        ( ( A
% 0.33/0.53          = ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_eq_less_subst
% 0.33/0.53  thf(fact_530_ord__less__eq__subst,axiom,
% 0.33/0.53      ! [A: real,B: real,F: real > real,C: real] :
% 0.33/0.53        ( ( ord_less_real @ A @ B )
% 0.33/0.53       => ( ( ( F @ B )
% 0.33/0.53            = C )
% 0.33/0.53         => ( ! [X4: real,Y3: real] :
% 0.33/0.53                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_less_eq_subst
% 0.33/0.53  thf(fact_531_ord__less__eq__subst,axiom,
% 0.33/0.53      ! [A: real,B: real,F: real > rat,C: rat] :
% 0.33/0.53        ( ( ord_less_real @ A @ B )
% 0.33/0.53       => ( ( ( F @ B )
% 0.33/0.53            = C )
% 0.33/0.53         => ( ! [X4: real,Y3: real] :
% 0.33/0.53                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_less_eq_subst
% 0.33/0.53  thf(fact_532_ord__less__eq__subst,axiom,
% 0.33/0.53      ! [A: real,B: real,F: real > num,C: num] :
% 0.33/0.53        ( ( ord_less_real @ A @ B )
% 0.33/0.53       => ( ( ( F @ B )
% 0.33/0.53            = C )
% 0.33/0.53         => ( ! [X4: real,Y3: real] :
% 0.33/0.53                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_less_eq_subst
% 0.33/0.53  thf(fact_533_ord__less__eq__subst,axiom,
% 0.33/0.53      ! [A: real,B: real,F: real > nat,C: nat] :
% 0.33/0.53        ( ( ord_less_real @ A @ B )
% 0.33/0.53       => ( ( ( F @ B )
% 0.33/0.53            = C )
% 0.33/0.53         => ( ! [X4: real,Y3: real] :
% 0.33/0.53                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_less_eq_subst
% 0.33/0.53  thf(fact_534_ord__less__eq__subst,axiom,
% 0.33/0.53      ! [A: real,B: real,F: real > int,C: int] :
% 0.33/0.53        ( ( ord_less_real @ A @ B )
% 0.33/0.53       => ( ( ( F @ B )
% 0.33/0.53            = C )
% 0.33/0.53         => ( ! [X4: real,Y3: real] :
% 0.33/0.53                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_less_eq_subst
% 0.33/0.53  thf(fact_535_ord__less__eq__subst,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.33/0.53        ( ( ord_less_rat @ A @ B )
% 0.33/0.53       => ( ( ( F @ B )
% 0.33/0.53            = C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_less_eq_subst
% 0.33/0.53  thf(fact_536_ord__less__eq__subst,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.33/0.53        ( ( ord_less_rat @ A @ B )
% 0.33/0.53       => ( ( ( F @ B )
% 0.33/0.53            = C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_less_eq_subst
% 0.33/0.53  thf(fact_537_ord__less__eq__subst,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.33/0.53        ( ( ord_less_rat @ A @ B )
% 0.33/0.53       => ( ( ( F @ B )
% 0.33/0.53            = C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_less_eq_subst
% 0.33/0.53  thf(fact_538_ord__less__eq__subst,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.33/0.53        ( ( ord_less_rat @ A @ B )
% 0.33/0.53       => ( ( ( F @ B )
% 0.33/0.53            = C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_less_eq_subst
% 0.33/0.53  thf(fact_539_ord__less__eq__subst,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.33/0.53        ( ( ord_less_rat @ A @ B )
% 0.33/0.53       => ( ( ( F @ B )
% 0.33/0.53            = C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % ord_less_eq_subst
% 0.33/0.53  thf(fact_540_order__less__irrefl,axiom,
% 0.33/0.53      ! [X: real] :
% 0.33/0.53        ~ ( ord_less_real @ X @ X ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_irrefl
% 0.33/0.53  thf(fact_541_order__less__irrefl,axiom,
% 0.33/0.53      ! [X: rat] :
% 0.33/0.53        ~ ( ord_less_rat @ X @ X ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_irrefl
% 0.33/0.53  thf(fact_542_order__less__irrefl,axiom,
% 0.33/0.53      ! [X: num] :
% 0.33/0.53        ~ ( ord_less_num @ X @ X ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_irrefl
% 0.33/0.53  thf(fact_543_order__less__irrefl,axiom,
% 0.33/0.53      ! [X: nat] :
% 0.33/0.53        ~ ( ord_less_nat @ X @ X ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_irrefl
% 0.33/0.53  thf(fact_544_order__less__irrefl,axiom,
% 0.33/0.53      ! [X: int] :
% 0.33/0.53        ~ ( ord_less_int @ X @ X ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_irrefl
% 0.33/0.53  thf(fact_545_order__less__subst1,axiom,
% 0.33/0.53      ! [A: real,F: real > real,B: real,C: real] :
% 0.33/0.53        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_real @ B @ C )
% 0.33/0.53         => ( ! [X4: real,Y3: real] :
% 0.33/0.53                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_subst1
% 0.33/0.53  thf(fact_546_order__less__subst1,axiom,
% 0.33/0.53      ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.33/0.53        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_subst1
% 0.33/0.53  thf(fact_547_order__less__subst1,axiom,
% 0.33/0.53      ! [A: real,F: num > real,B: num,C: num] :
% 0.33/0.53        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_num @ B @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_subst1
% 0.33/0.53  thf(fact_548_order__less__subst1,axiom,
% 0.33/0.53      ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.33/0.53        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_nat @ B @ C )
% 0.33/0.53         => ( ! [X4: nat,Y3: nat] :
% 0.33/0.53                ( ( ord_less_nat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_subst1
% 0.33/0.53  thf(fact_549_order__less__subst1,axiom,
% 0.33/0.53      ! [A: real,F: int > real,B: int,C: int] :
% 0.33/0.53        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_int @ B @ C )
% 0.33/0.53         => ( ! [X4: int,Y3: int] :
% 0.33/0.53                ( ( ord_less_int @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_subst1
% 0.33/0.53  thf(fact_550_order__less__subst1,axiom,
% 0.33/0.53      ! [A: rat,F: real > rat,B: real,C: real] :
% 0.33/0.53        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_real @ B @ C )
% 0.33/0.53         => ( ! [X4: real,Y3: real] :
% 0.33/0.53                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_subst1
% 0.33/0.53  thf(fact_551_order__less__subst1,axiom,
% 0.33/0.53      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.33/0.53        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_subst1
% 0.33/0.53  thf(fact_552_order__less__subst1,axiom,
% 0.33/0.53      ! [A: rat,F: num > rat,B: num,C: num] :
% 0.33/0.53        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_num @ B @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_subst1
% 0.33/0.53  thf(fact_553_order__less__subst1,axiom,
% 0.33/0.53      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.33/0.53        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_nat @ B @ C )
% 0.33/0.53         => ( ! [X4: nat,Y3: nat] :
% 0.33/0.53                ( ( ord_less_nat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_subst1
% 0.33/0.53  thf(fact_554_order__less__subst1,axiom,
% 0.33/0.53      ! [A: rat,F: int > rat,B: int,C: int] :
% 0.33/0.53        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_int @ B @ C )
% 0.33/0.53         => ( ! [X4: int,Y3: int] :
% 0.33/0.53                ( ( ord_less_int @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_subst1
% 0.33/0.53  thf(fact_555_order__less__subst2,axiom,
% 0.33/0.53      ! [A: real,B: real,F: real > real,C: real] :
% 0.33/0.53        ( ( ord_less_real @ A @ B )
% 0.33/0.53       => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: real,Y3: real] :
% 0.33/0.53                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_subst2
% 0.33/0.53  thf(fact_556_order__less__subst2,axiom,
% 0.33/0.53      ! [A: real,B: real,F: real > rat,C: rat] :
% 0.33/0.53        ( ( ord_less_real @ A @ B )
% 0.33/0.53       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: real,Y3: real] :
% 0.33/0.53                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_subst2
% 0.33/0.53  thf(fact_557_order__less__subst2,axiom,
% 0.33/0.53      ! [A: real,B: real,F: real > num,C: num] :
% 0.33/0.53        ( ( ord_less_real @ A @ B )
% 0.33/0.53       => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: real,Y3: real] :
% 0.33/0.53                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_subst2
% 0.33/0.53  thf(fact_558_order__less__subst2,axiom,
% 0.33/0.53      ! [A: real,B: real,F: real > nat,C: nat] :
% 0.33/0.53        ( ( ord_less_real @ A @ B )
% 0.33/0.53       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: real,Y3: real] :
% 0.33/0.53                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_subst2
% 0.33/0.53  thf(fact_559_order__less__subst2,axiom,
% 0.33/0.53      ! [A: real,B: real,F: real > int,C: int] :
% 0.33/0.53        ( ( ord_less_real @ A @ B )
% 0.33/0.53       => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: real,Y3: real] :
% 0.33/0.53                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_subst2
% 0.33/0.53  thf(fact_560_order__less__subst2,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.33/0.53        ( ( ord_less_rat @ A @ B )
% 0.33/0.53       => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_subst2
% 0.33/0.53  thf(fact_561_order__less__subst2,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.33/0.53        ( ( ord_less_rat @ A @ B )
% 0.33/0.53       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_subst2
% 0.33/0.53  thf(fact_562_order__less__subst2,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.33/0.53        ( ( ord_less_rat @ A @ B )
% 0.33/0.53       => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_subst2
% 0.33/0.53  thf(fact_563_order__less__subst2,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.33/0.53        ( ( ord_less_rat @ A @ B )
% 0.33/0.53       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_subst2
% 0.33/0.53  thf(fact_564_order__less__subst2,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.33/0.53        ( ( ord_less_rat @ A @ B )
% 0.33/0.53       => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_subst2
% 0.33/0.53  thf(fact_565_order__less__not__sym,axiom,
% 0.33/0.53      ! [X: real,Y: real] :
% 0.33/0.53        ( ( ord_less_real @ X @ Y )
% 0.33/0.53       => ~ ( ord_less_real @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_not_sym
% 0.33/0.53  thf(fact_566_order__less__not__sym,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ( ord_less_rat @ X @ Y )
% 0.33/0.53       => ~ ( ord_less_rat @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_not_sym
% 0.33/0.53  thf(fact_567_order__less__not__sym,axiom,
% 0.33/0.53      ! [X: num,Y: num] :
% 0.33/0.53        ( ( ord_less_num @ X @ Y )
% 0.33/0.53       => ~ ( ord_less_num @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_not_sym
% 0.33/0.53  thf(fact_568_order__less__not__sym,axiom,
% 0.33/0.53      ! [X: nat,Y: nat] :
% 0.33/0.53        ( ( ord_less_nat @ X @ Y )
% 0.33/0.53       => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_not_sym
% 0.33/0.53  thf(fact_569_order__less__not__sym,axiom,
% 0.33/0.53      ! [X: int,Y: int] :
% 0.33/0.53        ( ( ord_less_int @ X @ Y )
% 0.33/0.53       => ~ ( ord_less_int @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_not_sym
% 0.33/0.53  thf(fact_570_order__less__imp__triv,axiom,
% 0.33/0.53      ! [X: real,Y: real,P: $o] :
% 0.33/0.53        ( ( ord_less_real @ X @ Y )
% 0.33/0.53       => ( ( ord_less_real @ Y @ X )
% 0.33/0.53         => P ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_triv
% 0.33/0.53  thf(fact_571_order__less__imp__triv,axiom,
% 0.33/0.53      ! [X: rat,Y: rat,P: $o] :
% 0.33/0.53        ( ( ord_less_rat @ X @ Y )
% 0.33/0.53       => ( ( ord_less_rat @ Y @ X )
% 0.33/0.53         => P ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_triv
% 0.33/0.53  thf(fact_572_order__less__imp__triv,axiom,
% 0.33/0.53      ! [X: num,Y: num,P: $o] :
% 0.33/0.53        ( ( ord_less_num @ X @ Y )
% 0.33/0.53       => ( ( ord_less_num @ Y @ X )
% 0.33/0.53         => P ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_triv
% 0.33/0.53  thf(fact_573_order__less__imp__triv,axiom,
% 0.33/0.53      ! [X: nat,Y: nat,P: $o] :
% 0.33/0.53        ( ( ord_less_nat @ X @ Y )
% 0.33/0.53       => ( ( ord_less_nat @ Y @ X )
% 0.33/0.53         => P ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_triv
% 0.33/0.53  thf(fact_574_order__less__imp__triv,axiom,
% 0.33/0.53      ! [X: int,Y: int,P: $o] :
% 0.33/0.53        ( ( ord_less_int @ X @ Y )
% 0.33/0.53       => ( ( ord_less_int @ Y @ X )
% 0.33/0.53         => P ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_triv
% 0.33/0.53  thf(fact_575_linorder__less__linear,axiom,
% 0.33/0.53      ! [X: real,Y: real] :
% 0.33/0.53        ( ( ord_less_real @ X @ Y )
% 0.33/0.53        | ( X = Y )
% 0.33/0.53        | ( ord_less_real @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_less_linear
% 0.33/0.53  thf(fact_576_linorder__less__linear,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ( ord_less_rat @ X @ Y )
% 0.33/0.53        | ( X = Y )
% 0.33/0.53        | ( ord_less_rat @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_less_linear
% 0.33/0.53  thf(fact_577_linorder__less__linear,axiom,
% 0.33/0.53      ! [X: num,Y: num] :
% 0.33/0.53        ( ( ord_less_num @ X @ Y )
% 0.33/0.53        | ( X = Y )
% 0.33/0.53        | ( ord_less_num @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_less_linear
% 0.33/0.53  thf(fact_578_linorder__less__linear,axiom,
% 0.33/0.53      ! [X: nat,Y: nat] :
% 0.33/0.53        ( ( ord_less_nat @ X @ Y )
% 0.33/0.53        | ( X = Y )
% 0.33/0.53        | ( ord_less_nat @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_less_linear
% 0.33/0.53  thf(fact_579_linorder__less__linear,axiom,
% 0.33/0.53      ! [X: int,Y: int] :
% 0.33/0.53        ( ( ord_less_int @ X @ Y )
% 0.33/0.53        | ( X = Y )
% 0.33/0.53        | ( ord_less_int @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_less_linear
% 0.33/0.53  thf(fact_580_order__less__imp__not__eq,axiom,
% 0.33/0.53      ! [X: real,Y: real] :
% 0.33/0.53        ( ( ord_less_real @ X @ Y )
% 0.33/0.53       => ( X != Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_not_eq
% 0.33/0.53  thf(fact_581_order__less__imp__not__eq,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ( ord_less_rat @ X @ Y )
% 0.33/0.53       => ( X != Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_not_eq
% 0.33/0.53  thf(fact_582_order__less__imp__not__eq,axiom,
% 0.33/0.53      ! [X: num,Y: num] :
% 0.33/0.53        ( ( ord_less_num @ X @ Y )
% 0.33/0.53       => ( X != Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_not_eq
% 0.33/0.53  thf(fact_583_order__less__imp__not__eq,axiom,
% 0.33/0.53      ! [X: nat,Y: nat] :
% 0.33/0.53        ( ( ord_less_nat @ X @ Y )
% 0.33/0.53       => ( X != Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_not_eq
% 0.33/0.53  thf(fact_584_order__less__imp__not__eq,axiom,
% 0.33/0.53      ! [X: int,Y: int] :
% 0.33/0.53        ( ( ord_less_int @ X @ Y )
% 0.33/0.53       => ( X != Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_not_eq
% 0.33/0.53  thf(fact_585_order__less__imp__not__eq2,axiom,
% 0.33/0.53      ! [X: real,Y: real] :
% 0.33/0.53        ( ( ord_less_real @ X @ Y )
% 0.33/0.53       => ( Y != X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_not_eq2
% 0.33/0.53  thf(fact_586_order__less__imp__not__eq2,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ( ord_less_rat @ X @ Y )
% 0.33/0.53       => ( Y != X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_not_eq2
% 0.33/0.53  thf(fact_587_order__less__imp__not__eq2,axiom,
% 0.33/0.53      ! [X: num,Y: num] :
% 0.33/0.53        ( ( ord_less_num @ X @ Y )
% 0.33/0.53       => ( Y != X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_not_eq2
% 0.33/0.53  thf(fact_588_order__less__imp__not__eq2,axiom,
% 0.33/0.53      ! [X: nat,Y: nat] :
% 0.33/0.53        ( ( ord_less_nat @ X @ Y )
% 0.33/0.53       => ( Y != X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_not_eq2
% 0.33/0.53  thf(fact_589_order__less__imp__not__eq2,axiom,
% 0.33/0.53      ! [X: int,Y: int] :
% 0.33/0.53        ( ( ord_less_int @ X @ Y )
% 0.33/0.53       => ( Y != X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_not_eq2
% 0.33/0.53  thf(fact_590_order__less__imp__not__less,axiom,
% 0.33/0.53      ! [X: real,Y: real] :
% 0.33/0.53        ( ( ord_less_real @ X @ Y )
% 0.33/0.53       => ~ ( ord_less_real @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_not_less
% 0.33/0.53  thf(fact_591_order__less__imp__not__less,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ( ord_less_rat @ X @ Y )
% 0.33/0.53       => ~ ( ord_less_rat @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_not_less
% 0.33/0.53  thf(fact_592_order__less__imp__not__less,axiom,
% 0.33/0.53      ! [X: num,Y: num] :
% 0.33/0.53        ( ( ord_less_num @ X @ Y )
% 0.33/0.53       => ~ ( ord_less_num @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_not_less
% 0.33/0.53  thf(fact_593_order__less__imp__not__less,axiom,
% 0.33/0.53      ! [X: nat,Y: nat] :
% 0.33/0.53        ( ( ord_less_nat @ X @ Y )
% 0.33/0.53       => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_not_less
% 0.33/0.53  thf(fact_594_order__less__imp__not__less,axiom,
% 0.33/0.53      ! [X: int,Y: int] :
% 0.33/0.53        ( ( ord_less_int @ X @ Y )
% 0.33/0.53       => ~ ( ord_less_int @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_not_less
% 0.33/0.53  thf(fact_595_leD,axiom,
% 0.33/0.53      ! [Y: real,X: real] :
% 0.33/0.53        ( ( ord_less_eq_real @ Y @ X )
% 0.33/0.53       => ~ ( ord_less_real @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % leD
% 0.33/0.53  thf(fact_596_leD,axiom,
% 0.33/0.53      ! [Y: set_int,X: set_int] :
% 0.33/0.53        ( ( ord_less_eq_set_int @ Y @ X )
% 0.33/0.53       => ~ ( ord_less_set_int @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % leD
% 0.33/0.53  thf(fact_597_leD,axiom,
% 0.33/0.53      ! [Y: rat,X: rat] :
% 0.33/0.53        ( ( ord_less_eq_rat @ Y @ X )
% 0.33/0.53       => ~ ( ord_less_rat @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % leD
% 0.33/0.53  thf(fact_598_leD,axiom,
% 0.33/0.53      ! [Y: num,X: num] :
% 0.33/0.53        ( ( ord_less_eq_num @ Y @ X )
% 0.33/0.53       => ~ ( ord_less_num @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % leD
% 0.33/0.53  thf(fact_599_leD,axiom,
% 0.33/0.53      ! [Y: nat,X: nat] :
% 0.33/0.53        ( ( ord_less_eq_nat @ Y @ X )
% 0.33/0.53       => ~ ( ord_less_nat @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % leD
% 0.33/0.53  thf(fact_600_leD,axiom,
% 0.33/0.53      ! [Y: int,X: int] :
% 0.33/0.53        ( ( ord_less_eq_int @ Y @ X )
% 0.33/0.53       => ~ ( ord_less_int @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % leD
% 0.33/0.53  thf(fact_601_leI,axiom,
% 0.33/0.53      ! [X: real,Y: real] :
% 0.33/0.53        ( ~ ( ord_less_real @ X @ Y )
% 0.33/0.53       => ( ord_less_eq_real @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % leI
% 0.33/0.53  thf(fact_602_leI,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ~ ( ord_less_rat @ X @ Y )
% 0.33/0.53       => ( ord_less_eq_rat @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % leI
% 0.33/0.53  thf(fact_603_leI,axiom,
% 0.33/0.53      ! [X: num,Y: num] :
% 0.33/0.53        ( ~ ( ord_less_num @ X @ Y )
% 0.33/0.53       => ( ord_less_eq_num @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % leI
% 0.33/0.53  thf(fact_604_leI,axiom,
% 0.33/0.53      ! [X: nat,Y: nat] :
% 0.33/0.53        ( ~ ( ord_less_nat @ X @ Y )
% 0.33/0.53       => ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % leI
% 0.33/0.53  thf(fact_605_leI,axiom,
% 0.33/0.53      ! [X: int,Y: int] :
% 0.33/0.53        ( ~ ( ord_less_int @ X @ Y )
% 0.33/0.53       => ( ord_less_eq_int @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % leI
% 0.33/0.53  thf(fact_606_nless__le,axiom,
% 0.33/0.53      ! [A: real,B: real] :
% 0.33/0.53        ( ( ~ ( ord_less_real @ A @ B ) )
% 0.33/0.53        = ( ~ ( ord_less_eq_real @ A @ B )
% 0.33/0.53          | ( A = B ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % nless_le
% 0.33/0.53  thf(fact_607_nless__le,axiom,
% 0.33/0.53      ! [A: set_int,B: set_int] :
% 0.33/0.53        ( ( ~ ( ord_less_set_int @ A @ B ) )
% 0.33/0.53        = ( ~ ( ord_less_eq_set_int @ A @ B )
% 0.33/0.53          | ( A = B ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % nless_le
% 0.33/0.53  thf(fact_608_nless__le,axiom,
% 0.33/0.53      ! [A: rat,B: rat] :
% 0.33/0.53        ( ( ~ ( ord_less_rat @ A @ B ) )
% 0.33/0.53        = ( ~ ( ord_less_eq_rat @ A @ B )
% 0.33/0.53          | ( A = B ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % nless_le
% 0.33/0.53  thf(fact_609_nless__le,axiom,
% 0.33/0.53      ! [A: num,B: num] :
% 0.33/0.53        ( ( ~ ( ord_less_num @ A @ B ) )
% 0.33/0.53        = ( ~ ( ord_less_eq_num @ A @ B )
% 0.33/0.53          | ( A = B ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % nless_le
% 0.33/0.53  thf(fact_610_nless__le,axiom,
% 0.33/0.53      ! [A: nat,B: nat] :
% 0.33/0.53        ( ( ~ ( ord_less_nat @ A @ B ) )
% 0.33/0.53        = ( ~ ( ord_less_eq_nat @ A @ B )
% 0.33/0.53          | ( A = B ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % nless_le
% 0.33/0.53  thf(fact_611_nless__le,axiom,
% 0.33/0.53      ! [A: int,B: int] :
% 0.33/0.53        ( ( ~ ( ord_less_int @ A @ B ) )
% 0.33/0.53        = ( ~ ( ord_less_eq_int @ A @ B )
% 0.33/0.53          | ( A = B ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % nless_le
% 0.33/0.53  thf(fact_612_antisym__conv1,axiom,
% 0.33/0.53      ! [X: real,Y: real] :
% 0.33/0.53        ( ~ ( ord_less_real @ X @ Y )
% 0.33/0.53       => ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % antisym_conv1
% 0.33/0.53  thf(fact_613_antisym__conv1,axiom,
% 0.33/0.53      ! [X: set_int,Y: set_int] :
% 0.33/0.53        ( ~ ( ord_less_set_int @ X @ Y )
% 0.33/0.53       => ( ( ord_less_eq_set_int @ X @ Y )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % antisym_conv1
% 0.33/0.53  thf(fact_614_antisym__conv1,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ~ ( ord_less_rat @ X @ Y )
% 0.33/0.53       => ( ( ord_less_eq_rat @ X @ Y )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % antisym_conv1
% 0.33/0.53  thf(fact_615_antisym__conv1,axiom,
% 0.33/0.53      ! [X: num,Y: num] :
% 0.33/0.53        ( ~ ( ord_less_num @ X @ Y )
% 0.33/0.53       => ( ( ord_less_eq_num @ X @ Y )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % antisym_conv1
% 0.33/0.53  thf(fact_616_antisym__conv1,axiom,
% 0.33/0.53      ! [X: nat,Y: nat] :
% 0.33/0.53        ( ~ ( ord_less_nat @ X @ Y )
% 0.33/0.53       => ( ( ord_less_eq_nat @ X @ Y )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % antisym_conv1
% 0.33/0.53  thf(fact_617_antisym__conv1,axiom,
% 0.33/0.53      ! [X: int,Y: int] :
% 0.33/0.53        ( ~ ( ord_less_int @ X @ Y )
% 0.33/0.53       => ( ( ord_less_eq_int @ X @ Y )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % antisym_conv1
% 0.33/0.53  thf(fact_618_antisym__conv2,axiom,
% 0.33/0.53      ! [X: real,Y: real] :
% 0.33/0.53        ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.53       => ( ( ~ ( ord_less_real @ X @ Y ) )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % antisym_conv2
% 0.33/0.53  thf(fact_619_antisym__conv2,axiom,
% 0.33/0.53      ! [X: set_int,Y: set_int] :
% 0.33/0.53        ( ( ord_less_eq_set_int @ X @ Y )
% 0.33/0.53       => ( ( ~ ( ord_less_set_int @ X @ Y ) )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % antisym_conv2
% 0.33/0.53  thf(fact_620_antisym__conv2,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ( ord_less_eq_rat @ X @ Y )
% 0.33/0.53       => ( ( ~ ( ord_less_rat @ X @ Y ) )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % antisym_conv2
% 0.33/0.53  thf(fact_621_antisym__conv2,axiom,
% 0.33/0.53      ! [X: num,Y: num] :
% 0.33/0.53        ( ( ord_less_eq_num @ X @ Y )
% 0.33/0.53       => ( ( ~ ( ord_less_num @ X @ Y ) )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % antisym_conv2
% 0.33/0.53  thf(fact_622_antisym__conv2,axiom,
% 0.33/0.53      ! [X: nat,Y: nat] :
% 0.33/0.53        ( ( ord_less_eq_nat @ X @ Y )
% 0.33/0.53       => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % antisym_conv2
% 0.33/0.53  thf(fact_623_antisym__conv2,axiom,
% 0.33/0.53      ! [X: int,Y: int] :
% 0.33/0.53        ( ( ord_less_eq_int @ X @ Y )
% 0.33/0.53       => ( ( ~ ( ord_less_int @ X @ Y ) )
% 0.33/0.53          = ( X = Y ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % antisym_conv2
% 0.33/0.53  thf(fact_624_dense__ge,axiom,
% 0.33/0.53      ! [Z2: real,Y: real] :
% 0.33/0.53        ( ! [X4: real] :
% 0.33/0.53            ( ( ord_less_real @ Z2 @ X4 )
% 0.33/0.53           => ( ord_less_eq_real @ Y @ X4 ) )
% 0.33/0.53       => ( ord_less_eq_real @ Y @ Z2 ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dense_ge
% 0.33/0.53  thf(fact_625_dense__ge,axiom,
% 0.33/0.53      ! [Z2: rat,Y: rat] :
% 0.33/0.53        ( ! [X4: rat] :
% 0.33/0.53            ( ( ord_less_rat @ Z2 @ X4 )
% 0.33/0.53           => ( ord_less_eq_rat @ Y @ X4 ) )
% 0.33/0.53       => ( ord_less_eq_rat @ Y @ Z2 ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dense_ge
% 0.33/0.53  thf(fact_626_dense__le,axiom,
% 0.33/0.53      ! [Y: real,Z2: real] :
% 0.33/0.53        ( ! [X4: real] :
% 0.33/0.53            ( ( ord_less_real @ X4 @ Y )
% 0.33/0.53           => ( ord_less_eq_real @ X4 @ Z2 ) )
% 0.33/0.53       => ( ord_less_eq_real @ Y @ Z2 ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dense_le
% 0.33/0.53  thf(fact_627_dense__le,axiom,
% 0.33/0.53      ! [Y: rat,Z2: rat] :
% 0.33/0.53        ( ! [X4: rat] :
% 0.33/0.53            ( ( ord_less_rat @ X4 @ Y )
% 0.33/0.53           => ( ord_less_eq_rat @ X4 @ Z2 ) )
% 0.33/0.53       => ( ord_less_eq_rat @ Y @ Z2 ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dense_le
% 0.33/0.53  thf(fact_628_less__le__not__le,axiom,
% 0.33/0.53      ( ord_less_real
% 0.33/0.53      = ( ^ [X3: real,Y2: real] :
% 0.33/0.53            ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.33/0.53            & ~ ( ord_less_eq_real @ Y2 @ X3 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % less_le_not_le
% 0.33/0.53  thf(fact_629_less__le__not__le,axiom,
% 0.33/0.53      ( ord_less_set_int
% 0.33/0.53      = ( ^ [X3: set_int,Y2: set_int] :
% 0.33/0.53            ( ( ord_less_eq_set_int @ X3 @ Y2 )
% 0.33/0.53            & ~ ( ord_less_eq_set_int @ Y2 @ X3 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % less_le_not_le
% 0.33/0.53  thf(fact_630_less__le__not__le,axiom,
% 0.33/0.53      ( ord_less_rat
% 0.33/0.53      = ( ^ [X3: rat,Y2: rat] :
% 0.33/0.53            ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.33/0.53            & ~ ( ord_less_eq_rat @ Y2 @ X3 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % less_le_not_le
% 0.33/0.53  thf(fact_631_less__le__not__le,axiom,
% 0.33/0.53      ( ord_less_num
% 0.33/0.53      = ( ^ [X3: num,Y2: num] :
% 0.33/0.53            ( ( ord_less_eq_num @ X3 @ Y2 )
% 0.33/0.53            & ~ ( ord_less_eq_num @ Y2 @ X3 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % less_le_not_le
% 0.33/0.53  thf(fact_632_less__le__not__le,axiom,
% 0.33/0.53      ( ord_less_nat
% 0.33/0.53      = ( ^ [X3: nat,Y2: nat] :
% 0.33/0.53            ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.33/0.53            & ~ ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % less_le_not_le
% 0.33/0.53  thf(fact_633_less__le__not__le,axiom,
% 0.33/0.53      ( ord_less_int
% 0.33/0.53      = ( ^ [X3: int,Y2: int] :
% 0.33/0.53            ( ( ord_less_eq_int @ X3 @ Y2 )
% 0.33/0.53            & ~ ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % less_le_not_le
% 0.33/0.53  thf(fact_634_not__le__imp__less,axiom,
% 0.33/0.53      ! [Y: real,X: real] :
% 0.33/0.53        ( ~ ( ord_less_eq_real @ Y @ X )
% 0.33/0.53       => ( ord_less_real @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % not_le_imp_less
% 0.33/0.53  thf(fact_635_not__le__imp__less,axiom,
% 0.33/0.53      ! [Y: rat,X: rat] :
% 0.33/0.53        ( ~ ( ord_less_eq_rat @ Y @ X )
% 0.33/0.53       => ( ord_less_rat @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % not_le_imp_less
% 0.33/0.53  thf(fact_636_not__le__imp__less,axiom,
% 0.33/0.53      ! [Y: num,X: num] :
% 0.33/0.53        ( ~ ( ord_less_eq_num @ Y @ X )
% 0.33/0.53       => ( ord_less_num @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % not_le_imp_less
% 0.33/0.53  thf(fact_637_not__le__imp__less,axiom,
% 0.33/0.53      ! [Y: nat,X: nat] :
% 0.33/0.53        ( ~ ( ord_less_eq_nat @ Y @ X )
% 0.33/0.53       => ( ord_less_nat @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % not_le_imp_less
% 0.33/0.53  thf(fact_638_not__le__imp__less,axiom,
% 0.33/0.53      ! [Y: int,X: int] :
% 0.33/0.53        ( ~ ( ord_less_eq_int @ Y @ X )
% 0.33/0.53       => ( ord_less_int @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % not_le_imp_less
% 0.33/0.53  thf(fact_639_order_Oorder__iff__strict,axiom,
% 0.33/0.53      ( ord_less_eq_real
% 0.33/0.53      = ( ^ [A4: real,B4: real] :
% 0.33/0.53            ( ( ord_less_real @ A4 @ B4 )
% 0.33/0.53            | ( A4 = B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.order_iff_strict
% 0.33/0.53  thf(fact_640_order_Oorder__iff__strict,axiom,
% 0.33/0.53      ( ord_less_eq_set_int
% 0.33/0.53      = ( ^ [A4: set_int,B4: set_int] :
% 0.33/0.53            ( ( ord_less_set_int @ A4 @ B4 )
% 0.33/0.53            | ( A4 = B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.order_iff_strict
% 0.33/0.53  thf(fact_641_order_Oorder__iff__strict,axiom,
% 0.33/0.53      ( ord_less_eq_rat
% 0.33/0.53      = ( ^ [A4: rat,B4: rat] :
% 0.33/0.53            ( ( ord_less_rat @ A4 @ B4 )
% 0.33/0.53            | ( A4 = B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.order_iff_strict
% 0.33/0.53  thf(fact_642_order_Oorder__iff__strict,axiom,
% 0.33/0.53      ( ord_less_eq_num
% 0.33/0.53      = ( ^ [A4: num,B4: num] :
% 0.33/0.53            ( ( ord_less_num @ A4 @ B4 )
% 0.33/0.53            | ( A4 = B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.order_iff_strict
% 0.33/0.53  thf(fact_643_order_Oorder__iff__strict,axiom,
% 0.33/0.53      ( ord_less_eq_nat
% 0.33/0.53      = ( ^ [A4: nat,B4: nat] :
% 0.33/0.53            ( ( ord_less_nat @ A4 @ B4 )
% 0.33/0.53            | ( A4 = B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.order_iff_strict
% 0.33/0.53  thf(fact_644_order_Oorder__iff__strict,axiom,
% 0.33/0.53      ( ord_less_eq_int
% 0.33/0.53      = ( ^ [A4: int,B4: int] :
% 0.33/0.53            ( ( ord_less_int @ A4 @ B4 )
% 0.33/0.53            | ( A4 = B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.order_iff_strict
% 0.33/0.53  thf(fact_645_order_Ostrict__iff__order,axiom,
% 0.33/0.53      ( ord_less_real
% 0.33/0.53      = ( ^ [A4: real,B4: real] :
% 0.33/0.53            ( ( ord_less_eq_real @ A4 @ B4 )
% 0.33/0.53            & ( A4 != B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_iff_order
% 0.33/0.53  thf(fact_646_order_Ostrict__iff__order,axiom,
% 0.33/0.53      ( ord_less_set_int
% 0.33/0.53      = ( ^ [A4: set_int,B4: set_int] :
% 0.33/0.53            ( ( ord_less_eq_set_int @ A4 @ B4 )
% 0.33/0.53            & ( A4 != B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_iff_order
% 0.33/0.53  thf(fact_647_order_Ostrict__iff__order,axiom,
% 0.33/0.53      ( ord_less_rat
% 0.33/0.53      = ( ^ [A4: rat,B4: rat] :
% 0.33/0.53            ( ( ord_less_eq_rat @ A4 @ B4 )
% 0.33/0.53            & ( A4 != B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_iff_order
% 0.33/0.53  thf(fact_648_order_Ostrict__iff__order,axiom,
% 0.33/0.53      ( ord_less_num
% 0.33/0.53      = ( ^ [A4: num,B4: num] :
% 0.33/0.53            ( ( ord_less_eq_num @ A4 @ B4 )
% 0.33/0.53            & ( A4 != B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_iff_order
% 0.33/0.53  thf(fact_649_order_Ostrict__iff__order,axiom,
% 0.33/0.53      ( ord_less_nat
% 0.33/0.53      = ( ^ [A4: nat,B4: nat] :
% 0.33/0.53            ( ( ord_less_eq_nat @ A4 @ B4 )
% 0.33/0.53            & ( A4 != B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_iff_order
% 0.33/0.53  thf(fact_650_order_Ostrict__iff__order,axiom,
% 0.33/0.53      ( ord_less_int
% 0.33/0.53      = ( ^ [A4: int,B4: int] :
% 0.33/0.53            ( ( ord_less_eq_int @ A4 @ B4 )
% 0.33/0.53            & ( A4 != B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_iff_order
% 0.33/0.53  thf(fact_651_order_Ostrict__trans1,axiom,
% 0.33/0.53      ! [A: real,B: real,C: real] :
% 0.33/0.53        ( ( ord_less_eq_real @ A @ B )
% 0.33/0.53       => ( ( ord_less_real @ B @ C )
% 0.33/0.53         => ( ord_less_real @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_trans1
% 0.33/0.53  thf(fact_652_order_Ostrict__trans1,axiom,
% 0.33/0.53      ! [A: set_int,B: set_int,C: set_int] :
% 0.33/0.53        ( ( ord_less_eq_set_int @ A @ B )
% 0.33/0.53       => ( ( ord_less_set_int @ B @ C )
% 0.33/0.53         => ( ord_less_set_int @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_trans1
% 0.33/0.53  thf(fact_653_order_Ostrict__trans1,axiom,
% 0.33/0.53      ! [A: rat,B: rat,C: rat] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.53       => ( ( ord_less_rat @ B @ C )
% 0.33/0.53         => ( ord_less_rat @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_trans1
% 0.33/0.53  thf(fact_654_order_Ostrict__trans1,axiom,
% 0.33/0.53      ! [A: num,B: num,C: num] :
% 0.33/0.53        ( ( ord_less_eq_num @ A @ B )
% 0.33/0.53       => ( ( ord_less_num @ B @ C )
% 0.33/0.53         => ( ord_less_num @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_trans1
% 0.33/0.53  thf(fact_655_order_Ostrict__trans1,axiom,
% 0.33/0.53      ! [A: nat,B: nat,C: nat] :
% 0.33/0.53        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.53       => ( ( ord_less_nat @ B @ C )
% 0.33/0.53         => ( ord_less_nat @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_trans1
% 0.33/0.53  thf(fact_656_order_Ostrict__trans1,axiom,
% 0.33/0.53      ! [A: int,B: int,C: int] :
% 0.33/0.53        ( ( ord_less_eq_int @ A @ B )
% 0.33/0.53       => ( ( ord_less_int @ B @ C )
% 0.33/0.53         => ( ord_less_int @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_trans1
% 0.33/0.53  thf(fact_657_order_Ostrict__trans2,axiom,
% 0.33/0.53      ! [A: real,B: real,C: real] :
% 0.33/0.53        ( ( ord_less_real @ A @ B )
% 0.33/0.53       => ( ( ord_less_eq_real @ B @ C )
% 0.33/0.53         => ( ord_less_real @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_trans2
% 0.33/0.53  thf(fact_658_order_Ostrict__trans2,axiom,
% 0.33/0.53      ! [A: set_int,B: set_int,C: set_int] :
% 0.33/0.53        ( ( ord_less_set_int @ A @ B )
% 0.33/0.53       => ( ( ord_less_eq_set_int @ B @ C )
% 0.33/0.53         => ( ord_less_set_int @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_trans2
% 0.33/0.53  thf(fact_659_order_Ostrict__trans2,axiom,
% 0.33/0.53      ! [A: rat,B: rat,C: rat] :
% 0.33/0.53        ( ( ord_less_rat @ A @ B )
% 0.33/0.53       => ( ( ord_less_eq_rat @ B @ C )
% 0.33/0.53         => ( ord_less_rat @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_trans2
% 0.33/0.53  thf(fact_660_order_Ostrict__trans2,axiom,
% 0.33/0.53      ! [A: num,B: num,C: num] :
% 0.33/0.53        ( ( ord_less_num @ A @ B )
% 0.33/0.53       => ( ( ord_less_eq_num @ B @ C )
% 0.33/0.53         => ( ord_less_num @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_trans2
% 0.33/0.53  thf(fact_661_order_Ostrict__trans2,axiom,
% 0.33/0.53      ! [A: nat,B: nat,C: nat] :
% 0.33/0.53        ( ( ord_less_nat @ A @ B )
% 0.33/0.53       => ( ( ord_less_eq_nat @ B @ C )
% 0.33/0.53         => ( ord_less_nat @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_trans2
% 0.33/0.53  thf(fact_662_order_Ostrict__trans2,axiom,
% 0.33/0.53      ! [A: int,B: int,C: int] :
% 0.33/0.53        ( ( ord_less_int @ A @ B )
% 0.33/0.53       => ( ( ord_less_eq_int @ B @ C )
% 0.33/0.53         => ( ord_less_int @ A @ C ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_trans2
% 0.33/0.53  thf(fact_663_order_Ostrict__iff__not,axiom,
% 0.33/0.53      ( ord_less_real
% 0.33/0.53      = ( ^ [A4: real,B4: real] :
% 0.33/0.53            ( ( ord_less_eq_real @ A4 @ B4 )
% 0.33/0.53            & ~ ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_iff_not
% 0.33/0.53  thf(fact_664_order_Ostrict__iff__not,axiom,
% 0.33/0.53      ( ord_less_set_int
% 0.33/0.53      = ( ^ [A4: set_int,B4: set_int] :
% 0.33/0.53            ( ( ord_less_eq_set_int @ A4 @ B4 )
% 0.33/0.53            & ~ ( ord_less_eq_set_int @ B4 @ A4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_iff_not
% 0.33/0.53  thf(fact_665_order_Ostrict__iff__not,axiom,
% 0.33/0.53      ( ord_less_rat
% 0.33/0.53      = ( ^ [A4: rat,B4: rat] :
% 0.33/0.53            ( ( ord_less_eq_rat @ A4 @ B4 )
% 0.33/0.53            & ~ ( ord_less_eq_rat @ B4 @ A4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_iff_not
% 0.33/0.53  thf(fact_666_order_Ostrict__iff__not,axiom,
% 0.33/0.53      ( ord_less_num
% 0.33/0.53      = ( ^ [A4: num,B4: num] :
% 0.33/0.53            ( ( ord_less_eq_num @ A4 @ B4 )
% 0.33/0.53            & ~ ( ord_less_eq_num @ B4 @ A4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_iff_not
% 0.33/0.53  thf(fact_667_order_Ostrict__iff__not,axiom,
% 0.33/0.53      ( ord_less_nat
% 0.33/0.53      = ( ^ [A4: nat,B4: nat] :
% 0.33/0.53            ( ( ord_less_eq_nat @ A4 @ B4 )
% 0.33/0.53            & ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_iff_not
% 0.33/0.53  thf(fact_668_order_Ostrict__iff__not,axiom,
% 0.33/0.53      ( ord_less_int
% 0.33/0.53      = ( ^ [A4: int,B4: int] :
% 0.33/0.53            ( ( ord_less_eq_int @ A4 @ B4 )
% 0.33/0.53            & ~ ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_iff_not
% 0.33/0.53  thf(fact_669_dense__ge__bounded,axiom,
% 0.33/0.53      ! [Z2: real,X: real,Y: real] :
% 0.33/0.53        ( ( ord_less_real @ Z2 @ X )
% 0.33/0.53       => ( ! [W: real] :
% 0.33/0.53              ( ( ord_less_real @ Z2 @ W )
% 0.33/0.53             => ( ( ord_less_real @ W @ X )
% 0.33/0.53               => ( ord_less_eq_real @ Y @ W ) ) )
% 0.33/0.53         => ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dense_ge_bounded
% 0.33/0.53  thf(fact_670_dense__ge__bounded,axiom,
% 0.33/0.53      ! [Z2: rat,X: rat,Y: rat] :
% 0.33/0.53        ( ( ord_less_rat @ Z2 @ X )
% 0.33/0.53       => ( ! [W: rat] :
% 0.33/0.53              ( ( ord_less_rat @ Z2 @ W )
% 0.33/0.53             => ( ( ord_less_rat @ W @ X )
% 0.33/0.53               => ( ord_less_eq_rat @ Y @ W ) ) )
% 0.33/0.53         => ( ord_less_eq_rat @ Y @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dense_ge_bounded
% 0.33/0.53  thf(fact_671_dense__le__bounded,axiom,
% 0.33/0.53      ! [X: real,Y: real,Z2: real] :
% 0.33/0.53        ( ( ord_less_real @ X @ Y )
% 0.33/0.53       => ( ! [W: real] :
% 0.33/0.53              ( ( ord_less_real @ X @ W )
% 0.33/0.53             => ( ( ord_less_real @ W @ Y )
% 0.33/0.53               => ( ord_less_eq_real @ W @ Z2 ) ) )
% 0.33/0.53         => ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dense_le_bounded
% 0.33/0.53  thf(fact_672_dense__le__bounded,axiom,
% 0.33/0.53      ! [X: rat,Y: rat,Z2: rat] :
% 0.33/0.53        ( ( ord_less_rat @ X @ Y )
% 0.33/0.53       => ( ! [W: rat] :
% 0.33/0.53              ( ( ord_less_rat @ X @ W )
% 0.33/0.53             => ( ( ord_less_rat @ W @ Y )
% 0.33/0.53               => ( ord_less_eq_rat @ W @ Z2 ) ) )
% 0.33/0.53         => ( ord_less_eq_rat @ Y @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dense_le_bounded
% 0.33/0.53  thf(fact_673_dual__order_Oorder__iff__strict,axiom,
% 0.33/0.53      ( ord_less_eq_real
% 0.33/0.53      = ( ^ [B4: real,A4: real] :
% 0.33/0.53            ( ( ord_less_real @ B4 @ A4 )
% 0.33/0.53            | ( A4 = B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.order_iff_strict
% 0.33/0.53  thf(fact_674_dual__order_Oorder__iff__strict,axiom,
% 0.33/0.53      ( ord_less_eq_set_int
% 0.33/0.53      = ( ^ [B4: set_int,A4: set_int] :
% 0.33/0.53            ( ( ord_less_set_int @ B4 @ A4 )
% 0.33/0.53            | ( A4 = B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.order_iff_strict
% 0.33/0.53  thf(fact_675_dual__order_Oorder__iff__strict,axiom,
% 0.33/0.53      ( ord_less_eq_rat
% 0.33/0.53      = ( ^ [B4: rat,A4: rat] :
% 0.33/0.53            ( ( ord_less_rat @ B4 @ A4 )
% 0.33/0.53            | ( A4 = B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.order_iff_strict
% 0.33/0.53  thf(fact_676_dual__order_Oorder__iff__strict,axiom,
% 0.33/0.53      ( ord_less_eq_num
% 0.33/0.53      = ( ^ [B4: num,A4: num] :
% 0.33/0.53            ( ( ord_less_num @ B4 @ A4 )
% 0.33/0.53            | ( A4 = B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.order_iff_strict
% 0.33/0.53  thf(fact_677_dual__order_Oorder__iff__strict,axiom,
% 0.33/0.53      ( ord_less_eq_nat
% 0.33/0.53      = ( ^ [B4: nat,A4: nat] :
% 0.33/0.53            ( ( ord_less_nat @ B4 @ A4 )
% 0.33/0.53            | ( A4 = B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.order_iff_strict
% 0.33/0.53  thf(fact_678_dual__order_Oorder__iff__strict,axiom,
% 0.33/0.53      ( ord_less_eq_int
% 0.33/0.53      = ( ^ [B4: int,A4: int] :
% 0.33/0.53            ( ( ord_less_int @ B4 @ A4 )
% 0.33/0.53            | ( A4 = B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.order_iff_strict
% 0.33/0.53  thf(fact_679_dual__order_Ostrict__iff__order,axiom,
% 0.33/0.53      ( ord_less_real
% 0.33/0.53      = ( ^ [B4: real,A4: real] :
% 0.33/0.53            ( ( ord_less_eq_real @ B4 @ A4 )
% 0.33/0.53            & ( A4 != B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_iff_order
% 0.33/0.53  thf(fact_680_dual__order_Ostrict__iff__order,axiom,
% 0.33/0.53      ( ord_less_set_int
% 0.33/0.53      = ( ^ [B4: set_int,A4: set_int] :
% 0.33/0.53            ( ( ord_less_eq_set_int @ B4 @ A4 )
% 0.33/0.53            & ( A4 != B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_iff_order
% 0.33/0.53  thf(fact_681_dual__order_Ostrict__iff__order,axiom,
% 0.33/0.53      ( ord_less_rat
% 0.33/0.53      = ( ^ [B4: rat,A4: rat] :
% 0.33/0.53            ( ( ord_less_eq_rat @ B4 @ A4 )
% 0.33/0.53            & ( A4 != B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_iff_order
% 0.33/0.53  thf(fact_682_dual__order_Ostrict__iff__order,axiom,
% 0.33/0.53      ( ord_less_num
% 0.33/0.53      = ( ^ [B4: num,A4: num] :
% 0.33/0.53            ( ( ord_less_eq_num @ B4 @ A4 )
% 0.33/0.53            & ( A4 != B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_iff_order
% 0.33/0.53  thf(fact_683_dual__order_Ostrict__iff__order,axiom,
% 0.33/0.53      ( ord_less_nat
% 0.33/0.53      = ( ^ [B4: nat,A4: nat] :
% 0.33/0.53            ( ( ord_less_eq_nat @ B4 @ A4 )
% 0.33/0.53            & ( A4 != B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_iff_order
% 0.33/0.53  thf(fact_684_dual__order_Ostrict__iff__order,axiom,
% 0.33/0.53      ( ord_less_int
% 0.33/0.53      = ( ^ [B4: int,A4: int] :
% 0.33/0.53            ( ( ord_less_eq_int @ B4 @ A4 )
% 0.33/0.53            & ( A4 != B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_iff_order
% 0.33/0.53  thf(fact_685_dual__order_Ostrict__trans1,axiom,
% 0.33/0.53      ! [B: real,A: real,C: real] :
% 0.33/0.53        ( ( ord_less_eq_real @ B @ A )
% 0.33/0.53       => ( ( ord_less_real @ C @ B )
% 0.33/0.53         => ( ord_less_real @ C @ A ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_trans1
% 0.33/0.53  thf(fact_686_dual__order_Ostrict__trans1,axiom,
% 0.33/0.53      ! [B: set_int,A: set_int,C: set_int] :
% 0.33/0.53        ( ( ord_less_eq_set_int @ B @ A )
% 0.33/0.53       => ( ( ord_less_set_int @ C @ B )
% 0.33/0.53         => ( ord_less_set_int @ C @ A ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_trans1
% 0.33/0.53  thf(fact_687_dual__order_Ostrict__trans1,axiom,
% 0.33/0.53      ! [B: rat,A: rat,C: rat] :
% 0.33/0.53        ( ( ord_less_eq_rat @ B @ A )
% 0.33/0.53       => ( ( ord_less_rat @ C @ B )
% 0.33/0.53         => ( ord_less_rat @ C @ A ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_trans1
% 0.33/0.53  thf(fact_688_dual__order_Ostrict__trans1,axiom,
% 0.33/0.53      ! [B: num,A: num,C: num] :
% 0.33/0.53        ( ( ord_less_eq_num @ B @ A )
% 0.33/0.53       => ( ( ord_less_num @ C @ B )
% 0.33/0.53         => ( ord_less_num @ C @ A ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_trans1
% 0.33/0.53  thf(fact_689_dual__order_Ostrict__trans1,axiom,
% 0.33/0.53      ! [B: nat,A: nat,C: nat] :
% 0.33/0.53        ( ( ord_less_eq_nat @ B @ A )
% 0.33/0.53       => ( ( ord_less_nat @ C @ B )
% 0.33/0.53         => ( ord_less_nat @ C @ A ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_trans1
% 0.33/0.53  thf(fact_690_dual__order_Ostrict__trans1,axiom,
% 0.33/0.53      ! [B: int,A: int,C: int] :
% 0.33/0.53        ( ( ord_less_eq_int @ B @ A )
% 0.33/0.53       => ( ( ord_less_int @ C @ B )
% 0.33/0.53         => ( ord_less_int @ C @ A ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_trans1
% 0.33/0.53  thf(fact_691_dual__order_Ostrict__trans2,axiom,
% 0.33/0.53      ! [B: real,A: real,C: real] :
% 0.33/0.53        ( ( ord_less_real @ B @ A )
% 0.33/0.53       => ( ( ord_less_eq_real @ C @ B )
% 0.33/0.53         => ( ord_less_real @ C @ A ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_trans2
% 0.33/0.53  thf(fact_692_dual__order_Ostrict__trans2,axiom,
% 0.33/0.53      ! [B: set_int,A: set_int,C: set_int] :
% 0.33/0.53        ( ( ord_less_set_int @ B @ A )
% 0.33/0.53       => ( ( ord_less_eq_set_int @ C @ B )
% 0.33/0.53         => ( ord_less_set_int @ C @ A ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_trans2
% 0.33/0.53  thf(fact_693_dual__order_Ostrict__trans2,axiom,
% 0.33/0.53      ! [B: rat,A: rat,C: rat] :
% 0.33/0.53        ( ( ord_less_rat @ B @ A )
% 0.33/0.53       => ( ( ord_less_eq_rat @ C @ B )
% 0.33/0.53         => ( ord_less_rat @ C @ A ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_trans2
% 0.33/0.53  thf(fact_694_dual__order_Ostrict__trans2,axiom,
% 0.33/0.53      ! [B: num,A: num,C: num] :
% 0.33/0.53        ( ( ord_less_num @ B @ A )
% 0.33/0.53       => ( ( ord_less_eq_num @ C @ B )
% 0.33/0.53         => ( ord_less_num @ C @ A ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_trans2
% 0.33/0.53  thf(fact_695_dual__order_Ostrict__trans2,axiom,
% 0.33/0.53      ! [B: nat,A: nat,C: nat] :
% 0.33/0.53        ( ( ord_less_nat @ B @ A )
% 0.33/0.53       => ( ( ord_less_eq_nat @ C @ B )
% 0.33/0.53         => ( ord_less_nat @ C @ A ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_trans2
% 0.33/0.53  thf(fact_696_dual__order_Ostrict__trans2,axiom,
% 0.33/0.53      ! [B: int,A: int,C: int] :
% 0.33/0.53        ( ( ord_less_int @ B @ A )
% 0.33/0.53       => ( ( ord_less_eq_int @ C @ B )
% 0.33/0.53         => ( ord_less_int @ C @ A ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_trans2
% 0.33/0.53  thf(fact_697_dual__order_Ostrict__iff__not,axiom,
% 0.33/0.53      ( ord_less_real
% 0.33/0.53      = ( ^ [B4: real,A4: real] :
% 0.33/0.53            ( ( ord_less_eq_real @ B4 @ A4 )
% 0.33/0.53            & ~ ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_iff_not
% 0.33/0.53  thf(fact_698_dual__order_Ostrict__iff__not,axiom,
% 0.33/0.53      ( ord_less_set_int
% 0.33/0.53      = ( ^ [B4: set_int,A4: set_int] :
% 0.33/0.53            ( ( ord_less_eq_set_int @ B4 @ A4 )
% 0.33/0.53            & ~ ( ord_less_eq_set_int @ A4 @ B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_iff_not
% 0.33/0.53  thf(fact_699_dual__order_Ostrict__iff__not,axiom,
% 0.33/0.53      ( ord_less_rat
% 0.33/0.53      = ( ^ [B4: rat,A4: rat] :
% 0.33/0.53            ( ( ord_less_eq_rat @ B4 @ A4 )
% 0.33/0.53            & ~ ( ord_less_eq_rat @ A4 @ B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_iff_not
% 0.33/0.53  thf(fact_700_dual__order_Ostrict__iff__not,axiom,
% 0.33/0.53      ( ord_less_num
% 0.33/0.53      = ( ^ [B4: num,A4: num] :
% 0.33/0.53            ( ( ord_less_eq_num @ B4 @ A4 )
% 0.33/0.53            & ~ ( ord_less_eq_num @ A4 @ B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_iff_not
% 0.33/0.53  thf(fact_701_dual__order_Ostrict__iff__not,axiom,
% 0.33/0.53      ( ord_less_nat
% 0.33/0.53      = ( ^ [B4: nat,A4: nat] :
% 0.33/0.53            ( ( ord_less_eq_nat @ B4 @ A4 )
% 0.33/0.53            & ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_iff_not
% 0.33/0.53  thf(fact_702_dual__order_Ostrict__iff__not,axiom,
% 0.33/0.53      ( ord_less_int
% 0.33/0.53      = ( ^ [B4: int,A4: int] :
% 0.33/0.53            ( ( ord_less_eq_int @ B4 @ A4 )
% 0.33/0.53            & ~ ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_iff_not
% 0.33/0.53  thf(fact_703_order_Ostrict__implies__order,axiom,
% 0.33/0.53      ! [A: real,B: real] :
% 0.33/0.53        ( ( ord_less_real @ A @ B )
% 0.33/0.53       => ( ord_less_eq_real @ A @ B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_implies_order
% 0.33/0.53  thf(fact_704_order_Ostrict__implies__order,axiom,
% 0.33/0.53      ! [A: set_int,B: set_int] :
% 0.33/0.53        ( ( ord_less_set_int @ A @ B )
% 0.33/0.53       => ( ord_less_eq_set_int @ A @ B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_implies_order
% 0.33/0.53  thf(fact_705_order_Ostrict__implies__order,axiom,
% 0.33/0.53      ! [A: rat,B: rat] :
% 0.33/0.53        ( ( ord_less_rat @ A @ B )
% 0.33/0.53       => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_implies_order
% 0.33/0.53  thf(fact_706_order_Ostrict__implies__order,axiom,
% 0.33/0.53      ! [A: num,B: num] :
% 0.33/0.53        ( ( ord_less_num @ A @ B )
% 0.33/0.53       => ( ord_less_eq_num @ A @ B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_implies_order
% 0.33/0.53  thf(fact_707_order_Ostrict__implies__order,axiom,
% 0.33/0.53      ! [A: nat,B: nat] :
% 0.33/0.53        ( ( ord_less_nat @ A @ B )
% 0.33/0.53       => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_implies_order
% 0.33/0.53  thf(fact_708_order_Ostrict__implies__order,axiom,
% 0.33/0.53      ! [A: int,B: int] :
% 0.33/0.53        ( ( ord_less_int @ A @ B )
% 0.33/0.53       => ( ord_less_eq_int @ A @ B ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order.strict_implies_order
% 0.33/0.53  thf(fact_709_dual__order_Ostrict__implies__order,axiom,
% 0.33/0.53      ! [B: real,A: real] :
% 0.33/0.53        ( ( ord_less_real @ B @ A )
% 0.33/0.53       => ( ord_less_eq_real @ B @ A ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_implies_order
% 0.33/0.53  thf(fact_710_dual__order_Ostrict__implies__order,axiom,
% 0.33/0.53      ! [B: set_int,A: set_int] :
% 0.33/0.53        ( ( ord_less_set_int @ B @ A )
% 0.33/0.53       => ( ord_less_eq_set_int @ B @ A ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_implies_order
% 0.33/0.53  thf(fact_711_dual__order_Ostrict__implies__order,axiom,
% 0.33/0.53      ! [B: rat,A: rat] :
% 0.33/0.53        ( ( ord_less_rat @ B @ A )
% 0.33/0.53       => ( ord_less_eq_rat @ B @ A ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_implies_order
% 0.33/0.53  thf(fact_712_dual__order_Ostrict__implies__order,axiom,
% 0.33/0.53      ! [B: num,A: num] :
% 0.33/0.53        ( ( ord_less_num @ B @ A )
% 0.33/0.53       => ( ord_less_eq_num @ B @ A ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_implies_order
% 0.33/0.53  thf(fact_713_dual__order_Ostrict__implies__order,axiom,
% 0.33/0.53      ! [B: nat,A: nat] :
% 0.33/0.53        ( ( ord_less_nat @ B @ A )
% 0.33/0.53       => ( ord_less_eq_nat @ B @ A ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_implies_order
% 0.33/0.53  thf(fact_714_dual__order_Ostrict__implies__order,axiom,
% 0.33/0.53      ! [B: int,A: int] :
% 0.33/0.53        ( ( ord_less_int @ B @ A )
% 0.33/0.53       => ( ord_less_eq_int @ B @ A ) ) ).
% 0.33/0.53  
% 0.33/0.53  % dual_order.strict_implies_order
% 0.33/0.53  thf(fact_715_order__le__less,axiom,
% 0.33/0.53      ( ord_less_eq_real
% 0.33/0.53      = ( ^ [X3: real,Y2: real] :
% 0.33/0.53            ( ( ord_less_real @ X3 @ Y2 )
% 0.33/0.53            | ( X3 = Y2 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less
% 0.33/0.53  thf(fact_716_order__le__less,axiom,
% 0.33/0.53      ( ord_less_eq_set_int
% 0.33/0.53      = ( ^ [X3: set_int,Y2: set_int] :
% 0.33/0.53            ( ( ord_less_set_int @ X3 @ Y2 )
% 0.33/0.53            | ( X3 = Y2 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less
% 0.33/0.53  thf(fact_717_order__le__less,axiom,
% 0.33/0.53      ( ord_less_eq_rat
% 0.33/0.53      = ( ^ [X3: rat,Y2: rat] :
% 0.33/0.53            ( ( ord_less_rat @ X3 @ Y2 )
% 0.33/0.53            | ( X3 = Y2 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less
% 0.33/0.53  thf(fact_718_order__le__less,axiom,
% 0.33/0.53      ( ord_less_eq_num
% 0.33/0.53      = ( ^ [X3: num,Y2: num] :
% 0.33/0.53            ( ( ord_less_num @ X3 @ Y2 )
% 0.33/0.53            | ( X3 = Y2 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less
% 0.33/0.53  thf(fact_719_order__le__less,axiom,
% 0.33/0.53      ( ord_less_eq_nat
% 0.33/0.53      = ( ^ [X3: nat,Y2: nat] :
% 0.33/0.53            ( ( ord_less_nat @ X3 @ Y2 )
% 0.33/0.53            | ( X3 = Y2 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less
% 0.33/0.53  thf(fact_720_order__le__less,axiom,
% 0.33/0.53      ( ord_less_eq_int
% 0.33/0.53      = ( ^ [X3: int,Y2: int] :
% 0.33/0.53            ( ( ord_less_int @ X3 @ Y2 )
% 0.33/0.53            | ( X3 = Y2 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less
% 0.33/0.53  thf(fact_721_order__less__le,axiom,
% 0.33/0.53      ( ord_less_real
% 0.33/0.53      = ( ^ [X3: real,Y2: real] :
% 0.33/0.53            ( ( ord_less_eq_real @ X3 @ Y2 )
% 0.33/0.53            & ( X3 != Y2 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_le
% 0.33/0.53  thf(fact_722_order__less__le,axiom,
% 0.33/0.53      ( ord_less_set_int
% 0.33/0.53      = ( ^ [X3: set_int,Y2: set_int] :
% 0.33/0.53            ( ( ord_less_eq_set_int @ X3 @ Y2 )
% 0.33/0.53            & ( X3 != Y2 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_le
% 0.33/0.53  thf(fact_723_order__less__le,axiom,
% 0.33/0.53      ( ord_less_rat
% 0.33/0.53      = ( ^ [X3: rat,Y2: rat] :
% 0.33/0.53            ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.33/0.53            & ( X3 != Y2 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_le
% 0.33/0.53  thf(fact_724_order__less__le,axiom,
% 0.33/0.53      ( ord_less_num
% 0.33/0.53      = ( ^ [X3: num,Y2: num] :
% 0.33/0.53            ( ( ord_less_eq_num @ X3 @ Y2 )
% 0.33/0.53            & ( X3 != Y2 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_le
% 0.33/0.53  thf(fact_725_order__less__le,axiom,
% 0.33/0.53      ( ord_less_nat
% 0.33/0.53      = ( ^ [X3: nat,Y2: nat] :
% 0.33/0.53            ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.33/0.53            & ( X3 != Y2 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_le
% 0.33/0.53  thf(fact_726_order__less__le,axiom,
% 0.33/0.53      ( ord_less_int
% 0.33/0.53      = ( ^ [X3: int,Y2: int] :
% 0.33/0.53            ( ( ord_less_eq_int @ X3 @ Y2 )
% 0.33/0.53            & ( X3 != Y2 ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_le
% 0.33/0.53  thf(fact_727_linorder__not__le,axiom,
% 0.33/0.53      ! [X: real,Y: real] :
% 0.33/0.53        ( ( ~ ( ord_less_eq_real @ X @ Y ) )
% 0.33/0.53        = ( ord_less_real @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_not_le
% 0.33/0.53  thf(fact_728_linorder__not__le,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ( ~ ( ord_less_eq_rat @ X @ Y ) )
% 0.33/0.53        = ( ord_less_rat @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_not_le
% 0.33/0.53  thf(fact_729_linorder__not__le,axiom,
% 0.33/0.53      ! [X: num,Y: num] :
% 0.33/0.53        ( ( ~ ( ord_less_eq_num @ X @ Y ) )
% 0.33/0.53        = ( ord_less_num @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_not_le
% 0.33/0.53  thf(fact_730_linorder__not__le,axiom,
% 0.33/0.53      ! [X: nat,Y: nat] :
% 0.33/0.53        ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
% 0.33/0.53        = ( ord_less_nat @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_not_le
% 0.33/0.53  thf(fact_731_linorder__not__le,axiom,
% 0.33/0.53      ! [X: int,Y: int] :
% 0.33/0.53        ( ( ~ ( ord_less_eq_int @ X @ Y ) )
% 0.33/0.53        = ( ord_less_int @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_not_le
% 0.33/0.53  thf(fact_732_linorder__not__less,axiom,
% 0.33/0.53      ! [X: real,Y: real] :
% 0.33/0.53        ( ( ~ ( ord_less_real @ X @ Y ) )
% 0.33/0.53        = ( ord_less_eq_real @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_not_less
% 0.33/0.53  thf(fact_733_linorder__not__less,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ( ~ ( ord_less_rat @ X @ Y ) )
% 0.33/0.53        = ( ord_less_eq_rat @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_not_less
% 0.33/0.53  thf(fact_734_linorder__not__less,axiom,
% 0.33/0.53      ! [X: num,Y: num] :
% 0.33/0.53        ( ( ~ ( ord_less_num @ X @ Y ) )
% 0.33/0.53        = ( ord_less_eq_num @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_not_less
% 0.33/0.53  thf(fact_735_linorder__not__less,axiom,
% 0.33/0.53      ! [X: nat,Y: nat] :
% 0.33/0.53        ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.33/0.53        = ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_not_less
% 0.33/0.53  thf(fact_736_linorder__not__less,axiom,
% 0.33/0.53      ! [X: int,Y: int] :
% 0.33/0.53        ( ( ~ ( ord_less_int @ X @ Y ) )
% 0.33/0.53        = ( ord_less_eq_int @ Y @ X ) ) ).
% 0.33/0.53  
% 0.33/0.53  % linorder_not_less
% 0.33/0.53  thf(fact_737_order__less__imp__le,axiom,
% 0.33/0.53      ! [X: real,Y: real] :
% 0.33/0.53        ( ( ord_less_real @ X @ Y )
% 0.33/0.53       => ( ord_less_eq_real @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_le
% 0.33/0.53  thf(fact_738_order__less__imp__le,axiom,
% 0.33/0.53      ! [X: set_int,Y: set_int] :
% 0.33/0.53        ( ( ord_less_set_int @ X @ Y )
% 0.33/0.53       => ( ord_less_eq_set_int @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_le
% 0.33/0.53  thf(fact_739_order__less__imp__le,axiom,
% 0.33/0.53      ! [X: rat,Y: rat] :
% 0.33/0.53        ( ( ord_less_rat @ X @ Y )
% 0.33/0.53       => ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_le
% 0.33/0.53  thf(fact_740_order__less__imp__le,axiom,
% 0.33/0.53      ! [X: num,Y: num] :
% 0.33/0.53        ( ( ord_less_num @ X @ Y )
% 0.33/0.53       => ( ord_less_eq_num @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_le
% 0.33/0.53  thf(fact_741_order__less__imp__le,axiom,
% 0.33/0.53      ! [X: nat,Y: nat] :
% 0.33/0.53        ( ( ord_less_nat @ X @ Y )
% 0.33/0.53       => ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_le
% 0.33/0.53  thf(fact_742_order__less__imp__le,axiom,
% 0.33/0.53      ! [X: int,Y: int] :
% 0.33/0.53        ( ( ord_less_int @ X @ Y )
% 0.33/0.53       => ( ord_less_eq_int @ X @ Y ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_imp_le
% 0.33/0.53  thf(fact_743_order__le__neq__trans,axiom,
% 0.33/0.53      ! [A: real,B: real] :
% 0.33/0.53        ( ( ord_less_eq_real @ A @ B )
% 0.33/0.53       => ( ( A != B )
% 0.33/0.53         => ( ord_less_real @ A @ B ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_neq_trans
% 0.33/0.53  thf(fact_744_order__le__neq__trans,axiom,
% 0.33/0.53      ! [A: set_int,B: set_int] :
% 0.33/0.53        ( ( ord_less_eq_set_int @ A @ B )
% 0.33/0.53       => ( ( A != B )
% 0.33/0.53         => ( ord_less_set_int @ A @ B ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_neq_trans
% 0.33/0.53  thf(fact_745_order__le__neq__trans,axiom,
% 0.33/0.53      ! [A: rat,B: rat] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.53       => ( ( A != B )
% 0.33/0.53         => ( ord_less_rat @ A @ B ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_neq_trans
% 0.33/0.53  thf(fact_746_order__le__neq__trans,axiom,
% 0.33/0.53      ! [A: num,B: num] :
% 0.33/0.53        ( ( ord_less_eq_num @ A @ B )
% 0.33/0.53       => ( ( A != B )
% 0.33/0.53         => ( ord_less_num @ A @ B ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_neq_trans
% 0.33/0.53  thf(fact_747_order__le__neq__trans,axiom,
% 0.33/0.53      ! [A: nat,B: nat] :
% 0.33/0.53        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.53       => ( ( A != B )
% 0.33/0.53         => ( ord_less_nat @ A @ B ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_neq_trans
% 0.33/0.53  thf(fact_748_order__le__neq__trans,axiom,
% 0.33/0.53      ! [A: int,B: int] :
% 0.33/0.53        ( ( ord_less_eq_int @ A @ B )
% 0.33/0.53       => ( ( A != B )
% 0.33/0.53         => ( ord_less_int @ A @ B ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_neq_trans
% 0.33/0.53  thf(fact_749_order__neq__le__trans,axiom,
% 0.33/0.53      ! [A: real,B: real] :
% 0.33/0.53        ( ( A != B )
% 0.33/0.53       => ( ( ord_less_eq_real @ A @ B )
% 0.33/0.53         => ( ord_less_real @ A @ B ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_neq_le_trans
% 0.33/0.53  thf(fact_750_order__neq__le__trans,axiom,
% 0.33/0.53      ! [A: set_int,B: set_int] :
% 0.33/0.53        ( ( A != B )
% 0.33/0.53       => ( ( ord_less_eq_set_int @ A @ B )
% 0.33/0.53         => ( ord_less_set_int @ A @ B ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_neq_le_trans
% 0.33/0.53  thf(fact_751_order__neq__le__trans,axiom,
% 0.33/0.53      ! [A: rat,B: rat] :
% 0.33/0.53        ( ( A != B )
% 0.33/0.53       => ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.53         => ( ord_less_rat @ A @ B ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_neq_le_trans
% 0.33/0.53  thf(fact_752_order__neq__le__trans,axiom,
% 0.33/0.53      ! [A: num,B: num] :
% 0.33/0.53        ( ( A != B )
% 0.33/0.53       => ( ( ord_less_eq_num @ A @ B )
% 0.33/0.53         => ( ord_less_num @ A @ B ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_neq_le_trans
% 0.33/0.53  thf(fact_753_order__neq__le__trans,axiom,
% 0.33/0.53      ! [A: nat,B: nat] :
% 0.33/0.53        ( ( A != B )
% 0.33/0.53       => ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.53         => ( ord_less_nat @ A @ B ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_neq_le_trans
% 0.33/0.53  thf(fact_754_order__neq__le__trans,axiom,
% 0.33/0.53      ! [A: int,B: int] :
% 0.33/0.53        ( ( A != B )
% 0.33/0.53       => ( ( ord_less_eq_int @ A @ B )
% 0.33/0.53         => ( ord_less_int @ A @ B ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_neq_le_trans
% 0.33/0.53  thf(fact_755_order__le__less__trans,axiom,
% 0.33/0.53      ! [X: real,Y: real,Z2: real] :
% 0.33/0.53        ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.53       => ( ( ord_less_real @ Y @ Z2 )
% 0.33/0.53         => ( ord_less_real @ X @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_trans
% 0.33/0.53  thf(fact_756_order__le__less__trans,axiom,
% 0.33/0.53      ! [X: set_int,Y: set_int,Z2: set_int] :
% 0.33/0.53        ( ( ord_less_eq_set_int @ X @ Y )
% 0.33/0.53       => ( ( ord_less_set_int @ Y @ Z2 )
% 0.33/0.53         => ( ord_less_set_int @ X @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_trans
% 0.33/0.53  thf(fact_757_order__le__less__trans,axiom,
% 0.33/0.53      ! [X: rat,Y: rat,Z2: rat] :
% 0.33/0.53        ( ( ord_less_eq_rat @ X @ Y )
% 0.33/0.53       => ( ( ord_less_rat @ Y @ Z2 )
% 0.33/0.53         => ( ord_less_rat @ X @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_trans
% 0.33/0.53  thf(fact_758_order__le__less__trans,axiom,
% 0.33/0.53      ! [X: num,Y: num,Z2: num] :
% 0.33/0.53        ( ( ord_less_eq_num @ X @ Y )
% 0.33/0.53       => ( ( ord_less_num @ Y @ Z2 )
% 0.33/0.53         => ( ord_less_num @ X @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_trans
% 0.33/0.53  thf(fact_759_order__le__less__trans,axiom,
% 0.33/0.53      ! [X: nat,Y: nat,Z2: nat] :
% 0.33/0.53        ( ( ord_less_eq_nat @ X @ Y )
% 0.33/0.53       => ( ( ord_less_nat @ Y @ Z2 )
% 0.33/0.53         => ( ord_less_nat @ X @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_trans
% 0.33/0.53  thf(fact_760_order__le__less__trans,axiom,
% 0.33/0.53      ! [X: int,Y: int,Z2: int] :
% 0.33/0.53        ( ( ord_less_eq_int @ X @ Y )
% 0.33/0.53       => ( ( ord_less_int @ Y @ Z2 )
% 0.33/0.53         => ( ord_less_int @ X @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_trans
% 0.33/0.53  thf(fact_761_order__less__le__trans,axiom,
% 0.33/0.53      ! [X: real,Y: real,Z2: real] :
% 0.33/0.53        ( ( ord_less_real @ X @ Y )
% 0.33/0.53       => ( ( ord_less_eq_real @ Y @ Z2 )
% 0.33/0.53         => ( ord_less_real @ X @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_le_trans
% 0.33/0.53  thf(fact_762_order__less__le__trans,axiom,
% 0.33/0.53      ! [X: set_int,Y: set_int,Z2: set_int] :
% 0.33/0.53        ( ( ord_less_set_int @ X @ Y )
% 0.33/0.53       => ( ( ord_less_eq_set_int @ Y @ Z2 )
% 0.33/0.53         => ( ord_less_set_int @ X @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_le_trans
% 0.33/0.53  thf(fact_763_order__less__le__trans,axiom,
% 0.33/0.53      ! [X: rat,Y: rat,Z2: rat] :
% 0.33/0.53        ( ( ord_less_rat @ X @ Y )
% 0.33/0.53       => ( ( ord_less_eq_rat @ Y @ Z2 )
% 0.33/0.53         => ( ord_less_rat @ X @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_le_trans
% 0.33/0.53  thf(fact_764_order__less__le__trans,axiom,
% 0.33/0.53      ! [X: num,Y: num,Z2: num] :
% 0.33/0.53        ( ( ord_less_num @ X @ Y )
% 0.33/0.53       => ( ( ord_less_eq_num @ Y @ Z2 )
% 0.33/0.53         => ( ord_less_num @ X @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_le_trans
% 0.33/0.53  thf(fact_765_order__less__le__trans,axiom,
% 0.33/0.53      ! [X: nat,Y: nat,Z2: nat] :
% 0.33/0.53        ( ( ord_less_nat @ X @ Y )
% 0.33/0.53       => ( ( ord_less_eq_nat @ Y @ Z2 )
% 0.33/0.53         => ( ord_less_nat @ X @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_le_trans
% 0.33/0.53  thf(fact_766_order__less__le__trans,axiom,
% 0.33/0.53      ! [X: int,Y: int,Z2: int] :
% 0.33/0.53        ( ( ord_less_int @ X @ Y )
% 0.33/0.53       => ( ( ord_less_eq_int @ Y @ Z2 )
% 0.33/0.53         => ( ord_less_int @ X @ Z2 ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_le_trans
% 0.33/0.53  thf(fact_767_order__le__less__subst1,axiom,
% 0.33/0.53      ! [A: real,F: real > real,B: real,C: real] :
% 0.33/0.53        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_real @ B @ C )
% 0.33/0.53         => ( ! [X4: real,Y3: real] :
% 0.33/0.53                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_subst1
% 0.33/0.53  thf(fact_768_order__le__less__subst1,axiom,
% 0.33/0.53      ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.33/0.53        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_subst1
% 0.33/0.53  thf(fact_769_order__le__less__subst1,axiom,
% 0.33/0.53      ! [A: real,F: num > real,B: num,C: num] :
% 0.33/0.53        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_num @ B @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_subst1
% 0.33/0.53  thf(fact_770_order__le__less__subst1,axiom,
% 0.33/0.53      ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.33/0.53        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_nat @ B @ C )
% 0.33/0.53         => ( ! [X4: nat,Y3: nat] :
% 0.33/0.53                ( ( ord_less_nat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_subst1
% 0.33/0.53  thf(fact_771_order__le__less__subst1,axiom,
% 0.33/0.53      ! [A: real,F: int > real,B: int,C: int] :
% 0.33/0.53        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_int @ B @ C )
% 0.33/0.53         => ( ! [X4: int,Y3: int] :
% 0.33/0.53                ( ( ord_less_int @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_subst1
% 0.33/0.53  thf(fact_772_order__le__less__subst1,axiom,
% 0.33/0.53      ! [A: rat,F: real > rat,B: real,C: real] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_real @ B @ C )
% 0.33/0.53         => ( ! [X4: real,Y3: real] :
% 0.33/0.53                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_subst1
% 0.33/0.53  thf(fact_773_order__le__less__subst1,axiom,
% 0.33/0.53      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_subst1
% 0.33/0.53  thf(fact_774_order__le__less__subst1,axiom,
% 0.33/0.53      ! [A: rat,F: num > rat,B: num,C: num] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_num @ B @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_subst1
% 0.33/0.53  thf(fact_775_order__le__less__subst1,axiom,
% 0.33/0.53      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_nat @ B @ C )
% 0.33/0.53         => ( ! [X4: nat,Y3: nat] :
% 0.33/0.53                ( ( ord_less_nat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_subst1
% 0.33/0.53  thf(fact_776_order__le__less__subst1,axiom,
% 0.33/0.53      ! [A: rat,F: int > rat,B: int,C: int] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_int @ B @ C )
% 0.33/0.53         => ( ! [X4: int,Y3: int] :
% 0.33/0.53                ( ( ord_less_int @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_subst1
% 0.33/0.53  thf(fact_777_order__le__less__subst2,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.53       => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_subst2
% 0.33/0.53  thf(fact_778_order__le__less__subst2,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.53       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_subst2
% 0.33/0.53  thf(fact_779_order__le__less__subst2,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.53       => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_subst2
% 0.33/0.53  thf(fact_780_order__le__less__subst2,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.53       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_subst2
% 0.33/0.53  thf(fact_781_order__le__less__subst2,axiom,
% 0.33/0.53      ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.33/0.53        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.53       => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_subst2
% 0.33/0.53  thf(fact_782_order__le__less__subst2,axiom,
% 0.33/0.53      ! [A: num,B: num,F: num > real,C: real] :
% 0.33/0.53        ( ( ord_less_eq_num @ A @ B )
% 0.33/0.53       => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_subst2
% 0.33/0.53  thf(fact_783_order__le__less__subst2,axiom,
% 0.33/0.53      ! [A: num,B: num,F: num > rat,C: rat] :
% 0.33/0.53        ( ( ord_less_eq_num @ A @ B )
% 0.33/0.53       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_subst2
% 0.33/0.53  thf(fact_784_order__le__less__subst2,axiom,
% 0.33/0.53      ! [A: num,B: num,F: num > num,C: num] :
% 0.33/0.53        ( ( ord_less_eq_num @ A @ B )
% 0.33/0.53       => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_subst2
% 0.33/0.53  thf(fact_785_order__le__less__subst2,axiom,
% 0.33/0.53      ! [A: num,B: num,F: num > nat,C: nat] :
% 0.33/0.53        ( ( ord_less_eq_num @ A @ B )
% 0.33/0.53       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_subst2
% 0.33/0.53  thf(fact_786_order__le__less__subst2,axiom,
% 0.33/0.53      ! [A: num,B: num,F: num > int,C: int] :
% 0.33/0.53        ( ( ord_less_eq_num @ A @ B )
% 0.33/0.53       => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_le_less_subst2
% 0.33/0.53  thf(fact_787_order__less__le__subst1,axiom,
% 0.33/0.53      ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.33/0.53        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_le_subst1
% 0.33/0.53  thf(fact_788_order__less__le__subst1,axiom,
% 0.33/0.53      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.33/0.53        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_le_subst1
% 0.33/0.53  thf(fact_789_order__less__le__subst1,axiom,
% 0.33/0.53      ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.33/0.53        ( ( ord_less_num @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_le_subst1
% 0.33/0.53  thf(fact_790_order__less__le__subst1,axiom,
% 0.33/0.53      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.33/0.53        ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_le_subst1
% 0.33/0.53  thf(fact_791_order__less__le__subst1,axiom,
% 0.33/0.53      ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.33/0.53        ( ( ord_less_int @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_rat @ B @ C )
% 0.33/0.53         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.53                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_le_subst1
% 0.33/0.53  thf(fact_792_order__less__le__subst1,axiom,
% 0.33/0.53      ! [A: real,F: num > real,B: num,C: num] :
% 0.33/0.53        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_num @ B @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_le_subst1
% 0.33/0.53  thf(fact_793_order__less__le__subst1,axiom,
% 0.33/0.53      ! [A: rat,F: num > rat,B: num,C: num] :
% 0.33/0.53        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_num @ B @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_le_subst1
% 0.33/0.53  thf(fact_794_order__less__le__subst1,axiom,
% 0.33/0.53      ! [A: num,F: num > num,B: num,C: num] :
% 0.33/0.53        ( ( ord_less_num @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_num @ B @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.53               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.53           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.53  
% 0.33/0.53  % order_less_le_subst1
% 0.33/0.53  thf(fact_795_order__less__le__subst1,axiom,
% 0.33/0.53      ! [A: nat,F: num > nat,B: num,C: num] :
% 0.33/0.53        ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.33/0.53       => ( ( ord_less_eq_num @ B @ C )
% 0.33/0.53         => ( ! [X4: num,Y3: num] :
% 0.33/0.53                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.54               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.54           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % order_less_le_subst1
% 0.33/0.54  thf(fact_796_order__less__le__subst1,axiom,
% 0.33/0.54      ! [A: int,F: num > int,B: num,C: num] :
% 0.33/0.54        ( ( ord_less_int @ A @ ( F @ B ) )
% 0.33/0.54       => ( ( ord_less_eq_num @ B @ C )
% 0.33/0.54         => ( ! [X4: num,Y3: num] :
% 0.33/0.54                ( ( ord_less_eq_num @ X4 @ Y3 )
% 0.33/0.54               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.54           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % order_less_le_subst1
% 0.33/0.54  thf(fact_797_order__less__le__subst2,axiom,
% 0.33/0.54      ! [A: real,B: real,F: real > real,C: real] :
% 0.33/0.54        ( ( ord_less_real @ A @ B )
% 0.33/0.54       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.33/0.54         => ( ! [X4: real,Y3: real] :
% 0.33/0.54                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.54               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.54           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % order_less_le_subst2
% 0.33/0.54  thf(fact_798_order__less__le__subst2,axiom,
% 0.33/0.54      ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.33/0.54        ( ( ord_less_rat @ A @ B )
% 0.33/0.54       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.33/0.54         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.54                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.54               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.54           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % order_less_le_subst2
% 0.33/0.54  thf(fact_799_order__less__le__subst2,axiom,
% 0.33/0.54      ! [A: num,B: num,F: num > real,C: real] :
% 0.33/0.54        ( ( ord_less_num @ A @ B )
% 0.33/0.54       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.33/0.54         => ( ! [X4: num,Y3: num] :
% 0.33/0.54                ( ( ord_less_num @ X4 @ Y3 )
% 0.33/0.54               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.54           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % order_less_le_subst2
% 0.33/0.54  thf(fact_800_order__less__le__subst2,axiom,
% 0.33/0.54      ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.33/0.54        ( ( ord_less_nat @ A @ B )
% 0.33/0.54       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.33/0.54         => ( ! [X4: nat,Y3: nat] :
% 0.33/0.54                ( ( ord_less_nat @ X4 @ Y3 )
% 0.33/0.54               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.54           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % order_less_le_subst2
% 0.33/0.54  thf(fact_801_order__less__le__subst2,axiom,
% 0.33/0.54      ! [A: int,B: int,F: int > real,C: real] :
% 0.33/0.54        ( ( ord_less_int @ A @ B )
% 0.33/0.54       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.33/0.54         => ( ! [X4: int,Y3: int] :
% 0.33/0.54                ( ( ord_less_int @ X4 @ Y3 )
% 0.33/0.54               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.54           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % order_less_le_subst2
% 0.33/0.54  thf(fact_802_order__less__le__subst2,axiom,
% 0.33/0.54      ! [A: real,B: real,F: real > rat,C: rat] :
% 0.33/0.54        ( ( ord_less_real @ A @ B )
% 0.33/0.54       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.33/0.54         => ( ! [X4: real,Y3: real] :
% 0.33/0.54                ( ( ord_less_real @ X4 @ Y3 )
% 0.33/0.54               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.54           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % order_less_le_subst2
% 0.33/0.54  thf(fact_803_order__less__le__subst2,axiom,
% 0.33/0.54      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.33/0.54        ( ( ord_less_rat @ A @ B )
% 0.33/0.54       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.33/0.54         => ( ! [X4: rat,Y3: rat] :
% 0.33/0.54                ( ( ord_less_rat @ X4 @ Y3 )
% 0.33/0.54               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.54           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % order_less_le_subst2
% 0.33/0.54  thf(fact_804_order__less__le__subst2,axiom,
% 0.33/0.54      ! [A: num,B: num,F: num > rat,C: rat] :
% 0.33/0.54        ( ( ord_less_num @ A @ B )
% 0.33/0.54       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.33/0.54         => ( ! [X4: num,Y3: num] :
% 0.33/0.54                ( ( ord_less_num @ X4 @ Y3 )
% 0.33/0.54               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.54           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % order_less_le_subst2
% 0.33/0.54  thf(fact_805_order__less__le__subst2,axiom,
% 0.33/0.54      ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 0.33/0.54        ( ( ord_less_nat @ A @ B )
% 0.33/0.54       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.33/0.54         => ( ! [X4: nat,Y3: nat] :
% 0.33/0.54                ( ( ord_less_nat @ X4 @ Y3 )
% 0.33/0.54               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.54           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % order_less_le_subst2
% 0.33/0.54  thf(fact_806_order__less__le__subst2,axiom,
% 0.33/0.54      ! [A: int,B: int,F: int > rat,C: rat] :
% 0.33/0.54        ( ( ord_less_int @ A @ B )
% 0.33/0.54       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.33/0.54         => ( ! [X4: int,Y3: int] :
% 0.33/0.54                ( ( ord_less_int @ X4 @ Y3 )
% 0.33/0.54               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 0.33/0.54           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % order_less_le_subst2
% 0.33/0.54  thf(fact_807_linorder__le__less__linear,axiom,
% 0.33/0.54      ! [X: real,Y: real] :
% 0.33/0.54        ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.54        | ( ord_less_real @ Y @ X ) ) ).
% 0.33/0.54  
% 0.33/0.54  % linorder_le_less_linear
% 0.33/0.54  thf(fact_808_linorder__le__less__linear,axiom,
% 0.33/0.54      ! [X: rat,Y: rat] :
% 0.33/0.54        ( ( ord_less_eq_rat @ X @ Y )
% 0.33/0.54        | ( ord_less_rat @ Y @ X ) ) ).
% 0.33/0.54  
% 0.33/0.54  % linorder_le_less_linear
% 0.33/0.54  thf(fact_809_linorder__le__less__linear,axiom,
% 0.33/0.54      ! [X: num,Y: num] :
% 0.33/0.54        ( ( ord_less_eq_num @ X @ Y )
% 0.33/0.54        | ( ord_less_num @ Y @ X ) ) ).
% 0.33/0.54  
% 0.33/0.54  % linorder_le_less_linear
% 0.33/0.54  thf(fact_810_linorder__le__less__linear,axiom,
% 0.33/0.54      ! [X: nat,Y: nat] :
% 0.33/0.54        ( ( ord_less_eq_nat @ X @ Y )
% 0.33/0.54        | ( ord_less_nat @ Y @ X ) ) ).
% 0.33/0.54  
% 0.33/0.54  % linorder_le_less_linear
% 0.33/0.54  thf(fact_811_linorder__le__less__linear,axiom,
% 0.33/0.54      ! [X: int,Y: int] :
% 0.33/0.54        ( ( ord_less_eq_int @ X @ Y )
% 0.33/0.54        | ( ord_less_int @ Y @ X ) ) ).
% 0.33/0.54  
% 0.33/0.54  % linorder_le_less_linear
% 0.33/0.54  thf(fact_812_order__le__imp__less__or__eq,axiom,
% 0.33/0.54      ! [X: real,Y: real] :
% 0.33/0.54        ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.54       => ( ( ord_less_real @ X @ Y )
% 0.33/0.54          | ( X = Y ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % order_le_imp_less_or_eq
% 0.33/0.54  thf(fact_813_order__le__imp__less__or__eq,axiom,
% 0.33/0.54      ! [X: set_int,Y: set_int] :
% 0.33/0.54        ( ( ord_less_eq_set_int @ X @ Y )
% 0.33/0.54       => ( ( ord_less_set_int @ X @ Y )
% 0.33/0.54          | ( X = Y ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % order_le_imp_less_or_eq
% 0.33/0.54  thf(fact_814_order__le__imp__less__or__eq,axiom,
% 0.33/0.54      ! [X: rat,Y: rat] :
% 0.33/0.54        ( ( ord_less_eq_rat @ X @ Y )
% 0.33/0.54       => ( ( ord_less_rat @ X @ Y )
% 0.33/0.54          | ( X = Y ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % order_le_imp_less_or_eq
% 0.33/0.54  thf(fact_815_order__le__imp__less__or__eq,axiom,
% 0.33/0.54      ! [X: num,Y: num] :
% 0.33/0.54        ( ( ord_less_eq_num @ X @ Y )
% 0.33/0.54       => ( ( ord_less_num @ X @ Y )
% 0.33/0.54          | ( X = Y ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % order_le_imp_less_or_eq
% 0.33/0.54  thf(fact_816_order__le__imp__less__or__eq,axiom,
% 0.33/0.54      ! [X: nat,Y: nat] :
% 0.33/0.54        ( ( ord_less_eq_nat @ X @ Y )
% 0.33/0.54       => ( ( ord_less_nat @ X @ Y )
% 0.33/0.54          | ( X = Y ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % order_le_imp_less_or_eq
% 0.33/0.54  thf(fact_817_order__le__imp__less__or__eq,axiom,
% 0.33/0.54      ! [X: int,Y: int] :
% 0.33/0.54        ( ( ord_less_eq_int @ X @ Y )
% 0.33/0.54       => ( ( ord_less_int @ X @ Y )
% 0.33/0.54          | ( X = Y ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % order_le_imp_less_or_eq
% 0.33/0.54  thf(fact_818_greater__shift,axiom,
% 0.33/0.54      ( ord_less_nat
% 0.33/0.54      = ( ^ [Y2: nat,X3: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X3 ) @ ( some_nat @ Y2 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % greater_shift
% 0.33/0.54  thf(fact_819_less__shift,axiom,
% 0.33/0.54      ( ord_less_nat
% 0.33/0.54      = ( ^ [X3: nat,Y2: nat] : ( vEBT_VEBT_less @ ( some_nat @ X3 ) @ ( some_nat @ Y2 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % less_shift
% 0.33/0.54  thf(fact_820_mint__sound,axiom,
% 0.33/0.54      ! [T: vEBT_VEBT,N: nat,X: nat] :
% 0.33/0.54        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.54       => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
% 0.33/0.54         => ( ( vEBT_vebt_mint @ T )
% 0.33/0.54            = ( some_nat @ X ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % mint_sound
% 0.33/0.54  thf(fact_821_mint__corr,axiom,
% 0.33/0.54      ! [T: vEBT_VEBT,N: nat,X: nat] :
% 0.33/0.54        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.54       => ( ( ( vEBT_vebt_mint @ T )
% 0.33/0.54            = ( some_nat @ X ) )
% 0.33/0.54         => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % mint_corr
% 0.33/0.54  thf(fact_822_maxt__corr__help__empty,axiom,
% 0.33/0.54      ! [T: vEBT_VEBT,N: nat] :
% 0.33/0.54        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.54       => ( ( ( vEBT_vebt_maxt @ T )
% 0.33/0.54            = none_nat )
% 0.33/0.54         => ( ( vEBT_VEBT_set_vebt @ T )
% 0.33/0.54            = bot_bot_set_nat ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % maxt_corr_help_empty
% 0.33/0.54  thf(fact_823_mint__corr__help,axiom,
% 0.33/0.54      ! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
% 0.33/0.54        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.54       => ( ( ( vEBT_vebt_mint @ T )
% 0.33/0.54            = ( some_nat @ Mini ) )
% 0.33/0.54         => ( ( vEBT_vebt_member @ T @ X )
% 0.33/0.54           => ( ord_less_eq_nat @ Mini @ X ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % mint_corr_help
% 0.33/0.54  thf(fact_824_infinite__growing,axiom,
% 0.33/0.54      ! [X8: set_real] :
% 0.33/0.54        ( ( X8 != bot_bot_set_real )
% 0.33/0.54       => ( ! [X4: real] :
% 0.33/0.54              ( ( member_real @ X4 @ X8 )
% 0.33/0.54             => ? [Xa: real] :
% 0.33/0.54                  ( ( member_real @ Xa @ X8 )
% 0.33/0.54                  & ( ord_less_real @ X4 @ Xa ) ) )
% 0.33/0.54         => ~ ( finite_finite_real @ X8 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % infinite_growing
% 0.33/0.54  thf(fact_825_infinite__growing,axiom,
% 0.33/0.54      ! [X8: set_rat] :
% 0.33/0.54        ( ( X8 != bot_bot_set_rat )
% 0.33/0.54       => ( ! [X4: rat] :
% 0.33/0.54              ( ( member_rat @ X4 @ X8 )
% 0.33/0.54             => ? [Xa: rat] :
% 0.33/0.54                  ( ( member_rat @ Xa @ X8 )
% 0.33/0.54                  & ( ord_less_rat @ X4 @ Xa ) ) )
% 0.33/0.54         => ~ ( finite_finite_rat @ X8 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % infinite_growing
% 0.33/0.54  thf(fact_826_infinite__growing,axiom,
% 0.33/0.54      ! [X8: set_num] :
% 0.33/0.54        ( ( X8 != bot_bot_set_num )
% 0.33/0.54       => ( ! [X4: num] :
% 0.33/0.54              ( ( member_num @ X4 @ X8 )
% 0.33/0.54             => ? [Xa: num] :
% 0.33/0.54                  ( ( member_num @ Xa @ X8 )
% 0.33/0.54                  & ( ord_less_num @ X4 @ Xa ) ) )
% 0.33/0.54         => ~ ( finite_finite_num @ X8 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % infinite_growing
% 0.33/0.54  thf(fact_827_infinite__growing,axiom,
% 0.33/0.54      ! [X8: set_nat] :
% 0.33/0.54        ( ( X8 != bot_bot_set_nat )
% 0.33/0.54       => ( ! [X4: nat] :
% 0.33/0.54              ( ( member_nat @ X4 @ X8 )
% 0.33/0.54             => ? [Xa: nat] :
% 0.33/0.54                  ( ( member_nat @ Xa @ X8 )
% 0.33/0.54                  & ( ord_less_nat @ X4 @ Xa ) ) )
% 0.33/0.54         => ~ ( finite_finite_nat @ X8 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % infinite_growing
% 0.33/0.54  thf(fact_828_infinite__growing,axiom,
% 0.33/0.54      ! [X8: set_int] :
% 0.33/0.54        ( ( X8 != bot_bot_set_int )
% 0.33/0.54       => ( ! [X4: int] :
% 0.33/0.54              ( ( member_int @ X4 @ X8 )
% 0.33/0.54             => ? [Xa: int] :
% 0.33/0.54                  ( ( member_int @ Xa @ X8 )
% 0.33/0.54                  & ( ord_less_int @ X4 @ Xa ) ) )
% 0.33/0.54         => ~ ( finite_finite_int @ X8 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % infinite_growing
% 0.33/0.54  thf(fact_829_ex__min__if__finite,axiom,
% 0.33/0.54      ! [S2: set_real] :
% 0.33/0.54        ( ( finite_finite_real @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_real )
% 0.33/0.54         => ? [X4: real] :
% 0.33/0.54              ( ( member_real @ X4 @ S2 )
% 0.33/0.54              & ~ ? [Xa: real] :
% 0.33/0.54                    ( ( member_real @ Xa @ S2 )
% 0.33/0.54                    & ( ord_less_real @ Xa @ X4 ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % ex_min_if_finite
% 0.33/0.54  thf(fact_830_ex__min__if__finite,axiom,
% 0.33/0.54      ! [S2: set_rat] :
% 0.33/0.54        ( ( finite_finite_rat @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_rat )
% 0.33/0.54         => ? [X4: rat] :
% 0.33/0.54              ( ( member_rat @ X4 @ S2 )
% 0.33/0.54              & ~ ? [Xa: rat] :
% 0.33/0.54                    ( ( member_rat @ Xa @ S2 )
% 0.33/0.54                    & ( ord_less_rat @ Xa @ X4 ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % ex_min_if_finite
% 0.33/0.54  thf(fact_831_ex__min__if__finite,axiom,
% 0.33/0.54      ! [S2: set_num] :
% 0.33/0.54        ( ( finite_finite_num @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_num )
% 0.33/0.54         => ? [X4: num] :
% 0.33/0.54              ( ( member_num @ X4 @ S2 )
% 0.33/0.54              & ~ ? [Xa: num] :
% 0.33/0.54                    ( ( member_num @ Xa @ S2 )
% 0.33/0.54                    & ( ord_less_num @ Xa @ X4 ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % ex_min_if_finite
% 0.33/0.54  thf(fact_832_ex__min__if__finite,axiom,
% 0.33/0.54      ! [S2: set_nat] :
% 0.33/0.54        ( ( finite_finite_nat @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_nat )
% 0.33/0.54         => ? [X4: nat] :
% 0.33/0.54              ( ( member_nat @ X4 @ S2 )
% 0.33/0.54              & ~ ? [Xa: nat] :
% 0.33/0.54                    ( ( member_nat @ Xa @ S2 )
% 0.33/0.54                    & ( ord_less_nat @ Xa @ X4 ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % ex_min_if_finite
% 0.33/0.54  thf(fact_833_ex__min__if__finite,axiom,
% 0.33/0.54      ! [S2: set_int] :
% 0.33/0.54        ( ( finite_finite_int @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_int )
% 0.33/0.54         => ? [X4: int] :
% 0.33/0.54              ( ( member_int @ X4 @ S2 )
% 0.33/0.54              & ~ ? [Xa: int] :
% 0.33/0.54                    ( ( member_int @ Xa @ S2 )
% 0.33/0.54                    & ( ord_less_int @ Xa @ X4 ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % ex_min_if_finite
% 0.33/0.54  thf(fact_834_finite__has__minimal,axiom,
% 0.33/0.54      ! [A2: set_real] :
% 0.33/0.54        ( ( finite_finite_real @ A2 )
% 0.33/0.54       => ( ( A2 != bot_bot_set_real )
% 0.33/0.54         => ? [X4: real] :
% 0.33/0.54              ( ( member_real @ X4 @ A2 )
% 0.33/0.54              & ! [Xa: real] :
% 0.33/0.54                  ( ( member_real @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_real @ Xa @ X4 )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_minimal
% 0.33/0.54  thf(fact_835_finite__has__minimal,axiom,
% 0.33/0.54      ! [A2: set_set_int] :
% 0.33/0.54        ( ( finite6197958912794628473et_int @ A2 )
% 0.33/0.54       => ( ( A2 != bot_bot_set_set_int )
% 0.33/0.54         => ? [X4: set_int] :
% 0.33/0.54              ( ( member_set_int @ X4 @ A2 )
% 0.33/0.54              & ! [Xa: set_int] :
% 0.33/0.54                  ( ( member_set_int @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_set_int @ Xa @ X4 )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_minimal
% 0.33/0.54  thf(fact_836_finite__has__minimal,axiom,
% 0.33/0.54      ! [A2: set_rat] :
% 0.33/0.54        ( ( finite_finite_rat @ A2 )
% 0.33/0.54       => ( ( A2 != bot_bot_set_rat )
% 0.33/0.54         => ? [X4: rat] :
% 0.33/0.54              ( ( member_rat @ X4 @ A2 )
% 0.33/0.54              & ! [Xa: rat] :
% 0.33/0.54                  ( ( member_rat @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_rat @ Xa @ X4 )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_minimal
% 0.33/0.54  thf(fact_837_finite__has__minimal,axiom,
% 0.33/0.54      ! [A2: set_num] :
% 0.33/0.54        ( ( finite_finite_num @ A2 )
% 0.33/0.54       => ( ( A2 != bot_bot_set_num )
% 0.33/0.54         => ? [X4: num] :
% 0.33/0.54              ( ( member_num @ X4 @ A2 )
% 0.33/0.54              & ! [Xa: num] :
% 0.33/0.54                  ( ( member_num @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_num @ Xa @ X4 )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_minimal
% 0.33/0.54  thf(fact_838_finite__has__minimal,axiom,
% 0.33/0.54      ! [A2: set_nat] :
% 0.33/0.54        ( ( finite_finite_nat @ A2 )
% 0.33/0.54       => ( ( A2 != bot_bot_set_nat )
% 0.33/0.54         => ? [X4: nat] :
% 0.33/0.54              ( ( member_nat @ X4 @ A2 )
% 0.33/0.54              & ! [Xa: nat] :
% 0.33/0.54                  ( ( member_nat @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_nat @ Xa @ X4 )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_minimal
% 0.33/0.54  thf(fact_839_finite__has__minimal,axiom,
% 0.33/0.54      ! [A2: set_int] :
% 0.33/0.54        ( ( finite_finite_int @ A2 )
% 0.33/0.54       => ( ( A2 != bot_bot_set_int )
% 0.33/0.54         => ? [X4: int] :
% 0.33/0.54              ( ( member_int @ X4 @ A2 )
% 0.33/0.54              & ! [Xa: int] :
% 0.33/0.54                  ( ( member_int @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_int @ Xa @ X4 )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_minimal
% 0.33/0.54  thf(fact_840_finite__has__maximal,axiom,
% 0.33/0.54      ! [A2: set_real] :
% 0.33/0.54        ( ( finite_finite_real @ A2 )
% 0.33/0.54       => ( ( A2 != bot_bot_set_real )
% 0.33/0.54         => ? [X4: real] :
% 0.33/0.54              ( ( member_real @ X4 @ A2 )
% 0.33/0.54              & ! [Xa: real] :
% 0.33/0.54                  ( ( member_real @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_real @ X4 @ Xa )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_maximal
% 0.33/0.54  thf(fact_841_finite__has__maximal,axiom,
% 0.33/0.54      ! [A2: set_set_int] :
% 0.33/0.54        ( ( finite6197958912794628473et_int @ A2 )
% 0.33/0.54       => ( ( A2 != bot_bot_set_set_int )
% 0.33/0.54         => ? [X4: set_int] :
% 0.33/0.54              ( ( member_set_int @ X4 @ A2 )
% 0.33/0.54              & ! [Xa: set_int] :
% 0.33/0.54                  ( ( member_set_int @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_set_int @ X4 @ Xa )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_maximal
% 0.33/0.54  thf(fact_842_finite__has__maximal,axiom,
% 0.33/0.54      ! [A2: set_rat] :
% 0.33/0.54        ( ( finite_finite_rat @ A2 )
% 0.33/0.54       => ( ( A2 != bot_bot_set_rat )
% 0.33/0.54         => ? [X4: rat] :
% 0.33/0.54              ( ( member_rat @ X4 @ A2 )
% 0.33/0.54              & ! [Xa: rat] :
% 0.33/0.54                  ( ( member_rat @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_rat @ X4 @ Xa )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_maximal
% 0.33/0.54  thf(fact_843_finite__has__maximal,axiom,
% 0.33/0.54      ! [A2: set_num] :
% 0.33/0.54        ( ( finite_finite_num @ A2 )
% 0.33/0.54       => ( ( A2 != bot_bot_set_num )
% 0.33/0.54         => ? [X4: num] :
% 0.33/0.54              ( ( member_num @ X4 @ A2 )
% 0.33/0.54              & ! [Xa: num] :
% 0.33/0.54                  ( ( member_num @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_num @ X4 @ Xa )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_maximal
% 0.33/0.54  thf(fact_844_finite__has__maximal,axiom,
% 0.33/0.54      ! [A2: set_nat] :
% 0.33/0.54        ( ( finite_finite_nat @ A2 )
% 0.33/0.54       => ( ( A2 != bot_bot_set_nat )
% 0.33/0.54         => ? [X4: nat] :
% 0.33/0.54              ( ( member_nat @ X4 @ A2 )
% 0.33/0.54              & ! [Xa: nat] :
% 0.33/0.54                  ( ( member_nat @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_nat @ X4 @ Xa )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_maximal
% 0.33/0.54  thf(fact_845_finite__has__maximal,axiom,
% 0.33/0.54      ! [A2: set_int] :
% 0.33/0.54        ( ( finite_finite_int @ A2 )
% 0.33/0.54       => ( ( A2 != bot_bot_set_int )
% 0.33/0.54         => ? [X4: int] :
% 0.33/0.54              ( ( member_int @ X4 @ A2 )
% 0.33/0.54              & ! [Xa: int] :
% 0.33/0.54                  ( ( member_int @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_int @ X4 @ Xa )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_maximal
% 0.33/0.54  thf(fact_846_mint__member,axiom,
% 0.33/0.54      ! [T: vEBT_VEBT,N: nat,Maxi2: nat] :
% 0.33/0.54        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.54       => ( ( ( vEBT_vebt_mint @ T )
% 0.33/0.54            = ( some_nat @ Maxi2 ) )
% 0.33/0.54         => ( vEBT_vebt_member @ T @ Maxi2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % mint_member
% 0.33/0.54  thf(fact_847_option_Oinject,axiom,
% 0.33/0.54      ! [X22: nat,Y22: nat] :
% 0.33/0.54        ( ( ( some_nat @ X22 )
% 0.33/0.54          = ( some_nat @ Y22 ) )
% 0.33/0.54        = ( X22 = Y22 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % option.inject
% 0.33/0.54  thf(fact_848_option_Oinject,axiom,
% 0.33/0.54      ! [X22: product_prod_nat_nat,Y22: product_prod_nat_nat] :
% 0.33/0.54        ( ( ( some_P7363390416028606310at_nat @ X22 )
% 0.33/0.54          = ( some_P7363390416028606310at_nat @ Y22 ) )
% 0.33/0.54        = ( X22 = Y22 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % option.inject
% 0.33/0.54  thf(fact_849_option_Oinject,axiom,
% 0.33/0.54      ! [X22: num,Y22: num] :
% 0.33/0.54        ( ( ( some_num @ X22 )
% 0.33/0.54          = ( some_num @ Y22 ) )
% 0.33/0.54        = ( X22 = Y22 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % option.inject
% 0.33/0.54  thf(fact_850_minminNull,axiom,
% 0.33/0.54      ! [T: vEBT_VEBT] :
% 0.33/0.54        ( ( ( vEBT_vebt_mint @ T )
% 0.33/0.54          = none_nat )
% 0.33/0.54       => ( vEBT_VEBT_minNull @ T ) ) ).
% 0.33/0.54  
% 0.33/0.54  % minminNull
% 0.33/0.54  thf(fact_851_minNullmin,axiom,
% 0.33/0.54      ! [T: vEBT_VEBT] :
% 0.33/0.54        ( ( vEBT_VEBT_minNull @ T )
% 0.33/0.54       => ( ( vEBT_vebt_mint @ T )
% 0.33/0.54          = none_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % minNullmin
% 0.33/0.54  thf(fact_852_mint__corr__help__empty,axiom,
% 0.33/0.54      ! [T: vEBT_VEBT,N: nat] :
% 0.33/0.54        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.54       => ( ( ( vEBT_vebt_mint @ T )
% 0.33/0.54            = none_nat )
% 0.33/0.54         => ( ( vEBT_VEBT_set_vebt @ T )
% 0.33/0.54            = bot_bot_set_nat ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % mint_corr_help_empty
% 0.33/0.54  thf(fact_853_not__Some__eq,axiom,
% 0.33/0.54      ! [X: option_nat] :
% 0.33/0.54        ( ( ! [Y2: nat] :
% 0.33/0.54              ( X
% 0.33/0.54             != ( some_nat @ Y2 ) ) )
% 0.33/0.54        = ( X = none_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % not_Some_eq
% 0.33/0.54  thf(fact_854_not__Some__eq,axiom,
% 0.33/0.54      ! [X: option4927543243414619207at_nat] :
% 0.33/0.54        ( ( ! [Y2: product_prod_nat_nat] :
% 0.33/0.54              ( X
% 0.33/0.54             != ( some_P7363390416028606310at_nat @ Y2 ) ) )
% 0.33/0.54        = ( X = none_P5556105721700978146at_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % not_Some_eq
% 0.33/0.54  thf(fact_855_not__Some__eq,axiom,
% 0.33/0.54      ! [X: option_num] :
% 0.33/0.54        ( ( ! [Y2: num] :
% 0.33/0.54              ( X
% 0.33/0.54             != ( some_num @ Y2 ) ) )
% 0.33/0.54        = ( X = none_num ) ) ).
% 0.33/0.54  
% 0.33/0.54  % not_Some_eq
% 0.33/0.54  thf(fact_856_not__None__eq,axiom,
% 0.33/0.54      ! [X: option_nat] :
% 0.33/0.54        ( ( X != none_nat )
% 0.33/0.54        = ( ? [Y2: nat] :
% 0.33/0.54              ( X
% 0.33/0.54              = ( some_nat @ Y2 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % not_None_eq
% 0.33/0.54  thf(fact_857_not__None__eq,axiom,
% 0.33/0.54      ! [X: option4927543243414619207at_nat] :
% 0.33/0.54        ( ( X != none_P5556105721700978146at_nat )
% 0.33/0.54        = ( ? [Y2: product_prod_nat_nat] :
% 0.33/0.54              ( X
% 0.33/0.54              = ( some_P7363390416028606310at_nat @ Y2 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % not_None_eq
% 0.33/0.54  thf(fact_858_not__None__eq,axiom,
% 0.33/0.54      ! [X: option_num] :
% 0.33/0.54        ( ( X != none_num )
% 0.33/0.54        = ( ? [Y2: num] :
% 0.33/0.54              ( X
% 0.33/0.54              = ( some_num @ Y2 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % not_None_eq
% 0.33/0.54  thf(fact_859_combine__options__cases,axiom,
% 0.33/0.54      ! [X: option_nat,P: option_nat > option_nat > $o,Y: option_nat] :
% 0.33/0.54        ( ( ( X = none_nat )
% 0.33/0.54         => ( P @ X @ Y ) )
% 0.33/0.54       => ( ( ( Y = none_nat )
% 0.33/0.54           => ( P @ X @ Y ) )
% 0.33/0.54         => ( ! [A3: nat,B3: nat] :
% 0.33/0.54                ( ( X
% 0.33/0.54                  = ( some_nat @ A3 ) )
% 0.33/0.54               => ( ( Y
% 0.33/0.54                    = ( some_nat @ B3 ) )
% 0.33/0.54                 => ( P @ X @ Y ) ) )
% 0.33/0.54           => ( P @ X @ Y ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % combine_options_cases
% 0.33/0.54  thf(fact_860_combine__options__cases,axiom,
% 0.33/0.54      ! [X: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
% 0.33/0.54        ( ( ( X = none_nat )
% 0.33/0.54         => ( P @ X @ Y ) )
% 0.33/0.54       => ( ( ( Y = none_P5556105721700978146at_nat )
% 0.33/0.54           => ( P @ X @ Y ) )
% 0.33/0.54         => ( ! [A3: nat,B3: product_prod_nat_nat] :
% 0.33/0.54                ( ( X
% 0.33/0.54                  = ( some_nat @ A3 ) )
% 0.33/0.54               => ( ( Y
% 0.33/0.54                    = ( some_P7363390416028606310at_nat @ B3 ) )
% 0.33/0.54                 => ( P @ X @ Y ) ) )
% 0.33/0.54           => ( P @ X @ Y ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % combine_options_cases
% 0.33/0.54  thf(fact_861_combine__options__cases,axiom,
% 0.33/0.54      ! [X: option_nat,P: option_nat > option_num > $o,Y: option_num] :
% 0.33/0.54        ( ( ( X = none_nat )
% 0.33/0.54         => ( P @ X @ Y ) )
% 0.33/0.54       => ( ( ( Y = none_num )
% 0.33/0.54           => ( P @ X @ Y ) )
% 0.33/0.54         => ( ! [A3: nat,B3: num] :
% 0.33/0.54                ( ( X
% 0.33/0.54                  = ( some_nat @ A3 ) )
% 0.33/0.54               => ( ( Y
% 0.33/0.54                    = ( some_num @ B3 ) )
% 0.33/0.54                 => ( P @ X @ Y ) ) )
% 0.33/0.54           => ( P @ X @ Y ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % combine_options_cases
% 0.33/0.54  thf(fact_862_combine__options__cases,axiom,
% 0.33/0.54      ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y: option_nat] :
% 0.33/0.54        ( ( ( X = none_P5556105721700978146at_nat )
% 0.33/0.54         => ( P @ X @ Y ) )
% 0.33/0.54       => ( ( ( Y = none_nat )
% 0.33/0.54           => ( P @ X @ Y ) )
% 0.33/0.54         => ( ! [A3: product_prod_nat_nat,B3: nat] :
% 0.33/0.54                ( ( X
% 0.33/0.54                  = ( some_P7363390416028606310at_nat @ A3 ) )
% 0.33/0.54               => ( ( Y
% 0.33/0.54                    = ( some_nat @ B3 ) )
% 0.33/0.54                 => ( P @ X @ Y ) ) )
% 0.33/0.54           => ( P @ X @ Y ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % combine_options_cases
% 0.33/0.54  thf(fact_863_combine__options__cases,axiom,
% 0.33/0.54      ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
% 0.33/0.54        ( ( ( X = none_P5556105721700978146at_nat )
% 0.33/0.54         => ( P @ X @ Y ) )
% 0.33/0.54       => ( ( ( Y = none_P5556105721700978146at_nat )
% 0.33/0.54           => ( P @ X @ Y ) )
% 0.33/0.54         => ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
% 0.33/0.54                ( ( X
% 0.33/0.54                  = ( some_P7363390416028606310at_nat @ A3 ) )
% 0.33/0.54               => ( ( Y
% 0.33/0.54                    = ( some_P7363390416028606310at_nat @ B3 ) )
% 0.33/0.54                 => ( P @ X @ Y ) ) )
% 0.33/0.54           => ( P @ X @ Y ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % combine_options_cases
% 0.33/0.54  thf(fact_864_combine__options__cases,axiom,
% 0.33/0.54      ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y: option_num] :
% 0.33/0.54        ( ( ( X = none_P5556105721700978146at_nat )
% 0.33/0.54         => ( P @ X @ Y ) )
% 0.33/0.54       => ( ( ( Y = none_num )
% 0.33/0.54           => ( P @ X @ Y ) )
% 0.33/0.54         => ( ! [A3: product_prod_nat_nat,B3: num] :
% 0.33/0.54                ( ( X
% 0.33/0.54                  = ( some_P7363390416028606310at_nat @ A3 ) )
% 0.33/0.54               => ( ( Y
% 0.33/0.54                    = ( some_num @ B3 ) )
% 0.33/0.54                 => ( P @ X @ Y ) ) )
% 0.33/0.54           => ( P @ X @ Y ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % combine_options_cases
% 0.33/0.54  thf(fact_865_combine__options__cases,axiom,
% 0.33/0.54      ! [X: option_num,P: option_num > option_nat > $o,Y: option_nat] :
% 0.33/0.54        ( ( ( X = none_num )
% 0.33/0.54         => ( P @ X @ Y ) )
% 0.33/0.54       => ( ( ( Y = none_nat )
% 0.33/0.54           => ( P @ X @ Y ) )
% 0.33/0.54         => ( ! [A3: num,B3: nat] :
% 0.33/0.54                ( ( X
% 0.33/0.54                  = ( some_num @ A3 ) )
% 0.33/0.54               => ( ( Y
% 0.33/0.54                    = ( some_nat @ B3 ) )
% 0.33/0.54                 => ( P @ X @ Y ) ) )
% 0.33/0.54           => ( P @ X @ Y ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % combine_options_cases
% 0.33/0.54  thf(fact_866_combine__options__cases,axiom,
% 0.33/0.54      ! [X: option_num,P: option_num > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
% 0.33/0.54        ( ( ( X = none_num )
% 0.33/0.54         => ( P @ X @ Y ) )
% 0.33/0.54       => ( ( ( Y = none_P5556105721700978146at_nat )
% 0.33/0.54           => ( P @ X @ Y ) )
% 0.33/0.54         => ( ! [A3: num,B3: product_prod_nat_nat] :
% 0.33/0.54                ( ( X
% 0.33/0.54                  = ( some_num @ A3 ) )
% 0.33/0.54               => ( ( Y
% 0.33/0.54                    = ( some_P7363390416028606310at_nat @ B3 ) )
% 0.33/0.54                 => ( P @ X @ Y ) ) )
% 0.33/0.54           => ( P @ X @ Y ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % combine_options_cases
% 0.33/0.54  thf(fact_867_combine__options__cases,axiom,
% 0.33/0.54      ! [X: option_num,P: option_num > option_num > $o,Y: option_num] :
% 0.33/0.54        ( ( ( X = none_num )
% 0.33/0.54         => ( P @ X @ Y ) )
% 0.33/0.54       => ( ( ( Y = none_num )
% 0.33/0.54           => ( P @ X @ Y ) )
% 0.33/0.54         => ( ! [A3: num,B3: num] :
% 0.33/0.54                ( ( X
% 0.33/0.54                  = ( some_num @ A3 ) )
% 0.33/0.54               => ( ( Y
% 0.33/0.54                    = ( some_num @ B3 ) )
% 0.33/0.54                 => ( P @ X @ Y ) ) )
% 0.33/0.54           => ( P @ X @ Y ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % combine_options_cases
% 0.33/0.54  thf(fact_868_split__option__all,axiom,
% 0.33/0.54      ( ( ^ [P2: option_nat > $o] :
% 0.33/0.54          ! [X7: option_nat] : ( P2 @ X7 ) )
% 0.33/0.54      = ( ^ [P3: option_nat > $o] :
% 0.33/0.54            ( ( P3 @ none_nat )
% 0.33/0.54            & ! [X3: nat] : ( P3 @ ( some_nat @ X3 ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % split_option_all
% 0.33/0.54  thf(fact_869_split__option__all,axiom,
% 0.33/0.54      ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 0.33/0.54          ! [X7: option4927543243414619207at_nat] : ( P2 @ X7 ) )
% 0.33/0.54      = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 0.33/0.54            ( ( P3 @ none_P5556105721700978146at_nat )
% 0.33/0.54            & ! [X3: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X3 ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % split_option_all
% 0.33/0.54  thf(fact_870_split__option__all,axiom,
% 0.33/0.54      ( ( ^ [P2: option_num > $o] :
% 0.33/0.54          ! [X7: option_num] : ( P2 @ X7 ) )
% 0.33/0.54      = ( ^ [P3: option_num > $o] :
% 0.33/0.54            ( ( P3 @ none_num )
% 0.33/0.54            & ! [X3: num] : ( P3 @ ( some_num @ X3 ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % split_option_all
% 0.33/0.54  thf(fact_871_split__option__ex,axiom,
% 0.33/0.54      ( ( ^ [P2: option_nat > $o] :
% 0.33/0.54          ? [X7: option_nat] : ( P2 @ X7 ) )
% 0.33/0.54      = ( ^ [P3: option_nat > $o] :
% 0.33/0.54            ( ( P3 @ none_nat )
% 0.33/0.54            | ? [X3: nat] : ( P3 @ ( some_nat @ X3 ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % split_option_ex
% 0.33/0.54  thf(fact_872_split__option__ex,axiom,
% 0.33/0.54      ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 0.33/0.54          ? [X7: option4927543243414619207at_nat] : ( P2 @ X7 ) )
% 0.33/0.54      = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 0.33/0.54            ( ( P3 @ none_P5556105721700978146at_nat )
% 0.33/0.54            | ? [X3: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X3 ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % split_option_ex
% 0.33/0.54  thf(fact_873_split__option__ex,axiom,
% 0.33/0.54      ( ( ^ [P2: option_num > $o] :
% 0.33/0.54          ? [X7: option_num] : ( P2 @ X7 ) )
% 0.33/0.54      = ( ^ [P3: option_num > $o] :
% 0.33/0.54            ( ( P3 @ none_num )
% 0.33/0.54            | ? [X3: num] : ( P3 @ ( some_num @ X3 ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % split_option_ex
% 0.33/0.54  thf(fact_874_option_Oexhaust,axiom,
% 0.33/0.54      ! [Y: option_nat] :
% 0.33/0.54        ( ( Y != none_nat )
% 0.33/0.54       => ~ ! [X23: nat] :
% 0.33/0.54              ( Y
% 0.33/0.54             != ( some_nat @ X23 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % option.exhaust
% 0.33/0.54  thf(fact_875_option_Oexhaust,axiom,
% 0.33/0.54      ! [Y: option4927543243414619207at_nat] :
% 0.33/0.54        ( ( Y != none_P5556105721700978146at_nat )
% 0.33/0.54       => ~ ! [X23: product_prod_nat_nat] :
% 0.33/0.54              ( Y
% 0.33/0.54             != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % option.exhaust
% 0.33/0.54  thf(fact_876_option_Oexhaust,axiom,
% 0.33/0.54      ! [Y: option_num] :
% 0.33/0.54        ( ( Y != none_num )
% 0.33/0.54       => ~ ! [X23: num] :
% 0.33/0.54              ( Y
% 0.33/0.54             != ( some_num @ X23 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % option.exhaust
% 0.33/0.54  thf(fact_877_option_OdiscI,axiom,
% 0.33/0.54      ! [Option: option_nat,X22: nat] :
% 0.33/0.54        ( ( Option
% 0.33/0.54          = ( some_nat @ X22 ) )
% 0.33/0.54       => ( Option != none_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % option.discI
% 0.33/0.54  thf(fact_878_option_OdiscI,axiom,
% 0.33/0.54      ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
% 0.33/0.54        ( ( Option
% 0.33/0.54          = ( some_P7363390416028606310at_nat @ X22 ) )
% 0.33/0.54       => ( Option != none_P5556105721700978146at_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % option.discI
% 0.33/0.54  thf(fact_879_option_OdiscI,axiom,
% 0.33/0.54      ! [Option: option_num,X22: num] :
% 0.33/0.54        ( ( Option
% 0.33/0.54          = ( some_num @ X22 ) )
% 0.33/0.54       => ( Option != none_num ) ) ).
% 0.33/0.54  
% 0.33/0.54  % option.discI
% 0.33/0.54  thf(fact_880_option_Odistinct_I1_J,axiom,
% 0.33/0.54      ! [X22: nat] :
% 0.33/0.54        ( none_nat
% 0.33/0.54       != ( some_nat @ X22 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % option.distinct(1)
% 0.33/0.54  thf(fact_881_option_Odistinct_I1_J,axiom,
% 0.33/0.54      ! [X22: product_prod_nat_nat] :
% 0.33/0.54        ( none_P5556105721700978146at_nat
% 0.33/0.54       != ( some_P7363390416028606310at_nat @ X22 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % option.distinct(1)
% 0.33/0.54  thf(fact_882_option_Odistinct_I1_J,axiom,
% 0.33/0.54      ! [X22: num] :
% 0.33/0.54        ( none_num
% 0.33/0.54       != ( some_num @ X22 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % option.distinct(1)
% 0.33/0.54  thf(fact_883_bot__nat__def,axiom,
% 0.33/0.54      bot_bot_nat = zero_zero_nat ).
% 0.33/0.54  
% 0.33/0.54  % bot_nat_def
% 0.33/0.54  thf(fact_884_finite__maxlen,axiom,
% 0.33/0.54      ! [M4: set_list_VEBT_VEBT] :
% 0.33/0.54        ( ( finite3004134309566078307T_VEBT @ M4 )
% 0.33/0.54       => ? [N3: nat] :
% 0.33/0.54          ! [X2: list_VEBT_VEBT] :
% 0.33/0.54            ( ( member2936631157270082147T_VEBT @ X2 @ M4 )
% 0.33/0.54           => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X2 ) @ N3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_maxlen
% 0.33/0.54  thf(fact_885_finite__maxlen,axiom,
% 0.33/0.54      ! [M4: set_list_o] :
% 0.33/0.54        ( ( finite_finite_list_o @ M4 )
% 0.33/0.54       => ? [N3: nat] :
% 0.33/0.54          ! [X2: list_o] :
% 0.33/0.54            ( ( member_list_o @ X2 @ M4 )
% 0.33/0.54           => ( ord_less_nat @ ( size_size_list_o @ X2 ) @ N3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_maxlen
% 0.33/0.54  thf(fact_886_finite__maxlen,axiom,
% 0.33/0.54      ! [M4: set_list_nat] :
% 0.33/0.54        ( ( finite8100373058378681591st_nat @ M4 )
% 0.33/0.54       => ? [N3: nat] :
% 0.33/0.54          ! [X2: list_nat] :
% 0.33/0.54            ( ( member_list_nat @ X2 @ M4 )
% 0.33/0.54           => ( ord_less_nat @ ( size_size_list_nat @ X2 ) @ N3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_maxlen
% 0.33/0.54  thf(fact_887_finite__maxlen,axiom,
% 0.33/0.54      ! [M4: set_list_int] :
% 0.33/0.54        ( ( finite3922522038869484883st_int @ M4 )
% 0.33/0.54       => ? [N3: nat] :
% 0.33/0.54          ! [X2: list_int] :
% 0.33/0.54            ( ( member_list_int @ X2 @ M4 )
% 0.33/0.54           => ( ord_less_nat @ ( size_size_list_int @ X2 ) @ N3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_maxlen
% 0.33/0.54  thf(fact_888_finite__subset,axiom,
% 0.33/0.54      ! [A2: set_nat,B2: set_nat] :
% 0.33/0.54        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.33/0.54       => ( ( finite_finite_nat @ B2 )
% 0.33/0.54         => ( finite_finite_nat @ A2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_subset
% 0.33/0.54  thf(fact_889_finite__subset,axiom,
% 0.33/0.54      ! [A2: set_complex,B2: set_complex] :
% 0.33/0.54        ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.33/0.54       => ( ( finite3207457112153483333omplex @ B2 )
% 0.33/0.54         => ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_subset
% 0.33/0.54  thf(fact_890_finite__subset,axiom,
% 0.33/0.54      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.33/0.54        ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 0.33/0.54       => ( ( finite6177210948735845034at_nat @ B2 )
% 0.33/0.54         => ( finite6177210948735845034at_nat @ A2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_subset
% 0.33/0.54  thf(fact_891_finite__subset,axiom,
% 0.33/0.54      ! [A2: set_int,B2: set_int] :
% 0.33/0.54        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.33/0.54       => ( ( finite_finite_int @ B2 )
% 0.33/0.54         => ( finite_finite_int @ A2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_subset
% 0.33/0.54  thf(fact_892_infinite__super,axiom,
% 0.33/0.54      ! [S2: set_nat,T3: set_nat] :
% 0.33/0.54        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.33/0.54       => ( ~ ( finite_finite_nat @ S2 )
% 0.33/0.54         => ~ ( finite_finite_nat @ T3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % infinite_super
% 0.33/0.54  thf(fact_893_infinite__super,axiom,
% 0.33/0.54      ! [S2: set_complex,T3: set_complex] :
% 0.33/0.54        ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.33/0.54       => ( ~ ( finite3207457112153483333omplex @ S2 )
% 0.33/0.54         => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % infinite_super
% 0.33/0.54  thf(fact_894_infinite__super,axiom,
% 0.33/0.54      ! [S2: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
% 0.33/0.54        ( ( ord_le3146513528884898305at_nat @ S2 @ T3 )
% 0.33/0.54       => ( ~ ( finite6177210948735845034at_nat @ S2 )
% 0.33/0.54         => ~ ( finite6177210948735845034at_nat @ T3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % infinite_super
% 0.33/0.54  thf(fact_895_infinite__super,axiom,
% 0.33/0.54      ! [S2: set_int,T3: set_int] :
% 0.33/0.54        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.33/0.54       => ( ~ ( finite_finite_int @ S2 )
% 0.33/0.54         => ~ ( finite_finite_int @ T3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % infinite_super
% 0.33/0.54  thf(fact_896_rev__finite__subset,axiom,
% 0.33/0.54      ! [B2: set_nat,A2: set_nat] :
% 0.33/0.54        ( ( finite_finite_nat @ B2 )
% 0.33/0.54       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.33/0.54         => ( finite_finite_nat @ A2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % rev_finite_subset
% 0.33/0.54  thf(fact_897_rev__finite__subset,axiom,
% 0.33/0.54      ! [B2: set_complex,A2: set_complex] :
% 0.33/0.54        ( ( finite3207457112153483333omplex @ B2 )
% 0.33/0.54       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.33/0.54         => ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % rev_finite_subset
% 0.33/0.54  thf(fact_898_rev__finite__subset,axiom,
% 0.33/0.54      ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.33/0.54        ( ( finite6177210948735845034at_nat @ B2 )
% 0.33/0.54       => ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 0.33/0.54         => ( finite6177210948735845034at_nat @ A2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % rev_finite_subset
% 0.33/0.54  thf(fact_899_rev__finite__subset,axiom,
% 0.33/0.54      ! [B2: set_int,A2: set_int] :
% 0.33/0.54        ( ( finite_finite_int @ B2 )
% 0.33/0.54       => ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.33/0.54         => ( finite_finite_int @ A2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % rev_finite_subset
% 0.33/0.54  thf(fact_900_finite__psubset__induct,axiom,
% 0.33/0.54      ! [A2: set_nat,P: set_nat > $o] :
% 0.33/0.54        ( ( finite_finite_nat @ A2 )
% 0.33/0.54       => ( ! [A5: set_nat] :
% 0.33/0.54              ( ( finite_finite_nat @ A5 )
% 0.33/0.54             => ( ! [B5: set_nat] :
% 0.33/0.54                    ( ( ord_less_set_nat @ B5 @ A5 )
% 0.33/0.54                   => ( P @ B5 ) )
% 0.33/0.54               => ( P @ A5 ) ) )
% 0.33/0.54         => ( P @ A2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_psubset_induct
% 0.33/0.54  thf(fact_901_finite__psubset__induct,axiom,
% 0.33/0.54      ! [A2: set_int,P: set_int > $o] :
% 0.33/0.54        ( ( finite_finite_int @ A2 )
% 0.33/0.54       => ( ! [A5: set_int] :
% 0.33/0.54              ( ( finite_finite_int @ A5 )
% 0.33/0.54             => ( ! [B5: set_int] :
% 0.33/0.54                    ( ( ord_less_set_int @ B5 @ A5 )
% 0.33/0.54                   => ( P @ B5 ) )
% 0.33/0.54               => ( P @ A5 ) ) )
% 0.33/0.54         => ( P @ A2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_psubset_induct
% 0.33/0.54  thf(fact_902_finite__psubset__induct,axiom,
% 0.33/0.54      ! [A2: set_complex,P: set_complex > $o] :
% 0.33/0.54        ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.54       => ( ! [A5: set_complex] :
% 0.33/0.54              ( ( finite3207457112153483333omplex @ A5 )
% 0.33/0.54             => ( ! [B5: set_complex] :
% 0.33/0.54                    ( ( ord_less_set_complex @ B5 @ A5 )
% 0.33/0.54                   => ( P @ B5 ) )
% 0.33/0.54               => ( P @ A5 ) ) )
% 0.33/0.54         => ( P @ A2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_psubset_induct
% 0.33/0.54  thf(fact_903_finite__psubset__induct,axiom,
% 0.33/0.54      ! [A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.33/0.54        ( ( finite6177210948735845034at_nat @ A2 )
% 0.33/0.54       => ( ! [A5: set_Pr1261947904930325089at_nat] :
% 0.33/0.54              ( ( finite6177210948735845034at_nat @ A5 )
% 0.33/0.54             => ( ! [B5: set_Pr1261947904930325089at_nat] :
% 0.33/0.54                    ( ( ord_le7866589430770878221at_nat @ B5 @ A5 )
% 0.33/0.54                   => ( P @ B5 ) )
% 0.33/0.54               => ( P @ A5 ) ) )
% 0.33/0.54         => ( P @ A2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_psubset_induct
% 0.33/0.54  thf(fact_904_finite__has__minimal2,axiom,
% 0.33/0.54      ! [A2: set_real,A: real] :
% 0.33/0.54        ( ( finite_finite_real @ A2 )
% 0.33/0.54       => ( ( member_real @ A @ A2 )
% 0.33/0.54         => ? [X4: real] :
% 0.33/0.54              ( ( member_real @ X4 @ A2 )
% 0.33/0.54              & ( ord_less_eq_real @ X4 @ A )
% 0.33/0.54              & ! [Xa: real] :
% 0.33/0.54                  ( ( member_real @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_real @ Xa @ X4 )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_minimal2
% 0.33/0.54  thf(fact_905_finite__has__minimal2,axiom,
% 0.33/0.54      ! [A2: set_set_nat,A: set_nat] :
% 0.33/0.54        ( ( finite1152437895449049373et_nat @ A2 )
% 0.33/0.54       => ( ( member_set_nat @ A @ A2 )
% 0.33/0.54         => ? [X4: set_nat] :
% 0.33/0.54              ( ( member_set_nat @ X4 @ A2 )
% 0.33/0.54              & ( ord_less_eq_set_nat @ X4 @ A )
% 0.33/0.54              & ! [Xa: set_nat] :
% 0.33/0.54                  ( ( member_set_nat @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_set_nat @ Xa @ X4 )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_minimal2
% 0.33/0.54  thf(fact_906_finite__has__minimal2,axiom,
% 0.33/0.54      ! [A2: set_set_int,A: set_int] :
% 0.33/0.54        ( ( finite6197958912794628473et_int @ A2 )
% 0.33/0.54       => ( ( member_set_int @ A @ A2 )
% 0.33/0.54         => ? [X4: set_int] :
% 0.33/0.54              ( ( member_set_int @ X4 @ A2 )
% 0.33/0.54              & ( ord_less_eq_set_int @ X4 @ A )
% 0.33/0.54              & ! [Xa: set_int] :
% 0.33/0.54                  ( ( member_set_int @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_set_int @ Xa @ X4 )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_minimal2
% 0.33/0.54  thf(fact_907_finite__has__minimal2,axiom,
% 0.33/0.54      ! [A2: set_rat,A: rat] :
% 0.33/0.54        ( ( finite_finite_rat @ A2 )
% 0.33/0.54       => ( ( member_rat @ A @ A2 )
% 0.33/0.54         => ? [X4: rat] :
% 0.33/0.54              ( ( member_rat @ X4 @ A2 )
% 0.33/0.54              & ( ord_less_eq_rat @ X4 @ A )
% 0.33/0.54              & ! [Xa: rat] :
% 0.33/0.54                  ( ( member_rat @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_rat @ Xa @ X4 )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_minimal2
% 0.33/0.54  thf(fact_908_finite__has__minimal2,axiom,
% 0.33/0.54      ! [A2: set_num,A: num] :
% 0.33/0.54        ( ( finite_finite_num @ A2 )
% 0.33/0.54       => ( ( member_num @ A @ A2 )
% 0.33/0.54         => ? [X4: num] :
% 0.33/0.54              ( ( member_num @ X4 @ A2 )
% 0.33/0.54              & ( ord_less_eq_num @ X4 @ A )
% 0.33/0.54              & ! [Xa: num] :
% 0.33/0.54                  ( ( member_num @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_num @ Xa @ X4 )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_minimal2
% 0.33/0.54  thf(fact_909_finite__has__minimal2,axiom,
% 0.33/0.54      ! [A2: set_nat,A: nat] :
% 0.33/0.54        ( ( finite_finite_nat @ A2 )
% 0.33/0.54       => ( ( member_nat @ A @ A2 )
% 0.33/0.54         => ? [X4: nat] :
% 0.33/0.54              ( ( member_nat @ X4 @ A2 )
% 0.33/0.54              & ( ord_less_eq_nat @ X4 @ A )
% 0.33/0.54              & ! [Xa: nat] :
% 0.33/0.54                  ( ( member_nat @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_nat @ Xa @ X4 )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_minimal2
% 0.33/0.54  thf(fact_910_finite__has__minimal2,axiom,
% 0.33/0.54      ! [A2: set_int,A: int] :
% 0.33/0.54        ( ( finite_finite_int @ A2 )
% 0.33/0.54       => ( ( member_int @ A @ A2 )
% 0.33/0.54         => ? [X4: int] :
% 0.33/0.54              ( ( member_int @ X4 @ A2 )
% 0.33/0.54              & ( ord_less_eq_int @ X4 @ A )
% 0.33/0.54              & ! [Xa: int] :
% 0.33/0.54                  ( ( member_int @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_int @ Xa @ X4 )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_minimal2
% 0.33/0.54  thf(fact_911_finite__has__maximal2,axiom,
% 0.33/0.54      ! [A2: set_real,A: real] :
% 0.33/0.54        ( ( finite_finite_real @ A2 )
% 0.33/0.54       => ( ( member_real @ A @ A2 )
% 0.33/0.54         => ? [X4: real] :
% 0.33/0.54              ( ( member_real @ X4 @ A2 )
% 0.33/0.54              & ( ord_less_eq_real @ A @ X4 )
% 0.33/0.54              & ! [Xa: real] :
% 0.33/0.54                  ( ( member_real @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_real @ X4 @ Xa )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_maximal2
% 0.33/0.54  thf(fact_912_finite__has__maximal2,axiom,
% 0.33/0.54      ! [A2: set_set_nat,A: set_nat] :
% 0.33/0.54        ( ( finite1152437895449049373et_nat @ A2 )
% 0.33/0.54       => ( ( member_set_nat @ A @ A2 )
% 0.33/0.54         => ? [X4: set_nat] :
% 0.33/0.54              ( ( member_set_nat @ X4 @ A2 )
% 0.33/0.54              & ( ord_less_eq_set_nat @ A @ X4 )
% 0.33/0.54              & ! [Xa: set_nat] :
% 0.33/0.54                  ( ( member_set_nat @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_set_nat @ X4 @ Xa )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_maximal2
% 0.33/0.54  thf(fact_913_finite__has__maximal2,axiom,
% 0.33/0.54      ! [A2: set_set_int,A: set_int] :
% 0.33/0.54        ( ( finite6197958912794628473et_int @ A2 )
% 0.33/0.54       => ( ( member_set_int @ A @ A2 )
% 0.33/0.54         => ? [X4: set_int] :
% 0.33/0.54              ( ( member_set_int @ X4 @ A2 )
% 0.33/0.54              & ( ord_less_eq_set_int @ A @ X4 )
% 0.33/0.54              & ! [Xa: set_int] :
% 0.33/0.54                  ( ( member_set_int @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_set_int @ X4 @ Xa )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_maximal2
% 0.33/0.54  thf(fact_914_finite__has__maximal2,axiom,
% 0.33/0.54      ! [A2: set_rat,A: rat] :
% 0.33/0.54        ( ( finite_finite_rat @ A2 )
% 0.33/0.54       => ( ( member_rat @ A @ A2 )
% 0.33/0.54         => ? [X4: rat] :
% 0.33/0.54              ( ( member_rat @ X4 @ A2 )
% 0.33/0.54              & ( ord_less_eq_rat @ A @ X4 )
% 0.33/0.54              & ! [Xa: rat] :
% 0.33/0.54                  ( ( member_rat @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_rat @ X4 @ Xa )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_maximal2
% 0.33/0.54  thf(fact_915_finite__has__maximal2,axiom,
% 0.33/0.54      ! [A2: set_num,A: num] :
% 0.33/0.54        ( ( finite_finite_num @ A2 )
% 0.33/0.54       => ( ( member_num @ A @ A2 )
% 0.33/0.54         => ? [X4: num] :
% 0.33/0.54              ( ( member_num @ X4 @ A2 )
% 0.33/0.54              & ( ord_less_eq_num @ A @ X4 )
% 0.33/0.54              & ! [Xa: num] :
% 0.33/0.54                  ( ( member_num @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_num @ X4 @ Xa )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_maximal2
% 0.33/0.54  thf(fact_916_finite__has__maximal2,axiom,
% 0.33/0.54      ! [A2: set_nat,A: nat] :
% 0.33/0.54        ( ( finite_finite_nat @ A2 )
% 0.33/0.54       => ( ( member_nat @ A @ A2 )
% 0.33/0.54         => ? [X4: nat] :
% 0.33/0.54              ( ( member_nat @ X4 @ A2 )
% 0.33/0.54              & ( ord_less_eq_nat @ A @ X4 )
% 0.33/0.54              & ! [Xa: nat] :
% 0.33/0.54                  ( ( member_nat @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_nat @ X4 @ Xa )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_maximal2
% 0.33/0.54  thf(fact_917_finite__has__maximal2,axiom,
% 0.33/0.54      ! [A2: set_int,A: int] :
% 0.33/0.54        ( ( finite_finite_int @ A2 )
% 0.33/0.54       => ( ( member_int @ A @ A2 )
% 0.33/0.54         => ? [X4: int] :
% 0.33/0.54              ( ( member_int @ X4 @ A2 )
% 0.33/0.54              & ( ord_less_eq_int @ A @ X4 )
% 0.33/0.54              & ! [Xa: int] :
% 0.33/0.54                  ( ( member_int @ Xa @ A2 )
% 0.33/0.54                 => ( ( ord_less_eq_int @ X4 @ Xa )
% 0.33/0.54                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_has_maximal2
% 0.33/0.54  thf(fact_918_infinite__imp__nonempty,axiom,
% 0.33/0.54      ! [S2: set_complex] :
% 0.33/0.54        ( ~ ( finite3207457112153483333omplex @ S2 )
% 0.33/0.54       => ( S2 != bot_bot_set_complex ) ) ).
% 0.33/0.54  
% 0.33/0.54  % infinite_imp_nonempty
% 0.33/0.54  thf(fact_919_infinite__imp__nonempty,axiom,
% 0.33/0.54      ! [S2: set_Pr1261947904930325089at_nat] :
% 0.33/0.54        ( ~ ( finite6177210948735845034at_nat @ S2 )
% 0.33/0.54       => ( S2 != bot_bo2099793752762293965at_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % infinite_imp_nonempty
% 0.33/0.54  thf(fact_920_infinite__imp__nonempty,axiom,
% 0.33/0.54      ! [S2: set_real] :
% 0.33/0.54        ( ~ ( finite_finite_real @ S2 )
% 0.33/0.54       => ( S2 != bot_bot_set_real ) ) ).
% 0.33/0.54  
% 0.33/0.54  % infinite_imp_nonempty
% 0.33/0.54  thf(fact_921_infinite__imp__nonempty,axiom,
% 0.33/0.54      ! [S2: set_nat] :
% 0.33/0.54        ( ~ ( finite_finite_nat @ S2 )
% 0.33/0.54       => ( S2 != bot_bot_set_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % infinite_imp_nonempty
% 0.33/0.54  thf(fact_922_infinite__imp__nonempty,axiom,
% 0.33/0.54      ! [S2: set_int] :
% 0.33/0.54        ( ~ ( finite_finite_int @ S2 )
% 0.33/0.54       => ( S2 != bot_bot_set_int ) ) ).
% 0.33/0.54  
% 0.33/0.54  % infinite_imp_nonempty
% 0.33/0.54  thf(fact_923_finite_OemptyI,axiom,
% 0.33/0.54      finite3207457112153483333omplex @ bot_bot_set_complex ).
% 0.33/0.54  
% 0.33/0.54  % finite.emptyI
% 0.33/0.54  thf(fact_924_finite_OemptyI,axiom,
% 0.33/0.54      finite6177210948735845034at_nat @ bot_bo2099793752762293965at_nat ).
% 0.33/0.54  
% 0.33/0.54  % finite.emptyI
% 0.33/0.54  thf(fact_925_finite_OemptyI,axiom,
% 0.33/0.54      finite_finite_real @ bot_bot_set_real ).
% 0.33/0.54  
% 0.33/0.54  % finite.emptyI
% 0.33/0.54  thf(fact_926_finite_OemptyI,axiom,
% 0.33/0.54      finite_finite_nat @ bot_bot_set_nat ).
% 0.33/0.54  
% 0.33/0.54  % finite.emptyI
% 0.33/0.54  thf(fact_927_finite_OemptyI,axiom,
% 0.33/0.54      finite_finite_int @ bot_bot_set_int ).
% 0.33/0.54  
% 0.33/0.54  % finite.emptyI
% 0.33/0.54  thf(fact_928_pred__correct,axiom,
% 0.33/0.54      ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
% 0.33/0.54        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.54       => ( ( ( vEBT_vebt_pred @ T @ X )
% 0.33/0.54            = ( some_nat @ Sx ) )
% 0.33/0.54          = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % pred_correct
% 0.33/0.54  thf(fact_929_succ__correct,axiom,
% 0.33/0.54      ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
% 0.33/0.54        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.54       => ( ( ( vEBT_vebt_succ @ T @ X )
% 0.33/0.54            = ( some_nat @ Sx ) )
% 0.33/0.54          = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % succ_correct
% 0.33/0.54  thf(fact_930_pred__corr,axiom,
% 0.33/0.54      ! [T: vEBT_VEBT,N: nat,X: nat,Px: nat] :
% 0.33/0.54        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.54       => ( ( ( vEBT_vebt_pred @ T @ X )
% 0.33/0.54            = ( some_nat @ Px ) )
% 0.33/0.54          = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Px ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % pred_corr
% 0.33/0.54  thf(fact_931_succ__corr,axiom,
% 0.33/0.54      ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
% 0.33/0.54        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.54       => ( ( ( vEBT_vebt_succ @ T @ X )
% 0.33/0.54            = ( some_nat @ Sx ) )
% 0.33/0.54          = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % succ_corr
% 0.33/0.54  thf(fact_932_subset__empty,axiom,
% 0.33/0.54      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.33/0.54        ( ( ord_le3146513528884898305at_nat @ A2 @ bot_bo2099793752762293965at_nat )
% 0.33/0.54        = ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_empty
% 0.33/0.54  thf(fact_933_subset__empty,axiom,
% 0.33/0.54      ! [A2: set_real] :
% 0.33/0.54        ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
% 0.33/0.54        = ( A2 = bot_bot_set_real ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_empty
% 0.33/0.54  thf(fact_934_subset__empty,axiom,
% 0.33/0.54      ! [A2: set_nat] :
% 0.33/0.54        ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
% 0.33/0.54        = ( A2 = bot_bot_set_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_empty
% 0.33/0.54  thf(fact_935_subset__empty,axiom,
% 0.33/0.54      ! [A2: set_int] :
% 0.33/0.54        ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
% 0.33/0.54        = ( A2 = bot_bot_set_int ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_empty
% 0.33/0.54  thf(fact_936_empty__subsetI,axiom,
% 0.33/0.54      ! [A2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A2 ) ).
% 0.33/0.54  
% 0.33/0.54  % empty_subsetI
% 0.33/0.54  thf(fact_937_empty__subsetI,axiom,
% 0.33/0.54      ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% 0.33/0.54  
% 0.33/0.54  % empty_subsetI
% 0.33/0.54  thf(fact_938_empty__subsetI,axiom,
% 0.33/0.54      ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% 0.33/0.54  
% 0.33/0.54  % empty_subsetI
% 0.33/0.54  thf(fact_939_empty__subsetI,axiom,
% 0.33/0.54      ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% 0.33/0.54  
% 0.33/0.54  % empty_subsetI
% 0.33/0.54  thf(fact_940_is__succ__in__set__def,axiom,
% 0.33/0.54      ( vEBT_is_succ_in_set
% 0.33/0.54      = ( ^ [Xs2: set_nat,X3: nat,Y2: nat] :
% 0.33/0.54            ( ( member_nat @ Y2 @ Xs2 )
% 0.33/0.54            & ( ord_less_nat @ X3 @ Y2 )
% 0.33/0.54            & ! [Z3: nat] :
% 0.33/0.54                ( ( member_nat @ Z3 @ Xs2 )
% 0.33/0.54               => ( ( ord_less_nat @ X3 @ Z3 )
% 0.33/0.54                 => ( ord_less_eq_nat @ Y2 @ Z3 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % is_succ_in_set_def
% 0.33/0.54  thf(fact_941_is__pred__in__set__def,axiom,
% 0.33/0.54      ( vEBT_is_pred_in_set
% 0.33/0.54      = ( ^ [Xs2: set_nat,X3: nat,Y2: nat] :
% 0.33/0.54            ( ( member_nat @ Y2 @ Xs2 )
% 0.33/0.54            & ( ord_less_nat @ Y2 @ X3 )
% 0.33/0.54            & ! [Z3: nat] :
% 0.33/0.54                ( ( member_nat @ Z3 @ Xs2 )
% 0.33/0.54               => ( ( ord_less_nat @ Z3 @ X3 )
% 0.33/0.54                 => ( ord_less_eq_nat @ Z3 @ Y2 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % is_pred_in_set_def
% 0.33/0.54  thf(fact_942_mi__eq__ma__no__ch,axiom,
% 0.33/0.54      ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.33/0.54        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
% 0.33/0.54       => ( ( Mi = Ma )
% 0.33/0.54         => ( ! [X2: vEBT_VEBT] :
% 0.33/0.54                ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.33/0.54               => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) )
% 0.33/0.54            & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % mi_eq_ma_no_ch
% 0.33/0.54  thf(fact_943_vebt__member_Osimps_I3_J,axiom,
% 0.33/0.54      ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
% 0.33/0.54        ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X ) ).
% 0.33/0.54  
% 0.33/0.54  % vebt_member.simps(3)
% 0.33/0.54  thf(fact_944_psubsetI,axiom,
% 0.33/0.54      ! [A2: set_int,B2: set_int] :
% 0.33/0.54        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.33/0.54       => ( ( A2 != B2 )
% 0.33/0.54         => ( ord_less_set_int @ A2 @ B2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % psubsetI
% 0.33/0.54  thf(fact_945_empty__iff,axiom,
% 0.33/0.54      ! [C: set_nat] :
% 0.33/0.54        ~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% 0.33/0.54  
% 0.33/0.54  % empty_iff
% 0.33/0.54  thf(fact_946_empty__iff,axiom,
% 0.33/0.54      ! [C: product_prod_nat_nat] :
% 0.33/0.54        ~ ( member8440522571783428010at_nat @ C @ bot_bo2099793752762293965at_nat ) ).
% 0.33/0.54  
% 0.33/0.54  % empty_iff
% 0.33/0.54  thf(fact_947_empty__iff,axiom,
% 0.33/0.54      ! [C: real] :
% 0.33/0.54        ~ ( member_real @ C @ bot_bot_set_real ) ).
% 0.33/0.54  
% 0.33/0.54  % empty_iff
% 0.33/0.54  thf(fact_948_empty__iff,axiom,
% 0.33/0.54      ! [C: nat] :
% 0.33/0.54        ~ ( member_nat @ C @ bot_bot_set_nat ) ).
% 0.33/0.54  
% 0.33/0.54  % empty_iff
% 0.33/0.54  thf(fact_949_empty__iff,axiom,
% 0.33/0.54      ! [C: int] :
% 0.33/0.54        ~ ( member_int @ C @ bot_bot_set_int ) ).
% 0.33/0.54  
% 0.33/0.54  % empty_iff
% 0.33/0.54  thf(fact_950_all__not__in__conv,axiom,
% 0.33/0.54      ! [A2: set_set_nat] :
% 0.33/0.54        ( ( ! [X3: set_nat] :
% 0.33/0.54              ~ ( member_set_nat @ X3 @ A2 ) )
% 0.33/0.54        = ( A2 = bot_bot_set_set_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % all_not_in_conv
% 0.33/0.54  thf(fact_951_all__not__in__conv,axiom,
% 0.33/0.54      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.33/0.54        ( ( ! [X3: product_prod_nat_nat] :
% 0.33/0.54              ~ ( member8440522571783428010at_nat @ X3 @ A2 ) )
% 0.33/0.54        = ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % all_not_in_conv
% 0.33/0.54  thf(fact_952_all__not__in__conv,axiom,
% 0.33/0.54      ! [A2: set_real] :
% 0.33/0.54        ( ( ! [X3: real] :
% 0.33/0.54              ~ ( member_real @ X3 @ A2 ) )
% 0.33/0.54        = ( A2 = bot_bot_set_real ) ) ).
% 0.33/0.54  
% 0.33/0.54  % all_not_in_conv
% 0.33/0.54  thf(fact_953_all__not__in__conv,axiom,
% 0.33/0.54      ! [A2: set_nat] :
% 0.33/0.54        ( ( ! [X3: nat] :
% 0.33/0.54              ~ ( member_nat @ X3 @ A2 ) )
% 0.33/0.54        = ( A2 = bot_bot_set_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % all_not_in_conv
% 0.33/0.54  thf(fact_954_all__not__in__conv,axiom,
% 0.33/0.54      ! [A2: set_int] :
% 0.33/0.54        ( ( ! [X3: int] :
% 0.33/0.54              ~ ( member_int @ X3 @ A2 ) )
% 0.33/0.54        = ( A2 = bot_bot_set_int ) ) ).
% 0.33/0.54  
% 0.33/0.54  % all_not_in_conv
% 0.33/0.54  thf(fact_955_Collect__empty__eq,axiom,
% 0.33/0.54      ! [P: list_nat > $o] :
% 0.33/0.54        ( ( ( collect_list_nat @ P )
% 0.33/0.54          = bot_bot_set_list_nat )
% 0.33/0.54        = ( ! [X3: list_nat] :
% 0.33/0.54              ~ ( P @ X3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_empty_eq
% 0.33/0.54  thf(fact_956_Collect__empty__eq,axiom,
% 0.33/0.54      ! [P: set_nat > $o] :
% 0.33/0.54        ( ( ( collect_set_nat @ P )
% 0.33/0.54          = bot_bot_set_set_nat )
% 0.33/0.54        = ( ! [X3: set_nat] :
% 0.33/0.54              ~ ( P @ X3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_empty_eq
% 0.33/0.54  thf(fact_957_Collect__empty__eq,axiom,
% 0.33/0.54      ! [P: product_prod_nat_nat > $o] :
% 0.33/0.54        ( ( ( collec3392354462482085612at_nat @ P )
% 0.33/0.54          = bot_bo2099793752762293965at_nat )
% 0.33/0.54        = ( ! [X3: product_prod_nat_nat] :
% 0.33/0.54              ~ ( P @ X3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_empty_eq
% 0.33/0.54  thf(fact_958_Collect__empty__eq,axiom,
% 0.33/0.54      ! [P: real > $o] :
% 0.33/0.54        ( ( ( collect_real @ P )
% 0.33/0.54          = bot_bot_set_real )
% 0.33/0.54        = ( ! [X3: real] :
% 0.33/0.54              ~ ( P @ X3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_empty_eq
% 0.33/0.54  thf(fact_959_Collect__empty__eq,axiom,
% 0.33/0.54      ! [P: nat > $o] :
% 0.33/0.54        ( ( ( collect_nat @ P )
% 0.33/0.54          = bot_bot_set_nat )
% 0.33/0.54        = ( ! [X3: nat] :
% 0.33/0.54              ~ ( P @ X3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_empty_eq
% 0.33/0.54  thf(fact_960_Collect__empty__eq,axiom,
% 0.33/0.54      ! [P: int > $o] :
% 0.33/0.54        ( ( ( collect_int @ P )
% 0.33/0.54          = bot_bot_set_int )
% 0.33/0.54        = ( ! [X3: int] :
% 0.33/0.54              ~ ( P @ X3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_empty_eq
% 0.33/0.54  thf(fact_961_empty__Collect__eq,axiom,
% 0.33/0.54      ! [P: list_nat > $o] :
% 0.33/0.54        ( ( bot_bot_set_list_nat
% 0.33/0.54          = ( collect_list_nat @ P ) )
% 0.33/0.54        = ( ! [X3: list_nat] :
% 0.33/0.54              ~ ( P @ X3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % empty_Collect_eq
% 0.33/0.54  thf(fact_962_empty__Collect__eq,axiom,
% 0.33/0.54      ! [P: set_nat > $o] :
% 0.33/0.54        ( ( bot_bot_set_set_nat
% 0.33/0.54          = ( collect_set_nat @ P ) )
% 0.33/0.54        = ( ! [X3: set_nat] :
% 0.33/0.54              ~ ( P @ X3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % empty_Collect_eq
% 0.33/0.54  thf(fact_963_empty__Collect__eq,axiom,
% 0.33/0.54      ! [P: product_prod_nat_nat > $o] :
% 0.33/0.54        ( ( bot_bo2099793752762293965at_nat
% 0.33/0.54          = ( collec3392354462482085612at_nat @ P ) )
% 0.33/0.54        = ( ! [X3: product_prod_nat_nat] :
% 0.33/0.54              ~ ( P @ X3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % empty_Collect_eq
% 0.33/0.54  thf(fact_964_empty__Collect__eq,axiom,
% 0.33/0.54      ! [P: real > $o] :
% 0.33/0.54        ( ( bot_bot_set_real
% 0.33/0.54          = ( collect_real @ P ) )
% 0.33/0.54        = ( ! [X3: real] :
% 0.33/0.54              ~ ( P @ X3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % empty_Collect_eq
% 0.33/0.54  thf(fact_965_empty__Collect__eq,axiom,
% 0.33/0.54      ! [P: nat > $o] :
% 0.33/0.54        ( ( bot_bot_set_nat
% 0.33/0.54          = ( collect_nat @ P ) )
% 0.33/0.54        = ( ! [X3: nat] :
% 0.33/0.54              ~ ( P @ X3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % empty_Collect_eq
% 0.33/0.54  thf(fact_966_empty__Collect__eq,axiom,
% 0.33/0.54      ! [P: int > $o] :
% 0.33/0.54        ( ( bot_bot_set_int
% 0.33/0.54          = ( collect_int @ P ) )
% 0.33/0.54        = ( ! [X3: int] :
% 0.33/0.54              ~ ( P @ X3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % empty_Collect_eq
% 0.33/0.54  thf(fact_967_subsetI,axiom,
% 0.33/0.54      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.33/0.54        ( ! [X4: product_prod_nat_nat] :
% 0.33/0.54            ( ( member8440522571783428010at_nat @ X4 @ A2 )
% 0.33/0.54           => ( member8440522571783428010at_nat @ X4 @ B2 ) )
% 0.33/0.54       => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subsetI
% 0.33/0.54  thf(fact_968_subsetI,axiom,
% 0.33/0.54      ! [A2: set_real,B2: set_real] :
% 0.33/0.54        ( ! [X4: real] :
% 0.33/0.54            ( ( member_real @ X4 @ A2 )
% 0.33/0.54           => ( member_real @ X4 @ B2 ) )
% 0.33/0.54       => ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subsetI
% 0.33/0.54  thf(fact_969_subsetI,axiom,
% 0.33/0.54      ! [A2: set_set_nat,B2: set_set_nat] :
% 0.33/0.54        ( ! [X4: set_nat] :
% 0.33/0.54            ( ( member_set_nat @ X4 @ A2 )
% 0.33/0.54           => ( member_set_nat @ X4 @ B2 ) )
% 0.33/0.54       => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subsetI
% 0.33/0.54  thf(fact_970_subsetI,axiom,
% 0.33/0.54      ! [A2: set_nat,B2: set_nat] :
% 0.33/0.54        ( ! [X4: nat] :
% 0.33/0.54            ( ( member_nat @ X4 @ A2 )
% 0.33/0.54           => ( member_nat @ X4 @ B2 ) )
% 0.33/0.54       => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subsetI
% 0.33/0.54  thf(fact_971_subsetI,axiom,
% 0.33/0.54      ! [A2: set_int,B2: set_int] :
% 0.33/0.54        ( ! [X4: int] :
% 0.33/0.54            ( ( member_int @ X4 @ A2 )
% 0.33/0.54           => ( member_int @ X4 @ B2 ) )
% 0.33/0.54       => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subsetI
% 0.33/0.54  thf(fact_972_subset__antisym,axiom,
% 0.33/0.54      ! [A2: set_int,B2: set_int] :
% 0.33/0.54        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.33/0.54       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 0.33/0.54         => ( A2 = B2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_antisym
% 0.33/0.54  thf(fact_973_geqmaxNone,axiom,
% 0.33/0.54      ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
% 0.33/0.54        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 0.33/0.54       => ( ( ord_less_eq_nat @ Ma @ X )
% 0.33/0.54         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.33/0.54            = none_nat ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % geqmaxNone
% 0.33/0.54  thf(fact_974_psubsetD,axiom,
% 0.33/0.54      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
% 0.33/0.54        ( ( ord_le7866589430770878221at_nat @ A2 @ B2 )
% 0.33/0.54       => ( ( member8440522571783428010at_nat @ C @ A2 )
% 0.33/0.54         => ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % psubsetD
% 0.33/0.54  thf(fact_975_psubsetD,axiom,
% 0.33/0.54      ! [A2: set_real,B2: set_real,C: real] :
% 0.33/0.54        ( ( ord_less_set_real @ A2 @ B2 )
% 0.33/0.54       => ( ( member_real @ C @ A2 )
% 0.33/0.54         => ( member_real @ C @ B2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % psubsetD
% 0.33/0.54  thf(fact_976_psubsetD,axiom,
% 0.33/0.54      ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
% 0.33/0.54        ( ( ord_less_set_set_nat @ A2 @ B2 )
% 0.33/0.54       => ( ( member_set_nat @ C @ A2 )
% 0.33/0.54         => ( member_set_nat @ C @ B2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % psubsetD
% 0.33/0.54  thf(fact_977_psubsetD,axiom,
% 0.33/0.54      ! [A2: set_nat,B2: set_nat,C: nat] :
% 0.33/0.54        ( ( ord_less_set_nat @ A2 @ B2 )
% 0.33/0.54       => ( ( member_nat @ C @ A2 )
% 0.33/0.54         => ( member_nat @ C @ B2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % psubsetD
% 0.33/0.54  thf(fact_978_psubsetD,axiom,
% 0.33/0.54      ! [A2: set_int,B2: set_int,C: int] :
% 0.33/0.54        ( ( ord_less_set_int @ A2 @ B2 )
% 0.33/0.54       => ( ( member_int @ C @ A2 )
% 0.33/0.54         => ( member_int @ C @ B2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % psubsetD
% 0.33/0.54  thf(fact_979_vebt__succ_Osimps_I3_J,axiom,
% 0.33/0.54      ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
% 0.33/0.54        ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va )
% 0.33/0.54        = none_nat ) ).
% 0.33/0.54  
% 0.33/0.54  % vebt_succ.simps(3)
% 0.33/0.54  thf(fact_980_vebt__pred_Osimps_I4_J,axiom,
% 0.33/0.54      ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
% 0.33/0.54        ( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va ) @ Vb )
% 0.33/0.54        = none_nat ) ).
% 0.33/0.54  
% 0.33/0.54  % vebt_pred.simps(4)
% 0.33/0.54  thf(fact_981_vebt__member_Osimps_I2_J,axiom,
% 0.33/0.54      ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
% 0.33/0.54        ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X ) ).
% 0.33/0.54  
% 0.33/0.54  % vebt_member.simps(2)
% 0.33/0.54  thf(fact_982_VEBT__internal_OminNull_Osimps_I4_J,axiom,
% 0.33/0.54      ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).
% 0.33/0.54  
% 0.33/0.54  % VEBT_internal.minNull.simps(4)
% 0.33/0.54  thf(fact_983_vebt__delete_Osimps_I4_J,axiom,
% 0.33/0.54      ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
% 0.33/0.54        ( ( vEBT_vebt_delete @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Uu )
% 0.33/0.54        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) ) ).
% 0.33/0.54  
% 0.33/0.54  % vebt_delete.simps(4)
% 0.33/0.54  thf(fact_984_vebt__pred_Osimps_I5_J,axiom,
% 0.33/0.54      ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
% 0.33/0.54        ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
% 0.33/0.54        = none_nat ) ).
% 0.33/0.54  
% 0.33/0.54  % vebt_pred.simps(5)
% 0.33/0.54  thf(fact_985_vebt__succ_Osimps_I4_J,axiom,
% 0.33/0.54      ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
% 0.33/0.54        ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
% 0.33/0.54        = none_nat ) ).
% 0.33/0.54  
% 0.33/0.54  % vebt_succ.simps(4)
% 0.33/0.54  thf(fact_986_not__psubset__empty,axiom,
% 0.33/0.54      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.33/0.54        ~ ( ord_le7866589430770878221at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ).
% 0.33/0.54  
% 0.33/0.54  % not_psubset_empty
% 0.33/0.54  thf(fact_987_not__psubset__empty,axiom,
% 0.33/0.54      ! [A2: set_real] :
% 0.33/0.54        ~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).
% 0.33/0.54  
% 0.33/0.54  % not_psubset_empty
% 0.33/0.54  thf(fact_988_not__psubset__empty,axiom,
% 0.33/0.54      ! [A2: set_nat] :
% 0.33/0.54        ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% 0.33/0.54  
% 0.33/0.54  % not_psubset_empty
% 0.33/0.54  thf(fact_989_not__psubset__empty,axiom,
% 0.33/0.54      ! [A2: set_int] :
% 0.33/0.54        ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).
% 0.33/0.54  
% 0.33/0.54  % not_psubset_empty
% 0.33/0.54  thf(fact_990_emptyE,axiom,
% 0.33/0.54      ! [A: set_nat] :
% 0.33/0.54        ~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).
% 0.33/0.54  
% 0.33/0.54  % emptyE
% 0.33/0.54  thf(fact_991_emptyE,axiom,
% 0.33/0.54      ! [A: product_prod_nat_nat] :
% 0.33/0.54        ~ ( member8440522571783428010at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% 0.33/0.54  
% 0.33/0.54  % emptyE
% 0.33/0.54  thf(fact_992_emptyE,axiom,
% 0.33/0.54      ! [A: real] :
% 0.33/0.54        ~ ( member_real @ A @ bot_bot_set_real ) ).
% 0.33/0.54  
% 0.33/0.54  % emptyE
% 0.33/0.54  thf(fact_993_emptyE,axiom,
% 0.33/0.54      ! [A: nat] :
% 0.33/0.54        ~ ( member_nat @ A @ bot_bot_set_nat ) ).
% 0.33/0.54  
% 0.33/0.54  % emptyE
% 0.33/0.54  thf(fact_994_emptyE,axiom,
% 0.33/0.54      ! [A: int] :
% 0.33/0.54        ~ ( member_int @ A @ bot_bot_set_int ) ).
% 0.33/0.54  
% 0.33/0.54  % emptyE
% 0.33/0.54  thf(fact_995_equals0D,axiom,
% 0.33/0.54      ! [A2: set_set_nat,A: set_nat] :
% 0.33/0.54        ( ( A2 = bot_bot_set_set_nat )
% 0.33/0.54       => ~ ( member_set_nat @ A @ A2 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % equals0D
% 0.33/0.54  thf(fact_996_equals0D,axiom,
% 0.33/0.54      ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
% 0.33/0.54        ( ( A2 = bot_bo2099793752762293965at_nat )
% 0.33/0.54       => ~ ( member8440522571783428010at_nat @ A @ A2 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % equals0D
% 0.33/0.54  thf(fact_997_equals0D,axiom,
% 0.33/0.54      ! [A2: set_real,A: real] :
% 0.33/0.54        ( ( A2 = bot_bot_set_real )
% 0.33/0.54       => ~ ( member_real @ A @ A2 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % equals0D
% 0.33/0.54  thf(fact_998_equals0D,axiom,
% 0.33/0.54      ! [A2: set_nat,A: nat] :
% 0.33/0.54        ( ( A2 = bot_bot_set_nat )
% 0.33/0.54       => ~ ( member_nat @ A @ A2 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % equals0D
% 0.33/0.54  thf(fact_999_equals0D,axiom,
% 0.33/0.54      ! [A2: set_int,A: int] :
% 0.33/0.54        ( ( A2 = bot_bot_set_int )
% 0.33/0.54       => ~ ( member_int @ A @ A2 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % equals0D
% 0.33/0.54  thf(fact_1000_equals0I,axiom,
% 0.33/0.54      ! [A2: set_set_nat] :
% 0.33/0.54        ( ! [Y3: set_nat] :
% 0.33/0.54            ~ ( member_set_nat @ Y3 @ A2 )
% 0.33/0.54       => ( A2 = bot_bot_set_set_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % equals0I
% 0.33/0.54  thf(fact_1001_equals0I,axiom,
% 0.33/0.54      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.33/0.54        ( ! [Y3: product_prod_nat_nat] :
% 0.33/0.54            ~ ( member8440522571783428010at_nat @ Y3 @ A2 )
% 0.33/0.54       => ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % equals0I
% 0.33/0.54  thf(fact_1002_equals0I,axiom,
% 0.33/0.54      ! [A2: set_real] :
% 0.33/0.54        ( ! [Y3: real] :
% 0.33/0.54            ~ ( member_real @ Y3 @ A2 )
% 0.33/0.54       => ( A2 = bot_bot_set_real ) ) ).
% 0.33/0.54  
% 0.33/0.54  % equals0I
% 0.33/0.54  thf(fact_1003_equals0I,axiom,
% 0.33/0.54      ! [A2: set_nat] :
% 0.33/0.54        ( ! [Y3: nat] :
% 0.33/0.54            ~ ( member_nat @ Y3 @ A2 )
% 0.33/0.54       => ( A2 = bot_bot_set_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % equals0I
% 0.33/0.54  thf(fact_1004_equals0I,axiom,
% 0.33/0.54      ! [A2: set_int] :
% 0.33/0.54        ( ! [Y3: int] :
% 0.33/0.54            ~ ( member_int @ Y3 @ A2 )
% 0.33/0.54       => ( A2 = bot_bot_set_int ) ) ).
% 0.33/0.54  
% 0.33/0.54  % equals0I
% 0.33/0.54  thf(fact_1005_ex__in__conv,axiom,
% 0.33/0.54      ! [A2: set_set_nat] :
% 0.33/0.54        ( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
% 0.33/0.54        = ( A2 != bot_bot_set_set_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % ex_in_conv
% 0.33/0.54  thf(fact_1006_ex__in__conv,axiom,
% 0.33/0.54      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.33/0.54        ( ( ? [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A2 ) )
% 0.33/0.54        = ( A2 != bot_bo2099793752762293965at_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % ex_in_conv
% 0.33/0.54  thf(fact_1007_ex__in__conv,axiom,
% 0.33/0.54      ! [A2: set_real] :
% 0.33/0.54        ( ( ? [X3: real] : ( member_real @ X3 @ A2 ) )
% 0.33/0.54        = ( A2 != bot_bot_set_real ) ) ).
% 0.33/0.54  
% 0.33/0.54  % ex_in_conv
% 0.33/0.54  thf(fact_1008_ex__in__conv,axiom,
% 0.33/0.54      ! [A2: set_nat] :
% 0.33/0.54        ( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
% 0.33/0.54        = ( A2 != bot_bot_set_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % ex_in_conv
% 0.33/0.54  thf(fact_1009_ex__in__conv,axiom,
% 0.33/0.54      ! [A2: set_int] :
% 0.33/0.54        ( ( ? [X3: int] : ( member_int @ X3 @ A2 ) )
% 0.33/0.54        = ( A2 != bot_bot_set_int ) ) ).
% 0.33/0.54  
% 0.33/0.54  % ex_in_conv
% 0.33/0.54  thf(fact_1010_bot__set__def,axiom,
% 0.33/0.54      ( bot_bot_set_list_nat
% 0.33/0.54      = ( collect_list_nat @ bot_bot_list_nat_o ) ) ).
% 0.33/0.54  
% 0.33/0.54  % bot_set_def
% 0.33/0.54  thf(fact_1011_bot__set__def,axiom,
% 0.33/0.54      ( bot_bot_set_set_nat
% 0.33/0.54      = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% 0.33/0.54  
% 0.33/0.54  % bot_set_def
% 0.33/0.54  thf(fact_1012_bot__set__def,axiom,
% 0.33/0.54      ( bot_bo2099793752762293965at_nat
% 0.33/0.54      = ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) ) ).
% 0.33/0.54  
% 0.33/0.54  % bot_set_def
% 0.33/0.54  thf(fact_1013_bot__set__def,axiom,
% 0.33/0.54      ( bot_bot_set_real
% 0.33/0.54      = ( collect_real @ bot_bot_real_o ) ) ).
% 0.33/0.54  
% 0.33/0.54  % bot_set_def
% 0.33/0.54  thf(fact_1014_bot__set__def,axiom,
% 0.33/0.54      ( bot_bot_set_nat
% 0.33/0.54      = ( collect_nat @ bot_bot_nat_o ) ) ).
% 0.33/0.54  
% 0.33/0.54  % bot_set_def
% 0.33/0.54  thf(fact_1015_bot__set__def,axiom,
% 0.33/0.54      ( bot_bot_set_int
% 0.33/0.54      = ( collect_int @ bot_bot_int_o ) ) ).
% 0.33/0.54  
% 0.33/0.54  % bot_set_def
% 0.33/0.54  thf(fact_1016_vebt__delete_Osimps_I5_J,axiom,
% 0.33/0.54      ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X: nat] :
% 0.33/0.54        ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) @ X )
% 0.33/0.54        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) ) ).
% 0.33/0.54  
% 0.33/0.54  % vebt_delete.simps(5)
% 0.33/0.54  thf(fact_1017_in__mono,axiom,
% 0.33/0.54      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
% 0.33/0.54        ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 0.33/0.54       => ( ( member8440522571783428010at_nat @ X @ A2 )
% 0.33/0.54         => ( member8440522571783428010at_nat @ X @ B2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % in_mono
% 0.33/0.54  thf(fact_1018_in__mono,axiom,
% 0.33/0.54      ! [A2: set_real,B2: set_real,X: real] :
% 0.33/0.54        ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.33/0.54       => ( ( member_real @ X @ A2 )
% 0.33/0.54         => ( member_real @ X @ B2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % in_mono
% 0.33/0.54  thf(fact_1019_in__mono,axiom,
% 0.33/0.54      ! [A2: set_set_nat,B2: set_set_nat,X: set_nat] :
% 0.33/0.54        ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
% 0.33/0.54       => ( ( member_set_nat @ X @ A2 )
% 0.33/0.54         => ( member_set_nat @ X @ B2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % in_mono
% 0.33/0.54  thf(fact_1020_in__mono,axiom,
% 0.33/0.54      ! [A2: set_nat,B2: set_nat,X: nat] :
% 0.33/0.54        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.33/0.54       => ( ( member_nat @ X @ A2 )
% 0.33/0.54         => ( member_nat @ X @ B2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % in_mono
% 0.33/0.54  thf(fact_1021_in__mono,axiom,
% 0.33/0.54      ! [A2: set_int,B2: set_int,X: int] :
% 0.33/0.54        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.33/0.54       => ( ( member_int @ X @ A2 )
% 0.33/0.54         => ( member_int @ X @ B2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % in_mono
% 0.33/0.54  thf(fact_1022_subsetD,axiom,
% 0.33/0.54      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
% 0.33/0.54        ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 0.33/0.54       => ( ( member8440522571783428010at_nat @ C @ A2 )
% 0.33/0.54         => ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subsetD
% 0.33/0.54  thf(fact_1023_subsetD,axiom,
% 0.33/0.54      ! [A2: set_real,B2: set_real,C: real] :
% 0.33/0.54        ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.33/0.54       => ( ( member_real @ C @ A2 )
% 0.33/0.54         => ( member_real @ C @ B2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subsetD
% 0.33/0.54  thf(fact_1024_subsetD,axiom,
% 0.33/0.54      ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
% 0.33/0.54        ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
% 0.33/0.54       => ( ( member_set_nat @ C @ A2 )
% 0.33/0.54         => ( member_set_nat @ C @ B2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subsetD
% 0.33/0.54  thf(fact_1025_subsetD,axiom,
% 0.33/0.54      ! [A2: set_nat,B2: set_nat,C: nat] :
% 0.33/0.54        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.33/0.54       => ( ( member_nat @ C @ A2 )
% 0.33/0.54         => ( member_nat @ C @ B2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subsetD
% 0.33/0.54  thf(fact_1026_subsetD,axiom,
% 0.33/0.54      ! [A2: set_int,B2: set_int,C: int] :
% 0.33/0.54        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.33/0.54       => ( ( member_int @ C @ A2 )
% 0.33/0.54         => ( member_int @ C @ B2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subsetD
% 0.33/0.54  thf(fact_1027_equalityE,axiom,
% 0.33/0.54      ! [A2: set_int,B2: set_int] :
% 0.33/0.54        ( ( A2 = B2 )
% 0.33/0.54       => ~ ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.33/0.54           => ~ ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % equalityE
% 0.33/0.54  thf(fact_1028_subset__eq,axiom,
% 0.33/0.54      ( ord_le3146513528884898305at_nat
% 0.33/0.54      = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 0.33/0.54          ! [X3: product_prod_nat_nat] :
% 0.33/0.54            ( ( member8440522571783428010at_nat @ X3 @ A6 )
% 0.33/0.54           => ( member8440522571783428010at_nat @ X3 @ B6 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_eq
% 0.33/0.54  thf(fact_1029_subset__eq,axiom,
% 0.33/0.54      ( ord_less_eq_set_real
% 0.33/0.54      = ( ^ [A6: set_real,B6: set_real] :
% 0.33/0.54          ! [X3: real] :
% 0.33/0.54            ( ( member_real @ X3 @ A6 )
% 0.33/0.54           => ( member_real @ X3 @ B6 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_eq
% 0.33/0.54  thf(fact_1030_subset__eq,axiom,
% 0.33/0.54      ( ord_le6893508408891458716et_nat
% 0.33/0.54      = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 0.33/0.54          ! [X3: set_nat] :
% 0.33/0.54            ( ( member_set_nat @ X3 @ A6 )
% 0.33/0.54           => ( member_set_nat @ X3 @ B6 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_eq
% 0.33/0.54  thf(fact_1031_subset__eq,axiom,
% 0.33/0.54      ( ord_less_eq_set_nat
% 0.33/0.54      = ( ^ [A6: set_nat,B6: set_nat] :
% 0.33/0.54          ! [X3: nat] :
% 0.33/0.54            ( ( member_nat @ X3 @ A6 )
% 0.33/0.54           => ( member_nat @ X3 @ B6 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_eq
% 0.33/0.54  thf(fact_1032_subset__eq,axiom,
% 0.33/0.54      ( ord_less_eq_set_int
% 0.33/0.54      = ( ^ [A6: set_int,B6: set_int] :
% 0.33/0.54          ! [X3: int] :
% 0.33/0.54            ( ( member_int @ X3 @ A6 )
% 0.33/0.54           => ( member_int @ X3 @ B6 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_eq
% 0.33/0.54  thf(fact_1033_equalityD1,axiom,
% 0.33/0.54      ! [A2: set_int,B2: set_int] :
% 0.33/0.54        ( ( A2 = B2 )
% 0.33/0.54       => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % equalityD1
% 0.33/0.54  thf(fact_1034_equalityD2,axiom,
% 0.33/0.54      ! [A2: set_int,B2: set_int] :
% 0.33/0.54        ( ( A2 = B2 )
% 0.33/0.54       => ( ord_less_eq_set_int @ B2 @ A2 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % equalityD2
% 0.33/0.54  thf(fact_1035_subset__iff,axiom,
% 0.33/0.54      ( ord_le3146513528884898305at_nat
% 0.33/0.54      = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 0.33/0.54          ! [T2: product_prod_nat_nat] :
% 0.33/0.54            ( ( member8440522571783428010at_nat @ T2 @ A6 )
% 0.33/0.54           => ( member8440522571783428010at_nat @ T2 @ B6 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_iff
% 0.33/0.54  thf(fact_1036_subset__iff,axiom,
% 0.33/0.54      ( ord_less_eq_set_real
% 0.33/0.54      = ( ^ [A6: set_real,B6: set_real] :
% 0.33/0.54          ! [T2: real] :
% 0.33/0.54            ( ( member_real @ T2 @ A6 )
% 0.33/0.54           => ( member_real @ T2 @ B6 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_iff
% 0.33/0.54  thf(fact_1037_subset__iff,axiom,
% 0.33/0.54      ( ord_le6893508408891458716et_nat
% 0.33/0.54      = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 0.33/0.54          ! [T2: set_nat] :
% 0.33/0.54            ( ( member_set_nat @ T2 @ A6 )
% 0.33/0.54           => ( member_set_nat @ T2 @ B6 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_iff
% 0.33/0.54  thf(fact_1038_subset__iff,axiom,
% 0.33/0.54      ( ord_less_eq_set_nat
% 0.33/0.54      = ( ^ [A6: set_nat,B6: set_nat] :
% 0.33/0.54          ! [T2: nat] :
% 0.33/0.54            ( ( member_nat @ T2 @ A6 )
% 0.33/0.54           => ( member_nat @ T2 @ B6 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_iff
% 0.33/0.54  thf(fact_1039_subset__iff,axiom,
% 0.33/0.54      ( ord_less_eq_set_int
% 0.33/0.54      = ( ^ [A6: set_int,B6: set_int] :
% 0.33/0.54          ! [T2: int] :
% 0.33/0.54            ( ( member_int @ T2 @ A6 )
% 0.33/0.54           => ( member_int @ T2 @ B6 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_iff
% 0.33/0.54  thf(fact_1040_subset__refl,axiom,
% 0.33/0.54      ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_refl
% 0.33/0.54  thf(fact_1041_Collect__mono,axiom,
% 0.33/0.54      ! [P: real > $o,Q: real > $o] :
% 0.33/0.54        ( ! [X4: real] :
% 0.33/0.54            ( ( P @ X4 )
% 0.33/0.54           => ( Q @ X4 ) )
% 0.33/0.54       => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_mono
% 0.33/0.54  thf(fact_1042_Collect__mono,axiom,
% 0.33/0.54      ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.33/0.54        ( ! [X4: list_nat] :
% 0.33/0.54            ( ( P @ X4 )
% 0.33/0.54           => ( Q @ X4 ) )
% 0.33/0.54       => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_mono
% 0.33/0.54  thf(fact_1043_Collect__mono,axiom,
% 0.33/0.54      ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.33/0.54        ( ! [X4: set_nat] :
% 0.33/0.54            ( ( P @ X4 )
% 0.33/0.54           => ( Q @ X4 ) )
% 0.33/0.54       => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_mono
% 0.33/0.54  thf(fact_1044_Collect__mono,axiom,
% 0.33/0.54      ! [P: nat > $o,Q: nat > $o] :
% 0.33/0.54        ( ! [X4: nat] :
% 0.33/0.54            ( ( P @ X4 )
% 0.33/0.54           => ( Q @ X4 ) )
% 0.33/0.54       => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_mono
% 0.33/0.54  thf(fact_1045_Collect__mono,axiom,
% 0.33/0.54      ! [P: int > $o,Q: int > $o] :
% 0.33/0.54        ( ! [X4: int] :
% 0.33/0.54            ( ( P @ X4 )
% 0.33/0.54           => ( Q @ X4 ) )
% 0.33/0.54       => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_mono
% 0.33/0.54  thf(fact_1046_subset__trans,axiom,
% 0.33/0.54      ! [A2: set_int,B2: set_int,C2: set_int] :
% 0.33/0.54        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.33/0.54       => ( ( ord_less_eq_set_int @ B2 @ C2 )
% 0.33/0.54         => ( ord_less_eq_set_int @ A2 @ C2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_trans
% 0.33/0.54  thf(fact_1047_set__eq__subset,axiom,
% 0.33/0.54      ( ( ^ [Y5: set_int,Z: set_int] : ( Y5 = Z ) )
% 0.33/0.54      = ( ^ [A6: set_int,B6: set_int] :
% 0.33/0.54            ( ( ord_less_eq_set_int @ A6 @ B6 )
% 0.33/0.54            & ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % set_eq_subset
% 0.33/0.54  thf(fact_1048_Collect__mono__iff,axiom,
% 0.33/0.54      ! [P: real > $o,Q: real > $o] :
% 0.33/0.54        ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
% 0.33/0.54        = ( ! [X3: real] :
% 0.33/0.54              ( ( P @ X3 )
% 0.33/0.54             => ( Q @ X3 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_mono_iff
% 0.33/0.54  thf(fact_1049_Collect__mono__iff,axiom,
% 0.33/0.54      ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.33/0.54        ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
% 0.33/0.54        = ( ! [X3: list_nat] :
% 0.33/0.54              ( ( P @ X3 )
% 0.33/0.54             => ( Q @ X3 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_mono_iff
% 0.33/0.54  thf(fact_1050_Collect__mono__iff,axiom,
% 0.33/0.54      ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.33/0.54        ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
% 0.33/0.54        = ( ! [X3: set_nat] :
% 0.33/0.54              ( ( P @ X3 )
% 0.33/0.54             => ( Q @ X3 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_mono_iff
% 0.33/0.54  thf(fact_1051_Collect__mono__iff,axiom,
% 0.33/0.54      ! [P: nat > $o,Q: nat > $o] :
% 0.33/0.54        ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
% 0.33/0.54        = ( ! [X3: nat] :
% 0.33/0.54              ( ( P @ X3 )
% 0.33/0.54             => ( Q @ X3 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_mono_iff
% 0.33/0.54  thf(fact_1052_Collect__mono__iff,axiom,
% 0.33/0.54      ! [P: int > $o,Q: int > $o] :
% 0.33/0.54        ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
% 0.33/0.54        = ( ! [X3: int] :
% 0.33/0.54              ( ( P @ X3 )
% 0.33/0.54             => ( Q @ X3 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_mono_iff
% 0.33/0.54  thf(fact_1053_subset__iff__psubset__eq,axiom,
% 0.33/0.54      ( ord_less_eq_set_int
% 0.33/0.54      = ( ^ [A6: set_int,B6: set_int] :
% 0.33/0.54            ( ( ord_less_set_int @ A6 @ B6 )
% 0.33/0.54            | ( A6 = B6 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_iff_psubset_eq
% 0.33/0.54  thf(fact_1054_subset__psubset__trans,axiom,
% 0.33/0.54      ! [A2: set_int,B2: set_int,C2: set_int] :
% 0.33/0.54        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.33/0.54       => ( ( ord_less_set_int @ B2 @ C2 )
% 0.33/0.54         => ( ord_less_set_int @ A2 @ C2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_psubset_trans
% 0.33/0.54  thf(fact_1055_subset__not__subset__eq,axiom,
% 0.33/0.54      ( ord_less_set_int
% 0.33/0.54      = ( ^ [A6: set_int,B6: set_int] :
% 0.33/0.54            ( ( ord_less_eq_set_int @ A6 @ B6 )
% 0.33/0.54            & ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_not_subset_eq
% 0.33/0.54  thf(fact_1056_psubset__subset__trans,axiom,
% 0.33/0.54      ! [A2: set_int,B2: set_int,C2: set_int] :
% 0.33/0.54        ( ( ord_less_set_int @ A2 @ B2 )
% 0.33/0.54       => ( ( ord_less_eq_set_int @ B2 @ C2 )
% 0.33/0.54         => ( ord_less_set_int @ A2 @ C2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % psubset_subset_trans
% 0.33/0.54  thf(fact_1057_psubset__imp__subset,axiom,
% 0.33/0.54      ! [A2: set_int,B2: set_int] :
% 0.33/0.54        ( ( ord_less_set_int @ A2 @ B2 )
% 0.33/0.54       => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % psubset_imp_subset
% 0.33/0.54  thf(fact_1058_psubset__eq,axiom,
% 0.33/0.54      ( ord_less_set_int
% 0.33/0.54      = ( ^ [A6: set_int,B6: set_int] :
% 0.33/0.54            ( ( ord_less_eq_set_int @ A6 @ B6 )
% 0.33/0.54            & ( A6 != B6 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % psubset_eq
% 0.33/0.54  thf(fact_1059_psubsetE,axiom,
% 0.33/0.54      ! [A2: set_int,B2: set_int] :
% 0.33/0.54        ( ( ord_less_set_int @ A2 @ B2 )
% 0.33/0.54       => ~ ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.33/0.54           => ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % psubsetE
% 0.33/0.54  thf(fact_1060_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
% 0.33/0.54      ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
% 0.33/0.54        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X )
% 0.33/0.54        = ( ( X = Mi )
% 0.33/0.54          | ( X = Ma ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % VEBT_internal.membermima.simps(3)
% 0.33/0.54  thf(fact_1061_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
% 0.33/0.54      ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
% 0.33/0.54        ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).
% 0.33/0.54  
% 0.33/0.54  % VEBT_internal.membermima.simps(2)
% 0.33/0.54  thf(fact_1062_VEBT__internal_OminNull_Osimps_I5_J,axiom,
% 0.33/0.54      ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 0.33/0.54        ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).
% 0.33/0.54  
% 0.33/0.54  % VEBT_internal.minNull.simps(5)
% 0.33/0.54  thf(fact_1063_tvalid,axiom,
% 0.33/0.54      vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ deg ).
% 0.33/0.54  
% 0.33/0.54  % tvalid
% 0.33/0.54  thf(fact_1064_vebt__mint_Osimps_I3_J,axiom,
% 0.33/0.54      ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.33/0.54        ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 0.33/0.54        = ( some_nat @ Mi ) ) ).
% 0.33/0.54  
% 0.33/0.54  % vebt_mint.simps(3)
% 0.33/0.54  thf(fact_1065_vebt__maxt_Osimps_I3_J,axiom,
% 0.33/0.54      ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.33/0.54        ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 0.33/0.54        = ( some_nat @ Ma ) ) ).
% 0.33/0.54  
% 0.33/0.54  % vebt_maxt.simps(3)
% 0.33/0.54  thf(fact_1066_vebt__mint_Osimps_I2_J,axiom,
% 0.33/0.54      ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.33/0.54        ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 0.33/0.54        = none_nat ) ).
% 0.33/0.54  
% 0.33/0.54  % vebt_mint.simps(2)
% 0.33/0.54  thf(fact_1067_vebt__maxt_Osimps_I2_J,axiom,
% 0.33/0.54      ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.33/0.54        ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 0.33/0.54        = none_nat ) ).
% 0.33/0.54  
% 0.33/0.54  % vebt_maxt.simps(2)
% 0.33/0.54  thf(fact_1068_calculation,axiom,
% 0.33/0.54      ( ( ya = mi )
% 0.33/0.54     => ( ( xa != ya )
% 0.33/0.54        & ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ ya ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % calculation
% 0.33/0.54  thf(fact_1069_prod_Oinject,axiom,
% 0.33/0.54      ! [X1: int,X22: int,Y1: int,Y22: int] :
% 0.33/0.54        ( ( ( product_Pair_int_int @ X1 @ X22 )
% 0.33/0.54          = ( product_Pair_int_int @ Y1 @ Y22 ) )
% 0.33/0.54        = ( ( X1 = Y1 )
% 0.33/0.54          & ( X22 = Y22 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % prod.inject
% 0.33/0.54  thf(fact_1070_prod_Oinject,axiom,
% 0.33/0.54      ! [X1: code_integer > option6357759511663192854e_term,X22: produc8923325533196201883nteger,Y1: code_integer > option6357759511663192854e_term,Y22: produc8923325533196201883nteger] :
% 0.33/0.54        ( ( ( produc6137756002093451184nteger @ X1 @ X22 )
% 0.33/0.54          = ( produc6137756002093451184nteger @ Y1 @ Y22 ) )
% 0.33/0.54        = ( ( X1 = Y1 )
% 0.33/0.54          & ( X22 = Y22 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % prod.inject
% 0.33/0.54  thf(fact_1071_prod_Oinject,axiom,
% 0.33/0.54      ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X22: produc8923325533196201883nteger,Y1: produc6241069584506657477e_term > option6357759511663192854e_term,Y22: produc8923325533196201883nteger] :
% 0.33/0.54        ( ( ( produc8603105652947943368nteger @ X1 @ X22 )
% 0.33/0.54          = ( produc8603105652947943368nteger @ Y1 @ Y22 ) )
% 0.33/0.54        = ( ( X1 = Y1 )
% 0.33/0.54          & ( X22 = Y22 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % prod.inject
% 0.33/0.54  thf(fact_1072_prod_Oinject,axiom,
% 0.33/0.54      ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X22: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y22: product_prod_int_int] :
% 0.33/0.54        ( ( ( produc5700946648718959541nt_int @ X1 @ X22 )
% 0.33/0.54          = ( produc5700946648718959541nt_int @ Y1 @ Y22 ) )
% 0.33/0.54        = ( ( X1 = Y1 )
% 0.33/0.54          & ( X22 = Y22 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % prod.inject
% 0.33/0.54  thf(fact_1073_prod_Oinject,axiom,
% 0.33/0.54      ! [X1: int > option6357759511663192854e_term,X22: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y22: product_prod_int_int] :
% 0.33/0.54        ( ( ( produc4305682042979456191nt_int @ X1 @ X22 )
% 0.33/0.54          = ( produc4305682042979456191nt_int @ Y1 @ Y22 ) )
% 0.33/0.54        = ( ( X1 = Y1 )
% 0.33/0.54          & ( X22 = Y22 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % prod.inject
% 0.33/0.54  thf(fact_1074_old_Oprod_Oinject,axiom,
% 0.33/0.54      ! [A: int,B: int,A7: int,B7: int] :
% 0.33/0.54        ( ( ( product_Pair_int_int @ A @ B )
% 0.33/0.54          = ( product_Pair_int_int @ A7 @ B7 ) )
% 0.33/0.54        = ( ( A = A7 )
% 0.33/0.54          & ( B = B7 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % old.prod.inject
% 0.33/0.54  thf(fact_1075_old_Oprod_Oinject,axiom,
% 0.33/0.54      ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A7: code_integer > option6357759511663192854e_term,B7: produc8923325533196201883nteger] :
% 0.33/0.54        ( ( ( produc6137756002093451184nteger @ A @ B )
% 0.33/0.54          = ( produc6137756002093451184nteger @ A7 @ B7 ) )
% 0.33/0.54        = ( ( A = A7 )
% 0.33/0.54          & ( B = B7 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % old.prod.inject
% 0.33/0.54  thf(fact_1076_old_Oprod_Oinject,axiom,
% 0.33/0.54      ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A7: produc6241069584506657477e_term > option6357759511663192854e_term,B7: produc8923325533196201883nteger] :
% 0.33/0.54        ( ( ( produc8603105652947943368nteger @ A @ B )
% 0.33/0.54          = ( produc8603105652947943368nteger @ A7 @ B7 ) )
% 0.33/0.54        = ( ( A = A7 )
% 0.33/0.54          & ( B = B7 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % old.prod.inject
% 0.33/0.54  thf(fact_1077_old_Oprod_Oinject,axiom,
% 0.33/0.54      ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A7: produc8551481072490612790e_term > option6357759511663192854e_term,B7: product_prod_int_int] :
% 0.33/0.54        ( ( ( produc5700946648718959541nt_int @ A @ B )
% 0.33/0.54          = ( produc5700946648718959541nt_int @ A7 @ B7 ) )
% 0.33/0.54        = ( ( A = A7 )
% 0.33/0.54          & ( B = B7 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % old.prod.inject
% 0.33/0.54  thf(fact_1078_old_Oprod_Oinject,axiom,
% 0.33/0.54      ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A7: int > option6357759511663192854e_term,B7: product_prod_int_int] :
% 0.33/0.54        ( ( ( produc4305682042979456191nt_int @ A @ B )
% 0.33/0.54          = ( produc4305682042979456191nt_int @ A7 @ B7 ) )
% 0.33/0.54        = ( ( A = A7 )
% 0.33/0.54          & ( B = B7 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % old.prod.inject
% 0.33/0.54  thf(fact_1079_infinite__nat__iff__unbounded__le,axiom,
% 0.33/0.54      ! [S2: set_nat] :
% 0.33/0.54        ( ( ~ ( finite_finite_nat @ S2 ) )
% 0.33/0.54        = ( ! [M2: nat] :
% 0.33/0.54            ? [N2: nat] :
% 0.33/0.54              ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.33/0.54              & ( member_nat @ N2 @ S2 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % infinite_nat_iff_unbounded_le
% 0.33/0.54  thf(fact_1080_finite__nat__set__iff__bounded__le,axiom,
% 0.33/0.54      ( finite_finite_nat
% 0.33/0.54      = ( ^ [N4: set_nat] :
% 0.33/0.54          ? [M2: nat] :
% 0.33/0.54          ! [X3: nat] :
% 0.33/0.54            ( ( member_nat @ X3 @ N4 )
% 0.33/0.54           => ( ord_less_eq_nat @ X3 @ M2 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_nat_set_iff_bounded_le
% 0.33/0.54  thf(fact_1081_unbounded__k__infinite,axiom,
% 0.33/0.54      ! [K: nat,S2: set_nat] :
% 0.33/0.54        ( ! [M5: nat] :
% 0.33/0.54            ( ( ord_less_nat @ K @ M5 )
% 0.33/0.54           => ? [N5: nat] :
% 0.33/0.54                ( ( ord_less_nat @ M5 @ N5 )
% 0.33/0.54                & ( member_nat @ N5 @ S2 ) ) )
% 0.33/0.54       => ~ ( finite_finite_nat @ S2 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % unbounded_k_infinite
% 0.33/0.54  thf(fact_1082__C4_Ohyps_C_I4_J,axiom,
% 0.33/0.54      ( deg
% 0.33/0.54      = ( plus_plus_nat @ na @ m ) ) ).
% 0.33/0.54  
% 0.33/0.54  % "4.hyps"(4)
% 0.33/0.54  thf(fact_1083_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.33/0.54      ! [X: produc8306885398267862888on_nat] :
% 0.33/0.54        ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
% 0.33/0.54            ( X
% 0.33/0.54           != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 0.33/0.54       => ( ! [Uw2: nat > nat > nat,V2: nat] :
% 0.33/0.54              ( X
% 0.33/0.54             != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 0.33/0.54         => ~ ! [F2: nat > nat > nat,A3: nat,B3: nat] :
% 0.33/0.54                ( X
% 0.33/0.54               != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A3 ) @ ( some_nat @ B3 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % VEBT_internal.option_shift.cases
% 0.33/0.54  thf(fact_1084_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.33/0.54      ! [X: produc5542196010084753463at_nat] :
% 0.33/0.54        ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
% 0.33/0.54            ( X
% 0.33/0.54           != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 0.33/0.54       => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
% 0.33/0.54              ( X
% 0.33/0.54             != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 0.33/0.54         => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
% 0.33/0.54                ( X
% 0.33/0.54               != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B3 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % VEBT_internal.option_shift.cases
% 0.33/0.54  thf(fact_1085_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.33/0.54      ! [X: produc1193250871479095198on_num] :
% 0.33/0.54        ( ! [Uu2: num > num > num,Uv2: option_num] :
% 0.33/0.54            ( X
% 0.33/0.54           != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 0.33/0.54       => ( ! [Uw2: num > num > num,V2: num] :
% 0.33/0.54              ( X
% 0.33/0.54             != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 0.33/0.54         => ~ ! [F2: num > num > num,A3: num,B3: num] :
% 0.33/0.54                ( X
% 0.33/0.54               != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A3 ) @ ( some_num @ B3 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % VEBT_internal.option_shift.cases
% 0.33/0.54  thf(fact_1086_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.33/0.54      ! [X: produc2233624965454879586on_nat] :
% 0.33/0.54        ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
% 0.33/0.54            ( X
% 0.33/0.54           != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 0.33/0.54       => ( ! [Uw2: nat > nat > $o,V2: nat] :
% 0.33/0.54              ( X
% 0.33/0.54             != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 0.33/0.54         => ~ ! [F2: nat > nat > $o,X4: nat,Y3: nat] :
% 0.33/0.54                ( X
% 0.33/0.54               != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X4 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % VEBT_internal.option_comp_shift.cases
% 0.33/0.54  thf(fact_1087_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.33/0.54      ! [X: produc5491161045314408544at_nat] :
% 0.33/0.54        ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
% 0.33/0.54            ( X
% 0.33/0.54           != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 0.33/0.54       => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
% 0.33/0.54              ( X
% 0.33/0.54             != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 0.33/0.54         => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 0.33/0.54                ( X
% 0.33/0.54               != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X4 ) @ ( some_P7363390416028606310at_nat @ Y3 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % VEBT_internal.option_comp_shift.cases
% 0.33/0.54  thf(fact_1088_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.33/0.54      ! [X: produc7036089656553540234on_num] :
% 0.33/0.54        ( ! [Uu2: num > num > $o,Uv2: option_num] :
% 0.33/0.54            ( X
% 0.33/0.54           != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 0.33/0.54       => ( ! [Uw2: num > num > $o,V2: num] :
% 0.33/0.54              ( X
% 0.33/0.54             != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 0.33/0.54         => ~ ! [F2: num > num > $o,X4: num,Y3: num] :
% 0.33/0.54                ( X
% 0.33/0.54               != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X4 ) @ ( some_num @ Y3 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % VEBT_internal.option_comp_shift.cases
% 0.33/0.54  thf(fact_1089_prod__induct3,axiom,
% 0.33/0.54      ! [P: produc8763457246119570046nteger > $o,X: produc8763457246119570046nteger] :
% 0.33/0.54        ( ! [A3: code_integer > option6357759511663192854e_term,B3: code_integer,C3: code_integer] : ( P @ ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C3 ) ) )
% 0.33/0.54       => ( P @ X ) ) ).
% 0.33/0.54  
% 0.33/0.54  % prod_induct3
% 0.33/0.54  thf(fact_1090_prod__induct3,axiom,
% 0.33/0.54      ! [P: produc1908205239877642774nteger > $o,X: produc1908205239877642774nteger] :
% 0.33/0.54        ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: code_integer,C3: code_integer] : ( P @ ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C3 ) ) )
% 0.33/0.54       => ( P @ X ) ) ).
% 0.33/0.54  
% 0.33/0.54  % prod_induct3
% 0.33/0.54  thf(fact_1091_prod__induct3,axiom,
% 0.33/0.54      ! [P: produc2285326912895808259nt_int > $o,X: produc2285326912895808259nt_int] :
% 0.33/0.54        ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: int,C3: int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C3 ) ) )
% 0.33/0.54       => ( P @ X ) ) ).
% 0.33/0.54  
% 0.33/0.54  % prod_induct3
% 0.33/0.54  thf(fact_1092_prod__induct3,axiom,
% 0.33/0.54      ! [P: produc7773217078559923341nt_int > $o,X: produc7773217078559923341nt_int] :
% 0.33/0.54        ( ! [A3: int > option6357759511663192854e_term,B3: int,C3: int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C3 ) ) )
% 0.33/0.54       => ( P @ X ) ) ).
% 0.33/0.54  
% 0.33/0.54  % prod_induct3
% 0.33/0.54  thf(fact_1093_prod__cases3,axiom,
% 0.33/0.54      ! [Y: produc8763457246119570046nteger] :
% 0.33/0.54        ~ ! [A3: code_integer > option6357759511663192854e_term,B3: code_integer,C3: code_integer] :
% 0.33/0.54            ( Y
% 0.33/0.54           != ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % prod_cases3
% 0.33/0.54  thf(fact_1094_prod__cases3,axiom,
% 0.33/0.54      ! [Y: produc1908205239877642774nteger] :
% 0.33/0.54        ~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: code_integer,C3: code_integer] :
% 0.33/0.54            ( Y
% 0.33/0.54           != ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % prod_cases3
% 0.33/0.54  thf(fact_1095_prod__cases3,axiom,
% 0.33/0.54      ! [Y: produc2285326912895808259nt_int] :
% 0.33/0.54        ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: int,C3: int] :
% 0.33/0.54            ( Y
% 0.33/0.54           != ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % prod_cases3
% 0.33/0.54  thf(fact_1096_prod__cases3,axiom,
% 0.33/0.54      ! [Y: produc7773217078559923341nt_int] :
% 0.33/0.54        ~ ! [A3: int > option6357759511663192854e_term,B3: int,C3: int] :
% 0.33/0.54            ( Y
% 0.33/0.54           != ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C3 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % prod_cases3
% 0.33/0.54  thf(fact_1097_Pair__inject,axiom,
% 0.33/0.54      ! [A: int,B: int,A7: int,B7: int] :
% 0.33/0.54        ( ( ( product_Pair_int_int @ A @ B )
% 0.33/0.54          = ( product_Pair_int_int @ A7 @ B7 ) )
% 0.33/0.54       => ~ ( ( A = A7 )
% 0.33/0.54           => ( B != B7 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Pair_inject
% 0.33/0.54  thf(fact_1098_Pair__inject,axiom,
% 0.33/0.54      ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A7: code_integer > option6357759511663192854e_term,B7: produc8923325533196201883nteger] :
% 0.33/0.54        ( ( ( produc6137756002093451184nteger @ A @ B )
% 0.33/0.54          = ( produc6137756002093451184nteger @ A7 @ B7 ) )
% 0.33/0.54       => ~ ( ( A = A7 )
% 0.33/0.54           => ( B != B7 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Pair_inject
% 0.33/0.54  thf(fact_1099_Pair__inject,axiom,
% 0.33/0.54      ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A7: produc6241069584506657477e_term > option6357759511663192854e_term,B7: produc8923325533196201883nteger] :
% 0.33/0.54        ( ( ( produc8603105652947943368nteger @ A @ B )
% 0.33/0.54          = ( produc8603105652947943368nteger @ A7 @ B7 ) )
% 0.33/0.54       => ~ ( ( A = A7 )
% 0.33/0.54           => ( B != B7 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Pair_inject
% 0.33/0.54  thf(fact_1100_Pair__inject,axiom,
% 0.33/0.54      ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A7: produc8551481072490612790e_term > option6357759511663192854e_term,B7: product_prod_int_int] :
% 0.33/0.54        ( ( ( produc5700946648718959541nt_int @ A @ B )
% 0.33/0.54          = ( produc5700946648718959541nt_int @ A7 @ B7 ) )
% 0.33/0.54       => ~ ( ( A = A7 )
% 0.33/0.54           => ( B != B7 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Pair_inject
% 0.33/0.54  thf(fact_1101_Pair__inject,axiom,
% 0.33/0.54      ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A7: int > option6357759511663192854e_term,B7: product_prod_int_int] :
% 0.33/0.54        ( ( ( produc4305682042979456191nt_int @ A @ B )
% 0.33/0.54          = ( produc4305682042979456191nt_int @ A7 @ B7 ) )
% 0.33/0.54       => ~ ( ( A = A7 )
% 0.33/0.54           => ( B != B7 ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Pair_inject
% 0.33/0.54  thf(fact_1102_prod__cases,axiom,
% 0.33/0.54      ! [P: product_prod_int_int > $o,P4: product_prod_int_int] :
% 0.33/0.54        ( ! [A3: int,B3: int] : ( P @ ( product_Pair_int_int @ A3 @ B3 ) )
% 0.33/0.54       => ( P @ P4 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % prod_cases
% 0.33/0.54  thf(fact_1103_prod__cases,axiom,
% 0.33/0.54      ! [P: produc8763457246119570046nteger > $o,P4: produc8763457246119570046nteger] :
% 0.33/0.54        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] : ( P @ ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 0.33/0.54       => ( P @ P4 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % prod_cases
% 0.33/0.54  thf(fact_1104_prod__cases,axiom,
% 0.33/0.54      ! [P: produc1908205239877642774nteger > $o,P4: produc1908205239877642774nteger] :
% 0.33/0.54        ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] : ( P @ ( produc8603105652947943368nteger @ A3 @ B3 ) )
% 0.33/0.54       => ( P @ P4 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % prod_cases
% 0.33/0.54  thf(fact_1105_prod__cases,axiom,
% 0.33/0.54      ! [P: produc2285326912895808259nt_int > $o,P4: produc2285326912895808259nt_int] :
% 0.33/0.54        ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ B3 ) )
% 0.33/0.54       => ( P @ P4 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % prod_cases
% 0.33/0.54  thf(fact_1106_prod__cases,axiom,
% 0.33/0.54      ! [P: produc7773217078559923341nt_int > $o,P4: produc7773217078559923341nt_int] :
% 0.33/0.54        ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ B3 ) )
% 0.33/0.54       => ( P @ P4 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % prod_cases
% 0.33/0.54  thf(fact_1107_surj__pair,axiom,
% 0.33/0.54      ! [P4: product_prod_int_int] :
% 0.33/0.54      ? [X4: int,Y3: int] :
% 0.33/0.54        ( P4
% 0.33/0.54        = ( product_Pair_int_int @ X4 @ Y3 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % surj_pair
% 0.33/0.54  thf(fact_1108_surj__pair,axiom,
% 0.33/0.54      ! [P4: produc8763457246119570046nteger] :
% 0.33/0.54      ? [X4: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
% 0.33/0.54        ( P4
% 0.33/0.54        = ( produc6137756002093451184nteger @ X4 @ Y3 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % surj_pair
% 0.33/0.54  thf(fact_1109_surj__pair,axiom,
% 0.33/0.54      ! [P4: produc1908205239877642774nteger] :
% 0.33/0.54      ? [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
% 0.33/0.54        ( P4
% 0.33/0.54        = ( produc8603105652947943368nteger @ X4 @ Y3 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % surj_pair
% 0.33/0.54  thf(fact_1110_surj__pair,axiom,
% 0.33/0.54      ! [P4: produc2285326912895808259nt_int] :
% 0.33/0.54      ? [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
% 0.33/0.54        ( P4
% 0.33/0.54        = ( produc5700946648718959541nt_int @ X4 @ Y3 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % surj_pair
% 0.33/0.54  thf(fact_1111_surj__pair,axiom,
% 0.33/0.54      ! [P4: produc7773217078559923341nt_int] :
% 0.33/0.54      ? [X4: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
% 0.33/0.54        ( P4
% 0.33/0.54        = ( produc4305682042979456191nt_int @ X4 @ Y3 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % surj_pair
% 0.33/0.54  thf(fact_1112_old_Oprod_Oexhaust,axiom,
% 0.33/0.54      ! [Y: product_prod_int_int] :
% 0.33/0.54        ~ ! [A3: int,B3: int] :
% 0.33/0.54            ( Y
% 0.33/0.54           != ( product_Pair_int_int @ A3 @ B3 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % old.prod.exhaust
% 0.33/0.54  thf(fact_1113_old_Oprod_Oexhaust,axiom,
% 0.33/0.54      ! [Y: produc8763457246119570046nteger] :
% 0.33/0.54        ~ ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 0.33/0.54            ( Y
% 0.33/0.54           != ( produc6137756002093451184nteger @ A3 @ B3 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % old.prod.exhaust
% 0.33/0.54  thf(fact_1114_old_Oprod_Oexhaust,axiom,
% 0.33/0.54      ! [Y: produc1908205239877642774nteger] :
% 0.33/0.54        ~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 0.33/0.54            ( Y
% 0.33/0.54           != ( produc8603105652947943368nteger @ A3 @ B3 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % old.prod.exhaust
% 0.33/0.54  thf(fact_1115_old_Oprod_Oexhaust,axiom,
% 0.33/0.54      ! [Y: produc2285326912895808259nt_int] :
% 0.33/0.54        ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
% 0.33/0.54            ( Y
% 0.33/0.54           != ( produc5700946648718959541nt_int @ A3 @ B3 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % old.prod.exhaust
% 0.33/0.54  thf(fact_1116_old_Oprod_Oexhaust,axiom,
% 0.33/0.54      ! [Y: produc7773217078559923341nt_int] :
% 0.33/0.54        ~ ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
% 0.33/0.54            ( Y
% 0.33/0.54           != ( produc4305682042979456191nt_int @ A3 @ B3 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % old.prod.exhaust
% 0.33/0.54  thf(fact_1117_bounded__Max__nat,axiom,
% 0.33/0.54      ! [P: nat > $o,X: nat,M4: nat] :
% 0.33/0.54        ( ( P @ X )
% 0.33/0.54       => ( ! [X4: nat] :
% 0.33/0.54              ( ( P @ X4 )
% 0.33/0.54             => ( ord_less_eq_nat @ X4 @ M4 ) )
% 0.33/0.54         => ~ ! [M5: nat] :
% 0.33/0.54                ( ( P @ M5 )
% 0.33/0.54               => ~ ! [X2: nat] :
% 0.33/0.54                      ( ( P @ X2 )
% 0.33/0.54                     => ( ord_less_eq_nat @ X2 @ M5 ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % bounded_Max_nat
% 0.33/0.54  thf(fact_1118_finite__transitivity__chain,axiom,
% 0.33/0.54      ! [A2: set_set_nat,R: set_nat > set_nat > $o] :
% 0.33/0.54        ( ( finite1152437895449049373et_nat @ A2 )
% 0.33/0.54       => ( ! [X4: set_nat] :
% 0.33/0.54              ~ ( R @ X4 @ X4 )
% 0.33/0.54         => ( ! [X4: set_nat,Y3: set_nat,Z4: set_nat] :
% 0.33/0.54                ( ( R @ X4 @ Y3 )
% 0.33/0.54               => ( ( R @ Y3 @ Z4 )
% 0.33/0.54                 => ( R @ X4 @ Z4 ) ) )
% 0.33/0.54           => ( ! [X4: set_nat] :
% 0.33/0.54                  ( ( member_set_nat @ X4 @ A2 )
% 0.33/0.54                 => ? [Y4: set_nat] :
% 0.33/0.54                      ( ( member_set_nat @ Y4 @ A2 )
% 0.33/0.54                      & ( R @ X4 @ Y4 ) ) )
% 0.33/0.54             => ( A2 = bot_bot_set_set_nat ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_transitivity_chain
% 0.33/0.54  thf(fact_1119_finite__transitivity__chain,axiom,
% 0.33/0.54      ! [A2: set_complex,R: complex > complex > $o] :
% 0.33/0.54        ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.54       => ( ! [X4: complex] :
% 0.33/0.54              ~ ( R @ X4 @ X4 )
% 0.33/0.54         => ( ! [X4: complex,Y3: complex,Z4: complex] :
% 0.33/0.54                ( ( R @ X4 @ Y3 )
% 0.33/0.54               => ( ( R @ Y3 @ Z4 )
% 0.33/0.54                 => ( R @ X4 @ Z4 ) ) )
% 0.33/0.54           => ( ! [X4: complex] :
% 0.33/0.54                  ( ( member_complex @ X4 @ A2 )
% 0.33/0.54                 => ? [Y4: complex] :
% 0.33/0.54                      ( ( member_complex @ Y4 @ A2 )
% 0.33/0.54                      & ( R @ X4 @ Y4 ) ) )
% 0.33/0.54             => ( A2 = bot_bot_set_complex ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_transitivity_chain
% 0.33/0.54  thf(fact_1120_finite__transitivity__chain,axiom,
% 0.33/0.54      ! [A2: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 0.33/0.54        ( ( finite6177210948735845034at_nat @ A2 )
% 0.33/0.54       => ( ! [X4: product_prod_nat_nat] :
% 0.33/0.54              ~ ( R @ X4 @ X4 )
% 0.33/0.54         => ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat,Z4: product_prod_nat_nat] :
% 0.33/0.54                ( ( R @ X4 @ Y3 )
% 0.33/0.54               => ( ( R @ Y3 @ Z4 )
% 0.33/0.54                 => ( R @ X4 @ Z4 ) ) )
% 0.33/0.54           => ( ! [X4: product_prod_nat_nat] :
% 0.33/0.54                  ( ( member8440522571783428010at_nat @ X4 @ A2 )
% 0.33/0.54                 => ? [Y4: product_prod_nat_nat] :
% 0.33/0.54                      ( ( member8440522571783428010at_nat @ Y4 @ A2 )
% 0.33/0.54                      & ( R @ X4 @ Y4 ) ) )
% 0.33/0.54             => ( A2 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_transitivity_chain
% 0.33/0.54  thf(fact_1121_finite__transitivity__chain,axiom,
% 0.33/0.54      ! [A2: set_real,R: real > real > $o] :
% 0.33/0.54        ( ( finite_finite_real @ A2 )
% 0.33/0.54       => ( ! [X4: real] :
% 0.33/0.54              ~ ( R @ X4 @ X4 )
% 0.33/0.54         => ( ! [X4: real,Y3: real,Z4: real] :
% 0.33/0.54                ( ( R @ X4 @ Y3 )
% 0.33/0.54               => ( ( R @ Y3 @ Z4 )
% 0.33/0.54                 => ( R @ X4 @ Z4 ) ) )
% 0.33/0.54           => ( ! [X4: real] :
% 0.33/0.54                  ( ( member_real @ X4 @ A2 )
% 0.33/0.54                 => ? [Y4: real] :
% 0.33/0.54                      ( ( member_real @ Y4 @ A2 )
% 0.33/0.54                      & ( R @ X4 @ Y4 ) ) )
% 0.33/0.54             => ( A2 = bot_bot_set_real ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_transitivity_chain
% 0.33/0.54  thf(fact_1122_finite__transitivity__chain,axiom,
% 0.33/0.54      ! [A2: set_nat,R: nat > nat > $o] :
% 0.33/0.54        ( ( finite_finite_nat @ A2 )
% 0.33/0.54       => ( ! [X4: nat] :
% 0.33/0.54              ~ ( R @ X4 @ X4 )
% 0.33/0.54         => ( ! [X4: nat,Y3: nat,Z4: nat] :
% 0.33/0.54                ( ( R @ X4 @ Y3 )
% 0.33/0.54               => ( ( R @ Y3 @ Z4 )
% 0.33/0.54                 => ( R @ X4 @ Z4 ) ) )
% 0.33/0.54           => ( ! [X4: nat] :
% 0.33/0.54                  ( ( member_nat @ X4 @ A2 )
% 0.33/0.54                 => ? [Y4: nat] :
% 0.33/0.54                      ( ( member_nat @ Y4 @ A2 )
% 0.33/0.54                      & ( R @ X4 @ Y4 ) ) )
% 0.33/0.54             => ( A2 = bot_bot_set_nat ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_transitivity_chain
% 0.33/0.54  thf(fact_1123_finite__transitivity__chain,axiom,
% 0.33/0.54      ! [A2: set_int,R: int > int > $o] :
% 0.33/0.54        ( ( finite_finite_int @ A2 )
% 0.33/0.54       => ( ! [X4: int] :
% 0.33/0.54              ~ ( R @ X4 @ X4 )
% 0.33/0.54         => ( ! [X4: int,Y3: int,Z4: int] :
% 0.33/0.54                ( ( R @ X4 @ Y3 )
% 0.33/0.54               => ( ( R @ Y3 @ Z4 )
% 0.33/0.54                 => ( R @ X4 @ Z4 ) ) )
% 0.33/0.54           => ( ! [X4: int] :
% 0.33/0.54                  ( ( member_int @ X4 @ A2 )
% 0.33/0.54                 => ? [Y4: int] :
% 0.33/0.54                      ( ( member_int @ Y4 @ A2 )
% 0.33/0.54                      & ( R @ X4 @ Y4 ) ) )
% 0.33/0.54             => ( A2 = bot_bot_set_int ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_transitivity_chain
% 0.33/0.54  thf(fact_1124_finite__nat__set__iff__bounded,axiom,
% 0.33/0.54      ( finite_finite_nat
% 0.33/0.54      = ( ^ [N4: set_nat] :
% 0.33/0.54          ? [M2: nat] :
% 0.33/0.54          ! [X3: nat] :
% 0.33/0.54            ( ( member_nat @ X3 @ N4 )
% 0.33/0.54           => ( ord_less_nat @ X3 @ M2 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % finite_nat_set_iff_bounded
% 0.33/0.54  thf(fact_1125_infinite__nat__iff__unbounded,axiom,
% 0.33/0.54      ! [S2: set_nat] :
% 0.33/0.54        ( ( ~ ( finite_finite_nat @ S2 ) )
% 0.33/0.54        = ( ! [M2: nat] :
% 0.33/0.54            ? [N2: nat] :
% 0.33/0.54              ( ( ord_less_nat @ M2 @ N2 )
% 0.33/0.54              & ( member_nat @ N2 @ S2 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % infinite_nat_iff_unbounded
% 0.33/0.54  thf(fact_1126_bounded__nat__set__is__finite,axiom,
% 0.33/0.54      ! [N6: set_nat,N: nat] :
% 0.33/0.54        ( ! [X4: nat] :
% 0.33/0.54            ( ( member_nat @ X4 @ N6 )
% 0.33/0.54           => ( ord_less_nat @ X4 @ N ) )
% 0.33/0.54       => ( finite_finite_nat @ N6 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % bounded_nat_set_is_finite
% 0.33/0.54  thf(fact_1127_Collect__empty__eq__bot,axiom,
% 0.33/0.54      ! [P: list_nat > $o] :
% 0.33/0.54        ( ( ( collect_list_nat @ P )
% 0.33/0.54          = bot_bot_set_list_nat )
% 0.33/0.54        = ( P = bot_bot_list_nat_o ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_empty_eq_bot
% 0.33/0.54  thf(fact_1128_Collect__empty__eq__bot,axiom,
% 0.33/0.54      ! [P: set_nat > $o] :
% 0.33/0.54        ( ( ( collect_set_nat @ P )
% 0.33/0.54          = bot_bot_set_set_nat )
% 0.33/0.54        = ( P = bot_bot_set_nat_o ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_empty_eq_bot
% 0.33/0.54  thf(fact_1129_Collect__empty__eq__bot,axiom,
% 0.33/0.54      ! [P: product_prod_nat_nat > $o] :
% 0.33/0.54        ( ( ( collec3392354462482085612at_nat @ P )
% 0.33/0.54          = bot_bo2099793752762293965at_nat )
% 0.33/0.54        = ( P = bot_bo482883023278783056_nat_o ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_empty_eq_bot
% 0.33/0.54  thf(fact_1130_Collect__empty__eq__bot,axiom,
% 0.33/0.54      ! [P: real > $o] :
% 0.33/0.54        ( ( ( collect_real @ P )
% 0.33/0.54          = bot_bot_set_real )
% 0.33/0.54        = ( P = bot_bot_real_o ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_empty_eq_bot
% 0.33/0.54  thf(fact_1131_Collect__empty__eq__bot,axiom,
% 0.33/0.54      ! [P: nat > $o] :
% 0.33/0.54        ( ( ( collect_nat @ P )
% 0.33/0.54          = bot_bot_set_nat )
% 0.33/0.54        = ( P = bot_bot_nat_o ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_empty_eq_bot
% 0.33/0.54  thf(fact_1132_Collect__empty__eq__bot,axiom,
% 0.33/0.54      ! [P: int > $o] :
% 0.33/0.54        ( ( ( collect_int @ P )
% 0.33/0.54          = bot_bot_set_int )
% 0.33/0.54        = ( P = bot_bot_int_o ) ) ).
% 0.33/0.54  
% 0.33/0.54  % Collect_empty_eq_bot
% 0.33/0.54  thf(fact_1133_bot__empty__eq,axiom,
% 0.33/0.54      ( bot_bot_set_nat_o
% 0.33/0.54      = ( ^ [X3: set_nat] : ( member_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % bot_empty_eq
% 0.33/0.54  thf(fact_1134_bot__empty__eq,axiom,
% 0.33/0.54      ( bot_bo482883023278783056_nat_o
% 0.33/0.54      = ( ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % bot_empty_eq
% 0.33/0.54  thf(fact_1135_bot__empty__eq,axiom,
% 0.33/0.54      ( bot_bot_real_o
% 0.33/0.54      = ( ^ [X3: real] : ( member_real @ X3 @ bot_bot_set_real ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % bot_empty_eq
% 0.33/0.54  thf(fact_1136_bot__empty__eq,axiom,
% 0.33/0.54      ( bot_bot_nat_o
% 0.33/0.54      = ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % bot_empty_eq
% 0.33/0.54  thf(fact_1137_bot__empty__eq,axiom,
% 0.33/0.54      ( bot_bot_int_o
% 0.33/0.54      = ( ^ [X3: int] : ( member_int @ X3 @ bot_bot_set_int ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % bot_empty_eq
% 0.33/0.54  thf(fact_1138_arg__min__if__finite_I2_J,axiom,
% 0.33/0.54      ! [S2: set_complex,F: complex > real] :
% 0.33/0.54        ( ( finite3207457112153483333omplex @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_complex )
% 0.33/0.54         => ~ ? [X2: complex] :
% 0.33/0.54                ( ( member_complex @ X2 @ S2 )
% 0.33/0.54                & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic8794016678065449205x_real @ F @ S2 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % arg_min_if_finite(2)
% 0.33/0.54  thf(fact_1139_arg__min__if__finite_I2_J,axiom,
% 0.33/0.54      ! [S2: set_real,F: real > real] :
% 0.33/0.54        ( ( finite_finite_real @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_real )
% 0.33/0.54         => ~ ? [X2: real] :
% 0.33/0.54                ( ( member_real @ X2 @ S2 )
% 0.33/0.54                & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic8440615504127631091l_real @ F @ S2 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % arg_min_if_finite(2)
% 0.33/0.54  thf(fact_1140_arg__min__if__finite_I2_J,axiom,
% 0.33/0.54      ! [S2: set_nat,F: nat > real] :
% 0.33/0.54        ( ( finite_finite_nat @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_nat )
% 0.33/0.54         => ~ ? [X2: nat] :
% 0.33/0.54                ( ( member_nat @ X2 @ S2 )
% 0.33/0.54                & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic488527866317076247t_real @ F @ S2 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % arg_min_if_finite(2)
% 0.33/0.54  thf(fact_1141_arg__min__if__finite_I2_J,axiom,
% 0.33/0.54      ! [S2: set_int,F: int > real] :
% 0.33/0.54        ( ( finite_finite_int @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_int )
% 0.33/0.54         => ~ ? [X2: int] :
% 0.33/0.54                ( ( member_int @ X2 @ S2 )
% 0.33/0.54                & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic2675449441010098035t_real @ F @ S2 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % arg_min_if_finite(2)
% 0.33/0.54  thf(fact_1142_arg__min__if__finite_I2_J,axiom,
% 0.33/0.54      ! [S2: set_complex,F: complex > rat] :
% 0.33/0.54        ( ( finite3207457112153483333omplex @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_complex )
% 0.33/0.54         => ~ ? [X2: complex] :
% 0.33/0.54                ( ( member_complex @ X2 @ S2 )
% 0.33/0.54                & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S2 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % arg_min_if_finite(2)
% 0.33/0.54  thf(fact_1143_arg__min__if__finite_I2_J,axiom,
% 0.33/0.54      ! [S2: set_real,F: real > rat] :
% 0.33/0.54        ( ( finite_finite_real @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_real )
% 0.33/0.54         => ~ ? [X2: real] :
% 0.33/0.54                ( ( member_real @ X2 @ S2 )
% 0.33/0.54                & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic4420706379359479199al_rat @ F @ S2 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % arg_min_if_finite(2)
% 0.33/0.54  thf(fact_1144_arg__min__if__finite_I2_J,axiom,
% 0.33/0.54      ! [S2: set_nat,F: nat > rat] :
% 0.33/0.54        ( ( finite_finite_nat @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_nat )
% 0.33/0.54         => ~ ? [X2: nat] :
% 0.33/0.54                ( ( member_nat @ X2 @ S2 )
% 0.33/0.54                & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic6811802900495863747at_rat @ F @ S2 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % arg_min_if_finite(2)
% 0.33/0.54  thf(fact_1145_arg__min__if__finite_I2_J,axiom,
% 0.33/0.54      ! [S2: set_int,F: int > rat] :
% 0.33/0.54        ( ( finite_finite_int @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_int )
% 0.33/0.54         => ~ ? [X2: int] :
% 0.33/0.54                ( ( member_int @ X2 @ S2 )
% 0.33/0.54                & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S2 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % arg_min_if_finite(2)
% 0.33/0.54  thf(fact_1146_arg__min__if__finite_I2_J,axiom,
% 0.33/0.54      ! [S2: set_complex,F: complex > num] :
% 0.33/0.54        ( ( finite3207457112153483333omplex @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_complex )
% 0.33/0.54         => ~ ? [X2: complex] :
% 0.33/0.54                ( ( member_complex @ X2 @ S2 )
% 0.33/0.54                & ( ord_less_num @ ( F @ X2 ) @ ( F @ ( lattic1922116423962787043ex_num @ F @ S2 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % arg_min_if_finite(2)
% 0.33/0.54  thf(fact_1147_arg__min__if__finite_I2_J,axiom,
% 0.33/0.54      ! [S2: set_real,F: real > num] :
% 0.33/0.54        ( ( finite_finite_real @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_real )
% 0.33/0.54         => ~ ? [X2: real] :
% 0.33/0.54                ( ( member_real @ X2 @ S2 )
% 0.33/0.54                & ( ord_less_num @ ( F @ X2 ) @ ( F @ ( lattic1613168225601753569al_num @ F @ S2 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % arg_min_if_finite(2)
% 0.33/0.54  thf(fact_1148_arg__min__least,axiom,
% 0.33/0.54      ! [S2: set_complex,Y: complex,F: complex > rat] :
% 0.33/0.54        ( ( finite3207457112153483333omplex @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_complex )
% 0.33/0.54         => ( ( member_complex @ Y @ S2 )
% 0.33/0.54           => ( ord_less_eq_rat @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % arg_min_least
% 0.33/0.54  thf(fact_1149_arg__min__least,axiom,
% 0.33/0.54      ! [S2: set_real,Y: real,F: real > rat] :
% 0.33/0.54        ( ( finite_finite_real @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_real )
% 0.33/0.54         => ( ( member_real @ Y @ S2 )
% 0.33/0.54           => ( ord_less_eq_rat @ ( F @ ( lattic4420706379359479199al_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % arg_min_least
% 0.33/0.54  thf(fact_1150_arg__min__least,axiom,
% 0.33/0.54      ! [S2: set_nat,Y: nat,F: nat > rat] :
% 0.33/0.54        ( ( finite_finite_nat @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_nat )
% 0.33/0.54         => ( ( member_nat @ Y @ S2 )
% 0.33/0.54           => ( ord_less_eq_rat @ ( F @ ( lattic6811802900495863747at_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % arg_min_least
% 0.33/0.54  thf(fact_1151_arg__min__least,axiom,
% 0.33/0.54      ! [S2: set_int,Y: int,F: int > rat] :
% 0.33/0.54        ( ( finite_finite_int @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_int )
% 0.33/0.54         => ( ( member_int @ Y @ S2 )
% 0.33/0.54           => ( ord_less_eq_rat @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % arg_min_least
% 0.33/0.54  thf(fact_1152_arg__min__least,axiom,
% 0.33/0.54      ! [S2: set_complex,Y: complex,F: complex > num] :
% 0.33/0.54        ( ( finite3207457112153483333omplex @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_complex )
% 0.33/0.54         => ( ( member_complex @ Y @ S2 )
% 0.33/0.54           => ( ord_less_eq_num @ ( F @ ( lattic1922116423962787043ex_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % arg_min_least
% 0.33/0.54  thf(fact_1153_arg__min__least,axiom,
% 0.33/0.54      ! [S2: set_real,Y: real,F: real > num] :
% 0.33/0.54        ( ( finite_finite_real @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_real )
% 0.33/0.54         => ( ( member_real @ Y @ S2 )
% 0.33/0.54           => ( ord_less_eq_num @ ( F @ ( lattic1613168225601753569al_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % arg_min_least
% 0.33/0.54  thf(fact_1154_arg__min__least,axiom,
% 0.33/0.54      ! [S2: set_nat,Y: nat,F: nat > num] :
% 0.33/0.54        ( ( finite_finite_nat @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_nat )
% 0.33/0.54         => ( ( member_nat @ Y @ S2 )
% 0.33/0.54           => ( ord_less_eq_num @ ( F @ ( lattic4004264746738138117at_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % arg_min_least
% 0.33/0.54  thf(fact_1155_arg__min__least,axiom,
% 0.33/0.54      ! [S2: set_int,Y: int,F: int > num] :
% 0.33/0.54        ( ( finite_finite_int @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_int )
% 0.33/0.54         => ( ( member_int @ Y @ S2 )
% 0.33/0.54           => ( ord_less_eq_num @ ( F @ ( lattic5003618458639192673nt_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % arg_min_least
% 0.33/0.54  thf(fact_1156_arg__min__least,axiom,
% 0.33/0.54      ! [S2: set_complex,Y: complex,F: complex > nat] :
% 0.33/0.54        ( ( finite3207457112153483333omplex @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_complex )
% 0.33/0.54         => ( ( member_complex @ Y @ S2 )
% 0.33/0.54           => ( ord_less_eq_nat @ ( F @ ( lattic5364784637807008409ex_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % arg_min_least
% 0.33/0.54  thf(fact_1157_arg__min__least,axiom,
% 0.33/0.54      ! [S2: set_real,Y: real,F: real > nat] :
% 0.33/0.54        ( ( finite_finite_real @ S2 )
% 0.33/0.54       => ( ( S2 != bot_bot_set_real )
% 0.33/0.54         => ( ( member_real @ Y @ S2 )
% 0.33/0.54           => ( ord_less_eq_nat @ ( F @ ( lattic5055836439445974935al_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % arg_min_least
% 0.33/0.54  thf(fact_1158_nat__descend__induct,axiom,
% 0.33/0.54      ! [N: nat,P: nat > $o,M: nat] :
% 0.33/0.54        ( ! [K2: nat] :
% 0.33/0.54            ( ( ord_less_nat @ N @ K2 )
% 0.33/0.54           => ( P @ K2 ) )
% 0.33/0.54       => ( ! [K2: nat] :
% 0.33/0.54              ( ( ord_less_eq_nat @ K2 @ N )
% 0.33/0.54             => ( ! [I3: nat] :
% 0.33/0.54                    ( ( ord_less_nat @ K2 @ I3 )
% 0.33/0.54                   => ( P @ I3 ) )
% 0.33/0.54               => ( P @ K2 ) ) )
% 0.33/0.54         => ( P @ M ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % nat_descend_induct
% 0.33/0.54  thf(fact_1159_subset__emptyI,axiom,
% 0.33/0.54      ! [A2: set_set_nat] :
% 0.33/0.54        ( ! [X4: set_nat] :
% 0.33/0.54            ~ ( member_set_nat @ X4 @ A2 )
% 0.33/0.54       => ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_emptyI
% 0.33/0.54  thf(fact_1160_subset__emptyI,axiom,
% 0.33/0.54      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.33/0.54        ( ! [X4: product_prod_nat_nat] :
% 0.33/0.54            ~ ( member8440522571783428010at_nat @ X4 @ A2 )
% 0.33/0.54       => ( ord_le3146513528884898305at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_emptyI
% 0.33/0.54  thf(fact_1161_subset__emptyI,axiom,
% 0.33/0.54      ! [A2: set_real] :
% 0.33/0.54        ( ! [X4: real] :
% 0.33/0.54            ~ ( member_real @ X4 @ A2 )
% 0.33/0.54       => ( ord_less_eq_set_real @ A2 @ bot_bot_set_real ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_emptyI
% 0.33/0.54  thf(fact_1162_subset__emptyI,axiom,
% 0.33/0.54      ! [A2: set_nat] :
% 0.33/0.54        ( ! [X4: nat] :
% 0.33/0.54            ~ ( member_nat @ X4 @ A2 )
% 0.33/0.54       => ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_emptyI
% 0.33/0.54  thf(fact_1163_subset__emptyI,axiom,
% 0.33/0.54      ! [A2: set_int] :
% 0.33/0.54        ( ! [X4: int] :
% 0.33/0.54            ~ ( member_int @ X4 @ A2 )
% 0.33/0.54       => ( ord_less_eq_set_int @ A2 @ bot_bot_set_int ) ) ).
% 0.33/0.54  
% 0.33/0.54  % subset_emptyI
% 0.33/0.54  thf(fact_1164_field__lbound__gt__zero,axiom,
% 0.33/0.54      ! [D1: real,D2: real] :
% 0.33/0.54        ( ( ord_less_real @ zero_zero_real @ D1 )
% 0.33/0.54       => ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.33/0.54         => ? [E: real] :
% 0.33/0.54              ( ( ord_less_real @ zero_zero_real @ E )
% 0.33/0.54              & ( ord_less_real @ E @ D1 )
% 0.33/0.54              & ( ord_less_real @ E @ D2 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % field_lbound_gt_zero
% 0.33/0.54  thf(fact_1165_field__lbound__gt__zero,axiom,
% 0.33/0.54      ! [D1: rat,D2: rat] :
% 0.33/0.54        ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 0.33/0.54       => ( ( ord_less_rat @ zero_zero_rat @ D2 )
% 0.33/0.54         => ? [E: rat] :
% 0.33/0.54              ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.33/0.54              & ( ord_less_rat @ E @ D1 )
% 0.33/0.54              & ( ord_less_rat @ E @ D2 ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % field_lbound_gt_zero
% 0.33/0.54  thf(fact_1166_less__numeral__extra_I3_J,axiom,
% 0.33/0.54      ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 0.33/0.54  
% 0.33/0.54  % less_numeral_extra(3)
% 0.33/0.54  thf(fact_1167_less__numeral__extra_I3_J,axiom,
% 0.33/0.54      ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 0.33/0.54  
% 0.33/0.54  % less_numeral_extra(3)
% 0.33/0.54  thf(fact_1168_less__numeral__extra_I3_J,axiom,
% 0.33/0.54      ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.33/0.54  
% 0.33/0.54  % less_numeral_extra(3)
% 0.33/0.54  thf(fact_1169_less__numeral__extra_I3_J,axiom,
% 0.33/0.54      ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.33/0.54  
% 0.33/0.54  % less_numeral_extra(3)
% 0.33/0.54  thf(fact_1170_complete__interval,axiom,
% 0.33/0.54      ! [A: real,B: real,P: real > $o] :
% 0.33/0.54        ( ( ord_less_real @ A @ B )
% 0.33/0.54       => ( ( P @ A )
% 0.33/0.54         => ( ~ ( P @ B )
% 0.33/0.54           => ? [C3: real] :
% 0.33/0.54                ( ( ord_less_eq_real @ A @ C3 )
% 0.33/0.54                & ( ord_less_eq_real @ C3 @ B )
% 0.33/0.54                & ! [X2: real] :
% 0.33/0.54                    ( ( ( ord_less_eq_real @ A @ X2 )
% 0.33/0.54                      & ( ord_less_real @ X2 @ C3 ) )
% 0.33/0.54                   => ( P @ X2 ) )
% 0.33/0.54                & ! [D3: real] :
% 0.33/0.54                    ( ! [X4: real] :
% 0.33/0.54                        ( ( ( ord_less_eq_real @ A @ X4 )
% 0.33/0.54                          & ( ord_less_real @ X4 @ D3 ) )
% 0.33/0.54                       => ( P @ X4 ) )
% 0.33/0.54                   => ( ord_less_eq_real @ D3 @ C3 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % complete_interval
% 0.33/0.54  thf(fact_1171_complete__interval,axiom,
% 0.33/0.54      ! [A: nat,B: nat,P: nat > $o] :
% 0.33/0.54        ( ( ord_less_nat @ A @ B )
% 0.33/0.54       => ( ( P @ A )
% 0.33/0.54         => ( ~ ( P @ B )
% 0.33/0.54           => ? [C3: nat] :
% 0.33/0.54                ( ( ord_less_eq_nat @ A @ C3 )
% 0.33/0.54                & ( ord_less_eq_nat @ C3 @ B )
% 0.33/0.54                & ! [X2: nat] :
% 0.33/0.54                    ( ( ( ord_less_eq_nat @ A @ X2 )
% 0.33/0.54                      & ( ord_less_nat @ X2 @ C3 ) )
% 0.33/0.54                   => ( P @ X2 ) )
% 0.33/0.54                & ! [D3: nat] :
% 0.33/0.54                    ( ! [X4: nat] :
% 0.33/0.54                        ( ( ( ord_less_eq_nat @ A @ X4 )
% 0.33/0.54                          & ( ord_less_nat @ X4 @ D3 ) )
% 0.33/0.54                       => ( P @ X4 ) )
% 0.33/0.54                   => ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % complete_interval
% 0.33/0.54  thf(fact_1172_complete__interval,axiom,
% 0.33/0.54      ! [A: int,B: int,P: int > $o] :
% 0.33/0.54        ( ( ord_less_int @ A @ B )
% 0.33/0.54       => ( ( P @ A )
% 0.33/0.54         => ( ~ ( P @ B )
% 0.33/0.54           => ? [C3: int] :
% 0.33/0.54                ( ( ord_less_eq_int @ A @ C3 )
% 0.33/0.54                & ( ord_less_eq_int @ C3 @ B )
% 0.33/0.54                & ! [X2: int] :
% 0.33/0.54                    ( ( ( ord_less_eq_int @ A @ X2 )
% 0.33/0.54                      & ( ord_less_int @ X2 @ C3 ) )
% 0.33/0.54                   => ( P @ X2 ) )
% 0.33/0.54                & ! [D3: int] :
% 0.33/0.54                    ( ! [X4: int] :
% 0.33/0.54                        ( ( ( ord_less_eq_int @ A @ X4 )
% 0.33/0.54                          & ( ord_less_int @ X4 @ D3 ) )
% 0.33/0.54                       => ( P @ X4 ) )
% 0.33/0.54                   => ( ord_less_eq_int @ D3 @ C3 ) ) ) ) ) ) ).
% 0.33/0.54  
% 0.33/0.54  % complete_interval
% 0.33/0.54  thf(fact_1173_verit__comp__simplify1_I3_J,axiom,
% 0.33/0.54      ! [B7: real,A7: real] :
% 0.33/0.54        ( ( ~ ( ord_less_eq_real @ B7 @ A7 ) )
% 0.33/0.54        = ( ord_less_real @ A7 @ B7 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % verit_comp_simplify1(3)
% 0.33/0.54  thf(fact_1174_verit__comp__simplify1_I3_J,axiom,
% 0.33/0.54      ! [B7: rat,A7: rat] :
% 0.33/0.54        ( ( ~ ( ord_less_eq_rat @ B7 @ A7 ) )
% 0.33/0.54        = ( ord_less_rat @ A7 @ B7 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % verit_comp_simplify1(3)
% 0.33/0.54  thf(fact_1175_verit__comp__simplify1_I3_J,axiom,
% 0.33/0.54      ! [B7: num,A7: num] :
% 0.33/0.54        ( ( ~ ( ord_less_eq_num @ B7 @ A7 ) )
% 0.33/0.54        = ( ord_less_num @ A7 @ B7 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % verit_comp_simplify1(3)
% 0.33/0.54  thf(fact_1176_verit__comp__simplify1_I3_J,axiom,
% 0.33/0.54      ! [B7: nat,A7: nat] :
% 0.33/0.54        ( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
% 0.33/0.54        = ( ord_less_nat @ A7 @ B7 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % verit_comp_simplify1(3)
% 0.33/0.54  thf(fact_1177_verit__comp__simplify1_I3_J,axiom,
% 0.33/0.54      ! [B7: int,A7: int] :
% 0.33/0.54        ( ( ~ ( ord_less_eq_int @ B7 @ A7 ) )
% 0.33/0.54        = ( ord_less_int @ A7 @ B7 ) ) ).
% 0.33/0.54  
% 0.33/0.54  % verit_comp_simplify1(3)
% 0.33/0.54  thf(fact_1178_add__right__cancel,axiom,
% 0.33/0.54      ! [B: real,A: real,C: real] :
% 0.33/0.54        ( ( ( plus_plus_real @ B @ A )
% 0.33/0.54          = ( plus_plus_real @ C @ A ) )
% 0.33/0.54        = ( B = C ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_right_cancel
% 0.33/0.54  thf(fact_1179_add__right__cancel,axiom,
% 0.33/0.54      ! [B: rat,A: rat,C: rat] :
% 0.33/0.54        ( ( ( plus_plus_rat @ B @ A )
% 0.33/0.54          = ( plus_plus_rat @ C @ A ) )
% 0.33/0.54        = ( B = C ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_right_cancel
% 0.33/0.54  thf(fact_1180_add__right__cancel,axiom,
% 0.33/0.54      ! [B: nat,A: nat,C: nat] :
% 0.33/0.54        ( ( ( plus_plus_nat @ B @ A )
% 0.33/0.54          = ( plus_plus_nat @ C @ A ) )
% 0.33/0.54        = ( B = C ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_right_cancel
% 0.33/0.54  thf(fact_1181_add__right__cancel,axiom,
% 0.33/0.54      ! [B: int,A: int,C: int] :
% 0.33/0.54        ( ( ( plus_plus_int @ B @ A )
% 0.33/0.54          = ( plus_plus_int @ C @ A ) )
% 0.33/0.54        = ( B = C ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_right_cancel
% 0.33/0.54  thf(fact_1182_add__left__cancel,axiom,
% 0.33/0.54      ! [A: real,B: real,C: real] :
% 0.33/0.54        ( ( ( plus_plus_real @ A @ B )
% 0.33/0.54          = ( plus_plus_real @ A @ C ) )
% 0.33/0.54        = ( B = C ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_left_cancel
% 0.33/0.54  thf(fact_1183_add__left__cancel,axiom,
% 0.33/0.54      ! [A: rat,B: rat,C: rat] :
% 0.33/0.54        ( ( ( plus_plus_rat @ A @ B )
% 0.33/0.54          = ( plus_plus_rat @ A @ C ) )
% 0.33/0.54        = ( B = C ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_left_cancel
% 0.33/0.54  thf(fact_1184_add__left__cancel,axiom,
% 0.33/0.54      ! [A: nat,B: nat,C: nat] :
% 0.33/0.54        ( ( ( plus_plus_nat @ A @ B )
% 0.33/0.54          = ( plus_plus_nat @ A @ C ) )
% 0.33/0.54        = ( B = C ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_left_cancel
% 0.33/0.54  thf(fact_1185_add__left__cancel,axiom,
% 0.33/0.54      ! [A: int,B: int,C: int] :
% 0.33/0.54        ( ( ( plus_plus_int @ A @ B )
% 0.33/0.54          = ( plus_plus_int @ A @ C ) )
% 0.33/0.54        = ( B = C ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_left_cancel
% 0.33/0.54  thf(fact_1186_add__le__cancel__left,axiom,
% 0.33/0.54      ! [C: real,A: real,B: real] :
% 0.33/0.54        ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.33/0.54        = ( ord_less_eq_real @ A @ B ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_le_cancel_left
% 0.33/0.54  thf(fact_1187_add__le__cancel__left,axiom,
% 0.33/0.54      ! [C: rat,A: rat,B: rat] :
% 0.33/0.54        ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.33/0.54        = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_le_cancel_left
% 0.33/0.54  thf(fact_1188_add__le__cancel__left,axiom,
% 0.33/0.54      ! [C: nat,A: nat,B: nat] :
% 0.33/0.54        ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.33/0.54        = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_le_cancel_left
% 0.33/0.54  thf(fact_1189_add__le__cancel__left,axiom,
% 0.33/0.54      ! [C: int,A: int,B: int] :
% 0.33/0.54        ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.33/0.54        = ( ord_less_eq_int @ A @ B ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_le_cancel_left
% 0.33/0.54  thf(fact_1190_add__le__cancel__right,axiom,
% 0.33/0.54      ! [A: real,C: real,B: real] :
% 0.33/0.54        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.33/0.54        = ( ord_less_eq_real @ A @ B ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_le_cancel_right
% 0.33/0.54  thf(fact_1191_add__le__cancel__right,axiom,
% 0.33/0.54      ! [A: rat,C: rat,B: rat] :
% 0.33/0.54        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.33/0.54        = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_le_cancel_right
% 0.33/0.54  thf(fact_1192_add__le__cancel__right,axiom,
% 0.33/0.54      ! [A: nat,C: nat,B: nat] :
% 0.33/0.54        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.33/0.54        = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_le_cancel_right
% 0.33/0.54  thf(fact_1193_add__le__cancel__right,axiom,
% 0.33/0.54      ! [A: int,C: int,B: int] :
% 0.33/0.54        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.33/0.54        = ( ord_less_eq_int @ A @ B ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_le_cancel_right
% 0.33/0.54  thf(fact_1194_add_Oright__neutral,axiom,
% 0.33/0.54      ! [A: complex] :
% 0.33/0.54        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.33/0.54        = A ) ).
% 0.33/0.54  
% 0.33/0.54  % add.right_neutral
% 0.33/0.54  thf(fact_1195_add_Oright__neutral,axiom,
% 0.33/0.54      ! [A: real] :
% 0.33/0.54        ( ( plus_plus_real @ A @ zero_zero_real )
% 0.33/0.54        = A ) ).
% 0.33/0.54  
% 0.33/0.54  % add.right_neutral
% 0.33/0.54  thf(fact_1196_add_Oright__neutral,axiom,
% 0.33/0.54      ! [A: rat] :
% 0.33/0.54        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.33/0.54        = A ) ).
% 0.33/0.54  
% 0.33/0.54  % add.right_neutral
% 0.33/0.54  thf(fact_1197_add_Oright__neutral,axiom,
% 0.33/0.54      ! [A: nat] :
% 0.33/0.54        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.33/0.54        = A ) ).
% 0.33/0.54  
% 0.33/0.54  % add.right_neutral
% 0.33/0.54  thf(fact_1198_add_Oright__neutral,axiom,
% 0.33/0.54      ! [A: int] :
% 0.33/0.54        ( ( plus_plus_int @ A @ zero_zero_int )
% 0.33/0.54        = A ) ).
% 0.33/0.54  
% 0.33/0.54  % add.right_neutral
% 0.33/0.54  thf(fact_1199_double__zero__sym,axiom,
% 0.33/0.54      ! [A: real] :
% 0.33/0.54        ( ( zero_zero_real
% 0.33/0.54          = ( plus_plus_real @ A @ A ) )
% 0.33/0.54        = ( A = zero_zero_real ) ) ).
% 0.33/0.54  
% 0.33/0.54  % double_zero_sym
% 0.33/0.54  thf(fact_1200_double__zero__sym,axiom,
% 0.33/0.54      ! [A: rat] :
% 0.33/0.54        ( ( zero_zero_rat
% 0.33/0.54          = ( plus_plus_rat @ A @ A ) )
% 0.33/0.54        = ( A = zero_zero_rat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % double_zero_sym
% 0.33/0.54  thf(fact_1201_double__zero__sym,axiom,
% 0.33/0.54      ! [A: int] :
% 0.33/0.54        ( ( zero_zero_int
% 0.33/0.54          = ( plus_plus_int @ A @ A ) )
% 0.33/0.54        = ( A = zero_zero_int ) ) ).
% 0.33/0.54  
% 0.33/0.54  % double_zero_sym
% 0.33/0.54  thf(fact_1202_add__cancel__left__left,axiom,
% 0.33/0.54      ! [B: complex,A: complex] :
% 0.33/0.54        ( ( ( plus_plus_complex @ B @ A )
% 0.33/0.54          = A )
% 0.33/0.54        = ( B = zero_zero_complex ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_cancel_left_left
% 0.33/0.54  thf(fact_1203_add__cancel__left__left,axiom,
% 0.33/0.54      ! [B: real,A: real] :
% 0.33/0.54        ( ( ( plus_plus_real @ B @ A )
% 0.33/0.54          = A )
% 0.33/0.54        = ( B = zero_zero_real ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_cancel_left_left
% 0.33/0.54  thf(fact_1204_add__cancel__left__left,axiom,
% 0.33/0.54      ! [B: rat,A: rat] :
% 0.33/0.54        ( ( ( plus_plus_rat @ B @ A )
% 0.33/0.54          = A )
% 0.33/0.54        = ( B = zero_zero_rat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_cancel_left_left
% 0.33/0.54  thf(fact_1205_add__cancel__left__left,axiom,
% 0.33/0.54      ! [B: nat,A: nat] :
% 0.33/0.54        ( ( ( plus_plus_nat @ B @ A )
% 0.33/0.54          = A )
% 0.33/0.54        = ( B = zero_zero_nat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_cancel_left_left
% 0.33/0.54  thf(fact_1206_add__cancel__left__left,axiom,
% 0.33/0.54      ! [B: int,A: int] :
% 0.33/0.54        ( ( ( plus_plus_int @ B @ A )
% 0.33/0.54          = A )
% 0.33/0.54        = ( B = zero_zero_int ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_cancel_left_left
% 0.33/0.54  thf(fact_1207_add__cancel__left__right,axiom,
% 0.33/0.54      ! [A: complex,B: complex] :
% 0.33/0.54        ( ( ( plus_plus_complex @ A @ B )
% 0.33/0.54          = A )
% 0.33/0.54        = ( B = zero_zero_complex ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_cancel_left_right
% 0.33/0.54  thf(fact_1208_add__cancel__left__right,axiom,
% 0.33/0.54      ! [A: real,B: real] :
% 0.33/0.54        ( ( ( plus_plus_real @ A @ B )
% 0.33/0.54          = A )
% 0.33/0.54        = ( B = zero_zero_real ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_cancel_left_right
% 0.33/0.54  thf(fact_1209_add__cancel__left__right,axiom,
% 0.33/0.54      ! [A: rat,B: rat] :
% 0.33/0.54        ( ( ( plus_plus_rat @ A @ B )
% 0.33/0.54          = A )
% 0.33/0.54        = ( B = zero_zero_rat ) ) ).
% 0.33/0.54  
% 0.33/0.54  % add_cancel_left_right
% 0.33/0.54  thf(fact_1210_add__cancel__left__right,axiom,
% 0.33/0.54      ! [A: nat,B: nat] :
% 0.33/0.54        ( ( ( plus_plus_nat @ A @ B )
% 0.33/0.54          = A )
% 0.33/0.55        = ( B = zero_zero_nat ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_cancel_left_right
% 0.33/0.55  thf(fact_1211_add__cancel__left__right,axiom,
% 0.33/0.55      ! [A: int,B: int] :
% 0.33/0.55        ( ( ( plus_plus_int @ A @ B )
% 0.33/0.55          = A )
% 0.33/0.55        = ( B = zero_zero_int ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_cancel_left_right
% 0.33/0.55  thf(fact_1212_add__cancel__right__left,axiom,
% 0.33/0.55      ! [A: complex,B: complex] :
% 0.33/0.55        ( ( A
% 0.33/0.55          = ( plus_plus_complex @ B @ A ) )
% 0.33/0.55        = ( B = zero_zero_complex ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_cancel_right_left
% 0.33/0.55  thf(fact_1213_add__cancel__right__left,axiom,
% 0.33/0.55      ! [A: real,B: real] :
% 0.33/0.55        ( ( A
% 0.33/0.55          = ( plus_plus_real @ B @ A ) )
% 0.33/0.55        = ( B = zero_zero_real ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_cancel_right_left
% 0.33/0.55  thf(fact_1214_add__cancel__right__left,axiom,
% 0.33/0.55      ! [A: rat,B: rat] :
% 0.33/0.55        ( ( A
% 0.33/0.55          = ( plus_plus_rat @ B @ A ) )
% 0.33/0.55        = ( B = zero_zero_rat ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_cancel_right_left
% 0.33/0.55  thf(fact_1215_add__cancel__right__left,axiom,
% 0.33/0.55      ! [A: nat,B: nat] :
% 0.33/0.55        ( ( A
% 0.33/0.55          = ( plus_plus_nat @ B @ A ) )
% 0.33/0.55        = ( B = zero_zero_nat ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_cancel_right_left
% 0.33/0.55  thf(fact_1216_add__cancel__right__left,axiom,
% 0.33/0.55      ! [A: int,B: int] :
% 0.33/0.55        ( ( A
% 0.33/0.55          = ( plus_plus_int @ B @ A ) )
% 0.33/0.55        = ( B = zero_zero_int ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_cancel_right_left
% 0.33/0.55  thf(fact_1217_add__cancel__right__right,axiom,
% 0.33/0.55      ! [A: complex,B: complex] :
% 0.33/0.55        ( ( A
% 0.33/0.55          = ( plus_plus_complex @ A @ B ) )
% 0.33/0.55        = ( B = zero_zero_complex ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_cancel_right_right
% 0.33/0.55  thf(fact_1218_add__cancel__right__right,axiom,
% 0.33/0.55      ! [A: real,B: real] :
% 0.33/0.55        ( ( A
% 0.33/0.55          = ( plus_plus_real @ A @ B ) )
% 0.33/0.55        = ( B = zero_zero_real ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_cancel_right_right
% 0.33/0.55  thf(fact_1219_add__cancel__right__right,axiom,
% 0.33/0.55      ! [A: rat,B: rat] :
% 0.33/0.55        ( ( A
% 0.33/0.55          = ( plus_plus_rat @ A @ B ) )
% 0.33/0.55        = ( B = zero_zero_rat ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_cancel_right_right
% 0.33/0.55  thf(fact_1220_add__cancel__right__right,axiom,
% 0.33/0.55      ! [A: nat,B: nat] :
% 0.33/0.55        ( ( A
% 0.33/0.55          = ( plus_plus_nat @ A @ B ) )
% 0.33/0.55        = ( B = zero_zero_nat ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_cancel_right_right
% 0.33/0.55  thf(fact_1221_add__cancel__right__right,axiom,
% 0.33/0.55      ! [A: int,B: int] :
% 0.33/0.55        ( ( A
% 0.33/0.55          = ( plus_plus_int @ A @ B ) )
% 0.33/0.55        = ( B = zero_zero_int ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_cancel_right_right
% 0.33/0.55  thf(fact_1222_add__eq__0__iff__both__eq__0,axiom,
% 0.33/0.55      ! [X: nat,Y: nat] :
% 0.33/0.55        ( ( ( plus_plus_nat @ X @ Y )
% 0.33/0.55          = zero_zero_nat )
% 0.33/0.55        = ( ( X = zero_zero_nat )
% 0.33/0.55          & ( Y = zero_zero_nat ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_eq_0_iff_both_eq_0
% 0.33/0.55  thf(fact_1223_zero__eq__add__iff__both__eq__0,axiom,
% 0.33/0.55      ! [X: nat,Y: nat] :
% 0.33/0.55        ( ( zero_zero_nat
% 0.33/0.55          = ( plus_plus_nat @ X @ Y ) )
% 0.33/0.55        = ( ( X = zero_zero_nat )
% 0.33/0.55          & ( Y = zero_zero_nat ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % zero_eq_add_iff_both_eq_0
% 0.33/0.55  thf(fact_1224_add__0,axiom,
% 0.33/0.55      ! [A: complex] :
% 0.33/0.55        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % add_0
% 0.33/0.55  thf(fact_1225_add__0,axiom,
% 0.33/0.55      ! [A: real] :
% 0.33/0.55        ( ( plus_plus_real @ zero_zero_real @ A )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % add_0
% 0.33/0.55  thf(fact_1226_add__0,axiom,
% 0.33/0.55      ! [A: rat] :
% 0.33/0.55        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % add_0
% 0.33/0.55  thf(fact_1227_add__0,axiom,
% 0.33/0.55      ! [A: nat] :
% 0.33/0.55        ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % add_0
% 0.33/0.55  thf(fact_1228_add__0,axiom,
% 0.33/0.55      ! [A: int] :
% 0.33/0.55        ( ( plus_plus_int @ zero_zero_int @ A )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % add_0
% 0.33/0.55  thf(fact_1229_add__less__cancel__left,axiom,
% 0.33/0.55      ! [C: real,A: real,B: real] :
% 0.33/0.55        ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.33/0.55        = ( ord_less_real @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_cancel_left
% 0.33/0.55  thf(fact_1230_add__less__cancel__left,axiom,
% 0.33/0.55      ! [C: rat,A: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.33/0.55        = ( ord_less_rat @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_cancel_left
% 0.33/0.55  thf(fact_1231_add__less__cancel__left,axiom,
% 0.33/0.55      ! [C: nat,A: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.33/0.55        = ( ord_less_nat @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_cancel_left
% 0.33/0.55  thf(fact_1232_add__less__cancel__left,axiom,
% 0.33/0.55      ! [C: int,A: int,B: int] :
% 0.33/0.55        ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.33/0.55        = ( ord_less_int @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_cancel_left
% 0.33/0.55  thf(fact_1233_add__less__cancel__right,axiom,
% 0.33/0.55      ! [A: real,C: real,B: real] :
% 0.33/0.55        ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.33/0.55        = ( ord_less_real @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_cancel_right
% 0.33/0.55  thf(fact_1234_add__less__cancel__right,axiom,
% 0.33/0.55      ! [A: rat,C: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.33/0.55        = ( ord_less_rat @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_cancel_right
% 0.33/0.55  thf(fact_1235_add__less__cancel__right,axiom,
% 0.33/0.55      ! [A: nat,C: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.33/0.55        = ( ord_less_nat @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_cancel_right
% 0.33/0.55  thf(fact_1236_add__less__cancel__right,axiom,
% 0.33/0.55      ! [A: int,C: int,B: int] :
% 0.33/0.55        ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.33/0.55        = ( ord_less_int @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_cancel_right
% 0.33/0.55  thf(fact_1237_add__is__0,axiom,
% 0.33/0.55      ! [M: nat,N: nat] :
% 0.33/0.55        ( ( ( plus_plus_nat @ M @ N )
% 0.33/0.55          = zero_zero_nat )
% 0.33/0.55        = ( ( M = zero_zero_nat )
% 0.33/0.55          & ( N = zero_zero_nat ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_is_0
% 0.33/0.55  thf(fact_1238_Nat_Oadd__0__right,axiom,
% 0.33/0.55      ! [M: nat] :
% 0.33/0.55        ( ( plus_plus_nat @ M @ zero_zero_nat )
% 0.33/0.55        = M ) ).
% 0.33/0.55  
% 0.33/0.55  % Nat.add_0_right
% 0.33/0.55  thf(fact_1239_nat__add__left__cancel__less,axiom,
% 0.33/0.55      ! [K: nat,M: nat,N: nat] :
% 0.33/0.55        ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.33/0.55        = ( ord_less_nat @ M @ N ) ) ).
% 0.33/0.55  
% 0.33/0.55  % nat_add_left_cancel_less
% 0.33/0.55  thf(fact_1240_nat__add__left__cancel__le,axiom,
% 0.33/0.55      ! [K: nat,M: nat,N: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.33/0.55        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.33/0.55  
% 0.33/0.55  % nat_add_left_cancel_le
% 0.33/0.55  thf(fact_1241_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.33/0.55      ! [A: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.33/0.55        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.33/0.55  
% 0.33/0.55  % zero_le_double_add_iff_zero_le_single_add
% 0.33/0.55  thf(fact_1242_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.33/0.55      ! [A: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 0.33/0.55        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.33/0.55  
% 0.33/0.55  % zero_le_double_add_iff_zero_le_single_add
% 0.33/0.55  thf(fact_1243_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.33/0.55      ! [A: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.33/0.55        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.33/0.55  
% 0.33/0.55  % zero_le_double_add_iff_zero_le_single_add
% 0.33/0.55  thf(fact_1244_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.33/0.55      ! [A: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.33/0.55        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.33/0.55  
% 0.33/0.55  % double_add_le_zero_iff_single_add_le_zero
% 0.33/0.55  thf(fact_1245_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.33/0.55      ! [A: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 0.33/0.55        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.33/0.55  
% 0.33/0.55  % double_add_le_zero_iff_single_add_le_zero
% 0.33/0.55  thf(fact_1246_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.33/0.55      ! [A: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.33/0.55        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.33/0.55  
% 0.33/0.55  % double_add_le_zero_iff_single_add_le_zero
% 0.33/0.55  thf(fact_1247_le__add__same__cancel2,axiom,
% 0.33/0.55      ! [A: real,B: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.33/0.55        = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % le_add_same_cancel2
% 0.33/0.55  thf(fact_1248_le__add__same__cancel2,axiom,
% 0.33/0.55      ! [A: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.33/0.55        = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % le_add_same_cancel2
% 0.33/0.55  thf(fact_1249_le__add__same__cancel2,axiom,
% 0.33/0.55      ! [A: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.33/0.55        = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % le_add_same_cancel2
% 0.33/0.55  thf(fact_1250_le__add__same__cancel2,axiom,
% 0.33/0.55      ! [A: int,B: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.33/0.55        = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % le_add_same_cancel2
% 0.33/0.55  thf(fact_1251_le__add__same__cancel1,axiom,
% 0.33/0.55      ! [A: real,B: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.33/0.55        = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % le_add_same_cancel1
% 0.33/0.55  thf(fact_1252_le__add__same__cancel1,axiom,
% 0.33/0.55      ! [A: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.33/0.55        = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % le_add_same_cancel1
% 0.33/0.55  thf(fact_1253_le__add__same__cancel1,axiom,
% 0.33/0.55      ! [A: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.33/0.55        = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % le_add_same_cancel1
% 0.33/0.55  thf(fact_1254_le__add__same__cancel1,axiom,
% 0.33/0.55      ! [A: int,B: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.33/0.55        = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % le_add_same_cancel1
% 0.33/0.55  thf(fact_1255_add__le__same__cancel2,axiom,
% 0.33/0.55      ! [A: real,B: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.33/0.55        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_same_cancel2
% 0.33/0.55  thf(fact_1256_add__le__same__cancel2,axiom,
% 0.33/0.55      ! [A: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.33/0.55        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_same_cancel2
% 0.33/0.55  thf(fact_1257_add__le__same__cancel2,axiom,
% 0.33/0.55      ! [A: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.33/0.55        = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_same_cancel2
% 0.33/0.55  thf(fact_1258_add__le__same__cancel2,axiom,
% 0.33/0.55      ! [A: int,B: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.33/0.55        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_same_cancel2
% 0.33/0.55  thf(fact_1259_add__le__same__cancel1,axiom,
% 0.33/0.55      ! [B: real,A: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.33/0.55        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_same_cancel1
% 0.33/0.55  thf(fact_1260_add__le__same__cancel1,axiom,
% 0.33/0.55      ! [B: rat,A: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 0.33/0.55        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_same_cancel1
% 0.33/0.55  thf(fact_1261_add__le__same__cancel1,axiom,
% 0.33/0.55      ! [B: nat,A: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.33/0.55        = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_same_cancel1
% 0.33/0.55  thf(fact_1262_add__le__same__cancel1,axiom,
% 0.33/0.55      ! [B: int,A: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.33/0.55        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_same_cancel1
% 0.33/0.55  thf(fact_1263_add__less__same__cancel1,axiom,
% 0.33/0.55      ! [B: real,A: real] :
% 0.33/0.55        ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.33/0.55        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_same_cancel1
% 0.33/0.55  thf(fact_1264_add__less__same__cancel1,axiom,
% 0.33/0.55      ! [B: rat,A: rat] :
% 0.33/0.55        ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 0.33/0.55        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_same_cancel1
% 0.33/0.55  thf(fact_1265_add__less__same__cancel1,axiom,
% 0.33/0.55      ! [B: nat,A: nat] :
% 0.33/0.55        ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.33/0.55        = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_same_cancel1
% 0.33/0.55  thf(fact_1266_add__less__same__cancel1,axiom,
% 0.33/0.55      ! [B: int,A: int] :
% 0.33/0.55        ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.33/0.55        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_same_cancel1
% 0.33/0.55  thf(fact_1267_add__less__same__cancel2,axiom,
% 0.33/0.55      ! [A: real,B: real] :
% 0.33/0.55        ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.33/0.55        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_same_cancel2
% 0.33/0.55  thf(fact_1268_add__less__same__cancel2,axiom,
% 0.33/0.55      ! [A: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.33/0.55        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_same_cancel2
% 0.33/0.55  thf(fact_1269_add__less__same__cancel2,axiom,
% 0.33/0.55      ! [A: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.33/0.55        = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_same_cancel2
% 0.33/0.55  thf(fact_1270_add__less__same__cancel2,axiom,
% 0.33/0.55      ! [A: int,B: int] :
% 0.33/0.55        ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.33/0.55        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_same_cancel2
% 0.33/0.55  thf(fact_1271_less__add__same__cancel1,axiom,
% 0.33/0.55      ! [A: real,B: real] :
% 0.33/0.55        ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.33/0.55        = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % less_add_same_cancel1
% 0.33/0.55  thf(fact_1272_less__add__same__cancel1,axiom,
% 0.33/0.55      ! [A: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.33/0.55        = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % less_add_same_cancel1
% 0.33/0.55  thf(fact_1273_less__add__same__cancel1,axiom,
% 0.33/0.55      ! [A: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.33/0.55        = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % less_add_same_cancel1
% 0.33/0.55  thf(fact_1274_less__add__same__cancel1,axiom,
% 0.33/0.55      ! [A: int,B: int] :
% 0.33/0.55        ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.33/0.55        = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % less_add_same_cancel1
% 0.33/0.55  thf(fact_1275_less__add__same__cancel2,axiom,
% 0.33/0.55      ! [A: real,B: real] :
% 0.33/0.55        ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.33/0.55        = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % less_add_same_cancel2
% 0.33/0.55  thf(fact_1276_less__add__same__cancel2,axiom,
% 0.33/0.55      ! [A: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.33/0.55        = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % less_add_same_cancel2
% 0.33/0.55  thf(fact_1277_less__add__same__cancel2,axiom,
% 0.33/0.55      ! [A: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.33/0.55        = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % less_add_same_cancel2
% 0.33/0.55  thf(fact_1278_less__add__same__cancel2,axiom,
% 0.33/0.55      ! [A: int,B: int] :
% 0.33/0.55        ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.33/0.55        = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % less_add_same_cancel2
% 0.33/0.55  thf(fact_1279_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.33/0.55      ! [A: real] :
% 0.33/0.55        ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.33/0.55        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.33/0.55  
% 0.33/0.55  % double_add_less_zero_iff_single_add_less_zero
% 0.33/0.55  thf(fact_1280_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.33/0.55      ! [A: rat] :
% 0.33/0.55        ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 0.33/0.55        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.33/0.55  
% 0.33/0.55  % double_add_less_zero_iff_single_add_less_zero
% 0.33/0.55  thf(fact_1281_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.33/0.55      ! [A: int] :
% 0.33/0.55        ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.33/0.55        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.33/0.55  
% 0.33/0.55  % double_add_less_zero_iff_single_add_less_zero
% 0.33/0.55  thf(fact_1282_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.33/0.55      ! [A: real] :
% 0.33/0.55        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.33/0.55        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.33/0.55  
% 0.33/0.55  % zero_less_double_add_iff_zero_less_single_add
% 0.33/0.55  thf(fact_1283_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.33/0.55      ! [A: rat] :
% 0.33/0.55        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 0.33/0.55        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.33/0.55  
% 0.33/0.55  % zero_less_double_add_iff_zero_less_single_add
% 0.33/0.55  thf(fact_1284_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.33/0.55      ! [A: int] :
% 0.33/0.55        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.33/0.55        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.33/0.55  
% 0.33/0.55  % zero_less_double_add_iff_zero_less_single_add
% 0.33/0.55  thf(fact_1285_add__gr__0,axiom,
% 0.33/0.55      ! [M: nat,N: nat] :
% 0.33/0.55        ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 0.33/0.55        = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.55          | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_gr_0
% 0.33/0.55  thf(fact_1286_add__right__imp__eq,axiom,
% 0.33/0.55      ! [B: real,A: real,C: real] :
% 0.33/0.55        ( ( ( plus_plus_real @ B @ A )
% 0.33/0.55          = ( plus_plus_real @ C @ A ) )
% 0.33/0.55       => ( B = C ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_right_imp_eq
% 0.33/0.55  thf(fact_1287_add__right__imp__eq,axiom,
% 0.33/0.55      ! [B: rat,A: rat,C: rat] :
% 0.33/0.55        ( ( ( plus_plus_rat @ B @ A )
% 0.33/0.55          = ( plus_plus_rat @ C @ A ) )
% 0.33/0.55       => ( B = C ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_right_imp_eq
% 0.33/0.55  thf(fact_1288_add__right__imp__eq,axiom,
% 0.33/0.55      ! [B: nat,A: nat,C: nat] :
% 0.33/0.55        ( ( ( plus_plus_nat @ B @ A )
% 0.33/0.55          = ( plus_plus_nat @ C @ A ) )
% 0.33/0.55       => ( B = C ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_right_imp_eq
% 0.33/0.55  thf(fact_1289_add__right__imp__eq,axiom,
% 0.33/0.55      ! [B: int,A: int,C: int] :
% 0.33/0.55        ( ( ( plus_plus_int @ B @ A )
% 0.33/0.55          = ( plus_plus_int @ C @ A ) )
% 0.33/0.55       => ( B = C ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_right_imp_eq
% 0.33/0.55  thf(fact_1290_add__left__imp__eq,axiom,
% 0.33/0.55      ! [A: real,B: real,C: real] :
% 0.33/0.55        ( ( ( plus_plus_real @ A @ B )
% 0.33/0.55          = ( plus_plus_real @ A @ C ) )
% 0.33/0.55       => ( B = C ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_left_imp_eq
% 0.33/0.55  thf(fact_1291_add__left__imp__eq,axiom,
% 0.33/0.55      ! [A: rat,B: rat,C: rat] :
% 0.33/0.55        ( ( ( plus_plus_rat @ A @ B )
% 0.33/0.55          = ( plus_plus_rat @ A @ C ) )
% 0.33/0.55       => ( B = C ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_left_imp_eq
% 0.33/0.55  thf(fact_1292_add__left__imp__eq,axiom,
% 0.33/0.55      ! [A: nat,B: nat,C: nat] :
% 0.33/0.55        ( ( ( plus_plus_nat @ A @ B )
% 0.33/0.55          = ( plus_plus_nat @ A @ C ) )
% 0.33/0.55       => ( B = C ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_left_imp_eq
% 0.33/0.55  thf(fact_1293_add__left__imp__eq,axiom,
% 0.33/0.55      ! [A: int,B: int,C: int] :
% 0.33/0.55        ( ( ( plus_plus_int @ A @ B )
% 0.33/0.55          = ( plus_plus_int @ A @ C ) )
% 0.33/0.55       => ( B = C ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_left_imp_eq
% 0.33/0.55  thf(fact_1294_add_Oleft__commute,axiom,
% 0.33/0.55      ! [B: real,A: real,C: real] :
% 0.33/0.55        ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 0.33/0.55        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add.left_commute
% 0.33/0.55  thf(fact_1295_add_Oleft__commute,axiom,
% 0.33/0.55      ! [B: rat,A: rat,C: rat] :
% 0.33/0.55        ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
% 0.33/0.55        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add.left_commute
% 0.33/0.55  thf(fact_1296_add_Oleft__commute,axiom,
% 0.33/0.55      ! [B: nat,A: nat,C: nat] :
% 0.33/0.55        ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 0.33/0.55        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add.left_commute
% 0.33/0.55  thf(fact_1297_add_Oleft__commute,axiom,
% 0.33/0.55      ! [B: int,A: int,C: int] :
% 0.33/0.55        ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 0.33/0.55        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add.left_commute
% 0.33/0.55  thf(fact_1298_add_Ocommute,axiom,
% 0.33/0.55      ( plus_plus_real
% 0.33/0.55      = ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add.commute
% 0.33/0.55  thf(fact_1299_add_Ocommute,axiom,
% 0.33/0.55      ( plus_plus_rat
% 0.33/0.55      = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ B4 @ A4 ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add.commute
% 0.33/0.55  thf(fact_1300_add_Ocommute,axiom,
% 0.33/0.55      ( plus_plus_nat
% 0.33/0.55      = ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add.commute
% 0.33/0.55  thf(fact_1301_add_Ocommute,axiom,
% 0.33/0.55      ( plus_plus_int
% 0.33/0.55      = ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add.commute
% 0.33/0.55  thf(fact_1302_add_Oright__cancel,axiom,
% 0.33/0.55      ! [B: real,A: real,C: real] :
% 0.33/0.55        ( ( ( plus_plus_real @ B @ A )
% 0.33/0.55          = ( plus_plus_real @ C @ A ) )
% 0.33/0.55        = ( B = C ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add.right_cancel
% 0.33/0.55  thf(fact_1303_add_Oright__cancel,axiom,
% 0.33/0.55      ! [B: rat,A: rat,C: rat] :
% 0.33/0.55        ( ( ( plus_plus_rat @ B @ A )
% 0.33/0.55          = ( plus_plus_rat @ C @ A ) )
% 0.33/0.55        = ( B = C ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add.right_cancel
% 0.33/0.55  thf(fact_1304_add_Oright__cancel,axiom,
% 0.33/0.55      ! [B: int,A: int,C: int] :
% 0.33/0.55        ( ( ( plus_plus_int @ B @ A )
% 0.33/0.55          = ( plus_plus_int @ C @ A ) )
% 0.33/0.55        = ( B = C ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add.right_cancel
% 0.33/0.55  thf(fact_1305_add_Oleft__cancel,axiom,
% 0.33/0.55      ! [A: real,B: real,C: real] :
% 0.33/0.55        ( ( ( plus_plus_real @ A @ B )
% 0.33/0.55          = ( plus_plus_real @ A @ C ) )
% 0.33/0.55        = ( B = C ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add.left_cancel
% 0.33/0.55  thf(fact_1306_add_Oleft__cancel,axiom,
% 0.33/0.55      ! [A: rat,B: rat,C: rat] :
% 0.33/0.55        ( ( ( plus_plus_rat @ A @ B )
% 0.33/0.55          = ( plus_plus_rat @ A @ C ) )
% 0.33/0.55        = ( B = C ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add.left_cancel
% 0.33/0.55  thf(fact_1307_add_Oleft__cancel,axiom,
% 0.33/0.55      ! [A: int,B: int,C: int] :
% 0.33/0.55        ( ( ( plus_plus_int @ A @ B )
% 0.33/0.55          = ( plus_plus_int @ A @ C ) )
% 0.33/0.55        = ( B = C ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add.left_cancel
% 0.33/0.55  thf(fact_1308_add_Oassoc,axiom,
% 0.33/0.55      ! [A: real,B: real,C: real] :
% 0.33/0.55        ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.33/0.55        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add.assoc
% 0.33/0.55  thf(fact_1309_add_Oassoc,axiom,
% 0.33/0.55      ! [A: rat,B: rat,C: rat] :
% 0.33/0.55        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.33/0.55        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add.assoc
% 0.33/0.55  thf(fact_1310_add_Oassoc,axiom,
% 0.33/0.55      ! [A: nat,B: nat,C: nat] :
% 0.33/0.55        ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.33/0.55        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add.assoc
% 0.33/0.55  thf(fact_1311_add_Oassoc,axiom,
% 0.33/0.55      ! [A: int,B: int,C: int] :
% 0.33/0.55        ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.33/0.55        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add.assoc
% 0.33/0.55  thf(fact_1312_group__cancel_Oadd2,axiom,
% 0.33/0.55      ! [B2: real,K: real,B: real,A: real] :
% 0.33/0.55        ( ( B2
% 0.33/0.55          = ( plus_plus_real @ K @ B ) )
% 0.33/0.55       => ( ( plus_plus_real @ A @ B2 )
% 0.33/0.55          = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % group_cancel.add2
% 0.33/0.55  thf(fact_1313_group__cancel_Oadd2,axiom,
% 0.33/0.55      ! [B2: rat,K: rat,B: rat,A: rat] :
% 0.33/0.55        ( ( B2
% 0.33/0.55          = ( plus_plus_rat @ K @ B ) )
% 0.33/0.55       => ( ( plus_plus_rat @ A @ B2 )
% 0.33/0.55          = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % group_cancel.add2
% 0.33/0.55  thf(fact_1314_group__cancel_Oadd2,axiom,
% 0.33/0.55      ! [B2: nat,K: nat,B: nat,A: nat] :
% 0.33/0.55        ( ( B2
% 0.33/0.55          = ( plus_plus_nat @ K @ B ) )
% 0.33/0.55       => ( ( plus_plus_nat @ A @ B2 )
% 0.33/0.55          = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % group_cancel.add2
% 0.33/0.55  thf(fact_1315_group__cancel_Oadd2,axiom,
% 0.33/0.55      ! [B2: int,K: int,B: int,A: int] :
% 0.33/0.55        ( ( B2
% 0.33/0.55          = ( plus_plus_int @ K @ B ) )
% 0.33/0.55       => ( ( plus_plus_int @ A @ B2 )
% 0.33/0.55          = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % group_cancel.add2
% 0.33/0.55  thf(fact_1316_group__cancel_Oadd1,axiom,
% 0.33/0.55      ! [A2: real,K: real,A: real,B: real] :
% 0.33/0.55        ( ( A2
% 0.33/0.55          = ( plus_plus_real @ K @ A ) )
% 0.33/0.55       => ( ( plus_plus_real @ A2 @ B )
% 0.33/0.55          = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % group_cancel.add1
% 0.33/0.55  thf(fact_1317_group__cancel_Oadd1,axiom,
% 0.33/0.55      ! [A2: rat,K: rat,A: rat,B: rat] :
% 0.33/0.55        ( ( A2
% 0.33/0.55          = ( plus_plus_rat @ K @ A ) )
% 0.33/0.55       => ( ( plus_plus_rat @ A2 @ B )
% 0.33/0.55          = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % group_cancel.add1
% 0.33/0.55  thf(fact_1318_group__cancel_Oadd1,axiom,
% 0.33/0.55      ! [A2: nat,K: nat,A: nat,B: nat] :
% 0.33/0.55        ( ( A2
% 0.33/0.55          = ( plus_plus_nat @ K @ A ) )
% 0.33/0.55       => ( ( plus_plus_nat @ A2 @ B )
% 0.33/0.55          = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % group_cancel.add1
% 0.33/0.55  thf(fact_1319_group__cancel_Oadd1,axiom,
% 0.33/0.55      ! [A2: int,K: int,A: int,B: int] :
% 0.33/0.55        ( ( A2
% 0.33/0.55          = ( plus_plus_int @ K @ A ) )
% 0.33/0.55       => ( ( plus_plus_int @ A2 @ B )
% 0.33/0.55          = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % group_cancel.add1
% 0.33/0.55  thf(fact_1320_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.33/0.55      ! [I: real,J: real,K: real,L: real] :
% 0.33/0.55        ( ( ( I = J )
% 0.33/0.55          & ( K = L ) )
% 0.33/0.55       => ( ( plus_plus_real @ I @ K )
% 0.33/0.55          = ( plus_plus_real @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_semiring(4)
% 0.33/0.55  thf(fact_1321_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.33/0.55      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.33/0.55        ( ( ( I = J )
% 0.33/0.55          & ( K = L ) )
% 0.33/0.55       => ( ( plus_plus_rat @ I @ K )
% 0.33/0.55          = ( plus_plus_rat @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_semiring(4)
% 0.33/0.55  thf(fact_1322_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.33/0.55      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.33/0.55        ( ( ( I = J )
% 0.33/0.55          & ( K = L ) )
% 0.33/0.55       => ( ( plus_plus_nat @ I @ K )
% 0.33/0.55          = ( plus_plus_nat @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_semiring(4)
% 0.33/0.55  thf(fact_1323_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.33/0.55      ! [I: int,J: int,K: int,L: int] :
% 0.33/0.55        ( ( ( I = J )
% 0.33/0.55          & ( K = L ) )
% 0.33/0.55       => ( ( plus_plus_int @ I @ K )
% 0.33/0.55          = ( plus_plus_int @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_semiring(4)
% 0.33/0.55  thf(fact_1324_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.33/0.55      ! [A: real,B: real,C: real] :
% 0.33/0.55        ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.33/0.55        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % ab_semigroup_add_class.add_ac(1)
% 0.33/0.55  thf(fact_1325_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.33/0.55      ! [A: rat,B: rat,C: rat] :
% 0.33/0.55        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.33/0.55        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % ab_semigroup_add_class.add_ac(1)
% 0.33/0.55  thf(fact_1326_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.33/0.55      ! [A: nat,B: nat,C: nat] :
% 0.33/0.55        ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.33/0.55        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % ab_semigroup_add_class.add_ac(1)
% 0.33/0.55  thf(fact_1327_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.33/0.55      ! [A: int,B: int,C: int] :
% 0.33/0.55        ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.33/0.55        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % ab_semigroup_add_class.add_ac(1)
% 0.33/0.55  thf(fact_1328_verit__sum__simplify,axiom,
% 0.33/0.55      ! [A: complex] :
% 0.33/0.55        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % verit_sum_simplify
% 0.33/0.55  thf(fact_1329_verit__sum__simplify,axiom,
% 0.33/0.55      ! [A: real] :
% 0.33/0.55        ( ( plus_plus_real @ A @ zero_zero_real )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % verit_sum_simplify
% 0.33/0.55  thf(fact_1330_verit__sum__simplify,axiom,
% 0.33/0.55      ! [A: rat] :
% 0.33/0.55        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % verit_sum_simplify
% 0.33/0.55  thf(fact_1331_verit__sum__simplify,axiom,
% 0.33/0.55      ! [A: nat] :
% 0.33/0.55        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % verit_sum_simplify
% 0.33/0.55  thf(fact_1332_verit__sum__simplify,axiom,
% 0.33/0.55      ! [A: int] :
% 0.33/0.55        ( ( plus_plus_int @ A @ zero_zero_int )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % verit_sum_simplify
% 0.33/0.55  thf(fact_1333_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.33/0.55      ! [I: real,J: real,K: real,L: real] :
% 0.33/0.55        ( ( ( ord_less_eq_real @ I @ J )
% 0.33/0.55          & ( K = L ) )
% 0.33/0.55       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_semiring(3)
% 0.33/0.55  thf(fact_1334_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.33/0.55      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.33/0.55        ( ( ( ord_less_eq_rat @ I @ J )
% 0.33/0.55          & ( K = L ) )
% 0.33/0.55       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_semiring(3)
% 0.33/0.55  thf(fact_1335_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.33/0.55      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.33/0.55        ( ( ( ord_less_eq_nat @ I @ J )
% 0.33/0.55          & ( K = L ) )
% 0.33/0.55       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_semiring(3)
% 0.33/0.55  thf(fact_1336_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.33/0.55      ! [I: int,J: int,K: int,L: int] :
% 0.33/0.55        ( ( ( ord_less_eq_int @ I @ J )
% 0.33/0.55          & ( K = L ) )
% 0.33/0.55       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_semiring(3)
% 0.33/0.55  thf(fact_1337_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.33/0.55      ! [I: real,J: real,K: real,L: real] :
% 0.33/0.55        ( ( ( I = J )
% 0.33/0.55          & ( ord_less_eq_real @ K @ L ) )
% 0.33/0.55       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_semiring(2)
% 0.33/0.55  thf(fact_1338_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.33/0.55      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.33/0.55        ( ( ( I = J )
% 0.33/0.55          & ( ord_less_eq_rat @ K @ L ) )
% 0.33/0.55       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_semiring(2)
% 0.33/0.55  thf(fact_1339_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.33/0.55      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.33/0.55        ( ( ( I = J )
% 0.33/0.55          & ( ord_less_eq_nat @ K @ L ) )
% 0.33/0.55       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_semiring(2)
% 0.33/0.55  thf(fact_1340_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.33/0.55      ! [I: int,J: int,K: int,L: int] :
% 0.33/0.55        ( ( ( I = J )
% 0.33/0.55          & ( ord_less_eq_int @ K @ L ) )
% 0.33/0.55       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_semiring(2)
% 0.33/0.55  thf(fact_1341_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.33/0.55      ! [I: real,J: real,K: real,L: real] :
% 0.33/0.55        ( ( ( ord_less_eq_real @ I @ J )
% 0.33/0.55          & ( ord_less_eq_real @ K @ L ) )
% 0.33/0.55       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_semiring(1)
% 0.33/0.55  thf(fact_1342_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.33/0.55      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.33/0.55        ( ( ( ord_less_eq_rat @ I @ J )
% 0.33/0.55          & ( ord_less_eq_rat @ K @ L ) )
% 0.33/0.55       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_semiring(1)
% 0.33/0.55  thf(fact_1343_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.33/0.55      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.33/0.55        ( ( ( ord_less_eq_nat @ I @ J )
% 0.33/0.55          & ( ord_less_eq_nat @ K @ L ) )
% 0.33/0.55       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_semiring(1)
% 0.33/0.55  thf(fact_1344_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.33/0.55      ! [I: int,J: int,K: int,L: int] :
% 0.33/0.55        ( ( ( ord_less_eq_int @ I @ J )
% 0.33/0.55          & ( ord_less_eq_int @ K @ L ) )
% 0.33/0.55       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_semiring(1)
% 0.33/0.55  thf(fact_1345_add__mono,axiom,
% 0.33/0.55      ! [A: real,B: real,C: real,D: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ A @ B )
% 0.33/0.55       => ( ( ord_less_eq_real @ C @ D )
% 0.33/0.55         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono
% 0.33/0.55  thf(fact_1346_add__mono,axiom,
% 0.33/0.55      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.55       => ( ( ord_less_eq_rat @ C @ D )
% 0.33/0.55         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono
% 0.33/0.55  thf(fact_1347_add__mono,axiom,
% 0.33/0.55      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.55       => ( ( ord_less_eq_nat @ C @ D )
% 0.33/0.55         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono
% 0.33/0.55  thf(fact_1348_add__mono,axiom,
% 0.33/0.55      ! [A: int,B: int,C: int,D: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ A @ B )
% 0.33/0.55       => ( ( ord_less_eq_int @ C @ D )
% 0.33/0.55         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono
% 0.33/0.55  thf(fact_1349_add__left__mono,axiom,
% 0.33/0.55      ! [A: real,B: real,C: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ A @ B )
% 0.33/0.55       => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_left_mono
% 0.33/0.55  thf(fact_1350_add__left__mono,axiom,
% 0.33/0.55      ! [A: rat,B: rat,C: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.55       => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_left_mono
% 0.33/0.55  thf(fact_1351_add__left__mono,axiom,
% 0.33/0.55      ! [A: nat,B: nat,C: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.55       => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_left_mono
% 0.33/0.55  thf(fact_1352_add__left__mono,axiom,
% 0.33/0.55      ! [A: int,B: int,C: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ A @ B )
% 0.33/0.55       => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_left_mono
% 0.33/0.55  thf(fact_1353_less__eqE,axiom,
% 0.33/0.55      ! [A: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.55       => ~ ! [C3: nat] :
% 0.33/0.55              ( B
% 0.33/0.55             != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % less_eqE
% 0.33/0.55  thf(fact_1354_add__right__mono,axiom,
% 0.33/0.55      ! [A: real,B: real,C: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ A @ B )
% 0.33/0.55       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_right_mono
% 0.33/0.55  thf(fact_1355_add__right__mono,axiom,
% 0.33/0.55      ! [A: rat,B: rat,C: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.55       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_right_mono
% 0.33/0.55  thf(fact_1356_add__right__mono,axiom,
% 0.33/0.55      ! [A: nat,B: nat,C: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.55       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_right_mono
% 0.33/0.55  thf(fact_1357_add__right__mono,axiom,
% 0.33/0.55      ! [A: int,B: int,C: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ A @ B )
% 0.33/0.55       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_right_mono
% 0.33/0.55  thf(fact_1358_le__iff__add,axiom,
% 0.33/0.55      ( ord_less_eq_nat
% 0.33/0.55      = ( ^ [A4: nat,B4: nat] :
% 0.33/0.55          ? [C4: nat] :
% 0.33/0.55            ( B4
% 0.33/0.55            = ( plus_plus_nat @ A4 @ C4 ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % le_iff_add
% 0.33/0.55  thf(fact_1359_add__le__imp__le__left,axiom,
% 0.33/0.55      ! [C: real,A: real,B: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.33/0.55       => ( ord_less_eq_real @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_imp_le_left
% 0.33/0.55  thf(fact_1360_add__le__imp__le__left,axiom,
% 0.33/0.55      ! [C: rat,A: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.33/0.55       => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_imp_le_left
% 0.33/0.55  thf(fact_1361_add__le__imp__le__left,axiom,
% 0.33/0.55      ! [C: nat,A: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.33/0.55       => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_imp_le_left
% 0.33/0.55  thf(fact_1362_add__le__imp__le__left,axiom,
% 0.33/0.55      ! [C: int,A: int,B: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.33/0.55       => ( ord_less_eq_int @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_imp_le_left
% 0.33/0.55  thf(fact_1363_add__le__imp__le__right,axiom,
% 0.33/0.55      ! [A: real,C: real,B: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.33/0.55       => ( ord_less_eq_real @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_imp_le_right
% 0.33/0.55  thf(fact_1364_add__le__imp__le__right,axiom,
% 0.33/0.55      ! [A: rat,C: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.33/0.55       => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_imp_le_right
% 0.33/0.55  thf(fact_1365_add__le__imp__le__right,axiom,
% 0.33/0.55      ! [A: nat,C: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.33/0.55       => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_imp_le_right
% 0.33/0.55  thf(fact_1366_add__le__imp__le__right,axiom,
% 0.33/0.55      ! [A: int,C: int,B: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.33/0.55       => ( ord_less_eq_int @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_imp_le_right
% 0.33/0.55  thf(fact_1367_comm__monoid__add__class_Oadd__0,axiom,
% 0.33/0.55      ! [A: complex] :
% 0.33/0.55        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % comm_monoid_add_class.add_0
% 0.33/0.55  thf(fact_1368_comm__monoid__add__class_Oadd__0,axiom,
% 0.33/0.55      ! [A: real] :
% 0.33/0.55        ( ( plus_plus_real @ zero_zero_real @ A )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % comm_monoid_add_class.add_0
% 0.33/0.55  thf(fact_1369_comm__monoid__add__class_Oadd__0,axiom,
% 0.33/0.55      ! [A: rat] :
% 0.33/0.55        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % comm_monoid_add_class.add_0
% 0.33/0.55  thf(fact_1370_comm__monoid__add__class_Oadd__0,axiom,
% 0.33/0.55      ! [A: nat] :
% 0.33/0.55        ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % comm_monoid_add_class.add_0
% 0.33/0.55  thf(fact_1371_comm__monoid__add__class_Oadd__0,axiom,
% 0.33/0.55      ! [A: int] :
% 0.33/0.55        ( ( plus_plus_int @ zero_zero_int @ A )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % comm_monoid_add_class.add_0
% 0.33/0.55  thf(fact_1372_add_Ocomm__neutral,axiom,
% 0.33/0.55      ! [A: complex] :
% 0.33/0.55        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % add.comm_neutral
% 0.33/0.55  thf(fact_1373_add_Ocomm__neutral,axiom,
% 0.33/0.55      ! [A: real] :
% 0.33/0.55        ( ( plus_plus_real @ A @ zero_zero_real )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % add.comm_neutral
% 0.33/0.55  thf(fact_1374_add_Ocomm__neutral,axiom,
% 0.33/0.55      ! [A: rat] :
% 0.33/0.55        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % add.comm_neutral
% 0.33/0.55  thf(fact_1375_add_Ocomm__neutral,axiom,
% 0.33/0.55      ! [A: nat] :
% 0.33/0.55        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % add.comm_neutral
% 0.33/0.55  thf(fact_1376_add_Ocomm__neutral,axiom,
% 0.33/0.55      ! [A: int] :
% 0.33/0.55        ( ( plus_plus_int @ A @ zero_zero_int )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % add.comm_neutral
% 0.33/0.55  thf(fact_1377_add_Ogroup__left__neutral,axiom,
% 0.33/0.55      ! [A: complex] :
% 0.33/0.55        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % add.group_left_neutral
% 0.33/0.55  thf(fact_1378_add_Ogroup__left__neutral,axiom,
% 0.33/0.55      ! [A: real] :
% 0.33/0.55        ( ( plus_plus_real @ zero_zero_real @ A )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % add.group_left_neutral
% 0.33/0.55  thf(fact_1379_add_Ogroup__left__neutral,axiom,
% 0.33/0.55      ! [A: rat] :
% 0.33/0.55        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % add.group_left_neutral
% 0.33/0.55  thf(fact_1380_add_Ogroup__left__neutral,axiom,
% 0.33/0.55      ! [A: int] :
% 0.33/0.55        ( ( plus_plus_int @ zero_zero_int @ A )
% 0.33/0.55        = A ) ).
% 0.33/0.55  
% 0.33/0.55  % add.group_left_neutral
% 0.33/0.55  thf(fact_1381_add__mono__thms__linordered__field_I5_J,axiom,
% 0.33/0.55      ! [I: real,J: real,K: real,L: real] :
% 0.33/0.55        ( ( ( ord_less_real @ I @ J )
% 0.33/0.55          & ( ord_less_real @ K @ L ) )
% 0.33/0.55       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_field(5)
% 0.33/0.55  thf(fact_1382_add__mono__thms__linordered__field_I5_J,axiom,
% 0.33/0.55      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.33/0.55        ( ( ( ord_less_rat @ I @ J )
% 0.33/0.55          & ( ord_less_rat @ K @ L ) )
% 0.33/0.55       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_field(5)
% 0.33/0.55  thf(fact_1383_add__mono__thms__linordered__field_I5_J,axiom,
% 0.33/0.55      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.33/0.55        ( ( ( ord_less_nat @ I @ J )
% 0.33/0.55          & ( ord_less_nat @ K @ L ) )
% 0.33/0.55       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_field(5)
% 0.33/0.55  thf(fact_1384_add__mono__thms__linordered__field_I5_J,axiom,
% 0.33/0.55      ! [I: int,J: int,K: int,L: int] :
% 0.33/0.55        ( ( ( ord_less_int @ I @ J )
% 0.33/0.55          & ( ord_less_int @ K @ L ) )
% 0.33/0.55       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_field(5)
% 0.33/0.55  thf(fact_1385_add__mono__thms__linordered__field_I2_J,axiom,
% 0.33/0.55      ! [I: real,J: real,K: real,L: real] :
% 0.33/0.55        ( ( ( I = J )
% 0.33/0.55          & ( ord_less_real @ K @ L ) )
% 0.33/0.55       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_field(2)
% 0.33/0.55  thf(fact_1386_add__mono__thms__linordered__field_I2_J,axiom,
% 0.33/0.55      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.33/0.55        ( ( ( I = J )
% 0.33/0.55          & ( ord_less_rat @ K @ L ) )
% 0.33/0.55       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_field(2)
% 0.33/0.55  thf(fact_1387_add__mono__thms__linordered__field_I2_J,axiom,
% 0.33/0.55      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.33/0.55        ( ( ( I = J )
% 0.33/0.55          & ( ord_less_nat @ K @ L ) )
% 0.33/0.55       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_field(2)
% 0.33/0.55  thf(fact_1388_add__mono__thms__linordered__field_I2_J,axiom,
% 0.33/0.55      ! [I: int,J: int,K: int,L: int] :
% 0.33/0.55        ( ( ( I = J )
% 0.33/0.55          & ( ord_less_int @ K @ L ) )
% 0.33/0.55       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_field(2)
% 0.33/0.55  thf(fact_1389_add__mono__thms__linordered__field_I1_J,axiom,
% 0.33/0.55      ! [I: real,J: real,K: real,L: real] :
% 0.33/0.55        ( ( ( ord_less_real @ I @ J )
% 0.33/0.55          & ( K = L ) )
% 0.33/0.55       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_field(1)
% 0.33/0.55  thf(fact_1390_add__mono__thms__linordered__field_I1_J,axiom,
% 0.33/0.55      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.33/0.55        ( ( ( ord_less_rat @ I @ J )
% 0.33/0.55          & ( K = L ) )
% 0.33/0.55       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_field(1)
% 0.33/0.55  thf(fact_1391_add__mono__thms__linordered__field_I1_J,axiom,
% 0.33/0.55      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.33/0.55        ( ( ( ord_less_nat @ I @ J )
% 0.33/0.55          & ( K = L ) )
% 0.33/0.55       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_field(1)
% 0.33/0.55  thf(fact_1392_add__mono__thms__linordered__field_I1_J,axiom,
% 0.33/0.55      ! [I: int,J: int,K: int,L: int] :
% 0.33/0.55        ( ( ( ord_less_int @ I @ J )
% 0.33/0.55          & ( K = L ) )
% 0.33/0.55       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_field(1)
% 0.33/0.55  thf(fact_1393_add__strict__mono,axiom,
% 0.33/0.55      ! [A: real,B: real,C: real,D: real] :
% 0.33/0.55        ( ( ord_less_real @ A @ B )
% 0.33/0.55       => ( ( ord_less_real @ C @ D )
% 0.33/0.55         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_strict_mono
% 0.33/0.55  thf(fact_1394_add__strict__mono,axiom,
% 0.33/0.55      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.33/0.55        ( ( ord_less_rat @ A @ B )
% 0.33/0.55       => ( ( ord_less_rat @ C @ D )
% 0.33/0.55         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_strict_mono
% 0.33/0.55  thf(fact_1395_add__strict__mono,axiom,
% 0.33/0.55      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.33/0.55        ( ( ord_less_nat @ A @ B )
% 0.33/0.55       => ( ( ord_less_nat @ C @ D )
% 0.33/0.55         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_strict_mono
% 0.33/0.55  thf(fact_1396_add__strict__mono,axiom,
% 0.33/0.55      ! [A: int,B: int,C: int,D: int] :
% 0.33/0.55        ( ( ord_less_int @ A @ B )
% 0.33/0.55       => ( ( ord_less_int @ C @ D )
% 0.33/0.55         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_strict_mono
% 0.33/0.55  thf(fact_1397_add__strict__left__mono,axiom,
% 0.33/0.55      ! [A: real,B: real,C: real] :
% 0.33/0.55        ( ( ord_less_real @ A @ B )
% 0.33/0.55       => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_strict_left_mono
% 0.33/0.55  thf(fact_1398_add__strict__left__mono,axiom,
% 0.33/0.55      ! [A: rat,B: rat,C: rat] :
% 0.33/0.55        ( ( ord_less_rat @ A @ B )
% 0.33/0.55       => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_strict_left_mono
% 0.33/0.55  thf(fact_1399_add__strict__left__mono,axiom,
% 0.33/0.55      ! [A: nat,B: nat,C: nat] :
% 0.33/0.55        ( ( ord_less_nat @ A @ B )
% 0.33/0.55       => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_strict_left_mono
% 0.33/0.55  thf(fact_1400_add__strict__left__mono,axiom,
% 0.33/0.55      ! [A: int,B: int,C: int] :
% 0.33/0.55        ( ( ord_less_int @ A @ B )
% 0.33/0.55       => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_strict_left_mono
% 0.33/0.55  thf(fact_1401_add__strict__right__mono,axiom,
% 0.33/0.55      ! [A: real,B: real,C: real] :
% 0.33/0.55        ( ( ord_less_real @ A @ B )
% 0.33/0.55       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_strict_right_mono
% 0.33/0.55  thf(fact_1402_add__strict__right__mono,axiom,
% 0.33/0.55      ! [A: rat,B: rat,C: rat] :
% 0.33/0.55        ( ( ord_less_rat @ A @ B )
% 0.33/0.55       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_strict_right_mono
% 0.33/0.55  thf(fact_1403_add__strict__right__mono,axiom,
% 0.33/0.55      ! [A: nat,B: nat,C: nat] :
% 0.33/0.55        ( ( ord_less_nat @ A @ B )
% 0.33/0.55       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_strict_right_mono
% 0.33/0.55  thf(fact_1404_add__strict__right__mono,axiom,
% 0.33/0.55      ! [A: int,B: int,C: int] :
% 0.33/0.55        ( ( ord_less_int @ A @ B )
% 0.33/0.55       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_strict_right_mono
% 0.33/0.55  thf(fact_1405_add__less__imp__less__left,axiom,
% 0.33/0.55      ! [C: real,A: real,B: real] :
% 0.33/0.55        ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.33/0.55       => ( ord_less_real @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_imp_less_left
% 0.33/0.55  thf(fact_1406_add__less__imp__less__left,axiom,
% 0.33/0.55      ! [C: rat,A: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.33/0.55       => ( ord_less_rat @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_imp_less_left
% 0.33/0.55  thf(fact_1407_add__less__imp__less__left,axiom,
% 0.33/0.55      ! [C: nat,A: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.33/0.55       => ( ord_less_nat @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_imp_less_left
% 0.33/0.55  thf(fact_1408_add__less__imp__less__left,axiom,
% 0.33/0.55      ! [C: int,A: int,B: int] :
% 0.33/0.55        ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.33/0.55       => ( ord_less_int @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_imp_less_left
% 0.33/0.55  thf(fact_1409_add__less__imp__less__right,axiom,
% 0.33/0.55      ! [A: real,C: real,B: real] :
% 0.33/0.55        ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.33/0.55       => ( ord_less_real @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_imp_less_right
% 0.33/0.55  thf(fact_1410_add__less__imp__less__right,axiom,
% 0.33/0.55      ! [A: rat,C: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.33/0.55       => ( ord_less_rat @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_imp_less_right
% 0.33/0.55  thf(fact_1411_add__less__imp__less__right,axiom,
% 0.33/0.55      ! [A: nat,C: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.33/0.55       => ( ord_less_nat @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_imp_less_right
% 0.33/0.55  thf(fact_1412_add__less__imp__less__right,axiom,
% 0.33/0.55      ! [A: int,C: int,B: int] :
% 0.33/0.55        ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.33/0.55       => ( ord_less_int @ A @ B ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_imp_less_right
% 0.33/0.55  thf(fact_1413_plus__nat_Oadd__0,axiom,
% 0.33/0.55      ! [N: nat] :
% 0.33/0.55        ( ( plus_plus_nat @ zero_zero_nat @ N )
% 0.33/0.55        = N ) ).
% 0.33/0.55  
% 0.33/0.55  % plus_nat.add_0
% 0.33/0.55  thf(fact_1414_add__eq__self__zero,axiom,
% 0.33/0.55      ! [M: nat,N: nat] :
% 0.33/0.55        ( ( ( plus_plus_nat @ M @ N )
% 0.33/0.55          = M )
% 0.33/0.55       => ( N = zero_zero_nat ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_eq_self_zero
% 0.33/0.55  thf(fact_1415_add__lessD1,axiom,
% 0.33/0.55      ! [I: nat,J: nat,K: nat] :
% 0.33/0.55        ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.33/0.55       => ( ord_less_nat @ I @ K ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_lessD1
% 0.33/0.55  thf(fact_1416_add__less__mono,axiom,
% 0.33/0.55      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.33/0.55        ( ( ord_less_nat @ I @ J )
% 0.33/0.55       => ( ( ord_less_nat @ K @ L )
% 0.33/0.55         => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_mono
% 0.33/0.55  thf(fact_1417_not__add__less1,axiom,
% 0.33/0.55      ! [I: nat,J: nat] :
% 0.33/0.55        ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 0.33/0.55  
% 0.33/0.55  % not_add_less1
% 0.33/0.55  thf(fact_1418_not__add__less2,axiom,
% 0.33/0.55      ! [J: nat,I: nat] :
% 0.33/0.55        ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 0.33/0.55  
% 0.33/0.55  % not_add_less2
% 0.33/0.55  thf(fact_1419_add__less__mono1,axiom,
% 0.33/0.55      ! [I: nat,J: nat,K: nat] :
% 0.33/0.55        ( ( ord_less_nat @ I @ J )
% 0.33/0.55       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_mono1
% 0.33/0.55  thf(fact_1420_trans__less__add1,axiom,
% 0.33/0.55      ! [I: nat,J: nat,M: nat] :
% 0.33/0.55        ( ( ord_less_nat @ I @ J )
% 0.33/0.55       => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % trans_less_add1
% 0.33/0.55  thf(fact_1421_trans__less__add2,axiom,
% 0.33/0.55      ! [I: nat,J: nat,M: nat] :
% 0.33/0.55        ( ( ord_less_nat @ I @ J )
% 0.33/0.55       => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % trans_less_add2
% 0.33/0.55  thf(fact_1422_less__add__eq__less,axiom,
% 0.33/0.55      ! [K: nat,L: nat,M: nat,N: nat] :
% 0.33/0.55        ( ( ord_less_nat @ K @ L )
% 0.33/0.55       => ( ( ( plus_plus_nat @ M @ L )
% 0.33/0.55            = ( plus_plus_nat @ K @ N ) )
% 0.33/0.55         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % less_add_eq_less
% 0.33/0.55  thf(fact_1423_add__leE,axiom,
% 0.33/0.55      ! [M: nat,K: nat,N: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.33/0.55       => ~ ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.55           => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_leE
% 0.33/0.55  thf(fact_1424_le__add1,axiom,
% 0.33/0.55      ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 0.33/0.55  
% 0.33/0.55  % le_add1
% 0.33/0.55  thf(fact_1425_le__add2,axiom,
% 0.33/0.55      ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 0.33/0.55  
% 0.33/0.55  % le_add2
% 0.33/0.55  thf(fact_1426_add__leD1,axiom,
% 0.33/0.55      ! [M: nat,K: nat,N: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.33/0.55       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_leD1
% 0.33/0.55  thf(fact_1427_add__leD2,axiom,
% 0.33/0.55      ! [M: nat,K: nat,N: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.33/0.55       => ( ord_less_eq_nat @ K @ N ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_leD2
% 0.33/0.55  thf(fact_1428_le__Suc__ex,axiom,
% 0.33/0.55      ! [K: nat,L: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ K @ L )
% 0.33/0.55       => ? [N3: nat] :
% 0.33/0.55            ( L
% 0.33/0.55            = ( plus_plus_nat @ K @ N3 ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % le_Suc_ex
% 0.33/0.55  thf(fact_1429_add__le__mono,axiom,
% 0.33/0.55      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ I @ J )
% 0.33/0.55       => ( ( ord_less_eq_nat @ K @ L )
% 0.33/0.55         => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_mono
% 0.33/0.55  thf(fact_1430_add__le__mono1,axiom,
% 0.33/0.55      ! [I: nat,J: nat,K: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ I @ J )
% 0.33/0.55       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_mono1
% 0.33/0.55  thf(fact_1431_trans__le__add1,axiom,
% 0.33/0.55      ! [I: nat,J: nat,M: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ I @ J )
% 0.33/0.55       => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % trans_le_add1
% 0.33/0.55  thf(fact_1432_trans__le__add2,axiom,
% 0.33/0.55      ! [I: nat,J: nat,M: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ I @ J )
% 0.33/0.55       => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % trans_le_add2
% 0.33/0.55  thf(fact_1433_nat__le__iff__add,axiom,
% 0.33/0.55      ( ord_less_eq_nat
% 0.33/0.55      = ( ^ [M2: nat,N2: nat] :
% 0.33/0.55          ? [K3: nat] :
% 0.33/0.55            ( N2
% 0.33/0.55            = ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % nat_le_iff_add
% 0.33/0.55  thf(fact_1434_add__nonpos__eq__0__iff,axiom,
% 0.33/0.55      ! [X: real,Y: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.33/0.55       => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.33/0.55         => ( ( ( plus_plus_real @ X @ Y )
% 0.33/0.55              = zero_zero_real )
% 0.33/0.55            = ( ( X = zero_zero_real )
% 0.33/0.55              & ( Y = zero_zero_real ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonpos_eq_0_iff
% 0.33/0.55  thf(fact_1435_add__nonpos__eq__0__iff,axiom,
% 0.33/0.55      ! [X: rat,Y: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.33/0.55       => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 0.33/0.55         => ( ( ( plus_plus_rat @ X @ Y )
% 0.33/0.55              = zero_zero_rat )
% 0.33/0.55            = ( ( X = zero_zero_rat )
% 0.33/0.55              & ( Y = zero_zero_rat ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonpos_eq_0_iff
% 0.33/0.55  thf(fact_1436_add__nonpos__eq__0__iff,axiom,
% 0.33/0.55      ! [X: nat,Y: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ X @ zero_zero_nat )
% 0.33/0.55       => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
% 0.33/0.55         => ( ( ( plus_plus_nat @ X @ Y )
% 0.33/0.55              = zero_zero_nat )
% 0.33/0.55            = ( ( X = zero_zero_nat )
% 0.33/0.55              & ( Y = zero_zero_nat ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonpos_eq_0_iff
% 0.33/0.55  thf(fact_1437_add__nonpos__eq__0__iff,axiom,
% 0.33/0.55      ! [X: int,Y: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ X @ zero_zero_int )
% 0.33/0.55       => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 0.33/0.55         => ( ( ( plus_plus_int @ X @ Y )
% 0.33/0.55              = zero_zero_int )
% 0.33/0.55            = ( ( X = zero_zero_int )
% 0.33/0.55              & ( Y = zero_zero_int ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonpos_eq_0_iff
% 0.33/0.55  thf(fact_1438_add__nonneg__eq__0__iff,axiom,
% 0.33/0.55      ! [X: real,Y: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.55         => ( ( ( plus_plus_real @ X @ Y )
% 0.33/0.55              = zero_zero_real )
% 0.33/0.55            = ( ( X = zero_zero_real )
% 0.33/0.55              & ( Y = zero_zero_real ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonneg_eq_0_iff
% 0.33/0.55  thf(fact_1439_add__nonneg__eq__0__iff,axiom,
% 0.33/0.55      ! [X: rat,Y: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.33/0.55       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.33/0.55         => ( ( ( plus_plus_rat @ X @ Y )
% 0.33/0.55              = zero_zero_rat )
% 0.33/0.55            = ( ( X = zero_zero_rat )
% 0.33/0.55              & ( Y = zero_zero_rat ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonneg_eq_0_iff
% 0.33/0.55  thf(fact_1440_add__nonneg__eq__0__iff,axiom,
% 0.33/0.55      ! [X: nat,Y: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.33/0.55       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.33/0.55         => ( ( ( plus_plus_nat @ X @ Y )
% 0.33/0.55              = zero_zero_nat )
% 0.33/0.55            = ( ( X = zero_zero_nat )
% 0.33/0.55              & ( Y = zero_zero_nat ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonneg_eq_0_iff
% 0.33/0.55  thf(fact_1441_add__nonneg__eq__0__iff,axiom,
% 0.33/0.55      ! [X: int,Y: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.55       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.55         => ( ( ( plus_plus_int @ X @ Y )
% 0.33/0.55              = zero_zero_int )
% 0.33/0.55            = ( ( X = zero_zero_int )
% 0.33/0.55              & ( Y = zero_zero_int ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonneg_eq_0_iff
% 0.33/0.55  thf(fact_1442_add__nonpos__nonpos,axiom,
% 0.33/0.55      ! [A: real,B: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.33/0.55       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.33/0.55         => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonpos_nonpos
% 0.33/0.55  thf(fact_1443_add__nonpos__nonpos,axiom,
% 0.33/0.55      ! [A: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.33/0.55       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.33/0.55         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonpos_nonpos
% 0.33/0.55  thf(fact_1444_add__nonpos__nonpos,axiom,
% 0.33/0.55      ! [A: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.33/0.55       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.33/0.55         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonpos_nonpos
% 0.33/0.55  thf(fact_1445_add__nonpos__nonpos,axiom,
% 0.33/0.55      ! [A: int,B: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.33/0.55       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.33/0.55         => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonpos_nonpos
% 0.33/0.55  thf(fact_1446_add__nonneg__nonneg,axiom,
% 0.33/0.55      ! [A: real,B: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.33/0.55         => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonneg_nonneg
% 0.33/0.55  thf(fact_1447_add__nonneg__nonneg,axiom,
% 0.33/0.55      ! [A: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.55       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.33/0.55         => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonneg_nonneg
% 0.33/0.55  thf(fact_1448_add__nonneg__nonneg,axiom,
% 0.33/0.55      ! [A: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.33/0.55       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.33/0.55         => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonneg_nonneg
% 0.33/0.55  thf(fact_1449_add__nonneg__nonneg,axiom,
% 0.33/0.55      ! [A: int,B: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.55       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.33/0.55         => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonneg_nonneg
% 0.33/0.55  thf(fact_1450_add__increasing2,axiom,
% 0.33/0.55      ! [C: real,B: real,A: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.33/0.55       => ( ( ord_less_eq_real @ B @ A )
% 0.33/0.55         => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_increasing2
% 0.33/0.55  thf(fact_1451_add__increasing2,axiom,
% 0.33/0.55      ! [C: rat,B: rat,A: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.33/0.55       => ( ( ord_less_eq_rat @ B @ A )
% 0.33/0.55         => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_increasing2
% 0.33/0.55  thf(fact_1452_add__increasing2,axiom,
% 0.33/0.55      ! [C: nat,B: nat,A: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.33/0.55       => ( ( ord_less_eq_nat @ B @ A )
% 0.33/0.55         => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_increasing2
% 0.33/0.55  thf(fact_1453_add__increasing2,axiom,
% 0.33/0.55      ! [C: int,B: int,A: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.33/0.55       => ( ( ord_less_eq_int @ B @ A )
% 0.33/0.55         => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_increasing2
% 0.33/0.55  thf(fact_1454_add__decreasing2,axiom,
% 0.33/0.55      ! [C: real,A: real,B: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.33/0.55       => ( ( ord_less_eq_real @ A @ B )
% 0.33/0.55         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_decreasing2
% 0.33/0.55  thf(fact_1455_add__decreasing2,axiom,
% 0.33/0.55      ! [C: rat,A: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.33/0.55       => ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.55         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_decreasing2
% 0.33/0.55  thf(fact_1456_add__decreasing2,axiom,
% 0.33/0.55      ! [C: nat,A: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 0.33/0.55       => ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.55         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_decreasing2
% 0.33/0.55  thf(fact_1457_add__decreasing2,axiom,
% 0.33/0.55      ! [C: int,A: int,B: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.33/0.55       => ( ( ord_less_eq_int @ A @ B )
% 0.33/0.55         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_decreasing2
% 0.33/0.55  thf(fact_1458_add__increasing,axiom,
% 0.33/0.55      ! [A: real,B: real,C: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.55       => ( ( ord_less_eq_real @ B @ C )
% 0.33/0.55         => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_increasing
% 0.33/0.55  thf(fact_1459_add__increasing,axiom,
% 0.33/0.55      ! [A: rat,B: rat,C: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.55       => ( ( ord_less_eq_rat @ B @ C )
% 0.33/0.55         => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_increasing
% 0.33/0.55  thf(fact_1460_add__increasing,axiom,
% 0.33/0.55      ! [A: nat,B: nat,C: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.33/0.55       => ( ( ord_less_eq_nat @ B @ C )
% 0.33/0.55         => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_increasing
% 0.33/0.55  thf(fact_1461_add__increasing,axiom,
% 0.33/0.55      ! [A: int,B: int,C: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.55       => ( ( ord_less_eq_int @ B @ C )
% 0.33/0.55         => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_increasing
% 0.33/0.55  thf(fact_1462_add__decreasing,axiom,
% 0.33/0.55      ! [A: real,C: real,B: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.33/0.55       => ( ( ord_less_eq_real @ C @ B )
% 0.33/0.55         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_decreasing
% 0.33/0.55  thf(fact_1463_add__decreasing,axiom,
% 0.33/0.55      ! [A: rat,C: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.33/0.55       => ( ( ord_less_eq_rat @ C @ B )
% 0.33/0.55         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_decreasing
% 0.33/0.55  thf(fact_1464_add__decreasing,axiom,
% 0.33/0.55      ! [A: nat,C: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.33/0.55       => ( ( ord_less_eq_nat @ C @ B )
% 0.33/0.55         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_decreasing
% 0.33/0.55  thf(fact_1465_add__decreasing,axiom,
% 0.33/0.55      ! [A: int,C: int,B: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.33/0.55       => ( ( ord_less_eq_int @ C @ B )
% 0.33/0.55         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_decreasing
% 0.33/0.55  thf(fact_1466_add__less__le__mono,axiom,
% 0.33/0.55      ! [A: real,B: real,C: real,D: real] :
% 0.33/0.55        ( ( ord_less_real @ A @ B )
% 0.33/0.55       => ( ( ord_less_eq_real @ C @ D )
% 0.33/0.55         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_le_mono
% 0.33/0.55  thf(fact_1467_add__less__le__mono,axiom,
% 0.33/0.55      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.33/0.55        ( ( ord_less_rat @ A @ B )
% 0.33/0.55       => ( ( ord_less_eq_rat @ C @ D )
% 0.33/0.55         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_le_mono
% 0.33/0.55  thf(fact_1468_add__less__le__mono,axiom,
% 0.33/0.55      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.33/0.55        ( ( ord_less_nat @ A @ B )
% 0.33/0.55       => ( ( ord_less_eq_nat @ C @ D )
% 0.33/0.55         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_le_mono
% 0.33/0.55  thf(fact_1469_add__less__le__mono,axiom,
% 0.33/0.55      ! [A: int,B: int,C: int,D: int] :
% 0.33/0.55        ( ( ord_less_int @ A @ B )
% 0.33/0.55       => ( ( ord_less_eq_int @ C @ D )
% 0.33/0.55         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_le_mono
% 0.33/0.55  thf(fact_1470_add__le__less__mono,axiom,
% 0.33/0.55      ! [A: real,B: real,C: real,D: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ A @ B )
% 0.33/0.55       => ( ( ord_less_real @ C @ D )
% 0.33/0.55         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_less_mono
% 0.33/0.55  thf(fact_1471_add__le__less__mono,axiom,
% 0.33/0.55      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.55       => ( ( ord_less_rat @ C @ D )
% 0.33/0.55         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_less_mono
% 0.33/0.55  thf(fact_1472_add__le__less__mono,axiom,
% 0.33/0.55      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.55       => ( ( ord_less_nat @ C @ D )
% 0.33/0.55         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_less_mono
% 0.33/0.55  thf(fact_1473_add__le__less__mono,axiom,
% 0.33/0.55      ! [A: int,B: int,C: int,D: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ A @ B )
% 0.33/0.55       => ( ( ord_less_int @ C @ D )
% 0.33/0.55         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_le_less_mono
% 0.33/0.55  thf(fact_1474_add__mono__thms__linordered__field_I3_J,axiom,
% 0.33/0.55      ! [I: real,J: real,K: real,L: real] :
% 0.33/0.55        ( ( ( ord_less_real @ I @ J )
% 0.33/0.55          & ( ord_less_eq_real @ K @ L ) )
% 0.33/0.55       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_field(3)
% 0.33/0.55  thf(fact_1475_add__mono__thms__linordered__field_I3_J,axiom,
% 0.33/0.55      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.33/0.55        ( ( ( ord_less_rat @ I @ J )
% 0.33/0.55          & ( ord_less_eq_rat @ K @ L ) )
% 0.33/0.55       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_field(3)
% 0.33/0.55  thf(fact_1476_add__mono__thms__linordered__field_I3_J,axiom,
% 0.33/0.55      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.33/0.55        ( ( ( ord_less_nat @ I @ J )
% 0.33/0.55          & ( ord_less_eq_nat @ K @ L ) )
% 0.33/0.55       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_field(3)
% 0.33/0.55  thf(fact_1477_add__mono__thms__linordered__field_I3_J,axiom,
% 0.33/0.55      ! [I: int,J: int,K: int,L: int] :
% 0.33/0.55        ( ( ( ord_less_int @ I @ J )
% 0.33/0.55          & ( ord_less_eq_int @ K @ L ) )
% 0.33/0.55       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_field(3)
% 0.33/0.55  thf(fact_1478_add__mono__thms__linordered__field_I4_J,axiom,
% 0.33/0.55      ! [I: real,J: real,K: real,L: real] :
% 0.33/0.55        ( ( ( ord_less_eq_real @ I @ J )
% 0.33/0.55          & ( ord_less_real @ K @ L ) )
% 0.33/0.55       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_field(4)
% 0.33/0.55  thf(fact_1479_add__mono__thms__linordered__field_I4_J,axiom,
% 0.33/0.55      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.33/0.55        ( ( ( ord_less_eq_rat @ I @ J )
% 0.33/0.55          & ( ord_less_rat @ K @ L ) )
% 0.33/0.55       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_field(4)
% 0.33/0.55  thf(fact_1480_add__mono__thms__linordered__field_I4_J,axiom,
% 0.33/0.55      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.33/0.55        ( ( ( ord_less_eq_nat @ I @ J )
% 0.33/0.55          & ( ord_less_nat @ K @ L ) )
% 0.33/0.55       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_field(4)
% 0.33/0.55  thf(fact_1481_add__mono__thms__linordered__field_I4_J,axiom,
% 0.33/0.55      ! [I: int,J: int,K: int,L: int] :
% 0.33/0.55        ( ( ( ord_less_eq_int @ I @ J )
% 0.33/0.55          & ( ord_less_int @ K @ L ) )
% 0.33/0.55       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_mono_thms_linordered_field(4)
% 0.33/0.55  thf(fact_1482_add__neg__neg,axiom,
% 0.33/0.55      ! [A: real,B: real] :
% 0.33/0.55        ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.55       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.33/0.55         => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_neg_neg
% 0.33/0.55  thf(fact_1483_add__neg__neg,axiom,
% 0.33/0.55      ! [A: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.55       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.33/0.55         => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_neg_neg
% 0.33/0.55  thf(fact_1484_add__neg__neg,axiom,
% 0.33/0.55      ! [A: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.33/0.55       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.33/0.55         => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_neg_neg
% 0.33/0.55  thf(fact_1485_add__neg__neg,axiom,
% 0.33/0.55      ! [A: int,B: int] :
% 0.33/0.55        ( ( ord_less_int @ A @ zero_zero_int )
% 0.33/0.55       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.33/0.55         => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_neg_neg
% 0.33/0.55  thf(fact_1486_add__pos__pos,axiom,
% 0.33/0.55      ! [A: real,B: real] :
% 0.33/0.55        ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.55       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.55         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_pos_pos
% 0.33/0.55  thf(fact_1487_add__pos__pos,axiom,
% 0.33/0.55      ! [A: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.55       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.33/0.55         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_pos_pos
% 0.33/0.55  thf(fact_1488_add__pos__pos,axiom,
% 0.33/0.55      ! [A: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.33/0.55       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.33/0.55         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_pos_pos
% 0.33/0.55  thf(fact_1489_add__pos__pos,axiom,
% 0.33/0.55      ! [A: int,B: int] :
% 0.33/0.55        ( ( ord_less_int @ zero_zero_int @ A )
% 0.33/0.55       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.55         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_pos_pos
% 0.33/0.55  thf(fact_1490_canonically__ordered__monoid__add__class_OlessE,axiom,
% 0.33/0.55      ! [A: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_nat @ A @ B )
% 0.33/0.55       => ~ ! [C3: nat] :
% 0.33/0.55              ( ( B
% 0.33/0.55                = ( plus_plus_nat @ A @ C3 ) )
% 0.33/0.55             => ( C3 = zero_zero_nat ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % canonically_ordered_monoid_add_class.lessE
% 0.33/0.55  thf(fact_1491_pos__add__strict,axiom,
% 0.33/0.55      ! [A: real,B: real,C: real] :
% 0.33/0.55        ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.55       => ( ( ord_less_real @ B @ C )
% 0.33/0.55         => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pos_add_strict
% 0.33/0.55  thf(fact_1492_pos__add__strict,axiom,
% 0.33/0.55      ! [A: rat,B: rat,C: rat] :
% 0.33/0.55        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.55       => ( ( ord_less_rat @ B @ C )
% 0.33/0.55         => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pos_add_strict
% 0.33/0.55  thf(fact_1493_pos__add__strict,axiom,
% 0.33/0.55      ! [A: nat,B: nat,C: nat] :
% 0.33/0.55        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.33/0.55       => ( ( ord_less_nat @ B @ C )
% 0.33/0.55         => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pos_add_strict
% 0.33/0.55  thf(fact_1494_pos__add__strict,axiom,
% 0.33/0.55      ! [A: int,B: int,C: int] :
% 0.33/0.55        ( ( ord_less_int @ zero_zero_int @ A )
% 0.33/0.55       => ( ( ord_less_int @ B @ C )
% 0.33/0.55         => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pos_add_strict
% 0.33/0.55  thf(fact_1495_less__imp__add__positive,axiom,
% 0.33/0.55      ! [I: nat,J: nat] :
% 0.33/0.55        ( ( ord_less_nat @ I @ J )
% 0.33/0.55       => ? [K2: nat] :
% 0.33/0.55            ( ( ord_less_nat @ zero_zero_nat @ K2 )
% 0.33/0.55            & ( ( plus_plus_nat @ I @ K2 )
% 0.33/0.55              = J ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % less_imp_add_positive
% 0.33/0.55  thf(fact_1496_mono__nat__linear__lb,axiom,
% 0.33/0.55      ! [F: nat > nat,M: nat,K: nat] :
% 0.33/0.55        ( ! [M5: nat,N3: nat] :
% 0.33/0.55            ( ( ord_less_nat @ M5 @ N3 )
% 0.33/0.55           => ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
% 0.33/0.55       => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % mono_nat_linear_lb
% 0.33/0.55  thf(fact_1497_add__neg__nonpos,axiom,
% 0.33/0.55      ! [A: real,B: real] :
% 0.33/0.55        ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.55       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.33/0.55         => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_neg_nonpos
% 0.33/0.55  thf(fact_1498_add__neg__nonpos,axiom,
% 0.33/0.55      ! [A: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.55       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.33/0.55         => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_neg_nonpos
% 0.33/0.55  thf(fact_1499_add__neg__nonpos,axiom,
% 0.33/0.55      ! [A: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.33/0.55       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.33/0.55         => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_neg_nonpos
% 0.33/0.55  thf(fact_1500_add__neg__nonpos,axiom,
% 0.33/0.55      ! [A: int,B: int] :
% 0.33/0.55        ( ( ord_less_int @ A @ zero_zero_int )
% 0.33/0.55       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.33/0.55         => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_neg_nonpos
% 0.33/0.55  thf(fact_1501_add__nonneg__pos,axiom,
% 0.33/0.55      ! [A: real,B: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.55       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.55         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonneg_pos
% 0.33/0.55  thf(fact_1502_add__nonneg__pos,axiom,
% 0.33/0.55      ! [A: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.55       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.33/0.55         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonneg_pos
% 0.33/0.55  thf(fact_1503_add__nonneg__pos,axiom,
% 0.33/0.55      ! [A: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.33/0.55       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.33/0.55         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonneg_pos
% 0.33/0.55  thf(fact_1504_add__nonneg__pos,axiom,
% 0.33/0.55      ! [A: int,B: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.55       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.55         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonneg_pos
% 0.33/0.55  thf(fact_1505_add__nonpos__neg,axiom,
% 0.33/0.55      ! [A: real,B: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.33/0.55       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.33/0.55         => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonpos_neg
% 0.33/0.55  thf(fact_1506_add__nonpos__neg,axiom,
% 0.33/0.55      ! [A: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.33/0.55       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.33/0.55         => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonpos_neg
% 0.33/0.55  thf(fact_1507_add__nonpos__neg,axiom,
% 0.33/0.55      ! [A: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.33/0.55       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.33/0.55         => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonpos_neg
% 0.33/0.55  thf(fact_1508_add__nonpos__neg,axiom,
% 0.33/0.55      ! [A: int,B: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.33/0.55       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.33/0.55         => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_nonpos_neg
% 0.33/0.55  thf(fact_1509_add__pos__nonneg,axiom,
% 0.33/0.55      ! [A: real,B: real] :
% 0.33/0.55        ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.33/0.55         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_pos_nonneg
% 0.33/0.55  thf(fact_1510_add__pos__nonneg,axiom,
% 0.33/0.55      ! [A: rat,B: rat] :
% 0.33/0.55        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.55       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.33/0.55         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_pos_nonneg
% 0.33/0.55  thf(fact_1511_add__pos__nonneg,axiom,
% 0.33/0.55      ! [A: nat,B: nat] :
% 0.33/0.55        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.33/0.55       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.33/0.55         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_pos_nonneg
% 0.33/0.55  thf(fact_1512_add__pos__nonneg,axiom,
% 0.33/0.55      ! [A: int,B: int] :
% 0.33/0.55        ( ( ord_less_int @ zero_zero_int @ A )
% 0.33/0.55       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.33/0.55         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_pos_nonneg
% 0.33/0.55  thf(fact_1513_add__strict__increasing,axiom,
% 0.33/0.55      ! [A: real,B: real,C: real] :
% 0.33/0.55        ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.55       => ( ( ord_less_eq_real @ B @ C )
% 0.33/0.55         => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_strict_increasing
% 0.33/0.55  thf(fact_1514_add__strict__increasing,axiom,
% 0.33/0.55      ! [A: rat,B: rat,C: rat] :
% 0.33/0.55        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.55       => ( ( ord_less_eq_rat @ B @ C )
% 0.33/0.55         => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_strict_increasing
% 0.33/0.55  thf(fact_1515_add__strict__increasing,axiom,
% 0.33/0.55      ! [A: nat,B: nat,C: nat] :
% 0.33/0.55        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.33/0.55       => ( ( ord_less_eq_nat @ B @ C )
% 0.33/0.55         => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_strict_increasing
% 0.33/0.55  thf(fact_1516_add__strict__increasing,axiom,
% 0.33/0.55      ! [A: int,B: int,C: int] :
% 0.33/0.55        ( ( ord_less_int @ zero_zero_int @ A )
% 0.33/0.55       => ( ( ord_less_eq_int @ B @ C )
% 0.33/0.55         => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_strict_increasing
% 0.33/0.55  thf(fact_1517_add__strict__increasing2,axiom,
% 0.33/0.55      ! [A: real,B: real,C: real] :
% 0.33/0.55        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.55       => ( ( ord_less_real @ B @ C )
% 0.33/0.55         => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_strict_increasing2
% 0.33/0.55  thf(fact_1518_add__strict__increasing2,axiom,
% 0.33/0.55      ! [A: rat,B: rat,C: rat] :
% 0.33/0.55        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.55       => ( ( ord_less_rat @ B @ C )
% 0.33/0.55         => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_strict_increasing2
% 0.33/0.55  thf(fact_1519_add__strict__increasing2,axiom,
% 0.33/0.55      ! [A: nat,B: nat,C: nat] :
% 0.33/0.55        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.33/0.55       => ( ( ord_less_nat @ B @ C )
% 0.33/0.55         => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_strict_increasing2
% 0.33/0.55  thf(fact_1520_add__strict__increasing2,axiom,
% 0.33/0.55      ! [A: int,B: int,C: int] :
% 0.33/0.55        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.55       => ( ( ord_less_int @ B @ C )
% 0.33/0.55         => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_strict_increasing2
% 0.33/0.55  thf(fact_1521_verit__comp__simplify1_I2_J,axiom,
% 0.33/0.55      ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 0.33/0.55  
% 0.33/0.55  % verit_comp_simplify1(2)
% 0.33/0.55  thf(fact_1522_verit__comp__simplify1_I2_J,axiom,
% 0.33/0.55      ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 0.33/0.55  
% 0.33/0.55  % verit_comp_simplify1(2)
% 0.33/0.55  thf(fact_1523_verit__comp__simplify1_I2_J,axiom,
% 0.33/0.55      ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 0.33/0.55  
% 0.33/0.55  % verit_comp_simplify1(2)
% 0.33/0.55  thf(fact_1524_verit__comp__simplify1_I2_J,axiom,
% 0.33/0.55      ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.33/0.55  
% 0.33/0.55  % verit_comp_simplify1(2)
% 0.33/0.55  thf(fact_1525_verit__comp__simplify1_I2_J,axiom,
% 0.33/0.55      ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.33/0.55  
% 0.33/0.55  % verit_comp_simplify1(2)
% 0.33/0.55  thf(fact_1526_verit__la__disequality,axiom,
% 0.33/0.55      ! [A: rat,B: rat] :
% 0.33/0.55        ( ( A = B )
% 0.33/0.55        | ~ ( ord_less_eq_rat @ A @ B )
% 0.33/0.55        | ~ ( ord_less_eq_rat @ B @ A ) ) ).
% 0.33/0.55  
% 0.33/0.55  % verit_la_disequality
% 0.33/0.55  thf(fact_1527_verit__la__disequality,axiom,
% 0.33/0.55      ! [A: num,B: num] :
% 0.33/0.55        ( ( A = B )
% 0.33/0.55        | ~ ( ord_less_eq_num @ A @ B )
% 0.33/0.55        | ~ ( ord_less_eq_num @ B @ A ) ) ).
% 0.33/0.55  
% 0.33/0.55  % verit_la_disequality
% 0.33/0.55  thf(fact_1528_verit__la__disequality,axiom,
% 0.33/0.55      ! [A: nat,B: nat] :
% 0.33/0.55        ( ( A = B )
% 0.33/0.55        | ~ ( ord_less_eq_nat @ A @ B )
% 0.33/0.55        | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 0.33/0.55  
% 0.33/0.55  % verit_la_disequality
% 0.33/0.55  thf(fact_1529_verit__la__disequality,axiom,
% 0.33/0.55      ! [A: int,B: int] :
% 0.33/0.55        ( ( A = B )
% 0.33/0.55        | ~ ( ord_less_eq_int @ A @ B )
% 0.33/0.55        | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 0.33/0.55  
% 0.33/0.55  % verit_la_disequality
% 0.33/0.55  thf(fact_1530_verit__comp__simplify1_I1_J,axiom,
% 0.33/0.55      ! [A: real] :
% 0.33/0.55        ~ ( ord_less_real @ A @ A ) ).
% 0.33/0.55  
% 0.33/0.55  % verit_comp_simplify1(1)
% 0.33/0.55  thf(fact_1531_verit__comp__simplify1_I1_J,axiom,
% 0.33/0.55      ! [A: rat] :
% 0.33/0.55        ~ ( ord_less_rat @ A @ A ) ).
% 0.33/0.55  
% 0.33/0.55  % verit_comp_simplify1(1)
% 0.33/0.55  thf(fact_1532_verit__comp__simplify1_I1_J,axiom,
% 0.33/0.55      ! [A: num] :
% 0.33/0.55        ~ ( ord_less_num @ A @ A ) ).
% 0.33/0.55  
% 0.33/0.55  % verit_comp_simplify1(1)
% 0.33/0.55  thf(fact_1533_verit__comp__simplify1_I1_J,axiom,
% 0.33/0.55      ! [A: nat] :
% 0.33/0.55        ~ ( ord_less_nat @ A @ A ) ).
% 0.33/0.55  
% 0.33/0.55  % verit_comp_simplify1(1)
% 0.33/0.55  thf(fact_1534_verit__comp__simplify1_I1_J,axiom,
% 0.33/0.55      ! [A: int] :
% 0.33/0.55        ~ ( ord_less_int @ A @ A ) ).
% 0.33/0.55  
% 0.33/0.55  % verit_comp_simplify1(1)
% 0.33/0.55  thf(fact_1535_ex__gt__or__lt,axiom,
% 0.33/0.55      ! [A: real] :
% 0.33/0.55      ? [B3: real] :
% 0.33/0.55        ( ( ord_less_real @ A @ B3 )
% 0.33/0.55        | ( ord_less_real @ B3 @ A ) ) ).
% 0.33/0.55  
% 0.33/0.55  % ex_gt_or_lt
% 0.33/0.55  thf(fact_1536_subrelI,axiom,
% 0.33/0.55      ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
% 0.33/0.55        ( ! [X4: nat,Y3: nat] :
% 0.33/0.55            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R2 )
% 0.33/0.55           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ S ) )
% 0.33/0.55       => ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).
% 0.33/0.55  
% 0.33/0.55  % subrelI
% 0.33/0.55  thf(fact_1537_subrelI,axiom,
% 0.33/0.55      ! [R2: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
% 0.33/0.55        ( ! [X4: int,Y3: int] :
% 0.33/0.55            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ R2 )
% 0.33/0.55           => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ S ) )
% 0.33/0.55       => ( ord_le2843351958646193337nt_int @ R2 @ S ) ) ).
% 0.33/0.55  
% 0.33/0.55  % subrelI
% 0.33/0.55  thf(fact_1538_subrelI,axiom,
% 0.33/0.55      ! [R2: set_Pr8056137968301705908nteger,S: set_Pr8056137968301705908nteger] :
% 0.33/0.55        ( ! [X4: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
% 0.33/0.55            ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y3 ) @ R2 )
% 0.33/0.55           => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y3 ) @ S ) )
% 0.33/0.55       => ( ord_le3216752416896350996nteger @ R2 @ S ) ) ).
% 0.33/0.55  
% 0.33/0.55  % subrelI
% 0.33/0.55  thf(fact_1539_subrelI,axiom,
% 0.33/0.55      ! [R2: set_Pr1281608226676607948nteger,S: set_Pr1281608226676607948nteger] :
% 0.33/0.55        ( ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
% 0.33/0.55            ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y3 ) @ R2 )
% 0.33/0.55           => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y3 ) @ S ) )
% 0.33/0.55       => ( ord_le653643898420964396nteger @ R2 @ S ) ) ).
% 0.33/0.55  
% 0.33/0.55  % subrelI
% 0.33/0.55  thf(fact_1540_subrelI,axiom,
% 0.33/0.55      ! [R2: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
% 0.33/0.55        ( ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
% 0.33/0.55            ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y3 ) @ R2 )
% 0.33/0.55           => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y3 ) @ S ) )
% 0.33/0.55       => ( ord_le8725513860283290265nt_int @ R2 @ S ) ) ).
% 0.33/0.55  
% 0.33/0.55  % subrelI
% 0.33/0.55  thf(fact_1541_subrelI,axiom,
% 0.33/0.55      ! [R2: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
% 0.33/0.55        ( ! [X4: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
% 0.33/0.55            ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y3 ) @ R2 )
% 0.33/0.55           => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y3 ) @ S ) )
% 0.33/0.55       => ( ord_le135402666524580259nt_int @ R2 @ S ) ) ).
% 0.33/0.55  
% 0.33/0.55  % subrelI
% 0.33/0.55  thf(fact_1542_ssubst__Pair__rhs,axiom,
% 0.33/0.55      ! [R2: nat,S: nat,R: set_Pr1261947904930325089at_nat,S3: nat] :
% 0.33/0.55        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S ) @ R )
% 0.33/0.55       => ( ( S3 = S )
% 0.33/0.55         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S3 ) @ R ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % ssubst_Pair_rhs
% 0.33/0.55  thf(fact_1543_ssubst__Pair__rhs,axiom,
% 0.33/0.55      ! [R2: int,S: int,R: set_Pr958786334691620121nt_int,S3: int] :
% 0.33/0.55        ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S ) @ R )
% 0.33/0.55       => ( ( S3 = S )
% 0.33/0.55         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S3 ) @ R ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % ssubst_Pair_rhs
% 0.33/0.55  thf(fact_1544_ssubst__Pair__rhs,axiom,
% 0.33/0.55      ! [R2: code_integer > option6357759511663192854e_term,S: produc8923325533196201883nteger,R: set_Pr8056137968301705908nteger,S3: produc8923325533196201883nteger] :
% 0.33/0.55        ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ R2 @ S ) @ R )
% 0.33/0.55       => ( ( S3 = S )
% 0.33/0.55         => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ R2 @ S3 ) @ R ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % ssubst_Pair_rhs
% 0.33/0.55  thf(fact_1545_ssubst__Pair__rhs,axiom,
% 0.33/0.55      ! [R2: produc6241069584506657477e_term > option6357759511663192854e_term,S: produc8923325533196201883nteger,R: set_Pr1281608226676607948nteger,S3: produc8923325533196201883nteger] :
% 0.33/0.55        ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R2 @ S ) @ R )
% 0.33/0.55       => ( ( S3 = S )
% 0.33/0.55         => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R2 @ S3 ) @ R ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % ssubst_Pair_rhs
% 0.33/0.55  thf(fact_1546_ssubst__Pair__rhs,axiom,
% 0.33/0.55      ! [R2: produc8551481072490612790e_term > option6357759511663192854e_term,S: product_prod_int_int,R: set_Pr9222295170931077689nt_int,S3: product_prod_int_int] :
% 0.33/0.55        ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R2 @ S ) @ R )
% 0.33/0.55       => ( ( S3 = S )
% 0.33/0.55         => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R2 @ S3 ) @ R ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % ssubst_Pair_rhs
% 0.33/0.55  thf(fact_1547_ssubst__Pair__rhs,axiom,
% 0.33/0.55      ! [R2: int > option6357759511663192854e_term,S: product_prod_int_int,R: set_Pr1872883991513573699nt_int,S3: product_prod_int_int] :
% 0.33/0.55        ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R2 @ S ) @ R )
% 0.33/0.55       => ( ( S3 = S )
% 0.33/0.55         => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R2 @ S3 ) @ R ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % ssubst_Pair_rhs
% 0.33/0.55  thf(fact_1548_le__numeral__extra_I3_J,axiom,
% 0.33/0.55      ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.33/0.55  
% 0.33/0.55  % le_numeral_extra(3)
% 0.33/0.55  thf(fact_1549_le__numeral__extra_I3_J,axiom,
% 0.33/0.55      ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 0.33/0.55  
% 0.33/0.55  % le_numeral_extra(3)
% 0.33/0.55  thf(fact_1550_le__numeral__extra_I3_J,axiom,
% 0.33/0.55      ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.33/0.55  
% 0.33/0.55  % le_numeral_extra(3)
% 0.33/0.55  thf(fact_1551_le__numeral__extra_I3_J,axiom,
% 0.33/0.55      ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.33/0.55  
% 0.33/0.55  % le_numeral_extra(3)
% 0.33/0.55  thf(fact_1552_double__eq__0__iff,axiom,
% 0.33/0.55      ! [A: real] :
% 0.33/0.55        ( ( ( plus_plus_real @ A @ A )
% 0.33/0.55          = zero_zero_real )
% 0.33/0.55        = ( A = zero_zero_real ) ) ).
% 0.33/0.55  
% 0.33/0.55  % double_eq_0_iff
% 0.33/0.55  thf(fact_1553_double__eq__0__iff,axiom,
% 0.33/0.55      ! [A: rat] :
% 0.33/0.55        ( ( ( plus_plus_rat @ A @ A )
% 0.33/0.55          = zero_zero_rat )
% 0.33/0.55        = ( A = zero_zero_rat ) ) ).
% 0.33/0.55  
% 0.33/0.55  % double_eq_0_iff
% 0.33/0.55  thf(fact_1554_double__eq__0__iff,axiom,
% 0.33/0.55      ! [A: int] :
% 0.33/0.55        ( ( ( plus_plus_int @ A @ A )
% 0.33/0.55          = zero_zero_int )
% 0.33/0.55        = ( A = zero_zero_int ) ) ).
% 0.33/0.55  
% 0.33/0.55  % double_eq_0_iff
% 0.33/0.55  thf(fact_1555_add__shift,axiom,
% 0.33/0.55      ! [X: nat,Y: nat,Z2: nat] :
% 0.33/0.55        ( ( ( plus_plus_nat @ X @ Y )
% 0.33/0.55          = Z2 )
% 0.33/0.55        = ( ( vEBT_VEBT_add @ ( some_nat @ X ) @ ( some_nat @ Y ) )
% 0.33/0.55          = ( some_nat @ Z2 ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_shift
% 0.33/0.55  thf(fact_1556_field__le__epsilon,axiom,
% 0.33/0.55      ! [X: real,Y: real] :
% 0.33/0.55        ( ! [E: real] :
% 0.33/0.55            ( ( ord_less_real @ zero_zero_real @ E )
% 0.33/0.55           => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E ) ) )
% 0.33/0.55       => ( ord_less_eq_real @ X @ Y ) ) ).
% 0.33/0.55  
% 0.33/0.55  % field_le_epsilon
% 0.33/0.55  thf(fact_1557_field__le__epsilon,axiom,
% 0.33/0.55      ! [X: rat,Y: rat] :
% 0.33/0.55        ( ! [E: rat] :
% 0.33/0.55            ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.33/0.55           => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E ) ) )
% 0.33/0.55       => ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.33/0.55  
% 0.33/0.55  % field_le_epsilon
% 0.33/0.55  thf(fact_1558_add__less__zeroD,axiom,
% 0.33/0.55      ! [X: real,Y: real] :
% 0.33/0.55        ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 0.33/0.55       => ( ( ord_less_real @ X @ zero_zero_real )
% 0.33/0.55          | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_zeroD
% 0.33/0.55  thf(fact_1559_add__less__zeroD,axiom,
% 0.33/0.55      ! [X: rat,Y: rat] :
% 0.33/0.55        ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
% 0.33/0.55       => ( ( ord_less_rat @ X @ zero_zero_rat )
% 0.33/0.55          | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_zeroD
% 0.33/0.55  thf(fact_1560_add__less__zeroD,axiom,
% 0.33/0.55      ! [X: int,Y: int] :
% 0.33/0.55        ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
% 0.33/0.55       => ( ( ord_less_int @ X @ zero_zero_int )
% 0.33/0.55          | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_less_zeroD
% 0.33/0.55  thf(fact_1561_nth__enumerate__eq,axiom,
% 0.33/0.55      ! [M: nat,Xs: list_VEBT_VEBT,N: nat] :
% 0.33/0.55        ( ( ord_less_nat @ M @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.55       => ( ( nth_Pr744662078594809490T_VEBT @ ( enumerate_VEBT_VEBT @ N @ Xs ) @ M )
% 0.33/0.55          = ( produc599794634098209291T_VEBT @ ( plus_plus_nat @ N @ M ) @ ( nth_VEBT_VEBT @ Xs @ M ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % nth_enumerate_eq
% 0.33/0.55  thf(fact_1562_nth__enumerate__eq,axiom,
% 0.33/0.55      ! [M: nat,Xs: list_o,N: nat] :
% 0.33/0.55        ( ( ord_less_nat @ M @ ( size_size_list_o @ Xs ) )
% 0.33/0.55       => ( ( nth_Pr112076138515278198_nat_o @ ( enumerate_o @ N @ Xs ) @ M )
% 0.33/0.55          = ( product_Pair_nat_o @ ( plus_plus_nat @ N @ M ) @ ( nth_o @ Xs @ M ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % nth_enumerate_eq
% 0.33/0.55  thf(fact_1563_nth__enumerate__eq,axiom,
% 0.33/0.55      ! [M: nat,Xs: list_nat,N: nat] :
% 0.33/0.55        ( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
% 0.33/0.55       => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M )
% 0.33/0.55          = ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % nth_enumerate_eq
% 0.33/0.55  thf(fact_1564_nth__enumerate__eq,axiom,
% 0.33/0.55      ! [M: nat,Xs: list_int,N: nat] :
% 0.33/0.55        ( ( ord_less_nat @ M @ ( size_size_list_int @ Xs ) )
% 0.33/0.55       => ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N @ Xs ) @ M )
% 0.33/0.55          = ( product_Pair_nat_int @ ( plus_plus_nat @ N @ M ) @ ( nth_int @ Xs @ M ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % nth_enumerate_eq
% 0.33/0.55  thf(fact_1565_Euclid__induct,axiom,
% 0.33/0.55      ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.33/0.55        ( ! [A3: nat,B3: nat] :
% 0.33/0.55            ( ( P @ A3 @ B3 )
% 0.33/0.55            = ( P @ B3 @ A3 ) )
% 0.33/0.55       => ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
% 0.33/0.55         => ( ! [A3: nat,B3: nat] :
% 0.33/0.55                ( ( P @ A3 @ B3 )
% 0.33/0.55               => ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
% 0.33/0.55           => ( P @ A @ B ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % Euclid_induct
% 0.33/0.55  thf(fact_1566_add__0__iff,axiom,
% 0.33/0.55      ! [B: complex,A: complex] :
% 0.33/0.55        ( ( B
% 0.33/0.55          = ( plus_plus_complex @ B @ A ) )
% 0.33/0.55        = ( A = zero_zero_complex ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_0_iff
% 0.33/0.55  thf(fact_1567_add__0__iff,axiom,
% 0.33/0.55      ! [B: real,A: real] :
% 0.33/0.55        ( ( B
% 0.33/0.55          = ( plus_plus_real @ B @ A ) )
% 0.33/0.55        = ( A = zero_zero_real ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_0_iff
% 0.33/0.55  thf(fact_1568_add__0__iff,axiom,
% 0.33/0.55      ! [B: rat,A: rat] :
% 0.33/0.55        ( ( B
% 0.33/0.55          = ( plus_plus_rat @ B @ A ) )
% 0.33/0.55        = ( A = zero_zero_rat ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_0_iff
% 0.33/0.55  thf(fact_1569_add__0__iff,axiom,
% 0.33/0.55      ! [B: nat,A: nat] :
% 0.33/0.55        ( ( B
% 0.33/0.55          = ( plus_plus_nat @ B @ A ) )
% 0.33/0.55        = ( A = zero_zero_nat ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_0_iff
% 0.33/0.55  thf(fact_1570_add__0__iff,axiom,
% 0.33/0.55      ! [B: int,A: int] :
% 0.33/0.55        ( ( B
% 0.33/0.55          = ( plus_plus_int @ B @ A ) )
% 0.33/0.55        = ( A = zero_zero_int ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_0_iff
% 0.33/0.55  thf(fact_1571_minf_I8_J,axiom,
% 0.33/0.55      ! [T: real] :
% 0.33/0.55      ? [Z4: real] :
% 0.33/0.55      ! [X2: real] :
% 0.33/0.55        ( ( ord_less_real @ X2 @ Z4 )
% 0.33/0.55       => ~ ( ord_less_eq_real @ T @ X2 ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(8)
% 0.33/0.55  thf(fact_1572_minf_I8_J,axiom,
% 0.33/0.55      ! [T: rat] :
% 0.33/0.55      ? [Z4: rat] :
% 0.33/0.55      ! [X2: rat] :
% 0.33/0.55        ( ( ord_less_rat @ X2 @ Z4 )
% 0.33/0.55       => ~ ( ord_less_eq_rat @ T @ X2 ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(8)
% 0.33/0.55  thf(fact_1573_minf_I8_J,axiom,
% 0.33/0.55      ! [T: num] :
% 0.33/0.55      ? [Z4: num] :
% 0.33/0.55      ! [X2: num] :
% 0.33/0.55        ( ( ord_less_num @ X2 @ Z4 )
% 0.33/0.55       => ~ ( ord_less_eq_num @ T @ X2 ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(8)
% 0.33/0.55  thf(fact_1574_minf_I8_J,axiom,
% 0.33/0.55      ! [T: nat] :
% 0.33/0.55      ? [Z4: nat] :
% 0.33/0.55      ! [X2: nat] :
% 0.33/0.55        ( ( ord_less_nat @ X2 @ Z4 )
% 0.33/0.55       => ~ ( ord_less_eq_nat @ T @ X2 ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(8)
% 0.33/0.55  thf(fact_1575_minf_I8_J,axiom,
% 0.33/0.55      ! [T: int] :
% 0.33/0.55      ? [Z4: int] :
% 0.33/0.55      ! [X2: int] :
% 0.33/0.55        ( ( ord_less_int @ X2 @ Z4 )
% 0.33/0.55       => ~ ( ord_less_eq_int @ T @ X2 ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(8)
% 0.33/0.55  thf(fact_1576_minf_I6_J,axiom,
% 0.33/0.55      ! [T: real] :
% 0.33/0.55      ? [Z4: real] :
% 0.33/0.55      ! [X2: real] :
% 0.33/0.55        ( ( ord_less_real @ X2 @ Z4 )
% 0.33/0.55       => ( ord_less_eq_real @ X2 @ T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(6)
% 0.33/0.55  thf(fact_1577_minf_I6_J,axiom,
% 0.33/0.55      ! [T: rat] :
% 0.33/0.55      ? [Z4: rat] :
% 0.33/0.55      ! [X2: rat] :
% 0.33/0.55        ( ( ord_less_rat @ X2 @ Z4 )
% 0.33/0.55       => ( ord_less_eq_rat @ X2 @ T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(6)
% 0.33/0.55  thf(fact_1578_minf_I6_J,axiom,
% 0.33/0.55      ! [T: num] :
% 0.33/0.55      ? [Z4: num] :
% 0.33/0.55      ! [X2: num] :
% 0.33/0.55        ( ( ord_less_num @ X2 @ Z4 )
% 0.33/0.55       => ( ord_less_eq_num @ X2 @ T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(6)
% 0.33/0.55  thf(fact_1579_minf_I6_J,axiom,
% 0.33/0.55      ! [T: nat] :
% 0.33/0.55      ? [Z4: nat] :
% 0.33/0.55      ! [X2: nat] :
% 0.33/0.55        ( ( ord_less_nat @ X2 @ Z4 )
% 0.33/0.55       => ( ord_less_eq_nat @ X2 @ T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(6)
% 0.33/0.55  thf(fact_1580_minf_I6_J,axiom,
% 0.33/0.55      ! [T: int] :
% 0.33/0.55      ? [Z4: int] :
% 0.33/0.55      ! [X2: int] :
% 0.33/0.55        ( ( ord_less_int @ X2 @ Z4 )
% 0.33/0.55       => ( ord_less_eq_int @ X2 @ T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(6)
% 0.33/0.55  thf(fact_1581_pinf_I8_J,axiom,
% 0.33/0.55      ! [T: real] :
% 0.33/0.55      ? [Z4: real] :
% 0.33/0.55      ! [X2: real] :
% 0.33/0.55        ( ( ord_less_real @ Z4 @ X2 )
% 0.33/0.55       => ( ord_less_eq_real @ T @ X2 ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(8)
% 0.33/0.55  thf(fact_1582_pinf_I8_J,axiom,
% 0.33/0.55      ! [T: rat] :
% 0.33/0.55      ? [Z4: rat] :
% 0.33/0.55      ! [X2: rat] :
% 0.33/0.55        ( ( ord_less_rat @ Z4 @ X2 )
% 0.33/0.55       => ( ord_less_eq_rat @ T @ X2 ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(8)
% 0.33/0.55  thf(fact_1583_pinf_I8_J,axiom,
% 0.33/0.55      ! [T: num] :
% 0.33/0.55      ? [Z4: num] :
% 0.33/0.55      ! [X2: num] :
% 0.33/0.55        ( ( ord_less_num @ Z4 @ X2 )
% 0.33/0.55       => ( ord_less_eq_num @ T @ X2 ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(8)
% 0.33/0.55  thf(fact_1584_pinf_I8_J,axiom,
% 0.33/0.55      ! [T: nat] :
% 0.33/0.55      ? [Z4: nat] :
% 0.33/0.55      ! [X2: nat] :
% 0.33/0.55        ( ( ord_less_nat @ Z4 @ X2 )
% 0.33/0.55       => ( ord_less_eq_nat @ T @ X2 ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(8)
% 0.33/0.55  thf(fact_1585_pinf_I8_J,axiom,
% 0.33/0.55      ! [T: int] :
% 0.33/0.55      ? [Z4: int] :
% 0.33/0.55      ! [X2: int] :
% 0.33/0.55        ( ( ord_less_int @ Z4 @ X2 )
% 0.33/0.55       => ( ord_less_eq_int @ T @ X2 ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(8)
% 0.33/0.55  thf(fact_1586_pinf_I6_J,axiom,
% 0.33/0.55      ! [T: real] :
% 0.33/0.55      ? [Z4: real] :
% 0.33/0.55      ! [X2: real] :
% 0.33/0.55        ( ( ord_less_real @ Z4 @ X2 )
% 0.33/0.55       => ~ ( ord_less_eq_real @ X2 @ T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(6)
% 0.33/0.55  thf(fact_1587_pinf_I6_J,axiom,
% 0.33/0.55      ! [T: rat] :
% 0.33/0.55      ? [Z4: rat] :
% 0.33/0.55      ! [X2: rat] :
% 0.33/0.55        ( ( ord_less_rat @ Z4 @ X2 )
% 0.33/0.55       => ~ ( ord_less_eq_rat @ X2 @ T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(6)
% 0.33/0.55  thf(fact_1588_pinf_I6_J,axiom,
% 0.33/0.55      ! [T: num] :
% 0.33/0.55      ? [Z4: num] :
% 0.33/0.55      ! [X2: num] :
% 0.33/0.55        ( ( ord_less_num @ Z4 @ X2 )
% 0.33/0.55       => ~ ( ord_less_eq_num @ X2 @ T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(6)
% 0.33/0.55  thf(fact_1589_pinf_I6_J,axiom,
% 0.33/0.55      ! [T: nat] :
% 0.33/0.55      ? [Z4: nat] :
% 0.33/0.55      ! [X2: nat] :
% 0.33/0.55        ( ( ord_less_nat @ Z4 @ X2 )
% 0.33/0.55       => ~ ( ord_less_eq_nat @ X2 @ T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(6)
% 0.33/0.55  thf(fact_1590_pinf_I6_J,axiom,
% 0.33/0.55      ! [T: int] :
% 0.33/0.55      ? [Z4: int] :
% 0.33/0.55      ! [X2: int] :
% 0.33/0.55        ( ( ord_less_int @ Z4 @ X2 )
% 0.33/0.55       => ~ ( ord_less_eq_int @ X2 @ T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(6)
% 0.33/0.55  thf(fact_1591_length__enumerate,axiom,
% 0.33/0.55      ! [N: nat,Xs: list_VEBT_VEBT] :
% 0.33/0.55        ( ( size_s4762443039079500285T_VEBT @ ( enumerate_VEBT_VEBT @ N @ Xs ) )
% 0.33/0.55        = ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% 0.33/0.55  
% 0.33/0.55  % length_enumerate
% 0.33/0.55  thf(fact_1592_length__enumerate,axiom,
% 0.33/0.55      ! [N: nat,Xs: list_o] :
% 0.33/0.55        ( ( size_s6491369823275344609_nat_o @ ( enumerate_o @ N @ Xs ) )
% 0.33/0.55        = ( size_size_list_o @ Xs ) ) ).
% 0.33/0.55  
% 0.33/0.55  % length_enumerate
% 0.33/0.55  thf(fact_1593_length__enumerate,axiom,
% 0.33/0.55      ! [N: nat,Xs: list_nat] :
% 0.33/0.55        ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
% 0.33/0.55        = ( size_size_list_nat @ Xs ) ) ).
% 0.33/0.55  
% 0.33/0.55  % length_enumerate
% 0.33/0.55  thf(fact_1594_length__enumerate,axiom,
% 0.33/0.55      ! [N: nat,Xs: list_int] :
% 0.33/0.55        ( ( size_s2970893825323803983at_int @ ( enumerate_int @ N @ Xs ) )
% 0.33/0.55        = ( size_size_list_int @ Xs ) ) ).
% 0.33/0.55  
% 0.33/0.55  % length_enumerate
% 0.33/0.55  thf(fact_1595_add__def,axiom,
% 0.33/0.55      ( vEBT_VEBT_add
% 0.33/0.55      = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).
% 0.33/0.55  
% 0.33/0.55  % add_def
% 0.33/0.55  thf(fact_1596_less__by__empty,axiom,
% 0.33/0.55      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.33/0.55        ( ( A2 = bot_bo2099793752762293965at_nat )
% 0.33/0.55       => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ).
% 0.33/0.55  
% 0.33/0.55  % less_by_empty
% 0.33/0.55  thf(fact_1597_linorder__neqE__linordered__idom,axiom,
% 0.33/0.55      ! [X: real,Y: real] :
% 0.33/0.55        ( ( X != Y )
% 0.33/0.55       => ( ~ ( ord_less_real @ X @ Y )
% 0.33/0.55         => ( ord_less_real @ Y @ X ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % linorder_neqE_linordered_idom
% 0.33/0.55  thf(fact_1598_linorder__neqE__linordered__idom,axiom,
% 0.33/0.55      ! [X: rat,Y: rat] :
% 0.33/0.55        ( ( X != Y )
% 0.33/0.55       => ( ~ ( ord_less_rat @ X @ Y )
% 0.33/0.55         => ( ord_less_rat @ Y @ X ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % linorder_neqE_linordered_idom
% 0.33/0.55  thf(fact_1599_linorder__neqE__linordered__idom,axiom,
% 0.33/0.55      ! [X: int,Y: int] :
% 0.33/0.55        ( ( X != Y )
% 0.33/0.55       => ( ~ ( ord_less_int @ X @ Y )
% 0.33/0.55         => ( ord_less_int @ Y @ X ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % linorder_neqE_linordered_idom
% 0.33/0.55  thf(fact_1600_linordered__field__no__ub,axiom,
% 0.33/0.55      ! [X2: real] :
% 0.33/0.55      ? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% 0.33/0.55  
% 0.33/0.55  % linordered_field_no_ub
% 0.33/0.55  thf(fact_1601_linordered__field__no__ub,axiom,
% 0.33/0.55      ! [X2: rat] :
% 0.33/0.55      ? [X_1: rat] : ( ord_less_rat @ X2 @ X_1 ) ).
% 0.33/0.55  
% 0.33/0.55  % linordered_field_no_ub
% 0.33/0.55  thf(fact_1602_linordered__field__no__lb,axiom,
% 0.33/0.55      ! [X2: real] :
% 0.33/0.55      ? [Y3: real] : ( ord_less_real @ Y3 @ X2 ) ).
% 0.33/0.55  
% 0.33/0.55  % linordered_field_no_lb
% 0.33/0.55  thf(fact_1603_linordered__field__no__lb,axiom,
% 0.33/0.55      ! [X2: rat] :
% 0.33/0.55      ? [Y3: rat] : ( ord_less_rat @ Y3 @ X2 ) ).
% 0.33/0.55  
% 0.33/0.55  % linordered_field_no_lb
% 0.33/0.55  thf(fact_1604_pinf_I1_J,axiom,
% 0.33/0.55      ! [P: real > $o,P5: real > $o,Q: real > $o,Q2: real > $o] :
% 0.33/0.55        ( ? [Z5: real] :
% 0.33/0.55          ! [X4: real] :
% 0.33/0.55            ( ( ord_less_real @ Z5 @ X4 )
% 0.33/0.55           => ( ( P @ X4 )
% 0.33/0.55              = ( P5 @ X4 ) ) )
% 0.33/0.55       => ( ? [Z5: real] :
% 0.33/0.55            ! [X4: real] :
% 0.33/0.55              ( ( ord_less_real @ Z5 @ X4 )
% 0.33/0.55             => ( ( Q @ X4 )
% 0.33/0.55                = ( Q2 @ X4 ) ) )
% 0.33/0.55         => ? [Z4: real] :
% 0.33/0.55            ! [X2: real] :
% 0.33/0.55              ( ( ord_less_real @ Z4 @ X2 )
% 0.33/0.55             => ( ( ( P @ X2 )
% 0.33/0.55                  & ( Q @ X2 ) )
% 0.33/0.55                = ( ( P5 @ X2 )
% 0.33/0.55                  & ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(1)
% 0.33/0.55  thf(fact_1605_pinf_I1_J,axiom,
% 0.33/0.55      ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q2: rat > $o] :
% 0.33/0.55        ( ? [Z5: rat] :
% 0.33/0.55          ! [X4: rat] :
% 0.33/0.55            ( ( ord_less_rat @ Z5 @ X4 )
% 0.33/0.55           => ( ( P @ X4 )
% 0.33/0.55              = ( P5 @ X4 ) ) )
% 0.33/0.55       => ( ? [Z5: rat] :
% 0.33/0.55            ! [X4: rat] :
% 0.33/0.55              ( ( ord_less_rat @ Z5 @ X4 )
% 0.33/0.55             => ( ( Q @ X4 )
% 0.33/0.55                = ( Q2 @ X4 ) ) )
% 0.33/0.55         => ? [Z4: rat] :
% 0.33/0.55            ! [X2: rat] :
% 0.33/0.55              ( ( ord_less_rat @ Z4 @ X2 )
% 0.33/0.55             => ( ( ( P @ X2 )
% 0.33/0.55                  & ( Q @ X2 ) )
% 0.33/0.55                = ( ( P5 @ X2 )
% 0.33/0.55                  & ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(1)
% 0.33/0.55  thf(fact_1606_pinf_I1_J,axiom,
% 0.33/0.55      ! [P: num > $o,P5: num > $o,Q: num > $o,Q2: num > $o] :
% 0.33/0.55        ( ? [Z5: num] :
% 0.33/0.55          ! [X4: num] :
% 0.33/0.55            ( ( ord_less_num @ Z5 @ X4 )
% 0.33/0.55           => ( ( P @ X4 )
% 0.33/0.55              = ( P5 @ X4 ) ) )
% 0.33/0.55       => ( ? [Z5: num] :
% 0.33/0.55            ! [X4: num] :
% 0.33/0.55              ( ( ord_less_num @ Z5 @ X4 )
% 0.33/0.55             => ( ( Q @ X4 )
% 0.33/0.55                = ( Q2 @ X4 ) ) )
% 0.33/0.55         => ? [Z4: num] :
% 0.33/0.55            ! [X2: num] :
% 0.33/0.55              ( ( ord_less_num @ Z4 @ X2 )
% 0.33/0.55             => ( ( ( P @ X2 )
% 0.33/0.55                  & ( Q @ X2 ) )
% 0.33/0.55                = ( ( P5 @ X2 )
% 0.33/0.55                  & ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(1)
% 0.33/0.55  thf(fact_1607_pinf_I1_J,axiom,
% 0.33/0.55      ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.33/0.55        ( ? [Z5: nat] :
% 0.33/0.55          ! [X4: nat] :
% 0.33/0.55            ( ( ord_less_nat @ Z5 @ X4 )
% 0.33/0.55           => ( ( P @ X4 )
% 0.33/0.55              = ( P5 @ X4 ) ) )
% 0.33/0.55       => ( ? [Z5: nat] :
% 0.33/0.55            ! [X4: nat] :
% 0.33/0.55              ( ( ord_less_nat @ Z5 @ X4 )
% 0.33/0.55             => ( ( Q @ X4 )
% 0.33/0.55                = ( Q2 @ X4 ) ) )
% 0.33/0.55         => ? [Z4: nat] :
% 0.33/0.55            ! [X2: nat] :
% 0.33/0.55              ( ( ord_less_nat @ Z4 @ X2 )
% 0.33/0.55             => ( ( ( P @ X2 )
% 0.33/0.55                  & ( Q @ X2 ) )
% 0.33/0.55                = ( ( P5 @ X2 )
% 0.33/0.55                  & ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(1)
% 0.33/0.55  thf(fact_1608_pinf_I1_J,axiom,
% 0.33/0.55      ! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
% 0.33/0.55        ( ? [Z5: int] :
% 0.33/0.55          ! [X4: int] :
% 0.33/0.55            ( ( ord_less_int @ Z5 @ X4 )
% 0.33/0.55           => ( ( P @ X4 )
% 0.33/0.55              = ( P5 @ X4 ) ) )
% 0.33/0.55       => ( ? [Z5: int] :
% 0.33/0.55            ! [X4: int] :
% 0.33/0.55              ( ( ord_less_int @ Z5 @ X4 )
% 0.33/0.55             => ( ( Q @ X4 )
% 0.33/0.55                = ( Q2 @ X4 ) ) )
% 0.33/0.55         => ? [Z4: int] :
% 0.33/0.55            ! [X2: int] :
% 0.33/0.55              ( ( ord_less_int @ Z4 @ X2 )
% 0.33/0.55             => ( ( ( P @ X2 )
% 0.33/0.55                  & ( Q @ X2 ) )
% 0.33/0.55                = ( ( P5 @ X2 )
% 0.33/0.55                  & ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(1)
% 0.33/0.55  thf(fact_1609_pinf_I2_J,axiom,
% 0.33/0.55      ! [P: real > $o,P5: real > $o,Q: real > $o,Q2: real > $o] :
% 0.33/0.55        ( ? [Z5: real] :
% 0.33/0.55          ! [X4: real] :
% 0.33/0.55            ( ( ord_less_real @ Z5 @ X4 )
% 0.33/0.55           => ( ( P @ X4 )
% 0.33/0.55              = ( P5 @ X4 ) ) )
% 0.33/0.55       => ( ? [Z5: real] :
% 0.33/0.55            ! [X4: real] :
% 0.33/0.55              ( ( ord_less_real @ Z5 @ X4 )
% 0.33/0.55             => ( ( Q @ X4 )
% 0.33/0.55                = ( Q2 @ X4 ) ) )
% 0.33/0.55         => ? [Z4: real] :
% 0.33/0.55            ! [X2: real] :
% 0.33/0.55              ( ( ord_less_real @ Z4 @ X2 )
% 0.33/0.55             => ( ( ( P @ X2 )
% 0.33/0.55                  | ( Q @ X2 ) )
% 0.33/0.55                = ( ( P5 @ X2 )
% 0.33/0.55                  | ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(2)
% 0.33/0.55  thf(fact_1610_pinf_I2_J,axiom,
% 0.33/0.55      ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q2: rat > $o] :
% 0.33/0.55        ( ? [Z5: rat] :
% 0.33/0.55          ! [X4: rat] :
% 0.33/0.55            ( ( ord_less_rat @ Z5 @ X4 )
% 0.33/0.55           => ( ( P @ X4 )
% 0.33/0.55              = ( P5 @ X4 ) ) )
% 0.33/0.55       => ( ? [Z5: rat] :
% 0.33/0.55            ! [X4: rat] :
% 0.33/0.55              ( ( ord_less_rat @ Z5 @ X4 )
% 0.33/0.55             => ( ( Q @ X4 )
% 0.33/0.55                = ( Q2 @ X4 ) ) )
% 0.33/0.55         => ? [Z4: rat] :
% 0.33/0.55            ! [X2: rat] :
% 0.33/0.55              ( ( ord_less_rat @ Z4 @ X2 )
% 0.33/0.55             => ( ( ( P @ X2 )
% 0.33/0.55                  | ( Q @ X2 ) )
% 0.33/0.55                = ( ( P5 @ X2 )
% 0.33/0.55                  | ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(2)
% 0.33/0.55  thf(fact_1611_pinf_I2_J,axiom,
% 0.33/0.55      ! [P: num > $o,P5: num > $o,Q: num > $o,Q2: num > $o] :
% 0.33/0.55        ( ? [Z5: num] :
% 0.33/0.55          ! [X4: num] :
% 0.33/0.55            ( ( ord_less_num @ Z5 @ X4 )
% 0.33/0.55           => ( ( P @ X4 )
% 0.33/0.55              = ( P5 @ X4 ) ) )
% 0.33/0.55       => ( ? [Z5: num] :
% 0.33/0.55            ! [X4: num] :
% 0.33/0.55              ( ( ord_less_num @ Z5 @ X4 )
% 0.33/0.55             => ( ( Q @ X4 )
% 0.33/0.55                = ( Q2 @ X4 ) ) )
% 0.33/0.55         => ? [Z4: num] :
% 0.33/0.55            ! [X2: num] :
% 0.33/0.55              ( ( ord_less_num @ Z4 @ X2 )
% 0.33/0.55             => ( ( ( P @ X2 )
% 0.33/0.55                  | ( Q @ X2 ) )
% 0.33/0.55                = ( ( P5 @ X2 )
% 0.33/0.55                  | ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(2)
% 0.33/0.55  thf(fact_1612_pinf_I2_J,axiom,
% 0.33/0.55      ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.33/0.55        ( ? [Z5: nat] :
% 0.33/0.55          ! [X4: nat] :
% 0.33/0.55            ( ( ord_less_nat @ Z5 @ X4 )
% 0.33/0.55           => ( ( P @ X4 )
% 0.33/0.55              = ( P5 @ X4 ) ) )
% 0.33/0.55       => ( ? [Z5: nat] :
% 0.33/0.55            ! [X4: nat] :
% 0.33/0.55              ( ( ord_less_nat @ Z5 @ X4 )
% 0.33/0.55             => ( ( Q @ X4 )
% 0.33/0.55                = ( Q2 @ X4 ) ) )
% 0.33/0.55         => ? [Z4: nat] :
% 0.33/0.55            ! [X2: nat] :
% 0.33/0.55              ( ( ord_less_nat @ Z4 @ X2 )
% 0.33/0.55             => ( ( ( P @ X2 )
% 0.33/0.55                  | ( Q @ X2 ) )
% 0.33/0.55                = ( ( P5 @ X2 )
% 0.33/0.55                  | ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(2)
% 0.33/0.55  thf(fact_1613_pinf_I2_J,axiom,
% 0.33/0.55      ! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
% 0.33/0.55        ( ? [Z5: int] :
% 0.33/0.55          ! [X4: int] :
% 0.33/0.55            ( ( ord_less_int @ Z5 @ X4 )
% 0.33/0.55           => ( ( P @ X4 )
% 0.33/0.55              = ( P5 @ X4 ) ) )
% 0.33/0.55       => ( ? [Z5: int] :
% 0.33/0.55            ! [X4: int] :
% 0.33/0.55              ( ( ord_less_int @ Z5 @ X4 )
% 0.33/0.55             => ( ( Q @ X4 )
% 0.33/0.55                = ( Q2 @ X4 ) ) )
% 0.33/0.55         => ? [Z4: int] :
% 0.33/0.55            ! [X2: int] :
% 0.33/0.55              ( ( ord_less_int @ Z4 @ X2 )
% 0.33/0.55             => ( ( ( P @ X2 )
% 0.33/0.55                  | ( Q @ X2 ) )
% 0.33/0.55                = ( ( P5 @ X2 )
% 0.33/0.55                  | ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(2)
% 0.33/0.55  thf(fact_1614_pinf_I3_J,axiom,
% 0.33/0.55      ! [T: real] :
% 0.33/0.55      ? [Z4: real] :
% 0.33/0.55      ! [X2: real] :
% 0.33/0.55        ( ( ord_less_real @ Z4 @ X2 )
% 0.33/0.55       => ( X2 != T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(3)
% 0.33/0.55  thf(fact_1615_pinf_I3_J,axiom,
% 0.33/0.55      ! [T: rat] :
% 0.33/0.55      ? [Z4: rat] :
% 0.33/0.55      ! [X2: rat] :
% 0.33/0.55        ( ( ord_less_rat @ Z4 @ X2 )
% 0.33/0.55       => ( X2 != T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(3)
% 0.33/0.55  thf(fact_1616_pinf_I3_J,axiom,
% 0.33/0.55      ! [T: num] :
% 0.33/0.55      ? [Z4: num] :
% 0.33/0.55      ! [X2: num] :
% 0.33/0.55        ( ( ord_less_num @ Z4 @ X2 )
% 0.33/0.55       => ( X2 != T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(3)
% 0.33/0.55  thf(fact_1617_pinf_I3_J,axiom,
% 0.33/0.55      ! [T: nat] :
% 0.33/0.55      ? [Z4: nat] :
% 0.33/0.55      ! [X2: nat] :
% 0.33/0.55        ( ( ord_less_nat @ Z4 @ X2 )
% 0.33/0.55       => ( X2 != T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(3)
% 0.33/0.55  thf(fact_1618_pinf_I3_J,axiom,
% 0.33/0.55      ! [T: int] :
% 0.33/0.55      ? [Z4: int] :
% 0.33/0.55      ! [X2: int] :
% 0.33/0.55        ( ( ord_less_int @ Z4 @ X2 )
% 0.33/0.55       => ( X2 != T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(3)
% 0.33/0.55  thf(fact_1619_pinf_I4_J,axiom,
% 0.33/0.55      ! [T: real] :
% 0.33/0.55      ? [Z4: real] :
% 0.33/0.55      ! [X2: real] :
% 0.33/0.55        ( ( ord_less_real @ Z4 @ X2 )
% 0.33/0.55       => ( X2 != T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(4)
% 0.33/0.55  thf(fact_1620_pinf_I4_J,axiom,
% 0.33/0.55      ! [T: rat] :
% 0.33/0.55      ? [Z4: rat] :
% 0.33/0.55      ! [X2: rat] :
% 0.33/0.55        ( ( ord_less_rat @ Z4 @ X2 )
% 0.33/0.55       => ( X2 != T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(4)
% 0.33/0.55  thf(fact_1621_pinf_I4_J,axiom,
% 0.33/0.55      ! [T: num] :
% 0.33/0.55      ? [Z4: num] :
% 0.33/0.55      ! [X2: num] :
% 0.33/0.55        ( ( ord_less_num @ Z4 @ X2 )
% 0.33/0.55       => ( X2 != T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(4)
% 0.33/0.55  thf(fact_1622_pinf_I4_J,axiom,
% 0.33/0.55      ! [T: nat] :
% 0.33/0.55      ? [Z4: nat] :
% 0.33/0.55      ! [X2: nat] :
% 0.33/0.55        ( ( ord_less_nat @ Z4 @ X2 )
% 0.33/0.55       => ( X2 != T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(4)
% 0.33/0.55  thf(fact_1623_pinf_I4_J,axiom,
% 0.33/0.55      ! [T: int] :
% 0.33/0.55      ? [Z4: int] :
% 0.33/0.55      ! [X2: int] :
% 0.33/0.55        ( ( ord_less_int @ Z4 @ X2 )
% 0.33/0.55       => ( X2 != T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(4)
% 0.33/0.55  thf(fact_1624_pinf_I5_J,axiom,
% 0.33/0.55      ! [T: real] :
% 0.33/0.55      ? [Z4: real] :
% 0.33/0.55      ! [X2: real] :
% 0.33/0.55        ( ( ord_less_real @ Z4 @ X2 )
% 0.33/0.55       => ~ ( ord_less_real @ X2 @ T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(5)
% 0.33/0.55  thf(fact_1625_pinf_I5_J,axiom,
% 0.33/0.55      ! [T: rat] :
% 0.33/0.55      ? [Z4: rat] :
% 0.33/0.55      ! [X2: rat] :
% 0.33/0.55        ( ( ord_less_rat @ Z4 @ X2 )
% 0.33/0.55       => ~ ( ord_less_rat @ X2 @ T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(5)
% 0.33/0.55  thf(fact_1626_pinf_I5_J,axiom,
% 0.33/0.55      ! [T: num] :
% 0.33/0.55      ? [Z4: num] :
% 0.33/0.55      ! [X2: num] :
% 0.33/0.55        ( ( ord_less_num @ Z4 @ X2 )
% 0.33/0.55       => ~ ( ord_less_num @ X2 @ T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(5)
% 0.33/0.55  thf(fact_1627_pinf_I5_J,axiom,
% 0.33/0.55      ! [T: nat] :
% 0.33/0.55      ? [Z4: nat] :
% 0.33/0.55      ! [X2: nat] :
% 0.33/0.55        ( ( ord_less_nat @ Z4 @ X2 )
% 0.33/0.55       => ~ ( ord_less_nat @ X2 @ T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(5)
% 0.33/0.55  thf(fact_1628_pinf_I5_J,axiom,
% 0.33/0.55      ! [T: int] :
% 0.33/0.55      ? [Z4: int] :
% 0.33/0.55      ! [X2: int] :
% 0.33/0.55        ( ( ord_less_int @ Z4 @ X2 )
% 0.33/0.55       => ~ ( ord_less_int @ X2 @ T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(5)
% 0.33/0.55  thf(fact_1629_pinf_I7_J,axiom,
% 0.33/0.55      ! [T: real] :
% 0.33/0.55      ? [Z4: real] :
% 0.33/0.55      ! [X2: real] :
% 0.33/0.55        ( ( ord_less_real @ Z4 @ X2 )
% 0.33/0.55       => ( ord_less_real @ T @ X2 ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(7)
% 0.33/0.55  thf(fact_1630_pinf_I7_J,axiom,
% 0.33/0.55      ! [T: rat] :
% 0.33/0.55      ? [Z4: rat] :
% 0.33/0.55      ! [X2: rat] :
% 0.33/0.55        ( ( ord_less_rat @ Z4 @ X2 )
% 0.33/0.55       => ( ord_less_rat @ T @ X2 ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(7)
% 0.33/0.55  thf(fact_1631_pinf_I7_J,axiom,
% 0.33/0.55      ! [T: num] :
% 0.33/0.55      ? [Z4: num] :
% 0.33/0.55      ! [X2: num] :
% 0.33/0.55        ( ( ord_less_num @ Z4 @ X2 )
% 0.33/0.55       => ( ord_less_num @ T @ X2 ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(7)
% 0.33/0.55  thf(fact_1632_pinf_I7_J,axiom,
% 0.33/0.55      ! [T: nat] :
% 0.33/0.55      ? [Z4: nat] :
% 0.33/0.55      ! [X2: nat] :
% 0.33/0.55        ( ( ord_less_nat @ Z4 @ X2 )
% 0.33/0.55       => ( ord_less_nat @ T @ X2 ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(7)
% 0.33/0.55  thf(fact_1633_pinf_I7_J,axiom,
% 0.33/0.55      ! [T: int] :
% 0.33/0.55      ? [Z4: int] :
% 0.33/0.55      ! [X2: int] :
% 0.33/0.55        ( ( ord_less_int @ Z4 @ X2 )
% 0.33/0.55       => ( ord_less_int @ T @ X2 ) ) ).
% 0.33/0.55  
% 0.33/0.55  % pinf(7)
% 0.33/0.55  thf(fact_1634_minf_I1_J,axiom,
% 0.33/0.55      ! [P: real > $o,P5: real > $o,Q: real > $o,Q2: real > $o] :
% 0.33/0.55        ( ? [Z5: real] :
% 0.33/0.55          ! [X4: real] :
% 0.33/0.55            ( ( ord_less_real @ X4 @ Z5 )
% 0.33/0.55           => ( ( P @ X4 )
% 0.33/0.55              = ( P5 @ X4 ) ) )
% 0.33/0.55       => ( ? [Z5: real] :
% 0.33/0.55            ! [X4: real] :
% 0.33/0.55              ( ( ord_less_real @ X4 @ Z5 )
% 0.33/0.55             => ( ( Q @ X4 )
% 0.33/0.55                = ( Q2 @ X4 ) ) )
% 0.33/0.55         => ? [Z4: real] :
% 0.33/0.55            ! [X2: real] :
% 0.33/0.55              ( ( ord_less_real @ X2 @ Z4 )
% 0.33/0.55             => ( ( ( P @ X2 )
% 0.33/0.55                  & ( Q @ X2 ) )
% 0.33/0.55                = ( ( P5 @ X2 )
% 0.33/0.55                  & ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(1)
% 0.33/0.55  thf(fact_1635_minf_I1_J,axiom,
% 0.33/0.55      ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q2: rat > $o] :
% 0.33/0.55        ( ? [Z5: rat] :
% 0.33/0.55          ! [X4: rat] :
% 0.33/0.55            ( ( ord_less_rat @ X4 @ Z5 )
% 0.33/0.55           => ( ( P @ X4 )
% 0.33/0.55              = ( P5 @ X4 ) ) )
% 0.33/0.55       => ( ? [Z5: rat] :
% 0.33/0.55            ! [X4: rat] :
% 0.33/0.55              ( ( ord_less_rat @ X4 @ Z5 )
% 0.33/0.55             => ( ( Q @ X4 )
% 0.33/0.55                = ( Q2 @ X4 ) ) )
% 0.33/0.55         => ? [Z4: rat] :
% 0.33/0.55            ! [X2: rat] :
% 0.33/0.55              ( ( ord_less_rat @ X2 @ Z4 )
% 0.33/0.55             => ( ( ( P @ X2 )
% 0.33/0.55                  & ( Q @ X2 ) )
% 0.33/0.55                = ( ( P5 @ X2 )
% 0.33/0.55                  & ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(1)
% 0.33/0.55  thf(fact_1636_minf_I1_J,axiom,
% 0.33/0.55      ! [P: num > $o,P5: num > $o,Q: num > $o,Q2: num > $o] :
% 0.33/0.55        ( ? [Z5: num] :
% 0.33/0.55          ! [X4: num] :
% 0.33/0.55            ( ( ord_less_num @ X4 @ Z5 )
% 0.33/0.55           => ( ( P @ X4 )
% 0.33/0.55              = ( P5 @ X4 ) ) )
% 0.33/0.55       => ( ? [Z5: num] :
% 0.33/0.55            ! [X4: num] :
% 0.33/0.55              ( ( ord_less_num @ X4 @ Z5 )
% 0.33/0.55             => ( ( Q @ X4 )
% 0.33/0.55                = ( Q2 @ X4 ) ) )
% 0.33/0.55         => ? [Z4: num] :
% 0.33/0.55            ! [X2: num] :
% 0.33/0.55              ( ( ord_less_num @ X2 @ Z4 )
% 0.33/0.55             => ( ( ( P @ X2 )
% 0.33/0.55                  & ( Q @ X2 ) )
% 0.33/0.55                = ( ( P5 @ X2 )
% 0.33/0.55                  & ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(1)
% 0.33/0.55  thf(fact_1637_minf_I1_J,axiom,
% 0.33/0.55      ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.33/0.55        ( ? [Z5: nat] :
% 0.33/0.55          ! [X4: nat] :
% 0.33/0.55            ( ( ord_less_nat @ X4 @ Z5 )
% 0.33/0.55           => ( ( P @ X4 )
% 0.33/0.55              = ( P5 @ X4 ) ) )
% 0.33/0.55       => ( ? [Z5: nat] :
% 0.33/0.55            ! [X4: nat] :
% 0.33/0.55              ( ( ord_less_nat @ X4 @ Z5 )
% 0.33/0.55             => ( ( Q @ X4 )
% 0.33/0.55                = ( Q2 @ X4 ) ) )
% 0.33/0.55         => ? [Z4: nat] :
% 0.33/0.55            ! [X2: nat] :
% 0.33/0.55              ( ( ord_less_nat @ X2 @ Z4 )
% 0.33/0.55             => ( ( ( P @ X2 )
% 0.33/0.55                  & ( Q @ X2 ) )
% 0.33/0.55                = ( ( P5 @ X2 )
% 0.33/0.55                  & ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(1)
% 0.33/0.55  thf(fact_1638_minf_I1_J,axiom,
% 0.33/0.55      ! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
% 0.33/0.55        ( ? [Z5: int] :
% 0.33/0.55          ! [X4: int] :
% 0.33/0.55            ( ( ord_less_int @ X4 @ Z5 )
% 0.33/0.55           => ( ( P @ X4 )
% 0.33/0.55              = ( P5 @ X4 ) ) )
% 0.33/0.55       => ( ? [Z5: int] :
% 0.33/0.55            ! [X4: int] :
% 0.33/0.55              ( ( ord_less_int @ X4 @ Z5 )
% 0.33/0.55             => ( ( Q @ X4 )
% 0.33/0.55                = ( Q2 @ X4 ) ) )
% 0.33/0.55         => ? [Z4: int] :
% 0.33/0.55            ! [X2: int] :
% 0.33/0.55              ( ( ord_less_int @ X2 @ Z4 )
% 0.33/0.55             => ( ( ( P @ X2 )
% 0.33/0.55                  & ( Q @ X2 ) )
% 0.33/0.55                = ( ( P5 @ X2 )
% 0.33/0.55                  & ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(1)
% 0.33/0.55  thf(fact_1639_minf_I2_J,axiom,
% 0.33/0.55      ! [P: real > $o,P5: real > $o,Q: real > $o,Q2: real > $o] :
% 0.33/0.55        ( ? [Z5: real] :
% 0.33/0.55          ! [X4: real] :
% 0.33/0.55            ( ( ord_less_real @ X4 @ Z5 )
% 0.33/0.55           => ( ( P @ X4 )
% 0.33/0.55              = ( P5 @ X4 ) ) )
% 0.33/0.55       => ( ? [Z5: real] :
% 0.33/0.55            ! [X4: real] :
% 0.33/0.55              ( ( ord_less_real @ X4 @ Z5 )
% 0.33/0.55             => ( ( Q @ X4 )
% 0.33/0.55                = ( Q2 @ X4 ) ) )
% 0.33/0.55         => ? [Z4: real] :
% 0.33/0.55            ! [X2: real] :
% 0.33/0.55              ( ( ord_less_real @ X2 @ Z4 )
% 0.33/0.55             => ( ( ( P @ X2 )
% 0.33/0.55                  | ( Q @ X2 ) )
% 0.33/0.55                = ( ( P5 @ X2 )
% 0.33/0.55                  | ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(2)
% 0.33/0.55  thf(fact_1640_minf_I2_J,axiom,
% 0.33/0.55      ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q2: rat > $o] :
% 0.33/0.55        ( ? [Z5: rat] :
% 0.33/0.55          ! [X4: rat] :
% 0.33/0.55            ( ( ord_less_rat @ X4 @ Z5 )
% 0.33/0.55           => ( ( P @ X4 )
% 0.33/0.55              = ( P5 @ X4 ) ) )
% 0.33/0.55       => ( ? [Z5: rat] :
% 0.33/0.55            ! [X4: rat] :
% 0.33/0.55              ( ( ord_less_rat @ X4 @ Z5 )
% 0.33/0.55             => ( ( Q @ X4 )
% 0.33/0.55                = ( Q2 @ X4 ) ) )
% 0.33/0.55         => ? [Z4: rat] :
% 0.33/0.55            ! [X2: rat] :
% 0.33/0.55              ( ( ord_less_rat @ X2 @ Z4 )
% 0.33/0.55             => ( ( ( P @ X2 )
% 0.33/0.55                  | ( Q @ X2 ) )
% 0.33/0.55                = ( ( P5 @ X2 )
% 0.33/0.55                  | ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(2)
% 0.33/0.55  thf(fact_1641_minf_I2_J,axiom,
% 0.33/0.55      ! [P: num > $o,P5: num > $o,Q: num > $o,Q2: num > $o] :
% 0.33/0.55        ( ? [Z5: num] :
% 0.33/0.55          ! [X4: num] :
% 0.33/0.55            ( ( ord_less_num @ X4 @ Z5 )
% 0.33/0.55           => ( ( P @ X4 )
% 0.33/0.55              = ( P5 @ X4 ) ) )
% 0.33/0.55       => ( ? [Z5: num] :
% 0.33/0.55            ! [X4: num] :
% 0.33/0.55              ( ( ord_less_num @ X4 @ Z5 )
% 0.33/0.55             => ( ( Q @ X4 )
% 0.33/0.55                = ( Q2 @ X4 ) ) )
% 0.33/0.55         => ? [Z4: num] :
% 0.33/0.55            ! [X2: num] :
% 0.33/0.55              ( ( ord_less_num @ X2 @ Z4 )
% 0.33/0.55             => ( ( ( P @ X2 )
% 0.33/0.55                  | ( Q @ X2 ) )
% 0.33/0.55                = ( ( P5 @ X2 )
% 0.33/0.55                  | ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(2)
% 0.33/0.55  thf(fact_1642_minf_I2_J,axiom,
% 0.33/0.55      ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.33/0.55        ( ? [Z5: nat] :
% 0.33/0.55          ! [X4: nat] :
% 0.33/0.55            ( ( ord_less_nat @ X4 @ Z5 )
% 0.33/0.55           => ( ( P @ X4 )
% 0.33/0.55              = ( P5 @ X4 ) ) )
% 0.33/0.55       => ( ? [Z5: nat] :
% 0.33/0.55            ! [X4: nat] :
% 0.33/0.55              ( ( ord_less_nat @ X4 @ Z5 )
% 0.33/0.55             => ( ( Q @ X4 )
% 0.33/0.55                = ( Q2 @ X4 ) ) )
% 0.33/0.55         => ? [Z4: nat] :
% 0.33/0.55            ! [X2: nat] :
% 0.33/0.55              ( ( ord_less_nat @ X2 @ Z4 )
% 0.33/0.55             => ( ( ( P @ X2 )
% 0.33/0.55                  | ( Q @ X2 ) )
% 0.33/0.55                = ( ( P5 @ X2 )
% 0.33/0.55                  | ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(2)
% 0.33/0.55  thf(fact_1643_minf_I2_J,axiom,
% 0.33/0.55      ! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
% 0.33/0.55        ( ? [Z5: int] :
% 0.33/0.55          ! [X4: int] :
% 0.33/0.55            ( ( ord_less_int @ X4 @ Z5 )
% 0.33/0.55           => ( ( P @ X4 )
% 0.33/0.55              = ( P5 @ X4 ) ) )
% 0.33/0.55       => ( ? [Z5: int] :
% 0.33/0.55            ! [X4: int] :
% 0.33/0.55              ( ( ord_less_int @ X4 @ Z5 )
% 0.33/0.55             => ( ( Q @ X4 )
% 0.33/0.55                = ( Q2 @ X4 ) ) )
% 0.33/0.55         => ? [Z4: int] :
% 0.33/0.55            ! [X2: int] :
% 0.33/0.55              ( ( ord_less_int @ X2 @ Z4 )
% 0.33/0.55             => ( ( ( P @ X2 )
% 0.33/0.55                  | ( Q @ X2 ) )
% 0.33/0.55                = ( ( P5 @ X2 )
% 0.33/0.55                  | ( Q2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(2)
% 0.33/0.55  thf(fact_1644_minf_I3_J,axiom,
% 0.33/0.55      ! [T: real] :
% 0.33/0.55      ? [Z4: real] :
% 0.33/0.55      ! [X2: real] :
% 0.33/0.55        ( ( ord_less_real @ X2 @ Z4 )
% 0.33/0.55       => ( X2 != T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(3)
% 0.33/0.55  thf(fact_1645_minf_I3_J,axiom,
% 0.33/0.55      ! [T: rat] :
% 0.33/0.55      ? [Z4: rat] :
% 0.33/0.55      ! [X2: rat] :
% 0.33/0.55        ( ( ord_less_rat @ X2 @ Z4 )
% 0.33/0.55       => ( X2 != T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(3)
% 0.33/0.55  thf(fact_1646_minf_I3_J,axiom,
% 0.33/0.55      ! [T: num] :
% 0.33/0.55      ? [Z4: num] :
% 0.33/0.55      ! [X2: num] :
% 0.33/0.55        ( ( ord_less_num @ X2 @ Z4 )
% 0.33/0.55       => ( X2 != T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(3)
% 0.33/0.55  thf(fact_1647_minf_I3_J,axiom,
% 0.33/0.55      ! [T: nat] :
% 0.33/0.55      ? [Z4: nat] :
% 0.33/0.55      ! [X2: nat] :
% 0.33/0.55        ( ( ord_less_nat @ X2 @ Z4 )
% 0.33/0.55       => ( X2 != T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(3)
% 0.33/0.55  thf(fact_1648_minf_I3_J,axiom,
% 0.33/0.55      ! [T: int] :
% 0.33/0.55      ? [Z4: int] :
% 0.33/0.55      ! [X2: int] :
% 0.33/0.55        ( ( ord_less_int @ X2 @ Z4 )
% 0.33/0.55       => ( X2 != T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(3)
% 0.33/0.55  thf(fact_1649_minf_I4_J,axiom,
% 0.33/0.55      ! [T: real] :
% 0.33/0.55      ? [Z4: real] :
% 0.33/0.55      ! [X2: real] :
% 0.33/0.55        ( ( ord_less_real @ X2 @ Z4 )
% 0.33/0.55       => ( X2 != T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(4)
% 0.33/0.55  thf(fact_1650_minf_I4_J,axiom,
% 0.33/0.55      ! [T: rat] :
% 0.33/0.55      ? [Z4: rat] :
% 0.33/0.55      ! [X2: rat] :
% 0.33/0.55        ( ( ord_less_rat @ X2 @ Z4 )
% 0.33/0.55       => ( X2 != T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(4)
% 0.33/0.55  thf(fact_1651_minf_I4_J,axiom,
% 0.33/0.55      ! [T: num] :
% 0.33/0.55      ? [Z4: num] :
% 0.33/0.55      ! [X2: num] :
% 0.33/0.55        ( ( ord_less_num @ X2 @ Z4 )
% 0.33/0.55       => ( X2 != T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(4)
% 0.33/0.55  thf(fact_1652_minf_I4_J,axiom,
% 0.33/0.55      ! [T: nat] :
% 0.33/0.55      ? [Z4: nat] :
% 0.33/0.55      ! [X2: nat] :
% 0.33/0.55        ( ( ord_less_nat @ X2 @ Z4 )
% 0.33/0.55       => ( X2 != T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(4)
% 0.33/0.55  thf(fact_1653_minf_I4_J,axiom,
% 0.33/0.55      ! [T: int] :
% 0.33/0.55      ? [Z4: int] :
% 0.33/0.55      ! [X2: int] :
% 0.33/0.55        ( ( ord_less_int @ X2 @ Z4 )
% 0.33/0.55       => ( X2 != T ) ) ).
% 0.33/0.55  
% 0.33/0.55  % minf(4)
% 0.33/0.55  thf(fact_1654_minf_I5_J,axiom,
% 0.33/0.55      ! [T: real] :
% 0.33/0.55      ? [Z4: real] :
% 0.33/0.55      ! [X2: real] :
% 0.33/0.56        ( ( ord_less_real @ X2 @ Z4 )
% 0.33/0.56       => ( ord_less_real @ X2 @ T ) ) ).
% 0.33/0.56  
% 0.33/0.56  % minf(5)
% 0.33/0.56  thf(fact_1655_minf_I5_J,axiom,
% 0.33/0.56      ! [T: rat] :
% 0.33/0.56      ? [Z4: rat] :
% 0.33/0.56      ! [X2: rat] :
% 0.33/0.56        ( ( ord_less_rat @ X2 @ Z4 )
% 0.33/0.56       => ( ord_less_rat @ X2 @ T ) ) ).
% 0.33/0.56  
% 0.33/0.56  % minf(5)
% 0.33/0.56  thf(fact_1656_minf_I5_J,axiom,
% 0.33/0.56      ! [T: num] :
% 0.33/0.56      ? [Z4: num] :
% 0.33/0.56      ! [X2: num] :
% 0.33/0.56        ( ( ord_less_num @ X2 @ Z4 )
% 0.33/0.56       => ( ord_less_num @ X2 @ T ) ) ).
% 0.33/0.56  
% 0.33/0.56  % minf(5)
% 0.33/0.56  thf(fact_1657_minf_I5_J,axiom,
% 0.33/0.56      ! [T: nat] :
% 0.33/0.56      ? [Z4: nat] :
% 0.33/0.56      ! [X2: nat] :
% 0.33/0.56        ( ( ord_less_nat @ X2 @ Z4 )
% 0.33/0.56       => ( ord_less_nat @ X2 @ T ) ) ).
% 0.33/0.56  
% 0.33/0.56  % minf(5)
% 0.33/0.56  thf(fact_1658_minf_I5_J,axiom,
% 0.33/0.56      ! [T: int] :
% 0.33/0.56      ? [Z4: int] :
% 0.33/0.56      ! [X2: int] :
% 0.33/0.56        ( ( ord_less_int @ X2 @ Z4 )
% 0.33/0.56       => ( ord_less_int @ X2 @ T ) ) ).
% 0.33/0.56  
% 0.33/0.56  % minf(5)
% 0.33/0.56  thf(fact_1659_minf_I7_J,axiom,
% 0.33/0.56      ! [T: real] :
% 0.33/0.56      ? [Z4: real] :
% 0.33/0.56      ! [X2: real] :
% 0.33/0.56        ( ( ord_less_real @ X2 @ Z4 )
% 0.33/0.56       => ~ ( ord_less_real @ T @ X2 ) ) ).
% 0.33/0.56  
% 0.33/0.56  % minf(7)
% 0.33/0.56  thf(fact_1660_minf_I7_J,axiom,
% 0.33/0.56      ! [T: rat] :
% 0.33/0.56      ? [Z4: rat] :
% 0.33/0.56      ! [X2: rat] :
% 0.33/0.56        ( ( ord_less_rat @ X2 @ Z4 )
% 0.33/0.56       => ~ ( ord_less_rat @ T @ X2 ) ) ).
% 0.33/0.56  
% 0.33/0.56  % minf(7)
% 0.33/0.56  thf(fact_1661_minf_I7_J,axiom,
% 0.33/0.56      ! [T: num] :
% 0.33/0.56      ? [Z4: num] :
% 0.33/0.56      ! [X2: num] :
% 0.33/0.56        ( ( ord_less_num @ X2 @ Z4 )
% 0.33/0.56       => ~ ( ord_less_num @ T @ X2 ) ) ).
% 0.33/0.56  
% 0.33/0.56  % minf(7)
% 0.33/0.56  thf(fact_1662_minf_I7_J,axiom,
% 0.33/0.56      ! [T: nat] :
% 0.33/0.56      ? [Z4: nat] :
% 0.33/0.56      ! [X2: nat] :
% 0.33/0.56        ( ( ord_less_nat @ X2 @ Z4 )
% 0.33/0.56       => ~ ( ord_less_nat @ T @ X2 ) ) ).
% 0.33/0.56  
% 0.33/0.56  % minf(7)
% 0.33/0.56  thf(fact_1663_minf_I7_J,axiom,
% 0.33/0.56      ! [T: int] :
% 0.33/0.56      ? [Z4: int] :
% 0.33/0.56      ! [X2: int] :
% 0.33/0.56        ( ( ord_less_int @ X2 @ Z4 )
% 0.33/0.56       => ~ ( ord_less_int @ T @ X2 ) ) ).
% 0.33/0.56  
% 0.33/0.56  % minf(7)
% 0.33/0.56  thf(fact_1664_deg__SUcn__Node,axiom,
% 0.33/0.56      ! [Tree: vEBT_VEBT,N: nat] :
% 0.33/0.56        ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
% 0.33/0.56       => ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S4: vEBT_VEBT] :
% 0.33/0.56            ( Tree
% 0.33/0.56            = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList3 @ S4 ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % deg_SUcn_Node
% 0.33/0.56  thf(fact_1665_nth__zip,axiom,
% 0.33/0.56      ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.33/0.56        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.56       => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.33/0.56         => ( ( nth_Pr4953567300277697838T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) @ I )
% 0.33/0.56            = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nth_zip
% 0.33/0.56  thf(fact_1666_nth__zip,axiom,
% 0.33/0.56      ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
% 0.33/0.56        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.56       => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys ) )
% 0.33/0.56         => ( ( nth_Pr4606735188037164562VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys ) @ I )
% 0.33/0.56            = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_o @ Ys @ I ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nth_zip
% 0.33/0.56  thf(fact_1667_nth__zip,axiom,
% 0.33/0.56      ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
% 0.33/0.56        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.56       => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
% 0.33/0.56         => ( ( nth_Pr1791586995822124652BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) @ I )
% 0.33/0.56            = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nth_zip
% 0.33/0.56  thf(fact_1668_nth__zip,axiom,
% 0.33/0.56      ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
% 0.33/0.56        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.56       => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys ) )
% 0.33/0.56         => ( ( nth_Pr6837108013167703752BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) @ I )
% 0.33/0.56            = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_int @ Ys @ I ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nth_zip
% 0.33/0.56  thf(fact_1669_nth__zip,axiom,
% 0.33/0.56      ! [I: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
% 0.33/0.56        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.33/0.56       => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.33/0.56         => ( ( nth_Pr6777367263587873994T_VEBT @ ( zip_o_VEBT_VEBT @ Xs @ Ys ) @ I )
% 0.33/0.56            = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nth_zip
% 0.33/0.56  thf(fact_1670_nth__zip,axiom,
% 0.33/0.56      ! [I: nat,Xs: list_o,Ys: list_o] :
% 0.33/0.56        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.33/0.56       => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys ) )
% 0.33/0.56         => ( ( nth_Product_prod_o_o @ ( zip_o_o @ Xs @ Ys ) @ I )
% 0.33/0.56            = ( product_Pair_o_o @ ( nth_o @ Xs @ I ) @ ( nth_o @ Ys @ I ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nth_zip
% 0.33/0.56  thf(fact_1671_nth__zip,axiom,
% 0.33/0.56      ! [I: nat,Xs: list_o,Ys: list_nat] :
% 0.33/0.56        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.33/0.56       => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
% 0.33/0.56         => ( ( nth_Pr5826913651314560976_o_nat @ ( zip_o_nat @ Xs @ Ys ) @ I )
% 0.33/0.56            = ( product_Pair_o_nat @ ( nth_o @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nth_zip
% 0.33/0.56  thf(fact_1672_nth__zip,axiom,
% 0.33/0.56      ! [I: nat,Xs: list_o,Ys: list_int] :
% 0.33/0.56        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.33/0.56       => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys ) )
% 0.33/0.56         => ( ( nth_Pr1649062631805364268_o_int @ ( zip_o_int @ Xs @ Ys ) @ I )
% 0.33/0.56            = ( product_Pair_o_int @ ( nth_o @ Xs @ I ) @ ( nth_int @ Ys @ I ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nth_zip
% 0.33/0.56  thf(fact_1673_nth__zip,axiom,
% 0.33/0.56      ! [I: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
% 0.33/0.56        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 0.33/0.56       => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.33/0.56         => ( ( nth_Pr744662078594809490T_VEBT @ ( zip_nat_VEBT_VEBT @ Xs @ Ys ) @ I )
% 0.33/0.56            = ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nth_zip
% 0.33/0.56  thf(fact_1674_nth__zip,axiom,
% 0.33/0.56      ! [I: nat,Xs: list_nat,Ys: list_o] :
% 0.33/0.56        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 0.33/0.56       => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys ) )
% 0.33/0.56         => ( ( nth_Pr112076138515278198_nat_o @ ( zip_nat_o @ Xs @ Ys ) @ I )
% 0.33/0.56            = ( product_Pair_nat_o @ ( nth_nat @ Xs @ I ) @ ( nth_o @ Ys @ I ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nth_zip
% 0.33/0.56  thf(fact_1675_find__Some__iff2,axiom,
% 0.33/0.56      ! [X: product_prod_nat_nat,P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
% 0.33/0.56        ( ( ( some_P7363390416028606310at_nat @ X )
% 0.33/0.56          = ( find_P8199882355184865565at_nat @ P @ Xs ) )
% 0.33/0.56        = ( ? [I4: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 0.33/0.56              & ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
% 0.33/0.56              & ( X
% 0.33/0.56                = ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
% 0.33/0.56              & ! [J3: nat] :
% 0.33/0.56                  ( ( ord_less_nat @ J3 @ I4 )
% 0.33/0.56                 => ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_Some_iff2
% 0.33/0.56  thf(fact_1676_find__Some__iff2,axiom,
% 0.33/0.56      ! [X: num,P: num > $o,Xs: list_num] :
% 0.33/0.56        ( ( ( some_num @ X )
% 0.33/0.56          = ( find_num @ P @ Xs ) )
% 0.33/0.56        = ( ? [I4: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I4 @ ( size_size_list_num @ Xs ) )
% 0.33/0.56              & ( P @ ( nth_num @ Xs @ I4 ) )
% 0.33/0.56              & ( X
% 0.33/0.56                = ( nth_num @ Xs @ I4 ) )
% 0.33/0.56              & ! [J3: nat] :
% 0.33/0.56                  ( ( ord_less_nat @ J3 @ I4 )
% 0.33/0.56                 => ~ ( P @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_Some_iff2
% 0.33/0.56  thf(fact_1677_find__Some__iff2,axiom,
% 0.33/0.56      ! [X: vEBT_VEBT,P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 0.33/0.56        ( ( ( some_VEBT_VEBT @ X )
% 0.33/0.56          = ( find_VEBT_VEBT @ P @ Xs ) )
% 0.33/0.56        = ( ? [I4: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.56              & ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) )
% 0.33/0.56              & ( X
% 0.33/0.56                = ( nth_VEBT_VEBT @ Xs @ I4 ) )
% 0.33/0.56              & ! [J3: nat] :
% 0.33/0.56                  ( ( ord_less_nat @ J3 @ I4 )
% 0.33/0.56                 => ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_Some_iff2
% 0.33/0.56  thf(fact_1678_find__Some__iff2,axiom,
% 0.33/0.56      ! [X: $o,P: $o > $o,Xs: list_o] :
% 0.33/0.56        ( ( ( some_o @ X )
% 0.33/0.56          = ( find_o @ P @ Xs ) )
% 0.33/0.56        = ( ? [I4: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 0.33/0.56              & ( P @ ( nth_o @ Xs @ I4 ) )
% 0.33/0.56              & ( X
% 0.33/0.56                = ( nth_o @ Xs @ I4 ) )
% 0.33/0.56              & ! [J3: nat] :
% 0.33/0.56                  ( ( ord_less_nat @ J3 @ I4 )
% 0.33/0.56                 => ~ ( P @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_Some_iff2
% 0.33/0.56  thf(fact_1679_find__Some__iff2,axiom,
% 0.33/0.56      ! [X: nat,P: nat > $o,Xs: list_nat] :
% 0.33/0.56        ( ( ( some_nat @ X )
% 0.33/0.56          = ( find_nat @ P @ Xs ) )
% 0.33/0.56        = ( ? [I4: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 0.33/0.56              & ( P @ ( nth_nat @ Xs @ I4 ) )
% 0.33/0.56              & ( X
% 0.33/0.56                = ( nth_nat @ Xs @ I4 ) )
% 0.33/0.56              & ! [J3: nat] :
% 0.33/0.56                  ( ( ord_less_nat @ J3 @ I4 )
% 0.33/0.56                 => ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_Some_iff2
% 0.33/0.56  thf(fact_1680_find__Some__iff2,axiom,
% 0.33/0.56      ! [X: int,P: int > $o,Xs: list_int] :
% 0.33/0.56        ( ( ( some_int @ X )
% 0.33/0.56          = ( find_int @ P @ Xs ) )
% 0.33/0.56        = ( ? [I4: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 0.33/0.56              & ( P @ ( nth_int @ Xs @ I4 ) )
% 0.33/0.56              & ( X
% 0.33/0.56                = ( nth_int @ Xs @ I4 ) )
% 0.33/0.56              & ! [J3: nat] :
% 0.33/0.56                  ( ( ord_less_nat @ J3 @ I4 )
% 0.33/0.56                 => ~ ( P @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_Some_iff2
% 0.33/0.56  thf(fact_1681_find__Some__iff,axiom,
% 0.33/0.56      ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
% 0.33/0.56        ( ( ( find_P8199882355184865565at_nat @ P @ Xs )
% 0.33/0.56          = ( some_P7363390416028606310at_nat @ X ) )
% 0.33/0.56        = ( ? [I4: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 0.33/0.56              & ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
% 0.33/0.56              & ( X
% 0.33/0.56                = ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
% 0.33/0.56              & ! [J3: nat] :
% 0.33/0.56                  ( ( ord_less_nat @ J3 @ I4 )
% 0.33/0.56                 => ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_Some_iff
% 0.33/0.56  thf(fact_1682_find__Some__iff,axiom,
% 0.33/0.56      ! [P: num > $o,Xs: list_num,X: num] :
% 0.33/0.56        ( ( ( find_num @ P @ Xs )
% 0.33/0.56          = ( some_num @ X ) )
% 0.33/0.56        = ( ? [I4: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I4 @ ( size_size_list_num @ Xs ) )
% 0.33/0.56              & ( P @ ( nth_num @ Xs @ I4 ) )
% 0.33/0.56              & ( X
% 0.33/0.56                = ( nth_num @ Xs @ I4 ) )
% 0.33/0.56              & ! [J3: nat] :
% 0.33/0.56                  ( ( ord_less_nat @ J3 @ I4 )
% 0.33/0.56                 => ~ ( P @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_Some_iff
% 0.33/0.56  thf(fact_1683_find__Some__iff,axiom,
% 0.33/0.56      ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 0.33/0.56        ( ( ( find_VEBT_VEBT @ P @ Xs )
% 0.33/0.56          = ( some_VEBT_VEBT @ X ) )
% 0.33/0.56        = ( ? [I4: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.56              & ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) )
% 0.33/0.56              & ( X
% 0.33/0.56                = ( nth_VEBT_VEBT @ Xs @ I4 ) )
% 0.33/0.56              & ! [J3: nat] :
% 0.33/0.56                  ( ( ord_less_nat @ J3 @ I4 )
% 0.33/0.56                 => ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_Some_iff
% 0.33/0.56  thf(fact_1684_find__Some__iff,axiom,
% 0.33/0.56      ! [P: $o > $o,Xs: list_o,X: $o] :
% 0.33/0.56        ( ( ( find_o @ P @ Xs )
% 0.33/0.56          = ( some_o @ X ) )
% 0.33/0.56        = ( ? [I4: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 0.33/0.56              & ( P @ ( nth_o @ Xs @ I4 ) )
% 0.33/0.56              & ( X
% 0.33/0.56                = ( nth_o @ Xs @ I4 ) )
% 0.33/0.56              & ! [J3: nat] :
% 0.33/0.56                  ( ( ord_less_nat @ J3 @ I4 )
% 0.33/0.56                 => ~ ( P @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_Some_iff
% 0.33/0.56  thf(fact_1685_find__Some__iff,axiom,
% 0.33/0.56      ! [P: nat > $o,Xs: list_nat,X: nat] :
% 0.33/0.56        ( ( ( find_nat @ P @ Xs )
% 0.33/0.56          = ( some_nat @ X ) )
% 0.33/0.56        = ( ? [I4: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 0.33/0.56              & ( P @ ( nth_nat @ Xs @ I4 ) )
% 0.33/0.56              & ( X
% 0.33/0.56                = ( nth_nat @ Xs @ I4 ) )
% 0.33/0.56              & ! [J3: nat] :
% 0.33/0.56                  ( ( ord_less_nat @ J3 @ I4 )
% 0.33/0.56                 => ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_Some_iff
% 0.33/0.56  thf(fact_1686_find__Some__iff,axiom,
% 0.33/0.56      ! [P: int > $o,Xs: list_int,X: int] :
% 0.33/0.56        ( ( ( find_int @ P @ Xs )
% 0.33/0.56          = ( some_int @ X ) )
% 0.33/0.56        = ( ? [I4: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 0.33/0.56              & ( P @ ( nth_int @ Xs @ I4 ) )
% 0.33/0.56              & ( X
% 0.33/0.56                = ( nth_int @ Xs @ I4 ) )
% 0.33/0.56              & ! [J3: nat] :
% 0.33/0.56                  ( ( ord_less_nat @ J3 @ I4 )
% 0.33/0.56                 => ~ ( P @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_Some_iff
% 0.33/0.56  thf(fact_1687_count__notin,axiom,
% 0.33/0.56      ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
% 0.33/0.56        ( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.33/0.56       => ( ( count_4203492906077236349at_nat @ Xs @ X )
% 0.33/0.56          = zero_zero_nat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % count_notin
% 0.33/0.56  thf(fact_1688_count__notin,axiom,
% 0.33/0.56      ! [X: real,Xs: list_real] :
% 0.33/0.56        ( ~ ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.33/0.56       => ( ( count_list_real @ Xs @ X )
% 0.33/0.56          = zero_zero_nat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % count_notin
% 0.33/0.56  thf(fact_1689_count__notin,axiom,
% 0.33/0.56      ! [X: set_nat,Xs: list_set_nat] :
% 0.33/0.56        ( ~ ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 0.33/0.56       => ( ( count_list_set_nat @ Xs @ X )
% 0.33/0.56          = zero_zero_nat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % count_notin
% 0.33/0.56  thf(fact_1690_count__notin,axiom,
% 0.33/0.56      ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.33/0.56        ( ~ ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.56       => ( ( count_list_VEBT_VEBT @ Xs @ X )
% 0.33/0.56          = zero_zero_nat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % count_notin
% 0.33/0.56  thf(fact_1691_count__notin,axiom,
% 0.33/0.56      ! [X: int,Xs: list_int] :
% 0.33/0.56        ( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.33/0.56       => ( ( count_list_int @ Xs @ X )
% 0.33/0.56          = zero_zero_nat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % count_notin
% 0.33/0.56  thf(fact_1692_count__notin,axiom,
% 0.33/0.56      ! [X: nat,Xs: list_nat] :
% 0.33/0.56        ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.33/0.56       => ( ( count_list_nat @ Xs @ X )
% 0.33/0.56          = zero_zero_nat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % count_notin
% 0.33/0.56  thf(fact_1693__C4_Ohyps_C_I9_J,axiom,
% 0.33/0.56      ( ( mi != ma )
% 0.33/0.56     => ! [I3: nat] :
% 0.33/0.56          ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 0.33/0.56         => ( ( ( ( vEBT_VEBT_high @ ma @ na )
% 0.33/0.56                = I3 )
% 0.33/0.56             => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I3 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
% 0.33/0.56            & ! [X2: nat] :
% 0.33/0.56                ( ( ( ( vEBT_VEBT_high @ X2 @ na )
% 0.33/0.56                    = I3 )
% 0.33/0.56                  & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I3 ) @ ( vEBT_VEBT_low @ X2 @ na ) ) )
% 0.33/0.56               => ( ( ord_less_nat @ mi @ X2 )
% 0.33/0.56                  & ( ord_less_eq_nat @ X2 @ ma ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % "4.hyps"(9)
% 0.33/0.56  thf(fact_1694_listrel1__iff__update,axiom,
% 0.33/0.56      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
% 0.33/0.56        ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Xs @ Ys ) @ ( listrel1_VEBT_VEBT @ R2 ) )
% 0.33/0.56        = ( ? [Y2: vEBT_VEBT,N2: nat] :
% 0.33/0.56              ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ N2 ) @ Y2 ) @ R2 )
% 0.33/0.56              & ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.33/0.56              & ( Ys
% 0.33/0.56                = ( list_u1324408373059187874T_VEBT @ Xs @ N2 @ Y2 ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % listrel1_iff_update
% 0.33/0.56  thf(fact_1695_listrel1__iff__update,axiom,
% 0.33/0.56      ! [Xs: list_o,Ys: list_o,R2: set_Product_prod_o_o] :
% 0.33/0.56        ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( listrel1_o @ R2 ) )
% 0.33/0.56        = ( ? [Y2: $o,N2: nat] :
% 0.33/0.56              ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( nth_o @ Xs @ N2 ) @ Y2 ) @ R2 )
% 0.33/0.56              & ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
% 0.33/0.56              & ( Ys
% 0.33/0.56                = ( list_update_o @ Xs @ N2 @ Y2 ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % listrel1_iff_update
% 0.33/0.56  thf(fact_1696_listrel1__iff__update,axiom,
% 0.33/0.56      ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
% 0.33/0.56        ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
% 0.33/0.56        = ( ? [Y2: nat,N2: nat] :
% 0.33/0.56              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ Y2 ) @ R2 )
% 0.33/0.56              & ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
% 0.33/0.56              & ( Ys
% 0.33/0.56                = ( list_update_nat @ Xs @ N2 @ Y2 ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % listrel1_iff_update
% 0.33/0.56  thf(fact_1697_listrel1__iff__update,axiom,
% 0.33/0.56      ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
% 0.33/0.56        ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
% 0.33/0.56        = ( ? [Y2: int,N2: nat] :
% 0.33/0.56              ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N2 ) @ Y2 ) @ R2 )
% 0.33/0.56              & ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
% 0.33/0.56              & ( Ys
% 0.33/0.56                = ( list_update_int @ Xs @ N2 @ Y2 ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % listrel1_iff_update
% 0.33/0.56  thf(fact_1698_vebt__pred_Osimps_I6_J,axiom,
% 0.33/0.56      ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
% 0.33/0.56        ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
% 0.33/0.56        = none_nat ) ).
% 0.33/0.56  
% 0.33/0.56  % vebt_pred.simps(6)
% 0.33/0.56  thf(fact_1699_vebt__succ_Osimps_I5_J,axiom,
% 0.33/0.56      ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
% 0.33/0.56        ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
% 0.33/0.56        = none_nat ) ).
% 0.33/0.56  
% 0.33/0.56  % vebt_succ.simps(5)
% 0.33/0.56  thf(fact_1700_pair__lessI2,axiom,
% 0.33/0.56      ! [A: nat,B: nat,S: nat,T: nat] :
% 0.33/0.56        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.56       => ( ( ord_less_nat @ S @ T )
% 0.33/0.56         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % pair_lessI2
% 0.33/0.56  thf(fact_1701_even__odd__cases,axiom,
% 0.33/0.56      ! [X: nat] :
% 0.33/0.56        ( ! [N3: nat] :
% 0.33/0.56            ( X
% 0.33/0.56           != ( plus_plus_nat @ N3 @ N3 ) )
% 0.33/0.56       => ~ ! [N3: nat] :
% 0.33/0.56              ( X
% 0.33/0.56             != ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % even_odd_cases
% 0.33/0.56  thf(fact_1702_power__shift,axiom,
% 0.33/0.56      ! [X: nat,Y: nat,Z2: nat] :
% 0.33/0.56        ( ( ( power_power_nat @ X @ Y )
% 0.33/0.56          = Z2 )
% 0.33/0.56        = ( ( vEBT_VEBT_power @ ( some_nat @ X ) @ ( some_nat @ Y ) )
% 0.33/0.56          = ( some_nat @ Z2 ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_shift
% 0.33/0.56  thf(fact_1703__C4_Ohyps_C_I8_J,axiom,
% 0.33/0.56      ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 0.33/0.56  
% 0.33/0.56  % "4.hyps"(8)
% 0.33/0.56  thf(fact_1704__C4_Ohyps_C_I2_J,axiom,
% 0.33/0.56      ( ( size_s6755466524823107622T_VEBT @ treeList )
% 0.33/0.56      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% 0.33/0.56  
% 0.33/0.56  % "4.hyps"(2)
% 0.33/0.56  thf(fact_1705_old_Onat_Oinject,axiom,
% 0.33/0.56      ! [Nat: nat,Nat2: nat] :
% 0.33/0.56        ( ( ( suc @ Nat )
% 0.33/0.56          = ( suc @ Nat2 ) )
% 0.33/0.56        = ( Nat = Nat2 ) ) ).
% 0.33/0.56  
% 0.33/0.56  % old.nat.inject
% 0.33/0.56  thf(fact_1706_nat_Oinject,axiom,
% 0.33/0.56      ! [X22: nat,Y22: nat] :
% 0.33/0.56        ( ( ( suc @ X22 )
% 0.33/0.56          = ( suc @ Y22 ) )
% 0.33/0.56        = ( X22 = Y22 ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nat.inject
% 0.33/0.56  thf(fact_1707_local_Opower__def,axiom,
% 0.33/0.56      ( vEBT_VEBT_power
% 0.33/0.56      = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % local.power_def
% 0.33/0.56  thf(fact_1708_dp,axiom,
% 0.33/0.56      ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).
% 0.33/0.56  
% 0.33/0.56  % dp
% 0.33/0.56  thf(fact_1709__C4_Ohyps_C_I5_J,axiom,
% 0.33/0.56      ! [I3: nat] :
% 0.33/0.56        ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 0.33/0.56       => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I3 ) @ X6 ) )
% 0.33/0.56          = ( vEBT_V8194947554948674370ptions @ summary @ I3 ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % "4.hyps"(5)
% 0.33/0.56  thf(fact_1710_high__bound__aux,axiom,
% 0.33/0.56      ! [Ma: nat,N: nat,M: nat] :
% 0.33/0.56        ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.33/0.56       => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % high_bound_aux
% 0.33/0.56  thf(fact_1711_member__bound,axiom,
% 0.33/0.56      ! [Tree: vEBT_VEBT,X: nat,N: nat] :
% 0.33/0.56        ( ( vEBT_vebt_member @ Tree @ X )
% 0.33/0.56       => ( ( vEBT_invar_vebt @ Tree @ N )
% 0.33/0.56         => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % member_bound
% 0.33/0.56  thf(fact_1712_numeral__le__iff,axiom,
% 0.33/0.56      ! [M: num,N: num] :
% 0.33/0.56        ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.33/0.56        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % numeral_le_iff
% 0.33/0.56  thf(fact_1713_numeral__le__iff,axiom,
% 0.33/0.56      ! [M: num,N: num] :
% 0.33/0.56        ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.33/0.56        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % numeral_le_iff
% 0.33/0.56  thf(fact_1714_numeral__le__iff,axiom,
% 0.33/0.56      ! [M: num,N: num] :
% 0.33/0.56        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.33/0.56        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % numeral_le_iff
% 0.33/0.56  thf(fact_1715_numeral__le__iff,axiom,
% 0.33/0.56      ! [M: num,N: num] :
% 0.33/0.56        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.33/0.56        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % numeral_le_iff
% 0.33/0.56  thf(fact_1716_numeral__le__iff,axiom,
% 0.33/0.56      ! [M: num,N: num] :
% 0.33/0.56        ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.56        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % numeral_le_iff
% 0.33/0.56  thf(fact_1717_numeral__less__iff,axiom,
% 0.33/0.56      ! [M: num,N: num] :
% 0.33/0.56        ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.33/0.56        = ( ord_less_num @ M @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % numeral_less_iff
% 0.33/0.56  thf(fact_1718_numeral__less__iff,axiom,
% 0.33/0.56      ! [M: num,N: num] :
% 0.33/0.56        ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.33/0.56        = ( ord_less_num @ M @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % numeral_less_iff
% 0.33/0.56  thf(fact_1719_numeral__less__iff,axiom,
% 0.33/0.56      ! [M: num,N: num] :
% 0.33/0.56        ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.33/0.56        = ( ord_less_num @ M @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % numeral_less_iff
% 0.33/0.56  thf(fact_1720_numeral__less__iff,axiom,
% 0.33/0.56      ! [M: num,N: num] :
% 0.33/0.56        ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.33/0.56        = ( ord_less_num @ M @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % numeral_less_iff
% 0.33/0.56  thf(fact_1721_numeral__less__iff,axiom,
% 0.33/0.56      ! [M: num,N: num] :
% 0.33/0.56        ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.33/0.56        = ( ord_less_num @ M @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % numeral_less_iff
% 0.33/0.56  thf(fact_1722_lessI,axiom,
% 0.33/0.56      ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lessI
% 0.33/0.56  thf(fact_1723_Suc__mono,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ M @ N )
% 0.33/0.56       => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % Suc_mono
% 0.33/0.56  thf(fact_1724_Suc__less__eq,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.33/0.56        = ( ord_less_nat @ M @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % Suc_less_eq
% 0.33/0.56  thf(fact_1725_Suc__le__mono,axiom,
% 0.33/0.56      ! [N: nat,M: nat] :
% 0.33/0.56        ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.33/0.56        = ( ord_less_eq_nat @ N @ M ) ) ).
% 0.33/0.56  
% 0.33/0.56  % Suc_le_mono
% 0.33/0.56  thf(fact_1726_misiz,axiom,
% 0.33/0.56      ! [T: vEBT_VEBT,N: nat,M: nat] :
% 0.33/0.56        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.56       => ( ( ( some_nat @ M )
% 0.33/0.56            = ( vEBT_vebt_mint @ T ) )
% 0.33/0.56         => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % misiz
% 0.33/0.56  thf(fact_1727_add__Suc__right,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 0.33/0.56        = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % add_Suc_right
% 0.33/0.56  thf(fact_1728_helpyd,axiom,
% 0.33/0.56      ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
% 0.33/0.56        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.56       => ( ( ( vEBT_vebt_succ @ T @ X )
% 0.33/0.56            = ( some_nat @ Y ) )
% 0.33/0.56         => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % helpyd
% 0.33/0.56  thf(fact_1729_helpypredd,axiom,
% 0.33/0.56      ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
% 0.33/0.56        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.56       => ( ( ( vEBT_vebt_pred @ T @ X )
% 0.33/0.56            = ( some_nat @ Y ) )
% 0.33/0.56         => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % helpypredd
% 0.33/0.56  thf(fact_1730__092_060open_062x_A_060_A2_A_094_Adeg_092_060close_062,axiom,
% 0.33/0.56      ord_less_nat @ xa @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 0.33/0.56  
% 0.33/0.56  % \<open>x < 2 ^ deg\<close>
% 0.33/0.56  thf(fact_1731__092_060open_062ma_A_092_060le_062_A2_A_094_Adeg_092_060close_062,axiom,
% 0.33/0.56      ord_less_eq_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 0.33/0.56  
% 0.33/0.56  % \<open>ma \<le> 2 ^ deg\<close>
% 0.33/0.56  thf(fact_1732_delt__out__of__range,axiom,
% 0.33/0.56      ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.33/0.56        ( ( ( ord_less_nat @ X @ Mi )
% 0.33/0.56          | ( ord_less_nat @ Ma @ X ) )
% 0.33/0.56       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.33/0.56         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.33/0.56            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % delt_out_of_range
% 0.33/0.56  thf(fact_1733_del__single__cont,axiom,
% 0.33/0.56      ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.33/0.56        ( ( ( X = Mi )
% 0.33/0.56          & ( X = Ma ) )
% 0.33/0.56       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.33/0.56         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.33/0.56            = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % del_single_cont
% 0.33/0.56  thf(fact_1734_mi__ma__2__deg,axiom,
% 0.33/0.56      ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.33/0.56        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 0.33/0.56       => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.33/0.56          & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % mi_ma_2_deg
% 0.33/0.56  thf(fact_1735_pred__max,axiom,
% 0.33/0.56      ! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.33/0.56        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.33/0.56       => ( ( ord_less_nat @ Ma @ X )
% 0.33/0.56         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.33/0.56            = ( some_nat @ Ma ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % pred_max
% 0.33/0.56  thf(fact_1736_succ__min,axiom,
% 0.33/0.56      ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.33/0.56        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.33/0.56       => ( ( ord_less_nat @ X @ Mi )
% 0.33/0.56         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.33/0.56            = ( some_nat @ Mi ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % succ_min
% 0.33/0.56  thf(fact_1737_less__Suc0,axiom,
% 0.33/0.56      ! [N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.33/0.56        = ( N = zero_zero_nat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % less_Suc0
% 0.33/0.56  thf(fact_1738_zero__less__Suc,axiom,
% 0.33/0.56      ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_less_Suc
% 0.33/0.56  thf(fact_1739_nothprolist,axiom,
% 0.33/0.56      ! [I: nat] :
% 0.33/0.56        ( ( ( I
% 0.33/0.56           != ( vEBT_VEBT_high @ xa @ na ) )
% 0.33/0.56          & ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) )
% 0.33/0.56       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ I )
% 0.33/0.56          = ( nth_VEBT_VEBT @ treeList @ I ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nothprolist
% 0.33/0.56  thf(fact_1740_pair__less__iff1,axiom,
% 0.33/0.56      ! [X: nat,Y: nat,Z2: nat] :
% 0.33/0.56        ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z2 ) ) @ fun_pair_less )
% 0.33/0.56        = ( ord_less_nat @ Y @ Z2 ) ) ).
% 0.33/0.56  
% 0.33/0.56  % pair_less_iff1
% 0.33/0.56  thf(fact_1741_n__not__Suc__n,axiom,
% 0.33/0.56      ! [N: nat] :
% 0.33/0.56        ( N
% 0.33/0.56       != ( suc @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % n_not_Suc_n
% 0.33/0.56  thf(fact_1742_Suc__inject,axiom,
% 0.33/0.56      ! [X: nat,Y: nat] :
% 0.33/0.56        ( ( ( suc @ X )
% 0.33/0.56          = ( suc @ Y ) )
% 0.33/0.56       => ( X = Y ) ) ).
% 0.33/0.56  
% 0.33/0.56  % Suc_inject
% 0.33/0.56  thf(fact_1743_less__2__cases,axiom,
% 0.33/0.56      ! [N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.56       => ( ( N = zero_zero_nat )
% 0.33/0.56          | ( N
% 0.33/0.56            = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % less_2_cases
% 0.33/0.56  thf(fact_1744_less__2__cases__iff,axiom,
% 0.33/0.56      ! [N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.56        = ( ( N = zero_zero_nat )
% 0.33/0.56          | ( N
% 0.33/0.56            = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % less_2_cases_iff
% 0.33/0.56  thf(fact_1745_numeral__1__eq__Suc__0,axiom,
% 0.33/0.56      ( ( numeral_numeral_nat @ one )
% 0.33/0.56      = ( suc @ zero_zero_nat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % numeral_1_eq_Suc_0
% 0.33/0.56  thf(fact_1746_numeral__2__eq__2,axiom,
% 0.33/0.56      ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 0.33/0.56      = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % numeral_2_eq_2
% 0.33/0.56  thf(fact_1747_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
% 0.33/0.56      ! [X: nat,N: nat,M: nat] :
% 0.33/0.56        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.33/0.56       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.56         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.56           => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % VEBT_internal.exp_split_high_low(1)
% 0.33/0.56  thf(fact_1748_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
% 0.33/0.56      ! [X: nat,N: nat,M: nat] :
% 0.33/0.56        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.33/0.56       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.56         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.56           => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % VEBT_internal.exp_split_high_low(2)
% 0.33/0.56  thf(fact_1749_num_Osize_I5_J,axiom,
% 0.33/0.56      ! [X22: num] :
% 0.33/0.56        ( ( size_size_num @ ( bit0 @ X22 ) )
% 0.33/0.56        = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % num.size(5)
% 0.33/0.56  thf(fact_1750_zero__neq__numeral,axiom,
% 0.33/0.56      ! [N: num] :
% 0.33/0.56        ( zero_zero_complex
% 0.33/0.56       != ( numera6690914467698888265omplex @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_neq_numeral
% 0.33/0.56  thf(fact_1751_zero__neq__numeral,axiom,
% 0.33/0.56      ! [N: num] :
% 0.33/0.56        ( zero_z5237406670263579293d_enat
% 0.33/0.56       != ( numera1916890842035813515d_enat @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_neq_numeral
% 0.33/0.56  thf(fact_1752_zero__neq__numeral,axiom,
% 0.33/0.56      ! [N: num] :
% 0.33/0.56        ( zero_zero_real
% 0.33/0.56       != ( numeral_numeral_real @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_neq_numeral
% 0.33/0.56  thf(fact_1753_zero__neq__numeral,axiom,
% 0.33/0.56      ! [N: num] :
% 0.33/0.56        ( zero_zero_rat
% 0.33/0.56       != ( numeral_numeral_rat @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_neq_numeral
% 0.33/0.56  thf(fact_1754_zero__neq__numeral,axiom,
% 0.33/0.56      ! [N: num] :
% 0.33/0.56        ( zero_zero_nat
% 0.33/0.56       != ( numeral_numeral_nat @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_neq_numeral
% 0.33/0.56  thf(fact_1755_zero__neq__numeral,axiom,
% 0.33/0.56      ! [N: num] :
% 0.33/0.56        ( zero_zero_int
% 0.33/0.56       != ( numeral_numeral_int @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_neq_numeral
% 0.33/0.56  thf(fact_1756_vebt__buildup_Ocases,axiom,
% 0.33/0.56      ! [X: nat] :
% 0.33/0.56        ( ( X != zero_zero_nat )
% 0.33/0.56       => ( ( X
% 0.33/0.56           != ( suc @ zero_zero_nat ) )
% 0.33/0.56         => ~ ! [Va2: nat] :
% 0.33/0.56                ( X
% 0.33/0.56               != ( suc @ ( suc @ Va2 ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % vebt_buildup.cases
% 0.33/0.56  thf(fact_1757_nat_Odistinct_I1_J,axiom,
% 0.33/0.56      ! [X22: nat] :
% 0.33/0.56        ( zero_zero_nat
% 0.33/0.56       != ( suc @ X22 ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nat.distinct(1)
% 0.33/0.56  thf(fact_1758_old_Onat_Odistinct_I2_J,axiom,
% 0.33/0.56      ! [Nat2: nat] :
% 0.33/0.56        ( ( suc @ Nat2 )
% 0.33/0.56       != zero_zero_nat ) ).
% 0.33/0.56  
% 0.33/0.56  % old.nat.distinct(2)
% 0.33/0.56  thf(fact_1759_old_Onat_Odistinct_I1_J,axiom,
% 0.33/0.56      ! [Nat2: nat] :
% 0.33/0.56        ( zero_zero_nat
% 0.33/0.56       != ( suc @ Nat2 ) ) ).
% 0.33/0.56  
% 0.33/0.56  % old.nat.distinct(1)
% 0.33/0.56  thf(fact_1760_nat_OdiscI,axiom,
% 0.33/0.56      ! [Nat: nat,X22: nat] :
% 0.33/0.56        ( ( Nat
% 0.33/0.56          = ( suc @ X22 ) )
% 0.33/0.56       => ( Nat != zero_zero_nat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nat.discI
% 0.33/0.56  thf(fact_1761_old_Onat_Oexhaust,axiom,
% 0.33/0.56      ! [Y: nat] :
% 0.33/0.56        ( ( Y != zero_zero_nat )
% 0.33/0.56       => ~ ! [Nat3: nat] :
% 0.33/0.56              ( Y
% 0.33/0.56             != ( suc @ Nat3 ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % old.nat.exhaust
% 0.33/0.56  thf(fact_1762_nat__induct,axiom,
% 0.33/0.56      ! [P: nat > $o,N: nat] :
% 0.33/0.56        ( ( P @ zero_zero_nat )
% 0.33/0.56       => ( ! [N3: nat] :
% 0.33/0.56              ( ( P @ N3 )
% 0.33/0.56             => ( P @ ( suc @ N3 ) ) )
% 0.33/0.56         => ( P @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nat_induct
% 0.33/0.56  thf(fact_1763_diff__induct,axiom,
% 0.33/0.56      ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.33/0.56        ( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
% 0.33/0.56       => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
% 0.33/0.56         => ( ! [X4: nat,Y3: nat] :
% 0.33/0.56                ( ( P @ X4 @ Y3 )
% 0.33/0.56               => ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
% 0.33/0.56           => ( P @ M @ N ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % diff_induct
% 0.33/0.56  thf(fact_1764_zero__induct,axiom,
% 0.33/0.56      ! [P: nat > $o,K: nat] :
% 0.33/0.56        ( ( P @ K )
% 0.33/0.56       => ( ! [N3: nat] :
% 0.33/0.56              ( ( P @ ( suc @ N3 ) )
% 0.33/0.56             => ( P @ N3 ) )
% 0.33/0.56         => ( P @ zero_zero_nat ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_induct
% 0.33/0.56  thf(fact_1765_Suc__neq__Zero,axiom,
% 0.33/0.56      ! [M: nat] :
% 0.33/0.56        ( ( suc @ M )
% 0.33/0.56       != zero_zero_nat ) ).
% 0.33/0.56  
% 0.33/0.56  % Suc_neq_Zero
% 0.33/0.56  thf(fact_1766_Zero__neq__Suc,axiom,
% 0.33/0.56      ! [M: nat] :
% 0.33/0.56        ( zero_zero_nat
% 0.33/0.56       != ( suc @ M ) ) ).
% 0.33/0.56  
% 0.33/0.56  % Zero_neq_Suc
% 0.33/0.56  thf(fact_1767_Zero__not__Suc,axiom,
% 0.33/0.56      ! [M: nat] :
% 0.33/0.56        ( zero_zero_nat
% 0.33/0.56       != ( suc @ M ) ) ).
% 0.33/0.56  
% 0.33/0.56  % Zero_not_Suc
% 0.33/0.56  thf(fact_1768_not0__implies__Suc,axiom,
% 0.33/0.56      ! [N: nat] :
% 0.33/0.56        ( ( N != zero_zero_nat )
% 0.33/0.56       => ? [M5: nat] :
% 0.33/0.56            ( N
% 0.33/0.56            = ( suc @ M5 ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % not0_implies_Suc
% 0.33/0.56  thf(fact_1769_Nat_OlessE,axiom,
% 0.33/0.56      ! [I: nat,K: nat] :
% 0.33/0.56        ( ( ord_less_nat @ I @ K )
% 0.33/0.56       => ( ( K
% 0.33/0.56           != ( suc @ I ) )
% 0.33/0.56         => ~ ! [J2: nat] :
% 0.33/0.56                ( ( ord_less_nat @ I @ J2 )
% 0.33/0.56               => ( K
% 0.33/0.56                 != ( suc @ J2 ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % Nat.lessE
% 0.33/0.56  thf(fact_1770_Suc__lessD,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ ( suc @ M ) @ N )
% 0.33/0.56       => ( ord_less_nat @ M @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % Suc_lessD
% 0.33/0.56  thf(fact_1771_Suc__lessE,axiom,
% 0.33/0.56      ! [I: nat,K: nat] :
% 0.33/0.56        ( ( ord_less_nat @ ( suc @ I ) @ K )
% 0.33/0.56       => ~ ! [J2: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I @ J2 )
% 0.33/0.56             => ( K
% 0.33/0.56               != ( suc @ J2 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % Suc_lessE
% 0.33/0.56  thf(fact_1772_Suc__lessI,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ M @ N )
% 0.33/0.56       => ( ( ( suc @ M )
% 0.33/0.56           != N )
% 0.33/0.56         => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % Suc_lessI
% 0.33/0.56  thf(fact_1773_less__SucE,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.33/0.56       => ( ~ ( ord_less_nat @ M @ N )
% 0.33/0.56         => ( M = N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % less_SucE
% 0.33/0.56  thf(fact_1774_less__SucI,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ M @ N )
% 0.33/0.56       => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % less_SucI
% 0.33/0.56  thf(fact_1775_Ex__less__Suc,axiom,
% 0.33/0.56      ! [N: nat,P: nat > $o] :
% 0.33/0.56        ( ( ? [I4: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.33/0.56              & ( P @ I4 ) ) )
% 0.33/0.56        = ( ( P @ N )
% 0.33/0.56          | ? [I4: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I4 @ N )
% 0.33/0.56              & ( P @ I4 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % Ex_less_Suc
% 0.33/0.56  thf(fact_1776_less__Suc__eq,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.33/0.56        = ( ( ord_less_nat @ M @ N )
% 0.33/0.56          | ( M = N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % less_Suc_eq
% 0.33/0.56  thf(fact_1777_not__less__eq,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ~ ( ord_less_nat @ M @ N ) )
% 0.33/0.56        = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % not_less_eq
% 0.33/0.56  thf(fact_1778_All__less__Suc,axiom,
% 0.33/0.56      ! [N: nat,P: nat > $o] :
% 0.33/0.56        ( ( ! [I4: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.33/0.56             => ( P @ I4 ) ) )
% 0.33/0.56        = ( ( P @ N )
% 0.33/0.56          & ! [I4: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I4 @ N )
% 0.33/0.56             => ( P @ I4 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % All_less_Suc
% 0.33/0.56  thf(fact_1779_Suc__less__eq2,axiom,
% 0.33/0.56      ! [N: nat,M: nat] :
% 0.33/0.56        ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.33/0.56        = ( ? [M6: nat] :
% 0.33/0.56              ( ( M
% 0.33/0.56                = ( suc @ M6 ) )
% 0.33/0.56              & ( ord_less_nat @ N @ M6 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % Suc_less_eq2
% 0.33/0.56  thf(fact_1780_less__antisym,axiom,
% 0.33/0.56      ! [N: nat,M: nat] :
% 0.33/0.56        ( ~ ( ord_less_nat @ N @ M )
% 0.33/0.56       => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.33/0.56         => ( M = N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % less_antisym
% 0.33/0.56  thf(fact_1781_Suc__less__SucD,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.33/0.56       => ( ord_less_nat @ M @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % Suc_less_SucD
% 0.33/0.56  thf(fact_1782_less__trans__Suc,axiom,
% 0.33/0.56      ! [I: nat,J: nat,K: nat] :
% 0.33/0.56        ( ( ord_less_nat @ I @ J )
% 0.33/0.56       => ( ( ord_less_nat @ J @ K )
% 0.33/0.56         => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % less_trans_Suc
% 0.33/0.56  thf(fact_1783_less__Suc__induct,axiom,
% 0.33/0.56      ! [I: nat,J: nat,P: nat > nat > $o] :
% 0.33/0.56        ( ( ord_less_nat @ I @ J )
% 0.33/0.56       => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
% 0.33/0.56         => ( ! [I2: nat,J2: nat,K2: nat] :
% 0.33/0.56                ( ( ord_less_nat @ I2 @ J2 )
% 0.33/0.56               => ( ( ord_less_nat @ J2 @ K2 )
% 0.33/0.56                 => ( ( P @ I2 @ J2 )
% 0.33/0.56                   => ( ( P @ J2 @ K2 )
% 0.33/0.56                     => ( P @ I2 @ K2 ) ) ) ) )
% 0.33/0.56           => ( P @ I @ J ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % less_Suc_induct
% 0.33/0.56  thf(fact_1784_strict__inc__induct,axiom,
% 0.33/0.56      ! [I: nat,J: nat,P: nat > $o] :
% 0.33/0.56        ( ( ord_less_nat @ I @ J )
% 0.33/0.56       => ( ! [I2: nat] :
% 0.33/0.56              ( ( J
% 0.33/0.56                = ( suc @ I2 ) )
% 0.33/0.56             => ( P @ I2 ) )
% 0.33/0.56         => ( ! [I2: nat] :
% 0.33/0.56                ( ( ord_less_nat @ I2 @ J )
% 0.33/0.56               => ( ( P @ ( suc @ I2 ) )
% 0.33/0.56                 => ( P @ I2 ) ) )
% 0.33/0.56           => ( P @ I ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % strict_inc_induct
% 0.33/0.56  thf(fact_1785_not__less__less__Suc__eq,axiom,
% 0.33/0.56      ! [N: nat,M: nat] :
% 0.33/0.56        ( ~ ( ord_less_nat @ N @ M )
% 0.33/0.56       => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.33/0.56          = ( N = M ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % not_less_less_Suc_eq
% 0.33/0.56  thf(fact_1786_Suc__leD,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.33/0.56       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % Suc_leD
% 0.33/0.56  thf(fact_1787_le__SucE,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.33/0.56       => ( ~ ( ord_less_eq_nat @ M @ N )
% 0.33/0.56         => ( M
% 0.33/0.56            = ( suc @ N ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % le_SucE
% 0.33/0.56  thf(fact_1788_le__SucI,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.56       => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % le_SucI
% 0.33/0.56  thf(fact_1789_Suc__le__D,axiom,
% 0.33/0.56      ! [N: nat,M7: nat] :
% 0.33/0.56        ( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
% 0.33/0.56       => ? [M5: nat] :
% 0.33/0.56            ( M7
% 0.33/0.56            = ( suc @ M5 ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % Suc_le_D
% 0.33/0.56  thf(fact_1790_le__Suc__eq,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.33/0.56        = ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.56          | ( M
% 0.33/0.56            = ( suc @ N ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % le_Suc_eq
% 0.33/0.56  thf(fact_1791_Suc__n__not__le__n,axiom,
% 0.33/0.56      ! [N: nat] :
% 0.33/0.56        ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 0.33/0.56  
% 0.33/0.56  % Suc_n_not_le_n
% 0.33/0.56  thf(fact_1792_not__less__eq__eq,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 0.33/0.56        = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 0.33/0.56  
% 0.33/0.56  % not_less_eq_eq
% 0.33/0.56  thf(fact_1793_full__nat__induct,axiom,
% 0.33/0.56      ! [P: nat > $o,N: nat] :
% 0.33/0.56        ( ! [N3: nat] :
% 0.33/0.56            ( ! [M3: nat] :
% 0.33/0.56                ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
% 0.33/0.56               => ( P @ M3 ) )
% 0.33/0.56           => ( P @ N3 ) )
% 0.33/0.56       => ( P @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % full_nat_induct
% 0.33/0.56  thf(fact_1794_nat__induct__at__least,axiom,
% 0.33/0.56      ! [M: nat,N: nat,P: nat > $o] :
% 0.33/0.56        ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.56       => ( ( P @ M )
% 0.33/0.56         => ( ! [N3: nat] :
% 0.33/0.56                ( ( ord_less_eq_nat @ M @ N3 )
% 0.33/0.56               => ( ( P @ N3 )
% 0.33/0.56                 => ( P @ ( suc @ N3 ) ) ) )
% 0.33/0.56           => ( P @ N ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nat_induct_at_least
% 0.33/0.56  thf(fact_1795_transitive__stepwise__le,axiom,
% 0.33/0.56      ! [M: nat,N: nat,R: nat > nat > $o] :
% 0.33/0.56        ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.56       => ( ! [X4: nat] : ( R @ X4 @ X4 )
% 0.33/0.56         => ( ! [X4: nat,Y3: nat,Z4: nat] :
% 0.33/0.56                ( ( R @ X4 @ Y3 )
% 0.33/0.56               => ( ( R @ Y3 @ Z4 )
% 0.33/0.56                 => ( R @ X4 @ Z4 ) ) )
% 0.33/0.56           => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
% 0.33/0.56             => ( R @ M @ N ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % transitive_stepwise_le
% 0.33/0.56  thf(fact_1796_add__Suc__shift,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.33/0.56        = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % add_Suc_shift
% 0.33/0.56  thf(fact_1797_add__Suc,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.33/0.56        = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % add_Suc
% 0.33/0.56  thf(fact_1798_nat__arith_Osuc1,axiom,
% 0.33/0.56      ! [A2: nat,K: nat,A: nat] :
% 0.33/0.56        ( ( A2
% 0.33/0.56          = ( plus_plus_nat @ K @ A ) )
% 0.33/0.56       => ( ( suc @ A2 )
% 0.33/0.56          = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nat_arith.suc1
% 0.33/0.56  thf(fact_1799_num_Osize_I4_J,axiom,
% 0.33/0.56      ( ( size_size_num @ one )
% 0.33/0.56      = zero_zero_nat ) ).
% 0.33/0.56  
% 0.33/0.56  % num.size(4)
% 0.33/0.56  thf(fact_1800_listrel1__eq__len,axiom,
% 0.33/0.56      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
% 0.33/0.56        ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Xs @ Ys ) @ ( listrel1_VEBT_VEBT @ R2 ) )
% 0.33/0.56       => ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.33/0.56          = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % listrel1_eq_len
% 0.33/0.56  thf(fact_1801_listrel1__eq__len,axiom,
% 0.33/0.56      ! [Xs: list_o,Ys: list_o,R2: set_Product_prod_o_o] :
% 0.33/0.56        ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( listrel1_o @ R2 ) )
% 0.33/0.56       => ( ( size_size_list_o @ Xs )
% 0.33/0.56          = ( size_size_list_o @ Ys ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % listrel1_eq_len
% 0.33/0.56  thf(fact_1802_listrel1__eq__len,axiom,
% 0.33/0.56      ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
% 0.33/0.56        ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
% 0.33/0.56       => ( ( size_size_list_nat @ Xs )
% 0.33/0.56          = ( size_size_list_nat @ Ys ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % listrel1_eq_len
% 0.33/0.56  thf(fact_1803_listrel1__eq__len,axiom,
% 0.33/0.56      ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
% 0.33/0.56        ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
% 0.33/0.56       => ( ( size_size_list_int @ Xs )
% 0.33/0.56          = ( size_size_list_int @ Ys ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % listrel1_eq_len
% 0.33/0.56  thf(fact_1804_invar__vebt_Ointros_I3_J,axiom,
% 0.33/0.56      ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 0.33/0.56        ( ! [X4: vEBT_VEBT] :
% 0.33/0.56            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.33/0.56           => ( vEBT_invar_vebt @ X4 @ N ) )
% 0.33/0.56       => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.33/0.56         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.33/0.56              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.56           => ( ( M
% 0.33/0.56                = ( suc @ N ) )
% 0.33/0.56             => ( ( Deg
% 0.33/0.56                  = ( plus_plus_nat @ N @ M ) )
% 0.33/0.56               => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 0.33/0.56                 => ( ! [X4: vEBT_VEBT] :
% 0.33/0.56                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.33/0.56                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
% 0.33/0.56                   => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % invar_vebt.intros(3)
% 0.33/0.56  thf(fact_1805_invar__vebt_Ointros_I2_J,axiom,
% 0.33/0.56      ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 0.33/0.56        ( ! [X4: vEBT_VEBT] :
% 0.33/0.56            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.33/0.56           => ( vEBT_invar_vebt @ X4 @ N ) )
% 0.33/0.56       => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.33/0.56         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.33/0.56              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.56           => ( ( M = N )
% 0.33/0.56             => ( ( Deg
% 0.33/0.56                  = ( plus_plus_nat @ N @ M ) )
% 0.33/0.56               => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 0.33/0.56                 => ( ! [X4: vEBT_VEBT] :
% 0.33/0.56                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.33/0.56                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
% 0.33/0.56                   => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % invar_vebt.intros(2)
% 0.33/0.56  thf(fact_1806_find__cong,axiom,
% 0.33/0.56      ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 0.33/0.56        ( ( Xs = Ys )
% 0.33/0.56       => ( ! [X4: product_prod_nat_nat] :
% 0.33/0.56              ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Ys ) )
% 0.33/0.56             => ( ( P @ X4 )
% 0.33/0.56                = ( Q @ X4 ) ) )
% 0.33/0.56         => ( ( find_P8199882355184865565at_nat @ P @ Xs )
% 0.33/0.56            = ( find_P8199882355184865565at_nat @ Q @ Ys ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_cong
% 0.33/0.56  thf(fact_1807_find__cong,axiom,
% 0.33/0.56      ! [Xs: list_real,Ys: list_real,P: real > $o,Q: real > $o] :
% 0.33/0.56        ( ( Xs = Ys )
% 0.33/0.56       => ( ! [X4: real] :
% 0.33/0.56              ( ( member_real @ X4 @ ( set_real2 @ Ys ) )
% 0.33/0.56             => ( ( P @ X4 )
% 0.33/0.56                = ( Q @ X4 ) ) )
% 0.33/0.56         => ( ( find_real @ P @ Xs )
% 0.33/0.56            = ( find_real @ Q @ Ys ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_cong
% 0.33/0.56  thf(fact_1808_find__cong,axiom,
% 0.33/0.56      ! [Xs: list_set_nat,Ys: list_set_nat,P: set_nat > $o,Q: set_nat > $o] :
% 0.33/0.56        ( ( Xs = Ys )
% 0.33/0.56       => ( ! [X4: set_nat] :
% 0.33/0.56              ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Ys ) )
% 0.33/0.56             => ( ( P @ X4 )
% 0.33/0.56                = ( Q @ X4 ) ) )
% 0.33/0.56         => ( ( find_set_nat @ P @ Xs )
% 0.33/0.56            = ( find_set_nat @ Q @ Ys ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_cong
% 0.33/0.56  thf(fact_1809_find__cong,axiom,
% 0.33/0.56      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,P: vEBT_VEBT > $o,Q: vEBT_VEBT > $o] :
% 0.33/0.56        ( ( Xs = Ys )
% 0.33/0.56       => ( ! [X4: vEBT_VEBT] :
% 0.33/0.56              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Ys ) )
% 0.33/0.56             => ( ( P @ X4 )
% 0.33/0.56                = ( Q @ X4 ) ) )
% 0.33/0.56         => ( ( find_VEBT_VEBT @ P @ Xs )
% 0.33/0.56            = ( find_VEBT_VEBT @ Q @ Ys ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_cong
% 0.33/0.56  thf(fact_1810_find__cong,axiom,
% 0.33/0.56      ! [Xs: list_int,Ys: list_int,P: int > $o,Q: int > $o] :
% 0.33/0.56        ( ( Xs = Ys )
% 0.33/0.56       => ( ! [X4: int] :
% 0.33/0.56              ( ( member_int @ X4 @ ( set_int2 @ Ys ) )
% 0.33/0.56             => ( ( P @ X4 )
% 0.33/0.56                = ( Q @ X4 ) ) )
% 0.33/0.56         => ( ( find_int @ P @ Xs )
% 0.33/0.56            = ( find_int @ Q @ Ys ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_cong
% 0.33/0.56  thf(fact_1811_find__cong,axiom,
% 0.33/0.56      ! [Xs: list_nat,Ys: list_nat,P: nat > $o,Q: nat > $o] :
% 0.33/0.56        ( ( Xs = Ys )
% 0.33/0.56       => ( ! [X4: nat] :
% 0.33/0.56              ( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
% 0.33/0.56             => ( ( P @ X4 )
% 0.33/0.56                = ( Q @ X4 ) ) )
% 0.33/0.56         => ( ( find_nat @ P @ Xs )
% 0.33/0.56            = ( find_nat @ Q @ Ys ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_cong
% 0.33/0.56  thf(fact_1812_zip__same,axiom,
% 0.33/0.56      ! [A: product_prod_nat_nat,B: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
% 0.33/0.56        ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Xs ) ) )
% 0.33/0.56        = ( ( member8440522571783428010at_nat @ A @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.33/0.56          & ( A = B ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zip_same
% 0.33/0.56  thf(fact_1813_zip__same,axiom,
% 0.33/0.56      ! [A: real,B: real,Xs: list_real] :
% 0.33/0.56        ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ A @ B ) @ ( set_Pr5999470521830281550l_real @ ( zip_real_real @ Xs @ Xs ) ) )
% 0.33/0.56        = ( ( member_real @ A @ ( set_real2 @ Xs ) )
% 0.33/0.56          & ( A = B ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zip_same
% 0.33/0.56  thf(fact_1814_zip__same,axiom,
% 0.33/0.56      ! [A: set_nat,B: set_nat,Xs: list_set_nat] :
% 0.33/0.56        ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A @ B ) @ ( set_Pr9040384385603167362et_nat @ ( zip_set_nat_set_nat @ Xs @ Xs ) ) )
% 0.33/0.56        = ( ( member_set_nat @ A @ ( set_set_nat2 @ Xs ) )
% 0.33/0.56          & ( A = B ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zip_same
% 0.33/0.56  thf(fact_1815_zip__same,axiom,
% 0.33/0.56      ! [A: vEBT_VEBT,B: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.33/0.56        ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ A @ B ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Xs ) ) )
% 0.33/0.56        = ( ( member_VEBT_VEBT @ A @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.56          & ( A = B ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zip_same
% 0.33/0.56  thf(fact_1816_zip__same,axiom,
% 0.33/0.56      ! [A: nat,B: nat,Xs: list_nat] :
% 0.33/0.56        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
% 0.33/0.56        = ( ( member_nat @ A @ ( set_nat2 @ Xs ) )
% 0.33/0.56          & ( A = B ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zip_same
% 0.33/0.56  thf(fact_1817_zip__same,axiom,
% 0.33/0.56      ! [A: int,B: int,Xs: list_int] :
% 0.33/0.56        ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Xs ) ) )
% 0.33/0.56        = ( ( member_int @ A @ ( set_int2 @ Xs ) )
% 0.33/0.56          & ( A = B ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zip_same
% 0.33/0.56  thf(fact_1818_in__set__zipE,axiom,
% 0.33/0.56      ! [X: real,Y: real,Xs: list_real,Ys: list_real] :
% 0.33/0.56        ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X @ Y ) @ ( set_Pr5999470521830281550l_real @ ( zip_real_real @ Xs @ Ys ) ) )
% 0.33/0.56       => ~ ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.33/0.56           => ~ ( member_real @ Y @ ( set_real2 @ Ys ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_zipE
% 0.33/0.56  thf(fact_1819_in__set__zipE,axiom,
% 0.33/0.56      ! [X: real,Y: vEBT_VEBT,Xs: list_real,Ys: list_VEBT_VEBT] :
% 0.33/0.56        ( ( member7262085504369356948T_VEBT @ ( produc6931449550656315951T_VEBT @ X @ Y ) @ ( set_Pr8897343066327330088T_VEBT @ ( zip_real_VEBT_VEBT @ Xs @ Ys ) ) )
% 0.33/0.56       => ~ ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.33/0.56           => ~ ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_zipE
% 0.33/0.56  thf(fact_1820_in__set__zipE,axiom,
% 0.33/0.56      ! [X: real,Y: int,Xs: list_real,Ys: list_int] :
% 0.33/0.56        ( ( member1627681773268152802al_int @ ( produc3179012173361985393al_int @ X @ Y ) @ ( set_Pr8219819362198175822al_int @ ( zip_real_int @ Xs @ Ys ) ) )
% 0.33/0.56       => ~ ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.33/0.56           => ~ ( member_int @ Y @ ( set_int2 @ Ys ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_zipE
% 0.33/0.56  thf(fact_1821_in__set__zipE,axiom,
% 0.33/0.56      ! [X: real,Y: nat,Xs: list_real,Ys: list_nat] :
% 0.33/0.56        ( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X @ Y ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs @ Ys ) ) )
% 0.33/0.56       => ~ ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.33/0.56           => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_zipE
% 0.33/0.56  thf(fact_1822_in__set__zipE,axiom,
% 0.33/0.56      ! [X: vEBT_VEBT,Y: real,Xs: list_VEBT_VEBT,Ys: list_real] :
% 0.33/0.56        ( ( member8675245146396747942T_real @ ( produc8117437818029410057T_real @ X @ Y ) @ ( set_Pr1087130671499945274T_real @ ( zip_VEBT_VEBT_real @ Xs @ Ys ) ) )
% 0.33/0.56       => ~ ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.56           => ~ ( member_real @ Y @ ( set_real2 @ Ys ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_zipE
% 0.33/0.56  thf(fact_1823_in__set__zipE,axiom,
% 0.33/0.56      ! [X: vEBT_VEBT,Y: vEBT_VEBT,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.33/0.56        ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X @ Y ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) )
% 0.33/0.56       => ~ ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.56           => ~ ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_zipE
% 0.33/0.56  thf(fact_1824_in__set__zipE,axiom,
% 0.33/0.56      ! [X: vEBT_VEBT,Y: int,Xs: list_VEBT_VEBT,Ys: list_int] :
% 0.33/0.56        ( ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X @ Y ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) ) )
% 0.33/0.56       => ~ ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.56           => ~ ( member_int @ Y @ ( set_int2 @ Ys ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_zipE
% 0.33/0.56  thf(fact_1825_in__set__zipE,axiom,
% 0.33/0.56      ! [X: vEBT_VEBT,Y: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
% 0.33/0.56        ( ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X @ Y ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) )
% 0.33/0.56       => ~ ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.56           => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_zipE
% 0.33/0.56  thf(fact_1826_in__set__zipE,axiom,
% 0.33/0.56      ! [X: int,Y: real,Xs: list_int,Ys: list_real] :
% 0.33/0.56        ( ( member2744130022092475746t_real @ ( produc801115645435158769t_real @ X @ Y ) @ ( set_Pr112895574167722958t_real @ ( zip_int_real @ Xs @ Ys ) ) )
% 0.33/0.56       => ~ ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.33/0.56           => ~ ( member_real @ Y @ ( set_real2 @ Ys ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_zipE
% 0.33/0.56  thf(fact_1827_in__set__zipE,axiom,
% 0.33/0.56      ! [X: int,Y: vEBT_VEBT,Xs: list_int,Ys: list_VEBT_VEBT] :
% 0.33/0.56        ( ( member2056185340421749780T_VEBT @ ( produc3329399203697025711T_VEBT @ X @ Y ) @ ( set_Pr8714266321650254504T_VEBT @ ( zip_int_VEBT_VEBT @ Xs @ Ys ) ) )
% 0.33/0.56       => ~ ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.33/0.56           => ~ ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_zipE
% 0.33/0.56  thf(fact_1828_set__zip__leftD,axiom,
% 0.33/0.56      ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
% 0.33/0.56        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
% 0.33/0.56       => ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % set_zip_leftD
% 0.33/0.56  thf(fact_1829_set__zip__leftD,axiom,
% 0.33/0.56      ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
% 0.33/0.56        ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
% 0.33/0.56       => ( member_int @ X @ ( set_int2 @ Xs ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % set_zip_leftD
% 0.33/0.56  thf(fact_1830_set__zip__leftD,axiom,
% 0.33/0.56      ! [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger,Xs: list_C878401137130745250e_term,Ys: list_P5578671422887162913nteger] :
% 0.33/0.56        ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ ( set_Pr2999063419360598313nteger @ ( zip_Co8729459035503499408nteger @ Xs @ Ys ) ) )
% 0.33/0.56       => ( member1535805642427569193e_term @ X @ ( set_Co8062243466402858685e_term @ Xs ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % set_zip_leftD
% 0.33/0.56  thf(fact_1831_set__zip__leftD,axiom,
% 0.33/0.56      ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger,Xs: list_P1316552470764441098e_term,Ys: list_P5578671422887162913nteger] :
% 0.33/0.56        ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ ( set_Pr2135590979564877377nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys ) ) )
% 0.33/0.56       => ( member4242434998011752849e_term @ X @ ( set_Pr8342322266483756581e_term @ Xs ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % set_zip_leftD
% 0.33/0.56  thf(fact_1832_set__zip__leftD,axiom,
% 0.33/0.56      ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int] :
% 0.33/0.56        ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ ( set_Pr4943052134776177454nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) ) )
% 0.33/0.56       => ( member3222579708246209666e_term @ X @ ( set_Pr16608062948090134e_term @ Xs ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % set_zip_leftD
% 0.33/0.56  thf(fact_1833_set__zip__leftD,axiom,
% 0.33/0.56      ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int] :
% 0.33/0.56        ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ ( set_Pr1633835384712236856nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) ) )
% 0.33/0.56       => ( member8845023287901829240e_term @ X @ ( set_in5217446777445088012e_term @ Xs ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % set_zip_leftD
% 0.33/0.56  thf(fact_1834_set__zip__rightD,axiom,
% 0.33/0.56      ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
% 0.33/0.56        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
% 0.33/0.56       => ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % set_zip_rightD
% 0.33/0.56  thf(fact_1835_set__zip__rightD,axiom,
% 0.33/0.56      ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
% 0.33/0.56        ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
% 0.33/0.56       => ( member_int @ Y @ ( set_int2 @ Ys ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % set_zip_rightD
% 0.33/0.56  thf(fact_1836_set__zip__rightD,axiom,
% 0.33/0.56      ! [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger,Xs: list_C878401137130745250e_term,Ys: list_P5578671422887162913nteger] :
% 0.33/0.56        ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ ( set_Pr2999063419360598313nteger @ ( zip_Co8729459035503499408nteger @ Xs @ Ys ) ) )
% 0.33/0.56       => ( member157494554546826820nteger @ Y @ ( set_Pr920681315882439344nteger @ Ys ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % set_zip_rightD
% 0.33/0.56  thf(fact_1837_set__zip__rightD,axiom,
% 0.33/0.56      ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger,Xs: list_P1316552470764441098e_term,Ys: list_P5578671422887162913nteger] :
% 0.33/0.56        ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ ( set_Pr2135590979564877377nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys ) ) )
% 0.33/0.56       => ( member157494554546826820nteger @ Y @ ( set_Pr920681315882439344nteger @ Ys ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % set_zip_rightD
% 0.33/0.56  thf(fact_1838_set__zip__rightD,axiom,
% 0.33/0.56      ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int] :
% 0.33/0.56        ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ ( set_Pr4943052134776177454nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) ) )
% 0.33/0.56       => ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % set_zip_rightD
% 0.33/0.56  thf(fact_1839_set__zip__rightD,axiom,
% 0.33/0.56      ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int] :
% 0.33/0.56        ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ ( set_Pr1633835384712236856nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) ) )
% 0.33/0.56       => ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % set_zip_rightD
% 0.33/0.56  thf(fact_1840_zip__update,axiom,
% 0.33/0.56      ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
% 0.33/0.56        ( ( zip_VE537291747668921783T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Ys @ I @ Y ) )
% 0.33/0.56        = ( list_u6961636818849549845T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) @ I @ ( produc537772716801021591T_VEBT @ X @ Y ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zip_update
% 0.33/0.56  thf(fact_1841_zip__update,axiom,
% 0.33/0.56      ! [Xs: list_int,I: nat,X: int,Ys: list_int,Y: int] :
% 0.33/0.56        ( ( zip_int_int @ ( list_update_int @ Xs @ I @ X ) @ ( list_update_int @ Ys @ I @ Y ) )
% 0.33/0.56        = ( list_u3002344382305578791nt_int @ ( zip_int_int @ Xs @ Ys ) @ I @ ( product_Pair_int_int @ X @ Y ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zip_update
% 0.33/0.56  thf(fact_1842_zip__update,axiom,
% 0.33/0.56      ! [Xs: list_C878401137130745250e_term,I: nat,X: code_integer > option6357759511663192854e_term,Ys: list_P5578671422887162913nteger,Y: produc8923325533196201883nteger] :
% 0.33/0.56        ( ( zip_Co8729459035503499408nteger @ ( list_u4743598893156345252e_term @ Xs @ I @ X ) @ ( list_u2254550707601501961nteger @ Ys @ I @ Y ) )
% 0.33/0.56        = ( list_u1133519416628930960nteger @ ( zip_Co8729459035503499408nteger @ Xs @ Ys ) @ I @ ( produc6137756002093451184nteger @ X @ Y ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zip_update
% 0.33/0.56  thf(fact_1843_zip__update,axiom,
% 0.33/0.56      ! [Xs: list_P1316552470764441098e_term,I: nat,X: produc6241069584506657477e_term > option6357759511663192854e_term,Ys: list_P5578671422887162913nteger,Y: produc8923325533196201883nteger] :
% 0.33/0.56        ( ( zip_Pr8292346330294042792nteger @ ( list_u877304756163299468e_term @ Xs @ I @ X ) @ ( list_u2254550707601501961nteger @ Ys @ I @ Y ) )
% 0.33/0.56        = ( list_u234853988314817064nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys ) @ I @ ( produc8603105652947943368nteger @ X @ Y ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zip_update
% 0.33/0.56  thf(fact_1844_zip__update,axiom,
% 0.33/0.56      ! [Xs: list_P1743416141875011707e_term,I: nat,X: produc8551481072490612790e_term > option6357759511663192854e_term,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
% 0.33/0.56        ( ( zip_Pr4168994715204986005nt_int @ ( list_u3533491785856317309e_term @ Xs @ I @ X ) @ ( list_u3002344382305578791nt_int @ Ys @ I @ Y ) )
% 0.33/0.56        = ( list_u7736365598306452245nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) @ I @ ( produc5700946648718959541nt_int @ X @ Y ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zip_update
% 0.33/0.56  thf(fact_1845_zip__update,axiom,
% 0.33/0.56      ! [Xs: list_i8448526496819171953e_term,I: nat,X: int > option6357759511663192854e_term,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
% 0.33/0.56        ( ( zip_in8766932505889695135nt_int @ ( list_u8946639151299769843e_term @ Xs @ I @ X ) @ ( list_u3002344382305578791nt_int @ Ys @ I @ Y ) )
% 0.33/0.56        = ( list_u4780935413889332127nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) @ I @ ( produc4305682042979456191nt_int @ X @ Y ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zip_update
% 0.33/0.56  thf(fact_1846_not__numeral__le__zero,axiom,
% 0.33/0.56      ! [N: num] :
% 0.33/0.56        ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).
% 0.33/0.56  
% 0.33/0.56  % not_numeral_le_zero
% 0.33/0.56  thf(fact_1847_not__numeral__le__zero,axiom,
% 0.33/0.56      ! [N: num] :
% 0.33/0.56        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 0.33/0.56  
% 0.33/0.56  % not_numeral_le_zero
% 0.33/0.56  thf(fact_1848_not__numeral__le__zero,axiom,
% 0.33/0.56      ! [N: num] :
% 0.33/0.56        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 0.33/0.56  
% 0.33/0.56  % not_numeral_le_zero
% 0.33/0.56  thf(fact_1849_not__numeral__le__zero,axiom,
% 0.33/0.56      ! [N: num] :
% 0.33/0.56        ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 0.33/0.56  
% 0.33/0.56  % not_numeral_le_zero
% 0.33/0.56  thf(fact_1850_not__numeral__le__zero,axiom,
% 0.33/0.56      ! [N: num] :
% 0.33/0.56        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 0.33/0.56  
% 0.33/0.56  % not_numeral_le_zero
% 0.33/0.56  thf(fact_1851_zero__le__numeral,axiom,
% 0.33/0.56      ! [N: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_le_numeral
% 0.33/0.56  thf(fact_1852_zero__le__numeral,axiom,
% 0.33/0.56      ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_le_numeral
% 0.33/0.56  thf(fact_1853_zero__le__numeral,axiom,
% 0.33/0.56      ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_le_numeral
% 0.33/0.56  thf(fact_1854_zero__le__numeral,axiom,
% 0.33/0.56      ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_le_numeral
% 0.33/0.56  thf(fact_1855_zero__le__numeral,axiom,
% 0.33/0.56      ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_le_numeral
% 0.33/0.56  thf(fact_1856_zero__less__numeral,axiom,
% 0.33/0.56      ! [N: num] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_less_numeral
% 0.33/0.56  thf(fact_1857_zero__less__numeral,axiom,
% 0.33/0.56      ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_less_numeral
% 0.33/0.56  thf(fact_1858_zero__less__numeral,axiom,
% 0.33/0.56      ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_less_numeral
% 0.33/0.56  thf(fact_1859_zero__less__numeral,axiom,
% 0.33/0.56      ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_less_numeral
% 0.33/0.56  thf(fact_1860_zero__less__numeral,axiom,
% 0.33/0.56      ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_less_numeral
% 0.33/0.56  thf(fact_1861_not__numeral__less__zero,axiom,
% 0.33/0.56      ! [N: num] :
% 0.33/0.56        ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).
% 0.33/0.56  
% 0.33/0.56  % not_numeral_less_zero
% 0.33/0.56  thf(fact_1862_not__numeral__less__zero,axiom,
% 0.33/0.56      ! [N: num] :
% 0.33/0.56        ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 0.33/0.56  
% 0.33/0.56  % not_numeral_less_zero
% 0.33/0.56  thf(fact_1863_not__numeral__less__zero,axiom,
% 0.33/0.56      ! [N: num] :
% 0.33/0.56        ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 0.33/0.56  
% 0.33/0.56  % not_numeral_less_zero
% 0.33/0.56  thf(fact_1864_not__numeral__less__zero,axiom,
% 0.33/0.56      ! [N: num] :
% 0.33/0.56        ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 0.33/0.56  
% 0.33/0.56  % not_numeral_less_zero
% 0.33/0.56  thf(fact_1865_not__numeral__less__zero,axiom,
% 0.33/0.56      ! [N: num] :
% 0.33/0.56        ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 0.33/0.56  
% 0.33/0.56  % not_numeral_less_zero
% 0.33/0.56  thf(fact_1866_lift__Suc__mono__less,axiom,
% 0.33/0.56      ! [F: nat > real,N: nat,N7: nat] :
% 0.33/0.56        ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.33/0.56       => ( ( ord_less_nat @ N @ N7 )
% 0.33/0.56         => ( ord_less_real @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lift_Suc_mono_less
% 0.33/0.56  thf(fact_1867_lift__Suc__mono__less,axiom,
% 0.33/0.56      ! [F: nat > rat,N: nat,N7: nat] :
% 0.33/0.56        ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.33/0.56       => ( ( ord_less_nat @ N @ N7 )
% 0.33/0.56         => ( ord_less_rat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lift_Suc_mono_less
% 0.33/0.56  thf(fact_1868_lift__Suc__mono__less,axiom,
% 0.33/0.56      ! [F: nat > num,N: nat,N7: nat] :
% 0.33/0.56        ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.33/0.56       => ( ( ord_less_nat @ N @ N7 )
% 0.33/0.56         => ( ord_less_num @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lift_Suc_mono_less
% 0.33/0.56  thf(fact_1869_lift__Suc__mono__less,axiom,
% 0.33/0.56      ! [F: nat > nat,N: nat,N7: nat] :
% 0.33/0.56        ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.33/0.56       => ( ( ord_less_nat @ N @ N7 )
% 0.33/0.56         => ( ord_less_nat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lift_Suc_mono_less
% 0.33/0.56  thf(fact_1870_lift__Suc__mono__less,axiom,
% 0.33/0.56      ! [F: nat > int,N: nat,N7: nat] :
% 0.33/0.56        ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.33/0.56       => ( ( ord_less_nat @ N @ N7 )
% 0.33/0.56         => ( ord_less_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lift_Suc_mono_less
% 0.33/0.56  thf(fact_1871_lift__Suc__mono__less__iff,axiom,
% 0.33/0.56      ! [F: nat > real,N: nat,M: nat] :
% 0.33/0.56        ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.33/0.56       => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 0.33/0.56          = ( ord_less_nat @ N @ M ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lift_Suc_mono_less_iff
% 0.33/0.56  thf(fact_1872_lift__Suc__mono__less__iff,axiom,
% 0.33/0.56      ! [F: nat > rat,N: nat,M: nat] :
% 0.33/0.56        ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.33/0.56       => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
% 0.33/0.56          = ( ord_less_nat @ N @ M ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lift_Suc_mono_less_iff
% 0.33/0.56  thf(fact_1873_lift__Suc__mono__less__iff,axiom,
% 0.33/0.56      ! [F: nat > num,N: nat,M: nat] :
% 0.33/0.56        ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.33/0.56       => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
% 0.33/0.56          = ( ord_less_nat @ N @ M ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lift_Suc_mono_less_iff
% 0.33/0.56  thf(fact_1874_lift__Suc__mono__less__iff,axiom,
% 0.33/0.56      ! [F: nat > nat,N: nat,M: nat] :
% 0.33/0.56        ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.33/0.56       => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 0.33/0.56          = ( ord_less_nat @ N @ M ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lift_Suc_mono_less_iff
% 0.33/0.56  thf(fact_1875_lift__Suc__mono__less__iff,axiom,
% 0.33/0.56      ! [F: nat > int,N: nat,M: nat] :
% 0.33/0.56        ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.33/0.56       => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 0.33/0.56          = ( ord_less_nat @ N @ M ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lift_Suc_mono_less_iff
% 0.33/0.56  thf(fact_1876_lift__Suc__mono__le,axiom,
% 0.33/0.56      ! [F: nat > set_int,N: nat,N7: nat] :
% 0.33/0.56        ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.33/0.56       => ( ( ord_less_eq_nat @ N @ N7 )
% 0.33/0.56         => ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lift_Suc_mono_le
% 0.33/0.56  thf(fact_1877_lift__Suc__mono__le,axiom,
% 0.33/0.56      ! [F: nat > rat,N: nat,N7: nat] :
% 0.33/0.56        ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.33/0.56       => ( ( ord_less_eq_nat @ N @ N7 )
% 0.33/0.56         => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lift_Suc_mono_le
% 0.33/0.56  thf(fact_1878_lift__Suc__mono__le,axiom,
% 0.33/0.56      ! [F: nat > num,N: nat,N7: nat] :
% 0.33/0.56        ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.33/0.56       => ( ( ord_less_eq_nat @ N @ N7 )
% 0.33/0.56         => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lift_Suc_mono_le
% 0.33/0.56  thf(fact_1879_lift__Suc__mono__le,axiom,
% 0.33/0.56      ! [F: nat > nat,N: nat,N7: nat] :
% 0.33/0.56        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.33/0.56       => ( ( ord_less_eq_nat @ N @ N7 )
% 0.33/0.56         => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lift_Suc_mono_le
% 0.33/0.56  thf(fact_1880_lift__Suc__mono__le,axiom,
% 0.33/0.56      ! [F: nat > int,N: nat,N7: nat] :
% 0.33/0.56        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.33/0.56       => ( ( ord_less_eq_nat @ N @ N7 )
% 0.33/0.56         => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lift_Suc_mono_le
% 0.33/0.56  thf(fact_1881_lift__Suc__antimono__le,axiom,
% 0.33/0.56      ! [F: nat > set_int,N: nat,N7: nat] :
% 0.33/0.56        ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.33/0.56       => ( ( ord_less_eq_nat @ N @ N7 )
% 0.33/0.56         => ( ord_less_eq_set_int @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lift_Suc_antimono_le
% 0.33/0.56  thf(fact_1882_lift__Suc__antimono__le,axiom,
% 0.33/0.56      ! [F: nat > rat,N: nat,N7: nat] :
% 0.33/0.56        ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.33/0.56       => ( ( ord_less_eq_nat @ N @ N7 )
% 0.33/0.56         => ( ord_less_eq_rat @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lift_Suc_antimono_le
% 0.33/0.56  thf(fact_1883_lift__Suc__antimono__le,axiom,
% 0.33/0.56      ! [F: nat > num,N: nat,N7: nat] :
% 0.33/0.56        ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.33/0.56       => ( ( ord_less_eq_nat @ N @ N7 )
% 0.33/0.56         => ( ord_less_eq_num @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lift_Suc_antimono_le
% 0.33/0.56  thf(fact_1884_lift__Suc__antimono__le,axiom,
% 0.33/0.56      ! [F: nat > nat,N: nat,N7: nat] :
% 0.33/0.56        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.33/0.56       => ( ( ord_less_eq_nat @ N @ N7 )
% 0.33/0.56         => ( ord_less_eq_nat @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lift_Suc_antimono_le
% 0.33/0.56  thf(fact_1885_lift__Suc__antimono__le,axiom,
% 0.33/0.56      ! [F: nat > int,N: nat,N7: nat] :
% 0.33/0.56        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.33/0.56       => ( ( ord_less_eq_nat @ N @ N7 )
% 0.33/0.56         => ( ord_less_eq_int @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % lift_Suc_antimono_le
% 0.33/0.56  thf(fact_1886_Ex__less__Suc2,axiom,
% 0.33/0.56      ! [N: nat,P: nat > $o] :
% 0.33/0.56        ( ( ? [I4: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.33/0.56              & ( P @ I4 ) ) )
% 0.33/0.56        = ( ( P @ zero_zero_nat )
% 0.33/0.56          | ? [I4: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I4 @ N )
% 0.33/0.56              & ( P @ ( suc @ I4 ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % Ex_less_Suc2
% 0.33/0.56  thf(fact_1887_gr0__conv__Suc,axiom,
% 0.33/0.56      ! [N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.56        = ( ? [M2: nat] :
% 0.33/0.56              ( N
% 0.33/0.56              = ( suc @ M2 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % gr0_conv_Suc
% 0.33/0.56  thf(fact_1888_All__less__Suc2,axiom,
% 0.33/0.56      ! [N: nat,P: nat > $o] :
% 0.33/0.56        ( ( ! [I4: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.33/0.56             => ( P @ I4 ) ) )
% 0.33/0.56        = ( ( P @ zero_zero_nat )
% 0.33/0.56          & ! [I4: nat] :
% 0.33/0.56              ( ( ord_less_nat @ I4 @ N )
% 0.33/0.56             => ( P @ ( suc @ I4 ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % All_less_Suc2
% 0.33/0.56  thf(fact_1889_gr0__implies__Suc,axiom,
% 0.33/0.56      ! [N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.56       => ? [M5: nat] :
% 0.33/0.56            ( N
% 0.33/0.56            = ( suc @ M5 ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % gr0_implies_Suc
% 0.33/0.56  thf(fact_1890_less__Suc__eq__0__disj,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.33/0.56        = ( ( M = zero_zero_nat )
% 0.33/0.56          | ? [J3: nat] :
% 0.33/0.56              ( ( M
% 0.33/0.56                = ( suc @ J3 ) )
% 0.33/0.56              & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % less_Suc_eq_0_disj
% 0.33/0.56  thf(fact_1891_le__imp__less__Suc,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.56       => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % le_imp_less_Suc
% 0.33/0.56  thf(fact_1892_less__eq__Suc__le,axiom,
% 0.33/0.56      ( ord_less_nat
% 0.33/0.56      = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % less_eq_Suc_le
% 0.33/0.56  thf(fact_1893_less__Suc__eq__le,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.33/0.56        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % less_Suc_eq_le
% 0.33/0.56  thf(fact_1894_le__less__Suc__eq,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_nat @ M @ N )
% 0.33/0.56       => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.33/0.56          = ( N = M ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % le_less_Suc_eq
% 0.33/0.56  thf(fact_1895_Suc__le__lessD,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.33/0.56       => ( ord_less_nat @ M @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % Suc_le_lessD
% 0.33/0.56  thf(fact_1896_inc__induct,axiom,
% 0.33/0.56      ! [I: nat,J: nat,P: nat > $o] :
% 0.33/0.56        ( ( ord_less_eq_nat @ I @ J )
% 0.33/0.56       => ( ( P @ J )
% 0.33/0.56         => ( ! [N3: nat] :
% 0.33/0.56                ( ( ord_less_eq_nat @ I @ N3 )
% 0.33/0.56               => ( ( ord_less_nat @ N3 @ J )
% 0.33/0.56                 => ( ( P @ ( suc @ N3 ) )
% 0.33/0.56                   => ( P @ N3 ) ) ) )
% 0.33/0.56           => ( P @ I ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % inc_induct
% 0.33/0.56  thf(fact_1897_dec__induct,axiom,
% 0.33/0.56      ! [I: nat,J: nat,P: nat > $o] :
% 0.33/0.56        ( ( ord_less_eq_nat @ I @ J )
% 0.33/0.56       => ( ( P @ I )
% 0.33/0.56         => ( ! [N3: nat] :
% 0.33/0.56                ( ( ord_less_eq_nat @ I @ N3 )
% 0.33/0.56               => ( ( ord_less_nat @ N3 @ J )
% 0.33/0.56                 => ( ( P @ N3 )
% 0.33/0.56                   => ( P @ ( suc @ N3 ) ) ) ) )
% 0.33/0.56           => ( P @ J ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % dec_induct
% 0.33/0.56  thf(fact_1898_Suc__le__eq,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.33/0.56        = ( ord_less_nat @ M @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % Suc_le_eq
% 0.33/0.56  thf(fact_1899_Suc__leI,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ M @ N )
% 0.33/0.56       => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 0.33/0.56  
% 0.33/0.56  % Suc_leI
% 0.33/0.56  thf(fact_1900_one__is__add,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ( suc @ zero_zero_nat )
% 0.33/0.56          = ( plus_plus_nat @ M @ N ) )
% 0.33/0.56        = ( ( ( M
% 0.33/0.56              = ( suc @ zero_zero_nat ) )
% 0.33/0.56            & ( N = zero_zero_nat ) )
% 0.33/0.56          | ( ( M = zero_zero_nat )
% 0.33/0.56            & ( N
% 0.33/0.56              = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % one_is_add
% 0.33/0.56  thf(fact_1901_add__is__1,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ( plus_plus_nat @ M @ N )
% 0.33/0.56          = ( suc @ zero_zero_nat ) )
% 0.33/0.56        = ( ( ( M
% 0.33/0.56              = ( suc @ zero_zero_nat ) )
% 0.33/0.56            & ( N = zero_zero_nat ) )
% 0.33/0.56          | ( ( M = zero_zero_nat )
% 0.33/0.56            & ( N
% 0.33/0.56              = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % add_is_1
% 0.33/0.56  thf(fact_1902_less__imp__Suc__add,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ M @ N )
% 0.33/0.56       => ? [K2: nat] :
% 0.33/0.56            ( N
% 0.33/0.56            = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % less_imp_Suc_add
% 0.33/0.56  thf(fact_1903_less__iff__Suc__add,axiom,
% 0.33/0.56      ( ord_less_nat
% 0.33/0.56      = ( ^ [M2: nat,N2: nat] :
% 0.33/0.56          ? [K3: nat] :
% 0.33/0.56            ( N2
% 0.33/0.56            = ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % less_iff_Suc_add
% 0.33/0.56  thf(fact_1904_less__add__Suc2,axiom,
% 0.33/0.56      ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % less_add_Suc2
% 0.33/0.56  thf(fact_1905_less__add__Suc1,axiom,
% 0.33/0.56      ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % less_add_Suc1
% 0.33/0.56  thf(fact_1906_less__natE,axiom,
% 0.33/0.56      ! [M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ M @ N )
% 0.33/0.56       => ~ ! [Q3: nat] :
% 0.33/0.56              ( N
% 0.33/0.56             != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % less_natE
% 0.33/0.56  thf(fact_1907_invar__vebt_Ointros_I5_J,axiom,
% 0.33/0.56      ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 0.33/0.56        ( ! [X4: vEBT_VEBT] :
% 0.33/0.56            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.33/0.56           => ( vEBT_invar_vebt @ X4 @ N ) )
% 0.33/0.56       => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.33/0.56         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.33/0.56              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.56           => ( ( M
% 0.33/0.56                = ( suc @ N ) )
% 0.33/0.56             => ( ( Deg
% 0.33/0.56                  = ( plus_plus_nat @ N @ M ) )
% 0.33/0.56               => ( ! [I2: nat] :
% 0.33/0.56                      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.56                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
% 0.33/0.56                        = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 0.33/0.56                 => ( ( ( Mi = Ma )
% 0.33/0.56                     => ! [X4: vEBT_VEBT] :
% 0.33/0.56                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.33/0.56                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
% 0.33/0.56                   => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.33/0.56                     => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.33/0.56                       => ( ( ( Mi != Ma )
% 0.33/0.56                           => ! [I2: nat] :
% 0.33/0.56                                ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.56                               => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 0.33/0.56                                      = I2 )
% 0.33/0.56                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 0.33/0.56                                  & ! [X4: nat] :
% 0.33/0.56                                      ( ( ( ( vEBT_VEBT_high @ X4 @ N )
% 0.33/0.56                                          = I2 )
% 0.33/0.56                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
% 0.33/0.56                                     => ( ( ord_less_nat @ Mi @ X4 )
% 0.33/0.56                                        & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
% 0.33/0.56                         => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % invar_vebt.intros(5)
% 0.33/0.56  thf(fact_1908_in__set__impl__in__set__zip2,axiom,
% 0.33/0.56      ! [Xs: list_VEBT_VEBT,Ys: list_real,Y: real] :
% 0.33/0.56        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.33/0.56          = ( size_size_list_real @ Ys ) )
% 0.33/0.56       => ( ( member_real @ Y @ ( set_real2 @ Ys ) )
% 0.33/0.56         => ~ ! [X4: vEBT_VEBT] :
% 0.33/0.56                ~ ( member8675245146396747942T_real @ ( produc8117437818029410057T_real @ X4 @ Y ) @ ( set_Pr1087130671499945274T_real @ ( zip_VEBT_VEBT_real @ Xs @ Ys ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_impl_in_set_zip2
% 0.33/0.56  thf(fact_1909_in__set__impl__in__set__zip2,axiom,
% 0.33/0.56      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
% 0.33/0.56        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.33/0.56          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.33/0.56       => ( ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys ) )
% 0.33/0.56         => ~ ! [X4: vEBT_VEBT] :
% 0.33/0.56                ~ ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X4 @ Y ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_impl_in_set_zip2
% 0.33/0.56  thf(fact_1910_in__set__impl__in__set__zip2,axiom,
% 0.33/0.56      ! [Xs: list_VEBT_VEBT,Ys: list_o,Y: $o] :
% 0.33/0.56        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.33/0.56          = ( size_size_list_o @ Ys ) )
% 0.33/0.56       => ( ( member_o @ Y @ ( set_o2 @ Ys ) )
% 0.33/0.56         => ~ ! [X4: vEBT_VEBT] :
% 0.33/0.56                ~ ( member3307348790968139188VEBT_o @ ( produc8721562602347293563VEBT_o @ X4 @ Y ) @ ( set_Pr7708085864119495200VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_impl_in_set_zip2
% 0.33/0.56  thf(fact_1911_in__set__impl__in__set__zip2,axiom,
% 0.33/0.56      ! [Xs: list_VEBT_VEBT,Ys: list_nat,Y: nat] :
% 0.33/0.56        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.33/0.56          = ( size_size_list_nat @ Ys ) )
% 0.33/0.56       => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
% 0.33/0.56         => ~ ! [X4: vEBT_VEBT] :
% 0.33/0.56                ~ ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X4 @ Y ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_impl_in_set_zip2
% 0.33/0.56  thf(fact_1912_in__set__impl__in__set__zip2,axiom,
% 0.33/0.56      ! [Xs: list_VEBT_VEBT,Ys: list_int,Y: int] :
% 0.33/0.56        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.33/0.56          = ( size_size_list_int @ Ys ) )
% 0.33/0.56       => ( ( member_int @ Y @ ( set_int2 @ Ys ) )
% 0.33/0.56         => ~ ! [X4: vEBT_VEBT] :
% 0.33/0.56                ~ ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X4 @ Y ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_impl_in_set_zip2
% 0.33/0.56  thf(fact_1913_in__set__impl__in__set__zip2,axiom,
% 0.33/0.56      ! [Xs: list_o,Ys: list_real,Y: real] :
% 0.33/0.56        ( ( ( size_size_list_o @ Xs )
% 0.33/0.56          = ( size_size_list_real @ Ys ) )
% 0.33/0.56       => ( ( member_real @ Y @ ( set_real2 @ Ys ) )
% 0.33/0.56         => ~ ! [X4: $o] :
% 0.33/0.56                ~ ( member7400031367953476362o_real @ ( product_Pair_o_real @ X4 @ Y ) @ ( set_Pr2600826154070092190o_real @ ( zip_o_real @ Xs @ Ys ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_impl_in_set_zip2
% 0.33/0.56  thf(fact_1914_in__set__impl__in__set__zip2,axiom,
% 0.33/0.56      ! [Xs: list_o,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
% 0.33/0.56        ( ( ( size_size_list_o @ Xs )
% 0.33/0.56          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.33/0.56       => ( ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys ) )
% 0.33/0.56         => ~ ! [X4: $o] :
% 0.33/0.56                ~ ( member5477980866518848620T_VEBT @ ( produc2982872950893828659T_VEBT @ X4 @ Y ) @ ( set_Pr655345902815428824T_VEBT @ ( zip_o_VEBT_VEBT @ Xs @ Ys ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_impl_in_set_zip2
% 0.33/0.56  thf(fact_1915_in__set__impl__in__set__zip2,axiom,
% 0.33/0.56      ! [Xs: list_o,Ys: list_o,Y: $o] :
% 0.33/0.56        ( ( ( size_size_list_o @ Xs )
% 0.33/0.56          = ( size_size_list_o @ Ys ) )
% 0.33/0.56       => ( ( member_o @ Y @ ( set_o2 @ Ys ) )
% 0.33/0.56         => ~ ! [X4: $o] :
% 0.33/0.56                ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y ) @ ( set_Product_prod_o_o2 @ ( zip_o_o @ Xs @ Ys ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_impl_in_set_zip2
% 0.33/0.56  thf(fact_1916_in__set__impl__in__set__zip2,axiom,
% 0.33/0.56      ! [Xs: list_o,Ys: list_nat,Y: nat] :
% 0.33/0.56        ( ( ( size_size_list_o @ Xs )
% 0.33/0.56          = ( size_size_list_nat @ Ys ) )
% 0.33/0.56       => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
% 0.33/0.56         => ~ ! [X4: $o] :
% 0.33/0.56                ~ ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ X4 @ Y ) @ ( set_Pr7006799604034136130_o_nat @ ( zip_o_nat @ Xs @ Ys ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_impl_in_set_zip2
% 0.33/0.56  thf(fact_1917_in__set__impl__in__set__zip2,axiom,
% 0.33/0.56      ! [Xs: list_o,Ys: list_int,Y: int] :
% 0.33/0.56        ( ( ( size_size_list_o @ Xs )
% 0.33/0.56          = ( size_size_list_int @ Ys ) )
% 0.33/0.56       => ( ( member_int @ Y @ ( set_int2 @ Ys ) )
% 0.33/0.56         => ~ ! [X4: $o] :
% 0.33/0.56                ~ ( member7847949116333733898_o_int @ ( product_Pair_o_int @ X4 @ Y ) @ ( set_Pr2828948584524939422_o_int @ ( zip_o_int @ Xs @ Ys ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_impl_in_set_zip2
% 0.33/0.56  thf(fact_1918_in__set__impl__in__set__zip1,axiom,
% 0.33/0.56      ! [Xs: list_real,Ys: list_VEBT_VEBT,X: real] :
% 0.33/0.56        ( ( ( size_size_list_real @ Xs )
% 0.33/0.56          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.33/0.56       => ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.33/0.56         => ~ ! [Y3: vEBT_VEBT] :
% 0.33/0.56                ~ ( member7262085504369356948T_VEBT @ ( produc6931449550656315951T_VEBT @ X @ Y3 ) @ ( set_Pr8897343066327330088T_VEBT @ ( zip_real_VEBT_VEBT @ Xs @ Ys ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_impl_in_set_zip1
% 0.33/0.56  thf(fact_1919_in__set__impl__in__set__zip1,axiom,
% 0.33/0.56      ! [Xs: list_real,Ys: list_o,X: real] :
% 0.33/0.56        ( ( ( size_size_list_real @ Xs )
% 0.33/0.56          = ( size_size_list_o @ Ys ) )
% 0.33/0.56       => ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.33/0.56         => ~ ! [Y3: $o] :
% 0.33/0.56                ~ ( member772602641336174712real_o @ ( product_Pair_real_o @ X @ Y3 ) @ ( set_Pr5196769464307566348real_o @ ( zip_real_o @ Xs @ Ys ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_impl_in_set_zip1
% 0.33/0.56  thf(fact_1920_in__set__impl__in__set__zip1,axiom,
% 0.33/0.56      ! [Xs: list_real,Ys: list_nat,X: real] :
% 0.33/0.56        ( ( ( size_size_list_real @ Xs )
% 0.33/0.56          = ( size_size_list_nat @ Ys ) )
% 0.33/0.56       => ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.33/0.56         => ~ ! [Y3: nat] :
% 0.33/0.56                ~ ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X @ Y3 ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs @ Ys ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_impl_in_set_zip1
% 0.33/0.56  thf(fact_1921_in__set__impl__in__set__zip1,axiom,
% 0.33/0.56      ! [Xs: list_real,Ys: list_int,X: real] :
% 0.33/0.56        ( ( ( size_size_list_real @ Xs )
% 0.33/0.56          = ( size_size_list_int @ Ys ) )
% 0.33/0.56       => ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.33/0.56         => ~ ! [Y3: int] :
% 0.33/0.56                ~ ( member1627681773268152802al_int @ ( produc3179012173361985393al_int @ X @ Y3 ) @ ( set_Pr8219819362198175822al_int @ ( zip_real_int @ Xs @ Ys ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_impl_in_set_zip1
% 0.33/0.56  thf(fact_1922_in__set__impl__in__set__zip1,axiom,
% 0.33/0.56      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,X: vEBT_VEBT] :
% 0.33/0.56        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.33/0.56          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.33/0.56       => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.56         => ~ ! [Y3: vEBT_VEBT] :
% 0.33/0.56                ~ ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X @ Y3 ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_impl_in_set_zip1
% 0.33/0.56  thf(fact_1923_in__set__impl__in__set__zip1,axiom,
% 0.33/0.56      ! [Xs: list_VEBT_VEBT,Ys: list_o,X: vEBT_VEBT] :
% 0.33/0.56        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.33/0.56          = ( size_size_list_o @ Ys ) )
% 0.33/0.56       => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.56         => ~ ! [Y3: $o] :
% 0.33/0.56                ~ ( member3307348790968139188VEBT_o @ ( produc8721562602347293563VEBT_o @ X @ Y3 ) @ ( set_Pr7708085864119495200VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_impl_in_set_zip1
% 0.33/0.56  thf(fact_1924_in__set__impl__in__set__zip1,axiom,
% 0.33/0.56      ! [Xs: list_VEBT_VEBT,Ys: list_nat,X: vEBT_VEBT] :
% 0.33/0.56        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.33/0.56          = ( size_size_list_nat @ Ys ) )
% 0.33/0.56       => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.56         => ~ ! [Y3: nat] :
% 0.33/0.56                ~ ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X @ Y3 ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_impl_in_set_zip1
% 0.33/0.56  thf(fact_1925_in__set__impl__in__set__zip1,axiom,
% 0.33/0.56      ! [Xs: list_VEBT_VEBT,Ys: list_int,X: vEBT_VEBT] :
% 0.33/0.56        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.33/0.56          = ( size_size_list_int @ Ys ) )
% 0.33/0.56       => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.56         => ~ ! [Y3: int] :
% 0.33/0.56                ~ ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X @ Y3 ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_impl_in_set_zip1
% 0.33/0.56  thf(fact_1926_in__set__impl__in__set__zip1,axiom,
% 0.33/0.56      ! [Xs: list_o,Ys: list_VEBT_VEBT,X: $o] :
% 0.33/0.56        ( ( ( size_size_list_o @ Xs )
% 0.33/0.56          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.33/0.56       => ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 0.33/0.56         => ~ ! [Y3: vEBT_VEBT] :
% 0.33/0.56                ~ ( member5477980866518848620T_VEBT @ ( produc2982872950893828659T_VEBT @ X @ Y3 ) @ ( set_Pr655345902815428824T_VEBT @ ( zip_o_VEBT_VEBT @ Xs @ Ys ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_impl_in_set_zip1
% 0.33/0.56  thf(fact_1927_in__set__impl__in__set__zip1,axiom,
% 0.33/0.56      ! [Xs: list_o,Ys: list_o,X: $o] :
% 0.33/0.56        ( ( ( size_size_list_o @ Xs )
% 0.33/0.56          = ( size_size_list_o @ Ys ) )
% 0.33/0.56       => ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 0.33/0.56         => ~ ! [Y3: $o] :
% 0.33/0.56                ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y3 ) @ ( set_Product_prod_o_o2 @ ( zip_o_o @ Xs @ Ys ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % in_set_impl_in_set_zip1
% 0.33/0.56  thf(fact_1928_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 0.33/0.56      ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
% 0.33/0.56        ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) )
% 0.33/0.56        = ( some_P7363390416028606310at_nat @ ( F @ A @ B ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % VEBT_internal.option_shift.simps(3)
% 0.33/0.56  thf(fact_1929_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 0.33/0.56      ! [F: num > num > num,A: num,B: num] :
% 0.33/0.56        ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B ) )
% 0.33/0.56        = ( some_num @ ( F @ A @ B ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % VEBT_internal.option_shift.simps(3)
% 0.33/0.56  thf(fact_1930_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 0.33/0.56      ! [F: nat > nat > nat,A: nat,B: nat] :
% 0.33/0.56        ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
% 0.33/0.56        = ( some_nat @ ( F @ A @ B ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % VEBT_internal.option_shift.simps(3)
% 0.33/0.56  thf(fact_1931_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 0.33/0.56      ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
% 0.33/0.56        ( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
% 0.33/0.56        = none_P5556105721700978146at_nat ) ).
% 0.33/0.56  
% 0.33/0.56  % VEBT_internal.option_shift.simps(1)
% 0.33/0.56  thf(fact_1932_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 0.33/0.56      ! [Uu: num > num > num,Uv: option_num] :
% 0.33/0.56        ( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
% 0.33/0.56        = none_num ) ).
% 0.33/0.56  
% 0.33/0.56  % VEBT_internal.option_shift.simps(1)
% 0.33/0.56  thf(fact_1933_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 0.33/0.56      ! [Uu: nat > nat > nat,Uv: option_nat] :
% 0.33/0.56        ( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
% 0.33/0.56        = none_nat ) ).
% 0.33/0.56  
% 0.33/0.56  % VEBT_internal.option_shift.simps(1)
% 0.33/0.56  thf(fact_1934_ex__least__nat__less,axiom,
% 0.33/0.56      ! [P: nat > $o,N: nat] :
% 0.33/0.56        ( ( P @ N )
% 0.33/0.56       => ( ~ ( P @ zero_zero_nat )
% 0.33/0.56         => ? [K2: nat] :
% 0.33/0.56              ( ( ord_less_nat @ K2 @ N )
% 0.33/0.56              & ! [I3: nat] :
% 0.33/0.56                  ( ( ord_less_eq_nat @ I3 @ K2 )
% 0.33/0.56                 => ~ ( P @ I3 ) )
% 0.33/0.56              & ( P @ ( suc @ K2 ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % ex_least_nat_less
% 0.33/0.56  thf(fact_1935_find__None__iff,axiom,
% 0.33/0.56      ! [P: real > $o,Xs: list_real] :
% 0.33/0.56        ( ( ( find_real @ P @ Xs )
% 0.33/0.56          = none_real )
% 0.33/0.56        = ( ~ ? [X3: real] :
% 0.33/0.56                ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 0.33/0.56                & ( P @ X3 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_None_iff
% 0.33/0.56  thf(fact_1936_find__None__iff,axiom,
% 0.33/0.56      ! [P: set_nat > $o,Xs: list_set_nat] :
% 0.33/0.56        ( ( ( find_set_nat @ P @ Xs )
% 0.33/0.56          = none_set_nat )
% 0.33/0.56        = ( ~ ? [X3: set_nat] :
% 0.33/0.56                ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
% 0.33/0.56                & ( P @ X3 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_None_iff
% 0.33/0.56  thf(fact_1937_find__None__iff,axiom,
% 0.33/0.56      ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 0.33/0.56        ( ( ( find_VEBT_VEBT @ P @ Xs )
% 0.33/0.56          = none_VEBT_VEBT )
% 0.33/0.56        = ( ~ ? [X3: vEBT_VEBT] :
% 0.33/0.56                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.56                & ( P @ X3 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_None_iff
% 0.33/0.56  thf(fact_1938_find__None__iff,axiom,
% 0.33/0.56      ! [P: int > $o,Xs: list_int] :
% 0.33/0.56        ( ( ( find_int @ P @ Xs )
% 0.33/0.56          = none_int )
% 0.33/0.56        = ( ~ ? [X3: int] :
% 0.33/0.56                ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 0.33/0.56                & ( P @ X3 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_None_iff
% 0.33/0.56  thf(fact_1939_find__None__iff,axiom,
% 0.33/0.56      ! [P: nat > $o,Xs: list_nat] :
% 0.33/0.56        ( ( ( find_nat @ P @ Xs )
% 0.33/0.56          = none_nat )
% 0.33/0.56        = ( ~ ? [X3: nat] :
% 0.33/0.56                ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 0.33/0.56                & ( P @ X3 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_None_iff
% 0.33/0.56  thf(fact_1940_find__None__iff,axiom,
% 0.33/0.56      ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
% 0.33/0.56        ( ( ( find_P8199882355184865565at_nat @ P @ Xs )
% 0.33/0.56          = none_P5556105721700978146at_nat )
% 0.33/0.56        = ( ~ ? [X3: product_prod_nat_nat] :
% 0.33/0.56                ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.33/0.56                & ( P @ X3 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_None_iff
% 0.33/0.56  thf(fact_1941_find__None__iff,axiom,
% 0.33/0.56      ! [P: num > $o,Xs: list_num] :
% 0.33/0.56        ( ( ( find_num @ P @ Xs )
% 0.33/0.56          = none_num )
% 0.33/0.56        = ( ~ ? [X3: num] :
% 0.33/0.56                ( ( member_num @ X3 @ ( set_num2 @ Xs ) )
% 0.33/0.56                & ( P @ X3 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_None_iff
% 0.33/0.56  thf(fact_1942_find__None__iff2,axiom,
% 0.33/0.56      ! [P: real > $o,Xs: list_real] :
% 0.33/0.56        ( ( none_real
% 0.33/0.56          = ( find_real @ P @ Xs ) )
% 0.33/0.56        = ( ~ ? [X3: real] :
% 0.33/0.56                ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 0.33/0.56                & ( P @ X3 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_None_iff2
% 0.33/0.56  thf(fact_1943_find__None__iff2,axiom,
% 0.33/0.56      ! [P: set_nat > $o,Xs: list_set_nat] :
% 0.33/0.56        ( ( none_set_nat
% 0.33/0.56          = ( find_set_nat @ P @ Xs ) )
% 0.33/0.56        = ( ~ ? [X3: set_nat] :
% 0.33/0.56                ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
% 0.33/0.56                & ( P @ X3 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_None_iff2
% 0.33/0.56  thf(fact_1944_find__None__iff2,axiom,
% 0.33/0.56      ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 0.33/0.56        ( ( none_VEBT_VEBT
% 0.33/0.56          = ( find_VEBT_VEBT @ P @ Xs ) )
% 0.33/0.56        = ( ~ ? [X3: vEBT_VEBT] :
% 0.33/0.56                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.33/0.56                & ( P @ X3 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_None_iff2
% 0.33/0.56  thf(fact_1945_find__None__iff2,axiom,
% 0.33/0.56      ! [P: int > $o,Xs: list_int] :
% 0.33/0.56        ( ( none_int
% 0.33/0.56          = ( find_int @ P @ Xs ) )
% 0.33/0.56        = ( ~ ? [X3: int] :
% 0.33/0.56                ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 0.33/0.56                & ( P @ X3 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_None_iff2
% 0.33/0.56  thf(fact_1946_find__None__iff2,axiom,
% 0.33/0.56      ! [P: nat > $o,Xs: list_nat] :
% 0.33/0.56        ( ( none_nat
% 0.33/0.56          = ( find_nat @ P @ Xs ) )
% 0.33/0.56        = ( ~ ? [X3: nat] :
% 0.33/0.56                ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 0.33/0.56                & ( P @ X3 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_None_iff2
% 0.33/0.56  thf(fact_1947_find__None__iff2,axiom,
% 0.33/0.56      ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
% 0.33/0.56        ( ( none_P5556105721700978146at_nat
% 0.33/0.56          = ( find_P8199882355184865565at_nat @ P @ Xs ) )
% 0.33/0.56        = ( ~ ? [X3: product_prod_nat_nat] :
% 0.33/0.56                ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.33/0.56                & ( P @ X3 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_None_iff2
% 0.33/0.56  thf(fact_1948_find__None__iff2,axiom,
% 0.33/0.56      ! [P: num > $o,Xs: list_num] :
% 0.33/0.56        ( ( none_num
% 0.33/0.56          = ( find_num @ P @ Xs ) )
% 0.33/0.56        = ( ~ ? [X3: num] :
% 0.33/0.56                ( ( member_num @ X3 @ ( set_num2 @ Xs ) )
% 0.33/0.56                & ( P @ X3 ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % find_None_iff2
% 0.33/0.56  thf(fact_1949_invar__vebt_Ointros_I4_J,axiom,
% 0.33/0.56      ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 0.33/0.56        ( ! [X4: vEBT_VEBT] :
% 0.33/0.56            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.33/0.56           => ( vEBT_invar_vebt @ X4 @ N ) )
% 0.33/0.56       => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.33/0.56         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.33/0.56              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.56           => ( ( M = N )
% 0.33/0.56             => ( ( Deg
% 0.33/0.56                  = ( plus_plus_nat @ N @ M ) )
% 0.33/0.56               => ( ! [I2: nat] :
% 0.33/0.56                      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.56                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
% 0.33/0.56                        = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 0.33/0.56                 => ( ( ( Mi = Ma )
% 0.33/0.56                     => ! [X4: vEBT_VEBT] :
% 0.33/0.56                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.33/0.56                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
% 0.33/0.56                   => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.33/0.56                     => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.33/0.56                       => ( ( ( Mi != Ma )
% 0.33/0.56                           => ! [I2: nat] :
% 0.33/0.56                                ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.56                               => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 0.33/0.56                                      = I2 )
% 0.33/0.56                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 0.33/0.56                                  & ! [X4: nat] :
% 0.33/0.56                                      ( ( ( ( vEBT_VEBT_high @ X4 @ N )
% 0.33/0.56                                          = I2 )
% 0.33/0.56                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
% 0.33/0.56                                     => ( ( ord_less_nat @ Mi @ X4 )
% 0.33/0.56                                        & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
% 0.33/0.56                         => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % invar_vebt.intros(4)
% 0.33/0.56  thf(fact_1950_pair__lessI1,axiom,
% 0.33/0.56      ! [A: nat,B: nat,S: nat,T: nat] :
% 0.33/0.56        ( ( ord_less_nat @ A @ B )
% 0.33/0.56       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).
% 0.33/0.56  
% 0.33/0.56  % pair_lessI1
% 0.33/0.56  thf(fact_1951_count__le__length,axiom,
% 0.33/0.56      ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] : ( ord_less_eq_nat @ ( count_list_VEBT_VEBT @ Xs @ X ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% 0.33/0.56  
% 0.33/0.56  % count_le_length
% 0.33/0.56  thf(fact_1952_count__le__length,axiom,
% 0.33/0.56      ! [Xs: list_o,X: $o] : ( ord_less_eq_nat @ ( count_list_o @ Xs @ X ) @ ( size_size_list_o @ Xs ) ) ).
% 0.33/0.56  
% 0.33/0.56  % count_le_length
% 0.33/0.56  thf(fact_1953_count__le__length,axiom,
% 0.33/0.56      ! [Xs: list_nat,X: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X ) @ ( size_size_list_nat @ Xs ) ) ).
% 0.33/0.56  
% 0.33/0.56  % count_le_length
% 0.33/0.56  thf(fact_1954_count__le__length,axiom,
% 0.33/0.56      ! [Xs: list_int,X: int] : ( ord_less_eq_nat @ ( count_list_int @ Xs @ X ) @ ( size_size_list_int @ Xs ) ) ).
% 0.33/0.56  
% 0.33/0.56  % count_le_length
% 0.33/0.56  thf(fact_1955_option_Osize_I4_J,axiom,
% 0.33/0.56      ! [X22: nat] :
% 0.33/0.56        ( ( size_size_option_nat @ ( some_nat @ X22 ) )
% 0.33/0.56        = ( suc @ zero_zero_nat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % option.size(4)
% 0.33/0.56  thf(fact_1956_option_Osize_I4_J,axiom,
% 0.33/0.56      ! [X22: product_prod_nat_nat] :
% 0.33/0.56        ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 0.33/0.56        = ( suc @ zero_zero_nat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % option.size(4)
% 0.33/0.56  thf(fact_1957_option_Osize_I4_J,axiom,
% 0.33/0.56      ! [X22: num] :
% 0.33/0.56        ( ( size_size_option_num @ ( some_num @ X22 ) )
% 0.33/0.56        = ( suc @ zero_zero_nat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % option.size(4)
% 0.33/0.56  thf(fact_1958_option_Osize_I3_J,axiom,
% 0.33/0.56      ( ( size_size_option_nat @ none_nat )
% 0.33/0.56      = ( suc @ zero_zero_nat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % option.size(3)
% 0.33/0.56  thf(fact_1959_option_Osize_I3_J,axiom,
% 0.33/0.56      ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
% 0.33/0.56      = ( suc @ zero_zero_nat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % option.size(3)
% 0.33/0.56  thf(fact_1960_option_Osize_I3_J,axiom,
% 0.33/0.56      ( ( size_size_option_num @ none_num )
% 0.33/0.56      = ( suc @ zero_zero_nat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % option.size(3)
% 0.33/0.56  thf(fact_1961_vebt__member_Osimps_I4_J,axiom,
% 0.33/0.56      ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
% 0.33/0.56        ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X ) ).
% 0.33/0.56  
% 0.33/0.56  % vebt_member.simps(4)
% 0.33/0.56  thf(fact_1962_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.33/0.56      ! [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb2: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
% 0.33/0.56        ( ( ( vEBT_V1502963449132264192at_nat @ X @ Xa2 @ Xb2 )
% 0.33/0.56          = Y )
% 0.33/0.56       => ( ( ( Xa2 = none_P5556105721700978146at_nat )
% 0.33/0.56           => ( Y != none_P5556105721700978146at_nat ) )
% 0.33/0.56         => ( ( ? [V2: product_prod_nat_nat] :
% 0.33/0.56                  ( Xa2
% 0.33/0.56                  = ( some_P7363390416028606310at_nat @ V2 ) )
% 0.33/0.56             => ( ( Xb2 = none_P5556105721700978146at_nat )
% 0.33/0.56               => ( Y != none_P5556105721700978146at_nat ) ) )
% 0.33/0.56           => ~ ! [A3: product_prod_nat_nat] :
% 0.33/0.56                  ( ( Xa2
% 0.33/0.56                    = ( some_P7363390416028606310at_nat @ A3 ) )
% 0.33/0.56                 => ! [B3: product_prod_nat_nat] :
% 0.33/0.56                      ( ( Xb2
% 0.33/0.56                        = ( some_P7363390416028606310at_nat @ B3 ) )
% 0.33/0.56                     => ( Y
% 0.33/0.56                       != ( some_P7363390416028606310at_nat @ ( X @ A3 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % VEBT_internal.option_shift.elims
% 0.33/0.56  thf(fact_1963_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.33/0.56      ! [X: num > num > num,Xa2: option_num,Xb2: option_num,Y: option_num] :
% 0.33/0.56        ( ( ( vEBT_V819420779217536731ft_num @ X @ Xa2 @ Xb2 )
% 0.33/0.56          = Y )
% 0.33/0.56       => ( ( ( Xa2 = none_num )
% 0.33/0.56           => ( Y != none_num ) )
% 0.33/0.56         => ( ( ? [V2: num] :
% 0.33/0.56                  ( Xa2
% 0.33/0.56                  = ( some_num @ V2 ) )
% 0.33/0.56             => ( ( Xb2 = none_num )
% 0.33/0.56               => ( Y != none_num ) ) )
% 0.33/0.56           => ~ ! [A3: num] :
% 0.33/0.56                  ( ( Xa2
% 0.33/0.56                    = ( some_num @ A3 ) )
% 0.33/0.56                 => ! [B3: num] :
% 0.33/0.56                      ( ( Xb2
% 0.33/0.56                        = ( some_num @ B3 ) )
% 0.33/0.56                     => ( Y
% 0.33/0.56                       != ( some_num @ ( X @ A3 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % VEBT_internal.option_shift.elims
% 0.33/0.56  thf(fact_1964_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.33/0.56      ! [X: nat > nat > nat,Xa2: option_nat,Xb2: option_nat,Y: option_nat] :
% 0.33/0.56        ( ( ( vEBT_V4262088993061758097ft_nat @ X @ Xa2 @ Xb2 )
% 0.33/0.56          = Y )
% 0.33/0.56       => ( ( ( Xa2 = none_nat )
% 0.33/0.56           => ( Y != none_nat ) )
% 0.33/0.56         => ( ( ? [V2: nat] :
% 0.33/0.56                  ( Xa2
% 0.33/0.56                  = ( some_nat @ V2 ) )
% 0.33/0.56             => ( ( Xb2 = none_nat )
% 0.33/0.56               => ( Y != none_nat ) ) )
% 0.33/0.56           => ~ ! [A3: nat] :
% 0.33/0.56                  ( ( Xa2
% 0.33/0.56                    = ( some_nat @ A3 ) )
% 0.33/0.56                 => ! [B3: nat] :
% 0.33/0.56                      ( ( Xb2
% 0.33/0.56                        = ( some_nat @ B3 ) )
% 0.33/0.56                     => ( Y
% 0.33/0.56                       != ( some_nat @ ( X @ A3 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % VEBT_internal.option_shift.elims
% 0.33/0.56  thf(fact_1965_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 0.33/0.56      ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
% 0.33/0.56        ( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
% 0.33/0.56        = none_P5556105721700978146at_nat ) ).
% 0.33/0.56  
% 0.33/0.56  % VEBT_internal.option_shift.simps(2)
% 0.33/0.56  thf(fact_1966_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 0.33/0.56      ! [Uw: num > num > num,V: num] :
% 0.33/0.56        ( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
% 0.33/0.56        = none_num ) ).
% 0.33/0.56  
% 0.33/0.56  % VEBT_internal.option_shift.simps(2)
% 0.33/0.56  thf(fact_1967_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 0.33/0.56      ! [Uw: nat > nat > nat,V: nat] :
% 0.33/0.56        ( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
% 0.33/0.56        = none_nat ) ).
% 0.33/0.56  
% 0.33/0.56  % VEBT_internal.option_shift.simps(2)
% 0.33/0.56  thf(fact_1968_vebt__delete_Osimps_I6_J,axiom,
% 0.33/0.56      ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X: nat] :
% 0.33/0.56        ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ X )
% 0.33/0.56        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) ).
% 0.33/0.56  
% 0.33/0.56  % vebt_delete.simps(6)
% 0.33/0.56  thf(fact_1969_sum__power2__eq__zero__iff,axiom,
% 0.33/0.56      ! [X: rat,Y: rat] :
% 0.33/0.56        ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.56          = zero_zero_rat )
% 0.33/0.56        = ( ( X = zero_zero_rat )
% 0.33/0.56          & ( Y = zero_zero_rat ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % sum_power2_eq_zero_iff
% 0.33/0.56  thf(fact_1970_sum__power2__eq__zero__iff,axiom,
% 0.33/0.56      ! [X: real,Y: real] :
% 0.33/0.56        ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.56          = zero_zero_real )
% 0.33/0.56        = ( ( X = zero_zero_real )
% 0.33/0.56          & ( Y = zero_zero_real ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % sum_power2_eq_zero_iff
% 0.33/0.56  thf(fact_1971_sum__power2__eq__zero__iff,axiom,
% 0.33/0.56      ! [X: int,Y: int] :
% 0.33/0.56        ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.56          = zero_zero_int )
% 0.33/0.56        = ( ( X = zero_zero_int )
% 0.33/0.56          & ( Y = zero_zero_int ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % sum_power2_eq_zero_iff
% 0.33/0.56  thf(fact_1972_zero__less__power2,axiom,
% 0.33/0.56      ! [A: real] :
% 0.33/0.56        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.56        = ( A != zero_zero_real ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_less_power2
% 0.33/0.56  thf(fact_1973_zero__less__power2,axiom,
% 0.33/0.56      ! [A: rat] :
% 0.33/0.56        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.56        = ( A != zero_zero_rat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_less_power2
% 0.33/0.56  thf(fact_1974_zero__less__power2,axiom,
% 0.33/0.56      ! [A: int] :
% 0.33/0.56        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.56        = ( A != zero_zero_int ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_less_power2
% 0.33/0.56  thf(fact_1975_power2__eq__iff__nonneg,axiom,
% 0.33/0.56      ! [X: real,Y: real] :
% 0.33/0.56        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.56       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.56         => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.56              = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.56            = ( X = Y ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power2_eq_iff_nonneg
% 0.33/0.56  thf(fact_1976_power2__eq__iff__nonneg,axiom,
% 0.33/0.56      ! [X: rat,Y: rat] :
% 0.33/0.56        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.33/0.56       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.33/0.56         => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.56              = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.56            = ( X = Y ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power2_eq_iff_nonneg
% 0.33/0.56  thf(fact_1977_power2__eq__iff__nonneg,axiom,
% 0.33/0.56      ! [X: nat,Y: nat] :
% 0.33/0.56        ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.33/0.56       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.33/0.56         => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.56              = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.56            = ( X = Y ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power2_eq_iff_nonneg
% 0.33/0.56  thf(fact_1978_power2__eq__iff__nonneg,axiom,
% 0.33/0.56      ! [X: int,Y: int] :
% 0.33/0.56        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.56       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.56         => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.56              = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.56            = ( X = Y ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power2_eq_iff_nonneg
% 0.33/0.56  thf(fact_1979_power2__less__eq__zero__iff,axiom,
% 0.33/0.56      ! [A: real] :
% 0.33/0.56        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 0.33/0.56        = ( A = zero_zero_real ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power2_less_eq_zero_iff
% 0.33/0.56  thf(fact_1980_power2__less__eq__zero__iff,axiom,
% 0.33/0.56      ! [A: rat] :
% 0.33/0.56        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 0.33/0.56        = ( A = zero_zero_rat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power2_less_eq_zero_iff
% 0.33/0.56  thf(fact_1981_power2__less__eq__zero__iff,axiom,
% 0.33/0.56      ! [A: int] :
% 0.33/0.56        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.33/0.56        = ( A = zero_zero_int ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power2_less_eq_zero_iff
% 0.33/0.56  thf(fact_1982_post__member__pre__member,axiom,
% 0.33/0.56      ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
% 0.33/0.56        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.56       => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.56         => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.56           => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y )
% 0.33/0.56             => ( ( vEBT_vebt_member @ T @ Y )
% 0.33/0.56                | ( X = Y ) ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % post_member_pre_member
% 0.33/0.56  thf(fact_1983_valid__insert__both__member__options__add,axiom,
% 0.33/0.56      ! [T: vEBT_VEBT,N: nat,X: nat] :
% 0.33/0.56        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.56       => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.56         => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % valid_insert_both_member_options_add
% 0.33/0.56  thf(fact_1984_valid__insert__both__member__options__pres,axiom,
% 0.33/0.56      ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
% 0.33/0.56        ( ( vEBT_invar_vebt @ T @ N )
% 0.33/0.56       => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.56         => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.56           => ( ( vEBT_V8194947554948674370ptions @ T @ X )
% 0.33/0.56             => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % valid_insert_both_member_options_pres
% 0.33/0.56  thf(fact_1985_insert__simp__mima,axiom,
% 0.33/0.56      ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.33/0.56        ( ( ( X = Mi )
% 0.33/0.56          | ( X = Ma ) )
% 0.33/0.56       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.33/0.56         => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.33/0.56            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % insert_simp_mima
% 0.33/0.56  thf(fact_1986_power__mono__iff,axiom,
% 0.33/0.56      ! [A: real,B: real,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.56       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.33/0.56         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.56           => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 0.33/0.56              = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_mono_iff
% 0.33/0.56  thf(fact_1987_power__mono__iff,axiom,
% 0.33/0.56      ! [A: rat,B: rat,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.56       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.33/0.56         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.56           => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 0.33/0.56              = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_mono_iff
% 0.33/0.56  thf(fact_1988_power__mono__iff,axiom,
% 0.33/0.56      ! [A: nat,B: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.33/0.56       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.33/0.56         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.56           => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.33/0.56              = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_mono_iff
% 0.33/0.56  thf(fact_1989_power__mono__iff,axiom,
% 0.33/0.56      ! [A: int,B: int,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.56       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.33/0.56         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.56           => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.33/0.56              = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_mono_iff
% 0.33/0.56  thf(fact_1990_zero__eq__power2,axiom,
% 0.33/0.56      ! [A: rat] :
% 0.33/0.56        ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.56          = zero_zero_rat )
% 0.33/0.56        = ( A = zero_zero_rat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_eq_power2
% 0.33/0.56  thf(fact_1991_zero__eq__power2,axiom,
% 0.33/0.56      ! [A: nat] :
% 0.33/0.56        ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.56          = zero_zero_nat )
% 0.33/0.56        = ( A = zero_zero_nat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_eq_power2
% 0.33/0.56  thf(fact_1992_zero__eq__power2,axiom,
% 0.33/0.56      ! [A: real] :
% 0.33/0.56        ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.56          = zero_zero_real )
% 0.33/0.56        = ( A = zero_zero_real ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_eq_power2
% 0.33/0.56  thf(fact_1993_zero__eq__power2,axiom,
% 0.33/0.56      ! [A: int] :
% 0.33/0.56        ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.56          = zero_zero_int )
% 0.33/0.56        = ( A = zero_zero_int ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_eq_power2
% 0.33/0.56  thf(fact_1994_zero__eq__power2,axiom,
% 0.33/0.56      ! [A: complex] :
% 0.33/0.56        ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.56          = zero_zero_complex )
% 0.33/0.56        = ( A = zero_zero_complex ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_eq_power2
% 0.33/0.56  thf(fact_1995_power__0__Suc,axiom,
% 0.33/0.56      ! [N: nat] :
% 0.33/0.56        ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
% 0.33/0.56        = zero_zero_rat ) ).
% 0.33/0.56  
% 0.33/0.56  % power_0_Suc
% 0.33/0.56  thf(fact_1996_power__0__Suc,axiom,
% 0.33/0.56      ! [N: nat] :
% 0.33/0.56        ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.33/0.56        = zero_zero_nat ) ).
% 0.33/0.56  
% 0.33/0.56  % power_0_Suc
% 0.33/0.56  thf(fact_1997_power__0__Suc,axiom,
% 0.33/0.56      ! [N: nat] :
% 0.33/0.56        ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 0.33/0.56        = zero_zero_real ) ).
% 0.33/0.56  
% 0.33/0.56  % power_0_Suc
% 0.33/0.56  thf(fact_1998_power__0__Suc,axiom,
% 0.33/0.56      ! [N: nat] :
% 0.33/0.56        ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 0.33/0.56        = zero_zero_int ) ).
% 0.33/0.56  
% 0.33/0.56  % power_0_Suc
% 0.33/0.56  thf(fact_1999_power__0__Suc,axiom,
% 0.33/0.56      ! [N: nat] :
% 0.33/0.56        ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
% 0.33/0.56        = zero_zero_complex ) ).
% 0.33/0.56  
% 0.33/0.56  % power_0_Suc
% 0.33/0.56  thf(fact_2000_power__zero__numeral,axiom,
% 0.33/0.56      ! [K: num] :
% 0.33/0.56        ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 0.33/0.56        = zero_zero_rat ) ).
% 0.33/0.56  
% 0.33/0.56  % power_zero_numeral
% 0.33/0.56  thf(fact_2001_power__zero__numeral,axiom,
% 0.33/0.56      ! [K: num] :
% 0.33/0.56        ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 0.33/0.56        = zero_zero_nat ) ).
% 0.33/0.56  
% 0.33/0.56  % power_zero_numeral
% 0.33/0.56  thf(fact_2002_power__zero__numeral,axiom,
% 0.33/0.56      ! [K: num] :
% 0.33/0.56        ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 0.33/0.56        = zero_zero_real ) ).
% 0.33/0.56  
% 0.33/0.56  % power_zero_numeral
% 0.33/0.56  thf(fact_2003_power__zero__numeral,axiom,
% 0.33/0.56      ! [K: num] :
% 0.33/0.56        ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 0.33/0.56        = zero_zero_int ) ).
% 0.33/0.56  
% 0.33/0.56  % power_zero_numeral
% 0.33/0.56  thf(fact_2004_power__zero__numeral,axiom,
% 0.33/0.56      ! [K: num] :
% 0.33/0.56        ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 0.33/0.56        = zero_zero_complex ) ).
% 0.33/0.56  
% 0.33/0.56  % power_zero_numeral
% 0.33/0.56  thf(fact_2005_power__Suc0__right,axiom,
% 0.33/0.56      ! [A: nat] :
% 0.33/0.56        ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 0.33/0.56        = A ) ).
% 0.33/0.56  
% 0.33/0.56  % power_Suc0_right
% 0.33/0.56  thf(fact_2006_power__Suc0__right,axiom,
% 0.33/0.56      ! [A: real] :
% 0.33/0.56        ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 0.33/0.56        = A ) ).
% 0.33/0.56  
% 0.33/0.56  % power_Suc0_right
% 0.33/0.56  thf(fact_2007_power__Suc0__right,axiom,
% 0.33/0.56      ! [A: int] :
% 0.33/0.56        ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 0.33/0.56        = A ) ).
% 0.33/0.56  
% 0.33/0.56  % power_Suc0_right
% 0.33/0.56  thf(fact_2008_power__Suc0__right,axiom,
% 0.33/0.56      ! [A: complex] :
% 0.33/0.56        ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
% 0.33/0.56        = A ) ).
% 0.33/0.56  
% 0.33/0.56  % power_Suc0_right
% 0.33/0.56  thf(fact_2009_nat__power__eq__Suc__0__iff,axiom,
% 0.33/0.56      ! [X: nat,M: nat] :
% 0.33/0.56        ( ( ( power_power_nat @ X @ M )
% 0.33/0.56          = ( suc @ zero_zero_nat ) )
% 0.33/0.56        = ( ( M = zero_zero_nat )
% 0.33/0.56          | ( X
% 0.33/0.56            = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nat_power_eq_Suc_0_iff
% 0.33/0.56  thf(fact_2010_power__Suc__0,axiom,
% 0.33/0.56      ! [N: nat] :
% 0.33/0.56        ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.33/0.56        = ( suc @ zero_zero_nat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_Suc_0
% 0.33/0.56  thf(fact_2011_nat__zero__less__power__iff,axiom,
% 0.33/0.56      ! [X: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
% 0.33/0.56        = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.33/0.56          | ( N = zero_zero_nat ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nat_zero_less_power_iff
% 0.33/0.56  thf(fact_2012_power__eq__0__iff,axiom,
% 0.33/0.56      ! [A: rat,N: nat] :
% 0.33/0.56        ( ( ( power_power_rat @ A @ N )
% 0.33/0.56          = zero_zero_rat )
% 0.33/0.56        = ( ( A = zero_zero_rat )
% 0.33/0.56          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_eq_0_iff
% 0.33/0.56  thf(fact_2013_power__eq__0__iff,axiom,
% 0.33/0.56      ! [A: nat,N: nat] :
% 0.33/0.56        ( ( ( power_power_nat @ A @ N )
% 0.33/0.56          = zero_zero_nat )
% 0.33/0.56        = ( ( A = zero_zero_nat )
% 0.33/0.56          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_eq_0_iff
% 0.33/0.56  thf(fact_2014_power__eq__0__iff,axiom,
% 0.33/0.56      ! [A: real,N: nat] :
% 0.33/0.56        ( ( ( power_power_real @ A @ N )
% 0.33/0.56          = zero_zero_real )
% 0.33/0.56        = ( ( A = zero_zero_real )
% 0.33/0.56          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_eq_0_iff
% 0.33/0.56  thf(fact_2015_power__eq__0__iff,axiom,
% 0.33/0.56      ! [A: int,N: nat] :
% 0.33/0.56        ( ( ( power_power_int @ A @ N )
% 0.33/0.56          = zero_zero_int )
% 0.33/0.56        = ( ( A = zero_zero_int )
% 0.33/0.56          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_eq_0_iff
% 0.33/0.56  thf(fact_2016_power__eq__0__iff,axiom,
% 0.33/0.56      ! [A: complex,N: nat] :
% 0.33/0.56        ( ( ( power_power_complex @ A @ N )
% 0.33/0.56          = zero_zero_complex )
% 0.33/0.56        = ( ( A = zero_zero_complex )
% 0.33/0.56          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_eq_0_iff
% 0.33/0.56  thf(fact_2017_power__not__zero,axiom,
% 0.33/0.56      ! [A: rat,N: nat] :
% 0.33/0.56        ( ( A != zero_zero_rat )
% 0.33/0.56       => ( ( power_power_rat @ A @ N )
% 0.33/0.56         != zero_zero_rat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_not_zero
% 0.33/0.56  thf(fact_2018_power__not__zero,axiom,
% 0.33/0.56      ! [A: nat,N: nat] :
% 0.33/0.56        ( ( A != zero_zero_nat )
% 0.33/0.56       => ( ( power_power_nat @ A @ N )
% 0.33/0.56         != zero_zero_nat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_not_zero
% 0.33/0.56  thf(fact_2019_power__not__zero,axiom,
% 0.33/0.56      ! [A: real,N: nat] :
% 0.33/0.56        ( ( A != zero_zero_real )
% 0.33/0.56       => ( ( power_power_real @ A @ N )
% 0.33/0.56         != zero_zero_real ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_not_zero
% 0.33/0.56  thf(fact_2020_power__not__zero,axiom,
% 0.33/0.56      ! [A: int,N: nat] :
% 0.33/0.56        ( ( A != zero_zero_int )
% 0.33/0.56       => ( ( power_power_int @ A @ N )
% 0.33/0.56         != zero_zero_int ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_not_zero
% 0.33/0.56  thf(fact_2021_power__not__zero,axiom,
% 0.33/0.56      ! [A: complex,N: nat] :
% 0.33/0.56        ( ( A != zero_zero_complex )
% 0.33/0.56       => ( ( power_power_complex @ A @ N )
% 0.33/0.56         != zero_zero_complex ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_not_zero
% 0.33/0.56  thf(fact_2022_power__mono,axiom,
% 0.33/0.56      ! [A: real,B: real,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_real @ A @ B )
% 0.33/0.56       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.56         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_mono
% 0.33/0.56  thf(fact_2023_power__mono,axiom,
% 0.33/0.56      ! [A: rat,B: rat,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.56       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.56         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_mono
% 0.33/0.56  thf(fact_2024_power__mono,axiom,
% 0.33/0.56      ! [A: nat,B: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.56       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.33/0.56         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_mono
% 0.33/0.56  thf(fact_2025_power__mono,axiom,
% 0.33/0.56      ! [A: int,B: int,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_int @ A @ B )
% 0.33/0.56       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.56         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_mono
% 0.33/0.56  thf(fact_2026_zero__le__power,axiom,
% 0.33/0.56      ! [A: real,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.56       => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_le_power
% 0.33/0.56  thf(fact_2027_zero__le__power,axiom,
% 0.33/0.56      ! [A: rat,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.56       => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_le_power
% 0.33/0.56  thf(fact_2028_zero__le__power,axiom,
% 0.33/0.56      ! [A: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.33/0.56       => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_le_power
% 0.33/0.56  thf(fact_2029_zero__le__power,axiom,
% 0.33/0.56      ! [A: int,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.56       => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_le_power
% 0.33/0.56  thf(fact_2030_zero__less__power,axiom,
% 0.33/0.56      ! [A: real,N: nat] :
% 0.33/0.56        ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.56       => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_less_power
% 0.33/0.56  thf(fact_2031_zero__less__power,axiom,
% 0.33/0.56      ! [A: rat,N: nat] :
% 0.33/0.56        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.56       => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_less_power
% 0.33/0.56  thf(fact_2032_zero__less__power,axiom,
% 0.33/0.56      ! [A: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.33/0.56       => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_less_power
% 0.33/0.56  thf(fact_2033_zero__less__power,axiom,
% 0.33/0.56      ! [A: int,N: nat] :
% 0.33/0.56        ( ( ord_less_int @ zero_zero_int @ A )
% 0.33/0.56       => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_less_power
% 0.33/0.56  thf(fact_2034_nat__power__less__imp__less,axiom,
% 0.33/0.56      ! [I: nat,M: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ zero_zero_nat @ I )
% 0.33/0.56       => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 0.33/0.56         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nat_power_less_imp_less
% 0.33/0.56  thf(fact_2035_power__less__imp__less__base,axiom,
% 0.33/0.56      ! [A: real,N: nat,B: real] :
% 0.33/0.56        ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 0.33/0.56       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.33/0.56         => ( ord_less_real @ A @ B ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_less_imp_less_base
% 0.33/0.56  thf(fact_2036_power__less__imp__less__base,axiom,
% 0.33/0.56      ! [A: rat,N: nat,B: rat] :
% 0.33/0.56        ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 0.33/0.56       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.33/0.56         => ( ord_less_rat @ A @ B ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_less_imp_less_base
% 0.33/0.56  thf(fact_2037_power__less__imp__less__base,axiom,
% 0.33/0.56      ! [A: nat,N: nat,B: nat] :
% 0.33/0.56        ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.33/0.56       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.33/0.56         => ( ord_less_nat @ A @ B ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_less_imp_less_base
% 0.33/0.56  thf(fact_2038_power__less__imp__less__base,axiom,
% 0.33/0.56      ! [A: int,N: nat,B: int] :
% 0.33/0.56        ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.33/0.56       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.33/0.56         => ( ord_less_int @ A @ B ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_less_imp_less_base
% 0.33/0.56  thf(fact_2039_power__inject__base,axiom,
% 0.33/0.56      ! [A: real,N: nat,B: real] :
% 0.33/0.56        ( ( ( power_power_real @ A @ ( suc @ N ) )
% 0.33/0.56          = ( power_power_real @ B @ ( suc @ N ) ) )
% 0.33/0.56       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.56         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.33/0.56           => ( A = B ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_inject_base
% 0.33/0.56  thf(fact_2040_power__inject__base,axiom,
% 0.33/0.56      ! [A: rat,N: nat,B: rat] :
% 0.33/0.56        ( ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.33/0.56          = ( power_power_rat @ B @ ( suc @ N ) ) )
% 0.33/0.56       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.56         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.33/0.56           => ( A = B ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_inject_base
% 0.33/0.56  thf(fact_2041_power__inject__base,axiom,
% 0.33/0.56      ! [A: nat,N: nat,B: nat] :
% 0.33/0.56        ( ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.33/0.56          = ( power_power_nat @ B @ ( suc @ N ) ) )
% 0.33/0.56       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.33/0.56         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.33/0.56           => ( A = B ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_inject_base
% 0.33/0.56  thf(fact_2042_power__inject__base,axiom,
% 0.33/0.56      ! [A: int,N: nat,B: int] :
% 0.33/0.56        ( ( ( power_power_int @ A @ ( suc @ N ) )
% 0.33/0.56          = ( power_power_int @ B @ ( suc @ N ) ) )
% 0.33/0.56       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.56         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.33/0.56           => ( A = B ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_inject_base
% 0.33/0.56  thf(fact_2043_power__le__imp__le__base,axiom,
% 0.33/0.56      ! [A: real,N: nat,B: real] :
% 0.33/0.56        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
% 0.33/0.56       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.33/0.56         => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_le_imp_le_base
% 0.33/0.56  thf(fact_2044_power__le__imp__le__base,axiom,
% 0.33/0.56      ! [A: rat,N: nat,B: rat] :
% 0.33/0.56        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
% 0.33/0.56       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.33/0.56         => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_le_imp_le_base
% 0.33/0.56  thf(fact_2045_power__le__imp__le__base,axiom,
% 0.33/0.56      ! [A: nat,N: nat,B: nat] :
% 0.33/0.56        ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
% 0.33/0.56       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.33/0.56         => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_le_imp_le_base
% 0.33/0.56  thf(fact_2046_power__le__imp__le__base,axiom,
% 0.33/0.56      ! [A: int,N: nat,B: int] :
% 0.33/0.56        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
% 0.33/0.56       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.33/0.56         => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_le_imp_le_base
% 0.33/0.56  thf(fact_2047_zero__power,axiom,
% 0.33/0.56      ! [N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.56       => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.33/0.56          = zero_zero_rat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_power
% 0.33/0.56  thf(fact_2048_zero__power,axiom,
% 0.33/0.56      ! [N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.56       => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.33/0.56          = zero_zero_nat ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_power
% 0.33/0.56  thf(fact_2049_zero__power,axiom,
% 0.33/0.56      ! [N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.56       => ( ( power_power_real @ zero_zero_real @ N )
% 0.33/0.56          = zero_zero_real ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_power
% 0.33/0.56  thf(fact_2050_zero__power,axiom,
% 0.33/0.56      ! [N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.56       => ( ( power_power_int @ zero_zero_int @ N )
% 0.33/0.56          = zero_zero_int ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_power
% 0.33/0.56  thf(fact_2051_zero__power,axiom,
% 0.33/0.56      ! [N: nat] :
% 0.33/0.56        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.56       => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.33/0.56          = zero_zero_complex ) ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_power
% 0.33/0.56  thf(fact_2052_power__gt__expt,axiom,
% 0.33/0.56      ! [N: nat,K: nat] :
% 0.33/0.56        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.33/0.56       => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_gt_expt
% 0.33/0.56  thf(fact_2053_nat__one__le__power,axiom,
% 0.33/0.56      ! [I: nat,N: nat] :
% 0.33/0.56        ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 0.33/0.56       => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % nat_one_le_power
% 0.33/0.56  thf(fact_2054_zero__power2,axiom,
% 0.33/0.56      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.56      = zero_zero_rat ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_power2
% 0.33/0.56  thf(fact_2055_zero__power2,axiom,
% 0.33/0.56      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.56      = zero_zero_nat ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_power2
% 0.33/0.56  thf(fact_2056_zero__power2,axiom,
% 0.33/0.56      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.56      = zero_zero_real ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_power2
% 0.33/0.56  thf(fact_2057_zero__power2,axiom,
% 0.33/0.56      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.56      = zero_zero_int ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_power2
% 0.33/0.56  thf(fact_2058_zero__power2,axiom,
% 0.33/0.56      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.56      = zero_zero_complex ) ).
% 0.33/0.56  
% 0.33/0.56  % zero_power2
% 0.33/0.56  thf(fact_2059_power__eq__imp__eq__base,axiom,
% 0.33/0.56      ! [A: real,N: nat,B: real] :
% 0.33/0.56        ( ( ( power_power_real @ A @ N )
% 0.33/0.56          = ( power_power_real @ B @ N ) )
% 0.33/0.56       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.56         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.33/0.56           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.56             => ( A = B ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_eq_imp_eq_base
% 0.33/0.56  thf(fact_2060_power__eq__imp__eq__base,axiom,
% 0.33/0.56      ! [A: rat,N: nat,B: rat] :
% 0.33/0.56        ( ( ( power_power_rat @ A @ N )
% 0.33/0.56          = ( power_power_rat @ B @ N ) )
% 0.33/0.56       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.56         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.33/0.56           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.56             => ( A = B ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_eq_imp_eq_base
% 0.33/0.56  thf(fact_2061_power__eq__imp__eq__base,axiom,
% 0.33/0.56      ! [A: nat,N: nat,B: nat] :
% 0.33/0.56        ( ( ( power_power_nat @ A @ N )
% 0.33/0.56          = ( power_power_nat @ B @ N ) )
% 0.33/0.56       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.33/0.56         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.33/0.56           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.56             => ( A = B ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_eq_imp_eq_base
% 0.33/0.56  thf(fact_2062_power__eq__imp__eq__base,axiom,
% 0.33/0.56      ! [A: int,N: nat,B: int] :
% 0.33/0.56        ( ( ( power_power_int @ A @ N )
% 0.33/0.56          = ( power_power_int @ B @ N ) )
% 0.33/0.56       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.56         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.33/0.56           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.56             => ( A = B ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.56  % power_eq_imp_eq_base
% 0.33/0.56  thf(fact_2063_power__eq__iff__eq__base,axiom,
% 0.33/0.56      ! [N: nat,A: real,B: real] :
% 0.33/0.56        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.56       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.56         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.33/0.56           => ( ( ( power_power_real @ A @ N )
% 0.33/0.56                = ( power_power_real @ B @ N ) )
% 0.33/0.56              = ( A = B ) ) ) ) ) ).
% 0.33/0.56  
% 0.33/0.57  % power_eq_iff_eq_base
% 0.33/0.57  thf(fact_2064_power__eq__iff__eq__base,axiom,
% 0.33/0.57      ! [N: nat,A: rat,B: rat] :
% 0.33/0.57        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.57       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.57         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.33/0.57           => ( ( ( power_power_rat @ A @ N )
% 0.33/0.57                = ( power_power_rat @ B @ N ) )
% 0.33/0.57              = ( A = B ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_eq_iff_eq_base
% 0.33/0.57  thf(fact_2065_power__eq__iff__eq__base,axiom,
% 0.33/0.57      ! [N: nat,A: nat,B: nat] :
% 0.33/0.57        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.57       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.33/0.57         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.33/0.57           => ( ( ( power_power_nat @ A @ N )
% 0.33/0.57                = ( power_power_nat @ B @ N ) )
% 0.33/0.57              = ( A = B ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_eq_iff_eq_base
% 0.33/0.57  thf(fact_2066_power__eq__iff__eq__base,axiom,
% 0.33/0.57      ! [N: nat,A: int,B: int] :
% 0.33/0.57        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.57       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.57         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.33/0.57           => ( ( ( power_power_int @ A @ N )
% 0.33/0.57                = ( power_power_int @ B @ N ) )
% 0.33/0.57              = ( A = B ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_eq_iff_eq_base
% 0.33/0.57  thf(fact_2067_less__exp,axiom,
% 0.33/0.57      ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.33/0.57  
% 0.33/0.57  % less_exp
% 0.33/0.57  thf(fact_2068_power2__nat__le__imp__le,axiom,
% 0.33/0.57      ! [M: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 0.33/0.57       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power2_nat_le_imp_le
% 0.33/0.57  thf(fact_2069_power2__nat__le__eq__le,axiom,
% 0.33/0.57      ! [M: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.57        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power2_nat_le_eq_le
% 0.33/0.57  thf(fact_2070_self__le__ge2__pow,axiom,
% 0.33/0.57      ! [K: nat,M: nat] :
% 0.33/0.57        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.33/0.57       => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % self_le_ge2_pow
% 0.33/0.57  thf(fact_2071_zero__le__power2,axiom,
% 0.33/0.57      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % zero_le_power2
% 0.33/0.57  thf(fact_2072_zero__le__power2,axiom,
% 0.33/0.57      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % zero_le_power2
% 0.33/0.57  thf(fact_2073_zero__le__power2,axiom,
% 0.33/0.57      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % zero_le_power2
% 0.33/0.57  thf(fact_2074_power2__eq__imp__eq,axiom,
% 0.33/0.57      ! [X: real,Y: real] :
% 0.33/0.57        ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.57          = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.57       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.57         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.57           => ( X = Y ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power2_eq_imp_eq
% 0.33/0.57  thf(fact_2075_power2__eq__imp__eq,axiom,
% 0.33/0.57      ! [X: rat,Y: rat] :
% 0.33/0.57        ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.57          = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.57       => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.33/0.57         => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.33/0.57           => ( X = Y ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power2_eq_imp_eq
% 0.33/0.57  thf(fact_2076_power2__eq__imp__eq,axiom,
% 0.33/0.57      ! [X: nat,Y: nat] :
% 0.33/0.57        ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.57          = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.57       => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.33/0.57         => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.33/0.57           => ( X = Y ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power2_eq_imp_eq
% 0.33/0.57  thf(fact_2077_power2__eq__imp__eq,axiom,
% 0.33/0.57      ! [X: int,Y: int] :
% 0.33/0.57        ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.57          = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.57       => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.57         => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.57           => ( X = Y ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power2_eq_imp_eq
% 0.33/0.57  thf(fact_2078_power2__le__imp__le,axiom,
% 0.33/0.57      ! [X: real,Y: real] :
% 0.33/0.57        ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.57       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.57         => ( ord_less_eq_real @ X @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power2_le_imp_le
% 0.33/0.57  thf(fact_2079_power2__le__imp__le,axiom,
% 0.33/0.57      ! [X: rat,Y: rat] :
% 0.33/0.57        ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.57       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.33/0.57         => ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power2_le_imp_le
% 0.33/0.57  thf(fact_2080_power2__le__imp__le,axiom,
% 0.33/0.57      ! [X: nat,Y: nat] :
% 0.33/0.57        ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.57       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.33/0.57         => ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power2_le_imp_le
% 0.33/0.57  thf(fact_2081_power2__le__imp__le,axiom,
% 0.33/0.57      ! [X: int,Y: int] :
% 0.33/0.57        ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.57       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.57         => ( ord_less_eq_int @ X @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power2_le_imp_le
% 0.33/0.57  thf(fact_2082_power2__less__0,axiom,
% 0.33/0.57      ! [A: real] :
% 0.33/0.57        ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 0.33/0.57  
% 0.33/0.57  % power2_less_0
% 0.33/0.57  thf(fact_2083_power2__less__0,axiom,
% 0.33/0.57      ! [A: rat] :
% 0.33/0.57        ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 0.33/0.57  
% 0.33/0.57  % power2_less_0
% 0.33/0.57  thf(fact_2084_power2__less__0,axiom,
% 0.33/0.57      ! [A: int] :
% 0.33/0.57        ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 0.33/0.57  
% 0.33/0.57  % power2_less_0
% 0.33/0.57  thf(fact_2085_power__strict__mono,axiom,
% 0.33/0.57      ! [A: real,B: real,N: nat] :
% 0.33/0.57        ( ( ord_less_real @ A @ B )
% 0.33/0.57       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.57         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.57           => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_strict_mono
% 0.33/0.57  thf(fact_2086_power__strict__mono,axiom,
% 0.33/0.57      ! [A: rat,B: rat,N: nat] :
% 0.33/0.57        ( ( ord_less_rat @ A @ B )
% 0.33/0.57       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.57         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.57           => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_strict_mono
% 0.33/0.57  thf(fact_2087_power__strict__mono,axiom,
% 0.33/0.57      ! [A: nat,B: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_nat @ A @ B )
% 0.33/0.57       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.33/0.57         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.57           => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_strict_mono
% 0.33/0.57  thf(fact_2088_power__strict__mono,axiom,
% 0.33/0.57      ! [A: int,B: int,N: nat] :
% 0.33/0.57        ( ( ord_less_int @ A @ B )
% 0.33/0.57       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.57         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.33/0.57           => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_strict_mono
% 0.33/0.57  thf(fact_2089_power2__less__imp__less,axiom,
% 0.33/0.57      ! [X: real,Y: real] :
% 0.33/0.57        ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.57       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.57         => ( ord_less_real @ X @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power2_less_imp_less
% 0.33/0.57  thf(fact_2090_power2__less__imp__less,axiom,
% 0.33/0.57      ! [X: rat,Y: rat] :
% 0.33/0.57        ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.57       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.33/0.57         => ( ord_less_rat @ X @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power2_less_imp_less
% 0.33/0.57  thf(fact_2091_power2__less__imp__less,axiom,
% 0.33/0.57      ! [X: nat,Y: nat] :
% 0.33/0.57        ( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.57       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.33/0.57         => ( ord_less_nat @ X @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power2_less_imp_less
% 0.33/0.57  thf(fact_2092_power2__less__imp__less,axiom,
% 0.33/0.57      ! [X: int,Y: int] :
% 0.33/0.57        ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.57       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.57         => ( ord_less_int @ X @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power2_less_imp_less
% 0.33/0.57  thf(fact_2093_sum__power2__le__zero__iff,axiom,
% 0.33/0.57      ! [X: real,Y: real] :
% 0.33/0.57        ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
% 0.33/0.57        = ( ( X = zero_zero_real )
% 0.33/0.57          & ( Y = zero_zero_real ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % sum_power2_le_zero_iff
% 0.33/0.57  thf(fact_2094_sum__power2__le__zero__iff,axiom,
% 0.33/0.57      ! [X: rat,Y: rat] :
% 0.33/0.57        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
% 0.33/0.57        = ( ( X = zero_zero_rat )
% 0.33/0.57          & ( Y = zero_zero_rat ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % sum_power2_le_zero_iff
% 0.33/0.57  thf(fact_2095_sum__power2__le__zero__iff,axiom,
% 0.33/0.57      ! [X: int,Y: int] :
% 0.33/0.57        ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
% 0.33/0.57        = ( ( X = zero_zero_int )
% 0.33/0.57          & ( Y = zero_zero_int ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % sum_power2_le_zero_iff
% 0.33/0.57  thf(fact_2096_sum__power2__ge__zero,axiom,
% 0.33/0.57      ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % sum_power2_ge_zero
% 0.33/0.57  thf(fact_2097_sum__power2__ge__zero,axiom,
% 0.33/0.57      ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % sum_power2_ge_zero
% 0.33/0.57  thf(fact_2098_sum__power2__ge__zero,axiom,
% 0.33/0.57      ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % sum_power2_ge_zero
% 0.33/0.57  thf(fact_2099_not__sum__power2__lt__zero,axiom,
% 0.33/0.57      ! [X: real,Y: real] :
% 0.33/0.57        ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% 0.33/0.57  
% 0.33/0.57  % not_sum_power2_lt_zero
% 0.33/0.57  thf(fact_2100_not__sum__power2__lt__zero,axiom,
% 0.33/0.57      ! [X: rat,Y: rat] :
% 0.33/0.57        ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% 0.33/0.57  
% 0.33/0.57  % not_sum_power2_lt_zero
% 0.33/0.57  thf(fact_2101_not__sum__power2__lt__zero,axiom,
% 0.33/0.57      ! [X: int,Y: int] :
% 0.33/0.57        ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% 0.33/0.57  
% 0.33/0.57  % not_sum_power2_lt_zero
% 0.33/0.57  thf(fact_2102_sum__power2__gt__zero__iff,axiom,
% 0.33/0.57      ! [X: real,Y: real] :
% 0.33/0.57        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.57        = ( ( X != zero_zero_real )
% 0.33/0.57          | ( Y != zero_zero_real ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % sum_power2_gt_zero_iff
% 0.33/0.57  thf(fact_2103_sum__power2__gt__zero__iff,axiom,
% 0.33/0.57      ! [X: rat,Y: rat] :
% 0.33/0.57        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.57        = ( ( X != zero_zero_rat )
% 0.33/0.57          | ( Y != zero_zero_rat ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % sum_power2_gt_zero_iff
% 0.33/0.57  thf(fact_2104_sum__power2__gt__zero__iff,axiom,
% 0.33/0.57      ! [X: int,Y: int] :
% 0.33/0.57        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.57        = ( ( X != zero_zero_int )
% 0.33/0.57          | ( Y != zero_zero_int ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % sum_power2_gt_zero_iff
% 0.33/0.57  thf(fact_2105_member__inv,axiom,
% 0.33/0.57      ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 0.33/0.57        ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.33/0.57       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.33/0.57          & ( ( X = Mi )
% 0.33/0.57            | ( X = Ma )
% 0.33/0.57            | ( ( ord_less_nat @ X @ Ma )
% 0.33/0.57              & ( ord_less_nat @ Mi @ X )
% 0.33/0.57              & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.33/0.57              & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % member_inv
% 0.33/0.57  thf(fact_2106_pred__list__to__short,axiom,
% 0.33/0.57      ! [Deg: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
% 0.33/0.57        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.33/0.57       => ( ( ord_less_eq_nat @ X @ Ma )
% 0.33/0.57         => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.57           => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.33/0.57              = none_nat ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % pred_list_to_short
% 0.33/0.57  thf(fact_2107_succ__list__to__short,axiom,
% 0.33/0.57      ! [Deg: nat,Mi: nat,X: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 0.33/0.57        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.33/0.57       => ( ( ord_less_eq_nat @ Mi @ X )
% 0.33/0.57         => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.57           => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.33/0.57              = none_nat ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % succ_list_to_short
% 0.33/0.57  thf(fact_2108_both__member__options__ding,axiom,
% 0.33/0.57      ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
% 0.33/0.57        ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
% 0.33/0.57       => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.33/0.57         => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.57           => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % both_member_options_ding
% 0.33/0.57  thf(fact_2109_semiring__norm_I76_J,axiom,
% 0.33/0.57      ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% 0.33/0.57  
% 0.33/0.57  % semiring_norm(76)
% 0.33/0.57  thf(fact_2110_set__n__deg__not__0,axiom,
% 0.33/0.57      ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
% 0.33/0.57        ( ! [X4: vEBT_VEBT] :
% 0.33/0.57            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.33/0.57           => ( vEBT_invar_vebt @ X4 @ N ) )
% 0.33/0.57       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.33/0.57            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.33/0.57         => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % set_n_deg_not_0
% 0.33/0.57  thf(fact_2111_vebt__insert_Osimps_I4_J,axiom,
% 0.33/0.57      ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 0.33/0.57        ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
% 0.33/0.57        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) ) ).
% 0.33/0.57  
% 0.33/0.57  % vebt_insert.simps(4)
% 0.33/0.57  thf(fact_2112_exp__add__not__zero__imp__right,axiom,
% 0.33/0.57      ! [M: nat,N: nat] :
% 0.33/0.57        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.33/0.57         != zero_zero_nat )
% 0.33/0.57       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.57         != zero_zero_nat ) ) ).
% 0.33/0.57  
% 0.33/0.57  % exp_add_not_zero_imp_right
% 0.33/0.57  thf(fact_2113_exp__add__not__zero__imp__right,axiom,
% 0.33/0.57      ! [M: nat,N: nat] :
% 0.33/0.57        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.33/0.57         != zero_zero_int )
% 0.33/0.57       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.33/0.57         != zero_zero_int ) ) ).
% 0.33/0.57  
% 0.33/0.57  % exp_add_not_zero_imp_right
% 0.33/0.57  thf(fact_2114_exp__add__not__zero__imp__left,axiom,
% 0.33/0.57      ! [M: nat,N: nat] :
% 0.33/0.57        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.33/0.57         != zero_zero_nat )
% 0.33/0.57       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.33/0.57         != zero_zero_nat ) ) ).
% 0.33/0.57  
% 0.33/0.57  % exp_add_not_zero_imp_left
% 0.33/0.57  thf(fact_2115_exp__add__not__zero__imp__left,axiom,
% 0.33/0.57      ! [M: nat,N: nat] :
% 0.33/0.57        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.33/0.57         != zero_zero_int )
% 0.33/0.57       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.33/0.57         != zero_zero_int ) ) ).
% 0.33/0.57  
% 0.33/0.57  % exp_add_not_zero_imp_left
% 0.33/0.57  thf(fact_2116_bit__concat__def,axiom,
% 0.33/0.57      ( vEBT_VEBT_bit_concat
% 0.33/0.57      = ( ^ [H: nat,L2: nat,D4: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D4 ) ) @ L2 ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % bit_concat_def
% 0.33/0.57  thf(fact_2117_pow__sum,axiom,
% 0.33/0.57      ! [A: nat,B: nat] :
% 0.33/0.57        ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.33/0.57        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).
% 0.33/0.57  
% 0.33/0.57  % pow_sum
% 0.33/0.57  thf(fact_2118_high__def,axiom,
% 0.33/0.57      ( vEBT_VEBT_high
% 0.33/0.57      = ( ^ [X3: nat,N2: nat] : ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % high_def
% 0.33/0.57  thf(fact_2119__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
% 0.33/0.57      ( ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.57      = na ) ).
% 0.33/0.57  
% 0.33/0.57  % \<open>deg div 2 = n\<close>
% 0.33/0.57  thf(fact_2120_mult__zero__left,axiom,
% 0.33/0.57      ! [A: complex] :
% 0.33/0.57        ( ( times_times_complex @ zero_zero_complex @ A )
% 0.33/0.57        = zero_zero_complex ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_zero_left
% 0.33/0.57  thf(fact_2121_mult__zero__left,axiom,
% 0.33/0.57      ! [A: real] :
% 0.33/0.57        ( ( times_times_real @ zero_zero_real @ A )
% 0.33/0.57        = zero_zero_real ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_zero_left
% 0.33/0.57  thf(fact_2122_mult__zero__left,axiom,
% 0.33/0.57      ! [A: rat] :
% 0.33/0.57        ( ( times_times_rat @ zero_zero_rat @ A )
% 0.33/0.57        = zero_zero_rat ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_zero_left
% 0.33/0.57  thf(fact_2123_mult__zero__left,axiom,
% 0.33/0.57      ! [A: nat] :
% 0.33/0.57        ( ( times_times_nat @ zero_zero_nat @ A )
% 0.33/0.57        = zero_zero_nat ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_zero_left
% 0.33/0.57  thf(fact_2124_mult__zero__left,axiom,
% 0.33/0.57      ! [A: int] :
% 0.33/0.57        ( ( times_times_int @ zero_zero_int @ A )
% 0.33/0.57        = zero_zero_int ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_zero_left
% 0.33/0.57  thf(fact_2125_mult__zero__right,axiom,
% 0.33/0.57      ! [A: complex] :
% 0.33/0.57        ( ( times_times_complex @ A @ zero_zero_complex )
% 0.33/0.57        = zero_zero_complex ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_zero_right
% 0.33/0.57  thf(fact_2126_mult__zero__right,axiom,
% 0.33/0.57      ! [A: real] :
% 0.33/0.57        ( ( times_times_real @ A @ zero_zero_real )
% 0.33/0.57        = zero_zero_real ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_zero_right
% 0.33/0.57  thf(fact_2127_mult__zero__right,axiom,
% 0.33/0.57      ! [A: rat] :
% 0.33/0.57        ( ( times_times_rat @ A @ zero_zero_rat )
% 0.33/0.57        = zero_zero_rat ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_zero_right
% 0.33/0.57  thf(fact_2128_mult__zero__right,axiom,
% 0.33/0.57      ! [A: nat] :
% 0.33/0.57        ( ( times_times_nat @ A @ zero_zero_nat )
% 0.33/0.57        = zero_zero_nat ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_zero_right
% 0.33/0.57  thf(fact_2129_mult__zero__right,axiom,
% 0.33/0.57      ! [A: int] :
% 0.33/0.57        ( ( times_times_int @ A @ zero_zero_int )
% 0.33/0.57        = zero_zero_int ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_zero_right
% 0.33/0.57  thf(fact_2130_mult__eq__0__iff,axiom,
% 0.33/0.57      ! [A: complex,B: complex] :
% 0.33/0.57        ( ( ( times_times_complex @ A @ B )
% 0.33/0.57          = zero_zero_complex )
% 0.33/0.57        = ( ( A = zero_zero_complex )
% 0.33/0.57          | ( B = zero_zero_complex ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_eq_0_iff
% 0.33/0.57  thf(fact_2131_mult__eq__0__iff,axiom,
% 0.33/0.57      ! [A: real,B: real] :
% 0.33/0.57        ( ( ( times_times_real @ A @ B )
% 0.33/0.57          = zero_zero_real )
% 0.33/0.57        = ( ( A = zero_zero_real )
% 0.33/0.57          | ( B = zero_zero_real ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_eq_0_iff
% 0.33/0.57  thf(fact_2132_mult__eq__0__iff,axiom,
% 0.33/0.57      ! [A: rat,B: rat] :
% 0.33/0.57        ( ( ( times_times_rat @ A @ B )
% 0.33/0.57          = zero_zero_rat )
% 0.33/0.57        = ( ( A = zero_zero_rat )
% 0.33/0.57          | ( B = zero_zero_rat ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_eq_0_iff
% 0.33/0.57  thf(fact_2133_mult__eq__0__iff,axiom,
% 0.33/0.57      ! [A: nat,B: nat] :
% 0.33/0.57        ( ( ( times_times_nat @ A @ B )
% 0.33/0.57          = zero_zero_nat )
% 0.33/0.57        = ( ( A = zero_zero_nat )
% 0.33/0.57          | ( B = zero_zero_nat ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_eq_0_iff
% 0.33/0.57  thf(fact_2134_mult__eq__0__iff,axiom,
% 0.33/0.57      ! [A: int,B: int] :
% 0.33/0.57        ( ( ( times_times_int @ A @ B )
% 0.33/0.57          = zero_zero_int )
% 0.33/0.57        = ( ( A = zero_zero_int )
% 0.33/0.57          | ( B = zero_zero_int ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_eq_0_iff
% 0.33/0.57  thf(fact_2135_mult__cancel__left,axiom,
% 0.33/0.57      ! [C: complex,A: complex,B: complex] :
% 0.33/0.57        ( ( ( times_times_complex @ C @ A )
% 0.33/0.57          = ( times_times_complex @ C @ B ) )
% 0.33/0.57        = ( ( C = zero_zero_complex )
% 0.33/0.57          | ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_left
% 0.33/0.57  thf(fact_2136_mult__cancel__left,axiom,
% 0.33/0.57      ! [C: real,A: real,B: real] :
% 0.33/0.57        ( ( ( times_times_real @ C @ A )
% 0.33/0.57          = ( times_times_real @ C @ B ) )
% 0.33/0.57        = ( ( C = zero_zero_real )
% 0.33/0.57          | ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_left
% 0.33/0.57  thf(fact_2137_mult__cancel__left,axiom,
% 0.33/0.57      ! [C: rat,A: rat,B: rat] :
% 0.33/0.57        ( ( ( times_times_rat @ C @ A )
% 0.33/0.57          = ( times_times_rat @ C @ B ) )
% 0.33/0.57        = ( ( C = zero_zero_rat )
% 0.33/0.57          | ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_left
% 0.33/0.57  thf(fact_2138_mult__cancel__left,axiom,
% 0.33/0.57      ! [C: nat,A: nat,B: nat] :
% 0.33/0.57        ( ( ( times_times_nat @ C @ A )
% 0.33/0.57          = ( times_times_nat @ C @ B ) )
% 0.33/0.57        = ( ( C = zero_zero_nat )
% 0.33/0.57          | ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_left
% 0.33/0.57  thf(fact_2139_mult__cancel__left,axiom,
% 0.33/0.57      ! [C: int,A: int,B: int] :
% 0.33/0.57        ( ( ( times_times_int @ C @ A )
% 0.33/0.57          = ( times_times_int @ C @ B ) )
% 0.33/0.57        = ( ( C = zero_zero_int )
% 0.33/0.57          | ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_left
% 0.33/0.57  thf(fact_2140_mult__cancel__right,axiom,
% 0.33/0.57      ! [A: complex,C: complex,B: complex] :
% 0.33/0.57        ( ( ( times_times_complex @ A @ C )
% 0.33/0.57          = ( times_times_complex @ B @ C ) )
% 0.33/0.57        = ( ( C = zero_zero_complex )
% 0.33/0.57          | ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_right
% 0.33/0.57  thf(fact_2141_mult__cancel__right,axiom,
% 0.33/0.57      ! [A: real,C: real,B: real] :
% 0.33/0.57        ( ( ( times_times_real @ A @ C )
% 0.33/0.57          = ( times_times_real @ B @ C ) )
% 0.33/0.57        = ( ( C = zero_zero_real )
% 0.33/0.57          | ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_right
% 0.33/0.57  thf(fact_2142_mult__cancel__right,axiom,
% 0.33/0.57      ! [A: rat,C: rat,B: rat] :
% 0.33/0.57        ( ( ( times_times_rat @ A @ C )
% 0.33/0.57          = ( times_times_rat @ B @ C ) )
% 0.33/0.57        = ( ( C = zero_zero_rat )
% 0.33/0.57          | ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_right
% 0.33/0.57  thf(fact_2143_mult__cancel__right,axiom,
% 0.33/0.57      ! [A: nat,C: nat,B: nat] :
% 0.33/0.57        ( ( ( times_times_nat @ A @ C )
% 0.33/0.57          = ( times_times_nat @ B @ C ) )
% 0.33/0.57        = ( ( C = zero_zero_nat )
% 0.33/0.57          | ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_right
% 0.33/0.57  thf(fact_2144_mult__cancel__right,axiom,
% 0.33/0.57      ! [A: int,C: int,B: int] :
% 0.33/0.57        ( ( ( times_times_int @ A @ C )
% 0.33/0.57          = ( times_times_int @ B @ C ) )
% 0.33/0.57        = ( ( C = zero_zero_int )
% 0.33/0.57          | ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_right
% 0.33/0.57  thf(fact_2145_high__inv,axiom,
% 0.33/0.57      ! [X: nat,N: nat,Y: nat] :
% 0.33/0.57        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.57       => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
% 0.33/0.57          = Y ) ) ).
% 0.33/0.57  
% 0.33/0.57  % high_inv
% 0.33/0.57  thf(fact_2146_low__inv,axiom,
% 0.33/0.57      ! [X: nat,N: nat,Y: nat] :
% 0.33/0.57        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.33/0.57       => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
% 0.33/0.57          = X ) ) ).
% 0.33/0.57  
% 0.33/0.57  % low_inv
% 0.33/0.57  thf(fact_2147_bits__div__0,axiom,
% 0.33/0.57      ! [A: nat] :
% 0.33/0.57        ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.33/0.57        = zero_zero_nat ) ).
% 0.33/0.57  
% 0.33/0.57  % bits_div_0
% 0.33/0.57  thf(fact_2148_bits__div__0,axiom,
% 0.33/0.57      ! [A: int] :
% 0.33/0.57        ( ( divide_divide_int @ zero_zero_int @ A )
% 0.33/0.57        = zero_zero_int ) ).
% 0.33/0.57  
% 0.33/0.57  % bits_div_0
% 0.33/0.57  thf(fact_2149_bits__div__0,axiom,
% 0.33/0.57      ! [A: code_integer] :
% 0.33/0.57        ( ( divide6298287555418463151nteger @ zero_z3403309356797280102nteger @ A )
% 0.33/0.57        = zero_z3403309356797280102nteger ) ).
% 0.33/0.57  
% 0.33/0.57  % bits_div_0
% 0.33/0.57  thf(fact_2150_bits__div__by__0,axiom,
% 0.33/0.57      ! [A: nat] :
% 0.33/0.57        ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.33/0.57        = zero_zero_nat ) ).
% 0.33/0.57  
% 0.33/0.57  % bits_div_by_0
% 0.33/0.57  thf(fact_2151_bits__div__by__0,axiom,
% 0.33/0.57      ! [A: int] :
% 0.33/0.57        ( ( divide_divide_int @ A @ zero_zero_int )
% 0.33/0.57        = zero_zero_int ) ).
% 0.33/0.57  
% 0.33/0.57  % bits_div_by_0
% 0.33/0.57  thf(fact_2152_bits__div__by__0,axiom,
% 0.33/0.57      ! [A: code_integer] :
% 0.33/0.57        ( ( divide6298287555418463151nteger @ A @ zero_z3403309356797280102nteger )
% 0.33/0.57        = zero_z3403309356797280102nteger ) ).
% 0.33/0.57  
% 0.33/0.57  % bits_div_by_0
% 0.33/0.57  thf(fact_2153_div__0,axiom,
% 0.33/0.57      ! [A: complex] :
% 0.33/0.57        ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
% 0.33/0.57        = zero_zero_complex ) ).
% 0.33/0.57  
% 0.33/0.57  % div_0
% 0.33/0.57  thf(fact_2154_div__0,axiom,
% 0.33/0.57      ! [A: nat] :
% 0.33/0.57        ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.33/0.57        = zero_zero_nat ) ).
% 0.33/0.57  
% 0.33/0.57  % div_0
% 0.33/0.57  thf(fact_2155_div__0,axiom,
% 0.33/0.57      ! [A: int] :
% 0.33/0.57        ( ( divide_divide_int @ zero_zero_int @ A )
% 0.33/0.57        = zero_zero_int ) ).
% 0.33/0.57  
% 0.33/0.57  % div_0
% 0.33/0.57  thf(fact_2156_div__0,axiom,
% 0.33/0.57      ! [A: real] :
% 0.33/0.57        ( ( divide_divide_real @ zero_zero_real @ A )
% 0.33/0.57        = zero_zero_real ) ).
% 0.33/0.57  
% 0.33/0.57  % div_0
% 0.33/0.57  thf(fact_2157_div__0,axiom,
% 0.33/0.57      ! [A: rat] :
% 0.33/0.57        ( ( divide_divide_rat @ zero_zero_rat @ A )
% 0.33/0.57        = zero_zero_rat ) ).
% 0.33/0.57  
% 0.33/0.57  % div_0
% 0.33/0.57  thf(fact_2158_div__0,axiom,
% 0.33/0.57      ! [A: code_integer] :
% 0.33/0.57        ( ( divide6298287555418463151nteger @ zero_z3403309356797280102nteger @ A )
% 0.33/0.57        = zero_z3403309356797280102nteger ) ).
% 0.33/0.57  
% 0.33/0.57  % div_0
% 0.33/0.57  thf(fact_2159_divide__eq__0__iff,axiom,
% 0.33/0.57      ! [A: complex,B: complex] :
% 0.33/0.57        ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.33/0.57          = zero_zero_complex )
% 0.33/0.57        = ( ( A = zero_zero_complex )
% 0.33/0.57          | ( B = zero_zero_complex ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_eq_0_iff
% 0.33/0.57  thf(fact_2160_divide__eq__0__iff,axiom,
% 0.33/0.57      ! [A: real,B: real] :
% 0.33/0.57        ( ( ( divide_divide_real @ A @ B )
% 0.33/0.57          = zero_zero_real )
% 0.33/0.57        = ( ( A = zero_zero_real )
% 0.33/0.57          | ( B = zero_zero_real ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_eq_0_iff
% 0.33/0.57  thf(fact_2161_divide__eq__0__iff,axiom,
% 0.33/0.57      ! [A: rat,B: rat] :
% 0.33/0.57        ( ( ( divide_divide_rat @ A @ B )
% 0.33/0.57          = zero_zero_rat )
% 0.33/0.57        = ( ( A = zero_zero_rat )
% 0.33/0.57          | ( B = zero_zero_rat ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_eq_0_iff
% 0.33/0.57  thf(fact_2162_div__by__0,axiom,
% 0.33/0.57      ! [A: complex] :
% 0.33/0.57        ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 0.33/0.57        = zero_zero_complex ) ).
% 0.33/0.57  
% 0.33/0.57  % div_by_0
% 0.33/0.57  thf(fact_2163_div__by__0,axiom,
% 0.33/0.57      ! [A: nat] :
% 0.33/0.57        ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.33/0.57        = zero_zero_nat ) ).
% 0.33/0.57  
% 0.33/0.57  % div_by_0
% 0.33/0.57  thf(fact_2164_div__by__0,axiom,
% 0.33/0.57      ! [A: int] :
% 0.33/0.57        ( ( divide_divide_int @ A @ zero_zero_int )
% 0.33/0.57        = zero_zero_int ) ).
% 0.33/0.57  
% 0.33/0.57  % div_by_0
% 0.33/0.57  thf(fact_2165_div__by__0,axiom,
% 0.33/0.57      ! [A: real] :
% 0.33/0.57        ( ( divide_divide_real @ A @ zero_zero_real )
% 0.33/0.57        = zero_zero_real ) ).
% 0.33/0.57  
% 0.33/0.57  % div_by_0
% 0.33/0.57  thf(fact_2166_div__by__0,axiom,
% 0.33/0.57      ! [A: rat] :
% 0.33/0.57        ( ( divide_divide_rat @ A @ zero_zero_rat )
% 0.33/0.57        = zero_zero_rat ) ).
% 0.33/0.57  
% 0.33/0.57  % div_by_0
% 0.33/0.57  thf(fact_2167_div__by__0,axiom,
% 0.33/0.57      ! [A: code_integer] :
% 0.33/0.57        ( ( divide6298287555418463151nteger @ A @ zero_z3403309356797280102nteger )
% 0.33/0.57        = zero_z3403309356797280102nteger ) ).
% 0.33/0.57  
% 0.33/0.57  % div_by_0
% 0.33/0.57  thf(fact_2168_divide__cancel__left,axiom,
% 0.33/0.57      ! [C: complex,A: complex,B: complex] :
% 0.33/0.57        ( ( ( divide1717551699836669952omplex @ C @ A )
% 0.33/0.57          = ( divide1717551699836669952omplex @ C @ B ) )
% 0.33/0.57        = ( ( C = zero_zero_complex )
% 0.33/0.57          | ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_cancel_left
% 0.33/0.57  thf(fact_2169_divide__cancel__left,axiom,
% 0.33/0.57      ! [C: real,A: real,B: real] :
% 0.33/0.57        ( ( ( divide_divide_real @ C @ A )
% 0.33/0.57          = ( divide_divide_real @ C @ B ) )
% 0.33/0.57        = ( ( C = zero_zero_real )
% 0.33/0.57          | ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_cancel_left
% 0.33/0.57  thf(fact_2170_divide__cancel__left,axiom,
% 0.33/0.57      ! [C: rat,A: rat,B: rat] :
% 0.33/0.57        ( ( ( divide_divide_rat @ C @ A )
% 0.33/0.57          = ( divide_divide_rat @ C @ B ) )
% 0.33/0.57        = ( ( C = zero_zero_rat )
% 0.33/0.57          | ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_cancel_left
% 0.33/0.57  thf(fact_2171_divide__cancel__right,axiom,
% 0.33/0.57      ! [A: complex,C: complex,B: complex] :
% 0.33/0.57        ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.33/0.57          = ( divide1717551699836669952omplex @ B @ C ) )
% 0.33/0.57        = ( ( C = zero_zero_complex )
% 0.33/0.57          | ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_cancel_right
% 0.33/0.57  thf(fact_2172_divide__cancel__right,axiom,
% 0.33/0.57      ! [A: real,C: real,B: real] :
% 0.33/0.57        ( ( ( divide_divide_real @ A @ C )
% 0.33/0.57          = ( divide_divide_real @ B @ C ) )
% 0.33/0.57        = ( ( C = zero_zero_real )
% 0.33/0.57          | ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_cancel_right
% 0.33/0.57  thf(fact_2173_divide__cancel__right,axiom,
% 0.33/0.57      ! [A: rat,C: rat,B: rat] :
% 0.33/0.57        ( ( ( divide_divide_rat @ A @ C )
% 0.33/0.57          = ( divide_divide_rat @ B @ C ) )
% 0.33/0.57        = ( ( C = zero_zero_rat )
% 0.33/0.57          | ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_cancel_right
% 0.33/0.57  thf(fact_2174_division__ring__divide__zero,axiom,
% 0.33/0.57      ! [A: complex] :
% 0.33/0.57        ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 0.33/0.57        = zero_zero_complex ) ).
% 0.33/0.57  
% 0.33/0.57  % division_ring_divide_zero
% 0.33/0.57  thf(fact_2175_division__ring__divide__zero,axiom,
% 0.33/0.57      ! [A: real] :
% 0.33/0.57        ( ( divide_divide_real @ A @ zero_zero_real )
% 0.33/0.57        = zero_zero_real ) ).
% 0.33/0.57  
% 0.33/0.57  % division_ring_divide_zero
% 0.33/0.57  thf(fact_2176_division__ring__divide__zero,axiom,
% 0.33/0.57      ! [A: rat] :
% 0.33/0.57        ( ( divide_divide_rat @ A @ zero_zero_rat )
% 0.33/0.57        = zero_zero_rat ) ).
% 0.33/0.57  
% 0.33/0.57  % division_ring_divide_zero
% 0.33/0.57  thf(fact_2177_mult__1,axiom,
% 0.33/0.57      ! [A: complex] :
% 0.33/0.57        ( ( times_times_complex @ one_one_complex @ A )
% 0.33/0.57        = A ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_1
% 0.33/0.57  thf(fact_2178_mult__1,axiom,
% 0.33/0.57      ! [A: real] :
% 0.33/0.57        ( ( times_times_real @ one_one_real @ A )
% 0.33/0.57        = A ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_1
% 0.33/0.57  thf(fact_2179_mult__1,axiom,
% 0.33/0.57      ! [A: rat] :
% 0.33/0.57        ( ( times_times_rat @ one_one_rat @ A )
% 0.33/0.57        = A ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_1
% 0.33/0.57  thf(fact_2180_mult__1,axiom,
% 0.33/0.57      ! [A: nat] :
% 0.33/0.57        ( ( times_times_nat @ one_one_nat @ A )
% 0.33/0.57        = A ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_1
% 0.33/0.57  thf(fact_2181_mult__1,axiom,
% 0.33/0.57      ! [A: int] :
% 0.33/0.57        ( ( times_times_int @ one_one_int @ A )
% 0.33/0.57        = A ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_1
% 0.33/0.57  thf(fact_2182_mult_Oright__neutral,axiom,
% 0.33/0.57      ! [A: complex] :
% 0.33/0.57        ( ( times_times_complex @ A @ one_one_complex )
% 0.33/0.57        = A ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.right_neutral
% 0.33/0.57  thf(fact_2183_mult_Oright__neutral,axiom,
% 0.33/0.57      ! [A: real] :
% 0.33/0.57        ( ( times_times_real @ A @ one_one_real )
% 0.33/0.57        = A ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.right_neutral
% 0.33/0.57  thf(fact_2184_mult_Oright__neutral,axiom,
% 0.33/0.57      ! [A: rat] :
% 0.33/0.57        ( ( times_times_rat @ A @ one_one_rat )
% 0.33/0.57        = A ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.right_neutral
% 0.33/0.57  thf(fact_2185_mult_Oright__neutral,axiom,
% 0.33/0.57      ! [A: nat] :
% 0.33/0.57        ( ( times_times_nat @ A @ one_one_nat )
% 0.33/0.57        = A ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.right_neutral
% 0.33/0.57  thf(fact_2186_mult_Oright__neutral,axiom,
% 0.33/0.57      ! [A: int] :
% 0.33/0.57        ( ( times_times_int @ A @ one_one_int )
% 0.33/0.57        = A ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.right_neutral
% 0.33/0.57  thf(fact_2187_mult__is__0,axiom,
% 0.33/0.57      ! [M: nat,N: nat] :
% 0.33/0.57        ( ( ( times_times_nat @ M @ N )
% 0.33/0.57          = zero_zero_nat )
% 0.33/0.57        = ( ( M = zero_zero_nat )
% 0.33/0.57          | ( N = zero_zero_nat ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_is_0
% 0.33/0.57  thf(fact_2188_mult__0__right,axiom,
% 0.33/0.57      ! [M: nat] :
% 0.33/0.57        ( ( times_times_nat @ M @ zero_zero_nat )
% 0.33/0.57        = zero_zero_nat ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_0_right
% 0.33/0.57  thf(fact_2189_mult__cancel1,axiom,
% 0.33/0.57      ! [K: nat,M: nat,N: nat] :
% 0.33/0.57        ( ( ( times_times_nat @ K @ M )
% 0.33/0.57          = ( times_times_nat @ K @ N ) )
% 0.33/0.57        = ( ( M = N )
% 0.33/0.57          | ( K = zero_zero_nat ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel1
% 0.33/0.57  thf(fact_2190_mult__cancel2,axiom,
% 0.33/0.57      ! [M: nat,K: nat,N: nat] :
% 0.33/0.57        ( ( ( times_times_nat @ M @ K )
% 0.33/0.57          = ( times_times_nat @ N @ K ) )
% 0.33/0.57        = ( ( M = N )
% 0.33/0.57          | ( K = zero_zero_nat ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel2
% 0.33/0.57  thf(fact_2191_nat__mult__eq__1__iff,axiom,
% 0.33/0.57      ! [M: nat,N: nat] :
% 0.33/0.57        ( ( ( times_times_nat @ M @ N )
% 0.33/0.57          = one_one_nat )
% 0.33/0.57        = ( ( M = one_one_nat )
% 0.33/0.57          & ( N = one_one_nat ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nat_mult_eq_1_iff
% 0.33/0.57  thf(fact_2192_nat__1__eq__mult__iff,axiom,
% 0.33/0.57      ! [M: nat,N: nat] :
% 0.33/0.57        ( ( one_one_nat
% 0.33/0.57          = ( times_times_nat @ M @ N ) )
% 0.33/0.57        = ( ( M = one_one_nat )
% 0.33/0.57          & ( N = one_one_nat ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nat_1_eq_mult_iff
% 0.33/0.57  thf(fact_2193_semiring__norm_I78_J,axiom,
% 0.33/0.57      ! [M: num,N: num] :
% 0.33/0.57        ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.33/0.57        = ( ord_less_num @ M @ N ) ) ).
% 0.33/0.57  
% 0.33/0.57  % semiring_norm(78)
% 0.33/0.57  thf(fact_2194_semiring__norm_I75_J,axiom,
% 0.33/0.57      ! [M: num] :
% 0.33/0.57        ~ ( ord_less_num @ M @ one ) ).
% 0.33/0.57  
% 0.33/0.57  % semiring_norm(75)
% 0.33/0.57  thf(fact_2195_sum__squares__eq__zero__iff,axiom,
% 0.33/0.57      ! [X: real,Y: real] :
% 0.33/0.57        ( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
% 0.33/0.57          = zero_zero_real )
% 0.33/0.57        = ( ( X = zero_zero_real )
% 0.33/0.57          & ( Y = zero_zero_real ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % sum_squares_eq_zero_iff
% 0.33/0.57  thf(fact_2196_sum__squares__eq__zero__iff,axiom,
% 0.33/0.57      ! [X: rat,Y: rat] :
% 0.33/0.57        ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
% 0.33/0.57          = zero_zero_rat )
% 0.33/0.57        = ( ( X = zero_zero_rat )
% 0.33/0.57          & ( Y = zero_zero_rat ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % sum_squares_eq_zero_iff
% 0.33/0.57  thf(fact_2197_sum__squares__eq__zero__iff,axiom,
% 0.33/0.57      ! [X: int,Y: int] :
% 0.33/0.57        ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
% 0.33/0.57          = zero_zero_int )
% 0.33/0.57        = ( ( X = zero_zero_int )
% 0.33/0.57          & ( Y = zero_zero_int ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % sum_squares_eq_zero_iff
% 0.33/0.57  thf(fact_2198_mult__cancel__left1,axiom,
% 0.33/0.57      ! [C: complex,B: complex] :
% 0.33/0.57        ( ( C
% 0.33/0.57          = ( times_times_complex @ C @ B ) )
% 0.33/0.57        = ( ( C = zero_zero_complex )
% 0.33/0.57          | ( B = one_one_complex ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_left1
% 0.33/0.57  thf(fact_2199_mult__cancel__left1,axiom,
% 0.33/0.57      ! [C: real,B: real] :
% 0.33/0.57        ( ( C
% 0.33/0.57          = ( times_times_real @ C @ B ) )
% 0.33/0.57        = ( ( C = zero_zero_real )
% 0.33/0.57          | ( B = one_one_real ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_left1
% 0.33/0.57  thf(fact_2200_mult__cancel__left1,axiom,
% 0.33/0.57      ! [C: rat,B: rat] :
% 0.33/0.57        ( ( C
% 0.33/0.57          = ( times_times_rat @ C @ B ) )
% 0.33/0.57        = ( ( C = zero_zero_rat )
% 0.33/0.57          | ( B = one_one_rat ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_left1
% 0.33/0.57  thf(fact_2201_mult__cancel__left1,axiom,
% 0.33/0.57      ! [C: int,B: int] :
% 0.33/0.57        ( ( C
% 0.33/0.57          = ( times_times_int @ C @ B ) )
% 0.33/0.57        = ( ( C = zero_zero_int )
% 0.33/0.57          | ( B = one_one_int ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_left1
% 0.33/0.57  thf(fact_2202_mult__cancel__left2,axiom,
% 0.33/0.57      ! [C: complex,A: complex] :
% 0.33/0.57        ( ( ( times_times_complex @ C @ A )
% 0.33/0.57          = C )
% 0.33/0.57        = ( ( C = zero_zero_complex )
% 0.33/0.57          | ( A = one_one_complex ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_left2
% 0.33/0.57  thf(fact_2203_mult__cancel__left2,axiom,
% 0.33/0.57      ! [C: real,A: real] :
% 0.33/0.57        ( ( ( times_times_real @ C @ A )
% 0.33/0.57          = C )
% 0.33/0.57        = ( ( C = zero_zero_real )
% 0.33/0.57          | ( A = one_one_real ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_left2
% 0.33/0.57  thf(fact_2204_mult__cancel__left2,axiom,
% 0.33/0.57      ! [C: rat,A: rat] :
% 0.33/0.57        ( ( ( times_times_rat @ C @ A )
% 0.33/0.57          = C )
% 0.33/0.57        = ( ( C = zero_zero_rat )
% 0.33/0.57          | ( A = one_one_rat ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_left2
% 0.33/0.57  thf(fact_2205_mult__cancel__left2,axiom,
% 0.33/0.57      ! [C: int,A: int] :
% 0.33/0.57        ( ( ( times_times_int @ C @ A )
% 0.33/0.57          = C )
% 0.33/0.57        = ( ( C = zero_zero_int )
% 0.33/0.57          | ( A = one_one_int ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_left2
% 0.33/0.57  thf(fact_2206_mult__cancel__right1,axiom,
% 0.33/0.57      ! [C: complex,B: complex] :
% 0.33/0.57        ( ( C
% 0.33/0.57          = ( times_times_complex @ B @ C ) )
% 0.33/0.57        = ( ( C = zero_zero_complex )
% 0.33/0.57          | ( B = one_one_complex ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_right1
% 0.33/0.57  thf(fact_2207_mult__cancel__right1,axiom,
% 0.33/0.57      ! [C: real,B: real] :
% 0.33/0.57        ( ( C
% 0.33/0.57          = ( times_times_real @ B @ C ) )
% 0.33/0.57        = ( ( C = zero_zero_real )
% 0.33/0.57          | ( B = one_one_real ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_right1
% 0.33/0.57  thf(fact_2208_mult__cancel__right1,axiom,
% 0.33/0.57      ! [C: rat,B: rat] :
% 0.33/0.57        ( ( C
% 0.33/0.57          = ( times_times_rat @ B @ C ) )
% 0.33/0.57        = ( ( C = zero_zero_rat )
% 0.33/0.57          | ( B = one_one_rat ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_right1
% 0.33/0.57  thf(fact_2209_mult__cancel__right1,axiom,
% 0.33/0.57      ! [C: int,B: int] :
% 0.33/0.57        ( ( C
% 0.33/0.57          = ( times_times_int @ B @ C ) )
% 0.33/0.57        = ( ( C = zero_zero_int )
% 0.33/0.57          | ( B = one_one_int ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_right1
% 0.33/0.57  thf(fact_2210_mult__cancel__right2,axiom,
% 0.33/0.57      ! [A: complex,C: complex] :
% 0.33/0.57        ( ( ( times_times_complex @ A @ C )
% 0.33/0.57          = C )
% 0.33/0.57        = ( ( C = zero_zero_complex )
% 0.33/0.57          | ( A = one_one_complex ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_right2
% 0.33/0.57  thf(fact_2211_mult__cancel__right2,axiom,
% 0.33/0.57      ! [A: real,C: real] :
% 0.33/0.57        ( ( ( times_times_real @ A @ C )
% 0.33/0.57          = C )
% 0.33/0.57        = ( ( C = zero_zero_real )
% 0.33/0.57          | ( A = one_one_real ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_right2
% 0.33/0.57  thf(fact_2212_mult__cancel__right2,axiom,
% 0.33/0.57      ! [A: rat,C: rat] :
% 0.33/0.57        ( ( ( times_times_rat @ A @ C )
% 0.33/0.57          = C )
% 0.33/0.57        = ( ( C = zero_zero_rat )
% 0.33/0.57          | ( A = one_one_rat ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_right2
% 0.33/0.57  thf(fact_2213_mult__cancel__right2,axiom,
% 0.33/0.57      ! [A: int,C: int] :
% 0.33/0.57        ( ( ( times_times_int @ A @ C )
% 0.33/0.57          = C )
% 0.33/0.57        = ( ( C = zero_zero_int )
% 0.33/0.57          | ( A = one_one_int ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_cancel_right2
% 0.33/0.57  thf(fact_2214_mult__divide__mult__cancel__left__if,axiom,
% 0.33/0.57      ! [C: complex,A: complex,B: complex] :
% 0.33/0.57        ( ( ( C = zero_zero_complex )
% 0.33/0.57         => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.33/0.57            = zero_zero_complex ) )
% 0.33/0.57        & ( ( C != zero_zero_complex )
% 0.33/0.57         => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.33/0.57            = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_divide_mult_cancel_left_if
% 0.33/0.57  thf(fact_2215_mult__divide__mult__cancel__left__if,axiom,
% 0.33/0.57      ! [C: real,A: real,B: real] :
% 0.33/0.57        ( ( ( C = zero_zero_real )
% 0.33/0.57         => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.33/0.57            = zero_zero_real ) )
% 0.33/0.57        & ( ( C != zero_zero_real )
% 0.33/0.57         => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.33/0.57            = ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_divide_mult_cancel_left_if
% 0.33/0.57  thf(fact_2216_mult__divide__mult__cancel__left__if,axiom,
% 0.33/0.57      ! [C: rat,A: rat,B: rat] :
% 0.33/0.57        ( ( ( C = zero_zero_rat )
% 0.33/0.57         => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.33/0.57            = zero_zero_rat ) )
% 0.33/0.57        & ( ( C != zero_zero_rat )
% 0.33/0.57         => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.33/0.57            = ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_divide_mult_cancel_left_if
% 0.33/0.57  thf(fact_2217_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.33/0.57      ! [C: complex,A: complex,B: complex] :
% 0.33/0.57        ( ( C != zero_zero_complex )
% 0.33/0.57       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.33/0.57          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_divide_mult_cancel_left
% 0.33/0.57  thf(fact_2218_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.33/0.57      ! [C: real,A: real,B: real] :
% 0.33/0.57        ( ( C != zero_zero_real )
% 0.33/0.57       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.33/0.57          = ( divide_divide_real @ A @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_divide_mult_cancel_left
% 0.33/0.57  thf(fact_2219_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.33/0.57      ! [C: rat,A: rat,B: rat] :
% 0.33/0.57        ( ( C != zero_zero_rat )
% 0.33/0.57       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.33/0.57          = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_divide_mult_cancel_left
% 0.33/0.57  thf(fact_2220_nonzero__mult__div__cancel__left,axiom,
% 0.33/0.57      ! [A: complex,B: complex] :
% 0.33/0.57        ( ( A != zero_zero_complex )
% 0.33/0.57       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
% 0.33/0.57          = B ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_div_cancel_left
% 0.33/0.57  thf(fact_2221_nonzero__mult__div__cancel__left,axiom,
% 0.33/0.57      ! [A: nat,B: nat] :
% 0.33/0.57        ( ( A != zero_zero_nat )
% 0.33/0.57       => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
% 0.33/0.57          = B ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_div_cancel_left
% 0.33/0.57  thf(fact_2222_nonzero__mult__div__cancel__left,axiom,
% 0.33/0.57      ! [A: int,B: int] :
% 0.33/0.57        ( ( A != zero_zero_int )
% 0.33/0.57       => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
% 0.33/0.57          = B ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_div_cancel_left
% 0.33/0.57  thf(fact_2223_nonzero__mult__div__cancel__left,axiom,
% 0.33/0.57      ! [A: real,B: real] :
% 0.33/0.57        ( ( A != zero_zero_real )
% 0.33/0.57       => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
% 0.33/0.57          = B ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_div_cancel_left
% 0.33/0.57  thf(fact_2224_nonzero__mult__div__cancel__left,axiom,
% 0.33/0.57      ! [A: rat,B: rat] :
% 0.33/0.57        ( ( A != zero_zero_rat )
% 0.33/0.57       => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
% 0.33/0.57          = B ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_div_cancel_left
% 0.33/0.57  thf(fact_2225_nonzero__mult__div__cancel__left,axiom,
% 0.33/0.57      ! [A: code_integer,B: code_integer] :
% 0.33/0.57        ( ( A != zero_z3403309356797280102nteger )
% 0.33/0.57       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ A )
% 0.33/0.57          = B ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_div_cancel_left
% 0.33/0.57  thf(fact_2226_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.33/0.57      ! [C: complex,A: complex,B: complex] :
% 0.33/0.57        ( ( C != zero_zero_complex )
% 0.33/0.57       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
% 0.33/0.57          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_divide_mult_cancel_left2
% 0.33/0.57  thf(fact_2227_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.33/0.57      ! [C: real,A: real,B: real] :
% 0.33/0.57        ( ( C != zero_zero_real )
% 0.33/0.57       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
% 0.33/0.57          = ( divide_divide_real @ A @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_divide_mult_cancel_left2
% 0.33/0.57  thf(fact_2228_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.33/0.57      ! [C: rat,A: rat,B: rat] :
% 0.33/0.57        ( ( C != zero_zero_rat )
% 0.33/0.57       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
% 0.33/0.57          = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_divide_mult_cancel_left2
% 0.33/0.57  thf(fact_2229_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.33/0.57      ! [C: complex,A: complex,B: complex] :
% 0.33/0.57        ( ( C != zero_zero_complex )
% 0.33/0.57       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 0.33/0.57          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_divide_mult_cancel_right
% 0.33/0.57  thf(fact_2230_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.33/0.57      ! [C: real,A: real,B: real] :
% 0.33/0.57        ( ( C != zero_zero_real )
% 0.33/0.57       => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.33/0.57          = ( divide_divide_real @ A @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_divide_mult_cancel_right
% 0.33/0.57  thf(fact_2231_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.33/0.57      ! [C: rat,A: rat,B: rat] :
% 0.33/0.57        ( ( C != zero_zero_rat )
% 0.33/0.57       => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.33/0.57          = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_divide_mult_cancel_right
% 0.33/0.57  thf(fact_2232_nonzero__mult__div__cancel__right,axiom,
% 0.33/0.57      ! [B: complex,A: complex] :
% 0.33/0.57        ( ( B != zero_zero_complex )
% 0.33/0.57       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
% 0.33/0.57          = A ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_div_cancel_right
% 0.33/0.57  thf(fact_2233_nonzero__mult__div__cancel__right,axiom,
% 0.33/0.57      ! [B: nat,A: nat] :
% 0.33/0.57        ( ( B != zero_zero_nat )
% 0.33/0.57       => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
% 0.33/0.57          = A ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_div_cancel_right
% 0.33/0.57  thf(fact_2234_nonzero__mult__div__cancel__right,axiom,
% 0.33/0.57      ! [B: int,A: int] :
% 0.33/0.57        ( ( B != zero_zero_int )
% 0.33/0.57       => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
% 0.33/0.57          = A ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_div_cancel_right
% 0.33/0.57  thf(fact_2235_nonzero__mult__div__cancel__right,axiom,
% 0.33/0.57      ! [B: real,A: real] :
% 0.33/0.57        ( ( B != zero_zero_real )
% 0.33/0.57       => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
% 0.33/0.57          = A ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_div_cancel_right
% 0.33/0.57  thf(fact_2236_nonzero__mult__div__cancel__right,axiom,
% 0.33/0.57      ! [B: rat,A: rat] :
% 0.33/0.57        ( ( B != zero_zero_rat )
% 0.33/0.57       => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
% 0.33/0.57          = A ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_div_cancel_right
% 0.33/0.57  thf(fact_2237_nonzero__mult__div__cancel__right,axiom,
% 0.33/0.57      ! [B: code_integer,A: code_integer] :
% 0.33/0.57        ( ( B != zero_z3403309356797280102nteger )
% 0.33/0.57       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
% 0.33/0.57          = A ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_div_cancel_right
% 0.33/0.57  thf(fact_2238_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.33/0.57      ! [C: complex,A: complex,B: complex] :
% 0.33/0.57        ( ( C != zero_zero_complex )
% 0.33/0.57       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
% 0.33/0.57          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_divide_mult_cancel_right2
% 0.33/0.57  thf(fact_2239_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.33/0.57      ! [C: real,A: real,B: real] :
% 0.33/0.57        ( ( C != zero_zero_real )
% 0.33/0.57       => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
% 0.33/0.57          = ( divide_divide_real @ A @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_divide_mult_cancel_right2
% 0.33/0.57  thf(fact_2240_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.33/0.57      ! [C: rat,A: rat,B: rat] :
% 0.33/0.57        ( ( C != zero_zero_rat )
% 0.33/0.57       => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
% 0.33/0.57          = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_mult_divide_mult_cancel_right2
% 0.33/0.57  thf(fact_2241_divide__eq__1__iff,axiom,
% 0.33/0.57      ! [A: complex,B: complex] :
% 0.33/0.57        ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.33/0.57          = one_one_complex )
% 0.33/0.57        = ( ( B != zero_zero_complex )
% 0.33/0.57          & ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_eq_1_iff
% 0.33/0.57  thf(fact_2242_divide__eq__1__iff,axiom,
% 0.33/0.57      ! [A: real,B: real] :
% 0.33/0.57        ( ( ( divide_divide_real @ A @ B )
% 0.33/0.57          = one_one_real )
% 0.33/0.57        = ( ( B != zero_zero_real )
% 0.33/0.57          & ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_eq_1_iff
% 0.33/0.57  thf(fact_2243_divide__eq__1__iff,axiom,
% 0.33/0.57      ! [A: rat,B: rat] :
% 0.33/0.57        ( ( ( divide_divide_rat @ A @ B )
% 0.33/0.57          = one_one_rat )
% 0.33/0.57        = ( ( B != zero_zero_rat )
% 0.33/0.57          & ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_eq_1_iff
% 0.33/0.57  thf(fact_2244_div__self,axiom,
% 0.33/0.57      ! [A: complex] :
% 0.33/0.57        ( ( A != zero_zero_complex )
% 0.33/0.57       => ( ( divide1717551699836669952omplex @ A @ A )
% 0.33/0.57          = one_one_complex ) ) ).
% 0.33/0.57  
% 0.33/0.57  % div_self
% 0.33/0.57  thf(fact_2245_div__self,axiom,
% 0.33/0.57      ! [A: nat] :
% 0.33/0.57        ( ( A != zero_zero_nat )
% 0.33/0.57       => ( ( divide_divide_nat @ A @ A )
% 0.33/0.57          = one_one_nat ) ) ).
% 0.33/0.57  
% 0.33/0.57  % div_self
% 0.33/0.57  thf(fact_2246_div__self,axiom,
% 0.33/0.57      ! [A: int] :
% 0.33/0.57        ( ( A != zero_zero_int )
% 0.33/0.57       => ( ( divide_divide_int @ A @ A )
% 0.33/0.57          = one_one_int ) ) ).
% 0.33/0.57  
% 0.33/0.57  % div_self
% 0.33/0.57  thf(fact_2247_div__self,axiom,
% 0.33/0.57      ! [A: real] :
% 0.33/0.57        ( ( A != zero_zero_real )
% 0.33/0.57       => ( ( divide_divide_real @ A @ A )
% 0.33/0.57          = one_one_real ) ) ).
% 0.33/0.57  
% 0.33/0.57  % div_self
% 0.33/0.57  thf(fact_2248_div__self,axiom,
% 0.33/0.57      ! [A: rat] :
% 0.33/0.57        ( ( A != zero_zero_rat )
% 0.33/0.57       => ( ( divide_divide_rat @ A @ A )
% 0.33/0.57          = one_one_rat ) ) ).
% 0.33/0.57  
% 0.33/0.57  % div_self
% 0.33/0.57  thf(fact_2249_div__self,axiom,
% 0.33/0.57      ! [A: code_integer] :
% 0.33/0.57        ( ( A != zero_z3403309356797280102nteger )
% 0.33/0.57       => ( ( divide6298287555418463151nteger @ A @ A )
% 0.33/0.57          = one_one_Code_integer ) ) ).
% 0.33/0.57  
% 0.33/0.57  % div_self
% 0.33/0.57  thf(fact_2250_one__eq__divide__iff,axiom,
% 0.33/0.57      ! [A: complex,B: complex] :
% 0.33/0.57        ( ( one_one_complex
% 0.33/0.57          = ( divide1717551699836669952omplex @ A @ B ) )
% 0.33/0.57        = ( ( B != zero_zero_complex )
% 0.33/0.57          & ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % one_eq_divide_iff
% 0.33/0.57  thf(fact_2251_one__eq__divide__iff,axiom,
% 0.33/0.57      ! [A: real,B: real] :
% 0.33/0.57        ( ( one_one_real
% 0.33/0.57          = ( divide_divide_real @ A @ B ) )
% 0.33/0.57        = ( ( B != zero_zero_real )
% 0.33/0.57          & ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % one_eq_divide_iff
% 0.33/0.57  thf(fact_2252_one__eq__divide__iff,axiom,
% 0.33/0.57      ! [A: rat,B: rat] :
% 0.33/0.57        ( ( one_one_rat
% 0.33/0.57          = ( divide_divide_rat @ A @ B ) )
% 0.33/0.57        = ( ( B != zero_zero_rat )
% 0.33/0.57          & ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % one_eq_divide_iff
% 0.33/0.57  thf(fact_2253_divide__self,axiom,
% 0.33/0.57      ! [A: complex] :
% 0.33/0.57        ( ( A != zero_zero_complex )
% 0.33/0.57       => ( ( divide1717551699836669952omplex @ A @ A )
% 0.33/0.57          = one_one_complex ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_self
% 0.33/0.57  thf(fact_2254_divide__self,axiom,
% 0.33/0.57      ! [A: real] :
% 0.33/0.57        ( ( A != zero_zero_real )
% 0.33/0.57       => ( ( divide_divide_real @ A @ A )
% 0.33/0.57          = one_one_real ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_self
% 0.33/0.57  thf(fact_2255_divide__self,axiom,
% 0.33/0.57      ! [A: rat] :
% 0.33/0.57        ( ( A != zero_zero_rat )
% 0.33/0.57       => ( ( divide_divide_rat @ A @ A )
% 0.33/0.57          = one_one_rat ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_self
% 0.33/0.57  thf(fact_2256_divide__self__if,axiom,
% 0.33/0.57      ! [A: complex] :
% 0.33/0.57        ( ( ( A = zero_zero_complex )
% 0.33/0.57         => ( ( divide1717551699836669952omplex @ A @ A )
% 0.33/0.57            = zero_zero_complex ) )
% 0.33/0.57        & ( ( A != zero_zero_complex )
% 0.33/0.57         => ( ( divide1717551699836669952omplex @ A @ A )
% 0.33/0.57            = one_one_complex ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_self_if
% 0.33/0.57  thf(fact_2257_divide__self__if,axiom,
% 0.33/0.57      ! [A: real] :
% 0.33/0.57        ( ( ( A = zero_zero_real )
% 0.33/0.57         => ( ( divide_divide_real @ A @ A )
% 0.33/0.57            = zero_zero_real ) )
% 0.33/0.57        & ( ( A != zero_zero_real )
% 0.33/0.57         => ( ( divide_divide_real @ A @ A )
% 0.33/0.57            = one_one_real ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_self_if
% 0.33/0.57  thf(fact_2258_divide__self__if,axiom,
% 0.33/0.57      ! [A: rat] :
% 0.33/0.57        ( ( ( A = zero_zero_rat )
% 0.33/0.57         => ( ( divide_divide_rat @ A @ A )
% 0.33/0.57            = zero_zero_rat ) )
% 0.33/0.57        & ( ( A != zero_zero_rat )
% 0.33/0.57         => ( ( divide_divide_rat @ A @ A )
% 0.33/0.57            = one_one_rat ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_self_if
% 0.33/0.57  thf(fact_2259_divide__eq__eq__1,axiom,
% 0.33/0.57      ! [B: real,A: real] :
% 0.33/0.57        ( ( ( divide_divide_real @ B @ A )
% 0.33/0.57          = one_one_real )
% 0.33/0.57        = ( ( A != zero_zero_real )
% 0.33/0.57          & ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_eq_eq_1
% 0.33/0.57  thf(fact_2260_divide__eq__eq__1,axiom,
% 0.33/0.57      ! [B: rat,A: rat] :
% 0.33/0.57        ( ( ( divide_divide_rat @ B @ A )
% 0.33/0.57          = one_one_rat )
% 0.33/0.57        = ( ( A != zero_zero_rat )
% 0.33/0.57          & ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_eq_eq_1
% 0.33/0.57  thf(fact_2261_eq__divide__eq__1,axiom,
% 0.33/0.57      ! [B: real,A: real] :
% 0.33/0.57        ( ( one_one_real
% 0.33/0.57          = ( divide_divide_real @ B @ A ) )
% 0.33/0.57        = ( ( A != zero_zero_real )
% 0.33/0.57          & ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % eq_divide_eq_1
% 0.33/0.57  thf(fact_2262_eq__divide__eq__1,axiom,
% 0.33/0.57      ! [B: rat,A: rat] :
% 0.33/0.57        ( ( one_one_rat
% 0.33/0.57          = ( divide_divide_rat @ B @ A ) )
% 0.33/0.57        = ( ( A != zero_zero_rat )
% 0.33/0.57          & ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % eq_divide_eq_1
% 0.33/0.57  thf(fact_2263_one__divide__eq__0__iff,axiom,
% 0.33/0.57      ! [A: real] :
% 0.33/0.57        ( ( ( divide_divide_real @ one_one_real @ A )
% 0.33/0.57          = zero_zero_real )
% 0.33/0.57        = ( A = zero_zero_real ) ) ).
% 0.33/0.57  
% 0.33/0.57  % one_divide_eq_0_iff
% 0.33/0.57  thf(fact_2264_one__divide__eq__0__iff,axiom,
% 0.33/0.57      ! [A: rat] :
% 0.33/0.57        ( ( ( divide_divide_rat @ one_one_rat @ A )
% 0.33/0.57          = zero_zero_rat )
% 0.33/0.57        = ( A = zero_zero_rat ) ) ).
% 0.33/0.57  
% 0.33/0.57  % one_divide_eq_0_iff
% 0.33/0.57  thf(fact_2265_zero__eq__1__divide__iff,axiom,
% 0.33/0.57      ! [A: real] :
% 0.33/0.57        ( ( zero_zero_real
% 0.33/0.57          = ( divide_divide_real @ one_one_real @ A ) )
% 0.33/0.57        = ( A = zero_zero_real ) ) ).
% 0.33/0.57  
% 0.33/0.57  % zero_eq_1_divide_iff
% 0.33/0.57  thf(fact_2266_zero__eq__1__divide__iff,axiom,
% 0.33/0.57      ! [A: rat] :
% 0.33/0.57        ( ( zero_zero_rat
% 0.33/0.57          = ( divide_divide_rat @ one_one_rat @ A ) )
% 0.33/0.57        = ( A = zero_zero_rat ) ) ).
% 0.33/0.57  
% 0.33/0.57  % zero_eq_1_divide_iff
% 0.33/0.57  thf(fact_2267_power__inject__exp,axiom,
% 0.33/0.57      ! [A: real,M: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.57       => ( ( ( power_power_real @ A @ M )
% 0.33/0.57            = ( power_power_real @ A @ N ) )
% 0.33/0.57          = ( M = N ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_inject_exp
% 0.33/0.57  thf(fact_2268_power__inject__exp,axiom,
% 0.33/0.57      ! [A: rat,M: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_rat @ one_one_rat @ A )
% 0.33/0.57       => ( ( ( power_power_rat @ A @ M )
% 0.33/0.57            = ( power_power_rat @ A @ N ) )
% 0.33/0.57          = ( M = N ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_inject_exp
% 0.33/0.57  thf(fact_2269_power__inject__exp,axiom,
% 0.33/0.57      ! [A: nat,M: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_nat @ one_one_nat @ A )
% 0.33/0.57       => ( ( ( power_power_nat @ A @ M )
% 0.33/0.57            = ( power_power_nat @ A @ N ) )
% 0.33/0.57          = ( M = N ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_inject_exp
% 0.33/0.57  thf(fact_2270_power__inject__exp,axiom,
% 0.33/0.57      ! [A: int,M: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_int @ one_one_int @ A )
% 0.33/0.57       => ( ( ( power_power_int @ A @ M )
% 0.33/0.57            = ( power_power_int @ A @ N ) )
% 0.33/0.57          = ( M = N ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_inject_exp
% 0.33/0.57  thf(fact_2271_one__eq__mult__iff,axiom,
% 0.33/0.57      ! [M: nat,N: nat] :
% 0.33/0.57        ( ( ( suc @ zero_zero_nat )
% 0.33/0.57          = ( times_times_nat @ M @ N ) )
% 0.33/0.57        = ( ( M
% 0.33/0.57            = ( suc @ zero_zero_nat ) )
% 0.33/0.57          & ( N
% 0.33/0.57            = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % one_eq_mult_iff
% 0.33/0.57  thf(fact_2272_mult__eq__1__iff,axiom,
% 0.33/0.57      ! [M: nat,N: nat] :
% 0.33/0.57        ( ( ( times_times_nat @ M @ N )
% 0.33/0.57          = ( suc @ zero_zero_nat ) )
% 0.33/0.57        = ( ( M
% 0.33/0.57            = ( suc @ zero_zero_nat ) )
% 0.33/0.57          & ( N
% 0.33/0.57            = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_eq_1_iff
% 0.33/0.57  thf(fact_2273_nat__mult__less__cancel__disj,axiom,
% 0.33/0.57      ! [K: nat,M: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.33/0.57        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.57          & ( ord_less_nat @ M @ N ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nat_mult_less_cancel_disj
% 0.33/0.57  thf(fact_2274_mult__less__cancel2,axiom,
% 0.33/0.57      ! [M: nat,K: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.33/0.57        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.57          & ( ord_less_nat @ M @ N ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_less_cancel2
% 0.33/0.57  thf(fact_2275_nat__0__less__mult__iff,axiom,
% 0.33/0.57      ! [M: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 0.33/0.57        = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.57          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nat_0_less_mult_iff
% 0.33/0.57  thf(fact_2276_mult__Suc__right,axiom,
% 0.33/0.57      ! [M: nat,N: nat] :
% 0.33/0.57        ( ( times_times_nat @ M @ ( suc @ N ) )
% 0.33/0.57        = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_Suc_right
% 0.33/0.57  thf(fact_2277_less__one,axiom,
% 0.33/0.57      ! [N: nat] :
% 0.33/0.57        ( ( ord_less_nat @ N @ one_one_nat )
% 0.33/0.57        = ( N = zero_zero_nat ) ) ).
% 0.33/0.57  
% 0.33/0.57  % less_one
% 0.33/0.57  thf(fact_2278_nat__mult__div__cancel__disj,axiom,
% 0.33/0.57      ! [K: nat,M: nat,N: nat] :
% 0.33/0.57        ( ( ( K = zero_zero_nat )
% 0.33/0.57         => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.33/0.57            = zero_zero_nat ) )
% 0.33/0.57        & ( ( K != zero_zero_nat )
% 0.33/0.57         => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.33/0.57            = ( divide_divide_nat @ M @ N ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nat_mult_div_cancel_disj
% 0.33/0.57  thf(fact_2279_both__member__options__from__complete__tree__to__child,axiom,
% 0.33/0.57      ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 0.33/0.57        ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 0.33/0.57       => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.33/0.57         => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.57            | ( X = Mi )
% 0.33/0.57            | ( X = Ma ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % both_member_options_from_complete_tree_to_child
% 0.33/0.57  thf(fact_2280_zero__le__divide__1__iff,axiom,
% 0.33/0.57      ! [A: real] :
% 0.33/0.57        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.33/0.57        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.33/0.57  
% 0.33/0.57  % zero_le_divide_1_iff
% 0.33/0.57  thf(fact_2281_zero__le__divide__1__iff,axiom,
% 0.33/0.57      ! [A: rat] :
% 0.33/0.57        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 0.33/0.57        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.33/0.57  
% 0.33/0.57  % zero_le_divide_1_iff
% 0.33/0.57  thf(fact_2282_divide__le__0__1__iff,axiom,
% 0.33/0.57      ! [A: real] :
% 0.33/0.57        ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.33/0.57        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_le_0_1_iff
% 0.33/0.57  thf(fact_2283_divide__le__0__1__iff,axiom,
% 0.33/0.57      ! [A: rat] :
% 0.33/0.57        ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 0.33/0.57        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_le_0_1_iff
% 0.33/0.57  thf(fact_2284_divide__less__0__1__iff,axiom,
% 0.33/0.57      ! [A: real] :
% 0.33/0.57        ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.33/0.57        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_less_0_1_iff
% 0.33/0.57  thf(fact_2285_divide__less__0__1__iff,axiom,
% 0.33/0.57      ! [A: rat] :
% 0.33/0.57        ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 0.33/0.57        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_less_0_1_iff
% 0.33/0.57  thf(fact_2286_divide__less__eq__1__neg,axiom,
% 0.33/0.57      ! [A: real,B: real] :
% 0.33/0.57        ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.57       => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.33/0.57          = ( ord_less_real @ A @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_less_eq_1_neg
% 0.33/0.57  thf(fact_2287_divide__less__eq__1__neg,axiom,
% 0.33/0.57      ! [A: rat,B: rat] :
% 0.33/0.57        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.57       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.33/0.57          = ( ord_less_rat @ A @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_less_eq_1_neg
% 0.33/0.57  thf(fact_2288_divide__less__eq__1__pos,axiom,
% 0.33/0.57      ! [A: real,B: real] :
% 0.33/0.57        ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.57       => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.33/0.57          = ( ord_less_real @ B @ A ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_less_eq_1_pos
% 0.33/0.57  thf(fact_2289_divide__less__eq__1__pos,axiom,
% 0.33/0.57      ! [A: rat,B: rat] :
% 0.33/0.57        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.57       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.33/0.57          = ( ord_less_rat @ B @ A ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_less_eq_1_pos
% 0.33/0.57  thf(fact_2290_less__divide__eq__1__neg,axiom,
% 0.33/0.57      ! [A: real,B: real] :
% 0.33/0.57        ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.57       => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.33/0.57          = ( ord_less_real @ B @ A ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % less_divide_eq_1_neg
% 0.33/0.57  thf(fact_2291_less__divide__eq__1__neg,axiom,
% 0.33/0.57      ! [A: rat,B: rat] :
% 0.33/0.57        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.57       => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.33/0.57          = ( ord_less_rat @ B @ A ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % less_divide_eq_1_neg
% 0.33/0.57  thf(fact_2292_less__divide__eq__1__pos,axiom,
% 0.33/0.57      ! [A: real,B: real] :
% 0.33/0.57        ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.57       => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.33/0.57          = ( ord_less_real @ A @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % less_divide_eq_1_pos
% 0.33/0.57  thf(fact_2293_less__divide__eq__1__pos,axiom,
% 0.33/0.57      ! [A: rat,B: rat] :
% 0.33/0.57        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.57       => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.33/0.57          = ( ord_less_rat @ A @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % less_divide_eq_1_pos
% 0.33/0.57  thf(fact_2294_zero__less__divide__1__iff,axiom,
% 0.33/0.57      ! [A: real] :
% 0.33/0.57        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.33/0.57        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.33/0.57  
% 0.33/0.57  % zero_less_divide_1_iff
% 0.33/0.57  thf(fact_2295_zero__less__divide__1__iff,axiom,
% 0.33/0.57      ! [A: rat] :
% 0.33/0.57        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 0.33/0.57        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.33/0.57  
% 0.33/0.57  % zero_less_divide_1_iff
% 0.33/0.57  thf(fact_2296_le__divide__eq__numeral1_I1_J,axiom,
% 0.33/0.57      ! [A: real,B: real,W2: num] :
% 0.33/0.57        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
% 0.33/0.57        = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).
% 0.33/0.57  
% 0.33/0.57  % le_divide_eq_numeral1(1)
% 0.33/0.57  thf(fact_2297_le__divide__eq__numeral1_I1_J,axiom,
% 0.33/0.57      ! [A: rat,B: rat,W2: num] :
% 0.33/0.57        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
% 0.33/0.57        = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).
% 0.33/0.57  
% 0.33/0.57  % le_divide_eq_numeral1(1)
% 0.33/0.57  thf(fact_2298_divide__le__eq__numeral1_I1_J,axiom,
% 0.33/0.57      ! [B: real,W2: num,A: real] :
% 0.33/0.57        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
% 0.33/0.57        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_le_eq_numeral1(1)
% 0.33/0.57  thf(fact_2299_divide__le__eq__numeral1_I1_J,axiom,
% 0.33/0.57      ! [B: rat,W2: num,A: rat] :
% 0.33/0.57        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
% 0.33/0.57        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_le_eq_numeral1(1)
% 0.33/0.57  thf(fact_2300_eq__divide__eq__numeral1_I1_J,axiom,
% 0.33/0.57      ! [A: complex,B: complex,W2: num] :
% 0.33/0.57        ( ( A
% 0.33/0.57          = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) ) )
% 0.33/0.57        = ( ( ( ( numera6690914467698888265omplex @ W2 )
% 0.33/0.57             != zero_zero_complex )
% 0.33/0.57           => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) )
% 0.33/0.57              = B ) )
% 0.33/0.57          & ( ( ( numera6690914467698888265omplex @ W2 )
% 0.33/0.57              = zero_zero_complex )
% 0.33/0.57           => ( A = zero_zero_complex ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % eq_divide_eq_numeral1(1)
% 0.33/0.57  thf(fact_2301_eq__divide__eq__numeral1_I1_J,axiom,
% 0.33/0.57      ! [A: real,B: real,W2: num] :
% 0.33/0.57        ( ( A
% 0.33/0.57          = ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
% 0.33/0.57        = ( ( ( ( numeral_numeral_real @ W2 )
% 0.33/0.57             != zero_zero_real )
% 0.33/0.57           => ( ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) )
% 0.33/0.57              = B ) )
% 0.33/0.57          & ( ( ( numeral_numeral_real @ W2 )
% 0.33/0.57              = zero_zero_real )
% 0.33/0.57           => ( A = zero_zero_real ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % eq_divide_eq_numeral1(1)
% 0.33/0.57  thf(fact_2302_eq__divide__eq__numeral1_I1_J,axiom,
% 0.33/0.57      ! [A: rat,B: rat,W2: num] :
% 0.33/0.57        ( ( A
% 0.33/0.57          = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
% 0.33/0.57        = ( ( ( ( numeral_numeral_rat @ W2 )
% 0.33/0.57             != zero_zero_rat )
% 0.33/0.57           => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) )
% 0.33/0.57              = B ) )
% 0.33/0.57          & ( ( ( numeral_numeral_rat @ W2 )
% 0.33/0.57              = zero_zero_rat )
% 0.33/0.57           => ( A = zero_zero_rat ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % eq_divide_eq_numeral1(1)
% 0.33/0.57  thf(fact_2303_divide__eq__eq__numeral1_I1_J,axiom,
% 0.33/0.57      ! [B: complex,W2: num,A: complex] :
% 0.33/0.57        ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) )
% 0.33/0.57          = A )
% 0.33/0.57        = ( ( ( ( numera6690914467698888265omplex @ W2 )
% 0.33/0.57             != zero_zero_complex )
% 0.33/0.57           => ( B
% 0.33/0.57              = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) ) ) )
% 0.33/0.57          & ( ( ( numera6690914467698888265omplex @ W2 )
% 0.33/0.57              = zero_zero_complex )
% 0.33/0.57           => ( A = zero_zero_complex ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_eq_eq_numeral1(1)
% 0.33/0.57  thf(fact_2304_divide__eq__eq__numeral1_I1_J,axiom,
% 0.33/0.57      ! [B: real,W2: num,A: real] :
% 0.33/0.57        ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) )
% 0.33/0.57          = A )
% 0.33/0.57        = ( ( ( ( numeral_numeral_real @ W2 )
% 0.33/0.57             != zero_zero_real )
% 0.33/0.57           => ( B
% 0.33/0.57              = ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) )
% 0.33/0.57          & ( ( ( numeral_numeral_real @ W2 )
% 0.33/0.57              = zero_zero_real )
% 0.33/0.57           => ( A = zero_zero_real ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_eq_eq_numeral1(1)
% 0.33/0.57  thf(fact_2305_divide__eq__eq__numeral1_I1_J,axiom,
% 0.33/0.57      ! [B: rat,W2: num,A: rat] :
% 0.33/0.57        ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) )
% 0.33/0.57          = A )
% 0.33/0.57        = ( ( ( ( numeral_numeral_rat @ W2 )
% 0.33/0.57             != zero_zero_rat )
% 0.33/0.57           => ( B
% 0.33/0.57              = ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) )
% 0.33/0.57          & ( ( ( numeral_numeral_rat @ W2 )
% 0.33/0.57              = zero_zero_rat )
% 0.33/0.57           => ( A = zero_zero_rat ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_eq_eq_numeral1(1)
% 0.33/0.57  thf(fact_2306_divide__less__eq__numeral1_I1_J,axiom,
% 0.33/0.57      ! [B: real,W2: num,A: real] :
% 0.33/0.57        ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
% 0.33/0.57        = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_less_eq_numeral1(1)
% 0.33/0.57  thf(fact_2307_divide__less__eq__numeral1_I1_J,axiom,
% 0.33/0.57      ! [B: rat,W2: num,A: rat] :
% 0.33/0.57        ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
% 0.33/0.57        = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_less_eq_numeral1(1)
% 0.33/0.57  thf(fact_2308_less__divide__eq__numeral1_I1_J,axiom,
% 0.33/0.57      ! [A: real,B: real,W2: num] :
% 0.33/0.57        ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
% 0.33/0.57        = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).
% 0.33/0.57  
% 0.33/0.57  % less_divide_eq_numeral1(1)
% 0.33/0.57  thf(fact_2309_less__divide__eq__numeral1_I1_J,axiom,
% 0.33/0.57      ! [A: rat,B: rat,W2: num] :
% 0.33/0.57        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
% 0.33/0.57        = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).
% 0.33/0.57  
% 0.33/0.57  % less_divide_eq_numeral1(1)
% 0.33/0.57  thf(fact_2310_nonzero__divide__mult__cancel__right,axiom,
% 0.33/0.57      ! [B: complex,A: complex] :
% 0.33/0.57        ( ( B != zero_zero_complex )
% 0.33/0.57       => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
% 0.33/0.57          = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_divide_mult_cancel_right
% 0.33/0.57  thf(fact_2311_nonzero__divide__mult__cancel__right,axiom,
% 0.33/0.57      ! [B: real,A: real] :
% 0.33/0.57        ( ( B != zero_zero_real )
% 0.33/0.57       => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
% 0.33/0.57          = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_divide_mult_cancel_right
% 0.33/0.57  thf(fact_2312_nonzero__divide__mult__cancel__right,axiom,
% 0.33/0.57      ! [B: rat,A: rat] :
% 0.33/0.57        ( ( B != zero_zero_rat )
% 0.33/0.57       => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
% 0.33/0.57          = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_divide_mult_cancel_right
% 0.33/0.57  thf(fact_2313_nonzero__divide__mult__cancel__left,axiom,
% 0.33/0.57      ! [A: complex,B: complex] :
% 0.33/0.57        ( ( A != zero_zero_complex )
% 0.33/0.57       => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
% 0.33/0.57          = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_divide_mult_cancel_left
% 0.33/0.57  thf(fact_2314_nonzero__divide__mult__cancel__left,axiom,
% 0.33/0.57      ! [A: real,B: real] :
% 0.33/0.57        ( ( A != zero_zero_real )
% 0.33/0.57       => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
% 0.33/0.57          = ( divide_divide_real @ one_one_real @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_divide_mult_cancel_left
% 0.33/0.57  thf(fact_2315_nonzero__divide__mult__cancel__left,axiom,
% 0.33/0.57      ! [A: rat,B: rat] :
% 0.33/0.57        ( ( A != zero_zero_rat )
% 0.33/0.57       => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
% 0.33/0.57          = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_divide_mult_cancel_left
% 0.33/0.57  thf(fact_2316_power__strict__increasing__iff,axiom,
% 0.33/0.57      ! [B: real,X: nat,Y: nat] :
% 0.33/0.57        ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.57       => ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
% 0.33/0.57          = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_strict_increasing_iff
% 0.33/0.57  thf(fact_2317_power__strict__increasing__iff,axiom,
% 0.33/0.57      ! [B: rat,X: nat,Y: nat] :
% 0.33/0.57        ( ( ord_less_rat @ one_one_rat @ B )
% 0.33/0.57       => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
% 0.33/0.57          = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_strict_increasing_iff
% 0.33/0.57  thf(fact_2318_power__strict__increasing__iff,axiom,
% 0.33/0.57      ! [B: nat,X: nat,Y: nat] :
% 0.33/0.57        ( ( ord_less_nat @ one_one_nat @ B )
% 0.33/0.57       => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
% 0.33/0.57          = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_strict_increasing_iff
% 0.33/0.57  thf(fact_2319_power__strict__increasing__iff,axiom,
% 0.33/0.57      ! [B: int,X: nat,Y: nat] :
% 0.33/0.57        ( ( ord_less_int @ one_one_int @ B )
% 0.33/0.57       => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
% 0.33/0.57          = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_strict_increasing_iff
% 0.33/0.57  thf(fact_2320_both__member__options__from__chilf__to__complete__tree,axiom,
% 0.33/0.57      ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.33/0.57        ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.33/0.57       => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 0.33/0.57         => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.57           => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % both_member_options_from_chilf_to_complete_tree
% 0.33/0.57  thf(fact_2321_one__le__mult__iff,axiom,
% 0.33/0.57      ! [M: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 0.33/0.57        = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.33/0.57          & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % one_le_mult_iff
% 0.33/0.57  thf(fact_2322_nat__mult__le__cancel__disj,axiom,
% 0.33/0.57      ! [K: nat,M: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.33/0.57        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.57         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nat_mult_le_cancel_disj
% 0.33/0.57  thf(fact_2323_mult__le__cancel2,axiom,
% 0.33/0.57      ! [M: nat,K: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.33/0.57        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.57         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_le_cancel2
% 0.33/0.57  thf(fact_2324_divide__le__eq__1__neg,axiom,
% 0.33/0.57      ! [A: real,B: real] :
% 0.33/0.57        ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.57       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.33/0.57          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_le_eq_1_neg
% 0.33/0.57  thf(fact_2325_divide__le__eq__1__neg,axiom,
% 0.33/0.57      ! [A: rat,B: rat] :
% 0.33/0.57        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.57       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.33/0.57          = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_le_eq_1_neg
% 0.33/0.57  thf(fact_2326_divide__le__eq__1__pos,axiom,
% 0.33/0.57      ! [A: real,B: real] :
% 0.33/0.57        ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.57       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.33/0.57          = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_le_eq_1_pos
% 0.33/0.57  thf(fact_2327_divide__le__eq__1__pos,axiom,
% 0.33/0.57      ! [A: rat,B: rat] :
% 0.33/0.57        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.57       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.33/0.57          = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_le_eq_1_pos
% 0.33/0.57  thf(fact_2328_le__divide__eq__1__neg,axiom,
% 0.33/0.57      ! [A: real,B: real] :
% 0.33/0.57        ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.57       => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.33/0.57          = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % le_divide_eq_1_neg
% 0.33/0.57  thf(fact_2329_le__divide__eq__1__neg,axiom,
% 0.33/0.57      ! [A: rat,B: rat] :
% 0.33/0.57        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.57       => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.33/0.57          = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % le_divide_eq_1_neg
% 0.33/0.57  thf(fact_2330_le__divide__eq__1__pos,axiom,
% 0.33/0.57      ! [A: real,B: real] :
% 0.33/0.57        ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.57       => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.33/0.57          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % le_divide_eq_1_pos
% 0.33/0.57  thf(fact_2331_le__divide__eq__1__pos,axiom,
% 0.33/0.57      ! [A: rat,B: rat] :
% 0.33/0.57        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.57       => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.33/0.57          = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % le_divide_eq_1_pos
% 0.33/0.57  thf(fact_2332_power__strict__decreasing__iff,axiom,
% 0.33/0.57      ! [B: real,M: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.57       => ( ( ord_less_real @ B @ one_one_real )
% 0.33/0.57         => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 0.33/0.57            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_strict_decreasing_iff
% 0.33/0.57  thf(fact_2333_power__strict__decreasing__iff,axiom,
% 0.33/0.57      ! [B: rat,M: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.33/0.57       => ( ( ord_less_rat @ B @ one_one_rat )
% 0.33/0.57         => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 0.33/0.57            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_strict_decreasing_iff
% 0.33/0.57  thf(fact_2334_power__strict__decreasing__iff,axiom,
% 0.33/0.57      ! [B: nat,M: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.33/0.57       => ( ( ord_less_nat @ B @ one_one_nat )
% 0.33/0.57         => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 0.33/0.57            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_strict_decreasing_iff
% 0.33/0.57  thf(fact_2335_power__strict__decreasing__iff,axiom,
% 0.33/0.57      ! [B: int,M: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.57       => ( ( ord_less_int @ B @ one_one_int )
% 0.33/0.57         => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 0.33/0.57            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_strict_decreasing_iff
% 0.33/0.57  thf(fact_2336_power__increasing__iff,axiom,
% 0.33/0.57      ! [B: real,X: nat,Y: nat] :
% 0.33/0.57        ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.57       => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
% 0.33/0.57          = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_increasing_iff
% 0.33/0.57  thf(fact_2337_power__increasing__iff,axiom,
% 0.33/0.57      ! [B: rat,X: nat,Y: nat] :
% 0.33/0.57        ( ( ord_less_rat @ one_one_rat @ B )
% 0.33/0.57       => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
% 0.33/0.57          = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_increasing_iff
% 0.33/0.57  thf(fact_2338_power__increasing__iff,axiom,
% 0.33/0.57      ! [B: nat,X: nat,Y: nat] :
% 0.33/0.57        ( ( ord_less_nat @ one_one_nat @ B )
% 0.33/0.57       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
% 0.33/0.57          = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_increasing_iff
% 0.33/0.57  thf(fact_2339_power__increasing__iff,axiom,
% 0.33/0.57      ! [B: int,X: nat,Y: nat] :
% 0.33/0.57        ( ( ord_less_int @ one_one_int @ B )
% 0.33/0.57       => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
% 0.33/0.57          = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_increasing_iff
% 0.33/0.57  thf(fact_2340_numeral__le__one__iff,axiom,
% 0.33/0.57      ! [N: num] :
% 0.33/0.57        ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
% 0.33/0.57        = ( ord_less_eq_num @ N @ one ) ) ).
% 0.33/0.57  
% 0.33/0.57  % numeral_le_one_iff
% 0.33/0.57  thf(fact_2341_numeral__le__one__iff,axiom,
% 0.33/0.57      ! [N: num] :
% 0.33/0.57        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 0.33/0.57        = ( ord_less_eq_num @ N @ one ) ) ).
% 0.33/0.57  
% 0.33/0.57  % numeral_le_one_iff
% 0.33/0.57  thf(fact_2342_numeral__le__one__iff,axiom,
% 0.33/0.57      ! [N: num] :
% 0.33/0.57        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 0.33/0.57        = ( ord_less_eq_num @ N @ one ) ) ).
% 0.33/0.57  
% 0.33/0.57  % numeral_le_one_iff
% 0.33/0.57  thf(fact_2343_numeral__le__one__iff,axiom,
% 0.33/0.57      ! [N: num] :
% 0.33/0.57        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 0.33/0.57        = ( ord_less_eq_num @ N @ one ) ) ).
% 0.33/0.57  
% 0.33/0.57  % numeral_le_one_iff
% 0.33/0.57  thf(fact_2344_numeral__le__one__iff,axiom,
% 0.33/0.57      ! [N: num] :
% 0.33/0.57        ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 0.33/0.57        = ( ord_less_eq_num @ N @ one ) ) ).
% 0.33/0.57  
% 0.33/0.57  % numeral_le_one_iff
% 0.33/0.57  thf(fact_2345_one__less__numeral__iff,axiom,
% 0.33/0.57      ! [N: num] :
% 0.33/0.57        ( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
% 0.33/0.57        = ( ord_less_num @ one @ N ) ) ).
% 0.33/0.57  
% 0.33/0.57  % one_less_numeral_iff
% 0.33/0.57  thf(fact_2346_one__less__numeral__iff,axiom,
% 0.33/0.57      ! [N: num] :
% 0.33/0.57        ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 0.33/0.57        = ( ord_less_num @ one @ N ) ) ).
% 0.33/0.57  
% 0.33/0.57  % one_less_numeral_iff
% 0.33/0.57  thf(fact_2347_one__less__numeral__iff,axiom,
% 0.33/0.57      ! [N: num] :
% 0.33/0.57        ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 0.33/0.57        = ( ord_less_num @ one @ N ) ) ).
% 0.33/0.57  
% 0.33/0.57  % one_less_numeral_iff
% 0.33/0.57  thf(fact_2348_one__less__numeral__iff,axiom,
% 0.33/0.57      ! [N: num] :
% 0.33/0.57        ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 0.33/0.57        = ( ord_less_num @ one @ N ) ) ).
% 0.33/0.57  
% 0.33/0.57  % one_less_numeral_iff
% 0.33/0.57  thf(fact_2349_one__less__numeral__iff,axiom,
% 0.33/0.57      ! [N: num] :
% 0.33/0.57        ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 0.33/0.57        = ( ord_less_num @ one @ N ) ) ).
% 0.33/0.57  
% 0.33/0.57  % one_less_numeral_iff
% 0.33/0.57  thf(fact_2350_bits__1__div__2,axiom,
% 0.33/0.57      ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.57      = zero_zero_nat ) ).
% 0.33/0.57  
% 0.33/0.57  % bits_1_div_2
% 0.33/0.57  thf(fact_2351_bits__1__div__2,axiom,
% 0.33/0.57      ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.57      = zero_zero_int ) ).
% 0.33/0.57  
% 0.33/0.57  % bits_1_div_2
% 0.33/0.57  thf(fact_2352_bits__1__div__2,axiom,
% 0.33/0.57      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.33/0.57      = zero_z3403309356797280102nteger ) ).
% 0.33/0.57  
% 0.33/0.57  % bits_1_div_2
% 0.33/0.57  thf(fact_2353_power__decreasing__iff,axiom,
% 0.33/0.57      ! [B: real,M: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.57       => ( ( ord_less_real @ B @ one_one_real )
% 0.33/0.57         => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 0.33/0.57            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_decreasing_iff
% 0.33/0.57  thf(fact_2354_power__decreasing__iff,axiom,
% 0.33/0.57      ! [B: rat,M: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.33/0.57       => ( ( ord_less_rat @ B @ one_one_rat )
% 0.33/0.57         => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 0.33/0.57            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_decreasing_iff
% 0.33/0.57  thf(fact_2355_power__decreasing__iff,axiom,
% 0.33/0.57      ! [B: nat,M: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.33/0.57       => ( ( ord_less_nat @ B @ one_one_nat )
% 0.33/0.57         => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 0.33/0.57            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_decreasing_iff
% 0.33/0.57  thf(fact_2356_power__decreasing__iff,axiom,
% 0.33/0.57      ! [B: int,M: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.57       => ( ( ord_less_int @ B @ one_one_int )
% 0.33/0.57         => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 0.33/0.57            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % power_decreasing_iff
% 0.33/0.57  thf(fact_2357__092_060open_062y_A_061_A_Iif_Ax_A_061_Ama_Athen_Ahigh_Ax_An_A_K_A2_A_094_A_Ideg_Adiv_A2_J_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Ahigh_Ax_An_J_J_Aelse_Ama_J_092_060close_062,axiom,
% 0.33/0.57      ( ( ( xa = ma )
% 0.33/0.57       => ( ya
% 0.33/0.57          = ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ) ) )
% 0.33/0.57      & ( ( xa != ma )
% 0.33/0.57       => ( ya = ma ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % \<open>y = (if x = ma then high x n * 2 ^ (deg div 2) + the (vebt_maxt (treeList [high x n := vebt_delete (treeList ! high x n) (low x n)] ! high x n)) else ma)\<close>
% 0.33/0.57  thf(fact_2358_ssms,axiom,
% 0.33/0.57      vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ( if_nat @ ( xa = ma ) @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ) @ ma ) ) ) @ deg @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ summary ) @ ya ).
% 0.33/0.57  
% 0.33/0.57  % ssms
% 0.33/0.57  thf(fact_2359_right__inverse__eq,axiom,
% 0.33/0.57      ! [B: complex,A: complex] :
% 0.33/0.57        ( ( B != zero_zero_complex )
% 0.33/0.57       => ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.33/0.57            = one_one_complex )
% 0.33/0.57          = ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % right_inverse_eq
% 0.33/0.57  thf(fact_2360_right__inverse__eq,axiom,
% 0.33/0.57      ! [B: real,A: real] :
% 0.33/0.57        ( ( B != zero_zero_real )
% 0.33/0.57       => ( ( ( divide_divide_real @ A @ B )
% 0.33/0.57            = one_one_real )
% 0.33/0.57          = ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % right_inverse_eq
% 0.33/0.57  thf(fact_2361_right__inverse__eq,axiom,
% 0.33/0.57      ! [B: rat,A: rat] :
% 0.33/0.57        ( ( B != zero_zero_rat )
% 0.33/0.57       => ( ( ( divide_divide_rat @ A @ B )
% 0.33/0.57            = one_one_rat )
% 0.33/0.57          = ( A = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % right_inverse_eq
% 0.33/0.57  thf(fact_2362_less__1__mult,axiom,
% 0.33/0.57      ! [M: real,N: real] :
% 0.33/0.57        ( ( ord_less_real @ one_one_real @ M )
% 0.33/0.57       => ( ( ord_less_real @ one_one_real @ N )
% 0.33/0.57         => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % less_1_mult
% 0.33/0.57  thf(fact_2363_less__1__mult,axiom,
% 0.33/0.57      ! [M: rat,N: rat] :
% 0.33/0.57        ( ( ord_less_rat @ one_one_rat @ M )
% 0.33/0.57       => ( ( ord_less_rat @ one_one_rat @ N )
% 0.33/0.57         => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % less_1_mult
% 0.33/0.57  thf(fact_2364_less__1__mult,axiom,
% 0.33/0.57      ! [M: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_nat @ one_one_nat @ M )
% 0.33/0.57       => ( ( ord_less_nat @ one_one_nat @ N )
% 0.33/0.57         => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % less_1_mult
% 0.33/0.57  thf(fact_2365_less__1__mult,axiom,
% 0.33/0.57      ! [M: int,N: int] :
% 0.33/0.57        ( ( ord_less_int @ one_one_int @ M )
% 0.33/0.57       => ( ( ord_less_int @ one_one_int @ N )
% 0.33/0.57         => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % less_1_mult
% 0.33/0.57  thf(fact_2366_mult__eq__self__implies__10,axiom,
% 0.33/0.57      ! [M: nat,N: nat] :
% 0.33/0.57        ( ( M
% 0.33/0.57          = ( times_times_nat @ M @ N ) )
% 0.33/0.57       => ( ( N = one_one_nat )
% 0.33/0.57          | ( M = zero_zero_nat ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_eq_self_implies_10
% 0.33/0.57  thf(fact_2367_nonzero__eq__divide__eq,axiom,
% 0.33/0.57      ! [C: complex,A: complex,B: complex] :
% 0.33/0.57        ( ( C != zero_zero_complex )
% 0.33/0.57       => ( ( A
% 0.33/0.57            = ( divide1717551699836669952omplex @ B @ C ) )
% 0.33/0.57          = ( ( times_times_complex @ A @ C )
% 0.33/0.57            = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_eq_divide_eq
% 0.33/0.57  thf(fact_2368_nonzero__eq__divide__eq,axiom,
% 0.33/0.57      ! [C: real,A: real,B: real] :
% 0.33/0.57        ( ( C != zero_zero_real )
% 0.33/0.57       => ( ( A
% 0.33/0.57            = ( divide_divide_real @ B @ C ) )
% 0.33/0.57          = ( ( times_times_real @ A @ C )
% 0.33/0.57            = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_eq_divide_eq
% 0.33/0.57  thf(fact_2369_nonzero__eq__divide__eq,axiom,
% 0.33/0.57      ! [C: rat,A: rat,B: rat] :
% 0.33/0.57        ( ( C != zero_zero_rat )
% 0.33/0.57       => ( ( A
% 0.33/0.57            = ( divide_divide_rat @ B @ C ) )
% 0.33/0.57          = ( ( times_times_rat @ A @ C )
% 0.33/0.57            = B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_eq_divide_eq
% 0.33/0.57  thf(fact_2370_nonzero__divide__eq__eq,axiom,
% 0.33/0.57      ! [C: complex,B: complex,A: complex] :
% 0.33/0.57        ( ( C != zero_zero_complex )
% 0.33/0.57       => ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.33/0.57            = A )
% 0.33/0.57          = ( B
% 0.33/0.57            = ( times_times_complex @ A @ C ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_divide_eq_eq
% 0.33/0.57  thf(fact_2371_nonzero__divide__eq__eq,axiom,
% 0.33/0.57      ! [C: real,B: real,A: real] :
% 0.33/0.57        ( ( C != zero_zero_real )
% 0.33/0.57       => ( ( ( divide_divide_real @ B @ C )
% 0.33/0.57            = A )
% 0.33/0.57          = ( B
% 0.33/0.57            = ( times_times_real @ A @ C ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_divide_eq_eq
% 0.33/0.57  thf(fact_2372_nonzero__divide__eq__eq,axiom,
% 0.33/0.57      ! [C: rat,B: rat,A: rat] :
% 0.33/0.57        ( ( C != zero_zero_rat )
% 0.33/0.57       => ( ( ( divide_divide_rat @ B @ C )
% 0.33/0.57            = A )
% 0.33/0.57          = ( B
% 0.33/0.57            = ( times_times_rat @ A @ C ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nonzero_divide_eq_eq
% 0.33/0.57  thf(fact_2373_eq__divide__imp,axiom,
% 0.33/0.57      ! [C: complex,A: complex,B: complex] :
% 0.33/0.57        ( ( C != zero_zero_complex )
% 0.33/0.57       => ( ( ( times_times_complex @ A @ C )
% 0.33/0.57            = B )
% 0.33/0.57         => ( A
% 0.33/0.57            = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % eq_divide_imp
% 0.33/0.57  thf(fact_2374_eq__divide__imp,axiom,
% 0.33/0.57      ! [C: real,A: real,B: real] :
% 0.33/0.57        ( ( C != zero_zero_real )
% 0.33/0.57       => ( ( ( times_times_real @ A @ C )
% 0.33/0.57            = B )
% 0.33/0.57         => ( A
% 0.33/0.57            = ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % eq_divide_imp
% 0.33/0.57  thf(fact_2375_eq__divide__imp,axiom,
% 0.33/0.57      ! [C: rat,A: rat,B: rat] :
% 0.33/0.57        ( ( C != zero_zero_rat )
% 0.33/0.57       => ( ( ( times_times_rat @ A @ C )
% 0.33/0.57            = B )
% 0.33/0.57         => ( A
% 0.33/0.57            = ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % eq_divide_imp
% 0.33/0.57  thf(fact_2376_divide__eq__imp,axiom,
% 0.33/0.57      ! [C: complex,B: complex,A: complex] :
% 0.33/0.57        ( ( C != zero_zero_complex )
% 0.33/0.57       => ( ( B
% 0.33/0.57            = ( times_times_complex @ A @ C ) )
% 0.33/0.57         => ( ( divide1717551699836669952omplex @ B @ C )
% 0.33/0.57            = A ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_eq_imp
% 0.33/0.57  thf(fact_2377_divide__eq__imp,axiom,
% 0.33/0.57      ! [C: real,B: real,A: real] :
% 0.33/0.57        ( ( C != zero_zero_real )
% 0.33/0.57       => ( ( B
% 0.33/0.57            = ( times_times_real @ A @ C ) )
% 0.33/0.57         => ( ( divide_divide_real @ B @ C )
% 0.33/0.57            = A ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_eq_imp
% 0.33/0.57  thf(fact_2378_divide__eq__imp,axiom,
% 0.33/0.57      ! [C: rat,B: rat,A: rat] :
% 0.33/0.57        ( ( C != zero_zero_rat )
% 0.33/0.57       => ( ( B
% 0.33/0.57            = ( times_times_rat @ A @ C ) )
% 0.33/0.57         => ( ( divide_divide_rat @ B @ C )
% 0.33/0.57            = A ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_eq_imp
% 0.33/0.57  thf(fact_2379_eq__divide__eq,axiom,
% 0.33/0.57      ! [A: complex,B: complex,C: complex] :
% 0.33/0.57        ( ( A
% 0.33/0.57          = ( divide1717551699836669952omplex @ B @ C ) )
% 0.33/0.57        = ( ( ( C != zero_zero_complex )
% 0.33/0.57           => ( ( times_times_complex @ A @ C )
% 0.33/0.57              = B ) )
% 0.33/0.57          & ( ( C = zero_zero_complex )
% 0.33/0.57           => ( A = zero_zero_complex ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % eq_divide_eq
% 0.33/0.57  thf(fact_2380_eq__divide__eq,axiom,
% 0.33/0.57      ! [A: real,B: real,C: real] :
% 0.33/0.57        ( ( A
% 0.33/0.57          = ( divide_divide_real @ B @ C ) )
% 0.33/0.57        = ( ( ( C != zero_zero_real )
% 0.33/0.57           => ( ( times_times_real @ A @ C )
% 0.33/0.57              = B ) )
% 0.33/0.57          & ( ( C = zero_zero_real )
% 0.33/0.57           => ( A = zero_zero_real ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % eq_divide_eq
% 0.33/0.57  thf(fact_2381_eq__divide__eq,axiom,
% 0.33/0.57      ! [A: rat,B: rat,C: rat] :
% 0.33/0.57        ( ( A
% 0.33/0.57          = ( divide_divide_rat @ B @ C ) )
% 0.33/0.57        = ( ( ( C != zero_zero_rat )
% 0.33/0.57           => ( ( times_times_rat @ A @ C )
% 0.33/0.57              = B ) )
% 0.33/0.57          & ( ( C = zero_zero_rat )
% 0.33/0.57           => ( A = zero_zero_rat ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % eq_divide_eq
% 0.33/0.57  thf(fact_2382_divide__eq__eq,axiom,
% 0.33/0.57      ! [B: complex,C: complex,A: complex] :
% 0.33/0.57        ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.33/0.57          = A )
% 0.33/0.57        = ( ( ( C != zero_zero_complex )
% 0.33/0.57           => ( B
% 0.33/0.57              = ( times_times_complex @ A @ C ) ) )
% 0.33/0.57          & ( ( C = zero_zero_complex )
% 0.33/0.57           => ( A = zero_zero_complex ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_eq_eq
% 0.33/0.57  thf(fact_2383_divide__eq__eq,axiom,
% 0.33/0.57      ! [B: real,C: real,A: real] :
% 0.33/0.57        ( ( ( divide_divide_real @ B @ C )
% 0.33/0.57          = A )
% 0.33/0.57        = ( ( ( C != zero_zero_real )
% 0.33/0.57           => ( B
% 0.33/0.57              = ( times_times_real @ A @ C ) ) )
% 0.33/0.57          & ( ( C = zero_zero_real )
% 0.33/0.57           => ( A = zero_zero_real ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_eq_eq
% 0.33/0.57  thf(fact_2384_divide__eq__eq,axiom,
% 0.33/0.57      ! [B: rat,C: rat,A: rat] :
% 0.33/0.57        ( ( ( divide_divide_rat @ B @ C )
% 0.33/0.57          = A )
% 0.33/0.57        = ( ( ( C != zero_zero_rat )
% 0.33/0.57           => ( B
% 0.33/0.57              = ( times_times_rat @ A @ C ) ) )
% 0.33/0.57          & ( ( C = zero_zero_rat )
% 0.33/0.57           => ( A = zero_zero_rat ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_eq_eq
% 0.33/0.57  thf(fact_2385_frac__eq__eq,axiom,
% 0.33/0.57      ! [Y: complex,Z2: complex,X: complex,W2: complex] :
% 0.33/0.57        ( ( Y != zero_zero_complex )
% 0.33/0.57       => ( ( Z2 != zero_zero_complex )
% 0.33/0.57         => ( ( ( divide1717551699836669952omplex @ X @ Y )
% 0.33/0.57              = ( divide1717551699836669952omplex @ W2 @ Z2 ) )
% 0.33/0.57            = ( ( times_times_complex @ X @ Z2 )
% 0.33/0.57              = ( times_times_complex @ W2 @ Y ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % frac_eq_eq
% 0.33/0.57  thf(fact_2386_frac__eq__eq,axiom,
% 0.33/0.57      ! [Y: real,Z2: real,X: real,W2: real] :
% 0.33/0.57        ( ( Y != zero_zero_real )
% 0.33/0.57       => ( ( Z2 != zero_zero_real )
% 0.33/0.57         => ( ( ( divide_divide_real @ X @ Y )
% 0.33/0.57              = ( divide_divide_real @ W2 @ Z2 ) )
% 0.33/0.57            = ( ( times_times_real @ X @ Z2 )
% 0.33/0.57              = ( times_times_real @ W2 @ Y ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % frac_eq_eq
% 0.33/0.57  thf(fact_2387_frac__eq__eq,axiom,
% 0.33/0.57      ! [Y: rat,Z2: rat,X: rat,W2: rat] :
% 0.33/0.57        ( ( Y != zero_zero_rat )
% 0.33/0.57       => ( ( Z2 != zero_zero_rat )
% 0.33/0.57         => ( ( ( divide_divide_rat @ X @ Y )
% 0.33/0.57              = ( divide_divide_rat @ W2 @ Z2 ) )
% 0.33/0.57            = ( ( times_times_rat @ X @ Z2 )
% 0.33/0.57              = ( times_times_rat @ W2 @ Y ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % frac_eq_eq
% 0.33/0.57  thf(fact_2388_nat__mult__1__right,axiom,
% 0.33/0.57      ! [N: nat] :
% 0.33/0.57        ( ( times_times_nat @ N @ one_one_nat )
% 0.33/0.57        = N ) ).
% 0.33/0.57  
% 0.33/0.57  % nat_mult_1_right
% 0.33/0.57  thf(fact_2389_one__reorient,axiom,
% 0.33/0.57      ! [X: complex] :
% 0.33/0.57        ( ( one_one_complex = X )
% 0.33/0.57        = ( X = one_one_complex ) ) ).
% 0.33/0.57  
% 0.33/0.57  % one_reorient
% 0.33/0.57  thf(fact_2390_one__reorient,axiom,
% 0.33/0.57      ! [X: real] :
% 0.33/0.57        ( ( one_one_real = X )
% 0.33/0.57        = ( X = one_one_real ) ) ).
% 0.33/0.57  
% 0.33/0.57  % one_reorient
% 0.33/0.57  thf(fact_2391_one__reorient,axiom,
% 0.33/0.57      ! [X: rat] :
% 0.33/0.57        ( ( one_one_rat = X )
% 0.33/0.57        = ( X = one_one_rat ) ) ).
% 0.33/0.57  
% 0.33/0.57  % one_reorient
% 0.33/0.57  thf(fact_2392_one__reorient,axiom,
% 0.33/0.57      ! [X: nat] :
% 0.33/0.57        ( ( one_one_nat = X )
% 0.33/0.57        = ( X = one_one_nat ) ) ).
% 0.33/0.57  
% 0.33/0.57  % one_reorient
% 0.33/0.57  thf(fact_2393_one__reorient,axiom,
% 0.33/0.57      ! [X: int] :
% 0.33/0.57        ( ( one_one_int = X )
% 0.33/0.57        = ( X = one_one_int ) ) ).
% 0.33/0.57  
% 0.33/0.57  % one_reorient
% 0.33/0.57  thf(fact_2394_nat__mult__1,axiom,
% 0.33/0.57      ! [N: nat] :
% 0.33/0.57        ( ( times_times_nat @ one_one_nat @ N )
% 0.33/0.57        = N ) ).
% 0.33/0.57  
% 0.33/0.57  % nat_mult_1
% 0.33/0.57  thf(fact_2395_mult_Oleft__commute,axiom,
% 0.33/0.57      ! [B: complex,A: complex,C: complex] :
% 0.33/0.57        ( ( times_times_complex @ B @ ( times_times_complex @ A @ C ) )
% 0.33/0.57        = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.left_commute
% 0.33/0.57  thf(fact_2396_mult_Oleft__commute,axiom,
% 0.33/0.57      ! [B: real,A: real,C: real] :
% 0.33/0.57        ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
% 0.33/0.57        = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.left_commute
% 0.33/0.57  thf(fact_2397_mult_Oleft__commute,axiom,
% 0.33/0.57      ! [B: rat,A: rat,C: rat] :
% 0.33/0.57        ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
% 0.33/0.57        = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.left_commute
% 0.33/0.57  thf(fact_2398_mult_Oleft__commute,axiom,
% 0.33/0.57      ! [B: nat,A: nat,C: nat] :
% 0.33/0.57        ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 0.33/0.57        = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.left_commute
% 0.33/0.57  thf(fact_2399_mult_Oleft__commute,axiom,
% 0.33/0.57      ! [B: int,A: int,C: int] :
% 0.33/0.57        ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
% 0.33/0.57        = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.left_commute
% 0.33/0.57  thf(fact_2400_mult_Ocomm__neutral,axiom,
% 0.33/0.57      ! [A: complex] :
% 0.33/0.57        ( ( times_times_complex @ A @ one_one_complex )
% 0.33/0.57        = A ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.comm_neutral
% 0.33/0.57  thf(fact_2401_mult_Ocomm__neutral,axiom,
% 0.33/0.57      ! [A: real] :
% 0.33/0.57        ( ( times_times_real @ A @ one_one_real )
% 0.33/0.57        = A ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.comm_neutral
% 0.33/0.57  thf(fact_2402_mult_Ocomm__neutral,axiom,
% 0.33/0.57      ! [A: rat] :
% 0.33/0.57        ( ( times_times_rat @ A @ one_one_rat )
% 0.33/0.57        = A ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.comm_neutral
% 0.33/0.57  thf(fact_2403_mult_Ocomm__neutral,axiom,
% 0.33/0.57      ! [A: nat] :
% 0.33/0.57        ( ( times_times_nat @ A @ one_one_nat )
% 0.33/0.57        = A ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.comm_neutral
% 0.33/0.57  thf(fact_2404_mult_Ocomm__neutral,axiom,
% 0.33/0.57      ! [A: int] :
% 0.33/0.57        ( ( times_times_int @ A @ one_one_int )
% 0.33/0.57        = A ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.comm_neutral
% 0.33/0.57  thf(fact_2405_mult_Ocommute,axiom,
% 0.33/0.57      ( times_times_complex
% 0.33/0.57      = ( ^ [A4: complex,B4: complex] : ( times_times_complex @ B4 @ A4 ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.commute
% 0.33/0.57  thf(fact_2406_mult_Ocommute,axiom,
% 0.33/0.57      ( times_times_real
% 0.33/0.57      = ( ^ [A4: real,B4: real] : ( times_times_real @ B4 @ A4 ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.commute
% 0.33/0.57  thf(fact_2407_mult_Ocommute,axiom,
% 0.33/0.57      ( times_times_rat
% 0.33/0.57      = ( ^ [A4: rat,B4: rat] : ( times_times_rat @ B4 @ A4 ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.commute
% 0.33/0.57  thf(fact_2408_mult_Ocommute,axiom,
% 0.33/0.57      ( times_times_nat
% 0.33/0.57      = ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.commute
% 0.33/0.57  thf(fact_2409_mult_Ocommute,axiom,
% 0.33/0.57      ( times_times_int
% 0.33/0.57      = ( ^ [A4: int,B4: int] : ( times_times_int @ B4 @ A4 ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.commute
% 0.33/0.57  thf(fact_2410_mult_Oassoc,axiom,
% 0.33/0.57      ! [A: complex,B: complex,C: complex] :
% 0.33/0.57        ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
% 0.33/0.57        = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.assoc
% 0.33/0.57  thf(fact_2411_mult_Oassoc,axiom,
% 0.33/0.57      ! [A: real,B: real,C: real] :
% 0.33/0.57        ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 0.33/0.57        = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.assoc
% 0.33/0.57  thf(fact_2412_mult_Oassoc,axiom,
% 0.33/0.57      ! [A: rat,B: rat,C: rat] :
% 0.33/0.57        ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.33/0.57        = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.assoc
% 0.33/0.57  thf(fact_2413_mult_Oassoc,axiom,
% 0.33/0.57      ! [A: nat,B: nat,C: nat] :
% 0.33/0.57        ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.33/0.57        = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.assoc
% 0.33/0.57  thf(fact_2414_mult_Oassoc,axiom,
% 0.33/0.57      ! [A: int,B: int,C: int] :
% 0.33/0.57        ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 0.33/0.57        = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult.assoc
% 0.33/0.57  thf(fact_2415_comm__monoid__mult__class_Omult__1,axiom,
% 0.33/0.57      ! [A: complex] :
% 0.33/0.57        ( ( times_times_complex @ one_one_complex @ A )
% 0.33/0.57        = A ) ).
% 0.33/0.57  
% 0.33/0.57  % comm_monoid_mult_class.mult_1
% 0.33/0.57  thf(fact_2416_comm__monoid__mult__class_Omult__1,axiom,
% 0.33/0.57      ! [A: real] :
% 0.33/0.57        ( ( times_times_real @ one_one_real @ A )
% 0.33/0.57        = A ) ).
% 0.33/0.57  
% 0.33/0.57  % comm_monoid_mult_class.mult_1
% 0.33/0.57  thf(fact_2417_comm__monoid__mult__class_Omult__1,axiom,
% 0.33/0.57      ! [A: rat] :
% 0.33/0.57        ( ( times_times_rat @ one_one_rat @ A )
% 0.33/0.57        = A ) ).
% 0.33/0.57  
% 0.33/0.57  % comm_monoid_mult_class.mult_1
% 0.33/0.57  thf(fact_2418_comm__monoid__mult__class_Omult__1,axiom,
% 0.33/0.57      ! [A: nat] :
% 0.33/0.57        ( ( times_times_nat @ one_one_nat @ A )
% 0.33/0.57        = A ) ).
% 0.33/0.57  
% 0.33/0.57  % comm_monoid_mult_class.mult_1
% 0.33/0.57  thf(fact_2419_comm__monoid__mult__class_Omult__1,axiom,
% 0.33/0.57      ! [A: int] :
% 0.33/0.57        ( ( times_times_int @ one_one_int @ A )
% 0.33/0.57        = A ) ).
% 0.33/0.57  
% 0.33/0.57  % comm_monoid_mult_class.mult_1
% 0.33/0.57  thf(fact_2420_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.33/0.57      ! [A: complex,B: complex,C: complex] :
% 0.33/0.57        ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
% 0.33/0.57        = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % ab_semigroup_mult_class.mult_ac(1)
% 0.33/0.57  thf(fact_2421_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.33/0.57      ! [A: real,B: real,C: real] :
% 0.33/0.57        ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 0.33/0.57        = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % ab_semigroup_mult_class.mult_ac(1)
% 0.33/0.57  thf(fact_2422_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.33/0.57      ! [A: rat,B: rat,C: rat] :
% 0.33/0.57        ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.33/0.57        = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % ab_semigroup_mult_class.mult_ac(1)
% 0.33/0.57  thf(fact_2423_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.33/0.57      ! [A: nat,B: nat,C: nat] :
% 0.33/0.57        ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.33/0.57        = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % ab_semigroup_mult_class.mult_ac(1)
% 0.33/0.57  thf(fact_2424_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.33/0.57      ! [A: int,B: int,C: int] :
% 0.33/0.57        ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 0.33/0.57        = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % ab_semigroup_mult_class.mult_ac(1)
% 0.33/0.57  thf(fact_2425_nat__mult__eq__cancel__disj,axiom,
% 0.33/0.57      ! [K: nat,M: nat,N: nat] :
% 0.33/0.57        ( ( ( times_times_nat @ K @ M )
% 0.33/0.57          = ( times_times_nat @ K @ N ) )
% 0.33/0.57        = ( ( K = zero_zero_nat )
% 0.33/0.57          | ( M = N ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nat_mult_eq_cancel_disj
% 0.33/0.57  thf(fact_2426_nat__mult__div__cancel1,axiom,
% 0.33/0.57      ! [K: nat,M: nat,N: nat] :
% 0.33/0.57        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.57       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.33/0.57          = ( divide_divide_nat @ M @ N ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % nat_mult_div_cancel1
% 0.33/0.57  thf(fact_2427_divide__less__eq,axiom,
% 0.33/0.57      ! [B: real,C: real,A: real] :
% 0.33/0.57        ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.33/0.57        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.57           => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.33/0.57          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.57           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.57               => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.33/0.57              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.57               => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_less_eq
% 0.33/0.57  thf(fact_2428_divide__less__eq,axiom,
% 0.33/0.57      ! [B: rat,C: rat,A: rat] :
% 0.33/0.57        ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.33/0.57        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.57           => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.33/0.57          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.57           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.57               => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.33/0.57              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.57               => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_less_eq
% 0.33/0.57  thf(fact_2429_less__divide__eq,axiom,
% 0.33/0.57      ! [A: real,B: real,C: real] :
% 0.33/0.57        ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.33/0.57        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.57           => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.33/0.57          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.57           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.57               => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.33/0.57              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.57               => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % less_divide_eq
% 0.33/0.57  thf(fact_2430_less__divide__eq,axiom,
% 0.33/0.57      ! [A: rat,B: rat,C: rat] :
% 0.33/0.57        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.33/0.57        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.57           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.33/0.57          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.57           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.57               => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.33/0.57              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.57               => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % less_divide_eq
% 0.33/0.57  thf(fact_2431_neg__divide__less__eq,axiom,
% 0.33/0.57      ! [C: real,B: real,A: real] :
% 0.33/0.57        ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.57       => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.33/0.57          = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % neg_divide_less_eq
% 0.33/0.57  thf(fact_2432_neg__divide__less__eq,axiom,
% 0.33/0.57      ! [C: rat,B: rat,A: rat] :
% 0.33/0.57        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.57       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.33/0.57          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % neg_divide_less_eq
% 0.33/0.57  thf(fact_2433_neg__less__divide__eq,axiom,
% 0.33/0.57      ! [C: real,A: real,B: real] :
% 0.33/0.57        ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.57       => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.33/0.57          = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % neg_less_divide_eq
% 0.33/0.57  thf(fact_2434_neg__less__divide__eq,axiom,
% 0.33/0.57      ! [C: rat,A: rat,B: rat] :
% 0.33/0.57        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.57       => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.33/0.57          = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % neg_less_divide_eq
% 0.33/0.57  thf(fact_2435_pos__divide__less__eq,axiom,
% 0.33/0.57      ! [C: real,B: real,A: real] :
% 0.33/0.57        ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.57       => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.33/0.57          = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % pos_divide_less_eq
% 0.33/0.57  thf(fact_2436_pos__divide__less__eq,axiom,
% 0.33/0.57      ! [C: rat,B: rat,A: rat] :
% 0.33/0.57        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.57       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.33/0.57          = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % pos_divide_less_eq
% 0.33/0.57  thf(fact_2437_pos__less__divide__eq,axiom,
% 0.33/0.57      ! [C: real,A: real,B: real] :
% 0.33/0.57        ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.57       => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.33/0.57          = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % pos_less_divide_eq
% 0.33/0.57  thf(fact_2438_pos__less__divide__eq,axiom,
% 0.33/0.57      ! [C: rat,A: rat,B: rat] :
% 0.33/0.57        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.57       => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.33/0.57          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % pos_less_divide_eq
% 0.33/0.57  thf(fact_2439_mult__imp__div__pos__less,axiom,
% 0.33/0.57      ! [Y: real,X: real,Z2: real] :
% 0.33/0.57        ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.57       => ( ( ord_less_real @ X @ ( times_times_real @ Z2 @ Y ) )
% 0.33/0.57         => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z2 ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_imp_div_pos_less
% 0.33/0.57  thf(fact_2440_mult__imp__div__pos__less,axiom,
% 0.33/0.57      ! [Y: rat,X: rat,Z2: rat] :
% 0.33/0.57        ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.33/0.57       => ( ( ord_less_rat @ X @ ( times_times_rat @ Z2 @ Y ) )
% 0.33/0.57         => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_imp_div_pos_less
% 0.33/0.57  thf(fact_2441_mult__imp__less__div__pos,axiom,
% 0.33/0.57      ! [Y: real,Z2: real,X: real] :
% 0.33/0.57        ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.57       => ( ( ord_less_real @ ( times_times_real @ Z2 @ Y ) @ X )
% 0.33/0.57         => ( ord_less_real @ Z2 @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_imp_less_div_pos
% 0.33/0.57  thf(fact_2442_mult__imp__less__div__pos,axiom,
% 0.33/0.57      ! [Y: rat,Z2: rat,X: rat] :
% 0.33/0.57        ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.33/0.57       => ( ( ord_less_rat @ ( times_times_rat @ Z2 @ Y ) @ X )
% 0.33/0.57         => ( ord_less_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_imp_less_div_pos
% 0.33/0.57  thf(fact_2443_divide__strict__left__mono,axiom,
% 0.33/0.57      ! [B: real,A: real,C: real] :
% 0.33/0.57        ( ( ord_less_real @ B @ A )
% 0.33/0.57       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.57         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.33/0.57           => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_strict_left_mono
% 0.33/0.57  thf(fact_2444_divide__strict__left__mono,axiom,
% 0.33/0.57      ! [B: rat,A: rat,C: rat] :
% 0.33/0.57        ( ( ord_less_rat @ B @ A )
% 0.33/0.57       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.57         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.33/0.57           => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_strict_left_mono
% 0.33/0.57  thf(fact_2445_divide__strict__left__mono__neg,axiom,
% 0.33/0.57      ! [A: real,B: real,C: real] :
% 0.33/0.57        ( ( ord_less_real @ A @ B )
% 0.33/0.57       => ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.57         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.33/0.57           => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_strict_left_mono_neg
% 0.33/0.57  thf(fact_2446_divide__strict__left__mono__neg,axiom,
% 0.33/0.57      ! [A: rat,B: rat,C: rat] :
% 0.33/0.57        ( ( ord_less_rat @ A @ B )
% 0.33/0.57       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.57         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.33/0.57           => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_strict_left_mono_neg
% 0.33/0.57  thf(fact_2447_eq__divide__eq__numeral_I1_J,axiom,
% 0.33/0.57      ! [W2: num,B: complex,C: complex] :
% 0.33/0.57        ( ( ( numera6690914467698888265omplex @ W2 )
% 0.33/0.57          = ( divide1717551699836669952omplex @ B @ C ) )
% 0.33/0.57        = ( ( ( C != zero_zero_complex )
% 0.33/0.57           => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C )
% 0.33/0.57              = B ) )
% 0.33/0.57          & ( ( C = zero_zero_complex )
% 0.33/0.57           => ( ( numera6690914467698888265omplex @ W2 )
% 0.33/0.57              = zero_zero_complex ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % eq_divide_eq_numeral(1)
% 0.33/0.57  thf(fact_2448_eq__divide__eq__numeral_I1_J,axiom,
% 0.33/0.57      ! [W2: num,B: real,C: real] :
% 0.33/0.57        ( ( ( numeral_numeral_real @ W2 )
% 0.33/0.57          = ( divide_divide_real @ B @ C ) )
% 0.33/0.57        = ( ( ( C != zero_zero_real )
% 0.33/0.57           => ( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C )
% 0.33/0.57              = B ) )
% 0.33/0.57          & ( ( C = zero_zero_real )
% 0.33/0.57           => ( ( numeral_numeral_real @ W2 )
% 0.33/0.57              = zero_zero_real ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % eq_divide_eq_numeral(1)
% 0.33/0.57  thf(fact_2449_eq__divide__eq__numeral_I1_J,axiom,
% 0.33/0.57      ! [W2: num,B: rat,C: rat] :
% 0.33/0.57        ( ( ( numeral_numeral_rat @ W2 )
% 0.33/0.57          = ( divide_divide_rat @ B @ C ) )
% 0.33/0.57        = ( ( ( C != zero_zero_rat )
% 0.33/0.57           => ( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C )
% 0.33/0.57              = B ) )
% 0.33/0.57          & ( ( C = zero_zero_rat )
% 0.33/0.57           => ( ( numeral_numeral_rat @ W2 )
% 0.33/0.57              = zero_zero_rat ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % eq_divide_eq_numeral(1)
% 0.33/0.57  thf(fact_2450_divide__eq__eq__numeral_I1_J,axiom,
% 0.33/0.57      ! [B: complex,C: complex,W2: num] :
% 0.33/0.57        ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.33/0.57          = ( numera6690914467698888265omplex @ W2 ) )
% 0.33/0.57        = ( ( ( C != zero_zero_complex )
% 0.33/0.57           => ( B
% 0.33/0.57              = ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C ) ) )
% 0.33/0.57          & ( ( C = zero_zero_complex )
% 0.33/0.57           => ( ( numera6690914467698888265omplex @ W2 )
% 0.33/0.57              = zero_zero_complex ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_eq_eq_numeral(1)
% 0.33/0.57  thf(fact_2451_divide__eq__eq__numeral_I1_J,axiom,
% 0.33/0.57      ! [B: real,C: real,W2: num] :
% 0.33/0.57        ( ( ( divide_divide_real @ B @ C )
% 0.33/0.57          = ( numeral_numeral_real @ W2 ) )
% 0.33/0.57        = ( ( ( C != zero_zero_real )
% 0.33/0.57           => ( B
% 0.33/0.57              = ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 0.33/0.57          & ( ( C = zero_zero_real )
% 0.33/0.57           => ( ( numeral_numeral_real @ W2 )
% 0.33/0.57              = zero_zero_real ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_eq_eq_numeral(1)
% 0.33/0.57  thf(fact_2452_divide__eq__eq__numeral_I1_J,axiom,
% 0.33/0.57      ! [B: rat,C: rat,W2: num] :
% 0.33/0.57        ( ( ( divide_divide_rat @ B @ C )
% 0.33/0.57          = ( numeral_numeral_rat @ W2 ) )
% 0.33/0.57        = ( ( ( C != zero_zero_rat )
% 0.33/0.57           => ( B
% 0.33/0.57              = ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 0.33/0.57          & ( ( C = zero_zero_rat )
% 0.33/0.57           => ( ( numeral_numeral_rat @ W2 )
% 0.33/0.57              = zero_zero_rat ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_eq_eq_numeral(1)
% 0.33/0.57  thf(fact_2453_add__divide__eq__if__simps_I2_J,axiom,
% 0.33/0.57      ! [Z2: complex,A: complex,B: complex] :
% 0.33/0.57        ( ( ( Z2 = zero_zero_complex )
% 0.33/0.57         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 0.33/0.57            = B ) )
% 0.33/0.57        & ( ( Z2 != zero_zero_complex )
% 0.33/0.57         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 0.33/0.57            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % add_divide_eq_if_simps(2)
% 0.33/0.57  thf(fact_2454_add__divide__eq__if__simps_I2_J,axiom,
% 0.33/0.57      ! [Z2: real,A: real,B: real] :
% 0.33/0.57        ( ( ( Z2 = zero_zero_real )
% 0.33/0.57         => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 0.33/0.57            = B ) )
% 0.33/0.57        & ( ( Z2 != zero_zero_real )
% 0.33/0.57         => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 0.33/0.57            = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % add_divide_eq_if_simps(2)
% 0.33/0.57  thf(fact_2455_add__divide__eq__if__simps_I2_J,axiom,
% 0.33/0.57      ! [Z2: rat,A: rat,B: rat] :
% 0.33/0.57        ( ( ( Z2 = zero_zero_rat )
% 0.33/0.57         => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 0.33/0.57            = B ) )
% 0.33/0.57        & ( ( Z2 != zero_zero_rat )
% 0.33/0.57         => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 0.33/0.57            = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % add_divide_eq_if_simps(2)
% 0.33/0.57  thf(fact_2456_add__divide__eq__if__simps_I1_J,axiom,
% 0.33/0.57      ! [Z2: complex,A: complex,B: complex] :
% 0.33/0.57        ( ( ( Z2 = zero_zero_complex )
% 0.33/0.57         => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 0.33/0.57            = A ) )
% 0.33/0.57        & ( ( Z2 != zero_zero_complex )
% 0.33/0.57         => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 0.33/0.57            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % add_divide_eq_if_simps(1)
% 0.33/0.57  thf(fact_2457_add__divide__eq__if__simps_I1_J,axiom,
% 0.33/0.57      ! [Z2: real,A: real,B: real] :
% 0.33/0.57        ( ( ( Z2 = zero_zero_real )
% 0.33/0.57         => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 0.33/0.57            = A ) )
% 0.33/0.57        & ( ( Z2 != zero_zero_real )
% 0.33/0.57         => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 0.33/0.57            = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % add_divide_eq_if_simps(1)
% 0.33/0.57  thf(fact_2458_add__divide__eq__if__simps_I1_J,axiom,
% 0.33/0.57      ! [Z2: rat,A: rat,B: rat] :
% 0.33/0.57        ( ( ( Z2 = zero_zero_rat )
% 0.33/0.57         => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 0.33/0.57            = A ) )
% 0.33/0.57        & ( ( Z2 != zero_zero_rat )
% 0.33/0.57         => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 0.33/0.57            = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % add_divide_eq_if_simps(1)
% 0.33/0.57  thf(fact_2459_add__frac__eq,axiom,
% 0.33/0.57      ! [Y: complex,Z2: complex,X: complex,W2: complex] :
% 0.33/0.57        ( ( Y != zero_zero_complex )
% 0.33/0.57       => ( ( Z2 != zero_zero_complex )
% 0.33/0.57         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W2 @ Z2 ) )
% 0.33/0.57            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ W2 @ Y ) ) @ ( times_times_complex @ Y @ Z2 ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % add_frac_eq
% 0.33/0.57  thf(fact_2460_add__frac__eq,axiom,
% 0.33/0.57      ! [Y: real,Z2: real,X: real,W2: real] :
% 0.33/0.57        ( ( Y != zero_zero_real )
% 0.33/0.57       => ( ( Z2 != zero_zero_real )
% 0.33/0.57         => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z2 ) )
% 0.33/0.57            = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % add_frac_eq
% 0.33/0.57  thf(fact_2461_add__frac__eq,axiom,
% 0.33/0.57      ! [Y: rat,Z2: rat,X: rat,W2: rat] :
% 0.33/0.57        ( ( Y != zero_zero_rat )
% 0.33/0.57       => ( ( Z2 != zero_zero_rat )
% 0.33/0.57         => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z2 ) )
% 0.33/0.57            = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % add_frac_eq
% 0.33/0.57  thf(fact_2462_add__frac__num,axiom,
% 0.33/0.57      ! [Y: complex,X: complex,Z2: complex] :
% 0.33/0.57        ( ( Y != zero_zero_complex )
% 0.33/0.57       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z2 )
% 0.33/0.57          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z2 @ Y ) ) @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % add_frac_num
% 0.33/0.57  thf(fact_2463_add__frac__num,axiom,
% 0.33/0.57      ! [Y: real,X: real,Z2: real] :
% 0.33/0.57        ( ( Y != zero_zero_real )
% 0.33/0.57       => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z2 )
% 0.33/0.57          = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z2 @ Y ) ) @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % add_frac_num
% 0.33/0.57  thf(fact_2464_add__frac__num,axiom,
% 0.33/0.57      ! [Y: rat,X: rat,Z2: rat] :
% 0.33/0.57        ( ( Y != zero_zero_rat )
% 0.33/0.57       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 )
% 0.33/0.57          = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z2 @ Y ) ) @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % add_frac_num
% 0.33/0.57  thf(fact_2465_add__num__frac,axiom,
% 0.33/0.57      ! [Y: complex,Z2: complex,X: complex] :
% 0.33/0.57        ( ( Y != zero_zero_complex )
% 0.33/0.57       => ( ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ X @ Y ) )
% 0.33/0.57          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z2 @ Y ) ) @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % add_num_frac
% 0.33/0.57  thf(fact_2466_add__num__frac,axiom,
% 0.33/0.57      ! [Y: real,Z2: real,X: real] :
% 0.33/0.57        ( ( Y != zero_zero_real )
% 0.33/0.57       => ( ( plus_plus_real @ Z2 @ ( divide_divide_real @ X @ Y ) )
% 0.33/0.57          = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z2 @ Y ) ) @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % add_num_frac
% 0.33/0.57  thf(fact_2467_add__num__frac,axiom,
% 0.33/0.57      ! [Y: rat,Z2: rat,X: rat] :
% 0.33/0.57        ( ( Y != zero_zero_rat )
% 0.33/0.57       => ( ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) )
% 0.33/0.57          = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z2 @ Y ) ) @ Y ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % add_num_frac
% 0.33/0.57  thf(fact_2468_add__divide__eq__iff,axiom,
% 0.33/0.57      ! [Z2: complex,X: complex,Y: complex] :
% 0.33/0.57        ( ( Z2 != zero_zero_complex )
% 0.33/0.57       => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z2 ) )
% 0.33/0.57          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % add_divide_eq_iff
% 0.33/0.57  thf(fact_2469_add__divide__eq__iff,axiom,
% 0.33/0.57      ! [Z2: real,X: real,Y: real] :
% 0.33/0.57        ( ( Z2 != zero_zero_real )
% 0.33/0.57       => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z2 ) )
% 0.33/0.57          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % add_divide_eq_iff
% 0.33/0.57  thf(fact_2470_add__divide__eq__iff,axiom,
% 0.33/0.57      ! [Z2: rat,X: rat,Y: rat] :
% 0.33/0.57        ( ( Z2 != zero_zero_rat )
% 0.33/0.57       => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z2 ) )
% 0.33/0.57          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % add_divide_eq_iff
% 0.33/0.57  thf(fact_2471_divide__add__eq__iff,axiom,
% 0.33/0.57      ! [Z2: complex,X: complex,Y: complex] :
% 0.33/0.57        ( ( Z2 != zero_zero_complex )
% 0.33/0.57       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z2 ) @ Y )
% 0.33/0.57          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_add_eq_iff
% 0.33/0.57  thf(fact_2472_divide__add__eq__iff,axiom,
% 0.33/0.57      ! [Z2: real,X: real,Y: real] :
% 0.33/0.57        ( ( Z2 != zero_zero_real )
% 0.33/0.57       => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z2 ) @ Y )
% 0.33/0.57          = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_add_eq_iff
% 0.33/0.57  thf(fact_2473_divide__add__eq__iff,axiom,
% 0.33/0.57      ! [Z2: rat,X: rat,Y: rat] :
% 0.33/0.57        ( ( Z2 != zero_zero_rat )
% 0.33/0.57       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z2 ) @ Y )
% 0.33/0.57          = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_add_eq_iff
% 0.33/0.57  thf(fact_2474_divide__less__eq__1,axiom,
% 0.33/0.57      ! [B: real,A: real] :
% 0.33/0.57        ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.33/0.57        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.57            & ( ord_less_real @ B @ A ) )
% 0.33/0.57          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.57            & ( ord_less_real @ A @ B ) )
% 0.33/0.57          | ( A = zero_zero_real ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_less_eq_1
% 0.33/0.57  thf(fact_2475_divide__less__eq__1,axiom,
% 0.33/0.57      ! [B: rat,A: rat] :
% 0.33/0.57        ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.33/0.57        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.57            & ( ord_less_rat @ B @ A ) )
% 0.33/0.57          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.57            & ( ord_less_rat @ A @ B ) )
% 0.33/0.57          | ( A = zero_zero_rat ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % divide_less_eq_1
% 0.33/0.57  thf(fact_2476_less__divide__eq__1,axiom,
% 0.33/0.57      ! [B: real,A: real] :
% 0.33/0.57        ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.33/0.57        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.57            & ( ord_less_real @ A @ B ) )
% 0.33/0.57          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.57            & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % less_divide_eq_1
% 0.33/0.57  thf(fact_2477_less__divide__eq__1,axiom,
% 0.33/0.57      ! [B: rat,A: rat] :
% 0.33/0.57        ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.33/0.57        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.57            & ( ord_less_rat @ A @ B ) )
% 0.33/0.57          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.57            & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % less_divide_eq_1
% 0.33/0.57  thf(fact_2478_less__half__sum,axiom,
% 0.33/0.57      ! [A: real,B: real] :
% 0.33/0.57        ( ( ord_less_real @ A @ B )
% 0.33/0.57       => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % less_half_sum
% 0.33/0.57  thf(fact_2479_less__half__sum,axiom,
% 0.33/0.57      ! [A: rat,B: rat] :
% 0.33/0.57        ( ( ord_less_rat @ A @ B )
% 0.33/0.57       => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % less_half_sum
% 0.33/0.57  thf(fact_2480_gt__half__sum,axiom,
% 0.33/0.57      ! [A: real,B: real] :
% 0.33/0.57        ( ( ord_less_real @ A @ B )
% 0.33/0.57       => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% 0.33/0.57  
% 0.33/0.57  % gt_half_sum
% 0.33/0.57  thf(fact_2481_gt__half__sum,axiom,
% 0.33/0.57      ! [A: rat,B: rat] :
% 0.33/0.57        ( ( ord_less_rat @ A @ B )
% 0.33/0.57       => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% 0.33/0.57  
% 0.33/0.57  % gt_half_sum
% 0.33/0.57  thf(fact_2482_mult__left__le,axiom,
% 0.33/0.57      ! [C: real,A: real] :
% 0.33/0.57        ( ( ord_less_eq_real @ C @ one_one_real )
% 0.33/0.57       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.57         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_left_le
% 0.33/0.57  thf(fact_2483_mult__left__le,axiom,
% 0.33/0.57      ! [C: rat,A: rat] :
% 0.33/0.57        ( ( ord_less_eq_rat @ C @ one_one_rat )
% 0.33/0.57       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.57         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_left_le
% 0.33/0.57  thf(fact_2484_mult__left__le,axiom,
% 0.33/0.57      ! [C: nat,A: nat] :
% 0.33/0.57        ( ( ord_less_eq_nat @ C @ one_one_nat )
% 0.33/0.57       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.33/0.57         => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_left_le
% 0.33/0.57  thf(fact_2485_mult__left__le,axiom,
% 0.33/0.57      ! [C: int,A: int] :
% 0.33/0.57        ( ( ord_less_eq_int @ C @ one_one_int )
% 0.33/0.57       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.57         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_left_le
% 0.33/0.57  thf(fact_2486_mult__le__one,axiom,
% 0.33/0.57      ! [A: real,B: real] :
% 0.33/0.57        ( ( ord_less_eq_real @ A @ one_one_real )
% 0.33/0.57       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.33/0.57         => ( ( ord_less_eq_real @ B @ one_one_real )
% 0.33/0.57           => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_le_one
% 0.33/0.57  thf(fact_2487_mult__le__one,axiom,
% 0.33/0.57      ! [A: rat,B: rat] :
% 0.33/0.57        ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.33/0.57       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.33/0.57         => ( ( ord_less_eq_rat @ B @ one_one_rat )
% 0.33/0.57           => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_le_one
% 0.33/0.57  thf(fact_2488_mult__le__one,axiom,
% 0.33/0.57      ! [A: nat,B: nat] :
% 0.33/0.57        ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.33/0.57       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.33/0.57         => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 0.33/0.57           => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_le_one
% 0.33/0.57  thf(fact_2489_mult__le__one,axiom,
% 0.33/0.57      ! [A: int,B: int] :
% 0.33/0.57        ( ( ord_less_eq_int @ A @ one_one_int )
% 0.33/0.57       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.33/0.57         => ( ( ord_less_eq_int @ B @ one_one_int )
% 0.33/0.57           => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_le_one
% 0.33/0.57  thf(fact_2490_mult__right__le__one__le,axiom,
% 0.33/0.57      ! [X: real,Y: real] :
% 0.33/0.57        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.57       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.57         => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.57           => ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_right_le_one_le
% 0.33/0.57  thf(fact_2491_mult__right__le__one__le,axiom,
% 0.33/0.57      ! [X: rat,Y: rat] :
% 0.33/0.57        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.33/0.57       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.33/0.57         => ( ( ord_less_eq_rat @ Y @ one_one_rat )
% 0.33/0.57           => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_right_le_one_le
% 0.33/0.57  thf(fact_2492_mult__right__le__one__le,axiom,
% 0.33/0.57      ! [X: int,Y: int] :
% 0.33/0.57        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.57       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.57         => ( ( ord_less_eq_int @ Y @ one_one_int )
% 0.33/0.57           => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_right_le_one_le
% 0.33/0.57  thf(fact_2493_mult__left__le__one__le,axiom,
% 0.33/0.57      ! [X: real,Y: real] :
% 0.33/0.57        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.57       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.57         => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.57           => ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_left_le_one_le
% 0.33/0.57  thf(fact_2494_mult__left__le__one__le,axiom,
% 0.33/0.57      ! [X: rat,Y: rat] :
% 0.33/0.57        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.33/0.57       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.33/0.57         => ( ( ord_less_eq_rat @ Y @ one_one_rat )
% 0.33/0.57           => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).
% 0.33/0.57  
% 0.33/0.57  % mult_left_le_one_le
% 0.33/0.57  thf(fact_2495_mult__left__le__one__le,axiom,
% 0.33/0.57      ! [X: int,Y: int] :
% 0.33/0.57        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.57       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.57         => ( ( ord_less_eq_int @ Y @ one_one_int )
% 0.33/0.57           => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_left_le_one_le
% 0.33/0.58  thf(fact_2496_power__less__power__Suc,axiom,
% 0.33/0.58      ! [A: real,N: nat] :
% 0.33/0.58        ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.58       => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % power_less_power_Suc
% 0.33/0.58  thf(fact_2497_power__less__power__Suc,axiom,
% 0.33/0.58      ! [A: rat,N: nat] :
% 0.33/0.58        ( ( ord_less_rat @ one_one_rat @ A )
% 0.33/0.58       => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % power_less_power_Suc
% 0.33/0.58  thf(fact_2498_power__less__power__Suc,axiom,
% 0.33/0.58      ! [A: nat,N: nat] :
% 0.33/0.58        ( ( ord_less_nat @ one_one_nat @ A )
% 0.33/0.58       => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % power_less_power_Suc
% 0.33/0.58  thf(fact_2499_power__less__power__Suc,axiom,
% 0.33/0.58      ! [A: int,N: nat] :
% 0.33/0.58        ( ( ord_less_int @ one_one_int @ A )
% 0.33/0.58       => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % power_less_power_Suc
% 0.33/0.58  thf(fact_2500_power__gt1__lemma,axiom,
% 0.33/0.58      ! [A: real,N: nat] :
% 0.33/0.58        ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.58       => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % power_gt1_lemma
% 0.33/0.58  thf(fact_2501_power__gt1__lemma,axiom,
% 0.33/0.58      ! [A: rat,N: nat] :
% 0.33/0.58        ( ( ord_less_rat @ one_one_rat @ A )
% 0.33/0.58       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % power_gt1_lemma
% 0.33/0.58  thf(fact_2502_power__gt1__lemma,axiom,
% 0.33/0.58      ! [A: nat,N: nat] :
% 0.33/0.58        ( ( ord_less_nat @ one_one_nat @ A )
% 0.33/0.58       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % power_gt1_lemma
% 0.33/0.58  thf(fact_2503_power__gt1__lemma,axiom,
% 0.33/0.58      ! [A: int,N: nat] :
% 0.33/0.58        ( ( ord_less_int @ one_one_int @ A )
% 0.33/0.58       => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % power_gt1_lemma
% 0.33/0.58  thf(fact_2504_mult__not__zero,axiom,
% 0.33/0.58      ! [A: complex,B: complex] :
% 0.33/0.58        ( ( ( times_times_complex @ A @ B )
% 0.33/0.58         != zero_zero_complex )
% 0.33/0.58       => ( ( A != zero_zero_complex )
% 0.33/0.58          & ( B != zero_zero_complex ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_not_zero
% 0.33/0.58  thf(fact_2505_mult__not__zero,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ( times_times_real @ A @ B )
% 0.33/0.58         != zero_zero_real )
% 0.33/0.58       => ( ( A != zero_zero_real )
% 0.33/0.58          & ( B != zero_zero_real ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_not_zero
% 0.33/0.58  thf(fact_2506_mult__not__zero,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ( times_times_rat @ A @ B )
% 0.33/0.58         != zero_zero_rat )
% 0.33/0.58       => ( ( A != zero_zero_rat )
% 0.33/0.58          & ( B != zero_zero_rat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_not_zero
% 0.33/0.58  thf(fact_2507_mult__not__zero,axiom,
% 0.33/0.58      ! [A: nat,B: nat] :
% 0.33/0.58        ( ( ( times_times_nat @ A @ B )
% 0.33/0.58         != zero_zero_nat )
% 0.33/0.58       => ( ( A != zero_zero_nat )
% 0.33/0.58          & ( B != zero_zero_nat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_not_zero
% 0.33/0.58  thf(fact_2508_mult__not__zero,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( ( times_times_int @ A @ B )
% 0.33/0.58         != zero_zero_int )
% 0.33/0.58       => ( ( A != zero_zero_int )
% 0.33/0.58          & ( B != zero_zero_int ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_not_zero
% 0.33/0.58  thf(fact_2509_divisors__zero,axiom,
% 0.33/0.58      ! [A: complex,B: complex] :
% 0.33/0.58        ( ( ( times_times_complex @ A @ B )
% 0.33/0.58          = zero_zero_complex )
% 0.33/0.58       => ( ( A = zero_zero_complex )
% 0.33/0.58          | ( B = zero_zero_complex ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divisors_zero
% 0.33/0.58  thf(fact_2510_divisors__zero,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ( times_times_real @ A @ B )
% 0.33/0.58          = zero_zero_real )
% 0.33/0.58       => ( ( A = zero_zero_real )
% 0.33/0.58          | ( B = zero_zero_real ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divisors_zero
% 0.33/0.58  thf(fact_2511_divisors__zero,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ( times_times_rat @ A @ B )
% 0.33/0.58          = zero_zero_rat )
% 0.33/0.58       => ( ( A = zero_zero_rat )
% 0.33/0.58          | ( B = zero_zero_rat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divisors_zero
% 0.33/0.58  thf(fact_2512_divisors__zero,axiom,
% 0.33/0.58      ! [A: nat,B: nat] :
% 0.33/0.58        ( ( ( times_times_nat @ A @ B )
% 0.33/0.58          = zero_zero_nat )
% 0.33/0.58       => ( ( A = zero_zero_nat )
% 0.33/0.58          | ( B = zero_zero_nat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divisors_zero
% 0.33/0.58  thf(fact_2513_divisors__zero,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( ( times_times_int @ A @ B )
% 0.33/0.58          = zero_zero_int )
% 0.33/0.58       => ( ( A = zero_zero_int )
% 0.33/0.58          | ( B = zero_zero_int ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divisors_zero
% 0.33/0.58  thf(fact_2514_no__zero__divisors,axiom,
% 0.33/0.58      ! [A: complex,B: complex] :
% 0.33/0.58        ( ( A != zero_zero_complex )
% 0.33/0.58       => ( ( B != zero_zero_complex )
% 0.33/0.58         => ( ( times_times_complex @ A @ B )
% 0.33/0.58           != zero_zero_complex ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % no_zero_divisors
% 0.33/0.58  thf(fact_2515_no__zero__divisors,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( A != zero_zero_real )
% 0.33/0.58       => ( ( B != zero_zero_real )
% 0.33/0.58         => ( ( times_times_real @ A @ B )
% 0.33/0.58           != zero_zero_real ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % no_zero_divisors
% 0.33/0.58  thf(fact_2516_no__zero__divisors,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( A != zero_zero_rat )
% 0.33/0.58       => ( ( B != zero_zero_rat )
% 0.33/0.58         => ( ( times_times_rat @ A @ B )
% 0.33/0.58           != zero_zero_rat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % no_zero_divisors
% 0.33/0.58  thf(fact_2517_no__zero__divisors,axiom,
% 0.33/0.58      ! [A: nat,B: nat] :
% 0.33/0.58        ( ( A != zero_zero_nat )
% 0.33/0.58       => ( ( B != zero_zero_nat )
% 0.33/0.58         => ( ( times_times_nat @ A @ B )
% 0.33/0.58           != zero_zero_nat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % no_zero_divisors
% 0.33/0.58  thf(fact_2518_no__zero__divisors,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( A != zero_zero_int )
% 0.33/0.58       => ( ( B != zero_zero_int )
% 0.33/0.58         => ( ( times_times_int @ A @ B )
% 0.33/0.58           != zero_zero_int ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % no_zero_divisors
% 0.33/0.58  thf(fact_2519_mult__left__cancel,axiom,
% 0.33/0.58      ! [C: complex,A: complex,B: complex] :
% 0.33/0.58        ( ( C != zero_zero_complex )
% 0.33/0.58       => ( ( ( times_times_complex @ C @ A )
% 0.33/0.58            = ( times_times_complex @ C @ B ) )
% 0.33/0.58          = ( A = B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_left_cancel
% 0.33/0.58  thf(fact_2520_mult__left__cancel,axiom,
% 0.33/0.58      ! [C: real,A: real,B: real] :
% 0.33/0.58        ( ( C != zero_zero_real )
% 0.33/0.58       => ( ( ( times_times_real @ C @ A )
% 0.33/0.58            = ( times_times_real @ C @ B ) )
% 0.33/0.58          = ( A = B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_left_cancel
% 0.33/0.58  thf(fact_2521_mult__left__cancel,axiom,
% 0.33/0.58      ! [C: rat,A: rat,B: rat] :
% 0.33/0.58        ( ( C != zero_zero_rat )
% 0.33/0.58       => ( ( ( times_times_rat @ C @ A )
% 0.33/0.58            = ( times_times_rat @ C @ B ) )
% 0.33/0.58          = ( A = B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_left_cancel
% 0.33/0.58  thf(fact_2522_mult__left__cancel,axiom,
% 0.33/0.58      ! [C: nat,A: nat,B: nat] :
% 0.33/0.58        ( ( C != zero_zero_nat )
% 0.33/0.58       => ( ( ( times_times_nat @ C @ A )
% 0.33/0.58            = ( times_times_nat @ C @ B ) )
% 0.33/0.58          = ( A = B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_left_cancel
% 0.33/0.58  thf(fact_2523_mult__left__cancel,axiom,
% 0.33/0.58      ! [C: int,A: int,B: int] :
% 0.33/0.58        ( ( C != zero_zero_int )
% 0.33/0.58       => ( ( ( times_times_int @ C @ A )
% 0.33/0.58            = ( times_times_int @ C @ B ) )
% 0.33/0.58          = ( A = B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_left_cancel
% 0.33/0.58  thf(fact_2524_mult__right__cancel,axiom,
% 0.33/0.58      ! [C: complex,A: complex,B: complex] :
% 0.33/0.58        ( ( C != zero_zero_complex )
% 0.33/0.58       => ( ( ( times_times_complex @ A @ C )
% 0.33/0.58            = ( times_times_complex @ B @ C ) )
% 0.33/0.58          = ( A = B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_right_cancel
% 0.33/0.58  thf(fact_2525_mult__right__cancel,axiom,
% 0.33/0.58      ! [C: real,A: real,B: real] :
% 0.33/0.58        ( ( C != zero_zero_real )
% 0.33/0.58       => ( ( ( times_times_real @ A @ C )
% 0.33/0.58            = ( times_times_real @ B @ C ) )
% 0.33/0.58          = ( A = B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_right_cancel
% 0.33/0.58  thf(fact_2526_mult__right__cancel,axiom,
% 0.33/0.58      ! [C: rat,A: rat,B: rat] :
% 0.33/0.58        ( ( C != zero_zero_rat )
% 0.33/0.58       => ( ( ( times_times_rat @ A @ C )
% 0.33/0.58            = ( times_times_rat @ B @ C ) )
% 0.33/0.58          = ( A = B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_right_cancel
% 0.33/0.58  thf(fact_2527_mult__right__cancel,axiom,
% 0.33/0.58      ! [C: nat,A: nat,B: nat] :
% 0.33/0.58        ( ( C != zero_zero_nat )
% 0.33/0.58       => ( ( ( times_times_nat @ A @ C )
% 0.33/0.58            = ( times_times_nat @ B @ C ) )
% 0.33/0.58          = ( A = B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_right_cancel
% 0.33/0.58  thf(fact_2528_mult__right__cancel,axiom,
% 0.33/0.58      ! [C: int,A: int,B: int] :
% 0.33/0.58        ( ( C != zero_zero_int )
% 0.33/0.58       => ( ( ( times_times_int @ A @ C )
% 0.33/0.58            = ( times_times_int @ B @ C ) )
% 0.33/0.58          = ( A = B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_right_cancel
% 0.33/0.58  thf(fact_2529_Suc__mult__cancel1,axiom,
% 0.33/0.58      ! [K: nat,M: nat,N: nat] :
% 0.33/0.58        ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 0.33/0.58          = ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.33/0.58        = ( M = N ) ) ).
% 0.33/0.58  
% 0.33/0.58  % Suc_mult_cancel1
% 0.33/0.58  thf(fact_2530_mult__0,axiom,
% 0.33/0.58      ! [N: nat] :
% 0.33/0.58        ( ( times_times_nat @ zero_zero_nat @ N )
% 0.33/0.58        = zero_zero_nat ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_0
% 0.33/0.58  thf(fact_2531_le__numeral__extra_I4_J,axiom,
% 0.33/0.58      ord_less_eq_real @ one_one_real @ one_one_real ).
% 0.33/0.58  
% 0.33/0.58  % le_numeral_extra(4)
% 0.33/0.58  thf(fact_2532_le__numeral__extra_I4_J,axiom,
% 0.33/0.58      ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 0.33/0.58  
% 0.33/0.58  % le_numeral_extra(4)
% 0.33/0.58  thf(fact_2533_le__numeral__extra_I4_J,axiom,
% 0.33/0.58      ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 0.33/0.58  
% 0.33/0.58  % le_numeral_extra(4)
% 0.33/0.58  thf(fact_2534_le__numeral__extra_I4_J,axiom,
% 0.33/0.58      ord_less_eq_int @ one_one_int @ one_one_int ).
% 0.33/0.58  
% 0.33/0.58  % le_numeral_extra(4)
% 0.33/0.58  thf(fact_2535_zero__neq__one,axiom,
% 0.33/0.58      zero_zero_complex != one_one_complex ).
% 0.33/0.58  
% 0.33/0.58  % zero_neq_one
% 0.33/0.58  thf(fact_2536_zero__neq__one,axiom,
% 0.33/0.58      zero_zero_real != one_one_real ).
% 0.33/0.58  
% 0.33/0.58  % zero_neq_one
% 0.33/0.58  thf(fact_2537_zero__neq__one,axiom,
% 0.33/0.58      zero_zero_rat != one_one_rat ).
% 0.33/0.58  
% 0.33/0.58  % zero_neq_one
% 0.33/0.58  thf(fact_2538_zero__neq__one,axiom,
% 0.33/0.58      zero_zero_nat != one_one_nat ).
% 0.33/0.58  
% 0.33/0.58  % zero_neq_one
% 0.33/0.58  thf(fact_2539_zero__neq__one,axiom,
% 0.33/0.58      zero_zero_int != one_one_int ).
% 0.33/0.58  
% 0.33/0.58  % zero_neq_one
% 0.33/0.58  thf(fact_2540_less__numeral__extra_I4_J,axiom,
% 0.33/0.58      ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 0.33/0.58  
% 0.33/0.58  % less_numeral_extra(4)
% 0.33/0.58  thf(fact_2541_less__numeral__extra_I4_J,axiom,
% 0.33/0.58      ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 0.33/0.58  
% 0.33/0.58  % less_numeral_extra(4)
% 0.33/0.58  thf(fact_2542_less__numeral__extra_I4_J,axiom,
% 0.33/0.58      ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 0.33/0.58  
% 0.33/0.58  % less_numeral_extra(4)
% 0.33/0.58  thf(fact_2543_less__numeral__extra_I4_J,axiom,
% 0.33/0.58      ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 0.33/0.58  
% 0.33/0.58  % less_numeral_extra(4)
% 0.33/0.58  thf(fact_2544_le__cube,axiom,
% 0.33/0.58      ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % le_cube
% 0.33/0.58  thf(fact_2545_le__square,axiom,
% 0.33/0.58      ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 0.33/0.58  
% 0.33/0.58  % le_square
% 0.33/0.58  thf(fact_2546_mult__le__mono,axiom,
% 0.33/0.58      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.33/0.58        ( ( ord_less_eq_nat @ I @ J )
% 0.33/0.58       => ( ( ord_less_eq_nat @ K @ L )
% 0.33/0.58         => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_mono
% 0.33/0.58  thf(fact_2547_mult__le__mono1,axiom,
% 0.33/0.58      ! [I: nat,J: nat,K: nat] :
% 0.33/0.58        ( ( ord_less_eq_nat @ I @ J )
% 0.33/0.58       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_mono1
% 0.33/0.58  thf(fact_2548_mult__le__mono2,axiom,
% 0.33/0.58      ! [I: nat,J: nat,K: nat] :
% 0.33/0.58        ( ( ord_less_eq_nat @ I @ J )
% 0.33/0.58       => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_mono2
% 0.33/0.58  thf(fact_2549_add__mult__distrib2,axiom,
% 0.33/0.58      ! [K: nat,M: nat,N: nat] :
% 0.33/0.58        ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
% 0.33/0.58        = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % add_mult_distrib2
% 0.33/0.58  thf(fact_2550_add__mult__distrib,axiom,
% 0.33/0.58      ! [M: nat,N: nat,K: nat] :
% 0.33/0.58        ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 0.33/0.58        = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % add_mult_distrib
% 0.33/0.58  thf(fact_2551_nat__mult__eq__cancel1,axiom,
% 0.33/0.58      ! [K: nat,M: nat,N: nat] :
% 0.33/0.58        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.58       => ( ( ( times_times_nat @ K @ M )
% 0.33/0.58            = ( times_times_nat @ K @ N ) )
% 0.33/0.58          = ( M = N ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % nat_mult_eq_cancel1
% 0.33/0.58  thf(fact_2552_nat__mult__less__cancel1,axiom,
% 0.33/0.58      ! [K: nat,M: nat,N: nat] :
% 0.33/0.58        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.58       => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.33/0.58          = ( ord_less_nat @ M @ N ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % nat_mult_less_cancel1
% 0.33/0.58  thf(fact_2553_divide__le__eq,axiom,
% 0.33/0.58      ! [B: real,C: real,A: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.33/0.58          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58               => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.33/0.58              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58               => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_le_eq
% 0.33/0.58  thf(fact_2554_divide__le__eq,axiom,
% 0.33/0.58      ! [B: rat,C: rat,A: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.33/0.58          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58               => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.33/0.58              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58               => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_le_eq
% 0.33/0.58  thf(fact_2555_le__divide__eq,axiom,
% 0.33/0.58      ! [A: real,B: real,C: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.33/0.58          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58               => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.33/0.58              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58               => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % le_divide_eq
% 0.33/0.58  thf(fact_2556_le__divide__eq,axiom,
% 0.33/0.58      ! [A: rat,B: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.33/0.58          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58               => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.33/0.58              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58               => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % le_divide_eq
% 0.33/0.58  thf(fact_2557_divide__left__mono,axiom,
% 0.33/0.58      ! [B: real,A: real,C: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ B @ A )
% 0.33/0.58       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.33/0.58         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.33/0.58           => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_left_mono
% 0.33/0.58  thf(fact_2558_divide__left__mono,axiom,
% 0.33/0.58      ! [B: rat,A: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ B @ A )
% 0.33/0.58       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.33/0.58         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.33/0.58           => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_left_mono
% 0.33/0.58  thf(fact_2559_neg__divide__le__eq,axiom,
% 0.33/0.58      ! [C: real,B: real,A: real] :
% 0.33/0.58        ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.33/0.58          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % neg_divide_le_eq
% 0.33/0.58  thf(fact_2560_neg__divide__le__eq,axiom,
% 0.33/0.58      ! [C: rat,B: rat,A: rat] :
% 0.33/0.58        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.33/0.58          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % neg_divide_le_eq
% 0.33/0.58  thf(fact_2561_neg__le__divide__eq,axiom,
% 0.33/0.58      ! [C: real,A: real,B: real] :
% 0.33/0.58        ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58       => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.33/0.58          = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % neg_le_divide_eq
% 0.33/0.58  thf(fact_2562_neg__le__divide__eq,axiom,
% 0.33/0.58      ! [C: rat,A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58       => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.33/0.58          = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % neg_le_divide_eq
% 0.33/0.58  thf(fact_2563_pos__divide__le__eq,axiom,
% 0.33/0.58      ! [C: real,B: real,A: real] :
% 0.33/0.58        ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.33/0.58          = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % pos_divide_le_eq
% 0.33/0.58  thf(fact_2564_pos__divide__le__eq,axiom,
% 0.33/0.58      ! [C: rat,B: rat,A: rat] :
% 0.33/0.58        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.33/0.58          = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % pos_divide_le_eq
% 0.33/0.58  thf(fact_2565_pos__le__divide__eq,axiom,
% 0.33/0.58      ! [C: real,A: real,B: real] :
% 0.33/0.58        ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58       => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.33/0.58          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % pos_le_divide_eq
% 0.33/0.58  thf(fact_2566_pos__le__divide__eq,axiom,
% 0.33/0.58      ! [C: rat,A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58       => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.33/0.58          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % pos_le_divide_eq
% 0.33/0.58  thf(fact_2567_mult__imp__div__pos__le,axiom,
% 0.33/0.58      ! [Y: real,X: real,Z2: real] :
% 0.33/0.58        ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.58       => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z2 @ Y ) )
% 0.33/0.58         => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z2 ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_imp_div_pos_le
% 0.33/0.58  thf(fact_2568_mult__imp__div__pos__le,axiom,
% 0.33/0.58      ! [Y: rat,X: rat,Z2: rat] :
% 0.33/0.58        ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.33/0.58       => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z2 @ Y ) )
% 0.33/0.58         => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_imp_div_pos_le
% 0.33/0.58  thf(fact_2569_mult__imp__le__div__pos,axiom,
% 0.33/0.58      ! [Y: real,Z2: real,X: real] :
% 0.33/0.58        ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.58       => ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ Y ) @ X )
% 0.33/0.58         => ( ord_less_eq_real @ Z2 @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_imp_le_div_pos
% 0.33/0.58  thf(fact_2570_mult__imp__le__div__pos,axiom,
% 0.33/0.58      ! [Y: rat,Z2: rat,X: rat] :
% 0.33/0.58        ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.33/0.58       => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ Y ) @ X )
% 0.33/0.58         => ( ord_less_eq_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_imp_le_div_pos
% 0.33/0.58  thf(fact_2571_divide__left__mono__neg,axiom,
% 0.33/0.58      ! [A: real,B: real,C: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.33/0.58         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.33/0.58           => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_left_mono_neg
% 0.33/0.58  thf(fact_2572_divide__left__mono__neg,axiom,
% 0.33/0.58      ! [A: rat,B: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.33/0.58         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.33/0.58           => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_left_mono_neg
% 0.33/0.58  thf(fact_2573_divide__less__eq__numeral_I1_J,axiom,
% 0.33/0.58      ! [B: real,C: real,W2: num] :
% 0.33/0.58        ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 0.33/0.58          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58               => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
% 0.33/0.58              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58               => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_less_eq_numeral(1)
% 0.33/0.58  thf(fact_2574_divide__less__eq__numeral_I1_J,axiom,
% 0.33/0.58      ! [B: rat,C: rat,W2: num] :
% 0.33/0.58        ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W2 ) )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 0.33/0.58          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58               => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
% 0.33/0.58              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58               => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_less_eq_numeral(1)
% 0.33/0.58  thf(fact_2575_less__divide__eq__numeral_I1_J,axiom,
% 0.33/0.58      ! [W2: num,B: real,C: real] :
% 0.33/0.58        ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
% 0.33/0.58          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58               => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 0.33/0.58              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58               => ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % less_divide_eq_numeral(1)
% 0.33/0.58  thf(fact_2576_less__divide__eq__numeral_I1_J,axiom,
% 0.33/0.58      ! [W2: num,B: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
% 0.33/0.58          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58               => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 0.33/0.58              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58               => ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % less_divide_eq_numeral(1)
% 0.33/0.58  thf(fact_2577_divide__le__eq__1,axiom,
% 0.33/0.58      ! [B: real,A: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.58            & ( ord_less_eq_real @ B @ A ) )
% 0.33/0.58          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.58            & ( ord_less_eq_real @ A @ B ) )
% 0.33/0.58          | ( A = zero_zero_real ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_le_eq_1
% 0.33/0.58  thf(fact_2578_divide__le__eq__1,axiom,
% 0.33/0.58      ! [B: rat,A: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.58            & ( ord_less_eq_rat @ B @ A ) )
% 0.33/0.58          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.58            & ( ord_less_eq_rat @ A @ B ) )
% 0.33/0.58          | ( A = zero_zero_rat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_le_eq_1
% 0.33/0.58  thf(fact_2579_le__divide__eq__1,axiom,
% 0.33/0.58      ! [B: real,A: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.58            & ( ord_less_eq_real @ A @ B ) )
% 0.33/0.58          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.58            & ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % le_divide_eq_1
% 0.33/0.58  thf(fact_2580_le__divide__eq__1,axiom,
% 0.33/0.58      ! [B: rat,A: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.58            & ( ord_less_eq_rat @ A @ B ) )
% 0.33/0.58          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.58            & ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % le_divide_eq_1
% 0.33/0.58  thf(fact_2581_mult__le__cancel__left1,axiom,
% 0.33/0.58      ! [C: real,B: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ord_less_eq_real @ one_one_real @ B ) )
% 0.33/0.58          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58           => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_cancel_left1
% 0.33/0.58  thf(fact_2582_mult__le__cancel__left1,axiom,
% 0.33/0.58      ! [C: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 0.33/0.58          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58           => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_cancel_left1
% 0.33/0.58  thf(fact_2583_mult__le__cancel__left1,axiom,
% 0.33/0.58      ! [C: int,B: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
% 0.33/0.58        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.33/0.58           => ( ord_less_eq_int @ one_one_int @ B ) )
% 0.33/0.58          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.33/0.58           => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_cancel_left1
% 0.33/0.58  thf(fact_2584_mult__le__cancel__left2,axiom,
% 0.33/0.58      ! [C: real,A: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ord_less_eq_real @ A @ one_one_real ) )
% 0.33/0.58          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58           => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_cancel_left2
% 0.33/0.58  thf(fact_2585_mult__le__cancel__left2,axiom,
% 0.33/0.58      ! [C: rat,A: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 0.33/0.58          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58           => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_cancel_left2
% 0.33/0.58  thf(fact_2586_mult__le__cancel__left2,axiom,
% 0.33/0.58      ! [C: int,A: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
% 0.33/0.58        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.33/0.58           => ( ord_less_eq_int @ A @ one_one_int ) )
% 0.33/0.58          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.33/0.58           => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_cancel_left2
% 0.33/0.58  thf(fact_2587_mult__le__cancel__right1,axiom,
% 0.33/0.58      ! [C: real,B: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ord_less_eq_real @ one_one_real @ B ) )
% 0.33/0.58          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58           => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_cancel_right1
% 0.33/0.58  thf(fact_2588_mult__le__cancel__right1,axiom,
% 0.33/0.58      ! [C: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 0.33/0.58          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58           => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_cancel_right1
% 0.33/0.58  thf(fact_2589_mult__le__cancel__right1,axiom,
% 0.33/0.58      ! [C: int,B: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.33/0.58           => ( ord_less_eq_int @ one_one_int @ B ) )
% 0.33/0.58          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.33/0.58           => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_cancel_right1
% 0.33/0.58  thf(fact_2590_mult__le__cancel__right2,axiom,
% 0.33/0.58      ! [A: real,C: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ord_less_eq_real @ A @ one_one_real ) )
% 0.33/0.58          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58           => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_cancel_right2
% 0.33/0.58  thf(fact_2591_mult__le__cancel__right2,axiom,
% 0.33/0.58      ! [A: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 0.33/0.58          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58           => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_cancel_right2
% 0.33/0.58  thf(fact_2592_mult__le__cancel__right2,axiom,
% 0.33/0.58      ! [A: int,C: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
% 0.33/0.58        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.33/0.58           => ( ord_less_eq_int @ A @ one_one_int ) )
% 0.33/0.58          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.33/0.58           => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_cancel_right2
% 0.33/0.58  thf(fact_2593_mult__less__cancel__left1,axiom,
% 0.33/0.58      ! [C: real,B: real] :
% 0.33/0.58        ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
% 0.33/0.58        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ord_less_real @ one_one_real @ B ) )
% 0.33/0.58          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.33/0.58           => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_left1
% 0.33/0.58  thf(fact_2594_mult__less__cancel__left1,axiom,
% 0.33/0.58      ! [C: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
% 0.33/0.58        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ord_less_rat @ one_one_rat @ B ) )
% 0.33/0.58          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.33/0.58           => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_left1
% 0.33/0.58  thf(fact_2595_mult__less__cancel__left1,axiom,
% 0.33/0.58      ! [C: int,B: int] :
% 0.33/0.58        ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
% 0.33/0.58        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.33/0.58           => ( ord_less_int @ one_one_int @ B ) )
% 0.33/0.58          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.33/0.58           => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_left1
% 0.33/0.58  thf(fact_2596_mult__less__cancel__left2,axiom,
% 0.33/0.58      ! [C: real,A: real] :
% 0.33/0.58        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
% 0.33/0.58        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ord_less_real @ A @ one_one_real ) )
% 0.33/0.58          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.33/0.58           => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_left2
% 0.33/0.58  thf(fact_2597_mult__less__cancel__left2,axiom,
% 0.33/0.58      ! [C: rat,A: rat] :
% 0.33/0.58        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
% 0.33/0.58        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ord_less_rat @ A @ one_one_rat ) )
% 0.33/0.58          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.33/0.58           => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_left2
% 0.33/0.58  thf(fact_2598_mult__less__cancel__left2,axiom,
% 0.33/0.58      ! [C: int,A: int] :
% 0.33/0.58        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
% 0.33/0.58        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.33/0.58           => ( ord_less_int @ A @ one_one_int ) )
% 0.33/0.58          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.33/0.58           => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_left2
% 0.33/0.58  thf(fact_2599_mult__less__cancel__right1,axiom,
% 0.33/0.58      ! [C: real,B: real] :
% 0.33/0.58        ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ord_less_real @ one_one_real @ B ) )
% 0.33/0.58          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.33/0.58           => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_right1
% 0.33/0.58  thf(fact_2600_mult__less__cancel__right1,axiom,
% 0.33/0.58      ! [C: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ord_less_rat @ one_one_rat @ B ) )
% 0.33/0.58          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.33/0.58           => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_right1
% 0.33/0.58  thf(fact_2601_mult__less__cancel__right1,axiom,
% 0.33/0.58      ! [C: int,B: int] :
% 0.33/0.58        ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.33/0.58           => ( ord_less_int @ one_one_int @ B ) )
% 0.33/0.58          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.33/0.58           => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_right1
% 0.33/0.58  thf(fact_2602_mult__less__cancel__right2,axiom,
% 0.33/0.58      ! [A: real,C: real] :
% 0.33/0.58        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
% 0.33/0.58        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ord_less_real @ A @ one_one_real ) )
% 0.33/0.58          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.33/0.58           => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_right2
% 0.33/0.58  thf(fact_2603_mult__less__cancel__right2,axiom,
% 0.33/0.58      ! [A: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
% 0.33/0.58        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ord_less_rat @ A @ one_one_rat ) )
% 0.33/0.58          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.33/0.58           => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_right2
% 0.33/0.58  thf(fact_2604_mult__less__cancel__right2,axiom,
% 0.33/0.58      ! [A: int,C: int] :
% 0.33/0.58        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
% 0.33/0.58        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.33/0.58           => ( ord_less_int @ A @ one_one_int ) )
% 0.33/0.58          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.33/0.58           => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_right2
% 0.33/0.58  thf(fact_2605_field__le__mult__one__interval,axiom,
% 0.33/0.58      ! [X: real,Y: real] :
% 0.33/0.58        ( ! [Z4: real] :
% 0.33/0.58            ( ( ord_less_real @ zero_zero_real @ Z4 )
% 0.33/0.58           => ( ( ord_less_real @ Z4 @ one_one_real )
% 0.33/0.58             => ( ord_less_eq_real @ ( times_times_real @ Z4 @ X ) @ Y ) ) )
% 0.33/0.58       => ( ord_less_eq_real @ X @ Y ) ) ).
% 0.33/0.58  
% 0.33/0.58  % field_le_mult_one_interval
% 0.33/0.58  thf(fact_2606_field__le__mult__one__interval,axiom,
% 0.33/0.58      ! [X: rat,Y: rat] :
% 0.33/0.58        ( ! [Z4: rat] :
% 0.33/0.58            ( ( ord_less_rat @ zero_zero_rat @ Z4 )
% 0.33/0.58           => ( ( ord_less_rat @ Z4 @ one_one_rat )
% 0.33/0.58             => ( ord_less_eq_rat @ ( times_times_rat @ Z4 @ X ) @ Y ) ) )
% 0.33/0.58       => ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.33/0.58  
% 0.33/0.58  % field_le_mult_one_interval
% 0.33/0.58  thf(fact_2607_convex__bound__le,axiom,
% 0.33/0.58      ! [X: real,A: real,Y: real,U: real,V: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ X @ A )
% 0.33/0.58       => ( ( ord_less_eq_real @ Y @ A )
% 0.33/0.58         => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.33/0.58           => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.33/0.58             => ( ( ( plus_plus_real @ U @ V )
% 0.33/0.58                  = one_one_real )
% 0.33/0.58               => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % convex_bound_le
% 0.33/0.58  thf(fact_2608_convex__bound__le,axiom,
% 0.33/0.58      ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ X @ A )
% 0.33/0.58       => ( ( ord_less_eq_rat @ Y @ A )
% 0.33/0.58         => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.33/0.58           => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.33/0.58             => ( ( ( plus_plus_rat @ U @ V )
% 0.33/0.58                  = one_one_rat )
% 0.33/0.58               => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % convex_bound_le
% 0.33/0.58  thf(fact_2609_convex__bound__le,axiom,
% 0.33/0.58      ! [X: int,A: int,Y: int,U: int,V: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ X @ A )
% 0.33/0.58       => ( ( ord_less_eq_int @ Y @ A )
% 0.33/0.58         => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.33/0.58           => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.33/0.58             => ( ( ( plus_plus_int @ U @ V )
% 0.33/0.58                  = one_one_int )
% 0.33/0.58               => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % convex_bound_le
% 0.33/0.58  thf(fact_2610_power__Suc__less,axiom,
% 0.33/0.58      ! [A: real,N: nat] :
% 0.33/0.58        ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.58       => ( ( ord_less_real @ A @ one_one_real )
% 0.33/0.58         => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % power_Suc_less
% 0.33/0.58  thf(fact_2611_power__Suc__less,axiom,
% 0.33/0.58      ! [A: rat,N: nat] :
% 0.33/0.58        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.58       => ( ( ord_less_rat @ A @ one_one_rat )
% 0.33/0.58         => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % power_Suc_less
% 0.33/0.58  thf(fact_2612_power__Suc__less,axiom,
% 0.33/0.58      ! [A: nat,N: nat] :
% 0.33/0.58        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.33/0.58       => ( ( ord_less_nat @ A @ one_one_nat )
% 0.33/0.58         => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % power_Suc_less
% 0.33/0.58  thf(fact_2613_power__Suc__less,axiom,
% 0.33/0.58      ! [A: int,N: nat] :
% 0.33/0.58        ( ( ord_less_int @ zero_zero_int @ A )
% 0.33/0.58       => ( ( ord_less_int @ A @ one_one_int )
% 0.33/0.58         => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % power_Suc_less
% 0.33/0.58  thf(fact_2614_le__divide__eq__numeral_I1_J,axiom,
% 0.33/0.58      ! [W2: num,B: real,C: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
% 0.33/0.58          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58               => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 0.33/0.58              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58               => ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % le_divide_eq_numeral(1)
% 0.33/0.58  thf(fact_2615_le__divide__eq__numeral_I1_J,axiom,
% 0.33/0.58      ! [W2: num,B: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
% 0.33/0.58          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58               => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 0.33/0.58              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58               => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % le_divide_eq_numeral(1)
% 0.33/0.58  thf(fact_2616_divide__le__eq__numeral_I1_J,axiom,
% 0.33/0.58      ! [B: real,C: real,W2: num] :
% 0.33/0.58        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 0.33/0.58          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58               => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
% 0.33/0.58              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58               => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_le_eq_numeral(1)
% 0.33/0.58  thf(fact_2617_divide__le__eq__numeral_I1_J,axiom,
% 0.33/0.58      ! [B: rat,C: rat,W2: num] :
% 0.33/0.58        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W2 ) )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 0.33/0.58          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58               => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
% 0.33/0.58              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58               => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_le_eq_numeral(1)
% 0.33/0.58  thf(fact_2618_convex__bound__lt,axiom,
% 0.33/0.58      ! [X: real,A: real,Y: real,U: real,V: real] :
% 0.33/0.58        ( ( ord_less_real @ X @ A )
% 0.33/0.58       => ( ( ord_less_real @ Y @ A )
% 0.33/0.58         => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.33/0.58           => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.33/0.58             => ( ( ( plus_plus_real @ U @ V )
% 0.33/0.58                  = one_one_real )
% 0.33/0.58               => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % convex_bound_lt
% 0.33/0.58  thf(fact_2619_convex__bound__lt,axiom,
% 0.33/0.58      ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
% 0.33/0.58        ( ( ord_less_rat @ X @ A )
% 0.33/0.58       => ( ( ord_less_rat @ Y @ A )
% 0.33/0.58         => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.33/0.58           => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.33/0.58             => ( ( ( plus_plus_rat @ U @ V )
% 0.33/0.58                  = one_one_rat )
% 0.33/0.58               => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % convex_bound_lt
% 0.33/0.58  thf(fact_2620_convex__bound__lt,axiom,
% 0.33/0.58      ! [X: int,A: int,Y: int,U: int,V: int] :
% 0.33/0.58        ( ( ord_less_int @ X @ A )
% 0.33/0.58       => ( ( ord_less_int @ Y @ A )
% 0.33/0.58         => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.33/0.58           => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.33/0.58             => ( ( ( plus_plus_int @ U @ V )
% 0.33/0.58                  = one_one_int )
% 0.33/0.58               => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % convex_bound_lt
% 0.33/0.58  thf(fact_2621_divide__right__mono__neg,axiom,
% 0.33/0.58      ! [A: real,B: real,C: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.33/0.58         => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_right_mono_neg
% 0.33/0.58  thf(fact_2622_divide__right__mono__neg,axiom,
% 0.33/0.58      ! [A: rat,B: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.33/0.58         => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_right_mono_neg
% 0.33/0.58  thf(fact_2623_divide__nonpos__nonpos,axiom,
% 0.33/0.58      ! [X: real,Y: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.33/0.58       => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.33/0.58         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_nonpos_nonpos
% 0.33/0.58  thf(fact_2624_divide__nonpos__nonpos,axiom,
% 0.33/0.58      ! [X: rat,Y: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.33/0.58       => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 0.33/0.58         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_nonpos_nonpos
% 0.33/0.58  thf(fact_2625_divide__nonpos__nonneg,axiom,
% 0.33/0.58      ! [X: real,Y: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.33/0.58       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.58         => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_nonpos_nonneg
% 0.33/0.58  thf(fact_2626_divide__nonpos__nonneg,axiom,
% 0.33/0.58      ! [X: rat,Y: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.33/0.58       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.33/0.58         => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_nonpos_nonneg
% 0.33/0.58  thf(fact_2627_divide__nonneg__nonpos,axiom,
% 0.33/0.58      ! [X: real,Y: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.58       => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.33/0.58         => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_nonneg_nonpos
% 0.33/0.58  thf(fact_2628_divide__nonneg__nonpos,axiom,
% 0.33/0.58      ! [X: rat,Y: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.33/0.58       => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 0.33/0.58         => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_nonneg_nonpos
% 0.33/0.58  thf(fact_2629_divide__nonneg__nonneg,axiom,
% 0.33/0.58      ! [X: real,Y: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.58       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.58         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_nonneg_nonneg
% 0.33/0.58  thf(fact_2630_divide__nonneg__nonneg,axiom,
% 0.33/0.58      ! [X: rat,Y: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.33/0.58       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.33/0.58         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_nonneg_nonneg
% 0.33/0.58  thf(fact_2631_zero__le__divide__iff,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 0.33/0.58        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.58            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.33/0.58          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.33/0.58            & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_le_divide_iff
% 0.33/0.58  thf(fact_2632_zero__le__divide__iff,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 0.33/0.58        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.58            & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.33/0.58          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.33/0.58            & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_le_divide_iff
% 0.33/0.58  thf(fact_2633_divide__right__mono,axiom,
% 0.33/0.58      ! [A: real,B: real,C: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.33/0.58         => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_right_mono
% 0.33/0.58  thf(fact_2634_divide__right__mono,axiom,
% 0.33/0.58      ! [A: rat,B: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.33/0.58         => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_right_mono
% 0.33/0.58  thf(fact_2635_divide__le__0__iff,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 0.33/0.58        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.58            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.33/0.58          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.33/0.58            & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_le_0_iff
% 0.33/0.58  thf(fact_2636_divide__le__0__iff,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 0.33/0.58        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.58            & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.33/0.58          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.33/0.58            & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_le_0_iff
% 0.33/0.58  thf(fact_2637_divide__neg__neg,axiom,
% 0.33/0.58      ! [X: real,Y: real] :
% 0.33/0.58        ( ( ord_less_real @ X @ zero_zero_real )
% 0.33/0.58       => ( ( ord_less_real @ Y @ zero_zero_real )
% 0.33/0.58         => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_neg_neg
% 0.33/0.58  thf(fact_2638_divide__neg__neg,axiom,
% 0.33/0.58      ! [X: rat,Y: rat] :
% 0.33/0.58        ( ( ord_less_rat @ X @ zero_zero_rat )
% 0.33/0.58       => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 0.33/0.58         => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_neg_neg
% 0.33/0.58  thf(fact_2639_divide__neg__pos,axiom,
% 0.33/0.58      ! [X: real,Y: real] :
% 0.33/0.58        ( ( ord_less_real @ X @ zero_zero_real )
% 0.33/0.58       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.58         => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_neg_pos
% 0.33/0.58  thf(fact_2640_divide__neg__pos,axiom,
% 0.33/0.58      ! [X: rat,Y: rat] :
% 0.33/0.58        ( ( ord_less_rat @ X @ zero_zero_rat )
% 0.33/0.58       => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.33/0.58         => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_neg_pos
% 0.33/0.58  thf(fact_2641_divide__pos__neg,axiom,
% 0.33/0.58      ! [X: real,Y: real] :
% 0.33/0.58        ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58       => ( ( ord_less_real @ Y @ zero_zero_real )
% 0.33/0.58         => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_pos_neg
% 0.33/0.58  thf(fact_2642_divide__pos__neg,axiom,
% 0.33/0.58      ! [X: rat,Y: rat] :
% 0.33/0.58        ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.33/0.58       => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 0.33/0.58         => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_pos_neg
% 0.33/0.58  thf(fact_2643_divide__pos__pos,axiom,
% 0.33/0.58      ! [X: real,Y: real] :
% 0.33/0.58        ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.58         => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_pos_pos
% 0.33/0.58  thf(fact_2644_divide__pos__pos,axiom,
% 0.33/0.58      ! [X: rat,Y: rat] :
% 0.33/0.58        ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.33/0.58       => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.33/0.58         => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_pos_pos
% 0.33/0.58  thf(fact_2645_divide__less__0__iff,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.58            & ( ord_less_real @ B @ zero_zero_real ) )
% 0.33/0.58          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.58            & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_less_0_iff
% 0.33/0.58  thf(fact_2646_divide__less__0__iff,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.58            & ( ord_less_rat @ B @ zero_zero_rat ) )
% 0.33/0.58          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.58            & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_less_0_iff
% 0.33/0.58  thf(fact_2647_divide__less__cancel,axiom,
% 0.33/0.58      ! [A: real,C: real,B: real] :
% 0.33/0.58        ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ord_less_real @ A @ B ) )
% 0.33/0.58          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58           => ( ord_less_real @ B @ A ) )
% 0.33/0.58          & ( C != zero_zero_real ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_less_cancel
% 0.33/0.58  thf(fact_2648_divide__less__cancel,axiom,
% 0.33/0.58      ! [A: rat,C: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ord_less_rat @ A @ B ) )
% 0.33/0.58          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58           => ( ord_less_rat @ B @ A ) )
% 0.33/0.58          & ( C != zero_zero_rat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_less_cancel
% 0.33/0.58  thf(fact_2649_zero__less__divide__iff,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.58            & ( ord_less_real @ zero_zero_real @ B ) )
% 0.33/0.58          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.58            & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_divide_iff
% 0.33/0.58  thf(fact_2650_zero__less__divide__iff,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.58            & ( ord_less_rat @ zero_zero_rat @ B ) )
% 0.33/0.58          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.58            & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_divide_iff
% 0.33/0.58  thf(fact_2651_divide__strict__right__mono,axiom,
% 0.33/0.58      ! [A: real,B: real,C: real] :
% 0.33/0.58        ( ( ord_less_real @ A @ B )
% 0.33/0.58       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58         => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_strict_right_mono
% 0.33/0.58  thf(fact_2652_divide__strict__right__mono,axiom,
% 0.33/0.58      ! [A: rat,B: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_rat @ A @ B )
% 0.33/0.58       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58         => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_strict_right_mono
% 0.33/0.58  thf(fact_2653_divide__strict__right__mono__neg,axiom,
% 0.33/0.58      ! [B: real,A: real,C: real] :
% 0.33/0.58        ( ( ord_less_real @ B @ A )
% 0.33/0.58       => ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58         => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_strict_right_mono_neg
% 0.33/0.58  thf(fact_2654_divide__strict__right__mono__neg,axiom,
% 0.33/0.58      ! [B: rat,A: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_rat @ B @ A )
% 0.33/0.58       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58         => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_strict_right_mono_neg
% 0.33/0.58  thf(fact_2655_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.33/0.58      ! [A: real,B: real,C: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.33/0.58         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % ordered_comm_semiring_class.comm_mult_left_mono
% 0.33/0.58  thf(fact_2656_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.33/0.58      ! [A: rat,B: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.33/0.58         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % ordered_comm_semiring_class.comm_mult_left_mono
% 0.33/0.58  thf(fact_2657_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.33/0.58      ! [A: nat,B: nat,C: nat] :
% 0.33/0.58        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.33/0.58         => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % ordered_comm_semiring_class.comm_mult_left_mono
% 0.33/0.58  thf(fact_2658_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.33/0.58      ! [A: int,B: int,C: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.33/0.58         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % ordered_comm_semiring_class.comm_mult_left_mono
% 0.33/0.58  thf(fact_2659_zero__le__mult__iff,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.33/0.58        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.58            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.33/0.58          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.33/0.58            & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_le_mult_iff
% 0.33/0.58  thf(fact_2660_zero__le__mult__iff,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.33/0.58        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.58            & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.33/0.58          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.33/0.58            & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_le_mult_iff
% 0.33/0.58  thf(fact_2661_zero__le__mult__iff,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.33/0.58        = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.58            & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 0.33/0.58          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.33/0.58            & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_le_mult_iff
% 0.33/0.58  thf(fact_2662_mult__nonneg__nonpos2,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.58       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.33/0.58         => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_nonneg_nonpos2
% 0.33/0.58  thf(fact_2663_mult__nonneg__nonpos2,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.58       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.33/0.58         => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_nonneg_nonpos2
% 0.33/0.58  thf(fact_2664_mult__nonneg__nonpos2,axiom,
% 0.33/0.58      ! [A: nat,B: nat] :
% 0.33/0.58        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.33/0.58       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.33/0.58         => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_nonneg_nonpos2
% 0.33/0.58  thf(fact_2665_mult__nonneg__nonpos2,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.58       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.33/0.58         => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_nonneg_nonpos2
% 0.33/0.58  thf(fact_2666_mult__nonpos__nonneg,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.33/0.58       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.33/0.58         => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_nonpos_nonneg
% 0.33/0.58  thf(fact_2667_mult__nonpos__nonneg,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.33/0.58       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.33/0.58         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_nonpos_nonneg
% 0.33/0.58  thf(fact_2668_mult__nonpos__nonneg,axiom,
% 0.33/0.58      ! [A: nat,B: nat] :
% 0.33/0.58        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.33/0.58       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.33/0.58         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_nonpos_nonneg
% 0.33/0.58  thf(fact_2669_mult__nonpos__nonneg,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.33/0.58       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.33/0.58         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_nonpos_nonneg
% 0.33/0.58  thf(fact_2670_mult__nonneg__nonpos,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.58       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.33/0.58         => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_nonneg_nonpos
% 0.33/0.58  thf(fact_2671_mult__nonneg__nonpos,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.58       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.33/0.58         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_nonneg_nonpos
% 0.33/0.58  thf(fact_2672_mult__nonneg__nonpos,axiom,
% 0.33/0.58      ! [A: nat,B: nat] :
% 0.33/0.58        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.33/0.58       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.33/0.58         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_nonneg_nonpos
% 0.33/0.58  thf(fact_2673_mult__nonneg__nonpos,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.58       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.33/0.58         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_nonneg_nonpos
% 0.33/0.58  thf(fact_2674_mult__nonneg__nonneg,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.58       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.33/0.58         => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_nonneg_nonneg
% 0.33/0.58  thf(fact_2675_mult__nonneg__nonneg,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.58       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.33/0.58         => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_nonneg_nonneg
% 0.33/0.58  thf(fact_2676_mult__nonneg__nonneg,axiom,
% 0.33/0.58      ! [A: nat,B: nat] :
% 0.33/0.58        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.33/0.58       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.33/0.58         => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_nonneg_nonneg
% 0.33/0.58  thf(fact_2677_mult__nonneg__nonneg,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.58       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.33/0.58         => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_nonneg_nonneg
% 0.33/0.58  thf(fact_2678_split__mult__neg__le,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.58            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.33/0.58          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.33/0.58            & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
% 0.33/0.58       => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% 0.33/0.58  
% 0.33/0.58  % split_mult_neg_le
% 0.33/0.58  thf(fact_2679_split__mult__neg__le,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.58            & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.33/0.58          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.33/0.58            & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
% 0.33/0.58       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% 0.33/0.58  
% 0.33/0.58  % split_mult_neg_le
% 0.33/0.58  thf(fact_2680_split__mult__neg__le,axiom,
% 0.33/0.58      ! [A: nat,B: nat] :
% 0.33/0.58        ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.33/0.58            & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 0.33/0.58          | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.33/0.58            & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 0.33/0.58       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% 0.33/0.58  
% 0.33/0.58  % split_mult_neg_le
% 0.33/0.58  thf(fact_2681_split__mult__neg__le,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.58            & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 0.33/0.58          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.33/0.58            & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
% 0.33/0.58       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% 0.33/0.58  
% 0.33/0.58  % split_mult_neg_le
% 0.33/0.58  thf(fact_2682_mult__le__0__iff,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.33/0.58        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.58            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.33/0.58          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.33/0.58            & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_0_iff
% 0.33/0.58  thf(fact_2683_mult__le__0__iff,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.33/0.58        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.58            & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.33/0.58          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.33/0.58            & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_0_iff
% 0.33/0.58  thf(fact_2684_mult__le__0__iff,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 0.33/0.58        = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.58            & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 0.33/0.58          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.33/0.58            & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_0_iff
% 0.33/0.58  thf(fact_2685_mult__right__mono,axiom,
% 0.33/0.58      ! [A: real,B: real,C: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.33/0.58         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_right_mono
% 0.33/0.58  thf(fact_2686_mult__right__mono,axiom,
% 0.33/0.58      ! [A: rat,B: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.33/0.58         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_right_mono
% 0.33/0.58  thf(fact_2687_mult__right__mono,axiom,
% 0.33/0.58      ! [A: nat,B: nat,C: nat] :
% 0.33/0.58        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.33/0.58         => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_right_mono
% 0.33/0.58  thf(fact_2688_mult__right__mono,axiom,
% 0.33/0.58      ! [A: int,B: int,C: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.33/0.58         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_right_mono
% 0.33/0.58  thf(fact_2689_mult__right__mono__neg,axiom,
% 0.33/0.58      ! [B: real,A: real,C: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ B @ A )
% 0.33/0.58       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.33/0.58         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_right_mono_neg
% 0.33/0.58  thf(fact_2690_mult__right__mono__neg,axiom,
% 0.33/0.58      ! [B: rat,A: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ B @ A )
% 0.33/0.58       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.33/0.58         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_right_mono_neg
% 0.33/0.58  thf(fact_2691_mult__right__mono__neg,axiom,
% 0.33/0.58      ! [B: int,A: int,C: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ B @ A )
% 0.33/0.58       => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.33/0.58         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_right_mono_neg
% 0.33/0.58  thf(fact_2692_mult__left__mono,axiom,
% 0.33/0.58      ! [A: real,B: real,C: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.33/0.58         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_left_mono
% 0.33/0.58  thf(fact_2693_mult__left__mono,axiom,
% 0.33/0.58      ! [A: rat,B: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.33/0.58         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_left_mono
% 0.33/0.58  thf(fact_2694_mult__left__mono,axiom,
% 0.33/0.58      ! [A: nat,B: nat,C: nat] :
% 0.33/0.58        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.33/0.58         => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_left_mono
% 0.33/0.58  thf(fact_2695_mult__left__mono,axiom,
% 0.33/0.58      ! [A: int,B: int,C: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.33/0.58         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_left_mono
% 0.33/0.58  thf(fact_2696_mult__nonpos__nonpos,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.33/0.58       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.33/0.58         => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_nonpos_nonpos
% 0.33/0.58  thf(fact_2697_mult__nonpos__nonpos,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.33/0.58       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.33/0.58         => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_nonpos_nonpos
% 0.33/0.58  thf(fact_2698_mult__nonpos__nonpos,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.33/0.58       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.33/0.58         => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_nonpos_nonpos
% 0.33/0.58  thf(fact_2699_mult__left__mono__neg,axiom,
% 0.33/0.58      ! [B: real,A: real,C: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ B @ A )
% 0.33/0.58       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.33/0.58         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_left_mono_neg
% 0.33/0.58  thf(fact_2700_mult__left__mono__neg,axiom,
% 0.33/0.58      ! [B: rat,A: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ B @ A )
% 0.33/0.58       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.33/0.58         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_left_mono_neg
% 0.33/0.58  thf(fact_2701_mult__left__mono__neg,axiom,
% 0.33/0.58      ! [B: int,A: int,C: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ B @ A )
% 0.33/0.58       => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.33/0.58         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_left_mono_neg
% 0.33/0.58  thf(fact_2702_split__mult__pos__le,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.58            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.33/0.58          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.33/0.58            & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.33/0.58       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % split_mult_pos_le
% 0.33/0.58  thf(fact_2703_split__mult__pos__le,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.58            & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.33/0.58          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.33/0.58            & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 0.33/0.58       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % split_mult_pos_le
% 0.33/0.58  thf(fact_2704_split__mult__pos__le,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.58            & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 0.33/0.58          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.33/0.58            & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 0.33/0.58       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % split_mult_pos_le
% 0.33/0.58  thf(fact_2705_zero__le__square,axiom,
% 0.33/0.58      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_le_square
% 0.33/0.58  thf(fact_2706_zero__le__square,axiom,
% 0.33/0.58      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_le_square
% 0.33/0.58  thf(fact_2707_zero__le__square,axiom,
% 0.33/0.58      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_le_square
% 0.33/0.58  thf(fact_2708_mult__mono_H,axiom,
% 0.33/0.58      ! [A: real,B: real,C: real,D: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_real @ C @ D )
% 0.33/0.58         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.58           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.33/0.58             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_mono'
% 0.33/0.58  thf(fact_2709_mult__mono_H,axiom,
% 0.33/0.58      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_rat @ C @ D )
% 0.33/0.58         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.58           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.33/0.58             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_mono'
% 0.33/0.58  thf(fact_2710_mult__mono_H,axiom,
% 0.33/0.58      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.33/0.58        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_nat @ C @ D )
% 0.33/0.58         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.33/0.58           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.33/0.58             => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_mono'
% 0.33/0.58  thf(fact_2711_mult__mono_H,axiom,
% 0.33/0.58      ! [A: int,B: int,C: int,D: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_int @ C @ D )
% 0.33/0.58         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.58           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.33/0.58             => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_mono'
% 0.33/0.58  thf(fact_2712_mult__mono,axiom,
% 0.33/0.58      ! [A: real,B: real,C: real,D: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_real @ C @ D )
% 0.33/0.58         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.33/0.58           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.33/0.58             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_mono
% 0.33/0.58  thf(fact_2713_mult__mono,axiom,
% 0.33/0.58      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_rat @ C @ D )
% 0.33/0.58         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.33/0.58           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.33/0.58             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_mono
% 0.33/0.58  thf(fact_2714_mult__mono,axiom,
% 0.33/0.58      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.33/0.58        ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_nat @ C @ D )
% 0.33/0.58         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.33/0.58           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.33/0.58             => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_mono
% 0.33/0.58  thf(fact_2715_mult__mono,axiom,
% 0.33/0.58      ! [A: int,B: int,C: int,D: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ A @ B )
% 0.33/0.58       => ( ( ord_less_eq_int @ C @ D )
% 0.33/0.58         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.33/0.58           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.33/0.58             => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_mono
% 0.33/0.58  thf(fact_2716_mult__neg__neg,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.58       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.33/0.58         => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_neg_neg
% 0.33/0.58  thf(fact_2717_mult__neg__neg,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.58       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.33/0.58         => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_neg_neg
% 0.33/0.58  thf(fact_2718_mult__neg__neg,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( ord_less_int @ A @ zero_zero_int )
% 0.33/0.58       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.33/0.58         => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_neg_neg
% 0.33/0.58  thf(fact_2719_not__square__less__zero,axiom,
% 0.33/0.58      ! [A: real] :
% 0.33/0.58        ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% 0.33/0.58  
% 0.33/0.58  % not_square_less_zero
% 0.33/0.58  thf(fact_2720_not__square__less__zero,axiom,
% 0.33/0.58      ! [A: rat] :
% 0.33/0.58        ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% 0.33/0.58  
% 0.33/0.58  % not_square_less_zero
% 0.33/0.58  thf(fact_2721_not__square__less__zero,axiom,
% 0.33/0.58      ! [A: int] :
% 0.33/0.58        ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% 0.33/0.58  
% 0.33/0.58  % not_square_less_zero
% 0.33/0.58  thf(fact_2722_mult__less__0__iff,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.58            & ( ord_less_real @ B @ zero_zero_real ) )
% 0.33/0.58          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.58            & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_0_iff
% 0.33/0.58  thf(fact_2723_mult__less__0__iff,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.58            & ( ord_less_rat @ B @ zero_zero_rat ) )
% 0.33/0.58          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.58            & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_0_iff
% 0.33/0.58  thf(fact_2724_mult__less__0__iff,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 0.33/0.58        = ( ( ( ord_less_int @ zero_zero_int @ A )
% 0.33/0.58            & ( ord_less_int @ B @ zero_zero_int ) )
% 0.33/0.58          | ( ( ord_less_int @ A @ zero_zero_int )
% 0.33/0.58            & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_0_iff
% 0.33/0.58  thf(fact_2725_mult__neg__pos,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.58       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.58         => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_neg_pos
% 0.33/0.58  thf(fact_2726_mult__neg__pos,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.58       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.33/0.58         => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_neg_pos
% 0.33/0.58  thf(fact_2727_mult__neg__pos,axiom,
% 0.33/0.58      ! [A: nat,B: nat] :
% 0.33/0.58        ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.33/0.58       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.33/0.58         => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_neg_pos
% 0.33/0.58  thf(fact_2728_mult__neg__pos,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( ord_less_int @ A @ zero_zero_int )
% 0.33/0.58       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.58         => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_neg_pos
% 0.33/0.58  thf(fact_2729_mult__pos__neg,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.58       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.33/0.58         => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_pos_neg
% 0.33/0.58  thf(fact_2730_mult__pos__neg,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.58       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.33/0.58         => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_pos_neg
% 0.33/0.58  thf(fact_2731_mult__pos__neg,axiom,
% 0.33/0.58      ! [A: nat,B: nat] :
% 0.33/0.58        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.33/0.58       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.33/0.58         => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_pos_neg
% 0.33/0.58  thf(fact_2732_mult__pos__neg,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( ord_less_int @ zero_zero_int @ A )
% 0.33/0.58       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.33/0.58         => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_pos_neg
% 0.33/0.58  thf(fact_2733_mult__pos__pos,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.58       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.58         => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_pos_pos
% 0.33/0.58  thf(fact_2734_mult__pos__pos,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.58       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.33/0.58         => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_pos_pos
% 0.33/0.58  thf(fact_2735_mult__pos__pos,axiom,
% 0.33/0.58      ! [A: nat,B: nat] :
% 0.33/0.58        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.33/0.58       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.33/0.58         => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_pos_pos
% 0.33/0.58  thf(fact_2736_mult__pos__pos,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( ord_less_int @ zero_zero_int @ A )
% 0.33/0.58       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.58         => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_pos_pos
% 0.33/0.58  thf(fact_2737_mult__pos__neg2,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.58       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.33/0.58         => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_pos_neg2
% 0.33/0.58  thf(fact_2738_mult__pos__neg2,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.58       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.33/0.58         => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_pos_neg2
% 0.33/0.58  thf(fact_2739_mult__pos__neg2,axiom,
% 0.33/0.58      ! [A: nat,B: nat] :
% 0.33/0.58        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.33/0.58       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.33/0.58         => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_pos_neg2
% 0.33/0.58  thf(fact_2740_mult__pos__neg2,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( ord_less_int @ zero_zero_int @ A )
% 0.33/0.58       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.33/0.58         => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_pos_neg2
% 0.33/0.58  thf(fact_2741_zero__less__mult__iff,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.58            & ( ord_less_real @ zero_zero_real @ B ) )
% 0.33/0.58          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.58            & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_mult_iff
% 0.33/0.58  thf(fact_2742_zero__less__mult__iff,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.58            & ( ord_less_rat @ zero_zero_rat @ B ) )
% 0.33/0.58          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.58            & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_mult_iff
% 0.33/0.58  thf(fact_2743_zero__less__mult__iff,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.33/0.58        = ( ( ( ord_less_int @ zero_zero_int @ A )
% 0.33/0.58            & ( ord_less_int @ zero_zero_int @ B ) )
% 0.33/0.58          | ( ( ord_less_int @ A @ zero_zero_int )
% 0.33/0.58            & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_mult_iff
% 0.33/0.58  thf(fact_2744_zero__less__mult__pos,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.33/0.58       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.58         => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_mult_pos
% 0.33/0.58  thf(fact_2745_zero__less__mult__pos,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.33/0.58       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.58         => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_mult_pos
% 0.33/0.58  thf(fact_2746_zero__less__mult__pos,axiom,
% 0.33/0.58      ! [A: nat,B: nat] :
% 0.33/0.58        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
% 0.33/0.58       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.33/0.58         => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_mult_pos
% 0.33/0.58  thf(fact_2747_zero__less__mult__pos,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.33/0.58       => ( ( ord_less_int @ zero_zero_int @ A )
% 0.33/0.58         => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_mult_pos
% 0.33/0.58  thf(fact_2748_zero__less__mult__pos2,axiom,
% 0.33/0.58      ! [B: real,A: real] :
% 0.33/0.58        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
% 0.33/0.58       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.58         => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_mult_pos2
% 0.33/0.58  thf(fact_2749_zero__less__mult__pos2,axiom,
% 0.33/0.58      ! [B: rat,A: rat] :
% 0.33/0.58        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
% 0.33/0.58       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.58         => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_mult_pos2
% 0.33/0.58  thf(fact_2750_zero__less__mult__pos2,axiom,
% 0.33/0.58      ! [B: nat,A: nat] :
% 0.33/0.58        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
% 0.33/0.58       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.33/0.58         => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_mult_pos2
% 0.33/0.58  thf(fact_2751_zero__less__mult__pos2,axiom,
% 0.33/0.58      ! [B: int,A: int] :
% 0.33/0.58        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
% 0.33/0.58       => ( ( ord_less_int @ zero_zero_int @ A )
% 0.33/0.58         => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_mult_pos2
% 0.33/0.58  thf(fact_2752_mult__less__cancel__left__neg,axiom,
% 0.33/0.58      ! [C: real,A: real,B: real] :
% 0.33/0.58        ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58       => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.33/0.58          = ( ord_less_real @ B @ A ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_left_neg
% 0.33/0.58  thf(fact_2753_mult__less__cancel__left__neg,axiom,
% 0.33/0.58      ! [C: rat,A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58       => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.33/0.58          = ( ord_less_rat @ B @ A ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_left_neg
% 0.33/0.58  thf(fact_2754_mult__less__cancel__left__neg,axiom,
% 0.33/0.58      ! [C: int,A: int,B: int] :
% 0.33/0.58        ( ( ord_less_int @ C @ zero_zero_int )
% 0.33/0.58       => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.33/0.58          = ( ord_less_int @ B @ A ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_left_neg
% 0.33/0.58  thf(fact_2755_mult__less__cancel__left__pos,axiom,
% 0.33/0.58      ! [C: real,A: real,B: real] :
% 0.33/0.58        ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58       => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.33/0.58          = ( ord_less_real @ A @ B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_left_pos
% 0.33/0.58  thf(fact_2756_mult__less__cancel__left__pos,axiom,
% 0.33/0.58      ! [C: rat,A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58       => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.33/0.58          = ( ord_less_rat @ A @ B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_left_pos
% 0.33/0.58  thf(fact_2757_mult__less__cancel__left__pos,axiom,
% 0.33/0.58      ! [C: int,A: int,B: int] :
% 0.33/0.58        ( ( ord_less_int @ zero_zero_int @ C )
% 0.33/0.58       => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.33/0.58          = ( ord_less_int @ A @ B ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_left_pos
% 0.33/0.58  thf(fact_2758_mult__strict__left__mono__neg,axiom,
% 0.33/0.58      ! [B: real,A: real,C: real] :
% 0.33/0.58        ( ( ord_less_real @ B @ A )
% 0.33/0.58       => ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_strict_left_mono_neg
% 0.33/0.58  thf(fact_2759_mult__strict__left__mono__neg,axiom,
% 0.33/0.58      ! [B: rat,A: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_rat @ B @ A )
% 0.33/0.58       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_strict_left_mono_neg
% 0.33/0.58  thf(fact_2760_mult__strict__left__mono__neg,axiom,
% 0.33/0.58      ! [B: int,A: int,C: int] :
% 0.33/0.58        ( ( ord_less_int @ B @ A )
% 0.33/0.58       => ( ( ord_less_int @ C @ zero_zero_int )
% 0.33/0.58         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_strict_left_mono_neg
% 0.33/0.58  thf(fact_2761_mult__strict__left__mono,axiom,
% 0.33/0.58      ! [A: real,B: real,C: real] :
% 0.33/0.58        ( ( ord_less_real @ A @ B )
% 0.33/0.58       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_strict_left_mono
% 0.33/0.58  thf(fact_2762_mult__strict__left__mono,axiom,
% 0.33/0.58      ! [A: rat,B: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_rat @ A @ B )
% 0.33/0.58       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_strict_left_mono
% 0.33/0.58  thf(fact_2763_mult__strict__left__mono,axiom,
% 0.33/0.58      ! [A: nat,B: nat,C: nat] :
% 0.33/0.58        ( ( ord_less_nat @ A @ B )
% 0.33/0.58       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.33/0.58         => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_strict_left_mono
% 0.33/0.58  thf(fact_2764_mult__strict__left__mono,axiom,
% 0.33/0.58      ! [A: int,B: int,C: int] :
% 0.33/0.58        ( ( ord_less_int @ A @ B )
% 0.33/0.58       => ( ( ord_less_int @ zero_zero_int @ C )
% 0.33/0.58         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_strict_left_mono
% 0.33/0.58  thf(fact_2765_mult__less__cancel__left__disj,axiom,
% 0.33/0.58      ! [C: real,A: real,B: real] :
% 0.33/0.58        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58            & ( ord_less_real @ A @ B ) )
% 0.33/0.58          | ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58            & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_left_disj
% 0.33/0.58  thf(fact_2766_mult__less__cancel__left__disj,axiom,
% 0.33/0.58      ! [C: rat,A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58            & ( ord_less_rat @ A @ B ) )
% 0.33/0.58          | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58            & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_left_disj
% 0.33/0.58  thf(fact_2767_mult__less__cancel__left__disj,axiom,
% 0.33/0.58      ! [C: int,A: int,B: int] :
% 0.33/0.58        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.33/0.58        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.33/0.58            & ( ord_less_int @ A @ B ) )
% 0.33/0.58          | ( ( ord_less_int @ C @ zero_zero_int )
% 0.33/0.58            & ( ord_less_int @ B @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_left_disj
% 0.33/0.58  thf(fact_2768_mult__strict__right__mono__neg,axiom,
% 0.33/0.58      ! [B: real,A: real,C: real] :
% 0.33/0.58        ( ( ord_less_real @ B @ A )
% 0.33/0.58       => ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_strict_right_mono_neg
% 0.33/0.58  thf(fact_2769_mult__strict__right__mono__neg,axiom,
% 0.33/0.58      ! [B: rat,A: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_rat @ B @ A )
% 0.33/0.58       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_strict_right_mono_neg
% 0.33/0.58  thf(fact_2770_mult__strict__right__mono__neg,axiom,
% 0.33/0.58      ! [B: int,A: int,C: int] :
% 0.33/0.58        ( ( ord_less_int @ B @ A )
% 0.33/0.58       => ( ( ord_less_int @ C @ zero_zero_int )
% 0.33/0.58         => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_strict_right_mono_neg
% 0.33/0.58  thf(fact_2771_mult__strict__right__mono,axiom,
% 0.33/0.58      ! [A: real,B: real,C: real] :
% 0.33/0.58        ( ( ord_less_real @ A @ B )
% 0.33/0.58       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_strict_right_mono
% 0.33/0.58  thf(fact_2772_mult__strict__right__mono,axiom,
% 0.33/0.58      ! [A: rat,B: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_rat @ A @ B )
% 0.33/0.58       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_strict_right_mono
% 0.33/0.58  thf(fact_2773_mult__strict__right__mono,axiom,
% 0.33/0.58      ! [A: nat,B: nat,C: nat] :
% 0.33/0.58        ( ( ord_less_nat @ A @ B )
% 0.33/0.58       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.33/0.58         => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_strict_right_mono
% 0.33/0.58  thf(fact_2774_mult__strict__right__mono,axiom,
% 0.33/0.58      ! [A: int,B: int,C: int] :
% 0.33/0.58        ( ( ord_less_int @ A @ B )
% 0.33/0.58       => ( ( ord_less_int @ zero_zero_int @ C )
% 0.33/0.58         => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_strict_right_mono
% 0.33/0.58  thf(fact_2775_mult__less__cancel__right__disj,axiom,
% 0.33/0.58      ! [A: real,C: real,B: real] :
% 0.33/0.58        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58            & ( ord_less_real @ A @ B ) )
% 0.33/0.58          | ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58            & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_right_disj
% 0.33/0.58  thf(fact_2776_mult__less__cancel__right__disj,axiom,
% 0.33/0.58      ! [A: rat,C: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58            & ( ord_less_rat @ A @ B ) )
% 0.33/0.58          | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58            & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_right_disj
% 0.33/0.58  thf(fact_2777_mult__less__cancel__right__disj,axiom,
% 0.33/0.58      ! [A: int,C: int,B: int] :
% 0.33/0.58        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.33/0.58            & ( ord_less_int @ A @ B ) )
% 0.33/0.58          | ( ( ord_less_int @ C @ zero_zero_int )
% 0.33/0.58            & ( ord_less_int @ B @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_cancel_right_disj
% 0.33/0.58  thf(fact_2778_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.33/0.58      ! [A: real,B: real,C: real] :
% 0.33/0.58        ( ( ord_less_real @ A @ B )
% 0.33/0.58       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.33/0.58  thf(fact_2779_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.33/0.58      ! [A: rat,B: rat,C: rat] :
% 0.33/0.58        ( ( ord_less_rat @ A @ B )
% 0.33/0.58       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.33/0.58  thf(fact_2780_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.33/0.58      ! [A: nat,B: nat,C: nat] :
% 0.33/0.58        ( ( ord_less_nat @ A @ B )
% 0.33/0.58       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.33/0.58         => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.33/0.58  thf(fact_2781_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.33/0.58      ! [A: int,B: int,C: int] :
% 0.33/0.58        ( ( ord_less_int @ A @ B )
% 0.33/0.58       => ( ( ord_less_int @ zero_zero_int @ C )
% 0.33/0.58         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.33/0.58  thf(fact_2782_add__scale__eq__noteq,axiom,
% 0.33/0.58      ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
% 0.33/0.58        ( ( R2 != zero_zero_complex )
% 0.33/0.58       => ( ( ( A = B )
% 0.33/0.58            & ( C != D ) )
% 0.33/0.58         => ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C ) )
% 0.33/0.58           != ( plus_plus_complex @ B @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % add_scale_eq_noteq
% 0.33/0.58  thf(fact_2783_add__scale__eq__noteq,axiom,
% 0.33/0.58      ! [R2: real,A: real,B: real,C: real,D: real] :
% 0.33/0.58        ( ( R2 != zero_zero_real )
% 0.33/0.58       => ( ( ( A = B )
% 0.33/0.58            & ( C != D ) )
% 0.33/0.58         => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
% 0.33/0.58           != ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % add_scale_eq_noteq
% 0.33/0.58  thf(fact_2784_add__scale__eq__noteq,axiom,
% 0.33/0.58      ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
% 0.33/0.58        ( ( R2 != zero_zero_rat )
% 0.33/0.58       => ( ( ( A = B )
% 0.33/0.58            & ( C != D ) )
% 0.33/0.58         => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
% 0.33/0.58           != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % add_scale_eq_noteq
% 0.33/0.58  thf(fact_2785_add__scale__eq__noteq,axiom,
% 0.33/0.58      ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
% 0.33/0.58        ( ( R2 != zero_zero_nat )
% 0.33/0.58       => ( ( ( A = B )
% 0.33/0.58            & ( C != D ) )
% 0.33/0.58         => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
% 0.33/0.58           != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % add_scale_eq_noteq
% 0.33/0.58  thf(fact_2786_add__scale__eq__noteq,axiom,
% 0.33/0.58      ! [R2: int,A: int,B: int,C: int,D: int] :
% 0.33/0.58        ( ( R2 != zero_zero_int )
% 0.33/0.58       => ( ( ( A = B )
% 0.33/0.58            & ( C != D ) )
% 0.33/0.58         => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
% 0.33/0.58           != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % add_scale_eq_noteq
% 0.33/0.58  thf(fact_2787_nat__mult__le__cancel1,axiom,
% 0.33/0.58      ! [K: nat,M: nat,N: nat] :
% 0.33/0.58        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.58       => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.33/0.58          = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % nat_mult_le_cancel1
% 0.33/0.58  thf(fact_2788_Suc__mult__less__cancel1,axiom,
% 0.33/0.58      ! [K: nat,M: nat,N: nat] :
% 0.33/0.58        ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.33/0.58        = ( ord_less_nat @ M @ N ) ) ).
% 0.33/0.58  
% 0.33/0.58  % Suc_mult_less_cancel1
% 0.33/0.58  thf(fact_2789_not__one__le__zero,axiom,
% 0.33/0.58      ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 0.33/0.58  
% 0.33/0.58  % not_one_le_zero
% 0.33/0.58  thf(fact_2790_not__one__le__zero,axiom,
% 0.33/0.58      ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.33/0.58  
% 0.33/0.58  % not_one_le_zero
% 0.33/0.58  thf(fact_2791_not__one__le__zero,axiom,
% 0.33/0.58      ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.33/0.58  
% 0.33/0.58  % not_one_le_zero
% 0.33/0.58  thf(fact_2792_not__one__le__zero,axiom,
% 0.33/0.58      ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 0.33/0.58  
% 0.33/0.58  % not_one_le_zero
% 0.33/0.58  thf(fact_2793_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.33/0.58      ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.33/0.58  
% 0.33/0.58  % linordered_nonzero_semiring_class.zero_le_one
% 0.33/0.58  thf(fact_2794_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.33/0.58      ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.33/0.58  
% 0.33/0.58  % linordered_nonzero_semiring_class.zero_le_one
% 0.33/0.58  thf(fact_2795_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.33/0.58      ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.33/0.58  
% 0.33/0.58  % linordered_nonzero_semiring_class.zero_le_one
% 0.33/0.58  thf(fact_2796_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.33/0.58      ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.33/0.58  
% 0.33/0.58  % linordered_nonzero_semiring_class.zero_le_one
% 0.33/0.58  thf(fact_2797_zero__less__one__class_Ozero__le__one,axiom,
% 0.33/0.58      ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_one_class.zero_le_one
% 0.33/0.58  thf(fact_2798_zero__less__one__class_Ozero__le__one,axiom,
% 0.33/0.58      ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_one_class.zero_le_one
% 0.33/0.58  thf(fact_2799_zero__less__one__class_Ozero__le__one,axiom,
% 0.33/0.58      ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_one_class.zero_le_one
% 0.33/0.58  thf(fact_2800_zero__less__one__class_Ozero__le__one,axiom,
% 0.33/0.58      ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_one_class.zero_le_one
% 0.33/0.58  thf(fact_2801_less__numeral__extra_I1_J,axiom,
% 0.33/0.58      ord_less_real @ zero_zero_real @ one_one_real ).
% 0.33/0.58  
% 0.33/0.58  % less_numeral_extra(1)
% 0.33/0.58  thf(fact_2802_less__numeral__extra_I1_J,axiom,
% 0.33/0.58      ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.33/0.58  
% 0.33/0.58  % less_numeral_extra(1)
% 0.33/0.58  thf(fact_2803_less__numeral__extra_I1_J,axiom,
% 0.33/0.58      ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.33/0.58  
% 0.33/0.58  % less_numeral_extra(1)
% 0.33/0.58  thf(fact_2804_less__numeral__extra_I1_J,axiom,
% 0.33/0.58      ord_less_int @ zero_zero_int @ one_one_int ).
% 0.33/0.58  
% 0.33/0.58  % less_numeral_extra(1)
% 0.33/0.58  thf(fact_2805_zero__less__one,axiom,
% 0.33/0.58      ord_less_real @ zero_zero_real @ one_one_real ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_one
% 0.33/0.58  thf(fact_2806_zero__less__one,axiom,
% 0.33/0.58      ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_one
% 0.33/0.58  thf(fact_2807_zero__less__one,axiom,
% 0.33/0.58      ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_one
% 0.33/0.58  thf(fact_2808_zero__less__one,axiom,
% 0.33/0.58      ord_less_int @ zero_zero_int @ one_one_int ).
% 0.33/0.58  
% 0.33/0.58  % zero_less_one
% 0.33/0.58  thf(fact_2809_not__one__less__zero,axiom,
% 0.33/0.58      ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 0.33/0.58  
% 0.33/0.58  % not_one_less_zero
% 0.33/0.58  thf(fact_2810_not__one__less__zero,axiom,
% 0.33/0.58      ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.33/0.58  
% 0.33/0.58  % not_one_less_zero
% 0.33/0.58  thf(fact_2811_not__one__less__zero,axiom,
% 0.33/0.58      ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.33/0.58  
% 0.33/0.58  % not_one_less_zero
% 0.33/0.58  thf(fact_2812_not__one__less__zero,axiom,
% 0.33/0.58      ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 0.33/0.58  
% 0.33/0.58  % not_one_less_zero
% 0.33/0.58  thf(fact_2813_mult__less__mono1,axiom,
% 0.33/0.58      ! [I: nat,J: nat,K: nat] :
% 0.33/0.58        ( ( ord_less_nat @ I @ J )
% 0.33/0.58       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.58         => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_mono1
% 0.33/0.58  thf(fact_2814_mult__less__mono2,axiom,
% 0.33/0.58      ! [I: nat,J: nat,K: nat] :
% 0.33/0.58        ( ( ord_less_nat @ I @ J )
% 0.33/0.58       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.58         => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_less_mono2
% 0.33/0.58  thf(fact_2815_Suc__mult__le__cancel1,axiom,
% 0.33/0.58      ! [K: nat,M: nat,N: nat] :
% 0.33/0.58        ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.33/0.58        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.33/0.58  
% 0.33/0.58  % Suc_mult_le_cancel1
% 0.33/0.58  thf(fact_2816_one__le__numeral,axiom,
% 0.33/0.58      ! [N: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% 0.33/0.58  
% 0.33/0.58  % one_le_numeral
% 0.33/0.58  thf(fact_2817_one__le__numeral,axiom,
% 0.33/0.58      ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% 0.33/0.58  
% 0.33/0.58  % one_le_numeral
% 0.33/0.58  thf(fact_2818_one__le__numeral,axiom,
% 0.33/0.58      ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.33/0.58  
% 0.33/0.58  % one_le_numeral
% 0.33/0.58  thf(fact_2819_one__le__numeral,axiom,
% 0.33/0.58      ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.33/0.58  
% 0.33/0.58  % one_le_numeral
% 0.33/0.58  thf(fact_2820_one__le__numeral,axiom,
% 0.33/0.58      ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% 0.33/0.58  
% 0.33/0.58  % one_le_numeral
% 0.33/0.58  thf(fact_2821_not__numeral__less__one,axiom,
% 0.33/0.58      ! [N: num] :
% 0.33/0.58        ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat ) ).
% 0.33/0.58  
% 0.33/0.58  % not_numeral_less_one
% 0.33/0.58  thf(fact_2822_not__numeral__less__one,axiom,
% 0.33/0.58      ! [N: num] :
% 0.33/0.58        ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% 0.33/0.58  
% 0.33/0.58  % not_numeral_less_one
% 0.33/0.58  thf(fact_2823_not__numeral__less__one,axiom,
% 0.33/0.58      ! [N: num] :
% 0.33/0.58        ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% 0.33/0.58  
% 0.33/0.58  % not_numeral_less_one
% 0.33/0.58  thf(fact_2824_not__numeral__less__one,axiom,
% 0.33/0.58      ! [N: num] :
% 0.33/0.58        ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% 0.33/0.58  
% 0.33/0.58  % not_numeral_less_one
% 0.33/0.58  thf(fact_2825_not__numeral__less__one,axiom,
% 0.33/0.58      ! [N: num] :
% 0.33/0.58        ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% 0.33/0.58  
% 0.33/0.58  % not_numeral_less_one
% 0.33/0.58  thf(fact_2826_mult__Suc,axiom,
% 0.33/0.58      ! [M: nat,N: nat] :
% 0.33/0.58        ( ( times_times_nat @ ( suc @ M ) @ N )
% 0.33/0.58        = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_Suc
% 0.33/0.58  thf(fact_2827_add__mono1,axiom,
% 0.33/0.58      ! [A: real,B: real] :
% 0.33/0.58        ( ( ord_less_real @ A @ B )
% 0.33/0.58       => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % add_mono1
% 0.33/0.58  thf(fact_2828_add__mono1,axiom,
% 0.33/0.58      ! [A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_rat @ A @ B )
% 0.33/0.58       => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % add_mono1
% 0.33/0.58  thf(fact_2829_add__mono1,axiom,
% 0.33/0.58      ! [A: nat,B: nat] :
% 0.33/0.58        ( ( ord_less_nat @ A @ B )
% 0.33/0.58       => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % add_mono1
% 0.33/0.58  thf(fact_2830_add__mono1,axiom,
% 0.33/0.58      ! [A: int,B: int] :
% 0.33/0.58        ( ( ord_less_int @ A @ B )
% 0.33/0.58       => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % add_mono1
% 0.33/0.58  thf(fact_2831_less__add__one,axiom,
% 0.33/0.58      ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 0.33/0.58  
% 0.33/0.58  % less_add_one
% 0.33/0.58  thf(fact_2832_less__add__one,axiom,
% 0.33/0.58      ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% 0.33/0.58  
% 0.33/0.58  % less_add_one
% 0.33/0.58  thf(fact_2833_less__add__one,axiom,
% 0.33/0.58      ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 0.33/0.58  
% 0.33/0.58  % less_add_one
% 0.33/0.58  thf(fact_2834_less__add__one,axiom,
% 0.33/0.58      ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% 0.33/0.58  
% 0.33/0.58  % less_add_one
% 0.33/0.58  thf(fact_2835_one__le__power,axiom,
% 0.33/0.58      ! [A: real,N: nat] :
% 0.33/0.58        ( ( ord_less_eq_real @ one_one_real @ A )
% 0.33/0.58       => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % one_le_power
% 0.33/0.58  thf(fact_2836_one__le__power,axiom,
% 0.33/0.58      ! [A: rat,N: nat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.33/0.58       => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % one_le_power
% 0.33/0.58  thf(fact_2837_one__le__power,axiom,
% 0.33/0.58      ! [A: nat,N: nat] :
% 0.33/0.58        ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.33/0.58       => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % one_le_power
% 0.33/0.58  thf(fact_2838_one__le__power,axiom,
% 0.33/0.58      ! [A: int,N: nat] :
% 0.33/0.58        ( ( ord_less_eq_int @ one_one_int @ A )
% 0.33/0.58       => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % one_le_power
% 0.33/0.58  thf(fact_2839_power__0,axiom,
% 0.33/0.58      ! [A: rat] :
% 0.33/0.58        ( ( power_power_rat @ A @ zero_zero_nat )
% 0.33/0.58        = one_one_rat ) ).
% 0.33/0.58  
% 0.33/0.58  % power_0
% 0.33/0.58  thf(fact_2840_power__0,axiom,
% 0.33/0.58      ! [A: nat] :
% 0.33/0.58        ( ( power_power_nat @ A @ zero_zero_nat )
% 0.33/0.58        = one_one_nat ) ).
% 0.33/0.58  
% 0.33/0.58  % power_0
% 0.33/0.58  thf(fact_2841_power__0,axiom,
% 0.33/0.58      ! [A: real] :
% 0.33/0.58        ( ( power_power_real @ A @ zero_zero_nat )
% 0.33/0.58        = one_one_real ) ).
% 0.33/0.58  
% 0.33/0.58  % power_0
% 0.33/0.58  thf(fact_2842_power__0,axiom,
% 0.33/0.58      ! [A: int] :
% 0.33/0.58        ( ( power_power_int @ A @ zero_zero_nat )
% 0.33/0.58        = one_one_int ) ).
% 0.33/0.58  
% 0.33/0.58  % power_0
% 0.33/0.58  thf(fact_2843_power__0,axiom,
% 0.33/0.58      ! [A: complex] :
% 0.33/0.58        ( ( power_power_complex @ A @ zero_zero_nat )
% 0.33/0.58        = one_one_complex ) ).
% 0.33/0.58  
% 0.33/0.58  % power_0
% 0.33/0.58  thf(fact_2844_One__nat__def,axiom,
% 0.33/0.58      ( one_one_nat
% 0.33/0.58      = ( suc @ zero_zero_nat ) ) ).
% 0.33/0.58  
% 0.33/0.58  % One_nat_def
% 0.33/0.58  thf(fact_2845_Suc__eq__plus1__left,axiom,
% 0.33/0.58      ( suc
% 0.33/0.58      = ( plus_plus_nat @ one_one_nat ) ) ).
% 0.33/0.58  
% 0.33/0.58  % Suc_eq_plus1_left
% 0.33/0.58  thf(fact_2846_plus__1__eq__Suc,axiom,
% 0.33/0.58      ( ( plus_plus_nat @ one_one_nat )
% 0.33/0.58      = suc ) ).
% 0.33/0.58  
% 0.33/0.58  % plus_1_eq_Suc
% 0.33/0.58  thf(fact_2847_Suc__eq__plus1,axiom,
% 0.33/0.58      ( suc
% 0.33/0.58      = ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % Suc_eq_plus1
% 0.33/0.58  thf(fact_2848_frac__le,axiom,
% 0.33/0.58      ! [Y: real,X: real,W2: real,Z2: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.58       => ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.58         => ( ( ord_less_real @ zero_zero_real @ W2 )
% 0.33/0.58           => ( ( ord_less_eq_real @ W2 @ Z2 )
% 0.33/0.58             => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % frac_le
% 0.33/0.58  thf(fact_2849_frac__le,axiom,
% 0.33/0.58      ! [Y: rat,X: rat,W2: rat,Z2: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.33/0.58       => ( ( ord_less_eq_rat @ X @ Y )
% 0.33/0.58         => ( ( ord_less_rat @ zero_zero_rat @ W2 )
% 0.33/0.58           => ( ( ord_less_eq_rat @ W2 @ Z2 )
% 0.33/0.58             => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % frac_le
% 0.33/0.58  thf(fact_2850_frac__less,axiom,
% 0.33/0.58      ! [X: real,Y: real,W2: real,Z2: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.58       => ( ( ord_less_real @ X @ Y )
% 0.33/0.58         => ( ( ord_less_real @ zero_zero_real @ W2 )
% 0.33/0.58           => ( ( ord_less_eq_real @ W2 @ Z2 )
% 0.33/0.58             => ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % frac_less
% 0.33/0.58  thf(fact_2851_frac__less,axiom,
% 0.33/0.58      ! [X: rat,Y: rat,W2: rat,Z2: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.33/0.58       => ( ( ord_less_rat @ X @ Y )
% 0.33/0.58         => ( ( ord_less_rat @ zero_zero_rat @ W2 )
% 0.33/0.58           => ( ( ord_less_eq_rat @ W2 @ Z2 )
% 0.33/0.58             => ( ord_less_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % frac_less
% 0.33/0.58  thf(fact_2852_frac__less2,axiom,
% 0.33/0.58      ! [X: real,Y: real,W2: real,Z2: real] :
% 0.33/0.58        ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.58       => ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.58         => ( ( ord_less_real @ zero_zero_real @ W2 )
% 0.33/0.58           => ( ( ord_less_real @ W2 @ Z2 )
% 0.33/0.58             => ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % frac_less2
% 0.33/0.58  thf(fact_2853_frac__less2,axiom,
% 0.33/0.58      ! [X: rat,Y: rat,W2: rat,Z2: rat] :
% 0.33/0.58        ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.33/0.58       => ( ( ord_less_eq_rat @ X @ Y )
% 0.33/0.58         => ( ( ord_less_rat @ zero_zero_rat @ W2 )
% 0.33/0.58           => ( ( ord_less_rat @ W2 @ Z2 )
% 0.33/0.58             => ( ord_less_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % frac_less2
% 0.33/0.58  thf(fact_2854_divide__le__cancel,axiom,
% 0.33/0.58      ! [A: real,C: real,B: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ord_less_eq_real @ A @ B ) )
% 0.33/0.58          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58           => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_le_cancel
% 0.33/0.58  thf(fact_2855_divide__le__cancel,axiom,
% 0.33/0.58      ! [A: rat,C: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ord_less_eq_rat @ A @ B ) )
% 0.33/0.58          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58           => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_le_cancel
% 0.33/0.58  thf(fact_2856_divide__nonneg__neg,axiom,
% 0.33/0.58      ! [X: real,Y: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.58       => ( ( ord_less_real @ Y @ zero_zero_real )
% 0.33/0.58         => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_nonneg_neg
% 0.33/0.58  thf(fact_2857_divide__nonneg__neg,axiom,
% 0.33/0.58      ! [X: rat,Y: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.33/0.58       => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 0.33/0.58         => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_nonneg_neg
% 0.33/0.58  thf(fact_2858_divide__nonneg__pos,axiom,
% 0.33/0.58      ! [X: real,Y: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.58       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.58         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_nonneg_pos
% 0.33/0.58  thf(fact_2859_divide__nonneg__pos,axiom,
% 0.33/0.58      ! [X: rat,Y: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.33/0.58       => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.33/0.58         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_nonneg_pos
% 0.33/0.58  thf(fact_2860_divide__nonpos__neg,axiom,
% 0.33/0.58      ! [X: real,Y: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.33/0.58       => ( ( ord_less_real @ Y @ zero_zero_real )
% 0.33/0.58         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_nonpos_neg
% 0.33/0.58  thf(fact_2861_divide__nonpos__neg,axiom,
% 0.33/0.58      ! [X: rat,Y: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.33/0.58       => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 0.33/0.58         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_nonpos_neg
% 0.33/0.58  thf(fact_2862_divide__nonpos__pos,axiom,
% 0.33/0.58      ! [X: real,Y: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.33/0.58       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.58         => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_nonpos_pos
% 0.33/0.58  thf(fact_2863_divide__nonpos__pos,axiom,
% 0.33/0.58      ! [X: rat,Y: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.33/0.58       => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.33/0.58         => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % divide_nonpos_pos
% 0.33/0.58  thf(fact_2864_mult__le__cancel__left,axiom,
% 0.33/0.58      ! [C: real,A: real,B: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ord_less_eq_real @ A @ B ) )
% 0.33/0.58          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58           => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_cancel_left
% 0.33/0.58  thf(fact_2865_mult__le__cancel__left,axiom,
% 0.33/0.58      ! [C: rat,A: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ord_less_eq_rat @ A @ B ) )
% 0.33/0.58          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58           => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_cancel_left
% 0.33/0.58  thf(fact_2866_mult__le__cancel__left,axiom,
% 0.33/0.58      ! [C: int,A: int,B: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.33/0.58        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.33/0.58           => ( ord_less_eq_int @ A @ B ) )
% 0.33/0.58          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.33/0.58           => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_cancel_left
% 0.33/0.58  thf(fact_2867_mult__le__cancel__right,axiom,
% 0.33/0.58      ! [A: real,C: real,B: real] :
% 0.33/0.58        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.58           => ( ord_less_eq_real @ A @ B ) )
% 0.33/0.58          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.58           => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_cancel_right
% 0.33/0.58  thf(fact_2868_mult__le__cancel__right,axiom,
% 0.33/0.58      ! [A: rat,C: rat,B: rat] :
% 0.33/0.58        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.58           => ( ord_less_eq_rat @ A @ B ) )
% 0.33/0.58          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.58           => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_cancel_right
% 0.33/0.58  thf(fact_2869_mult__le__cancel__right,axiom,
% 0.33/0.58      ! [A: int,C: int,B: int] :
% 0.33/0.58        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.33/0.58        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.33/0.58           => ( ord_less_eq_int @ A @ B ) )
% 0.33/0.58          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.33/0.58           => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 0.33/0.58  
% 0.33/0.58  % mult_le_cancel_right
% 0.33/0.58  thf(fact_2870_mult__left__less__imp__less,axiom,
% 0.33/0.58      ! [C: real,A: real,B: real] :
% 0.35/0.58        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.35/0.58       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.58         => ( ord_less_real @ A @ B ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_left_less_imp_less
% 0.35/0.58  thf(fact_2871_mult__left__less__imp__less,axiom,
% 0.35/0.58      ! [C: rat,A: rat,B: rat] :
% 0.35/0.58        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.35/0.58       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.58         => ( ord_less_rat @ A @ B ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_left_less_imp_less
% 0.35/0.58  thf(fact_2872_mult__left__less__imp__less,axiom,
% 0.35/0.58      ! [C: nat,A: nat,B: nat] :
% 0.35/0.58        ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.35/0.58       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.35/0.58         => ( ord_less_nat @ A @ B ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_left_less_imp_less
% 0.35/0.58  thf(fact_2873_mult__left__less__imp__less,axiom,
% 0.35/0.58      ! [C: int,A: int,B: int] :
% 0.35/0.58        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.35/0.58       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.58         => ( ord_less_int @ A @ B ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_left_less_imp_less
% 0.35/0.58  thf(fact_2874_mult__strict__mono,axiom,
% 0.35/0.58      ! [A: real,B: real,C: real,D: real] :
% 0.35/0.58        ( ( ord_less_real @ A @ B )
% 0.35/0.58       => ( ( ord_less_real @ C @ D )
% 0.35/0.58         => ( ( ord_less_real @ zero_zero_real @ B )
% 0.35/0.58           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.58             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_strict_mono
% 0.35/0.58  thf(fact_2875_mult__strict__mono,axiom,
% 0.35/0.58      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.35/0.58        ( ( ord_less_rat @ A @ B )
% 0.35/0.58       => ( ( ord_less_rat @ C @ D )
% 0.35/0.58         => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.35/0.58           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.58             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_strict_mono
% 0.35/0.58  thf(fact_2876_mult__strict__mono,axiom,
% 0.35/0.58      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.35/0.58        ( ( ord_less_nat @ A @ B )
% 0.35/0.58       => ( ( ord_less_nat @ C @ D )
% 0.35/0.58         => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.35/0.58           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.35/0.58             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_strict_mono
% 0.35/0.58  thf(fact_2877_mult__strict__mono,axiom,
% 0.35/0.58      ! [A: int,B: int,C: int,D: int] :
% 0.35/0.58        ( ( ord_less_int @ A @ B )
% 0.35/0.58       => ( ( ord_less_int @ C @ D )
% 0.35/0.58         => ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.58           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.58             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_strict_mono
% 0.35/0.58  thf(fact_2878_mult__less__cancel__left,axiom,
% 0.35/0.58      ! [C: real,A: real,B: real] :
% 0.35/0.58        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.35/0.58        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.58           => ( ord_less_real @ A @ B ) )
% 0.35/0.58          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.35/0.58           => ( ord_less_real @ B @ A ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_less_cancel_left
% 0.35/0.58  thf(fact_2879_mult__less__cancel__left,axiom,
% 0.35/0.58      ! [C: rat,A: rat,B: rat] :
% 0.35/0.58        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.35/0.58        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.58           => ( ord_less_rat @ A @ B ) )
% 0.35/0.58          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.35/0.58           => ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_less_cancel_left
% 0.35/0.58  thf(fact_2880_mult__less__cancel__left,axiom,
% 0.35/0.58      ! [C: int,A: int,B: int] :
% 0.35/0.58        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.35/0.58        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.58           => ( ord_less_int @ A @ B ) )
% 0.35/0.58          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.35/0.58           => ( ord_less_int @ B @ A ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_less_cancel_left
% 0.35/0.58  thf(fact_2881_mult__right__less__imp__less,axiom,
% 0.35/0.58      ! [A: real,C: real,B: real] :
% 0.35/0.58        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.35/0.58       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.58         => ( ord_less_real @ A @ B ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_right_less_imp_less
% 0.35/0.58  thf(fact_2882_mult__right__less__imp__less,axiom,
% 0.35/0.58      ! [A: rat,C: rat,B: rat] :
% 0.35/0.58        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.35/0.58       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.58         => ( ord_less_rat @ A @ B ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_right_less_imp_less
% 0.35/0.58  thf(fact_2883_mult__right__less__imp__less,axiom,
% 0.35/0.58      ! [A: nat,C: nat,B: nat] :
% 0.35/0.58        ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.35/0.58       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.35/0.58         => ( ord_less_nat @ A @ B ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_right_less_imp_less
% 0.35/0.58  thf(fact_2884_mult__right__less__imp__less,axiom,
% 0.35/0.58      ! [A: int,C: int,B: int] :
% 0.35/0.58        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.35/0.58       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.58         => ( ord_less_int @ A @ B ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_right_less_imp_less
% 0.35/0.58  thf(fact_2885_mult__strict__mono_H,axiom,
% 0.35/0.58      ! [A: real,B: real,C: real,D: real] :
% 0.35/0.58        ( ( ord_less_real @ A @ B )
% 0.35/0.58       => ( ( ord_less_real @ C @ D )
% 0.35/0.58         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.35/0.58           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.58             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_strict_mono'
% 0.35/0.58  thf(fact_2886_mult__strict__mono_H,axiom,
% 0.35/0.58      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.35/0.58        ( ( ord_less_rat @ A @ B )
% 0.35/0.58       => ( ( ord_less_rat @ C @ D )
% 0.35/0.58         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.35/0.58           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.58             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_strict_mono'
% 0.35/0.58  thf(fact_2887_mult__strict__mono_H,axiom,
% 0.35/0.58      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.35/0.58        ( ( ord_less_nat @ A @ B )
% 0.35/0.58       => ( ( ord_less_nat @ C @ D )
% 0.35/0.58         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.35/0.58           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.35/0.58             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_strict_mono'
% 0.35/0.58  thf(fact_2888_mult__strict__mono_H,axiom,
% 0.35/0.58      ! [A: int,B: int,C: int,D: int] :
% 0.35/0.58        ( ( ord_less_int @ A @ B )
% 0.35/0.58       => ( ( ord_less_int @ C @ D )
% 0.35/0.58         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.35/0.58           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.58             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_strict_mono'
% 0.35/0.58  thf(fact_2889_mult__less__cancel__right,axiom,
% 0.35/0.58      ! [A: real,C: real,B: real] :
% 0.35/0.58        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.35/0.58        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.58           => ( ord_less_real @ A @ B ) )
% 0.35/0.58          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.35/0.58           => ( ord_less_real @ B @ A ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_less_cancel_right
% 0.35/0.58  thf(fact_2890_mult__less__cancel__right,axiom,
% 0.35/0.58      ! [A: rat,C: rat,B: rat] :
% 0.35/0.58        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.35/0.58        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.58           => ( ord_less_rat @ A @ B ) )
% 0.35/0.58          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.35/0.58           => ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_less_cancel_right
% 0.35/0.58  thf(fact_2891_mult__less__cancel__right,axiom,
% 0.35/0.58      ! [A: int,C: int,B: int] :
% 0.35/0.58        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.35/0.58        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.58           => ( ord_less_int @ A @ B ) )
% 0.35/0.58          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.35/0.58           => ( ord_less_int @ B @ A ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_less_cancel_right
% 0.35/0.58  thf(fact_2892_mult__le__cancel__left__neg,axiom,
% 0.35/0.58      ! [C: real,A: real,B: real] :
% 0.35/0.58        ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.58       => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.35/0.58          = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_le_cancel_left_neg
% 0.35/0.58  thf(fact_2893_mult__le__cancel__left__neg,axiom,
% 0.35/0.58      ! [C: rat,A: rat,B: rat] :
% 0.35/0.58        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.58       => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.35/0.58          = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_le_cancel_left_neg
% 0.35/0.58  thf(fact_2894_mult__le__cancel__left__neg,axiom,
% 0.35/0.58      ! [C: int,A: int,B: int] :
% 0.35/0.58        ( ( ord_less_int @ C @ zero_zero_int )
% 0.35/0.58       => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.35/0.58          = ( ord_less_eq_int @ B @ A ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_le_cancel_left_neg
% 0.35/0.58  thf(fact_2895_mult__le__cancel__left__pos,axiom,
% 0.35/0.58      ! [C: real,A: real,B: real] :
% 0.35/0.58        ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.58       => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.35/0.58          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_le_cancel_left_pos
% 0.35/0.58  thf(fact_2896_mult__le__cancel__left__pos,axiom,
% 0.35/0.58      ! [C: rat,A: rat,B: rat] :
% 0.35/0.58        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.58       => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.35/0.58          = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_le_cancel_left_pos
% 0.35/0.58  thf(fact_2897_mult__le__cancel__left__pos,axiom,
% 0.35/0.58      ! [C: int,A: int,B: int] :
% 0.35/0.58        ( ( ord_less_int @ zero_zero_int @ C )
% 0.35/0.58       => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.35/0.58          = ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_le_cancel_left_pos
% 0.35/0.58  thf(fact_2898_mult__left__le__imp__le,axiom,
% 0.35/0.58      ! [C: real,A: real,B: real] :
% 0.35/0.58        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.35/0.58       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.58         => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_left_le_imp_le
% 0.35/0.58  thf(fact_2899_mult__left__le__imp__le,axiom,
% 0.35/0.58      ! [C: rat,A: rat,B: rat] :
% 0.35/0.58        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.35/0.58       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.58         => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_left_le_imp_le
% 0.35/0.58  thf(fact_2900_mult__left__le__imp__le,axiom,
% 0.35/0.58      ! [C: nat,A: nat,B: nat] :
% 0.35/0.58        ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.35/0.58       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.35/0.58         => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_left_le_imp_le
% 0.35/0.58  thf(fact_2901_mult__left__le__imp__le,axiom,
% 0.35/0.58      ! [C: int,A: int,B: int] :
% 0.35/0.58        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.35/0.58       => ( ( ord_less_int @ zero_zero_int @ C )
% 0.35/0.58         => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_left_le_imp_le
% 0.35/0.58  thf(fact_2902_mult__right__le__imp__le,axiom,
% 0.35/0.58      ! [A: real,C: real,B: real] :
% 0.35/0.58        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.35/0.58       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.58         => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_right_le_imp_le
% 0.35/0.58  thf(fact_2903_mult__right__le__imp__le,axiom,
% 0.35/0.58      ! [A: rat,C: rat,B: rat] :
% 0.35/0.58        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.35/0.58       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.58         => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_right_le_imp_le
% 0.35/0.58  thf(fact_2904_mult__right__le__imp__le,axiom,
% 0.35/0.58      ! [A: nat,C: nat,B: nat] :
% 0.35/0.58        ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.35/0.58       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.35/0.58         => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_right_le_imp_le
% 0.35/0.58  thf(fact_2905_mult__right__le__imp__le,axiom,
% 0.35/0.58      ! [A: int,C: int,B: int] :
% 0.35/0.58        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.35/0.58       => ( ( ord_less_int @ zero_zero_int @ C )
% 0.35/0.58         => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_right_le_imp_le
% 0.35/0.58  thf(fact_2906_mult__le__less__imp__less,axiom,
% 0.35/0.58      ! [A: real,B: real,C: real,D: real] :
% 0.35/0.58        ( ( ord_less_eq_real @ A @ B )
% 0.35/0.58       => ( ( ord_less_real @ C @ D )
% 0.35/0.58         => ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.58           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.58             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_le_less_imp_less
% 0.35/0.58  thf(fact_2907_mult__le__less__imp__less,axiom,
% 0.35/0.58      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.35/0.58        ( ( ord_less_eq_rat @ A @ B )
% 0.35/0.58       => ( ( ord_less_rat @ C @ D )
% 0.35/0.58         => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.35/0.58           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.58             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_le_less_imp_less
% 0.35/0.58  thf(fact_2908_mult__le__less__imp__less,axiom,
% 0.35/0.58      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.35/0.58        ( ( ord_less_eq_nat @ A @ B )
% 0.35/0.58       => ( ( ord_less_nat @ C @ D )
% 0.35/0.58         => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.35/0.58           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.35/0.58             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_le_less_imp_less
% 0.35/0.58  thf(fact_2909_mult__le__less__imp__less,axiom,
% 0.35/0.58      ! [A: int,B: int,C: int,D: int] :
% 0.35/0.58        ( ( ord_less_eq_int @ A @ B )
% 0.35/0.58       => ( ( ord_less_int @ C @ D )
% 0.35/0.58         => ( ( ord_less_int @ zero_zero_int @ A )
% 0.35/0.58           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.58             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_le_less_imp_less
% 0.35/0.58  thf(fact_2910_mult__less__le__imp__less,axiom,
% 0.35/0.58      ! [A: real,B: real,C: real,D: real] :
% 0.35/0.58        ( ( ord_less_real @ A @ B )
% 0.35/0.58       => ( ( ord_less_eq_real @ C @ D )
% 0.35/0.58         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.35/0.58           => ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.58             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_less_le_imp_less
% 0.35/0.58  thf(fact_2911_mult__less__le__imp__less,axiom,
% 0.35/0.58      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.35/0.58        ( ( ord_less_rat @ A @ B )
% 0.35/0.58       => ( ( ord_less_eq_rat @ C @ D )
% 0.35/0.58         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.35/0.58           => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.58             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_less_le_imp_less
% 0.35/0.58  thf(fact_2912_mult__less__le__imp__less,axiom,
% 0.35/0.58      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.35/0.58        ( ( ord_less_nat @ A @ B )
% 0.35/0.58       => ( ( ord_less_eq_nat @ C @ D )
% 0.35/0.58         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.35/0.58           => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.35/0.58             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_less_le_imp_less
% 0.35/0.58  thf(fact_2913_mult__less__le__imp__less,axiom,
% 0.35/0.58      ! [A: int,B: int,C: int,D: int] :
% 0.35/0.58        ( ( ord_less_int @ A @ B )
% 0.35/0.58       => ( ( ord_less_eq_int @ C @ D )
% 0.35/0.58         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.35/0.58           => ( ( ord_less_int @ zero_zero_int @ C )
% 0.35/0.58             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % mult_less_le_imp_less
% 0.35/0.58  thf(fact_2914_sum__squares__ge__zero,axiom,
% 0.35/0.58      ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % sum_squares_ge_zero
% 0.35/0.58  thf(fact_2915_sum__squares__ge__zero,axiom,
% 0.35/0.58      ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % sum_squares_ge_zero
% 0.35/0.58  thf(fact_2916_sum__squares__ge__zero,axiom,
% 0.35/0.58      ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % sum_squares_ge_zero
% 0.35/0.58  thf(fact_2917_sum__squares__le__zero__iff,axiom,
% 0.35/0.58      ! [X: real,Y: real] :
% 0.35/0.58        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
% 0.35/0.58        = ( ( X = zero_zero_real )
% 0.35/0.58          & ( Y = zero_zero_real ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % sum_squares_le_zero_iff
% 0.35/0.58  thf(fact_2918_sum__squares__le__zero__iff,axiom,
% 0.35/0.58      ! [X: rat,Y: rat] :
% 0.35/0.58        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
% 0.35/0.58        = ( ( X = zero_zero_rat )
% 0.35/0.58          & ( Y = zero_zero_rat ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % sum_squares_le_zero_iff
% 0.35/0.58  thf(fact_2919_sum__squares__le__zero__iff,axiom,
% 0.35/0.58      ! [X: int,Y: int] :
% 0.35/0.58        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
% 0.35/0.58        = ( ( X = zero_zero_int )
% 0.35/0.58          & ( Y = zero_zero_int ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % sum_squares_le_zero_iff
% 0.35/0.58  thf(fact_2920_not__sum__squares__lt__zero,axiom,
% 0.35/0.58      ! [X: real,Y: real] :
% 0.35/0.58        ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% 0.35/0.58  
% 0.35/0.58  % not_sum_squares_lt_zero
% 0.35/0.58  thf(fact_2921_not__sum__squares__lt__zero,axiom,
% 0.35/0.58      ! [X: rat,Y: rat] :
% 0.35/0.58        ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).
% 0.35/0.58  
% 0.35/0.58  % not_sum_squares_lt_zero
% 0.35/0.58  thf(fact_2922_not__sum__squares__lt__zero,axiom,
% 0.35/0.58      ! [X: int,Y: int] :
% 0.35/0.58        ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% 0.35/0.58  
% 0.35/0.58  % not_sum_squares_lt_zero
% 0.35/0.58  thf(fact_2923_sum__squares__gt__zero__iff,axiom,
% 0.35/0.58      ! [X: real,Y: real] :
% 0.35/0.58        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
% 0.35/0.58        = ( ( X != zero_zero_real )
% 0.35/0.58          | ( Y != zero_zero_real ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % sum_squares_gt_zero_iff
% 0.35/0.58  thf(fact_2924_sum__squares__gt__zero__iff,axiom,
% 0.35/0.58      ! [X: rat,Y: rat] :
% 0.35/0.58        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
% 0.35/0.58        = ( ( X != zero_zero_rat )
% 0.35/0.58          | ( Y != zero_zero_rat ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % sum_squares_gt_zero_iff
% 0.35/0.58  thf(fact_2925_sum__squares__gt__zero__iff,axiom,
% 0.35/0.58      ! [X: int,Y: int] :
% 0.35/0.58        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
% 0.35/0.58        = ( ( X != zero_zero_int )
% 0.35/0.58          | ( Y != zero_zero_int ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % sum_squares_gt_zero_iff
% 0.35/0.58  thf(fact_2926_one__less__mult,axiom,
% 0.35/0.58      ! [N: nat,M: nat] :
% 0.35/0.58        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.58       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.35/0.58         => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % one_less_mult
% 0.35/0.58  thf(fact_2927_n__less__m__mult__n,axiom,
% 0.35/0.58      ! [N: nat,M: nat] :
% 0.35/0.58        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.58       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.35/0.58         => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.35/0.58  
% 0.35/0.58  % n_less_m_mult_n
% 0.35/0.58  thf(fact_2928_n__less__n__mult__m,axiom,
% 0.35/0.58      ! [N: nat,M: nat] :
% 0.35/0.58        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.58       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.35/0.59         => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % n_less_n_mult_m
% 0.35/0.59  thf(fact_2929_zero__less__two,axiom,
% 0.35/0.59      ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 0.35/0.59  
% 0.35/0.59  % zero_less_two
% 0.35/0.59  thf(fact_2930_zero__less__two,axiom,
% 0.35/0.59      ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 0.35/0.59  
% 0.35/0.59  % zero_less_two
% 0.35/0.59  thf(fact_2931_zero__less__two,axiom,
% 0.35/0.59      ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 0.35/0.59  
% 0.35/0.59  % zero_less_two
% 0.35/0.59  thf(fact_2932_zero__less__two,axiom,
% 0.35/0.59      ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 0.35/0.59  
% 0.35/0.59  % zero_less_two
% 0.35/0.59  thf(fact_2933_power__le__one,axiom,
% 0.35/0.59      ! [A: real,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.35/0.59       => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.35/0.59         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_le_one
% 0.35/0.59  thf(fact_2934_power__le__one,axiom,
% 0.35/0.59      ! [A: rat,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.35/0.59       => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.35/0.59         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_le_one
% 0.35/0.59  thf(fact_2935_power__le__one,axiom,
% 0.35/0.59      ! [A: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.35/0.59       => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.35/0.59         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_le_one
% 0.35/0.59  thf(fact_2936_power__le__one,axiom,
% 0.35/0.59      ! [A: int,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.35/0.59       => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.35/0.59         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_le_one
% 0.35/0.59  thf(fact_2937_power__gt1,axiom,
% 0.35/0.59      ! [A: real,N: nat] :
% 0.35/0.59        ( ( ord_less_real @ one_one_real @ A )
% 0.35/0.59       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_gt1
% 0.35/0.59  thf(fact_2938_power__gt1,axiom,
% 0.35/0.59      ! [A: rat,N: nat] :
% 0.35/0.59        ( ( ord_less_rat @ one_one_rat @ A )
% 0.35/0.59       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_gt1
% 0.35/0.59  thf(fact_2939_power__gt1,axiom,
% 0.35/0.59      ! [A: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_nat @ one_one_nat @ A )
% 0.35/0.59       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_gt1
% 0.35/0.59  thf(fact_2940_power__gt1,axiom,
% 0.35/0.59      ! [A: int,N: nat] :
% 0.35/0.59        ( ( ord_less_int @ one_one_int @ A )
% 0.35/0.59       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_gt1
% 0.35/0.59  thf(fact_2941_power__0__left,axiom,
% 0.35/0.59      ! [N: nat] :
% 0.35/0.59        ( ( ( N = zero_zero_nat )
% 0.35/0.59         => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.35/0.59            = one_one_rat ) )
% 0.35/0.59        & ( ( N != zero_zero_nat )
% 0.35/0.59         => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.35/0.59            = zero_zero_rat ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_0_left
% 0.35/0.59  thf(fact_2942_power__0__left,axiom,
% 0.35/0.59      ! [N: nat] :
% 0.35/0.59        ( ( ( N = zero_zero_nat )
% 0.35/0.59         => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.35/0.59            = one_one_nat ) )
% 0.35/0.59        & ( ( N != zero_zero_nat )
% 0.35/0.59         => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.35/0.59            = zero_zero_nat ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_0_left
% 0.35/0.59  thf(fact_2943_power__0__left,axiom,
% 0.35/0.59      ! [N: nat] :
% 0.35/0.59        ( ( ( N = zero_zero_nat )
% 0.35/0.59         => ( ( power_power_real @ zero_zero_real @ N )
% 0.35/0.59            = one_one_real ) )
% 0.35/0.59        & ( ( N != zero_zero_nat )
% 0.35/0.59         => ( ( power_power_real @ zero_zero_real @ N )
% 0.35/0.59            = zero_zero_real ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_0_left
% 0.35/0.59  thf(fact_2944_power__0__left,axiom,
% 0.35/0.59      ! [N: nat] :
% 0.35/0.59        ( ( ( N = zero_zero_nat )
% 0.35/0.59         => ( ( power_power_int @ zero_zero_int @ N )
% 0.35/0.59            = one_one_int ) )
% 0.35/0.59        & ( ( N != zero_zero_nat )
% 0.35/0.59         => ( ( power_power_int @ zero_zero_int @ N )
% 0.35/0.59            = zero_zero_int ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_0_left
% 0.35/0.59  thf(fact_2945_power__0__left,axiom,
% 0.35/0.59      ! [N: nat] :
% 0.35/0.59        ( ( ( N = zero_zero_nat )
% 0.35/0.59         => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.35/0.59            = one_one_complex ) )
% 0.35/0.59        & ( ( N != zero_zero_nat )
% 0.35/0.59         => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.35/0.59            = zero_zero_complex ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_0_left
% 0.35/0.59  thf(fact_2946_power__less__imp__less__exp,axiom,
% 0.35/0.59      ! [A: real,M: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_real @ one_one_real @ A )
% 0.35/0.59       => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 0.35/0.59         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_less_imp_less_exp
% 0.35/0.59  thf(fact_2947_power__less__imp__less__exp,axiom,
% 0.35/0.59      ! [A: rat,M: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_rat @ one_one_rat @ A )
% 0.35/0.59       => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 0.35/0.59         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_less_imp_less_exp
% 0.35/0.59  thf(fact_2948_power__less__imp__less__exp,axiom,
% 0.35/0.59      ! [A: nat,M: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_nat @ one_one_nat @ A )
% 0.35/0.59       => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.35/0.59         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_less_imp_less_exp
% 0.35/0.59  thf(fact_2949_power__less__imp__less__exp,axiom,
% 0.35/0.59      ! [A: int,M: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_int @ one_one_int @ A )
% 0.35/0.59       => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.35/0.59         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_less_imp_less_exp
% 0.35/0.59  thf(fact_2950_power__strict__increasing,axiom,
% 0.35/0.59      ! [N: nat,N6: nat,A: real] :
% 0.35/0.59        ( ( ord_less_nat @ N @ N6 )
% 0.35/0.59       => ( ( ord_less_real @ one_one_real @ A )
% 0.35/0.59         => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N6 ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_strict_increasing
% 0.35/0.59  thf(fact_2951_power__strict__increasing,axiom,
% 0.35/0.59      ! [N: nat,N6: nat,A: rat] :
% 0.35/0.59        ( ( ord_less_nat @ N @ N6 )
% 0.35/0.59       => ( ( ord_less_rat @ one_one_rat @ A )
% 0.35/0.59         => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N6 ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_strict_increasing
% 0.35/0.59  thf(fact_2952_power__strict__increasing,axiom,
% 0.35/0.59      ! [N: nat,N6: nat,A: nat] :
% 0.35/0.59        ( ( ord_less_nat @ N @ N6 )
% 0.35/0.59       => ( ( ord_less_nat @ one_one_nat @ A )
% 0.35/0.59         => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_strict_increasing
% 0.35/0.59  thf(fact_2953_power__strict__increasing,axiom,
% 0.35/0.59      ! [N: nat,N6: nat,A: int] :
% 0.35/0.59        ( ( ord_less_nat @ N @ N6 )
% 0.35/0.59       => ( ( ord_less_int @ one_one_int @ A )
% 0.35/0.59         => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N6 ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_strict_increasing
% 0.35/0.59  thf(fact_2954_power__increasing,axiom,
% 0.35/0.59      ! [N: nat,N6: nat,A: real] :
% 0.35/0.59        ( ( ord_less_eq_nat @ N @ N6 )
% 0.35/0.59       => ( ( ord_less_eq_real @ one_one_real @ A )
% 0.35/0.59         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N6 ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_increasing
% 0.35/0.59  thf(fact_2955_power__increasing,axiom,
% 0.35/0.59      ! [N: nat,N6: nat,A: rat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ N @ N6 )
% 0.35/0.59       => ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.35/0.59         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N6 ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_increasing
% 0.35/0.59  thf(fact_2956_power__increasing,axiom,
% 0.35/0.59      ! [N: nat,N6: nat,A: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ N @ N6 )
% 0.35/0.59       => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.35/0.59         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_increasing
% 0.35/0.59  thf(fact_2957_power__increasing,axiom,
% 0.35/0.59      ! [N: nat,N6: nat,A: int] :
% 0.35/0.59        ( ( ord_less_eq_nat @ N @ N6 )
% 0.35/0.59       => ( ( ord_less_eq_int @ one_one_int @ A )
% 0.35/0.59         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N6 ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_increasing
% 0.35/0.59  thf(fact_2958_nat__induct__non__zero,axiom,
% 0.35/0.59      ! [N: nat,P: nat > $o] :
% 0.35/0.59        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59       => ( ( P @ one_one_nat )
% 0.35/0.59         => ( ! [N3: nat] :
% 0.35/0.59                ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.35/0.59               => ( ( P @ N3 )
% 0.35/0.59                 => ( P @ ( suc @ N3 ) ) ) )
% 0.35/0.59           => ( P @ N ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % nat_induct_non_zero
% 0.35/0.59  thf(fact_2959_power__Suc__le__self,axiom,
% 0.35/0.59      ! [A: real,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.35/0.59       => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.35/0.59         => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_Suc_le_self
% 0.35/0.59  thf(fact_2960_power__Suc__le__self,axiom,
% 0.35/0.59      ! [A: rat,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.35/0.59       => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.35/0.59         => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_Suc_le_self
% 0.35/0.59  thf(fact_2961_power__Suc__le__self,axiom,
% 0.35/0.59      ! [A: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.35/0.59       => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.35/0.59         => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_Suc_le_self
% 0.35/0.59  thf(fact_2962_power__Suc__le__self,axiom,
% 0.35/0.59      ! [A: int,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.35/0.59       => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.35/0.59         => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_Suc_le_self
% 0.35/0.59  thf(fact_2963_power__Suc__less__one,axiom,
% 0.35/0.59      ! [A: real,N: nat] :
% 0.35/0.59        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.59       => ( ( ord_less_real @ A @ one_one_real )
% 0.35/0.59         => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_Suc_less_one
% 0.35/0.59  thf(fact_2964_power__Suc__less__one,axiom,
% 0.35/0.59      ! [A: rat,N: nat] :
% 0.35/0.59        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.35/0.59       => ( ( ord_less_rat @ A @ one_one_rat )
% 0.35/0.59         => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_Suc_less_one
% 0.35/0.59  thf(fact_2965_power__Suc__less__one,axiom,
% 0.35/0.59      ! [A: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.35/0.59       => ( ( ord_less_nat @ A @ one_one_nat )
% 0.35/0.59         => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_Suc_less_one
% 0.35/0.59  thf(fact_2966_power__Suc__less__one,axiom,
% 0.35/0.59      ! [A: int,N: nat] :
% 0.35/0.59        ( ( ord_less_int @ zero_zero_int @ A )
% 0.35/0.59       => ( ( ord_less_int @ A @ one_one_int )
% 0.35/0.59         => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_Suc_less_one
% 0.35/0.59  thf(fact_2967_power__strict__decreasing,axiom,
% 0.35/0.59      ! [N: nat,N6: nat,A: real] :
% 0.35/0.59        ( ( ord_less_nat @ N @ N6 )
% 0.35/0.59       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.59         => ( ( ord_less_real @ A @ one_one_real )
% 0.35/0.59           => ( ord_less_real @ ( power_power_real @ A @ N6 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_strict_decreasing
% 0.35/0.59  thf(fact_2968_power__strict__decreasing,axiom,
% 0.35/0.59      ! [N: nat,N6: nat,A: rat] :
% 0.35/0.59        ( ( ord_less_nat @ N @ N6 )
% 0.35/0.59       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.35/0.59         => ( ( ord_less_rat @ A @ one_one_rat )
% 0.35/0.59           => ( ord_less_rat @ ( power_power_rat @ A @ N6 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_strict_decreasing
% 0.35/0.59  thf(fact_2969_power__strict__decreasing,axiom,
% 0.35/0.59      ! [N: nat,N6: nat,A: nat] :
% 0.35/0.59        ( ( ord_less_nat @ N @ N6 )
% 0.35/0.59       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.35/0.59         => ( ( ord_less_nat @ A @ one_one_nat )
% 0.35/0.59           => ( ord_less_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_strict_decreasing
% 0.35/0.59  thf(fact_2970_power__strict__decreasing,axiom,
% 0.35/0.59      ! [N: nat,N6: nat,A: int] :
% 0.35/0.59        ( ( ord_less_nat @ N @ N6 )
% 0.35/0.59       => ( ( ord_less_int @ zero_zero_int @ A )
% 0.35/0.59         => ( ( ord_less_int @ A @ one_one_int )
% 0.35/0.59           => ( ord_less_int @ ( power_power_int @ A @ N6 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_strict_decreasing
% 0.35/0.59  thf(fact_2971_power__decreasing,axiom,
% 0.35/0.59      ! [N: nat,N6: nat,A: real] :
% 0.35/0.59        ( ( ord_less_eq_nat @ N @ N6 )
% 0.35/0.59       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.35/0.59         => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.35/0.59           => ( ord_less_eq_real @ ( power_power_real @ A @ N6 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_decreasing
% 0.35/0.59  thf(fact_2972_power__decreasing,axiom,
% 0.35/0.59      ! [N: nat,N6: nat,A: rat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ N @ N6 )
% 0.35/0.59       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.35/0.59         => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.35/0.59           => ( ord_less_eq_rat @ ( power_power_rat @ A @ N6 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_decreasing
% 0.35/0.59  thf(fact_2973_power__decreasing,axiom,
% 0.35/0.59      ! [N: nat,N6: nat,A: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ N @ N6 )
% 0.35/0.59       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.35/0.59         => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.35/0.59           => ( ord_less_eq_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_decreasing
% 0.35/0.59  thf(fact_2974_power__decreasing,axiom,
% 0.35/0.59      ! [N: nat,N6: nat,A: int] :
% 0.35/0.59        ( ( ord_less_eq_nat @ N @ N6 )
% 0.35/0.59       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.35/0.59         => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.35/0.59           => ( ord_less_eq_int @ ( power_power_int @ A @ N6 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_decreasing
% 0.35/0.59  thf(fact_2975_power__le__imp__le__exp,axiom,
% 0.35/0.59      ! [A: real,M: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_real @ one_one_real @ A )
% 0.35/0.59       => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 0.35/0.59         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_le_imp_le_exp
% 0.35/0.59  thf(fact_2976_power__le__imp__le__exp,axiom,
% 0.35/0.59      ! [A: rat,M: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_rat @ one_one_rat @ A )
% 0.35/0.59       => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 0.35/0.59         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_le_imp_le_exp
% 0.35/0.59  thf(fact_2977_power__le__imp__le__exp,axiom,
% 0.35/0.59      ! [A: nat,M: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_nat @ one_one_nat @ A )
% 0.35/0.59       => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.35/0.59         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_le_imp_le_exp
% 0.35/0.59  thf(fact_2978_power__le__imp__le__exp,axiom,
% 0.35/0.59      ! [A: int,M: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_int @ one_one_int @ A )
% 0.35/0.59       => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.35/0.59         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_le_imp_le_exp
% 0.35/0.59  thf(fact_2979_self__le__power,axiom,
% 0.35/0.59      ! [A: real,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_real @ one_one_real @ A )
% 0.35/0.59       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59         => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % self_le_power
% 0.35/0.59  thf(fact_2980_self__le__power,axiom,
% 0.35/0.59      ! [A: rat,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.35/0.59       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59         => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % self_le_power
% 0.35/0.59  thf(fact_2981_self__le__power,axiom,
% 0.35/0.59      ! [A: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.35/0.59       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59         => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % self_le_power
% 0.35/0.59  thf(fact_2982_self__le__power,axiom,
% 0.35/0.59      ! [A: int,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_int @ one_one_int @ A )
% 0.35/0.59       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59         => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % self_le_power
% 0.35/0.59  thf(fact_2983_one__less__power,axiom,
% 0.35/0.59      ! [A: real,N: nat] :
% 0.35/0.59        ( ( ord_less_real @ one_one_real @ A )
% 0.35/0.59       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59         => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % one_less_power
% 0.35/0.59  thf(fact_2984_one__less__power,axiom,
% 0.35/0.59      ! [A: rat,N: nat] :
% 0.35/0.59        ( ( ord_less_rat @ one_one_rat @ A )
% 0.35/0.59       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59         => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % one_less_power
% 0.35/0.59  thf(fact_2985_one__less__power,axiom,
% 0.35/0.59      ! [A: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_nat @ one_one_nat @ A )
% 0.35/0.59       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59         => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % one_less_power
% 0.35/0.59  thf(fact_2986_one__less__power,axiom,
% 0.35/0.59      ! [A: int,N: nat] :
% 0.35/0.59        ( ( ord_less_int @ one_one_int @ A )
% 0.35/0.59       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59         => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % one_less_power
% 0.35/0.59  thf(fact_2987_nat__bit__induct,axiom,
% 0.35/0.59      ! [P: nat > $o,N: nat] :
% 0.35/0.59        ( ( P @ zero_zero_nat )
% 0.35/0.59       => ( ! [N3: nat] :
% 0.35/0.59              ( ( P @ N3 )
% 0.35/0.59             => ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.35/0.59               => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 0.35/0.59         => ( ! [N3: nat] :
% 0.35/0.59                ( ( P @ N3 )
% 0.35/0.59               => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 0.35/0.59           => ( P @ N ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % nat_bit_induct
% 0.35/0.59  thf(fact_2988_half__gt__zero,axiom,
% 0.35/0.59      ! [A: real] :
% 0.35/0.59        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.59       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % half_gt_zero
% 0.35/0.59  thf(fact_2989_half__gt__zero,axiom,
% 0.35/0.59      ! [A: rat] :
% 0.35/0.59        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.35/0.59       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % half_gt_zero
% 0.35/0.59  thf(fact_2990_half__gt__zero__iff,axiom,
% 0.35/0.59      ! [A: real] :
% 0.35/0.59        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.59        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % half_gt_zero_iff
% 0.35/0.59  thf(fact_2991_half__gt__zero__iff,axiom,
% 0.35/0.59      ! [A: rat] :
% 0.35/0.59        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.35/0.59        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % half_gt_zero_iff
% 0.35/0.59  thf(fact_2992_field__less__half__sum,axiom,
% 0.35/0.59      ! [X: real,Y: real] :
% 0.35/0.59        ( ( ord_less_real @ X @ Y )
% 0.35/0.59       => ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % field_less_half_sum
% 0.35/0.59  thf(fact_2993_field__less__half__sum,axiom,
% 0.35/0.59      ! [X: rat,Y: rat] :
% 0.35/0.59        ( ( ord_less_rat @ X @ Y )
% 0.35/0.59       => ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % field_less_half_sum
% 0.35/0.59  thf(fact_2994_zero__le__even__power_H,axiom,
% 0.35/0.59      ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % zero_le_even_power'
% 0.35/0.59  thf(fact_2995_zero__le__even__power_H,axiom,
% 0.35/0.59      ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % zero_le_even_power'
% 0.35/0.59  thf(fact_2996_zero__le__even__power_H,axiom,
% 0.35/0.59      ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % zero_le_even_power'
% 0.35/0.59  thf(fact_2997_odd__0__le__power__imp__0__le,axiom,
% 0.35/0.59      ! [A: real,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.35/0.59       => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % odd_0_le_power_imp_0_le
% 0.35/0.59  thf(fact_2998_odd__0__le__power__imp__0__le,axiom,
% 0.35/0.59      ! [A: rat,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.35/0.59       => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % odd_0_le_power_imp_0_le
% 0.35/0.59  thf(fact_2999_odd__0__le__power__imp__0__le,axiom,
% 0.35/0.59      ! [A: int,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.35/0.59       => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % odd_0_le_power_imp_0_le
% 0.35/0.59  thf(fact_3000_odd__power__less__zero,axiom,
% 0.35/0.59      ! [A: real,N: nat] :
% 0.35/0.59        ( ( ord_less_real @ A @ zero_zero_real )
% 0.35/0.59       => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% 0.35/0.59  
% 0.35/0.59  % odd_power_less_zero
% 0.35/0.59  thf(fact_3001_odd__power__less__zero,axiom,
% 0.35/0.59      ! [A: rat,N: nat] :
% 0.35/0.59        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.35/0.59       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% 0.35/0.59  
% 0.35/0.59  % odd_power_less_zero
% 0.35/0.59  thf(fact_3002_odd__power__less__zero,axiom,
% 0.35/0.59      ! [A: int,N: nat] :
% 0.35/0.59        ( ( ord_less_int @ A @ zero_zero_int )
% 0.35/0.59       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% 0.35/0.59  
% 0.35/0.59  % odd_power_less_zero
% 0.35/0.59  thf(fact_3003_ex__power__ivl2,axiom,
% 0.35/0.59      ! [B: nat,K: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.35/0.59       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.35/0.59         => ? [N3: nat] :
% 0.35/0.59              ( ( ord_less_nat @ ( power_power_nat @ B @ N3 ) @ K )
% 0.35/0.59              & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % ex_power_ivl2
% 0.35/0.59  thf(fact_3004_ex__power__ivl1,axiom,
% 0.35/0.59      ! [B: nat,K: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.35/0.59       => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 0.35/0.59         => ? [N3: nat] :
% 0.35/0.59              ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N3 ) @ K )
% 0.35/0.59              & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % ex_power_ivl1
% 0.35/0.59  thf(fact_3005_vebt__insert_Osimps_I2_J,axiom,
% 0.35/0.59      ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
% 0.35/0.59        ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) @ X )
% 0.35/0.59        = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) ) ).
% 0.35/0.59  
% 0.35/0.59  % vebt_insert.simps(2)
% 0.35/0.59  thf(fact_3006_vebt__insert_Osimps_I3_J,axiom,
% 0.35/0.59      ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
% 0.35/0.59        ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X )
% 0.35/0.59        = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) ).
% 0.35/0.59  
% 0.35/0.59  % vebt_insert.simps(3)
% 0.35/0.59  thf(fact_3007_aaaa,axiom,
% 0.35/0.59      ( ( ya = mi )
% 0.35/0.59      | ( ( ( xa = ma )
% 0.35/0.59         => ( ya
% 0.35/0.59            = ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ) ) )
% 0.35/0.59        & ( ( xa != ma )
% 0.35/0.59         => ( ya = ma ) ) )
% 0.35/0.59      | ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( vEBT_VEBT_high @ ya @ na ) ) @ ( vEBT_VEBT_low @ ya @ na ) )
% 0.35/0.59        & ( ord_less_nat @ ( vEBT_VEBT_high @ ya @ na ) @ ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % aaaa
% 0.35/0.59  thf(fact_3008__092_060open_062vebt__delete_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_061_ANode_A_ISome_A_Imi_M_Aif_Ax_A_061_Ama_Athen_Ahigh_Ax_An_A_K_A2_A_094_A_Ideg_Adiv_A2_J_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Ahigh_Ax_An_J_J_Aelse_Ama_J_J_Adeg_A_ItreeList_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_J_Asummary_092_060close_062,axiom,
% 0.35/0.59      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.35/0.59      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ( if_nat @ ( xa = ma ) @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ) @ ma ) ) ) @ deg @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ summary ) ) ).
% 0.35/0.59  
% 0.35/0.59  % \<open>vebt_delete (Node (Some (mi, ma)) deg treeList summary) x = Node (Some (mi, if x = ma then high x n * 2 ^ (deg div 2) + the (vebt_maxt (treeList [high x n := vebt_delete (treeList ! high x n) (low x n)] ! high x n)) else ma)) deg (treeList[high x n := vebt_delete (treeList ! high x n) (low x n)]) summary\<close>
% 0.35/0.59  thf(fact_3009_one__div__two__eq__zero,axiom,
% 0.35/0.59      ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.59      = zero_zero_nat ) ).
% 0.35/0.59  
% 0.35/0.59  % one_div_two_eq_zero
% 0.35/0.59  thf(fact_3010_one__div__two__eq__zero,axiom,
% 0.35/0.59      ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.59      = zero_zero_int ) ).
% 0.35/0.59  
% 0.35/0.59  % one_div_two_eq_zero
% 0.35/0.59  thf(fact_3011_one__div__two__eq__zero,axiom,
% 0.35/0.59      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.59      = zero_z3403309356797280102nteger ) ).
% 0.35/0.59  
% 0.35/0.59  % one_div_two_eq_zero
% 0.35/0.59  thf(fact_3012__092_060open_062_Iif_Ax_A_061_Ama_Athen_Ahigh_Ax_An_A_K_A2_A_094_A_Ideg_Adiv_A2_J_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Ahigh_Ax_An_J_J_Aelse_Ama_J_A_061_Ahigh_Ax_An_A_K_A2_A_094_A_Ideg_Adiv_A2_J_A_L_Athe_A_Ivebt__maxt_A_ItreeList_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Ahigh_Ax_An_J_J_092_060close_062,axiom,
% 0.35/0.59      ( ( xa != ma )
% 0.35/0.59     => ( ma
% 0.35/0.59        = ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % \<open>(if x = ma then high x n * 2 ^ (deg div 2) + the (vebt_maxt (treeList [high x n := vebt_delete (treeList ! high x n) (low x n)] ! high x n)) else ma) = high x n * 2 ^ (deg div 2) + the (vebt_maxt (treeList[high x n := vebt_delete (treeList ! high x n) (low x n)] ! high x n))\<close>
% 0.35/0.59  thf(fact_3013_del__x__mi__lets__in__not__minNull,axiom,
% 0.35/0.59      ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
% 0.35/0.59        ( ( ( X = Mi )
% 0.35/0.59          & ( ord_less_nat @ X @ Ma ) )
% 0.35/0.59       => ( ( Mi != Ma )
% 0.35/0.59         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.59           => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.59                = H2 )
% 0.35/0.59             => ( ( Xn
% 0.35/0.59                  = ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
% 0.35/0.59               => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.59                    = L )
% 0.35/0.59                 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.59                   => ( ( Newnode
% 0.35/0.59                        = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
% 0.35/0.59                     => ( ( Newlist
% 0.35/0.59                          = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
% 0.35/0.59                       => ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 0.35/0.59                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.35/0.59                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % del_x_mi_lets_in_not_minNull
% 0.35/0.59  thf(fact_3014_del__x__not__mi__newnode__not__nil,axiom,
% 0.35/0.59      ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.59        ( ( ( ord_less_nat @ Mi @ X )
% 0.35/0.59          & ( ord_less_eq_nat @ X @ Ma ) )
% 0.35/0.59       => ( ( Mi != Ma )
% 0.35/0.59         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.59           => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.59                = H2 )
% 0.35/0.59             => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.59                  = L )
% 0.35/0.59               => ( ( Newnode
% 0.35/0.59                    = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
% 0.35/0.59                 => ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 0.35/0.59                   => ( ( Newlist
% 0.35/0.59                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
% 0.35/0.59                     => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.59                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.35/0.59                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % del_x_not_mi_newnode_not_nil
% 0.35/0.59  thf(fact_3015_insert__simp__excp,axiom,
% 0.35/0.59      ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.35/0.59        ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.59       => ( ( ord_less_nat @ X @ Mi )
% 0.35/0.59         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.59           => ( ( X != Ma )
% 0.35/0.59             => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.35/0.59                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % insert_simp_excp
% 0.35/0.59  thf(fact_3016_insert__simp__norm,axiom,
% 0.35/0.59      ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.35/0.59        ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.59       => ( ( ord_less_nat @ Mi @ X )
% 0.35/0.59         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.59           => ( ( X != Ma )
% 0.35/0.59             => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.35/0.59                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % insert_simp_norm
% 0.35/0.59  thf(fact_3017_nested__mint,axiom,
% 0.35/0.59      ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
% 0.35/0.59        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 0.35/0.59       => ( ( N
% 0.35/0.59            = ( suc @ ( suc @ Va ) ) )
% 0.35/0.59         => ( ~ ( ord_less_nat @ Ma @ Mi )
% 0.35/0.59           => ( ( Ma != Mi )
% 0.35/0.59             => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % nested_mint
% 0.35/0.59  thf(fact_3018_div__mult__self__is__m,axiom,
% 0.35/0.59      ! [N: nat,M: nat] :
% 0.35/0.59        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59       => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
% 0.35/0.59          = M ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_self_is_m
% 0.35/0.59  thf(fact_3019_half__negative__int__iff,axiom,
% 0.35/0.59      ! [K: int] :
% 0.35/0.59        ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.35/0.59        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.35/0.59  
% 0.35/0.59  % half_negative_int_iff
% 0.35/0.59  thf(fact_3020_max__bot2,axiom,
% 0.35/0.59      ! [X: set_Pr1261947904930325089at_nat] :
% 0.35/0.59        ( ( ord_ma7524802468073614006at_nat @ X @ bot_bo2099793752762293965at_nat )
% 0.35/0.59        = X ) ).
% 0.35/0.59  
% 0.35/0.59  % max_bot2
% 0.35/0.59  thf(fact_3021_max__bot2,axiom,
% 0.35/0.59      ! [X: set_real] :
% 0.35/0.59        ( ( ord_max_set_real @ X @ bot_bot_set_real )
% 0.35/0.59        = X ) ).
% 0.35/0.59  
% 0.35/0.59  % max_bot2
% 0.35/0.59  thf(fact_3022_max__bot2,axiom,
% 0.35/0.59      ! [X: set_nat] :
% 0.35/0.59        ( ( ord_max_set_nat @ X @ bot_bot_set_nat )
% 0.35/0.59        = X ) ).
% 0.35/0.59  
% 0.35/0.59  % max_bot2
% 0.35/0.59  thf(fact_3023_max__bot2,axiom,
% 0.35/0.59      ! [X: set_int] :
% 0.35/0.59        ( ( ord_max_set_int @ X @ bot_bot_set_int )
% 0.35/0.59        = X ) ).
% 0.35/0.59  
% 0.35/0.59  % max_bot2
% 0.35/0.59  thf(fact_3024_max__bot2,axiom,
% 0.35/0.59      ! [X: nat] :
% 0.35/0.59        ( ( ord_max_nat @ X @ bot_bot_nat )
% 0.35/0.59        = X ) ).
% 0.35/0.59  
% 0.35/0.59  % max_bot2
% 0.35/0.59  thf(fact_3025_max__bot,axiom,
% 0.35/0.59      ! [X: set_Pr1261947904930325089at_nat] :
% 0.35/0.59        ( ( ord_ma7524802468073614006at_nat @ bot_bo2099793752762293965at_nat @ X )
% 0.35/0.59        = X ) ).
% 0.35/0.59  
% 0.35/0.59  % max_bot
% 0.35/0.59  thf(fact_3026_max__bot,axiom,
% 0.35/0.59      ! [X: set_real] :
% 0.35/0.59        ( ( ord_max_set_real @ bot_bot_set_real @ X )
% 0.35/0.59        = X ) ).
% 0.35/0.59  
% 0.35/0.59  % max_bot
% 0.35/0.59  thf(fact_3027_max__bot,axiom,
% 0.35/0.59      ! [X: set_nat] :
% 0.35/0.59        ( ( ord_max_set_nat @ bot_bot_set_nat @ X )
% 0.35/0.59        = X ) ).
% 0.35/0.59  
% 0.35/0.59  % max_bot
% 0.35/0.59  thf(fact_3028_max__bot,axiom,
% 0.35/0.59      ! [X: set_int] :
% 0.35/0.59        ( ( ord_max_set_int @ bot_bot_set_int @ X )
% 0.35/0.59        = X ) ).
% 0.35/0.59  
% 0.35/0.59  % max_bot
% 0.35/0.59  thf(fact_3029_max__bot,axiom,
% 0.35/0.59      ! [X: nat] :
% 0.35/0.59        ( ( ord_max_nat @ bot_bot_nat @ X )
% 0.35/0.59        = X ) ).
% 0.35/0.59  
% 0.35/0.59  % max_bot
% 0.35/0.59  thf(fact_3030_max__Suc__Suc,axiom,
% 0.35/0.59      ! [M: nat,N: nat] :
% 0.35/0.59        ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.35/0.59        = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_Suc_Suc
% 0.35/0.59  thf(fact_3031_max__0R,axiom,
% 0.35/0.59      ! [N: nat] :
% 0.35/0.59        ( ( ord_max_nat @ N @ zero_zero_nat )
% 0.35/0.59        = N ) ).
% 0.35/0.59  
% 0.35/0.59  % max_0R
% 0.35/0.59  thf(fact_3032_max__0L,axiom,
% 0.35/0.59      ! [N: nat] :
% 0.35/0.59        ( ( ord_max_nat @ zero_zero_nat @ N )
% 0.35/0.59        = N ) ).
% 0.35/0.59  
% 0.35/0.59  % max_0L
% 0.35/0.59  thf(fact_3033_max__nat_Oright__neutral,axiom,
% 0.35/0.59      ! [A: nat] :
% 0.35/0.59        ( ( ord_max_nat @ A @ zero_zero_nat )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % max_nat.right_neutral
% 0.35/0.59  thf(fact_3034_max__nat_Oneutr__eq__iff,axiom,
% 0.35/0.59      ! [A: nat,B: nat] :
% 0.35/0.59        ( ( zero_zero_nat
% 0.35/0.59          = ( ord_max_nat @ A @ B ) )
% 0.35/0.59        = ( ( A = zero_zero_nat )
% 0.35/0.59          & ( B = zero_zero_nat ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_nat.neutr_eq_iff
% 0.35/0.59  thf(fact_3035_max__nat_Oleft__neutral,axiom,
% 0.35/0.59      ! [A: nat] :
% 0.35/0.59        ( ( ord_max_nat @ zero_zero_nat @ A )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % max_nat.left_neutral
% 0.35/0.59  thf(fact_3036_max__nat_Oeq__neutr__iff,axiom,
% 0.35/0.59      ! [A: nat,B: nat] :
% 0.35/0.59        ( ( ( ord_max_nat @ A @ B )
% 0.35/0.59          = zero_zero_nat )
% 0.35/0.59        = ( ( A = zero_zero_nat )
% 0.35/0.59          & ( B = zero_zero_nat ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_nat.eq_neutr_iff
% 0.35/0.59  thf(fact_3037_summaxma,axiom,
% 0.35/0.59      ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.59        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
% 0.35/0.59       => ( ( Mi != Ma )
% 0.35/0.59         => ( ( the_nat2 @ ( vEBT_vebt_maxt @ Summary ) )
% 0.35/0.59            = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % summaxma
% 0.35/0.59  thf(fact_3038_div__mult__mult1__if,axiom,
% 0.35/0.59      ! [C: nat,A: nat,B: nat] :
% 0.35/0.59        ( ( ( C = zero_zero_nat )
% 0.35/0.59         => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.35/0.59            = zero_zero_nat ) )
% 0.35/0.59        & ( ( C != zero_zero_nat )
% 0.35/0.59         => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.35/0.59            = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_mult1_if
% 0.35/0.59  thf(fact_3039_div__mult__mult1__if,axiom,
% 0.35/0.59      ! [C: int,A: int,B: int] :
% 0.35/0.59        ( ( ( C = zero_zero_int )
% 0.35/0.59         => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.35/0.59            = zero_zero_int ) )
% 0.35/0.59        & ( ( C != zero_zero_int )
% 0.35/0.59         => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.35/0.59            = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_mult1_if
% 0.35/0.59  thf(fact_3040_div__mult__mult1__if,axiom,
% 0.35/0.59      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.35/0.59        ( ( ( C = zero_z3403309356797280102nteger )
% 0.35/0.59         => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.35/0.59            = zero_z3403309356797280102nteger ) )
% 0.35/0.59        & ( ( C != zero_z3403309356797280102nteger )
% 0.35/0.59         => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.35/0.59            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_mult1_if
% 0.35/0.59  thf(fact_3041_div__mult__mult2,axiom,
% 0.35/0.59      ! [C: nat,A: nat,B: nat] :
% 0.35/0.59        ( ( C != zero_zero_nat )
% 0.35/0.59       => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.35/0.59          = ( divide_divide_nat @ A @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_mult2
% 0.35/0.59  thf(fact_3042_div__mult__mult2,axiom,
% 0.35/0.59      ! [C: int,A: int,B: int] :
% 0.35/0.59        ( ( C != zero_zero_int )
% 0.35/0.59       => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.35/0.59          = ( divide_divide_int @ A @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_mult2
% 0.35/0.59  thf(fact_3043_div__mult__mult2,axiom,
% 0.35/0.59      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.35/0.59        ( ( C != zero_z3403309356797280102nteger )
% 0.35/0.59       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.35/0.59          = ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_mult2
% 0.35/0.59  thf(fact_3044_div__mult__mult1,axiom,
% 0.35/0.59      ! [C: nat,A: nat,B: nat] :
% 0.35/0.59        ( ( C != zero_zero_nat )
% 0.35/0.59       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.35/0.59          = ( divide_divide_nat @ A @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_mult1
% 0.35/0.59  thf(fact_3045_div__mult__mult1,axiom,
% 0.35/0.59      ! [C: int,A: int,B: int] :
% 0.35/0.59        ( ( C != zero_zero_int )
% 0.35/0.59       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.35/0.59          = ( divide_divide_int @ A @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_mult1
% 0.35/0.59  thf(fact_3046_div__mult__mult1,axiom,
% 0.35/0.59      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.35/0.59        ( ( C != zero_z3403309356797280102nteger )
% 0.35/0.59       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.35/0.59          = ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_mult1
% 0.35/0.59  thf(fact_3047_max__number__of_I1_J,axiom,
% 0.35/0.59      ! [U: num,V: num] :
% 0.35/0.59        ( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.35/0.59         => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.35/0.59            = ( numera1916890842035813515d_enat @ V ) ) )
% 0.35/0.59        & ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.35/0.59         => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.35/0.59            = ( numera1916890842035813515d_enat @ U ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_number_of(1)
% 0.35/0.59  thf(fact_3048_max__number__of_I1_J,axiom,
% 0.35/0.59      ! [U: num,V: num] :
% 0.35/0.59        ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.35/0.59         => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.35/0.59            = ( numeral_numeral_real @ V ) ) )
% 0.35/0.59        & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.35/0.59         => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.35/0.59            = ( numeral_numeral_real @ U ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_number_of(1)
% 0.35/0.59  thf(fact_3049_max__number__of_I1_J,axiom,
% 0.35/0.59      ! [U: num,V: num] :
% 0.35/0.59        ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.35/0.59         => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.35/0.59            = ( numeral_numeral_rat @ V ) ) )
% 0.35/0.59        & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.35/0.59         => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.35/0.59            = ( numeral_numeral_rat @ U ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_number_of(1)
% 0.35/0.59  thf(fact_3050_max__number__of_I1_J,axiom,
% 0.35/0.59      ! [U: num,V: num] :
% 0.35/0.59        ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.35/0.59         => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.35/0.59            = ( numeral_numeral_nat @ V ) ) )
% 0.35/0.59        & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.35/0.59         => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.35/0.59            = ( numeral_numeral_nat @ U ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_number_of(1)
% 0.35/0.59  thf(fact_3051_max__number__of_I1_J,axiom,
% 0.35/0.59      ! [U: num,V: num] :
% 0.35/0.59        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.59         => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.59            = ( numeral_numeral_int @ V ) ) )
% 0.35/0.59        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.59         => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.59            = ( numeral_numeral_int @ U ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_number_of(1)
% 0.35/0.59  thf(fact_3052_max__0__1_I3_J,axiom,
% 0.35/0.59      ! [X: num] :
% 0.35/0.59        ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X ) )
% 0.35/0.59        = ( numera1916890842035813515d_enat @ X ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_0_1(3)
% 0.35/0.59  thf(fact_3053_max__0__1_I3_J,axiom,
% 0.35/0.59      ! [X: num] :
% 0.35/0.59        ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X ) )
% 0.35/0.59        = ( numeral_numeral_real @ X ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_0_1(3)
% 0.35/0.59  thf(fact_3054_max__0__1_I3_J,axiom,
% 0.35/0.59      ! [X: num] :
% 0.35/0.59        ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
% 0.35/0.59        = ( numeral_numeral_rat @ X ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_0_1(3)
% 0.35/0.59  thf(fact_3055_max__0__1_I3_J,axiom,
% 0.35/0.59      ! [X: num] :
% 0.35/0.59        ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
% 0.35/0.59        = ( numeral_numeral_nat @ X ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_0_1(3)
% 0.35/0.59  thf(fact_3056_max__0__1_I3_J,axiom,
% 0.35/0.59      ! [X: num] :
% 0.35/0.59        ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
% 0.35/0.59        = ( numeral_numeral_int @ X ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_0_1(3)
% 0.35/0.59  thf(fact_3057_max__0__1_I4_J,axiom,
% 0.35/0.59      ! [X: num] :
% 0.35/0.59        ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ zero_z5237406670263579293d_enat )
% 0.35/0.59        = ( numera1916890842035813515d_enat @ X ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_0_1(4)
% 0.35/0.59  thf(fact_3058_max__0__1_I4_J,axiom,
% 0.35/0.59      ! [X: num] :
% 0.35/0.59        ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ zero_zero_real )
% 0.35/0.59        = ( numeral_numeral_real @ X ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_0_1(4)
% 0.35/0.59  thf(fact_3059_max__0__1_I4_J,axiom,
% 0.35/0.59      ! [X: num] :
% 0.35/0.59        ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
% 0.35/0.59        = ( numeral_numeral_rat @ X ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_0_1(4)
% 0.35/0.59  thf(fact_3060_max__0__1_I4_J,axiom,
% 0.35/0.59      ! [X: num] :
% 0.35/0.59        ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
% 0.35/0.59        = ( numeral_numeral_nat @ X ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_0_1(4)
% 0.35/0.59  thf(fact_3061_max__0__1_I4_J,axiom,
% 0.35/0.59      ! [X: num] :
% 0.35/0.59        ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
% 0.35/0.59        = ( numeral_numeral_int @ X ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_0_1(4)
% 0.35/0.59  thf(fact_3062_max__0__1_I2_J,axiom,
% 0.35/0.59      ( ( ord_max_real @ one_one_real @ zero_zero_real )
% 0.35/0.59      = one_one_real ) ).
% 0.35/0.59  
% 0.35/0.59  % max_0_1(2)
% 0.35/0.59  thf(fact_3063_max__0__1_I2_J,axiom,
% 0.35/0.59      ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
% 0.35/0.59      = one_one_rat ) ).
% 0.35/0.59  
% 0.35/0.59  % max_0_1(2)
% 0.35/0.59  thf(fact_3064_max__0__1_I2_J,axiom,
% 0.35/0.59      ( ( ord_max_int @ one_one_int @ zero_zero_int )
% 0.35/0.59      = one_one_int ) ).
% 0.35/0.59  
% 0.35/0.59  % max_0_1(2)
% 0.35/0.59  thf(fact_3065_max__0__1_I2_J,axiom,
% 0.35/0.59      ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
% 0.35/0.59      = one_one_nat ) ).
% 0.35/0.59  
% 0.35/0.59  % max_0_1(2)
% 0.35/0.59  thf(fact_3066_max__0__1_I1_J,axiom,
% 0.35/0.59      ( ( ord_max_real @ zero_zero_real @ one_one_real )
% 0.35/0.59      = one_one_real ) ).
% 0.35/0.59  
% 0.35/0.59  % max_0_1(1)
% 0.35/0.59  thf(fact_3067_max__0__1_I1_J,axiom,
% 0.35/0.59      ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
% 0.35/0.59      = one_one_rat ) ).
% 0.35/0.59  
% 0.35/0.59  % max_0_1(1)
% 0.35/0.59  thf(fact_3068_max__0__1_I1_J,axiom,
% 0.35/0.59      ( ( ord_max_int @ zero_zero_int @ one_one_int )
% 0.35/0.59      = one_one_int ) ).
% 0.35/0.59  
% 0.35/0.59  % max_0_1(1)
% 0.35/0.59  thf(fact_3069_max__0__1_I1_J,axiom,
% 0.35/0.59      ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
% 0.35/0.59      = one_one_nat ) ).
% 0.35/0.59  
% 0.35/0.59  % max_0_1(1)
% 0.35/0.59  thf(fact_3070_div__by__Suc__0,axiom,
% 0.35/0.59      ! [M: nat] :
% 0.35/0.59        ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.35/0.59        = M ) ).
% 0.35/0.59  
% 0.35/0.59  % div_by_Suc_0
% 0.35/0.59  thf(fact_3071_div__less,axiom,
% 0.35/0.59      ! [M: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_nat @ M @ N )
% 0.35/0.59       => ( ( divide_divide_nat @ M @ N )
% 0.35/0.59          = zero_zero_nat ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_less
% 0.35/0.59  thf(fact_3072_option_Ocollapse,axiom,
% 0.35/0.59      ! [Option: option_nat] :
% 0.35/0.59        ( ( Option != none_nat )
% 0.35/0.59       => ( ( some_nat @ ( the_nat2 @ Option ) )
% 0.35/0.59          = Option ) ) ).
% 0.35/0.59  
% 0.35/0.59  % option.collapse
% 0.35/0.59  thf(fact_3073_option_Ocollapse,axiom,
% 0.35/0.59      ! [Option: option4927543243414619207at_nat] :
% 0.35/0.59        ( ( Option != none_P5556105721700978146at_nat )
% 0.35/0.59       => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
% 0.35/0.59          = Option ) ) ).
% 0.35/0.59  
% 0.35/0.59  % option.collapse
% 0.35/0.59  thf(fact_3074_option_Ocollapse,axiom,
% 0.35/0.59      ! [Option: option_num] :
% 0.35/0.59        ( ( Option != none_num )
% 0.35/0.59       => ( ( some_num @ ( the_num @ Option ) )
% 0.35/0.59          = Option ) ) ).
% 0.35/0.59  
% 0.35/0.59  % option.collapse
% 0.35/0.59  thf(fact_3075_div__mult__self4,axiom,
% 0.35/0.59      ! [B: nat,C: nat,A: nat] :
% 0.35/0.59        ( ( B != zero_zero_nat )
% 0.35/0.59       => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 0.35/0.59          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_self4
% 0.35/0.59  thf(fact_3076_div__mult__self4,axiom,
% 0.35/0.59      ! [B: int,C: int,A: int] :
% 0.35/0.59        ( ( B != zero_zero_int )
% 0.35/0.59       => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 0.35/0.59          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_self4
% 0.35/0.59  thf(fact_3077_div__mult__self4,axiom,
% 0.35/0.59      ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.35/0.59        ( ( B != zero_z3403309356797280102nteger )
% 0.35/0.59       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
% 0.35/0.59          = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_self4
% 0.35/0.59  thf(fact_3078_div__mult__self3,axiom,
% 0.35/0.59      ! [B: nat,C: nat,A: nat] :
% 0.35/0.59        ( ( B != zero_zero_nat )
% 0.35/0.59       => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 0.35/0.59          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_self3
% 0.35/0.59  thf(fact_3079_div__mult__self3,axiom,
% 0.35/0.59      ! [B: int,C: int,A: int] :
% 0.35/0.59        ( ( B != zero_zero_int )
% 0.35/0.59       => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 0.35/0.59          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_self3
% 0.35/0.59  thf(fact_3080_div__mult__self3,axiom,
% 0.35/0.59      ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.35/0.59        ( ( B != zero_z3403309356797280102nteger )
% 0.35/0.59       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
% 0.35/0.59          = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_self3
% 0.35/0.59  thf(fact_3081_div__mult__self2,axiom,
% 0.35/0.59      ! [B: nat,A: nat,C: nat] :
% 0.35/0.59        ( ( B != zero_zero_nat )
% 0.35/0.59       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 0.35/0.59          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_self2
% 0.35/0.59  thf(fact_3082_div__mult__self2,axiom,
% 0.35/0.59      ! [B: int,A: int,C: int] :
% 0.35/0.59        ( ( B != zero_zero_int )
% 0.35/0.59       => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 0.35/0.59          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_self2
% 0.35/0.59  thf(fact_3083_div__mult__self2,axiom,
% 0.35/0.59      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.35/0.59        ( ( B != zero_z3403309356797280102nteger )
% 0.35/0.59       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
% 0.35/0.59          = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_self2
% 0.35/0.59  thf(fact_3084_div__mult__self1,axiom,
% 0.35/0.59      ! [B: nat,A: nat,C: nat] :
% 0.35/0.59        ( ( B != zero_zero_nat )
% 0.35/0.59       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 0.35/0.59          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_self1
% 0.35/0.59  thf(fact_3085_div__mult__self1,axiom,
% 0.35/0.59      ! [B: int,A: int,C: int] :
% 0.35/0.59        ( ( B != zero_zero_int )
% 0.35/0.59       => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 0.35/0.59          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_self1
% 0.35/0.59  thf(fact_3086_div__mult__self1,axiom,
% 0.35/0.59      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.35/0.59        ( ( B != zero_z3403309356797280102nteger )
% 0.35/0.59       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
% 0.35/0.59          = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_self1
% 0.35/0.59  thf(fact_3087_div__mult__self1__is__m,axiom,
% 0.35/0.59      ! [N: nat,M: nat] :
% 0.35/0.59        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59       => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
% 0.35/0.59          = M ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_mult_self1_is_m
% 0.35/0.59  thf(fact_3088_max__absorb2,axiom,
% 0.35/0.59      ! [X: set_int,Y: set_int] :
% 0.35/0.59        ( ( ord_less_eq_set_int @ X @ Y )
% 0.35/0.59       => ( ( ord_max_set_int @ X @ Y )
% 0.35/0.59          = Y ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_absorb2
% 0.35/0.59  thf(fact_3089_max__absorb2,axiom,
% 0.35/0.59      ! [X: rat,Y: rat] :
% 0.35/0.59        ( ( ord_less_eq_rat @ X @ Y )
% 0.35/0.59       => ( ( ord_max_rat @ X @ Y )
% 0.35/0.59          = Y ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_absorb2
% 0.35/0.59  thf(fact_3090_max__absorb2,axiom,
% 0.35/0.59      ! [X: num,Y: num] :
% 0.35/0.59        ( ( ord_less_eq_num @ X @ Y )
% 0.35/0.59       => ( ( ord_max_num @ X @ Y )
% 0.35/0.59          = Y ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_absorb2
% 0.35/0.59  thf(fact_3091_max__absorb2,axiom,
% 0.35/0.59      ! [X: nat,Y: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ X @ Y )
% 0.35/0.59       => ( ( ord_max_nat @ X @ Y )
% 0.35/0.59          = Y ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_absorb2
% 0.35/0.59  thf(fact_3092_max__absorb2,axiom,
% 0.35/0.59      ! [X: int,Y: int] :
% 0.35/0.59        ( ( ord_less_eq_int @ X @ Y )
% 0.35/0.59       => ( ( ord_max_int @ X @ Y )
% 0.35/0.59          = Y ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_absorb2
% 0.35/0.59  thf(fact_3093_max__absorb1,axiom,
% 0.35/0.59      ! [Y: set_int,X: set_int] :
% 0.35/0.59        ( ( ord_less_eq_set_int @ Y @ X )
% 0.35/0.59       => ( ( ord_max_set_int @ X @ Y )
% 0.35/0.59          = X ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_absorb1
% 0.35/0.59  thf(fact_3094_max__absorb1,axiom,
% 0.35/0.59      ! [Y: rat,X: rat] :
% 0.35/0.59        ( ( ord_less_eq_rat @ Y @ X )
% 0.35/0.59       => ( ( ord_max_rat @ X @ Y )
% 0.35/0.59          = X ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_absorb1
% 0.35/0.59  thf(fact_3095_max__absorb1,axiom,
% 0.35/0.59      ! [Y: num,X: num] :
% 0.35/0.59        ( ( ord_less_eq_num @ Y @ X )
% 0.35/0.59       => ( ( ord_max_num @ X @ Y )
% 0.35/0.59          = X ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_absorb1
% 0.35/0.59  thf(fact_3096_max__absorb1,axiom,
% 0.35/0.59      ! [Y: nat,X: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ Y @ X )
% 0.35/0.59       => ( ( ord_max_nat @ X @ Y )
% 0.35/0.59          = X ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_absorb1
% 0.35/0.59  thf(fact_3097_max__absorb1,axiom,
% 0.35/0.59      ! [Y: int,X: int] :
% 0.35/0.59        ( ( ord_less_eq_int @ Y @ X )
% 0.35/0.59       => ( ( ord_max_int @ X @ Y )
% 0.35/0.59          = X ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_absorb1
% 0.35/0.59  thf(fact_3098_max__def,axiom,
% 0.35/0.59      ( ord_max_set_int
% 0.35/0.59      = ( ^ [A4: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_def
% 0.35/0.59  thf(fact_3099_max__def,axiom,
% 0.35/0.59      ( ord_max_rat
% 0.35/0.59      = ( ^ [A4: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_def
% 0.35/0.59  thf(fact_3100_max__def,axiom,
% 0.35/0.59      ( ord_max_num
% 0.35/0.59      = ( ^ [A4: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_def
% 0.35/0.59  thf(fact_3101_max__def,axiom,
% 0.35/0.59      ( ord_max_nat
% 0.35/0.59      = ( ^ [A4: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_def
% 0.35/0.59  thf(fact_3102_max__def,axiom,
% 0.35/0.59      ( ord_max_int
% 0.35/0.59      = ( ^ [A4: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_def
% 0.35/0.59  thf(fact_3103_max__add__distrib__left,axiom,
% 0.35/0.59      ! [X: real,Y: real,Z2: real] :
% 0.35/0.59        ( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z2 )
% 0.35/0.59        = ( ord_max_real @ ( plus_plus_real @ X @ Z2 ) @ ( plus_plus_real @ Y @ Z2 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_add_distrib_left
% 0.35/0.59  thf(fact_3104_max__add__distrib__left,axiom,
% 0.35/0.59      ! [X: rat,Y: rat,Z2: rat] :
% 0.35/0.59        ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z2 )
% 0.35/0.59        = ( ord_max_rat @ ( plus_plus_rat @ X @ Z2 ) @ ( plus_plus_rat @ Y @ Z2 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_add_distrib_left
% 0.35/0.59  thf(fact_3105_max__add__distrib__left,axiom,
% 0.35/0.59      ! [X: int,Y: int,Z2: int] :
% 0.35/0.59        ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z2 )
% 0.35/0.59        = ( ord_max_int @ ( plus_plus_int @ X @ Z2 ) @ ( plus_plus_int @ Y @ Z2 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_add_distrib_left
% 0.35/0.59  thf(fact_3106_max__add__distrib__left,axiom,
% 0.35/0.59      ! [X: nat,Y: nat,Z2: nat] :
% 0.35/0.59        ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z2 )
% 0.35/0.59        = ( ord_max_nat @ ( plus_plus_nat @ X @ Z2 ) @ ( plus_plus_nat @ Y @ Z2 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_add_distrib_left
% 0.35/0.59  thf(fact_3107_max__add__distrib__right,axiom,
% 0.35/0.59      ! [X: real,Y: real,Z2: real] :
% 0.35/0.59        ( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z2 ) )
% 0.35/0.59        = ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z2 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_add_distrib_right
% 0.35/0.59  thf(fact_3108_max__add__distrib__right,axiom,
% 0.35/0.59      ! [X: rat,Y: rat,Z2: rat] :
% 0.35/0.59        ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z2 ) )
% 0.35/0.59        = ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z2 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_add_distrib_right
% 0.35/0.59  thf(fact_3109_max__add__distrib__right,axiom,
% 0.35/0.59      ! [X: int,Y: int,Z2: int] :
% 0.35/0.59        ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z2 ) )
% 0.35/0.59        = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z2 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_add_distrib_right
% 0.35/0.59  thf(fact_3110_max__add__distrib__right,axiom,
% 0.35/0.59      ! [X: nat,Y: nat,Z2: nat] :
% 0.35/0.59        ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z2 ) )
% 0.35/0.59        = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z2 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_add_distrib_right
% 0.35/0.59  thf(fact_3111_nat__add__max__left,axiom,
% 0.35/0.59      ! [M: nat,N: nat,Q4: nat] :
% 0.35/0.59        ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q4 )
% 0.35/0.59        = ( ord_max_nat @ ( plus_plus_nat @ M @ Q4 ) @ ( plus_plus_nat @ N @ Q4 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % nat_add_max_left
% 0.35/0.59  thf(fact_3112_nat__add__max__right,axiom,
% 0.35/0.59      ! [M: nat,N: nat,Q4: nat] :
% 0.35/0.59        ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q4 ) )
% 0.35/0.59        = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q4 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % nat_add_max_right
% 0.35/0.59  thf(fact_3113_nat__mult__max__left,axiom,
% 0.35/0.59      ! [M: nat,N: nat,Q4: nat] :
% 0.35/0.59        ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q4 )
% 0.35/0.59        = ( ord_max_nat @ ( times_times_nat @ M @ Q4 ) @ ( times_times_nat @ N @ Q4 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % nat_mult_max_left
% 0.35/0.59  thf(fact_3114_nat__mult__max__right,axiom,
% 0.35/0.59      ! [M: nat,N: nat,Q4: nat] :
% 0.35/0.59        ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q4 ) )
% 0.35/0.59        = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q4 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % nat_mult_max_right
% 0.35/0.59  thf(fact_3115_option_Osel,axiom,
% 0.35/0.59      ! [X22: nat] :
% 0.35/0.59        ( ( the_nat2 @ ( some_nat @ X22 ) )
% 0.35/0.59        = X22 ) ).
% 0.35/0.59  
% 0.35/0.59  % option.sel
% 0.35/0.59  thf(fact_3116_option_Osel,axiom,
% 0.35/0.59      ! [X22: product_prod_nat_nat] :
% 0.35/0.59        ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 0.35/0.59        = X22 ) ).
% 0.35/0.59  
% 0.35/0.59  % option.sel
% 0.35/0.59  thf(fact_3117_option_Osel,axiom,
% 0.35/0.59      ! [X22: num] :
% 0.35/0.59        ( ( the_num @ ( some_num @ X22 ) )
% 0.35/0.59        = X22 ) ).
% 0.35/0.59  
% 0.35/0.59  % option.sel
% 0.35/0.59  thf(fact_3118_option_Oexpand,axiom,
% 0.35/0.59      ! [Option: option_nat,Option2: option_nat] :
% 0.35/0.59        ( ( ( Option = none_nat )
% 0.35/0.59          = ( Option2 = none_nat ) )
% 0.35/0.59       => ( ( ( Option != none_nat )
% 0.35/0.59           => ( ( Option2 != none_nat )
% 0.35/0.59             => ( ( the_nat2 @ Option )
% 0.35/0.59                = ( the_nat2 @ Option2 ) ) ) )
% 0.35/0.59         => ( Option = Option2 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % option.expand
% 0.35/0.59  thf(fact_3119_option_Oexpand,axiom,
% 0.35/0.59      ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
% 0.35/0.59        ( ( ( Option = none_P5556105721700978146at_nat )
% 0.35/0.59          = ( Option2 = none_P5556105721700978146at_nat ) )
% 0.35/0.59       => ( ( ( Option != none_P5556105721700978146at_nat )
% 0.35/0.59           => ( ( Option2 != none_P5556105721700978146at_nat )
% 0.35/0.59             => ( ( the_Pr8591224930841456533at_nat @ Option )
% 0.35/0.59                = ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
% 0.35/0.59         => ( Option = Option2 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % option.expand
% 0.35/0.59  thf(fact_3120_option_Oexpand,axiom,
% 0.35/0.59      ! [Option: option_num,Option2: option_num] :
% 0.35/0.59        ( ( ( Option = none_num )
% 0.35/0.59          = ( Option2 = none_num ) )
% 0.35/0.59       => ( ( ( Option != none_num )
% 0.35/0.59           => ( ( Option2 != none_num )
% 0.35/0.59             => ( ( the_num @ Option )
% 0.35/0.59                = ( the_num @ Option2 ) ) ) )
% 0.35/0.59         => ( Option = Option2 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % option.expand
% 0.35/0.59  thf(fact_3121_option_Oexhaust__sel,axiom,
% 0.35/0.59      ! [Option: option_nat] :
% 0.35/0.59        ( ( Option != none_nat )
% 0.35/0.59       => ( Option
% 0.35/0.59          = ( some_nat @ ( the_nat2 @ Option ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % option.exhaust_sel
% 0.35/0.59  thf(fact_3122_option_Oexhaust__sel,axiom,
% 0.35/0.59      ! [Option: option4927543243414619207at_nat] :
% 0.35/0.59        ( ( Option != none_P5556105721700978146at_nat )
% 0.35/0.59       => ( Option
% 0.35/0.59          = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % option.exhaust_sel
% 0.35/0.59  thf(fact_3123_option_Oexhaust__sel,axiom,
% 0.35/0.59      ! [Option: option_num] :
% 0.35/0.59        ( ( Option != none_num )
% 0.35/0.59       => ( Option
% 0.35/0.59          = ( some_num @ ( the_num @ Option ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % option.exhaust_sel
% 0.35/0.59  thf(fact_3124_div__le__dividend,axiom,
% 0.35/0.59      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 0.35/0.59  
% 0.35/0.59  % div_le_dividend
% 0.35/0.59  thf(fact_3125_div__le__mono,axiom,
% 0.35/0.59      ! [M: nat,N: nat,K: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.59       => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_le_mono
% 0.35/0.59  thf(fact_3126_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 0.35/0.59      ! [M: nat,N: nat] :
% 0.35/0.59        ( ( ( divide_divide_nat @ M @ N )
% 0.35/0.59          = zero_zero_nat )
% 0.35/0.59        = ( ( ord_less_nat @ M @ N )
% 0.35/0.59          | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % Euclidean_Division.div_eq_0_iff
% 0.35/0.59  thf(fact_3127_Suc__div__le__mono,axiom,
% 0.35/0.59      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% 0.35/0.59  
% 0.35/0.59  % Suc_div_le_mono
% 0.35/0.59  thf(fact_3128_less__mult__imp__div__less,axiom,
% 0.35/0.59      ! [M: nat,I: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
% 0.35/0.59       => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% 0.35/0.59  
% 0.35/0.59  % less_mult_imp_div_less
% 0.35/0.59  thf(fact_3129_div__times__less__eq__dividend,axiom,
% 0.35/0.59      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% 0.35/0.59  
% 0.35/0.59  % div_times_less_eq_dividend
% 0.35/0.59  thf(fact_3130_times__div__less__eq__dividend,axiom,
% 0.35/0.59      ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% 0.35/0.59  
% 0.35/0.59  % times_div_less_eq_dividend
% 0.35/0.59  thf(fact_3131_div__add__self2,axiom,
% 0.35/0.59      ! [B: nat,A: nat] :
% 0.35/0.59        ( ( B != zero_zero_nat )
% 0.35/0.59       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.35/0.59          = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_add_self2
% 0.35/0.59  thf(fact_3132_div__add__self2,axiom,
% 0.35/0.59      ! [B: int,A: int] :
% 0.35/0.59        ( ( B != zero_zero_int )
% 0.35/0.59       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.35/0.59          = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_add_self2
% 0.35/0.59  thf(fact_3133_div__add__self2,axiom,
% 0.35/0.59      ! [B: code_integer,A: code_integer] :
% 0.35/0.59        ( ( B != zero_z3403309356797280102nteger )
% 0.35/0.59       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
% 0.35/0.59          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_add_self2
% 0.35/0.59  thf(fact_3134_div__add__self1,axiom,
% 0.35/0.59      ! [B: nat,A: nat] :
% 0.35/0.59        ( ( B != zero_zero_nat )
% 0.35/0.59       => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.35/0.59          = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_add_self1
% 0.35/0.59  thf(fact_3135_div__add__self1,axiom,
% 0.35/0.59      ! [B: int,A: int] :
% 0.35/0.59        ( ( B != zero_zero_int )
% 0.35/0.59       => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.35/0.59          = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_add_self1
% 0.35/0.59  thf(fact_3136_div__add__self1,axiom,
% 0.35/0.59      ! [B: code_integer,A: code_integer] :
% 0.35/0.59        ( ( B != zero_z3403309356797280102nteger )
% 0.35/0.59       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
% 0.35/0.59          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_add_self1
% 0.35/0.59  thf(fact_3137_div__greater__zero__iff,axiom,
% 0.35/0.59      ! [M: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
% 0.35/0.59        = ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.59          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_greater_zero_iff
% 0.35/0.59  thf(fact_3138_div__le__mono2,axiom,
% 0.35/0.59      ! [M: nat,N: nat,K: nat] :
% 0.35/0.59        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.59       => ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.59         => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_le_mono2
% 0.35/0.59  thf(fact_3139_div__less__iff__less__mult,axiom,
% 0.35/0.59      ! [Q4: nat,M: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_nat @ zero_zero_nat @ Q4 )
% 0.35/0.59       => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q4 ) @ N )
% 0.35/0.59          = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q4 ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_less_iff_less_mult
% 0.35/0.59  thf(fact_3140_div__eq__dividend__iff,axiom,
% 0.35/0.59      ! [M: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.59       => ( ( ( divide_divide_nat @ M @ N )
% 0.35/0.59            = M )
% 0.35/0.59          = ( N = one_one_nat ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_eq_dividend_iff
% 0.35/0.59  thf(fact_3141_div__less__dividend,axiom,
% 0.35/0.59      ! [N: nat,M: nat] :
% 0.35/0.59        ( ( ord_less_nat @ one_one_nat @ N )
% 0.35/0.59       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.59         => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_less_dividend
% 0.35/0.59  thf(fact_3142_div__nat__eqI,axiom,
% 0.35/0.59      ! [N: nat,Q4: nat,M: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
% 0.35/0.59       => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
% 0.35/0.59         => ( ( divide_divide_nat @ M @ N )
% 0.35/0.59            = Q4 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_nat_eqI
% 0.35/0.59  thf(fact_3143_less__eq__div__iff__mult__less__eq,axiom,
% 0.35/0.59      ! [Q4: nat,M: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_nat @ zero_zero_nat @ Q4 )
% 0.35/0.59       => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q4 ) )
% 0.35/0.59          = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q4 ) @ N ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % less_eq_div_iff_mult_less_eq
% 0.35/0.59  thf(fact_3144_dividend__less__times__div,axiom,
% 0.35/0.59      ! [N: nat,M: nat] :
% 0.35/0.59        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59       => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % dividend_less_times_div
% 0.35/0.59  thf(fact_3145_dividend__less__div__times,axiom,
% 0.35/0.59      ! [N: nat,M: nat] :
% 0.35/0.59        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59       => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % dividend_less_div_times
% 0.35/0.59  thf(fact_3146_split__div,axiom,
% 0.35/0.59      ! [P: nat > $o,M: nat,N: nat] :
% 0.35/0.59        ( ( P @ ( divide_divide_nat @ M @ N ) )
% 0.35/0.59        = ( ( ( N = zero_zero_nat )
% 0.35/0.59           => ( P @ zero_zero_nat ) )
% 0.35/0.59          & ( ( N != zero_zero_nat )
% 0.35/0.59           => ! [I4: nat,J3: nat] :
% 0.35/0.59                ( ( ord_less_nat @ J3 @ N )
% 0.35/0.59               => ( ( M
% 0.35/0.59                    = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 0.35/0.59                 => ( P @ I4 ) ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % split_div
% 0.35/0.59  thf(fact_3147_split__div_H,axiom,
% 0.35/0.59      ! [P: nat > $o,M: nat,N: nat] :
% 0.35/0.59        ( ( P @ ( divide_divide_nat @ M @ N ) )
% 0.35/0.59        = ( ( ( N = zero_zero_nat )
% 0.35/0.59            & ( P @ zero_zero_nat ) )
% 0.35/0.59          | ? [Q5: nat] :
% 0.35/0.59              ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q5 ) @ M )
% 0.35/0.59              & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q5 ) ) )
% 0.35/0.59              & ( P @ Q5 ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % split_div'
% 0.35/0.59  thf(fact_3148_div__2__gt__zero,axiom,
% 0.35/0.59      ! [N: nat] :
% 0.35/0.59        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.59       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_2_gt_zero
% 0.35/0.59  thf(fact_3149_Suc__n__div__2__gt__zero,axiom,
% 0.35/0.59      ! [N: nat] :
% 0.35/0.59        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % Suc_n_div_2_gt_zero
% 0.35/0.59  thf(fact_3150_arith__geo__mean,axiom,
% 0.35/0.59      ! [U: real,X: real,Y: real] :
% 0.35/0.59        ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.59          = ( times_times_real @ X @ Y ) )
% 0.35/0.59       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.59         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.35/0.59           => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % arith_geo_mean
% 0.35/0.59  thf(fact_3151_arith__geo__mean,axiom,
% 0.35/0.59      ! [U: rat,X: rat,Y: rat] :
% 0.35/0.59        ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.59          = ( times_times_rat @ X @ Y ) )
% 0.35/0.59       => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.35/0.59         => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.35/0.59           => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % arith_geo_mean
% 0.35/0.59  thf(fact_3152_mintlistlength,axiom,
% 0.35/0.59      ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.35/0.59        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 0.35/0.59       => ( ( Mi != Ma )
% 0.35/0.59         => ( ( ord_less_nat @ Mi @ Ma )
% 0.35/0.59            & ? [M5: nat] :
% 0.35/0.59                ( ( ( some_nat @ M5 )
% 0.35/0.59                  = ( vEBT_vebt_mint @ Summary ) )
% 0.35/0.59                & ( ord_less_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % mintlistlength
% 0.35/0.59  thf(fact_3153_sum__squares__bound,axiom,
% 0.35/0.59      ! [X: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % sum_squares_bound
% 0.35/0.59  thf(fact_3154_sum__squares__bound,axiom,
% 0.35/0.59      ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % sum_squares_bound
% 0.35/0.59  thf(fact_3155_mul__def,axiom,
% 0.35/0.59      ( vEBT_VEBT_mul
% 0.35/0.59      = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).
% 0.35/0.59  
% 0.35/0.59  % mul_def
% 0.35/0.59  thf(fact_3156_nat__induct2,axiom,
% 0.35/0.59      ! [P: nat > $o,N: nat] :
% 0.35/0.59        ( ( P @ zero_zero_nat )
% 0.35/0.59       => ( ( P @ one_one_nat )
% 0.35/0.59         => ( ! [N3: nat] :
% 0.35/0.59                ( ( P @ N3 )
% 0.35/0.59               => ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.59           => ( P @ N ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % nat_induct2
% 0.35/0.59  thf(fact_3157_set__bit__0,axiom,
% 0.35/0.59      ! [A: nat] :
% 0.35/0.59        ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
% 0.35/0.59        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % set_bit_0
% 0.35/0.59  thf(fact_3158_set__bit__0,axiom,
% 0.35/0.59      ! [A: code_integer] :
% 0.35/0.59        ( ( bit_se2793503036327961859nteger @ zero_zero_nat @ A )
% 0.35/0.59        = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % set_bit_0
% 0.35/0.59  thf(fact_3159_set__bit__0,axiom,
% 0.35/0.59      ! [A: int] :
% 0.35/0.59        ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
% 0.35/0.59        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % set_bit_0
% 0.35/0.59  thf(fact_3160_invar__vebt_Osimps,axiom,
% 0.35/0.59      ( vEBT_invar_vebt
% 0.35/0.59      = ( ^ [A1: vEBT_VEBT,A22: nat] :
% 0.35/0.59            ( ( ? [A4: $o,B4: $o] :
% 0.35/0.59                  ( A1
% 0.35/0.59                  = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.35/0.59              & ( A22
% 0.35/0.59                = ( suc @ zero_zero_nat ) ) )
% 0.35/0.59            | ? [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT] :
% 0.35/0.59                ( ( A1
% 0.35/0.59                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList2 @ Summary2 ) )
% 0.35/0.59                & ! [X3: vEBT_VEBT] :
% 0.35/0.59                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.35/0.59                   => ( vEBT_invar_vebt @ X3 @ N2 ) )
% 0.35/0.59                & ( vEBT_invar_vebt @ Summary2 @ N2 )
% 0.35/0.59                & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.35/0.59                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.35/0.59                & ( A22
% 0.35/0.59                  = ( plus_plus_nat @ N2 @ N2 ) )
% 0.35/0.59                & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 0.35/0.59                & ! [X3: vEBT_VEBT] :
% 0.35/0.59                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.35/0.59                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 0.35/0.59            | ? [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT] :
% 0.35/0.59                ( ( A1
% 0.35/0.59                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList2 @ Summary2 ) )
% 0.35/0.59                & ! [X3: vEBT_VEBT] :
% 0.35/0.59                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.35/0.59                   => ( vEBT_invar_vebt @ X3 @ N2 ) )
% 0.35/0.59                & ( vEBT_invar_vebt @ Summary2 @ ( suc @ N2 ) )
% 0.35/0.59                & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.35/0.59                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 0.35/0.59                & ( A22
% 0.35/0.59                  = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
% 0.35/0.59                & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 0.35/0.59                & ! [X3: vEBT_VEBT] :
% 0.35/0.59                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.35/0.59                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 0.35/0.59            | ? [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
% 0.35/0.59                ( ( A1
% 0.35/0.59                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ A22 @ TreeList2 @ Summary2 ) )
% 0.35/0.59                & ! [X3: vEBT_VEBT] :
% 0.35/0.59                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.35/0.59                   => ( vEBT_invar_vebt @ X3 @ N2 ) )
% 0.35/0.59                & ( vEBT_invar_vebt @ Summary2 @ N2 )
% 0.35/0.59                & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.35/0.59                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.35/0.59                & ( A22
% 0.35/0.59                  = ( plus_plus_nat @ N2 @ N2 ) )
% 0.35/0.59                & ! [I4: nat] :
% 0.35/0.59                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.35/0.59                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
% 0.35/0.59                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.35/0.59                & ( ( Mi2 = Ma2 )
% 0.35/0.59                 => ! [X3: vEBT_VEBT] :
% 0.35/0.59                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.35/0.59                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 0.35/0.59                & ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.35/0.59                & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 0.35/0.59                & ( ( Mi2 != Ma2 )
% 0.35/0.59                 => ! [I4: nat] :
% 0.35/0.59                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.35/0.59                     => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
% 0.35/0.59                            = I4 )
% 0.35/0.59                         => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
% 0.35/0.59                        & ! [X3: nat] :
% 0.35/0.59                            ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
% 0.35/0.59                                = I4 )
% 0.35/0.59                              & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
% 0.35/0.59                           => ( ( ord_less_nat @ Mi2 @ X3 )
% 0.35/0.59                              & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) )
% 0.35/0.59            | ? [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
% 0.35/0.59                ( ( A1
% 0.35/0.59                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ A22 @ TreeList2 @ Summary2 ) )
% 0.35/0.59                & ! [X3: vEBT_VEBT] :
% 0.35/0.59                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.35/0.59                   => ( vEBT_invar_vebt @ X3 @ N2 ) )
% 0.35/0.59                & ( vEBT_invar_vebt @ Summary2 @ ( suc @ N2 ) )
% 0.35/0.59                & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.35/0.59                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 0.35/0.59                & ( A22
% 0.35/0.59                  = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
% 0.35/0.59                & ! [I4: nat] :
% 0.35/0.59                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 0.35/0.59                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
% 0.35/0.59                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.35/0.59                & ( ( Mi2 = Ma2 )
% 0.35/0.59                 => ! [X3: vEBT_VEBT] :
% 0.35/0.59                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.35/0.59                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 0.35/0.59                & ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.35/0.59                & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 0.35/0.59                & ( ( Mi2 != Ma2 )
% 0.35/0.59                 => ! [I4: nat] :
% 0.35/0.59                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 0.35/0.59                     => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
% 0.35/0.59                            = I4 )
% 0.35/0.59                         => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
% 0.35/0.59                        & ! [X3: nat] :
% 0.35/0.59                            ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
% 0.35/0.59                                = I4 )
% 0.35/0.59                              & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
% 0.35/0.59                           => ( ( ord_less_nat @ Mi2 @ X3 )
% 0.35/0.59                              & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % invar_vebt.simps
% 0.35/0.59  thf(fact_3161_Leaf__0__not,axiom,
% 0.35/0.59      ! [A: $o,B: $o] :
% 0.35/0.59        ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).
% 0.35/0.59  
% 0.35/0.59  % Leaf_0_not
% 0.35/0.59  thf(fact_3162_deg__1__Leafy,axiom,
% 0.35/0.59      ! [T: vEBT_VEBT,N: nat] :
% 0.35/0.59        ( ( vEBT_invar_vebt @ T @ N )
% 0.35/0.59       => ( ( N = one_one_nat )
% 0.35/0.59         => ? [A3: $o,B3: $o] :
% 0.35/0.59              ( T
% 0.35/0.59              = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % deg_1_Leafy
% 0.35/0.59  thf(fact_3163_deg__1__Leaf,axiom,
% 0.35/0.59      ! [T: vEBT_VEBT] :
% 0.35/0.59        ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 0.35/0.59       => ? [A3: $o,B3: $o] :
% 0.35/0.59            ( T
% 0.35/0.59            = ( vEBT_Leaf @ A3 @ B3 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % deg_1_Leaf
% 0.35/0.59  thf(fact_3164_deg1Leaf,axiom,
% 0.35/0.59      ! [T: vEBT_VEBT] :
% 0.35/0.59        ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 0.35/0.59        = ( ? [A4: $o,B4: $o] :
% 0.35/0.59              ( T
% 0.35/0.59              = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % deg1Leaf
% 0.35/0.59  thf(fact_3165_mul__shift,axiom,
% 0.35/0.59      ! [X: nat,Y: nat,Z2: nat] :
% 0.35/0.59        ( ( ( times_times_nat @ X @ Y )
% 0.35/0.59          = Z2 )
% 0.35/0.59        = ( ( vEBT_VEBT_mul @ ( some_nat @ X ) @ ( some_nat @ Y ) )
% 0.35/0.59          = ( some_nat @ Z2 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % mul_shift
% 0.35/0.59  thf(fact_3166_zle__add1__eq__le,axiom,
% 0.35/0.59      ! [W2: int,Z2: int] :
% 0.35/0.59        ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
% 0.35/0.59        = ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% 0.35/0.59  
% 0.35/0.59  % zle_add1_eq_le
% 0.35/0.59  thf(fact_3167_VEBT_Oinject_I2_J,axiom,
% 0.35/0.59      ! [X21: $o,X222: $o,Y21: $o,Y222: $o] :
% 0.35/0.59        ( ( ( vEBT_Leaf @ X21 @ X222 )
% 0.35/0.59          = ( vEBT_Leaf @ Y21 @ Y222 ) )
% 0.35/0.59        = ( ( X21 = Y21 )
% 0.35/0.59          & ( X222 = Y222 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % VEBT.inject(2)
% 0.35/0.59  thf(fact_3168_power__minus__is__div,axiom,
% 0.35/0.59      ! [B: nat,A: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ B @ A )
% 0.35/0.59       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B ) )
% 0.35/0.59          = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % power_minus_is_div
% 0.35/0.59  thf(fact_3169_diff__self,axiom,
% 0.35/0.59      ! [A: complex] :
% 0.35/0.59        ( ( minus_minus_complex @ A @ A )
% 0.35/0.59        = zero_zero_complex ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_self
% 0.35/0.59  thf(fact_3170_diff__self,axiom,
% 0.35/0.59      ! [A: real] :
% 0.35/0.59        ( ( minus_minus_real @ A @ A )
% 0.35/0.59        = zero_zero_real ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_self
% 0.35/0.59  thf(fact_3171_diff__self,axiom,
% 0.35/0.59      ! [A: rat] :
% 0.35/0.59        ( ( minus_minus_rat @ A @ A )
% 0.35/0.59        = zero_zero_rat ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_self
% 0.35/0.59  thf(fact_3172_diff__self,axiom,
% 0.35/0.59      ! [A: int] :
% 0.35/0.59        ( ( minus_minus_int @ A @ A )
% 0.35/0.59        = zero_zero_int ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_self
% 0.35/0.59  thf(fact_3173_diff__0__right,axiom,
% 0.35/0.59      ! [A: complex] :
% 0.35/0.59        ( ( minus_minus_complex @ A @ zero_zero_complex )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_0_right
% 0.35/0.59  thf(fact_3174_diff__0__right,axiom,
% 0.35/0.59      ! [A: real] :
% 0.35/0.59        ( ( minus_minus_real @ A @ zero_zero_real )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_0_right
% 0.35/0.59  thf(fact_3175_diff__0__right,axiom,
% 0.35/0.59      ! [A: rat] :
% 0.35/0.59        ( ( minus_minus_rat @ A @ zero_zero_rat )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_0_right
% 0.35/0.59  thf(fact_3176_diff__0__right,axiom,
% 0.35/0.59      ! [A: int] :
% 0.35/0.59        ( ( minus_minus_int @ A @ zero_zero_int )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_0_right
% 0.35/0.59  thf(fact_3177_zero__diff,axiom,
% 0.35/0.59      ! [A: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ zero_zero_nat @ A )
% 0.35/0.59        = zero_zero_nat ) ).
% 0.35/0.59  
% 0.35/0.59  % zero_diff
% 0.35/0.59  thf(fact_3178_diff__zero,axiom,
% 0.35/0.59      ! [A: complex] :
% 0.35/0.59        ( ( minus_minus_complex @ A @ zero_zero_complex )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_zero
% 0.35/0.59  thf(fact_3179_diff__zero,axiom,
% 0.35/0.59      ! [A: real] :
% 0.35/0.59        ( ( minus_minus_real @ A @ zero_zero_real )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_zero
% 0.35/0.59  thf(fact_3180_diff__zero,axiom,
% 0.35/0.59      ! [A: rat] :
% 0.35/0.59        ( ( minus_minus_rat @ A @ zero_zero_rat )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_zero
% 0.35/0.59  thf(fact_3181_diff__zero,axiom,
% 0.35/0.59      ! [A: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ A @ zero_zero_nat )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_zero
% 0.35/0.59  thf(fact_3182_diff__zero,axiom,
% 0.35/0.59      ! [A: int] :
% 0.35/0.59        ( ( minus_minus_int @ A @ zero_zero_int )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_zero
% 0.35/0.59  thf(fact_3183_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.35/0.59      ! [A: complex] :
% 0.35/0.59        ( ( minus_minus_complex @ A @ A )
% 0.35/0.59        = zero_zero_complex ) ).
% 0.35/0.59  
% 0.35/0.59  % cancel_comm_monoid_add_class.diff_cancel
% 0.35/0.59  thf(fact_3184_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.35/0.59      ! [A: real] :
% 0.35/0.59        ( ( minus_minus_real @ A @ A )
% 0.35/0.59        = zero_zero_real ) ).
% 0.35/0.59  
% 0.35/0.59  % cancel_comm_monoid_add_class.diff_cancel
% 0.35/0.59  thf(fact_3185_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.35/0.59      ! [A: rat] :
% 0.35/0.59        ( ( minus_minus_rat @ A @ A )
% 0.35/0.59        = zero_zero_rat ) ).
% 0.35/0.59  
% 0.35/0.59  % cancel_comm_monoid_add_class.diff_cancel
% 0.35/0.59  thf(fact_3186_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.35/0.59      ! [A: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ A @ A )
% 0.35/0.59        = zero_zero_nat ) ).
% 0.35/0.59  
% 0.35/0.59  % cancel_comm_monoid_add_class.diff_cancel
% 0.35/0.59  thf(fact_3187_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.35/0.59      ! [A: int] :
% 0.35/0.59        ( ( minus_minus_int @ A @ A )
% 0.35/0.59        = zero_zero_int ) ).
% 0.35/0.59  
% 0.35/0.59  % cancel_comm_monoid_add_class.diff_cancel
% 0.35/0.59  thf(fact_3188_add__diff__cancel__right_H,axiom,
% 0.35/0.59      ! [A: real,B: real] :
% 0.35/0.59        ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_cancel_right'
% 0.35/0.59  thf(fact_3189_add__diff__cancel__right_H,axiom,
% 0.35/0.59      ! [A: rat,B: rat] :
% 0.35/0.59        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_cancel_right'
% 0.35/0.59  thf(fact_3190_add__diff__cancel__right_H,axiom,
% 0.35/0.59      ! [A: nat,B: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_cancel_right'
% 0.35/0.59  thf(fact_3191_add__diff__cancel__right_H,axiom,
% 0.35/0.59      ! [A: int,B: int] :
% 0.35/0.59        ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_cancel_right'
% 0.35/0.59  thf(fact_3192_add__diff__cancel__right,axiom,
% 0.35/0.59      ! [A: real,C: real,B: real] :
% 0.35/0.59        ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.35/0.59        = ( minus_minus_real @ A @ B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_cancel_right
% 0.35/0.59  thf(fact_3193_add__diff__cancel__right,axiom,
% 0.35/0.59      ! [A: rat,C: rat,B: rat] :
% 0.35/0.59        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.35/0.59        = ( minus_minus_rat @ A @ B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_cancel_right
% 0.35/0.59  thf(fact_3194_add__diff__cancel__right,axiom,
% 0.35/0.59      ! [A: nat,C: nat,B: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.35/0.59        = ( minus_minus_nat @ A @ B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_cancel_right
% 0.35/0.59  thf(fact_3195_add__diff__cancel__right,axiom,
% 0.35/0.59      ! [A: int,C: int,B: int] :
% 0.35/0.59        ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.35/0.59        = ( minus_minus_int @ A @ B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_cancel_right
% 0.35/0.59  thf(fact_3196_add__diff__cancel__left_H,axiom,
% 0.35/0.59      ! [A: real,B: real] :
% 0.35/0.59        ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
% 0.35/0.59        = B ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_cancel_left'
% 0.35/0.59  thf(fact_3197_add__diff__cancel__left_H,axiom,
% 0.35/0.59      ! [A: rat,B: rat] :
% 0.35/0.59        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
% 0.35/0.59        = B ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_cancel_left'
% 0.35/0.59  thf(fact_3198_add__diff__cancel__left_H,axiom,
% 0.35/0.59      ! [A: nat,B: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 0.35/0.59        = B ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_cancel_left'
% 0.35/0.59  thf(fact_3199_add__diff__cancel__left_H,axiom,
% 0.35/0.59      ! [A: int,B: int] :
% 0.35/0.59        ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 0.35/0.59        = B ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_cancel_left'
% 0.35/0.59  thf(fact_3200_add__diff__cancel__left,axiom,
% 0.35/0.59      ! [C: real,A: real,B: real] :
% 0.35/0.59        ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.35/0.59        = ( minus_minus_real @ A @ B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_cancel_left
% 0.35/0.59  thf(fact_3201_add__diff__cancel__left,axiom,
% 0.35/0.59      ! [C: rat,A: rat,B: rat] :
% 0.35/0.59        ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.35/0.59        = ( minus_minus_rat @ A @ B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_cancel_left
% 0.35/0.59  thf(fact_3202_add__diff__cancel__left,axiom,
% 0.35/0.59      ! [C: nat,A: nat,B: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.35/0.59        = ( minus_minus_nat @ A @ B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_cancel_left
% 0.35/0.59  thf(fact_3203_add__diff__cancel__left,axiom,
% 0.35/0.59      ! [C: int,A: int,B: int] :
% 0.35/0.59        ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.35/0.59        = ( minus_minus_int @ A @ B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_cancel_left
% 0.35/0.59  thf(fact_3204_diff__add__cancel,axiom,
% 0.35/0.59      ! [A: real,B: real] :
% 0.35/0.59        ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_add_cancel
% 0.35/0.59  thf(fact_3205_diff__add__cancel,axiom,
% 0.35/0.59      ! [A: rat,B: rat] :
% 0.35/0.59        ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_add_cancel
% 0.35/0.59  thf(fact_3206_diff__add__cancel,axiom,
% 0.35/0.59      ! [A: int,B: int] :
% 0.35/0.59        ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_add_cancel
% 0.35/0.59  thf(fact_3207_add__diff__cancel,axiom,
% 0.35/0.59      ! [A: real,B: real] :
% 0.35/0.59        ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_cancel
% 0.35/0.59  thf(fact_3208_add__diff__cancel,axiom,
% 0.35/0.59      ! [A: rat,B: rat] :
% 0.35/0.59        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_cancel
% 0.35/0.59  thf(fact_3209_add__diff__cancel,axiom,
% 0.35/0.59      ! [A: int,B: int] :
% 0.35/0.59        ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.35/0.59        = A ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_cancel
% 0.35/0.59  thf(fact_3210_Suc__diff__diff,axiom,
% 0.35/0.59      ! [M: nat,N: nat,K: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 0.35/0.59        = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 0.35/0.59  
% 0.35/0.59  % Suc_diff_diff
% 0.35/0.59  thf(fact_3211_diff__Suc__Suc,axiom,
% 0.35/0.59      ! [M: nat,N: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.35/0.59        = ( minus_minus_nat @ M @ N ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_Suc_Suc
% 0.35/0.59  thf(fact_3212_diff__0__eq__0,axiom,
% 0.35/0.59      ! [N: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ zero_zero_nat @ N )
% 0.35/0.59        = zero_zero_nat ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_0_eq_0
% 0.35/0.59  thf(fact_3213_diff__self__eq__0,axiom,
% 0.35/0.59      ! [M: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ M @ M )
% 0.35/0.59        = zero_zero_nat ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_self_eq_0
% 0.35/0.59  thf(fact_3214_diff__diff__cancel,axiom,
% 0.35/0.59      ! [I: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ I @ N )
% 0.35/0.59       => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 0.35/0.59          = I ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_diff_cancel
% 0.35/0.59  thf(fact_3215_diff__diff__left,axiom,
% 0.35/0.59      ! [I: nat,J: nat,K: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.35/0.59        = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_diff_left
% 0.35/0.59  thf(fact_3216_set__bit__negative__int__iff,axiom,
% 0.35/0.59      ! [N: nat,K: int] :
% 0.35/0.59        ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
% 0.35/0.59        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.35/0.59  
% 0.35/0.59  % set_bit_negative_int_iff
% 0.35/0.59  thf(fact_3217_diff__ge__0__iff__ge,axiom,
% 0.35/0.59      ! [A: real,B: real] :
% 0.35/0.59        ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.35/0.59        = ( ord_less_eq_real @ B @ A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_ge_0_iff_ge
% 0.35/0.59  thf(fact_3218_diff__ge__0__iff__ge,axiom,
% 0.35/0.59      ! [A: rat,B: rat] :
% 0.35/0.59        ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 0.35/0.59        = ( ord_less_eq_rat @ B @ A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_ge_0_iff_ge
% 0.35/0.59  thf(fact_3219_diff__ge__0__iff__ge,axiom,
% 0.35/0.59      ! [A: int,B: int] :
% 0.35/0.59        ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 0.35/0.59        = ( ord_less_eq_int @ B @ A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_ge_0_iff_ge
% 0.35/0.59  thf(fact_3220_diff__gt__0__iff__gt,axiom,
% 0.35/0.59      ! [A: real,B: real] :
% 0.35/0.59        ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.35/0.59        = ( ord_less_real @ B @ A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_gt_0_iff_gt
% 0.35/0.59  thf(fact_3221_diff__gt__0__iff__gt,axiom,
% 0.35/0.59      ! [A: rat,B: rat] :
% 0.35/0.59        ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 0.35/0.59        = ( ord_less_rat @ B @ A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_gt_0_iff_gt
% 0.35/0.59  thf(fact_3222_diff__gt__0__iff__gt,axiom,
% 0.35/0.59      ! [A: int,B: int] :
% 0.35/0.59        ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 0.35/0.59        = ( ord_less_int @ B @ A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_gt_0_iff_gt
% 0.35/0.59  thf(fact_3223_le__add__diff__inverse2,axiom,
% 0.35/0.59      ! [B: real,A: real] :
% 0.35/0.59        ( ( ord_less_eq_real @ B @ A )
% 0.35/0.59       => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 0.35/0.59          = A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % le_add_diff_inverse2
% 0.35/0.59  thf(fact_3224_le__add__diff__inverse2,axiom,
% 0.35/0.59      ! [B: rat,A: rat] :
% 0.35/0.59        ( ( ord_less_eq_rat @ B @ A )
% 0.35/0.59       => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 0.35/0.59          = A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % le_add_diff_inverse2
% 0.35/0.59  thf(fact_3225_le__add__diff__inverse2,axiom,
% 0.35/0.59      ! [B: nat,A: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ B @ A )
% 0.35/0.59       => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
% 0.35/0.59          = A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % le_add_diff_inverse2
% 0.35/0.59  thf(fact_3226_le__add__diff__inverse2,axiom,
% 0.35/0.59      ! [B: int,A: int] :
% 0.35/0.59        ( ( ord_less_eq_int @ B @ A )
% 0.35/0.59       => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.35/0.59          = A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % le_add_diff_inverse2
% 0.35/0.59  thf(fact_3227_le__add__diff__inverse,axiom,
% 0.35/0.59      ! [B: real,A: real] :
% 0.35/0.59        ( ( ord_less_eq_real @ B @ A )
% 0.35/0.59       => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 0.35/0.59          = A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % le_add_diff_inverse
% 0.35/0.59  thf(fact_3228_le__add__diff__inverse,axiom,
% 0.35/0.59      ! [B: rat,A: rat] :
% 0.35/0.59        ( ( ord_less_eq_rat @ B @ A )
% 0.35/0.59       => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 0.35/0.59          = A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % le_add_diff_inverse
% 0.35/0.59  thf(fact_3229_le__add__diff__inverse,axiom,
% 0.35/0.59      ! [B: nat,A: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ B @ A )
% 0.35/0.59       => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 0.35/0.59          = A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % le_add_diff_inverse
% 0.35/0.59  thf(fact_3230_le__add__diff__inverse,axiom,
% 0.35/0.59      ! [B: int,A: int] :
% 0.35/0.59        ( ( ord_less_eq_int @ B @ A )
% 0.35/0.59       => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 0.35/0.59          = A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % le_add_diff_inverse
% 0.35/0.59  thf(fact_3231_diff__add__zero,axiom,
% 0.35/0.59      ! [A: nat,B: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.35/0.59        = zero_zero_nat ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_add_zero
% 0.35/0.59  thf(fact_3232_diff__numeral__special_I9_J,axiom,
% 0.35/0.59      ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 0.35/0.59      = zero_zero_complex ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_numeral_special(9)
% 0.35/0.59  thf(fact_3233_diff__numeral__special_I9_J,axiom,
% 0.35/0.59      ( ( minus_minus_real @ one_one_real @ one_one_real )
% 0.35/0.59      = zero_zero_real ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_numeral_special(9)
% 0.35/0.59  thf(fact_3234_diff__numeral__special_I9_J,axiom,
% 0.35/0.59      ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 0.35/0.59      = zero_zero_rat ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_numeral_special(9)
% 0.35/0.59  thf(fact_3235_diff__numeral__special_I9_J,axiom,
% 0.35/0.59      ( ( minus_minus_int @ one_one_int @ one_one_int )
% 0.35/0.59      = zero_zero_int ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_numeral_special(9)
% 0.35/0.59  thf(fact_3236_zero__less__diff,axiom,
% 0.35/0.59      ! [N: nat,M: nat] :
% 0.35/0.59        ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 0.35/0.59        = ( ord_less_nat @ M @ N ) ) ).
% 0.35/0.59  
% 0.35/0.59  % zero_less_diff
% 0.35/0.59  thf(fact_3237_diff__is__0__eq_H,axiom,
% 0.35/0.59      ! [M: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.59       => ( ( minus_minus_nat @ M @ N )
% 0.35/0.59          = zero_zero_nat ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_is_0_eq'
% 0.35/0.59  thf(fact_3238_diff__is__0__eq,axiom,
% 0.35/0.59      ! [M: nat,N: nat] :
% 0.35/0.59        ( ( ( minus_minus_nat @ M @ N )
% 0.35/0.59          = zero_zero_nat )
% 0.35/0.59        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_is_0_eq
% 0.35/0.59  thf(fact_3239_Nat_Oadd__diff__assoc,axiom,
% 0.35/0.59      ! [K: nat,J: nat,I: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ K @ J )
% 0.35/0.59       => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.35/0.59          = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % Nat.add_diff_assoc
% 0.35/0.59  thf(fact_3240_Nat_Oadd__diff__assoc2,axiom,
% 0.35/0.59      ! [K: nat,J: nat,I: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ K @ J )
% 0.35/0.59       => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.35/0.59          = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % Nat.add_diff_assoc2
% 0.35/0.59  thf(fact_3241_Nat_Odiff__diff__right,axiom,
% 0.35/0.59      ! [K: nat,J: nat,I: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ K @ J )
% 0.35/0.59       => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.35/0.59          = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % Nat.diff_diff_right
% 0.35/0.59  thf(fact_3242_diff__Suc__1,axiom,
% 0.35/0.59      ! [N: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
% 0.35/0.59        = N ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_Suc_1
% 0.35/0.59  thf(fact_3243_div__pos__pos__trivial,axiom,
% 0.35/0.59      ! [K: int,L: int] :
% 0.35/0.59        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.59       => ( ( ord_less_int @ K @ L )
% 0.35/0.59         => ( ( divide_divide_int @ K @ L )
% 0.35/0.59            = zero_zero_int ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_pos_pos_trivial
% 0.35/0.59  thf(fact_3244_div__neg__neg__trivial,axiom,
% 0.35/0.59      ! [K: int,L: int] :
% 0.35/0.59        ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.35/0.59       => ( ( ord_less_int @ L @ K )
% 0.35/0.59         => ( ( divide_divide_int @ K @ L )
% 0.35/0.59            = zero_zero_int ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % div_neg_neg_trivial
% 0.35/0.59  thf(fact_3245_Suc__pred,axiom,
% 0.35/0.59      ! [N: nat] :
% 0.35/0.59        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59       => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.35/0.59          = N ) ) ).
% 0.35/0.59  
% 0.35/0.59  % Suc_pred
% 0.35/0.59  thf(fact_3246_diff__Suc__diff__eq2,axiom,
% 0.35/0.59      ! [K: nat,J: nat,I: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ K @ J )
% 0.35/0.59       => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
% 0.35/0.59          = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_Suc_diff_eq2
% 0.35/0.59  thf(fact_3247_diff__Suc__diff__eq1,axiom,
% 0.35/0.59      ! [K: nat,J: nat,I: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ K @ J )
% 0.35/0.59       => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 0.35/0.59          = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_Suc_diff_eq1
% 0.35/0.59  thf(fact_3248_Suc__diff__1,axiom,
% 0.35/0.59      ! [N: nat] :
% 0.35/0.59        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59       => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.35/0.59          = N ) ) ).
% 0.35/0.59  
% 0.35/0.59  % Suc_diff_1
% 0.35/0.59  thf(fact_3249_pos__zmult__eq__1__iff,axiom,
% 0.35/0.59      ! [M: int,N: int] :
% 0.35/0.59        ( ( ord_less_int @ zero_zero_int @ M )
% 0.35/0.59       => ( ( ( times_times_int @ M @ N )
% 0.35/0.59            = one_one_int )
% 0.35/0.59          = ( ( M = one_one_int )
% 0.35/0.59            & ( N = one_one_int ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % pos_zmult_eq_1_iff
% 0.35/0.59  thf(fact_3250_zmult__zless__mono2,axiom,
% 0.35/0.59      ! [I: int,J: int,K: int] :
% 0.35/0.59        ( ( ord_less_int @ I @ J )
% 0.35/0.59       => ( ( ord_less_int @ zero_zero_int @ K )
% 0.35/0.59         => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % zmult_zless_mono2
% 0.35/0.59  thf(fact_3251_int__one__le__iff__zero__less,axiom,
% 0.35/0.59      ! [Z2: int] :
% 0.35/0.59        ( ( ord_less_eq_int @ one_one_int @ Z2 )
% 0.35/0.59        = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 0.35/0.59  
% 0.35/0.59  % int_one_le_iff_zero_less
% 0.35/0.59  thf(fact_3252_add1__zle__eq,axiom,
% 0.35/0.59      ! [W2: int,Z2: int] :
% 0.35/0.59        ( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 )
% 0.35/0.59        = ( ord_less_int @ W2 @ Z2 ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add1_zle_eq
% 0.35/0.59  thf(fact_3253_le__imp__0__less,axiom,
% 0.35/0.59      ! [Z2: int] :
% 0.35/0.59        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.35/0.59       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % le_imp_0_less
% 0.35/0.59  thf(fact_3254_zless__imp__add1__zle,axiom,
% 0.35/0.59      ! [W2: int,Z2: int] :
% 0.35/0.59        ( ( ord_less_int @ W2 @ Z2 )
% 0.35/0.59       => ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 ) ) ).
% 0.35/0.59  
% 0.35/0.59  % zless_imp_add1_zle
% 0.35/0.59  thf(fact_3255_incr__mult__lemma,axiom,
% 0.35/0.59      ! [D: int,P: int > $o,K: int] :
% 0.35/0.59        ( ( ord_less_int @ zero_zero_int @ D )
% 0.35/0.59       => ( ! [X4: int] :
% 0.35/0.59              ( ( P @ X4 )
% 0.35/0.59             => ( P @ ( plus_plus_int @ X4 @ D ) ) )
% 0.35/0.59         => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.59           => ! [X2: int] :
% 0.35/0.59                ( ( P @ X2 )
% 0.35/0.59               => ( P @ ( plus_plus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % incr_mult_lemma
% 0.35/0.59  thf(fact_3256_int__gr__induct,axiom,
% 0.35/0.59      ! [K: int,I: int,P: int > $o] :
% 0.35/0.59        ( ( ord_less_int @ K @ I )
% 0.35/0.59       => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 0.35/0.59         => ( ! [I2: int] :
% 0.35/0.59                ( ( ord_less_int @ K @ I2 )
% 0.35/0.59               => ( ( P @ I2 )
% 0.35/0.59                 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 0.35/0.59           => ( P @ I ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % int_gr_induct
% 0.35/0.59  thf(fact_3257_zless__add1__eq,axiom,
% 0.35/0.59      ! [W2: int,Z2: int] :
% 0.35/0.59        ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
% 0.35/0.59        = ( ( ord_less_int @ W2 @ Z2 )
% 0.35/0.59          | ( W2 = Z2 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % zless_add1_eq
% 0.35/0.59  thf(fact_3258_odd__less__0__iff,axiom,
% 0.35/0.59      ! [Z2: int] :
% 0.35/0.59        ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
% 0.35/0.59        = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 0.35/0.59  
% 0.35/0.59  % odd_less_0_iff
% 0.35/0.59  thf(fact_3259_less__int__code_I1_J,axiom,
% 0.35/0.59      ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.35/0.59  
% 0.35/0.59  % less_int_code(1)
% 0.35/0.59  thf(fact_3260_diff__commute,axiom,
% 0.35/0.59      ! [I: nat,J: nat,K: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.35/0.59        = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_commute
% 0.35/0.59  thf(fact_3261_diff__right__commute,axiom,
% 0.35/0.59      ! [A: real,C: real,B: real] :
% 0.35/0.59        ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 0.35/0.59        = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_right_commute
% 0.35/0.59  thf(fact_3262_diff__right__commute,axiom,
% 0.35/0.59      ! [A: rat,C: rat,B: rat] :
% 0.35/0.59        ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
% 0.35/0.59        = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_right_commute
% 0.35/0.59  thf(fact_3263_diff__right__commute,axiom,
% 0.35/0.59      ! [A: nat,C: nat,B: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.35/0.59        = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_right_commute
% 0.35/0.59  thf(fact_3264_diff__right__commute,axiom,
% 0.35/0.59      ! [A: int,C: int,B: int] :
% 0.35/0.59        ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 0.35/0.59        = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_right_commute
% 0.35/0.59  thf(fact_3265_diff__eq__diff__eq,axiom,
% 0.35/0.59      ! [A: real,B: real,C: real,D: real] :
% 0.35/0.59        ( ( ( minus_minus_real @ A @ B )
% 0.35/0.59          = ( minus_minus_real @ C @ D ) )
% 0.35/0.59       => ( ( A = B )
% 0.35/0.59          = ( C = D ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_eq_diff_eq
% 0.35/0.59  thf(fact_3266_diff__eq__diff__eq,axiom,
% 0.35/0.59      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.35/0.59        ( ( ( minus_minus_rat @ A @ B )
% 0.35/0.59          = ( minus_minus_rat @ C @ D ) )
% 0.35/0.59       => ( ( A = B )
% 0.35/0.59          = ( C = D ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_eq_diff_eq
% 0.35/0.59  thf(fact_3267_diff__eq__diff__eq,axiom,
% 0.35/0.59      ! [A: int,B: int,C: int,D: int] :
% 0.35/0.59        ( ( ( minus_minus_int @ A @ B )
% 0.35/0.59          = ( minus_minus_int @ C @ D ) )
% 0.35/0.59       => ( ( A = B )
% 0.35/0.59          = ( C = D ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_eq_diff_eq
% 0.35/0.59  thf(fact_3268_VEBT_Osize_I4_J,axiom,
% 0.35/0.59      ! [X21: $o,X222: $o] :
% 0.35/0.59        ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 0.35/0.59        = zero_zero_nat ) ).
% 0.35/0.59  
% 0.35/0.59  % VEBT.size(4)
% 0.35/0.59  thf(fact_3269_diff__mono,axiom,
% 0.35/0.59      ! [A: real,B: real,D: real,C: real] :
% 0.35/0.59        ( ( ord_less_eq_real @ A @ B )
% 0.35/0.59       => ( ( ord_less_eq_real @ D @ C )
% 0.35/0.59         => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_mono
% 0.35/0.59  thf(fact_3270_diff__mono,axiom,
% 0.35/0.59      ! [A: rat,B: rat,D: rat,C: rat] :
% 0.35/0.59        ( ( ord_less_eq_rat @ A @ B )
% 0.35/0.59       => ( ( ord_less_eq_rat @ D @ C )
% 0.35/0.59         => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_mono
% 0.35/0.59  thf(fact_3271_diff__mono,axiom,
% 0.35/0.59      ! [A: int,B: int,D: int,C: int] :
% 0.35/0.59        ( ( ord_less_eq_int @ A @ B )
% 0.35/0.59       => ( ( ord_less_eq_int @ D @ C )
% 0.35/0.59         => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_mono
% 0.35/0.59  thf(fact_3272_diff__left__mono,axiom,
% 0.35/0.59      ! [B: real,A: real,C: real] :
% 0.35/0.59        ( ( ord_less_eq_real @ B @ A )
% 0.35/0.59       => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_left_mono
% 0.35/0.59  thf(fact_3273_diff__left__mono,axiom,
% 0.35/0.59      ! [B: rat,A: rat,C: rat] :
% 0.35/0.59        ( ( ord_less_eq_rat @ B @ A )
% 0.35/0.59       => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_left_mono
% 0.35/0.59  thf(fact_3274_diff__left__mono,axiom,
% 0.35/0.59      ! [B: int,A: int,C: int] :
% 0.35/0.59        ( ( ord_less_eq_int @ B @ A )
% 0.35/0.59       => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_left_mono
% 0.35/0.59  thf(fact_3275_diff__right__mono,axiom,
% 0.35/0.59      ! [A: real,B: real,C: real] :
% 0.35/0.59        ( ( ord_less_eq_real @ A @ B )
% 0.35/0.59       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_right_mono
% 0.35/0.59  thf(fact_3276_diff__right__mono,axiom,
% 0.35/0.59      ! [A: rat,B: rat,C: rat] :
% 0.35/0.59        ( ( ord_less_eq_rat @ A @ B )
% 0.35/0.59       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_right_mono
% 0.35/0.59  thf(fact_3277_diff__right__mono,axiom,
% 0.35/0.59      ! [A: int,B: int,C: int] :
% 0.35/0.59        ( ( ord_less_eq_int @ A @ B )
% 0.35/0.59       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_right_mono
% 0.35/0.59  thf(fact_3278_diff__eq__diff__less__eq,axiom,
% 0.35/0.59      ! [A: real,B: real,C: real,D: real] :
% 0.35/0.59        ( ( ( minus_minus_real @ A @ B )
% 0.35/0.59          = ( minus_minus_real @ C @ D ) )
% 0.35/0.59       => ( ( ord_less_eq_real @ A @ B )
% 0.35/0.59          = ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_eq_diff_less_eq
% 0.35/0.59  thf(fact_3279_diff__eq__diff__less__eq,axiom,
% 0.35/0.59      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.35/0.59        ( ( ( minus_minus_rat @ A @ B )
% 0.35/0.59          = ( minus_minus_rat @ C @ D ) )
% 0.35/0.59       => ( ( ord_less_eq_rat @ A @ B )
% 0.35/0.59          = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_eq_diff_less_eq
% 0.35/0.59  thf(fact_3280_diff__eq__diff__less__eq,axiom,
% 0.35/0.59      ! [A: int,B: int,C: int,D: int] :
% 0.35/0.59        ( ( ( minus_minus_int @ A @ B )
% 0.35/0.59          = ( minus_minus_int @ C @ D ) )
% 0.35/0.59       => ( ( ord_less_eq_int @ A @ B )
% 0.35/0.59          = ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_eq_diff_less_eq
% 0.35/0.59  thf(fact_3281_eq__iff__diff__eq__0,axiom,
% 0.35/0.59      ( ( ^ [Y5: complex,Z: complex] : ( Y5 = Z ) )
% 0.35/0.59      = ( ^ [A4: complex,B4: complex] :
% 0.35/0.59            ( ( minus_minus_complex @ A4 @ B4 )
% 0.35/0.59            = zero_zero_complex ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % eq_iff_diff_eq_0
% 0.35/0.59  thf(fact_3282_eq__iff__diff__eq__0,axiom,
% 0.35/0.59      ( ( ^ [Y5: real,Z: real] : ( Y5 = Z ) )
% 0.35/0.59      = ( ^ [A4: real,B4: real] :
% 0.35/0.59            ( ( minus_minus_real @ A4 @ B4 )
% 0.35/0.59            = zero_zero_real ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % eq_iff_diff_eq_0
% 0.35/0.59  thf(fact_3283_eq__iff__diff__eq__0,axiom,
% 0.35/0.59      ( ( ^ [Y5: rat,Z: rat] : ( Y5 = Z ) )
% 0.35/0.59      = ( ^ [A4: rat,B4: rat] :
% 0.35/0.59            ( ( minus_minus_rat @ A4 @ B4 )
% 0.35/0.59            = zero_zero_rat ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % eq_iff_diff_eq_0
% 0.35/0.59  thf(fact_3284_eq__iff__diff__eq__0,axiom,
% 0.35/0.59      ( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
% 0.35/0.59      = ( ^ [A4: int,B4: int] :
% 0.35/0.59            ( ( minus_minus_int @ A4 @ B4 )
% 0.35/0.59            = zero_zero_int ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % eq_iff_diff_eq_0
% 0.35/0.59  thf(fact_3285_VEBT_Odistinct_I1_J,axiom,
% 0.35/0.59      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
% 0.35/0.59        ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 0.35/0.59       != ( vEBT_Leaf @ X21 @ X222 ) ) ).
% 0.35/0.59  
% 0.35/0.59  % VEBT.distinct(1)
% 0.35/0.59  thf(fact_3286_VEBT_Oexhaust,axiom,
% 0.35/0.59      ! [Y: vEBT_VEBT] :
% 0.35/0.59        ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
% 0.35/0.59            ( Y
% 0.35/0.59           != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
% 0.35/0.59       => ~ ! [X212: $o,X223: $o] :
% 0.35/0.59              ( Y
% 0.35/0.59             != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % VEBT.exhaust
% 0.35/0.59  thf(fact_3287_VEBT__internal_Ovalid_H_Ocases,axiom,
% 0.35/0.59      ! [X: produc9072475918466114483BT_nat] :
% 0.35/0.59        ( ! [Uu2: $o,Uv2: $o,D5: nat] :
% 0.35/0.59            ( X
% 0.35/0.59           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D5 ) )
% 0.35/0.59       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,Deg3: nat] :
% 0.35/0.59              ( X
% 0.35/0.59             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) @ Deg3 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % VEBT_internal.valid'.cases
% 0.35/0.59  thf(fact_3288_diff__strict__mono,axiom,
% 0.35/0.59      ! [A: real,B: real,D: real,C: real] :
% 0.35/0.59        ( ( ord_less_real @ A @ B )
% 0.35/0.59       => ( ( ord_less_real @ D @ C )
% 0.35/0.59         => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_strict_mono
% 0.35/0.59  thf(fact_3289_diff__strict__mono,axiom,
% 0.35/0.59      ! [A: rat,B: rat,D: rat,C: rat] :
% 0.35/0.59        ( ( ord_less_rat @ A @ B )
% 0.35/0.59       => ( ( ord_less_rat @ D @ C )
% 0.35/0.59         => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_strict_mono
% 0.35/0.59  thf(fact_3290_diff__strict__mono,axiom,
% 0.35/0.59      ! [A: int,B: int,D: int,C: int] :
% 0.35/0.59        ( ( ord_less_int @ A @ B )
% 0.35/0.59       => ( ( ord_less_int @ D @ C )
% 0.35/0.59         => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_strict_mono
% 0.35/0.59  thf(fact_3291_diff__eq__diff__less,axiom,
% 0.35/0.59      ! [A: real,B: real,C: real,D: real] :
% 0.35/0.59        ( ( ( minus_minus_real @ A @ B )
% 0.35/0.59          = ( minus_minus_real @ C @ D ) )
% 0.35/0.59       => ( ( ord_less_real @ A @ B )
% 0.35/0.59          = ( ord_less_real @ C @ D ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_eq_diff_less
% 0.35/0.59  thf(fact_3292_diff__eq__diff__less,axiom,
% 0.35/0.59      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.35/0.59        ( ( ( minus_minus_rat @ A @ B )
% 0.35/0.59          = ( minus_minus_rat @ C @ D ) )
% 0.35/0.59       => ( ( ord_less_rat @ A @ B )
% 0.35/0.59          = ( ord_less_rat @ C @ D ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_eq_diff_less
% 0.35/0.59  thf(fact_3293_diff__eq__diff__less,axiom,
% 0.35/0.59      ! [A: int,B: int,C: int,D: int] :
% 0.35/0.59        ( ( ( minus_minus_int @ A @ B )
% 0.35/0.59          = ( minus_minus_int @ C @ D ) )
% 0.35/0.59       => ( ( ord_less_int @ A @ B )
% 0.35/0.59          = ( ord_less_int @ C @ D ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_eq_diff_less
% 0.35/0.59  thf(fact_3294_diff__strict__left__mono,axiom,
% 0.35/0.59      ! [B: real,A: real,C: real] :
% 0.35/0.59        ( ( ord_less_real @ B @ A )
% 0.35/0.59       => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_strict_left_mono
% 0.35/0.59  thf(fact_3295_diff__strict__left__mono,axiom,
% 0.35/0.59      ! [B: rat,A: rat,C: rat] :
% 0.35/0.59        ( ( ord_less_rat @ B @ A )
% 0.35/0.59       => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_strict_left_mono
% 0.35/0.59  thf(fact_3296_diff__strict__left__mono,axiom,
% 0.35/0.59      ! [B: int,A: int,C: int] :
% 0.35/0.59        ( ( ord_less_int @ B @ A )
% 0.35/0.59       => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_strict_left_mono
% 0.35/0.59  thf(fact_3297_diff__strict__right__mono,axiom,
% 0.35/0.59      ! [A: real,B: real,C: real] :
% 0.35/0.59        ( ( ord_less_real @ A @ B )
% 0.35/0.59       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_strict_right_mono
% 0.35/0.59  thf(fact_3298_diff__strict__right__mono,axiom,
% 0.35/0.59      ! [A: rat,B: rat,C: rat] :
% 0.35/0.59        ( ( ord_less_rat @ A @ B )
% 0.35/0.59       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_strict_right_mono
% 0.35/0.59  thf(fact_3299_diff__strict__right__mono,axiom,
% 0.35/0.59      ! [A: int,B: int,C: int] :
% 0.35/0.59        ( ( ord_less_int @ A @ B )
% 0.35/0.59       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_strict_right_mono
% 0.35/0.59  thf(fact_3300_diff__diff__eq,axiom,
% 0.35/0.59      ! [A: real,B: real,C: real] :
% 0.35/0.59        ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.35/0.59        = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_diff_eq
% 0.35/0.59  thf(fact_3301_diff__diff__eq,axiom,
% 0.35/0.59      ! [A: rat,B: rat,C: rat] :
% 0.35/0.59        ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.35/0.59        = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_diff_eq
% 0.35/0.59  thf(fact_3302_diff__diff__eq,axiom,
% 0.35/0.59      ! [A: nat,B: nat,C: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 0.35/0.59        = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_diff_eq
% 0.35/0.59  thf(fact_3303_diff__diff__eq,axiom,
% 0.35/0.59      ! [A: int,B: int,C: int] :
% 0.35/0.59        ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.35/0.59        = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_diff_eq
% 0.35/0.59  thf(fact_3304_add__implies__diff,axiom,
% 0.35/0.59      ! [C: real,B: real,A: real] :
% 0.35/0.59        ( ( ( plus_plus_real @ C @ B )
% 0.35/0.59          = A )
% 0.35/0.59       => ( C
% 0.35/0.59          = ( minus_minus_real @ A @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_implies_diff
% 0.35/0.59  thf(fact_3305_add__implies__diff,axiom,
% 0.35/0.59      ! [C: rat,B: rat,A: rat] :
% 0.35/0.59        ( ( ( plus_plus_rat @ C @ B )
% 0.35/0.59          = A )
% 0.35/0.59       => ( C
% 0.35/0.59          = ( minus_minus_rat @ A @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_implies_diff
% 0.35/0.59  thf(fact_3306_add__implies__diff,axiom,
% 0.35/0.59      ! [C: nat,B: nat,A: nat] :
% 0.35/0.59        ( ( ( plus_plus_nat @ C @ B )
% 0.35/0.59          = A )
% 0.35/0.59       => ( C
% 0.35/0.59          = ( minus_minus_nat @ A @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_implies_diff
% 0.35/0.59  thf(fact_3307_add__implies__diff,axiom,
% 0.35/0.59      ! [C: int,B: int,A: int] :
% 0.35/0.59        ( ( ( plus_plus_int @ C @ B )
% 0.35/0.59          = A )
% 0.35/0.59       => ( C
% 0.35/0.59          = ( minus_minus_int @ A @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_implies_diff
% 0.35/0.59  thf(fact_3308_diff__add__eq__diff__diff__swap,axiom,
% 0.35/0.59      ! [A: real,B: real,C: real] :
% 0.35/0.59        ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.35/0.59        = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_add_eq_diff_diff_swap
% 0.35/0.59  thf(fact_3309_diff__add__eq__diff__diff__swap,axiom,
% 0.35/0.59      ! [A: rat,B: rat,C: rat] :
% 0.35/0.59        ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.35/0.59        = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_add_eq_diff_diff_swap
% 0.35/0.59  thf(fact_3310_diff__add__eq__diff__diff__swap,axiom,
% 0.35/0.59      ! [A: int,B: int,C: int] :
% 0.35/0.59        ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.35/0.59        = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_add_eq_diff_diff_swap
% 0.35/0.59  thf(fact_3311_diff__add__eq,axiom,
% 0.35/0.59      ! [A: real,B: real,C: real] :
% 0.35/0.59        ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.35/0.59        = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_add_eq
% 0.35/0.59  thf(fact_3312_diff__add__eq,axiom,
% 0.35/0.59      ! [A: rat,B: rat,C: rat] :
% 0.35/0.59        ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.35/0.59        = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_add_eq
% 0.35/0.59  thf(fact_3313_diff__add__eq,axiom,
% 0.35/0.59      ! [A: int,B: int,C: int] :
% 0.35/0.59        ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.35/0.59        = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_add_eq
% 0.35/0.59  thf(fact_3314_diff__diff__eq2,axiom,
% 0.35/0.59      ! [A: real,B: real,C: real] :
% 0.35/0.59        ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.35/0.59        = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_diff_eq2
% 0.35/0.59  thf(fact_3315_diff__diff__eq2,axiom,
% 0.35/0.59      ! [A: rat,B: rat,C: rat] :
% 0.35/0.59        ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.35/0.59        = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_diff_eq2
% 0.35/0.59  thf(fact_3316_diff__diff__eq2,axiom,
% 0.35/0.59      ! [A: int,B: int,C: int] :
% 0.35/0.59        ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.35/0.59        = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_diff_eq2
% 0.35/0.59  thf(fact_3317_add__diff__eq,axiom,
% 0.35/0.59      ! [A: real,B: real,C: real] :
% 0.35/0.59        ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.35/0.59        = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_eq
% 0.35/0.59  thf(fact_3318_add__diff__eq,axiom,
% 0.35/0.59      ! [A: rat,B: rat,C: rat] :
% 0.35/0.59        ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.35/0.59        = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_eq
% 0.35/0.59  thf(fact_3319_add__diff__eq,axiom,
% 0.35/0.59      ! [A: int,B: int,C: int] :
% 0.35/0.59        ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.35/0.59        = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_diff_eq
% 0.35/0.59  thf(fact_3320_eq__diff__eq,axiom,
% 0.35/0.59      ! [A: real,C: real,B: real] :
% 0.35/0.59        ( ( A
% 0.35/0.59          = ( minus_minus_real @ C @ B ) )
% 0.35/0.59        = ( ( plus_plus_real @ A @ B )
% 0.35/0.59          = C ) ) ).
% 0.35/0.59  
% 0.35/0.59  % eq_diff_eq
% 0.35/0.59  thf(fact_3321_eq__diff__eq,axiom,
% 0.35/0.59      ! [A: rat,C: rat,B: rat] :
% 0.35/0.59        ( ( A
% 0.35/0.59          = ( minus_minus_rat @ C @ B ) )
% 0.35/0.59        = ( ( plus_plus_rat @ A @ B )
% 0.35/0.59          = C ) ) ).
% 0.35/0.59  
% 0.35/0.59  % eq_diff_eq
% 0.35/0.59  thf(fact_3322_eq__diff__eq,axiom,
% 0.35/0.59      ! [A: int,C: int,B: int] :
% 0.35/0.59        ( ( A
% 0.35/0.59          = ( minus_minus_int @ C @ B ) )
% 0.35/0.59        = ( ( plus_plus_int @ A @ B )
% 0.35/0.59          = C ) ) ).
% 0.35/0.59  
% 0.35/0.59  % eq_diff_eq
% 0.35/0.59  thf(fact_3323_diff__eq__eq,axiom,
% 0.35/0.59      ! [A: real,B: real,C: real] :
% 0.35/0.59        ( ( ( minus_minus_real @ A @ B )
% 0.35/0.59          = C )
% 0.35/0.59        = ( A
% 0.35/0.59          = ( plus_plus_real @ C @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_eq_eq
% 0.35/0.59  thf(fact_3324_diff__eq__eq,axiom,
% 0.35/0.59      ! [A: rat,B: rat,C: rat] :
% 0.35/0.59        ( ( ( minus_minus_rat @ A @ B )
% 0.35/0.59          = C )
% 0.35/0.59        = ( A
% 0.35/0.59          = ( plus_plus_rat @ C @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_eq_eq
% 0.35/0.59  thf(fact_3325_diff__eq__eq,axiom,
% 0.35/0.59      ! [A: int,B: int,C: int] :
% 0.35/0.59        ( ( ( minus_minus_int @ A @ B )
% 0.35/0.59          = C )
% 0.35/0.59        = ( A
% 0.35/0.59          = ( plus_plus_int @ C @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_eq_eq
% 0.35/0.59  thf(fact_3326_group__cancel_Osub1,axiom,
% 0.35/0.59      ! [A2: real,K: real,A: real,B: real] :
% 0.35/0.59        ( ( A2
% 0.35/0.59          = ( plus_plus_real @ K @ A ) )
% 0.35/0.59       => ( ( minus_minus_real @ A2 @ B )
% 0.35/0.59          = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % group_cancel.sub1
% 0.35/0.59  thf(fact_3327_group__cancel_Osub1,axiom,
% 0.35/0.59      ! [A2: rat,K: rat,A: rat,B: rat] :
% 0.35/0.59        ( ( A2
% 0.35/0.59          = ( plus_plus_rat @ K @ A ) )
% 0.35/0.59       => ( ( minus_minus_rat @ A2 @ B )
% 0.35/0.59          = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % group_cancel.sub1
% 0.35/0.59  thf(fact_3328_group__cancel_Osub1,axiom,
% 0.35/0.59      ! [A2: int,K: int,A: int,B: int] :
% 0.35/0.59        ( ( A2
% 0.35/0.59          = ( plus_plus_int @ K @ A ) )
% 0.35/0.59       => ( ( minus_minus_int @ A2 @ B )
% 0.35/0.59          = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % group_cancel.sub1
% 0.35/0.59  thf(fact_3329_zero__induct__lemma,axiom,
% 0.35/0.59      ! [P: nat > $o,K: nat,I: nat] :
% 0.35/0.59        ( ( P @ K )
% 0.35/0.59       => ( ! [N3: nat] :
% 0.35/0.59              ( ( P @ ( suc @ N3 ) )
% 0.35/0.59             => ( P @ N3 ) )
% 0.35/0.59         => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % zero_induct_lemma
% 0.35/0.59  thf(fact_3330_minus__nat_Odiff__0,axiom,
% 0.35/0.59      ! [M: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ M @ zero_zero_nat )
% 0.35/0.59        = M ) ).
% 0.35/0.59  
% 0.35/0.59  % minus_nat.diff_0
% 0.35/0.59  thf(fact_3331_diffs0__imp__equal,axiom,
% 0.35/0.59      ! [M: nat,N: nat] :
% 0.35/0.59        ( ( ( minus_minus_nat @ M @ N )
% 0.35/0.59          = zero_zero_nat )
% 0.35/0.59       => ( ( ( minus_minus_nat @ N @ M )
% 0.35/0.59            = zero_zero_nat )
% 0.35/0.59         => ( M = N ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diffs0_imp_equal
% 0.35/0.59  thf(fact_3332_diff__less__mono2,axiom,
% 0.35/0.59      ! [M: nat,N: nat,L: nat] :
% 0.35/0.59        ( ( ord_less_nat @ M @ N )
% 0.35/0.59       => ( ( ord_less_nat @ M @ L )
% 0.35/0.59         => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_less_mono2
% 0.35/0.59  thf(fact_3333_less__imp__diff__less,axiom,
% 0.35/0.59      ! [J: nat,K: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_nat @ J @ K )
% 0.35/0.59       => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 0.35/0.59  
% 0.35/0.59  % less_imp_diff_less
% 0.35/0.59  thf(fact_3334_eq__diff__iff,axiom,
% 0.35/0.59      ! [K: nat,M: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ K @ M )
% 0.35/0.59       => ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.59         => ( ( ( minus_minus_nat @ M @ K )
% 0.35/0.59              = ( minus_minus_nat @ N @ K ) )
% 0.35/0.59            = ( M = N ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % eq_diff_iff
% 0.35/0.59  thf(fact_3335_le__diff__iff,axiom,
% 0.35/0.59      ! [K: nat,M: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ K @ M )
% 0.35/0.59       => ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.59         => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.35/0.59            = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % le_diff_iff
% 0.35/0.59  thf(fact_3336_Nat_Odiff__diff__eq,axiom,
% 0.35/0.59      ! [K: nat,M: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ K @ M )
% 0.35/0.59       => ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.59         => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.35/0.59            = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % Nat.diff_diff_eq
% 0.35/0.59  thf(fact_3337_diff__le__mono,axiom,
% 0.35/0.59      ! [M: nat,N: nat,L: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.59       => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_le_mono
% 0.35/0.59  thf(fact_3338_diff__le__self,axiom,
% 0.35/0.59      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_le_self
% 0.35/0.59  thf(fact_3339_le__diff__iff_H,axiom,
% 0.35/0.59      ! [A: nat,C: nat,B: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ A @ C )
% 0.35/0.59       => ( ( ord_less_eq_nat @ B @ C )
% 0.35/0.59         => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 0.35/0.59            = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % le_diff_iff'
% 0.35/0.59  thf(fact_3340_diff__le__mono2,axiom,
% 0.35/0.59      ! [M: nat,N: nat,L: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.59       => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_le_mono2
% 0.35/0.59  thf(fact_3341_diff__add__inverse2,axiom,
% 0.35/0.59      ! [M: nat,N: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 0.35/0.59        = M ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_add_inverse2
% 0.35/0.59  thf(fact_3342_diff__add__inverse,axiom,
% 0.35/0.59      ! [N: nat,M: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 0.35/0.59        = M ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_add_inverse
% 0.35/0.59  thf(fact_3343_diff__cancel2,axiom,
% 0.35/0.59      ! [M: nat,K: nat,N: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 0.35/0.59        = ( minus_minus_nat @ M @ N ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_cancel2
% 0.35/0.59  thf(fact_3344_Nat_Odiff__cancel,axiom,
% 0.35/0.59      ! [K: nat,M: nat,N: nat] :
% 0.35/0.59        ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.35/0.59        = ( minus_minus_nat @ M @ N ) ) ).
% 0.35/0.59  
% 0.35/0.59  % Nat.diff_cancel
% 0.35/0.59  thf(fact_3345_diff__mult__distrib2,axiom,
% 0.35/0.59      ! [K: nat,M: nat,N: nat] :
% 0.35/0.59        ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.59        = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_mult_distrib2
% 0.35/0.59  thf(fact_3346_diff__mult__distrib,axiom,
% 0.35/0.59      ! [M: nat,N: nat,K: nat] :
% 0.35/0.59        ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 0.35/0.59        = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_mult_distrib
% 0.35/0.59  thf(fact_3347_max__diff__distrib__left,axiom,
% 0.35/0.59      ! [X: real,Y: real,Z2: real] :
% 0.35/0.59        ( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z2 )
% 0.35/0.59        = ( ord_max_real @ ( minus_minus_real @ X @ Z2 ) @ ( minus_minus_real @ Y @ Z2 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_diff_distrib_left
% 0.35/0.59  thf(fact_3348_max__diff__distrib__left,axiom,
% 0.35/0.59      ! [X: rat,Y: rat,Z2: rat] :
% 0.35/0.59        ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z2 )
% 0.35/0.59        = ( ord_max_rat @ ( minus_minus_rat @ X @ Z2 ) @ ( minus_minus_rat @ Y @ Z2 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_diff_distrib_left
% 0.35/0.59  thf(fact_3349_max__diff__distrib__left,axiom,
% 0.35/0.59      ! [X: int,Y: int,Z2: int] :
% 0.35/0.59        ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z2 )
% 0.35/0.59        = ( ord_max_int @ ( minus_minus_int @ X @ Z2 ) @ ( minus_minus_int @ Y @ Z2 ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % max_diff_distrib_left
% 0.35/0.59  thf(fact_3350_VEBT__internal_OminNull_Osimps_I3_J,axiom,
% 0.35/0.59      ! [Uu: $o] :
% 0.35/0.59        ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% 0.35/0.59  
% 0.35/0.59  % VEBT_internal.minNull.simps(3)
% 0.35/0.59  thf(fact_3351_VEBT__internal_OminNull_Osimps_I2_J,axiom,
% 0.35/0.59      ! [Uv: $o] :
% 0.35/0.59        ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% 0.35/0.59  
% 0.35/0.59  % VEBT_internal.minNull.simps(2)
% 0.35/0.59  thf(fact_3352_VEBT__internal_OminNull_Osimps_I1_J,axiom,
% 0.35/0.59      vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% 0.35/0.59  
% 0.35/0.59  % VEBT_internal.minNull.simps(1)
% 0.35/0.59  thf(fact_3353_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
% 0.35/0.59      ! [Uu: $o,Uv: $o,Uw: nat] :
% 0.35/0.59        ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% 0.35/0.59  
% 0.35/0.59  % VEBT_internal.membermima.simps(1)
% 0.35/0.59  thf(fact_3354_le__iff__diff__le__0,axiom,
% 0.35/0.59      ( ord_less_eq_real
% 0.35/0.59      = ( ^ [A4: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % le_iff_diff_le_0
% 0.35/0.59  thf(fact_3355_le__iff__diff__le__0,axiom,
% 0.35/0.59      ( ord_less_eq_rat
% 0.35/0.59      = ( ^ [A4: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A4 @ B4 ) @ zero_zero_rat ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % le_iff_diff_le_0
% 0.35/0.59  thf(fact_3356_le__iff__diff__le__0,axiom,
% 0.35/0.59      ( ord_less_eq_int
% 0.35/0.59      = ( ^ [A4: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % le_iff_diff_le_0
% 0.35/0.59  thf(fact_3357_less__iff__diff__less__0,axiom,
% 0.35/0.59      ( ord_less_real
% 0.35/0.59      = ( ^ [A4: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % less_iff_diff_less_0
% 0.35/0.59  thf(fact_3358_less__iff__diff__less__0,axiom,
% 0.35/0.59      ( ord_less_rat
% 0.35/0.59      = ( ^ [A4: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A4 @ B4 ) @ zero_zero_rat ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % less_iff_diff_less_0
% 0.35/0.59  thf(fact_3359_less__iff__diff__less__0,axiom,
% 0.35/0.59      ( ord_less_int
% 0.35/0.59      = ( ^ [A4: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % less_iff_diff_less_0
% 0.35/0.59  thf(fact_3360_diff__le__eq,axiom,
% 0.35/0.59      ! [A: real,B: real,C: real] :
% 0.35/0.59        ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.35/0.59        = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_le_eq
% 0.35/0.59  thf(fact_3361_diff__le__eq,axiom,
% 0.35/0.59      ! [A: rat,B: rat,C: rat] :
% 0.35/0.59        ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.35/0.59        = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_le_eq
% 0.35/0.59  thf(fact_3362_diff__le__eq,axiom,
% 0.35/0.59      ! [A: int,B: int,C: int] :
% 0.35/0.59        ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.35/0.59        = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_le_eq
% 0.35/0.59  thf(fact_3363_le__diff__eq,axiom,
% 0.35/0.59      ! [A: real,C: real,B: real] :
% 0.35/0.59        ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
% 0.35/0.59        = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.35/0.59  
% 0.35/0.59  % le_diff_eq
% 0.35/0.59  thf(fact_3364_le__diff__eq,axiom,
% 0.35/0.59      ! [A: rat,C: rat,B: rat] :
% 0.35/0.59        ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 0.35/0.59        = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.35/0.59  
% 0.35/0.59  % le_diff_eq
% 0.35/0.59  thf(fact_3365_le__diff__eq,axiom,
% 0.35/0.59      ! [A: int,C: int,B: int] :
% 0.35/0.59        ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.35/0.59        = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.35/0.59  
% 0.35/0.59  % le_diff_eq
% 0.35/0.59  thf(fact_3366_diff__add,axiom,
% 0.35/0.59      ! [A: nat,B: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ A @ B )
% 0.35/0.59       => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 0.35/0.59          = B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_add
% 0.35/0.59  thf(fact_3367_le__add__diff,axiom,
% 0.35/0.59      ! [A: nat,B: nat,C: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ A @ B )
% 0.35/0.59       => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % le_add_diff
% 0.35/0.59  thf(fact_3368_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.35/0.59      ! [A: nat,B: nat,C: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ A @ B )
% 0.35/0.59       => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.35/0.59          = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.35/0.59  thf(fact_3369_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.35/0.59      ! [A: nat,B: nat,C: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ A @ B )
% 0.35/0.59       => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.35/0.59          = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.35/0.59  thf(fact_3370_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.35/0.59      ! [A: nat,B: nat,C: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ A @ B )
% 0.35/0.59       => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 0.35/0.59          = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.35/0.59  thf(fact_3371_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.35/0.59      ! [A: nat,B: nat,C: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ A @ B )
% 0.35/0.59       => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 0.35/0.59          = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.35/0.59  thf(fact_3372_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.35/0.59      ! [A: nat,B: nat,C: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ A @ B )
% 0.35/0.59       => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 0.35/0.59          = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.35/0.59  thf(fact_3373_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.35/0.59      ! [A: nat,B: nat,C: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ A @ B )
% 0.35/0.59       => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.35/0.59          = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.35/0.59  thf(fact_3374_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.35/0.59      ! [A: nat,B: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ A @ B )
% 0.35/0.59       => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 0.35/0.59          = B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.35/0.59  thf(fact_3375_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.35/0.59      ! [A: nat,B: nat,C: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ A @ B )
% 0.35/0.59       => ( ( ord_less_eq_nat @ A @ B )
% 0.35/0.59         => ( ( ( minus_minus_nat @ B @ A )
% 0.35/0.59              = C )
% 0.35/0.59            = ( B
% 0.35/0.59              = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.35/0.59  thf(fact_3376_add__le__add__imp__diff__le,axiom,
% 0.35/0.59      ! [I: real,K: real,N: real,J: real] :
% 0.35/0.59        ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.35/0.59       => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.35/0.59         => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.35/0.59           => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.35/0.59             => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_le_add_imp_diff_le
% 0.35/0.59  thf(fact_3377_add__le__add__imp__diff__le,axiom,
% 0.35/0.59      ! [I: rat,K: rat,N: rat,J: rat] :
% 0.35/0.59        ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.35/0.59       => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 0.35/0.59         => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.35/0.59           => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 0.35/0.59             => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_le_add_imp_diff_le
% 0.35/0.59  thf(fact_3378_add__le__add__imp__diff__le,axiom,
% 0.35/0.59      ! [I: nat,K: nat,N: nat,J: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.35/0.59       => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.35/0.59         => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.35/0.59           => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.35/0.59             => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_le_add_imp_diff_le
% 0.35/0.59  thf(fact_3379_add__le__add__imp__diff__le,axiom,
% 0.35/0.59      ! [I: int,K: int,N: int,J: int] :
% 0.35/0.59        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.35/0.59       => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.35/0.59         => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.35/0.59           => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.35/0.59             => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_le_add_imp_diff_le
% 0.35/0.59  thf(fact_3380_add__le__imp__le__diff,axiom,
% 0.35/0.59      ! [I: real,K: real,N: real] :
% 0.35/0.59        ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.35/0.59       => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_le_imp_le_diff
% 0.35/0.59  thf(fact_3381_add__le__imp__le__diff,axiom,
% 0.35/0.59      ! [I: rat,K: rat,N: rat] :
% 0.35/0.59        ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.35/0.59       => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_le_imp_le_diff
% 0.35/0.59  thf(fact_3382_add__le__imp__le__diff,axiom,
% 0.35/0.59      ! [I: nat,K: nat,N: nat] :
% 0.35/0.59        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.35/0.59       => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_le_imp_le_diff
% 0.35/0.59  thf(fact_3383_add__le__imp__le__diff,axiom,
% 0.35/0.59      ! [I: int,K: int,N: int] :
% 0.35/0.59        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.35/0.59       => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % add_le_imp_le_diff
% 0.35/0.59  thf(fact_3384_less__diff__eq,axiom,
% 0.35/0.59      ! [A: real,C: real,B: real] :
% 0.35/0.59        ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
% 0.35/0.59        = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.35/0.59  
% 0.35/0.59  % less_diff_eq
% 0.35/0.59  thf(fact_3385_less__diff__eq,axiom,
% 0.35/0.59      ! [A: rat,C: rat,B: rat] :
% 0.35/0.59        ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 0.35/0.59        = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.35/0.59  
% 0.35/0.59  % less_diff_eq
% 0.35/0.59  thf(fact_3386_less__diff__eq,axiom,
% 0.35/0.59      ! [A: int,C: int,B: int] :
% 0.35/0.59        ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.35/0.59        = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.35/0.59  
% 0.35/0.59  % less_diff_eq
% 0.35/0.59  thf(fact_3387_diff__less__eq,axiom,
% 0.35/0.59      ! [A: real,B: real,C: real] :
% 0.35/0.59        ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.35/0.59        = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_less_eq
% 0.35/0.59  thf(fact_3388_diff__less__eq,axiom,
% 0.35/0.59      ! [A: rat,B: rat,C: rat] :
% 0.35/0.59        ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.35/0.59        = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_less_eq
% 0.35/0.59  thf(fact_3389_diff__less__eq,axiom,
% 0.35/0.59      ! [A: int,B: int,C: int] :
% 0.35/0.59        ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.35/0.59        = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_less_eq
% 0.35/0.59  thf(fact_3390_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.35/0.59      ! [A: real,B: real] :
% 0.35/0.59        ( ~ ( ord_less_real @ A @ B )
% 0.35/0.59       => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 0.35/0.59          = A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % linordered_semidom_class.add_diff_inverse
% 0.35/0.59  thf(fact_3391_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.35/0.59      ! [A: rat,B: rat] :
% 0.35/0.59        ( ~ ( ord_less_rat @ A @ B )
% 0.35/0.59       => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 0.35/0.59          = A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % linordered_semidom_class.add_diff_inverse
% 0.35/0.59  thf(fact_3392_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.35/0.59      ! [A: nat,B: nat] :
% 0.35/0.59        ( ~ ( ord_less_nat @ A @ B )
% 0.35/0.59       => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 0.35/0.59          = A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % linordered_semidom_class.add_diff_inverse
% 0.35/0.59  thf(fact_3393_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.35/0.59      ! [A: int,B: int] :
% 0.35/0.59        ( ~ ( ord_less_int @ A @ B )
% 0.35/0.59       => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 0.35/0.59          = A ) ) ).
% 0.35/0.59  
% 0.35/0.59  % linordered_semidom_class.add_diff_inverse
% 0.35/0.59  thf(fact_3394_vebt__delete_Osimps_I3_J,axiom,
% 0.35/0.59      ! [A: $o,B: $o,N: nat] :
% 0.35/0.59        ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N ) ) )
% 0.35/0.59        = ( vEBT_Leaf @ A @ B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % vebt_delete.simps(3)
% 0.35/0.59  thf(fact_3395_vebt__delete_Osimps_I1_J,axiom,
% 0.35/0.59      ! [A: $o,B: $o] :
% 0.35/0.59        ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat )
% 0.35/0.59        = ( vEBT_Leaf @ $false @ B ) ) ).
% 0.35/0.59  
% 0.35/0.59  % vebt_delete.simps(1)
% 0.35/0.59  thf(fact_3396_diff__less__Suc,axiom,
% 0.35/0.59      ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 0.35/0.59  
% 0.35/0.59  % diff_less_Suc
% 0.35/0.59  thf(fact_3397_Suc__diff__Suc,axiom,
% 0.35/0.59      ! [N: nat,M: nat] :
% 0.35/0.59        ( ( ord_less_nat @ N @ M )
% 0.35/0.59       => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
% 0.35/0.60          = ( minus_minus_nat @ M @ N ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Suc_diff_Suc
% 0.35/0.60  thf(fact_3398_diff__less,axiom,
% 0.35/0.60      ! [N: nat,M: nat] :
% 0.35/0.60        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.60         => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % diff_less
% 0.35/0.60  thf(fact_3399_Suc__diff__le,axiom,
% 0.35/0.60      ! [N: nat,M: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.60       => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.35/0.60          = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Suc_diff_le
% 0.35/0.60  thf(fact_3400_diff__less__mono,axiom,
% 0.35/0.60      ! [A: nat,B: nat,C: nat] :
% 0.35/0.60        ( ( ord_less_nat @ A @ B )
% 0.35/0.60       => ( ( ord_less_eq_nat @ C @ A )
% 0.35/0.60         => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % diff_less_mono
% 0.35/0.60  thf(fact_3401_less__diff__iff,axiom,
% 0.35/0.60      ! [K: nat,M: nat,N: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ K @ M )
% 0.35/0.60       => ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.60         => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.35/0.60            = ( ord_less_nat @ M @ N ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % less_diff_iff
% 0.35/0.60  thf(fact_3402_diff__add__0,axiom,
% 0.35/0.60      ! [N: nat,M: nat] :
% 0.35/0.60        ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 0.35/0.60        = zero_zero_nat ) ).
% 0.35/0.60  
% 0.35/0.60  % diff_add_0
% 0.35/0.60  thf(fact_3403_add__diff__inverse__nat,axiom,
% 0.35/0.60      ! [M: nat,N: nat] :
% 0.35/0.60        ( ~ ( ord_less_nat @ M @ N )
% 0.35/0.60       => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.60          = M ) ) ).
% 0.35/0.60  
% 0.35/0.60  % add_diff_inverse_nat
% 0.35/0.60  thf(fact_3404_less__diff__conv,axiom,
% 0.35/0.60      ! [I: nat,J: nat,K: nat] :
% 0.35/0.60        ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.35/0.60        = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% 0.35/0.60  
% 0.35/0.60  % less_diff_conv
% 0.35/0.60  thf(fact_3405_le__diff__conv,axiom,
% 0.35/0.60      ! [J: nat,K: nat,I: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.35/0.60        = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % le_diff_conv
% 0.35/0.60  thf(fact_3406_Nat_Ole__diff__conv2,axiom,
% 0.35/0.60      ! [K: nat,J: nat,I: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ K @ J )
% 0.35/0.60       => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.35/0.60          = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Nat.le_diff_conv2
% 0.35/0.60  thf(fact_3407_Nat_Odiff__add__assoc,axiom,
% 0.35/0.60      ! [K: nat,J: nat,I: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ K @ J )
% 0.35/0.60       => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.35/0.60          = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Nat.diff_add_assoc
% 0.35/0.60  thf(fact_3408_Nat_Odiff__add__assoc2,axiom,
% 0.35/0.60      ! [K: nat,J: nat,I: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ K @ J )
% 0.35/0.60       => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 0.35/0.60          = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Nat.diff_add_assoc2
% 0.35/0.60  thf(fact_3409_Nat_Ole__imp__diff__is__add,axiom,
% 0.35/0.60      ! [I: nat,J: nat,K: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.60       => ( ( ( minus_minus_nat @ J @ I )
% 0.35/0.60            = K )
% 0.35/0.60          = ( J
% 0.35/0.60            = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Nat.le_imp_diff_is_add
% 0.35/0.60  thf(fact_3410_diff__Suc__eq__diff__pred,axiom,
% 0.35/0.60      ! [M: nat,N: nat] :
% 0.35/0.60        ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 0.35/0.60        = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% 0.35/0.60  
% 0.35/0.60  % diff_Suc_eq_diff_pred
% 0.35/0.60  thf(fact_3411_vebt__buildup_Osimps_I1_J,axiom,
% 0.35/0.60      ( ( vEBT_vebt_buildup @ zero_zero_nat )
% 0.35/0.60      = ( vEBT_Leaf @ $false @ $false ) ) ).
% 0.35/0.60  
% 0.35/0.60  % vebt_buildup.simps(1)
% 0.35/0.60  thf(fact_3412_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
% 0.35/0.60      ! [Uu: $o,Uv: $o,D: nat] :
% 0.35/0.60        ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
% 0.35/0.60        = ( D = one_one_nat ) ) ).
% 0.35/0.60  
% 0.35/0.60  % VEBT_internal.valid'.simps(1)
% 0.35/0.60  thf(fact_3413_nat__minus__add__max,axiom,
% 0.35/0.60      ! [N: nat,M: nat] :
% 0.35/0.60        ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
% 0.35/0.60        = ( ord_max_nat @ N @ M ) ) ).
% 0.35/0.60  
% 0.35/0.60  % nat_minus_add_max
% 0.35/0.60  thf(fact_3414_VEBT__internal_Onaive__member_Ocases,axiom,
% 0.35/0.60      ! [X: produc9072475918466114483BT_nat] :
% 0.35/0.60        ( ! [A3: $o,B3: $o,X4: nat] :
% 0.35/0.60            ( X
% 0.35/0.60           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X4 ) )
% 0.35/0.60       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
% 0.35/0.60              ( X
% 0.35/0.60             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
% 0.35/0.60         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S4: vEBT_VEBT,X4: nat] :
% 0.35/0.60                ( X
% 0.35/0.60               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S4 ) @ X4 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % VEBT_internal.naive_member.cases
% 0.35/0.60  thf(fact_3415_ordered__ring__class_Ole__add__iff2,axiom,
% 0.35/0.60      ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.35/0.60        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.35/0.60        = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % ordered_ring_class.le_add_iff2
% 0.35/0.60  thf(fact_3416_ordered__ring__class_Ole__add__iff2,axiom,
% 0.35/0.60      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.35/0.60        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.35/0.60        = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % ordered_ring_class.le_add_iff2
% 0.35/0.60  thf(fact_3417_ordered__ring__class_Ole__add__iff2,axiom,
% 0.35/0.60      ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.35/0.60        = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % ordered_ring_class.le_add_iff2
% 0.35/0.60  thf(fact_3418_ordered__ring__class_Ole__add__iff1,axiom,
% 0.35/0.60      ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.35/0.60        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.35/0.60        = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.35/0.60  
% 0.35/0.60  % ordered_ring_class.le_add_iff1
% 0.35/0.60  thf(fact_3419_ordered__ring__class_Ole__add__iff1,axiom,
% 0.35/0.60      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.35/0.60        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.35/0.60        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.35/0.60  
% 0.35/0.60  % ordered_ring_class.le_add_iff1
% 0.35/0.60  thf(fact_3420_ordered__ring__class_Ole__add__iff1,axiom,
% 0.35/0.60      ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.35/0.60        = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.35/0.60  
% 0.35/0.60  % ordered_ring_class.le_add_iff1
% 0.35/0.60  thf(fact_3421_less__add__iff2,axiom,
% 0.35/0.60      ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.35/0.60        ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.35/0.60        = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % less_add_iff2
% 0.35/0.60  thf(fact_3422_less__add__iff2,axiom,
% 0.35/0.60      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.35/0.60        ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.35/0.60        = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % less_add_iff2
% 0.35/0.60  thf(fact_3423_less__add__iff2,axiom,
% 0.35/0.60      ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.35/0.60        ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.35/0.60        = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % less_add_iff2
% 0.35/0.60  thf(fact_3424_less__add__iff1,axiom,
% 0.35/0.60      ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.35/0.60        ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.35/0.60        = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.35/0.60  
% 0.35/0.60  % less_add_iff1
% 0.35/0.60  thf(fact_3425_less__add__iff1,axiom,
% 0.35/0.60      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.35/0.60        ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.35/0.60        = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.35/0.60  
% 0.35/0.60  % less_add_iff1
% 0.35/0.60  thf(fact_3426_less__add__iff1,axiom,
% 0.35/0.60      ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.35/0.60        ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.35/0.60        = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.35/0.60  
% 0.35/0.60  % less_add_iff1
% 0.35/0.60  thf(fact_3427_invar__vebt_Ointros_I1_J,axiom,
% 0.35/0.60      ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).
% 0.35/0.60  
% 0.35/0.60  % invar_vebt.intros(1)
% 0.35/0.60  thf(fact_3428_divide__diff__eq__iff,axiom,
% 0.35/0.60      ! [Z2: complex,X: complex,Y: complex] :
% 0.35/0.60        ( ( Z2 != zero_zero_complex )
% 0.35/0.60       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z2 ) @ Y )
% 0.35/0.60          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % divide_diff_eq_iff
% 0.35/0.60  thf(fact_3429_divide__diff__eq__iff,axiom,
% 0.35/0.60      ! [Z2: real,X: real,Y: real] :
% 0.35/0.60        ( ( Z2 != zero_zero_real )
% 0.35/0.60       => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z2 ) @ Y )
% 0.35/0.60          = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % divide_diff_eq_iff
% 0.35/0.60  thf(fact_3430_divide__diff__eq__iff,axiom,
% 0.35/0.60      ! [Z2: rat,X: rat,Y: rat] :
% 0.35/0.60        ( ( Z2 != zero_zero_rat )
% 0.35/0.60       => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z2 ) @ Y )
% 0.35/0.60          = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % divide_diff_eq_iff
% 0.35/0.60  thf(fact_3431_diff__divide__eq__iff,axiom,
% 0.35/0.60      ! [Z2: complex,X: complex,Y: complex] :
% 0.35/0.60        ( ( Z2 != zero_zero_complex )
% 0.35/0.60       => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z2 ) )
% 0.35/0.60          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % diff_divide_eq_iff
% 0.35/0.60  thf(fact_3432_diff__divide__eq__iff,axiom,
% 0.35/0.60      ! [Z2: real,X: real,Y: real] :
% 0.35/0.60        ( ( Z2 != zero_zero_real )
% 0.35/0.60       => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z2 ) )
% 0.35/0.60          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % diff_divide_eq_iff
% 0.35/0.60  thf(fact_3433_diff__divide__eq__iff,axiom,
% 0.35/0.60      ! [Z2: rat,X: rat,Y: rat] :
% 0.35/0.60        ( ( Z2 != zero_zero_rat )
% 0.35/0.60       => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z2 ) )
% 0.35/0.60          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % diff_divide_eq_iff
% 0.35/0.60  thf(fact_3434_diff__frac__eq,axiom,
% 0.35/0.60      ! [Y: complex,Z2: complex,X: complex,W2: complex] :
% 0.35/0.60        ( ( Y != zero_zero_complex )
% 0.35/0.60       => ( ( Z2 != zero_zero_complex )
% 0.35/0.60         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W2 @ Z2 ) )
% 0.35/0.60            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ W2 @ Y ) ) @ ( times_times_complex @ Y @ Z2 ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % diff_frac_eq
% 0.35/0.60  thf(fact_3435_diff__frac__eq,axiom,
% 0.35/0.60      ! [Y: real,Z2: real,X: real,W2: real] :
% 0.35/0.60        ( ( Y != zero_zero_real )
% 0.35/0.60       => ( ( Z2 != zero_zero_real )
% 0.35/0.60         => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z2 ) )
% 0.35/0.60            = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % diff_frac_eq
% 0.35/0.60  thf(fact_3436_diff__frac__eq,axiom,
% 0.35/0.60      ! [Y: rat,Z2: rat,X: rat,W2: rat] :
% 0.35/0.60        ( ( Y != zero_zero_rat )
% 0.35/0.60       => ( ( Z2 != zero_zero_rat )
% 0.35/0.60         => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z2 ) )
% 0.35/0.60            = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % diff_frac_eq
% 0.35/0.60  thf(fact_3437_add__divide__eq__if__simps_I4_J,axiom,
% 0.35/0.60      ! [Z2: complex,A: complex,B: complex] :
% 0.35/0.60        ( ( ( Z2 = zero_zero_complex )
% 0.35/0.60         => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 0.35/0.60            = A ) )
% 0.35/0.60        & ( ( Z2 != zero_zero_complex )
% 0.35/0.60         => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 0.35/0.60            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % add_divide_eq_if_simps(4)
% 0.35/0.60  thf(fact_3438_add__divide__eq__if__simps_I4_J,axiom,
% 0.35/0.60      ! [Z2: real,A: real,B: real] :
% 0.35/0.60        ( ( ( Z2 = zero_zero_real )
% 0.35/0.60         => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 0.35/0.60            = A ) )
% 0.35/0.60        & ( ( Z2 != zero_zero_real )
% 0.35/0.60         => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 0.35/0.60            = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % add_divide_eq_if_simps(4)
% 0.35/0.60  thf(fact_3439_add__divide__eq__if__simps_I4_J,axiom,
% 0.35/0.60      ! [Z2: rat,A: rat,B: rat] :
% 0.35/0.60        ( ( ( Z2 = zero_zero_rat )
% 0.35/0.60         => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 0.35/0.60            = A ) )
% 0.35/0.60        & ( ( Z2 != zero_zero_rat )
% 0.35/0.60         => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 0.35/0.60            = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % add_divide_eq_if_simps(4)
% 0.35/0.60  thf(fact_3440_vebt__delete_Osimps_I2_J,axiom,
% 0.35/0.60      ! [A: $o,B: $o] :
% 0.35/0.60        ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) )
% 0.35/0.60        = ( vEBT_Leaf @ A @ $false ) ) ).
% 0.35/0.60  
% 0.35/0.60  % vebt_delete.simps(2)
% 0.35/0.60  thf(fact_3441_VEBT__internal_OminNull_Ocases,axiom,
% 0.35/0.60      ! [X: vEBT_VEBT] :
% 0.35/0.60        ( ( X
% 0.35/0.60         != ( vEBT_Leaf @ $false @ $false ) )
% 0.35/0.60       => ( ! [Uv2: $o] :
% 0.35/0.60              ( X
% 0.35/0.60             != ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.35/0.60         => ( ! [Uu2: $o] :
% 0.35/0.60                ( X
% 0.35/0.60               != ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.35/0.60           => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.35/0.60                  ( X
% 0.35/0.60                 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.35/0.60             => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.60                    ( X
% 0.35/0.60                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % VEBT_internal.minNull.cases
% 0.35/0.60  thf(fact_3442_diff__Suc__less,axiom,
% 0.35/0.60      ! [N: nat,I: nat] :
% 0.35/0.60        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60       => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 0.35/0.60  
% 0.35/0.60  % diff_Suc_less
% 0.35/0.60  thf(fact_3443_vebt__member_Osimps_I1_J,axiom,
% 0.35/0.60      ! [A: $o,B: $o,X: nat] :
% 0.35/0.60        ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X )
% 0.35/0.60        = ( ( ( X = zero_zero_nat )
% 0.35/0.60           => A )
% 0.35/0.60          & ( ( X != zero_zero_nat )
% 0.35/0.60           => ( ( ( X = one_one_nat )
% 0.35/0.60               => B )
% 0.35/0.60              & ( X = one_one_nat ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % vebt_member.simps(1)
% 0.35/0.60  thf(fact_3444_vebt__buildup_Osimps_I2_J,axiom,
% 0.35/0.60      ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
% 0.35/0.60      = ( vEBT_Leaf @ $false @ $false ) ) ).
% 0.35/0.60  
% 0.35/0.60  % vebt_buildup.simps(2)
% 0.35/0.60  thf(fact_3445_nat__diff__split,axiom,
% 0.35/0.60      ! [P: nat > $o,A: nat,B: nat] :
% 0.35/0.60        ( ( P @ ( minus_minus_nat @ A @ B ) )
% 0.35/0.60        = ( ( ( ord_less_nat @ A @ B )
% 0.35/0.60           => ( P @ zero_zero_nat ) )
% 0.35/0.60          & ! [D4: nat] :
% 0.35/0.60              ( ( A
% 0.35/0.60                = ( plus_plus_nat @ B @ D4 ) )
% 0.35/0.60             => ( P @ D4 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % nat_diff_split
% 0.35/0.60  thf(fact_3446_nat__diff__split__asm,axiom,
% 0.35/0.60      ! [P: nat > $o,A: nat,B: nat] :
% 0.35/0.60        ( ( P @ ( minus_minus_nat @ A @ B ) )
% 0.35/0.60        = ( ~ ( ( ( ord_less_nat @ A @ B )
% 0.35/0.60                & ~ ( P @ zero_zero_nat ) )
% 0.35/0.60              | ? [D4: nat] :
% 0.35/0.60                  ( ( A
% 0.35/0.60                    = ( plus_plus_nat @ B @ D4 ) )
% 0.35/0.60                  & ~ ( P @ D4 ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % nat_diff_split_asm
% 0.35/0.60  thf(fact_3447_less__diff__conv2,axiom,
% 0.35/0.60      ! [K: nat,J: nat,I: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ K @ J )
% 0.35/0.60       => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.35/0.60          = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % less_diff_conv2
% 0.35/0.60  thf(fact_3448_VEBT__internal_OminNull_Oelims_I3_J,axiom,
% 0.35/0.60      ! [X: vEBT_VEBT] :
% 0.35/0.60        ( ~ ( vEBT_VEBT_minNull @ X )
% 0.35/0.60       => ( ! [Uv2: $o] :
% 0.35/0.60              ( X
% 0.35/0.60             != ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.35/0.60         => ( ! [Uu2: $o] :
% 0.35/0.60                ( X
% 0.35/0.60               != ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.35/0.60           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.60                  ( X
% 0.35/0.60                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % VEBT_internal.minNull.elims(3)
% 0.35/0.60  thf(fact_3449_vebt__insert_Osimps_I1_J,axiom,
% 0.35/0.60      ! [X: nat,A: $o,B: $o] :
% 0.35/0.60        ( ( ( X = zero_zero_nat )
% 0.35/0.60         => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
% 0.35/0.60            = ( vEBT_Leaf @ $true @ B ) ) )
% 0.35/0.60        & ( ( X != zero_zero_nat )
% 0.35/0.60         => ( ( ( X = one_one_nat )
% 0.35/0.60             => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
% 0.35/0.60                = ( vEBT_Leaf @ A @ $true ) ) )
% 0.35/0.60            & ( ( X != one_one_nat )
% 0.35/0.60             => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
% 0.35/0.60                = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % vebt_insert.simps(1)
% 0.35/0.60  thf(fact_3450_nat__diff__add__eq2,axiom,
% 0.35/0.60      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.60       => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.35/0.60          = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % nat_diff_add_eq2
% 0.35/0.60  thf(fact_3451_nat__diff__add__eq1,axiom,
% 0.35/0.60      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ J @ I )
% 0.35/0.60       => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.35/0.60          = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % nat_diff_add_eq1
% 0.35/0.60  thf(fact_3452_nat__le__add__iff2,axiom,
% 0.35/0.60      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.60       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.35/0.60          = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % nat_le_add_iff2
% 0.35/0.60  thf(fact_3453_nat__le__add__iff1,axiom,
% 0.35/0.60      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ J @ I )
% 0.35/0.60       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.35/0.60          = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % nat_le_add_iff1
% 0.35/0.60  thf(fact_3454_nat__eq__add__iff2,axiom,
% 0.35/0.60      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.60       => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 0.35/0.60            = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.35/0.60          = ( M
% 0.35/0.60            = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % nat_eq_add_iff2
% 0.35/0.60  thf(fact_3455_nat__eq__add__iff1,axiom,
% 0.35/0.60      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ J @ I )
% 0.35/0.60       => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 0.35/0.60            = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.35/0.60          = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
% 0.35/0.60            = N ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % nat_eq_add_iff1
% 0.35/0.60  thf(fact_3456_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
% 0.35/0.60      ! [A: $o,B: $o,X: nat] :
% 0.35/0.60        ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
% 0.35/0.60        = ( ( ( X = zero_zero_nat )
% 0.35/0.60           => A )
% 0.35/0.60          & ( ( X != zero_zero_nat )
% 0.35/0.60           => ( ( ( X = one_one_nat )
% 0.35/0.60               => B )
% 0.35/0.60              & ( X = one_one_nat ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % VEBT_internal.naive_member.simps(1)
% 0.35/0.60  thf(fact_3457_vebt__succ_Osimps_I2_J,axiom,
% 0.35/0.60      ! [Uv: $o,Uw: $o,N: nat] :
% 0.35/0.60        ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
% 0.35/0.60        = none_nat ) ).
% 0.35/0.60  
% 0.35/0.60  % vebt_succ.simps(2)
% 0.35/0.60  thf(fact_3458_vebt__pred_Osimps_I1_J,axiom,
% 0.35/0.60      ! [Uu: $o,Uv: $o] :
% 0.35/0.60        ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
% 0.35/0.60        = none_nat ) ).
% 0.35/0.60  
% 0.35/0.60  % vebt_pred.simps(1)
% 0.35/0.60  thf(fact_3459_VEBT__internal_OminNull_Oelims_I2_J,axiom,
% 0.35/0.60      ! [X: vEBT_VEBT] :
% 0.35/0.60        ( ( vEBT_VEBT_minNull @ X )
% 0.35/0.60       => ( ( X
% 0.35/0.60           != ( vEBT_Leaf @ $false @ $false ) )
% 0.35/0.60         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.35/0.60                ( X
% 0.35/0.60               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % VEBT_internal.minNull.elims(2)
% 0.35/0.60  thf(fact_3460_realpow__pos__nth2,axiom,
% 0.35/0.60      ! [A: real,N: nat] :
% 0.35/0.60        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.60       => ? [R3: real] :
% 0.35/0.60            ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.35/0.60            & ( ( power_power_real @ R3 @ ( suc @ N ) )
% 0.35/0.60              = A ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % realpow_pos_nth2
% 0.35/0.60  thf(fact_3461_frac__le__eq,axiom,
% 0.35/0.60      ! [Y: real,Z2: real,X: real,W2: real] :
% 0.35/0.60        ( ( Y != zero_zero_real )
% 0.35/0.60       => ( ( Z2 != zero_zero_real )
% 0.35/0.60         => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z2 ) )
% 0.35/0.60            = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % frac_le_eq
% 0.35/0.60  thf(fact_3462_frac__le__eq,axiom,
% 0.35/0.60      ! [Y: rat,Z2: rat,X: rat,W2: rat] :
% 0.35/0.60        ( ( Y != zero_zero_rat )
% 0.35/0.60       => ( ( Z2 != zero_zero_rat )
% 0.35/0.60         => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z2 ) )
% 0.35/0.60            = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % frac_le_eq
% 0.35/0.60  thf(fact_3463_frac__less__eq,axiom,
% 0.35/0.60      ! [Y: real,Z2: real,X: real,W2: real] :
% 0.35/0.60        ( ( Y != zero_zero_real )
% 0.35/0.60       => ( ( Z2 != zero_zero_real )
% 0.35/0.60         => ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z2 ) )
% 0.35/0.60            = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % frac_less_eq
% 0.35/0.60  thf(fact_3464_frac__less__eq,axiom,
% 0.35/0.60      ! [Y: rat,Z2: rat,X: rat,W2: rat] :
% 0.35/0.60        ( ( Y != zero_zero_rat )
% 0.35/0.60       => ( ( Z2 != zero_zero_rat )
% 0.35/0.60         => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z2 ) )
% 0.35/0.60            = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % frac_less_eq
% 0.35/0.60  thf(fact_3465_power__diff,axiom,
% 0.35/0.60      ! [A: complex,N: nat,M: nat] :
% 0.35/0.60        ( ( A != zero_zero_complex )
% 0.35/0.60       => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.60         => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.60            = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % power_diff
% 0.35/0.60  thf(fact_3466_power__diff,axiom,
% 0.35/0.60      ! [A: nat,N: nat,M: nat] :
% 0.35/0.60        ( ( A != zero_zero_nat )
% 0.35/0.60       => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.60         => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.60            = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % power_diff
% 0.35/0.60  thf(fact_3467_power__diff,axiom,
% 0.35/0.60      ! [A: int,N: nat,M: nat] :
% 0.35/0.60        ( ( A != zero_zero_int )
% 0.35/0.60       => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.60         => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.60            = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % power_diff
% 0.35/0.60  thf(fact_3468_power__diff,axiom,
% 0.35/0.60      ! [A: real,N: nat,M: nat] :
% 0.35/0.60        ( ( A != zero_zero_real )
% 0.35/0.60       => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.60         => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.60            = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % power_diff
% 0.35/0.60  thf(fact_3469_power__diff,axiom,
% 0.35/0.60      ! [A: rat,N: nat,M: nat] :
% 0.35/0.60        ( ( A != zero_zero_rat )
% 0.35/0.60       => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.60         => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.60            = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % power_diff
% 0.35/0.60  thf(fact_3470_power__diff,axiom,
% 0.35/0.60      ! [A: code_integer,N: nat,M: nat] :
% 0.35/0.60        ( ( A != zero_z3403309356797280102nteger )
% 0.35/0.60       => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.60         => ( ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.60            = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % power_diff
% 0.35/0.60  thf(fact_3471_vebt__mint_Ocases,axiom,
% 0.35/0.60      ! [X: vEBT_VEBT] :
% 0.35/0.60        ( ! [A3: $o,B3: $o] :
% 0.35/0.60            ( X
% 0.35/0.60           != ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.60       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.60              ( X
% 0.35/0.60             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.35/0.60         => ~ ! [Mi3: nat,Ma3: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.35/0.60                ( X
% 0.35/0.60               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % vebt_mint.cases
% 0.35/0.60  thf(fact_3472_div__if,axiom,
% 0.35/0.60      ( divide_divide_nat
% 0.35/0.60      = ( ^ [M2: nat,N2: nat] :
% 0.35/0.60            ( if_nat
% 0.35/0.60            @ ( ( ord_less_nat @ M2 @ N2 )
% 0.35/0.60              | ( N2 = zero_zero_nat ) )
% 0.35/0.60            @ zero_zero_nat
% 0.35/0.60            @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % div_if
% 0.35/0.60  thf(fact_3473_Suc__pred_H,axiom,
% 0.35/0.60      ! [N: nat] :
% 0.35/0.60        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60       => ( N
% 0.35/0.60          = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Suc_pred'
% 0.35/0.60  thf(fact_3474_Suc__diff__eq__diff__pred,axiom,
% 0.35/0.60      ! [N: nat,M: nat] :
% 0.35/0.60        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60       => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.35/0.60          = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Suc_diff_eq_diff_pred
% 0.35/0.60  thf(fact_3475_add__eq__if,axiom,
% 0.35/0.60      ( plus_plus_nat
% 0.35/0.60      = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % add_eq_if
% 0.35/0.60  thf(fact_3476_nat__less__add__iff2,axiom,
% 0.35/0.60      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.60       => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.35/0.60          = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % nat_less_add_iff2
% 0.35/0.60  thf(fact_3477_nat__less__add__iff1,axiom,
% 0.35/0.60      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ J @ I )
% 0.35/0.60       => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.35/0.60          = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % nat_less_add_iff1
% 0.35/0.60  thf(fact_3478_mult__eq__if,axiom,
% 0.35/0.60      ( times_times_nat
% 0.35/0.60      = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % mult_eq_if
% 0.35/0.60  thf(fact_3479_VEBT__internal_OminNull_Oelims_I1_J,axiom,
% 0.35/0.60      ! [X: vEBT_VEBT,Y: $o] :
% 0.35/0.60        ( ( ( vEBT_VEBT_minNull @ X )
% 0.35/0.60          = Y )
% 0.35/0.60       => ( ( ( X
% 0.35/0.60              = ( vEBT_Leaf @ $false @ $false ) )
% 0.35/0.60           => ~ Y )
% 0.35/0.60         => ( ( ? [Uv2: $o] :
% 0.35/0.60                  ( X
% 0.35/0.60                  = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.35/0.60             => Y )
% 0.35/0.60           => ( ( ? [Uu2: $o] :
% 0.35/0.60                    ( X
% 0.35/0.60                    = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.35/0.60               => Y )
% 0.35/0.60             => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.35/0.60                      ( X
% 0.35/0.60                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.35/0.60                 => ~ Y )
% 0.35/0.60               => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.60                        ( X
% 0.35/0.60                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 0.35/0.60                   => Y ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % VEBT_internal.minNull.elims(1)
% 0.35/0.60  thf(fact_3480_scaling__mono,axiom,
% 0.35/0.60      ! [U: real,V: real,R2: real,S: real] :
% 0.35/0.60        ( ( ord_less_eq_real @ U @ V )
% 0.35/0.60       => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 0.35/0.60         => ( ( ord_less_eq_real @ R2 @ S )
% 0.35/0.60           => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % scaling_mono
% 0.35/0.60  thf(fact_3481_scaling__mono,axiom,
% 0.35/0.60      ! [U: rat,V: rat,R2: rat,S: rat] :
% 0.35/0.60        ( ( ord_less_eq_rat @ U @ V )
% 0.35/0.60       => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 0.35/0.60         => ( ( ord_less_eq_rat @ R2 @ S )
% 0.35/0.60           => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % scaling_mono
% 0.35/0.60  thf(fact_3482_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.35/0.60      ! [N: nat,M: nat] :
% 0.35/0.60        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.60         != zero_zero_nat )
% 0.35/0.60       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 0.35/0.60         != zero_zero_nat ) ) ).
% 0.35/0.60  
% 0.35/0.60  % exp_not_zero_imp_exp_diff_not_zero
% 0.35/0.60  thf(fact_3483_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.35/0.60      ! [N: nat,M: nat] :
% 0.35/0.60        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.35/0.60         != zero_zero_int )
% 0.35/0.60       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 0.35/0.60         != zero_zero_int ) ) ).
% 0.35/0.60  
% 0.35/0.60  % exp_not_zero_imp_exp_diff_not_zero
% 0.35/0.60  thf(fact_3484_power__diff__power__eq,axiom,
% 0.35/0.60      ! [A: nat,N: nat,M: nat] :
% 0.35/0.60        ( ( A != zero_zero_nat )
% 0.35/0.60       => ( ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.60           => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.35/0.60              = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 0.35/0.60          & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.35/0.60           => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.35/0.60              = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % power_diff_power_eq
% 0.35/0.60  thf(fact_3485_power__diff__power__eq,axiom,
% 0.35/0.60      ! [A: int,N: nat,M: nat] :
% 0.35/0.60        ( ( A != zero_zero_int )
% 0.35/0.60       => ( ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.60           => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.35/0.60              = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 0.35/0.60          & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.35/0.60           => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.35/0.60              = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % power_diff_power_eq
% 0.35/0.60  thf(fact_3486_power__diff__power__eq,axiom,
% 0.35/0.60      ! [A: code_integer,N: nat,M: nat] :
% 0.35/0.60        ( ( A != zero_z3403309356797280102nteger )
% 0.35/0.60       => ( ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.60           => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) )
% 0.35/0.60              = ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 0.35/0.60          & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.35/0.60           => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) )
% 0.35/0.60              = ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % power_diff_power_eq
% 0.35/0.60  thf(fact_3487_power__eq__if,axiom,
% 0.35/0.60      ( power_power_complex
% 0.35/0.60      = ( ^ [P6: complex,M2: nat] : ( if_complex @ ( M2 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P6 @ ( power_power_complex @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % power_eq_if
% 0.35/0.60  thf(fact_3488_power__eq__if,axiom,
% 0.35/0.60      ( power_power_real
% 0.35/0.60      = ( ^ [P6: real,M2: nat] : ( if_real @ ( M2 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P6 @ ( power_power_real @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % power_eq_if
% 0.35/0.60  thf(fact_3489_power__eq__if,axiom,
% 0.35/0.60      ( power_power_rat
% 0.35/0.60      = ( ^ [P6: rat,M2: nat] : ( if_rat @ ( M2 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P6 @ ( power_power_rat @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % power_eq_if
% 0.35/0.60  thf(fact_3490_power__eq__if,axiom,
% 0.35/0.60      ( power_power_nat
% 0.35/0.60      = ( ^ [P6: nat,M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P6 @ ( power_power_nat @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % power_eq_if
% 0.35/0.60  thf(fact_3491_power__eq__if,axiom,
% 0.35/0.60      ( power_power_int
% 0.35/0.60      = ( ^ [P6: int,M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P6 @ ( power_power_int @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % power_eq_if
% 0.35/0.60  thf(fact_3492_vebt__maxt_Osimps_I1_J,axiom,
% 0.35/0.60      ! [B: $o,A: $o] :
% 0.35/0.60        ( ( B
% 0.35/0.60         => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 0.35/0.60            = ( some_nat @ one_one_nat ) ) )
% 0.35/0.60        & ( ~ B
% 0.35/0.60         => ( ( A
% 0.35/0.60             => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 0.35/0.60                = ( some_nat @ zero_zero_nat ) ) )
% 0.35/0.60            & ( ~ A
% 0.35/0.60             => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 0.35/0.60                = none_nat ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % vebt_maxt.simps(1)
% 0.35/0.60  thf(fact_3493_power__minus__mult,axiom,
% 0.35/0.60      ! [N: nat,A: complex] :
% 0.35/0.60        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60       => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.35/0.60          = ( power_power_complex @ A @ N ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % power_minus_mult
% 0.35/0.60  thf(fact_3494_power__minus__mult,axiom,
% 0.35/0.60      ! [N: nat,A: real] :
% 0.35/0.60        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60       => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.35/0.60          = ( power_power_real @ A @ N ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % power_minus_mult
% 0.35/0.60  thf(fact_3495_power__minus__mult,axiom,
% 0.35/0.60      ! [N: nat,A: rat] :
% 0.35/0.60        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60       => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.35/0.60          = ( power_power_rat @ A @ N ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % power_minus_mult
% 0.35/0.60  thf(fact_3496_power__minus__mult,axiom,
% 0.35/0.60      ! [N: nat,A: nat] :
% 0.35/0.60        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60       => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.35/0.60          = ( power_power_nat @ A @ N ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % power_minus_mult
% 0.35/0.60  thf(fact_3497_power__minus__mult,axiom,
% 0.35/0.60      ! [N: nat,A: int] :
% 0.35/0.60        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60       => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.35/0.60          = ( power_power_int @ A @ N ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % power_minus_mult
% 0.35/0.60  thf(fact_3498_vebt__mint_Osimps_I1_J,axiom,
% 0.35/0.60      ! [A: $o,B: $o] :
% 0.35/0.60        ( ( A
% 0.35/0.60         => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 0.35/0.60            = ( some_nat @ zero_zero_nat ) ) )
% 0.35/0.60        & ( ~ A
% 0.35/0.60         => ( ( B
% 0.35/0.60             => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 0.35/0.60                = ( some_nat @ one_one_nat ) ) )
% 0.35/0.60            & ( ~ B
% 0.35/0.60             => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 0.35/0.60                = none_nat ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % vebt_mint.simps(1)
% 0.35/0.60  thf(fact_3499_diff__le__diff__pow,axiom,
% 0.35/0.60      ! [K: nat,M: nat,N: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.35/0.60       => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % diff_le_diff_pow
% 0.35/0.60  thf(fact_3500_le__div__geq,axiom,
% 0.35/0.60      ! [N: nat,M: nat] :
% 0.35/0.60        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60       => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.60         => ( ( divide_divide_nat @ M @ N )
% 0.35/0.60            = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % le_div_geq
% 0.35/0.60  thf(fact_3501_vebt__pred_Osimps_I2_J,axiom,
% 0.35/0.60      ! [A: $o,Uw: $o] :
% 0.35/0.60        ( ( A
% 0.35/0.60         => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
% 0.35/0.60            = ( some_nat @ zero_zero_nat ) ) )
% 0.35/0.60        & ( ~ A
% 0.35/0.60         => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
% 0.35/0.60            = none_nat ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % vebt_pred.simps(2)
% 0.35/0.60  thf(fact_3502_vebt__succ_Osimps_I1_J,axiom,
% 0.35/0.60      ! [B: $o,Uu: $o] :
% 0.35/0.60        ( ( B
% 0.35/0.60         => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
% 0.35/0.60            = ( some_nat @ one_one_nat ) ) )
% 0.35/0.60        & ( ~ B
% 0.35/0.60         => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
% 0.35/0.60            = none_nat ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % vebt_succ.simps(1)
% 0.35/0.60  thf(fact_3503_vebt__insert_Ocases,axiom,
% 0.35/0.60      ! [X: produc9072475918466114483BT_nat] :
% 0.35/0.60        ( ! [A3: $o,B3: $o,X4: nat] :
% 0.35/0.60            ( X
% 0.35/0.60           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X4 ) )
% 0.35/0.60       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT,X4: nat] :
% 0.35/0.60              ( X
% 0.35/0.60             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S4 ) @ X4 ) )
% 0.35/0.60         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT,X4: nat] :
% 0.35/0.60                ( X
% 0.35/0.60               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S4 ) @ X4 ) )
% 0.35/0.60           => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,X4: nat] :
% 0.35/0.60                  ( X
% 0.35/0.60                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary3 ) @ X4 ) )
% 0.35/0.60             => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,X4: nat] :
% 0.35/0.60                    ( X
% 0.35/0.60                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ X4 ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % vebt_insert.cases
% 0.35/0.60  thf(fact_3504_vebt__pred_Ocases,axiom,
% 0.35/0.60      ! [X: produc9072475918466114483BT_nat] :
% 0.35/0.60        ( ! [Uu2: $o,Uv2: $o] :
% 0.35/0.60            ( X
% 0.35/0.60           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
% 0.35/0.60       => ( ! [A3: $o,Uw2: $o] :
% 0.35/0.60              ( X
% 0.35/0.60             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
% 0.35/0.60         => ( ! [A3: $o,B3: $o,Va2: nat] :
% 0.35/0.60                ( X
% 0.35/0.60               != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va2 ) ) ) )
% 0.35/0.60           => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
% 0.35/0.60                  ( X
% 0.35/0.60                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Vb2 ) )
% 0.35/0.60             => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
% 0.35/0.60                    ( X
% 0.35/0.60                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 ) )
% 0.35/0.60               => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
% 0.35/0.60                      ( X
% 0.35/0.60                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 ) )
% 0.35/0.60                 => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,X4: nat] :
% 0.35/0.60                        ( X
% 0.35/0.60                       != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ X4 ) ) ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % vebt_pred.cases
% 0.35/0.60  thf(fact_3505_vebt__succ_Ocases,axiom,
% 0.35/0.60      ! [X: produc9072475918466114483BT_nat] :
% 0.35/0.60        ( ! [Uu2: $o,B3: $o] :
% 0.35/0.60            ( X
% 0.35/0.60           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat ) )
% 0.35/0.60       => ( ! [Uv2: $o,Uw2: $o,N3: nat] :
% 0.35/0.60              ( X
% 0.35/0.60             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) ) )
% 0.35/0.60         => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] :
% 0.35/0.60                ( X
% 0.35/0.60               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va3 ) )
% 0.35/0.60           => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
% 0.35/0.60                  ( X
% 0.35/0.60                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 ) )
% 0.35/0.60             => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
% 0.35/0.60                    ( X
% 0.35/0.60                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 ) )
% 0.35/0.60               => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,X4: nat] :
% 0.35/0.60                      ( X
% 0.35/0.60                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ X4 ) ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % vebt_succ.cases
% 0.35/0.60  thf(fact_3506_vebt__delete_Ocases,axiom,
% 0.35/0.60      ! [X: produc9072475918466114483BT_nat] :
% 0.35/0.60        ( ! [A3: $o,B3: $o] :
% 0.35/0.60            ( X
% 0.35/0.60           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat ) )
% 0.35/0.60       => ( ! [A3: $o,B3: $o] :
% 0.35/0.60              ( X
% 0.35/0.60             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) ) )
% 0.35/0.60         => ( ! [A3: $o,B3: $o,N3: nat] :
% 0.35/0.60                ( X
% 0.35/0.60               != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N3 ) ) ) )
% 0.35/0.60           => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,Uu2: nat] :
% 0.35/0.60                  ( X
% 0.35/0.60                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary3 ) @ Uu2 ) )
% 0.35/0.60             => ( ! [Mi3: nat,Ma3: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,X4: nat] :
% 0.35/0.60                    ( X
% 0.35/0.60                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ X4 ) )
% 0.35/0.60               => ( ! [Mi3: nat,Ma3: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,X4: nat] :
% 0.35/0.60                      ( X
% 0.35/0.60                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ X4 ) )
% 0.35/0.60                 => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,X4: nat] :
% 0.35/0.60                        ( X
% 0.35/0.60                       != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ X4 ) ) ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % vebt_delete.cases
% 0.35/0.60  thf(fact_3507_vebt__member_Ocases,axiom,
% 0.35/0.60      ! [X: produc9072475918466114483BT_nat] :
% 0.35/0.60        ( ! [A3: $o,B3: $o,X4: nat] :
% 0.35/0.60            ( X
% 0.35/0.60           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X4 ) )
% 0.35/0.60       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] :
% 0.35/0.60              ( X
% 0.35/0.60             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X4 ) )
% 0.35/0.60         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] :
% 0.35/0.60                ( X
% 0.35/0.60               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X4 ) )
% 0.35/0.60           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
% 0.35/0.60                  ( X
% 0.35/0.60                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X4 ) )
% 0.35/0.60             => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,X4: nat] :
% 0.35/0.60                    ( X
% 0.35/0.60                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ X4 ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % vebt_member.cases
% 0.35/0.60  thf(fact_3508_VEBT__internal_Omembermima_Ocases,axiom,
% 0.35/0.60      ! [X: produc9072475918466114483BT_nat] :
% 0.35/0.60        ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
% 0.35/0.60            ( X
% 0.35/0.60           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
% 0.35/0.60       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
% 0.35/0.60              ( X
% 0.35/0.60             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
% 0.35/0.60         => ( ! [Mi3: nat,Ma3: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] :
% 0.35/0.60                ( X
% 0.35/0.60               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X4 ) )
% 0.35/0.60           => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
% 0.35/0.60                  ( X
% 0.35/0.60                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ X4 ) )
% 0.35/0.60             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT,X4: nat] :
% 0.35/0.60                    ( X
% 0.35/0.60                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ X4 ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % VEBT_internal.membermima.cases
% 0.35/0.60  thf(fact_3509_vebt__pred_Osimps_I3_J,axiom,
% 0.35/0.60      ! [B: $o,A: $o,Va: nat] :
% 0.35/0.60        ( ( B
% 0.35/0.60         => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
% 0.35/0.60            = ( some_nat @ one_one_nat ) ) )
% 0.35/0.60        & ( ~ B
% 0.35/0.60         => ( ( A
% 0.35/0.60             => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
% 0.35/0.60                = ( some_nat @ zero_zero_nat ) ) )
% 0.35/0.60            & ( ~ A
% 0.35/0.60             => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
% 0.35/0.60                = none_nat ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % vebt_pred.simps(3)
% 0.35/0.60  thf(fact_3510_realpow__pos__nth,axiom,
% 0.35/0.60      ! [N: nat,A: real] :
% 0.35/0.60        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.60         => ? [R3: real] :
% 0.35/0.60              ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.35/0.60              & ( ( power_power_real @ R3 @ N )
% 0.35/0.60                = A ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % realpow_pos_nth
% 0.35/0.60  thf(fact_3511_realpow__pos__nth__unique,axiom,
% 0.35/0.60      ! [N: nat,A: real] :
% 0.35/0.60        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.60         => ? [X4: real] :
% 0.35/0.60              ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.35/0.60              & ( ( power_power_real @ X4 @ N )
% 0.35/0.60                = A )
% 0.35/0.60              & ! [Y4: real] :
% 0.35/0.60                  ( ( ( ord_less_real @ zero_zero_real @ Y4 )
% 0.35/0.60                    & ( ( power_power_real @ Y4 @ N )
% 0.35/0.60                      = A ) )
% 0.35/0.60                 => ( Y4 = X4 ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % realpow_pos_nth_unique
% 0.35/0.60  thf(fact_3512_vebt__maxt_Oelims,axiom,
% 0.35/0.60      ! [X: vEBT_VEBT,Y: option_nat] :
% 0.35/0.60        ( ( ( vEBT_vebt_maxt @ X )
% 0.35/0.60          = Y )
% 0.35/0.60       => ( ! [A3: $o,B3: $o] :
% 0.35/0.60              ( ( X
% 0.35/0.60                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.60             => ~ ( ( B3
% 0.35/0.60                   => ( Y
% 0.35/0.60                      = ( some_nat @ one_one_nat ) ) )
% 0.35/0.60                  & ( ~ B3
% 0.35/0.60                   => ( ( A3
% 0.35/0.60                       => ( Y
% 0.35/0.60                          = ( some_nat @ zero_zero_nat ) ) )
% 0.35/0.60                      & ( ~ A3
% 0.35/0.60                       => ( Y = none_nat ) ) ) ) ) )
% 0.35/0.60         => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.60                  ( X
% 0.35/0.60                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.35/0.60             => ( Y != none_nat ) )
% 0.35/0.60           => ~ ! [Mi3: nat,Ma3: nat] :
% 0.35/0.60                  ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.35/0.60                      ( X
% 0.35/0.60                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.35/0.60                 => ( Y
% 0.35/0.60                   != ( some_nat @ Ma3 ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % vebt_maxt.elims
% 0.35/0.60  thf(fact_3513_vebt__mint_Oelims,axiom,
% 0.35/0.60      ! [X: vEBT_VEBT,Y: option_nat] :
% 0.35/0.60        ( ( ( vEBT_vebt_mint @ X )
% 0.35/0.60          = Y )
% 0.35/0.60       => ( ! [A3: $o,B3: $o] :
% 0.35/0.60              ( ( X
% 0.35/0.60                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.60             => ~ ( ( A3
% 0.35/0.60                   => ( Y
% 0.35/0.60                      = ( some_nat @ zero_zero_nat ) ) )
% 0.35/0.60                  & ( ~ A3
% 0.35/0.60                   => ( ( B3
% 0.35/0.60                       => ( Y
% 0.35/0.60                          = ( some_nat @ one_one_nat ) ) )
% 0.35/0.60                      & ( ~ B3
% 0.35/0.60                       => ( Y = none_nat ) ) ) ) ) )
% 0.35/0.60         => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.60                  ( X
% 0.35/0.60                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.35/0.60             => ( Y != none_nat ) )
% 0.35/0.60           => ~ ! [Mi3: nat] :
% 0.35/0.60                  ( ? [Ma3: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.35/0.60                      ( X
% 0.35/0.60                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.35/0.60                 => ( Y
% 0.35/0.60                   != ( some_nat @ Mi3 ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % vebt_mint.elims
% 0.35/0.60  thf(fact_3514_pos2,axiom,
% 0.35/0.60      ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 0.35/0.60  
% 0.35/0.60  % pos2
% 0.35/0.60  thf(fact_3515_invar__vebt_Ocases,axiom,
% 0.35/0.60      ! [A12: vEBT_VEBT,A23: nat] :
% 0.35/0.60        ( ( vEBT_invar_vebt @ A12 @ A23 )
% 0.35/0.60       => ( ( ? [A3: $o,B3: $o] :
% 0.35/0.60                ( A12
% 0.35/0.60                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.60           => ( A23
% 0.35/0.60             != ( suc @ zero_zero_nat ) ) )
% 0.35/0.60         => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat] :
% 0.35/0.60                ( ( A12
% 0.35/0.60                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary3 ) )
% 0.35/0.60               => ( ( A23 = Deg2 )
% 0.35/0.60                 => ( ! [X2: vEBT_VEBT] :
% 0.35/0.60                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.60                       => ( vEBT_invar_vebt @ X2 @ N3 ) )
% 0.35/0.60                   => ( ( vEBT_invar_vebt @ Summary3 @ M5 )
% 0.35/0.60                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.35/0.60                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 0.35/0.60                       => ( ( M5 = N3 )
% 0.35/0.60                         => ( ( Deg2
% 0.35/0.60                              = ( plus_plus_nat @ N3 @ M5 ) )
% 0.35/0.60                           => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X_12 )
% 0.35/0.60                             => ~ ! [X2: vEBT_VEBT] :
% 0.35/0.60                                    ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.60                                   => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ) ) ) ) ) ) )
% 0.35/0.60           => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat] :
% 0.35/0.60                  ( ( A12
% 0.35/0.60                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary3 ) )
% 0.35/0.60                 => ( ( A23 = Deg2 )
% 0.35/0.60                   => ( ! [X2: vEBT_VEBT] :
% 0.35/0.60                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.60                         => ( vEBT_invar_vebt @ X2 @ N3 ) )
% 0.35/0.60                     => ( ( vEBT_invar_vebt @ Summary3 @ M5 )
% 0.35/0.60                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.35/0.60                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 0.35/0.60                         => ( ( M5
% 0.35/0.60                              = ( suc @ N3 ) )
% 0.35/0.60                           => ( ( Deg2
% 0.35/0.60                                = ( plus_plus_nat @ N3 @ M5 ) )
% 0.35/0.60                             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X_12 )
% 0.35/0.60                               => ~ ! [X2: vEBT_VEBT] :
% 0.35/0.60                                      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.60                                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ) ) ) ) ) ) )
% 0.35/0.60             => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat,Mi3: nat,Ma3: nat] :
% 0.35/0.60                    ( ( A12
% 0.35/0.60                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg2 @ TreeList3 @ Summary3 ) )
% 0.35/0.60                   => ( ( A23 = Deg2 )
% 0.35/0.60                     => ( ! [X2: vEBT_VEBT] :
% 0.35/0.60                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.60                           => ( vEBT_invar_vebt @ X2 @ N3 ) )
% 0.35/0.60                       => ( ( vEBT_invar_vebt @ Summary3 @ M5 )
% 0.35/0.60                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.35/0.60                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 0.35/0.60                           => ( ( M5 = N3 )
% 0.35/0.60                             => ( ( Deg2
% 0.35/0.60                                  = ( plus_plus_nat @ N3 @ M5 ) )
% 0.35/0.60                               => ( ! [I3: nat] :
% 0.35/0.60                                      ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 0.35/0.60                                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
% 0.35/0.60                                        = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
% 0.35/0.60                                 => ( ( ( Mi3 = Ma3 )
% 0.35/0.60                                     => ! [X2: vEBT_VEBT] :
% 0.35/0.60                                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.60                                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) )
% 0.35/0.60                                   => ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.35/0.60                                     => ( ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.60                                       => ~ ( ( Mi3 != Ma3 )
% 0.35/0.60                                           => ! [I3: nat] :
% 0.35/0.60                                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 0.35/0.60                                               => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
% 0.35/0.60                                                      = I3 )
% 0.35/0.60                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
% 0.35/0.60                                                  & ! [X2: nat] :
% 0.35/0.60                                                      ( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
% 0.35/0.60                                                          = I3 )
% 0.35/0.60                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
% 0.35/0.60                                                     => ( ( ord_less_nat @ Mi3 @ X2 )
% 0.35/0.60                                                        & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.60               => ~ ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat,Mi3: nat,Ma3: nat] :
% 0.35/0.60                      ( ( A12
% 0.35/0.60                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg2 @ TreeList3 @ Summary3 ) )
% 0.35/0.60                     => ( ( A23 = Deg2 )
% 0.35/0.60                       => ( ! [X2: vEBT_VEBT] :
% 0.35/0.60                              ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.60                             => ( vEBT_invar_vebt @ X2 @ N3 ) )
% 0.35/0.60                         => ( ( vEBT_invar_vebt @ Summary3 @ M5 )
% 0.35/0.60                           => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.35/0.60                                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 0.35/0.60                             => ( ( M5
% 0.35/0.60                                  = ( suc @ N3 ) )
% 0.35/0.60                               => ( ( Deg2
% 0.35/0.60                                    = ( plus_plus_nat @ N3 @ M5 ) )
% 0.35/0.60                                 => ( ! [I3: nat] :
% 0.35/0.60                                        ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 0.35/0.60                                       => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
% 0.35/0.60                                          = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
% 0.35/0.60                                   => ( ( ( Mi3 = Ma3 )
% 0.35/0.60                                       => ! [X2: vEBT_VEBT] :
% 0.35/0.60                                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.60                                           => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) )
% 0.35/0.60                                     => ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.35/0.60                                       => ( ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.60                                         => ~ ( ( Mi3 != Ma3 )
% 0.35/0.60                                             => ! [I3: nat] :
% 0.35/0.60                                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 0.35/0.60                                                 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
% 0.35/0.60                                                        = I3 )
% 0.35/0.60                                                     => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
% 0.35/0.60                                                    & ! [X2: nat] :
% 0.35/0.60                                                        ( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
% 0.35/0.60                                                            = I3 )
% 0.35/0.60                                                          & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
% 0.35/0.60                                                       => ( ( ord_less_nat @ Mi3 @ X2 )
% 0.35/0.60                                                          & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % invar_vebt.cases
% 0.35/0.60  thf(fact_3516_int__power__div__base,axiom,
% 0.35/0.60      ! [M: nat,K: int] :
% 0.35/0.60        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.60       => ( ( ord_less_int @ zero_zero_int @ K )
% 0.35/0.60         => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 0.35/0.60            = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % int_power_div_base
% 0.35/0.60  thf(fact_3517_inrange,axiom,
% 0.35/0.60      ! [T: vEBT_VEBT,N: nat] :
% 0.35/0.60        ( ( vEBT_invar_vebt @ T @ N )
% 0.35/0.60       => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % inrange
% 0.35/0.60  thf(fact_3518_max_Oabsorb3,axiom,
% 0.35/0.60      ! [B: real,A: real] :
% 0.35/0.60        ( ( ord_less_real @ B @ A )
% 0.35/0.60       => ( ( ord_max_real @ A @ B )
% 0.35/0.60          = A ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb3
% 0.35/0.60  thf(fact_3519_max_Oabsorb3,axiom,
% 0.35/0.60      ! [B: rat,A: rat] :
% 0.35/0.60        ( ( ord_less_rat @ B @ A )
% 0.35/0.60       => ( ( ord_max_rat @ A @ B )
% 0.35/0.60          = A ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb3
% 0.35/0.60  thf(fact_3520_max_Oabsorb3,axiom,
% 0.35/0.60      ! [B: num,A: num] :
% 0.35/0.60        ( ( ord_less_num @ B @ A )
% 0.35/0.60       => ( ( ord_max_num @ A @ B )
% 0.35/0.60          = A ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb3
% 0.35/0.60  thf(fact_3521_max_Oabsorb3,axiom,
% 0.35/0.60      ! [B: nat,A: nat] :
% 0.35/0.60        ( ( ord_less_nat @ B @ A )
% 0.35/0.60       => ( ( ord_max_nat @ A @ B )
% 0.35/0.60          = A ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb3
% 0.35/0.60  thf(fact_3522_max_Oabsorb3,axiom,
% 0.35/0.60      ! [B: int,A: int] :
% 0.35/0.60        ( ( ord_less_int @ B @ A )
% 0.35/0.60       => ( ( ord_max_int @ A @ B )
% 0.35/0.60          = A ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb3
% 0.35/0.60  thf(fact_3523_max_Oabsorb4,axiom,
% 0.35/0.60      ! [A: real,B: real] :
% 0.35/0.60        ( ( ord_less_real @ A @ B )
% 0.35/0.60       => ( ( ord_max_real @ A @ B )
% 0.35/0.60          = B ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb4
% 0.35/0.60  thf(fact_3524_max_Oabsorb4,axiom,
% 0.35/0.60      ! [A: rat,B: rat] :
% 0.35/0.60        ( ( ord_less_rat @ A @ B )
% 0.35/0.60       => ( ( ord_max_rat @ A @ B )
% 0.35/0.60          = B ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb4
% 0.35/0.60  thf(fact_3525_max_Oabsorb4,axiom,
% 0.35/0.60      ! [A: num,B: num] :
% 0.35/0.60        ( ( ord_less_num @ A @ B )
% 0.35/0.60       => ( ( ord_max_num @ A @ B )
% 0.35/0.60          = B ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb4
% 0.35/0.60  thf(fact_3526_max_Oabsorb4,axiom,
% 0.35/0.60      ! [A: nat,B: nat] :
% 0.35/0.60        ( ( ord_less_nat @ A @ B )
% 0.35/0.60       => ( ( ord_max_nat @ A @ B )
% 0.35/0.60          = B ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb4
% 0.35/0.60  thf(fact_3527_max_Oabsorb4,axiom,
% 0.35/0.60      ! [A: int,B: int] :
% 0.35/0.60        ( ( ord_less_int @ A @ B )
% 0.35/0.60       => ( ( ord_max_int @ A @ B )
% 0.35/0.60          = B ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb4
% 0.35/0.60  thf(fact_3528_max__less__iff__conj,axiom,
% 0.35/0.60      ! [X: real,Y: real,Z2: real] :
% 0.35/0.60        ( ( ord_less_real @ ( ord_max_real @ X @ Y ) @ Z2 )
% 0.35/0.60        = ( ( ord_less_real @ X @ Z2 )
% 0.35/0.60          & ( ord_less_real @ Y @ Z2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max_less_iff_conj
% 0.35/0.60  thf(fact_3529_max__less__iff__conj,axiom,
% 0.35/0.60      ! [X: rat,Y: rat,Z2: rat] :
% 0.35/0.60        ( ( ord_less_rat @ ( ord_max_rat @ X @ Y ) @ Z2 )
% 0.35/0.60        = ( ( ord_less_rat @ X @ Z2 )
% 0.35/0.60          & ( ord_less_rat @ Y @ Z2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max_less_iff_conj
% 0.35/0.60  thf(fact_3530_max__less__iff__conj,axiom,
% 0.35/0.60      ! [X: num,Y: num,Z2: num] :
% 0.35/0.60        ( ( ord_less_num @ ( ord_max_num @ X @ Y ) @ Z2 )
% 0.35/0.60        = ( ( ord_less_num @ X @ Z2 )
% 0.35/0.60          & ( ord_less_num @ Y @ Z2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max_less_iff_conj
% 0.35/0.60  thf(fact_3531_max__less__iff__conj,axiom,
% 0.35/0.60      ! [X: nat,Y: nat,Z2: nat] :
% 0.35/0.60        ( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z2 )
% 0.35/0.60        = ( ( ord_less_nat @ X @ Z2 )
% 0.35/0.60          & ( ord_less_nat @ Y @ Z2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max_less_iff_conj
% 0.35/0.60  thf(fact_3532_max__less__iff__conj,axiom,
% 0.35/0.60      ! [X: int,Y: int,Z2: int] :
% 0.35/0.60        ( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z2 )
% 0.35/0.60        = ( ( ord_less_int @ X @ Z2 )
% 0.35/0.60          & ( ord_less_int @ Y @ Z2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max_less_iff_conj
% 0.35/0.60  thf(fact_3533_max_Oabsorb1,axiom,
% 0.35/0.60      ! [B: rat,A: rat] :
% 0.35/0.60        ( ( ord_less_eq_rat @ B @ A )
% 0.35/0.60       => ( ( ord_max_rat @ A @ B )
% 0.35/0.60          = A ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb1
% 0.35/0.60  thf(fact_3534_max_Oabsorb1,axiom,
% 0.35/0.60      ! [B: num,A: num] :
% 0.35/0.60        ( ( ord_less_eq_num @ B @ A )
% 0.35/0.60       => ( ( ord_max_num @ A @ B )
% 0.35/0.60          = A ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb1
% 0.35/0.60  thf(fact_3535_max_Oabsorb1,axiom,
% 0.35/0.60      ! [B: nat,A: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ B @ A )
% 0.35/0.60       => ( ( ord_max_nat @ A @ B )
% 0.35/0.60          = A ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb1
% 0.35/0.60  thf(fact_3536_max_Oabsorb1,axiom,
% 0.35/0.60      ! [B: int,A: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ B @ A )
% 0.35/0.60       => ( ( ord_max_int @ A @ B )
% 0.35/0.60          = A ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb1
% 0.35/0.60  thf(fact_3537_max_Oabsorb2,axiom,
% 0.35/0.60      ! [A: rat,B: rat] :
% 0.35/0.60        ( ( ord_less_eq_rat @ A @ B )
% 0.35/0.60       => ( ( ord_max_rat @ A @ B )
% 0.35/0.60          = B ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb2
% 0.35/0.60  thf(fact_3538_max_Oabsorb2,axiom,
% 0.35/0.60      ! [A: num,B: num] :
% 0.35/0.60        ( ( ord_less_eq_num @ A @ B )
% 0.35/0.60       => ( ( ord_max_num @ A @ B )
% 0.35/0.60          = B ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb2
% 0.35/0.60  thf(fact_3539_max_Oabsorb2,axiom,
% 0.35/0.60      ! [A: nat,B: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ A @ B )
% 0.35/0.60       => ( ( ord_max_nat @ A @ B )
% 0.35/0.60          = B ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb2
% 0.35/0.60  thf(fact_3540_max_Oabsorb2,axiom,
% 0.35/0.60      ! [A: int,B: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ A @ B )
% 0.35/0.60       => ( ( ord_max_int @ A @ B )
% 0.35/0.60          = B ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb2
% 0.35/0.60  thf(fact_3541_max_Obounded__iff,axiom,
% 0.35/0.60      ! [B: rat,C: rat,A: rat] :
% 0.35/0.60        ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 0.35/0.60        = ( ( ord_less_eq_rat @ B @ A )
% 0.35/0.60          & ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.bounded_iff
% 0.35/0.60  thf(fact_3542_max_Obounded__iff,axiom,
% 0.35/0.60      ! [B: num,C: num,A: num] :
% 0.35/0.60        ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 0.35/0.60        = ( ( ord_less_eq_num @ B @ A )
% 0.35/0.60          & ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.bounded_iff
% 0.35/0.60  thf(fact_3543_max_Obounded__iff,axiom,
% 0.35/0.60      ! [B: nat,C: nat,A: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 0.35/0.60        = ( ( ord_less_eq_nat @ B @ A )
% 0.35/0.60          & ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.bounded_iff
% 0.35/0.60  thf(fact_3544_max_Obounded__iff,axiom,
% 0.35/0.60      ! [B: int,C: int,A: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 0.35/0.60        = ( ( ord_less_eq_int @ B @ A )
% 0.35/0.60          & ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.bounded_iff
% 0.35/0.60  thf(fact_3545_div__geq,axiom,
% 0.35/0.60      ! [N: nat,M: nat] :
% 0.35/0.60        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60       => ( ~ ( ord_less_nat @ M @ N )
% 0.35/0.60         => ( ( divide_divide_nat @ M @ N )
% 0.35/0.60            = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % div_geq
% 0.35/0.60  thf(fact_3546_Diff__cancel,axiom,
% 0.35/0.60      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.60        ( ( minus_1356011639430497352at_nat @ A2 @ A2 )
% 0.35/0.60        = bot_bo2099793752762293965at_nat ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_cancel
% 0.35/0.60  thf(fact_3547_Diff__cancel,axiom,
% 0.35/0.60      ! [A2: set_real] :
% 0.35/0.60        ( ( minus_minus_set_real @ A2 @ A2 )
% 0.35/0.60        = bot_bot_set_real ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_cancel
% 0.35/0.60  thf(fact_3548_Diff__cancel,axiom,
% 0.35/0.60      ! [A2: set_int] :
% 0.35/0.60        ( ( minus_minus_set_int @ A2 @ A2 )
% 0.35/0.60        = bot_bot_set_int ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_cancel
% 0.35/0.60  thf(fact_3549_Diff__cancel,axiom,
% 0.35/0.60      ! [A2: set_nat] :
% 0.35/0.60        ( ( minus_minus_set_nat @ A2 @ A2 )
% 0.35/0.60        = bot_bot_set_nat ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_cancel
% 0.35/0.60  thf(fact_3550_empty__Diff,axiom,
% 0.35/0.60      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.60        ( ( minus_1356011639430497352at_nat @ bot_bo2099793752762293965at_nat @ A2 )
% 0.35/0.60        = bot_bo2099793752762293965at_nat ) ).
% 0.35/0.60  
% 0.35/0.60  % empty_Diff
% 0.35/0.60  thf(fact_3551_empty__Diff,axiom,
% 0.35/0.60      ! [A2: set_real] :
% 0.35/0.60        ( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
% 0.35/0.60        = bot_bot_set_real ) ).
% 0.35/0.60  
% 0.35/0.60  % empty_Diff
% 0.35/0.60  thf(fact_3552_empty__Diff,axiom,
% 0.35/0.60      ! [A2: set_int] :
% 0.35/0.60        ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
% 0.35/0.60        = bot_bot_set_int ) ).
% 0.35/0.60  
% 0.35/0.60  % empty_Diff
% 0.35/0.60  thf(fact_3553_empty__Diff,axiom,
% 0.35/0.60      ! [A2: set_nat] :
% 0.35/0.60        ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
% 0.35/0.60        = bot_bot_set_nat ) ).
% 0.35/0.60  
% 0.35/0.60  % empty_Diff
% 0.35/0.60  thf(fact_3554_Diff__empty,axiom,
% 0.35/0.60      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.60        ( ( minus_1356011639430497352at_nat @ A2 @ bot_bo2099793752762293965at_nat )
% 0.35/0.60        = A2 ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_empty
% 0.35/0.60  thf(fact_3555_Diff__empty,axiom,
% 0.35/0.60      ! [A2: set_real] :
% 0.35/0.60        ( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
% 0.35/0.60        = A2 ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_empty
% 0.35/0.60  thf(fact_3556_Diff__empty,axiom,
% 0.35/0.60      ! [A2: set_int] :
% 0.35/0.60        ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
% 0.35/0.60        = A2 ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_empty
% 0.35/0.60  thf(fact_3557_Diff__empty,axiom,
% 0.35/0.60      ! [A2: set_nat] :
% 0.35/0.60        ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
% 0.35/0.60        = A2 ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_empty
% 0.35/0.60  thf(fact_3558_finite__Diff2,axiom,
% 0.35/0.60      ! [B2: set_int,A2: set_int] :
% 0.35/0.60        ( ( finite_finite_int @ B2 )
% 0.35/0.60       => ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.35/0.60          = ( finite_finite_int @ A2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % finite_Diff2
% 0.35/0.60  thf(fact_3559_finite__Diff2,axiom,
% 0.35/0.60      ! [B2: set_complex,A2: set_complex] :
% 0.35/0.60        ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.60       => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 0.35/0.60          = ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % finite_Diff2
% 0.35/0.60  thf(fact_3560_finite__Diff2,axiom,
% 0.35/0.60      ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.60        ( ( finite6177210948735845034at_nat @ B2 )
% 0.35/0.60       => ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 0.35/0.60          = ( finite6177210948735845034at_nat @ A2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % finite_Diff2
% 0.35/0.60  thf(fact_3561_finite__Diff2,axiom,
% 0.35/0.60      ! [B2: set_nat,A2: set_nat] :
% 0.35/0.60        ( ( finite_finite_nat @ B2 )
% 0.35/0.60       => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.35/0.60          = ( finite_finite_nat @ A2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % finite_Diff2
% 0.35/0.60  thf(fact_3562_finite__Diff,axiom,
% 0.35/0.60      ! [A2: set_int,B2: set_int] :
% 0.35/0.60        ( ( finite_finite_int @ A2 )
% 0.35/0.60       => ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % finite_Diff
% 0.35/0.60  thf(fact_3563_finite__Diff,axiom,
% 0.35/0.60      ! [A2: set_complex,B2: set_complex] :
% 0.35/0.60        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.60       => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % finite_Diff
% 0.35/0.60  thf(fact_3564_finite__Diff,axiom,
% 0.35/0.60      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.60        ( ( finite6177210948735845034at_nat @ A2 )
% 0.35/0.60       => ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % finite_Diff
% 0.35/0.60  thf(fact_3565_finite__Diff,axiom,
% 0.35/0.60      ! [A2: set_nat,B2: set_nat] :
% 0.35/0.60        ( ( finite_finite_nat @ A2 )
% 0.35/0.60       => ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % finite_Diff
% 0.35/0.60  thf(fact_3566_Diff__eq__empty__iff,axiom,
% 0.35/0.60      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.60        ( ( ( minus_1356011639430497352at_nat @ A2 @ B2 )
% 0.35/0.60          = bot_bo2099793752762293965at_nat )
% 0.35/0.60        = ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_eq_empty_iff
% 0.35/0.60  thf(fact_3567_Diff__eq__empty__iff,axiom,
% 0.35/0.60      ! [A2: set_real,B2: set_real] :
% 0.35/0.60        ( ( ( minus_minus_set_real @ A2 @ B2 )
% 0.35/0.60          = bot_bot_set_real )
% 0.35/0.60        = ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_eq_empty_iff
% 0.35/0.60  thf(fact_3568_Diff__eq__empty__iff,axiom,
% 0.35/0.60      ! [A2: set_nat,B2: set_nat] :
% 0.35/0.60        ( ( ( minus_minus_set_nat @ A2 @ B2 )
% 0.35/0.60          = bot_bot_set_nat )
% 0.35/0.60        = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_eq_empty_iff
% 0.35/0.60  thf(fact_3569_Diff__eq__empty__iff,axiom,
% 0.35/0.60      ! [A2: set_int,B2: set_int] :
% 0.35/0.60        ( ( ( minus_minus_set_int @ A2 @ B2 )
% 0.35/0.60          = bot_bot_set_int )
% 0.35/0.60        = ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_eq_empty_iff
% 0.35/0.60  thf(fact_3570_atLeastAtMost__iff,axiom,
% 0.35/0.60      ! [I: set_nat,L: set_nat,U: set_nat] :
% 0.35/0.60        ( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
% 0.35/0.60        = ( ( ord_less_eq_set_nat @ L @ I )
% 0.35/0.60          & ( ord_less_eq_set_nat @ I @ U ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastAtMost_iff
% 0.35/0.60  thf(fact_3571_atLeastAtMost__iff,axiom,
% 0.35/0.60      ! [I: set_int,L: set_int,U: set_int] :
% 0.35/0.60        ( ( member_set_int @ I @ ( set_or370866239135849197et_int @ L @ U ) )
% 0.35/0.60        = ( ( ord_less_eq_set_int @ L @ I )
% 0.35/0.60          & ( ord_less_eq_set_int @ I @ U ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastAtMost_iff
% 0.35/0.60  thf(fact_3572_atLeastAtMost__iff,axiom,
% 0.35/0.60      ! [I: rat,L: rat,U: rat] :
% 0.35/0.60        ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
% 0.35/0.60        = ( ( ord_less_eq_rat @ L @ I )
% 0.35/0.60          & ( ord_less_eq_rat @ I @ U ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastAtMost_iff
% 0.35/0.60  thf(fact_3573_atLeastAtMost__iff,axiom,
% 0.35/0.60      ! [I: num,L: num,U: num] :
% 0.35/0.60        ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L @ U ) )
% 0.35/0.60        = ( ( ord_less_eq_num @ L @ I )
% 0.35/0.60          & ( ord_less_eq_num @ I @ U ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastAtMost_iff
% 0.35/0.60  thf(fact_3574_atLeastAtMost__iff,axiom,
% 0.35/0.60      ! [I: nat,L: nat,U: nat] :
% 0.35/0.60        ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 0.35/0.60        = ( ( ord_less_eq_nat @ L @ I )
% 0.35/0.60          & ( ord_less_eq_nat @ I @ U ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastAtMost_iff
% 0.35/0.60  thf(fact_3575_atLeastAtMost__iff,axiom,
% 0.35/0.60      ! [I: int,L: int,U: int] :
% 0.35/0.60        ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
% 0.35/0.60        = ( ( ord_less_eq_int @ L @ I )
% 0.35/0.60          & ( ord_less_eq_int @ I @ U ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastAtMost_iff
% 0.35/0.60  thf(fact_3576_atLeastAtMost__iff,axiom,
% 0.35/0.60      ! [I: real,L: real,U: real] :
% 0.35/0.60        ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
% 0.35/0.60        = ( ( ord_less_eq_real @ L @ I )
% 0.35/0.60          & ( ord_less_eq_real @ I @ U ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastAtMost_iff
% 0.35/0.60  thf(fact_3577_Icc__eq__Icc,axiom,
% 0.35/0.60      ! [L: set_int,H2: set_int,L3: set_int,H3: set_int] :
% 0.35/0.60        ( ( ( set_or370866239135849197et_int @ L @ H2 )
% 0.35/0.60          = ( set_or370866239135849197et_int @ L3 @ H3 ) )
% 0.35/0.60        = ( ( ( L = L3 )
% 0.35/0.60            & ( H2 = H3 ) )
% 0.35/0.60          | ( ~ ( ord_less_eq_set_int @ L @ H2 )
% 0.35/0.60            & ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Icc_eq_Icc
% 0.35/0.60  thf(fact_3578_Icc__eq__Icc,axiom,
% 0.35/0.60      ! [L: rat,H2: rat,L3: rat,H3: rat] :
% 0.35/0.60        ( ( ( set_or633870826150836451st_rat @ L @ H2 )
% 0.35/0.60          = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
% 0.35/0.60        = ( ( ( L = L3 )
% 0.35/0.60            & ( H2 = H3 ) )
% 0.35/0.60          | ( ~ ( ord_less_eq_rat @ L @ H2 )
% 0.35/0.60            & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Icc_eq_Icc
% 0.35/0.60  thf(fact_3579_Icc__eq__Icc,axiom,
% 0.35/0.60      ! [L: num,H2: num,L3: num,H3: num] :
% 0.35/0.60        ( ( ( set_or7049704709247886629st_num @ L @ H2 )
% 0.35/0.60          = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
% 0.35/0.60        = ( ( ( L = L3 )
% 0.35/0.60            & ( H2 = H3 ) )
% 0.35/0.60          | ( ~ ( ord_less_eq_num @ L @ H2 )
% 0.35/0.60            & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Icc_eq_Icc
% 0.35/0.60  thf(fact_3580_Icc__eq__Icc,axiom,
% 0.35/0.60      ! [L: nat,H2: nat,L3: nat,H3: nat] :
% 0.35/0.60        ( ( ( set_or1269000886237332187st_nat @ L @ H2 )
% 0.35/0.60          = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
% 0.35/0.60        = ( ( ( L = L3 )
% 0.35/0.60            & ( H2 = H3 ) )
% 0.35/0.60          | ( ~ ( ord_less_eq_nat @ L @ H2 )
% 0.35/0.60            & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Icc_eq_Icc
% 0.35/0.60  thf(fact_3581_Icc__eq__Icc,axiom,
% 0.35/0.60      ! [L: int,H2: int,L3: int,H3: int] :
% 0.35/0.60        ( ( ( set_or1266510415728281911st_int @ L @ H2 )
% 0.35/0.60          = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
% 0.35/0.60        = ( ( ( L = L3 )
% 0.35/0.60            & ( H2 = H3 ) )
% 0.35/0.60          | ( ~ ( ord_less_eq_int @ L @ H2 )
% 0.35/0.60            & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Icc_eq_Icc
% 0.35/0.60  thf(fact_3582_Icc__eq__Icc,axiom,
% 0.35/0.60      ! [L: real,H2: real,L3: real,H3: real] :
% 0.35/0.60        ( ( ( set_or1222579329274155063t_real @ L @ H2 )
% 0.35/0.60          = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
% 0.35/0.60        = ( ( ( L = L3 )
% 0.35/0.60            & ( H2 = H3 ) )
% 0.35/0.60          | ( ~ ( ord_less_eq_real @ L @ H2 )
% 0.35/0.60            & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Icc_eq_Icc
% 0.35/0.60  thf(fact_3583_not__real__square__gt__zero,axiom,
% 0.35/0.60      ! [X: real] :
% 0.35/0.60        ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
% 0.35/0.60        = ( X = zero_zero_real ) ) ).
% 0.35/0.60  
% 0.35/0.60  % not_real_square_gt_zero
% 0.35/0.60  thf(fact_3584_finite__atLeastAtMost,axiom,
% 0.35/0.60      ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% 0.35/0.60  
% 0.35/0.60  % finite_atLeastAtMost
% 0.35/0.60  thf(fact_3585_atLeastatMost__empty__iff,axiom,
% 0.35/0.60      ! [A: set_int,B: set_int] :
% 0.35/0.60        ( ( ( set_or370866239135849197et_int @ A @ B )
% 0.35/0.60          = bot_bot_set_set_int )
% 0.35/0.60        = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_empty_iff
% 0.35/0.60  thf(fact_3586_atLeastatMost__empty__iff,axiom,
% 0.35/0.60      ! [A: rat,B: rat] :
% 0.35/0.60        ( ( ( set_or633870826150836451st_rat @ A @ B )
% 0.35/0.60          = bot_bot_set_rat )
% 0.35/0.60        = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_empty_iff
% 0.35/0.60  thf(fact_3587_atLeastatMost__empty__iff,axiom,
% 0.35/0.60      ! [A: num,B: num] :
% 0.35/0.60        ( ( ( set_or7049704709247886629st_num @ A @ B )
% 0.35/0.60          = bot_bot_set_num )
% 0.35/0.60        = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_empty_iff
% 0.35/0.60  thf(fact_3588_atLeastatMost__empty__iff,axiom,
% 0.35/0.60      ! [A: nat,B: nat] :
% 0.35/0.60        ( ( ( set_or1269000886237332187st_nat @ A @ B )
% 0.35/0.60          = bot_bot_set_nat )
% 0.35/0.60        = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_empty_iff
% 0.35/0.60  thf(fact_3589_atLeastatMost__empty__iff,axiom,
% 0.35/0.60      ! [A: int,B: int] :
% 0.35/0.60        ( ( ( set_or1266510415728281911st_int @ A @ B )
% 0.35/0.60          = bot_bot_set_int )
% 0.35/0.60        = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_empty_iff
% 0.35/0.60  thf(fact_3590_atLeastatMost__empty__iff,axiom,
% 0.35/0.60      ! [A: real,B: real] :
% 0.35/0.60        ( ( ( set_or1222579329274155063t_real @ A @ B )
% 0.35/0.60          = bot_bot_set_real )
% 0.35/0.60        = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_empty_iff
% 0.35/0.60  thf(fact_3591_atLeastatMost__empty__iff2,axiom,
% 0.35/0.60      ! [A: set_int,B: set_int] :
% 0.35/0.60        ( ( bot_bot_set_set_int
% 0.35/0.60          = ( set_or370866239135849197et_int @ A @ B ) )
% 0.35/0.60        = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_empty_iff2
% 0.35/0.60  thf(fact_3592_atLeastatMost__empty__iff2,axiom,
% 0.35/0.60      ! [A: rat,B: rat] :
% 0.35/0.60        ( ( bot_bot_set_rat
% 0.35/0.60          = ( set_or633870826150836451st_rat @ A @ B ) )
% 0.35/0.60        = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_empty_iff2
% 0.35/0.60  thf(fact_3593_atLeastatMost__empty__iff2,axiom,
% 0.35/0.60      ! [A: num,B: num] :
% 0.35/0.60        ( ( bot_bot_set_num
% 0.35/0.60          = ( set_or7049704709247886629st_num @ A @ B ) )
% 0.35/0.60        = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_empty_iff2
% 0.35/0.60  thf(fact_3594_atLeastatMost__empty__iff2,axiom,
% 0.35/0.60      ! [A: nat,B: nat] :
% 0.35/0.60        ( ( bot_bot_set_nat
% 0.35/0.60          = ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.35/0.60        = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_empty_iff2
% 0.35/0.60  thf(fact_3595_atLeastatMost__empty__iff2,axiom,
% 0.35/0.60      ! [A: int,B: int] :
% 0.35/0.60        ( ( bot_bot_set_int
% 0.35/0.60          = ( set_or1266510415728281911st_int @ A @ B ) )
% 0.35/0.60        = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_empty_iff2
% 0.35/0.60  thf(fact_3596_atLeastatMost__empty__iff2,axiom,
% 0.35/0.60      ! [A: real,B: real] :
% 0.35/0.60        ( ( bot_bot_set_real
% 0.35/0.60          = ( set_or1222579329274155063t_real @ A @ B ) )
% 0.35/0.60        = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_empty_iff2
% 0.35/0.60  thf(fact_3597_atLeastatMost__subset__iff,axiom,
% 0.35/0.60      ! [A: set_int,B: set_int,C: set_int,D: set_int] :
% 0.35/0.60        ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 0.35/0.60        = ( ~ ( ord_less_eq_set_int @ A @ B )
% 0.35/0.60          | ( ( ord_less_eq_set_int @ C @ A )
% 0.35/0.60            & ( ord_less_eq_set_int @ B @ D ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_subset_iff
% 0.35/0.60  thf(fact_3598_atLeastatMost__subset__iff,axiom,
% 0.35/0.60      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.35/0.60        ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 0.35/0.60        = ( ~ ( ord_less_eq_rat @ A @ B )
% 0.35/0.60          | ( ( ord_less_eq_rat @ C @ A )
% 0.35/0.60            & ( ord_less_eq_rat @ B @ D ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_subset_iff
% 0.35/0.60  thf(fact_3599_atLeastatMost__subset__iff,axiom,
% 0.35/0.60      ! [A: num,B: num,C: num,D: num] :
% 0.35/0.60        ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 0.35/0.60        = ( ~ ( ord_less_eq_num @ A @ B )
% 0.35/0.60          | ( ( ord_less_eq_num @ C @ A )
% 0.35/0.60            & ( ord_less_eq_num @ B @ D ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_subset_iff
% 0.35/0.60  thf(fact_3600_atLeastatMost__subset__iff,axiom,
% 0.35/0.60      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.35/0.60        ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 0.35/0.60        = ( ~ ( ord_less_eq_nat @ A @ B )
% 0.35/0.60          | ( ( ord_less_eq_nat @ C @ A )
% 0.35/0.60            & ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_subset_iff
% 0.35/0.60  thf(fact_3601_atLeastatMost__subset__iff,axiom,
% 0.35/0.60      ! [A: int,B: int,C: int,D: int] :
% 0.35/0.60        ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 0.35/0.60        = ( ~ ( ord_less_eq_int @ A @ B )
% 0.35/0.60          | ( ( ord_less_eq_int @ C @ A )
% 0.35/0.60            & ( ord_less_eq_int @ B @ D ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_subset_iff
% 0.35/0.60  thf(fact_3602_atLeastatMost__subset__iff,axiom,
% 0.35/0.60      ! [A: real,B: real,C: real,D: real] :
% 0.35/0.60        ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 0.35/0.60        = ( ~ ( ord_less_eq_real @ A @ B )
% 0.35/0.60          | ( ( ord_less_eq_real @ C @ A )
% 0.35/0.60            & ( ord_less_eq_real @ B @ D ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_subset_iff
% 0.35/0.60  thf(fact_3603_atLeastatMost__empty,axiom,
% 0.35/0.60      ! [B: rat,A: rat] :
% 0.35/0.60        ( ( ord_less_rat @ B @ A )
% 0.35/0.60       => ( ( set_or633870826150836451st_rat @ A @ B )
% 0.35/0.60          = bot_bot_set_rat ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_empty
% 0.35/0.60  thf(fact_3604_atLeastatMost__empty,axiom,
% 0.35/0.60      ! [B: num,A: num] :
% 0.35/0.60        ( ( ord_less_num @ B @ A )
% 0.35/0.60       => ( ( set_or7049704709247886629st_num @ A @ B )
% 0.35/0.60          = bot_bot_set_num ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_empty
% 0.35/0.60  thf(fact_3605_atLeastatMost__empty,axiom,
% 0.35/0.60      ! [B: nat,A: nat] :
% 0.35/0.60        ( ( ord_less_nat @ B @ A )
% 0.35/0.60       => ( ( set_or1269000886237332187st_nat @ A @ B )
% 0.35/0.60          = bot_bot_set_nat ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_empty
% 0.35/0.60  thf(fact_3606_atLeastatMost__empty,axiom,
% 0.35/0.60      ! [B: int,A: int] :
% 0.35/0.60        ( ( ord_less_int @ B @ A )
% 0.35/0.60       => ( ( set_or1266510415728281911st_int @ A @ B )
% 0.35/0.60          = bot_bot_set_int ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_empty
% 0.35/0.60  thf(fact_3607_atLeastatMost__empty,axiom,
% 0.35/0.60      ! [B: real,A: real] :
% 0.35/0.60        ( ( ord_less_real @ B @ A )
% 0.35/0.60       => ( ( set_or1222579329274155063t_real @ A @ B )
% 0.35/0.60          = bot_bot_set_real ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_empty
% 0.35/0.60  thf(fact_3608_infinite__Icc__iff,axiom,
% 0.35/0.60      ! [A: rat,B: rat] :
% 0.35/0.60        ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
% 0.35/0.60        = ( ord_less_rat @ A @ B ) ) ).
% 0.35/0.60  
% 0.35/0.60  % infinite_Icc_iff
% 0.35/0.60  thf(fact_3609_infinite__Icc__iff,axiom,
% 0.35/0.60      ! [A: real,B: real] :
% 0.35/0.60        ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
% 0.35/0.60        = ( ord_less_real @ A @ B ) ) ).
% 0.35/0.60  
% 0.35/0.60  % infinite_Icc_iff
% 0.35/0.60  thf(fact_3610_zle__diff1__eq,axiom,
% 0.35/0.60      ! [W2: int,Z2: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z2 @ one_one_int ) )
% 0.35/0.60        = ( ord_less_int @ W2 @ Z2 ) ) ).
% 0.35/0.60  
% 0.35/0.60  % zle_diff1_eq
% 0.35/0.60  thf(fact_3611_real__arch__pow,axiom,
% 0.35/0.60      ! [X: real,Y: real] :
% 0.35/0.60        ( ( ord_less_real @ one_one_real @ X )
% 0.35/0.60       => ? [N3: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N3 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % real_arch_pow
% 0.35/0.60  thf(fact_3612_real__arch__pow__inv,axiom,
% 0.35/0.60      ! [Y: real,X: real] :
% 0.35/0.60        ( ( ord_less_real @ zero_zero_real @ Y )
% 0.35/0.60       => ( ( ord_less_real @ X @ one_one_real )
% 0.35/0.60         => ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X @ N3 ) @ Y ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % real_arch_pow_inv
% 0.35/0.60  thf(fact_3613_less__eq__real__def,axiom,
% 0.35/0.60      ( ord_less_eq_real
% 0.35/0.60      = ( ^ [X3: real,Y2: real] :
% 0.35/0.60            ( ( ord_less_real @ X3 @ Y2 )
% 0.35/0.60            | ( X3 = Y2 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % less_eq_real_def
% 0.35/0.60  thf(fact_3614_Diff__infinite__finite,axiom,
% 0.35/0.60      ! [T3: set_int,S2: set_int] :
% 0.35/0.60        ( ( finite_finite_int @ T3 )
% 0.35/0.60       => ( ~ ( finite_finite_int @ S2 )
% 0.35/0.60         => ~ ( finite_finite_int @ ( minus_minus_set_int @ S2 @ T3 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_infinite_finite
% 0.35/0.60  thf(fact_3615_Diff__infinite__finite,axiom,
% 0.35/0.60      ! [T3: set_complex,S2: set_complex] :
% 0.35/0.60        ( ( finite3207457112153483333omplex @ T3 )
% 0.35/0.60       => ( ~ ( finite3207457112153483333omplex @ S2 )
% 0.35/0.60         => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S2 @ T3 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_infinite_finite
% 0.35/0.60  thf(fact_3616_Diff__infinite__finite,axiom,
% 0.35/0.60      ! [T3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 0.35/0.60        ( ( finite6177210948735845034at_nat @ T3 )
% 0.35/0.60       => ( ~ ( finite6177210948735845034at_nat @ S2 )
% 0.35/0.60         => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S2 @ T3 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_infinite_finite
% 0.35/0.60  thf(fact_3617_Diff__infinite__finite,axiom,
% 0.35/0.60      ! [T3: set_nat,S2: set_nat] :
% 0.35/0.60        ( ( finite_finite_nat @ T3 )
% 0.35/0.60       => ( ~ ( finite_finite_nat @ S2 )
% 0.35/0.60         => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T3 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_infinite_finite
% 0.35/0.60  thf(fact_3618_double__diff,axiom,
% 0.35/0.60      ! [A2: set_nat,B2: set_nat,C2: set_nat] :
% 0.35/0.60        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.35/0.60       => ( ( ord_less_eq_set_nat @ B2 @ C2 )
% 0.35/0.60         => ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C2 @ A2 ) )
% 0.35/0.60            = A2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % double_diff
% 0.35/0.60  thf(fact_3619_double__diff,axiom,
% 0.35/0.60      ! [A2: set_int,B2: set_int,C2: set_int] :
% 0.35/0.60        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.35/0.60       => ( ( ord_less_eq_set_int @ B2 @ C2 )
% 0.35/0.60         => ( ( minus_minus_set_int @ B2 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 0.35/0.60            = A2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % double_diff
% 0.35/0.60  thf(fact_3620_Diff__subset,axiom,
% 0.35/0.60      ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_subset
% 0.35/0.60  thf(fact_3621_Diff__subset,axiom,
% 0.35/0.60      ! [A2: set_int,B2: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ A2 ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_subset
% 0.35/0.60  thf(fact_3622_Diff__mono,axiom,
% 0.35/0.60      ! [A2: set_nat,C2: set_nat,D6: set_nat,B2: set_nat] :
% 0.35/0.60        ( ( ord_less_eq_set_nat @ A2 @ C2 )
% 0.35/0.60       => ( ( ord_less_eq_set_nat @ D6 @ B2 )
% 0.35/0.60         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C2 @ D6 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_mono
% 0.35/0.60  thf(fact_3623_Diff__mono,axiom,
% 0.35/0.60      ! [A2: set_int,C2: set_int,D6: set_int,B2: set_int] :
% 0.35/0.60        ( ( ord_less_eq_set_int @ A2 @ C2 )
% 0.35/0.60       => ( ( ord_less_eq_set_int @ D6 @ B2 )
% 0.35/0.60         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ ( minus_minus_set_int @ C2 @ D6 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_mono
% 0.35/0.60  thf(fact_3624_int__less__induct,axiom,
% 0.35/0.60      ! [I: int,K: int,P: int > $o] :
% 0.35/0.60        ( ( ord_less_int @ I @ K )
% 0.35/0.60       => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 0.35/0.60         => ( ! [I2: int] :
% 0.35/0.60                ( ( ord_less_int @ I2 @ K )
% 0.35/0.60               => ( ( P @ I2 )
% 0.35/0.60                 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 0.35/0.60           => ( P @ I ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % int_less_induct
% 0.35/0.60  thf(fact_3625_psubset__imp__ex__mem,axiom,
% 0.35/0.60      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.60        ( ( ord_le7866589430770878221at_nat @ A2 @ B2 )
% 0.35/0.60       => ? [B3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ B2 @ A2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % psubset_imp_ex_mem
% 0.35/0.60  thf(fact_3626_psubset__imp__ex__mem,axiom,
% 0.35/0.60      ! [A2: set_real,B2: set_real] :
% 0.35/0.60        ( ( ord_less_set_real @ A2 @ B2 )
% 0.35/0.60       => ? [B3: real] : ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % psubset_imp_ex_mem
% 0.35/0.60  thf(fact_3627_psubset__imp__ex__mem,axiom,
% 0.35/0.60      ! [A2: set_set_nat,B2: set_set_nat] :
% 0.35/0.60        ( ( ord_less_set_set_nat @ A2 @ B2 )
% 0.35/0.60       => ? [B3: set_nat] : ( member_set_nat @ B3 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % psubset_imp_ex_mem
% 0.35/0.60  thf(fact_3628_psubset__imp__ex__mem,axiom,
% 0.35/0.60      ! [A2: set_int,B2: set_int] :
% 0.35/0.60        ( ( ord_less_set_int @ A2 @ B2 )
% 0.35/0.60       => ? [B3: int] : ( member_int @ B3 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % psubset_imp_ex_mem
% 0.35/0.60  thf(fact_3629_psubset__imp__ex__mem,axiom,
% 0.35/0.60      ! [A2: set_nat,B2: set_nat] :
% 0.35/0.60        ( ( ord_less_set_nat @ A2 @ B2 )
% 0.35/0.60       => ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % psubset_imp_ex_mem
% 0.35/0.60  thf(fact_3630_infinite__Icc,axiom,
% 0.35/0.60      ! [A: rat,B: rat] :
% 0.35/0.60        ( ( ord_less_rat @ A @ B )
% 0.35/0.60       => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % infinite_Icc
% 0.35/0.60  thf(fact_3631_infinite__Icc,axiom,
% 0.35/0.60      ! [A: real,B: real] :
% 0.35/0.60        ( ( ord_less_real @ A @ B )
% 0.35/0.60       => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % infinite_Icc
% 0.35/0.60  thf(fact_3632_ex__nat__less,axiom,
% 0.35/0.60      ! [N: nat,P: nat > $o] :
% 0.35/0.60        ( ( ? [M2: nat] :
% 0.35/0.60              ( ( ord_less_eq_nat @ M2 @ N )
% 0.35/0.60              & ( P @ M2 ) ) )
% 0.35/0.60        = ( ? [X3: nat] :
% 0.35/0.60              ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.60              & ( P @ X3 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % ex_nat_less
% 0.35/0.60  thf(fact_3633_all__nat__less,axiom,
% 0.35/0.60      ! [N: nat,P: nat > $o] :
% 0.35/0.60        ( ( ! [M2: nat] :
% 0.35/0.60              ( ( ord_less_eq_nat @ M2 @ N )
% 0.35/0.60             => ( P @ M2 ) ) )
% 0.35/0.60        = ( ! [X3: nat] :
% 0.35/0.60              ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.60             => ( P @ X3 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % all_nat_less
% 0.35/0.60  thf(fact_3634_subset__eq__atLeast0__atMost__finite,axiom,
% 0.35/0.60      ! [N6: set_nat,N: nat] :
% 0.35/0.60        ( ( ord_less_eq_set_nat @ N6 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.60       => ( finite_finite_nat @ N6 ) ) ).
% 0.35/0.60  
% 0.35/0.60  % subset_eq_atLeast0_atMost_finite
% 0.35/0.60  thf(fact_3635_decr__mult__lemma,axiom,
% 0.35/0.60      ! [D: int,P: int > $o,K: int] :
% 0.35/0.60        ( ( ord_less_int @ zero_zero_int @ D )
% 0.35/0.60       => ( ! [X4: int] :
% 0.35/0.60              ( ( P @ X4 )
% 0.35/0.60             => ( P @ ( minus_minus_int @ X4 @ D ) ) )
% 0.35/0.60         => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.60           => ! [X2: int] :
% 0.35/0.60                ( ( P @ X2 )
% 0.35/0.60               => ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % decr_mult_lemma
% 0.35/0.60  thf(fact_3636_plusinfinity,axiom,
% 0.35/0.60      ! [D: int,P5: int > $o,P: int > $o] :
% 0.35/0.60        ( ( ord_less_int @ zero_zero_int @ D )
% 0.35/0.60       => ( ! [X4: int,K2: int] :
% 0.35/0.60              ( ( P5 @ X4 )
% 0.35/0.60              = ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 0.35/0.60         => ( ? [Z5: int] :
% 0.35/0.60              ! [X4: int] :
% 0.35/0.60                ( ( ord_less_int @ Z5 @ X4 )
% 0.35/0.60               => ( ( P @ X4 )
% 0.35/0.60                  = ( P5 @ X4 ) ) )
% 0.35/0.60           => ( ? [X_12: int] : ( P5 @ X_12 )
% 0.35/0.60             => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % plusinfinity
% 0.35/0.60  thf(fact_3637_minusinfinity,axiom,
% 0.35/0.60      ! [D: int,P1: int > $o,P: int > $o] :
% 0.35/0.60        ( ( ord_less_int @ zero_zero_int @ D )
% 0.35/0.60       => ( ! [X4: int,K2: int] :
% 0.35/0.60              ( ( P1 @ X4 )
% 0.35/0.60              = ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 0.35/0.60         => ( ? [Z5: int] :
% 0.35/0.60              ! [X4: int] :
% 0.35/0.60                ( ( ord_less_int @ X4 @ Z5 )
% 0.35/0.60               => ( ( P @ X4 )
% 0.35/0.60                  = ( P1 @ X4 ) ) )
% 0.35/0.60           => ( ? [X_12: int] : ( P1 @ X_12 )
% 0.35/0.60             => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % minusinfinity
% 0.35/0.60  thf(fact_3638_div__pos__geq,axiom,
% 0.35/0.60      ! [L: int,K: int] :
% 0.35/0.60        ( ( ord_less_int @ zero_zero_int @ L )
% 0.35/0.60       => ( ( ord_less_eq_int @ L @ K )
% 0.35/0.60         => ( ( divide_divide_int @ K @ L )
% 0.35/0.60            = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % div_pos_geq
% 0.35/0.60  thf(fact_3639_atLeastatMost__psubset__iff,axiom,
% 0.35/0.60      ! [A: set_int,B: set_int,C: set_int,D: set_int] :
% 0.35/0.60        ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 0.35/0.60        = ( ( ~ ( ord_less_eq_set_int @ A @ B )
% 0.35/0.60            | ( ( ord_less_eq_set_int @ C @ A )
% 0.35/0.60              & ( ord_less_eq_set_int @ B @ D )
% 0.35/0.60              & ( ( ord_less_set_int @ C @ A )
% 0.35/0.60                | ( ord_less_set_int @ B @ D ) ) ) )
% 0.35/0.60          & ( ord_less_eq_set_int @ C @ D ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_psubset_iff
% 0.35/0.60  thf(fact_3640_atLeastatMost__psubset__iff,axiom,
% 0.35/0.60      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.35/0.60        ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 0.35/0.60        = ( ( ~ ( ord_less_eq_rat @ A @ B )
% 0.35/0.60            | ( ( ord_less_eq_rat @ C @ A )
% 0.35/0.60              & ( ord_less_eq_rat @ B @ D )
% 0.35/0.60              & ( ( ord_less_rat @ C @ A )
% 0.35/0.60                | ( ord_less_rat @ B @ D ) ) ) )
% 0.35/0.60          & ( ord_less_eq_rat @ C @ D ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_psubset_iff
% 0.35/0.60  thf(fact_3641_atLeastatMost__psubset__iff,axiom,
% 0.35/0.60      ! [A: num,B: num,C: num,D: num] :
% 0.35/0.60        ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 0.35/0.60        = ( ( ~ ( ord_less_eq_num @ A @ B )
% 0.35/0.60            | ( ( ord_less_eq_num @ C @ A )
% 0.35/0.60              & ( ord_less_eq_num @ B @ D )
% 0.35/0.60              & ( ( ord_less_num @ C @ A )
% 0.35/0.60                | ( ord_less_num @ B @ D ) ) ) )
% 0.35/0.60          & ( ord_less_eq_num @ C @ D ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_psubset_iff
% 0.35/0.60  thf(fact_3642_atLeastatMost__psubset__iff,axiom,
% 0.35/0.60      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.35/0.60        ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 0.35/0.60        = ( ( ~ ( ord_less_eq_nat @ A @ B )
% 0.35/0.60            | ( ( ord_less_eq_nat @ C @ A )
% 0.35/0.60              & ( ord_less_eq_nat @ B @ D )
% 0.35/0.60              & ( ( ord_less_nat @ C @ A )
% 0.35/0.60                | ( ord_less_nat @ B @ D ) ) ) )
% 0.35/0.60          & ( ord_less_eq_nat @ C @ D ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_psubset_iff
% 0.35/0.60  thf(fact_3643_atLeastatMost__psubset__iff,axiom,
% 0.35/0.60      ! [A: int,B: int,C: int,D: int] :
% 0.35/0.60        ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 0.35/0.60        = ( ( ~ ( ord_less_eq_int @ A @ B )
% 0.35/0.60            | ( ( ord_less_eq_int @ C @ A )
% 0.35/0.60              & ( ord_less_eq_int @ B @ D )
% 0.35/0.60              & ( ( ord_less_int @ C @ A )
% 0.35/0.60                | ( ord_less_int @ B @ D ) ) ) )
% 0.35/0.60          & ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_psubset_iff
% 0.35/0.60  thf(fact_3644_atLeastatMost__psubset__iff,axiom,
% 0.35/0.60      ! [A: real,B: real,C: real,D: real] :
% 0.35/0.60        ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 0.35/0.60        = ( ( ~ ( ord_less_eq_real @ A @ B )
% 0.35/0.60            | ( ( ord_less_eq_real @ C @ A )
% 0.35/0.60              & ( ord_less_eq_real @ B @ D )
% 0.35/0.60              & ( ( ord_less_real @ C @ A )
% 0.35/0.60                | ( ord_less_real @ B @ D ) ) ) )
% 0.35/0.60          & ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % atLeastatMost_psubset_iff
% 0.35/0.60  thf(fact_3645_max_Omono,axiom,
% 0.35/0.60      ! [C: rat,A: rat,D: rat,B: rat] :
% 0.35/0.60        ( ( ord_less_eq_rat @ C @ A )
% 0.35/0.60       => ( ( ord_less_eq_rat @ D @ B )
% 0.35/0.60         => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.mono
% 0.35/0.60  thf(fact_3646_max_Omono,axiom,
% 0.35/0.60      ! [C: num,A: num,D: num,B: num] :
% 0.35/0.60        ( ( ord_less_eq_num @ C @ A )
% 0.35/0.60       => ( ( ord_less_eq_num @ D @ B )
% 0.35/0.60         => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.mono
% 0.35/0.60  thf(fact_3647_max_Omono,axiom,
% 0.35/0.60      ! [C: nat,A: nat,D: nat,B: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ C @ A )
% 0.35/0.60       => ( ( ord_less_eq_nat @ D @ B )
% 0.35/0.60         => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.mono
% 0.35/0.60  thf(fact_3648_max_Omono,axiom,
% 0.35/0.60      ! [C: int,A: int,D: int,B: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ C @ A )
% 0.35/0.60       => ( ( ord_less_eq_int @ D @ B )
% 0.35/0.60         => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.mono
% 0.35/0.60  thf(fact_3649_max_OorderE,axiom,
% 0.35/0.60      ! [B: rat,A: rat] :
% 0.35/0.60        ( ( ord_less_eq_rat @ B @ A )
% 0.35/0.60       => ( A
% 0.35/0.60          = ( ord_max_rat @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.orderE
% 0.35/0.60  thf(fact_3650_max_OorderE,axiom,
% 0.35/0.60      ! [B: num,A: num] :
% 0.35/0.60        ( ( ord_less_eq_num @ B @ A )
% 0.35/0.60       => ( A
% 0.35/0.60          = ( ord_max_num @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.orderE
% 0.35/0.60  thf(fact_3651_max_OorderE,axiom,
% 0.35/0.60      ! [B: nat,A: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ B @ A )
% 0.35/0.60       => ( A
% 0.35/0.60          = ( ord_max_nat @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.orderE
% 0.35/0.60  thf(fact_3652_max_OorderE,axiom,
% 0.35/0.60      ! [B: int,A: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ B @ A )
% 0.35/0.60       => ( A
% 0.35/0.60          = ( ord_max_int @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.orderE
% 0.35/0.60  thf(fact_3653_max_OorderI,axiom,
% 0.35/0.60      ! [A: rat,B: rat] :
% 0.35/0.60        ( ( A
% 0.35/0.60          = ( ord_max_rat @ A @ B ) )
% 0.35/0.60       => ( ord_less_eq_rat @ B @ A ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.orderI
% 0.35/0.60  thf(fact_3654_max_OorderI,axiom,
% 0.35/0.60      ! [A: num,B: num] :
% 0.35/0.60        ( ( A
% 0.35/0.60          = ( ord_max_num @ A @ B ) )
% 0.35/0.60       => ( ord_less_eq_num @ B @ A ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.orderI
% 0.35/0.60  thf(fact_3655_max_OorderI,axiom,
% 0.35/0.60      ! [A: nat,B: nat] :
% 0.35/0.60        ( ( A
% 0.35/0.60          = ( ord_max_nat @ A @ B ) )
% 0.35/0.60       => ( ord_less_eq_nat @ B @ A ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.orderI
% 0.35/0.60  thf(fact_3656_max_OorderI,axiom,
% 0.35/0.60      ! [A: int,B: int] :
% 0.35/0.60        ( ( A
% 0.35/0.60          = ( ord_max_int @ A @ B ) )
% 0.35/0.60       => ( ord_less_eq_int @ B @ A ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.orderI
% 0.35/0.60  thf(fact_3657_max_OboundedE,axiom,
% 0.35/0.60      ! [B: rat,C: rat,A: rat] :
% 0.35/0.60        ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 0.35/0.60       => ~ ( ( ord_less_eq_rat @ B @ A )
% 0.35/0.60           => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.boundedE
% 0.35/0.60  thf(fact_3658_max_OboundedE,axiom,
% 0.35/0.60      ! [B: num,C: num,A: num] :
% 0.35/0.60        ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 0.35/0.60       => ~ ( ( ord_less_eq_num @ B @ A )
% 0.35/0.60           => ~ ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.boundedE
% 0.35/0.60  thf(fact_3659_max_OboundedE,axiom,
% 0.35/0.60      ! [B: nat,C: nat,A: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 0.35/0.60       => ~ ( ( ord_less_eq_nat @ B @ A )
% 0.35/0.60           => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.boundedE
% 0.35/0.60  thf(fact_3660_max_OboundedE,axiom,
% 0.35/0.60      ! [B: int,C: int,A: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 0.35/0.60       => ~ ( ( ord_less_eq_int @ B @ A )
% 0.35/0.60           => ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.boundedE
% 0.35/0.60  thf(fact_3661_max_OboundedI,axiom,
% 0.35/0.60      ! [B: rat,A: rat,C: rat] :
% 0.35/0.60        ( ( ord_less_eq_rat @ B @ A )
% 0.35/0.60       => ( ( ord_less_eq_rat @ C @ A )
% 0.35/0.60         => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.boundedI
% 0.35/0.60  thf(fact_3662_max_OboundedI,axiom,
% 0.35/0.60      ! [B: num,A: num,C: num] :
% 0.35/0.60        ( ( ord_less_eq_num @ B @ A )
% 0.35/0.60       => ( ( ord_less_eq_num @ C @ A )
% 0.35/0.60         => ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.boundedI
% 0.35/0.60  thf(fact_3663_max_OboundedI,axiom,
% 0.35/0.60      ! [B: nat,A: nat,C: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ B @ A )
% 0.35/0.60       => ( ( ord_less_eq_nat @ C @ A )
% 0.35/0.60         => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.boundedI
% 0.35/0.60  thf(fact_3664_max_OboundedI,axiom,
% 0.35/0.60      ! [B: int,A: int,C: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ B @ A )
% 0.35/0.60       => ( ( ord_less_eq_int @ C @ A )
% 0.35/0.60         => ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.boundedI
% 0.35/0.60  thf(fact_3665_max_Oorder__iff,axiom,
% 0.35/0.60      ( ord_less_eq_rat
% 0.35/0.60      = ( ^ [B4: rat,A4: rat] :
% 0.35/0.60            ( A4
% 0.35/0.60            = ( ord_max_rat @ A4 @ B4 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.order_iff
% 0.35/0.60  thf(fact_3666_max_Oorder__iff,axiom,
% 0.35/0.60      ( ord_less_eq_num
% 0.35/0.60      = ( ^ [B4: num,A4: num] :
% 0.35/0.60            ( A4
% 0.35/0.60            = ( ord_max_num @ A4 @ B4 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.order_iff
% 0.35/0.60  thf(fact_3667_max_Oorder__iff,axiom,
% 0.35/0.60      ( ord_less_eq_nat
% 0.35/0.60      = ( ^ [B4: nat,A4: nat] :
% 0.35/0.60            ( A4
% 0.35/0.60            = ( ord_max_nat @ A4 @ B4 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.order_iff
% 0.35/0.60  thf(fact_3668_max_Oorder__iff,axiom,
% 0.35/0.60      ( ord_less_eq_int
% 0.35/0.60      = ( ^ [B4: int,A4: int] :
% 0.35/0.60            ( A4
% 0.35/0.60            = ( ord_max_int @ A4 @ B4 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.order_iff
% 0.35/0.60  thf(fact_3669_max_Ocobounded1,axiom,
% 0.35/0.60      ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.cobounded1
% 0.35/0.60  thf(fact_3670_max_Ocobounded1,axiom,
% 0.35/0.60      ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.cobounded1
% 0.35/0.60  thf(fact_3671_max_Ocobounded1,axiom,
% 0.35/0.60      ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.cobounded1
% 0.35/0.60  thf(fact_3672_max_Ocobounded1,axiom,
% 0.35/0.60      ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.cobounded1
% 0.35/0.60  thf(fact_3673_max_Ocobounded2,axiom,
% 0.35/0.60      ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.cobounded2
% 0.35/0.60  thf(fact_3674_max_Ocobounded2,axiom,
% 0.35/0.60      ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.cobounded2
% 0.35/0.60  thf(fact_3675_max_Ocobounded2,axiom,
% 0.35/0.60      ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.cobounded2
% 0.35/0.60  thf(fact_3676_max_Ocobounded2,axiom,
% 0.35/0.60      ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.cobounded2
% 0.35/0.60  thf(fact_3677_le__max__iff__disj,axiom,
% 0.35/0.60      ! [Z2: rat,X: rat,Y: rat] :
% 0.35/0.60        ( ( ord_less_eq_rat @ Z2 @ ( ord_max_rat @ X @ Y ) )
% 0.35/0.60        = ( ( ord_less_eq_rat @ Z2 @ X )
% 0.35/0.60          | ( ord_less_eq_rat @ Z2 @ Y ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % le_max_iff_disj
% 0.35/0.60  thf(fact_3678_le__max__iff__disj,axiom,
% 0.35/0.60      ! [Z2: num,X: num,Y: num] :
% 0.35/0.60        ( ( ord_less_eq_num @ Z2 @ ( ord_max_num @ X @ Y ) )
% 0.35/0.60        = ( ( ord_less_eq_num @ Z2 @ X )
% 0.35/0.60          | ( ord_less_eq_num @ Z2 @ Y ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % le_max_iff_disj
% 0.35/0.60  thf(fact_3679_le__max__iff__disj,axiom,
% 0.35/0.60      ! [Z2: nat,X: nat,Y: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ Z2 @ ( ord_max_nat @ X @ Y ) )
% 0.35/0.60        = ( ( ord_less_eq_nat @ Z2 @ X )
% 0.35/0.60          | ( ord_less_eq_nat @ Z2 @ Y ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % le_max_iff_disj
% 0.35/0.60  thf(fact_3680_le__max__iff__disj,axiom,
% 0.35/0.60      ! [Z2: int,X: int,Y: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ Z2 @ ( ord_max_int @ X @ Y ) )
% 0.35/0.60        = ( ( ord_less_eq_int @ Z2 @ X )
% 0.35/0.60          | ( ord_less_eq_int @ Z2 @ Y ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % le_max_iff_disj
% 0.35/0.60  thf(fact_3681_max_Oabsorb__iff1,axiom,
% 0.35/0.60      ( ord_less_eq_rat
% 0.35/0.60      = ( ^ [B4: rat,A4: rat] :
% 0.35/0.60            ( ( ord_max_rat @ A4 @ B4 )
% 0.35/0.60            = A4 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb_iff1
% 0.35/0.60  thf(fact_3682_max_Oabsorb__iff1,axiom,
% 0.35/0.60      ( ord_less_eq_num
% 0.35/0.60      = ( ^ [B4: num,A4: num] :
% 0.35/0.60            ( ( ord_max_num @ A4 @ B4 )
% 0.35/0.60            = A4 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb_iff1
% 0.35/0.60  thf(fact_3683_max_Oabsorb__iff1,axiom,
% 0.35/0.60      ( ord_less_eq_nat
% 0.35/0.60      = ( ^ [B4: nat,A4: nat] :
% 0.35/0.60            ( ( ord_max_nat @ A4 @ B4 )
% 0.35/0.60            = A4 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb_iff1
% 0.35/0.60  thf(fact_3684_max_Oabsorb__iff1,axiom,
% 0.35/0.60      ( ord_less_eq_int
% 0.35/0.60      = ( ^ [B4: int,A4: int] :
% 0.35/0.60            ( ( ord_max_int @ A4 @ B4 )
% 0.35/0.60            = A4 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb_iff1
% 0.35/0.60  thf(fact_3685_max_Oabsorb__iff2,axiom,
% 0.35/0.60      ( ord_less_eq_rat
% 0.35/0.60      = ( ^ [A4: rat,B4: rat] :
% 0.35/0.60            ( ( ord_max_rat @ A4 @ B4 )
% 0.35/0.60            = B4 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb_iff2
% 0.35/0.60  thf(fact_3686_max_Oabsorb__iff2,axiom,
% 0.35/0.60      ( ord_less_eq_num
% 0.35/0.60      = ( ^ [A4: num,B4: num] :
% 0.35/0.60            ( ( ord_max_num @ A4 @ B4 )
% 0.35/0.60            = B4 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb_iff2
% 0.35/0.60  thf(fact_3687_max_Oabsorb__iff2,axiom,
% 0.35/0.60      ( ord_less_eq_nat
% 0.35/0.60      = ( ^ [A4: nat,B4: nat] :
% 0.35/0.60            ( ( ord_max_nat @ A4 @ B4 )
% 0.35/0.60            = B4 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb_iff2
% 0.35/0.60  thf(fact_3688_max_Oabsorb__iff2,axiom,
% 0.35/0.60      ( ord_less_eq_int
% 0.35/0.60      = ( ^ [A4: int,B4: int] :
% 0.35/0.60            ( ( ord_max_int @ A4 @ B4 )
% 0.35/0.60            = B4 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.absorb_iff2
% 0.35/0.60  thf(fact_3689_max_OcoboundedI1,axiom,
% 0.35/0.60      ! [C: rat,A: rat,B: rat] :
% 0.35/0.60        ( ( ord_less_eq_rat @ C @ A )
% 0.35/0.60       => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.coboundedI1
% 0.35/0.60  thf(fact_3690_max_OcoboundedI1,axiom,
% 0.35/0.60      ! [C: num,A: num,B: num] :
% 0.35/0.60        ( ( ord_less_eq_num @ C @ A )
% 0.35/0.60       => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.coboundedI1
% 0.35/0.60  thf(fact_3691_max_OcoboundedI1,axiom,
% 0.35/0.60      ! [C: nat,A: nat,B: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ C @ A )
% 0.35/0.60       => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.coboundedI1
% 0.35/0.60  thf(fact_3692_max_OcoboundedI1,axiom,
% 0.35/0.60      ! [C: int,A: int,B: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ C @ A )
% 0.35/0.60       => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.coboundedI1
% 0.35/0.60  thf(fact_3693_max_OcoboundedI2,axiom,
% 0.35/0.60      ! [C: rat,B: rat,A: rat] :
% 0.35/0.60        ( ( ord_less_eq_rat @ C @ B )
% 0.35/0.60       => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.coboundedI2
% 0.35/0.60  thf(fact_3694_max_OcoboundedI2,axiom,
% 0.35/0.60      ! [C: num,B: num,A: num] :
% 0.35/0.60        ( ( ord_less_eq_num @ C @ B )
% 0.35/0.60       => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.coboundedI2
% 0.35/0.60  thf(fact_3695_max_OcoboundedI2,axiom,
% 0.35/0.60      ! [C: nat,B: nat,A: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ C @ B )
% 0.35/0.60       => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.coboundedI2
% 0.35/0.60  thf(fact_3696_max_OcoboundedI2,axiom,
% 0.35/0.60      ! [C: int,B: int,A: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ C @ B )
% 0.35/0.60       => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.coboundedI2
% 0.35/0.60  thf(fact_3697_less__max__iff__disj,axiom,
% 0.35/0.60      ! [Z2: real,X: real,Y: real] :
% 0.35/0.60        ( ( ord_less_real @ Z2 @ ( ord_max_real @ X @ Y ) )
% 0.35/0.60        = ( ( ord_less_real @ Z2 @ X )
% 0.35/0.60          | ( ord_less_real @ Z2 @ Y ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % less_max_iff_disj
% 0.35/0.60  thf(fact_3698_less__max__iff__disj,axiom,
% 0.35/0.60      ! [Z2: rat,X: rat,Y: rat] :
% 0.35/0.60        ( ( ord_less_rat @ Z2 @ ( ord_max_rat @ X @ Y ) )
% 0.35/0.60        = ( ( ord_less_rat @ Z2 @ X )
% 0.35/0.60          | ( ord_less_rat @ Z2 @ Y ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % less_max_iff_disj
% 0.35/0.60  thf(fact_3699_less__max__iff__disj,axiom,
% 0.35/0.60      ! [Z2: num,X: num,Y: num] :
% 0.35/0.60        ( ( ord_less_num @ Z2 @ ( ord_max_num @ X @ Y ) )
% 0.35/0.60        = ( ( ord_less_num @ Z2 @ X )
% 0.35/0.60          | ( ord_less_num @ Z2 @ Y ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % less_max_iff_disj
% 0.35/0.60  thf(fact_3700_less__max__iff__disj,axiom,
% 0.35/0.60      ! [Z2: nat,X: nat,Y: nat] :
% 0.35/0.60        ( ( ord_less_nat @ Z2 @ ( ord_max_nat @ X @ Y ) )
% 0.35/0.60        = ( ( ord_less_nat @ Z2 @ X )
% 0.35/0.60          | ( ord_less_nat @ Z2 @ Y ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % less_max_iff_disj
% 0.35/0.60  thf(fact_3701_less__max__iff__disj,axiom,
% 0.35/0.60      ! [Z2: int,X: int,Y: int] :
% 0.35/0.60        ( ( ord_less_int @ Z2 @ ( ord_max_int @ X @ Y ) )
% 0.35/0.60        = ( ( ord_less_int @ Z2 @ X )
% 0.35/0.60          | ( ord_less_int @ Z2 @ Y ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % less_max_iff_disj
% 0.35/0.60  thf(fact_3702_max_Ostrict__boundedE,axiom,
% 0.35/0.60      ! [B: real,C: real,A: real] :
% 0.35/0.60        ( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
% 0.35/0.60       => ~ ( ( ord_less_real @ B @ A )
% 0.35/0.60           => ~ ( ord_less_real @ C @ A ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.strict_boundedE
% 0.35/0.60  thf(fact_3703_max_Ostrict__boundedE,axiom,
% 0.35/0.60      ! [B: rat,C: rat,A: rat] :
% 0.35/0.60        ( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
% 0.35/0.60       => ~ ( ( ord_less_rat @ B @ A )
% 0.35/0.60           => ~ ( ord_less_rat @ C @ A ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.strict_boundedE
% 0.35/0.60  thf(fact_3704_max_Ostrict__boundedE,axiom,
% 0.35/0.60      ! [B: num,C: num,A: num] :
% 0.35/0.60        ( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
% 0.35/0.60       => ~ ( ( ord_less_num @ B @ A )
% 0.35/0.60           => ~ ( ord_less_num @ C @ A ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.strict_boundedE
% 0.35/0.60  thf(fact_3705_max_Ostrict__boundedE,axiom,
% 0.35/0.60      ! [B: nat,C: nat,A: nat] :
% 0.35/0.60        ( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
% 0.35/0.60       => ~ ( ( ord_less_nat @ B @ A )
% 0.35/0.60           => ~ ( ord_less_nat @ C @ A ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.strict_boundedE
% 0.35/0.60  thf(fact_3706_max_Ostrict__boundedE,axiom,
% 0.35/0.60      ! [B: int,C: int,A: int] :
% 0.35/0.60        ( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
% 0.35/0.60       => ~ ( ( ord_less_int @ B @ A )
% 0.35/0.60           => ~ ( ord_less_int @ C @ A ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.strict_boundedE
% 0.35/0.60  thf(fact_3707_max_Ostrict__order__iff,axiom,
% 0.35/0.60      ( ord_less_real
% 0.35/0.60      = ( ^ [B4: real,A4: real] :
% 0.35/0.60            ( ( A4
% 0.35/0.60              = ( ord_max_real @ A4 @ B4 ) )
% 0.35/0.60            & ( A4 != B4 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.strict_order_iff
% 0.35/0.60  thf(fact_3708_max_Ostrict__order__iff,axiom,
% 0.35/0.60      ( ord_less_rat
% 0.35/0.60      = ( ^ [B4: rat,A4: rat] :
% 0.35/0.60            ( ( A4
% 0.35/0.60              = ( ord_max_rat @ A4 @ B4 ) )
% 0.35/0.60            & ( A4 != B4 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.strict_order_iff
% 0.35/0.60  thf(fact_3709_max_Ostrict__order__iff,axiom,
% 0.35/0.60      ( ord_less_num
% 0.35/0.60      = ( ^ [B4: num,A4: num] :
% 0.35/0.60            ( ( A4
% 0.35/0.60              = ( ord_max_num @ A4 @ B4 ) )
% 0.35/0.60            & ( A4 != B4 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.strict_order_iff
% 0.35/0.60  thf(fact_3710_max_Ostrict__order__iff,axiom,
% 0.35/0.60      ( ord_less_nat
% 0.35/0.60      = ( ^ [B4: nat,A4: nat] :
% 0.35/0.60            ( ( A4
% 0.35/0.60              = ( ord_max_nat @ A4 @ B4 ) )
% 0.35/0.60            & ( A4 != B4 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.strict_order_iff
% 0.35/0.60  thf(fact_3711_max_Ostrict__order__iff,axiom,
% 0.35/0.60      ( ord_less_int
% 0.35/0.60      = ( ^ [B4: int,A4: int] :
% 0.35/0.60            ( ( A4
% 0.35/0.60              = ( ord_max_int @ A4 @ B4 ) )
% 0.35/0.60            & ( A4 != B4 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.strict_order_iff
% 0.35/0.60  thf(fact_3712_max_Ostrict__coboundedI1,axiom,
% 0.35/0.60      ! [C: real,A: real,B: real] :
% 0.35/0.60        ( ( ord_less_real @ C @ A )
% 0.35/0.60       => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.strict_coboundedI1
% 0.35/0.60  thf(fact_3713_max_Ostrict__coboundedI1,axiom,
% 0.35/0.60      ! [C: rat,A: rat,B: rat] :
% 0.35/0.60        ( ( ord_less_rat @ C @ A )
% 0.35/0.60       => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.strict_coboundedI1
% 0.35/0.60  thf(fact_3714_max_Ostrict__coboundedI1,axiom,
% 0.35/0.60      ! [C: num,A: num,B: num] :
% 0.35/0.60        ( ( ord_less_num @ C @ A )
% 0.35/0.60       => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.strict_coboundedI1
% 0.35/0.60  thf(fact_3715_max_Ostrict__coboundedI1,axiom,
% 0.35/0.60      ! [C: nat,A: nat,B: nat] :
% 0.35/0.60        ( ( ord_less_nat @ C @ A )
% 0.35/0.60       => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.strict_coboundedI1
% 0.35/0.60  thf(fact_3716_max_Ostrict__coboundedI1,axiom,
% 0.35/0.60      ! [C: int,A: int,B: int] :
% 0.35/0.60        ( ( ord_less_int @ C @ A )
% 0.35/0.60       => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.strict_coboundedI1
% 0.35/0.60  thf(fact_3717_max_Ostrict__coboundedI2,axiom,
% 0.35/0.60      ! [C: real,B: real,A: real] :
% 0.35/0.60        ( ( ord_less_real @ C @ B )
% 0.35/0.60       => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.strict_coboundedI2
% 0.35/0.60  thf(fact_3718_max_Ostrict__coboundedI2,axiom,
% 0.35/0.60      ! [C: rat,B: rat,A: rat] :
% 0.35/0.60        ( ( ord_less_rat @ C @ B )
% 0.35/0.60       => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.strict_coboundedI2
% 0.35/0.60  thf(fact_3719_max_Ostrict__coboundedI2,axiom,
% 0.35/0.60      ! [C: num,B: num,A: num] :
% 0.35/0.60        ( ( ord_less_num @ C @ B )
% 0.35/0.60       => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.strict_coboundedI2
% 0.35/0.60  thf(fact_3720_max_Ostrict__coboundedI2,axiom,
% 0.35/0.60      ! [C: nat,B: nat,A: nat] :
% 0.35/0.60        ( ( ord_less_nat @ C @ B )
% 0.35/0.60       => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.strict_coboundedI2
% 0.35/0.60  thf(fact_3721_max_Ostrict__coboundedI2,axiom,
% 0.35/0.60      ! [C: int,B: int,A: int] :
% 0.35/0.60        ( ( ord_less_int @ C @ B )
% 0.35/0.60       => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % max.strict_coboundedI2
% 0.35/0.60  thf(fact_3722_unique__quotient__lemma__neg,axiom,
% 0.35/0.60      ! [B: int,Q6: int,R4: int,Q4: int,R2: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 0.35/0.60       => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.35/0.60         => ( ( ord_less_int @ B @ R2 )
% 0.35/0.60           => ( ( ord_less_int @ B @ R4 )
% 0.35/0.60             => ( ord_less_eq_int @ Q4 @ Q6 ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % unique_quotient_lemma_neg
% 0.35/0.60  thf(fact_3723_unique__quotient__lemma,axiom,
% 0.35/0.60      ! [B: int,Q6: int,R4: int,Q4: int,R2: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 0.35/0.60       => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 0.35/0.60         => ( ( ord_less_int @ R4 @ B )
% 0.35/0.60           => ( ( ord_less_int @ R2 @ B )
% 0.35/0.60             => ( ord_less_eq_int @ Q6 @ Q4 ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % unique_quotient_lemma
% 0.35/0.60  thf(fact_3724_zdiv__mono2__neg__lemma,axiom,
% 0.35/0.60      ! [B: int,Q4: int,R2: int,B7: int,Q6: int,R4: int] :
% 0.35/0.60        ( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
% 0.35/0.60          = ( plus_plus_int @ ( times_times_int @ B7 @ Q6 ) @ R4 ) )
% 0.35/0.60       => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q6 ) @ R4 ) @ zero_zero_int )
% 0.35/0.60         => ( ( ord_less_int @ R2 @ B )
% 0.35/0.60           => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 0.35/0.60             => ( ( ord_less_int @ zero_zero_int @ B7 )
% 0.35/0.60               => ( ( ord_less_eq_int @ B7 @ B )
% 0.35/0.60                 => ( ord_less_eq_int @ Q6 @ Q4 ) ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % zdiv_mono2_neg_lemma
% 0.35/0.60  thf(fact_3725_zdiv__mono2__lemma,axiom,
% 0.35/0.60      ! [B: int,Q4: int,R2: int,B7: int,Q6: int,R4: int] :
% 0.35/0.60        ( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
% 0.35/0.60          = ( plus_plus_int @ ( times_times_int @ B7 @ Q6 ) @ R4 ) )
% 0.35/0.60       => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q6 ) @ R4 ) )
% 0.35/0.60         => ( ( ord_less_int @ R4 @ B7 )
% 0.35/0.60           => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.35/0.60             => ( ( ord_less_int @ zero_zero_int @ B7 )
% 0.35/0.60               => ( ( ord_less_eq_int @ B7 @ B )
% 0.35/0.60                 => ( ord_less_eq_int @ Q4 @ Q6 ) ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % zdiv_mono2_lemma
% 0.35/0.60  thf(fact_3726_q__pos__lemma,axiom,
% 0.35/0.60      ! [B7: int,Q6: int,R4: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q6 ) @ R4 ) )
% 0.35/0.60       => ( ( ord_less_int @ R4 @ B7 )
% 0.35/0.60         => ( ( ord_less_int @ zero_zero_int @ B7 )
% 0.35/0.60           => ( ord_less_eq_int @ zero_zero_int @ Q6 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % q_pos_lemma
% 0.35/0.60  thf(fact_3727_int__div__pos__eq,axiom,
% 0.35/0.60      ! [A: int,B: int,Q4: int,R2: int] :
% 0.35/0.60        ( ( A
% 0.35/0.60          = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 0.35/0.60       => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.35/0.60         => ( ( ord_less_int @ R2 @ B )
% 0.35/0.60           => ( ( divide_divide_int @ A @ B )
% 0.35/0.60              = Q4 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % int_div_pos_eq
% 0.35/0.60  thf(fact_3728_int__div__neg__eq,axiom,
% 0.35/0.60      ! [A: int,B: int,Q4: int,R2: int] :
% 0.35/0.60        ( ( A
% 0.35/0.60          = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 0.35/0.60       => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.35/0.60         => ( ( ord_less_int @ B @ R2 )
% 0.35/0.60           => ( ( divide_divide_int @ A @ B )
% 0.35/0.60              = Q4 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % int_div_neg_eq
% 0.35/0.60  thf(fact_3729_split__zdiv,axiom,
% 0.35/0.60      ! [P: int > $o,N: int,K: int] :
% 0.35/0.60        ( ( P @ ( divide_divide_int @ N @ K ) )
% 0.35/0.60        = ( ( ( K = zero_zero_int )
% 0.35/0.60           => ( P @ zero_zero_int ) )
% 0.35/0.60          & ( ( ord_less_int @ zero_zero_int @ K )
% 0.35/0.60           => ! [I4: int,J3: int] :
% 0.35/0.60                ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.35/0.60                  & ( ord_less_int @ J3 @ K )
% 0.35/0.60                  & ( N
% 0.35/0.60                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.35/0.60               => ( P @ I4 ) ) )
% 0.35/0.60          & ( ( ord_less_int @ K @ zero_zero_int )
% 0.35/0.60           => ! [I4: int,J3: int] :
% 0.35/0.60                ( ( ( ord_less_int @ K @ J3 )
% 0.35/0.60                  & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.35/0.60                  & ( N
% 0.35/0.60                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.35/0.60               => ( P @ I4 ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % split_zdiv
% 0.35/0.60  thf(fact_3730_zdiv__mono1,axiom,
% 0.35/0.60      ! [A: int,A7: int,B: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ A @ A7 )
% 0.35/0.60       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.60         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A7 @ B ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % zdiv_mono1
% 0.35/0.60  thf(fact_3731_zdiv__mono2,axiom,
% 0.35/0.60      ! [A: int,B7: int,B: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.35/0.60       => ( ( ord_less_int @ zero_zero_int @ B7 )
% 0.35/0.60         => ( ( ord_less_eq_int @ B7 @ B )
% 0.35/0.60           => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B7 ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % zdiv_mono2
% 0.35/0.60  thf(fact_3732_zdiv__eq__0__iff,axiom,
% 0.35/0.60      ! [I: int,K: int] :
% 0.35/0.60        ( ( ( divide_divide_int @ I @ K )
% 0.35/0.60          = zero_zero_int )
% 0.35/0.60        = ( ( K = zero_zero_int )
% 0.35/0.60          | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.35/0.60            & ( ord_less_int @ I @ K ) )
% 0.35/0.60          | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 0.35/0.60            & ( ord_less_int @ K @ I ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % zdiv_eq_0_iff
% 0.35/0.60  thf(fact_3733_zdiv__mono1__neg,axiom,
% 0.35/0.60      ! [A: int,A7: int,B: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ A @ A7 )
% 0.35/0.60       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.35/0.60         => ( ord_less_eq_int @ ( divide_divide_int @ A7 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % zdiv_mono1_neg
% 0.35/0.60  thf(fact_3734_zdiv__mono2__neg,axiom,
% 0.35/0.60      ! [A: int,B7: int,B: int] :
% 0.35/0.60        ( ( ord_less_int @ A @ zero_zero_int )
% 0.35/0.60       => ( ( ord_less_int @ zero_zero_int @ B7 )
% 0.35/0.60         => ( ( ord_less_eq_int @ B7 @ B )
% 0.35/0.60           => ( ord_less_eq_int @ ( divide_divide_int @ A @ B7 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % zdiv_mono2_neg
% 0.35/0.60  thf(fact_3735_div__int__pos__iff,axiom,
% 0.35/0.60      ! [K: int,L: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
% 0.35/0.60        = ( ( K = zero_zero_int )
% 0.35/0.60          | ( L = zero_zero_int )
% 0.35/0.60          | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.60            & ( ord_less_eq_int @ zero_zero_int @ L ) )
% 0.35/0.60          | ( ( ord_less_int @ K @ zero_zero_int )
% 0.35/0.60            & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % div_int_pos_iff
% 0.35/0.60  thf(fact_3736_div__positive__int,axiom,
% 0.35/0.60      ! [L: int,K: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ L @ K )
% 0.35/0.60       => ( ( ord_less_int @ zero_zero_int @ L )
% 0.35/0.60         => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % div_positive_int
% 0.35/0.60  thf(fact_3737_div__nonneg__neg__le0,axiom,
% 0.35/0.60      ! [A: int,B: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.35/0.60       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.35/0.60         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % div_nonneg_neg_le0
% 0.35/0.60  thf(fact_3738_div__nonpos__pos__le0,axiom,
% 0.35/0.60      ! [A: int,B: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.35/0.60       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.60         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % div_nonpos_pos_le0
% 0.35/0.60  thf(fact_3739_pos__imp__zdiv__pos__iff,axiom,
% 0.35/0.60      ! [K: int,I: int] :
% 0.35/0.60        ( ( ord_less_int @ zero_zero_int @ K )
% 0.35/0.60       => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
% 0.35/0.60          = ( ord_less_eq_int @ K @ I ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % pos_imp_zdiv_pos_iff
% 0.35/0.60  thf(fact_3740_neg__imp__zdiv__nonneg__iff,axiom,
% 0.35/0.60      ! [B: int,A: int] :
% 0.35/0.60        ( ( ord_less_int @ B @ zero_zero_int )
% 0.35/0.60       => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.35/0.60          = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % neg_imp_zdiv_nonneg_iff
% 0.35/0.60  thf(fact_3741_pos__imp__zdiv__nonneg__iff,axiom,
% 0.35/0.60      ! [B: int,A: int] :
% 0.35/0.60        ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.60       => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.35/0.60          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % pos_imp_zdiv_nonneg_iff
% 0.35/0.60  thf(fact_3742_nonneg1__imp__zdiv__pos__iff,axiom,
% 0.35/0.60      ! [A: int,B: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.35/0.60       => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.35/0.60          = ( ( ord_less_eq_int @ B @ A )
% 0.35/0.60            & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % nonneg1_imp_zdiv_pos_iff
% 0.35/0.60  thf(fact_3743_pos__imp__zdiv__neg__iff,axiom,
% 0.35/0.60      ! [B: int,A: int] :
% 0.35/0.60        ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.60       => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 0.35/0.60          = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % pos_imp_zdiv_neg_iff
% 0.35/0.60  thf(fact_3744_neg__imp__zdiv__neg__iff,axiom,
% 0.35/0.60      ! [B: int,A: int] :
% 0.35/0.60        ( ( ord_less_int @ B @ zero_zero_int )
% 0.35/0.60       => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 0.35/0.60          = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % neg_imp_zdiv_neg_iff
% 0.35/0.60  thf(fact_3745_int__div__less__self,axiom,
% 0.35/0.60      ! [X: int,K: int] :
% 0.35/0.60        ( ( ord_less_int @ zero_zero_int @ X )
% 0.35/0.60       => ( ( ord_less_int @ one_one_int @ K )
% 0.35/0.60         => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % int_div_less_self
% 0.35/0.60  thf(fact_3746_div__neg__pos__less0,axiom,
% 0.35/0.60      ! [A: int,B: int] :
% 0.35/0.60        ( ( ord_less_int @ A @ zero_zero_int )
% 0.35/0.60       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.60         => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % div_neg_pos_less0
% 0.35/0.60  thf(fact_3747_div__positive,axiom,
% 0.35/0.60      ! [B: code_integer,A: code_integer] :
% 0.35/0.60        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.35/0.60       => ( ( ord_le3102999989581377725nteger @ B @ A )
% 0.35/0.60         => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % div_positive
% 0.35/0.60  thf(fact_3748_div__positive,axiom,
% 0.35/0.60      ! [B: nat,A: nat] :
% 0.35/0.60        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.35/0.60       => ( ( ord_less_eq_nat @ B @ A )
% 0.35/0.60         => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % div_positive
% 0.35/0.60  thf(fact_3749_div__positive,axiom,
% 0.35/0.60      ! [B: int,A: int] :
% 0.35/0.60        ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.60       => ( ( ord_less_eq_int @ B @ A )
% 0.35/0.60         => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % div_positive
% 0.35/0.60  thf(fact_3750_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.35/0.60      ! [A: code_integer,B: code_integer] :
% 0.35/0.60        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.35/0.60       => ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.35/0.60         => ( ( divide6298287555418463151nteger @ A @ B )
% 0.35/0.60            = zero_z3403309356797280102nteger ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % unique_euclidean_semiring_numeral_class.div_less
% 0.35/0.60  thf(fact_3751_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.35/0.60      ! [A: nat,B: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.35/0.60       => ( ( ord_less_nat @ A @ B )
% 0.35/0.60         => ( ( divide_divide_nat @ A @ B )
% 0.35/0.60            = zero_zero_nat ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % unique_euclidean_semiring_numeral_class.div_less
% 0.35/0.60  thf(fact_3752_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.35/0.60      ! [A: int,B: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.35/0.60       => ( ( ord_less_int @ A @ B )
% 0.35/0.60         => ( ( divide_divide_int @ A @ B )
% 0.35/0.60            = zero_zero_int ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % unique_euclidean_semiring_numeral_class.div_less
% 0.35/0.60  thf(fact_3753_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.35/0.60      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.35/0.60        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.35/0.60       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.35/0.60          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.35/0.60  thf(fact_3754_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.35/0.60      ! [C: nat,A: nat,B: nat] :
% 0.35/0.60        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.35/0.60       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.35/0.60          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.35/0.60  thf(fact_3755_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.35/0.60      ! [C: int,A: int,B: int] :
% 0.35/0.60        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.60       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.35/0.60          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.35/0.60  thf(fact_3756_discrete,axiom,
% 0.35/0.60      ( ord_less_nat
% 0.35/0.60      = ( ^ [A4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ one_one_nat ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % discrete
% 0.35/0.60  thf(fact_3757_discrete,axiom,
% 0.35/0.60      ( ord_less_int
% 0.35/0.60      = ( ^ [A4: int] : ( ord_less_eq_int @ ( plus_plus_int @ A4 @ one_one_int ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % discrete
% 0.35/0.60  thf(fact_3758_divmod__step__eq,axiom,
% 0.35/0.60      ! [L: num,R2: nat,Q4: nat] :
% 0.35/0.60        ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
% 0.35/0.60         => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
% 0.35/0.60            = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
% 0.35/0.60        & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
% 0.35/0.60         => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
% 0.35/0.60            = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % divmod_step_eq
% 0.35/0.60  thf(fact_3759_divmod__step__eq,axiom,
% 0.35/0.60      ! [L: num,R2: int,Q4: int] :
% 0.35/0.60        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
% 0.35/0.60         => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.35/0.60            = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
% 0.35/0.60        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
% 0.35/0.60         => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.35/0.60            = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % divmod_step_eq
% 0.35/0.60  thf(fact_3760_divmod__step__eq,axiom,
% 0.35/0.60      ! [L: num,R2: code_integer,Q4: code_integer] :
% 0.35/0.60        ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
% 0.35/0.60         => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
% 0.35/0.60            = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
% 0.35/0.60        & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
% 0.35/0.60         => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
% 0.35/0.60            = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % divmod_step_eq
% 0.35/0.60  thf(fact_3761_enat__ord__number_I1_J,axiom,
% 0.35/0.60      ! [M: num,N: num] :
% 0.35/0.60        ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.35/0.60        = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % enat_ord_number(1)
% 0.35/0.60  thf(fact_3762_enat__ord__number_I2_J,axiom,
% 0.35/0.60      ! [M: num,N: num] :
% 0.35/0.60        ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.35/0.60        = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % enat_ord_number(2)
% 0.35/0.60  thf(fact_3763_unset__bit__0,axiom,
% 0.35/0.60      ! [A: nat] :
% 0.35/0.60        ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
% 0.35/0.60        = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % unset_bit_0
% 0.35/0.60  thf(fact_3764_unset__bit__0,axiom,
% 0.35/0.60      ! [A: code_integer] :
% 0.35/0.60        ( ( bit_se8260200283734997820nteger @ zero_zero_nat @ A )
% 0.35/0.60        = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % unset_bit_0
% 0.35/0.60  thf(fact_3765_unset__bit__0,axiom,
% 0.35/0.60      ! [A: int] :
% 0.35/0.60        ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
% 0.35/0.60        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % unset_bit_0
% 0.35/0.60  thf(fact_3766_succ__less__length__list,axiom,
% 0.35/0.60      ! [Deg: nat,Mi: nat,X: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 0.35/0.60        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.60       => ( ( ord_less_eq_nat @ Mi @ X )
% 0.35/0.60         => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.60           => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.35/0.60              = ( if_option_nat
% 0.35/0.60                @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.60                   != none_nat )
% 0.35/0.60                  & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.60                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.60                @ ( if_option_nat
% 0.35/0.60                  @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.60                    = none_nat )
% 0.35/0.60                  @ none_nat
% 0.35/0.60                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % succ_less_length_list
% 0.35/0.60  thf(fact_3767_succ__greatereq__min,axiom,
% 0.35/0.60      ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.60        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.60       => ( ( ord_less_eq_nat @ Mi @ X )
% 0.35/0.60         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.35/0.60            = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.60              @ ( if_option_nat
% 0.35/0.60                @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.60                   != none_nat )
% 0.35/0.60                  & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.60                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.60                @ ( if_option_nat
% 0.35/0.60                  @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.60                    = none_nat )
% 0.35/0.60                  @ none_nat
% 0.35/0.60                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.60              @ none_nat ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % succ_greatereq_min
% 0.35/0.60  thf(fact_3768_pred__lesseq__max,axiom,
% 0.35/0.60      ! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.60        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.60       => ( ( ord_less_eq_nat @ X @ Ma )
% 0.35/0.60         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.35/0.60            = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.60              @ ( if_option_nat
% 0.35/0.60                @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.60                   != none_nat )
% 0.35/0.60                  & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.60                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.60                @ ( if_option_nat
% 0.35/0.60                  @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.60                    = none_nat )
% 0.35/0.60                  @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
% 0.35/0.60                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.60              @ none_nat ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % pred_lesseq_max
% 0.35/0.60  thf(fact_3769_pred__less__length__list,axiom,
% 0.35/0.60      ! [Deg: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
% 0.35/0.60        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.60       => ( ( ord_less_eq_nat @ X @ Ma )
% 0.35/0.60         => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.60           => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.35/0.60              = ( if_option_nat
% 0.35/0.60                @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.60                   != none_nat )
% 0.35/0.60                  & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.60                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.60                @ ( if_option_nat
% 0.35/0.60                  @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.60                    = none_nat )
% 0.35/0.60                  @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
% 0.35/0.60                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % pred_less_length_list
% 0.35/0.60  thf(fact_3770_set__vebt_H__def,axiom,
% 0.35/0.60      ( vEBT_VEBT_set_vebt
% 0.35/0.60      = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % set_vebt'_def
% 0.35/0.60  thf(fact_3771_Diff__idemp,axiom,
% 0.35/0.60      ! [A2: set_nat,B2: set_nat] :
% 0.35/0.60        ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ B2 )
% 0.35/0.60        = ( minus_minus_set_nat @ A2 @ B2 ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_idemp
% 0.35/0.60  thf(fact_3772_Diff__iff,axiom,
% 0.35/0.60      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.60        ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 0.35/0.60        = ( ( member8440522571783428010at_nat @ C @ A2 )
% 0.35/0.60          & ~ ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_iff
% 0.35/0.60  thf(fact_3773_Diff__iff,axiom,
% 0.35/0.60      ! [C: real,A2: set_real,B2: set_real] :
% 0.35/0.60        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 0.35/0.60        = ( ( member_real @ C @ A2 )
% 0.35/0.60          & ~ ( member_real @ C @ B2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_iff
% 0.35/0.60  thf(fact_3774_Diff__iff,axiom,
% 0.35/0.60      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.35/0.60        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 0.35/0.60        = ( ( member_set_nat @ C @ A2 )
% 0.35/0.60          & ~ ( member_set_nat @ C @ B2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_iff
% 0.35/0.60  thf(fact_3775_Diff__iff,axiom,
% 0.35/0.60      ! [C: int,A2: set_int,B2: set_int] :
% 0.35/0.60        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.35/0.60        = ( ( member_int @ C @ A2 )
% 0.35/0.60          & ~ ( member_int @ C @ B2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_iff
% 0.35/0.60  thf(fact_3776_Diff__iff,axiom,
% 0.35/0.60      ! [C: nat,A2: set_nat,B2: set_nat] :
% 0.35/0.60        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.35/0.60        = ( ( member_nat @ C @ A2 )
% 0.35/0.60          & ~ ( member_nat @ C @ B2 ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % Diff_iff
% 0.35/0.60  thf(fact_3777_DiffI,axiom,
% 0.35/0.60      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.60        ( ( member8440522571783428010at_nat @ C @ A2 )
% 0.35/0.60       => ( ~ ( member8440522571783428010at_nat @ C @ B2 )
% 0.35/0.60         => ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % DiffI
% 0.35/0.60  thf(fact_3778_DiffI,axiom,
% 0.35/0.60      ! [C: real,A2: set_real,B2: set_real] :
% 0.35/0.60        ( ( member_real @ C @ A2 )
% 0.35/0.60       => ( ~ ( member_real @ C @ B2 )
% 0.35/0.60         => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % DiffI
% 0.35/0.60  thf(fact_3779_DiffI,axiom,
% 0.35/0.60      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.35/0.60        ( ( member_set_nat @ C @ A2 )
% 0.35/0.60       => ( ~ ( member_set_nat @ C @ B2 )
% 0.35/0.60         => ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % DiffI
% 0.35/0.60  thf(fact_3780_DiffI,axiom,
% 0.35/0.60      ! [C: int,A2: set_int,B2: set_int] :
% 0.35/0.60        ( ( member_int @ C @ A2 )
% 0.35/0.60       => ( ~ ( member_int @ C @ B2 )
% 0.35/0.60         => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % DiffI
% 0.35/0.60  thf(fact_3781_DiffI,axiom,
% 0.35/0.60      ! [C: nat,A2: set_nat,B2: set_nat] :
% 0.35/0.60        ( ( member_nat @ C @ A2 )
% 0.35/0.60       => ( ~ ( member_nat @ C @ B2 )
% 0.35/0.60         => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % DiffI
% 0.35/0.60  thf(fact_3782_i0__less,axiom,
% 0.35/0.60      ! [N: extended_enat] :
% 0.35/0.60        ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.35/0.60        = ( N != zero_z5237406670263579293d_enat ) ) ).
% 0.35/0.60  
% 0.35/0.60  % i0_less
% 0.35/0.60  thf(fact_3783_finite__Collect__conjI,axiom,
% 0.35/0.60      ! [P: real > $o,Q: real > $o] :
% 0.35/0.60        ( ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.35/0.60          | ( finite_finite_real @ ( collect_real @ Q ) ) )
% 0.35/0.60       => ( finite_finite_real
% 0.35/0.60          @ ( collect_real
% 0.35/0.60            @ ^ [X3: real] :
% 0.35/0.60                ( ( P @ X3 )
% 0.35/0.60                & ( Q @ X3 ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % finite_Collect_conjI
% 0.35/0.60  thf(fact_3784_finite__Collect__conjI,axiom,
% 0.35/0.60      ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.35/0.60        ( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
% 0.35/0.60          | ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) )
% 0.35/0.60       => ( finite8100373058378681591st_nat
% 0.35/0.60          @ ( collect_list_nat
% 0.35/0.60            @ ^ [X3: list_nat] :
% 0.35/0.60                ( ( P @ X3 )
% 0.35/0.60                & ( Q @ X3 ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % finite_Collect_conjI
% 0.35/0.60  thf(fact_3785_finite__Collect__conjI,axiom,
% 0.35/0.60      ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.35/0.60        ( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 0.35/0.60          | ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
% 0.35/0.60       => ( finite1152437895449049373et_nat
% 0.35/0.60          @ ( collect_set_nat
% 0.35/0.60            @ ^ [X3: set_nat] :
% 0.35/0.60                ( ( P @ X3 )
% 0.35/0.60                & ( Q @ X3 ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % finite_Collect_conjI
% 0.35/0.60  thf(fact_3786_finite__Collect__conjI,axiom,
% 0.35/0.60      ! [P: nat > $o,Q: nat > $o] :
% 0.35/0.60        ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.35/0.60          | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
% 0.35/0.60       => ( finite_finite_nat
% 0.35/0.60          @ ( collect_nat
% 0.35/0.60            @ ^ [X3: nat] :
% 0.35/0.60                ( ( P @ X3 )
% 0.35/0.60                & ( Q @ X3 ) ) ) ) ) ).
% 0.35/0.60  
% 0.35/0.60  % finite_Collect_conjI
% 0.35/0.60  thf(fact_3787_finite__Collect__conjI,axiom,
% 0.35/0.60      ! [P: int > $o,Q: int > $o] :
% 0.35/0.60        ( ( ( finite_finite_int @ ( collect_int @ P ) )
% 0.35/0.60          | ( finite_finite_int @ ( collect_int @ Q ) ) )
% 0.35/0.60       => ( finite_finite_int
% 0.35/0.60          @ ( collect_int
% 0.35/0.61            @ ^ [X3: int] :
% 0.35/0.61                ( ( P @ X3 )
% 0.35/0.61                & ( Q @ X3 ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_Collect_conjI
% 0.35/0.61  thf(fact_3788_finite__Collect__conjI,axiom,
% 0.35/0.61      ! [P: complex > $o,Q: complex > $o] :
% 0.35/0.61        ( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 0.35/0.61          | ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
% 0.35/0.61       => ( finite3207457112153483333omplex
% 0.35/0.61          @ ( collect_complex
% 0.35/0.61            @ ^ [X3: complex] :
% 0.35/0.61                ( ( P @ X3 )
% 0.35/0.61                & ( Q @ X3 ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_Collect_conjI
% 0.35/0.61  thf(fact_3789_finite__Collect__conjI,axiom,
% 0.35/0.61      ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 0.35/0.61        ( ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
% 0.35/0.61          | ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) )
% 0.35/0.61       => ( finite6177210948735845034at_nat
% 0.35/0.61          @ ( collec3392354462482085612at_nat
% 0.35/0.61            @ ^ [X3: product_prod_nat_nat] :
% 0.35/0.61                ( ( P @ X3 )
% 0.35/0.61                & ( Q @ X3 ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_Collect_conjI
% 0.35/0.61  thf(fact_3790_finite__Collect__disjI,axiom,
% 0.35/0.61      ! [P: real > $o,Q: real > $o] :
% 0.35/0.61        ( ( finite_finite_real
% 0.35/0.61          @ ( collect_real
% 0.35/0.61            @ ^ [X3: real] :
% 0.35/0.61                ( ( P @ X3 )
% 0.35/0.61                | ( Q @ X3 ) ) ) )
% 0.35/0.61        = ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.35/0.61          & ( finite_finite_real @ ( collect_real @ Q ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_Collect_disjI
% 0.35/0.61  thf(fact_3791_finite__Collect__disjI,axiom,
% 0.35/0.61      ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.35/0.61        ( ( finite8100373058378681591st_nat
% 0.35/0.61          @ ( collect_list_nat
% 0.35/0.61            @ ^ [X3: list_nat] :
% 0.35/0.61                ( ( P @ X3 )
% 0.35/0.61                | ( Q @ X3 ) ) ) )
% 0.35/0.61        = ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
% 0.35/0.61          & ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_Collect_disjI
% 0.35/0.61  thf(fact_3792_finite__Collect__disjI,axiom,
% 0.35/0.61      ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.35/0.61        ( ( finite1152437895449049373et_nat
% 0.35/0.61          @ ( collect_set_nat
% 0.35/0.61            @ ^ [X3: set_nat] :
% 0.35/0.61                ( ( P @ X3 )
% 0.35/0.61                | ( Q @ X3 ) ) ) )
% 0.35/0.61        = ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 0.35/0.61          & ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_Collect_disjI
% 0.35/0.61  thf(fact_3793_finite__Collect__disjI,axiom,
% 0.35/0.61      ! [P: nat > $o,Q: nat > $o] :
% 0.35/0.61        ( ( finite_finite_nat
% 0.35/0.61          @ ( collect_nat
% 0.35/0.61            @ ^ [X3: nat] :
% 0.35/0.61                ( ( P @ X3 )
% 0.35/0.61                | ( Q @ X3 ) ) ) )
% 0.35/0.61        = ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.35/0.61          & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_Collect_disjI
% 0.35/0.61  thf(fact_3794_finite__Collect__disjI,axiom,
% 0.35/0.61      ! [P: int > $o,Q: int > $o] :
% 0.35/0.61        ( ( finite_finite_int
% 0.35/0.61          @ ( collect_int
% 0.35/0.61            @ ^ [X3: int] :
% 0.35/0.61                ( ( P @ X3 )
% 0.35/0.61                | ( Q @ X3 ) ) ) )
% 0.35/0.61        = ( ( finite_finite_int @ ( collect_int @ P ) )
% 0.35/0.61          & ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_Collect_disjI
% 0.35/0.61  thf(fact_3795_finite__Collect__disjI,axiom,
% 0.35/0.61      ! [P: complex > $o,Q: complex > $o] :
% 0.35/0.61        ( ( finite3207457112153483333omplex
% 0.35/0.61          @ ( collect_complex
% 0.35/0.61            @ ^ [X3: complex] :
% 0.35/0.61                ( ( P @ X3 )
% 0.35/0.61                | ( Q @ X3 ) ) ) )
% 0.35/0.61        = ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 0.35/0.61          & ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_Collect_disjI
% 0.35/0.61  thf(fact_3796_finite__Collect__disjI,axiom,
% 0.35/0.61      ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 0.35/0.61        ( ( finite6177210948735845034at_nat
% 0.35/0.61          @ ( collec3392354462482085612at_nat
% 0.35/0.61            @ ^ [X3: product_prod_nat_nat] :
% 0.35/0.61                ( ( P @ X3 )
% 0.35/0.61                | ( Q @ X3 ) ) ) )
% 0.35/0.61        = ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
% 0.35/0.61          & ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_Collect_disjI
% 0.35/0.61  thf(fact_3797_finite__interval__int1,axiom,
% 0.35/0.61      ! [A: int,B: int] :
% 0.35/0.61        ( finite_finite_int
% 0.35/0.61        @ ( collect_int
% 0.35/0.61          @ ^ [I4: int] :
% 0.35/0.61              ( ( ord_less_eq_int @ A @ I4 )
% 0.35/0.61              & ( ord_less_eq_int @ I4 @ B ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_interval_int1
% 0.35/0.61  thf(fact_3798_finite__interval__int4,axiom,
% 0.35/0.61      ! [A: int,B: int] :
% 0.35/0.61        ( finite_finite_int
% 0.35/0.61        @ ( collect_int
% 0.35/0.61          @ ^ [I4: int] :
% 0.35/0.61              ( ( ord_less_int @ A @ I4 )
% 0.35/0.61              & ( ord_less_int @ I4 @ B ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_interval_int4
% 0.35/0.61  thf(fact_3799_pred__empty,axiom,
% 0.35/0.61      ! [T: vEBT_VEBT,N: nat,X: nat] :
% 0.35/0.61        ( ( vEBT_invar_vebt @ T @ N )
% 0.35/0.61       => ( ( ( vEBT_vebt_pred @ T @ X )
% 0.35/0.61            = none_nat )
% 0.35/0.61          = ( ( collect_nat
% 0.35/0.61              @ ^ [Y2: nat] :
% 0.35/0.61                  ( ( vEBT_vebt_member @ T @ Y2 )
% 0.35/0.61                  & ( ord_less_nat @ Y2 @ X ) ) )
% 0.35/0.61            = bot_bot_set_nat ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pred_empty
% 0.35/0.61  thf(fact_3800_succ__empty,axiom,
% 0.35/0.61      ! [T: vEBT_VEBT,N: nat,X: nat] :
% 0.35/0.61        ( ( vEBT_invar_vebt @ T @ N )
% 0.35/0.61       => ( ( ( vEBT_vebt_succ @ T @ X )
% 0.35/0.61            = none_nat )
% 0.35/0.61          = ( ( collect_nat
% 0.35/0.61              @ ^ [Y2: nat] :
% 0.35/0.61                  ( ( vEBT_vebt_member @ T @ Y2 )
% 0.35/0.61                  & ( ord_less_nat @ X @ Y2 ) ) )
% 0.35/0.61            = bot_bot_set_nat ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % succ_empty
% 0.35/0.61  thf(fact_3801_unset__bit__negative__int__iff,axiom,
% 0.35/0.61      ! [N: nat,K: int] :
% 0.35/0.61        ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
% 0.35/0.61        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.35/0.61  
% 0.35/0.61  % unset_bit_negative_int_iff
% 0.35/0.61  thf(fact_3802_finite__Collect__subsets,axiom,
% 0.35/0.61      ! [A2: set_nat] :
% 0.35/0.61        ( ( finite_finite_nat @ A2 )
% 0.35/0.61       => ( finite1152437895449049373et_nat
% 0.35/0.61          @ ( collect_set_nat
% 0.35/0.61            @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A2 ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_Collect_subsets
% 0.35/0.61  thf(fact_3803_finite__Collect__subsets,axiom,
% 0.35/0.61      ! [A2: set_complex] :
% 0.35/0.61        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.61       => ( finite6551019134538273531omplex
% 0.35/0.61          @ ( collect_set_complex
% 0.35/0.61            @ ^ [B6: set_complex] : ( ord_le211207098394363844omplex @ B6 @ A2 ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_Collect_subsets
% 0.35/0.61  thf(fact_3804_finite__Collect__subsets,axiom,
% 0.35/0.61      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.61        ( ( finite6177210948735845034at_nat @ A2 )
% 0.35/0.61       => ( finite9047747110432174090at_nat
% 0.35/0.61          @ ( collec5514110066124741708at_nat
% 0.35/0.61            @ ^ [B6: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ B6 @ A2 ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_Collect_subsets
% 0.35/0.61  thf(fact_3805_finite__Collect__subsets,axiom,
% 0.35/0.61      ! [A2: set_int] :
% 0.35/0.61        ( ( finite_finite_int @ A2 )
% 0.35/0.61       => ( finite6197958912794628473et_int
% 0.35/0.61          @ ( collect_set_int
% 0.35/0.61            @ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A2 ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_Collect_subsets
% 0.35/0.61  thf(fact_3806_finite__interval__int2,axiom,
% 0.35/0.61      ! [A: int,B: int] :
% 0.35/0.61        ( finite_finite_int
% 0.35/0.61        @ ( collect_int
% 0.35/0.61          @ ^ [I4: int] :
% 0.35/0.61              ( ( ord_less_eq_int @ A @ I4 )
% 0.35/0.61              & ( ord_less_int @ I4 @ B ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_interval_int2
% 0.35/0.61  thf(fact_3807_finite__interval__int3,axiom,
% 0.35/0.61      ! [A: int,B: int] :
% 0.35/0.61        ( finite_finite_int
% 0.35/0.61        @ ( collect_int
% 0.35/0.61          @ ^ [I4: int] :
% 0.35/0.61              ( ( ord_less_int @ A @ I4 )
% 0.35/0.61              & ( ord_less_eq_int @ I4 @ B ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_interval_int3
% 0.35/0.61  thf(fact_3808_finite__Collect__less__nat,axiom,
% 0.35/0.61      ! [K: nat] :
% 0.35/0.61        ( finite_finite_nat
% 0.35/0.61        @ ( collect_nat
% 0.35/0.61          @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_Collect_less_nat
% 0.35/0.61  thf(fact_3809_finite__Collect__le__nat,axiom,
% 0.35/0.61      ! [K: nat] :
% 0.35/0.61        ( finite_finite_nat
% 0.35/0.61        @ ( collect_nat
% 0.35/0.61          @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_Collect_le_nat
% 0.35/0.61  thf(fact_3810_del__x__not__mi,axiom,
% 0.35/0.61      ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.61        ( ( ( ord_less_nat @ Mi @ X )
% 0.35/0.61          & ( ord_less_eq_nat @ X @ Ma ) )
% 0.35/0.61       => ( ( Mi != Ma )
% 0.35/0.61         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.61           => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.61                = H2 )
% 0.35/0.61             => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.61                  = L )
% 0.35/0.61               => ( ( Newnode
% 0.35/0.61                    = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
% 0.35/0.61                 => ( ( Newlist
% 0.35/0.61                      = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
% 0.35/0.61                   => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.61                     => ( ( ( vEBT_VEBT_minNull @ Newnode )
% 0.35/0.61                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.35/0.61                            = ( vEBT_Node
% 0.35/0.61                              @ ( some_P7363390416028606310at_nat
% 0.35/0.61                                @ ( product_Pair_nat_nat @ Mi
% 0.35/0.61                                  @ ( if_nat @ ( X = Ma )
% 0.35/0.61                                    @ ( if_nat
% 0.35/0.61                                      @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.35/0.61                                        = none_nat )
% 0.35/0.61                                      @ Mi
% 0.35/0.61                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
% 0.35/0.61                                    @ Ma ) ) )
% 0.35/0.61                              @ Deg
% 0.35/0.61                              @ Newlist
% 0.35/0.61                              @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
% 0.35/0.61                        & ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 0.35/0.61                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.35/0.61                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % del_x_not_mi
% 0.35/0.61  thf(fact_3811_del__x__not__mi__new__node__nil,axiom,
% 0.35/0.61      ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
% 0.35/0.61        ( ( ( ord_less_nat @ Mi @ X )
% 0.35/0.61          & ( ord_less_eq_nat @ X @ Ma ) )
% 0.35/0.61       => ( ( Mi != Ma )
% 0.35/0.61         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.61           => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.61                = H2 )
% 0.35/0.61             => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.61                  = L )
% 0.35/0.61               => ( ( Newnode
% 0.35/0.61                    = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
% 0.35/0.61                 => ( ( vEBT_VEBT_minNull @ Newnode )
% 0.35/0.61                   => ( ( Sn
% 0.35/0.61                        = ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.35/0.61                     => ( ( Newlist
% 0.35/0.61                          = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
% 0.35/0.61                       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.61                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.35/0.61                            = ( vEBT_Node
% 0.35/0.61                              @ ( some_P7363390416028606310at_nat
% 0.35/0.61                                @ ( product_Pair_nat_nat @ Mi
% 0.35/0.61                                  @ ( if_nat @ ( X = Ma )
% 0.35/0.61                                    @ ( if_nat
% 0.35/0.61                                      @ ( ( vEBT_vebt_maxt @ Sn )
% 0.35/0.61                                        = none_nat )
% 0.35/0.61                                      @ Mi
% 0.35/0.61                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat2 @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
% 0.35/0.61                                    @ Ma ) ) )
% 0.35/0.61                              @ Deg
% 0.35/0.61                              @ Newlist
% 0.35/0.61                              @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % del_x_not_mi_new_node_nil
% 0.35/0.61  thf(fact_3812_del__x__not__mia,axiom,
% 0.35/0.61      ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.61        ( ( ( ord_less_nat @ Mi @ X )
% 0.35/0.61          & ( ord_less_eq_nat @ X @ Ma ) )
% 0.35/0.61       => ( ( Mi != Ma )
% 0.35/0.61         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.61           => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.61                = H2 )
% 0.35/0.61             => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.61                  = L )
% 0.35/0.61               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.61                 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.35/0.61                    = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
% 0.35/0.61                      @ ( vEBT_Node
% 0.35/0.61                        @ ( some_P7363390416028606310at_nat
% 0.35/0.61                          @ ( product_Pair_nat_nat @ Mi
% 0.35/0.61                            @ ( if_nat @ ( X = Ma )
% 0.35/0.61                              @ ( if_nat
% 0.35/0.61                                @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.35/0.61                                  = none_nat )
% 0.35/0.61                                @ Mi
% 0.35/0.61                                @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
% 0.35/0.61                              @ Ma ) ) )
% 0.35/0.61                        @ Deg
% 0.35/0.61                        @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
% 0.35/0.61                        @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.35/0.61                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % del_x_not_mia
% 0.35/0.61  thf(fact_3813_del__in__range,axiom,
% 0.35/0.61      ! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.61        ( ( ( ord_less_eq_nat @ Mi @ X )
% 0.35/0.61          & ( ord_less_eq_nat @ X @ Ma ) )
% 0.35/0.61       => ( ( Mi != Ma )
% 0.35/0.61         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.61           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.35/0.61              = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.61                @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                  @ ( vEBT_Node
% 0.35/0.61                    @ ( some_P7363390416028606310at_nat
% 0.35/0.61                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
% 0.35/0.61                        @ ( if_nat
% 0.35/0.61                          @ ( ( ( X = Mi )
% 0.35/0.61                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 0.35/0.61                                = Ma ) )
% 0.35/0.61                            & ( ( X != Mi )
% 0.35/0.61                             => ( X = Ma ) ) )
% 0.35/0.61                          @ ( if_nat
% 0.35/0.61                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                              = none_nat )
% 0.35/0.61                            @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
% 0.35/0.61                            @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.61                          @ Ma ) ) )
% 0.35/0.61                    @ Deg
% 0.35/0.61                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                    @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                  @ ( vEBT_Node
% 0.35/0.61                    @ ( some_P7363390416028606310at_nat
% 0.35/0.61                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
% 0.35/0.61                        @ ( if_nat
% 0.35/0.61                          @ ( ( ( X = Mi )
% 0.35/0.61                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 0.35/0.61                                = Ma ) )
% 0.35/0.61                            & ( ( X != Mi )
% 0.35/0.61                             => ( X = Ma ) ) )
% 0.35/0.61                          @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.61                          @ Ma ) ) )
% 0.35/0.61                    @ Deg
% 0.35/0.61                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                    @ Summary ) )
% 0.35/0.61                @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % del_in_range
% 0.35/0.61  thf(fact_3814_del__x__mi,axiom,
% 0.35/0.61      ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat] :
% 0.35/0.61        ( ( ( X = Mi )
% 0.35/0.61          & ( ord_less_nat @ X @ Ma ) )
% 0.35/0.61       => ( ( Mi != Ma )
% 0.35/0.61         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.61           => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.61                = H2 )
% 0.35/0.61             => ( ( Xn
% 0.35/0.61                  = ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
% 0.35/0.61               => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.61                    = L )
% 0.35/0.61                 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.61                   => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.35/0.61                      = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
% 0.35/0.61                        @ ( vEBT_Node
% 0.35/0.61                          @ ( some_P7363390416028606310at_nat
% 0.35/0.61                            @ ( product_Pair_nat_nat @ Xn
% 0.35/0.61                              @ ( if_nat @ ( Xn = Ma )
% 0.35/0.61                                @ ( if_nat
% 0.35/0.61                                  @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.35/0.61                                    = none_nat )
% 0.35/0.61                                  @ Xn
% 0.35/0.61                                  @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
% 0.35/0.61                                @ Ma ) ) )
% 0.35/0.61                          @ Deg
% 0.35/0.61                          @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
% 0.35/0.61                          @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.35/0.61                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % del_x_mi
% 0.35/0.61  thf(fact_3815_del__x__mi__lets__in,axiom,
% 0.35/0.61      ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
% 0.35/0.61        ( ( ( X = Mi )
% 0.35/0.61          & ( ord_less_nat @ X @ Ma ) )
% 0.35/0.61       => ( ( Mi != Ma )
% 0.35/0.61         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.61           => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.61                = H2 )
% 0.35/0.61             => ( ( Xn
% 0.35/0.61                  = ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
% 0.35/0.61               => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.61                    = L )
% 0.35/0.61                 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.61                   => ( ( Newnode
% 0.35/0.61                        = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
% 0.35/0.61                     => ( ( Newlist
% 0.35/0.61                          = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
% 0.35/0.61                       => ( ( ( vEBT_VEBT_minNull @ Newnode )
% 0.35/0.61                           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.35/0.61                              = ( vEBT_Node
% 0.35/0.61                                @ ( some_P7363390416028606310at_nat
% 0.35/0.61                                  @ ( product_Pair_nat_nat @ Xn
% 0.35/0.61                                    @ ( if_nat @ ( Xn = Ma )
% 0.35/0.61                                      @ ( if_nat
% 0.35/0.61                                        @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.35/0.61                                          = none_nat )
% 0.35/0.61                                        @ Xn
% 0.35/0.61                                        @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
% 0.35/0.61                                      @ Ma ) ) )
% 0.35/0.61                                @ Deg
% 0.35/0.61                                @ Newlist
% 0.35/0.61                                @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
% 0.35/0.61                          & ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 0.35/0.61                           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.35/0.61                              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % del_x_mi_lets_in
% 0.35/0.61  thf(fact_3816_del__x__mi__lets__in__minNull,axiom,
% 0.35/0.61      ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
% 0.35/0.61        ( ( ( X = Mi )
% 0.35/0.61          & ( ord_less_nat @ X @ Ma ) )
% 0.35/0.61       => ( ( Mi != Ma )
% 0.35/0.61         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.61           => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.61                = H2 )
% 0.35/0.61             => ( ( Xn
% 0.35/0.61                  = ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
% 0.35/0.61               => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.61                    = L )
% 0.35/0.61                 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.61                   => ( ( Newnode
% 0.35/0.61                        = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
% 0.35/0.61                     => ( ( Newlist
% 0.35/0.61                          = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
% 0.35/0.61                       => ( ( vEBT_VEBT_minNull @ Newnode )
% 0.35/0.61                         => ( ( Sn
% 0.35/0.61                              = ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.35/0.61                           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.35/0.61                              = ( vEBT_Node
% 0.35/0.61                                @ ( some_P7363390416028606310at_nat
% 0.35/0.61                                  @ ( product_Pair_nat_nat @ Xn
% 0.35/0.61                                    @ ( if_nat @ ( Xn = Ma )
% 0.35/0.61                                      @ ( if_nat
% 0.35/0.61                                        @ ( ( vEBT_vebt_maxt @ Sn )
% 0.35/0.61                                          = none_nat )
% 0.35/0.61                                        @ Xn
% 0.35/0.61                                        @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat2 @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
% 0.35/0.61                                      @ Ma ) ) )
% 0.35/0.61                                @ Deg
% 0.35/0.61                                @ Newlist
% 0.35/0.61                                @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % del_x_mi_lets_in_minNull
% 0.35/0.61  thf(fact_3817_del__x__mia,axiom,
% 0.35/0.61      ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.61        ( ( ( X = Mi )
% 0.35/0.61          & ( ord_less_nat @ X @ Ma ) )
% 0.35/0.61       => ( ( Mi != Ma )
% 0.35/0.61         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.61           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 0.35/0.61              = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.61                @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                  @ ( vEBT_Node
% 0.35/0.61                    @ ( some_P7363390416028606310at_nat
% 0.35/0.61                      @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 0.35/0.61                        @ ( if_nat
% 0.35/0.61                          @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 0.35/0.61                            = Ma )
% 0.35/0.61                          @ ( if_nat
% 0.35/0.61                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                              = none_nat )
% 0.35/0.61                            @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 0.35/0.61                            @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.61                          @ Ma ) ) )
% 0.35/0.61                    @ Deg
% 0.35/0.61                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                    @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                  @ ( vEBT_Node
% 0.35/0.61                    @ ( some_P7363390416028606310at_nat
% 0.35/0.61                      @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 0.35/0.61                        @ ( if_nat
% 0.35/0.61                          @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 0.35/0.61                            = Ma )
% 0.35/0.61                          @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.61                          @ Ma ) ) )
% 0.35/0.61                    @ Deg
% 0.35/0.61                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                    @ Summary ) )
% 0.35/0.61                @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % del_x_mia
% 0.35/0.61  thf(fact_3818_enat__0__less__mult__iff,axiom,
% 0.35/0.61      ! [M: extended_enat,N: extended_enat] :
% 0.35/0.61        ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
% 0.35/0.61        = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 0.35/0.61          & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % enat_0_less_mult_iff
% 0.35/0.61  thf(fact_3819_not__iless0,axiom,
% 0.35/0.61      ! [N: extended_enat] :
% 0.35/0.61        ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% 0.35/0.61  
% 0.35/0.61  % not_iless0
% 0.35/0.61  thf(fact_3820_enat__less__induct,axiom,
% 0.35/0.61      ! [P: extended_enat > $o,N: extended_enat] :
% 0.35/0.61        ( ! [N3: extended_enat] :
% 0.35/0.61            ( ! [M3: extended_enat] :
% 0.35/0.61                ( ( ord_le72135733267957522d_enat @ M3 @ N3 )
% 0.35/0.61               => ( P @ M3 ) )
% 0.35/0.61           => ( P @ N3 ) )
% 0.35/0.61       => ( P @ N ) ) ).
% 0.35/0.61  
% 0.35/0.61  % enat_less_induct
% 0.35/0.61  thf(fact_3821_less__set__def,axiom,
% 0.35/0.61      ( ord_le7866589430770878221at_nat
% 0.35/0.61      = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 0.35/0.61            ( ord_le549003669493604880_nat_o
% 0.35/0.61            @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A6 )
% 0.35/0.61            @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ B6 ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % less_set_def
% 0.35/0.61  thf(fact_3822_less__set__def,axiom,
% 0.35/0.61      ( ord_less_set_real
% 0.35/0.61      = ( ^ [A6: set_real,B6: set_real] :
% 0.35/0.61            ( ord_less_real_o
% 0.35/0.61            @ ^ [X3: real] : ( member_real @ X3 @ A6 )
% 0.35/0.61            @ ^ [X3: real] : ( member_real @ X3 @ B6 ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % less_set_def
% 0.35/0.61  thf(fact_3823_less__set__def,axiom,
% 0.35/0.61      ( ord_less_set_set_nat
% 0.35/0.61      = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 0.35/0.61            ( ord_less_set_nat_o
% 0.35/0.61            @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 )
% 0.35/0.61            @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B6 ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % less_set_def
% 0.35/0.61  thf(fact_3824_less__set__def,axiom,
% 0.35/0.61      ( ord_less_set_nat
% 0.35/0.61      = ( ^ [A6: set_nat,B6: set_nat] :
% 0.35/0.61            ( ord_less_nat_o
% 0.35/0.61            @ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
% 0.35/0.61            @ ^ [X3: nat] : ( member_nat @ X3 @ B6 ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % less_set_def
% 0.35/0.61  thf(fact_3825_less__set__def,axiom,
% 0.35/0.61      ( ord_less_set_int
% 0.35/0.61      = ( ^ [A6: set_int,B6: set_int] :
% 0.35/0.61            ( ord_less_int_o
% 0.35/0.61            @ ^ [X3: int] : ( member_int @ X3 @ A6 )
% 0.35/0.61            @ ^ [X3: int] : ( member_int @ X3 @ B6 ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % less_set_def
% 0.35/0.61  thf(fact_3826_minus__set__def,axiom,
% 0.35/0.61      ( minus_1356011639430497352at_nat
% 0.35/0.61      = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 0.35/0.61            ( collec3392354462482085612at_nat
% 0.35/0.61            @ ( minus_2270307095948843157_nat_o
% 0.35/0.61              @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A6 )
% 0.35/0.61              @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ B6 ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % minus_set_def
% 0.35/0.61  thf(fact_3827_minus__set__def,axiom,
% 0.35/0.61      ( minus_minus_set_real
% 0.35/0.61      = ( ^ [A6: set_real,B6: set_real] :
% 0.35/0.61            ( collect_real
% 0.35/0.61            @ ( minus_minus_real_o
% 0.35/0.61              @ ^ [X3: real] : ( member_real @ X3 @ A6 )
% 0.35/0.61              @ ^ [X3: real] : ( member_real @ X3 @ B6 ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % minus_set_def
% 0.35/0.61  thf(fact_3828_minus__set__def,axiom,
% 0.35/0.61      ( minus_7954133019191499631st_nat
% 0.35/0.61      = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 0.35/0.61            ( collect_list_nat
% 0.35/0.61            @ ( minus_1139252259498527702_nat_o
% 0.35/0.61              @ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A6 )
% 0.35/0.61              @ ^ [X3: list_nat] : ( member_list_nat @ X3 @ B6 ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % minus_set_def
% 0.35/0.61  thf(fact_3829_minus__set__def,axiom,
% 0.35/0.61      ( minus_2163939370556025621et_nat
% 0.35/0.61      = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 0.35/0.61            ( collect_set_nat
% 0.35/0.61            @ ( minus_6910147592129066416_nat_o
% 0.35/0.61              @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 )
% 0.35/0.61              @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B6 ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % minus_set_def
% 0.35/0.61  thf(fact_3830_minus__set__def,axiom,
% 0.35/0.61      ( minus_minus_set_int
% 0.35/0.61      = ( ^ [A6: set_int,B6: set_int] :
% 0.35/0.61            ( collect_int
% 0.35/0.61            @ ( minus_minus_int_o
% 0.35/0.61              @ ^ [X3: int] : ( member_int @ X3 @ A6 )
% 0.35/0.61              @ ^ [X3: int] : ( member_int @ X3 @ B6 ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % minus_set_def
% 0.35/0.61  thf(fact_3831_minus__set__def,axiom,
% 0.35/0.61      ( minus_minus_set_nat
% 0.35/0.61      = ( ^ [A6: set_nat,B6: set_nat] :
% 0.35/0.61            ( collect_nat
% 0.35/0.61            @ ( minus_minus_nat_o
% 0.35/0.61              @ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
% 0.35/0.61              @ ^ [X3: nat] : ( member_nat @ X3 @ B6 ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % minus_set_def
% 0.35/0.61  thf(fact_3832_set__diff__eq,axiom,
% 0.35/0.61      ( minus_1356011639430497352at_nat
% 0.35/0.61      = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 0.35/0.61            ( collec3392354462482085612at_nat
% 0.35/0.61            @ ^ [X3: product_prod_nat_nat] :
% 0.35/0.61                ( ( member8440522571783428010at_nat @ X3 @ A6 )
% 0.35/0.61                & ~ ( member8440522571783428010at_nat @ X3 @ B6 ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % set_diff_eq
% 0.35/0.61  thf(fact_3833_set__diff__eq,axiom,
% 0.35/0.61      ( minus_minus_set_real
% 0.35/0.61      = ( ^ [A6: set_real,B6: set_real] :
% 0.35/0.61            ( collect_real
% 0.35/0.61            @ ^ [X3: real] :
% 0.35/0.61                ( ( member_real @ X3 @ A6 )
% 0.35/0.61                & ~ ( member_real @ X3 @ B6 ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % set_diff_eq
% 0.35/0.61  thf(fact_3834_set__diff__eq,axiom,
% 0.35/0.61      ( minus_7954133019191499631st_nat
% 0.35/0.61      = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 0.35/0.61            ( collect_list_nat
% 0.35/0.61            @ ^ [X3: list_nat] :
% 0.35/0.61                ( ( member_list_nat @ X3 @ A6 )
% 0.35/0.61                & ~ ( member_list_nat @ X3 @ B6 ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % set_diff_eq
% 0.35/0.61  thf(fact_3835_set__diff__eq,axiom,
% 0.35/0.61      ( minus_2163939370556025621et_nat
% 0.35/0.61      = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 0.35/0.61            ( collect_set_nat
% 0.35/0.61            @ ^ [X3: set_nat] :
% 0.35/0.61                ( ( member_set_nat @ X3 @ A6 )
% 0.35/0.61                & ~ ( member_set_nat @ X3 @ B6 ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % set_diff_eq
% 0.35/0.61  thf(fact_3836_set__diff__eq,axiom,
% 0.35/0.61      ( minus_minus_set_int
% 0.35/0.61      = ( ^ [A6: set_int,B6: set_int] :
% 0.35/0.61            ( collect_int
% 0.35/0.61            @ ^ [X3: int] :
% 0.35/0.61                ( ( member_int @ X3 @ A6 )
% 0.35/0.61                & ~ ( member_int @ X3 @ B6 ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % set_diff_eq
% 0.35/0.61  thf(fact_3837_set__diff__eq,axiom,
% 0.35/0.61      ( minus_minus_set_nat
% 0.35/0.61      = ( ^ [A6: set_nat,B6: set_nat] :
% 0.35/0.61            ( collect_nat
% 0.35/0.61            @ ^ [X3: nat] :
% 0.35/0.61                ( ( member_nat @ X3 @ A6 )
% 0.35/0.61                & ~ ( member_nat @ X3 @ B6 ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % set_diff_eq
% 0.35/0.61  thf(fact_3838_DiffD2,axiom,
% 0.35/0.61      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.61        ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 0.35/0.61       => ~ ( member8440522571783428010at_nat @ C @ B2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % DiffD2
% 0.35/0.61  thf(fact_3839_DiffD2,axiom,
% 0.35/0.61      ! [C: real,A2: set_real,B2: set_real] :
% 0.35/0.61        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 0.35/0.61       => ~ ( member_real @ C @ B2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % DiffD2
% 0.35/0.61  thf(fact_3840_DiffD2,axiom,
% 0.35/0.61      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.35/0.61        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 0.35/0.61       => ~ ( member_set_nat @ C @ B2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % DiffD2
% 0.35/0.61  thf(fact_3841_DiffD2,axiom,
% 0.35/0.61      ! [C: int,A2: set_int,B2: set_int] :
% 0.35/0.61        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.35/0.61       => ~ ( member_int @ C @ B2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % DiffD2
% 0.35/0.61  thf(fact_3842_DiffD2,axiom,
% 0.35/0.61      ! [C: nat,A2: set_nat,B2: set_nat] :
% 0.35/0.61        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.35/0.61       => ~ ( member_nat @ C @ B2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % DiffD2
% 0.35/0.61  thf(fact_3843_DiffD1,axiom,
% 0.35/0.61      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.61        ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 0.35/0.61       => ( member8440522571783428010at_nat @ C @ A2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % DiffD1
% 0.35/0.61  thf(fact_3844_DiffD1,axiom,
% 0.35/0.61      ! [C: real,A2: set_real,B2: set_real] :
% 0.35/0.61        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 0.35/0.61       => ( member_real @ C @ A2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % DiffD1
% 0.35/0.61  thf(fact_3845_DiffD1,axiom,
% 0.35/0.61      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.35/0.61        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 0.35/0.61       => ( member_set_nat @ C @ A2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % DiffD1
% 0.35/0.61  thf(fact_3846_DiffD1,axiom,
% 0.35/0.61      ! [C: int,A2: set_int,B2: set_int] :
% 0.35/0.61        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.35/0.61       => ( member_int @ C @ A2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % DiffD1
% 0.35/0.61  thf(fact_3847_DiffD1,axiom,
% 0.35/0.61      ! [C: nat,A2: set_nat,B2: set_nat] :
% 0.35/0.61        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.35/0.61       => ( member_nat @ C @ A2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % DiffD1
% 0.35/0.61  thf(fact_3848_DiffE,axiom,
% 0.35/0.61      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.61        ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 0.35/0.61       => ~ ( ( member8440522571783428010at_nat @ C @ A2 )
% 0.35/0.61           => ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % DiffE
% 0.35/0.61  thf(fact_3849_DiffE,axiom,
% 0.35/0.61      ! [C: real,A2: set_real,B2: set_real] :
% 0.35/0.61        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 0.35/0.61       => ~ ( ( member_real @ C @ A2 )
% 0.35/0.61           => ( member_real @ C @ B2 ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % DiffE
% 0.35/0.61  thf(fact_3850_DiffE,axiom,
% 0.35/0.61      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.35/0.61        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 0.35/0.61       => ~ ( ( member_set_nat @ C @ A2 )
% 0.35/0.61           => ( member_set_nat @ C @ B2 ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % DiffE
% 0.35/0.61  thf(fact_3851_DiffE,axiom,
% 0.35/0.61      ! [C: int,A2: set_int,B2: set_int] :
% 0.35/0.61        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.35/0.61       => ~ ( ( member_int @ C @ A2 )
% 0.35/0.61           => ( member_int @ C @ B2 ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % DiffE
% 0.35/0.61  thf(fact_3852_DiffE,axiom,
% 0.35/0.61      ! [C: nat,A2: set_nat,B2: set_nat] :
% 0.35/0.61        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.35/0.61       => ~ ( ( member_nat @ C @ A2 )
% 0.35/0.61           => ( member_nat @ C @ B2 ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % DiffE
% 0.35/0.61  thf(fact_3853_pred__equals__eq2,axiom,
% 0.35/0.61      ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 0.35/0.61        ( ( ( ^ [X3: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R ) )
% 0.35/0.61          = ( ^ [X3: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ S2 ) ) )
% 0.35/0.61        = ( R = S2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pred_equals_eq2
% 0.35/0.61  thf(fact_3854_pred__equals__eq2,axiom,
% 0.35/0.61      ! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 0.35/0.61        ( ( ( ^ [X3: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R ) )
% 0.35/0.61          = ( ^ [X3: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ S2 ) ) )
% 0.35/0.61        = ( R = S2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pred_equals_eq2
% 0.35/0.61  thf(fact_3855_pred__equals__eq2,axiom,
% 0.35/0.61      ! [R: set_Pr8056137968301705908nteger,S2: set_Pr8056137968301705908nteger] :
% 0.35/0.61        ( ( ( ^ [X3: code_integer > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y2 ) @ R ) )
% 0.35/0.61          = ( ^ [X3: code_integer > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y2 ) @ S2 ) ) )
% 0.35/0.61        = ( R = S2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pred_equals_eq2
% 0.35/0.61  thf(fact_3856_pred__equals__eq2,axiom,
% 0.35/0.61      ! [R: set_Pr1281608226676607948nteger,S2: set_Pr1281608226676607948nteger] :
% 0.35/0.61        ( ( ( ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y2 ) @ R ) )
% 0.35/0.61          = ( ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y2 ) @ S2 ) ) )
% 0.35/0.61        = ( R = S2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pred_equals_eq2
% 0.35/0.61  thf(fact_3857_pred__equals__eq2,axiom,
% 0.35/0.61      ! [R: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
% 0.35/0.61        ( ( ( ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y2 ) @ R ) )
% 0.35/0.61          = ( ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y2 ) @ S2 ) ) )
% 0.35/0.61        = ( R = S2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pred_equals_eq2
% 0.35/0.61  thf(fact_3858_pred__equals__eq2,axiom,
% 0.35/0.61      ! [R: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
% 0.35/0.61        ( ( ( ^ [X3: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y2 ) @ R ) )
% 0.35/0.61          = ( ^ [X3: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y2 ) @ S2 ) ) )
% 0.35/0.61        = ( R = S2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pred_equals_eq2
% 0.35/0.61  thf(fact_3859_Collect__restrict,axiom,
% 0.35/0.61      ! [X8: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
% 0.35/0.61        ( ord_le3146513528884898305at_nat
% 0.35/0.61        @ ( collec3392354462482085612at_nat
% 0.35/0.61          @ ^ [X3: product_prod_nat_nat] :
% 0.35/0.61              ( ( member8440522571783428010at_nat @ X3 @ X8 )
% 0.35/0.61              & ( P @ X3 ) ) )
% 0.35/0.61        @ X8 ) ).
% 0.35/0.61  
% 0.35/0.61  % Collect_restrict
% 0.35/0.61  thf(fact_3860_Collect__restrict,axiom,
% 0.35/0.61      ! [X8: set_real,P: real > $o] :
% 0.35/0.61        ( ord_less_eq_set_real
% 0.35/0.61        @ ( collect_real
% 0.35/0.61          @ ^ [X3: real] :
% 0.35/0.61              ( ( member_real @ X3 @ X8 )
% 0.35/0.61              & ( P @ X3 ) ) )
% 0.35/0.61        @ X8 ) ).
% 0.35/0.61  
% 0.35/0.61  % Collect_restrict
% 0.35/0.61  thf(fact_3861_Collect__restrict,axiom,
% 0.35/0.61      ! [X8: set_list_nat,P: list_nat > $o] :
% 0.35/0.61        ( ord_le6045566169113846134st_nat
% 0.35/0.61        @ ( collect_list_nat
% 0.35/0.61          @ ^ [X3: list_nat] :
% 0.35/0.61              ( ( member_list_nat @ X3 @ X8 )
% 0.35/0.61              & ( P @ X3 ) ) )
% 0.35/0.61        @ X8 ) ).
% 0.35/0.61  
% 0.35/0.61  % Collect_restrict
% 0.35/0.61  thf(fact_3862_Collect__restrict,axiom,
% 0.35/0.61      ! [X8: set_set_nat,P: set_nat > $o] :
% 0.35/0.61        ( ord_le6893508408891458716et_nat
% 0.35/0.61        @ ( collect_set_nat
% 0.35/0.61          @ ^ [X3: set_nat] :
% 0.35/0.61              ( ( member_set_nat @ X3 @ X8 )
% 0.35/0.61              & ( P @ X3 ) ) )
% 0.35/0.61        @ X8 ) ).
% 0.35/0.61  
% 0.35/0.61  % Collect_restrict
% 0.35/0.61  thf(fact_3863_Collect__restrict,axiom,
% 0.35/0.61      ! [X8: set_nat,P: nat > $o] :
% 0.35/0.61        ( ord_less_eq_set_nat
% 0.35/0.61        @ ( collect_nat
% 0.35/0.61          @ ^ [X3: nat] :
% 0.35/0.61              ( ( member_nat @ X3 @ X8 )
% 0.35/0.61              & ( P @ X3 ) ) )
% 0.35/0.61        @ X8 ) ).
% 0.35/0.61  
% 0.35/0.61  % Collect_restrict
% 0.35/0.61  thf(fact_3864_Collect__restrict,axiom,
% 0.35/0.61      ! [X8: set_int,P: int > $o] :
% 0.35/0.61        ( ord_less_eq_set_int
% 0.35/0.61        @ ( collect_int
% 0.35/0.61          @ ^ [X3: int] :
% 0.35/0.61              ( ( member_int @ X3 @ X8 )
% 0.35/0.61              & ( P @ X3 ) ) )
% 0.35/0.61        @ X8 ) ).
% 0.35/0.61  
% 0.35/0.61  % Collect_restrict
% 0.35/0.61  thf(fact_3865_prop__restrict,axiom,
% 0.35/0.61      ! [X: product_prod_nat_nat,Z6: set_Pr1261947904930325089at_nat,X8: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
% 0.35/0.61        ( ( member8440522571783428010at_nat @ X @ Z6 )
% 0.35/0.61       => ( ( ord_le3146513528884898305at_nat @ Z6
% 0.35/0.61            @ ( collec3392354462482085612at_nat
% 0.35/0.61              @ ^ [X3: product_prod_nat_nat] :
% 0.35/0.61                  ( ( member8440522571783428010at_nat @ X3 @ X8 )
% 0.35/0.61                  & ( P @ X3 ) ) ) )
% 0.35/0.61         => ( P @ X ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % prop_restrict
% 0.35/0.61  thf(fact_3866_prop__restrict,axiom,
% 0.35/0.61      ! [X: real,Z6: set_real,X8: set_real,P: real > $o] :
% 0.35/0.61        ( ( member_real @ X @ Z6 )
% 0.35/0.61       => ( ( ord_less_eq_set_real @ Z6
% 0.35/0.61            @ ( collect_real
% 0.35/0.61              @ ^ [X3: real] :
% 0.35/0.61                  ( ( member_real @ X3 @ X8 )
% 0.35/0.61                  & ( P @ X3 ) ) ) )
% 0.35/0.61         => ( P @ X ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % prop_restrict
% 0.35/0.61  thf(fact_3867_prop__restrict,axiom,
% 0.35/0.61      ! [X: list_nat,Z6: set_list_nat,X8: set_list_nat,P: list_nat > $o] :
% 0.35/0.61        ( ( member_list_nat @ X @ Z6 )
% 0.35/0.61       => ( ( ord_le6045566169113846134st_nat @ Z6
% 0.35/0.61            @ ( collect_list_nat
% 0.35/0.61              @ ^ [X3: list_nat] :
% 0.35/0.61                  ( ( member_list_nat @ X3 @ X8 )
% 0.35/0.61                  & ( P @ X3 ) ) ) )
% 0.35/0.61         => ( P @ X ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % prop_restrict
% 0.35/0.61  thf(fact_3868_prop__restrict,axiom,
% 0.35/0.61      ! [X: set_nat,Z6: set_set_nat,X8: set_set_nat,P: set_nat > $o] :
% 0.35/0.61        ( ( member_set_nat @ X @ Z6 )
% 0.35/0.61       => ( ( ord_le6893508408891458716et_nat @ Z6
% 0.35/0.61            @ ( collect_set_nat
% 0.35/0.61              @ ^ [X3: set_nat] :
% 0.35/0.61                  ( ( member_set_nat @ X3 @ X8 )
% 0.35/0.61                  & ( P @ X3 ) ) ) )
% 0.35/0.61         => ( P @ X ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % prop_restrict
% 0.35/0.61  thf(fact_3869_prop__restrict,axiom,
% 0.35/0.61      ! [X: nat,Z6: set_nat,X8: set_nat,P: nat > $o] :
% 0.35/0.61        ( ( member_nat @ X @ Z6 )
% 0.35/0.61       => ( ( ord_less_eq_set_nat @ Z6
% 0.35/0.61            @ ( collect_nat
% 0.35/0.61              @ ^ [X3: nat] :
% 0.35/0.61                  ( ( member_nat @ X3 @ X8 )
% 0.35/0.61                  & ( P @ X3 ) ) ) )
% 0.35/0.61         => ( P @ X ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % prop_restrict
% 0.35/0.61  thf(fact_3870_prop__restrict,axiom,
% 0.35/0.61      ! [X: int,Z6: set_int,X8: set_int,P: int > $o] :
% 0.35/0.61        ( ( member_int @ X @ Z6 )
% 0.35/0.61       => ( ( ord_less_eq_set_int @ Z6
% 0.35/0.61            @ ( collect_int
% 0.35/0.61              @ ^ [X3: int] :
% 0.35/0.61                  ( ( member_int @ X3 @ X8 )
% 0.35/0.61                  & ( P @ X3 ) ) ) )
% 0.35/0.61         => ( P @ X ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % prop_restrict
% 0.35/0.61  thf(fact_3871_pred__subset__eq,axiom,
% 0.35/0.61      ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 0.35/0.61        ( ( ord_le704812498762024988_nat_o
% 0.35/0.61          @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ R )
% 0.35/0.61          @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ S2 ) )
% 0.35/0.61        = ( ord_le3146513528884898305at_nat @ R @ S2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pred_subset_eq
% 0.35/0.61  thf(fact_3872_pred__subset__eq,axiom,
% 0.35/0.61      ! [R: set_real,S2: set_real] :
% 0.35/0.61        ( ( ord_less_eq_real_o
% 0.35/0.61          @ ^ [X3: real] : ( member_real @ X3 @ R )
% 0.35/0.61          @ ^ [X3: real] : ( member_real @ X3 @ S2 ) )
% 0.35/0.61        = ( ord_less_eq_set_real @ R @ S2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pred_subset_eq
% 0.35/0.61  thf(fact_3873_pred__subset__eq,axiom,
% 0.35/0.61      ! [R: set_set_nat,S2: set_set_nat] :
% 0.35/0.61        ( ( ord_le3964352015994296041_nat_o
% 0.35/0.61          @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ R )
% 0.35/0.61          @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ S2 ) )
% 0.35/0.61        = ( ord_le6893508408891458716et_nat @ R @ S2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pred_subset_eq
% 0.35/0.61  thf(fact_3874_pred__subset__eq,axiom,
% 0.35/0.61      ! [R: set_nat,S2: set_nat] :
% 0.35/0.61        ( ( ord_less_eq_nat_o
% 0.35/0.61          @ ^ [X3: nat] : ( member_nat @ X3 @ R )
% 0.35/0.61          @ ^ [X3: nat] : ( member_nat @ X3 @ S2 ) )
% 0.35/0.61        = ( ord_less_eq_set_nat @ R @ S2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pred_subset_eq
% 0.35/0.61  thf(fact_3875_pred__subset__eq,axiom,
% 0.35/0.61      ! [R: set_int,S2: set_int] :
% 0.35/0.61        ( ( ord_less_eq_int_o
% 0.35/0.61          @ ^ [X3: int] : ( member_int @ X3 @ R )
% 0.35/0.61          @ ^ [X3: int] : ( member_int @ X3 @ S2 ) )
% 0.35/0.61        = ( ord_less_eq_set_int @ R @ S2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pred_subset_eq
% 0.35/0.61  thf(fact_3876_less__eq__set__def,axiom,
% 0.35/0.61      ( ord_le3146513528884898305at_nat
% 0.35/0.61      = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 0.35/0.61            ( ord_le704812498762024988_nat_o
% 0.35/0.61            @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A6 )
% 0.35/0.61            @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ B6 ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % less_eq_set_def
% 0.35/0.61  thf(fact_3877_less__eq__set__def,axiom,
% 0.35/0.61      ( ord_less_eq_set_real
% 0.35/0.61      = ( ^ [A6: set_real,B6: set_real] :
% 0.35/0.61            ( ord_less_eq_real_o
% 0.35/0.61            @ ^ [X3: real] : ( member_real @ X3 @ A6 )
% 0.35/0.61            @ ^ [X3: real] : ( member_real @ X3 @ B6 ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % less_eq_set_def
% 0.35/0.61  thf(fact_3878_less__eq__set__def,axiom,
% 0.35/0.61      ( ord_le6893508408891458716et_nat
% 0.35/0.61      = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 0.35/0.61            ( ord_le3964352015994296041_nat_o
% 0.35/0.61            @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 )
% 0.35/0.61            @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B6 ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % less_eq_set_def
% 0.35/0.61  thf(fact_3879_less__eq__set__def,axiom,
% 0.35/0.61      ( ord_less_eq_set_nat
% 0.35/0.61      = ( ^ [A6: set_nat,B6: set_nat] :
% 0.35/0.61            ( ord_less_eq_nat_o
% 0.35/0.61            @ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
% 0.35/0.61            @ ^ [X3: nat] : ( member_nat @ X3 @ B6 ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % less_eq_set_def
% 0.35/0.61  thf(fact_3880_less__eq__set__def,axiom,
% 0.35/0.61      ( ord_less_eq_set_int
% 0.35/0.61      = ( ^ [A6: set_int,B6: set_int] :
% 0.35/0.61            ( ord_less_eq_int_o
% 0.35/0.61            @ ^ [X3: int] : ( member_int @ X3 @ A6 )
% 0.35/0.61            @ ^ [X3: int] : ( member_int @ X3 @ B6 ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % less_eq_set_def
% 0.35/0.61  thf(fact_3881_Collect__subset,axiom,
% 0.35/0.61      ! [A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
% 0.35/0.61        ( ord_le3146513528884898305at_nat
% 0.35/0.61        @ ( collec3392354462482085612at_nat
% 0.35/0.61          @ ^ [X3: product_prod_nat_nat] :
% 0.35/0.61              ( ( member8440522571783428010at_nat @ X3 @ A2 )
% 0.35/0.61              & ( P @ X3 ) ) )
% 0.35/0.61        @ A2 ) ).
% 0.35/0.61  
% 0.35/0.61  % Collect_subset
% 0.35/0.61  thf(fact_3882_Collect__subset,axiom,
% 0.35/0.61      ! [A2: set_real,P: real > $o] :
% 0.35/0.61        ( ord_less_eq_set_real
% 0.35/0.61        @ ( collect_real
% 0.35/0.61          @ ^ [X3: real] :
% 0.35/0.61              ( ( member_real @ X3 @ A2 )
% 0.35/0.61              & ( P @ X3 ) ) )
% 0.35/0.61        @ A2 ) ).
% 0.35/0.61  
% 0.35/0.61  % Collect_subset
% 0.35/0.61  thf(fact_3883_Collect__subset,axiom,
% 0.35/0.61      ! [A2: set_list_nat,P: list_nat > $o] :
% 0.35/0.61        ( ord_le6045566169113846134st_nat
% 0.35/0.61        @ ( collect_list_nat
% 0.35/0.61          @ ^ [X3: list_nat] :
% 0.35/0.61              ( ( member_list_nat @ X3 @ A2 )
% 0.35/0.61              & ( P @ X3 ) ) )
% 0.35/0.61        @ A2 ) ).
% 0.35/0.61  
% 0.35/0.61  % Collect_subset
% 0.35/0.61  thf(fact_3884_Collect__subset,axiom,
% 0.35/0.61      ! [A2: set_set_nat,P: set_nat > $o] :
% 0.35/0.61        ( ord_le6893508408891458716et_nat
% 0.35/0.61        @ ( collect_set_nat
% 0.35/0.61          @ ^ [X3: set_nat] :
% 0.35/0.61              ( ( member_set_nat @ X3 @ A2 )
% 0.35/0.61              & ( P @ X3 ) ) )
% 0.35/0.61        @ A2 ) ).
% 0.35/0.61  
% 0.35/0.61  % Collect_subset
% 0.35/0.61  thf(fact_3885_Collect__subset,axiom,
% 0.35/0.61      ! [A2: set_nat,P: nat > $o] :
% 0.35/0.61        ( ord_less_eq_set_nat
% 0.35/0.61        @ ( collect_nat
% 0.35/0.61          @ ^ [X3: nat] :
% 0.35/0.61              ( ( member_nat @ X3 @ A2 )
% 0.35/0.61              & ( P @ X3 ) ) )
% 0.35/0.61        @ A2 ) ).
% 0.35/0.61  
% 0.35/0.61  % Collect_subset
% 0.35/0.61  thf(fact_3886_Collect__subset,axiom,
% 0.35/0.61      ! [A2: set_int,P: int > $o] :
% 0.35/0.61        ( ord_less_eq_set_int
% 0.35/0.61        @ ( collect_int
% 0.35/0.61          @ ^ [X3: int] :
% 0.35/0.61              ( ( member_int @ X3 @ A2 )
% 0.35/0.61              & ( P @ X3 ) ) )
% 0.35/0.61        @ A2 ) ).
% 0.35/0.61  
% 0.35/0.61  % Collect_subset
% 0.35/0.61  thf(fact_3887_pigeonhole__infinite__rel,axiom,
% 0.35/0.61      ! [A2: set_real,B2: set_nat,R: real > nat > $o] :
% 0.35/0.61        ( ~ ( finite_finite_real @ A2 )
% 0.35/0.61       => ( ( finite_finite_nat @ B2 )
% 0.35/0.61         => ( ! [X4: real] :
% 0.35/0.61                ( ( member_real @ X4 @ A2 )
% 0.35/0.61               => ? [Xa: nat] :
% 0.35/0.61                    ( ( member_nat @ Xa @ B2 )
% 0.35/0.61                    & ( R @ X4 @ Xa ) ) )
% 0.35/0.61           => ? [X4: nat] :
% 0.35/0.61                ( ( member_nat @ X4 @ B2 )
% 0.35/0.61                & ~ ( finite_finite_real
% 0.35/0.61                    @ ( collect_real
% 0.35/0.61                      @ ^ [A4: real] :
% 0.35/0.61                          ( ( member_real @ A4 @ A2 )
% 0.35/0.61                          & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pigeonhole_infinite_rel
% 0.35/0.61  thf(fact_3888_pigeonhole__infinite__rel,axiom,
% 0.35/0.61      ! [A2: set_real,B2: set_int,R: real > int > $o] :
% 0.35/0.61        ( ~ ( finite_finite_real @ A2 )
% 0.35/0.61       => ( ( finite_finite_int @ B2 )
% 0.35/0.61         => ( ! [X4: real] :
% 0.35/0.61                ( ( member_real @ X4 @ A2 )
% 0.35/0.61               => ? [Xa: int] :
% 0.35/0.61                    ( ( member_int @ Xa @ B2 )
% 0.35/0.61                    & ( R @ X4 @ Xa ) ) )
% 0.35/0.61           => ? [X4: int] :
% 0.35/0.61                ( ( member_int @ X4 @ B2 )
% 0.35/0.61                & ~ ( finite_finite_real
% 0.35/0.61                    @ ( collect_real
% 0.35/0.61                      @ ^ [A4: real] :
% 0.35/0.61                          ( ( member_real @ A4 @ A2 )
% 0.35/0.61                          & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pigeonhole_infinite_rel
% 0.35/0.61  thf(fact_3889_pigeonhole__infinite__rel,axiom,
% 0.35/0.61      ! [A2: set_real,B2: set_complex,R: real > complex > $o] :
% 0.35/0.61        ( ~ ( finite_finite_real @ A2 )
% 0.35/0.61       => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.61         => ( ! [X4: real] :
% 0.35/0.61                ( ( member_real @ X4 @ A2 )
% 0.35/0.61               => ? [Xa: complex] :
% 0.35/0.61                    ( ( member_complex @ Xa @ B2 )
% 0.35/0.61                    & ( R @ X4 @ Xa ) ) )
% 0.35/0.61           => ? [X4: complex] :
% 0.35/0.61                ( ( member_complex @ X4 @ B2 )
% 0.35/0.61                & ~ ( finite_finite_real
% 0.35/0.61                    @ ( collect_real
% 0.35/0.61                      @ ^ [A4: real] :
% 0.35/0.61                          ( ( member_real @ A4 @ A2 )
% 0.35/0.61                          & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pigeonhole_infinite_rel
% 0.35/0.61  thf(fact_3890_pigeonhole__infinite__rel,axiom,
% 0.35/0.61      ! [A2: set_nat,B2: set_nat,R: nat > nat > $o] :
% 0.35/0.61        ( ~ ( finite_finite_nat @ A2 )
% 0.35/0.61       => ( ( finite_finite_nat @ B2 )
% 0.35/0.61         => ( ! [X4: nat] :
% 0.35/0.61                ( ( member_nat @ X4 @ A2 )
% 0.35/0.61               => ? [Xa: nat] :
% 0.35/0.61                    ( ( member_nat @ Xa @ B2 )
% 0.35/0.61                    & ( R @ X4 @ Xa ) ) )
% 0.35/0.61           => ? [X4: nat] :
% 0.35/0.61                ( ( member_nat @ X4 @ B2 )
% 0.35/0.61                & ~ ( finite_finite_nat
% 0.35/0.61                    @ ( collect_nat
% 0.35/0.61                      @ ^ [A4: nat] :
% 0.35/0.61                          ( ( member_nat @ A4 @ A2 )
% 0.35/0.61                          & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pigeonhole_infinite_rel
% 0.35/0.61  thf(fact_3891_pigeonhole__infinite__rel,axiom,
% 0.35/0.61      ! [A2: set_nat,B2: set_int,R: nat > int > $o] :
% 0.35/0.61        ( ~ ( finite_finite_nat @ A2 )
% 0.35/0.61       => ( ( finite_finite_int @ B2 )
% 0.35/0.61         => ( ! [X4: nat] :
% 0.35/0.61                ( ( member_nat @ X4 @ A2 )
% 0.35/0.61               => ? [Xa: int] :
% 0.35/0.61                    ( ( member_int @ Xa @ B2 )
% 0.35/0.61                    & ( R @ X4 @ Xa ) ) )
% 0.35/0.61           => ? [X4: int] :
% 0.35/0.61                ( ( member_int @ X4 @ B2 )
% 0.35/0.61                & ~ ( finite_finite_nat
% 0.35/0.61                    @ ( collect_nat
% 0.35/0.61                      @ ^ [A4: nat] :
% 0.35/0.61                          ( ( member_nat @ A4 @ A2 )
% 0.35/0.61                          & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pigeonhole_infinite_rel
% 0.35/0.61  thf(fact_3892_pigeonhole__infinite__rel,axiom,
% 0.35/0.61      ! [A2: set_nat,B2: set_complex,R: nat > complex > $o] :
% 0.35/0.61        ( ~ ( finite_finite_nat @ A2 )
% 0.35/0.61       => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.61         => ( ! [X4: nat] :
% 0.35/0.61                ( ( member_nat @ X4 @ A2 )
% 0.35/0.61               => ? [Xa: complex] :
% 0.35/0.61                    ( ( member_complex @ Xa @ B2 )
% 0.35/0.61                    & ( R @ X4 @ Xa ) ) )
% 0.35/0.61           => ? [X4: complex] :
% 0.35/0.61                ( ( member_complex @ X4 @ B2 )
% 0.35/0.61                & ~ ( finite_finite_nat
% 0.35/0.61                    @ ( collect_nat
% 0.35/0.61                      @ ^ [A4: nat] :
% 0.35/0.61                          ( ( member_nat @ A4 @ A2 )
% 0.35/0.61                          & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pigeonhole_infinite_rel
% 0.35/0.61  thf(fact_3893_pigeonhole__infinite__rel,axiom,
% 0.35/0.61      ! [A2: set_int,B2: set_nat,R: int > nat > $o] :
% 0.35/0.61        ( ~ ( finite_finite_int @ A2 )
% 0.35/0.61       => ( ( finite_finite_nat @ B2 )
% 0.35/0.61         => ( ! [X4: int] :
% 0.35/0.61                ( ( member_int @ X4 @ A2 )
% 0.35/0.61               => ? [Xa: nat] :
% 0.35/0.61                    ( ( member_nat @ Xa @ B2 )
% 0.35/0.61                    & ( R @ X4 @ Xa ) ) )
% 0.35/0.61           => ? [X4: nat] :
% 0.35/0.61                ( ( member_nat @ X4 @ B2 )
% 0.35/0.61                & ~ ( finite_finite_int
% 0.35/0.61                    @ ( collect_int
% 0.35/0.61                      @ ^ [A4: int] :
% 0.35/0.61                          ( ( member_int @ A4 @ A2 )
% 0.35/0.61                          & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pigeonhole_infinite_rel
% 0.35/0.61  thf(fact_3894_pigeonhole__infinite__rel,axiom,
% 0.35/0.61      ! [A2: set_int,B2: set_int,R: int > int > $o] :
% 0.35/0.61        ( ~ ( finite_finite_int @ A2 )
% 0.35/0.61       => ( ( finite_finite_int @ B2 )
% 0.35/0.61         => ( ! [X4: int] :
% 0.35/0.61                ( ( member_int @ X4 @ A2 )
% 0.35/0.61               => ? [Xa: int] :
% 0.35/0.61                    ( ( member_int @ Xa @ B2 )
% 0.35/0.61                    & ( R @ X4 @ Xa ) ) )
% 0.35/0.61           => ? [X4: int] :
% 0.35/0.61                ( ( member_int @ X4 @ B2 )
% 0.35/0.61                & ~ ( finite_finite_int
% 0.35/0.61                    @ ( collect_int
% 0.35/0.61                      @ ^ [A4: int] :
% 0.35/0.61                          ( ( member_int @ A4 @ A2 )
% 0.35/0.61                          & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pigeonhole_infinite_rel
% 0.35/0.61  thf(fact_3895_pigeonhole__infinite__rel,axiom,
% 0.35/0.61      ! [A2: set_int,B2: set_complex,R: int > complex > $o] :
% 0.35/0.61        ( ~ ( finite_finite_int @ A2 )
% 0.35/0.61       => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.61         => ( ! [X4: int] :
% 0.35/0.61                ( ( member_int @ X4 @ A2 )
% 0.35/0.61               => ? [Xa: complex] :
% 0.35/0.61                    ( ( member_complex @ Xa @ B2 )
% 0.35/0.61                    & ( R @ X4 @ Xa ) ) )
% 0.35/0.61           => ? [X4: complex] :
% 0.35/0.61                ( ( member_complex @ X4 @ B2 )
% 0.35/0.61                & ~ ( finite_finite_int
% 0.35/0.61                    @ ( collect_int
% 0.35/0.61                      @ ^ [A4: int] :
% 0.35/0.61                          ( ( member_int @ A4 @ A2 )
% 0.35/0.61                          & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pigeonhole_infinite_rel
% 0.35/0.61  thf(fact_3896_pigeonhole__infinite__rel,axiom,
% 0.35/0.61      ! [A2: set_complex,B2: set_nat,R: complex > nat > $o] :
% 0.35/0.61        ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.35/0.61       => ( ( finite_finite_nat @ B2 )
% 0.35/0.61         => ( ! [X4: complex] :
% 0.35/0.61                ( ( member_complex @ X4 @ A2 )
% 0.35/0.61               => ? [Xa: nat] :
% 0.35/0.61                    ( ( member_nat @ Xa @ B2 )
% 0.35/0.61                    & ( R @ X4 @ Xa ) ) )
% 0.35/0.61           => ? [X4: nat] :
% 0.35/0.61                ( ( member_nat @ X4 @ B2 )
% 0.35/0.61                & ~ ( finite3207457112153483333omplex
% 0.35/0.61                    @ ( collect_complex
% 0.35/0.61                      @ ^ [A4: complex] :
% 0.35/0.61                          ( ( member_complex @ A4 @ A2 )
% 0.35/0.61                          & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pigeonhole_infinite_rel
% 0.35/0.61  thf(fact_3897_not__finite__existsD,axiom,
% 0.35/0.61      ! [P: real > $o] :
% 0.35/0.61        ( ~ ( finite_finite_real @ ( collect_real @ P ) )
% 0.35/0.61       => ? [X_1: real] : ( P @ X_1 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % not_finite_existsD
% 0.35/0.61  thf(fact_3898_not__finite__existsD,axiom,
% 0.35/0.61      ! [P: list_nat > $o] :
% 0.35/0.61        ( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
% 0.35/0.61       => ? [X_1: list_nat] : ( P @ X_1 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % not_finite_existsD
% 0.35/0.61  thf(fact_3899_not__finite__existsD,axiom,
% 0.35/0.61      ! [P: set_nat > $o] :
% 0.35/0.61        ( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 0.35/0.61       => ? [X_1: set_nat] : ( P @ X_1 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % not_finite_existsD
% 0.35/0.61  thf(fact_3900_not__finite__existsD,axiom,
% 0.35/0.61      ! [P: nat > $o] :
% 0.35/0.61        ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.35/0.61       => ? [X_1: nat] : ( P @ X_1 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % not_finite_existsD
% 0.35/0.61  thf(fact_3901_not__finite__existsD,axiom,
% 0.35/0.61      ! [P: int > $o] :
% 0.35/0.61        ( ~ ( finite_finite_int @ ( collect_int @ P ) )
% 0.35/0.61       => ? [X_1: int] : ( P @ X_1 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % not_finite_existsD
% 0.35/0.61  thf(fact_3902_not__finite__existsD,axiom,
% 0.35/0.61      ! [P: complex > $o] :
% 0.35/0.61        ( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 0.35/0.61       => ? [X_1: complex] : ( P @ X_1 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % not_finite_existsD
% 0.35/0.61  thf(fact_3903_not__finite__existsD,axiom,
% 0.35/0.61      ! [P: product_prod_nat_nat > $o] :
% 0.35/0.61        ( ~ ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
% 0.35/0.61       => ? [X_1: product_prod_nat_nat] : ( P @ X_1 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % not_finite_existsD
% 0.35/0.61  thf(fact_3904_empty__def,axiom,
% 0.35/0.61      ( bot_bot_set_list_nat
% 0.35/0.61      = ( collect_list_nat
% 0.35/0.61        @ ^ [X3: list_nat] : $false ) ) ).
% 0.35/0.61  
% 0.35/0.61  % empty_def
% 0.35/0.61  thf(fact_3905_empty__def,axiom,
% 0.35/0.61      ( bot_bot_set_set_nat
% 0.35/0.61      = ( collect_set_nat
% 0.35/0.61        @ ^ [X3: set_nat] : $false ) ) ).
% 0.35/0.61  
% 0.35/0.61  % empty_def
% 0.35/0.61  thf(fact_3906_empty__def,axiom,
% 0.35/0.61      ( bot_bo2099793752762293965at_nat
% 0.35/0.61      = ( collec3392354462482085612at_nat
% 0.35/0.61        @ ^ [X3: product_prod_nat_nat] : $false ) ) ).
% 0.35/0.61  
% 0.35/0.61  % empty_def
% 0.35/0.61  thf(fact_3907_empty__def,axiom,
% 0.35/0.61      ( bot_bot_set_real
% 0.35/0.61      = ( collect_real
% 0.35/0.61        @ ^ [X3: real] : $false ) ) ).
% 0.35/0.61  
% 0.35/0.61  % empty_def
% 0.35/0.61  thf(fact_3908_empty__def,axiom,
% 0.35/0.61      ( bot_bot_set_nat
% 0.35/0.61      = ( collect_nat
% 0.35/0.61        @ ^ [X3: nat] : $false ) ) ).
% 0.35/0.61  
% 0.35/0.61  % empty_def
% 0.35/0.61  thf(fact_3909_empty__def,axiom,
% 0.35/0.61      ( bot_bot_set_int
% 0.35/0.61      = ( collect_int
% 0.35/0.61        @ ^ [X3: int] : $false ) ) ).
% 0.35/0.61  
% 0.35/0.61  % empty_def
% 0.35/0.61  thf(fact_3910_lambda__zero,axiom,
% 0.35/0.61      ( ( ^ [H: complex] : zero_zero_complex )
% 0.35/0.61      = ( times_times_complex @ zero_zero_complex ) ) ).
% 0.35/0.61  
% 0.35/0.61  % lambda_zero
% 0.35/0.61  thf(fact_3911_lambda__zero,axiom,
% 0.35/0.61      ( ( ^ [H: real] : zero_zero_real )
% 0.35/0.61      = ( times_times_real @ zero_zero_real ) ) ).
% 0.35/0.61  
% 0.35/0.61  % lambda_zero
% 0.35/0.61  thf(fact_3912_lambda__zero,axiom,
% 0.35/0.61      ( ( ^ [H: rat] : zero_zero_rat )
% 0.35/0.61      = ( times_times_rat @ zero_zero_rat ) ) ).
% 0.35/0.61  
% 0.35/0.61  % lambda_zero
% 0.35/0.61  thf(fact_3913_lambda__zero,axiom,
% 0.35/0.61      ( ( ^ [H: nat] : zero_zero_nat )
% 0.35/0.61      = ( times_times_nat @ zero_zero_nat ) ) ).
% 0.35/0.61  
% 0.35/0.61  % lambda_zero
% 0.35/0.61  thf(fact_3914_lambda__zero,axiom,
% 0.35/0.61      ( ( ^ [H: int] : zero_zero_int )
% 0.35/0.61      = ( times_times_int @ zero_zero_int ) ) ).
% 0.35/0.61  
% 0.35/0.61  % lambda_zero
% 0.35/0.61  thf(fact_3915_max__def__raw,axiom,
% 0.35/0.61      ( ord_max_set_int
% 0.35/0.61      = ( ^ [A4: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % max_def_raw
% 0.35/0.61  thf(fact_3916_max__def__raw,axiom,
% 0.35/0.61      ( ord_max_rat
% 0.35/0.61      = ( ^ [A4: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % max_def_raw
% 0.35/0.61  thf(fact_3917_max__def__raw,axiom,
% 0.35/0.61      ( ord_max_num
% 0.35/0.61      = ( ^ [A4: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % max_def_raw
% 0.35/0.61  thf(fact_3918_max__def__raw,axiom,
% 0.35/0.61      ( ord_max_nat
% 0.35/0.61      = ( ^ [A4: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % max_def_raw
% 0.35/0.61  thf(fact_3919_max__def__raw,axiom,
% 0.35/0.61      ( ord_max_int
% 0.35/0.61      = ( ^ [A4: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % max_def_raw
% 0.35/0.61  thf(fact_3920_pred__subset__eq2,axiom,
% 0.35/0.61      ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 0.35/0.61        ( ( ord_le2646555220125990790_nat_o
% 0.35/0.61          @ ^ [X3: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R )
% 0.35/0.61          @ ^ [X3: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ S2 ) )
% 0.35/0.61        = ( ord_le3146513528884898305at_nat @ R @ S2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pred_subset_eq2
% 0.35/0.61  thf(fact_3921_pred__subset__eq2,axiom,
% 0.35/0.61      ! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 0.35/0.61        ( ( ord_le6741204236512500942_int_o
% 0.35/0.61          @ ^ [X3: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R )
% 0.35/0.61          @ ^ [X3: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ S2 ) )
% 0.35/0.61        = ( ord_le2843351958646193337nt_int @ R @ S2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pred_subset_eq2
% 0.35/0.61  thf(fact_3922_pred__subset__eq2,axiom,
% 0.35/0.61      ! [R: set_Pr8056137968301705908nteger,S2: set_Pr8056137968301705908nteger] :
% 0.35/0.61        ( ( ord_le3636971675376928563eger_o
% 0.35/0.61          @ ^ [X3: code_integer > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y2 ) @ R )
% 0.35/0.61          @ ^ [X3: code_integer > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y2 ) @ S2 ) )
% 0.35/0.61        = ( ord_le3216752416896350996nteger @ R @ S2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pred_subset_eq2
% 0.35/0.61  thf(fact_3923_pred__subset__eq2,axiom,
% 0.35/0.61      ! [R: set_Pr1281608226676607948nteger,S2: set_Pr1281608226676607948nteger] :
% 0.35/0.61        ( ( ord_le4340812435750786203eger_o
% 0.35/0.61          @ ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y2 ) @ R )
% 0.35/0.61          @ ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y2 ) @ S2 ) )
% 0.35/0.61        = ( ord_le653643898420964396nteger @ R @ S2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pred_subset_eq2
% 0.35/0.61  thf(fact_3924_pred__subset__eq2,axiom,
% 0.35/0.61      ! [R: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
% 0.35/0.61        ( ( ord_le5643404153117327598_int_o
% 0.35/0.61          @ ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y2 ) @ R )
% 0.35/0.61          @ ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y2 ) @ S2 ) )
% 0.35/0.61        = ( ord_le8725513860283290265nt_int @ R @ S2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pred_subset_eq2
% 0.35/0.61  thf(fact_3925_pred__subset__eq2,axiom,
% 0.35/0.61      ! [R: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
% 0.35/0.61        ( ( ord_le2124322318746777828_int_o
% 0.35/0.61          @ ^ [X3: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y2 ) @ R )
% 0.35/0.61          @ ^ [X3: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y2 ) @ S2 ) )
% 0.35/0.61        = ( ord_le135402666524580259nt_int @ R @ S2 ) ) ).
% 0.35/0.61  
% 0.35/0.61  % pred_subset_eq2
% 0.35/0.61  thf(fact_3926_bot__empty__eq2,axiom,
% 0.35/0.61      ( bot_bot_int_int_o
% 0.35/0.61      = ( ^ [X3: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ bot_bo1796632182523588997nt_int ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % bot_empty_eq2
% 0.35/0.61  thf(fact_3927_bot__empty__eq2,axiom,
% 0.35/0.61      ( bot_bo5358457235160185703eger_o
% 0.35/0.61      = ( ^ [X3: code_integer > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y2 ) @ bot_bo3145834390647256904nteger ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % bot_empty_eq2
% 0.35/0.61  thf(fact_3928_bot__empty__eq2,axiom,
% 0.35/0.61      ( bot_bo3000040243691356879eger_o
% 0.35/0.61      = ( ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y2 ) @ bot_bo5443222936135328352nteger ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % bot_empty_eq2
% 0.35/0.61  thf(fact_3929_bot__empty__eq2,axiom,
% 0.35/0.61      ( bot_bo8662317086119403298_int_o
% 0.35/0.61      = ( ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y2 ) @ bot_bo572930865798478029nt_int ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % bot_empty_eq2
% 0.35/0.61  thf(fact_3930_bot__empty__eq2,axiom,
% 0.35/0.61      ( bot_bo1403522918969695512_int_o
% 0.35/0.61      = ( ^ [X3: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y2 ) @ bot_bo4508923176915781079nt_int ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % bot_empty_eq2
% 0.35/0.61  thf(fact_3931_bot__empty__eq2,axiom,
% 0.35/0.61      ( bot_bot_nat_nat_o
% 0.35/0.61      = ( ^ [X3: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ bot_bo2099793752762293965at_nat ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % bot_empty_eq2
% 0.35/0.61  thf(fact_3932_finite__M__bounded__by__nat,axiom,
% 0.35/0.61      ! [P: nat > $o,I: nat] :
% 0.35/0.61        ( finite_finite_nat
% 0.35/0.61        @ ( collect_nat
% 0.35/0.61          @ ^ [K3: nat] :
% 0.35/0.61              ( ( P @ K3 )
% 0.35/0.61              & ( ord_less_nat @ K3 @ I ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_M_bounded_by_nat
% 0.35/0.61  thf(fact_3933_finite__less__ub,axiom,
% 0.35/0.61      ! [F: nat > nat,U: nat] :
% 0.35/0.61        ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
% 0.35/0.61       => ( finite_finite_nat
% 0.35/0.61          @ ( collect_nat
% 0.35/0.61            @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_less_ub
% 0.35/0.61  thf(fact_3934_set__vebt__def,axiom,
% 0.35/0.61      ( vEBT_set_vebt
% 0.35/0.61      = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % set_vebt_def
% 0.35/0.61  thf(fact_3935_finite__lists__length__eq,axiom,
% 0.35/0.61      ! [A2: set_complex,N: nat] :
% 0.35/0.61        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.61       => ( finite8712137658972009173omplex
% 0.35/0.61          @ ( collect_list_complex
% 0.35/0.61            @ ^ [Xs2: list_complex] :
% 0.35/0.61                ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
% 0.35/0.61                & ( ( size_s3451745648224563538omplex @ Xs2 )
% 0.35/0.61                  = N ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_lists_length_eq
% 0.35/0.61  thf(fact_3936_finite__lists__length__eq,axiom,
% 0.35/0.61      ! [A2: set_Pr1261947904930325089at_nat,N: nat] :
% 0.35/0.61        ( ( finite6177210948735845034at_nat @ A2 )
% 0.35/0.61       => ( finite500796754983035824at_nat
% 0.35/0.61          @ ( collec3343600615725829874at_nat
% 0.35/0.61            @ ^ [Xs2: list_P6011104703257516679at_nat] :
% 0.35/0.61                ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 )
% 0.35/0.61                & ( ( size_s5460976970255530739at_nat @ Xs2 )
% 0.35/0.61                  = N ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_lists_length_eq
% 0.35/0.61  thf(fact_3937_finite__lists__length__eq,axiom,
% 0.35/0.61      ! [A2: set_VEBT_VEBT,N: nat] :
% 0.35/0.61        ( ( finite5795047828879050333T_VEBT @ A2 )
% 0.35/0.61       => ( finite3004134309566078307T_VEBT
% 0.35/0.61          @ ( collec5608196760682091941T_VEBT
% 0.35/0.61            @ ^ [Xs2: list_VEBT_VEBT] :
% 0.35/0.61                ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
% 0.35/0.61                & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.35/0.61                  = N ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_lists_length_eq
% 0.35/0.61  thf(fact_3938_finite__lists__length__eq,axiom,
% 0.35/0.61      ! [A2: set_o,N: nat] :
% 0.35/0.61        ( ( finite_finite_o @ A2 )
% 0.35/0.61       => ( finite_finite_list_o
% 0.35/0.61          @ ( collect_list_o
% 0.35/0.61            @ ^ [Xs2: list_o] :
% 0.35/0.61                ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A2 )
% 0.35/0.61                & ( ( size_size_list_o @ Xs2 )
% 0.35/0.61                  = N ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_lists_length_eq
% 0.35/0.61  thf(fact_3939_finite__lists__length__eq,axiom,
% 0.35/0.61      ! [A2: set_nat,N: nat] :
% 0.35/0.61        ( ( finite_finite_nat @ A2 )
% 0.35/0.61       => ( finite8100373058378681591st_nat
% 0.35/0.61          @ ( collect_list_nat
% 0.35/0.61            @ ^ [Xs2: list_nat] :
% 0.35/0.61                ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
% 0.35/0.61                & ( ( size_size_list_nat @ Xs2 )
% 0.35/0.61                  = N ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_lists_length_eq
% 0.35/0.61  thf(fact_3940_finite__lists__length__eq,axiom,
% 0.35/0.61      ! [A2: set_int,N: nat] :
% 0.35/0.61        ( ( finite_finite_int @ A2 )
% 0.35/0.61       => ( finite3922522038869484883st_int
% 0.35/0.61          @ ( collect_list_int
% 0.35/0.61            @ ^ [Xs2: list_int] :
% 0.35/0.61                ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
% 0.35/0.61                & ( ( size_size_list_int @ Xs2 )
% 0.35/0.61                  = N ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_lists_length_eq
% 0.35/0.61  thf(fact_3941_finite__lists__length__le,axiom,
% 0.35/0.61      ! [A2: set_complex,N: nat] :
% 0.35/0.61        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.61       => ( finite8712137658972009173omplex
% 0.35/0.61          @ ( collect_list_complex
% 0.35/0.61            @ ^ [Xs2: list_complex] :
% 0.35/0.61                ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
% 0.35/0.61                & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs2 ) @ N ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_lists_length_le
% 0.35/0.61  thf(fact_3942_finite__lists__length__le,axiom,
% 0.35/0.61      ! [A2: set_Pr1261947904930325089at_nat,N: nat] :
% 0.35/0.61        ( ( finite6177210948735845034at_nat @ A2 )
% 0.35/0.61       => ( finite500796754983035824at_nat
% 0.35/0.61          @ ( collec3343600615725829874at_nat
% 0.35/0.61            @ ^ [Xs2: list_P6011104703257516679at_nat] :
% 0.35/0.61                ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 )
% 0.35/0.61                & ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) @ N ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_lists_length_le
% 0.35/0.61  thf(fact_3943_finite__lists__length__le,axiom,
% 0.35/0.61      ! [A2: set_VEBT_VEBT,N: nat] :
% 0.35/0.61        ( ( finite5795047828879050333T_VEBT @ A2 )
% 0.35/0.61       => ( finite3004134309566078307T_VEBT
% 0.35/0.61          @ ( collec5608196760682091941T_VEBT
% 0.35/0.61            @ ^ [Xs2: list_VEBT_VEBT] :
% 0.35/0.61                ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
% 0.35/0.61                & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ N ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_lists_length_le
% 0.35/0.61  thf(fact_3944_finite__lists__length__le,axiom,
% 0.35/0.61      ! [A2: set_o,N: nat] :
% 0.35/0.61        ( ( finite_finite_o @ A2 )
% 0.35/0.61       => ( finite_finite_list_o
% 0.35/0.61          @ ( collect_list_o
% 0.35/0.61            @ ^ [Xs2: list_o] :
% 0.35/0.61                ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A2 )
% 0.35/0.61                & ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ N ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_lists_length_le
% 0.35/0.61  thf(fact_3945_finite__lists__length__le,axiom,
% 0.35/0.61      ! [A2: set_nat,N: nat] :
% 0.35/0.61        ( ( finite_finite_nat @ A2 )
% 0.35/0.61       => ( finite8100373058378681591st_nat
% 0.35/0.61          @ ( collect_list_nat
% 0.35/0.61            @ ^ [Xs2: list_nat] :
% 0.35/0.61                ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
% 0.35/0.61                & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_lists_length_le
% 0.35/0.61  thf(fact_3946_finite__lists__length__le,axiom,
% 0.35/0.61      ! [A2: set_int,N: nat] :
% 0.35/0.61        ( ( finite_finite_int @ A2 )
% 0.35/0.61       => ( finite3922522038869484883st_int
% 0.35/0.61          @ ( collect_list_int
% 0.35/0.61            @ ^ [Xs2: list_int] :
% 0.35/0.61                ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
% 0.35/0.61                & ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_lists_length_le
% 0.35/0.61  thf(fact_3947_periodic__finite__ex,axiom,
% 0.35/0.61      ! [D: int,P: int > $o] :
% 0.35/0.61        ( ( ord_less_int @ zero_zero_int @ D )
% 0.35/0.61       => ( ! [X4: int,K2: int] :
% 0.35/0.61              ( ( P @ X4 )
% 0.35/0.61              = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 0.35/0.61         => ( ( ? [X6: int] : ( P @ X6 ) )
% 0.35/0.61            = ( ? [X3: int] :
% 0.35/0.61                  ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 0.35/0.61                  & ( P @ X3 ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % periodic_finite_ex
% 0.35/0.61  thf(fact_3948_aset_I7_J,axiom,
% 0.35/0.61      ! [D6: int,A2: set_int,T: int] :
% 0.35/0.61        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.35/0.61       => ! [X2: int] :
% 0.35/0.61            ( ! [Xa3: int] :
% 0.35/0.61                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.35/0.61               => ! [Xb3: int] :
% 0.35/0.61                    ( ( member_int @ Xb3 @ A2 )
% 0.35/0.61                   => ( X2
% 0.35/0.61                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.35/0.61           => ( ( ord_less_int @ T @ X2 )
% 0.35/0.61             => ( ord_less_int @ T @ ( plus_plus_int @ X2 @ D6 ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % aset(7)
% 0.35/0.61  thf(fact_3949_aset_I5_J,axiom,
% 0.35/0.61      ! [D6: int,T: int,A2: set_int] :
% 0.35/0.61        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.35/0.61       => ( ( member_int @ T @ A2 )
% 0.35/0.61         => ! [X2: int] :
% 0.35/0.61              ( ! [Xa3: int] :
% 0.35/0.61                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.35/0.61                 => ! [Xb3: int] :
% 0.35/0.61                      ( ( member_int @ Xb3 @ A2 )
% 0.35/0.61                     => ( X2
% 0.35/0.61                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.35/0.61             => ( ( ord_less_int @ X2 @ T )
% 0.35/0.61               => ( ord_less_int @ ( plus_plus_int @ X2 @ D6 ) @ T ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % aset(5)
% 0.35/0.61  thf(fact_3950_aset_I4_J,axiom,
% 0.35/0.61      ! [D6: int,T: int,A2: set_int] :
% 0.35/0.61        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.35/0.61       => ( ( member_int @ T @ A2 )
% 0.35/0.61         => ! [X2: int] :
% 0.35/0.61              ( ! [Xa3: int] :
% 0.35/0.61                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.35/0.61                 => ! [Xb3: int] :
% 0.35/0.61                      ( ( member_int @ Xb3 @ A2 )
% 0.35/0.61                     => ( X2
% 0.35/0.61                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.35/0.61             => ( ( X2 != T )
% 0.35/0.61               => ( ( plus_plus_int @ X2 @ D6 )
% 0.35/0.61                 != T ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % aset(4)
% 0.35/0.61  thf(fact_3951_aset_I3_J,axiom,
% 0.35/0.61      ! [D6: int,T: int,A2: set_int] :
% 0.35/0.61        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.35/0.61       => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 0.35/0.61         => ! [X2: int] :
% 0.35/0.61              ( ! [Xa3: int] :
% 0.35/0.61                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.35/0.61                 => ! [Xb3: int] :
% 0.35/0.61                      ( ( member_int @ Xb3 @ A2 )
% 0.35/0.61                     => ( X2
% 0.35/0.61                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.35/0.61             => ( ( X2 = T )
% 0.35/0.61               => ( ( plus_plus_int @ X2 @ D6 )
% 0.35/0.61                  = T ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % aset(3)
% 0.35/0.61  thf(fact_3952_bset_I7_J,axiom,
% 0.35/0.61      ! [D6: int,T: int,B2: set_int] :
% 0.35/0.61        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.35/0.61       => ( ( member_int @ T @ B2 )
% 0.35/0.61         => ! [X2: int] :
% 0.35/0.61              ( ! [Xa3: int] :
% 0.35/0.61                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.35/0.61                 => ! [Xb3: int] :
% 0.35/0.61                      ( ( member_int @ Xb3 @ B2 )
% 0.35/0.61                     => ( X2
% 0.35/0.61                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.35/0.61             => ( ( ord_less_int @ T @ X2 )
% 0.35/0.61               => ( ord_less_int @ T @ ( minus_minus_int @ X2 @ D6 ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % bset(7)
% 0.35/0.61  thf(fact_3953_bset_I5_J,axiom,
% 0.35/0.61      ! [D6: int,B2: set_int,T: int] :
% 0.35/0.61        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.35/0.61       => ! [X2: int] :
% 0.35/0.61            ( ! [Xa3: int] :
% 0.35/0.61                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.35/0.61               => ! [Xb3: int] :
% 0.35/0.61                    ( ( member_int @ Xb3 @ B2 )
% 0.35/0.61                   => ( X2
% 0.35/0.61                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.35/0.61           => ( ( ord_less_int @ X2 @ T )
% 0.35/0.61             => ( ord_less_int @ ( minus_minus_int @ X2 @ D6 ) @ T ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % bset(5)
% 0.35/0.61  thf(fact_3954_bset_I4_J,axiom,
% 0.35/0.61      ! [D6: int,T: int,B2: set_int] :
% 0.35/0.61        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.35/0.61       => ( ( member_int @ T @ B2 )
% 0.35/0.61         => ! [X2: int] :
% 0.35/0.61              ( ! [Xa3: int] :
% 0.35/0.61                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.35/0.61                 => ! [Xb3: int] :
% 0.35/0.61                      ( ( member_int @ Xb3 @ B2 )
% 0.35/0.61                     => ( X2
% 0.35/0.61                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.35/0.61             => ( ( X2 != T )
% 0.35/0.61               => ( ( minus_minus_int @ X2 @ D6 )
% 0.35/0.61                 != T ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % bset(4)
% 0.35/0.61  thf(fact_3955_bset_I3_J,axiom,
% 0.35/0.61      ! [D6: int,T: int,B2: set_int] :
% 0.35/0.61        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.35/0.61       => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
% 0.35/0.61         => ! [X2: int] :
% 0.35/0.61              ( ! [Xa3: int] :
% 0.35/0.61                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.35/0.61                 => ! [Xb3: int] :
% 0.35/0.61                      ( ( member_int @ Xb3 @ B2 )
% 0.35/0.61                     => ( X2
% 0.35/0.61                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.35/0.61             => ( ( X2 = T )
% 0.35/0.61               => ( ( minus_minus_int @ X2 @ D6 )
% 0.35/0.61                  = T ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % bset(3)
% 0.35/0.61  thf(fact_3956_aset_I8_J,axiom,
% 0.35/0.61      ! [D6: int,A2: set_int,T: int] :
% 0.35/0.61        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.35/0.61       => ! [X2: int] :
% 0.35/0.61            ( ! [Xa3: int] :
% 0.35/0.61                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.35/0.61               => ! [Xb3: int] :
% 0.35/0.61                    ( ( member_int @ Xb3 @ A2 )
% 0.35/0.61                   => ( X2
% 0.35/0.61                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.35/0.61           => ( ( ord_less_eq_int @ T @ X2 )
% 0.35/0.61             => ( ord_less_eq_int @ T @ ( plus_plus_int @ X2 @ D6 ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % aset(8)
% 0.35/0.61  thf(fact_3957_aset_I6_J,axiom,
% 0.35/0.61      ! [D6: int,T: int,A2: set_int] :
% 0.35/0.61        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.35/0.61       => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 0.35/0.61         => ! [X2: int] :
% 0.35/0.61              ( ! [Xa3: int] :
% 0.35/0.61                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.35/0.61                 => ! [Xb3: int] :
% 0.35/0.61                      ( ( member_int @ Xb3 @ A2 )
% 0.35/0.61                     => ( X2
% 0.35/0.61                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.35/0.61             => ( ( ord_less_eq_int @ X2 @ T )
% 0.35/0.61               => ( ord_less_eq_int @ ( plus_plus_int @ X2 @ D6 ) @ T ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % aset(6)
% 0.35/0.61  thf(fact_3958_bset_I8_J,axiom,
% 0.35/0.61      ! [D6: int,T: int,B2: set_int] :
% 0.35/0.61        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.35/0.61       => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
% 0.35/0.61         => ! [X2: int] :
% 0.35/0.61              ( ! [Xa3: int] :
% 0.35/0.61                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.35/0.61                 => ! [Xb3: int] :
% 0.35/0.61                      ( ( member_int @ Xb3 @ B2 )
% 0.35/0.61                     => ( X2
% 0.35/0.61                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.35/0.61             => ( ( ord_less_eq_int @ T @ X2 )
% 0.35/0.61               => ( ord_less_eq_int @ T @ ( minus_minus_int @ X2 @ D6 ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % bset(8)
% 0.35/0.61  thf(fact_3959_bset_I6_J,axiom,
% 0.35/0.61      ! [D6: int,B2: set_int,T: int] :
% 0.35/0.61        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.35/0.61       => ! [X2: int] :
% 0.35/0.61            ( ! [Xa3: int] :
% 0.35/0.61                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.35/0.61               => ! [Xb3: int] :
% 0.35/0.61                    ( ( member_int @ Xb3 @ B2 )
% 0.35/0.61                   => ( X2
% 0.35/0.61                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.35/0.61           => ( ( ord_less_eq_int @ X2 @ T )
% 0.35/0.61             => ( ord_less_eq_int @ ( minus_minus_int @ X2 @ D6 ) @ T ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % bset(6)
% 0.35/0.61  thf(fact_3960_cpmi,axiom,
% 0.35/0.61      ! [D6: int,P: int > $o,P5: int > $o,B2: set_int] :
% 0.35/0.61        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.35/0.61       => ( ? [Z5: int] :
% 0.35/0.61            ! [X4: int] :
% 0.35/0.61              ( ( ord_less_int @ X4 @ Z5 )
% 0.35/0.61             => ( ( P @ X4 )
% 0.35/0.61                = ( P5 @ X4 ) ) )
% 0.35/0.61         => ( ! [X4: int] :
% 0.35/0.61                ( ! [Xa: int] :
% 0.35/0.61                    ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.35/0.61                   => ! [Xb: int] :
% 0.35/0.61                        ( ( member_int @ Xb @ B2 )
% 0.35/0.61                       => ( X4
% 0.35/0.61                         != ( plus_plus_int @ Xb @ Xa ) ) ) )
% 0.35/0.61               => ( ( P @ X4 )
% 0.35/0.61                 => ( P @ ( minus_minus_int @ X4 @ D6 ) ) ) )
% 0.35/0.61           => ( ! [X4: int,K2: int] :
% 0.35/0.61                  ( ( P5 @ X4 )
% 0.35/0.61                  = ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D6 ) ) ) )
% 0.35/0.61             => ( ( ? [X6: int] : ( P @ X6 ) )
% 0.35/0.61                = ( ? [X3: int] :
% 0.35/0.61                      ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.35/0.61                      & ( P5 @ X3 ) )
% 0.35/0.61                  | ? [X3: int] :
% 0.35/0.61                      ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.35/0.61                      & ? [Y2: int] :
% 0.35/0.61                          ( ( member_int @ Y2 @ B2 )
% 0.35/0.61                          & ( P @ ( plus_plus_int @ Y2 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % cpmi
% 0.35/0.61  thf(fact_3961_cppi,axiom,
% 0.35/0.61      ! [D6: int,P: int > $o,P5: int > $o,A2: set_int] :
% 0.35/0.61        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.35/0.61       => ( ? [Z5: int] :
% 0.35/0.61            ! [X4: int] :
% 0.35/0.61              ( ( ord_less_int @ Z5 @ X4 )
% 0.35/0.61             => ( ( P @ X4 )
% 0.35/0.61                = ( P5 @ X4 ) ) )
% 0.35/0.61         => ( ! [X4: int] :
% 0.35/0.61                ( ! [Xa: int] :
% 0.35/0.61                    ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.35/0.61                   => ! [Xb: int] :
% 0.35/0.61                        ( ( member_int @ Xb @ A2 )
% 0.35/0.61                       => ( X4
% 0.35/0.61                         != ( minus_minus_int @ Xb @ Xa ) ) ) )
% 0.35/0.61               => ( ( P @ X4 )
% 0.35/0.61                 => ( P @ ( plus_plus_int @ X4 @ D6 ) ) ) )
% 0.35/0.61           => ( ! [X4: int,K2: int] :
% 0.35/0.61                  ( ( P5 @ X4 )
% 0.35/0.61                  = ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D6 ) ) ) )
% 0.35/0.61             => ( ( ? [X6: int] : ( P @ X6 ) )
% 0.35/0.61                = ( ? [X3: int] :
% 0.35/0.61                      ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.35/0.61                      & ( P5 @ X3 ) )
% 0.35/0.61                  | ? [X3: int] :
% 0.35/0.61                      ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.35/0.61                      & ? [Y2: int] :
% 0.35/0.61                          ( ( member_int @ Y2 @ A2 )
% 0.35/0.61                          & ( P @ ( minus_minus_int @ Y2 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % cppi
% 0.35/0.61  thf(fact_3962_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 0.35/0.61      ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
% 0.35/0.61        ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) @ X )
% 0.35/0.61        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.61           => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61          & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % VEBT_internal.naive_member.simps(3)
% 0.35/0.61  thf(fact_3963_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 0.35/0.61      ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
% 0.35/0.61        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) @ X )
% 0.35/0.61        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.61           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61          & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % VEBT_internal.membermima.simps(5)
% 0.35/0.61  thf(fact_3964_vebt__member_Osimps_I5_J,axiom,
% 0.35/0.61      ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 0.35/0.61        ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
% 0.35/0.61        = ( ( X != Mi )
% 0.35/0.61         => ( ( X != Ma )
% 0.35/0.61           => ( ~ ( ord_less_nat @ X @ Mi )
% 0.35/0.61              & ( ~ ( ord_less_nat @ X @ Mi )
% 0.35/0.61               => ( ~ ( ord_less_nat @ Ma @ X )
% 0.35/0.61                  & ( ~ ( ord_less_nat @ Ma @ X )
% 0.35/0.61                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.61                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                      & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % vebt_member.simps(5)
% 0.35/0.61  thf(fact_3965_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 0.35/0.61      ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
% 0.35/0.61        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc ) @ X )
% 0.35/0.61        = ( ( X = Mi )
% 0.35/0.61          | ( X = Ma )
% 0.35/0.61          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.61             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61            & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % VEBT_internal.membermima.simps(4)
% 0.35/0.61  thf(fact_3966_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat] :
% 0.35/0.61        ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 0.35/0.61       => ( ! [A3: $o,B3: $o] :
% 0.35/0.61              ( ( X
% 0.35/0.61                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61             => ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.61                 => A3 )
% 0.35/0.61                & ( ( Xa2 != zero_zero_nat )
% 0.35/0.61                 => ( ( ( Xa2 = one_one_nat )
% 0.35/0.61                     => B3 )
% 0.35/0.61                    & ( Xa2 = one_one_nat ) ) ) ) )
% 0.35/0.61         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.61                ( X
% 0.35/0.61               != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.35/0.61           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.35/0.61                  ( ? [S4: vEBT_VEBT] :
% 0.35/0.61                      ( X
% 0.35/0.61                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S4 ) )
% 0.35/0.61                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                     => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % VEBT_internal.naive_member.elims(3)
% 0.35/0.61  thf(fact_3967_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat] :
% 0.35/0.61        ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 0.35/0.61       => ( ! [A3: $o,B3: $o] :
% 0.35/0.61              ( ( X
% 0.35/0.61                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61             => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.61                   => A3 )
% 0.35/0.61                  & ( ( Xa2 != zero_zero_nat )
% 0.35/0.61                   => ( ( ( Xa2 = one_one_nat )
% 0.35/0.61                       => B3 )
% 0.35/0.61                      & ( Xa2 = one_one_nat ) ) ) ) )
% 0.35/0.61         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.35/0.61                ( ? [S4: vEBT_VEBT] :
% 0.35/0.61                    ( X
% 0.35/0.61                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S4 ) )
% 0.35/0.61               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                     => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % VEBT_internal.naive_member.elims(2)
% 0.35/0.61  thf(fact_3968_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 0.35/0.61        ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 0.35/0.61          = Y )
% 0.35/0.61       => ( ! [A3: $o,B3: $o] :
% 0.35/0.61              ( ( X
% 0.35/0.61                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61             => ( Y
% 0.35/0.61                = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.61                       => A3 )
% 0.35/0.61                      & ( ( Xa2 != zero_zero_nat )
% 0.35/0.61                       => ( ( ( Xa2 = one_one_nat )
% 0.35/0.61                           => B3 )
% 0.35/0.61                          & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 0.35/0.61         => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.61                  ( X
% 0.35/0.61                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.35/0.61             => Y )
% 0.35/0.61           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.35/0.61                  ( ? [S4: vEBT_VEBT] :
% 0.35/0.61                      ( X
% 0.35/0.61                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S4 ) )
% 0.35/0.61                 => ( Y
% 0.35/0.61                    = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                           => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % VEBT_internal.naive_member.elims(1)
% 0.35/0.61  thf(fact_3969_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat] :
% 0.35/0.61        ( ( vEBT_VEBT_membermima @ X @ Xa2 )
% 0.35/0.61       => ( ! [Mi3: nat,Ma3: nat] :
% 0.35/0.61              ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.35/0.61                  ( X
% 0.35/0.61                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.35/0.61             => ~ ( ( Xa2 = Mi3 )
% 0.35/0.61                  | ( Xa2 = Ma3 ) ) )
% 0.35/0.61         => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.35/0.61                ( ? [Vc2: vEBT_VEBT] :
% 0.35/0.61                    ( X
% 0.35/0.61                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.35/0.61               => ~ ( ( Xa2 = Mi3 )
% 0.35/0.61                    | ( Xa2 = Ma3 )
% 0.35/0.61                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.35/0.61           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.35/0.61                  ( ? [Vd2: vEBT_VEBT] :
% 0.35/0.61                      ( X
% 0.35/0.61                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.35/0.61                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % VEBT_internal.membermima.elims(2)
% 0.35/0.61  thf(fact_3970_vebt__member_Oelims_I2_J,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat] :
% 0.35/0.61        ( ( vEBT_vebt_member @ X @ Xa2 )
% 0.35/0.61       => ( ! [A3: $o,B3: $o] :
% 0.35/0.61              ( ( X
% 0.35/0.61                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61             => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.61                   => A3 )
% 0.35/0.61                  & ( ( Xa2 != zero_zero_nat )
% 0.35/0.61                   => ( ( ( Xa2 = one_one_nat )
% 0.35/0.61                       => B3 )
% 0.35/0.61                      & ( Xa2 = one_one_nat ) ) ) ) )
% 0.35/0.61         => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 0.35/0.61                ( ? [Summary3: vEBT_VEBT] :
% 0.35/0.61                    ( X
% 0.35/0.61                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 0.35/0.61               => ~ ( ( Xa2 != Mi3 )
% 0.35/0.61                   => ( ( Xa2 != Ma3 )
% 0.35/0.61                     => ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 0.35/0.61                        & ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 0.35/0.61                         => ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 0.35/0.61                            & ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 0.35/0.61                             => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                                 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % vebt_member.elims(2)
% 0.35/0.61  thf(fact_3971_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat] :
% 0.35/0.61        ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
% 0.35/0.61       => ( ! [Uu2: $o,Uv2: $o] :
% 0.35/0.61              ( X
% 0.35/0.61             != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.61         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.35/0.61                ( X
% 0.35/0.61               != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.35/0.61           => ( ! [Mi3: nat,Ma3: nat] :
% 0.35/0.61                  ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.35/0.61                      ( X
% 0.35/0.61                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.35/0.61                 => ( ( Xa2 = Mi3 )
% 0.35/0.61                    | ( Xa2 = Ma3 ) ) )
% 0.35/0.61             => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.35/0.61                    ( ? [Vc2: vEBT_VEBT] :
% 0.35/0.61                        ( X
% 0.35/0.61                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.35/0.61                   => ( ( Xa2 = Mi3 )
% 0.35/0.61                      | ( Xa2 = Ma3 )
% 0.35/0.61                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.35/0.61               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.35/0.61                      ( ? [Vd2: vEBT_VEBT] :
% 0.35/0.61                          ( X
% 0.35/0.61                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.35/0.61                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % VEBT_internal.membermima.elims(3)
% 0.35/0.61  thf(fact_3972_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 0.35/0.61        ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
% 0.35/0.61          = Y )
% 0.35/0.61       => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.35/0.61                ( X
% 0.35/0.61                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.61           => Y )
% 0.35/0.61         => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.35/0.61                  ( X
% 0.35/0.61                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.35/0.61             => Y )
% 0.35/0.61           => ( ! [Mi3: nat,Ma3: nat] :
% 0.35/0.61                  ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.35/0.61                      ( X
% 0.35/0.61                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.35/0.61                 => ( Y
% 0.35/0.61                    = ( ~ ( ( Xa2 = Mi3 )
% 0.35/0.61                          | ( Xa2 = Ma3 ) ) ) ) )
% 0.35/0.61             => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.35/0.61                    ( ? [Vc2: vEBT_VEBT] :
% 0.35/0.61                        ( X
% 0.35/0.61                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.35/0.61                   => ( Y
% 0.35/0.61                      = ( ~ ( ( Xa2 = Mi3 )
% 0.35/0.61                            | ( Xa2 = Ma3 )
% 0.35/0.61                            | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                               => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
% 0.35/0.61               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.35/0.61                      ( ? [Vd2: vEBT_VEBT] :
% 0.35/0.61                          ( X
% 0.35/0.61                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.35/0.61                     => ( Y
% 0.35/0.61                        = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                               => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % VEBT_internal.membermima.elims(1)
% 0.35/0.61  thf(fact_3973_vebt__insert_Osimps_I5_J,axiom,
% 0.35/0.61      ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 0.35/0.61        ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
% 0.35/0.61        = ( if_VEBT_VEBT
% 0.35/0.61          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.61            & ~ ( ( X = Mi )
% 0.35/0.61                | ( X = Ma ) ) )
% 0.35/0.61          @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
% 0.35/0.61          @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % vebt_insert.simps(5)
% 0.35/0.61  thf(fact_3974_vebt__member_Oelims_I3_J,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat] :
% 0.35/0.61        ( ~ ( vEBT_vebt_member @ X @ Xa2 )
% 0.35/0.61       => ( ! [A3: $o,B3: $o] :
% 0.35/0.61              ( ( X
% 0.35/0.61                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61             => ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.61                 => A3 )
% 0.35/0.61                & ( ( Xa2 != zero_zero_nat )
% 0.35/0.61                 => ( ( ( Xa2 = one_one_nat )
% 0.35/0.61                     => B3 )
% 0.35/0.61                    & ( Xa2 = one_one_nat ) ) ) ) )
% 0.35/0.61         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.61                ( X
% 0.35/0.61               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.35/0.61           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.35/0.61                  ( X
% 0.35/0.61                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.35/0.61             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.61                    ( X
% 0.35/0.61                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.35/0.61               => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 0.35/0.61                      ( ? [Summary3: vEBT_VEBT] :
% 0.35/0.61                          ( X
% 0.35/0.61                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 0.35/0.61                     => ( ( Xa2 != Mi3 )
% 0.35/0.61                       => ( ( Xa2 != Ma3 )
% 0.35/0.61                         => ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 0.35/0.61                            & ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 0.35/0.61                             => ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 0.35/0.61                                & ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 0.35/0.61                                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % vebt_member.elims(3)
% 0.35/0.61  thf(fact_3975_vebt__member_Oelims_I1_J,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 0.35/0.61        ( ( ( vEBT_vebt_member @ X @ Xa2 )
% 0.35/0.61          = Y )
% 0.35/0.61       => ( ! [A3: $o,B3: $o] :
% 0.35/0.61              ( ( X
% 0.35/0.61                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61             => ( Y
% 0.35/0.61                = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.61                       => A3 )
% 0.35/0.61                      & ( ( Xa2 != zero_zero_nat )
% 0.35/0.61                       => ( ( ( Xa2 = one_one_nat )
% 0.35/0.61                           => B3 )
% 0.35/0.61                          & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 0.35/0.61         => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.61                  ( X
% 0.35/0.61                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.35/0.61             => Y )
% 0.35/0.61           => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.35/0.61                    ( X
% 0.35/0.61                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.35/0.61               => Y )
% 0.35/0.61             => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.61                      ( X
% 0.35/0.61                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.35/0.61                 => Y )
% 0.35/0.61               => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 0.35/0.61                      ( ? [Summary3: vEBT_VEBT] :
% 0.35/0.61                          ( X
% 0.35/0.61                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 0.35/0.61                     => ( Y
% 0.35/0.61                        = ( ~ ( ( Xa2 != Mi3 )
% 0.35/0.61                             => ( ( Xa2 != Ma3 )
% 0.35/0.61                               => ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 0.35/0.61                                  & ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 0.35/0.61                                   => ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 0.35/0.61                                      & ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 0.35/0.61                                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                                           => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % vebt_member.elims(1)
% 0.35/0.61  thf(fact_3976_vebt__insert_Oelims,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
% 0.35/0.61        ( ( ( vEBT_vebt_insert @ X @ Xa2 )
% 0.35/0.61          = Y )
% 0.35/0.61       => ( ! [A3: $o,B3: $o] :
% 0.35/0.61              ( ( X
% 0.35/0.61                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61             => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.61                   => ( Y
% 0.35/0.61                      = ( vEBT_Leaf @ $true @ B3 ) ) )
% 0.35/0.61                  & ( ( Xa2 != zero_zero_nat )
% 0.35/0.61                   => ( ( ( Xa2 = one_one_nat )
% 0.35/0.61                       => ( Y
% 0.35/0.61                          = ( vEBT_Leaf @ A3 @ $true ) ) )
% 0.35/0.61                      & ( ( Xa2 != one_one_nat )
% 0.35/0.61                       => ( Y
% 0.35/0.61                          = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) ) )
% 0.35/0.61         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT] :
% 0.35/0.61                ( ( X
% 0.35/0.61                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S4 ) )
% 0.35/0.61               => ( Y
% 0.35/0.61                 != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S4 ) ) )
% 0.35/0.61           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT] :
% 0.35/0.61                  ( ( X
% 0.35/0.61                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S4 ) )
% 0.35/0.61                 => ( Y
% 0.35/0.61                   != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S4 ) ) )
% 0.35/0.61             => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.61                    ( ( X
% 0.35/0.61                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary3 ) )
% 0.35/0.61                   => ( Y
% 0.35/0.61                     != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary3 ) ) )
% 0.35/0.61               => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.61                      ( ( X
% 0.35/0.61                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 0.35/0.61                     => ( Y
% 0.35/0.61                       != ( if_VEBT_VEBT
% 0.35/0.61                          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                            & ~ ( ( Xa2 = Mi3 )
% 0.35/0.61                                | ( Xa2 = Ma3 ) ) )
% 0.35/0.61                          @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Xa2 @ Mi3 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ Ma3 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary3 ) )
% 0.35/0.61                          @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % vebt_insert.elims
% 0.35/0.61  thf(fact_3977_vebt__succ_Osimps_I6_J,axiom,
% 0.35/0.61      ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.61        ( ( ( ord_less_nat @ X @ Mi )
% 0.35/0.61         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
% 0.35/0.61            = ( some_nat @ Mi ) ) )
% 0.35/0.61        & ( ~ ( ord_less_nat @ X @ Mi )
% 0.35/0.61         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
% 0.35/0.61            = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.61              @ ( if_option_nat
% 0.35/0.61                @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                   != none_nat )
% 0.35/0.61                  & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.61                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                @ ( if_option_nat
% 0.35/0.61                  @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.61                    = none_nat )
% 0.35/0.61                  @ none_nat
% 0.35/0.61                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.61              @ none_nat ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % vebt_succ.simps(6)
% 0.35/0.61  thf(fact_3978_vebt__pred_Osimps_I7_J,axiom,
% 0.35/0.61      ! [Ma: nat,X: nat,Mi: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.61        ( ( ( ord_less_nat @ Ma @ X )
% 0.35/0.61         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
% 0.35/0.61            = ( some_nat @ Ma ) ) )
% 0.35/0.61        & ( ~ ( ord_less_nat @ Ma @ X )
% 0.35/0.61         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
% 0.35/0.61            = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.61              @ ( if_option_nat
% 0.35/0.61                @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                   != none_nat )
% 0.35/0.61                  & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.61                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                @ ( if_option_nat
% 0.35/0.61                  @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.61                    = none_nat )
% 0.35/0.61                  @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
% 0.35/0.61                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.61              @ none_nat ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % vebt_pred.simps(7)
% 0.35/0.61  thf(fact_3979_vebt__delete_Osimps_I7_J,axiom,
% 0.35/0.61      ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.61        ( ( ( ( ord_less_nat @ X @ Mi )
% 0.35/0.61            | ( ord_less_nat @ Ma @ X ) )
% 0.35/0.61         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
% 0.35/0.61            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
% 0.35/0.61        & ( ~ ( ( ord_less_nat @ X @ Mi )
% 0.35/0.61              | ( ord_less_nat @ Ma @ X ) )
% 0.35/0.61         => ( ( ( ( X = Mi )
% 0.35/0.61                & ( X = Ma ) )
% 0.35/0.61             => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
% 0.35/0.61                = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
% 0.35/0.61            & ( ~ ( ( X = Mi )
% 0.35/0.61                  & ( X = Ma ) )
% 0.35/0.61             => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
% 0.35/0.61                = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.61                  @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                    @ ( vEBT_Node
% 0.35/0.61                      @ ( some_P7363390416028606310at_nat
% 0.35/0.61                        @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
% 0.35/0.61                          @ ( if_nat
% 0.35/0.61                            @ ( ( ( X = Mi )
% 0.35/0.61                               => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 0.35/0.61                                  = Ma ) )
% 0.35/0.61                              & ( ( X != Mi )
% 0.35/0.61                               => ( X = Ma ) ) )
% 0.35/0.61                            @ ( if_nat
% 0.35/0.61                              @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                = none_nat )
% 0.35/0.61                              @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
% 0.35/0.61                              @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.61                            @ Ma ) ) )
% 0.35/0.61                      @ ( suc @ ( suc @ Va ) )
% 0.35/0.61                      @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                      @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                    @ ( vEBT_Node
% 0.35/0.61                      @ ( some_P7363390416028606310at_nat
% 0.35/0.61                        @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
% 0.35/0.61                          @ ( if_nat
% 0.35/0.61                            @ ( ( ( X = Mi )
% 0.35/0.61                               => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 0.35/0.61                                  = Ma ) )
% 0.35/0.61                              & ( ( X != Mi )
% 0.35/0.61                               => ( X = Ma ) ) )
% 0.35/0.61                            @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.61                            @ Ma ) ) )
% 0.35/0.61                      @ ( suc @ ( suc @ Va ) )
% 0.35/0.61                      @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                      @ Summary ) )
% 0.35/0.61                  @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % vebt_delete.simps(7)
% 0.35/0.61  thf(fact_3980_vebt__delete_Oelims,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
% 0.35/0.61        ( ( ( vEBT_vebt_delete @ X @ Xa2 )
% 0.35/0.61          = Y )
% 0.35/0.61       => ( ! [A3: $o,B3: $o] :
% 0.35/0.61              ( ( X
% 0.35/0.61                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61             => ( ( Xa2 = zero_zero_nat )
% 0.35/0.61               => ( Y
% 0.35/0.61                 != ( vEBT_Leaf @ $false @ B3 ) ) ) )
% 0.35/0.61         => ( ! [A3: $o] :
% 0.35/0.61                ( ? [B3: $o] :
% 0.35/0.61                    ( X
% 0.35/0.61                    = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61               => ( ( Xa2
% 0.35/0.61                    = ( suc @ zero_zero_nat ) )
% 0.35/0.61                 => ( Y
% 0.35/0.61                   != ( vEBT_Leaf @ A3 @ $false ) ) ) )
% 0.35/0.61           => ( ! [A3: $o,B3: $o] :
% 0.35/0.61                  ( ( X
% 0.35/0.61                    = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61                 => ( ? [N3: nat] :
% 0.35/0.61                        ( Xa2
% 0.35/0.61                        = ( suc @ ( suc @ N3 ) ) )
% 0.35/0.61                   => ( Y
% 0.35/0.61                     != ( vEBT_Leaf @ A3 @ B3 ) ) ) )
% 0.35/0.61             => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.61                    ( ( X
% 0.35/0.61                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary3 ) )
% 0.35/0.61                   => ( Y
% 0.35/0.61                     != ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary3 ) ) )
% 0.35/0.61               => ( ! [Mi3: nat,Ma3: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
% 0.35/0.61                      ( ( X
% 0.35/0.61                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
% 0.35/0.61                     => ( Y
% 0.35/0.61                       != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) ) )
% 0.35/0.61                 => ( ! [Mi3: nat,Ma3: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
% 0.35/0.61                        ( ( X
% 0.35/0.61                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
% 0.35/0.61                       => ( Y
% 0.35/0.61                         != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) ) )
% 0.35/0.61                   => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.61                          ( ( X
% 0.35/0.61                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 0.35/0.61                         => ~ ( ( ( ( ord_less_nat @ Xa2 @ Mi3 )
% 0.35/0.61                                  | ( ord_less_nat @ Ma3 @ Xa2 ) )
% 0.35/0.61                               => ( Y
% 0.35/0.61                                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) ) )
% 0.35/0.61                              & ( ~ ( ( ord_less_nat @ Xa2 @ Mi3 )
% 0.35/0.61                                    | ( ord_less_nat @ Ma3 @ Xa2 ) )
% 0.35/0.61                               => ( ( ( ( Xa2 = Mi3 )
% 0.35/0.61                                      & ( Xa2 = Ma3 ) )
% 0.35/0.61                                   => ( Y
% 0.35/0.61                                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) ) )
% 0.35/0.61                                  & ( ~ ( ( Xa2 = Mi3 )
% 0.35/0.61                                        & ( Xa2 = Ma3 ) )
% 0.35/0.61                                   => ( Y
% 0.35/0.61                                      = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                                        @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                          @ ( vEBT_Node
% 0.35/0.61                                            @ ( some_P7363390416028606310at_nat
% 0.35/0.61                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi3 ) @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi3 )
% 0.35/0.61                                                @ ( if_nat
% 0.35/0.61                                                  @ ( ( ( Xa2 = Mi3 )
% 0.35/0.61                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) )
% 0.35/0.61                                                        = Ma3 ) )
% 0.35/0.61                                                    & ( ( Xa2 != Mi3 )
% 0.35/0.61                                                     => ( Xa2 = Ma3 ) ) )
% 0.35/0.61                                                  @ ( if_nat
% 0.35/0.61                                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                                      = none_nat )
% 0.35/0.61                                                    @ ( if_nat @ ( Xa2 = Mi3 ) @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi3 )
% 0.35/0.61                                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.61                                                  @ Ma3 ) ) )
% 0.35/0.61                                            @ ( suc @ ( suc @ Va2 ) )
% 0.35/0.61                                            @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                            @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                          @ ( vEBT_Node
% 0.35/0.61                                            @ ( some_P7363390416028606310at_nat
% 0.35/0.61                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi3 ) @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi3 )
% 0.35/0.61                                                @ ( if_nat
% 0.35/0.61                                                  @ ( ( ( Xa2 = Mi3 )
% 0.35/0.61                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) )
% 0.35/0.61                                                        = Ma3 ) )
% 0.35/0.61                                                    & ( ( Xa2 != Mi3 )
% 0.35/0.61                                                     => ( Xa2 = Ma3 ) ) )
% 0.35/0.61                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.61                                                  @ Ma3 ) ) )
% 0.35/0.61                                            @ ( suc @ ( suc @ Va2 ) )
% 0.35/0.61                                            @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                            @ Summary3 ) )
% 0.35/0.61                                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % vebt_delete.elims
% 0.35/0.61  thf(fact_3981_vebt__succ_Oelims,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
% 0.35/0.61        ( ( ( vEBT_vebt_succ @ X @ Xa2 )
% 0.35/0.61          = Y )
% 0.35/0.61       => ( ! [Uu2: $o,B3: $o] :
% 0.35/0.61              ( ( X
% 0.35/0.61                = ( vEBT_Leaf @ Uu2 @ B3 ) )
% 0.35/0.61             => ( ( Xa2 = zero_zero_nat )
% 0.35/0.61               => ~ ( ( B3
% 0.35/0.61                     => ( Y
% 0.35/0.61                        = ( some_nat @ one_one_nat ) ) )
% 0.35/0.61                    & ( ~ B3
% 0.35/0.61                     => ( Y = none_nat ) ) ) ) )
% 0.35/0.61         => ( ( ? [Uv2: $o,Uw2: $o] :
% 0.35/0.61                  ( X
% 0.35/0.61                  = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
% 0.35/0.61             => ( ? [N3: nat] :
% 0.35/0.61                    ( Xa2
% 0.35/0.61                    = ( suc @ N3 ) )
% 0.35/0.61               => ( Y != none_nat ) ) )
% 0.35/0.61           => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.35/0.61                    ( X
% 0.35/0.61                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
% 0.35/0.61               => ( Y != none_nat ) )
% 0.35/0.61             => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.35/0.61                      ( X
% 0.35/0.61                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 0.35/0.61                 => ( Y != none_nat ) )
% 0.35/0.61               => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
% 0.35/0.61                        ( X
% 0.35/0.61                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
% 0.35/0.61                   => ( Y != none_nat ) )
% 0.35/0.61                 => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.61                        ( ( X
% 0.35/0.61                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 0.35/0.61                       => ~ ( ( ( ord_less_nat @ Xa2 @ Mi3 )
% 0.35/0.61                             => ( Y
% 0.35/0.61                                = ( some_nat @ Mi3 ) ) )
% 0.35/0.61                            & ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 0.35/0.61                             => ( Y
% 0.35/0.61                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                                  @ ( if_option_nat
% 0.35/0.61                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                       != none_nat )
% 0.35/0.61                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.61                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                    @ ( if_option_nat
% 0.35/0.61                                      @ ( ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.61                                        = none_nat )
% 0.35/0.61                                      @ none_nat
% 0.35/0.61                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.61                                  @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % vebt_succ.elims
% 0.35/0.61  thf(fact_3982_vebt__pred_Oelims,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
% 0.35/0.61        ( ( ( vEBT_vebt_pred @ X @ Xa2 )
% 0.35/0.61          = Y )
% 0.35/0.61       => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.35/0.61                ( X
% 0.35/0.61                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.61           => ( ( Xa2 = zero_zero_nat )
% 0.35/0.61             => ( Y != none_nat ) ) )
% 0.35/0.61         => ( ! [A3: $o] :
% 0.35/0.61                ( ? [Uw2: $o] :
% 0.35/0.61                    ( X
% 0.35/0.61                    = ( vEBT_Leaf @ A3 @ Uw2 ) )
% 0.35/0.61               => ( ( Xa2
% 0.35/0.61                    = ( suc @ zero_zero_nat ) )
% 0.35/0.61                 => ~ ( ( A3
% 0.35/0.61                       => ( Y
% 0.35/0.61                          = ( some_nat @ zero_zero_nat ) ) )
% 0.35/0.61                      & ( ~ A3
% 0.35/0.61                       => ( Y = none_nat ) ) ) ) )
% 0.35/0.61           => ( ! [A3: $o,B3: $o] :
% 0.35/0.61                  ( ( X
% 0.35/0.61                    = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61                 => ( ? [Va2: nat] :
% 0.35/0.61                        ( Xa2
% 0.35/0.61                        = ( suc @ ( suc @ Va2 ) ) )
% 0.35/0.61                   => ~ ( ( B3
% 0.35/0.61                         => ( Y
% 0.35/0.61                            = ( some_nat @ one_one_nat ) ) )
% 0.35/0.61                        & ( ~ B3
% 0.35/0.61                         => ( ( A3
% 0.35/0.61                             => ( Y
% 0.35/0.61                                = ( some_nat @ zero_zero_nat ) ) )
% 0.35/0.61                            & ( ~ A3
% 0.35/0.61                             => ( Y = none_nat ) ) ) ) ) ) )
% 0.35/0.61             => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
% 0.35/0.61                      ( X
% 0.35/0.61                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
% 0.35/0.61                 => ( Y != none_nat ) )
% 0.35/0.61               => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
% 0.35/0.61                        ( X
% 0.35/0.61                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
% 0.35/0.61                   => ( Y != none_nat ) )
% 0.35/0.61                 => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
% 0.35/0.61                          ( X
% 0.35/0.61                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
% 0.35/0.61                     => ( Y != none_nat ) )
% 0.35/0.61                   => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.61                          ( ( X
% 0.35/0.61                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 0.35/0.61                         => ~ ( ( ( ord_less_nat @ Ma3 @ Xa2 )
% 0.35/0.61                               => ( Y
% 0.35/0.61                                  = ( some_nat @ Ma3 ) ) )
% 0.35/0.61                              & ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 0.35/0.61                               => ( Y
% 0.35/0.61                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                                    @ ( if_option_nat
% 0.35/0.61                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                         != none_nat )
% 0.35/0.61                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.61                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                      @ ( if_option_nat
% 0.35/0.61                                        @ ( ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.61                                          = none_nat )
% 0.35/0.61                                        @ ( if_option_nat @ ( ord_less_nat @ Mi3 @ Xa2 ) @ ( some_nat @ Mi3 ) @ none_nat )
% 0.35/0.61                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.61                                    @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % vebt_pred.elims
% 0.35/0.61  thf(fact_3983_finite__nth__roots,axiom,
% 0.35/0.61      ! [N: nat,C: complex] :
% 0.35/0.61        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61       => ( finite3207457112153483333omplex
% 0.35/0.61          @ ( collect_complex
% 0.35/0.61            @ ^ [Z3: complex] :
% 0.35/0.61                ( ( power_power_complex @ Z3 @ N )
% 0.35/0.61                = C ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_nth_roots
% 0.35/0.61  thf(fact_3984_finite__roots__unity,axiom,
% 0.35/0.61      ! [N: nat] :
% 0.35/0.61        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.35/0.61       => ( finite_finite_real
% 0.35/0.61          @ ( collect_real
% 0.35/0.61            @ ^ [Z3: real] :
% 0.35/0.61                ( ( power_power_real @ Z3 @ N )
% 0.35/0.61                = one_one_real ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_roots_unity
% 0.35/0.61  thf(fact_3985_finite__roots__unity,axiom,
% 0.35/0.61      ! [N: nat] :
% 0.35/0.61        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.35/0.61       => ( finite3207457112153483333omplex
% 0.35/0.61          @ ( collect_complex
% 0.35/0.61            @ ^ [Z3: complex] :
% 0.35/0.61                ( ( power_power_complex @ Z3 @ N )
% 0.35/0.61                = one_one_complex ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_roots_unity
% 0.35/0.61  thf(fact_3986_vebt__succ_Opelims,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
% 0.35/0.61        ( ( ( vEBT_vebt_succ @ X @ Xa2 )
% 0.35/0.61          = Y )
% 0.35/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.35/0.61         => ( ! [Uu2: $o,B3: $o] :
% 0.35/0.61                ( ( X
% 0.35/0.61                  = ( vEBT_Leaf @ Uu2 @ B3 ) )
% 0.35/0.61               => ( ( Xa2 = zero_zero_nat )
% 0.35/0.61                 => ( ( ( B3
% 0.35/0.61                       => ( Y
% 0.35/0.61                          = ( some_nat @ one_one_nat ) ) )
% 0.35/0.61                      & ( ~ B3
% 0.35/0.61                       => ( Y = none_nat ) ) )
% 0.35/0.61                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat ) ) ) ) )
% 0.35/0.61           => ( ! [Uv2: $o,Uw2: $o] :
% 0.35/0.61                  ( ( X
% 0.35/0.61                    = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
% 0.35/0.61                 => ! [N3: nat] :
% 0.35/0.61                      ( ( Xa2
% 0.35/0.61                        = ( suc @ N3 ) )
% 0.35/0.61                     => ( ( Y = none_nat )
% 0.35/0.61                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) ) ) ) ) )
% 0.35/0.61             => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.35/0.61                    ( ( X
% 0.35/0.61                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
% 0.35/0.61                   => ( ( Y = none_nat )
% 0.35/0.61                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 0.35/0.61               => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.35/0.61                      ( ( X
% 0.35/0.61                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 0.35/0.61                     => ( ( Y = none_nat )
% 0.35/0.61                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa2 ) ) ) )
% 0.35/0.61                 => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
% 0.35/0.61                        ( ( X
% 0.35/0.61                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
% 0.35/0.61                       => ( ( Y = none_nat )
% 0.35/0.61                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa2 ) ) ) )
% 0.35/0.61                   => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.61                          ( ( X
% 0.35/0.61                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 0.35/0.61                         => ( ( ( ( ord_less_nat @ Xa2 @ Mi3 )
% 0.35/0.61                               => ( Y
% 0.35/0.61                                  = ( some_nat @ Mi3 ) ) )
% 0.35/0.61                              & ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 0.35/0.61                               => ( Y
% 0.35/0.61                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                                    @ ( if_option_nat
% 0.35/0.61                                      @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                         != none_nat )
% 0.35/0.61                                        & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.61                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                      @ ( if_option_nat
% 0.35/0.61                                        @ ( ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.61                                          = none_nat )
% 0.35/0.61                                        @ none_nat
% 0.35/0.61                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.61                                    @ none_nat ) ) ) )
% 0.35/0.61                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % vebt_succ.pelims
% 0.35/0.61  thf(fact_3987_vebt__pred_Opelims,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
% 0.35/0.61        ( ( ( vEBT_vebt_pred @ X @ Xa2 )
% 0.35/0.61          = Y )
% 0.35/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.35/0.61         => ( ! [Uu2: $o,Uv2: $o] :
% 0.35/0.61                ( ( X
% 0.35/0.61                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.61               => ( ( Xa2 = zero_zero_nat )
% 0.35/0.61                 => ( ( Y = none_nat )
% 0.35/0.61                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
% 0.35/0.61           => ( ! [A3: $o,Uw2: $o] :
% 0.35/0.61                  ( ( X
% 0.35/0.61                    = ( vEBT_Leaf @ A3 @ Uw2 ) )
% 0.35/0.61                 => ( ( Xa2
% 0.35/0.61                      = ( suc @ zero_zero_nat ) )
% 0.35/0.61                   => ( ( ( A3
% 0.35/0.61                         => ( Y
% 0.35/0.61                            = ( some_nat @ zero_zero_nat ) ) )
% 0.35/0.61                        & ( ~ A3
% 0.35/0.61                         => ( Y = none_nat ) ) )
% 0.35/0.61                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
% 0.35/0.61             => ( ! [A3: $o,B3: $o] :
% 0.35/0.61                    ( ( X
% 0.35/0.61                      = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61                   => ! [Va2: nat] :
% 0.35/0.61                        ( ( Xa2
% 0.35/0.61                          = ( suc @ ( suc @ Va2 ) ) )
% 0.35/0.61                       => ( ( ( B3
% 0.35/0.61                             => ( Y
% 0.35/0.61                                = ( some_nat @ one_one_nat ) ) )
% 0.35/0.61                            & ( ~ B3
% 0.35/0.61                             => ( ( A3
% 0.35/0.61                                 => ( Y
% 0.35/0.61                                    = ( some_nat @ zero_zero_nat ) ) )
% 0.35/0.61                                & ( ~ A3
% 0.35/0.61                                 => ( Y = none_nat ) ) ) ) )
% 0.35/0.61                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) )
% 0.35/0.61               => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
% 0.35/0.61                      ( ( X
% 0.35/0.61                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
% 0.35/0.61                     => ( ( Y = none_nat )
% 0.35/0.61                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
% 0.35/0.61                 => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
% 0.35/0.61                        ( ( X
% 0.35/0.61                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
% 0.35/0.61                       => ( ( Y = none_nat )
% 0.35/0.61                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa2 ) ) ) )
% 0.35/0.61                   => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
% 0.35/0.61                          ( ( X
% 0.35/0.61                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
% 0.35/0.61                         => ( ( Y = none_nat )
% 0.35/0.61                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa2 ) ) ) )
% 0.35/0.61                     => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.61                            ( ( X
% 0.35/0.61                              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 0.35/0.61                           => ( ( ( ( ord_less_nat @ Ma3 @ Xa2 )
% 0.35/0.61                                 => ( Y
% 0.35/0.61                                    = ( some_nat @ Ma3 ) ) )
% 0.35/0.61                                & ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 0.35/0.61                                 => ( Y
% 0.35/0.61                                    = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                                      @ ( if_option_nat
% 0.35/0.61                                        @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                           != none_nat )
% 0.35/0.61                                          & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.61                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                        @ ( if_option_nat
% 0.35/0.61                                          @ ( ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.61                                            = none_nat )
% 0.35/0.61                                          @ ( if_option_nat @ ( ord_less_nat @ Mi3 @ Xa2 ) @ ( some_nat @ Mi3 ) @ none_nat )
% 0.35/0.61                                          @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.61                                      @ none_nat ) ) ) )
% 0.35/0.61                             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % vebt_pred.pelims
% 0.35/0.61  thf(fact_3988_vebt__delete_Opelims,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
% 0.35/0.61        ( ( ( vEBT_vebt_delete @ X @ Xa2 )
% 0.35/0.61          = Y )
% 0.35/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.35/0.61         => ( ! [A3: $o,B3: $o] :
% 0.35/0.61                ( ( X
% 0.35/0.61                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61               => ( ( Xa2 = zero_zero_nat )
% 0.35/0.61                 => ( ( Y
% 0.35/0.61                      = ( vEBT_Leaf @ $false @ B3 ) )
% 0.35/0.61                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat ) ) ) ) )
% 0.35/0.61           => ( ! [A3: $o,B3: $o] :
% 0.35/0.61                  ( ( X
% 0.35/0.61                    = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61                 => ( ( Xa2
% 0.35/0.61                      = ( suc @ zero_zero_nat ) )
% 0.35/0.61                   => ( ( Y
% 0.35/0.61                        = ( vEBT_Leaf @ A3 @ $false ) )
% 0.35/0.61                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
% 0.35/0.61             => ( ! [A3: $o,B3: $o] :
% 0.35/0.61                    ( ( X
% 0.35/0.61                      = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61                   => ! [N3: nat] :
% 0.35/0.61                        ( ( Xa2
% 0.35/0.61                          = ( suc @ ( suc @ N3 ) ) )
% 0.35/0.61                       => ( ( Y
% 0.35/0.61                            = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N3 ) ) ) ) ) ) )
% 0.35/0.61               => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.61                      ( ( X
% 0.35/0.61                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary3 ) )
% 0.35/0.61                     => ( ( Y
% 0.35/0.61                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary3 ) )
% 0.35/0.61                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary3 ) @ Xa2 ) ) ) )
% 0.35/0.61                 => ( ! [Mi3: nat,Ma3: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
% 0.35/0.61                        ( ( X
% 0.35/0.61                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
% 0.35/0.61                       => ( ( Y
% 0.35/0.61                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
% 0.35/0.61                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ Xa2 ) ) ) )
% 0.35/0.61                   => ( ! [Mi3: nat,Ma3: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
% 0.35/0.61                          ( ( X
% 0.35/0.61                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
% 0.35/0.61                         => ( ( Y
% 0.35/0.61                              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
% 0.35/0.61                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ Xa2 ) ) ) )
% 0.35/0.61                     => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.61                            ( ( X
% 0.35/0.61                              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 0.35/0.61                           => ( ( ( ( ( ord_less_nat @ Xa2 @ Mi3 )
% 0.35/0.61                                    | ( ord_less_nat @ Ma3 @ Xa2 ) )
% 0.35/0.61                                 => ( Y
% 0.35/0.61                                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) ) )
% 0.35/0.61                                & ( ~ ( ( ord_less_nat @ Xa2 @ Mi3 )
% 0.35/0.61                                      | ( ord_less_nat @ Ma3 @ Xa2 ) )
% 0.35/0.61                                 => ( ( ( ( Xa2 = Mi3 )
% 0.35/0.61                                        & ( Xa2 = Ma3 ) )
% 0.35/0.61                                     => ( Y
% 0.35/0.61                                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) ) )
% 0.35/0.61                                    & ( ~ ( ( Xa2 = Mi3 )
% 0.35/0.61                                          & ( Xa2 = Ma3 ) )
% 0.35/0.61                                     => ( Y
% 0.35/0.61                                        = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                                          @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                            @ ( vEBT_Node
% 0.35/0.61                                              @ ( some_P7363390416028606310at_nat
% 0.35/0.61                                                @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi3 ) @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi3 )
% 0.35/0.61                                                  @ ( if_nat
% 0.35/0.61                                                    @ ( ( ( Xa2 = Mi3 )
% 0.35/0.61                                                       => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) )
% 0.35/0.61                                                          = Ma3 ) )
% 0.35/0.61                                                      & ( ( Xa2 != Mi3 )
% 0.35/0.61                                                       => ( Xa2 = Ma3 ) ) )
% 0.35/0.61                                                    @ ( if_nat
% 0.35/0.61                                                      @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                                        = none_nat )
% 0.35/0.61                                                      @ ( if_nat @ ( Xa2 = Mi3 ) @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi3 )
% 0.35/0.61                                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.61                                                    @ Ma3 ) ) )
% 0.35/0.61                                              @ ( suc @ ( suc @ Va2 ) )
% 0.35/0.61                                              @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                              @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                            @ ( vEBT_Node
% 0.35/0.61                                              @ ( some_P7363390416028606310at_nat
% 0.35/0.61                                                @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi3 ) @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi3 )
% 0.35/0.61                                                  @ ( if_nat
% 0.35/0.61                                                    @ ( ( ( Xa2 = Mi3 )
% 0.35/0.61                                                       => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) )
% 0.35/0.61                                                          = Ma3 ) )
% 0.35/0.61                                                      & ( ( Xa2 != Mi3 )
% 0.35/0.61                                                       => ( Xa2 = Ma3 ) ) )
% 0.35/0.61                                                    @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.61                                                    @ Ma3 ) ) )
% 0.35/0.61                                              @ ( suc @ ( suc @ Va2 ) )
% 0.35/0.61                                              @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                              @ Summary3 ) )
% 0.35/0.61                                          @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) ) ) ) ) ) )
% 0.35/0.61                             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % vebt_delete.pelims
% 0.35/0.61  thf(fact_3989_vebt__insert_Opelims,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
% 0.35/0.61        ( ( ( vEBT_vebt_insert @ X @ Xa2 )
% 0.35/0.61          = Y )
% 0.35/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.35/0.61         => ( ! [A3: $o,B3: $o] :
% 0.35/0.61                ( ( X
% 0.35/0.61                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61               => ( ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.61                     => ( Y
% 0.35/0.61                        = ( vEBT_Leaf @ $true @ B3 ) ) )
% 0.35/0.61                    & ( ( Xa2 != zero_zero_nat )
% 0.35/0.61                     => ( ( ( Xa2 = one_one_nat )
% 0.35/0.61                         => ( Y
% 0.35/0.61                            = ( vEBT_Leaf @ A3 @ $true ) ) )
% 0.35/0.61                        & ( ( Xa2 != one_one_nat )
% 0.35/0.61                         => ( Y
% 0.35/0.61                            = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) )
% 0.35/0.61                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
% 0.35/0.61           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT] :
% 0.35/0.61                  ( ( X
% 0.35/0.61                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S4 ) )
% 0.35/0.61                 => ( ( Y
% 0.35/0.61                      = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S4 ) )
% 0.35/0.61                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S4 ) @ Xa2 ) ) ) )
% 0.35/0.61             => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT] :
% 0.35/0.61                    ( ( X
% 0.35/0.61                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S4 ) )
% 0.35/0.61                   => ( ( Y
% 0.35/0.61                        = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S4 ) )
% 0.35/0.61                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S4 ) @ Xa2 ) ) ) )
% 0.35/0.61               => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.61                      ( ( X
% 0.35/0.61                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary3 ) )
% 0.35/0.61                     => ( ( Y
% 0.35/0.61                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary3 ) )
% 0.35/0.61                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary3 ) @ Xa2 ) ) ) )
% 0.35/0.61                 => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.61                        ( ( X
% 0.35/0.61                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 0.35/0.61                       => ( ( Y
% 0.35/0.61                            = ( if_VEBT_VEBT
% 0.35/0.61                              @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                                & ~ ( ( Xa2 = Mi3 )
% 0.35/0.61                                    | ( Xa2 = Ma3 ) ) )
% 0.35/0.61                              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Xa2 @ Mi3 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ Ma3 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary3 ) )
% 0.35/0.61                              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) ) )
% 0.35/0.61                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % vebt_insert.pelims
% 0.35/0.61  thf(fact_3990_finite__atLeastAtMost__int,axiom,
% 0.35/0.61      ! [L: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L @ U ) ) ).
% 0.35/0.61  
% 0.35/0.61  % finite_atLeastAtMost_int
% 0.35/0.61  thf(fact_3991_bot__enat__def,axiom,
% 0.35/0.61      bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% 0.35/0.61  
% 0.35/0.61  % bot_enat_def
% 0.35/0.61  thf(fact_3992_vebt__member_Opelims_I1_J,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 0.35/0.61        ( ( ( vEBT_vebt_member @ X @ Xa2 )
% 0.35/0.61          = Y )
% 0.35/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.35/0.61         => ( ! [A3: $o,B3: $o] :
% 0.35/0.61                ( ( X
% 0.35/0.61                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61               => ( ( Y
% 0.35/0.61                    = ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.61                       => A3 )
% 0.35/0.61                      & ( ( Xa2 != zero_zero_nat )
% 0.35/0.61                       => ( ( ( Xa2 = one_one_nat )
% 0.35/0.61                           => B3 )
% 0.35/0.61                          & ( Xa2 = one_one_nat ) ) ) ) )
% 0.35/0.61                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
% 0.35/0.61           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.61                  ( ( X
% 0.35/0.61                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.35/0.61                 => ( ~ Y
% 0.35/0.61                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 0.35/0.61             => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.35/0.61                    ( ( X
% 0.35/0.61                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.35/0.61                   => ( ~ Y
% 0.35/0.61                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 0.35/0.61               => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.61                      ( ( X
% 0.35/0.61                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.35/0.61                     => ( ~ Y
% 0.35/0.61                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
% 0.35/0.61                 => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.61                        ( ( X
% 0.35/0.61                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 0.35/0.61                       => ( ( Y
% 0.35/0.61                            = ( ( Xa2 != Mi3 )
% 0.35/0.61                             => ( ( Xa2 != Ma3 )
% 0.35/0.61                               => ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 0.35/0.61                                  & ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 0.35/0.61                                   => ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 0.35/0.61                                      & ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 0.35/0.61                                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                                           => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) )
% 0.35/0.61                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % vebt_member.pelims(1)
% 0.35/0.61  thf(fact_3993_vebt__member_Opelims_I3_J,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat] :
% 0.35/0.61        ( ~ ( vEBT_vebt_member @ X @ Xa2 )
% 0.35/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.35/0.61         => ( ! [A3: $o,B3: $o] :
% 0.35/0.61                ( ( X
% 0.35/0.61                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
% 0.35/0.61                 => ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.61                     => A3 )
% 0.35/0.61                    & ( ( Xa2 != zero_zero_nat )
% 0.35/0.61                     => ( ( ( Xa2 = one_one_nat )
% 0.35/0.61                         => B3 )
% 0.35/0.61                        & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.35/0.61           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.61                  ( ( X
% 0.35/0.61                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.35/0.61                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 0.35/0.61             => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.35/0.61                    ( ( X
% 0.35/0.61                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.35/0.61                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
% 0.35/0.61               => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.61                      ( ( X
% 0.35/0.61                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.35/0.61                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
% 0.35/0.61                 => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.61                        ( ( X
% 0.35/0.61                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 0.35/0.61                       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ Xa2 ) )
% 0.35/0.61                         => ( ( Xa2 != Mi3 )
% 0.35/0.61                           => ( ( Xa2 != Ma3 )
% 0.35/0.61                             => ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 0.35/0.61                                & ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 0.35/0.61                                 => ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 0.35/0.61                                    & ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 0.35/0.61                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % vebt_member.pelims(3)
% 0.35/0.61  thf(fact_3994_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat] :
% 0.35/0.61        ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 0.35/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.35/0.61         => ( ! [A3: $o,B3: $o] :
% 0.35/0.61                ( ( X
% 0.35/0.61                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
% 0.35/0.61                 => ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.61                     => A3 )
% 0.35/0.61                    & ( ( Xa2 != zero_zero_nat )
% 0.35/0.61                     => ( ( ( Xa2 = one_one_nat )
% 0.35/0.61                         => B3 )
% 0.35/0.61                        & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.35/0.61           => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.61                  ( ( X
% 0.35/0.61                    = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.35/0.61                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 0.35/0.61             => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S4: vEBT_VEBT] :
% 0.35/0.61                    ( ( X
% 0.35/0.61                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S4 ) )
% 0.35/0.61                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S4 ) @ Xa2 ) )
% 0.35/0.61                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.61                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61  
% 0.35/0.61  % VEBT_internal.naive_member.pelims(3)
% 0.35/0.61  thf(fact_3995_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 0.35/0.61      ! [X: vEBT_VEBT,Xa2: nat] :
% 0.35/0.61        ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 0.35/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.35/0.61         => ( ! [A3: $o,B3: $o] :
% 0.35/0.61                ( ( X
% 0.35/0.61                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.61               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
% 0.35/0.61                 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.61                       => A3 )
% 0.35/0.61                      & ( ( Xa2 != zero_zero_nat )
% 0.35/0.61                       => ( ( ( Xa2 = one_one_nat )
% 0.35/0.61                           => B3 )
% 0.35/0.61                          & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.35/0.61           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S4: vEBT_VEBT] :
% 0.35/0.61                  ( ( X
% 0.35/0.61                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S4 ) )
% 0.35/0.61                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S4 ) @ Xa2 ) )
% 0.35/0.61                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.61                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.62                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % VEBT_internal.naive_member.pelims(2)
% 0.35/0.62  thf(fact_3996_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 0.35/0.62      ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 0.35/0.62        ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 0.35/0.62          = Y )
% 0.35/0.62       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.35/0.62         => ( ! [A3: $o,B3: $o] :
% 0.35/0.62                ( ( X
% 0.35/0.62                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.62               => ( ( Y
% 0.35/0.62                    = ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.62                       => A3 )
% 0.35/0.62                      & ( ( Xa2 != zero_zero_nat )
% 0.35/0.62                       => ( ( ( Xa2 = one_one_nat )
% 0.35/0.62                           => B3 )
% 0.35/0.62                          & ( Xa2 = one_one_nat ) ) ) ) )
% 0.35/0.62                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
% 0.35/0.62           => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.62                  ( ( X
% 0.35/0.62                    = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.35/0.62                 => ( ~ Y
% 0.35/0.62                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 0.35/0.62             => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S4: vEBT_VEBT] :
% 0.35/0.62                    ( ( X
% 0.35/0.62                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S4 ) )
% 0.35/0.62                   => ( ( Y
% 0.35/0.62                        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.62                           => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.62                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 0.35/0.62                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S4 ) @ Xa2 ) ) ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % VEBT_internal.naive_member.pelims(1)
% 0.35/0.62  thf(fact_3997_vebt__member_Opelims_I2_J,axiom,
% 0.35/0.62      ! [X: vEBT_VEBT,Xa2: nat] :
% 0.35/0.62        ( ( vEBT_vebt_member @ X @ Xa2 )
% 0.35/0.62       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.35/0.62         => ( ! [A3: $o,B3: $o] :
% 0.35/0.62                ( ( X
% 0.35/0.62                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.62               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
% 0.35/0.62                 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.62                       => A3 )
% 0.35/0.62                      & ( ( Xa2 != zero_zero_nat )
% 0.35/0.62                       => ( ( ( Xa2 = one_one_nat )
% 0.35/0.62                           => B3 )
% 0.35/0.62                          & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.35/0.62           => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.62                  ( ( X
% 0.35/0.62                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 0.35/0.62                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ Xa2 ) )
% 0.35/0.62                   => ~ ( ( Xa2 != Mi3 )
% 0.35/0.62                       => ( ( Xa2 != Ma3 )
% 0.35/0.62                         => ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 0.35/0.62                            & ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 0.35/0.62                             => ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 0.35/0.62                                & ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 0.35/0.62                                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.62                                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.62                                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % vebt_member.pelims(2)
% 0.35/0.62  thf(fact_3998_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 0.35/0.62      ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 0.35/0.62        ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
% 0.35/0.62          = Y )
% 0.35/0.62       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.35/0.62         => ( ! [Uu2: $o,Uv2: $o] :
% 0.35/0.62                ( ( X
% 0.35/0.62                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.62               => ( ~ Y
% 0.35/0.62                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 0.35/0.62           => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.35/0.62                  ( ( X
% 0.35/0.62                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.35/0.62                 => ( ~ Y
% 0.35/0.62                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
% 0.35/0.62             => ( ! [Mi3: nat,Ma3: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.35/0.62                    ( ( X
% 0.35/0.62                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.35/0.62                   => ( ( Y
% 0.35/0.62                        = ( ( Xa2 = Mi3 )
% 0.35/0.62                          | ( Xa2 = Ma3 ) ) )
% 0.35/0.62                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
% 0.35/0.62               => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.62                      ( ( X
% 0.35/0.62                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.35/0.62                     => ( ( Y
% 0.35/0.62                          = ( ( Xa2 = Mi3 )
% 0.35/0.62                            | ( Xa2 = Ma3 )
% 0.35/0.62                            | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.62                               => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.62                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.35/0.62                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) ) )
% 0.35/0.62                 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.35/0.62                        ( ( X
% 0.35/0.62                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.35/0.62                       => ( ( Y
% 0.35/0.62                            = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.62                               => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.62                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 0.35/0.62                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % VEBT_internal.membermima.pelims(1)
% 0.35/0.62  thf(fact_3999_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 0.35/0.62      ! [X: vEBT_VEBT,Xa2: nat] :
% 0.35/0.62        ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
% 0.35/0.62       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.35/0.62         => ( ! [Uu2: $o,Uv2: $o] :
% 0.35/0.62                ( ( X
% 0.35/0.62                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.62               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
% 0.35/0.62           => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.35/0.62                  ( ( X
% 0.35/0.62                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.35/0.62                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
% 0.35/0.62             => ( ! [Mi3: nat,Ma3: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.35/0.62                    ( ( X
% 0.35/0.62                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.35/0.62                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
% 0.35/0.62                     => ( ( Xa2 = Mi3 )
% 0.35/0.62                        | ( Xa2 = Ma3 ) ) ) )
% 0.35/0.62               => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.62                      ( ( X
% 0.35/0.62                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.35/0.62                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
% 0.35/0.62                       => ( ( Xa2 = Mi3 )
% 0.35/0.62                          | ( Xa2 = Ma3 )
% 0.35/0.62                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.62                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.62                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 0.35/0.62                 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.35/0.62                        ( ( X
% 0.35/0.62                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.35/0.62                       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
% 0.35/0.62                         => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.62                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.62                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % VEBT_internal.membermima.pelims(3)
% 0.35/0.62  thf(fact_4000_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 0.35/0.62      ! [X: vEBT_VEBT,Xa2: nat] :
% 0.35/0.62        ( ( vEBT_VEBT_membermima @ X @ Xa2 )
% 0.35/0.62       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.35/0.62         => ( ! [Mi3: nat,Ma3: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.35/0.62                ( ( X
% 0.35/0.62                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.35/0.62               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
% 0.35/0.62                 => ~ ( ( Xa2 = Mi3 )
% 0.35/0.62                      | ( Xa2 = Ma3 ) ) ) )
% 0.35/0.62           => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.62                  ( ( X
% 0.35/0.62                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.35/0.62                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
% 0.35/0.62                   => ~ ( ( Xa2 = Mi3 )
% 0.35/0.62                        | ( Xa2 = Ma3 )
% 0.35/0.62                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.62                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.62                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 0.35/0.62             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.35/0.62                    ( ( X
% 0.35/0.62                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.35/0.62                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
% 0.35/0.62                     => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.35/0.62                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.62                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % VEBT_internal.membermima.pelims(2)
% 0.35/0.62  thf(fact_4001_arcosh__1,axiom,
% 0.35/0.62      ( ( arcosh_real @ one_one_real )
% 0.35/0.62      = zero_zero_real ) ).
% 0.35/0.62  
% 0.35/0.62  % arcosh_1
% 0.35/0.62  thf(fact_4002_mult__le__cancel__iff1,axiom,
% 0.35/0.62      ! [Z2: real,X: real,Y: real] :
% 0.35/0.62        ( ( ord_less_real @ zero_zero_real @ Z2 )
% 0.35/0.62       => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
% 0.35/0.62          = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mult_le_cancel_iff1
% 0.35/0.62  thf(fact_4003_mult__le__cancel__iff1,axiom,
% 0.35/0.62      ! [Z2: rat,X: rat,Y: rat] :
% 0.35/0.62        ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 0.35/0.62       => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y @ Z2 ) )
% 0.35/0.62          = ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mult_le_cancel_iff1
% 0.35/0.62  thf(fact_4004_mult__le__cancel__iff1,axiom,
% 0.35/0.62      ! [Z2: int,X: int,Y: int] :
% 0.35/0.62        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.35/0.62       => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
% 0.35/0.62          = ( ord_less_eq_int @ X @ Y ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mult_le_cancel_iff1
% 0.35/0.62  thf(fact_4005_mult__le__cancel__iff2,axiom,
% 0.35/0.62      ! [Z2: real,X: real,Y: real] :
% 0.35/0.62        ( ( ord_less_real @ zero_zero_real @ Z2 )
% 0.35/0.62       => ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ ( times_times_real @ Z2 @ Y ) )
% 0.35/0.62          = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mult_le_cancel_iff2
% 0.35/0.62  thf(fact_4006_mult__le__cancel__iff2,axiom,
% 0.35/0.62      ! [Z2: rat,X: rat,Y: rat] :
% 0.35/0.62        ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 0.35/0.62       => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X ) @ ( times_times_rat @ Z2 @ Y ) )
% 0.35/0.62          = ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mult_le_cancel_iff2
% 0.35/0.62  thf(fact_4007_mult__le__cancel__iff2,axiom,
% 0.35/0.62      ! [Z2: int,X: int,Y: int] :
% 0.35/0.62        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.35/0.62       => ( ( ord_less_eq_int @ ( times_times_int @ Z2 @ X ) @ ( times_times_int @ Z2 @ Y ) )
% 0.35/0.62          = ( ord_less_eq_int @ X @ Y ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mult_le_cancel_iff2
% 0.35/0.62  thf(fact_4008_divides__aux__eq,axiom,
% 0.35/0.62      ! [Q4: nat,R2: nat] :
% 0.35/0.62        ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
% 0.35/0.62        = ( R2 = zero_zero_nat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % divides_aux_eq
% 0.35/0.62  thf(fact_4009_divides__aux__eq,axiom,
% 0.35/0.62      ! [Q4: int,R2: int] :
% 0.35/0.62        ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.35/0.62        = ( R2 = zero_zero_int ) ) ).
% 0.35/0.62  
% 0.35/0.62  % divides_aux_eq
% 0.35/0.62  thf(fact_4010_arsinh__0,axiom,
% 0.35/0.62      ( ( arsinh_real @ zero_zero_real )
% 0.35/0.62      = zero_zero_real ) ).
% 0.35/0.62  
% 0.35/0.62  % arsinh_0
% 0.35/0.62  thf(fact_4011_artanh__0,axiom,
% 0.35/0.62      ( ( artanh_real @ zero_zero_real )
% 0.35/0.62      = zero_zero_real ) ).
% 0.35/0.62  
% 0.35/0.62  % artanh_0
% 0.35/0.62  thf(fact_4012_neg__eucl__rel__int__mult__2,axiom,
% 0.35/0.62      ! [B: int,A: int,Q4: int,R2: int] :
% 0.35/0.62        ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.35/0.62       => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.35/0.62         => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % neg_eucl_rel_int_mult_2
% 0.35/0.62  thf(fact_4013_low__def,axiom,
% 0.35/0.62      ( vEBT_VEBT_low
% 0.35/0.62      = ( ^ [X3: nat,N2: nat] : ( modulo_modulo_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % low_def
% 0.35/0.62  thf(fact_4014_mod__self,axiom,
% 0.35/0.62      ! [A: nat] :
% 0.35/0.62        ( ( modulo_modulo_nat @ A @ A )
% 0.35/0.62        = zero_zero_nat ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_self
% 0.35/0.62  thf(fact_4015_mod__self,axiom,
% 0.35/0.62      ! [A: int] :
% 0.35/0.62        ( ( modulo_modulo_int @ A @ A )
% 0.35/0.62        = zero_zero_int ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_self
% 0.35/0.62  thf(fact_4016_mod__self,axiom,
% 0.35/0.62      ! [A: code_integer] :
% 0.35/0.62        ( ( modulo364778990260209775nteger @ A @ A )
% 0.35/0.62        = zero_z3403309356797280102nteger ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_self
% 0.35/0.62  thf(fact_4017_mod__by__0,axiom,
% 0.35/0.62      ! [A: nat] :
% 0.35/0.62        ( ( modulo_modulo_nat @ A @ zero_zero_nat )
% 0.35/0.62        = A ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_by_0
% 0.35/0.62  thf(fact_4018_mod__by__0,axiom,
% 0.35/0.62      ! [A: int] :
% 0.35/0.62        ( ( modulo_modulo_int @ A @ zero_zero_int )
% 0.35/0.62        = A ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_by_0
% 0.35/0.62  thf(fact_4019_mod__by__0,axiom,
% 0.35/0.62      ! [A: code_integer] :
% 0.35/0.62        ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
% 0.35/0.62        = A ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_by_0
% 0.35/0.62  thf(fact_4020_mod__0,axiom,
% 0.35/0.62      ! [A: nat] :
% 0.35/0.62        ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 0.35/0.62        = zero_zero_nat ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_0
% 0.35/0.62  thf(fact_4021_mod__0,axiom,
% 0.35/0.62      ! [A: int] :
% 0.35/0.62        ( ( modulo_modulo_int @ zero_zero_int @ A )
% 0.35/0.62        = zero_zero_int ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_0
% 0.35/0.62  thf(fact_4022_mod__0,axiom,
% 0.35/0.62      ! [A: code_integer] :
% 0.35/0.62        ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 0.35/0.62        = zero_z3403309356797280102nteger ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_0
% 0.35/0.62  thf(fact_4023_bits__mod__0,axiom,
% 0.35/0.62      ! [A: nat] :
% 0.35/0.62        ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 0.35/0.62        = zero_zero_nat ) ).
% 0.35/0.62  
% 0.35/0.62  % bits_mod_0
% 0.35/0.62  thf(fact_4024_bits__mod__0,axiom,
% 0.35/0.62      ! [A: int] :
% 0.35/0.62        ( ( modulo_modulo_int @ zero_zero_int @ A )
% 0.35/0.62        = zero_zero_int ) ).
% 0.35/0.62  
% 0.35/0.62  % bits_mod_0
% 0.35/0.62  thf(fact_4025_bits__mod__0,axiom,
% 0.35/0.62      ! [A: code_integer] :
% 0.35/0.62        ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 0.35/0.62        = zero_z3403309356797280102nteger ) ).
% 0.35/0.62  
% 0.35/0.62  % bits_mod_0
% 0.35/0.62  thf(fact_4026_mod__less,axiom,
% 0.35/0.62      ! [M: nat,N: nat] :
% 0.35/0.62        ( ( ord_less_nat @ M @ N )
% 0.35/0.62       => ( ( modulo_modulo_nat @ M @ N )
% 0.35/0.62          = M ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_less
% 0.35/0.62  thf(fact_4027_mod__mult__self2__is__0,axiom,
% 0.35/0.62      ! [A: nat,B: nat] :
% 0.35/0.62        ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
% 0.35/0.62        = zero_zero_nat ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_mult_self2_is_0
% 0.35/0.62  thf(fact_4028_mod__mult__self2__is__0,axiom,
% 0.35/0.62      ! [A: int,B: int] :
% 0.35/0.62        ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
% 0.35/0.62        = zero_zero_int ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_mult_self2_is_0
% 0.35/0.62  thf(fact_4029_mod__mult__self2__is__0,axiom,
% 0.35/0.62      ! [A: code_integer,B: code_integer] :
% 0.35/0.62        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
% 0.35/0.62        = zero_z3403309356797280102nteger ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_mult_self2_is_0
% 0.35/0.62  thf(fact_4030_mod__mult__self1__is__0,axiom,
% 0.35/0.62      ! [B: nat,A: nat] :
% 0.35/0.62        ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
% 0.35/0.62        = zero_zero_nat ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_mult_self1_is_0
% 0.35/0.62  thf(fact_4031_mod__mult__self1__is__0,axiom,
% 0.35/0.62      ! [B: int,A: int] :
% 0.35/0.62        ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
% 0.35/0.62        = zero_zero_int ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_mult_self1_is_0
% 0.35/0.62  thf(fact_4032_mod__mult__self1__is__0,axiom,
% 0.35/0.62      ! [B: code_integer,A: code_integer] :
% 0.35/0.62        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
% 0.35/0.62        = zero_z3403309356797280102nteger ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_mult_self1_is_0
% 0.35/0.62  thf(fact_4033_mod__by__1,axiom,
% 0.35/0.62      ! [A: nat] :
% 0.35/0.62        ( ( modulo_modulo_nat @ A @ one_one_nat )
% 0.35/0.62        = zero_zero_nat ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_by_1
% 0.35/0.62  thf(fact_4034_mod__by__1,axiom,
% 0.35/0.62      ! [A: int] :
% 0.35/0.62        ( ( modulo_modulo_int @ A @ one_one_int )
% 0.35/0.62        = zero_zero_int ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_by_1
% 0.35/0.62  thf(fact_4035_mod__by__1,axiom,
% 0.35/0.62      ! [A: code_integer] :
% 0.35/0.62        ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 0.35/0.62        = zero_z3403309356797280102nteger ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_by_1
% 0.35/0.62  thf(fact_4036_bits__mod__by__1,axiom,
% 0.35/0.62      ! [A: nat] :
% 0.35/0.62        ( ( modulo_modulo_nat @ A @ one_one_nat )
% 0.35/0.62        = zero_zero_nat ) ).
% 0.35/0.62  
% 0.35/0.62  % bits_mod_by_1
% 0.35/0.62  thf(fact_4037_bits__mod__by__1,axiom,
% 0.35/0.62      ! [A: int] :
% 0.35/0.62        ( ( modulo_modulo_int @ A @ one_one_int )
% 0.35/0.62        = zero_zero_int ) ).
% 0.35/0.62  
% 0.35/0.62  % bits_mod_by_1
% 0.35/0.62  thf(fact_4038_bits__mod__by__1,axiom,
% 0.35/0.62      ! [A: code_integer] :
% 0.35/0.62        ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 0.35/0.62        = zero_z3403309356797280102nteger ) ).
% 0.35/0.62  
% 0.35/0.62  % bits_mod_by_1
% 0.35/0.62  thf(fact_4039_mod__div__trivial,axiom,
% 0.35/0.62      ! [A: nat,B: nat] :
% 0.35/0.62        ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.35/0.62        = zero_zero_nat ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_div_trivial
% 0.35/0.62  thf(fact_4040_mod__div__trivial,axiom,
% 0.35/0.62      ! [A: int,B: int] :
% 0.35/0.62        ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.35/0.62        = zero_zero_int ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_div_trivial
% 0.35/0.62  thf(fact_4041_mod__div__trivial,axiom,
% 0.35/0.62      ! [A: code_integer,B: code_integer] :
% 0.35/0.62        ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.35/0.62        = zero_z3403309356797280102nteger ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_div_trivial
% 0.35/0.62  thf(fact_4042_bits__mod__div__trivial,axiom,
% 0.35/0.62      ! [A: nat,B: nat] :
% 0.35/0.62        ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.35/0.62        = zero_zero_nat ) ).
% 0.35/0.62  
% 0.35/0.62  % bits_mod_div_trivial
% 0.35/0.62  thf(fact_4043_bits__mod__div__trivial,axiom,
% 0.35/0.62      ! [A: int,B: int] :
% 0.35/0.62        ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.35/0.62        = zero_zero_int ) ).
% 0.35/0.62  
% 0.35/0.62  % bits_mod_div_trivial
% 0.35/0.62  thf(fact_4044_bits__mod__div__trivial,axiom,
% 0.35/0.62      ! [A: code_integer,B: code_integer] :
% 0.35/0.62        ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.35/0.62        = zero_z3403309356797280102nteger ) ).
% 0.35/0.62  
% 0.35/0.62  % bits_mod_div_trivial
% 0.35/0.62  thf(fact_4045_mod__by__Suc__0,axiom,
% 0.35/0.62      ! [M: nat] :
% 0.35/0.62        ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.35/0.62        = zero_zero_nat ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_by_Suc_0
% 0.35/0.62  thf(fact_4046_not__mod__2__eq__1__eq__0,axiom,
% 0.35/0.62      ! [A: nat] :
% 0.35/0.62        ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62         != one_one_nat )
% 0.35/0.62        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62          = zero_zero_nat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % not_mod_2_eq_1_eq_0
% 0.35/0.62  thf(fact_4047_not__mod__2__eq__1__eq__0,axiom,
% 0.35/0.62      ! [A: int] :
% 0.35/0.62        ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.62         != one_one_int )
% 0.35/0.62        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.62          = zero_zero_int ) ) ).
% 0.35/0.62  
% 0.35/0.62  % not_mod_2_eq_1_eq_0
% 0.35/0.62  thf(fact_4048_not__mod__2__eq__1__eq__0,axiom,
% 0.35/0.62      ! [A: code_integer] :
% 0.35/0.62        ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.62         != one_one_Code_integer )
% 0.35/0.62        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.62          = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.62  
% 0.35/0.62  % not_mod_2_eq_1_eq_0
% 0.35/0.62  thf(fact_4049_not__mod__2__eq__0__eq__1,axiom,
% 0.35/0.62      ! [A: nat] :
% 0.35/0.62        ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62         != zero_zero_nat )
% 0.35/0.62        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62          = one_one_nat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % not_mod_2_eq_0_eq_1
% 0.35/0.62  thf(fact_4050_not__mod__2__eq__0__eq__1,axiom,
% 0.35/0.62      ! [A: int] :
% 0.35/0.62        ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.62         != zero_zero_int )
% 0.35/0.62        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.62          = one_one_int ) ) ).
% 0.35/0.62  
% 0.35/0.62  % not_mod_2_eq_0_eq_1
% 0.35/0.62  thf(fact_4051_not__mod__2__eq__0__eq__1,axiom,
% 0.35/0.62      ! [A: code_integer] :
% 0.35/0.62        ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.62         != zero_z3403309356797280102nteger )
% 0.35/0.62        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.62          = one_one_Code_integer ) ) ).
% 0.35/0.62  
% 0.35/0.62  % not_mod_2_eq_0_eq_1
% 0.35/0.62  thf(fact_4052_not__mod2__eq__Suc__0__eq__0,axiom,
% 0.35/0.62      ! [N: nat] :
% 0.35/0.62        ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62         != ( suc @ zero_zero_nat ) )
% 0.35/0.62        = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62          = zero_zero_nat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % not_mod2_eq_Suc_0_eq_0
% 0.35/0.62  thf(fact_4053_add__self__mod__2,axiom,
% 0.35/0.62      ! [M: nat] :
% 0.35/0.62        ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62        = zero_zero_nat ) ).
% 0.35/0.62  
% 0.35/0.62  % add_self_mod_2
% 0.35/0.62  thf(fact_4054_mod2__gr__0,axiom,
% 0.35/0.62      ! [M: nat] :
% 0.35/0.62        ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62        = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62          = one_one_nat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod2_gr_0
% 0.35/0.62  thf(fact_4055_unique__quotient,axiom,
% 0.35/0.62      ! [A: int,B: int,Q4: int,R2: int,Q6: int,R4: int] :
% 0.35/0.62        ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.35/0.62       => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R4 ) )
% 0.35/0.62         => ( Q4 = Q6 ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unique_quotient
% 0.35/0.62  thf(fact_4056_unique__remainder,axiom,
% 0.35/0.62      ! [A: int,B: int,Q4: int,R2: int,Q6: int,R4: int] :
% 0.35/0.62        ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.35/0.62       => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R4 ) )
% 0.35/0.62         => ( R2 = R4 ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unique_remainder
% 0.35/0.62  thf(fact_4057_mod__less__eq__dividend,axiom,
% 0.35/0.62      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_less_eq_dividend
% 0.35/0.62  thf(fact_4058_eucl__rel__int__by0,axiom,
% 0.35/0.62      ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).
% 0.35/0.62  
% 0.35/0.62  % eucl_rel_int_by0
% 0.35/0.62  thf(fact_4059_div__int__unique,axiom,
% 0.35/0.62      ! [K: int,L: int,Q4: int,R2: int] :
% 0.35/0.62        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.35/0.62       => ( ( divide_divide_int @ K @ L )
% 0.35/0.62          = Q4 ) ) ).
% 0.35/0.62  
% 0.35/0.62  % div_int_unique
% 0.35/0.62  thf(fact_4060_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.35/0.62      ! [A: code_integer,B: code_integer] :
% 0.35/0.62        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.35/0.62       => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.35/0.62  thf(fact_4061_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.35/0.62      ! [A: nat,B: nat] :
% 0.35/0.62        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.35/0.62       => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.35/0.62  thf(fact_4062_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.35/0.62      ! [A: int,B: int] :
% 0.35/0.62        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.35/0.62       => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.35/0.62  thf(fact_4063_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.35/0.62      ! [B: nat,A: nat] :
% 0.35/0.62        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.35/0.62       => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.35/0.62  thf(fact_4064_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.35/0.62      ! [B: int,A: int] :
% 0.35/0.62        ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.62       => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.35/0.62  thf(fact_4065_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.35/0.62      ! [B: code_integer,A: code_integer] :
% 0.35/0.62        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.35/0.62       => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.35/0.62  thf(fact_4066_mod__eq__self__iff__div__eq__0,axiom,
% 0.35/0.62      ! [A: nat,B: nat] :
% 0.35/0.62        ( ( ( modulo_modulo_nat @ A @ B )
% 0.35/0.62          = A )
% 0.35/0.62        = ( ( divide_divide_nat @ A @ B )
% 0.35/0.62          = zero_zero_nat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_eq_self_iff_div_eq_0
% 0.35/0.62  thf(fact_4067_mod__eq__self__iff__div__eq__0,axiom,
% 0.35/0.62      ! [A: int,B: int] :
% 0.35/0.62        ( ( ( modulo_modulo_int @ A @ B )
% 0.35/0.62          = A )
% 0.35/0.62        = ( ( divide_divide_int @ A @ B )
% 0.35/0.62          = zero_zero_int ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_eq_self_iff_div_eq_0
% 0.35/0.62  thf(fact_4068_mod__eq__self__iff__div__eq__0,axiom,
% 0.35/0.62      ! [A: code_integer,B: code_integer] :
% 0.35/0.62        ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.35/0.62          = A )
% 0.35/0.62        = ( ( divide6298287555418463151nteger @ A @ B )
% 0.35/0.62          = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_eq_self_iff_div_eq_0
% 0.35/0.62  thf(fact_4069_mod__Suc,axiom,
% 0.35/0.62      ! [M: nat,N: nat] :
% 0.35/0.62        ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 0.35/0.62            = N )
% 0.35/0.62         => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 0.35/0.62            = zero_zero_nat ) )
% 0.35/0.62        & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 0.35/0.62           != N )
% 0.35/0.62         => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 0.35/0.62            = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_Suc
% 0.35/0.62  thf(fact_4070_mod__induct,axiom,
% 0.35/0.62      ! [P: nat > $o,N: nat,P4: nat,M: nat] :
% 0.35/0.62        ( ( P @ N )
% 0.35/0.62       => ( ( ord_less_nat @ N @ P4 )
% 0.35/0.62         => ( ( ord_less_nat @ M @ P4 )
% 0.35/0.62           => ( ! [N3: nat] :
% 0.35/0.62                  ( ( ord_less_nat @ N3 @ P4 )
% 0.35/0.62                 => ( ( P @ N3 )
% 0.35/0.62                   => ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P4 ) ) ) )
% 0.35/0.62             => ( P @ M ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_induct
% 0.35/0.62  thf(fact_4071_mod__less__divisor,axiom,
% 0.35/0.62      ! [N: nat,M: nat] :
% 0.35/0.62        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.62       => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_less_divisor
% 0.35/0.62  thf(fact_4072_gcd__nat__induct,axiom,
% 0.35/0.62      ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.35/0.62        ( ! [M5: nat] : ( P @ M5 @ zero_zero_nat )
% 0.35/0.62       => ( ! [M5: nat,N3: nat] :
% 0.35/0.62              ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.35/0.62             => ( ( P @ N3 @ ( modulo_modulo_nat @ M5 @ N3 ) )
% 0.35/0.62               => ( P @ M5 @ N3 ) ) )
% 0.35/0.62         => ( P @ M @ N ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % gcd_nat_induct
% 0.35/0.62  thf(fact_4073_mod__Suc__le__divisor,axiom,
% 0.35/0.62      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_Suc_le_divisor
% 0.35/0.62  thf(fact_4074_mod__eq__0D,axiom,
% 0.35/0.62      ! [M: nat,D: nat] :
% 0.35/0.62        ( ( ( modulo_modulo_nat @ M @ D )
% 0.35/0.62          = zero_zero_nat )
% 0.35/0.62       => ? [Q3: nat] :
% 0.35/0.62            ( M
% 0.35/0.62            = ( times_times_nat @ D @ Q3 ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_eq_0D
% 0.35/0.62  thf(fact_4075_mod__geq,axiom,
% 0.35/0.62      ! [M: nat,N: nat] :
% 0.35/0.62        ( ~ ( ord_less_nat @ M @ N )
% 0.35/0.62       => ( ( modulo_modulo_nat @ M @ N )
% 0.35/0.62          = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_geq
% 0.35/0.62  thf(fact_4076_mod__if,axiom,
% 0.35/0.62      ( modulo_modulo_nat
% 0.35/0.62      = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M2 @ N2 ) @ M2 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_if
% 0.35/0.62  thf(fact_4077_le__mod__geq,axiom,
% 0.35/0.62      ! [N: nat,M: nat] :
% 0.35/0.62        ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.62       => ( ( modulo_modulo_nat @ M @ N )
% 0.35/0.62          = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % le_mod_geq
% 0.35/0.62  thf(fact_4078_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.35/0.62      ! [B: code_integer,A: code_integer] :
% 0.35/0.62        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.35/0.62       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.35/0.62  thf(fact_4079_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.35/0.62      ! [B: nat,A: nat] :
% 0.35/0.62        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.35/0.62       => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.35/0.62  thf(fact_4080_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.35/0.62      ! [B: int,A: int] :
% 0.35/0.62        ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.62       => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.35/0.62  thf(fact_4081_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.35/0.62      ! [A: code_integer,B: code_integer] :
% 0.35/0.62        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.35/0.62       => ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.35/0.62         => ( ( modulo364778990260209775nteger @ A @ B )
% 0.35/0.62            = A ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unique_euclidean_semiring_numeral_class.mod_less
% 0.35/0.62  thf(fact_4082_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.35/0.62      ! [A: nat,B: nat] :
% 0.35/0.62        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.35/0.62       => ( ( ord_less_nat @ A @ B )
% 0.35/0.62         => ( ( modulo_modulo_nat @ A @ B )
% 0.35/0.62            = A ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unique_euclidean_semiring_numeral_class.mod_less
% 0.35/0.62  thf(fact_4083_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.35/0.62      ! [A: int,B: int] :
% 0.35/0.62        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.35/0.62       => ( ( ord_less_int @ A @ B )
% 0.35/0.62         => ( ( modulo_modulo_int @ A @ B )
% 0.35/0.62            = A ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unique_euclidean_semiring_numeral_class.mod_less
% 0.35/0.62  thf(fact_4084_cong__exp__iff__simps_I2_J,axiom,
% 0.35/0.62      ! [N: num,Q4: num] :
% 0.35/0.62        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
% 0.35/0.62          = zero_zero_nat )
% 0.35/0.62        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q4 ) )
% 0.35/0.62          = zero_zero_nat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % cong_exp_iff_simps(2)
% 0.35/0.62  thf(fact_4085_cong__exp__iff__simps_I2_J,axiom,
% 0.35/0.62      ! [N: num,Q4: num] :
% 0.35/0.62        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
% 0.35/0.62          = zero_zero_int )
% 0.35/0.62        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q4 ) )
% 0.35/0.62          = zero_zero_int ) ) ).
% 0.35/0.62  
% 0.35/0.62  % cong_exp_iff_simps(2)
% 0.35/0.62  thf(fact_4086_cong__exp__iff__simps_I2_J,axiom,
% 0.35/0.62      ! [N: num,Q4: num] :
% 0.35/0.62        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
% 0.35/0.62          = zero_z3403309356797280102nteger )
% 0.35/0.62        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q4 ) )
% 0.35/0.62          = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.62  
% 0.35/0.62  % cong_exp_iff_simps(2)
% 0.35/0.62  thf(fact_4087_cong__exp__iff__simps_I1_J,axiom,
% 0.35/0.62      ! [N: num] :
% 0.35/0.62        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
% 0.35/0.62        = zero_zero_nat ) ).
% 0.35/0.62  
% 0.35/0.62  % cong_exp_iff_simps(1)
% 0.35/0.62  thf(fact_4088_cong__exp__iff__simps_I1_J,axiom,
% 0.35/0.62      ! [N: num] :
% 0.35/0.62        ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
% 0.35/0.62        = zero_zero_int ) ).
% 0.35/0.62  
% 0.35/0.62  % cong_exp_iff_simps(1)
% 0.35/0.62  thf(fact_4089_cong__exp__iff__simps_I1_J,axiom,
% 0.35/0.62      ! [N: num] :
% 0.35/0.62        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
% 0.35/0.62        = zero_z3403309356797280102nteger ) ).
% 0.35/0.62  
% 0.35/0.62  % cong_exp_iff_simps(1)
% 0.35/0.62  thf(fact_4090_mod__le__divisor,axiom,
% 0.35/0.62      ! [N: nat,M: nat] :
% 0.35/0.62        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.62       => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_le_divisor
% 0.35/0.62  thf(fact_4091_div__less__mono,axiom,
% 0.35/0.62      ! [A2: nat,B2: nat,N: nat] :
% 0.35/0.62        ( ( ord_less_nat @ A2 @ B2 )
% 0.35/0.62       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.62         => ( ( ( modulo_modulo_nat @ A2 @ N )
% 0.35/0.62              = zero_zero_nat )
% 0.35/0.62           => ( ( ( modulo_modulo_nat @ B2 @ N )
% 0.35/0.62                = zero_zero_nat )
% 0.35/0.62             => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N ) @ ( divide_divide_nat @ B2 @ N ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % div_less_mono
% 0.35/0.62  thf(fact_4092_mod__eq__nat1E,axiom,
% 0.35/0.62      ! [M: nat,Q4: nat,N: nat] :
% 0.35/0.62        ( ( ( modulo_modulo_nat @ M @ Q4 )
% 0.35/0.62          = ( modulo_modulo_nat @ N @ Q4 ) )
% 0.35/0.62       => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.62         => ~ ! [S4: nat] :
% 0.35/0.62                ( M
% 0.35/0.62               != ( plus_plus_nat @ N @ ( times_times_nat @ Q4 @ S4 ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_eq_nat1E
% 0.35/0.62  thf(fact_4093_mod__eq__nat2E,axiom,
% 0.35/0.62      ! [M: nat,Q4: nat,N: nat] :
% 0.35/0.62        ( ( ( modulo_modulo_nat @ M @ Q4 )
% 0.35/0.62          = ( modulo_modulo_nat @ N @ Q4 ) )
% 0.35/0.62       => ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.62         => ~ ! [S4: nat] :
% 0.35/0.62                ( N
% 0.35/0.62               != ( plus_plus_nat @ M @ ( times_times_nat @ Q4 @ S4 ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_eq_nat2E
% 0.35/0.62  thf(fact_4094_nat__mod__eq__lemma,axiom,
% 0.35/0.62      ! [X: nat,N: nat,Y: nat] :
% 0.35/0.62        ( ( ( modulo_modulo_nat @ X @ N )
% 0.35/0.62          = ( modulo_modulo_nat @ Y @ N ) )
% 0.35/0.62       => ( ( ord_less_eq_nat @ Y @ X )
% 0.35/0.62         => ? [Q3: nat] :
% 0.35/0.62              ( X
% 0.35/0.62              = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q3 ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % nat_mod_eq_lemma
% 0.35/0.62  thf(fact_4095_eucl__rel__int__dividesI,axiom,
% 0.35/0.62      ! [L: int,K: int,Q4: int] :
% 0.35/0.62        ( ( L != zero_zero_int )
% 0.35/0.62       => ( ( K
% 0.35/0.62            = ( times_times_int @ Q4 @ L ) )
% 0.35/0.62         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ zero_zero_int ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % eucl_rel_int_dividesI
% 0.35/0.62  thf(fact_4096_split__mod,axiom,
% 0.35/0.62      ! [P: nat > $o,M: nat,N: nat] :
% 0.35/0.62        ( ( P @ ( modulo_modulo_nat @ M @ N ) )
% 0.35/0.62        = ( ( ( N = zero_zero_nat )
% 0.35/0.62           => ( P @ M ) )
% 0.35/0.62          & ( ( N != zero_zero_nat )
% 0.35/0.62           => ! [I4: nat,J3: nat] :
% 0.35/0.62                ( ( ord_less_nat @ J3 @ N )
% 0.35/0.62               => ( ( M
% 0.35/0.62                    = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 0.35/0.62                 => ( P @ J3 ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % split_mod
% 0.35/0.62  thf(fact_4097_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.35/0.62      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.35/0.62        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.35/0.62       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.35/0.62          = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.35/0.62  thf(fact_4098_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.35/0.62      ! [C: nat,A: nat,B: nat] :
% 0.35/0.62        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.35/0.62       => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.35/0.62          = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.35/0.62  thf(fact_4099_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.35/0.62      ! [C: int,A: int,B: int] :
% 0.35/0.62        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.62       => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 0.35/0.62          = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.35/0.62  thf(fact_4100_Suc__times__mod__eq,axiom,
% 0.35/0.62      ! [M: nat,N: nat] :
% 0.35/0.62        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.35/0.62       => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
% 0.35/0.62          = one_one_nat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % Suc_times_mod_eq
% 0.35/0.62  thf(fact_4101_divmod__digit__0_I2_J,axiom,
% 0.35/0.62      ! [B: nat,A: nat] :
% 0.35/0.62        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.35/0.62       => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.35/0.62         => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
% 0.35/0.62            = ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % divmod_digit_0(2)
% 0.35/0.62  thf(fact_4102_divmod__digit__0_I2_J,axiom,
% 0.35/0.62      ! [B: int,A: int] :
% 0.35/0.62        ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.62       => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.35/0.62         => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
% 0.35/0.62            = ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % divmod_digit_0(2)
% 0.35/0.62  thf(fact_4103_divmod__digit__0_I2_J,axiom,
% 0.35/0.62      ! [B: code_integer,A: code_integer] :
% 0.35/0.62        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.35/0.62       => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.35/0.62         => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
% 0.35/0.62            = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % divmod_digit_0(2)
% 0.35/0.62  thf(fact_4104_bits__stable__imp__add__self,axiom,
% 0.35/0.62      ! [A: nat] :
% 0.35/0.62        ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62          = A )
% 0.35/0.62       => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62          = zero_zero_nat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % bits_stable_imp_add_self
% 0.35/0.62  thf(fact_4105_bits__stable__imp__add__self,axiom,
% 0.35/0.62      ! [A: int] :
% 0.35/0.62        ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.62          = A )
% 0.35/0.62       => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.35/0.62          = zero_zero_int ) ) ).
% 0.35/0.62  
% 0.35/0.62  % bits_stable_imp_add_self
% 0.35/0.62  thf(fact_4106_bits__stable__imp__add__self,axiom,
% 0.35/0.62      ! [A: code_integer] :
% 0.35/0.62        ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.62          = A )
% 0.35/0.62       => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.35/0.62          = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.62  
% 0.35/0.62  % bits_stable_imp_add_self
% 0.35/0.62  thf(fact_4107_verit__le__mono__div,axiom,
% 0.35/0.62      ! [A2: nat,B2: nat,N: nat] :
% 0.35/0.62        ( ( ord_less_nat @ A2 @ B2 )
% 0.35/0.62       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.62         => ( ord_less_eq_nat
% 0.35/0.62            @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N )
% 0.35/0.62              @ ( if_nat
% 0.35/0.62                @ ( ( modulo_modulo_nat @ B2 @ N )
% 0.35/0.62                  = zero_zero_nat )
% 0.35/0.62                @ one_one_nat
% 0.35/0.62                @ zero_zero_nat ) )
% 0.35/0.62            @ ( divide_divide_nat @ B2 @ N ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % verit_le_mono_div
% 0.35/0.62  thf(fact_4108_divmod__digit__0_I1_J,axiom,
% 0.35/0.62      ! [B: nat,A: nat] :
% 0.35/0.62        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.35/0.62       => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.35/0.62         => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.35/0.62            = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % divmod_digit_0(1)
% 0.35/0.62  thf(fact_4109_divmod__digit__0_I1_J,axiom,
% 0.35/0.62      ! [B: int,A: int] :
% 0.35/0.62        ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.62       => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.35/0.62         => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.35/0.62            = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % divmod_digit_0(1)
% 0.35/0.62  thf(fact_4110_divmod__digit__0_I1_J,axiom,
% 0.35/0.62      ! [B: code_integer,A: code_integer] :
% 0.35/0.62        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.35/0.62       => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.35/0.62         => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.35/0.62            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % divmod_digit_0(1)
% 0.35/0.62  thf(fact_4111_mult__exp__mod__exp__eq,axiom,
% 0.35/0.62      ! [M: nat,N: nat,A: nat] :
% 0.35/0.62        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.62       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.62          = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mult_exp_mod_exp_eq
% 0.35/0.62  thf(fact_4112_mult__exp__mod__exp__eq,axiom,
% 0.35/0.62      ! [M: nat,N: nat,A: int] :
% 0.35/0.62        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.62       => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.62          = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mult_exp_mod_exp_eq
% 0.35/0.62  thf(fact_4113_mult__exp__mod__exp__eq,axiom,
% 0.35/0.62      ! [M: nat,N: nat,A: code_integer] :
% 0.35/0.62        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.62       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.62          = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mult_exp_mod_exp_eq
% 0.35/0.62  thf(fact_4114_eucl__rel__int__iff,axiom,
% 0.35/0.62      ! [K: int,L: int,Q4: int,R2: int] :
% 0.35/0.62        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.35/0.62        = ( ( K
% 0.35/0.62            = ( plus_plus_int @ ( times_times_int @ L @ Q4 ) @ R2 ) )
% 0.35/0.62          & ( ( ord_less_int @ zero_zero_int @ L )
% 0.35/0.62           => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.35/0.62              & ( ord_less_int @ R2 @ L ) ) )
% 0.35/0.62          & ( ~ ( ord_less_int @ zero_zero_int @ L )
% 0.35/0.62           => ( ( ( ord_less_int @ L @ zero_zero_int )
% 0.35/0.62               => ( ( ord_less_int @ L @ R2 )
% 0.35/0.62                  & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
% 0.35/0.62              & ( ~ ( ord_less_int @ L @ zero_zero_int )
% 0.35/0.62               => ( Q4 = zero_zero_int ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % eucl_rel_int_iff
% 0.35/0.62  thf(fact_4115_mod__double__modulus,axiom,
% 0.35/0.62      ! [M: code_integer,X: code_integer] :
% 0.35/0.62        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 0.35/0.62       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 0.35/0.62         => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.62              = ( modulo364778990260209775nteger @ X @ M ) )
% 0.35/0.62            | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.62              = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_double_modulus
% 0.35/0.62  thf(fact_4116_mod__double__modulus,axiom,
% 0.35/0.62      ! [M: nat,X: nat] :
% 0.35/0.62        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.62       => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.35/0.62         => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.62              = ( modulo_modulo_nat @ X @ M ) )
% 0.35/0.62            | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.62              = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_double_modulus
% 0.35/0.62  thf(fact_4117_mod__double__modulus,axiom,
% 0.35/0.62      ! [M: int,X: int] :
% 0.35/0.62        ( ( ord_less_int @ zero_zero_int @ M )
% 0.35/0.62       => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.35/0.62         => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.62              = ( modulo_modulo_int @ X @ M ) )
% 0.35/0.62            | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.62              = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_double_modulus
% 0.35/0.62  thf(fact_4118_divmod__digit__1_I2_J,axiom,
% 0.35/0.62      ! [A: code_integer,B: code_integer] :
% 0.35/0.62        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.35/0.62       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.35/0.62         => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.35/0.62           => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.35/0.62              = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % divmod_digit_1(2)
% 0.35/0.62  thf(fact_4119_divmod__digit__1_I2_J,axiom,
% 0.35/0.62      ! [A: nat,B: nat] :
% 0.35/0.62        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.35/0.62       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.35/0.62         => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.35/0.62           => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.35/0.62              = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % divmod_digit_1(2)
% 0.35/0.62  thf(fact_4120_divmod__digit__1_I2_J,axiom,
% 0.35/0.62      ! [A: int,B: int] :
% 0.35/0.62        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.35/0.62       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.62         => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.35/0.62           => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.35/0.62              = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % divmod_digit_1(2)
% 0.35/0.62  thf(fact_4121_divmod__digit__1_I1_J,axiom,
% 0.35/0.62      ! [A: code_integer,B: code_integer] :
% 0.35/0.62        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.35/0.62       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.35/0.62         => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.35/0.62           => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
% 0.35/0.62              = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % divmod_digit_1(1)
% 0.35/0.62  thf(fact_4122_divmod__digit__1_I1_J,axiom,
% 0.35/0.62      ! [A: nat,B: nat] :
% 0.35/0.62        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.35/0.62       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.35/0.62         => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.35/0.62           => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
% 0.35/0.62              = ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % divmod_digit_1(1)
% 0.35/0.62  thf(fact_4123_divmod__digit__1_I1_J,axiom,
% 0.35/0.62      ! [A: int,B: int] :
% 0.35/0.62        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.35/0.62       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.62         => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.35/0.62           => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
% 0.35/0.62              = ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % divmod_digit_1(1)
% 0.35/0.62  thf(fact_4124_mult__less__iff1,axiom,
% 0.35/0.62      ! [Z2: real,X: real,Y: real] :
% 0.35/0.62        ( ( ord_less_real @ zero_zero_real @ Z2 )
% 0.35/0.62       => ( ( ord_less_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
% 0.35/0.62          = ( ord_less_real @ X @ Y ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mult_less_iff1
% 0.35/0.62  thf(fact_4125_mult__less__iff1,axiom,
% 0.35/0.62      ! [Z2: rat,X: rat,Y: rat] :
% 0.35/0.62        ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 0.35/0.62       => ( ( ord_less_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y @ Z2 ) )
% 0.35/0.62          = ( ord_less_rat @ X @ Y ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mult_less_iff1
% 0.35/0.62  thf(fact_4126_mult__less__iff1,axiom,
% 0.35/0.62      ! [Z2: int,X: int,Y: int] :
% 0.35/0.62        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.35/0.62       => ( ( ord_less_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
% 0.35/0.62          = ( ord_less_int @ X @ Y ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mult_less_iff1
% 0.35/0.62  thf(fact_4127_pos__eucl__rel__int__mult__2,axiom,
% 0.35/0.62      ! [B: int,A: int,Q4: int,R2: int] :
% 0.35/0.62        ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.35/0.62       => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.35/0.62         => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % pos_eucl_rel_int_mult_2
% 0.35/0.62  thf(fact_4128_product__nth,axiom,
% 0.35/0.62      ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.35/0.62        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 0.35/0.62       => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) @ N )
% 0.35/0.62          = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % product_nth
% 0.35/0.62  thf(fact_4129_product__nth,axiom,
% 0.35/0.62      ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
% 0.35/0.62        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) )
% 0.35/0.62       => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) @ N )
% 0.35/0.62          = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % product_nth
% 0.35/0.62  thf(fact_4130_product__nth,axiom,
% 0.35/0.62      ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
% 0.35/0.62        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
% 0.35/0.62       => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys ) @ N )
% 0.35/0.62          = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % product_nth
% 0.35/0.62  thf(fact_4131_product__nth,axiom,
% 0.35/0.62      ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
% 0.35/0.62        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) )
% 0.35/0.62       => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) @ N )
% 0.35/0.62          = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % product_nth
% 0.35/0.62  thf(fact_4132_product__nth,axiom,
% 0.35/0.62      ! [N: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
% 0.35/0.62        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 0.35/0.62       => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys ) @ N )
% 0.35/0.62          = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % product_nth
% 0.35/0.62  thf(fact_4133_product__nth,axiom,
% 0.35/0.62      ! [N: nat,Xs: list_o,Ys: list_o] :
% 0.35/0.62        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) )
% 0.35/0.62       => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs @ Ys ) @ N )
% 0.35/0.62          = ( product_Pair_o_o @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % product_nth
% 0.35/0.62  thf(fact_4134_product__nth,axiom,
% 0.35/0.62      ! [N: nat,Xs: list_o,Ys: list_nat] :
% 0.35/0.62        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
% 0.35/0.62       => ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs @ Ys ) @ N )
% 0.35/0.62          = ( product_Pair_o_nat @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % product_nth
% 0.35/0.62  thf(fact_4135_product__nth,axiom,
% 0.35/0.62      ! [N: nat,Xs: list_o,Ys: list_int] :
% 0.35/0.62        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) )
% 0.35/0.62       => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs @ Ys ) @ N )
% 0.35/0.62          = ( product_Pair_o_int @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % product_nth
% 0.35/0.62  thf(fact_4136_product__nth,axiom,
% 0.35/0.62      ! [N: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
% 0.35/0.62        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 0.35/0.62       => ( ( nth_Pr744662078594809490T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys ) @ N )
% 0.35/0.62          = ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % product_nth
% 0.35/0.62  thf(fact_4137_product__nth,axiom,
% 0.35/0.62      ! [N: nat,Xs: list_nat,Ys: list_o] :
% 0.35/0.62        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys ) ) )
% 0.35/0.62       => ( ( nth_Pr112076138515278198_nat_o @ ( product_nat_o @ Xs @ Ys ) @ N )
% 0.35/0.62          = ( product_Pair_nat_o @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % product_nth
% 0.35/0.62  thf(fact_4138_prod_Ofinite__Collect__op,axiom,
% 0.35/0.62      ! [I6: set_real,X: real > complex,Y: real > complex] :
% 0.35/0.62        ( ( finite_finite_real
% 0.35/0.62          @ ( collect_real
% 0.35/0.62            @ ^ [I4: real] :
% 0.35/0.62                ( ( member_real @ I4 @ I6 )
% 0.35/0.62                & ( ( X @ I4 )
% 0.35/0.62                 != one_one_complex ) ) ) )
% 0.35/0.62       => ( ( finite_finite_real
% 0.35/0.62            @ ( collect_real
% 0.35/0.62              @ ^ [I4: real] :
% 0.35/0.62                  ( ( member_real @ I4 @ I6 )
% 0.35/0.62                  & ( ( Y @ I4 )
% 0.35/0.62                   != one_one_complex ) ) ) )
% 0.35/0.62         => ( finite_finite_real
% 0.35/0.62            @ ( collect_real
% 0.35/0.62              @ ^ [I4: real] :
% 0.35/0.62                  ( ( member_real @ I4 @ I6 )
% 0.35/0.62                  & ( ( times_times_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
% 0.35/0.62                   != one_one_complex ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % prod.finite_Collect_op
% 0.35/0.62  thf(fact_4139_prod_Ofinite__Collect__op,axiom,
% 0.35/0.62      ! [I6: set_nat,X: nat > complex,Y: nat > complex] :
% 0.35/0.62        ( ( finite_finite_nat
% 0.35/0.62          @ ( collect_nat
% 0.35/0.62            @ ^ [I4: nat] :
% 0.35/0.62                ( ( member_nat @ I4 @ I6 )
% 0.35/0.62                & ( ( X @ I4 )
% 0.35/0.62                 != one_one_complex ) ) ) )
% 0.35/0.62       => ( ( finite_finite_nat
% 0.35/0.62            @ ( collect_nat
% 0.35/0.62              @ ^ [I4: nat] :
% 0.35/0.62                  ( ( member_nat @ I4 @ I6 )
% 0.35/0.62                  & ( ( Y @ I4 )
% 0.35/0.62                   != one_one_complex ) ) ) )
% 0.35/0.62         => ( finite_finite_nat
% 0.35/0.62            @ ( collect_nat
% 0.35/0.62              @ ^ [I4: nat] :
% 0.35/0.62                  ( ( member_nat @ I4 @ I6 )
% 0.35/0.62                  & ( ( times_times_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
% 0.35/0.62                   != one_one_complex ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % prod.finite_Collect_op
% 0.35/0.62  thf(fact_4140_prod_Ofinite__Collect__op,axiom,
% 0.35/0.62      ! [I6: set_int,X: int > complex,Y: int > complex] :
% 0.35/0.62        ( ( finite_finite_int
% 0.35/0.62          @ ( collect_int
% 0.35/0.62            @ ^ [I4: int] :
% 0.35/0.62                ( ( member_int @ I4 @ I6 )
% 0.35/0.62                & ( ( X @ I4 )
% 0.35/0.62                 != one_one_complex ) ) ) )
% 0.35/0.62       => ( ( finite_finite_int
% 0.35/0.62            @ ( collect_int
% 0.35/0.62              @ ^ [I4: int] :
% 0.35/0.62                  ( ( member_int @ I4 @ I6 )
% 0.35/0.62                  & ( ( Y @ I4 )
% 0.35/0.62                   != one_one_complex ) ) ) )
% 0.35/0.62         => ( finite_finite_int
% 0.35/0.62            @ ( collect_int
% 0.35/0.62              @ ^ [I4: int] :
% 0.35/0.62                  ( ( member_int @ I4 @ I6 )
% 0.35/0.62                  & ( ( times_times_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
% 0.35/0.62                   != one_one_complex ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % prod.finite_Collect_op
% 0.35/0.62  thf(fact_4141_prod_Ofinite__Collect__op,axiom,
% 0.35/0.62      ! [I6: set_complex,X: complex > complex,Y: complex > complex] :
% 0.35/0.62        ( ( finite3207457112153483333omplex
% 0.35/0.62          @ ( collect_complex
% 0.35/0.62            @ ^ [I4: complex] :
% 0.35/0.62                ( ( member_complex @ I4 @ I6 )
% 0.35/0.62                & ( ( X @ I4 )
% 0.35/0.62                 != one_one_complex ) ) ) )
% 0.35/0.62       => ( ( finite3207457112153483333omplex
% 0.35/0.62            @ ( collect_complex
% 0.35/0.62              @ ^ [I4: complex] :
% 0.35/0.62                  ( ( member_complex @ I4 @ I6 )
% 0.35/0.62                  & ( ( Y @ I4 )
% 0.35/0.62                   != one_one_complex ) ) ) )
% 0.35/0.62         => ( finite3207457112153483333omplex
% 0.35/0.62            @ ( collect_complex
% 0.35/0.62              @ ^ [I4: complex] :
% 0.35/0.62                  ( ( member_complex @ I4 @ I6 )
% 0.35/0.62                  & ( ( times_times_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
% 0.35/0.62                   != one_one_complex ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % prod.finite_Collect_op
% 0.35/0.62  thf(fact_4142_prod_Ofinite__Collect__op,axiom,
% 0.35/0.62      ! [I6: set_real,X: real > real,Y: real > real] :
% 0.35/0.62        ( ( finite_finite_real
% 0.35/0.62          @ ( collect_real
% 0.35/0.62            @ ^ [I4: real] :
% 0.35/0.62                ( ( member_real @ I4 @ I6 )
% 0.35/0.62                & ( ( X @ I4 )
% 0.35/0.62                 != one_one_real ) ) ) )
% 0.35/0.62       => ( ( finite_finite_real
% 0.35/0.62            @ ( collect_real
% 0.35/0.62              @ ^ [I4: real] :
% 0.35/0.62                  ( ( member_real @ I4 @ I6 )
% 0.35/0.62                  & ( ( Y @ I4 )
% 0.35/0.62                   != one_one_real ) ) ) )
% 0.35/0.62         => ( finite_finite_real
% 0.35/0.62            @ ( collect_real
% 0.35/0.62              @ ^ [I4: real] :
% 0.35/0.62                  ( ( member_real @ I4 @ I6 )
% 0.35/0.62                  & ( ( times_times_real @ ( X @ I4 ) @ ( Y @ I4 ) )
% 0.35/0.62                   != one_one_real ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % prod.finite_Collect_op
% 0.35/0.62  thf(fact_4143_prod_Ofinite__Collect__op,axiom,
% 0.35/0.62      ! [I6: set_nat,X: nat > real,Y: nat > real] :
% 0.35/0.62        ( ( finite_finite_nat
% 0.35/0.62          @ ( collect_nat
% 0.35/0.62            @ ^ [I4: nat] :
% 0.35/0.62                ( ( member_nat @ I4 @ I6 )
% 0.35/0.62                & ( ( X @ I4 )
% 0.35/0.62                 != one_one_real ) ) ) )
% 0.35/0.62       => ( ( finite_finite_nat
% 0.35/0.62            @ ( collect_nat
% 0.35/0.62              @ ^ [I4: nat] :
% 0.35/0.62                  ( ( member_nat @ I4 @ I6 )
% 0.35/0.62                  & ( ( Y @ I4 )
% 0.35/0.62                   != one_one_real ) ) ) )
% 0.35/0.62         => ( finite_finite_nat
% 0.35/0.62            @ ( collect_nat
% 0.35/0.62              @ ^ [I4: nat] :
% 0.35/0.62                  ( ( member_nat @ I4 @ I6 )
% 0.35/0.62                  & ( ( times_times_real @ ( X @ I4 ) @ ( Y @ I4 ) )
% 0.35/0.62                   != one_one_real ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % prod.finite_Collect_op
% 0.35/0.62  thf(fact_4144_prod_Ofinite__Collect__op,axiom,
% 0.35/0.62      ! [I6: set_int,X: int > real,Y: int > real] :
% 0.35/0.62        ( ( finite_finite_int
% 0.35/0.62          @ ( collect_int
% 0.35/0.62            @ ^ [I4: int] :
% 0.35/0.62                ( ( member_int @ I4 @ I6 )
% 0.35/0.62                & ( ( X @ I4 )
% 0.35/0.62                 != one_one_real ) ) ) )
% 0.35/0.62       => ( ( finite_finite_int
% 0.35/0.62            @ ( collect_int
% 0.35/0.62              @ ^ [I4: int] :
% 0.35/0.62                  ( ( member_int @ I4 @ I6 )
% 0.35/0.62                  & ( ( Y @ I4 )
% 0.35/0.62                   != one_one_real ) ) ) )
% 0.35/0.62         => ( finite_finite_int
% 0.35/0.62            @ ( collect_int
% 0.35/0.62              @ ^ [I4: int] :
% 0.35/0.62                  ( ( member_int @ I4 @ I6 )
% 0.35/0.62                  & ( ( times_times_real @ ( X @ I4 ) @ ( Y @ I4 ) )
% 0.35/0.62                   != one_one_real ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % prod.finite_Collect_op
% 0.35/0.62  thf(fact_4145_prod_Ofinite__Collect__op,axiom,
% 0.35/0.62      ! [I6: set_complex,X: complex > real,Y: complex > real] :
% 0.35/0.62        ( ( finite3207457112153483333omplex
% 0.35/0.62          @ ( collect_complex
% 0.35/0.62            @ ^ [I4: complex] :
% 0.35/0.62                ( ( member_complex @ I4 @ I6 )
% 0.35/0.62                & ( ( X @ I4 )
% 0.35/0.62                 != one_one_real ) ) ) )
% 0.35/0.62       => ( ( finite3207457112153483333omplex
% 0.35/0.62            @ ( collect_complex
% 0.35/0.62              @ ^ [I4: complex] :
% 0.35/0.62                  ( ( member_complex @ I4 @ I6 )
% 0.35/0.62                  & ( ( Y @ I4 )
% 0.35/0.62                   != one_one_real ) ) ) )
% 0.35/0.62         => ( finite3207457112153483333omplex
% 0.35/0.62            @ ( collect_complex
% 0.35/0.62              @ ^ [I4: complex] :
% 0.35/0.62                  ( ( member_complex @ I4 @ I6 )
% 0.35/0.62                  & ( ( times_times_real @ ( X @ I4 ) @ ( Y @ I4 ) )
% 0.35/0.62                   != one_one_real ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % prod.finite_Collect_op
% 0.35/0.62  thf(fact_4146_prod_Ofinite__Collect__op,axiom,
% 0.35/0.62      ! [I6: set_real,X: real > rat,Y: real > rat] :
% 0.35/0.62        ( ( finite_finite_real
% 0.35/0.62          @ ( collect_real
% 0.35/0.62            @ ^ [I4: real] :
% 0.35/0.62                ( ( member_real @ I4 @ I6 )
% 0.35/0.62                & ( ( X @ I4 )
% 0.35/0.62                 != one_one_rat ) ) ) )
% 0.35/0.62       => ( ( finite_finite_real
% 0.35/0.62            @ ( collect_real
% 0.35/0.62              @ ^ [I4: real] :
% 0.35/0.62                  ( ( member_real @ I4 @ I6 )
% 0.35/0.62                  & ( ( Y @ I4 )
% 0.35/0.62                   != one_one_rat ) ) ) )
% 0.35/0.62         => ( finite_finite_real
% 0.35/0.62            @ ( collect_real
% 0.35/0.62              @ ^ [I4: real] :
% 0.35/0.62                  ( ( member_real @ I4 @ I6 )
% 0.35/0.62                  & ( ( times_times_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
% 0.35/0.62                   != one_one_rat ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % prod.finite_Collect_op
% 0.35/0.62  thf(fact_4147_prod_Ofinite__Collect__op,axiom,
% 0.35/0.62      ! [I6: set_nat,X: nat > rat,Y: nat > rat] :
% 0.35/0.62        ( ( finite_finite_nat
% 0.35/0.62          @ ( collect_nat
% 0.35/0.62            @ ^ [I4: nat] :
% 0.35/0.62                ( ( member_nat @ I4 @ I6 )
% 0.35/0.62                & ( ( X @ I4 )
% 0.35/0.62                 != one_one_rat ) ) ) )
% 0.35/0.62       => ( ( finite_finite_nat
% 0.35/0.62            @ ( collect_nat
% 0.35/0.62              @ ^ [I4: nat] :
% 0.35/0.62                  ( ( member_nat @ I4 @ I6 )
% 0.35/0.62                  & ( ( Y @ I4 )
% 0.35/0.62                   != one_one_rat ) ) ) )
% 0.35/0.62         => ( finite_finite_nat
% 0.35/0.62            @ ( collect_nat
% 0.35/0.62              @ ^ [I4: nat] :
% 0.35/0.62                  ( ( member_nat @ I4 @ I6 )
% 0.35/0.62                  & ( ( times_times_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
% 0.35/0.62                   != one_one_rat ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % prod.finite_Collect_op
% 0.35/0.62  thf(fact_4148_sum_Ofinite__Collect__op,axiom,
% 0.35/0.62      ! [I6: set_real,X: real > complex,Y: real > complex] :
% 0.35/0.62        ( ( finite_finite_real
% 0.35/0.62          @ ( collect_real
% 0.35/0.62            @ ^ [I4: real] :
% 0.35/0.62                ( ( member_real @ I4 @ I6 )
% 0.35/0.62                & ( ( X @ I4 )
% 0.35/0.62                 != zero_zero_complex ) ) ) )
% 0.35/0.62       => ( ( finite_finite_real
% 0.35/0.62            @ ( collect_real
% 0.35/0.62              @ ^ [I4: real] :
% 0.35/0.62                  ( ( member_real @ I4 @ I6 )
% 0.35/0.62                  & ( ( Y @ I4 )
% 0.35/0.62                   != zero_zero_complex ) ) ) )
% 0.35/0.62         => ( finite_finite_real
% 0.35/0.62            @ ( collect_real
% 0.35/0.62              @ ^ [I4: real] :
% 0.35/0.62                  ( ( member_real @ I4 @ I6 )
% 0.35/0.62                  & ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
% 0.35/0.62                   != zero_zero_complex ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % sum.finite_Collect_op
% 0.35/0.62  thf(fact_4149_sum_Ofinite__Collect__op,axiom,
% 0.35/0.62      ! [I6: set_nat,X: nat > complex,Y: nat > complex] :
% 0.35/0.62        ( ( finite_finite_nat
% 0.35/0.62          @ ( collect_nat
% 0.35/0.62            @ ^ [I4: nat] :
% 0.35/0.62                ( ( member_nat @ I4 @ I6 )
% 0.35/0.62                & ( ( X @ I4 )
% 0.35/0.62                 != zero_zero_complex ) ) ) )
% 0.35/0.62       => ( ( finite_finite_nat
% 0.35/0.62            @ ( collect_nat
% 0.35/0.62              @ ^ [I4: nat] :
% 0.35/0.62                  ( ( member_nat @ I4 @ I6 )
% 0.35/0.62                  & ( ( Y @ I4 )
% 0.35/0.62                   != zero_zero_complex ) ) ) )
% 0.35/0.62         => ( finite_finite_nat
% 0.35/0.62            @ ( collect_nat
% 0.35/0.62              @ ^ [I4: nat] :
% 0.35/0.62                  ( ( member_nat @ I4 @ I6 )
% 0.35/0.62                  & ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
% 0.35/0.62                   != zero_zero_complex ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % sum.finite_Collect_op
% 0.35/0.62  thf(fact_4150_sum_Ofinite__Collect__op,axiom,
% 0.35/0.62      ! [I6: set_int,X: int > complex,Y: int > complex] :
% 0.35/0.62        ( ( finite_finite_int
% 0.35/0.62          @ ( collect_int
% 0.35/0.62            @ ^ [I4: int] :
% 0.35/0.62                ( ( member_int @ I4 @ I6 )
% 0.35/0.62                & ( ( X @ I4 )
% 0.35/0.62                 != zero_zero_complex ) ) ) )
% 0.35/0.62       => ( ( finite_finite_int
% 0.35/0.62            @ ( collect_int
% 0.35/0.62              @ ^ [I4: int] :
% 0.35/0.62                  ( ( member_int @ I4 @ I6 )
% 0.35/0.62                  & ( ( Y @ I4 )
% 0.35/0.62                   != zero_zero_complex ) ) ) )
% 0.35/0.62         => ( finite_finite_int
% 0.35/0.62            @ ( collect_int
% 0.35/0.62              @ ^ [I4: int] :
% 0.35/0.62                  ( ( member_int @ I4 @ I6 )
% 0.35/0.62                  & ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
% 0.35/0.62                   != zero_zero_complex ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % sum.finite_Collect_op
% 0.35/0.62  thf(fact_4151_sum_Ofinite__Collect__op,axiom,
% 0.35/0.62      ! [I6: set_complex,X: complex > complex,Y: complex > complex] :
% 0.35/0.62        ( ( finite3207457112153483333omplex
% 0.35/0.62          @ ( collect_complex
% 0.35/0.62            @ ^ [I4: complex] :
% 0.35/0.62                ( ( member_complex @ I4 @ I6 )
% 0.35/0.62                & ( ( X @ I4 )
% 0.35/0.62                 != zero_zero_complex ) ) ) )
% 0.35/0.62       => ( ( finite3207457112153483333omplex
% 0.35/0.62            @ ( collect_complex
% 0.35/0.62              @ ^ [I4: complex] :
% 0.35/0.62                  ( ( member_complex @ I4 @ I6 )
% 0.35/0.62                  & ( ( Y @ I4 )
% 0.35/0.62                   != zero_zero_complex ) ) ) )
% 0.35/0.62         => ( finite3207457112153483333omplex
% 0.35/0.62            @ ( collect_complex
% 0.35/0.62              @ ^ [I4: complex] :
% 0.35/0.62                  ( ( member_complex @ I4 @ I6 )
% 0.35/0.62                  & ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
% 0.35/0.62                   != zero_zero_complex ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % sum.finite_Collect_op
% 0.35/0.62  thf(fact_4152_sum_Ofinite__Collect__op,axiom,
% 0.35/0.62      ! [I6: set_real,X: real > real,Y: real > real] :
% 0.35/0.62        ( ( finite_finite_real
% 0.35/0.62          @ ( collect_real
% 0.35/0.62            @ ^ [I4: real] :
% 0.35/0.62                ( ( member_real @ I4 @ I6 )
% 0.35/0.62                & ( ( X @ I4 )
% 0.35/0.62                 != zero_zero_real ) ) ) )
% 0.35/0.62       => ( ( finite_finite_real
% 0.35/0.62            @ ( collect_real
% 0.35/0.62              @ ^ [I4: real] :
% 0.35/0.62                  ( ( member_real @ I4 @ I6 )
% 0.35/0.62                  & ( ( Y @ I4 )
% 0.35/0.62                   != zero_zero_real ) ) ) )
% 0.35/0.62         => ( finite_finite_real
% 0.35/0.62            @ ( collect_real
% 0.35/0.62              @ ^ [I4: real] :
% 0.35/0.62                  ( ( member_real @ I4 @ I6 )
% 0.35/0.62                  & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y @ I4 ) )
% 0.35/0.62                   != zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % sum.finite_Collect_op
% 0.35/0.62  thf(fact_4153_sum_Ofinite__Collect__op,axiom,
% 0.35/0.62      ! [I6: set_nat,X: nat > real,Y: nat > real] :
% 0.35/0.62        ( ( finite_finite_nat
% 0.35/0.62          @ ( collect_nat
% 0.35/0.62            @ ^ [I4: nat] :
% 0.35/0.62                ( ( member_nat @ I4 @ I6 )
% 0.35/0.62                & ( ( X @ I4 )
% 0.35/0.62                 != zero_zero_real ) ) ) )
% 0.35/0.62       => ( ( finite_finite_nat
% 0.35/0.62            @ ( collect_nat
% 0.35/0.62              @ ^ [I4: nat] :
% 0.35/0.62                  ( ( member_nat @ I4 @ I6 )
% 0.35/0.62                  & ( ( Y @ I4 )
% 0.35/0.62                   != zero_zero_real ) ) ) )
% 0.35/0.62         => ( finite_finite_nat
% 0.35/0.62            @ ( collect_nat
% 0.35/0.62              @ ^ [I4: nat] :
% 0.35/0.62                  ( ( member_nat @ I4 @ I6 )
% 0.35/0.62                  & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y @ I4 ) )
% 0.35/0.62                   != zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % sum.finite_Collect_op
% 0.35/0.62  thf(fact_4154_sum_Ofinite__Collect__op,axiom,
% 0.35/0.62      ! [I6: set_int,X: int > real,Y: int > real] :
% 0.35/0.62        ( ( finite_finite_int
% 0.35/0.62          @ ( collect_int
% 0.35/0.62            @ ^ [I4: int] :
% 0.35/0.62                ( ( member_int @ I4 @ I6 )
% 0.35/0.62                & ( ( X @ I4 )
% 0.35/0.62                 != zero_zero_real ) ) ) )
% 0.35/0.62       => ( ( finite_finite_int
% 0.35/0.62            @ ( collect_int
% 0.35/0.62              @ ^ [I4: int] :
% 0.35/0.62                  ( ( member_int @ I4 @ I6 )
% 0.35/0.62                  & ( ( Y @ I4 )
% 0.35/0.62                   != zero_zero_real ) ) ) )
% 0.35/0.62         => ( finite_finite_int
% 0.35/0.62            @ ( collect_int
% 0.35/0.62              @ ^ [I4: int] :
% 0.35/0.62                  ( ( member_int @ I4 @ I6 )
% 0.35/0.62                  & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y @ I4 ) )
% 0.35/0.62                   != zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % sum.finite_Collect_op
% 0.35/0.62  thf(fact_4155_sum_Ofinite__Collect__op,axiom,
% 0.35/0.62      ! [I6: set_complex,X: complex > real,Y: complex > real] :
% 0.35/0.62        ( ( finite3207457112153483333omplex
% 0.35/0.62          @ ( collect_complex
% 0.35/0.62            @ ^ [I4: complex] :
% 0.35/0.62                ( ( member_complex @ I4 @ I6 )
% 0.35/0.62                & ( ( X @ I4 )
% 0.35/0.62                 != zero_zero_real ) ) ) )
% 0.35/0.62       => ( ( finite3207457112153483333omplex
% 0.35/0.62            @ ( collect_complex
% 0.35/0.62              @ ^ [I4: complex] :
% 0.35/0.62                  ( ( member_complex @ I4 @ I6 )
% 0.35/0.62                  & ( ( Y @ I4 )
% 0.35/0.62                   != zero_zero_real ) ) ) )
% 0.35/0.62         => ( finite3207457112153483333omplex
% 0.35/0.62            @ ( collect_complex
% 0.35/0.62              @ ^ [I4: complex] :
% 0.35/0.62                  ( ( member_complex @ I4 @ I6 )
% 0.35/0.62                  & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y @ I4 ) )
% 0.35/0.62                   != zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % sum.finite_Collect_op
% 0.35/0.62  thf(fact_4156_sum_Ofinite__Collect__op,axiom,
% 0.35/0.62      ! [I6: set_real,X: real > rat,Y: real > rat] :
% 0.35/0.62        ( ( finite_finite_real
% 0.35/0.62          @ ( collect_real
% 0.35/0.62            @ ^ [I4: real] :
% 0.35/0.62                ( ( member_real @ I4 @ I6 )
% 0.35/0.62                & ( ( X @ I4 )
% 0.35/0.62                 != zero_zero_rat ) ) ) )
% 0.35/0.62       => ( ( finite_finite_real
% 0.35/0.62            @ ( collect_real
% 0.35/0.62              @ ^ [I4: real] :
% 0.35/0.62                  ( ( member_real @ I4 @ I6 )
% 0.35/0.62                  & ( ( Y @ I4 )
% 0.35/0.62                   != zero_zero_rat ) ) ) )
% 0.35/0.62         => ( finite_finite_real
% 0.35/0.62            @ ( collect_real
% 0.35/0.62              @ ^ [I4: real] :
% 0.35/0.62                  ( ( member_real @ I4 @ I6 )
% 0.35/0.62                  & ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
% 0.35/0.62                   != zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % sum.finite_Collect_op
% 0.35/0.62  thf(fact_4157_sum_Ofinite__Collect__op,axiom,
% 0.35/0.62      ! [I6: set_nat,X: nat > rat,Y: nat > rat] :
% 0.35/0.62        ( ( finite_finite_nat
% 0.35/0.62          @ ( collect_nat
% 0.35/0.62            @ ^ [I4: nat] :
% 0.35/0.62                ( ( member_nat @ I4 @ I6 )
% 0.35/0.62                & ( ( X @ I4 )
% 0.35/0.62                 != zero_zero_rat ) ) ) )
% 0.35/0.62       => ( ( finite_finite_nat
% 0.35/0.62            @ ( collect_nat
% 0.35/0.62              @ ^ [I4: nat] :
% 0.35/0.62                  ( ( member_nat @ I4 @ I6 )
% 0.35/0.62                  & ( ( Y @ I4 )
% 0.35/0.62                   != zero_zero_rat ) ) ) )
% 0.35/0.62         => ( finite_finite_nat
% 0.35/0.62            @ ( collect_nat
% 0.35/0.62              @ ^ [I4: nat] :
% 0.35/0.62                  ( ( member_nat @ I4 @ I6 )
% 0.35/0.62                  & ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
% 0.35/0.62                   != zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % sum.finite_Collect_op
% 0.35/0.62  thf(fact_4158_even__succ__mod__exp,axiom,
% 0.35/0.62      ! [A: nat,N: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.35/0.62       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.62         => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.62            = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % even_succ_mod_exp
% 0.35/0.62  thf(fact_4159_even__succ__mod__exp,axiom,
% 0.35/0.62      ! [A: int,N: nat] :
% 0.35/0.62        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.35/0.62       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.62         => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.62            = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % even_succ_mod_exp
% 0.35/0.62  thf(fact_4160_even__succ__mod__exp,axiom,
% 0.35/0.62      ! [A: code_integer,N: nat] :
% 0.35/0.62        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.35/0.62       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.62         => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.62            = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % even_succ_mod_exp
% 0.35/0.62  thf(fact_4161_Bolzano,axiom,
% 0.35/0.62      ! [A: real,B: real,P: real > real > $o] :
% 0.35/0.62        ( ( ord_less_eq_real @ A @ B )
% 0.35/0.62       => ( ! [A3: real,B3: real,C3: real] :
% 0.35/0.62              ( ( P @ A3 @ B3 )
% 0.35/0.62             => ( ( P @ B3 @ C3 )
% 0.35/0.62               => ( ( ord_less_eq_real @ A3 @ B3 )
% 0.35/0.62                 => ( ( ord_less_eq_real @ B3 @ C3 )
% 0.35/0.62                   => ( P @ A3 @ C3 ) ) ) ) )
% 0.35/0.62         => ( ! [X4: real] :
% 0.35/0.62                ( ( ord_less_eq_real @ A @ X4 )
% 0.35/0.62               => ( ( ord_less_eq_real @ X4 @ B )
% 0.35/0.62                 => ? [D3: real] :
% 0.35/0.62                      ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.35/0.62                      & ! [A3: real,B3: real] :
% 0.35/0.62                          ( ( ( ord_less_eq_real @ A3 @ X4 )
% 0.35/0.62                            & ( ord_less_eq_real @ X4 @ B3 )
% 0.35/0.62                            & ( ord_less_real @ ( minus_minus_real @ B3 @ A3 ) @ D3 ) )
% 0.35/0.62                         => ( P @ A3 @ B3 ) ) ) ) )
% 0.35/0.62           => ( P @ A @ B ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % Bolzano
% 0.35/0.62  thf(fact_4162_nat__dvd__1__iff__1,axiom,
% 0.35/0.62      ! [M: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ M @ one_one_nat )
% 0.35/0.62        = ( M = one_one_nat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % nat_dvd_1_iff_1
% 0.35/0.62  thf(fact_4163_dvd__0__right,axiom,
% 0.35/0.62      ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_0_right
% 0.35/0.62  thf(fact_4164_dvd__0__right,axiom,
% 0.35/0.62      ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_0_right
% 0.35/0.62  thf(fact_4165_dvd__0__right,axiom,
% 0.35/0.62      ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_0_right
% 0.35/0.62  thf(fact_4166_dvd__0__right,axiom,
% 0.35/0.62      ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_0_right
% 0.35/0.62  thf(fact_4167_dvd__0__right,axiom,
% 0.35/0.62      ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_0_right
% 0.35/0.62  thf(fact_4168_dvd__0__right,axiom,
% 0.35/0.62      ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_0_right
% 0.35/0.62  thf(fact_4169_dvd__0__left__iff,axiom,
% 0.35/0.62      ! [A: code_integer] :
% 0.35/0.62        ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 0.35/0.62        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_0_left_iff
% 0.35/0.62  thf(fact_4170_dvd__0__left__iff,axiom,
% 0.35/0.62      ! [A: complex] :
% 0.35/0.62        ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 0.35/0.62        = ( A = zero_zero_complex ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_0_left_iff
% 0.35/0.62  thf(fact_4171_dvd__0__left__iff,axiom,
% 0.35/0.62      ! [A: real] :
% 0.35/0.62        ( ( dvd_dvd_real @ zero_zero_real @ A )
% 0.35/0.62        = ( A = zero_zero_real ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_0_left_iff
% 0.35/0.62  thf(fact_4172_dvd__0__left__iff,axiom,
% 0.35/0.62      ! [A: rat] :
% 0.35/0.62        ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 0.35/0.62        = ( A = zero_zero_rat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_0_left_iff
% 0.35/0.62  thf(fact_4173_dvd__0__left__iff,axiom,
% 0.35/0.62      ! [A: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.35/0.62        = ( A = zero_zero_nat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_0_left_iff
% 0.35/0.62  thf(fact_4174_dvd__0__left__iff,axiom,
% 0.35/0.62      ! [A: int] :
% 0.35/0.62        ( ( dvd_dvd_int @ zero_zero_int @ A )
% 0.35/0.62        = ( A = zero_zero_int ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_0_left_iff
% 0.35/0.62  thf(fact_4175_dvd__1__iff__1,axiom,
% 0.35/0.62      ! [M: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.35/0.62        = ( M
% 0.35/0.62          = ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_1_iff_1
% 0.35/0.62  thf(fact_4176_dvd__1__left,axiom,
% 0.35/0.62      ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_1_left
% 0.35/0.62  thf(fact_4177_nat__mult__dvd__cancel__disj,axiom,
% 0.35/0.62      ! [K: nat,M: nat,N: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.35/0.62        = ( ( K = zero_zero_nat )
% 0.35/0.62          | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % nat_mult_dvd_cancel_disj
% 0.35/0.62  thf(fact_4178_dvd__mult__cancel__left,axiom,
% 0.35/0.62      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.35/0.62        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.35/0.62        = ( ( C = zero_z3403309356797280102nteger )
% 0.35/0.62          | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_mult_cancel_left
% 0.35/0.62  thf(fact_4179_dvd__mult__cancel__left,axiom,
% 0.35/0.62      ! [C: complex,A: complex,B: complex] :
% 0.35/0.62        ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.35/0.62        = ( ( C = zero_zero_complex )
% 0.35/0.62          | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_mult_cancel_left
% 0.35/0.62  thf(fact_4180_dvd__mult__cancel__left,axiom,
% 0.35/0.62      ! [C: real,A: real,B: real] :
% 0.35/0.62        ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.35/0.62        = ( ( C = zero_zero_real )
% 0.35/0.62          | ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_mult_cancel_left
% 0.35/0.62  thf(fact_4181_dvd__mult__cancel__left,axiom,
% 0.35/0.62      ! [C: rat,A: rat,B: rat] :
% 0.35/0.62        ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.35/0.62        = ( ( C = zero_zero_rat )
% 0.35/0.62          | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_mult_cancel_left
% 0.35/0.62  thf(fact_4182_dvd__mult__cancel__left,axiom,
% 0.35/0.62      ! [C: int,A: int,B: int] :
% 0.35/0.62        ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.35/0.62        = ( ( C = zero_zero_int )
% 0.35/0.62          | ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_mult_cancel_left
% 0.35/0.62  thf(fact_4183_dvd__mult__cancel__right,axiom,
% 0.35/0.62      ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.35/0.62        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.35/0.62        = ( ( C = zero_z3403309356797280102nteger )
% 0.35/0.62          | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_mult_cancel_right
% 0.35/0.62  thf(fact_4184_dvd__mult__cancel__right,axiom,
% 0.35/0.62      ! [A: complex,C: complex,B: complex] :
% 0.35/0.62        ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 0.35/0.62        = ( ( C = zero_zero_complex )
% 0.35/0.62          | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_mult_cancel_right
% 0.35/0.62  thf(fact_4185_dvd__mult__cancel__right,axiom,
% 0.35/0.62      ! [A: real,C: real,B: real] :
% 0.35/0.62        ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.35/0.62        = ( ( C = zero_zero_real )
% 0.35/0.62          | ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_mult_cancel_right
% 0.35/0.62  thf(fact_4186_dvd__mult__cancel__right,axiom,
% 0.35/0.62      ! [A: rat,C: rat,B: rat] :
% 0.35/0.62        ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.35/0.62        = ( ( C = zero_zero_rat )
% 0.35/0.62          | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_mult_cancel_right
% 0.35/0.62  thf(fact_4187_dvd__mult__cancel__right,axiom,
% 0.35/0.62      ! [A: int,C: int,B: int] :
% 0.35/0.62        ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.35/0.62        = ( ( C = zero_zero_int )
% 0.35/0.62          | ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_mult_cancel_right
% 0.35/0.62  thf(fact_4188_dvd__times__left__cancel__iff,axiom,
% 0.35/0.62      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.35/0.62        ( ( A != zero_z3403309356797280102nteger )
% 0.35/0.62       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
% 0.35/0.62          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_times_left_cancel_iff
% 0.35/0.62  thf(fact_4189_dvd__times__left__cancel__iff,axiom,
% 0.35/0.62      ! [A: nat,B: nat,C: nat] :
% 0.35/0.62        ( ( A != zero_zero_nat )
% 0.35/0.62       => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
% 0.35/0.62          = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_times_left_cancel_iff
% 0.35/0.62  thf(fact_4190_dvd__times__left__cancel__iff,axiom,
% 0.35/0.62      ! [A: int,B: int,C: int] :
% 0.35/0.62        ( ( A != zero_zero_int )
% 0.35/0.62       => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
% 0.35/0.62          = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_times_left_cancel_iff
% 0.35/0.62  thf(fact_4191_dvd__times__right__cancel__iff,axiom,
% 0.35/0.62      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.35/0.62        ( ( A != zero_z3403309356797280102nteger )
% 0.35/0.62       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
% 0.35/0.62          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_times_right_cancel_iff
% 0.35/0.62  thf(fact_4192_dvd__times__right__cancel__iff,axiom,
% 0.35/0.62      ! [A: nat,B: nat,C: nat] :
% 0.35/0.62        ( ( A != zero_zero_nat )
% 0.35/0.62       => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
% 0.35/0.62          = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_times_right_cancel_iff
% 0.35/0.62  thf(fact_4193_dvd__times__right__cancel__iff,axiom,
% 0.35/0.62      ! [A: int,B: int,C: int] :
% 0.35/0.62        ( ( A != zero_zero_int )
% 0.35/0.62       => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
% 0.35/0.62          = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_times_right_cancel_iff
% 0.35/0.62  thf(fact_4194_dvd__imp__mod__0,axiom,
% 0.35/0.62      ! [A: nat,B: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ A @ B )
% 0.35/0.62       => ( ( modulo_modulo_nat @ B @ A )
% 0.35/0.62          = zero_zero_nat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_imp_mod_0
% 0.35/0.62  thf(fact_4195_dvd__imp__mod__0,axiom,
% 0.35/0.62      ! [A: int,B: int] :
% 0.35/0.62        ( ( dvd_dvd_int @ A @ B )
% 0.35/0.62       => ( ( modulo_modulo_int @ B @ A )
% 0.35/0.62          = zero_zero_int ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_imp_mod_0
% 0.35/0.62  thf(fact_4196_dvd__imp__mod__0,axiom,
% 0.35/0.62      ! [A: code_integer,B: code_integer] :
% 0.35/0.62        ( ( dvd_dvd_Code_integer @ A @ B )
% 0.35/0.62       => ( ( modulo364778990260209775nteger @ B @ A )
% 0.35/0.62          = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_imp_mod_0
% 0.35/0.62  thf(fact_4197_mod__neg__neg__trivial,axiom,
% 0.35/0.62      ! [K: int,L: int] :
% 0.35/0.62        ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.35/0.62       => ( ( ord_less_int @ L @ K )
% 0.35/0.62         => ( ( modulo_modulo_int @ K @ L )
% 0.35/0.62            = K ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_neg_neg_trivial
% 0.35/0.62  thf(fact_4198_mod__pos__pos__trivial,axiom,
% 0.35/0.62      ! [K: int,L: int] :
% 0.35/0.62        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.62       => ( ( ord_less_int @ K @ L )
% 0.35/0.62         => ( ( modulo_modulo_int @ K @ L )
% 0.35/0.62            = K ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_pos_pos_trivial
% 0.35/0.62  thf(fact_4199_pow__divides__pow__iff,axiom,
% 0.35/0.62      ! [N: nat,A: nat,B: nat] :
% 0.35/0.62        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.62       => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.35/0.62          = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % pow_divides_pow_iff
% 0.35/0.62  thf(fact_4200_pow__divides__pow__iff,axiom,
% 0.35/0.62      ! [N: nat,A: int,B: int] :
% 0.35/0.62        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.62       => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.35/0.62          = ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % pow_divides_pow_iff
% 0.35/0.62  thf(fact_4201_length__product,axiom,
% 0.35/0.62      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.35/0.62        ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) )
% 0.35/0.62        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % length_product
% 0.35/0.62  thf(fact_4202_length__product,axiom,
% 0.35/0.62      ! [Xs: list_VEBT_VEBT,Ys: list_o] :
% 0.35/0.62        ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) )
% 0.35/0.62        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % length_product
% 0.35/0.62  thf(fact_4203_length__product,axiom,
% 0.35/0.62      ! [Xs: list_VEBT_VEBT,Ys: list_nat] :
% 0.35/0.62        ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys ) )
% 0.35/0.62        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % length_product
% 0.35/0.62  thf(fact_4204_length__product,axiom,
% 0.35/0.62      ! [Xs: list_VEBT_VEBT,Ys: list_int] :
% 0.35/0.62        ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) )
% 0.35/0.62        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % length_product
% 0.35/0.62  thf(fact_4205_length__product,axiom,
% 0.35/0.62      ! [Xs: list_o,Ys: list_VEBT_VEBT] :
% 0.35/0.62        ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys ) )
% 0.35/0.62        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % length_product
% 0.35/0.62  thf(fact_4206_length__product,axiom,
% 0.35/0.62      ! [Xs: list_o,Ys: list_o] :
% 0.35/0.62        ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs @ Ys ) )
% 0.35/0.62        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % length_product
% 0.35/0.62  thf(fact_4207_length__product,axiom,
% 0.35/0.62      ! [Xs: list_o,Ys: list_nat] :
% 0.35/0.62        ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs @ Ys ) )
% 0.35/0.62        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % length_product
% 0.35/0.62  thf(fact_4208_length__product,axiom,
% 0.35/0.62      ! [Xs: list_o,Ys: list_int] :
% 0.35/0.62        ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs @ Ys ) )
% 0.35/0.62        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % length_product
% 0.35/0.62  thf(fact_4209_length__product,axiom,
% 0.35/0.62      ! [Xs: list_nat,Ys: list_VEBT_VEBT] :
% 0.35/0.62        ( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys ) )
% 0.35/0.62        = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % length_product
% 0.35/0.62  thf(fact_4210_length__product,axiom,
% 0.35/0.62      ! [Xs: list_nat,Ys: list_o] :
% 0.35/0.62        ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs @ Ys ) )
% 0.35/0.62        = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % length_product
% 0.35/0.62  thf(fact_4211_zero__le__power__eq__numeral,axiom,
% 0.35/0.62      ! [A: real,W2: num] :
% 0.35/0.62        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 0.35/0.62        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62            & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % zero_le_power_eq_numeral
% 0.35/0.62  thf(fact_4212_zero__le__power__eq__numeral,axiom,
% 0.35/0.62      ! [A: rat,W2: num] :
% 0.35/0.62        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 0.35/0.62        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62            & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % zero_le_power_eq_numeral
% 0.35/0.62  thf(fact_4213_zero__le__power__eq__numeral,axiom,
% 0.35/0.62      ! [A: int,W2: num] :
% 0.35/0.62        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 0.35/0.62        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62            & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % zero_le_power_eq_numeral
% 0.35/0.62  thf(fact_4214_power__less__zero__eq__numeral,axiom,
% 0.35/0.62      ! [A: real,W2: num] :
% 0.35/0.62        ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
% 0.35/0.62        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62          & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % power_less_zero_eq_numeral
% 0.35/0.62  thf(fact_4215_power__less__zero__eq__numeral,axiom,
% 0.35/0.62      ! [A: rat,W2: num] :
% 0.35/0.62        ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
% 0.35/0.62        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62          & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % power_less_zero_eq_numeral
% 0.35/0.62  thf(fact_4216_power__less__zero__eq__numeral,axiom,
% 0.35/0.62      ! [A: int,W2: num] :
% 0.35/0.62        ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
% 0.35/0.62        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62          & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % power_less_zero_eq_numeral
% 0.35/0.62  thf(fact_4217_power__less__zero__eq,axiom,
% 0.35/0.62      ! [A: real,N: nat] :
% 0.35/0.62        ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 0.35/0.62        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.62          & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % power_less_zero_eq
% 0.35/0.62  thf(fact_4218_power__less__zero__eq,axiom,
% 0.35/0.62      ! [A: rat,N: nat] :
% 0.35/0.62        ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 0.35/0.62        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.62          & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % power_less_zero_eq
% 0.35/0.62  thf(fact_4219_power__less__zero__eq,axiom,
% 0.35/0.62      ! [A: int,N: nat] :
% 0.35/0.62        ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 0.35/0.62        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.62          & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % power_less_zero_eq
% 0.35/0.62  thf(fact_4220_odd__Suc__minus__one,axiom,
% 0.35/0.62      ! [N: nat] :
% 0.35/0.62        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.62       => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.35/0.62          = N ) ) ).
% 0.35/0.62  
% 0.35/0.62  % odd_Suc_minus_one
% 0.35/0.62  thf(fact_4221_even__diff__nat,axiom,
% 0.35/0.62      ! [M: nat,N: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.62        = ( ( ord_less_nat @ M @ N )
% 0.35/0.62          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % even_diff_nat
% 0.35/0.62  thf(fact_4222_zero__less__power__eq__numeral,axiom,
% 0.35/0.62      ! [A: real,W2: num] :
% 0.35/0.62        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 0.35/0.62        = ( ( ( numeral_numeral_nat @ W2 )
% 0.35/0.62            = zero_zero_nat )
% 0.35/0.62          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62            & ( A != zero_zero_real ) )
% 0.35/0.62          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62            & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % zero_less_power_eq_numeral
% 0.35/0.62  thf(fact_4223_zero__less__power__eq__numeral,axiom,
% 0.35/0.62      ! [A: rat,W2: num] :
% 0.35/0.62        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 0.35/0.62        = ( ( ( numeral_numeral_nat @ W2 )
% 0.35/0.62            = zero_zero_nat )
% 0.35/0.62          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62            & ( A != zero_zero_rat ) )
% 0.35/0.62          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62            & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % zero_less_power_eq_numeral
% 0.35/0.62  thf(fact_4224_zero__less__power__eq__numeral,axiom,
% 0.35/0.62      ! [A: int,W2: num] :
% 0.35/0.62        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 0.35/0.62        = ( ( ( numeral_numeral_nat @ W2 )
% 0.35/0.62            = zero_zero_nat )
% 0.35/0.62          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62            & ( A != zero_zero_int ) )
% 0.35/0.62          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62            & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % zero_less_power_eq_numeral
% 0.35/0.62  thf(fact_4225_even__power,axiom,
% 0.35/0.62      ! [A: code_integer,N: nat] :
% 0.35/0.62        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
% 0.35/0.62        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.35/0.62          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % even_power
% 0.35/0.62  thf(fact_4226_even__power,axiom,
% 0.35/0.62      ! [A: nat,N: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
% 0.35/0.62        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.35/0.62          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % even_power
% 0.35/0.62  thf(fact_4227_even__power,axiom,
% 0.35/0.62      ! [A: int,N: nat] :
% 0.35/0.62        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
% 0.35/0.62        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.35/0.62          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % even_power
% 0.35/0.62  thf(fact_4228_power__le__zero__eq__numeral,axiom,
% 0.35/0.62      ! [A: real,W2: num] :
% 0.35/0.62        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
% 0.35/0.62        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62              & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.35/0.62            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62              & ( A = zero_zero_real ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % power_le_zero_eq_numeral
% 0.35/0.62  thf(fact_4229_power__le__zero__eq__numeral,axiom,
% 0.35/0.62      ! [A: rat,W2: num] :
% 0.35/0.62        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
% 0.35/0.62        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62              & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.35/0.62            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62              & ( A = zero_zero_rat ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % power_le_zero_eq_numeral
% 0.35/0.62  thf(fact_4230_power__le__zero__eq__numeral,axiom,
% 0.35/0.62      ! [A: int,W2: num] :
% 0.35/0.62        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
% 0.35/0.62        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62              & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.35/0.62            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.35/0.62              & ( A = zero_zero_int ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % power_le_zero_eq_numeral
% 0.35/0.62  thf(fact_4231_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.35/0.62      ! [N: nat] :
% 0.35/0.62        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
% 0.35/0.62        = ( N = zero_zero_nat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % semiring_parity_class.even_mask_iff
% 0.35/0.62  thf(fact_4232_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.35/0.62      ! [N: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
% 0.35/0.62        = ( N = zero_zero_nat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % semiring_parity_class.even_mask_iff
% 0.35/0.62  thf(fact_4233_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.35/0.62      ! [N: nat] :
% 0.35/0.62        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 0.35/0.62        = ( N = zero_zero_nat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % semiring_parity_class.even_mask_iff
% 0.35/0.62  thf(fact_4234_even__succ__div__exp,axiom,
% 0.35/0.62      ! [A: nat,N: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.35/0.62       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.62         => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.62            = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % even_succ_div_exp
% 0.35/0.62  thf(fact_4235_even__succ__div__exp,axiom,
% 0.35/0.62      ! [A: int,N: nat] :
% 0.35/0.62        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.35/0.62       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.62         => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.62            = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % even_succ_div_exp
% 0.35/0.62  thf(fact_4236_even__succ__div__exp,axiom,
% 0.35/0.62      ! [A: code_integer,N: nat] :
% 0.35/0.62        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.35/0.62       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.62         => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.62            = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % even_succ_div_exp
% 0.35/0.62  thf(fact_4237_dvd__field__iff,axiom,
% 0.35/0.62      ( dvd_dvd_complex
% 0.35/0.62      = ( ^ [A4: complex,B4: complex] :
% 0.35/0.62            ( ( A4 = zero_zero_complex )
% 0.35/0.62           => ( B4 = zero_zero_complex ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_field_iff
% 0.35/0.62  thf(fact_4238_dvd__field__iff,axiom,
% 0.35/0.62      ( dvd_dvd_real
% 0.35/0.62      = ( ^ [A4: real,B4: real] :
% 0.35/0.62            ( ( A4 = zero_zero_real )
% 0.35/0.62           => ( B4 = zero_zero_real ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_field_iff
% 0.35/0.62  thf(fact_4239_dvd__field__iff,axiom,
% 0.35/0.62      ( dvd_dvd_rat
% 0.35/0.62      = ( ^ [A4: rat,B4: rat] :
% 0.35/0.62            ( ( A4 = zero_zero_rat )
% 0.35/0.62           => ( B4 = zero_zero_rat ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_field_iff
% 0.35/0.62  thf(fact_4240_dvd__0__left,axiom,
% 0.35/0.62      ! [A: code_integer] :
% 0.35/0.62        ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 0.35/0.62       => ( A = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_0_left
% 0.35/0.62  thf(fact_4241_dvd__0__left,axiom,
% 0.35/0.62      ! [A: complex] :
% 0.35/0.62        ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 0.35/0.62       => ( A = zero_zero_complex ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_0_left
% 0.35/0.62  thf(fact_4242_dvd__0__left,axiom,
% 0.35/0.62      ! [A: real] :
% 0.35/0.62        ( ( dvd_dvd_real @ zero_zero_real @ A )
% 0.35/0.62       => ( A = zero_zero_real ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_0_left
% 0.35/0.62  thf(fact_4243_dvd__0__left,axiom,
% 0.35/0.62      ! [A: rat] :
% 0.35/0.62        ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 0.35/0.62       => ( A = zero_zero_rat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_0_left
% 0.35/0.62  thf(fact_4244_dvd__0__left,axiom,
% 0.35/0.62      ! [A: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.35/0.62       => ( A = zero_zero_nat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_0_left
% 0.35/0.62  thf(fact_4245_dvd__0__left,axiom,
% 0.35/0.62      ! [A: int] :
% 0.35/0.62        ( ( dvd_dvd_int @ zero_zero_int @ A )
% 0.35/0.62       => ( A = zero_zero_int ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_0_left
% 0.35/0.62  thf(fact_4246_gcd__nat_Oextremum,axiom,
% 0.35/0.62      ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 0.35/0.62  
% 0.35/0.62  % gcd_nat.extremum
% 0.35/0.62  thf(fact_4247_gcd__nat_Oextremum__strict,axiom,
% 0.35/0.62      ! [A: nat] :
% 0.35/0.62        ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.35/0.62          & ( zero_zero_nat != A ) ) ).
% 0.35/0.62  
% 0.35/0.62  % gcd_nat.extremum_strict
% 0.35/0.62  thf(fact_4248_gcd__nat_Oextremum__unique,axiom,
% 0.35/0.62      ! [A: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.35/0.62        = ( A = zero_zero_nat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % gcd_nat.extremum_unique
% 0.35/0.62  thf(fact_4249_gcd__nat_Onot__eq__extremum,axiom,
% 0.35/0.62      ! [A: nat] :
% 0.35/0.62        ( ( A != zero_zero_nat )
% 0.35/0.62        = ( ( dvd_dvd_nat @ A @ zero_zero_nat )
% 0.35/0.62          & ( A != zero_zero_nat ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % gcd_nat.not_eq_extremum
% 0.35/0.62  thf(fact_4250_gcd__nat_Oextremum__uniqueI,axiom,
% 0.35/0.62      ! [A: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.35/0.62       => ( A = zero_zero_nat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % gcd_nat.extremum_uniqueI
% 0.35/0.62  thf(fact_4251_dvd__diff__nat,axiom,
% 0.35/0.62      ! [K: nat,M: nat,N: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ K @ M )
% 0.35/0.62       => ( ( dvd_dvd_nat @ K @ N )
% 0.35/0.62         => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_diff_nat
% 0.35/0.62  thf(fact_4252_mod__int__pos__iff,axiom,
% 0.35/0.62      ! [K: int,L: int] :
% 0.35/0.62        ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
% 0.35/0.62        = ( ( dvd_dvd_int @ L @ K )
% 0.35/0.62          | ( ( L = zero_zero_int )
% 0.35/0.62            & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 0.35/0.62          | ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_int_pos_iff
% 0.35/0.62  thf(fact_4253_subset__divisors__dvd,axiom,
% 0.35/0.62      ! [A: real,B: real] :
% 0.35/0.62        ( ( ord_less_eq_set_real
% 0.35/0.62          @ ( collect_real
% 0.35/0.62            @ ^ [C4: real] : ( dvd_dvd_real @ C4 @ A ) )
% 0.35/0.62          @ ( collect_real
% 0.35/0.62            @ ^ [C4: real] : ( dvd_dvd_real @ C4 @ B ) ) )
% 0.35/0.62        = ( dvd_dvd_real @ A @ B ) ) ).
% 0.35/0.62  
% 0.35/0.62  % subset_divisors_dvd
% 0.35/0.62  thf(fact_4254_subset__divisors__dvd,axiom,
% 0.35/0.62      ! [A: nat,B: nat] :
% 0.35/0.62        ( ( ord_less_eq_set_nat
% 0.35/0.62          @ ( collect_nat
% 0.35/0.62            @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ A ) )
% 0.35/0.62          @ ( collect_nat
% 0.35/0.62            @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ B ) ) )
% 0.35/0.62        = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.35/0.62  
% 0.35/0.62  % subset_divisors_dvd
% 0.35/0.62  thf(fact_4255_subset__divisors__dvd,axiom,
% 0.35/0.62      ! [A: code_integer,B: code_integer] :
% 0.35/0.62        ( ( ord_le7084787975880047091nteger
% 0.35/0.62          @ ( collect_Code_integer
% 0.35/0.62            @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ A ) )
% 0.35/0.62          @ ( collect_Code_integer
% 0.35/0.62            @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ B ) ) )
% 0.35/0.62        = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.35/0.62  
% 0.35/0.62  % subset_divisors_dvd
% 0.35/0.62  thf(fact_4256_subset__divisors__dvd,axiom,
% 0.35/0.62      ! [A: int,B: int] :
% 0.35/0.62        ( ( ord_less_eq_set_int
% 0.35/0.62          @ ( collect_int
% 0.35/0.62            @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ A ) )
% 0.35/0.62          @ ( collect_int
% 0.35/0.62            @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ B ) ) )
% 0.35/0.62        = ( dvd_dvd_int @ A @ B ) ) ).
% 0.35/0.62  
% 0.35/0.62  % subset_divisors_dvd
% 0.35/0.62  thf(fact_4257_strict__subset__divisors__dvd,axiom,
% 0.35/0.62      ! [A: real,B: real] :
% 0.35/0.62        ( ( ord_less_set_real
% 0.35/0.62          @ ( collect_real
% 0.35/0.62            @ ^ [C4: real] : ( dvd_dvd_real @ C4 @ A ) )
% 0.35/0.62          @ ( collect_real
% 0.35/0.62            @ ^ [C4: real] : ( dvd_dvd_real @ C4 @ B ) ) )
% 0.35/0.62        = ( ( dvd_dvd_real @ A @ B )
% 0.35/0.62          & ~ ( dvd_dvd_real @ B @ A ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % strict_subset_divisors_dvd
% 0.35/0.62  thf(fact_4258_strict__subset__divisors__dvd,axiom,
% 0.35/0.62      ! [A: nat,B: nat] :
% 0.35/0.62        ( ( ord_less_set_nat
% 0.35/0.62          @ ( collect_nat
% 0.35/0.62            @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ A ) )
% 0.35/0.62          @ ( collect_nat
% 0.35/0.62            @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ B ) ) )
% 0.35/0.62        = ( ( dvd_dvd_nat @ A @ B )
% 0.35/0.62          & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % strict_subset_divisors_dvd
% 0.35/0.62  thf(fact_4259_strict__subset__divisors__dvd,axiom,
% 0.35/0.62      ! [A: int,B: int] :
% 0.35/0.62        ( ( ord_less_set_int
% 0.35/0.62          @ ( collect_int
% 0.35/0.62            @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ A ) )
% 0.35/0.62          @ ( collect_int
% 0.35/0.62            @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ B ) ) )
% 0.35/0.62        = ( ( dvd_dvd_int @ A @ B )
% 0.35/0.62          & ~ ( dvd_dvd_int @ B @ A ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % strict_subset_divisors_dvd
% 0.35/0.62  thf(fact_4260_strict__subset__divisors__dvd,axiom,
% 0.35/0.62      ! [A: code_integer,B: code_integer] :
% 0.35/0.62        ( ( ord_le1307284697595431911nteger
% 0.35/0.62          @ ( collect_Code_integer
% 0.35/0.62            @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ A ) )
% 0.35/0.62          @ ( collect_Code_integer
% 0.35/0.62            @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ B ) ) )
% 0.35/0.62        = ( ( dvd_dvd_Code_integer @ A @ B )
% 0.35/0.62          & ~ ( dvd_dvd_Code_integer @ B @ A ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % strict_subset_divisors_dvd
% 0.35/0.62  thf(fact_4261_not__is__unit__0,axiom,
% 0.35/0.62      ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 0.35/0.62  
% 0.35/0.62  % not_is_unit_0
% 0.35/0.62  thf(fact_4262_not__is__unit__0,axiom,
% 0.35/0.62      ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 0.35/0.62  
% 0.35/0.62  % not_is_unit_0
% 0.35/0.62  thf(fact_4263_not__is__unit__0,axiom,
% 0.35/0.62      ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 0.35/0.62  
% 0.35/0.62  % not_is_unit_0
% 0.35/0.62  thf(fact_4264_pinf_I9_J,axiom,
% 0.35/0.62      ! [D: code_integer,S: code_integer] :
% 0.35/0.62      ? [Z4: code_integer] :
% 0.35/0.62      ! [X2: code_integer] :
% 0.35/0.62        ( ( ord_le6747313008572928689nteger @ Z4 @ X2 )
% 0.35/0.62       => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) )
% 0.35/0.62          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % pinf(9)
% 0.35/0.62  thf(fact_4265_pinf_I9_J,axiom,
% 0.35/0.62      ! [D: real,S: real] :
% 0.35/0.62      ? [Z4: real] :
% 0.35/0.62      ! [X2: real] :
% 0.35/0.62        ( ( ord_less_real @ Z4 @ X2 )
% 0.35/0.62       => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) )
% 0.35/0.62          = ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % pinf(9)
% 0.35/0.62  thf(fact_4266_pinf_I9_J,axiom,
% 0.35/0.62      ! [D: rat,S: rat] :
% 0.35/0.62      ? [Z4: rat] :
% 0.35/0.62      ! [X2: rat] :
% 0.35/0.62        ( ( ord_less_rat @ Z4 @ X2 )
% 0.35/0.62       => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) )
% 0.35/0.62          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % pinf(9)
% 0.35/0.62  thf(fact_4267_pinf_I9_J,axiom,
% 0.35/0.62      ! [D: nat,S: nat] :
% 0.35/0.62      ? [Z4: nat] :
% 0.35/0.62      ! [X2: nat] :
% 0.35/0.62        ( ( ord_less_nat @ Z4 @ X2 )
% 0.35/0.62       => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) )
% 0.35/0.62          = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % pinf(9)
% 0.35/0.62  thf(fact_4268_pinf_I9_J,axiom,
% 0.35/0.62      ! [D: int,S: int] :
% 0.35/0.62      ? [Z4: int] :
% 0.35/0.62      ! [X2: int] :
% 0.35/0.62        ( ( ord_less_int @ Z4 @ X2 )
% 0.35/0.62       => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) )
% 0.35/0.62          = ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % pinf(9)
% 0.35/0.62  thf(fact_4269_pinf_I10_J,axiom,
% 0.35/0.62      ! [D: code_integer,S: code_integer] :
% 0.35/0.62      ? [Z4: code_integer] :
% 0.35/0.62      ! [X2: code_integer] :
% 0.35/0.62        ( ( ord_le6747313008572928689nteger @ Z4 @ X2 )
% 0.35/0.62       => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) )
% 0.35/0.62          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % pinf(10)
% 0.35/0.62  thf(fact_4270_pinf_I10_J,axiom,
% 0.35/0.62      ! [D: real,S: real] :
% 0.35/0.62      ? [Z4: real] :
% 0.35/0.62      ! [X2: real] :
% 0.35/0.62        ( ( ord_less_real @ Z4 @ X2 )
% 0.35/0.62       => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) )
% 0.35/0.62          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % pinf(10)
% 0.35/0.62  thf(fact_4271_pinf_I10_J,axiom,
% 0.35/0.62      ! [D: rat,S: rat] :
% 0.35/0.62      ? [Z4: rat] :
% 0.35/0.62      ! [X2: rat] :
% 0.35/0.62        ( ( ord_less_rat @ Z4 @ X2 )
% 0.35/0.62       => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) )
% 0.35/0.62          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % pinf(10)
% 0.35/0.62  thf(fact_4272_pinf_I10_J,axiom,
% 0.35/0.62      ! [D: nat,S: nat] :
% 0.35/0.62      ? [Z4: nat] :
% 0.35/0.62      ! [X2: nat] :
% 0.35/0.62        ( ( ord_less_nat @ Z4 @ X2 )
% 0.35/0.62       => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) )
% 0.35/0.62          = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % pinf(10)
% 0.35/0.62  thf(fact_4273_pinf_I10_J,axiom,
% 0.35/0.62      ! [D: int,S: int] :
% 0.35/0.62      ? [Z4: int] :
% 0.35/0.62      ! [X2: int] :
% 0.35/0.62        ( ( ord_less_int @ Z4 @ X2 )
% 0.35/0.62       => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) )
% 0.35/0.62          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % pinf(10)
% 0.35/0.62  thf(fact_4274_minf_I9_J,axiom,
% 0.35/0.62      ! [D: code_integer,S: code_integer] :
% 0.35/0.62      ? [Z4: code_integer] :
% 0.35/0.62      ! [X2: code_integer] :
% 0.35/0.62        ( ( ord_le6747313008572928689nteger @ X2 @ Z4 )
% 0.35/0.62       => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) )
% 0.35/0.62          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % minf(9)
% 0.35/0.62  thf(fact_4275_minf_I9_J,axiom,
% 0.35/0.62      ! [D: real,S: real] :
% 0.35/0.62      ? [Z4: real] :
% 0.35/0.62      ! [X2: real] :
% 0.35/0.62        ( ( ord_less_real @ X2 @ Z4 )
% 0.35/0.62       => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) )
% 0.35/0.62          = ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % minf(9)
% 0.35/0.62  thf(fact_4276_minf_I9_J,axiom,
% 0.35/0.62      ! [D: rat,S: rat] :
% 0.35/0.62      ? [Z4: rat] :
% 0.35/0.62      ! [X2: rat] :
% 0.35/0.62        ( ( ord_less_rat @ X2 @ Z4 )
% 0.35/0.62       => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) )
% 0.35/0.62          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % minf(9)
% 0.35/0.62  thf(fact_4277_minf_I9_J,axiom,
% 0.35/0.62      ! [D: nat,S: nat] :
% 0.35/0.62      ? [Z4: nat] :
% 0.35/0.62      ! [X2: nat] :
% 0.35/0.62        ( ( ord_less_nat @ X2 @ Z4 )
% 0.35/0.62       => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) )
% 0.35/0.62          = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % minf(9)
% 0.35/0.62  thf(fact_4278_minf_I9_J,axiom,
% 0.35/0.62      ! [D: int,S: int] :
% 0.35/0.62      ? [Z4: int] :
% 0.35/0.62      ! [X2: int] :
% 0.35/0.62        ( ( ord_less_int @ X2 @ Z4 )
% 0.35/0.62       => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) )
% 0.35/0.62          = ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % minf(9)
% 0.35/0.62  thf(fact_4279_minf_I10_J,axiom,
% 0.35/0.62      ! [D: code_integer,S: code_integer] :
% 0.35/0.62      ? [Z4: code_integer] :
% 0.35/0.62      ! [X2: code_integer] :
% 0.35/0.62        ( ( ord_le6747313008572928689nteger @ X2 @ Z4 )
% 0.35/0.62       => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) )
% 0.35/0.62          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % minf(10)
% 0.35/0.62  thf(fact_4280_minf_I10_J,axiom,
% 0.35/0.62      ! [D: real,S: real] :
% 0.35/0.62      ? [Z4: real] :
% 0.35/0.62      ! [X2: real] :
% 0.35/0.62        ( ( ord_less_real @ X2 @ Z4 )
% 0.35/0.62       => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) )
% 0.35/0.62          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % minf(10)
% 0.35/0.62  thf(fact_4281_minf_I10_J,axiom,
% 0.35/0.62      ! [D: rat,S: rat] :
% 0.35/0.62      ? [Z4: rat] :
% 0.35/0.62      ! [X2: rat] :
% 0.35/0.62        ( ( ord_less_rat @ X2 @ Z4 )
% 0.35/0.62       => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) )
% 0.35/0.62          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % minf(10)
% 0.35/0.62  thf(fact_4282_minf_I10_J,axiom,
% 0.35/0.62      ! [D: nat,S: nat] :
% 0.35/0.62      ? [Z4: nat] :
% 0.35/0.62      ! [X2: nat] :
% 0.35/0.62        ( ( ord_less_nat @ X2 @ Z4 )
% 0.35/0.62       => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) )
% 0.35/0.62          = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % minf(10)
% 0.35/0.62  thf(fact_4283_minf_I10_J,axiom,
% 0.35/0.62      ! [D: int,S: int] :
% 0.35/0.62      ? [Z4: int] :
% 0.35/0.62      ! [X2: int] :
% 0.35/0.62        ( ( ord_less_int @ X2 @ Z4 )
% 0.35/0.62       => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) )
% 0.35/0.62          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % minf(10)
% 0.35/0.62  thf(fact_4284_dvd__div__eq__0__iff,axiom,
% 0.35/0.62      ! [B: complex,A: complex] :
% 0.35/0.62        ( ( dvd_dvd_complex @ B @ A )
% 0.35/0.62       => ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.35/0.62            = zero_zero_complex )
% 0.35/0.62          = ( A = zero_zero_complex ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_div_eq_0_iff
% 0.35/0.62  thf(fact_4285_dvd__div__eq__0__iff,axiom,
% 0.35/0.62      ! [B: nat,A: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ B @ A )
% 0.35/0.62       => ( ( ( divide_divide_nat @ A @ B )
% 0.35/0.62            = zero_zero_nat )
% 0.35/0.62          = ( A = zero_zero_nat ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_div_eq_0_iff
% 0.35/0.62  thf(fact_4286_dvd__div__eq__0__iff,axiom,
% 0.35/0.62      ! [B: int,A: int] :
% 0.35/0.62        ( ( dvd_dvd_int @ B @ A )
% 0.35/0.62       => ( ( ( divide_divide_int @ A @ B )
% 0.35/0.62            = zero_zero_int )
% 0.35/0.62          = ( A = zero_zero_int ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_div_eq_0_iff
% 0.35/0.62  thf(fact_4287_dvd__div__eq__0__iff,axiom,
% 0.35/0.62      ! [B: real,A: real] :
% 0.35/0.62        ( ( dvd_dvd_real @ B @ A )
% 0.35/0.62       => ( ( ( divide_divide_real @ A @ B )
% 0.35/0.62            = zero_zero_real )
% 0.35/0.62          = ( A = zero_zero_real ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_div_eq_0_iff
% 0.35/0.62  thf(fact_4288_dvd__div__eq__0__iff,axiom,
% 0.35/0.62      ! [B: rat,A: rat] :
% 0.35/0.62        ( ( dvd_dvd_rat @ B @ A )
% 0.35/0.62       => ( ( ( divide_divide_rat @ A @ B )
% 0.35/0.62            = zero_zero_rat )
% 0.35/0.62          = ( A = zero_zero_rat ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_div_eq_0_iff
% 0.35/0.62  thf(fact_4289_dvd__div__eq__0__iff,axiom,
% 0.35/0.62      ! [B: code_integer,A: code_integer] :
% 0.35/0.62        ( ( dvd_dvd_Code_integer @ B @ A )
% 0.35/0.62       => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.35/0.62            = zero_z3403309356797280102nteger )
% 0.35/0.62          = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_div_eq_0_iff
% 0.35/0.62  thf(fact_4290_mod__0__imp__dvd,axiom,
% 0.35/0.62      ! [A: nat,B: nat] :
% 0.35/0.62        ( ( ( modulo_modulo_nat @ A @ B )
% 0.35/0.62          = zero_zero_nat )
% 0.35/0.62       => ( dvd_dvd_nat @ B @ A ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_0_imp_dvd
% 0.35/0.62  thf(fact_4291_mod__0__imp__dvd,axiom,
% 0.35/0.62      ! [A: int,B: int] :
% 0.35/0.62        ( ( ( modulo_modulo_int @ A @ B )
% 0.35/0.62          = zero_zero_int )
% 0.35/0.62       => ( dvd_dvd_int @ B @ A ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_0_imp_dvd
% 0.35/0.62  thf(fact_4292_mod__0__imp__dvd,axiom,
% 0.35/0.62      ! [A: code_integer,B: code_integer] :
% 0.35/0.62        ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.35/0.62          = zero_z3403309356797280102nteger )
% 0.35/0.62       => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_0_imp_dvd
% 0.35/0.62  thf(fact_4293_dvd__eq__mod__eq__0,axiom,
% 0.35/0.62      ( dvd_dvd_nat
% 0.35/0.62      = ( ^ [A4: nat,B4: nat] :
% 0.35/0.62            ( ( modulo_modulo_nat @ B4 @ A4 )
% 0.35/0.62            = zero_zero_nat ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_eq_mod_eq_0
% 0.35/0.62  thf(fact_4294_dvd__eq__mod__eq__0,axiom,
% 0.35/0.62      ( dvd_dvd_int
% 0.35/0.62      = ( ^ [A4: int,B4: int] :
% 0.35/0.62            ( ( modulo_modulo_int @ B4 @ A4 )
% 0.35/0.62            = zero_zero_int ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_eq_mod_eq_0
% 0.35/0.62  thf(fact_4295_dvd__eq__mod__eq__0,axiom,
% 0.35/0.62      ( dvd_dvd_Code_integer
% 0.35/0.62      = ( ^ [A4: code_integer,B4: code_integer] :
% 0.35/0.62            ( ( modulo364778990260209775nteger @ B4 @ A4 )
% 0.35/0.62            = zero_z3403309356797280102nteger ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_eq_mod_eq_0
% 0.35/0.62  thf(fact_4296_mod__eq__0__iff__dvd,axiom,
% 0.35/0.62      ! [A: nat,B: nat] :
% 0.35/0.62        ( ( ( modulo_modulo_nat @ A @ B )
% 0.35/0.62          = zero_zero_nat )
% 0.35/0.62        = ( dvd_dvd_nat @ B @ A ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_eq_0_iff_dvd
% 0.35/0.62  thf(fact_4297_mod__eq__0__iff__dvd,axiom,
% 0.35/0.62      ! [A: int,B: int] :
% 0.35/0.62        ( ( ( modulo_modulo_int @ A @ B )
% 0.35/0.62          = zero_zero_int )
% 0.35/0.62        = ( dvd_dvd_int @ B @ A ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_eq_0_iff_dvd
% 0.35/0.62  thf(fact_4298_mod__eq__0__iff__dvd,axiom,
% 0.35/0.62      ! [A: code_integer,B: code_integer] :
% 0.35/0.62        ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.35/0.62          = zero_z3403309356797280102nteger )
% 0.35/0.62        = ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_eq_0_iff_dvd
% 0.35/0.62  thf(fact_4299_dvd__power__le,axiom,
% 0.35/0.62      ! [X: code_integer,Y: code_integer,N: nat,M: nat] :
% 0.35/0.62        ( ( dvd_dvd_Code_integer @ X @ Y )
% 0.35/0.62       => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.62         => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ M ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_power_le
% 0.35/0.62  thf(fact_4300_dvd__power__le,axiom,
% 0.35/0.62      ! [X: nat,Y: nat,N: nat,M: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ X @ Y )
% 0.35/0.62       => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.62         => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ M ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_power_le
% 0.35/0.62  thf(fact_4301_dvd__power__le,axiom,
% 0.35/0.62      ! [X: real,Y: real,N: nat,M: nat] :
% 0.35/0.62        ( ( dvd_dvd_real @ X @ Y )
% 0.35/0.62       => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.62         => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ M ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_power_le
% 0.35/0.62  thf(fact_4302_dvd__power__le,axiom,
% 0.35/0.62      ! [X: int,Y: int,N: nat,M: nat] :
% 0.35/0.62        ( ( dvd_dvd_int @ X @ Y )
% 0.35/0.62       => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.62         => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ M ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_power_le
% 0.35/0.62  thf(fact_4303_dvd__power__le,axiom,
% 0.35/0.62      ! [X: complex,Y: complex,N: nat,M: nat] :
% 0.35/0.62        ( ( dvd_dvd_complex @ X @ Y )
% 0.35/0.62       => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.62         => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ M ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_power_le
% 0.35/0.62  thf(fact_4304_power__le__dvd,axiom,
% 0.35/0.62      ! [A: code_integer,N: nat,B: code_integer,M: nat] :
% 0.35/0.62        ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
% 0.35/0.62       => ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.62         => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % power_le_dvd
% 0.35/0.62  thf(fact_4305_power__le__dvd,axiom,
% 0.35/0.62      ! [A: nat,N: nat,B: nat,M: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
% 0.35/0.62       => ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.62         => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % power_le_dvd
% 0.35/0.62  thf(fact_4306_power__le__dvd,axiom,
% 0.35/0.62      ! [A: real,N: nat,B: real,M: nat] :
% 0.35/0.62        ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
% 0.35/0.62       => ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.62         => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % power_le_dvd
% 0.35/0.62  thf(fact_4307_power__le__dvd,axiom,
% 0.35/0.62      ! [A: int,N: nat,B: int,M: nat] :
% 0.35/0.62        ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
% 0.35/0.62       => ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.62         => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % power_le_dvd
% 0.35/0.62  thf(fact_4308_power__le__dvd,axiom,
% 0.35/0.62      ! [A: complex,N: nat,B: complex,M: nat] :
% 0.35/0.62        ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
% 0.35/0.62       => ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.62         => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % power_le_dvd
% 0.35/0.62  thf(fact_4309_le__imp__power__dvd,axiom,
% 0.35/0.62      ! [M: nat,N: nat,A: code_integer] :
% 0.35/0.62        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.62       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % le_imp_power_dvd
% 0.35/0.62  thf(fact_4310_le__imp__power__dvd,axiom,
% 0.35/0.62      ! [M: nat,N: nat,A: nat] :
% 0.35/0.62        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.62       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % le_imp_power_dvd
% 0.35/0.62  thf(fact_4311_le__imp__power__dvd,axiom,
% 0.35/0.62      ! [M: nat,N: nat,A: real] :
% 0.35/0.62        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.62       => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % le_imp_power_dvd
% 0.35/0.62  thf(fact_4312_le__imp__power__dvd,axiom,
% 0.35/0.62      ! [M: nat,N: nat,A: int] :
% 0.35/0.62        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.62       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % le_imp_power_dvd
% 0.35/0.62  thf(fact_4313_le__imp__power__dvd,axiom,
% 0.35/0.62      ! [M: nat,N: nat,A: complex] :
% 0.35/0.62        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.62       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % le_imp_power_dvd
% 0.35/0.62  thf(fact_4314_Euclidean__Division_Opos__mod__bound,axiom,
% 0.35/0.62      ! [L: int,K: int] :
% 0.35/0.62        ( ( ord_less_int @ zero_zero_int @ L )
% 0.35/0.62       => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% 0.35/0.62  
% 0.35/0.62  % Euclidean_Division.pos_mod_bound
% 0.35/0.62  thf(fact_4315_neg__mod__bound,axiom,
% 0.35/0.62      ! [L: int,K: int] :
% 0.35/0.62        ( ( ord_less_int @ L @ zero_zero_int )
% 0.35/0.62       => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % neg_mod_bound
% 0.35/0.62  thf(fact_4316_nat__dvd__not__less,axiom,
% 0.35/0.62      ! [M: nat,N: nat] :
% 0.35/0.62        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.62       => ( ( ord_less_nat @ M @ N )
% 0.35/0.62         => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % nat_dvd_not_less
% 0.35/0.62  thf(fact_4317_dvd__pos__nat,axiom,
% 0.35/0.62      ! [N: nat,M: nat] :
% 0.35/0.62        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.62       => ( ( dvd_dvd_nat @ M @ N )
% 0.35/0.62         => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_pos_nat
% 0.35/0.62  thf(fact_4318_dvd__minus__self,axiom,
% 0.35/0.62      ! [M: nat,N: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
% 0.35/0.62        = ( ( ord_less_nat @ N @ M )
% 0.35/0.62          | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_minus_self
% 0.35/0.62  thf(fact_4319_less__eq__dvd__minus,axiom,
% 0.35/0.62      ! [M: nat,N: nat] :
% 0.35/0.62        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.62       => ( ( dvd_dvd_nat @ M @ N )
% 0.35/0.62          = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % less_eq_dvd_minus
% 0.35/0.62  thf(fact_4320_dvd__diffD1,axiom,
% 0.35/0.62      ! [K: nat,M: nat,N: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.62       => ( ( dvd_dvd_nat @ K @ M )
% 0.35/0.62         => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.62           => ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_diffD1
% 0.35/0.62  thf(fact_4321_dvd__diffD,axiom,
% 0.35/0.62      ! [K: nat,M: nat,N: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.62       => ( ( dvd_dvd_nat @ K @ N )
% 0.35/0.62         => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.62           => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_diffD
% 0.35/0.62  thf(fact_4322_zdvd__not__zless,axiom,
% 0.35/0.62      ! [M: int,N: int] :
% 0.35/0.62        ( ( ord_less_int @ zero_zero_int @ M )
% 0.35/0.62       => ( ( ord_less_int @ M @ N )
% 0.35/0.62         => ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % zdvd_not_zless
% 0.35/0.62  thf(fact_4323_mod__int__unique,axiom,
% 0.35/0.62      ! [K: int,L: int,Q4: int,R2: int] :
% 0.35/0.62        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.35/0.62       => ( ( modulo_modulo_int @ K @ L )
% 0.35/0.62          = R2 ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_int_unique
% 0.35/0.62  thf(fact_4324_finite__divisors__int,axiom,
% 0.35/0.62      ! [I: int] :
% 0.35/0.62        ( ( I != zero_zero_int )
% 0.35/0.62       => ( finite_finite_int
% 0.35/0.62          @ ( collect_int
% 0.35/0.62            @ ^ [D4: int] : ( dvd_dvd_int @ D4 @ I ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % finite_divisors_int
% 0.35/0.62  thf(fact_4325_unity__coeff__ex,axiom,
% 0.35/0.62      ! [P: code_integer > $o,L: code_integer] :
% 0.35/0.62        ( ( ? [X3: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X3 ) ) )
% 0.35/0.62        = ( ? [X3: code_integer] :
% 0.35/0.62              ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X3 @ zero_z3403309356797280102nteger ) )
% 0.35/0.62              & ( P @ X3 ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unity_coeff_ex
% 0.35/0.62  thf(fact_4326_unity__coeff__ex,axiom,
% 0.35/0.62      ! [P: complex > $o,L: complex] :
% 0.35/0.62        ( ( ? [X3: complex] : ( P @ ( times_times_complex @ L @ X3 ) ) )
% 0.35/0.62        = ( ? [X3: complex] :
% 0.35/0.62              ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X3 @ zero_zero_complex ) )
% 0.35/0.62              & ( P @ X3 ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unity_coeff_ex
% 0.35/0.62  thf(fact_4327_unity__coeff__ex,axiom,
% 0.35/0.62      ! [P: real > $o,L: real] :
% 0.35/0.62        ( ( ? [X3: real] : ( P @ ( times_times_real @ L @ X3 ) ) )
% 0.35/0.62        = ( ? [X3: real] :
% 0.35/0.62              ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X3 @ zero_zero_real ) )
% 0.35/0.62              & ( P @ X3 ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unity_coeff_ex
% 0.35/0.62  thf(fact_4328_unity__coeff__ex,axiom,
% 0.35/0.62      ! [P: rat > $o,L: rat] :
% 0.35/0.62        ( ( ? [X3: rat] : ( P @ ( times_times_rat @ L @ X3 ) ) )
% 0.35/0.62        = ( ? [X3: rat] :
% 0.35/0.62              ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X3 @ zero_zero_rat ) )
% 0.35/0.62              & ( P @ X3 ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unity_coeff_ex
% 0.35/0.62  thf(fact_4329_unity__coeff__ex,axiom,
% 0.35/0.62      ! [P: nat > $o,L: nat] :
% 0.35/0.62        ( ( ? [X3: nat] : ( P @ ( times_times_nat @ L @ X3 ) ) )
% 0.35/0.62        = ( ? [X3: nat] :
% 0.35/0.62              ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X3 @ zero_zero_nat ) )
% 0.35/0.62              & ( P @ X3 ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unity_coeff_ex
% 0.35/0.62  thf(fact_4330_unity__coeff__ex,axiom,
% 0.35/0.62      ! [P: int > $o,L: int] :
% 0.35/0.62        ( ( ? [X3: int] : ( P @ ( times_times_int @ L @ X3 ) ) )
% 0.35/0.62        = ( ? [X3: int] :
% 0.35/0.62              ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X3 @ zero_zero_int ) )
% 0.35/0.62              & ( P @ X3 ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unity_coeff_ex
% 0.35/0.62  thf(fact_4331_unit__dvdE,axiom,
% 0.35/0.62      ! [A: code_integer,B: code_integer] :
% 0.35/0.62        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.35/0.62       => ~ ( ( A != zero_z3403309356797280102nteger )
% 0.35/0.62           => ! [C3: code_integer] :
% 0.35/0.62                ( B
% 0.35/0.62               != ( times_3573771949741848930nteger @ A @ C3 ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unit_dvdE
% 0.35/0.62  thf(fact_4332_unit__dvdE,axiom,
% 0.35/0.62      ! [A: nat,B: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.35/0.62       => ~ ( ( A != zero_zero_nat )
% 0.35/0.62           => ! [C3: nat] :
% 0.35/0.62                ( B
% 0.35/0.62               != ( times_times_nat @ A @ C3 ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unit_dvdE
% 0.35/0.62  thf(fact_4333_unit__dvdE,axiom,
% 0.35/0.62      ! [A: int,B: int] :
% 0.35/0.62        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.35/0.62       => ~ ( ( A != zero_zero_int )
% 0.35/0.62           => ! [C3: int] :
% 0.35/0.62                ( B
% 0.35/0.62               != ( times_times_int @ A @ C3 ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unit_dvdE
% 0.35/0.62  thf(fact_4334_dvd__div__div__eq__mult,axiom,
% 0.35/0.62      ! [A: nat,C: nat,B: nat,D: nat] :
% 0.35/0.62        ( ( A != zero_zero_nat )
% 0.35/0.62       => ( ( C != zero_zero_nat )
% 0.35/0.62         => ( ( dvd_dvd_nat @ A @ B )
% 0.35/0.62           => ( ( dvd_dvd_nat @ C @ D )
% 0.35/0.62             => ( ( ( divide_divide_nat @ B @ A )
% 0.35/0.62                  = ( divide_divide_nat @ D @ C ) )
% 0.35/0.62                = ( ( times_times_nat @ B @ C )
% 0.35/0.62                  = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_div_div_eq_mult
% 0.35/0.62  thf(fact_4335_dvd__div__div__eq__mult,axiom,
% 0.35/0.62      ! [A: int,C: int,B: int,D: int] :
% 0.35/0.62        ( ( A != zero_zero_int )
% 0.35/0.62       => ( ( C != zero_zero_int )
% 0.35/0.62         => ( ( dvd_dvd_int @ A @ B )
% 0.35/0.62           => ( ( dvd_dvd_int @ C @ D )
% 0.35/0.62             => ( ( ( divide_divide_int @ B @ A )
% 0.35/0.62                  = ( divide_divide_int @ D @ C ) )
% 0.35/0.62                = ( ( times_times_int @ B @ C )
% 0.35/0.62                  = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_div_div_eq_mult
% 0.35/0.62  thf(fact_4336_dvd__div__div__eq__mult,axiom,
% 0.35/0.62      ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 0.35/0.62        ( ( A != zero_z3403309356797280102nteger )
% 0.35/0.62       => ( ( C != zero_z3403309356797280102nteger )
% 0.35/0.62         => ( ( dvd_dvd_Code_integer @ A @ B )
% 0.35/0.62           => ( ( dvd_dvd_Code_integer @ C @ D )
% 0.35/0.62             => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.35/0.62                  = ( divide6298287555418463151nteger @ D @ C ) )
% 0.35/0.62                = ( ( times_3573771949741848930nteger @ B @ C )
% 0.35/0.62                  = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_div_div_eq_mult
% 0.35/0.62  thf(fact_4337_dvd__div__iff__mult,axiom,
% 0.35/0.62      ! [C: nat,B: nat,A: nat] :
% 0.35/0.62        ( ( C != zero_zero_nat )
% 0.35/0.62       => ( ( dvd_dvd_nat @ C @ B )
% 0.35/0.62         => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.35/0.62            = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_div_iff_mult
% 0.35/0.62  thf(fact_4338_dvd__div__iff__mult,axiom,
% 0.35/0.62      ! [C: int,B: int,A: int] :
% 0.35/0.62        ( ( C != zero_zero_int )
% 0.35/0.62       => ( ( dvd_dvd_int @ C @ B )
% 0.35/0.62         => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.35/0.62            = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_div_iff_mult
% 0.35/0.62  thf(fact_4339_dvd__div__iff__mult,axiom,
% 0.35/0.62      ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.35/0.62        ( ( C != zero_z3403309356797280102nteger )
% 0.35/0.62       => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.35/0.62         => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.35/0.62            = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_div_iff_mult
% 0.35/0.62  thf(fact_4340_div__dvd__iff__mult,axiom,
% 0.35/0.62      ! [B: nat,A: nat,C: nat] :
% 0.35/0.62        ( ( B != zero_zero_nat )
% 0.35/0.62       => ( ( dvd_dvd_nat @ B @ A )
% 0.35/0.62         => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.35/0.62            = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % div_dvd_iff_mult
% 0.35/0.62  thf(fact_4341_div__dvd__iff__mult,axiom,
% 0.35/0.62      ! [B: int,A: int,C: int] :
% 0.35/0.62        ( ( B != zero_zero_int )
% 0.35/0.62       => ( ( dvd_dvd_int @ B @ A )
% 0.35/0.62         => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.35/0.62            = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % div_dvd_iff_mult
% 0.35/0.62  thf(fact_4342_div__dvd__iff__mult,axiom,
% 0.35/0.62      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.35/0.62        ( ( B != zero_z3403309356797280102nteger )
% 0.35/0.62       => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.35/0.62         => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.35/0.62            = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % div_dvd_iff_mult
% 0.35/0.62  thf(fact_4343_dvd__div__eq__mult,axiom,
% 0.35/0.62      ! [A: nat,B: nat,C: nat] :
% 0.35/0.62        ( ( A != zero_zero_nat )
% 0.35/0.62       => ( ( dvd_dvd_nat @ A @ B )
% 0.35/0.62         => ( ( ( divide_divide_nat @ B @ A )
% 0.35/0.62              = C )
% 0.35/0.62            = ( B
% 0.35/0.62              = ( times_times_nat @ C @ A ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_div_eq_mult
% 0.35/0.62  thf(fact_4344_dvd__div__eq__mult,axiom,
% 0.35/0.62      ! [A: int,B: int,C: int] :
% 0.35/0.62        ( ( A != zero_zero_int )
% 0.35/0.62       => ( ( dvd_dvd_int @ A @ B )
% 0.35/0.62         => ( ( ( divide_divide_int @ B @ A )
% 0.35/0.62              = C )
% 0.35/0.62            = ( B
% 0.35/0.62              = ( times_times_int @ C @ A ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_div_eq_mult
% 0.35/0.62  thf(fact_4345_dvd__div__eq__mult,axiom,
% 0.35/0.62      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.35/0.62        ( ( A != zero_z3403309356797280102nteger )
% 0.35/0.62       => ( ( dvd_dvd_Code_integer @ A @ B )
% 0.35/0.62         => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.35/0.62              = C )
% 0.35/0.62            = ( B
% 0.35/0.62              = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_div_eq_mult
% 0.35/0.62  thf(fact_4346_unit__div__eq__0__iff,axiom,
% 0.35/0.62      ! [B: nat,A: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.35/0.62       => ( ( ( divide_divide_nat @ A @ B )
% 0.35/0.62            = zero_zero_nat )
% 0.35/0.62          = ( A = zero_zero_nat ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unit_div_eq_0_iff
% 0.35/0.62  thf(fact_4347_unit__div__eq__0__iff,axiom,
% 0.35/0.62      ! [B: int,A: int] :
% 0.35/0.62        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.35/0.62       => ( ( ( divide_divide_int @ A @ B )
% 0.35/0.62            = zero_zero_int )
% 0.35/0.62          = ( A = zero_zero_int ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unit_div_eq_0_iff
% 0.35/0.62  thf(fact_4348_unit__div__eq__0__iff,axiom,
% 0.35/0.62      ! [B: code_integer,A: code_integer] :
% 0.35/0.62        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.35/0.62       => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.35/0.62            = zero_z3403309356797280102nteger )
% 0.35/0.62          = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unit_div_eq_0_iff
% 0.35/0.62  thf(fact_4349_is__unit__power__iff,axiom,
% 0.35/0.62      ! [A: code_integer,N: nat] :
% 0.35/0.62        ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
% 0.35/0.62        = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.35/0.62          | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % is_unit_power_iff
% 0.35/0.62  thf(fact_4350_is__unit__power__iff,axiom,
% 0.35/0.62      ! [A: nat,N: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
% 0.35/0.62        = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.35/0.62          | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % is_unit_power_iff
% 0.35/0.62  thf(fact_4351_is__unit__power__iff,axiom,
% 0.35/0.62      ! [A: int,N: nat] :
% 0.35/0.62        ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
% 0.35/0.62        = ( ( dvd_dvd_int @ A @ one_one_int )
% 0.35/0.62          | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % is_unit_power_iff
% 0.35/0.62  thf(fact_4352_unit__imp__mod__eq__0,axiom,
% 0.35/0.62      ! [B: nat,A: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.35/0.62       => ( ( modulo_modulo_nat @ A @ B )
% 0.35/0.62          = zero_zero_nat ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unit_imp_mod_eq_0
% 0.35/0.62  thf(fact_4353_unit__imp__mod__eq__0,axiom,
% 0.35/0.62      ! [B: int,A: int] :
% 0.35/0.62        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.35/0.62       => ( ( modulo_modulo_int @ A @ B )
% 0.35/0.62          = zero_zero_int ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unit_imp_mod_eq_0
% 0.35/0.62  thf(fact_4354_unit__imp__mod__eq__0,axiom,
% 0.35/0.62      ! [B: code_integer,A: code_integer] :
% 0.35/0.62        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.35/0.62       => ( ( modulo364778990260209775nteger @ A @ B )
% 0.35/0.62          = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.62  
% 0.35/0.62  % unit_imp_mod_eq_0
% 0.35/0.62  thf(fact_4355_neg__mod__sign,axiom,
% 0.35/0.62      ! [L: int,K: int] :
% 0.35/0.62        ( ( ord_less_int @ L @ zero_zero_int )
% 0.35/0.62       => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% 0.35/0.62  
% 0.35/0.62  % neg_mod_sign
% 0.35/0.62  thf(fact_4356_Euclidean__Division_Opos__mod__sign,axiom,
% 0.35/0.62      ! [L: int,K: int] :
% 0.35/0.62        ( ( ord_less_int @ zero_zero_int @ L )
% 0.35/0.62       => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % Euclidean_Division.pos_mod_sign
% 0.35/0.62  thf(fact_4357_zmod__trivial__iff,axiom,
% 0.35/0.62      ! [I: int,K: int] :
% 0.35/0.62        ( ( ( modulo_modulo_int @ I @ K )
% 0.35/0.62          = I )
% 0.35/0.62        = ( ( K = zero_zero_int )
% 0.35/0.62          | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.35/0.62            & ( ord_less_int @ I @ K ) )
% 0.35/0.62          | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 0.35/0.62            & ( ord_less_int @ K @ I ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % zmod_trivial_iff
% 0.35/0.62  thf(fact_4358_pos__mod__conj,axiom,
% 0.35/0.62      ! [B: int,A: int] :
% 0.35/0.62        ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.62       => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
% 0.35/0.62          & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % pos_mod_conj
% 0.35/0.62  thf(fact_4359_neg__mod__conj,axiom,
% 0.35/0.62      ! [B: int,A: int] :
% 0.35/0.62        ( ( ord_less_int @ B @ zero_zero_int )
% 0.35/0.62       => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
% 0.35/0.62          & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % neg_mod_conj
% 0.35/0.62  thf(fact_4360_dvd__imp__le,axiom,
% 0.35/0.62      ! [K: nat,N: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ K @ N )
% 0.35/0.62       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.62         => ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_imp_le
% 0.35/0.62  thf(fact_4361_nat__mult__dvd__cancel1,axiom,
% 0.35/0.62      ! [K: nat,M: nat,N: nat] :
% 0.35/0.62        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.62       => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.35/0.62          = ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % nat_mult_dvd_cancel1
% 0.35/0.62  thf(fact_4362_dvd__mult__cancel,axiom,
% 0.35/0.62      ! [K: nat,M: nat,N: nat] :
% 0.35/0.62        ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.35/0.62       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.62         => ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % dvd_mult_cancel
% 0.35/0.62  thf(fact_4363_zdiv__mono__strict,axiom,
% 0.35/0.62      ! [A2: int,B2: int,N: int] :
% 0.35/0.62        ( ( ord_less_int @ A2 @ B2 )
% 0.35/0.62       => ( ( ord_less_int @ zero_zero_int @ N )
% 0.35/0.62         => ( ( ( modulo_modulo_int @ A2 @ N )
% 0.35/0.62              = zero_zero_int )
% 0.35/0.62           => ( ( ( modulo_modulo_int @ B2 @ N )
% 0.35/0.62                = zero_zero_int )
% 0.35/0.62             => ( ord_less_int @ ( divide_divide_int @ A2 @ N ) @ ( divide_divide_int @ B2 @ N ) ) ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % zdiv_mono_strict
% 0.35/0.62  thf(fact_4364_bezout__add__strong__nat,axiom,
% 0.35/0.62      ! [A: nat,B: nat] :
% 0.35/0.62        ( ( A != zero_zero_nat )
% 0.35/0.62       => ? [D5: nat,X4: nat,Y3: nat] :
% 0.35/0.62            ( ( dvd_dvd_nat @ D5 @ A )
% 0.35/0.62            & ( dvd_dvd_nat @ D5 @ B )
% 0.35/0.62            & ( ( times_times_nat @ A @ X4 )
% 0.35/0.62              = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D5 ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % bezout_add_strong_nat
% 0.35/0.62  thf(fact_4365_zdvd__imp__le,axiom,
% 0.35/0.62      ! [Z2: int,N: int] :
% 0.35/0.62        ( ( dvd_dvd_int @ Z2 @ N )
% 0.35/0.62       => ( ( ord_less_int @ zero_zero_int @ N )
% 0.35/0.62         => ( ord_less_eq_int @ Z2 @ N ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % zdvd_imp_le
% 0.35/0.62  thf(fact_4366_mod__greater__zero__iff__not__dvd,axiom,
% 0.35/0.62      ! [M: nat,N: nat] :
% 0.35/0.62        ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
% 0.35/0.62        = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_greater_zero_iff_not_dvd
% 0.35/0.62  thf(fact_4367_mod__eq__dvd__iff__nat,axiom,
% 0.35/0.62      ! [N: nat,M: nat,Q4: nat] :
% 0.35/0.62        ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.62       => ( ( ( modulo_modulo_nat @ M @ Q4 )
% 0.35/0.62            = ( modulo_modulo_nat @ N @ Q4 ) )
% 0.35/0.62          = ( dvd_dvd_nat @ Q4 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % mod_eq_dvd_iff_nat
% 0.35/0.62  thf(fact_4368_eucl__rel__int,axiom,
% 0.35/0.62      ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % eucl_rel_int
% 0.35/0.62  thf(fact_4369_prod__decode__aux_Ocases,axiom,
% 0.35/0.62      ! [X: product_prod_nat_nat] :
% 0.35/0.62        ~ ! [K2: nat,M5: nat] :
% 0.35/0.62            ( X
% 0.35/0.62           != ( product_Pair_nat_nat @ K2 @ M5 ) ) ).
% 0.35/0.62  
% 0.35/0.62  % prod_decode_aux.cases
% 0.35/0.62  thf(fact_4370_finite__divisors__nat,axiom,
% 0.35/0.62      ! [M: nat] :
% 0.35/0.62        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.62       => ( finite_finite_nat
% 0.35/0.62          @ ( collect_nat
% 0.35/0.62            @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % finite_divisors_nat
% 0.35/0.62  thf(fact_4371_full__exhaustive__int_H_Ocases,axiom,
% 0.35/0.62      ! [X: produc2285326912895808259nt_int] :
% 0.35/0.62        ~ ! [F2: produc8551481072490612790e_term > option6357759511663192854e_term,D5: int,I2: int] :
% 0.35/0.62            ( X
% 0.35/0.62           != ( produc5700946648718959541nt_int @ F2 @ ( product_Pair_int_int @ D5 @ I2 ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % full_exhaustive_int'.cases
% 0.35/0.62  thf(fact_4372_exhaustive__int_H_Ocases,axiom,
% 0.35/0.62      ! [X: produc7773217078559923341nt_int] :
% 0.35/0.62        ~ ! [F2: int > option6357759511663192854e_term,D5: int,I2: int] :
% 0.35/0.62            ( X
% 0.35/0.62           != ( produc4305682042979456191nt_int @ F2 @ ( product_Pair_int_int @ D5 @ I2 ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % exhaustive_int'.cases
% 0.35/0.62  thf(fact_4373_small__lazy_H_Ocases,axiom,
% 0.35/0.62      ! [X: product_prod_int_int] :
% 0.35/0.62        ~ ! [D5: int,I2: int] :
% 0.35/0.62            ( X
% 0.35/0.62           != ( product_Pair_int_int @ D5 @ I2 ) ) ).
% 0.35/0.62  
% 0.35/0.62  % small_lazy'.cases
% 0.35/0.62  thf(fact_4374_even__zero,axiom,
% 0.35/0.62      dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% 0.35/0.62  
% 0.35/0.62  % even_zero
% 0.35/0.62  thf(fact_4375_even__zero,axiom,
% 0.35/0.62      dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% 0.35/0.62  
% 0.35/0.62  % even_zero
% 0.35/0.62  thf(fact_4376_even__zero,axiom,
% 0.35/0.62      dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% 0.35/0.62  
% 0.35/0.62  % even_zero
% 0.35/0.62  thf(fact_4377_is__unit__div__mult__cancel__right,axiom,
% 0.35/0.62      ! [A: nat,B: nat] :
% 0.35/0.62        ( ( A != zero_zero_nat )
% 0.35/0.62       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.35/0.62         => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
% 0.35/0.62            = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % is_unit_div_mult_cancel_right
% 0.35/0.62  thf(fact_4378_is__unit__div__mult__cancel__right,axiom,
% 0.35/0.62      ! [A: int,B: int] :
% 0.35/0.62        ( ( A != zero_zero_int )
% 0.35/0.62       => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.35/0.62         => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
% 0.35/0.62            = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 0.35/0.62  
% 0.35/0.62  % is_unit_div_mult_cancel_right
% 0.35/0.62  thf(fact_4379_is__unit__div__mult__cancel__right,axiom,
% 0.35/0.62      ! [A: code_integer,B: code_integer] :
% 0.35/0.63        ( ( A != zero_z3403309356797280102nteger )
% 0.35/0.63       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.35/0.63         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
% 0.35/0.63            = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % is_unit_div_mult_cancel_right
% 0.35/0.63  thf(fact_4380_is__unit__div__mult__cancel__left,axiom,
% 0.35/0.63      ! [A: nat,B: nat] :
% 0.35/0.63        ( ( A != zero_zero_nat )
% 0.35/0.63       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.35/0.63         => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
% 0.35/0.63            = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % is_unit_div_mult_cancel_left
% 0.35/0.63  thf(fact_4381_is__unit__div__mult__cancel__left,axiom,
% 0.35/0.63      ! [A: int,B: int] :
% 0.35/0.63        ( ( A != zero_zero_int )
% 0.35/0.63       => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.35/0.63         => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
% 0.35/0.63            = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % is_unit_div_mult_cancel_left
% 0.35/0.63  thf(fact_4382_is__unit__div__mult__cancel__left,axiom,
% 0.35/0.63      ! [A: code_integer,B: code_integer] :
% 0.35/0.63        ( ( A != zero_z3403309356797280102nteger )
% 0.35/0.63       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.35/0.63         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.35/0.63            = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % is_unit_div_mult_cancel_left
% 0.35/0.63  thf(fact_4383_is__unitE,axiom,
% 0.35/0.63      ! [A: nat,C: nat] :
% 0.35/0.63        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.35/0.63       => ~ ( ( A != zero_zero_nat )
% 0.35/0.63           => ! [B3: nat] :
% 0.35/0.63                ( ( B3 != zero_zero_nat )
% 0.35/0.63               => ( ( dvd_dvd_nat @ B3 @ one_one_nat )
% 0.35/0.63                 => ( ( ( divide_divide_nat @ one_one_nat @ A )
% 0.35/0.63                      = B3 )
% 0.35/0.63                   => ( ( ( divide_divide_nat @ one_one_nat @ B3 )
% 0.35/0.63                        = A )
% 0.35/0.63                     => ( ( ( times_times_nat @ A @ B3 )
% 0.35/0.63                          = one_one_nat )
% 0.35/0.63                       => ( ( divide_divide_nat @ C @ A )
% 0.35/0.63                         != ( times_times_nat @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % is_unitE
% 0.35/0.63  thf(fact_4384_is__unitE,axiom,
% 0.35/0.63      ! [A: int,C: int] :
% 0.35/0.63        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.35/0.63       => ~ ( ( A != zero_zero_int )
% 0.35/0.63           => ! [B3: int] :
% 0.35/0.63                ( ( B3 != zero_zero_int )
% 0.35/0.63               => ( ( dvd_dvd_int @ B3 @ one_one_int )
% 0.35/0.63                 => ( ( ( divide_divide_int @ one_one_int @ A )
% 0.35/0.63                      = B3 )
% 0.35/0.63                   => ( ( ( divide_divide_int @ one_one_int @ B3 )
% 0.35/0.63                        = A )
% 0.35/0.63                     => ( ( ( times_times_int @ A @ B3 )
% 0.35/0.63                          = one_one_int )
% 0.35/0.63                       => ( ( divide_divide_int @ C @ A )
% 0.35/0.63                         != ( times_times_int @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % is_unitE
% 0.35/0.63  thf(fact_4385_is__unitE,axiom,
% 0.35/0.63      ! [A: code_integer,C: code_integer] :
% 0.35/0.63        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.35/0.63       => ~ ( ( A != zero_z3403309356797280102nteger )
% 0.35/0.63           => ! [B3: code_integer] :
% 0.35/0.63                ( ( B3 != zero_z3403309356797280102nteger )
% 0.35/0.63               => ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
% 0.35/0.63                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
% 0.35/0.63                      = B3 )
% 0.35/0.63                   => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B3 )
% 0.35/0.63                        = A )
% 0.35/0.63                     => ( ( ( times_3573771949741848930nteger @ A @ B3 )
% 0.35/0.63                          = one_one_Code_integer )
% 0.35/0.63                       => ( ( divide6298287555418463151nteger @ C @ A )
% 0.35/0.63                         != ( times_3573771949741848930nteger @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % is_unitE
% 0.35/0.63  thf(fact_4386_dvd__power__iff,axiom,
% 0.35/0.63      ! [X: code_integer,M: nat,N: nat] :
% 0.35/0.63        ( ( X != zero_z3403309356797280102nteger )
% 0.35/0.63       => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N ) )
% 0.35/0.63          = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
% 0.35/0.63            | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % dvd_power_iff
% 0.35/0.63  thf(fact_4387_dvd__power__iff,axiom,
% 0.35/0.63      ! [X: nat,M: nat,N: nat] :
% 0.35/0.63        ( ( X != zero_zero_nat )
% 0.35/0.63       => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
% 0.35/0.63          = ( ( dvd_dvd_nat @ X @ one_one_nat )
% 0.35/0.63            | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % dvd_power_iff
% 0.35/0.63  thf(fact_4388_dvd__power__iff,axiom,
% 0.35/0.63      ! [X: int,M: nat,N: nat] :
% 0.35/0.63        ( ( X != zero_zero_int )
% 0.35/0.63       => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
% 0.35/0.63          = ( ( dvd_dvd_int @ X @ one_one_int )
% 0.35/0.63            | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % dvd_power_iff
% 0.35/0.63  thf(fact_4389_dvd__power,axiom,
% 0.35/0.63      ! [N: nat,X: code_integer] :
% 0.35/0.63        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.63          | ( X = one_one_Code_integer ) )
% 0.35/0.63       => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % dvd_power
% 0.35/0.63  thf(fact_4390_dvd__power,axiom,
% 0.35/0.63      ! [N: nat,X: rat] :
% 0.35/0.63        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.63          | ( X = one_one_rat ) )
% 0.35/0.63       => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % dvd_power
% 0.35/0.63  thf(fact_4391_dvd__power,axiom,
% 0.35/0.63      ! [N: nat,X: nat] :
% 0.35/0.63        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.63          | ( X = one_one_nat ) )
% 0.35/0.63       => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % dvd_power
% 0.35/0.63  thf(fact_4392_dvd__power,axiom,
% 0.35/0.63      ! [N: nat,X: real] :
% 0.35/0.63        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.63          | ( X = one_one_real ) )
% 0.35/0.63       => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % dvd_power
% 0.35/0.63  thf(fact_4393_dvd__power,axiom,
% 0.35/0.63      ! [N: nat,X: int] :
% 0.35/0.63        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.63          | ( X = one_one_int ) )
% 0.35/0.63       => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % dvd_power
% 0.35/0.63  thf(fact_4394_dvd__power,axiom,
% 0.35/0.63      ! [N: nat,X: complex] :
% 0.35/0.63        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.63          | ( X = one_one_complex ) )
% 0.35/0.63       => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % dvd_power
% 0.35/0.63  thf(fact_4395_mod__pos__neg__trivial,axiom,
% 0.35/0.63      ! [K: int,L: int] :
% 0.35/0.63        ( ( ord_less_int @ zero_zero_int @ K )
% 0.35/0.63       => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 0.35/0.63         => ( ( modulo_modulo_int @ K @ L )
% 0.35/0.63            = ( plus_plus_int @ K @ L ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mod_pos_neg_trivial
% 0.35/0.63  thf(fact_4396_mod__pos__geq,axiom,
% 0.35/0.63      ! [L: int,K: int] :
% 0.35/0.63        ( ( ord_less_int @ zero_zero_int @ L )
% 0.35/0.63       => ( ( ord_less_eq_int @ L @ K )
% 0.35/0.63         => ( ( modulo_modulo_int @ K @ L )
% 0.35/0.63            = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mod_pos_geq
% 0.35/0.63  thf(fact_4397_dvd__mult__cancel1,axiom,
% 0.35/0.63      ! [M: nat,N: nat] :
% 0.35/0.63        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.63       => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
% 0.35/0.63          = ( N = one_one_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % dvd_mult_cancel1
% 0.35/0.63  thf(fact_4398_dvd__mult__cancel2,axiom,
% 0.35/0.63      ! [M: nat,N: nat] :
% 0.35/0.63        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.63       => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
% 0.35/0.63          = ( N = one_one_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % dvd_mult_cancel2
% 0.35/0.63  thf(fact_4399_dvd__minus__add,axiom,
% 0.35/0.63      ! [Q4: nat,N: nat,R2: nat,M: nat] :
% 0.35/0.63        ( ( ord_less_eq_nat @ Q4 @ N )
% 0.35/0.63       => ( ( ord_less_eq_nat @ Q4 @ ( times_times_nat @ R2 @ M ) )
% 0.35/0.63         => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q4 ) )
% 0.35/0.63            = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q4 ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % dvd_minus_add
% 0.35/0.63  thf(fact_4400_power__dvd__imp__le,axiom,
% 0.35/0.63      ! [I: nat,M: nat,N: nat] :
% 0.35/0.63        ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 0.35/0.63       => ( ( ord_less_nat @ one_one_nat @ I )
% 0.35/0.63         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % power_dvd_imp_le
% 0.35/0.63  thf(fact_4401_mod__nat__eqI,axiom,
% 0.35/0.63      ! [R2: nat,N: nat,M: nat] :
% 0.35/0.63        ( ( ord_less_nat @ R2 @ N )
% 0.35/0.63       => ( ( ord_less_eq_nat @ R2 @ M )
% 0.35/0.63         => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R2 ) )
% 0.35/0.63           => ( ( modulo_modulo_nat @ M @ N )
% 0.35/0.63              = R2 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mod_nat_eqI
% 0.35/0.63  thf(fact_4402_even__iff__mod__2__eq__zero,axiom,
% 0.35/0.63      ! [A: nat] :
% 0.35/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.35/0.63        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.63          = zero_zero_nat ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_iff_mod_2_eq_zero
% 0.35/0.63  thf(fact_4403_even__iff__mod__2__eq__zero,axiom,
% 0.35/0.63      ! [A: int] :
% 0.35/0.63        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.35/0.63        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.63          = zero_zero_int ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_iff_mod_2_eq_zero
% 0.35/0.63  thf(fact_4404_even__iff__mod__2__eq__zero,axiom,
% 0.35/0.63      ! [A: code_integer] :
% 0.35/0.63        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.35/0.63        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.63          = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_iff_mod_2_eq_zero
% 0.35/0.63  thf(fact_4405_power__mono__odd,axiom,
% 0.35/0.63      ! [N: nat,A: real,B: real] :
% 0.35/0.63        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63       => ( ( ord_less_eq_real @ A @ B )
% 0.35/0.63         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % power_mono_odd
% 0.35/0.63  thf(fact_4406_power__mono__odd,axiom,
% 0.35/0.63      ! [N: nat,A: rat,B: rat] :
% 0.35/0.63        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63       => ( ( ord_less_eq_rat @ A @ B )
% 0.35/0.63         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % power_mono_odd
% 0.35/0.63  thf(fact_4407_power__mono__odd,axiom,
% 0.35/0.63      ! [N: nat,A: int,B: int] :
% 0.35/0.63        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63       => ( ( ord_less_eq_int @ A @ B )
% 0.35/0.63         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % power_mono_odd
% 0.35/0.63  thf(fact_4408_odd__pos,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.35/0.63  
% 0.35/0.63  % odd_pos
% 0.35/0.63  thf(fact_4409_dvd__power__iff__le,axiom,
% 0.35/0.63      ! [K: nat,M: nat,N: nat] :
% 0.35/0.63        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.35/0.63       => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
% 0.35/0.63          = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % dvd_power_iff_le
% 0.35/0.63  thf(fact_4410_split__zmod,axiom,
% 0.35/0.63      ! [P: int > $o,N: int,K: int] :
% 0.35/0.63        ( ( P @ ( modulo_modulo_int @ N @ K ) )
% 0.35/0.63        = ( ( ( K = zero_zero_int )
% 0.35/0.63           => ( P @ N ) )
% 0.35/0.63          & ( ( ord_less_int @ zero_zero_int @ K )
% 0.35/0.63           => ! [I4: int,J3: int] :
% 0.35/0.63                ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.35/0.63                  & ( ord_less_int @ J3 @ K )
% 0.35/0.63                  & ( N
% 0.35/0.63                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.35/0.63               => ( P @ J3 ) ) )
% 0.35/0.63          & ( ( ord_less_int @ K @ zero_zero_int )
% 0.35/0.63           => ! [I4: int,J3: int] :
% 0.35/0.63                ( ( ( ord_less_int @ K @ J3 )
% 0.35/0.63                  & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.35/0.63                  & ( N
% 0.35/0.63                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.35/0.63               => ( P @ J3 ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % split_zmod
% 0.35/0.63  thf(fact_4411_int__mod__neg__eq,axiom,
% 0.35/0.63      ! [A: int,B: int,Q4: int,R2: int] :
% 0.35/0.63        ( ( A
% 0.35/0.63          = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 0.35/0.63       => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.35/0.63         => ( ( ord_less_int @ B @ R2 )
% 0.35/0.63           => ( ( modulo_modulo_int @ A @ B )
% 0.35/0.63              = R2 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % int_mod_neg_eq
% 0.35/0.63  thf(fact_4412_int__mod__pos__eq,axiom,
% 0.35/0.63      ! [A: int,B: int,Q4: int,R2: int] :
% 0.35/0.63        ( ( A
% 0.35/0.63          = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 0.35/0.63       => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.35/0.63         => ( ( ord_less_int @ R2 @ B )
% 0.35/0.63           => ( ( modulo_modulo_int @ A @ B )
% 0.35/0.63              = R2 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % int_mod_pos_eq
% 0.35/0.63  thf(fact_4413_even__unset__bit__iff,axiom,
% 0.35/0.63      ! [M: nat,A: code_integer] :
% 0.35/0.63        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 0.35/0.63        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.35/0.63          | ( M = zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_unset_bit_iff
% 0.35/0.63  thf(fact_4414_even__unset__bit__iff,axiom,
% 0.35/0.63      ! [M: nat,A: nat] :
% 0.35/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.35/0.63        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.35/0.63          | ( M = zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_unset_bit_iff
% 0.35/0.63  thf(fact_4415_even__unset__bit__iff,axiom,
% 0.35/0.63      ! [M: nat,A: int] :
% 0.35/0.63        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.35/0.63        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.35/0.63          | ( M = zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_unset_bit_iff
% 0.35/0.63  thf(fact_4416_even__set__bit__iff,axiom,
% 0.35/0.63      ! [M: nat,A: code_integer] :
% 0.35/0.63        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 0.35/0.63        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.35/0.63          & ( M != zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_set_bit_iff
% 0.35/0.63  thf(fact_4417_even__set__bit__iff,axiom,
% 0.35/0.63      ! [M: nat,A: nat] :
% 0.35/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.35/0.63        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.35/0.63          & ( M != zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_set_bit_iff
% 0.35/0.63  thf(fact_4418_even__set__bit__iff,axiom,
% 0.35/0.63      ! [M: nat,A: int] :
% 0.35/0.63        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.35/0.63        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.35/0.63          & ( M != zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_set_bit_iff
% 0.35/0.63  thf(fact_4419_mod2__eq__if,axiom,
% 0.35/0.63      ! [A: nat] :
% 0.35/0.63        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.35/0.63         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.63            = zero_zero_nat ) )
% 0.35/0.63        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.35/0.63         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.63            = one_one_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mod2_eq_if
% 0.35/0.63  thf(fact_4420_mod2__eq__if,axiom,
% 0.35/0.63      ! [A: int] :
% 0.35/0.63        ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.35/0.63         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.63            = zero_zero_int ) )
% 0.35/0.63        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.35/0.63         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.63            = one_one_int ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mod2_eq_if
% 0.35/0.63  thf(fact_4421_mod2__eq__if,axiom,
% 0.35/0.63      ! [A: code_integer] :
% 0.35/0.63        ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.35/0.63         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.63            = zero_z3403309356797280102nteger ) )
% 0.35/0.63        & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.35/0.63         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.63            = one_one_Code_integer ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mod2_eq_if
% 0.35/0.63  thf(fact_4422_parity__cases,axiom,
% 0.35/0.63      ! [A: nat] :
% 0.35/0.63        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.35/0.63         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.63           != zero_zero_nat ) )
% 0.35/0.63       => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.35/0.63           => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.63             != one_one_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % parity_cases
% 0.35/0.63  thf(fact_4423_parity__cases,axiom,
% 0.35/0.63      ! [A: int] :
% 0.35/0.63        ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.35/0.63         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.63           != zero_zero_int ) )
% 0.35/0.63       => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.35/0.63           => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.63             != one_one_int ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % parity_cases
% 0.35/0.63  thf(fact_4424_parity__cases,axiom,
% 0.35/0.63      ! [A: code_integer] :
% 0.35/0.63        ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.35/0.63         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.63           != zero_z3403309356797280102nteger ) )
% 0.35/0.63       => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.35/0.63           => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.63             != one_one_Code_integer ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % parity_cases
% 0.35/0.63  thf(fact_4425_zero__le__even__power,axiom,
% 0.35/0.63      ! [N: nat,A: real] :
% 0.35/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63       => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % zero_le_even_power
% 0.35/0.63  thf(fact_4426_zero__le__even__power,axiom,
% 0.35/0.63      ! [N: nat,A: rat] :
% 0.35/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63       => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % zero_le_even_power
% 0.35/0.63  thf(fact_4427_zero__le__even__power,axiom,
% 0.35/0.63      ! [N: nat,A: int] :
% 0.35/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63       => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % zero_le_even_power
% 0.35/0.63  thf(fact_4428_zero__le__odd__power,axiom,
% 0.35/0.63      ! [N: nat,A: real] :
% 0.35/0.63        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63       => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.35/0.63          = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % zero_le_odd_power
% 0.35/0.63  thf(fact_4429_zero__le__odd__power,axiom,
% 0.35/0.63      ! [N: nat,A: rat] :
% 0.35/0.63        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63       => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.35/0.63          = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % zero_le_odd_power
% 0.35/0.63  thf(fact_4430_zero__le__odd__power,axiom,
% 0.35/0.63      ! [N: nat,A: int] :
% 0.35/0.63        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63       => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.35/0.63          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % zero_le_odd_power
% 0.35/0.63  thf(fact_4431_zero__le__power__eq,axiom,
% 0.35/0.63      ! [A: real,N: nat] :
% 0.35/0.63        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.35/0.63        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63            & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % zero_le_power_eq
% 0.35/0.63  thf(fact_4432_zero__le__power__eq,axiom,
% 0.35/0.63      ! [A: rat,N: nat] :
% 0.35/0.63        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.35/0.63        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63            & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % zero_le_power_eq
% 0.35/0.63  thf(fact_4433_zero__le__power__eq,axiom,
% 0.35/0.63      ! [A: int,N: nat] :
% 0.35/0.63        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.35/0.63        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63            & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % zero_le_power_eq
% 0.35/0.63  thf(fact_4434_verit__le__mono__div__int,axiom,
% 0.35/0.63      ! [A2: int,B2: int,N: int] :
% 0.35/0.63        ( ( ord_less_int @ A2 @ B2 )
% 0.35/0.63       => ( ( ord_less_int @ zero_zero_int @ N )
% 0.35/0.63         => ( ord_less_eq_int
% 0.35/0.63            @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N )
% 0.35/0.63              @ ( if_int
% 0.35/0.63                @ ( ( modulo_modulo_int @ B2 @ N )
% 0.35/0.63                  = zero_zero_int )
% 0.35/0.63                @ one_one_int
% 0.35/0.63                @ zero_zero_int ) )
% 0.35/0.63            @ ( divide_divide_int @ B2 @ N ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % verit_le_mono_div_int
% 0.35/0.63  thf(fact_4435_split__neg__lemma,axiom,
% 0.35/0.63      ! [K: int,P: int > int > $o,N: int] :
% 0.35/0.63        ( ( ord_less_int @ K @ zero_zero_int )
% 0.35/0.63       => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 0.35/0.63          = ( ! [I4: int,J3: int] :
% 0.35/0.63                ( ( ( ord_less_int @ K @ J3 )
% 0.35/0.63                  & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.35/0.63                  & ( N
% 0.35/0.63                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.35/0.63               => ( P @ I4 @ J3 ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % split_neg_lemma
% 0.35/0.63  thf(fact_4436_split__pos__lemma,axiom,
% 0.35/0.63      ! [K: int,P: int > int > $o,N: int] :
% 0.35/0.63        ( ( ord_less_int @ zero_zero_int @ K )
% 0.35/0.63       => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 0.35/0.63          = ( ! [I4: int,J3: int] :
% 0.35/0.63                ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.35/0.63                  & ( ord_less_int @ J3 @ K )
% 0.35/0.63                  & ( N
% 0.35/0.63                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.35/0.63               => ( P @ I4 @ J3 ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % split_pos_lemma
% 0.35/0.63  thf(fact_4437_list__decode_Ocases,axiom,
% 0.35/0.63      ! [X: nat] :
% 0.35/0.63        ( ( X != zero_zero_nat )
% 0.35/0.63       => ~ ! [N3: nat] :
% 0.35/0.63              ( X
% 0.35/0.63             != ( suc @ N3 ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % list_decode.cases
% 0.35/0.63  thf(fact_4438_zero__less__power__eq,axiom,
% 0.35/0.63      ! [A: real,N: nat] :
% 0.35/0.63        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.35/0.63        = ( ( N = zero_zero_nat )
% 0.35/0.63          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63            & ( A != zero_zero_real ) )
% 0.35/0.63          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63            & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % zero_less_power_eq
% 0.35/0.63  thf(fact_4439_zero__less__power__eq,axiom,
% 0.35/0.63      ! [A: rat,N: nat] :
% 0.35/0.63        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.35/0.63        = ( ( N = zero_zero_nat )
% 0.35/0.63          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63            & ( A != zero_zero_rat ) )
% 0.35/0.63          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63            & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % zero_less_power_eq
% 0.35/0.63  thf(fact_4440_zero__less__power__eq,axiom,
% 0.35/0.63      ! [A: int,N: nat] :
% 0.35/0.63        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.35/0.63        = ( ( N = zero_zero_nat )
% 0.35/0.63          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63            & ( A != zero_zero_int ) )
% 0.35/0.63          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63            & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % zero_less_power_eq
% 0.35/0.63  thf(fact_4441_even__mask__div__iff_H,axiom,
% 0.35/0.63      ! [M: nat,N: nat] :
% 0.35/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.35/0.63        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_mask_div_iff'
% 0.35/0.63  thf(fact_4442_even__mask__div__iff_H,axiom,
% 0.35/0.63      ! [M: nat,N: nat] :
% 0.35/0.63        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.35/0.63        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_mask_div_iff'
% 0.35/0.63  thf(fact_4443_even__mask__div__iff_H,axiom,
% 0.35/0.63      ! [M: nat,N: nat] :
% 0.35/0.63        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 0.35/0.63        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_mask_div_iff'
% 0.35/0.63  thf(fact_4444_power__le__zero__eq,axiom,
% 0.35/0.63      ! [A: real,N: nat] :
% 0.35/0.63        ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 0.35/0.63        = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.63          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63              & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.35/0.63            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63              & ( A = zero_zero_real ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % power_le_zero_eq
% 0.35/0.63  thf(fact_4445_power__le__zero__eq,axiom,
% 0.35/0.63      ! [A: rat,N: nat] :
% 0.35/0.63        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 0.35/0.63        = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.63          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63              & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.35/0.63            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63              & ( A = zero_zero_rat ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % power_le_zero_eq
% 0.35/0.63  thf(fact_4446_power__le__zero__eq,axiom,
% 0.35/0.63      ! [A: int,N: nat] :
% 0.35/0.63        ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 0.35/0.63        = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.63          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63              & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.35/0.63            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63              & ( A = zero_zero_int ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % power_le_zero_eq
% 0.35/0.63  thf(fact_4447_even__mod__4__div__2,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.35/0.63          = ( suc @ zero_zero_nat ) )
% 0.35/0.63       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_mod_4_div_2
% 0.35/0.63  thf(fact_4448_even__mask__div__iff,axiom,
% 0.35/0.63      ! [M: nat,N: nat] :
% 0.35/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.35/0.63        = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63            = zero_zero_nat )
% 0.35/0.63          | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_mask_div_iff
% 0.35/0.63  thf(fact_4449_even__mask__div__iff,axiom,
% 0.35/0.63      ! [M: nat,N: nat] :
% 0.35/0.63        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.35/0.63        = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.35/0.63            = zero_zero_int )
% 0.35/0.63          | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_mask_div_iff
% 0.35/0.63  thf(fact_4450_even__mask__div__iff,axiom,
% 0.35/0.63      ! [M: nat,N: nat] :
% 0.35/0.63        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 0.35/0.63        = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 0.35/0.63            = zero_z3403309356797280102nteger )
% 0.35/0.63          | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_mask_div_iff
% 0.35/0.63  thf(fact_4451_even__mult__exp__div__exp__iff,axiom,
% 0.35/0.63      ! [A: nat,M: nat,N: nat] :
% 0.35/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.35/0.63        = ( ( ord_less_nat @ N @ M )
% 0.35/0.63          | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63            = zero_zero_nat )
% 0.35/0.63          | ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.63            & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_mult_exp_div_exp_iff
% 0.35/0.63  thf(fact_4452_even__mult__exp__div__exp__iff,axiom,
% 0.35/0.63      ! [A: int,M: nat,N: nat] :
% 0.35/0.63        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.35/0.63        = ( ( ord_less_nat @ N @ M )
% 0.35/0.63          | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.35/0.63            = zero_zero_int )
% 0.35/0.63          | ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.63            & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_mult_exp_div_exp_iff
% 0.35/0.63  thf(fact_4453_even__mult__exp__div__exp__iff,axiom,
% 0.35/0.63      ! [A: code_integer,M: nat,N: nat] :
% 0.35/0.63        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 0.35/0.63        = ( ( ord_less_nat @ N @ M )
% 0.35/0.63          | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 0.35/0.63            = zero_z3403309356797280102nteger )
% 0.35/0.63          | ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.63            & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_mult_exp_div_exp_iff
% 0.35/0.63  thf(fact_4454_vebt__buildup_Oelims,axiom,
% 0.35/0.63      ! [X: nat,Y: vEBT_VEBT] :
% 0.35/0.63        ( ( ( vEBT_vebt_buildup @ X )
% 0.35/0.63          = Y )
% 0.35/0.63       => ( ( ( X = zero_zero_nat )
% 0.35/0.63           => ( Y
% 0.35/0.63             != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.35/0.63         => ( ( ( X
% 0.35/0.63                = ( suc @ zero_zero_nat ) )
% 0.35/0.63             => ( Y
% 0.35/0.63               != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.35/0.63           => ~ ! [Va2: nat] :
% 0.35/0.63                  ( ( X
% 0.35/0.63                    = ( suc @ ( suc @ Va2 ) ) )
% 0.35/0.63                 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.35/0.63                       => ( Y
% 0.35/0.63                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 0.35/0.63                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.35/0.63                       => ( Y
% 0.35/0.63                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % vebt_buildup.elims
% 0.35/0.63  thf(fact_4455_option_Osize__gen_I2_J,axiom,
% 0.35/0.63      ! [X: nat > nat,X22: nat] :
% 0.35/0.63        ( ( size_option_nat @ X @ ( some_nat @ X22 ) )
% 0.35/0.63        = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % option.size_gen(2)
% 0.35/0.63  thf(fact_4456_option_Osize__gen_I2_J,axiom,
% 0.35/0.63      ! [X: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
% 0.35/0.63        ( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X22 ) )
% 0.35/0.63        = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % option.size_gen(2)
% 0.35/0.63  thf(fact_4457_option_Osize__gen_I2_J,axiom,
% 0.35/0.63      ! [X: num > nat,X22: num] :
% 0.35/0.63        ( ( size_option_num @ X @ ( some_num @ X22 ) )
% 0.35/0.63        = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % option.size_gen(2)
% 0.35/0.63  thf(fact_4458_diff__shunt__var,axiom,
% 0.35/0.63      ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
% 0.35/0.63        ( ( ( minus_1356011639430497352at_nat @ X @ Y )
% 0.35/0.63          = bot_bo2099793752762293965at_nat )
% 0.35/0.63        = ( ord_le3146513528884898305at_nat @ X @ Y ) ) ).
% 0.35/0.63  
% 0.35/0.63  % diff_shunt_var
% 0.35/0.63  thf(fact_4459_diff__shunt__var,axiom,
% 0.35/0.63      ! [X: set_real,Y: set_real] :
% 0.35/0.63        ( ( ( minus_minus_set_real @ X @ Y )
% 0.35/0.63          = bot_bot_set_real )
% 0.35/0.63        = ( ord_less_eq_set_real @ X @ Y ) ) ).
% 0.35/0.63  
% 0.35/0.63  % diff_shunt_var
% 0.35/0.63  thf(fact_4460_diff__shunt__var,axiom,
% 0.35/0.63      ! [X: set_nat,Y: set_nat] :
% 0.35/0.63        ( ( ( minus_minus_set_nat @ X @ Y )
% 0.35/0.63          = bot_bot_set_nat )
% 0.35/0.63        = ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 0.35/0.63  
% 0.35/0.63  % diff_shunt_var
% 0.35/0.63  thf(fact_4461_diff__shunt__var,axiom,
% 0.35/0.63      ! [X: set_int,Y: set_int] :
% 0.35/0.63        ( ( ( minus_minus_set_int @ X @ Y )
% 0.35/0.63          = bot_bot_set_int )
% 0.35/0.63        = ( ord_less_eq_set_int @ X @ Y ) ) ).
% 0.35/0.63  
% 0.35/0.63  % diff_shunt_var
% 0.35/0.63  thf(fact_4462_divmod__step__def,axiom,
% 0.35/0.63      ( unique5026877609467782581ep_nat
% 0.35/0.63      = ( ^ [L2: num] :
% 0.35/0.63            ( produc2626176000494625587at_nat
% 0.35/0.63            @ ^ [Q5: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_step_def
% 0.35/0.63  thf(fact_4463_divmod__step__def,axiom,
% 0.35/0.63      ( unique5024387138958732305ep_int
% 0.35/0.63      = ( ^ [L2: num] :
% 0.35/0.63            ( produc4245557441103728435nt_int
% 0.35/0.63            @ ^ [Q5: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_step_def
% 0.35/0.63  thf(fact_4464_divmod__step__def,axiom,
% 0.35/0.63      ( unique4921790084139445826nteger
% 0.35/0.63      = ( ^ [L2: num] :
% 0.35/0.63            ( produc6916734918728496179nteger
% 0.35/0.63            @ ^ [Q5: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_step_def
% 0.35/0.63  thf(fact_4465_take__bit__rec,axiom,
% 0.35/0.63      ( bit_se1745604003318907178nteger
% 0.35/0.63      = ( ^ [N2: nat,A4: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_rec
% 0.35/0.63  thf(fact_4466_take__bit__rec,axiom,
% 0.35/0.63      ( bit_se2925701944663578781it_nat
% 0.35/0.63      = ( ^ [N2: nat,A4: nat] : ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_rec
% 0.35/0.63  thf(fact_4467_take__bit__rec,axiom,
% 0.35/0.63      ( bit_se2923211474154528505it_int
% 0.35/0.63      = ( ^ [N2: nat,A4: int] : ( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_rec
% 0.35/0.63  thf(fact_4468_num_Osize__gen_I2_J,axiom,
% 0.35/0.63      ! [X22: num] :
% 0.35/0.63        ( ( size_num @ ( bit0 @ X22 ) )
% 0.35/0.63        = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % num.size_gen(2)
% 0.35/0.63  thf(fact_4469_vebt__buildup_Osimps_I3_J,axiom,
% 0.35/0.63      ! [Va: nat] :
% 0.35/0.63        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.35/0.63         => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 0.35/0.63            = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 0.35/0.63        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.35/0.63         => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 0.35/0.63            = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % vebt_buildup.simps(3)
% 0.35/0.63  thf(fact_4470_even__flip__bit__iff,axiom,
% 0.35/0.63      ! [M: nat,A: code_integer] :
% 0.35/0.63        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 0.35/0.63        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.35/0.63         != ( M = zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_flip_bit_iff
% 0.35/0.63  thf(fact_4471_even__flip__bit__iff,axiom,
% 0.35/0.63      ! [M: nat,A: nat] :
% 0.35/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.35/0.63        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.35/0.63         != ( M = zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_flip_bit_iff
% 0.35/0.63  thf(fact_4472_even__flip__bit__iff,axiom,
% 0.35/0.63      ! [M: nat,A: int] :
% 0.35/0.63        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.35/0.63        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.35/0.63         != ( M = zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_flip_bit_iff
% 0.35/0.63  thf(fact_4473_intind,axiom,
% 0.35/0.63      ! [I: nat,N: nat,P: nat > $o,X: nat] :
% 0.35/0.63        ( ( ord_less_nat @ I @ N )
% 0.35/0.63       => ( ( P @ X )
% 0.35/0.63         => ( P @ ( nth_nat @ ( replicate_nat @ N @ X ) @ I ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % intind
% 0.35/0.63  thf(fact_4474_intind,axiom,
% 0.35/0.63      ! [I: nat,N: nat,P: int > $o,X: int] :
% 0.35/0.63        ( ( ord_less_nat @ I @ N )
% 0.35/0.63       => ( ( P @ X )
% 0.35/0.63         => ( P @ ( nth_int @ ( replicate_int @ N @ X ) @ I ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % intind
% 0.35/0.63  thf(fact_4475_intind,axiom,
% 0.35/0.63      ! [I: nat,N: nat,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
% 0.35/0.63        ( ( ord_less_nat @ I @ N )
% 0.35/0.63       => ( ( P @ X )
% 0.35/0.63         => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % intind
% 0.35/0.63  thf(fact_4476_take__bit__of__0,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
% 0.35/0.63        = zero_zero_nat ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_of_0
% 0.35/0.63  thf(fact_4477_take__bit__of__0,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
% 0.35/0.63        = zero_zero_int ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_of_0
% 0.35/0.63  thf(fact_4478_flip__bit__negative__int__iff,axiom,
% 0.35/0.63      ! [N: nat,K: int] :
% 0.35/0.63        ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
% 0.35/0.63        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.35/0.63  
% 0.35/0.63  % flip_bit_negative_int_iff
% 0.35/0.63  thf(fact_4479_replicate__eq__replicate,axiom,
% 0.35/0.63      ! [M: nat,X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
% 0.35/0.63        ( ( ( replicate_VEBT_VEBT @ M @ X )
% 0.35/0.63          = ( replicate_VEBT_VEBT @ N @ Y ) )
% 0.35/0.63        = ( ( M = N )
% 0.35/0.63          & ( ( M != zero_zero_nat )
% 0.35/0.63           => ( X = Y ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % replicate_eq_replicate
% 0.35/0.63  thf(fact_4480_length__replicate,axiom,
% 0.35/0.63      ! [N: nat,X: vEBT_VEBT] :
% 0.35/0.63        ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) )
% 0.35/0.63        = N ) ).
% 0.35/0.63  
% 0.35/0.63  % length_replicate
% 0.35/0.63  thf(fact_4481_length__replicate,axiom,
% 0.35/0.63      ! [N: nat,X: $o] :
% 0.35/0.63        ( ( size_size_list_o @ ( replicate_o @ N @ X ) )
% 0.35/0.63        = N ) ).
% 0.35/0.63  
% 0.35/0.63  % length_replicate
% 0.35/0.63  thf(fact_4482_length__replicate,axiom,
% 0.35/0.63      ! [N: nat,X: nat] :
% 0.35/0.63        ( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
% 0.35/0.63        = N ) ).
% 0.35/0.63  
% 0.35/0.63  % length_replicate
% 0.35/0.63  thf(fact_4483_length__replicate,axiom,
% 0.35/0.63      ! [N: nat,X: int] :
% 0.35/0.63        ( ( size_size_list_int @ ( replicate_int @ N @ X ) )
% 0.35/0.63        = N ) ).
% 0.35/0.63  
% 0.35/0.63  % length_replicate
% 0.35/0.63  thf(fact_4484_case__prod__conv,axiom,
% 0.35/0.63      ! [F: nat > nat > $o,A: nat,B: nat] :
% 0.35/0.63        ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 0.35/0.63        = ( F @ A @ B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prod_conv
% 0.35/0.63  thf(fact_4485_case__prod__conv,axiom,
% 0.35/0.63      ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
% 0.35/0.63        ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 0.35/0.63        = ( F @ A @ B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prod_conv
% 0.35/0.63  thf(fact_4486_case__prod__conv,axiom,
% 0.35/0.63      ! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
% 0.35/0.63        ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 0.35/0.63        = ( F @ A @ B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prod_conv
% 0.35/0.63  thf(fact_4487_case__prod__conv,axiom,
% 0.35/0.63      ! [F: int > int > product_prod_int_int,A: int,B: int] :
% 0.35/0.63        ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
% 0.35/0.63        = ( F @ A @ B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prod_conv
% 0.35/0.63  thf(fact_4488_case__prod__conv,axiom,
% 0.35/0.63      ! [F: int > int > $o,A: int,B: int] :
% 0.35/0.63        ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
% 0.35/0.63        = ( F @ A @ B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prod_conv
% 0.35/0.63  thf(fact_4489_take__bit__0,axiom,
% 0.35/0.63      ! [A: nat] :
% 0.35/0.63        ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
% 0.35/0.63        = zero_zero_nat ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_0
% 0.35/0.63  thf(fact_4490_take__bit__0,axiom,
% 0.35/0.63      ! [A: int] :
% 0.35/0.63        ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
% 0.35/0.63        = zero_zero_int ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_0
% 0.35/0.63  thf(fact_4491_in__set__replicate,axiom,
% 0.35/0.63      ! [X: product_prod_nat_nat,N: nat,Y: product_prod_nat_nat] :
% 0.35/0.63        ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ Y ) ) )
% 0.35/0.63        = ( ( X = Y )
% 0.35/0.63          & ( N != zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % in_set_replicate
% 0.35/0.63  thf(fact_4492_in__set__replicate,axiom,
% 0.35/0.63      ! [X: real,N: nat,Y: real] :
% 0.35/0.63        ( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N @ Y ) ) )
% 0.35/0.63        = ( ( X = Y )
% 0.35/0.63          & ( N != zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % in_set_replicate
% 0.35/0.63  thf(fact_4493_in__set__replicate,axiom,
% 0.35/0.63      ! [X: set_nat,N: nat,Y: set_nat] :
% 0.35/0.63        ( ( member_set_nat @ X @ ( set_set_nat2 @ ( replicate_set_nat @ N @ Y ) ) )
% 0.35/0.63        = ( ( X = Y )
% 0.35/0.63          & ( N != zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % in_set_replicate
% 0.35/0.63  thf(fact_4494_in__set__replicate,axiom,
% 0.35/0.63      ! [X: int,N: nat,Y: int] :
% 0.35/0.63        ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ Y ) ) )
% 0.35/0.63        = ( ( X = Y )
% 0.35/0.63          & ( N != zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % in_set_replicate
% 0.35/0.63  thf(fact_4495_in__set__replicate,axiom,
% 0.35/0.63      ! [X: nat,N: nat,Y: nat] :
% 0.35/0.63        ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
% 0.35/0.63        = ( ( X = Y )
% 0.35/0.63          & ( N != zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % in_set_replicate
% 0.35/0.63  thf(fact_4496_in__set__replicate,axiom,
% 0.35/0.63      ! [X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
% 0.35/0.63        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y ) ) )
% 0.35/0.63        = ( ( X = Y )
% 0.35/0.63          & ( N != zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % in_set_replicate
% 0.35/0.63  thf(fact_4497_Bex__set__replicate,axiom,
% 0.35/0.63      ! [N: nat,A: int,P: int > $o] :
% 0.35/0.63        ( ( ? [X3: int] :
% 0.35/0.63              ( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
% 0.35/0.63              & ( P @ X3 ) ) )
% 0.35/0.63        = ( ( P @ A )
% 0.35/0.63          & ( N != zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % Bex_set_replicate
% 0.35/0.63  thf(fact_4498_Bex__set__replicate,axiom,
% 0.35/0.63      ! [N: nat,A: nat,P: nat > $o] :
% 0.35/0.63        ( ( ? [X3: nat] :
% 0.35/0.63              ( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
% 0.35/0.63              & ( P @ X3 ) ) )
% 0.35/0.63        = ( ( P @ A )
% 0.35/0.63          & ( N != zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % Bex_set_replicate
% 0.35/0.63  thf(fact_4499_Bex__set__replicate,axiom,
% 0.35/0.63      ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.35/0.63        ( ( ? [X3: vEBT_VEBT] :
% 0.35/0.63              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
% 0.35/0.63              & ( P @ X3 ) ) )
% 0.35/0.63        = ( ( P @ A )
% 0.35/0.63          & ( N != zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % Bex_set_replicate
% 0.35/0.63  thf(fact_4500_Ball__set__replicate,axiom,
% 0.35/0.63      ! [N: nat,A: int,P: int > $o] :
% 0.35/0.63        ( ( ! [X3: int] :
% 0.35/0.63              ( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
% 0.35/0.63             => ( P @ X3 ) ) )
% 0.35/0.63        = ( ( P @ A )
% 0.35/0.63          | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % Ball_set_replicate
% 0.35/0.63  thf(fact_4501_Ball__set__replicate,axiom,
% 0.35/0.63      ! [N: nat,A: nat,P: nat > $o] :
% 0.35/0.63        ( ( ! [X3: nat] :
% 0.35/0.63              ( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
% 0.35/0.63             => ( P @ X3 ) ) )
% 0.35/0.63        = ( ( P @ A )
% 0.35/0.63          | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % Ball_set_replicate
% 0.35/0.63  thf(fact_4502_Ball__set__replicate,axiom,
% 0.35/0.63      ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.35/0.63        ( ( ! [X3: vEBT_VEBT] :
% 0.35/0.63              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
% 0.35/0.63             => ( P @ X3 ) ) )
% 0.35/0.63        = ( ( P @ A )
% 0.35/0.63          | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % Ball_set_replicate
% 0.35/0.63  thf(fact_4503_nth__replicate,axiom,
% 0.35/0.63      ! [I: nat,N: nat,X: nat] :
% 0.35/0.63        ( ( ord_less_nat @ I @ N )
% 0.35/0.63       => ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I )
% 0.35/0.63          = X ) ) ).
% 0.35/0.63  
% 0.35/0.63  % nth_replicate
% 0.35/0.63  thf(fact_4504_nth__replicate,axiom,
% 0.35/0.63      ! [I: nat,N: nat,X: int] :
% 0.35/0.63        ( ( ord_less_nat @ I @ N )
% 0.35/0.63       => ( ( nth_int @ ( replicate_int @ N @ X ) @ I )
% 0.35/0.63          = X ) ) ).
% 0.35/0.63  
% 0.35/0.63  % nth_replicate
% 0.35/0.63  thf(fact_4505_nth__replicate,axiom,
% 0.35/0.63      ! [I: nat,N: nat,X: vEBT_VEBT] :
% 0.35/0.63        ( ( ord_less_nat @ I @ N )
% 0.35/0.63       => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I )
% 0.35/0.63          = X ) ) ).
% 0.35/0.63  
% 0.35/0.63  % nth_replicate
% 0.35/0.63  thf(fact_4506_take__bit__of__1__eq__0__iff,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 0.35/0.63          = zero_zero_nat )
% 0.35/0.63        = ( N = zero_zero_nat ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_of_1_eq_0_iff
% 0.35/0.63  thf(fact_4507_take__bit__of__1__eq__0__iff,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 0.35/0.63          = zero_zero_int )
% 0.35/0.63        = ( N = zero_zero_nat ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_of_1_eq_0_iff
% 0.35/0.63  thf(fact_4508_even__take__bit__eq,axiom,
% 0.35/0.63      ! [N: nat,A: code_integer] :
% 0.35/0.63        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N @ A ) )
% 0.35/0.63        = ( ( N = zero_zero_nat )
% 0.35/0.63          | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_take_bit_eq
% 0.35/0.63  thf(fact_4509_even__take__bit__eq,axiom,
% 0.35/0.63      ! [N: nat,A: nat] :
% 0.35/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 0.35/0.63        = ( ( N = zero_zero_nat )
% 0.35/0.63          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_take_bit_eq
% 0.35/0.63  thf(fact_4510_even__take__bit__eq,axiom,
% 0.35/0.63      ! [N: nat,A: int] :
% 0.35/0.63        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 0.35/0.63        = ( ( N = zero_zero_nat )
% 0.35/0.63          | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_take_bit_eq
% 0.35/0.63  thf(fact_4511_take__bit__Suc__0,axiom,
% 0.35/0.63      ! [A: code_integer] :
% 0.35/0.63        ( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A )
% 0.35/0.63        = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_Suc_0
% 0.35/0.63  thf(fact_4512_take__bit__Suc__0,axiom,
% 0.35/0.63      ! [A: nat] :
% 0.35/0.63        ( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
% 0.35/0.63        = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_Suc_0
% 0.35/0.63  thf(fact_4513_take__bit__Suc__0,axiom,
% 0.35/0.63      ! [A: int] :
% 0.35/0.63        ( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
% 0.35/0.63        = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_Suc_0
% 0.35/0.63  thf(fact_4514_dvd__antisym,axiom,
% 0.35/0.63      ! [M: nat,N: nat] :
% 0.35/0.63        ( ( dvd_dvd_nat @ M @ N )
% 0.35/0.63       => ( ( dvd_dvd_nat @ N @ M )
% 0.35/0.63         => ( M = N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % dvd_antisym
% 0.35/0.63  thf(fact_4515_take__bit__flip__bit__eq,axiom,
% 0.35/0.63      ! [N: nat,M: nat,A: nat] :
% 0.35/0.63        ( ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.63         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.35/0.63            = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 0.35/0.63        & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.35/0.63         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.35/0.63            = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_flip_bit_eq
% 0.35/0.63  thf(fact_4516_take__bit__flip__bit__eq,axiom,
% 0.35/0.63      ! [N: nat,M: nat,A: int] :
% 0.35/0.63        ( ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.63         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.35/0.63            = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.35/0.63        & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.35/0.63         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.35/0.63            = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_flip_bit_eq
% 0.35/0.63  thf(fact_4517_take__bit__tightened,axiom,
% 0.35/0.63      ! [N: nat,A: nat,B: nat,M: nat] :
% 0.35/0.63        ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.35/0.63          = ( bit_se2925701944663578781it_nat @ N @ B ) )
% 0.35/0.63       => ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.63         => ( ( bit_se2925701944663578781it_nat @ M @ A )
% 0.35/0.63            = ( bit_se2925701944663578781it_nat @ M @ B ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_tightened
% 0.35/0.63  thf(fact_4518_take__bit__tightened,axiom,
% 0.35/0.63      ! [N: nat,A: int,B: int,M: nat] :
% 0.35/0.63        ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.35/0.63          = ( bit_se2923211474154528505it_int @ N @ B ) )
% 0.35/0.63       => ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.63         => ( ( bit_se2923211474154528505it_int @ M @ A )
% 0.35/0.63            = ( bit_se2923211474154528505it_int @ M @ B ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_tightened
% 0.35/0.63  thf(fact_4519_take__bit__tightened__less__eq__nat,axiom,
% 0.35/0.63      ! [M: nat,N: nat,Q4: nat] :
% 0.35/0.63        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.63       => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q4 ) @ ( bit_se2925701944663578781it_nat @ N @ Q4 ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_tightened_less_eq_nat
% 0.35/0.63  thf(fact_4520_take__bit__nat__less__eq__self,axiom,
% 0.35/0.63      ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_nat_less_eq_self
% 0.35/0.63  thf(fact_4521_old_Oprod_Ocase,axiom,
% 0.35/0.63      ! [F: nat > nat > $o,X1: nat,X22: nat] :
% 0.35/0.63        ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 0.35/0.63        = ( F @ X1 @ X22 ) ) ).
% 0.35/0.63  
% 0.35/0.63  % old.prod.case
% 0.35/0.63  thf(fact_4522_old_Oprod_Ocase,axiom,
% 0.35/0.63      ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X22: nat] :
% 0.35/0.63        ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 0.35/0.63        = ( F @ X1 @ X22 ) ) ).
% 0.35/0.63  
% 0.35/0.63  % old.prod.case
% 0.35/0.63  thf(fact_4523_old_Oprod_Ocase,axiom,
% 0.35/0.63      ! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X22: nat] :
% 0.35/0.63        ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 0.35/0.63        = ( F @ X1 @ X22 ) ) ).
% 0.35/0.63  
% 0.35/0.63  % old.prod.case
% 0.35/0.63  thf(fact_4524_old_Oprod_Ocase,axiom,
% 0.35/0.63      ! [F: int > int > product_prod_int_int,X1: int,X22: int] :
% 0.35/0.63        ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
% 0.35/0.63        = ( F @ X1 @ X22 ) ) ).
% 0.35/0.63  
% 0.35/0.63  % old.prod.case
% 0.35/0.63  thf(fact_4525_old_Oprod_Ocase,axiom,
% 0.35/0.63      ! [F: int > int > $o,X1: int,X22: int] :
% 0.35/0.63        ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
% 0.35/0.63        = ( F @ X1 @ X22 ) ) ).
% 0.35/0.63  
% 0.35/0.63  % old.prod.case
% 0.35/0.63  thf(fact_4526_split__cong,axiom,
% 0.35/0.63      ! [Q4: product_prod_nat_nat,F: nat > nat > $o,G: nat > nat > $o,P4: product_prod_nat_nat] :
% 0.35/0.63        ( ! [X4: nat,Y3: nat] :
% 0.35/0.63            ( ( ( product_Pair_nat_nat @ X4 @ Y3 )
% 0.35/0.63              = Q4 )
% 0.35/0.63           => ( ( F @ X4 @ Y3 )
% 0.35/0.63              = ( G @ X4 @ Y3 ) ) )
% 0.35/0.63       => ( ( P4 = Q4 )
% 0.35/0.63         => ( ( produc6081775807080527818_nat_o @ F @ P4 )
% 0.35/0.63            = ( produc6081775807080527818_nat_o @ G @ Q4 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % split_cong
% 0.35/0.63  thf(fact_4527_split__cong,axiom,
% 0.35/0.63      ! [Q4: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: nat > nat > product_prod_nat_nat > product_prod_nat_nat,P4: product_prod_nat_nat] :
% 0.35/0.63        ( ! [X4: nat,Y3: nat] :
% 0.35/0.63            ( ( ( product_Pair_nat_nat @ X4 @ Y3 )
% 0.35/0.63              = Q4 )
% 0.35/0.63           => ( ( F @ X4 @ Y3 )
% 0.35/0.63              = ( G @ X4 @ Y3 ) ) )
% 0.35/0.63       => ( ( P4 = Q4 )
% 0.35/0.63         => ( ( produc27273713700761075at_nat @ F @ P4 )
% 0.35/0.63            = ( produc27273713700761075at_nat @ G @ Q4 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % split_cong
% 0.35/0.63  thf(fact_4528_split__cong,axiom,
% 0.35/0.63      ! [Q4: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > $o,G: nat > nat > product_prod_nat_nat > $o,P4: product_prod_nat_nat] :
% 0.35/0.63        ( ! [X4: nat,Y3: nat] :
% 0.35/0.63            ( ( ( product_Pair_nat_nat @ X4 @ Y3 )
% 0.35/0.63              = Q4 )
% 0.35/0.63           => ( ( F @ X4 @ Y3 )
% 0.35/0.63              = ( G @ X4 @ Y3 ) ) )
% 0.35/0.63       => ( ( P4 = Q4 )
% 0.35/0.63         => ( ( produc8739625826339149834_nat_o @ F @ P4 )
% 0.35/0.63            = ( produc8739625826339149834_nat_o @ G @ Q4 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % split_cong
% 0.35/0.63  thf(fact_4529_split__cong,axiom,
% 0.35/0.63      ! [Q4: product_prod_int_int,F: int > int > product_prod_int_int,G: int > int > product_prod_int_int,P4: product_prod_int_int] :
% 0.35/0.63        ( ! [X4: int,Y3: int] :
% 0.35/0.63            ( ( ( product_Pair_int_int @ X4 @ Y3 )
% 0.35/0.63              = Q4 )
% 0.35/0.63           => ( ( F @ X4 @ Y3 )
% 0.35/0.63              = ( G @ X4 @ Y3 ) ) )
% 0.35/0.63       => ( ( P4 = Q4 )
% 0.35/0.63         => ( ( produc4245557441103728435nt_int @ F @ P4 )
% 0.35/0.63            = ( produc4245557441103728435nt_int @ G @ Q4 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % split_cong
% 0.35/0.63  thf(fact_4530_split__cong,axiom,
% 0.35/0.63      ! [Q4: product_prod_int_int,F: int > int > $o,G: int > int > $o,P4: product_prod_int_int] :
% 0.35/0.63        ( ! [X4: int,Y3: int] :
% 0.35/0.63            ( ( ( product_Pair_int_int @ X4 @ Y3 )
% 0.35/0.63              = Q4 )
% 0.35/0.63           => ( ( F @ X4 @ Y3 )
% 0.35/0.63              = ( G @ X4 @ Y3 ) ) )
% 0.35/0.63       => ( ( P4 = Q4 )
% 0.35/0.63         => ( ( produc4947309494688390418_int_o @ F @ P4 )
% 0.35/0.63            = ( produc4947309494688390418_int_o @ G @ Q4 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % split_cong
% 0.35/0.63  thf(fact_4531_case__prodE2,axiom,
% 0.35/0.63      ! [Q: $o > $o,P: nat > nat > $o,Z2: product_prod_nat_nat] :
% 0.35/0.63        ( ( Q @ ( produc6081775807080527818_nat_o @ P @ Z2 ) )
% 0.35/0.63       => ~ ! [X4: nat,Y3: nat] :
% 0.35/0.63              ( ( Z2
% 0.35/0.63                = ( product_Pair_nat_nat @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( Q @ ( P @ X4 @ Y3 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodE2
% 0.35/0.63  thf(fact_4532_case__prodE2,axiom,
% 0.35/0.63      ! [Q: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z2: product_prod_nat_nat] :
% 0.35/0.63        ( ( Q @ ( produc27273713700761075at_nat @ P @ Z2 ) )
% 0.35/0.63       => ~ ! [X4: nat,Y3: nat] :
% 0.35/0.63              ( ( Z2
% 0.35/0.63                = ( product_Pair_nat_nat @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( Q @ ( P @ X4 @ Y3 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodE2
% 0.35/0.63  thf(fact_4533_case__prodE2,axiom,
% 0.35/0.63      ! [Q: ( product_prod_nat_nat > $o ) > $o,P: nat > nat > product_prod_nat_nat > $o,Z2: product_prod_nat_nat] :
% 0.35/0.63        ( ( Q @ ( produc8739625826339149834_nat_o @ P @ Z2 ) )
% 0.35/0.63       => ~ ! [X4: nat,Y3: nat] :
% 0.35/0.63              ( ( Z2
% 0.35/0.63                = ( product_Pair_nat_nat @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( Q @ ( P @ X4 @ Y3 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodE2
% 0.35/0.63  thf(fact_4534_case__prodE2,axiom,
% 0.35/0.63      ! [Q: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z2: product_prod_int_int] :
% 0.35/0.63        ( ( Q @ ( produc4245557441103728435nt_int @ P @ Z2 ) )
% 0.35/0.63       => ~ ! [X4: int,Y3: int] :
% 0.35/0.63              ( ( Z2
% 0.35/0.63                = ( product_Pair_int_int @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( Q @ ( P @ X4 @ Y3 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodE2
% 0.35/0.63  thf(fact_4535_case__prodE2,axiom,
% 0.35/0.63      ! [Q: $o > $o,P: int > int > $o,Z2: product_prod_int_int] :
% 0.35/0.63        ( ( Q @ ( produc4947309494688390418_int_o @ P @ Z2 ) )
% 0.35/0.63       => ~ ! [X4: int,Y3: int] :
% 0.35/0.63              ( ( Z2
% 0.35/0.63                = ( product_Pair_int_int @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( Q @ ( P @ X4 @ Y3 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodE2
% 0.35/0.63  thf(fact_4536_case__prod__eta,axiom,
% 0.35/0.63      ! [F: product_prod_nat_nat > $o] :
% 0.35/0.63        ( ( produc6081775807080527818_nat_o
% 0.35/0.63          @ ^ [X3: nat,Y2: nat] : ( F @ ( product_Pair_nat_nat @ X3 @ Y2 ) ) )
% 0.35/0.63        = F ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prod_eta
% 0.35/0.63  thf(fact_4537_case__prod__eta,axiom,
% 0.35/0.63      ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 0.35/0.63        ( ( produc27273713700761075at_nat
% 0.35/0.63          @ ^ [X3: nat,Y2: nat] : ( F @ ( product_Pair_nat_nat @ X3 @ Y2 ) ) )
% 0.35/0.63        = F ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prod_eta
% 0.35/0.63  thf(fact_4538_case__prod__eta,axiom,
% 0.35/0.63      ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 0.35/0.63        ( ( produc8739625826339149834_nat_o
% 0.35/0.63          @ ^ [X3: nat,Y2: nat] : ( F @ ( product_Pair_nat_nat @ X3 @ Y2 ) ) )
% 0.35/0.63        = F ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prod_eta
% 0.35/0.63  thf(fact_4539_case__prod__eta,axiom,
% 0.35/0.63      ! [F: product_prod_int_int > product_prod_int_int] :
% 0.35/0.63        ( ( produc4245557441103728435nt_int
% 0.35/0.63          @ ^ [X3: int,Y2: int] : ( F @ ( product_Pair_int_int @ X3 @ Y2 ) ) )
% 0.35/0.63        = F ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prod_eta
% 0.35/0.63  thf(fact_4540_case__prod__eta,axiom,
% 0.35/0.63      ! [F: product_prod_int_int > $o] :
% 0.35/0.63        ( ( produc4947309494688390418_int_o
% 0.35/0.63          @ ^ [X3: int,Y2: int] : ( F @ ( product_Pair_int_int @ X3 @ Y2 ) ) )
% 0.35/0.63        = F ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prod_eta
% 0.35/0.63  thf(fact_4541_cond__case__prod__eta,axiom,
% 0.35/0.63      ! [F: nat > nat > $o,G: product_prod_nat_nat > $o] :
% 0.35/0.63        ( ! [X4: nat,Y3: nat] :
% 0.35/0.63            ( ( F @ X4 @ Y3 )
% 0.35/0.63            = ( G @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
% 0.35/0.63       => ( ( produc6081775807080527818_nat_o @ F )
% 0.35/0.63          = G ) ) ).
% 0.35/0.63  
% 0.35/0.63  % cond_case_prod_eta
% 0.35/0.63  thf(fact_4542_cond__case__prod__eta,axiom,
% 0.35/0.63      ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 0.35/0.63        ( ! [X4: nat,Y3: nat] :
% 0.35/0.63            ( ( F @ X4 @ Y3 )
% 0.35/0.63            = ( G @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
% 0.35/0.63       => ( ( produc27273713700761075at_nat @ F )
% 0.35/0.63          = G ) ) ).
% 0.35/0.63  
% 0.35/0.63  % cond_case_prod_eta
% 0.35/0.63  thf(fact_4543_cond__case__prod__eta,axiom,
% 0.35/0.63      ! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 0.35/0.63        ( ! [X4: nat,Y3: nat] :
% 0.35/0.63            ( ( F @ X4 @ Y3 )
% 0.35/0.63            = ( G @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
% 0.35/0.63       => ( ( produc8739625826339149834_nat_o @ F )
% 0.35/0.63          = G ) ) ).
% 0.35/0.63  
% 0.35/0.63  % cond_case_prod_eta
% 0.35/0.63  thf(fact_4544_cond__case__prod__eta,axiom,
% 0.35/0.63      ! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
% 0.35/0.63        ( ! [X4: int,Y3: int] :
% 0.35/0.63            ( ( F @ X4 @ Y3 )
% 0.35/0.63            = ( G @ ( product_Pair_int_int @ X4 @ Y3 ) ) )
% 0.35/0.63       => ( ( produc4245557441103728435nt_int @ F )
% 0.35/0.63          = G ) ) ).
% 0.35/0.63  
% 0.35/0.63  % cond_case_prod_eta
% 0.35/0.63  thf(fact_4545_cond__case__prod__eta,axiom,
% 0.35/0.63      ! [F: int > int > $o,G: product_prod_int_int > $o] :
% 0.35/0.63        ( ! [X4: int,Y3: int] :
% 0.35/0.63            ( ( F @ X4 @ Y3 )
% 0.35/0.63            = ( G @ ( product_Pair_int_int @ X4 @ Y3 ) ) )
% 0.35/0.63       => ( ( produc4947309494688390418_int_o @ F )
% 0.35/0.63          = G ) ) ).
% 0.35/0.63  
% 0.35/0.63  % cond_case_prod_eta
% 0.35/0.63  thf(fact_4546_take__bit__tightened__less__eq__int,axiom,
% 0.35/0.63      ! [M: nat,N: nat,K: int] :
% 0.35/0.63        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.63       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_tightened_less_eq_int
% 0.35/0.63  thf(fact_4547_not__take__bit__negative,axiom,
% 0.35/0.63      ! [N: nat,K: int] :
% 0.35/0.63        ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).
% 0.35/0.63  
% 0.35/0.63  % not_take_bit_negative
% 0.35/0.63  thf(fact_4548_take__bit__int__greater__self__iff,axiom,
% 0.35/0.63      ! [K: int,N: nat] :
% 0.35/0.63        ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.35/0.63        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_int_greater_self_iff
% 0.35/0.63  thf(fact_4549_take__bit__unset__bit__eq,axiom,
% 0.35/0.63      ! [N: nat,M: nat,A: nat] :
% 0.35/0.63        ( ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.63         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.35/0.63            = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 0.35/0.63        & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.35/0.63         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.35/0.63            = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_unset_bit_eq
% 0.35/0.63  thf(fact_4550_take__bit__unset__bit__eq,axiom,
% 0.35/0.63      ! [N: nat,M: nat,A: int] :
% 0.35/0.63        ( ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.63         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.35/0.63            = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.35/0.63        & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.35/0.63         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.35/0.63            = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_unset_bit_eq
% 0.35/0.63  thf(fact_4551_take__bit__set__bit__eq,axiom,
% 0.35/0.63      ! [N: nat,M: nat,A: nat] :
% 0.35/0.63        ( ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.63         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.35/0.63            = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 0.35/0.63        & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.35/0.63         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.35/0.63            = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_set_bit_eq
% 0.35/0.63  thf(fact_4552_take__bit__set__bit__eq,axiom,
% 0.35/0.63      ! [N: nat,M: nat,A: int] :
% 0.35/0.63        ( ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.63         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.35/0.63            = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.35/0.63        & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.35/0.63         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.35/0.63            = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_set_bit_eq
% 0.35/0.63  thf(fact_4553_replicate__eqI,axiom,
% 0.35/0.63      ! [Xs: list_P6011104703257516679at_nat,N: nat,X: product_prod_nat_nat] :
% 0.35/0.63        ( ( ( size_s5460976970255530739at_nat @ Xs )
% 0.35/0.63          = N )
% 0.35/0.63       => ( ! [Y3: product_prod_nat_nat] :
% 0.35/0.63              ( ( member8440522571783428010at_nat @ Y3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.35/0.63             => ( Y3 = X ) )
% 0.35/0.63         => ( Xs
% 0.35/0.63            = ( replic4235873036481779905at_nat @ N @ X ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % replicate_eqI
% 0.35/0.63  thf(fact_4554_replicate__eqI,axiom,
% 0.35/0.63      ! [Xs: list_real,N: nat,X: real] :
% 0.35/0.63        ( ( ( size_size_list_real @ Xs )
% 0.35/0.63          = N )
% 0.35/0.63       => ( ! [Y3: real] :
% 0.35/0.63              ( ( member_real @ Y3 @ ( set_real2 @ Xs ) )
% 0.35/0.63             => ( Y3 = X ) )
% 0.35/0.63         => ( Xs
% 0.35/0.63            = ( replicate_real @ N @ X ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % replicate_eqI
% 0.35/0.63  thf(fact_4555_replicate__eqI,axiom,
% 0.35/0.63      ! [Xs: list_set_nat,N: nat,X: set_nat] :
% 0.35/0.63        ( ( ( size_s3254054031482475050et_nat @ Xs )
% 0.35/0.63          = N )
% 0.35/0.63       => ( ! [Y3: set_nat] :
% 0.35/0.63              ( ( member_set_nat @ Y3 @ ( set_set_nat2 @ Xs ) )
% 0.35/0.63             => ( Y3 = X ) )
% 0.35/0.63         => ( Xs
% 0.35/0.63            = ( replicate_set_nat @ N @ X ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % replicate_eqI
% 0.35/0.63  thf(fact_4556_replicate__eqI,axiom,
% 0.35/0.63      ! [Xs: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
% 0.35/0.63        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.35/0.63          = N )
% 0.35/0.63       => ( ! [Y3: vEBT_VEBT] :
% 0.35/0.63              ( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.35/0.63             => ( Y3 = X ) )
% 0.35/0.63         => ( Xs
% 0.35/0.63            = ( replicate_VEBT_VEBT @ N @ X ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % replicate_eqI
% 0.35/0.63  thf(fact_4557_replicate__eqI,axiom,
% 0.35/0.63      ! [Xs: list_o,N: nat,X: $o] :
% 0.35/0.63        ( ( ( size_size_list_o @ Xs )
% 0.35/0.63          = N )
% 0.35/0.63       => ( ! [Y3: $o] :
% 0.35/0.63              ( ( member_o @ Y3 @ ( set_o2 @ Xs ) )
% 0.35/0.63             => ( Y3 = X ) )
% 0.35/0.63         => ( Xs
% 0.35/0.63            = ( replicate_o @ N @ X ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % replicate_eqI
% 0.35/0.63  thf(fact_4558_replicate__eqI,axiom,
% 0.35/0.63      ! [Xs: list_nat,N: nat,X: nat] :
% 0.35/0.63        ( ( ( size_size_list_nat @ Xs )
% 0.35/0.63          = N )
% 0.35/0.63       => ( ! [Y3: nat] :
% 0.35/0.63              ( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
% 0.35/0.63             => ( Y3 = X ) )
% 0.35/0.63         => ( Xs
% 0.35/0.63            = ( replicate_nat @ N @ X ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % replicate_eqI
% 0.35/0.63  thf(fact_4559_replicate__eqI,axiom,
% 0.35/0.63      ! [Xs: list_int,N: nat,X: int] :
% 0.35/0.63        ( ( ( size_size_list_int @ Xs )
% 0.35/0.63          = N )
% 0.35/0.63       => ( ! [Y3: int] :
% 0.35/0.63              ( ( member_int @ Y3 @ ( set_int2 @ Xs ) )
% 0.35/0.63             => ( Y3 = X ) )
% 0.35/0.63         => ( Xs
% 0.35/0.63            = ( replicate_int @ N @ X ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % replicate_eqI
% 0.35/0.63  thf(fact_4560_replicate__length__same,axiom,
% 0.35/0.63      ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 0.35/0.63        ( ! [X4: vEBT_VEBT] :
% 0.35/0.63            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.35/0.63           => ( X4 = X ) )
% 0.35/0.63       => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs ) @ X )
% 0.35/0.63          = Xs ) ) ).
% 0.35/0.63  
% 0.35/0.63  % replicate_length_same
% 0.35/0.63  thf(fact_4561_replicate__length__same,axiom,
% 0.35/0.63      ! [Xs: list_o,X: $o] :
% 0.35/0.63        ( ! [X4: $o] :
% 0.35/0.63            ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 0.35/0.63           => ( X4 = X ) )
% 0.35/0.63       => ( ( replicate_o @ ( size_size_list_o @ Xs ) @ X )
% 0.35/0.63          = Xs ) ) ).
% 0.35/0.63  
% 0.35/0.63  % replicate_length_same
% 0.35/0.63  thf(fact_4562_replicate__length__same,axiom,
% 0.35/0.63      ! [Xs: list_nat,X: nat] :
% 0.35/0.63        ( ! [X4: nat] :
% 0.35/0.63            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 0.35/0.63           => ( X4 = X ) )
% 0.35/0.63       => ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X )
% 0.35/0.63          = Xs ) ) ).
% 0.35/0.63  
% 0.35/0.63  % replicate_length_same
% 0.35/0.63  thf(fact_4563_replicate__length__same,axiom,
% 0.35/0.63      ! [Xs: list_int,X: int] :
% 0.35/0.63        ( ! [X4: int] :
% 0.35/0.63            ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 0.35/0.63           => ( X4 = X ) )
% 0.35/0.63       => ( ( replicate_int @ ( size_size_list_int @ Xs ) @ X )
% 0.35/0.63          = Xs ) ) ).
% 0.35/0.63  
% 0.35/0.63  % replicate_length_same
% 0.35/0.63  thf(fact_4564_take__bit__nat__eq__self,axiom,
% 0.35/0.63      ! [M: nat,N: nat] :
% 0.35/0.63        ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63       => ( ( bit_se2925701944663578781it_nat @ N @ M )
% 0.35/0.63          = M ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_nat_eq_self
% 0.35/0.63  thf(fact_4565_take__bit__nat__less__exp,axiom,
% 0.35/0.63      ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_nat_less_exp
% 0.35/0.63  thf(fact_4566_take__bit__nat__eq__self__iff,axiom,
% 0.35/0.63      ! [N: nat,M: nat] :
% 0.35/0.63        ( ( ( bit_se2925701944663578781it_nat @ N @ M )
% 0.35/0.63          = M )
% 0.35/0.63        = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_nat_eq_self_iff
% 0.35/0.63  thf(fact_4567_take__bit__int__less__exp,axiom,
% 0.35/0.63      ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_int_less_exp
% 0.35/0.63  thf(fact_4568_num_Osize__gen_I1_J,axiom,
% 0.35/0.63      ( ( size_num @ one )
% 0.35/0.63      = zero_zero_nat ) ).
% 0.35/0.63  
% 0.35/0.63  % num.size_gen(1)
% 0.35/0.63  thf(fact_4569_take__bit__eq__0__iff,axiom,
% 0.35/0.63      ! [N: nat,A: code_integer] :
% 0.35/0.63        ( ( ( bit_se1745604003318907178nteger @ N @ A )
% 0.35/0.63          = zero_z3403309356797280102nteger )
% 0.35/0.63        = ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_eq_0_iff
% 0.35/0.63  thf(fact_4570_take__bit__eq__0__iff,axiom,
% 0.35/0.63      ! [N: nat,A: nat] :
% 0.35/0.63        ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.35/0.63          = zero_zero_nat )
% 0.35/0.63        = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_eq_0_iff
% 0.35/0.63  thf(fact_4571_take__bit__eq__0__iff,axiom,
% 0.35/0.63      ! [N: nat,A: int] :
% 0.35/0.63        ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.35/0.63          = zero_zero_int )
% 0.35/0.63        = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_eq_0_iff
% 0.35/0.63  thf(fact_4572_take__bit__nat__less__self__iff,axiom,
% 0.35/0.63      ! [N: nat,M: nat] :
% 0.35/0.63        ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
% 0.35/0.63        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_nat_less_self_iff
% 0.35/0.63  thf(fact_4573_take__bit__int__less__self__iff,axiom,
% 0.35/0.63      ! [N: nat,K: int] :
% 0.35/0.63        ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 0.35/0.63        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_int_less_self_iff
% 0.35/0.63  thf(fact_4574_take__bit__int__greater__eq__self__iff,axiom,
% 0.35/0.63      ! [K: int,N: nat] :
% 0.35/0.63        ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.35/0.63        = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_int_greater_eq_self_iff
% 0.35/0.63  thf(fact_4575_take__bit__int__eq__self,axiom,
% 0.35/0.63      ! [K: int,N: nat] :
% 0.35/0.63        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.63       => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63         => ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.35/0.63            = K ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_int_eq_self
% 0.35/0.63  thf(fact_4576_take__bit__int__eq__self__iff,axiom,
% 0.35/0.63      ! [N: nat,K: int] :
% 0.35/0.63        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.35/0.63          = K )
% 0.35/0.63        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.63          & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_int_eq_self_iff
% 0.35/0.63  thf(fact_4577_take__bit__int__less__eq,axiom,
% 0.35/0.63      ! [N: nat,K: int] :
% 0.35/0.63        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 0.35/0.63       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.63         => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_int_less_eq
% 0.35/0.63  thf(fact_4578_take__bit__int__greater__eq,axiom,
% 0.35/0.63      ! [K: int,N: nat] :
% 0.35/0.63        ( ( ord_less_int @ K @ zero_zero_int )
% 0.35/0.63       => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_int_greater_eq
% 0.35/0.63  thf(fact_4579_stable__imp__take__bit__eq,axiom,
% 0.35/0.63      ! [A: code_integer,N: nat] :
% 0.35/0.63        ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.63          = A )
% 0.35/0.63       => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.35/0.63           => ( ( bit_se1745604003318907178nteger @ N @ A )
% 0.35/0.63              = zero_z3403309356797280102nteger ) )
% 0.35/0.63          & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.35/0.63           => ( ( bit_se1745604003318907178nteger @ N @ A )
% 0.35/0.63              = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % stable_imp_take_bit_eq
% 0.35/0.63  thf(fact_4580_stable__imp__take__bit__eq,axiom,
% 0.35/0.63      ! [A: nat,N: nat] :
% 0.35/0.63        ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.63          = A )
% 0.35/0.63       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.35/0.63           => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.35/0.63              = zero_zero_nat ) )
% 0.35/0.63          & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.35/0.63           => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.35/0.63              = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % stable_imp_take_bit_eq
% 0.35/0.63  thf(fact_4581_stable__imp__take__bit__eq,axiom,
% 0.35/0.63      ! [A: int,N: nat] :
% 0.35/0.63        ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.63          = A )
% 0.35/0.63       => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.35/0.63           => ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.35/0.63              = zero_zero_int ) )
% 0.35/0.63          & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.35/0.63           => ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.35/0.63              = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % stable_imp_take_bit_eq
% 0.35/0.63  thf(fact_4582_option_Osize__gen_I1_J,axiom,
% 0.35/0.63      ! [X: nat > nat] :
% 0.35/0.63        ( ( size_option_nat @ X @ none_nat )
% 0.35/0.63        = ( suc @ zero_zero_nat ) ) ).
% 0.35/0.63  
% 0.35/0.63  % option.size_gen(1)
% 0.35/0.63  thf(fact_4583_option_Osize__gen_I1_J,axiom,
% 0.35/0.63      ! [X: product_prod_nat_nat > nat] :
% 0.35/0.63        ( ( size_o8335143837870341156at_nat @ X @ none_P5556105721700978146at_nat )
% 0.35/0.63        = ( suc @ zero_zero_nat ) ) ).
% 0.35/0.63  
% 0.35/0.63  % option.size_gen(1)
% 0.35/0.63  thf(fact_4584_option_Osize__gen_I1_J,axiom,
% 0.35/0.63      ! [X: num > nat] :
% 0.35/0.63        ( ( size_option_num @ X @ none_num )
% 0.35/0.63        = ( suc @ zero_zero_nat ) ) ).
% 0.35/0.63  
% 0.35/0.63  % option.size_gen(1)
% 0.35/0.63  thf(fact_4585_divmod__step__nat__def,axiom,
% 0.35/0.63      ( unique5026877609467782581ep_nat
% 0.35/0.63      = ( ^ [L2: num] :
% 0.35/0.63            ( produc2626176000494625587at_nat
% 0.35/0.63            @ ^ [Q5: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_step_nat_def
% 0.35/0.63  thf(fact_4586_divmod__step__int__def,axiom,
% 0.35/0.63      ( unique5024387138958732305ep_int
% 0.35/0.63      = ( ^ [L2: num] :
% 0.35/0.63            ( produc4245557441103728435nt_int
% 0.35/0.63            @ ^ [Q5: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_step_int_def
% 0.35/0.63  thf(fact_4587_divmod__algorithm__code_I5_J,axiom,
% 0.35/0.63      ! [M: num,N: num] :
% 0.35/0.63        ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.35/0.63        = ( produc4245557441103728435nt_int
% 0.35/0.63          @ ^ [Q5: int,R5: int] : ( product_Pair_int_int @ Q5 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
% 0.35/0.63          @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_algorithm_code(5)
% 0.35/0.63  thf(fact_4588_divmod__algorithm__code_I5_J,axiom,
% 0.35/0.63      ! [M: num,N: num] :
% 0.35/0.63        ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.35/0.63        = ( produc2626176000494625587at_nat
% 0.35/0.63          @ ^ [Q5: nat,R5: nat] : ( product_Pair_nat_nat @ Q5 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
% 0.35/0.63          @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_algorithm_code(5)
% 0.35/0.63  thf(fact_4589_divmod__algorithm__code_I5_J,axiom,
% 0.35/0.63      ! [M: num,N: num] :
% 0.35/0.63        ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.35/0.63        = ( produc6916734918728496179nteger
% 0.35/0.63          @ ^ [Q5: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
% 0.35/0.63          @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_algorithm_code(5)
% 0.35/0.63  thf(fact_4590_divmod__nat__if,axiom,
% 0.35/0.63      ( divmod_nat
% 0.35/0.63      = ( ^ [M2: nat,N2: nat] :
% 0.35/0.63            ( if_Pro6206227464963214023at_nat
% 0.35/0.63            @ ( ( N2 = zero_zero_nat )
% 0.35/0.63              | ( ord_less_nat @ M2 @ N2 ) )
% 0.35/0.63            @ ( product_Pair_nat_nat @ zero_zero_nat @ M2 )
% 0.35/0.63            @ ( produc2626176000494625587at_nat
% 0.35/0.63              @ ^ [Q5: nat] : ( product_Pair_nat_nat @ ( suc @ Q5 ) )
% 0.35/0.63              @ ( divmod_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_nat_if
% 0.35/0.63  thf(fact_4591_vebt__buildup_Opelims,axiom,
% 0.35/0.63      ! [X: nat,Y: vEBT_VEBT] :
% 0.35/0.63        ( ( ( vEBT_vebt_buildup @ X )
% 0.35/0.63          = Y )
% 0.35/0.63       => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
% 0.35/0.63         => ( ( ( X = zero_zero_nat )
% 0.35/0.63             => ( ( Y
% 0.35/0.63                  = ( vEBT_Leaf @ $false @ $false ) )
% 0.35/0.63               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 0.35/0.63           => ( ( ( X
% 0.35/0.63                  = ( suc @ zero_zero_nat ) )
% 0.35/0.63               => ( ( Y
% 0.35/0.63                    = ( vEBT_Leaf @ $false @ $false ) )
% 0.35/0.63                 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 0.35/0.63             => ~ ! [Va2: nat] :
% 0.35/0.63                    ( ( X
% 0.35/0.63                      = ( suc @ ( suc @ Va2 ) ) )
% 0.35/0.63                   => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.35/0.63                         => ( Y
% 0.35/0.63                            = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 0.35/0.63                        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.35/0.63                         => ( Y
% 0.35/0.63                            = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.35/0.63                     => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % vebt_buildup.pelims
% 0.35/0.63  thf(fact_4592_flip__bit__0,axiom,
% 0.35/0.63      ! [A: code_integer] :
% 0.35/0.63        ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
% 0.35/0.63        = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % flip_bit_0
% 0.35/0.63  thf(fact_4593_flip__bit__0,axiom,
% 0.35/0.63      ! [A: nat] :
% 0.35/0.63        ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
% 0.35/0.63        = ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % flip_bit_0
% 0.35/0.63  thf(fact_4594_flip__bit__0,axiom,
% 0.35/0.63      ! [A: int] :
% 0.35/0.63        ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
% 0.35/0.63        = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % flip_bit_0
% 0.35/0.63  thf(fact_4595_set__decode__0,axiom,
% 0.35/0.63      ! [X: nat] :
% 0.35/0.63        ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
% 0.35/0.63        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % set_decode_0
% 0.35/0.63  thf(fact_4596_even__set__encode__iff,axiom,
% 0.35/0.63      ! [A2: set_nat] :
% 0.35/0.63        ( ( finite_finite_nat @ A2 )
% 0.35/0.63       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
% 0.35/0.63          = ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % even_set_encode_iff
% 0.35/0.63  thf(fact_4597_one__mod__2__pow__eq,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63        = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % one_mod_2_pow_eq
% 0.35/0.63  thf(fact_4598_one__mod__2__pow__eq,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % one_mod_2_pow_eq
% 0.35/0.63  thf(fact_4599_one__mod__2__pow__eq,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63        = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % one_mod_2_pow_eq
% 0.35/0.63  thf(fact_4600_case__prodI2,axiom,
% 0.35/0.63      ! [P4: produc8763457246119570046nteger,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
% 0.35/0.63        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 0.35/0.63            ( ( P4
% 0.35/0.63              = ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 0.35/0.63           => ( C @ A3 @ B3 ) )
% 0.35/0.63       => ( produc127349428274296955eger_o @ C @ P4 ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodI2
% 0.35/0.63  thf(fact_4601_case__prodI2,axiom,
% 0.35/0.63      ! [P4: produc1908205239877642774nteger,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
% 0.35/0.63        ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 0.35/0.63            ( ( P4
% 0.35/0.63              = ( produc8603105652947943368nteger @ A3 @ B3 ) )
% 0.35/0.63           => ( C @ A3 @ B3 ) )
% 0.35/0.63       => ( produc6253627499356882019eger_o @ C @ P4 ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodI2
% 0.35/0.63  thf(fact_4602_case__prodI2,axiom,
% 0.35/0.63      ! [P4: produc2285326912895808259nt_int,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
% 0.35/0.63        ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
% 0.35/0.63            ( ( P4
% 0.35/0.63              = ( produc5700946648718959541nt_int @ A3 @ B3 ) )
% 0.35/0.63           => ( C @ A3 @ B3 ) )
% 0.35/0.63       => ( produc1573362020775583542_int_o @ C @ P4 ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodI2
% 0.35/0.63  thf(fact_4603_case__prodI2,axiom,
% 0.35/0.63      ! [P4: produc7773217078559923341nt_int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
% 0.35/0.63        ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
% 0.35/0.63            ( ( P4
% 0.35/0.63              = ( produc4305682042979456191nt_int @ A3 @ B3 ) )
% 0.35/0.63           => ( C @ A3 @ B3 ) )
% 0.35/0.63       => ( produc2558449545302689196_int_o @ C @ P4 ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodI2
% 0.35/0.63  thf(fact_4604_case__prodI2,axiom,
% 0.35/0.63      ! [P4: product_prod_nat_nat,C: nat > nat > $o] :
% 0.35/0.63        ( ! [A3: nat,B3: nat] :
% 0.35/0.63            ( ( P4
% 0.35/0.63              = ( product_Pair_nat_nat @ A3 @ B3 ) )
% 0.35/0.63           => ( C @ A3 @ B3 ) )
% 0.35/0.63       => ( produc6081775807080527818_nat_o @ C @ P4 ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodI2
% 0.35/0.63  thf(fact_4605_case__prodI2,axiom,
% 0.35/0.63      ! [P4: product_prod_int_int,C: int > int > $o] :
% 0.35/0.63        ( ! [A3: int,B3: int] :
% 0.35/0.63            ( ( P4
% 0.35/0.63              = ( product_Pair_int_int @ A3 @ B3 ) )
% 0.35/0.63           => ( C @ A3 @ B3 ) )
% 0.35/0.63       => ( produc4947309494688390418_int_o @ C @ P4 ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodI2
% 0.35/0.63  thf(fact_4606_case__prodI,axiom,
% 0.35/0.63      ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 0.35/0.63        ( ( F @ A @ B )
% 0.35/0.63       => ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodI
% 0.35/0.63  thf(fact_4607_case__prodI,axiom,
% 0.35/0.63      ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 0.35/0.63        ( ( F @ A @ B )
% 0.35/0.63       => ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodI
% 0.35/0.63  thf(fact_4608_case__prodI,axiom,
% 0.35/0.63      ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
% 0.35/0.63        ( ( F @ A @ B )
% 0.35/0.63       => ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodI
% 0.35/0.63  thf(fact_4609_case__prodI,axiom,
% 0.35/0.63      ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
% 0.35/0.63        ( ( F @ A @ B )
% 0.35/0.63       => ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodI
% 0.35/0.63  thf(fact_4610_case__prodI,axiom,
% 0.35/0.63      ! [F: nat > nat > $o,A: nat,B: nat] :
% 0.35/0.63        ( ( F @ A @ B )
% 0.35/0.63       => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodI
% 0.35/0.63  thf(fact_4611_case__prodI,axiom,
% 0.35/0.63      ! [F: int > int > $o,A: int,B: int] :
% 0.35/0.63        ( ( F @ A @ B )
% 0.35/0.63       => ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodI
% 0.35/0.63  thf(fact_4612_mem__case__prodI2,axiom,
% 0.35/0.63      ! [P4: product_prod_int_int,Z2: real,C: int > int > set_real] :
% 0.35/0.63        ( ! [A3: int,B3: int] :
% 0.35/0.63            ( ( P4
% 0.35/0.63              = ( product_Pair_int_int @ A3 @ B3 ) )
% 0.35/0.63           => ( member_real @ Z2 @ ( C @ A3 @ B3 ) ) )
% 0.35/0.63       => ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C @ P4 ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodI2
% 0.35/0.63  thf(fact_4613_mem__case__prodI2,axiom,
% 0.35/0.63      ! [P4: product_prod_int_int,Z2: nat,C: int > int > set_nat] :
% 0.35/0.63        ( ! [A3: int,B3: int] :
% 0.35/0.63            ( ( P4
% 0.35/0.63              = ( product_Pair_int_int @ A3 @ B3 ) )
% 0.35/0.63           => ( member_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
% 0.35/0.63       => ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ P4 ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodI2
% 0.35/0.63  thf(fact_4614_mem__case__prodI2,axiom,
% 0.35/0.63      ! [P4: product_prod_int_int,Z2: int,C: int > int > set_int] :
% 0.35/0.63        ( ! [A3: int,B3: int] :
% 0.35/0.63            ( ( P4
% 0.35/0.63              = ( product_Pair_int_int @ A3 @ B3 ) )
% 0.35/0.63           => ( member_int @ Z2 @ ( C @ A3 @ B3 ) ) )
% 0.35/0.63       => ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ P4 ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodI2
% 0.35/0.63  thf(fact_4615_mem__case__prodI2,axiom,
% 0.35/0.63      ! [P4: product_prod_int_int,Z2: set_nat,C: int > int > set_set_nat] :
% 0.35/0.63        ( ! [A3: int,B3: int] :
% 0.35/0.63            ( ( P4
% 0.35/0.63              = ( product_Pair_int_int @ A3 @ B3 ) )
% 0.35/0.63           => ( member_set_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
% 0.35/0.63       => ( member_set_nat @ Z2 @ ( produc5233655623923918146et_nat @ C @ P4 ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodI2
% 0.35/0.63  thf(fact_4616_mem__case__prodI2,axiom,
% 0.35/0.63      ! [P4: product_prod_int_int,Z2: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat] :
% 0.35/0.63        ( ! [A3: int,B3: int] :
% 0.35/0.63            ( ( P4
% 0.35/0.63              = ( product_Pair_int_int @ A3 @ B3 ) )
% 0.35/0.63           => ( member8440522571783428010at_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
% 0.35/0.63       => ( member8440522571783428010at_nat @ Z2 @ ( produc1656060378719767003at_nat @ C @ P4 ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodI2
% 0.35/0.63  thf(fact_4617_mem__case__prodI2,axiom,
% 0.35/0.63      ! [P4: produc8763457246119570046nteger,Z2: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real] :
% 0.35/0.63        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 0.35/0.63            ( ( P4
% 0.35/0.63              = ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 0.35/0.63           => ( member_real @ Z2 @ ( C @ A3 @ B3 ) ) )
% 0.35/0.63       => ( member_real @ Z2 @ ( produc815715089573277247t_real @ C @ P4 ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodI2
% 0.35/0.63  thf(fact_4618_mem__case__prodI2,axiom,
% 0.35/0.63      ! [P4: produc8763457246119570046nteger,Z2: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat] :
% 0.35/0.63        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 0.35/0.63            ( ( P4
% 0.35/0.63              = ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 0.35/0.63           => ( member_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
% 0.35/0.63       => ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C @ P4 ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodI2
% 0.35/0.63  thf(fact_4619_mem__case__prodI2,axiom,
% 0.35/0.63      ! [P4: produc8763457246119570046nteger,Z2: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int] :
% 0.35/0.63        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 0.35/0.63            ( ( P4
% 0.35/0.63              = ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 0.35/0.63           => ( member_int @ Z2 @ ( C @ A3 @ B3 ) ) )
% 0.35/0.63       => ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C @ P4 ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodI2
% 0.35/0.63  thf(fact_4620_mem__case__prodI2,axiom,
% 0.35/0.63      ! [P4: produc7773217078559923341nt_int,Z2: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real] :
% 0.35/0.63        ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
% 0.35/0.63            ( ( P4
% 0.35/0.63              = ( produc4305682042979456191nt_int @ A3 @ B3 ) )
% 0.35/0.63           => ( member_real @ Z2 @ ( C @ A3 @ B3 ) ) )
% 0.35/0.63       => ( member_real @ Z2 @ ( produc8709739885379107790t_real @ C @ P4 ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodI2
% 0.35/0.63  thf(fact_4621_mem__case__prodI2,axiom,
% 0.35/0.63      ! [P4: produc7773217078559923341nt_int,Z2: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat] :
% 0.35/0.63        ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
% 0.35/0.63            ( ( P4
% 0.35/0.63              = ( produc4305682042979456191nt_int @ A3 @ B3 ) )
% 0.35/0.63           => ( member_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
% 0.35/0.63       => ( member_nat @ Z2 @ ( produc8289552606927098482et_nat @ C @ P4 ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodI2
% 0.35/0.63  thf(fact_4622_mem__case__prodI,axiom,
% 0.35/0.63      ! [Z2: real,C: int > int > set_real,A: int,B: int] :
% 0.35/0.63        ( ( member_real @ Z2 @ ( C @ A @ B ) )
% 0.35/0.63       => ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodI
% 0.35/0.63  thf(fact_4623_mem__case__prodI,axiom,
% 0.35/0.63      ! [Z2: nat,C: int > int > set_nat,A: int,B: int] :
% 0.35/0.63        ( ( member_nat @ Z2 @ ( C @ A @ B ) )
% 0.35/0.63       => ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodI
% 0.35/0.63  thf(fact_4624_mem__case__prodI,axiom,
% 0.35/0.63      ! [Z2: int,C: int > int > set_int,A: int,B: int] :
% 0.35/0.63        ( ( member_int @ Z2 @ ( C @ A @ B ) )
% 0.35/0.63       => ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodI
% 0.35/0.63  thf(fact_4625_mem__case__prodI,axiom,
% 0.35/0.63      ! [Z2: set_nat,C: int > int > set_set_nat,A: int,B: int] :
% 0.35/0.63        ( ( member_set_nat @ Z2 @ ( C @ A @ B ) )
% 0.35/0.63       => ( member_set_nat @ Z2 @ ( produc5233655623923918146et_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodI
% 0.35/0.63  thf(fact_4626_mem__case__prodI,axiom,
% 0.35/0.63      ! [Z2: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat,A: int,B: int] :
% 0.35/0.63        ( ( member8440522571783428010at_nat @ Z2 @ ( C @ A @ B ) )
% 0.35/0.63       => ( member8440522571783428010at_nat @ Z2 @ ( produc1656060378719767003at_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodI
% 0.35/0.63  thf(fact_4627_mem__case__prodI,axiom,
% 0.35/0.63      ! [Z2: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 0.35/0.63        ( ( member_real @ Z2 @ ( C @ A @ B ) )
% 0.35/0.63       => ( member_real @ Z2 @ ( produc815715089573277247t_real @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodI
% 0.35/0.63  thf(fact_4628_mem__case__prodI,axiom,
% 0.35/0.63      ! [Z2: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 0.35/0.63        ( ( member_nat @ Z2 @ ( C @ A @ B ) )
% 0.35/0.63       => ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodI
% 0.35/0.63  thf(fact_4629_mem__case__prodI,axiom,
% 0.35/0.63      ! [Z2: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 0.35/0.63        ( ( member_int @ Z2 @ ( C @ A @ B ) )
% 0.35/0.63       => ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodI
% 0.35/0.63  thf(fact_4630_mem__case__prodI,axiom,
% 0.35/0.63      ! [Z2: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
% 0.35/0.63        ( ( member_real @ Z2 @ ( C @ A @ B ) )
% 0.35/0.63       => ( member_real @ Z2 @ ( produc8709739885379107790t_real @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodI
% 0.35/0.63  thf(fact_4631_mem__case__prodI,axiom,
% 0.35/0.63      ! [Z2: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
% 0.35/0.63        ( ( member_nat @ Z2 @ ( C @ A @ B ) )
% 0.35/0.63       => ( member_nat @ Z2 @ ( produc8289552606927098482et_nat @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodI
% 0.35/0.63  thf(fact_4632_case__prodI2_H,axiom,
% 0.35/0.63      ! [P4: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
% 0.35/0.63        ( ! [A3: nat,B3: nat] :
% 0.35/0.63            ( ( ( product_Pair_nat_nat @ A3 @ B3 )
% 0.35/0.63              = P4 )
% 0.35/0.63           => ( C @ A3 @ B3 @ X ) )
% 0.35/0.63       => ( produc8739625826339149834_nat_o @ C @ P4 @ X ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodI2'
% 0.35/0.63  thf(fact_4633_of__bool__less__eq__iff,axiom,
% 0.35/0.63      ! [P: $o,Q: $o] :
% 0.35/0.63        ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 0.35/0.63        = ( P
% 0.35/0.63         => Q ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_less_eq_iff
% 0.35/0.63  thf(fact_4634_of__bool__less__eq__iff,axiom,
% 0.35/0.63      ! [P: $o,Q: $o] :
% 0.35/0.63        ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 0.35/0.63        = ( P
% 0.35/0.63         => Q ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_less_eq_iff
% 0.35/0.63  thf(fact_4635_of__bool__less__eq__iff,axiom,
% 0.35/0.63      ! [P: $o,Q: $o] :
% 0.35/0.63        ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 0.35/0.63        = ( P
% 0.35/0.63         => Q ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_less_eq_iff
% 0.35/0.63  thf(fact_4636_of__bool__eq_I1_J,axiom,
% 0.35/0.63      ( ( zero_n1201886186963655149omplex @ $false )
% 0.35/0.63      = zero_zero_complex ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_eq(1)
% 0.35/0.63  thf(fact_4637_of__bool__eq_I1_J,axiom,
% 0.35/0.63      ( ( zero_n3304061248610475627l_real @ $false )
% 0.35/0.63      = zero_zero_real ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_eq(1)
% 0.35/0.63  thf(fact_4638_of__bool__eq_I1_J,axiom,
% 0.35/0.63      ( ( zero_n2052037380579107095ol_rat @ $false )
% 0.35/0.63      = zero_zero_rat ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_eq(1)
% 0.35/0.63  thf(fact_4639_of__bool__eq_I1_J,axiom,
% 0.35/0.63      ( ( zero_n2687167440665602831ol_nat @ $false )
% 0.35/0.63      = zero_zero_nat ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_eq(1)
% 0.35/0.63  thf(fact_4640_of__bool__eq_I1_J,axiom,
% 0.35/0.63      ( ( zero_n2684676970156552555ol_int @ $false )
% 0.35/0.63      = zero_zero_int ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_eq(1)
% 0.35/0.63  thf(fact_4641_of__bool__eq__0__iff,axiom,
% 0.35/0.63      ! [P: $o] :
% 0.35/0.63        ( ( ( zero_n1201886186963655149omplex @ P )
% 0.35/0.63          = zero_zero_complex )
% 0.35/0.63        = ( ~ P ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_eq_0_iff
% 0.35/0.63  thf(fact_4642_of__bool__eq__0__iff,axiom,
% 0.35/0.63      ! [P: $o] :
% 0.35/0.63        ( ( ( zero_n3304061248610475627l_real @ P )
% 0.35/0.63          = zero_zero_real )
% 0.35/0.63        = ( ~ P ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_eq_0_iff
% 0.35/0.63  thf(fact_4643_of__bool__eq__0__iff,axiom,
% 0.35/0.63      ! [P: $o] :
% 0.35/0.63        ( ( ( zero_n2052037380579107095ol_rat @ P )
% 0.35/0.63          = zero_zero_rat )
% 0.35/0.63        = ( ~ P ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_eq_0_iff
% 0.35/0.63  thf(fact_4644_of__bool__eq__0__iff,axiom,
% 0.35/0.63      ! [P: $o] :
% 0.35/0.63        ( ( ( zero_n2687167440665602831ol_nat @ P )
% 0.35/0.63          = zero_zero_nat )
% 0.35/0.63        = ( ~ P ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_eq_0_iff
% 0.35/0.63  thf(fact_4645_of__bool__eq__0__iff,axiom,
% 0.35/0.63      ! [P: $o] :
% 0.35/0.63        ( ( ( zero_n2684676970156552555ol_int @ P )
% 0.35/0.63          = zero_zero_int )
% 0.35/0.63        = ( ~ P ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_eq_0_iff
% 0.35/0.63  thf(fact_4646_of__bool__less__iff,axiom,
% 0.35/0.63      ! [P: $o,Q: $o] :
% 0.35/0.63        ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 0.35/0.63        = ( ~ P
% 0.35/0.63          & Q ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_less_iff
% 0.35/0.63  thf(fact_4647_of__bool__less__iff,axiom,
% 0.35/0.63      ! [P: $o,Q: $o] :
% 0.35/0.63        ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 0.35/0.63        = ( ~ P
% 0.35/0.63          & Q ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_less_iff
% 0.35/0.63  thf(fact_4648_of__bool__less__iff,axiom,
% 0.35/0.63      ! [P: $o,Q: $o] :
% 0.35/0.63        ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 0.35/0.63        = ( ~ P
% 0.35/0.63          & Q ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_less_iff
% 0.35/0.63  thf(fact_4649_of__bool__less__iff,axiom,
% 0.35/0.63      ! [P: $o,Q: $o] :
% 0.35/0.63        ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 0.35/0.63        = ( ~ P
% 0.35/0.63          & Q ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_less_iff
% 0.35/0.63  thf(fact_4650_zero__less__of__bool__iff,axiom,
% 0.35/0.63      ! [P: $o] :
% 0.35/0.63        ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
% 0.35/0.63        = P ) ).
% 0.35/0.63  
% 0.35/0.63  % zero_less_of_bool_iff
% 0.35/0.63  thf(fact_4651_zero__less__of__bool__iff,axiom,
% 0.35/0.63      ! [P: $o] :
% 0.35/0.63        ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 0.35/0.63        = P ) ).
% 0.35/0.63  
% 0.35/0.63  % zero_less_of_bool_iff
% 0.35/0.63  thf(fact_4652_zero__less__of__bool__iff,axiom,
% 0.35/0.63      ! [P: $o] :
% 0.35/0.63        ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.35/0.63        = P ) ).
% 0.35/0.63  
% 0.35/0.63  % zero_less_of_bool_iff
% 0.35/0.63  thf(fact_4653_zero__less__of__bool__iff,axiom,
% 0.35/0.63      ! [P: $o] :
% 0.35/0.63        ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.35/0.63        = P ) ).
% 0.35/0.63  
% 0.35/0.63  % zero_less_of_bool_iff
% 0.35/0.63  thf(fact_4654_of__bool__less__one__iff,axiom,
% 0.35/0.63      ! [P: $o] :
% 0.35/0.63        ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
% 0.35/0.63        = ( ~ P ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_less_one_iff
% 0.35/0.63  thf(fact_4655_of__bool__less__one__iff,axiom,
% 0.35/0.63      ! [P: $o] :
% 0.35/0.63        ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
% 0.35/0.63        = ( ~ P ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_less_one_iff
% 0.35/0.63  thf(fact_4656_of__bool__less__one__iff,axiom,
% 0.35/0.63      ! [P: $o] :
% 0.35/0.63        ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
% 0.35/0.63        = ( ~ P ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_less_one_iff
% 0.35/0.63  thf(fact_4657_of__bool__less__one__iff,axiom,
% 0.35/0.63      ! [P: $o] :
% 0.35/0.63        ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
% 0.35/0.63        = ( ~ P ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_less_one_iff
% 0.35/0.63  thf(fact_4658_Suc__0__mod__eq,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.63        = ( zero_n2687167440665602831ol_nat
% 0.35/0.63          @ ( N
% 0.35/0.63           != ( suc @ zero_zero_nat ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % Suc_0_mod_eq
% 0.35/0.63  thf(fact_4659_set__decode__zero,axiom,
% 0.35/0.63      ( ( nat_set_decode @ zero_zero_nat )
% 0.35/0.63      = bot_bot_set_nat ) ).
% 0.35/0.63  
% 0.35/0.63  % set_decode_zero
% 0.35/0.63  thf(fact_4660_set__encode__empty,axiom,
% 0.35/0.63      ( ( nat_set_encode @ bot_bot_set_nat )
% 0.35/0.63      = zero_zero_nat ) ).
% 0.35/0.63  
% 0.35/0.63  % set_encode_empty
% 0.35/0.63  thf(fact_4661_set__encode__inverse,axiom,
% 0.35/0.63      ! [A2: set_nat] :
% 0.35/0.63        ( ( finite_finite_nat @ A2 )
% 0.35/0.63       => ( ( nat_set_decode @ ( nat_set_encode @ A2 ) )
% 0.35/0.63          = A2 ) ) ).
% 0.35/0.63  
% 0.35/0.63  % set_encode_inverse
% 0.35/0.63  thf(fact_4662_take__bit__of__Suc__0,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.35/0.63        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_of_Suc_0
% 0.35/0.63  thf(fact_4663_divmod__algorithm__code_I2_J,axiom,
% 0.35/0.63      ! [M: num] :
% 0.35/0.63        ( ( unique5052692396658037445od_int @ M @ one )
% 0.35/0.63        = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_algorithm_code(2)
% 0.35/0.63  thf(fact_4664_divmod__algorithm__code_I2_J,axiom,
% 0.35/0.63      ! [M: num] :
% 0.35/0.63        ( ( unique5055182867167087721od_nat @ M @ one )
% 0.35/0.63        = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_algorithm_code(2)
% 0.35/0.63  thf(fact_4665_divmod__algorithm__code_I2_J,axiom,
% 0.35/0.63      ! [M: num] :
% 0.35/0.63        ( ( unique3479559517661332726nteger @ M @ one )
% 0.35/0.63        = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_algorithm_code(2)
% 0.35/0.63  thf(fact_4666_take__bit__of__1,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 0.35/0.63        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_of_1
% 0.35/0.63  thf(fact_4667_take__bit__of__1,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 0.35/0.63        = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_of_1
% 0.35/0.63  thf(fact_4668_of__bool__half__eq__0,axiom,
% 0.35/0.63      ! [B: $o] :
% 0.35/0.63        ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.63        = zero_z3403309356797280102nteger ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_half_eq_0
% 0.35/0.63  thf(fact_4669_of__bool__half__eq__0,axiom,
% 0.35/0.63      ! [B: $o] :
% 0.35/0.63        ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.63        = zero_zero_nat ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_half_eq_0
% 0.35/0.63  thf(fact_4670_of__bool__half__eq__0,axiom,
% 0.35/0.63      ! [B: $o] :
% 0.35/0.63        ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.63        = zero_zero_int ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_half_eq_0
% 0.35/0.63  thf(fact_4671_divmod__algorithm__code_I3_J,axiom,
% 0.35/0.63      ! [N: num] :
% 0.35/0.63        ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
% 0.35/0.63        = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_algorithm_code(3)
% 0.35/0.63  thf(fact_4672_divmod__algorithm__code_I3_J,axiom,
% 0.35/0.63      ! [N: num] :
% 0.35/0.63        ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
% 0.35/0.63        = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_algorithm_code(3)
% 0.35/0.63  thf(fact_4673_divmod__algorithm__code_I3_J,axiom,
% 0.35/0.63      ! [N: num] :
% 0.35/0.63        ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
% 0.35/0.63        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_algorithm_code(3)
% 0.35/0.63  thf(fact_4674_bits__1__div__exp,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63        = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % bits_1_div_exp
% 0.35/0.63  thf(fact_4675_bits__1__div__exp,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63        = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % bits_1_div_exp
% 0.35/0.63  thf(fact_4676_bits__1__div__exp,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63        = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % bits_1_div_exp
% 0.35/0.63  thf(fact_4677_one__div__2__pow__eq,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63        = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % one_div_2_pow_eq
% 0.35/0.63  thf(fact_4678_one__div__2__pow__eq,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63        = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % one_div_2_pow_eq
% 0.35/0.63  thf(fact_4679_one__div__2__pow__eq,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63        = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % one_div_2_pow_eq
% 0.35/0.63  thf(fact_4680_take__bit__of__exp,axiom,
% 0.35/0.63      ! [M: nat,N: nat] :
% 0.35/0.63        ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_of_exp
% 0.35/0.63  thf(fact_4681_take__bit__of__exp,axiom,
% 0.35/0.63      ! [M: nat,N: nat] :
% 0.35/0.63        ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_of_exp
% 0.35/0.63  thf(fact_4682_take__bit__of__2,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.63        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_of_2
% 0.35/0.63  thf(fact_4683_take__bit__of__2,axiom,
% 0.35/0.63      ! [N: nat] :
% 0.35/0.63        ( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.63        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_of_2
% 0.35/0.63  thf(fact_4684_mem__case__prodE,axiom,
% 0.35/0.63      ! [Z2: real,C: int > int > set_real,P4: product_prod_int_int] :
% 0.35/0.63        ( ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C @ P4 ) )
% 0.35/0.63       => ~ ! [X4: int,Y3: int] :
% 0.35/0.63              ( ( P4
% 0.35/0.63                = ( product_Pair_int_int @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( member_real @ Z2 @ ( C @ X4 @ Y3 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodE
% 0.35/0.63  thf(fact_4685_mem__case__prodE,axiom,
% 0.35/0.63      ! [Z2: nat,C: int > int > set_nat,P4: product_prod_int_int] :
% 0.35/0.63        ( ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ P4 ) )
% 0.35/0.63       => ~ ! [X4: int,Y3: int] :
% 0.35/0.63              ( ( P4
% 0.35/0.63                = ( product_Pair_int_int @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( member_nat @ Z2 @ ( C @ X4 @ Y3 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodE
% 0.35/0.63  thf(fact_4686_mem__case__prodE,axiom,
% 0.35/0.63      ! [Z2: int,C: int > int > set_int,P4: product_prod_int_int] :
% 0.35/0.63        ( ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ P4 ) )
% 0.35/0.63       => ~ ! [X4: int,Y3: int] :
% 0.35/0.63              ( ( P4
% 0.35/0.63                = ( product_Pair_int_int @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( member_int @ Z2 @ ( C @ X4 @ Y3 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodE
% 0.35/0.63  thf(fact_4687_mem__case__prodE,axiom,
% 0.35/0.63      ! [Z2: set_nat,C: int > int > set_set_nat,P4: product_prod_int_int] :
% 0.35/0.63        ( ( member_set_nat @ Z2 @ ( produc5233655623923918146et_nat @ C @ P4 ) )
% 0.35/0.63       => ~ ! [X4: int,Y3: int] :
% 0.35/0.63              ( ( P4
% 0.35/0.63                = ( product_Pair_int_int @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( member_set_nat @ Z2 @ ( C @ X4 @ Y3 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodE
% 0.35/0.63  thf(fact_4688_mem__case__prodE,axiom,
% 0.35/0.63      ! [Z2: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat,P4: product_prod_int_int] :
% 0.35/0.63        ( ( member8440522571783428010at_nat @ Z2 @ ( produc1656060378719767003at_nat @ C @ P4 ) )
% 0.35/0.63       => ~ ! [X4: int,Y3: int] :
% 0.35/0.63              ( ( P4
% 0.35/0.63                = ( product_Pair_int_int @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( member8440522571783428010at_nat @ Z2 @ ( C @ X4 @ Y3 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodE
% 0.35/0.63  thf(fact_4689_mem__case__prodE,axiom,
% 0.35/0.63      ! [Z2: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,P4: produc8763457246119570046nteger] :
% 0.35/0.63        ( ( member_real @ Z2 @ ( produc815715089573277247t_real @ C @ P4 ) )
% 0.35/0.63       => ~ ! [X4: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
% 0.35/0.63              ( ( P4
% 0.35/0.63                = ( produc6137756002093451184nteger @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( member_real @ Z2 @ ( C @ X4 @ Y3 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodE
% 0.35/0.63  thf(fact_4690_mem__case__prodE,axiom,
% 0.35/0.63      ! [Z2: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,P4: produc8763457246119570046nteger] :
% 0.35/0.63        ( ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C @ P4 ) )
% 0.35/0.63       => ~ ! [X4: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
% 0.35/0.63              ( ( P4
% 0.35/0.63                = ( produc6137756002093451184nteger @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( member_nat @ Z2 @ ( C @ X4 @ Y3 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodE
% 0.35/0.63  thf(fact_4691_mem__case__prodE,axiom,
% 0.35/0.63      ! [Z2: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,P4: produc8763457246119570046nteger] :
% 0.35/0.63        ( ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C @ P4 ) )
% 0.35/0.63       => ~ ! [X4: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
% 0.35/0.63              ( ( P4
% 0.35/0.63                = ( produc6137756002093451184nteger @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( member_int @ Z2 @ ( C @ X4 @ Y3 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodE
% 0.35/0.63  thf(fact_4692_mem__case__prodE,axiom,
% 0.35/0.63      ! [Z2: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real,P4: produc7773217078559923341nt_int] :
% 0.35/0.63        ( ( member_real @ Z2 @ ( produc8709739885379107790t_real @ C @ P4 ) )
% 0.35/0.63       => ~ ! [X4: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
% 0.35/0.63              ( ( P4
% 0.35/0.63                = ( produc4305682042979456191nt_int @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( member_real @ Z2 @ ( C @ X4 @ Y3 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodE
% 0.35/0.63  thf(fact_4693_mem__case__prodE,axiom,
% 0.35/0.63      ! [Z2: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,P4: produc7773217078559923341nt_int] :
% 0.35/0.63        ( ( member_nat @ Z2 @ ( produc8289552606927098482et_nat @ C @ P4 ) )
% 0.35/0.63       => ~ ! [X4: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
% 0.35/0.63              ( ( P4
% 0.35/0.63                = ( produc4305682042979456191nt_int @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( member_nat @ Z2 @ ( C @ X4 @ Y3 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % mem_case_prodE
% 0.35/0.63  thf(fact_4694_case__prodE,axiom,
% 0.35/0.63      ! [C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc8763457246119570046nteger] :
% 0.35/0.63        ( ( produc127349428274296955eger_o @ C @ P4 )
% 0.35/0.63       => ~ ! [X4: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
% 0.35/0.63              ( ( P4
% 0.35/0.63                = ( produc6137756002093451184nteger @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( C @ X4 @ Y3 ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodE
% 0.35/0.63  thf(fact_4695_case__prodE,axiom,
% 0.35/0.63      ! [C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc1908205239877642774nteger] :
% 0.35/0.63        ( ( produc6253627499356882019eger_o @ C @ P4 )
% 0.35/0.63       => ~ ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
% 0.35/0.63              ( ( P4
% 0.35/0.63                = ( produc8603105652947943368nteger @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( C @ X4 @ Y3 ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodE
% 0.35/0.63  thf(fact_4696_case__prodE,axiom,
% 0.35/0.63      ! [C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc2285326912895808259nt_int] :
% 0.35/0.63        ( ( produc1573362020775583542_int_o @ C @ P4 )
% 0.35/0.63       => ~ ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
% 0.35/0.63              ( ( P4
% 0.35/0.63                = ( produc5700946648718959541nt_int @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( C @ X4 @ Y3 ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodE
% 0.35/0.63  thf(fact_4697_case__prodE,axiom,
% 0.35/0.63      ! [C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc7773217078559923341nt_int] :
% 0.35/0.63        ( ( produc2558449545302689196_int_o @ C @ P4 )
% 0.35/0.63       => ~ ! [X4: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
% 0.35/0.63              ( ( P4
% 0.35/0.63                = ( produc4305682042979456191nt_int @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( C @ X4 @ Y3 ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodE
% 0.35/0.63  thf(fact_4698_case__prodE,axiom,
% 0.35/0.63      ! [C: nat > nat > $o,P4: product_prod_nat_nat] :
% 0.35/0.63        ( ( produc6081775807080527818_nat_o @ C @ P4 )
% 0.35/0.63       => ~ ! [X4: nat,Y3: nat] :
% 0.35/0.63              ( ( P4
% 0.35/0.63                = ( product_Pair_nat_nat @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( C @ X4 @ Y3 ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodE
% 0.35/0.63  thf(fact_4699_case__prodE,axiom,
% 0.35/0.63      ! [C: int > int > $o,P4: product_prod_int_int] :
% 0.35/0.63        ( ( produc4947309494688390418_int_o @ C @ P4 )
% 0.35/0.63       => ~ ! [X4: int,Y3: int] :
% 0.35/0.63              ( ( P4
% 0.35/0.63                = ( product_Pair_int_int @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( C @ X4 @ Y3 ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodE
% 0.35/0.63  thf(fact_4700_case__prodD,axiom,
% 0.35/0.63      ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 0.35/0.63        ( ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B ) )
% 0.35/0.63       => ( F @ A @ B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodD
% 0.35/0.63  thf(fact_4701_case__prodD,axiom,
% 0.35/0.63      ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 0.35/0.63        ( ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) )
% 0.35/0.63       => ( F @ A @ B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodD
% 0.35/0.63  thf(fact_4702_case__prodD,axiom,
% 0.35/0.63      ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
% 0.35/0.63        ( ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) )
% 0.35/0.63       => ( F @ A @ B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodD
% 0.35/0.63  thf(fact_4703_case__prodD,axiom,
% 0.35/0.63      ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
% 0.35/0.63        ( ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) )
% 0.35/0.63       => ( F @ A @ B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodD
% 0.35/0.63  thf(fact_4704_case__prodD,axiom,
% 0.35/0.63      ! [F: nat > nat > $o,A: nat,B: nat] :
% 0.35/0.63        ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 0.35/0.63       => ( F @ A @ B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodD
% 0.35/0.63  thf(fact_4705_case__prodD,axiom,
% 0.35/0.63      ! [F: int > int > $o,A: int,B: int] :
% 0.35/0.63        ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
% 0.35/0.63       => ( F @ A @ B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodD
% 0.35/0.63  thf(fact_4706_case__prodE_H,axiom,
% 0.35/0.63      ! [C: nat > nat > product_prod_nat_nat > $o,P4: product_prod_nat_nat,Z2: product_prod_nat_nat] :
% 0.35/0.63        ( ( produc8739625826339149834_nat_o @ C @ P4 @ Z2 )
% 0.35/0.63       => ~ ! [X4: nat,Y3: nat] :
% 0.35/0.63              ( ( P4
% 0.35/0.63                = ( product_Pair_nat_nat @ X4 @ Y3 ) )
% 0.35/0.63             => ~ ( C @ X4 @ Y3 @ Z2 ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodE'
% 0.35/0.63  thf(fact_4707_case__prodD_H,axiom,
% 0.35/0.63      ! [R: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
% 0.35/0.63        ( ( produc8739625826339149834_nat_o @ R @ ( product_Pair_nat_nat @ A @ B ) @ C )
% 0.35/0.63       => ( R @ A @ B @ C ) ) ).
% 0.35/0.63  
% 0.35/0.63  % case_prodD'
% 0.35/0.63  thf(fact_4708_zero__less__eq__of__bool,axiom,
% 0.35/0.63      ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.35/0.63  
% 0.35/0.63  % zero_less_eq_of_bool
% 0.35/0.63  thf(fact_4709_zero__less__eq__of__bool,axiom,
% 0.35/0.63      ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.35/0.63  
% 0.35/0.63  % zero_less_eq_of_bool
% 0.35/0.63  thf(fact_4710_zero__less__eq__of__bool,axiom,
% 0.35/0.63      ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.35/0.63  
% 0.35/0.63  % zero_less_eq_of_bool
% 0.35/0.63  thf(fact_4711_zero__less__eq__of__bool,axiom,
% 0.35/0.63      ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.35/0.63  
% 0.35/0.63  % zero_less_eq_of_bool
% 0.35/0.63  thf(fact_4712_of__bool__less__eq__one,axiom,
% 0.35/0.63      ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_less_eq_one
% 0.35/0.63  thf(fact_4713_of__bool__less__eq__one,axiom,
% 0.35/0.63      ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_less_eq_one
% 0.35/0.63  thf(fact_4714_of__bool__less__eq__one,axiom,
% 0.35/0.63      ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_less_eq_one
% 0.35/0.63  thf(fact_4715_of__bool__less__eq__one,axiom,
% 0.35/0.63      ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_less_eq_one
% 0.35/0.63  thf(fact_4716_split__of__bool__asm,axiom,
% 0.35/0.63      ! [P: complex > $o,P4: $o] :
% 0.35/0.63        ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
% 0.35/0.63        = ( ~ ( ( P4
% 0.35/0.63                & ~ ( P @ one_one_complex ) )
% 0.35/0.63              | ( ~ P4
% 0.35/0.63                & ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % split_of_bool_asm
% 0.35/0.63  thf(fact_4717_split__of__bool__asm,axiom,
% 0.35/0.63      ! [P: real > $o,P4: $o] :
% 0.35/0.63        ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
% 0.35/0.63        = ( ~ ( ( P4
% 0.35/0.63                & ~ ( P @ one_one_real ) )
% 0.35/0.63              | ( ~ P4
% 0.35/0.63                & ~ ( P @ zero_zero_real ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % split_of_bool_asm
% 0.35/0.63  thf(fact_4718_split__of__bool__asm,axiom,
% 0.35/0.63      ! [P: rat > $o,P4: $o] :
% 0.35/0.63        ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
% 0.35/0.63        = ( ~ ( ( P4
% 0.35/0.63                & ~ ( P @ one_one_rat ) )
% 0.35/0.63              | ( ~ P4
% 0.35/0.63                & ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % split_of_bool_asm
% 0.35/0.63  thf(fact_4719_split__of__bool__asm,axiom,
% 0.35/0.63      ! [P: nat > $o,P4: $o] :
% 0.35/0.63        ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
% 0.35/0.63        = ( ~ ( ( P4
% 0.35/0.63                & ~ ( P @ one_one_nat ) )
% 0.35/0.63              | ( ~ P4
% 0.35/0.63                & ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % split_of_bool_asm
% 0.35/0.63  thf(fact_4720_split__of__bool__asm,axiom,
% 0.35/0.63      ! [P: int > $o,P4: $o] :
% 0.35/0.63        ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
% 0.35/0.63        = ( ~ ( ( P4
% 0.35/0.63                & ~ ( P @ one_one_int ) )
% 0.35/0.63              | ( ~ P4
% 0.35/0.63                & ~ ( P @ zero_zero_int ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % split_of_bool_asm
% 0.35/0.63  thf(fact_4721_split__of__bool,axiom,
% 0.35/0.63      ! [P: complex > $o,P4: $o] :
% 0.35/0.63        ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
% 0.35/0.63        = ( ( P4
% 0.35/0.63           => ( P @ one_one_complex ) )
% 0.35/0.63          & ( ~ P4
% 0.35/0.63           => ( P @ zero_zero_complex ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % split_of_bool
% 0.35/0.63  thf(fact_4722_split__of__bool,axiom,
% 0.35/0.63      ! [P: real > $o,P4: $o] :
% 0.35/0.63        ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
% 0.35/0.63        = ( ( P4
% 0.35/0.63           => ( P @ one_one_real ) )
% 0.35/0.63          & ( ~ P4
% 0.35/0.63           => ( P @ zero_zero_real ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % split_of_bool
% 0.35/0.63  thf(fact_4723_split__of__bool,axiom,
% 0.35/0.63      ! [P: rat > $o,P4: $o] :
% 0.35/0.63        ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
% 0.35/0.63        = ( ( P4
% 0.35/0.63           => ( P @ one_one_rat ) )
% 0.35/0.63          & ( ~ P4
% 0.35/0.63           => ( P @ zero_zero_rat ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % split_of_bool
% 0.35/0.63  thf(fact_4724_split__of__bool,axiom,
% 0.35/0.63      ! [P: nat > $o,P4: $o] :
% 0.35/0.63        ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
% 0.35/0.63        = ( ( P4
% 0.35/0.63           => ( P @ one_one_nat ) )
% 0.35/0.63          & ( ~ P4
% 0.35/0.63           => ( P @ zero_zero_nat ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % split_of_bool
% 0.35/0.63  thf(fact_4725_split__of__bool,axiom,
% 0.35/0.63      ! [P: int > $o,P4: $o] :
% 0.35/0.63        ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
% 0.35/0.63        = ( ( P4
% 0.35/0.63           => ( P @ one_one_int ) )
% 0.35/0.63          & ( ~ P4
% 0.35/0.63           => ( P @ zero_zero_int ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % split_of_bool
% 0.35/0.63  thf(fact_4726_of__bool__def,axiom,
% 0.35/0.63      ( zero_n1201886186963655149omplex
% 0.35/0.63      = ( ^ [P6: $o] : ( if_complex @ P6 @ one_one_complex @ zero_zero_complex ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_def
% 0.35/0.63  thf(fact_4727_of__bool__def,axiom,
% 0.35/0.63      ( zero_n3304061248610475627l_real
% 0.35/0.63      = ( ^ [P6: $o] : ( if_real @ P6 @ one_one_real @ zero_zero_real ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_def
% 0.35/0.63  thf(fact_4728_of__bool__def,axiom,
% 0.35/0.63      ( zero_n2052037380579107095ol_rat
% 0.35/0.63      = ( ^ [P6: $o] : ( if_rat @ P6 @ one_one_rat @ zero_zero_rat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_def
% 0.35/0.63  thf(fact_4729_of__bool__def,axiom,
% 0.35/0.63      ( zero_n2687167440665602831ol_nat
% 0.35/0.63      = ( ^ [P6: $o] : ( if_nat @ P6 @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_def
% 0.35/0.63  thf(fact_4730_of__bool__def,axiom,
% 0.35/0.63      ( zero_n2684676970156552555ol_int
% 0.35/0.63      = ( ^ [P6: $o] : ( if_int @ P6 @ one_one_int @ zero_zero_int ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % of_bool_def
% 0.35/0.63  thf(fact_4731_finite__set__decode,axiom,
% 0.35/0.63      ! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).
% 0.35/0.63  
% 0.35/0.63  % finite_set_decode
% 0.35/0.63  thf(fact_4732_set__encode__eq,axiom,
% 0.35/0.63      ! [A2: set_nat,B2: set_nat] :
% 0.35/0.63        ( ( finite_finite_nat @ A2 )
% 0.35/0.63       => ( ( finite_finite_nat @ B2 )
% 0.35/0.63         => ( ( ( nat_set_encode @ A2 )
% 0.35/0.63              = ( nat_set_encode @ B2 ) )
% 0.35/0.63            = ( A2 = B2 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % set_encode_eq
% 0.35/0.63  thf(fact_4733_set__encode__inf,axiom,
% 0.35/0.63      ! [A2: set_nat] :
% 0.35/0.63        ( ~ ( finite_finite_nat @ A2 )
% 0.35/0.63       => ( ( nat_set_encode @ A2 )
% 0.35/0.63          = zero_zero_nat ) ) ).
% 0.35/0.63  
% 0.35/0.63  % set_encode_inf
% 0.35/0.63  thf(fact_4734_subset__decode__imp__le,axiom,
% 0.35/0.63      ! [M: nat,N: nat] :
% 0.35/0.63        ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
% 0.35/0.63       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.35/0.63  
% 0.35/0.63  % subset_decode_imp_le
% 0.35/0.63  thf(fact_4735_divmod__int__def,axiom,
% 0.35/0.63      ( unique5052692396658037445od_int
% 0.35/0.63      = ( ^ [M2: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_int_def
% 0.35/0.63  thf(fact_4736_divmod__def,axiom,
% 0.35/0.63      ( unique5052692396658037445od_int
% 0.35/0.63      = ( ^ [M2: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_def
% 0.35/0.63  thf(fact_4737_divmod__def,axiom,
% 0.35/0.63      ( unique5055182867167087721od_nat
% 0.35/0.63      = ( ^ [M2: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_def
% 0.35/0.63  thf(fact_4738_divmod__def,axiom,
% 0.35/0.63      ( unique3479559517661332726nteger
% 0.35/0.63      = ( ^ [M2: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_def
% 0.35/0.63  thf(fact_4739_divmod_H__nat__def,axiom,
% 0.35/0.63      ( unique5055182867167087721od_nat
% 0.35/0.63      = ( ^ [M2: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod'_nat_def
% 0.35/0.63  thf(fact_4740_exp__mod__exp,axiom,
% 0.35/0.63      ! [M: nat,N: nat] :
% 0.35/0.63        ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63        = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % exp_mod_exp
% 0.35/0.63  thf(fact_4741_exp__mod__exp,axiom,
% 0.35/0.63      ! [M: nat,N: nat] :
% 0.35/0.63        ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % exp_mod_exp
% 0.35/0.63  thf(fact_4742_exp__mod__exp,axiom,
% 0.35/0.63      ! [M: nat,N: nat] :
% 0.35/0.63        ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % exp_mod_exp
% 0.35/0.63  thf(fact_4743_divmod__nat__def,axiom,
% 0.35/0.63      ( divmod_nat
% 0.35/0.63      = ( ^ [M2: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M2 @ N2 ) @ ( modulo_modulo_nat @ M2 @ N2 ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_nat_def
% 0.35/0.63  thf(fact_4744_exp__div__exp__eq,axiom,
% 0.35/0.63      ! [M: nat,N: nat] :
% 0.35/0.63        ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63        = ( times_3573771949741848930nteger
% 0.35/0.63          @ ( zero_n356916108424825756nteger
% 0.35/0.63            @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 0.35/0.63               != zero_z3403309356797280102nteger )
% 0.35/0.63              & ( ord_less_eq_nat @ N @ M ) ) )
% 0.35/0.63          @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % exp_div_exp_eq
% 0.35/0.63  thf(fact_4745_exp__div__exp__eq,axiom,
% 0.35/0.63      ! [M: nat,N: nat] :
% 0.35/0.63        ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63        = ( times_times_nat
% 0.35/0.63          @ ( zero_n2687167440665602831ol_nat
% 0.35/0.63            @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.35/0.63               != zero_zero_nat )
% 0.35/0.63              & ( ord_less_eq_nat @ N @ M ) ) )
% 0.35/0.63          @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % exp_div_exp_eq
% 0.35/0.63  thf(fact_4746_exp__div__exp__eq,axiom,
% 0.35/0.63      ! [M: nat,N: nat] :
% 0.35/0.63        ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63        = ( times_times_int
% 0.35/0.63          @ ( zero_n2684676970156552555ol_int
% 0.35/0.63            @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.35/0.63               != zero_zero_int )
% 0.35/0.63              & ( ord_less_eq_nat @ N @ M ) ) )
% 0.35/0.63          @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % exp_div_exp_eq
% 0.35/0.63  thf(fact_4747_divmod__divmod__step,axiom,
% 0.35/0.63      ( unique5055182867167087721od_nat
% 0.35/0.63      = ( ^ [M2: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M2 @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M2 ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_divmod_step
% 0.35/0.63  thf(fact_4748_divmod__divmod__step,axiom,
% 0.35/0.63      ( unique5052692396658037445od_int
% 0.35/0.63      = ( ^ [M2: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M2 @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M2 ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_divmod_step
% 0.35/0.63  thf(fact_4749_divmod__divmod__step,axiom,
% 0.35/0.63      ( unique3479559517661332726nteger
% 0.35/0.63      = ( ^ [M2: num,N2: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M2 @ N2 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( unique4921790084139445826nteger @ N2 @ ( unique3479559517661332726nteger @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_divmod_step
% 0.35/0.63  thf(fact_4750_listrel1p__def,axiom,
% 0.35/0.63      ( listrel1p_nat
% 0.35/0.63      = ( ^ [R5: nat > nat > $o,Xs2: list_nat,Ys3: list_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys3 ) @ ( listrel1_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R5 ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % listrel1p_def
% 0.35/0.63  thf(fact_4751_listrel1p__def,axiom,
% 0.35/0.63      ( listrel1p_int
% 0.35/0.63      = ( ^ [R5: int > int > $o,Xs2: list_int,Ys3: list_int] : ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys3 ) @ ( listrel1_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R5 ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % listrel1p_def
% 0.35/0.63  thf(fact_4752_divmod__algorithm__code_I6_J,axiom,
% 0.35/0.63      ! [M: num,N: num] :
% 0.35/0.63        ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.35/0.63        = ( produc4245557441103728435nt_int
% 0.35/0.63          @ ^ [Q5: int,R5: int] : ( product_Pair_int_int @ Q5 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
% 0.35/0.63          @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_algorithm_code(6)
% 0.35/0.63  thf(fact_4753_divmod__algorithm__code_I6_J,axiom,
% 0.35/0.63      ! [M: num,N: num] :
% 0.35/0.63        ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.35/0.63        = ( produc2626176000494625587at_nat
% 0.35/0.63          @ ^ [Q5: nat,R5: nat] : ( product_Pair_nat_nat @ Q5 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
% 0.35/0.63          @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_algorithm_code(6)
% 0.35/0.63  thf(fact_4754_divmod__algorithm__code_I6_J,axiom,
% 0.35/0.63      ! [M: num,N: num] :
% 0.35/0.63        ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.35/0.63        = ( produc6916734918728496179nteger
% 0.35/0.63          @ ^ [Q5: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
% 0.35/0.63          @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_algorithm_code(6)
% 0.35/0.63  thf(fact_4755_Divides_Oadjust__div__eq,axiom,
% 0.35/0.63      ! [Q4: int,R2: int] :
% 0.35/0.63        ( ( adjust_div @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.35/0.63        = ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % Divides.adjust_div_eq
% 0.35/0.63  thf(fact_4756_int__ge__less__than2__def,axiom,
% 0.35/0.63      ( int_ge_less_than2
% 0.35/0.63      = ( ^ [D4: int] :
% 0.35/0.63            ( collec213857154873943460nt_int
% 0.35/0.63            @ ( produc4947309494688390418_int_o
% 0.35/0.63              @ ^ [Z7: int,Z3: int] :
% 0.35/0.63                  ( ( ord_less_eq_int @ D4 @ Z3 )
% 0.35/0.63                  & ( ord_less_int @ Z7 @ Z3 ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % int_ge_less_than2_def
% 0.35/0.63  thf(fact_4757_int__ge__less__than__def,axiom,
% 0.35/0.63      ( int_ge_less_than
% 0.35/0.63      = ( ^ [D4: int] :
% 0.35/0.63            ( collec213857154873943460nt_int
% 0.35/0.63            @ ( produc4947309494688390418_int_o
% 0.35/0.63              @ ^ [Z7: int,Z3: int] :
% 0.35/0.63                  ( ( ord_less_eq_int @ D4 @ Z7 )
% 0.35/0.63                  & ( ord_less_int @ Z7 @ Z3 ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % int_ge_less_than_def
% 0.35/0.63  thf(fact_4758_take__bit__minus__small__eq,axiom,
% 0.35/0.63      ! [K: int,N: nat] :
% 0.35/0.63        ( ( ord_less_int @ zero_zero_int @ K )
% 0.35/0.63       => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63         => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
% 0.35/0.63            = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % take_bit_minus_small_eq
% 0.35/0.63  thf(fact_4759_divmod__algorithm__code_I7_J,axiom,
% 0.35/0.63      ! [M: num,N: num] :
% 0.35/0.63        ( ( ( ord_less_eq_num @ M @ N )
% 0.35/0.63         => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.35/0.63            = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
% 0.35/0.63        & ( ~ ( ord_less_eq_num @ M @ N )
% 0.35/0.63         => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.35/0.63            = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_algorithm_code(7)
% 0.35/0.63  thf(fact_4760_divmod__algorithm__code_I7_J,axiom,
% 0.35/0.63      ! [M: num,N: num] :
% 0.35/0.63        ( ( ( ord_less_eq_num @ M @ N )
% 0.35/0.63         => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.35/0.63            = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
% 0.35/0.63        & ( ~ ( ord_less_eq_num @ M @ N )
% 0.35/0.63         => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.35/0.63            = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_algorithm_code(7)
% 0.35/0.63  thf(fact_4761_divmod__algorithm__code_I7_J,axiom,
% 0.35/0.63      ! [M: num,N: num] :
% 0.35/0.63        ( ( ( ord_less_eq_num @ M @ N )
% 0.35/0.63         => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.35/0.63            = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
% 0.35/0.63        & ( ~ ( ord_less_eq_num @ M @ N )
% 0.35/0.63         => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.35/0.63            = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.35/0.63  
% 0.35/0.63  % divmod_algorithm_code(7)
% 0.35/0.63  thf(fact_4762_add_Oinverse__inverse,axiom,
% 0.35/0.63      ! [A: int] :
% 0.35/0.63        ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 0.35/0.63        = A ) ).
% 0.35/0.63  
% 0.35/0.63  % add.inverse_inverse
% 0.35/0.63  thf(fact_4763_add_Oinverse__inverse,axiom,
% 0.35/0.63      ! [A: real] :
% 0.35/0.63        ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 0.35/0.63        = A ) ).
% 0.35/0.63  
% 0.35/0.63  % add.inverse_inverse
% 0.35/0.63  thf(fact_4764_add_Oinverse__inverse,axiom,
% 0.35/0.63      ! [A: rat] :
% 0.35/0.63        ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
% 0.35/0.63        = A ) ).
% 0.35/0.63  
% 0.35/0.63  % add.inverse_inverse
% 0.35/0.63  thf(fact_4765_add_Oinverse__inverse,axiom,
% 0.35/0.63      ! [A: code_integer] :
% 0.35/0.63        ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.35/0.63        = A ) ).
% 0.35/0.63  
% 0.35/0.63  % add.inverse_inverse
% 0.35/0.63  thf(fact_4766_neg__equal__iff__equal,axiom,
% 0.35/0.63      ! [A: int,B: int] :
% 0.35/0.63        ( ( ( uminus_uminus_int @ A )
% 0.35/0.63          = ( uminus_uminus_int @ B ) )
% 0.35/0.63        = ( A = B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_equal_iff_equal
% 0.35/0.63  thf(fact_4767_neg__equal__iff__equal,axiom,
% 0.35/0.63      ! [A: real,B: real] :
% 0.35/0.63        ( ( ( uminus_uminus_real @ A )
% 0.35/0.63          = ( uminus_uminus_real @ B ) )
% 0.35/0.63        = ( A = B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_equal_iff_equal
% 0.35/0.63  thf(fact_4768_neg__equal__iff__equal,axiom,
% 0.35/0.63      ! [A: rat,B: rat] :
% 0.35/0.63        ( ( ( uminus_uminus_rat @ A )
% 0.35/0.63          = ( uminus_uminus_rat @ B ) )
% 0.35/0.63        = ( A = B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_equal_iff_equal
% 0.35/0.63  thf(fact_4769_neg__equal__iff__equal,axiom,
% 0.35/0.63      ! [A: code_integer,B: code_integer] :
% 0.35/0.63        ( ( ( uminus1351360451143612070nteger @ A )
% 0.35/0.63          = ( uminus1351360451143612070nteger @ B ) )
% 0.35/0.63        = ( A = B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_equal_iff_equal
% 0.35/0.63  thf(fact_4770_compl__le__compl__iff,axiom,
% 0.35/0.63      ! [X: set_int,Y: set_int] :
% 0.35/0.63        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( uminus1532241313380277803et_int @ Y ) )
% 0.35/0.63        = ( ord_less_eq_set_int @ Y @ X ) ) ).
% 0.35/0.63  
% 0.35/0.63  % compl_le_compl_iff
% 0.35/0.63  thf(fact_4771_neg__le__iff__le,axiom,
% 0.35/0.63      ! [B: real,A: real] :
% 0.35/0.63        ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.35/0.63        = ( ord_less_eq_real @ A @ B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_le_iff_le
% 0.35/0.63  thf(fact_4772_neg__le__iff__le,axiom,
% 0.35/0.63      ! [B: code_integer,A: code_integer] :
% 0.35/0.63        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.35/0.63        = ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_le_iff_le
% 0.35/0.63  thf(fact_4773_neg__le__iff__le,axiom,
% 0.35/0.63      ! [B: rat,A: rat] :
% 0.35/0.63        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 0.35/0.63        = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_le_iff_le
% 0.35/0.63  thf(fact_4774_neg__le__iff__le,axiom,
% 0.35/0.63      ! [B: int,A: int] :
% 0.35/0.63        ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 0.35/0.63        = ( ord_less_eq_int @ A @ B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_le_iff_le
% 0.35/0.63  thf(fact_4775_neg__equal__zero,axiom,
% 0.35/0.63      ! [A: int] :
% 0.35/0.63        ( ( ( uminus_uminus_int @ A )
% 0.35/0.63          = A )
% 0.35/0.63        = ( A = zero_zero_int ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_equal_zero
% 0.35/0.63  thf(fact_4776_neg__equal__zero,axiom,
% 0.35/0.63      ! [A: real] :
% 0.35/0.63        ( ( ( uminus_uminus_real @ A )
% 0.35/0.63          = A )
% 0.35/0.63        = ( A = zero_zero_real ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_equal_zero
% 0.35/0.63  thf(fact_4777_neg__equal__zero,axiom,
% 0.35/0.63      ! [A: rat] :
% 0.35/0.63        ( ( ( uminus_uminus_rat @ A )
% 0.35/0.63          = A )
% 0.35/0.63        = ( A = zero_zero_rat ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_equal_zero
% 0.35/0.63  thf(fact_4778_neg__equal__zero,axiom,
% 0.35/0.63      ! [A: code_integer] :
% 0.35/0.63        ( ( ( uminus1351360451143612070nteger @ A )
% 0.35/0.63          = A )
% 0.35/0.63        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_equal_zero
% 0.35/0.63  thf(fact_4779_equal__neg__zero,axiom,
% 0.35/0.63      ! [A: int] :
% 0.35/0.63        ( ( A
% 0.35/0.63          = ( uminus_uminus_int @ A ) )
% 0.35/0.63        = ( A = zero_zero_int ) ) ).
% 0.35/0.63  
% 0.35/0.63  % equal_neg_zero
% 0.35/0.63  thf(fact_4780_equal__neg__zero,axiom,
% 0.35/0.63      ! [A: real] :
% 0.35/0.63        ( ( A
% 0.35/0.63          = ( uminus_uminus_real @ A ) )
% 0.35/0.63        = ( A = zero_zero_real ) ) ).
% 0.35/0.63  
% 0.35/0.63  % equal_neg_zero
% 0.35/0.63  thf(fact_4781_equal__neg__zero,axiom,
% 0.35/0.63      ! [A: rat] :
% 0.35/0.63        ( ( A
% 0.35/0.63          = ( uminus_uminus_rat @ A ) )
% 0.35/0.63        = ( A = zero_zero_rat ) ) ).
% 0.35/0.63  
% 0.35/0.63  % equal_neg_zero
% 0.35/0.63  thf(fact_4782_equal__neg__zero,axiom,
% 0.35/0.63      ! [A: code_integer] :
% 0.35/0.63        ( ( A
% 0.35/0.63          = ( uminus1351360451143612070nteger @ A ) )
% 0.35/0.63        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.63  
% 0.35/0.63  % equal_neg_zero
% 0.35/0.63  thf(fact_4783_neg__equal__0__iff__equal,axiom,
% 0.35/0.63      ! [A: complex] :
% 0.35/0.63        ( ( ( uminus1482373934393186551omplex @ A )
% 0.35/0.63          = zero_zero_complex )
% 0.35/0.63        = ( A = zero_zero_complex ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_equal_0_iff_equal
% 0.35/0.63  thf(fact_4784_neg__equal__0__iff__equal,axiom,
% 0.35/0.63      ! [A: int] :
% 0.35/0.63        ( ( ( uminus_uminus_int @ A )
% 0.35/0.63          = zero_zero_int )
% 0.35/0.63        = ( A = zero_zero_int ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_equal_0_iff_equal
% 0.35/0.63  thf(fact_4785_neg__equal__0__iff__equal,axiom,
% 0.35/0.63      ! [A: real] :
% 0.35/0.63        ( ( ( uminus_uminus_real @ A )
% 0.35/0.63          = zero_zero_real )
% 0.35/0.63        = ( A = zero_zero_real ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_equal_0_iff_equal
% 0.35/0.63  thf(fact_4786_neg__equal__0__iff__equal,axiom,
% 0.35/0.63      ! [A: rat] :
% 0.35/0.63        ( ( ( uminus_uminus_rat @ A )
% 0.35/0.63          = zero_zero_rat )
% 0.35/0.63        = ( A = zero_zero_rat ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_equal_0_iff_equal
% 0.35/0.63  thf(fact_4787_neg__equal__0__iff__equal,axiom,
% 0.35/0.63      ! [A: code_integer] :
% 0.35/0.63        ( ( ( uminus1351360451143612070nteger @ A )
% 0.35/0.63          = zero_z3403309356797280102nteger )
% 0.35/0.63        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_equal_0_iff_equal
% 0.35/0.63  thf(fact_4788_neg__0__equal__iff__equal,axiom,
% 0.35/0.63      ! [A: complex] :
% 0.35/0.63        ( ( zero_zero_complex
% 0.35/0.63          = ( uminus1482373934393186551omplex @ A ) )
% 0.35/0.63        = ( zero_zero_complex = A ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_0_equal_iff_equal
% 0.35/0.63  thf(fact_4789_neg__0__equal__iff__equal,axiom,
% 0.35/0.63      ! [A: int] :
% 0.35/0.63        ( ( zero_zero_int
% 0.35/0.63          = ( uminus_uminus_int @ A ) )
% 0.35/0.63        = ( zero_zero_int = A ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_0_equal_iff_equal
% 0.35/0.63  thf(fact_4790_neg__0__equal__iff__equal,axiom,
% 0.35/0.63      ! [A: real] :
% 0.35/0.63        ( ( zero_zero_real
% 0.35/0.63          = ( uminus_uminus_real @ A ) )
% 0.35/0.63        = ( zero_zero_real = A ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_0_equal_iff_equal
% 0.35/0.63  thf(fact_4791_neg__0__equal__iff__equal,axiom,
% 0.35/0.63      ! [A: rat] :
% 0.35/0.63        ( ( zero_zero_rat
% 0.35/0.63          = ( uminus_uminus_rat @ A ) )
% 0.35/0.63        = ( zero_zero_rat = A ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_0_equal_iff_equal
% 0.35/0.63  thf(fact_4792_neg__0__equal__iff__equal,axiom,
% 0.35/0.63      ! [A: code_integer] :
% 0.35/0.63        ( ( zero_z3403309356797280102nteger
% 0.35/0.63          = ( uminus1351360451143612070nteger @ A ) )
% 0.35/0.63        = ( zero_z3403309356797280102nteger = A ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_0_equal_iff_equal
% 0.35/0.63  thf(fact_4793_add_Oinverse__neutral,axiom,
% 0.35/0.63      ( ( uminus1482373934393186551omplex @ zero_zero_complex )
% 0.35/0.63      = zero_zero_complex ) ).
% 0.35/0.63  
% 0.35/0.63  % add.inverse_neutral
% 0.35/0.63  thf(fact_4794_add_Oinverse__neutral,axiom,
% 0.35/0.63      ( ( uminus_uminus_int @ zero_zero_int )
% 0.35/0.63      = zero_zero_int ) ).
% 0.35/0.63  
% 0.35/0.63  % add.inverse_neutral
% 0.35/0.63  thf(fact_4795_add_Oinverse__neutral,axiom,
% 0.35/0.63      ( ( uminus_uminus_real @ zero_zero_real )
% 0.35/0.63      = zero_zero_real ) ).
% 0.35/0.63  
% 0.35/0.63  % add.inverse_neutral
% 0.35/0.63  thf(fact_4796_add_Oinverse__neutral,axiom,
% 0.35/0.63      ( ( uminus_uminus_rat @ zero_zero_rat )
% 0.35/0.63      = zero_zero_rat ) ).
% 0.35/0.63  
% 0.35/0.63  % add.inverse_neutral
% 0.35/0.63  thf(fact_4797_add_Oinverse__neutral,axiom,
% 0.35/0.63      ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 0.35/0.63      = zero_z3403309356797280102nteger ) ).
% 0.35/0.63  
% 0.35/0.63  % add.inverse_neutral
% 0.35/0.63  thf(fact_4798_neg__less__iff__less,axiom,
% 0.35/0.63      ! [B: int,A: int] :
% 0.35/0.63        ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 0.35/0.63        = ( ord_less_int @ A @ B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_less_iff_less
% 0.35/0.63  thf(fact_4799_neg__less__iff__less,axiom,
% 0.35/0.63      ! [B: real,A: real] :
% 0.35/0.63        ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.35/0.63        = ( ord_less_real @ A @ B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_less_iff_less
% 0.35/0.63  thf(fact_4800_neg__less__iff__less,axiom,
% 0.35/0.63      ! [B: rat,A: rat] :
% 0.35/0.63        ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 0.35/0.63        = ( ord_less_rat @ A @ B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_less_iff_less
% 0.35/0.63  thf(fact_4801_neg__less__iff__less,axiom,
% 0.35/0.63      ! [B: code_integer,A: code_integer] :
% 0.35/0.63        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.35/0.63        = ( ord_le6747313008572928689nteger @ A @ B ) ) ).
% 0.35/0.63  
% 0.35/0.63  % neg_less_iff_less
% 0.35/0.63  thf(fact_4802_add__minus__cancel,axiom,
% 0.35/0.63      ! [A: int,B: int] :
% 0.35/0.63        ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 0.35/0.63        = B ) ).
% 0.35/0.63  
% 0.35/0.63  % add_minus_cancel
% 0.35/0.63  thf(fact_4803_add__minus__cancel,axiom,
% 0.35/0.63      ! [A: real,B: real] :
% 0.35/0.63        ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
% 0.35/0.63        = B ) ).
% 0.35/0.63  
% 0.35/0.63  % add_minus_cancel
% 0.35/0.63  thf(fact_4804_add__minus__cancel,axiom,
% 0.35/0.63      ! [A: rat,B: rat] :
% 0.35/0.63        ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
% 0.35/0.63        = B ) ).
% 0.35/0.63  
% 0.35/0.63  % add_minus_cancel
% 0.35/0.63  thf(fact_4805_add__minus__cancel,axiom,
% 0.35/0.63      ! [A: code_integer,B: code_integer] :
% 0.35/0.63        ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
% 0.35/0.63        = B ) ).
% 0.35/0.63  
% 0.35/0.63  % add_minus_cancel
% 0.35/0.63  thf(fact_4806_minus__add__cancel,axiom,
% 0.35/0.63      ! [A: int,B: int] :
% 0.35/0.63        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 0.35/0.63        = B ) ).
% 0.35/0.63  
% 0.35/0.63  % minus_add_cancel
% 0.35/0.63  thf(fact_4807_minus__add__cancel,axiom,
% 0.35/0.63      ! [A: real,B: real] :
% 0.35/0.63        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
% 0.35/0.63        = B ) ).
% 0.35/0.63  
% 0.35/0.63  % minus_add_cancel
% 0.35/0.63  thf(fact_4808_minus__add__cancel,axiom,
% 0.35/0.63      ! [A: rat,B: rat] :
% 0.35/0.63        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
% 0.35/0.63        = B ) ).
% 0.35/0.63  
% 0.35/0.63  % minus_add_cancel
% 0.35/0.63  thf(fact_4809_minus__add__cancel,axiom,
% 0.35/0.63      ! [A: code_integer,B: code_integer] :
% 0.35/0.63        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.35/0.63        = B ) ).
% 0.35/0.63  
% 0.35/0.63  % minus_add_cancel
% 0.35/0.63  thf(fact_4810_minus__add__distrib,axiom,
% 0.35/0.63      ! [A: int,B: int] :
% 0.35/0.64        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.35/0.64        = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_add_distrib
% 0.35/0.64  thf(fact_4811_minus__add__distrib,axiom,
% 0.35/0.64      ! [A: real,B: real] :
% 0.35/0.64        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.35/0.64        = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_add_distrib
% 0.35/0.64  thf(fact_4812_minus__add__distrib,axiom,
% 0.35/0.64      ! [A: rat,B: rat] :
% 0.35/0.64        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.35/0.64        = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_add_distrib
% 0.35/0.64  thf(fact_4813_minus__add__distrib,axiom,
% 0.35/0.64      ! [A: code_integer,B: code_integer] :
% 0.35/0.64        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.35/0.64        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_add_distrib
% 0.35/0.64  thf(fact_4814_minus__diff__eq,axiom,
% 0.35/0.64      ! [A: int,B: int] :
% 0.35/0.64        ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 0.35/0.64        = ( minus_minus_int @ B @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_diff_eq
% 0.35/0.64  thf(fact_4815_minus__diff__eq,axiom,
% 0.35/0.64      ! [A: real,B: real] :
% 0.35/0.64        ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
% 0.35/0.64        = ( minus_minus_real @ B @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_diff_eq
% 0.35/0.64  thf(fact_4816_minus__diff__eq,axiom,
% 0.35/0.64      ! [A: rat,B: rat] :
% 0.35/0.64        ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
% 0.35/0.64        = ( minus_minus_rat @ B @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_diff_eq
% 0.35/0.64  thf(fact_4817_minus__diff__eq,axiom,
% 0.35/0.64      ! [A: code_integer,B: code_integer] :
% 0.35/0.64        ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.35/0.64        = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_diff_eq
% 0.35/0.64  thf(fact_4818_semiring__norm_I80_J,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.35/0.64        = ( ord_less_num @ M @ N ) ) ).
% 0.35/0.64  
% 0.35/0.64  % semiring_norm(80)
% 0.35/0.64  thf(fact_4819_neg__0__le__iff__le,axiom,
% 0.35/0.64      ! [A: real] :
% 0.35/0.64        ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.35/0.64        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_0_le_iff_le
% 0.35/0.64  thf(fact_4820_neg__0__le__iff__le,axiom,
% 0.35/0.64      ! [A: code_integer] :
% 0.35/0.64        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.35/0.64        = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_0_le_iff_le
% 0.35/0.64  thf(fact_4821_neg__0__le__iff__le,axiom,
% 0.35/0.64      ! [A: rat] :
% 0.35/0.64        ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 0.35/0.64        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_0_le_iff_le
% 0.35/0.64  thf(fact_4822_neg__0__le__iff__le,axiom,
% 0.35/0.64      ! [A: int] :
% 0.35/0.64        ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.35/0.64        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_0_le_iff_le
% 0.35/0.64  thf(fact_4823_neg__le__0__iff__le,axiom,
% 0.35/0.64      ! [A: real] :
% 0.35/0.64        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.35/0.64        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_le_0_iff_le
% 0.35/0.64  thf(fact_4824_neg__le__0__iff__le,axiom,
% 0.35/0.64      ! [A: code_integer] :
% 0.35/0.64        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 0.35/0.64        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_le_0_iff_le
% 0.35/0.64  thf(fact_4825_neg__le__0__iff__le,axiom,
% 0.35/0.64      ! [A: rat] :
% 0.35/0.64        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 0.35/0.64        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_le_0_iff_le
% 0.35/0.64  thf(fact_4826_neg__le__0__iff__le,axiom,
% 0.35/0.64      ! [A: int] :
% 0.35/0.64        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.35/0.64        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_le_0_iff_le
% 0.35/0.64  thf(fact_4827_less__eq__neg__nonpos,axiom,
% 0.35/0.64      ! [A: real] :
% 0.35/0.64        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 0.35/0.64        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_eq_neg_nonpos
% 0.35/0.64  thf(fact_4828_less__eq__neg__nonpos,axiom,
% 0.35/0.64      ! [A: code_integer] :
% 0.35/0.64        ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.35/0.64        = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_eq_neg_nonpos
% 0.35/0.64  thf(fact_4829_less__eq__neg__nonpos,axiom,
% 0.35/0.64      ! [A: rat] :
% 0.35/0.64        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.35/0.64        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_eq_neg_nonpos
% 0.35/0.64  thf(fact_4830_less__eq__neg__nonpos,axiom,
% 0.35/0.64      ! [A: int] :
% 0.35/0.64        ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 0.35/0.64        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_eq_neg_nonpos
% 0.35/0.64  thf(fact_4831_neg__less__eq__nonneg,axiom,
% 0.35/0.64      ! [A: real] :
% 0.35/0.64        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 0.35/0.64        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_less_eq_nonneg
% 0.35/0.64  thf(fact_4832_neg__less__eq__nonneg,axiom,
% 0.35/0.64      ! [A: code_integer] :
% 0.35/0.64        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.35/0.64        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_less_eq_nonneg
% 0.35/0.64  thf(fact_4833_neg__less__eq__nonneg,axiom,
% 0.35/0.64      ! [A: rat] :
% 0.35/0.64        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.35/0.64        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_less_eq_nonneg
% 0.35/0.64  thf(fact_4834_neg__less__eq__nonneg,axiom,
% 0.35/0.64      ! [A: int] :
% 0.35/0.64        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 0.35/0.64        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_less_eq_nonneg
% 0.35/0.64  thf(fact_4835_neg__less__0__iff__less,axiom,
% 0.35/0.64      ! [A: int] :
% 0.35/0.64        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.35/0.64        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_less_0_iff_less
% 0.35/0.64  thf(fact_4836_neg__less__0__iff__less,axiom,
% 0.35/0.64      ! [A: real] :
% 0.35/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.35/0.64        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_less_0_iff_less
% 0.35/0.64  thf(fact_4837_neg__less__0__iff__less,axiom,
% 0.35/0.64      ! [A: rat] :
% 0.35/0.64        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 0.35/0.64        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_less_0_iff_less
% 0.35/0.64  thf(fact_4838_neg__less__0__iff__less,axiom,
% 0.35/0.64      ! [A: code_integer] :
% 0.35/0.64        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 0.35/0.64        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_less_0_iff_less
% 0.35/0.64  thf(fact_4839_neg__0__less__iff__less,axiom,
% 0.35/0.64      ! [A: int] :
% 0.35/0.64        ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.35/0.64        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_0_less_iff_less
% 0.35/0.64  thf(fact_4840_neg__0__less__iff__less,axiom,
% 0.35/0.64      ! [A: real] :
% 0.35/0.64        ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.35/0.64        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_0_less_iff_less
% 0.35/0.64  thf(fact_4841_neg__0__less__iff__less,axiom,
% 0.35/0.64      ! [A: rat] :
% 0.35/0.64        ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 0.35/0.64        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_0_less_iff_less
% 0.35/0.64  thf(fact_4842_neg__0__less__iff__less,axiom,
% 0.35/0.64      ! [A: code_integer] :
% 0.35/0.64        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.35/0.64        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_0_less_iff_less
% 0.35/0.64  thf(fact_4843_neg__less__pos,axiom,
% 0.35/0.64      ! [A: int] :
% 0.35/0.64        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 0.35/0.64        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_less_pos
% 0.35/0.64  thf(fact_4844_neg__less__pos,axiom,
% 0.35/0.64      ! [A: real] :
% 0.35/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 0.35/0.64        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_less_pos
% 0.35/0.64  thf(fact_4845_neg__less__pos,axiom,
% 0.35/0.64      ! [A: rat] :
% 0.35/0.64        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.35/0.64        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_less_pos
% 0.35/0.64  thf(fact_4846_neg__less__pos,axiom,
% 0.35/0.64      ! [A: code_integer] :
% 0.35/0.64        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.35/0.64        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_less_pos
% 0.35/0.64  thf(fact_4847_less__neg__neg,axiom,
% 0.35/0.64      ! [A: int] :
% 0.35/0.64        ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 0.35/0.64        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_neg_neg
% 0.35/0.64  thf(fact_4848_less__neg__neg,axiom,
% 0.35/0.64      ! [A: real] :
% 0.35/0.64        ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 0.35/0.64        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_neg_neg
% 0.35/0.64  thf(fact_4849_less__neg__neg,axiom,
% 0.35/0.64      ! [A: rat] :
% 0.35/0.64        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.35/0.64        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_neg_neg
% 0.35/0.64  thf(fact_4850_less__neg__neg,axiom,
% 0.35/0.64      ! [A: code_integer] :
% 0.35/0.64        ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.35/0.64        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_neg_neg
% 0.35/0.64  thf(fact_4851_add_Oright__inverse,axiom,
% 0.35/0.64      ! [A: complex] :
% 0.35/0.64        ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
% 0.35/0.64        = zero_zero_complex ) ).
% 0.35/0.64  
% 0.35/0.64  % add.right_inverse
% 0.35/0.64  thf(fact_4852_add_Oright__inverse,axiom,
% 0.35/0.64      ! [A: int] :
% 0.35/0.64        ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
% 0.35/0.64        = zero_zero_int ) ).
% 0.35/0.64  
% 0.35/0.64  % add.right_inverse
% 0.35/0.64  thf(fact_4853_add_Oright__inverse,axiom,
% 0.35/0.64      ! [A: real] :
% 0.35/0.64        ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 0.35/0.64        = zero_zero_real ) ).
% 0.35/0.64  
% 0.35/0.64  % add.right_inverse
% 0.35/0.64  thf(fact_4854_add_Oright__inverse,axiom,
% 0.35/0.64      ! [A: rat] :
% 0.35/0.64        ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.35/0.64        = zero_zero_rat ) ).
% 0.35/0.64  
% 0.35/0.64  % add.right_inverse
% 0.35/0.64  thf(fact_4855_add_Oright__inverse,axiom,
% 0.35/0.64      ! [A: code_integer] :
% 0.35/0.64        ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.35/0.64        = zero_z3403309356797280102nteger ) ).
% 0.35/0.64  
% 0.35/0.64  % add.right_inverse
% 0.35/0.64  thf(fact_4856_ab__left__minus,axiom,
% 0.35/0.64      ! [A: complex] :
% 0.35/0.64        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 0.35/0.64        = zero_zero_complex ) ).
% 0.35/0.64  
% 0.35/0.64  % ab_left_minus
% 0.35/0.64  thf(fact_4857_ab__left__minus,axiom,
% 0.35/0.64      ! [A: int] :
% 0.35/0.64        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 0.35/0.64        = zero_zero_int ) ).
% 0.35/0.64  
% 0.35/0.64  % ab_left_minus
% 0.35/0.64  thf(fact_4858_ab__left__minus,axiom,
% 0.35/0.64      ! [A: real] :
% 0.35/0.64        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.35/0.64        = zero_zero_real ) ).
% 0.35/0.64  
% 0.35/0.64  % ab_left_minus
% 0.35/0.64  thf(fact_4859_ab__left__minus,axiom,
% 0.35/0.64      ! [A: rat] :
% 0.35/0.64        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.35/0.64        = zero_zero_rat ) ).
% 0.35/0.64  
% 0.35/0.64  % ab_left_minus
% 0.35/0.64  thf(fact_4860_ab__left__minus,axiom,
% 0.35/0.64      ! [A: code_integer] :
% 0.35/0.64        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.35/0.64        = zero_z3403309356797280102nteger ) ).
% 0.35/0.64  
% 0.35/0.64  % ab_left_minus
% 0.35/0.64  thf(fact_4861_verit__minus__simplify_I3_J,axiom,
% 0.35/0.64      ! [B: complex] :
% 0.35/0.64        ( ( minus_minus_complex @ zero_zero_complex @ B )
% 0.35/0.64        = ( uminus1482373934393186551omplex @ B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % verit_minus_simplify(3)
% 0.35/0.64  thf(fact_4862_verit__minus__simplify_I3_J,axiom,
% 0.35/0.64      ! [B: int] :
% 0.35/0.64        ( ( minus_minus_int @ zero_zero_int @ B )
% 0.35/0.64        = ( uminus_uminus_int @ B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % verit_minus_simplify(3)
% 0.35/0.64  thf(fact_4863_verit__minus__simplify_I3_J,axiom,
% 0.35/0.64      ! [B: real] :
% 0.35/0.64        ( ( minus_minus_real @ zero_zero_real @ B )
% 0.35/0.64        = ( uminus_uminus_real @ B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % verit_minus_simplify(3)
% 0.35/0.64  thf(fact_4864_verit__minus__simplify_I3_J,axiom,
% 0.35/0.64      ! [B: rat] :
% 0.35/0.64        ( ( minus_minus_rat @ zero_zero_rat @ B )
% 0.35/0.64        = ( uminus_uminus_rat @ B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % verit_minus_simplify(3)
% 0.35/0.64  thf(fact_4865_verit__minus__simplify_I3_J,axiom,
% 0.35/0.64      ! [B: code_integer] :
% 0.35/0.64        ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
% 0.35/0.64        = ( uminus1351360451143612070nteger @ B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % verit_minus_simplify(3)
% 0.35/0.64  thf(fact_4866_diff__0,axiom,
% 0.35/0.64      ! [A: complex] :
% 0.35/0.64        ( ( minus_minus_complex @ zero_zero_complex @ A )
% 0.35/0.64        = ( uminus1482373934393186551omplex @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % diff_0
% 0.35/0.64  thf(fact_4867_diff__0,axiom,
% 0.35/0.64      ! [A: int] :
% 0.35/0.64        ( ( minus_minus_int @ zero_zero_int @ A )
% 0.35/0.64        = ( uminus_uminus_int @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % diff_0
% 0.35/0.64  thf(fact_4868_diff__0,axiom,
% 0.35/0.64      ! [A: real] :
% 0.35/0.64        ( ( minus_minus_real @ zero_zero_real @ A )
% 0.35/0.64        = ( uminus_uminus_real @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % diff_0
% 0.35/0.64  thf(fact_4869_diff__0,axiom,
% 0.35/0.64      ! [A: rat] :
% 0.35/0.64        ( ( minus_minus_rat @ zero_zero_rat @ A )
% 0.35/0.64        = ( uminus_uminus_rat @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % diff_0
% 0.35/0.64  thf(fact_4870_diff__0,axiom,
% 0.35/0.64      ! [A: code_integer] :
% 0.35/0.64        ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
% 0.35/0.64        = ( uminus1351360451143612070nteger @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % diff_0
% 0.35/0.64  thf(fact_4871_uminus__add__conv__diff,axiom,
% 0.35/0.64      ! [A: int,B: int] :
% 0.35/0.64        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 0.35/0.64        = ( minus_minus_int @ B @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % uminus_add_conv_diff
% 0.35/0.64  thf(fact_4872_uminus__add__conv__diff,axiom,
% 0.35/0.64      ! [A: real,B: real] :
% 0.35/0.64        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
% 0.35/0.64        = ( minus_minus_real @ B @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % uminus_add_conv_diff
% 0.35/0.64  thf(fact_4873_uminus__add__conv__diff,axiom,
% 0.35/0.64      ! [A: rat,B: rat] :
% 0.35/0.64        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.35/0.64        = ( minus_minus_rat @ B @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % uminus_add_conv_diff
% 0.35/0.64  thf(fact_4874_uminus__add__conv__diff,axiom,
% 0.35/0.64      ! [A: code_integer,B: code_integer] :
% 0.35/0.64        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.35/0.64        = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % uminus_add_conv_diff
% 0.35/0.64  thf(fact_4875_diff__minus__eq__add,axiom,
% 0.35/0.64      ! [A: int,B: int] :
% 0.35/0.64        ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 0.35/0.64        = ( plus_plus_int @ A @ B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % diff_minus_eq_add
% 0.35/0.64  thf(fact_4876_diff__minus__eq__add,axiom,
% 0.35/0.64      ! [A: real,B: real] :
% 0.35/0.64        ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
% 0.35/0.64        = ( plus_plus_real @ A @ B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % diff_minus_eq_add
% 0.35/0.64  thf(fact_4877_diff__minus__eq__add,axiom,
% 0.35/0.64      ! [A: rat,B: rat] :
% 0.35/0.64        ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.35/0.64        = ( plus_plus_rat @ A @ B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % diff_minus_eq_add
% 0.35/0.64  thf(fact_4878_diff__minus__eq__add,axiom,
% 0.35/0.64      ! [A: code_integer,B: code_integer] :
% 0.35/0.64        ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.35/0.64        = ( plus_p5714425477246183910nteger @ A @ B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % diff_minus_eq_add
% 0.35/0.64  thf(fact_4879_semiring__norm_I81_J,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.35/0.64        = ( ord_less_num @ M @ N ) ) ).
% 0.35/0.64  
% 0.35/0.64  % semiring_norm(81)
% 0.35/0.64  thf(fact_4880_semiring__norm_I77_J,axiom,
% 0.35/0.64      ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% 0.35/0.64  
% 0.35/0.64  % semiring_norm(77)
% 0.35/0.64  thf(fact_4881_add__neg__numeral__special_I7_J,axiom,
% 0.35/0.64      ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.35/0.64      = zero_zero_complex ) ).
% 0.35/0.64  
% 0.35/0.64  % add_neg_numeral_special(7)
% 0.35/0.64  thf(fact_4882_add__neg__numeral__special_I7_J,axiom,
% 0.35/0.64      ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.35/0.64      = zero_zero_int ) ).
% 0.35/0.64  
% 0.35/0.64  % add_neg_numeral_special(7)
% 0.35/0.64  thf(fact_4883_add__neg__numeral__special_I7_J,axiom,
% 0.35/0.64      ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.35/0.64      = zero_zero_real ) ).
% 0.35/0.64  
% 0.35/0.64  % add_neg_numeral_special(7)
% 0.35/0.64  thf(fact_4884_add__neg__numeral__special_I7_J,axiom,
% 0.35/0.64      ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.35/0.64      = zero_zero_rat ) ).
% 0.35/0.64  
% 0.35/0.64  % add_neg_numeral_special(7)
% 0.35/0.64  thf(fact_4885_add__neg__numeral__special_I7_J,axiom,
% 0.35/0.64      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.35/0.64      = zero_z3403309356797280102nteger ) ).
% 0.35/0.64  
% 0.35/0.64  % add_neg_numeral_special(7)
% 0.35/0.64  thf(fact_4886_add__neg__numeral__special_I8_J,axiom,
% 0.35/0.64      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 0.35/0.64      = zero_zero_complex ) ).
% 0.35/0.64  
% 0.35/0.64  % add_neg_numeral_special(8)
% 0.35/0.64  thf(fact_4887_add__neg__numeral__special_I8_J,axiom,
% 0.35/0.64      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.35/0.64      = zero_zero_int ) ).
% 0.35/0.64  
% 0.35/0.64  % add_neg_numeral_special(8)
% 0.35/0.64  thf(fact_4888_add__neg__numeral__special_I8_J,axiom,
% 0.35/0.64      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.35/0.64      = zero_zero_real ) ).
% 0.35/0.64  
% 0.35/0.64  % add_neg_numeral_special(8)
% 0.35/0.64  thf(fact_4889_add__neg__numeral__special_I8_J,axiom,
% 0.35/0.64      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 0.35/0.64      = zero_zero_rat ) ).
% 0.35/0.64  
% 0.35/0.64  % add_neg_numeral_special(8)
% 0.35/0.64  thf(fact_4890_add__neg__numeral__special_I8_J,axiom,
% 0.35/0.64      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 0.35/0.64      = zero_z3403309356797280102nteger ) ).
% 0.35/0.64  
% 0.35/0.64  % add_neg_numeral_special(8)
% 0.35/0.64  thf(fact_4891_diff__numeral__special_I12_J,axiom,
% 0.35/0.64      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.35/0.64      = zero_zero_complex ) ).
% 0.35/0.64  
% 0.35/0.64  % diff_numeral_special(12)
% 0.35/0.64  thf(fact_4892_diff__numeral__special_I12_J,axiom,
% 0.35/0.64      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.35/0.64      = zero_zero_int ) ).
% 0.35/0.64  
% 0.35/0.64  % diff_numeral_special(12)
% 0.35/0.64  thf(fact_4893_diff__numeral__special_I12_J,axiom,
% 0.35/0.64      ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.35/0.64      = zero_zero_real ) ).
% 0.35/0.64  
% 0.35/0.64  % diff_numeral_special(12)
% 0.35/0.64  thf(fact_4894_diff__numeral__special_I12_J,axiom,
% 0.35/0.64      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.35/0.64      = zero_zero_rat ) ).
% 0.35/0.64  
% 0.35/0.64  % diff_numeral_special(12)
% 0.35/0.64  thf(fact_4895_diff__numeral__special_I12_J,axiom,
% 0.35/0.64      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.35/0.64      = zero_z3403309356797280102nteger ) ).
% 0.35/0.64  
% 0.35/0.64  % diff_numeral_special(12)
% 0.35/0.64  thf(fact_4896_mod__minus1__right,axiom,
% 0.35/0.64      ! [A: int] :
% 0.35/0.64        ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.35/0.64        = zero_zero_int ) ).
% 0.35/0.64  
% 0.35/0.64  % mod_minus1_right
% 0.35/0.64  thf(fact_4897_mod__minus1__right,axiom,
% 0.35/0.64      ! [A: code_integer] :
% 0.35/0.64        ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.35/0.64        = zero_z3403309356797280102nteger ) ).
% 0.35/0.64  
% 0.35/0.64  % mod_minus1_right
% 0.35/0.64  thf(fact_4898_max__number__of_I4_J,axiom,
% 0.35/0.64      ! [U: num,V: num] :
% 0.35/0.64        ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.35/0.64         => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.35/0.64            = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 0.35/0.64        & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.35/0.64         => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.35/0.64            = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % max_number_of(4)
% 0.35/0.64  thf(fact_4899_max__number__of_I4_J,axiom,
% 0.35/0.64      ! [U: num,V: num] :
% 0.35/0.64        ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.35/0.64         => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.35/0.64            = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 0.35/0.64        & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.35/0.64         => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.35/0.64            = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % max_number_of(4)
% 0.35/0.64  thf(fact_4900_max__number__of_I4_J,axiom,
% 0.35/0.64      ! [U: num,V: num] :
% 0.35/0.64        ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.35/0.64         => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.35/0.64            = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 0.35/0.64        & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.35/0.64         => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.35/0.64            = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % max_number_of(4)
% 0.35/0.64  thf(fact_4901_max__number__of_I4_J,axiom,
% 0.35/0.64      ! [U: num,V: num] :
% 0.35/0.64        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.64         => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.64            = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 0.35/0.64        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.64         => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.64            = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % max_number_of(4)
% 0.35/0.64  thf(fact_4902_max__number__of_I3_J,axiom,
% 0.35/0.64      ! [U: num,V: num] :
% 0.35/0.64        ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.35/0.64         => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.35/0.64            = ( numeral_numeral_real @ V ) ) )
% 0.35/0.64        & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.35/0.64         => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.35/0.64            = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % max_number_of(3)
% 0.35/0.64  thf(fact_4903_max__number__of_I3_J,axiom,
% 0.35/0.64      ! [U: num,V: num] :
% 0.35/0.64        ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.35/0.64         => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.35/0.64            = ( numera6620942414471956472nteger @ V ) ) )
% 0.35/0.64        & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.35/0.64         => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.35/0.64            = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % max_number_of(3)
% 0.35/0.64  thf(fact_4904_max__number__of_I3_J,axiom,
% 0.35/0.64      ! [U: num,V: num] :
% 0.35/0.64        ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.35/0.64         => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.35/0.64            = ( numeral_numeral_rat @ V ) ) )
% 0.35/0.64        & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.35/0.64         => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.35/0.64            = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % max_number_of(3)
% 0.35/0.64  thf(fact_4905_max__number__of_I3_J,axiom,
% 0.35/0.64      ! [U: num,V: num] :
% 0.35/0.64        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.64         => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.64            = ( numeral_numeral_int @ V ) ) )
% 0.35/0.64        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.64         => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.64            = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % max_number_of(3)
% 0.35/0.64  thf(fact_4906_max__number__of_I2_J,axiom,
% 0.35/0.64      ! [U: num,V: num] :
% 0.35/0.64        ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.35/0.64         => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.35/0.64            = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 0.35/0.64        & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.35/0.64         => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.35/0.64            = ( numeral_numeral_real @ U ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % max_number_of(2)
% 0.35/0.64  thf(fact_4907_max__number__of_I2_J,axiom,
% 0.35/0.64      ! [U: num,V: num] :
% 0.35/0.64        ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.35/0.64         => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.35/0.64            = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 0.35/0.64        & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.35/0.64         => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.35/0.64            = ( numera6620942414471956472nteger @ U ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % max_number_of(2)
% 0.35/0.64  thf(fact_4908_max__number__of_I2_J,axiom,
% 0.35/0.64      ! [U: num,V: num] :
% 0.35/0.64        ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.35/0.64         => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.35/0.64            = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 0.35/0.64        & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.35/0.64         => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.35/0.64            = ( numeral_numeral_rat @ U ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % max_number_of(2)
% 0.35/0.64  thf(fact_4909_max__number__of_I2_J,axiom,
% 0.35/0.64      ! [U: num,V: num] :
% 0.35/0.64        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.64         => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.64            = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 0.35/0.64        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.64         => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.64            = ( numeral_numeral_int @ U ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % max_number_of(2)
% 0.35/0.64  thf(fact_4910_neg__numeral__le__iff,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.35/0.64        = ( ord_less_eq_num @ N @ M ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_le_iff
% 0.35/0.64  thf(fact_4911_neg__numeral__le__iff,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.35/0.64        = ( ord_less_eq_num @ N @ M ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_le_iff
% 0.35/0.64  thf(fact_4912_neg__numeral__le__iff,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.35/0.64        = ( ord_less_eq_num @ N @ M ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_le_iff
% 0.35/0.64  thf(fact_4913_neg__numeral__le__iff,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.64        = ( ord_less_eq_num @ N @ M ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_le_iff
% 0.35/0.64  thf(fact_4914_neg__numeral__less__iff,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.64        = ( ord_less_num @ N @ M ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_less_iff
% 0.35/0.64  thf(fact_4915_neg__numeral__less__iff,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.35/0.64        = ( ord_less_num @ N @ M ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_less_iff
% 0.35/0.64  thf(fact_4916_neg__numeral__less__iff,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.35/0.64        = ( ord_less_num @ N @ M ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_less_iff
% 0.35/0.64  thf(fact_4917_neg__numeral__less__iff,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.35/0.64        = ( ord_less_num @ N @ M ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_less_iff
% 0.35/0.64  thf(fact_4918_semiring__norm_I74_J,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.35/0.64        = ( ord_less_num @ M @ N ) ) ).
% 0.35/0.64  
% 0.35/0.64  % semiring_norm(74)
% 0.35/0.64  thf(fact_4919_semiring__norm_I79_J,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.35/0.64        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.35/0.64  
% 0.35/0.64  % semiring_norm(79)
% 0.35/0.64  thf(fact_4920_not__neg__one__le__neg__numeral__iff,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 0.35/0.64        = ( M != one ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_neg_one_le_neg_numeral_iff
% 0.35/0.64  thf(fact_4921_not__neg__one__le__neg__numeral__iff,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 0.35/0.64        = ( M != one ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_neg_one_le_neg_numeral_iff
% 0.35/0.64  thf(fact_4922_not__neg__one__le__neg__numeral__iff,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 0.35/0.64        = ( M != one ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_neg_one_le_neg_numeral_iff
% 0.35/0.64  thf(fact_4923_not__neg__one__le__neg__numeral__iff,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 0.35/0.64        = ( M != one ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_neg_one_le_neg_numeral_iff
% 0.35/0.64  thf(fact_4924_divide__le__eq__numeral1_I2_J,axiom,
% 0.35/0.64      ! [B: real,W2: num,A: real] :
% 0.35/0.64        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
% 0.35/0.64        = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divide_le_eq_numeral1(2)
% 0.35/0.64  thf(fact_4925_divide__le__eq__numeral1_I2_J,axiom,
% 0.35/0.64      ! [B: rat,W2: num,A: rat] :
% 0.35/0.64        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
% 0.35/0.64        = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divide_le_eq_numeral1(2)
% 0.35/0.64  thf(fact_4926_le__divide__eq__numeral1_I2_J,axiom,
% 0.35/0.64      ! [A: real,B: real,W2: num] :
% 0.35/0.64        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
% 0.35/0.64        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_divide_eq_numeral1(2)
% 0.35/0.64  thf(fact_4927_le__divide__eq__numeral1_I2_J,axiom,
% 0.35/0.64      ! [A: rat,B: rat,W2: num] :
% 0.35/0.64        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
% 0.35/0.64        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_divide_eq_numeral1(2)
% 0.35/0.64  thf(fact_4928_divide__eq__eq__numeral1_I2_J,axiom,
% 0.35/0.64      ! [B: complex,W2: num,A: complex] :
% 0.35/0.64        ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
% 0.35/0.64          = A )
% 0.35/0.64        = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.35/0.64             != zero_zero_complex )
% 0.35/0.64           => ( B
% 0.35/0.64              = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) )
% 0.35/0.64          & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.35/0.64              = zero_zero_complex )
% 0.35/0.64           => ( A = zero_zero_complex ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divide_eq_eq_numeral1(2)
% 0.35/0.64  thf(fact_4929_divide__eq__eq__numeral1_I2_J,axiom,
% 0.35/0.64      ! [B: real,W2: num,A: real] :
% 0.35/0.64        ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 0.35/0.64          = A )
% 0.35/0.64        = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.35/0.64             != zero_zero_real )
% 0.35/0.64           => ( B
% 0.35/0.64              = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) )
% 0.35/0.64          & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.35/0.64              = zero_zero_real )
% 0.35/0.64           => ( A = zero_zero_real ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divide_eq_eq_numeral1(2)
% 0.35/0.64  thf(fact_4930_divide__eq__eq__numeral1_I2_J,axiom,
% 0.35/0.64      ! [B: rat,W2: num,A: rat] :
% 0.35/0.64        ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 0.35/0.64          = A )
% 0.35/0.64        = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.35/0.64             != zero_zero_rat )
% 0.35/0.64           => ( B
% 0.35/0.64              = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) )
% 0.35/0.64          & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.35/0.64              = zero_zero_rat )
% 0.35/0.64           => ( A = zero_zero_rat ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divide_eq_eq_numeral1(2)
% 0.35/0.64  thf(fact_4931_eq__divide__eq__numeral1_I2_J,axiom,
% 0.35/0.64      ! [A: complex,B: complex,W2: num] :
% 0.35/0.64        ( ( A
% 0.35/0.64          = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) )
% 0.35/0.64        = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.35/0.64             != zero_zero_complex )
% 0.35/0.64           => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
% 0.35/0.64              = B ) )
% 0.35/0.64          & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.35/0.64              = zero_zero_complex )
% 0.35/0.64           => ( A = zero_zero_complex ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % eq_divide_eq_numeral1(2)
% 0.35/0.64  thf(fact_4932_eq__divide__eq__numeral1_I2_J,axiom,
% 0.35/0.64      ! [A: real,B: real,W2: num] :
% 0.35/0.64        ( ( A
% 0.35/0.64          = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
% 0.35/0.64        = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.35/0.64             != zero_zero_real )
% 0.35/0.64           => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 0.35/0.64              = B ) )
% 0.35/0.64          & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.35/0.64              = zero_zero_real )
% 0.35/0.64           => ( A = zero_zero_real ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % eq_divide_eq_numeral1(2)
% 0.35/0.64  thf(fact_4933_eq__divide__eq__numeral1_I2_J,axiom,
% 0.35/0.64      ! [A: rat,B: rat,W2: num] :
% 0.35/0.64        ( ( A
% 0.35/0.64          = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
% 0.35/0.64        = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.35/0.64             != zero_zero_rat )
% 0.35/0.64           => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 0.35/0.64              = B ) )
% 0.35/0.64          & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.35/0.64              = zero_zero_rat )
% 0.35/0.64           => ( A = zero_zero_rat ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % eq_divide_eq_numeral1(2)
% 0.35/0.64  thf(fact_4934_neg__numeral__less__neg__one__iff,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.35/0.64        = ( M != one ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_less_neg_one_iff
% 0.35/0.64  thf(fact_4935_neg__numeral__less__neg__one__iff,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.35/0.64        = ( M != one ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_less_neg_one_iff
% 0.35/0.64  thf(fact_4936_neg__numeral__less__neg__one__iff,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.35/0.64        = ( M != one ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_less_neg_one_iff
% 0.35/0.64  thf(fact_4937_neg__numeral__less__neg__one__iff,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.35/0.64        = ( M != one ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_less_neg_one_iff
% 0.35/0.64  thf(fact_4938_less__divide__eq__numeral1_I2_J,axiom,
% 0.35/0.64      ! [A: real,B: real,W2: num] :
% 0.35/0.64        ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
% 0.35/0.64        = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_divide_eq_numeral1(2)
% 0.35/0.64  thf(fact_4939_less__divide__eq__numeral1_I2_J,axiom,
% 0.35/0.64      ! [A: rat,B: rat,W2: num] :
% 0.35/0.64        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
% 0.35/0.64        = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_divide_eq_numeral1(2)
% 0.35/0.64  thf(fact_4940_divide__less__eq__numeral1_I2_J,axiom,
% 0.35/0.64      ! [B: real,W2: num,A: real] :
% 0.35/0.64        ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
% 0.35/0.64        = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divide_less_eq_numeral1(2)
% 0.35/0.64  thf(fact_4941_divide__less__eq__numeral1_I2_J,axiom,
% 0.35/0.64      ! [B: rat,W2: num,A: rat] :
% 0.35/0.64        ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
% 0.35/0.64        = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divide_less_eq_numeral1(2)
% 0.35/0.64  thf(fact_4942_divmod__algorithm__code_I4_J,axiom,
% 0.35/0.64      ! [N: num] :
% 0.35/0.64        ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
% 0.35/0.64        = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divmod_algorithm_code(4)
% 0.35/0.64  thf(fact_4943_divmod__algorithm__code_I4_J,axiom,
% 0.35/0.64      ! [N: num] :
% 0.35/0.64        ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
% 0.35/0.64        = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divmod_algorithm_code(4)
% 0.35/0.64  thf(fact_4944_divmod__algorithm__code_I4_J,axiom,
% 0.35/0.64      ! [N: num] :
% 0.35/0.64        ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
% 0.35/0.64        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divmod_algorithm_code(4)
% 0.35/0.64  thf(fact_4945_divmod__algorithm__code_I8_J,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ( ( ( ord_less_num @ M @ N )
% 0.35/0.64         => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.35/0.64            = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
% 0.35/0.64        & ( ~ ( ord_less_num @ M @ N )
% 0.35/0.64         => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.35/0.64            = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divmod_algorithm_code(8)
% 0.35/0.64  thf(fact_4946_divmod__algorithm__code_I8_J,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ( ( ( ord_less_num @ M @ N )
% 0.35/0.64         => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.35/0.64            = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
% 0.35/0.64        & ( ~ ( ord_less_num @ M @ N )
% 0.35/0.64         => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.35/0.64            = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divmod_algorithm_code(8)
% 0.35/0.64  thf(fact_4947_divmod__algorithm__code_I8_J,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ( ( ( ord_less_num @ M @ N )
% 0.35/0.64         => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.35/0.64            = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
% 0.35/0.64        & ( ~ ( ord_less_num @ M @ N )
% 0.35/0.64         => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.35/0.64            = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divmod_algorithm_code(8)
% 0.35/0.64  thf(fact_4948_equation__minus__iff,axiom,
% 0.35/0.64      ! [A: int,B: int] :
% 0.35/0.64        ( ( A
% 0.35/0.64          = ( uminus_uminus_int @ B ) )
% 0.35/0.64        = ( B
% 0.35/0.64          = ( uminus_uminus_int @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % equation_minus_iff
% 0.35/0.64  thf(fact_4949_equation__minus__iff,axiom,
% 0.35/0.64      ! [A: real,B: real] :
% 0.35/0.64        ( ( A
% 0.35/0.64          = ( uminus_uminus_real @ B ) )
% 0.35/0.64        = ( B
% 0.35/0.64          = ( uminus_uminus_real @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % equation_minus_iff
% 0.35/0.64  thf(fact_4950_equation__minus__iff,axiom,
% 0.35/0.64      ! [A: rat,B: rat] :
% 0.35/0.64        ( ( A
% 0.35/0.64          = ( uminus_uminus_rat @ B ) )
% 0.35/0.64        = ( B
% 0.35/0.64          = ( uminus_uminus_rat @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % equation_minus_iff
% 0.35/0.64  thf(fact_4951_equation__minus__iff,axiom,
% 0.35/0.64      ! [A: code_integer,B: code_integer] :
% 0.35/0.64        ( ( A
% 0.35/0.64          = ( uminus1351360451143612070nteger @ B ) )
% 0.35/0.64        = ( B
% 0.35/0.64          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % equation_minus_iff
% 0.35/0.64  thf(fact_4952_minus__equation__iff,axiom,
% 0.35/0.64      ! [A: int,B: int] :
% 0.35/0.64        ( ( ( uminus_uminus_int @ A )
% 0.35/0.64          = B )
% 0.35/0.64        = ( ( uminus_uminus_int @ B )
% 0.35/0.64          = A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_equation_iff
% 0.35/0.64  thf(fact_4953_minus__equation__iff,axiom,
% 0.35/0.64      ! [A: real,B: real] :
% 0.35/0.64        ( ( ( uminus_uminus_real @ A )
% 0.35/0.64          = B )
% 0.35/0.64        = ( ( uminus_uminus_real @ B )
% 0.35/0.64          = A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_equation_iff
% 0.35/0.64  thf(fact_4954_minus__equation__iff,axiom,
% 0.35/0.64      ! [A: rat,B: rat] :
% 0.35/0.64        ( ( ( uminus_uminus_rat @ A )
% 0.35/0.64          = B )
% 0.35/0.64        = ( ( uminus_uminus_rat @ B )
% 0.35/0.64          = A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_equation_iff
% 0.35/0.64  thf(fact_4955_minus__equation__iff,axiom,
% 0.35/0.64      ! [A: code_integer,B: code_integer] :
% 0.35/0.64        ( ( ( uminus1351360451143612070nteger @ A )
% 0.35/0.64          = B )
% 0.35/0.64        = ( ( uminus1351360451143612070nteger @ B )
% 0.35/0.64          = A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_equation_iff
% 0.35/0.64  thf(fact_4956_compl__le__swap2,axiom,
% 0.35/0.64      ! [Y: set_int,X: set_int] :
% 0.35/0.64        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ X )
% 0.35/0.64       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) ) ).
% 0.35/0.64  
% 0.35/0.64  % compl_le_swap2
% 0.35/0.64  thf(fact_4957_compl__le__swap1,axiom,
% 0.35/0.64      ! [Y: set_int,X: set_int] :
% 0.35/0.64        ( ( ord_less_eq_set_int @ Y @ ( uminus1532241313380277803et_int @ X ) )
% 0.35/0.64       => ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % compl_le_swap1
% 0.35/0.64  thf(fact_4958_compl__mono,axiom,
% 0.35/0.64      ! [X: set_int,Y: set_int] :
% 0.35/0.64        ( ( ord_less_eq_set_int @ X @ Y )
% 0.35/0.64       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ ( uminus1532241313380277803et_int @ X ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % compl_mono
% 0.35/0.64  thf(fact_4959_le__imp__neg__le,axiom,
% 0.35/0.64      ! [A: real,B: real] :
% 0.35/0.64        ( ( ord_less_eq_real @ A @ B )
% 0.35/0.64       => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_imp_neg_le
% 0.35/0.64  thf(fact_4960_le__imp__neg__le,axiom,
% 0.35/0.64      ! [A: code_integer,B: code_integer] :
% 0.35/0.64        ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.35/0.64       => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_imp_neg_le
% 0.35/0.64  thf(fact_4961_le__imp__neg__le,axiom,
% 0.35/0.64      ! [A: rat,B: rat] :
% 0.35/0.64        ( ( ord_less_eq_rat @ A @ B )
% 0.35/0.64       => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_imp_neg_le
% 0.35/0.64  thf(fact_4962_le__imp__neg__le,axiom,
% 0.35/0.64      ! [A: int,B: int] :
% 0.35/0.64        ( ( ord_less_eq_int @ A @ B )
% 0.35/0.64       => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_imp_neg_le
% 0.35/0.64  thf(fact_4963_minus__le__iff,axiom,
% 0.35/0.64      ! [A: real,B: real] :
% 0.35/0.64        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 0.35/0.64        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_le_iff
% 0.35/0.64  thf(fact_4964_minus__le__iff,axiom,
% 0.35/0.64      ! [A: code_integer,B: code_integer] :
% 0.35/0.64        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.35/0.64        = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_le_iff
% 0.35/0.64  thf(fact_4965_minus__le__iff,axiom,
% 0.35/0.64      ! [A: rat,B: rat] :
% 0.35/0.64        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.35/0.64        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_le_iff
% 0.35/0.64  thf(fact_4966_minus__le__iff,axiom,
% 0.35/0.64      ! [A: int,B: int] :
% 0.35/0.64        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 0.35/0.64        = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_le_iff
% 0.35/0.64  thf(fact_4967_le__minus__iff,axiom,
% 0.35/0.64      ! [A: real,B: real] :
% 0.35/0.64        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 0.35/0.64        = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_iff
% 0.35/0.64  thf(fact_4968_le__minus__iff,axiom,
% 0.35/0.64      ! [A: code_integer,B: code_integer] :
% 0.35/0.64        ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.35/0.64        = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_iff
% 0.35/0.64  thf(fact_4969_le__minus__iff,axiom,
% 0.35/0.64      ! [A: rat,B: rat] :
% 0.35/0.64        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.35/0.64        = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_iff
% 0.35/0.64  thf(fact_4970_le__minus__iff,axiom,
% 0.35/0.64      ! [A: int,B: int] :
% 0.35/0.64        ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 0.35/0.64        = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_iff
% 0.35/0.64  thf(fact_4971_minus__less__iff,axiom,
% 0.35/0.64      ! [A: int,B: int] :
% 0.35/0.64        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
% 0.35/0.64        = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_less_iff
% 0.35/0.64  thf(fact_4972_minus__less__iff,axiom,
% 0.35/0.64      ! [A: real,B: real] :
% 0.35/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
% 0.35/0.64        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_less_iff
% 0.35/0.64  thf(fact_4973_minus__less__iff,axiom,
% 0.35/0.64      ! [A: rat,B: rat] :
% 0.35/0.64        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.35/0.64        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_less_iff
% 0.35/0.64  thf(fact_4974_minus__less__iff,axiom,
% 0.35/0.64      ! [A: code_integer,B: code_integer] :
% 0.35/0.64        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.35/0.64        = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_less_iff
% 0.35/0.64  thf(fact_4975_less__minus__iff,axiom,
% 0.35/0.64      ! [A: int,B: int] :
% 0.35/0.64        ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
% 0.35/0.64        = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_iff
% 0.35/0.64  thf(fact_4976_less__minus__iff,axiom,
% 0.35/0.64      ! [A: real,B: real] :
% 0.35/0.64        ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
% 0.35/0.64        = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_iff
% 0.35/0.64  thf(fact_4977_less__minus__iff,axiom,
% 0.35/0.64      ! [A: rat,B: rat] :
% 0.35/0.64        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.35/0.64        = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_iff
% 0.35/0.64  thf(fact_4978_less__minus__iff,axiom,
% 0.35/0.64      ! [A: code_integer,B: code_integer] :
% 0.35/0.64        ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.35/0.64        = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_iff
% 0.35/0.64  thf(fact_4979_verit__negate__coefficient_I2_J,axiom,
% 0.35/0.64      ! [A: int,B: int] :
% 0.35/0.64        ( ( ord_less_int @ A @ B )
% 0.35/0.64       => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % verit_negate_coefficient(2)
% 0.35/0.64  thf(fact_4980_verit__negate__coefficient_I2_J,axiom,
% 0.35/0.64      ! [A: real,B: real] :
% 0.35/0.64        ( ( ord_less_real @ A @ B )
% 0.35/0.64       => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % verit_negate_coefficient(2)
% 0.35/0.64  thf(fact_4981_verit__negate__coefficient_I2_J,axiom,
% 0.35/0.64      ! [A: rat,B: rat] :
% 0.35/0.64        ( ( ord_less_rat @ A @ B )
% 0.35/0.64       => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % verit_negate_coefficient(2)
% 0.35/0.64  thf(fact_4982_verit__negate__coefficient_I2_J,axiom,
% 0.35/0.64      ! [A: code_integer,B: code_integer] :
% 0.35/0.64        ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.35/0.64       => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % verit_negate_coefficient(2)
% 0.35/0.64  thf(fact_4983_group__cancel_Oneg1,axiom,
% 0.35/0.64      ! [A2: int,K: int,A: int] :
% 0.35/0.64        ( ( A2
% 0.35/0.64          = ( plus_plus_int @ K @ A ) )
% 0.35/0.64       => ( ( uminus_uminus_int @ A2 )
% 0.35/0.64          = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % group_cancel.neg1
% 0.35/0.64  thf(fact_4984_group__cancel_Oneg1,axiom,
% 0.35/0.64      ! [A2: real,K: real,A: real] :
% 0.35/0.64        ( ( A2
% 0.35/0.64          = ( plus_plus_real @ K @ A ) )
% 0.35/0.64       => ( ( uminus_uminus_real @ A2 )
% 0.35/0.64          = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % group_cancel.neg1
% 0.35/0.64  thf(fact_4985_group__cancel_Oneg1,axiom,
% 0.35/0.64      ! [A2: rat,K: rat,A: rat] :
% 0.35/0.64        ( ( A2
% 0.35/0.64          = ( plus_plus_rat @ K @ A ) )
% 0.35/0.64       => ( ( uminus_uminus_rat @ A2 )
% 0.35/0.64          = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % group_cancel.neg1
% 0.35/0.64  thf(fact_4986_group__cancel_Oneg1,axiom,
% 0.35/0.64      ! [A2: code_integer,K: code_integer,A: code_integer] :
% 0.35/0.64        ( ( A2
% 0.35/0.64          = ( plus_p5714425477246183910nteger @ K @ A ) )
% 0.35/0.64       => ( ( uminus1351360451143612070nteger @ A2 )
% 0.35/0.64          = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % group_cancel.neg1
% 0.35/0.64  thf(fact_4987_add_Oinverse__distrib__swap,axiom,
% 0.35/0.64      ! [A: int,B: int] :
% 0.35/0.64        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.35/0.64        = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add.inverse_distrib_swap
% 0.35/0.64  thf(fact_4988_add_Oinverse__distrib__swap,axiom,
% 0.35/0.64      ! [A: real,B: real] :
% 0.35/0.64        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.35/0.64        = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add.inverse_distrib_swap
% 0.35/0.64  thf(fact_4989_add_Oinverse__distrib__swap,axiom,
% 0.35/0.64      ! [A: rat,B: rat] :
% 0.35/0.64        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.35/0.64        = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add.inverse_distrib_swap
% 0.35/0.64  thf(fact_4990_add_Oinverse__distrib__swap,axiom,
% 0.35/0.64      ! [A: code_integer,B: code_integer] :
% 0.35/0.64        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.35/0.64        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add.inverse_distrib_swap
% 0.35/0.64  thf(fact_4991_minus__diff__commute,axiom,
% 0.35/0.64      ! [B: int,A: int] :
% 0.35/0.64        ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 0.35/0.64        = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_diff_commute
% 0.35/0.64  thf(fact_4992_minus__diff__commute,axiom,
% 0.35/0.64      ! [B: real,A: real] :
% 0.35/0.64        ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
% 0.35/0.64        = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_diff_commute
% 0.35/0.64  thf(fact_4993_minus__diff__commute,axiom,
% 0.35/0.64      ! [B: rat,A: rat] :
% 0.35/0.64        ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
% 0.35/0.64        = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_diff_commute
% 0.35/0.64  thf(fact_4994_minus__diff__commute,axiom,
% 0.35/0.64      ! [B: code_integer,A: code_integer] :
% 0.35/0.64        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
% 0.35/0.64        = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_diff_commute
% 0.35/0.64  thf(fact_4995_not__numeral__le__neg__numeral,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_numeral_le_neg_numeral
% 0.35/0.64  thf(fact_4996_not__numeral__le__neg__numeral,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_numeral_le_neg_numeral
% 0.35/0.64  thf(fact_4997_not__numeral__le__neg__numeral,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_numeral_le_neg_numeral
% 0.35/0.64  thf(fact_4998_not__numeral__le__neg__numeral,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_numeral_le_neg_numeral
% 0.35/0.64  thf(fact_4999_neg__numeral__le__numeral,axiom,
% 0.35/0.64      ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_le_numeral
% 0.35/0.64  thf(fact_5000_neg__numeral__le__numeral,axiom,
% 0.35/0.64      ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_le_numeral
% 0.35/0.64  thf(fact_5001_neg__numeral__le__numeral,axiom,
% 0.35/0.64      ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_le_numeral
% 0.35/0.64  thf(fact_5002_neg__numeral__le__numeral,axiom,
% 0.35/0.64      ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_le_numeral
% 0.35/0.64  thf(fact_5003_zero__neq__neg__numeral,axiom,
% 0.35/0.64      ! [N: num] :
% 0.35/0.64        ( zero_zero_complex
% 0.35/0.64       != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % zero_neq_neg_numeral
% 0.35/0.64  thf(fact_5004_zero__neq__neg__numeral,axiom,
% 0.35/0.64      ! [N: num] :
% 0.35/0.64        ( zero_zero_int
% 0.35/0.64       != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % zero_neq_neg_numeral
% 0.35/0.64  thf(fact_5005_zero__neq__neg__numeral,axiom,
% 0.35/0.64      ! [N: num] :
% 0.35/0.64        ( zero_zero_real
% 0.35/0.64       != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % zero_neq_neg_numeral
% 0.35/0.64  thf(fact_5006_zero__neq__neg__numeral,axiom,
% 0.35/0.64      ! [N: num] :
% 0.35/0.64        ( zero_zero_rat
% 0.35/0.64       != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % zero_neq_neg_numeral
% 0.35/0.64  thf(fact_5007_zero__neq__neg__numeral,axiom,
% 0.35/0.64      ! [N: num] :
% 0.35/0.64        ( zero_z3403309356797280102nteger
% 0.35/0.64       != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % zero_neq_neg_numeral
% 0.35/0.64  thf(fact_5008_not__numeral__less__neg__numeral,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_numeral_less_neg_numeral
% 0.35/0.64  thf(fact_5009_not__numeral__less__neg__numeral,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_numeral_less_neg_numeral
% 0.35/0.64  thf(fact_5010_not__numeral__less__neg__numeral,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_numeral_less_neg_numeral
% 0.35/0.64  thf(fact_5011_not__numeral__less__neg__numeral,axiom,
% 0.35/0.64      ! [M: num,N: num] :
% 0.35/0.64        ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_numeral_less_neg_numeral
% 0.35/0.64  thf(fact_5012_neg__numeral__less__numeral,axiom,
% 0.35/0.64      ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_less_numeral
% 0.35/0.64  thf(fact_5013_neg__numeral__less__numeral,axiom,
% 0.35/0.64      ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_less_numeral
% 0.35/0.64  thf(fact_5014_neg__numeral__less__numeral,axiom,
% 0.35/0.64      ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_less_numeral
% 0.35/0.64  thf(fact_5015_neg__numeral__less__numeral,axiom,
% 0.35/0.64      ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_less_numeral
% 0.35/0.64  thf(fact_5016_xor__num_Ocases,axiom,
% 0.35/0.64      ! [X: product_prod_num_num] :
% 0.35/0.64        ( ( X
% 0.35/0.64         != ( product_Pair_num_num @ one @ one ) )
% 0.35/0.64       => ( ! [N3: num] :
% 0.35/0.64              ( X
% 0.35/0.64             != ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) )
% 0.35/0.64         => ( ! [N3: num] :
% 0.35/0.64                ( X
% 0.35/0.64               != ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) )
% 0.35/0.64           => ( ! [M5: num] :
% 0.35/0.64                  ( X
% 0.35/0.64                 != ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) )
% 0.35/0.64             => ( ! [M5: num,N3: num] :
% 0.35/0.64                    ( X
% 0.35/0.64                   != ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) )
% 0.35/0.64               => ( ! [M5: num,N3: num] :
% 0.35/0.64                      ( X
% 0.35/0.64                     != ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) )
% 0.35/0.64                 => ( ! [M5: num] :
% 0.35/0.64                        ( X
% 0.35/0.64                       != ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) )
% 0.35/0.64                   => ( ! [M5: num,N3: num] :
% 0.35/0.64                          ( X
% 0.35/0.64                         != ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) )
% 0.35/0.64                     => ~ ! [M5: num,N3: num] :
% 0.35/0.64                            ( X
% 0.35/0.64                           != ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % xor_num.cases
% 0.35/0.64  thf(fact_5017_le__minus__one__simps_I2_J,axiom,
% 0.35/0.64      ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_one_simps(2)
% 0.35/0.64  thf(fact_5018_le__minus__one__simps_I2_J,axiom,
% 0.35/0.64      ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_one_simps(2)
% 0.35/0.64  thf(fact_5019_le__minus__one__simps_I2_J,axiom,
% 0.35/0.64      ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_one_simps(2)
% 0.35/0.64  thf(fact_5020_le__minus__one__simps_I2_J,axiom,
% 0.35/0.64      ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_one_simps(2)
% 0.35/0.64  thf(fact_5021_le__minus__one__simps_I4_J,axiom,
% 0.35/0.64      ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_one_simps(4)
% 0.35/0.64  thf(fact_5022_le__minus__one__simps_I4_J,axiom,
% 0.35/0.64      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_one_simps(4)
% 0.35/0.64  thf(fact_5023_le__minus__one__simps_I4_J,axiom,
% 0.35/0.64      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_one_simps(4)
% 0.35/0.64  thf(fact_5024_le__minus__one__simps_I4_J,axiom,
% 0.35/0.64      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_one_simps(4)
% 0.35/0.64  thf(fact_5025_add__eq__0__iff,axiom,
% 0.35/0.64      ! [A: complex,B: complex] :
% 0.35/0.64        ( ( ( plus_plus_complex @ A @ B )
% 0.35/0.64          = zero_zero_complex )
% 0.35/0.64        = ( B
% 0.35/0.64          = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add_eq_0_iff
% 0.35/0.64  thf(fact_5026_add__eq__0__iff,axiom,
% 0.35/0.64      ! [A: int,B: int] :
% 0.35/0.64        ( ( ( plus_plus_int @ A @ B )
% 0.35/0.64          = zero_zero_int )
% 0.35/0.64        = ( B
% 0.35/0.64          = ( uminus_uminus_int @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add_eq_0_iff
% 0.35/0.64  thf(fact_5027_add__eq__0__iff,axiom,
% 0.35/0.64      ! [A: real,B: real] :
% 0.35/0.64        ( ( ( plus_plus_real @ A @ B )
% 0.35/0.64          = zero_zero_real )
% 0.35/0.64        = ( B
% 0.35/0.64          = ( uminus_uminus_real @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add_eq_0_iff
% 0.35/0.64  thf(fact_5028_add__eq__0__iff,axiom,
% 0.35/0.64      ! [A: rat,B: rat] :
% 0.35/0.64        ( ( ( plus_plus_rat @ A @ B )
% 0.35/0.64          = zero_zero_rat )
% 0.35/0.64        = ( B
% 0.35/0.64          = ( uminus_uminus_rat @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add_eq_0_iff
% 0.35/0.64  thf(fact_5029_add__eq__0__iff,axiom,
% 0.35/0.64      ! [A: code_integer,B: code_integer] :
% 0.35/0.64        ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.35/0.64          = zero_z3403309356797280102nteger )
% 0.35/0.64        = ( B
% 0.35/0.64          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add_eq_0_iff
% 0.35/0.64  thf(fact_5030_ab__group__add__class_Oab__left__minus,axiom,
% 0.35/0.64      ! [A: complex] :
% 0.35/0.64        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 0.35/0.64        = zero_zero_complex ) ).
% 0.35/0.64  
% 0.35/0.64  % ab_group_add_class.ab_left_minus
% 0.35/0.64  thf(fact_5031_ab__group__add__class_Oab__left__minus,axiom,
% 0.35/0.64      ! [A: int] :
% 0.35/0.64        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 0.35/0.64        = zero_zero_int ) ).
% 0.35/0.64  
% 0.35/0.64  % ab_group_add_class.ab_left_minus
% 0.35/0.64  thf(fact_5032_ab__group__add__class_Oab__left__minus,axiom,
% 0.35/0.64      ! [A: real] :
% 0.35/0.64        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.35/0.64        = zero_zero_real ) ).
% 0.35/0.64  
% 0.35/0.64  % ab_group_add_class.ab_left_minus
% 0.35/0.64  thf(fact_5033_ab__group__add__class_Oab__left__minus,axiom,
% 0.35/0.64      ! [A: rat] :
% 0.35/0.64        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.35/0.64        = zero_zero_rat ) ).
% 0.35/0.64  
% 0.35/0.64  % ab_group_add_class.ab_left_minus
% 0.35/0.64  thf(fact_5034_ab__group__add__class_Oab__left__minus,axiom,
% 0.35/0.64      ! [A: code_integer] :
% 0.35/0.64        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.35/0.64        = zero_z3403309356797280102nteger ) ).
% 0.35/0.64  
% 0.35/0.64  % ab_group_add_class.ab_left_minus
% 0.35/0.64  thf(fact_5035_add_Oinverse__unique,axiom,
% 0.35/0.64      ! [A: complex,B: complex] :
% 0.35/0.64        ( ( ( plus_plus_complex @ A @ B )
% 0.35/0.64          = zero_zero_complex )
% 0.35/0.64       => ( ( uminus1482373934393186551omplex @ A )
% 0.35/0.64          = B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add.inverse_unique
% 0.35/0.64  thf(fact_5036_add_Oinverse__unique,axiom,
% 0.35/0.64      ! [A: int,B: int] :
% 0.35/0.64        ( ( ( plus_plus_int @ A @ B )
% 0.35/0.64          = zero_zero_int )
% 0.35/0.64       => ( ( uminus_uminus_int @ A )
% 0.35/0.64          = B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add.inverse_unique
% 0.35/0.64  thf(fact_5037_add_Oinverse__unique,axiom,
% 0.35/0.64      ! [A: real,B: real] :
% 0.35/0.64        ( ( ( plus_plus_real @ A @ B )
% 0.35/0.64          = zero_zero_real )
% 0.35/0.64       => ( ( uminus_uminus_real @ A )
% 0.35/0.64          = B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add.inverse_unique
% 0.35/0.64  thf(fact_5038_add_Oinverse__unique,axiom,
% 0.35/0.64      ! [A: rat,B: rat] :
% 0.35/0.64        ( ( ( plus_plus_rat @ A @ B )
% 0.35/0.64          = zero_zero_rat )
% 0.35/0.64       => ( ( uminus_uminus_rat @ A )
% 0.35/0.64          = B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add.inverse_unique
% 0.35/0.64  thf(fact_5039_add_Oinverse__unique,axiom,
% 0.35/0.64      ! [A: code_integer,B: code_integer] :
% 0.35/0.64        ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.35/0.64          = zero_z3403309356797280102nteger )
% 0.35/0.64       => ( ( uminus1351360451143612070nteger @ A )
% 0.35/0.64          = B ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add.inverse_unique
% 0.35/0.64  thf(fact_5040_eq__neg__iff__add__eq__0,axiom,
% 0.35/0.64      ! [A: complex,B: complex] :
% 0.35/0.64        ( ( A
% 0.35/0.64          = ( uminus1482373934393186551omplex @ B ) )
% 0.35/0.64        = ( ( plus_plus_complex @ A @ B )
% 0.35/0.64          = zero_zero_complex ) ) ).
% 0.35/0.64  
% 0.35/0.64  % eq_neg_iff_add_eq_0
% 0.35/0.64  thf(fact_5041_eq__neg__iff__add__eq__0,axiom,
% 0.35/0.64      ! [A: int,B: int] :
% 0.35/0.64        ( ( A
% 0.35/0.64          = ( uminus_uminus_int @ B ) )
% 0.35/0.64        = ( ( plus_plus_int @ A @ B )
% 0.35/0.64          = zero_zero_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % eq_neg_iff_add_eq_0
% 0.35/0.64  thf(fact_5042_eq__neg__iff__add__eq__0,axiom,
% 0.35/0.64      ! [A: real,B: real] :
% 0.35/0.64        ( ( A
% 0.35/0.64          = ( uminus_uminus_real @ B ) )
% 0.35/0.64        = ( ( plus_plus_real @ A @ B )
% 0.35/0.64          = zero_zero_real ) ) ).
% 0.35/0.64  
% 0.35/0.64  % eq_neg_iff_add_eq_0
% 0.35/0.64  thf(fact_5043_eq__neg__iff__add__eq__0,axiom,
% 0.35/0.64      ! [A: rat,B: rat] :
% 0.35/0.64        ( ( A
% 0.35/0.64          = ( uminus_uminus_rat @ B ) )
% 0.35/0.64        = ( ( plus_plus_rat @ A @ B )
% 0.35/0.64          = zero_zero_rat ) ) ).
% 0.35/0.64  
% 0.35/0.64  % eq_neg_iff_add_eq_0
% 0.35/0.64  thf(fact_5044_eq__neg__iff__add__eq__0,axiom,
% 0.35/0.64      ! [A: code_integer,B: code_integer] :
% 0.35/0.64        ( ( A
% 0.35/0.64          = ( uminus1351360451143612070nteger @ B ) )
% 0.35/0.64        = ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.35/0.64          = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.64  
% 0.35/0.64  % eq_neg_iff_add_eq_0
% 0.35/0.64  thf(fact_5045_neg__eq__iff__add__eq__0,axiom,
% 0.35/0.64      ! [A: complex,B: complex] :
% 0.35/0.64        ( ( ( uminus1482373934393186551omplex @ A )
% 0.35/0.64          = B )
% 0.35/0.64        = ( ( plus_plus_complex @ A @ B )
% 0.35/0.64          = zero_zero_complex ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_eq_iff_add_eq_0
% 0.35/0.64  thf(fact_5046_neg__eq__iff__add__eq__0,axiom,
% 0.35/0.64      ! [A: int,B: int] :
% 0.35/0.64        ( ( ( uminus_uminus_int @ A )
% 0.35/0.64          = B )
% 0.35/0.64        = ( ( plus_plus_int @ A @ B )
% 0.35/0.64          = zero_zero_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_eq_iff_add_eq_0
% 0.35/0.64  thf(fact_5047_neg__eq__iff__add__eq__0,axiom,
% 0.35/0.64      ! [A: real,B: real] :
% 0.35/0.64        ( ( ( uminus_uminus_real @ A )
% 0.35/0.64          = B )
% 0.35/0.64        = ( ( plus_plus_real @ A @ B )
% 0.35/0.64          = zero_zero_real ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_eq_iff_add_eq_0
% 0.35/0.64  thf(fact_5048_neg__eq__iff__add__eq__0,axiom,
% 0.35/0.64      ! [A: rat,B: rat] :
% 0.35/0.64        ( ( ( uminus_uminus_rat @ A )
% 0.35/0.64          = B )
% 0.35/0.64        = ( ( plus_plus_rat @ A @ B )
% 0.35/0.64          = zero_zero_rat ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_eq_iff_add_eq_0
% 0.35/0.64  thf(fact_5049_neg__eq__iff__add__eq__0,axiom,
% 0.35/0.64      ! [A: code_integer,B: code_integer] :
% 0.35/0.64        ( ( ( uminus1351360451143612070nteger @ A )
% 0.35/0.64          = B )
% 0.35/0.64        = ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.35/0.64          = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_eq_iff_add_eq_0
% 0.35/0.64  thf(fact_5050_zero__neq__neg__one,axiom,
% 0.35/0.64      ( zero_zero_complex
% 0.35/0.64     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.35/0.64  
% 0.35/0.64  % zero_neq_neg_one
% 0.35/0.64  thf(fact_5051_zero__neq__neg__one,axiom,
% 0.35/0.64      ( zero_zero_int
% 0.35/0.64     != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % zero_neq_neg_one
% 0.35/0.64  thf(fact_5052_zero__neq__neg__one,axiom,
% 0.35/0.64      ( zero_zero_real
% 0.35/0.64     != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.35/0.64  
% 0.35/0.64  % zero_neq_neg_one
% 0.35/0.64  thf(fact_5053_zero__neq__neg__one,axiom,
% 0.35/0.64      ( zero_zero_rat
% 0.35/0.64     != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.35/0.64  
% 0.35/0.64  % zero_neq_neg_one
% 0.35/0.64  thf(fact_5054_zero__neq__neg__one,axiom,
% 0.35/0.64      ( zero_z3403309356797280102nteger
% 0.35/0.64     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.35/0.64  
% 0.35/0.64  % zero_neq_neg_one
% 0.35/0.64  thf(fact_5055_less__minus__one__simps_I2_J,axiom,
% 0.35/0.64      ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_one_simps(2)
% 0.35/0.64  thf(fact_5056_less__minus__one__simps_I2_J,axiom,
% 0.35/0.64      ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_one_simps(2)
% 0.35/0.64  thf(fact_5057_less__minus__one__simps_I2_J,axiom,
% 0.35/0.64      ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_one_simps(2)
% 0.35/0.64  thf(fact_5058_less__minus__one__simps_I2_J,axiom,
% 0.35/0.64      ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_one_simps(2)
% 0.35/0.64  thf(fact_5059_less__minus__one__simps_I4_J,axiom,
% 0.35/0.64      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_one_simps(4)
% 0.35/0.64  thf(fact_5060_less__minus__one__simps_I4_J,axiom,
% 0.35/0.64      ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_one_simps(4)
% 0.35/0.64  thf(fact_5061_less__minus__one__simps_I4_J,axiom,
% 0.35/0.64      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_one_simps(4)
% 0.35/0.64  thf(fact_5062_less__minus__one__simps_I4_J,axiom,
% 0.35/0.64      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_one_simps(4)
% 0.35/0.64  thf(fact_5063_nonzero__minus__divide__right,axiom,
% 0.35/0.64      ! [B: complex,A: complex] :
% 0.35/0.64        ( ( B != zero_zero_complex )
% 0.35/0.64       => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.35/0.64          = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % nonzero_minus_divide_right
% 0.35/0.64  thf(fact_5064_nonzero__minus__divide__right,axiom,
% 0.35/0.64      ! [B: real,A: real] :
% 0.35/0.64        ( ( B != zero_zero_real )
% 0.35/0.64       => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.35/0.64          = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % nonzero_minus_divide_right
% 0.35/0.64  thf(fact_5065_nonzero__minus__divide__right,axiom,
% 0.35/0.64      ! [B: rat,A: rat] :
% 0.35/0.64        ( ( B != zero_zero_rat )
% 0.35/0.64       => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.35/0.64          = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % nonzero_minus_divide_right
% 0.35/0.64  thf(fact_5066_nonzero__minus__divide__divide,axiom,
% 0.35/0.64      ! [B: complex,A: complex] :
% 0.35/0.64        ( ( B != zero_zero_complex )
% 0.35/0.64       => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.35/0.64          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % nonzero_minus_divide_divide
% 0.35/0.64  thf(fact_5067_nonzero__minus__divide__divide,axiom,
% 0.35/0.64      ! [B: real,A: real] :
% 0.35/0.64        ( ( B != zero_zero_real )
% 0.35/0.64       => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.35/0.64          = ( divide_divide_real @ A @ B ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % nonzero_minus_divide_divide
% 0.35/0.64  thf(fact_5068_nonzero__minus__divide__divide,axiom,
% 0.35/0.64      ! [B: rat,A: rat] :
% 0.35/0.64        ( ( B != zero_zero_rat )
% 0.35/0.64       => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.35/0.64          = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % nonzero_minus_divide_divide
% 0.35/0.64  thf(fact_5069_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.35/0.64      ( minus_minus_int
% 0.35/0.64      = ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % ab_group_add_class.ab_diff_conv_add_uminus
% 0.35/0.64  thf(fact_5070_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.35/0.64      ( minus_minus_real
% 0.35/0.64      = ( ^ [A4: real,B4: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % ab_group_add_class.ab_diff_conv_add_uminus
% 0.35/0.64  thf(fact_5071_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.35/0.64      ( minus_minus_rat
% 0.35/0.64      = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % ab_group_add_class.ab_diff_conv_add_uminus
% 0.35/0.64  thf(fact_5072_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.35/0.64      ( minus_8373710615458151222nteger
% 0.35/0.64      = ( ^ [A4: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % ab_group_add_class.ab_diff_conv_add_uminus
% 0.35/0.64  thf(fact_5073_diff__conv__add__uminus,axiom,
% 0.35/0.64      ( minus_minus_int
% 0.35/0.64      = ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % diff_conv_add_uminus
% 0.35/0.64  thf(fact_5074_diff__conv__add__uminus,axiom,
% 0.35/0.64      ( minus_minus_real
% 0.35/0.64      = ( ^ [A4: real,B4: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % diff_conv_add_uminus
% 0.35/0.64  thf(fact_5075_diff__conv__add__uminus,axiom,
% 0.35/0.64      ( minus_minus_rat
% 0.35/0.64      = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % diff_conv_add_uminus
% 0.35/0.64  thf(fact_5076_diff__conv__add__uminus,axiom,
% 0.35/0.64      ( minus_8373710615458151222nteger
% 0.35/0.64      = ( ^ [A4: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % diff_conv_add_uminus
% 0.35/0.64  thf(fact_5077_group__cancel_Osub2,axiom,
% 0.35/0.64      ! [B2: int,K: int,B: int,A: int] :
% 0.35/0.64        ( ( B2
% 0.35/0.64          = ( plus_plus_int @ K @ B ) )
% 0.35/0.64       => ( ( minus_minus_int @ A @ B2 )
% 0.35/0.64          = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % group_cancel.sub2
% 0.35/0.64  thf(fact_5078_group__cancel_Osub2,axiom,
% 0.35/0.64      ! [B2: real,K: real,B: real,A: real] :
% 0.35/0.64        ( ( B2
% 0.35/0.64          = ( plus_plus_real @ K @ B ) )
% 0.35/0.64       => ( ( minus_minus_real @ A @ B2 )
% 0.35/0.64          = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % group_cancel.sub2
% 0.35/0.64  thf(fact_5079_group__cancel_Osub2,axiom,
% 0.35/0.64      ! [B2: rat,K: rat,B: rat,A: rat] :
% 0.35/0.64        ( ( B2
% 0.35/0.64          = ( plus_plus_rat @ K @ B ) )
% 0.35/0.64       => ( ( minus_minus_rat @ A @ B2 )
% 0.35/0.64          = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % group_cancel.sub2
% 0.35/0.64  thf(fact_5080_group__cancel_Osub2,axiom,
% 0.35/0.64      ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
% 0.35/0.64        ( ( B2
% 0.35/0.64          = ( plus_p5714425477246183910nteger @ K @ B ) )
% 0.35/0.64       => ( ( minus_8373710615458151222nteger @ A @ B2 )
% 0.35/0.64          = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % group_cancel.sub2
% 0.35/0.64  thf(fact_5081_neg__numeral__le__zero,axiom,
% 0.35/0.64      ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_le_zero
% 0.35/0.64  thf(fact_5082_neg__numeral__le__zero,axiom,
% 0.35/0.64      ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_le_zero
% 0.35/0.64  thf(fact_5083_neg__numeral__le__zero,axiom,
% 0.35/0.64      ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_le_zero
% 0.35/0.64  thf(fact_5084_neg__numeral__le__zero,axiom,
% 0.35/0.64      ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_le_zero
% 0.35/0.64  thf(fact_5085_not__zero__le__neg__numeral,axiom,
% 0.35/0.64      ! [N: num] :
% 0.35/0.64        ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_zero_le_neg_numeral
% 0.35/0.64  thf(fact_5086_not__zero__le__neg__numeral,axiom,
% 0.35/0.64      ! [N: num] :
% 0.35/0.64        ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_zero_le_neg_numeral
% 0.35/0.64  thf(fact_5087_not__zero__le__neg__numeral,axiom,
% 0.35/0.64      ! [N: num] :
% 0.35/0.64        ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_zero_le_neg_numeral
% 0.35/0.64  thf(fact_5088_not__zero__le__neg__numeral,axiom,
% 0.35/0.64      ! [N: num] :
% 0.35/0.64        ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_zero_le_neg_numeral
% 0.35/0.64  thf(fact_5089_not__zero__less__neg__numeral,axiom,
% 0.35/0.64      ! [N: num] :
% 0.35/0.64        ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_zero_less_neg_numeral
% 0.35/0.64  thf(fact_5090_not__zero__less__neg__numeral,axiom,
% 0.35/0.64      ! [N: num] :
% 0.35/0.64        ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_zero_less_neg_numeral
% 0.35/0.64  thf(fact_5091_not__zero__less__neg__numeral,axiom,
% 0.35/0.64      ! [N: num] :
% 0.35/0.64        ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_zero_less_neg_numeral
% 0.35/0.64  thf(fact_5092_not__zero__less__neg__numeral,axiom,
% 0.35/0.64      ! [N: num] :
% 0.35/0.64        ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_zero_less_neg_numeral
% 0.35/0.64  thf(fact_5093_neg__numeral__less__zero,axiom,
% 0.35/0.64      ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_less_zero
% 0.35/0.64  thf(fact_5094_neg__numeral__less__zero,axiom,
% 0.35/0.64      ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_less_zero
% 0.35/0.64  thf(fact_5095_neg__numeral__less__zero,axiom,
% 0.35/0.64      ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_less_zero
% 0.35/0.64  thf(fact_5096_neg__numeral__less__zero,axiom,
% 0.35/0.64      ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_less_zero
% 0.35/0.64  thf(fact_5097_le__minus__one__simps_I3_J,axiom,
% 0.35/0.64      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_one_simps(3)
% 0.35/0.64  thf(fact_5098_le__minus__one__simps_I3_J,axiom,
% 0.35/0.64      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_one_simps(3)
% 0.35/0.64  thf(fact_5099_le__minus__one__simps_I3_J,axiom,
% 0.35/0.64      ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_one_simps(3)
% 0.35/0.64  thf(fact_5100_le__minus__one__simps_I3_J,axiom,
% 0.35/0.64      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_one_simps(3)
% 0.35/0.64  thf(fact_5101_le__minus__one__simps_I1_J,axiom,
% 0.35/0.64      ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_one_simps(1)
% 0.35/0.64  thf(fact_5102_le__minus__one__simps_I1_J,axiom,
% 0.35/0.64      ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_one_simps(1)
% 0.35/0.64  thf(fact_5103_le__minus__one__simps_I1_J,axiom,
% 0.35/0.64      ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_one_simps(1)
% 0.35/0.64  thf(fact_5104_le__minus__one__simps_I1_J,axiom,
% 0.35/0.64      ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_one_simps(1)
% 0.35/0.64  thf(fact_5105_less__minus__one__simps_I1_J,axiom,
% 0.35/0.64      ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_one_simps(1)
% 0.35/0.64  thf(fact_5106_less__minus__one__simps_I1_J,axiom,
% 0.35/0.64      ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_one_simps(1)
% 0.35/0.64  thf(fact_5107_less__minus__one__simps_I1_J,axiom,
% 0.35/0.64      ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_one_simps(1)
% 0.35/0.64  thf(fact_5108_less__minus__one__simps_I1_J,axiom,
% 0.35/0.64      ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_one_simps(1)
% 0.35/0.64  thf(fact_5109_less__minus__one__simps_I3_J,axiom,
% 0.35/0.64      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_one_simps(3)
% 0.35/0.64  thf(fact_5110_less__minus__one__simps_I3_J,axiom,
% 0.35/0.64      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_one_simps(3)
% 0.35/0.64  thf(fact_5111_less__minus__one__simps_I3_J,axiom,
% 0.35/0.64      ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_one_simps(3)
% 0.35/0.64  thf(fact_5112_less__minus__one__simps_I3_J,axiom,
% 0.35/0.64      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_one_simps(3)
% 0.35/0.64  thf(fact_5113_neg__numeral__le__one,axiom,
% 0.35/0.64      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_le_one
% 0.35/0.64  thf(fact_5114_neg__numeral__le__one,axiom,
% 0.35/0.64      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_le_one
% 0.35/0.64  thf(fact_5115_neg__numeral__le__one,axiom,
% 0.35/0.64      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_le_one
% 0.35/0.64  thf(fact_5116_neg__numeral__le__one,axiom,
% 0.35/0.64      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_le_one
% 0.35/0.64  thf(fact_5117_neg__one__le__numeral,axiom,
% 0.35/0.64      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_one_le_numeral
% 0.35/0.64  thf(fact_5118_neg__one__le__numeral,axiom,
% 0.35/0.64      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_one_le_numeral
% 0.35/0.64  thf(fact_5119_neg__one__le__numeral,axiom,
% 0.35/0.64      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_one_le_numeral
% 0.35/0.64  thf(fact_5120_neg__one__le__numeral,axiom,
% 0.35/0.64      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_one_le_numeral
% 0.35/0.64  thf(fact_5121_neg__numeral__le__neg__one,axiom,
% 0.35/0.64      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_le_neg_one
% 0.35/0.64  thf(fact_5122_neg__numeral__le__neg__one,axiom,
% 0.35/0.64      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_le_neg_one
% 0.35/0.64  thf(fact_5123_neg__numeral__le__neg__one,axiom,
% 0.35/0.64      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_le_neg_one
% 0.35/0.64  thf(fact_5124_neg__numeral__le__neg__one,axiom,
% 0.35/0.64      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_le_neg_one
% 0.35/0.64  thf(fact_5125_not__numeral__le__neg__one,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_numeral_le_neg_one
% 0.35/0.64  thf(fact_5126_not__numeral__le__neg__one,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_numeral_le_neg_one
% 0.35/0.64  thf(fact_5127_not__numeral__le__neg__one,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_numeral_le_neg_one
% 0.35/0.64  thf(fact_5128_not__numeral__le__neg__one,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_numeral_le_neg_one
% 0.35/0.64  thf(fact_5129_not__one__le__neg__numeral,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_one_le_neg_numeral
% 0.35/0.64  thf(fact_5130_not__one__le__neg__numeral,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_one_le_neg_numeral
% 0.35/0.64  thf(fact_5131_not__one__le__neg__numeral,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_one_le_neg_numeral
% 0.35/0.64  thf(fact_5132_not__one__le__neg__numeral,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_one_le_neg_numeral
% 0.35/0.64  thf(fact_5133_not__neg__one__less__neg__numeral,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_neg_one_less_neg_numeral
% 0.35/0.64  thf(fact_5134_not__neg__one__less__neg__numeral,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_neg_one_less_neg_numeral
% 0.35/0.64  thf(fact_5135_not__neg__one__less__neg__numeral,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_neg_one_less_neg_numeral
% 0.35/0.64  thf(fact_5136_not__neg__one__less__neg__numeral,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_neg_one_less_neg_numeral
% 0.35/0.64  thf(fact_5137_not__one__less__neg__numeral,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_one_less_neg_numeral
% 0.35/0.64  thf(fact_5138_not__one__less__neg__numeral,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_one_less_neg_numeral
% 0.35/0.64  thf(fact_5139_not__one__less__neg__numeral,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_one_less_neg_numeral
% 0.35/0.64  thf(fact_5140_not__one__less__neg__numeral,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_one_less_neg_numeral
% 0.35/0.64  thf(fact_5141_not__numeral__less__neg__one,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_numeral_less_neg_one
% 0.35/0.64  thf(fact_5142_not__numeral__less__neg__one,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_numeral_less_neg_one
% 0.35/0.64  thf(fact_5143_not__numeral__less__neg__one,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_numeral_less_neg_one
% 0.35/0.64  thf(fact_5144_not__numeral__less__neg__one,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.35/0.64  
% 0.35/0.64  % not_numeral_less_neg_one
% 0.35/0.64  thf(fact_5145_neg__one__less__numeral,axiom,
% 0.35/0.64      ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_one_less_numeral
% 0.35/0.64  thf(fact_5146_neg__one__less__numeral,axiom,
% 0.35/0.64      ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_one_less_numeral
% 0.35/0.64  thf(fact_5147_neg__one__less__numeral,axiom,
% 0.35/0.64      ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_one_less_numeral
% 0.35/0.64  thf(fact_5148_neg__one__less__numeral,axiom,
% 0.35/0.64      ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_one_less_numeral
% 0.35/0.64  thf(fact_5149_neg__numeral__less__one,axiom,
% 0.35/0.64      ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_less_one
% 0.35/0.64  thf(fact_5150_neg__numeral__less__one,axiom,
% 0.35/0.64      ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_less_one
% 0.35/0.64  thf(fact_5151_neg__numeral__less__one,axiom,
% 0.35/0.64      ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_less_one
% 0.35/0.64  thf(fact_5152_neg__numeral__less__one,axiom,
% 0.35/0.64      ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_numeral_less_one
% 0.35/0.64  thf(fact_5153_eq__minus__divide__eq,axiom,
% 0.35/0.64      ! [A: complex,B: complex,C: complex] :
% 0.35/0.64        ( ( A
% 0.35/0.64          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
% 0.35/0.64        = ( ( ( C != zero_zero_complex )
% 0.35/0.64           => ( ( times_times_complex @ A @ C )
% 0.35/0.64              = ( uminus1482373934393186551omplex @ B ) ) )
% 0.35/0.64          & ( ( C = zero_zero_complex )
% 0.35/0.64           => ( A = zero_zero_complex ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % eq_minus_divide_eq
% 0.35/0.64  thf(fact_5154_eq__minus__divide__eq,axiom,
% 0.35/0.64      ! [A: real,B: real,C: real] :
% 0.35/0.64        ( ( A
% 0.35/0.64          = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.35/0.64        = ( ( ( C != zero_zero_real )
% 0.35/0.64           => ( ( times_times_real @ A @ C )
% 0.35/0.64              = ( uminus_uminus_real @ B ) ) )
% 0.35/0.64          & ( ( C = zero_zero_real )
% 0.35/0.64           => ( A = zero_zero_real ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % eq_minus_divide_eq
% 0.35/0.64  thf(fact_5155_eq__minus__divide__eq,axiom,
% 0.35/0.64      ! [A: rat,B: rat,C: rat] :
% 0.35/0.64        ( ( A
% 0.35/0.64          = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.35/0.64        = ( ( ( C != zero_zero_rat )
% 0.35/0.64           => ( ( times_times_rat @ A @ C )
% 0.35/0.64              = ( uminus_uminus_rat @ B ) ) )
% 0.35/0.64          & ( ( C = zero_zero_rat )
% 0.35/0.64           => ( A = zero_zero_rat ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % eq_minus_divide_eq
% 0.35/0.64  thf(fact_5156_minus__divide__eq__eq,axiom,
% 0.35/0.64      ! [B: complex,C: complex,A: complex] :
% 0.35/0.64        ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.35/0.64          = A )
% 0.35/0.64        = ( ( ( C != zero_zero_complex )
% 0.35/0.64           => ( ( uminus1482373934393186551omplex @ B )
% 0.35/0.64              = ( times_times_complex @ A @ C ) ) )
% 0.35/0.64          & ( ( C = zero_zero_complex )
% 0.35/0.64           => ( A = zero_zero_complex ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_divide_eq_eq
% 0.35/0.64  thf(fact_5157_minus__divide__eq__eq,axiom,
% 0.35/0.64      ! [B: real,C: real,A: real] :
% 0.35/0.64        ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
% 0.35/0.64          = A )
% 0.35/0.64        = ( ( ( C != zero_zero_real )
% 0.35/0.64           => ( ( uminus_uminus_real @ B )
% 0.35/0.64              = ( times_times_real @ A @ C ) ) )
% 0.35/0.64          & ( ( C = zero_zero_real )
% 0.35/0.64           => ( A = zero_zero_real ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_divide_eq_eq
% 0.35/0.64  thf(fact_5158_minus__divide__eq__eq,axiom,
% 0.35/0.64      ! [B: rat,C: rat,A: rat] :
% 0.35/0.64        ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
% 0.35/0.64          = A )
% 0.35/0.64        = ( ( ( C != zero_zero_rat )
% 0.35/0.64           => ( ( uminus_uminus_rat @ B )
% 0.35/0.64              = ( times_times_rat @ A @ C ) ) )
% 0.35/0.64          & ( ( C = zero_zero_rat )
% 0.35/0.64           => ( A = zero_zero_rat ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_divide_eq_eq
% 0.35/0.64  thf(fact_5159_nonzero__neg__divide__eq__eq,axiom,
% 0.35/0.64      ! [B: complex,A: complex,C: complex] :
% 0.35/0.64        ( ( B != zero_zero_complex )
% 0.35/0.64       => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.35/0.64            = C )
% 0.35/0.64          = ( ( uminus1482373934393186551omplex @ A )
% 0.35/0.64            = ( times_times_complex @ C @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % nonzero_neg_divide_eq_eq
% 0.35/0.64  thf(fact_5160_nonzero__neg__divide__eq__eq,axiom,
% 0.35/0.64      ! [B: real,A: real,C: real] :
% 0.35/0.64        ( ( B != zero_zero_real )
% 0.35/0.64       => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.35/0.64            = C )
% 0.35/0.64          = ( ( uminus_uminus_real @ A )
% 0.35/0.64            = ( times_times_real @ C @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % nonzero_neg_divide_eq_eq
% 0.35/0.64  thf(fact_5161_nonzero__neg__divide__eq__eq,axiom,
% 0.35/0.64      ! [B: rat,A: rat,C: rat] :
% 0.35/0.64        ( ( B != zero_zero_rat )
% 0.35/0.64       => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.35/0.64            = C )
% 0.35/0.64          = ( ( uminus_uminus_rat @ A )
% 0.35/0.64            = ( times_times_rat @ C @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % nonzero_neg_divide_eq_eq
% 0.35/0.64  thf(fact_5162_nonzero__neg__divide__eq__eq2,axiom,
% 0.35/0.64      ! [B: complex,C: complex,A: complex] :
% 0.35/0.64        ( ( B != zero_zero_complex )
% 0.35/0.64       => ( ( C
% 0.35/0.64            = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.35/0.64          = ( ( times_times_complex @ C @ B )
% 0.35/0.64            = ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % nonzero_neg_divide_eq_eq2
% 0.35/0.64  thf(fact_5163_nonzero__neg__divide__eq__eq2,axiom,
% 0.35/0.64      ! [B: real,C: real,A: real] :
% 0.35/0.64        ( ( B != zero_zero_real )
% 0.35/0.64       => ( ( C
% 0.35/0.64            = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
% 0.35/0.64          = ( ( times_times_real @ C @ B )
% 0.35/0.64            = ( uminus_uminus_real @ A ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % nonzero_neg_divide_eq_eq2
% 0.35/0.64  thf(fact_5164_nonzero__neg__divide__eq__eq2,axiom,
% 0.35/0.64      ! [B: rat,C: rat,A: rat] :
% 0.35/0.64        ( ( B != zero_zero_rat )
% 0.35/0.64       => ( ( C
% 0.35/0.64            = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
% 0.35/0.64          = ( ( times_times_rat @ C @ B )
% 0.35/0.64            = ( uminus_uminus_rat @ A ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % nonzero_neg_divide_eq_eq2
% 0.35/0.64  thf(fact_5165_divide__eq__minus__1__iff,axiom,
% 0.35/0.64      ! [A: complex,B: complex] :
% 0.35/0.64        ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.35/0.64          = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.35/0.64        = ( ( B != zero_zero_complex )
% 0.35/0.64          & ( A
% 0.35/0.64            = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divide_eq_minus_1_iff
% 0.35/0.64  thf(fact_5166_divide__eq__minus__1__iff,axiom,
% 0.35/0.64      ! [A: real,B: real] :
% 0.35/0.64        ( ( ( divide_divide_real @ A @ B )
% 0.35/0.64          = ( uminus_uminus_real @ one_one_real ) )
% 0.35/0.64        = ( ( B != zero_zero_real )
% 0.35/0.64          & ( A
% 0.35/0.64            = ( uminus_uminus_real @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divide_eq_minus_1_iff
% 0.35/0.64  thf(fact_5167_divide__eq__minus__1__iff,axiom,
% 0.35/0.64      ! [A: rat,B: rat] :
% 0.35/0.64        ( ( ( divide_divide_rat @ A @ B )
% 0.35/0.64          = ( uminus_uminus_rat @ one_one_rat ) )
% 0.35/0.64        = ( ( B != zero_zero_rat )
% 0.35/0.64          & ( A
% 0.35/0.64            = ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divide_eq_minus_1_iff
% 0.35/0.64  thf(fact_5168_pos__minus__divide__less__eq,axiom,
% 0.35/0.64      ! [C: real,B: real,A: real] :
% 0.35/0.64        ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.64       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.35/0.64          = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % pos_minus_divide_less_eq
% 0.35/0.64  thf(fact_5169_pos__minus__divide__less__eq,axiom,
% 0.35/0.64      ! [C: rat,B: rat,A: rat] :
% 0.35/0.64        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.64       => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.35/0.64          = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % pos_minus_divide_less_eq
% 0.35/0.64  thf(fact_5170_pos__less__minus__divide__eq,axiom,
% 0.35/0.64      ! [C: real,A: real,B: real] :
% 0.35/0.64        ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.64       => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.35/0.64          = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % pos_less_minus_divide_eq
% 0.35/0.64  thf(fact_5171_pos__less__minus__divide__eq,axiom,
% 0.35/0.64      ! [C: rat,A: rat,B: rat] :
% 0.35/0.64        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.64       => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.35/0.64          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % pos_less_minus_divide_eq
% 0.35/0.64  thf(fact_5172_neg__minus__divide__less__eq,axiom,
% 0.35/0.64      ! [C: real,B: real,A: real] :
% 0.35/0.64        ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.64       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.35/0.64          = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_minus_divide_less_eq
% 0.35/0.64  thf(fact_5173_neg__minus__divide__less__eq,axiom,
% 0.35/0.64      ! [C: rat,B: rat,A: rat] :
% 0.35/0.64        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.64       => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.35/0.64          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_minus_divide_less_eq
% 0.35/0.64  thf(fact_5174_neg__less__minus__divide__eq,axiom,
% 0.35/0.64      ! [C: real,A: real,B: real] :
% 0.35/0.64        ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.64       => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.35/0.64          = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_less_minus_divide_eq
% 0.35/0.64  thf(fact_5175_neg__less__minus__divide__eq,axiom,
% 0.35/0.64      ! [C: rat,A: rat,B: rat] :
% 0.35/0.64        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.64       => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.35/0.64          = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_less_minus_divide_eq
% 0.35/0.64  thf(fact_5176_minus__divide__less__eq,axiom,
% 0.35/0.64      ! [B: real,C: real,A: real] :
% 0.35/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.35/0.64        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.64           => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.35/0.64          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.64           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.64               => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.35/0.64              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.64               => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_divide_less_eq
% 0.35/0.64  thf(fact_5177_minus__divide__less__eq,axiom,
% 0.35/0.64      ! [B: rat,C: rat,A: rat] :
% 0.35/0.64        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.35/0.64        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.64           => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.35/0.64          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.64           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.64               => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.35/0.64              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.64               => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_divide_less_eq
% 0.35/0.64  thf(fact_5178_less__minus__divide__eq,axiom,
% 0.35/0.64      ! [A: real,B: real,C: real] :
% 0.35/0.64        ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.35/0.64        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.64           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.35/0.64          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.64           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.64               => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.35/0.64              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.64               => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_divide_eq
% 0.35/0.64  thf(fact_5179_less__minus__divide__eq,axiom,
% 0.35/0.64      ! [A: rat,B: rat,C: rat] :
% 0.35/0.64        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.35/0.64        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.64           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.35/0.64          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.64           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.64               => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.35/0.64              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.64               => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_minus_divide_eq
% 0.35/0.64  thf(fact_5180_divide__eq__eq__numeral_I2_J,axiom,
% 0.35/0.64      ! [B: complex,C: complex,W2: num] :
% 0.35/0.64        ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.35/0.64          = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
% 0.35/0.64        = ( ( ( C != zero_zero_complex )
% 0.35/0.64           => ( B
% 0.35/0.64              = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C ) ) )
% 0.35/0.64          & ( ( C = zero_zero_complex )
% 0.35/0.64           => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.35/0.64              = zero_zero_complex ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divide_eq_eq_numeral(2)
% 0.35/0.64  thf(fact_5181_divide__eq__eq__numeral_I2_J,axiom,
% 0.35/0.64      ! [B: real,C: real,W2: num] :
% 0.35/0.64        ( ( ( divide_divide_real @ B @ C )
% 0.35/0.64          = ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 0.35/0.64        = ( ( ( C != zero_zero_real )
% 0.35/0.64           => ( B
% 0.35/0.64              = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 0.35/0.64          & ( ( C = zero_zero_real )
% 0.35/0.64           => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.35/0.64              = zero_zero_real ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divide_eq_eq_numeral(2)
% 0.35/0.64  thf(fact_5182_divide__eq__eq__numeral_I2_J,axiom,
% 0.35/0.64      ! [B: rat,C: rat,W2: num] :
% 0.35/0.64        ( ( ( divide_divide_rat @ B @ C )
% 0.35/0.64          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 0.35/0.64        = ( ( ( C != zero_zero_rat )
% 0.35/0.64           => ( B
% 0.35/0.64              = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 0.35/0.64          & ( ( C = zero_zero_rat )
% 0.35/0.64           => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.35/0.64              = zero_zero_rat ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divide_eq_eq_numeral(2)
% 0.35/0.64  thf(fact_5183_eq__divide__eq__numeral_I2_J,axiom,
% 0.35/0.64      ! [W2: num,B: complex,C: complex] :
% 0.35/0.64        ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.35/0.64          = ( divide1717551699836669952omplex @ B @ C ) )
% 0.35/0.64        = ( ( ( C != zero_zero_complex )
% 0.35/0.64           => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C )
% 0.35/0.64              = B ) )
% 0.35/0.64          & ( ( C = zero_zero_complex )
% 0.35/0.64           => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.35/0.64              = zero_zero_complex ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % eq_divide_eq_numeral(2)
% 0.35/0.64  thf(fact_5184_eq__divide__eq__numeral_I2_J,axiom,
% 0.35/0.64      ! [W2: num,B: real,C: real] :
% 0.35/0.64        ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.35/0.64          = ( divide_divide_real @ B @ C ) )
% 0.35/0.64        = ( ( ( C != zero_zero_real )
% 0.35/0.64           => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C )
% 0.35/0.64              = B ) )
% 0.35/0.64          & ( ( C = zero_zero_real )
% 0.35/0.64           => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.35/0.64              = zero_zero_real ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % eq_divide_eq_numeral(2)
% 0.35/0.64  thf(fact_5185_eq__divide__eq__numeral_I2_J,axiom,
% 0.35/0.64      ! [W2: num,B: rat,C: rat] :
% 0.35/0.64        ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.35/0.64          = ( divide_divide_rat @ B @ C ) )
% 0.35/0.64        = ( ( ( C != zero_zero_rat )
% 0.35/0.64           => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C )
% 0.35/0.64              = B ) )
% 0.35/0.64          & ( ( C = zero_zero_rat )
% 0.35/0.64           => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.35/0.64              = zero_zero_rat ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % eq_divide_eq_numeral(2)
% 0.35/0.64  thf(fact_5186_cong__exp__iff__simps_I3_J,axiom,
% 0.35/0.64      ! [N: num,Q4: num] :
% 0.35/0.64        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
% 0.35/0.64       != zero_zero_nat ) ).
% 0.35/0.64  
% 0.35/0.64  % cong_exp_iff_simps(3)
% 0.35/0.64  thf(fact_5187_cong__exp__iff__simps_I3_J,axiom,
% 0.35/0.64      ! [N: num,Q4: num] :
% 0.35/0.64        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
% 0.35/0.64       != zero_zero_int ) ).
% 0.35/0.64  
% 0.35/0.64  % cong_exp_iff_simps(3)
% 0.35/0.64  thf(fact_5188_cong__exp__iff__simps_I3_J,axiom,
% 0.35/0.64      ! [N: num,Q4: num] :
% 0.35/0.64        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
% 0.35/0.64       != zero_z3403309356797280102nteger ) ).
% 0.35/0.64  
% 0.35/0.64  % cong_exp_iff_simps(3)
% 0.35/0.64  thf(fact_5189_add__divide__eq__if__simps_I3_J,axiom,
% 0.35/0.64      ! [Z2: complex,A: complex,B: complex] :
% 0.35/0.64        ( ( ( Z2 = zero_zero_complex )
% 0.35/0.64         => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 0.35/0.64            = B ) )
% 0.35/0.64        & ( ( Z2 != zero_zero_complex )
% 0.35/0.64         => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 0.35/0.64            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add_divide_eq_if_simps(3)
% 0.35/0.64  thf(fact_5190_add__divide__eq__if__simps_I3_J,axiom,
% 0.35/0.64      ! [Z2: real,A: real,B: real] :
% 0.35/0.64        ( ( ( Z2 = zero_zero_real )
% 0.35/0.64         => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 0.35/0.64            = B ) )
% 0.35/0.64        & ( ( Z2 != zero_zero_real )
% 0.35/0.64         => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 0.35/0.64            = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add_divide_eq_if_simps(3)
% 0.35/0.64  thf(fact_5191_add__divide__eq__if__simps_I3_J,axiom,
% 0.35/0.64      ! [Z2: rat,A: rat,B: rat] :
% 0.35/0.64        ( ( ( Z2 = zero_zero_rat )
% 0.35/0.64         => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 0.35/0.64            = B ) )
% 0.35/0.64        & ( ( Z2 != zero_zero_rat )
% 0.35/0.64         => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 0.35/0.64            = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add_divide_eq_if_simps(3)
% 0.35/0.64  thf(fact_5192_minus__divide__add__eq__iff,axiom,
% 0.35/0.64      ! [Z2: complex,X: complex,Y: complex] :
% 0.35/0.64        ( ( Z2 != zero_zero_complex )
% 0.35/0.64       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z2 ) ) @ Y )
% 0.35/0.64          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_divide_add_eq_iff
% 0.35/0.64  thf(fact_5193_minus__divide__add__eq__iff,axiom,
% 0.35/0.64      ! [Z2: real,X: real,Y: real] :
% 0.35/0.64        ( ( Z2 != zero_zero_real )
% 0.35/0.64       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z2 ) ) @ Y )
% 0.35/0.64          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_divide_add_eq_iff
% 0.35/0.64  thf(fact_5194_minus__divide__add__eq__iff,axiom,
% 0.35/0.64      ! [Z2: rat,X: rat,Y: rat] :
% 0.35/0.64        ( ( Z2 != zero_zero_rat )
% 0.35/0.64       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z2 ) ) @ Y )
% 0.35/0.64          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_divide_add_eq_iff
% 0.35/0.64  thf(fact_5195_add__divide__eq__if__simps_I6_J,axiom,
% 0.35/0.64      ! [Z2: complex,A: complex,B: complex] :
% 0.35/0.64        ( ( ( Z2 = zero_zero_complex )
% 0.35/0.64         => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 0.35/0.64            = ( uminus1482373934393186551omplex @ B ) ) )
% 0.35/0.64        & ( ( Z2 != zero_zero_complex )
% 0.35/0.64         => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 0.35/0.64            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add_divide_eq_if_simps(6)
% 0.35/0.64  thf(fact_5196_add__divide__eq__if__simps_I6_J,axiom,
% 0.35/0.64      ! [Z2: real,A: real,B: real] :
% 0.35/0.64        ( ( ( Z2 = zero_zero_real )
% 0.35/0.64         => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 0.35/0.64            = ( uminus_uminus_real @ B ) ) )
% 0.35/0.64        & ( ( Z2 != zero_zero_real )
% 0.35/0.64         => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 0.35/0.64            = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add_divide_eq_if_simps(6)
% 0.35/0.64  thf(fact_5197_add__divide__eq__if__simps_I6_J,axiom,
% 0.35/0.64      ! [Z2: rat,A: rat,B: rat] :
% 0.35/0.64        ( ( ( Z2 = zero_zero_rat )
% 0.35/0.64         => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 0.35/0.64            = ( uminus_uminus_rat @ B ) ) )
% 0.35/0.64        & ( ( Z2 != zero_zero_rat )
% 0.35/0.64         => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 0.35/0.64            = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add_divide_eq_if_simps(6)
% 0.35/0.64  thf(fact_5198_add__divide__eq__if__simps_I5_J,axiom,
% 0.35/0.64      ! [Z2: complex,A: complex,B: complex] :
% 0.35/0.64        ( ( ( Z2 = zero_zero_complex )
% 0.35/0.64         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 0.35/0.64            = ( uminus1482373934393186551omplex @ B ) ) )
% 0.35/0.64        & ( ( Z2 != zero_zero_complex )
% 0.35/0.64         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 0.35/0.64            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add_divide_eq_if_simps(5)
% 0.35/0.64  thf(fact_5199_add__divide__eq__if__simps_I5_J,axiom,
% 0.35/0.64      ! [Z2: real,A: real,B: real] :
% 0.35/0.64        ( ( ( Z2 = zero_zero_real )
% 0.35/0.64         => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 0.35/0.64            = ( uminus_uminus_real @ B ) ) )
% 0.35/0.64        & ( ( Z2 != zero_zero_real )
% 0.35/0.64         => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 0.35/0.64            = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add_divide_eq_if_simps(5)
% 0.35/0.64  thf(fact_5200_add__divide__eq__if__simps_I5_J,axiom,
% 0.35/0.64      ! [Z2: rat,A: rat,B: rat] :
% 0.35/0.64        ( ( ( Z2 = zero_zero_rat )
% 0.35/0.64         => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 0.35/0.64            = ( uminus_uminus_rat @ B ) ) )
% 0.35/0.64        & ( ( Z2 != zero_zero_rat )
% 0.35/0.64         => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 0.35/0.64            = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % add_divide_eq_if_simps(5)
% 0.35/0.64  thf(fact_5201_minus__divide__diff__eq__iff,axiom,
% 0.35/0.64      ! [Z2: complex,X: complex,Y: complex] :
% 0.35/0.64        ( ( Z2 != zero_zero_complex )
% 0.35/0.64       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z2 ) ) @ Y )
% 0.35/0.64          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_divide_diff_eq_iff
% 0.35/0.64  thf(fact_5202_minus__divide__diff__eq__iff,axiom,
% 0.35/0.64      ! [Z2: real,X: real,Y: real] :
% 0.35/0.64        ( ( Z2 != zero_zero_real )
% 0.35/0.64       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z2 ) ) @ Y )
% 0.35/0.64          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_divide_diff_eq_iff
% 0.35/0.64  thf(fact_5203_minus__divide__diff__eq__iff,axiom,
% 0.35/0.64      ! [Z2: rat,X: rat,Y: rat] :
% 0.35/0.64        ( ( Z2 != zero_zero_rat )
% 0.35/0.64       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z2 ) ) @ Y )
% 0.35/0.64          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_divide_diff_eq_iff
% 0.35/0.64  thf(fact_5204_numeral__3__eq__3,axiom,
% 0.35/0.64      ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 0.35/0.64      = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % numeral_3_eq_3
% 0.35/0.64  thf(fact_5205_verit__less__mono__div__int2,axiom,
% 0.35/0.64      ! [A2: int,B2: int,N: int] :
% 0.35/0.64        ( ( ord_less_eq_int @ A2 @ B2 )
% 0.35/0.64       => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
% 0.35/0.64         => ( ord_less_eq_int @ ( divide_divide_int @ B2 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % verit_less_mono_div_int2
% 0.35/0.64  thf(fact_5206_div__eq__minus1,axiom,
% 0.35/0.64      ! [B: int] :
% 0.35/0.64        ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.64       => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 0.35/0.64          = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % div_eq_minus1
% 0.35/0.64  thf(fact_5207_num_Osize__gen_I3_J,axiom,
% 0.35/0.64      ! [X32: num] :
% 0.35/0.64        ( ( size_num @ ( bit1 @ X32 ) )
% 0.35/0.64        = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % num.size_gen(3)
% 0.35/0.64  thf(fact_5208_num_Osize_I6_J,axiom,
% 0.35/0.64      ! [X32: num] :
% 0.35/0.64        ( ( size_size_num @ ( bit1 @ X32 ) )
% 0.35/0.64        = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % num.size(6)
% 0.35/0.64  thf(fact_5209_le__minus__divide__eq,axiom,
% 0.35/0.64      ! [A: real,B: real,C: real] :
% 0.35/0.64        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.35/0.64        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.64           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.35/0.64          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.64           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.64               => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.35/0.64              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.64               => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_divide_eq
% 0.35/0.64  thf(fact_5210_le__minus__divide__eq,axiom,
% 0.35/0.64      ! [A: rat,B: rat,C: rat] :
% 0.35/0.64        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.35/0.64        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.64           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.35/0.64          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.64           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.64               => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.35/0.64              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.64               => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_minus_divide_eq
% 0.35/0.64  thf(fact_5211_minus__divide__le__eq,axiom,
% 0.35/0.64      ! [B: real,C: real,A: real] :
% 0.35/0.64        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.35/0.64        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.64           => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.35/0.64          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.64           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.64               => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.35/0.64              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.64               => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_divide_le_eq
% 0.35/0.64  thf(fact_5212_minus__divide__le__eq,axiom,
% 0.35/0.64      ! [B: rat,C: rat,A: rat] :
% 0.35/0.64        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.35/0.64        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.64           => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.35/0.64          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.64           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.64               => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.35/0.64              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.64               => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % minus_divide_le_eq
% 0.35/0.64  thf(fact_5213_neg__le__minus__divide__eq,axiom,
% 0.35/0.64      ! [C: real,A: real,B: real] :
% 0.35/0.64        ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.64       => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.35/0.64          = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_le_minus_divide_eq
% 0.35/0.64  thf(fact_5214_neg__le__minus__divide__eq,axiom,
% 0.35/0.64      ! [C: rat,A: rat,B: rat] :
% 0.35/0.64        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.64       => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.35/0.64          = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_le_minus_divide_eq
% 0.35/0.64  thf(fact_5215_neg__minus__divide__le__eq,axiom,
% 0.35/0.64      ! [C: real,B: real,A: real] :
% 0.35/0.64        ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.64       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.35/0.64          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_minus_divide_le_eq
% 0.35/0.64  thf(fact_5216_neg__minus__divide__le__eq,axiom,
% 0.35/0.64      ! [C: rat,B: rat,A: rat] :
% 0.35/0.64        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.64       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.35/0.64          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_minus_divide_le_eq
% 0.35/0.64  thf(fact_5217_pos__le__minus__divide__eq,axiom,
% 0.35/0.64      ! [C: real,A: real,B: real] :
% 0.35/0.64        ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.64       => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.35/0.64          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % pos_le_minus_divide_eq
% 0.35/0.64  thf(fact_5218_pos__le__minus__divide__eq,axiom,
% 0.35/0.64      ! [C: rat,A: rat,B: rat] :
% 0.35/0.64        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.64       => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.35/0.64          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % pos_le_minus_divide_eq
% 0.35/0.64  thf(fact_5219_pos__minus__divide__le__eq,axiom,
% 0.35/0.64      ! [C: real,B: real,A: real] :
% 0.35/0.64        ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.64       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.35/0.64          = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % pos_minus_divide_le_eq
% 0.35/0.64  thf(fact_5220_pos__minus__divide__le__eq,axiom,
% 0.35/0.64      ! [C: rat,B: rat,A: rat] :
% 0.35/0.64        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.64       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.35/0.64          = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % pos_minus_divide_le_eq
% 0.35/0.64  thf(fact_5221_less__divide__eq__numeral_I2_J,axiom,
% 0.35/0.64      ! [W2: num,B: real,C: real] :
% 0.35/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C ) )
% 0.35/0.64        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.64           => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
% 0.35/0.64          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.64           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.64               => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 0.35/0.64              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.64               => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_divide_eq_numeral(2)
% 0.35/0.64  thf(fact_5222_less__divide__eq__numeral_I2_J,axiom,
% 0.35/0.64      ! [W2: num,B: rat,C: rat] :
% 0.35/0.64        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C ) )
% 0.35/0.64        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.64           => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
% 0.35/0.64          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.64           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.64               => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 0.35/0.64              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.64               => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % less_divide_eq_numeral(2)
% 0.35/0.64  thf(fact_5223_divide__less__eq__numeral_I2_J,axiom,
% 0.35/0.64      ! [B: real,C: real,W2: num] :
% 0.35/0.64        ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 0.35/0.64        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.64           => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 0.35/0.64          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.64           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.64               => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
% 0.35/0.64              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.64               => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divide_less_eq_numeral(2)
% 0.35/0.64  thf(fact_5224_divide__less__eq__numeral_I2_J,axiom,
% 0.35/0.64      ! [B: rat,C: rat,W2: num] :
% 0.35/0.64        ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 0.35/0.64        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.64           => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 0.35/0.64          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.64           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.64               => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
% 0.35/0.64              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.64               => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divide_less_eq_numeral(2)
% 0.35/0.64  thf(fact_5225_cong__exp__iff__simps_I7_J,axiom,
% 0.35/0.64      ! [Q4: num,N: num] :
% 0.35/0.64        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
% 0.35/0.64          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) )
% 0.35/0.64        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q4 ) )
% 0.35/0.64          = zero_zero_nat ) ) ).
% 0.35/0.64  
% 0.35/0.64  % cong_exp_iff_simps(7)
% 0.35/0.64  thf(fact_5226_cong__exp__iff__simps_I7_J,axiom,
% 0.35/0.64      ! [Q4: num,N: num] :
% 0.35/0.64        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
% 0.35/0.64          = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) )
% 0.35/0.64        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q4 ) )
% 0.35/0.64          = zero_zero_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % cong_exp_iff_simps(7)
% 0.35/0.64  thf(fact_5227_cong__exp__iff__simps_I7_J,axiom,
% 0.35/0.64      ! [Q4: num,N: num] :
% 0.35/0.64        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
% 0.35/0.64          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) )
% 0.35/0.64        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q4 ) )
% 0.35/0.64          = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.64  
% 0.35/0.64  % cong_exp_iff_simps(7)
% 0.35/0.64  thf(fact_5228_cong__exp__iff__simps_I11_J,axiom,
% 0.35/0.64      ! [M: num,Q4: num] :
% 0.35/0.64        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
% 0.35/0.64          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) )
% 0.35/0.64        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q4 ) )
% 0.35/0.64          = zero_zero_nat ) ) ).
% 0.35/0.64  
% 0.35/0.64  % cong_exp_iff_simps(11)
% 0.35/0.64  thf(fact_5229_cong__exp__iff__simps_I11_J,axiom,
% 0.35/0.64      ! [M: num,Q4: num] :
% 0.35/0.64        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
% 0.35/0.64          = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) )
% 0.35/0.64        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q4 ) )
% 0.35/0.64          = zero_zero_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % cong_exp_iff_simps(11)
% 0.35/0.64  thf(fact_5230_cong__exp__iff__simps_I11_J,axiom,
% 0.35/0.64      ! [M: num,Q4: num] :
% 0.35/0.64        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
% 0.35/0.64          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) )
% 0.35/0.64        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q4 ) )
% 0.35/0.64          = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.64  
% 0.35/0.64  % cong_exp_iff_simps(11)
% 0.35/0.64  thf(fact_5231_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.35/0.64      ! [K: nat,N: nat] :
% 0.35/0.64        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.64       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
% 0.35/0.64          = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_one_power_add_eq_neg_one_power_diff
% 0.35/0.64  thf(fact_5232_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.35/0.64      ! [K: nat,N: nat] :
% 0.35/0.64        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.64       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
% 0.35/0.64          = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_one_power_add_eq_neg_one_power_diff
% 0.35/0.64  thf(fact_5233_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.35/0.64      ! [K: nat,N: nat] :
% 0.35/0.64        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.64       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
% 0.35/0.64          = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_one_power_add_eq_neg_one_power_diff
% 0.35/0.64  thf(fact_5234_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.35/0.64      ! [K: nat,N: nat] :
% 0.35/0.64        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.64       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
% 0.35/0.64          = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_one_power_add_eq_neg_one_power_diff
% 0.35/0.64  thf(fact_5235_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.35/0.64      ! [K: nat,N: nat] :
% 0.35/0.64        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.64       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
% 0.35/0.64          = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % neg_one_power_add_eq_neg_one_power_diff
% 0.35/0.64  thf(fact_5236_zmod__minus1,axiom,
% 0.35/0.64      ! [B: int] :
% 0.35/0.64        ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.64       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 0.35/0.64          = ( minus_minus_int @ B @ one_one_int ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % zmod_minus1
% 0.35/0.64  thf(fact_5237_zminus1__lemma,axiom,
% 0.35/0.64      ! [A: int,B: int,Q4: int,R2: int] :
% 0.35/0.64        ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.35/0.64       => ( ( B != zero_zero_int )
% 0.35/0.64         => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q4 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q4 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R2 ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % zminus1_lemma
% 0.35/0.64  thf(fact_5238_divide__le__eq__numeral_I2_J,axiom,
% 0.35/0.64      ! [B: real,C: real,W2: num] :
% 0.35/0.64        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 0.35/0.64        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.64           => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 0.35/0.64          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.64           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.64               => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
% 0.35/0.64              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.64               => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divide_le_eq_numeral(2)
% 0.35/0.64  thf(fact_5239_divide__le__eq__numeral_I2_J,axiom,
% 0.35/0.64      ! [B: rat,C: rat,W2: num] :
% 0.35/0.64        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 0.35/0.64        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.64           => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 0.35/0.64          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.64           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.64               => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
% 0.35/0.64              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.64               => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divide_le_eq_numeral(2)
% 0.35/0.64  thf(fact_5240_le__divide__eq__numeral_I2_J,axiom,
% 0.35/0.64      ! [W2: num,B: real,C: real] :
% 0.35/0.64        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C ) )
% 0.35/0.64        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.64           => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
% 0.35/0.64          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.64           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.64               => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 0.35/0.64              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.64               => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_divide_eq_numeral(2)
% 0.35/0.64  thf(fact_5241_le__divide__eq__numeral_I2_J,axiom,
% 0.35/0.64      ! [W2: num,B: rat,C: rat] :
% 0.35/0.64        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C ) )
% 0.35/0.64        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.64           => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
% 0.35/0.64          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.64           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.64               => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 0.35/0.64              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.64               => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % le_divide_eq_numeral(2)
% 0.35/0.64  thf(fact_5242_square__le__1,axiom,
% 0.35/0.64      ! [X: real] :
% 0.35/0.64        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.35/0.64       => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.35/0.64         => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % square_le_1
% 0.35/0.64  thf(fact_5243_square__le__1,axiom,
% 0.35/0.64      ! [X: code_integer] :
% 0.35/0.64        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 0.35/0.64       => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
% 0.35/0.64         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % square_le_1
% 0.35/0.64  thf(fact_5244_square__le__1,axiom,
% 0.35/0.64      ! [X: rat] :
% 0.35/0.64        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
% 0.35/0.64       => ( ( ord_less_eq_rat @ X @ one_one_rat )
% 0.35/0.64         => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % square_le_1
% 0.35/0.64  thf(fact_5245_square__le__1,axiom,
% 0.35/0.64      ! [X: int] :
% 0.35/0.64        ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 0.35/0.64       => ( ( ord_less_eq_int @ X @ one_one_int )
% 0.35/0.64         => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % square_le_1
% 0.35/0.64  thf(fact_5246_div__pos__neg__trivial,axiom,
% 0.35/0.64      ! [K: int,L: int] :
% 0.35/0.64        ( ( ord_less_int @ zero_zero_int @ K )
% 0.35/0.64       => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 0.35/0.64         => ( ( divide_divide_int @ K @ L )
% 0.35/0.64            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % div_pos_neg_trivial
% 0.35/0.64  thf(fact_5247_odd__mod__4__div__2,axiom,
% 0.35/0.64      ! [N: nat] :
% 0.35/0.64        ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.35/0.64          = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.35/0.64       => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % odd_mod_4_div_2
% 0.35/0.64  thf(fact_5248_mod__exhaust__less__4,axiom,
% 0.35/0.64      ! [M: nat] :
% 0.35/0.64        ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.35/0.64          = zero_zero_nat )
% 0.35/0.64        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.35/0.64          = one_one_nat )
% 0.35/0.64        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.35/0.64          = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.64        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.35/0.64          = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % mod_exhaust_less_4
% 0.35/0.64  thf(fact_5249_signed__take__bit__rec,axiom,
% 0.35/0.64      ( bit_ri6519982836138164636nteger
% 0.35/0.64      = ( ^ [N2: nat,A4: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % signed_take_bit_rec
% 0.35/0.64  thf(fact_5250_signed__take__bit__rec,axiom,
% 0.35/0.64      ( bit_ri631733984087533419it_int
% 0.35/0.64      = ( ^ [N2: nat,A4: int] : ( if_int @ ( N2 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % signed_take_bit_rec
% 0.35/0.64  thf(fact_5251_divmod__BitM__2__eq,axiom,
% 0.35/0.64      ! [M: num] :
% 0.35/0.64        ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
% 0.35/0.64        = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % divmod_BitM_2_eq
% 0.35/0.64  thf(fact_5252_of__int__code__if,axiom,
% 0.35/0.64      ( ring_17405671764205052669omplex
% 0.35/0.64      = ( ^ [K3: int] :
% 0.35/0.64            ( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
% 0.35/0.64            @ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
% 0.35/0.64              @ ( if_complex
% 0.35/0.64                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.64                  = zero_zero_int )
% 0.35/0.64                @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.35/0.64                @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_code_if
% 0.35/0.64  thf(fact_5253_of__int__code__if,axiom,
% 0.35/0.64      ( ring_1_of_int_int
% 0.35/0.64      = ( ^ [K3: int] :
% 0.35/0.64            ( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
% 0.35/0.64            @ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
% 0.35/0.64              @ ( if_int
% 0.35/0.64                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.64                  = zero_zero_int )
% 0.35/0.64                @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.35/0.64                @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_code_if
% 0.35/0.64  thf(fact_5254_of__int__code__if,axiom,
% 0.35/0.64      ( ring_1_of_int_real
% 0.35/0.64      = ( ^ [K3: int] :
% 0.35/0.64            ( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
% 0.35/0.64            @ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
% 0.35/0.64              @ ( if_real
% 0.35/0.64                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.64                  = zero_zero_int )
% 0.35/0.64                @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.35/0.64                @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_code_if
% 0.35/0.64  thf(fact_5255_of__int__code__if,axiom,
% 0.35/0.64      ( ring_1_of_int_rat
% 0.35/0.64      = ( ^ [K3: int] :
% 0.35/0.64            ( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
% 0.35/0.64            @ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
% 0.35/0.64              @ ( if_rat
% 0.35/0.64                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.64                  = zero_zero_int )
% 0.35/0.64                @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.35/0.64                @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_code_if
% 0.35/0.64  thf(fact_5256_of__int__code__if,axiom,
% 0.35/0.64      ( ring_18347121197199848620nteger
% 0.35/0.64      = ( ^ [K3: int] :
% 0.35/0.64            ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 0.35/0.64            @ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
% 0.35/0.64              @ ( if_Code_integer
% 0.35/0.64                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.64                  = zero_zero_int )
% 0.35/0.64                @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.35/0.64                @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_code_if
% 0.35/0.64  thf(fact_5257_signed__take__bit__0,axiom,
% 0.35/0.64      ! [A: code_integer] :
% 0.35/0.64        ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
% 0.35/0.64        = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % signed_take_bit_0
% 0.35/0.64  thf(fact_5258_signed__take__bit__0,axiom,
% 0.35/0.64      ! [A: int] :
% 0.35/0.64        ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
% 0.35/0.64        = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % signed_take_bit_0
% 0.35/0.64  thf(fact_5259_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.35/0.64      ! [A: int,X: num,N: nat] :
% 0.35/0.64        ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 0.35/0.64        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_less_neg_numeral_power_cancel_iff
% 0.35/0.64  thf(fact_5260_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.35/0.64      ! [A: int,X: num,N: nat] :
% 0.35/0.64        ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 0.35/0.64        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_less_neg_numeral_power_cancel_iff
% 0.35/0.64  thf(fact_5261_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.35/0.64      ! [A: int,X: num,N: nat] :
% 0.35/0.64        ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 0.35/0.64        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_less_neg_numeral_power_cancel_iff
% 0.35/0.64  thf(fact_5262_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.35/0.64      ! [A: int,X: num,N: nat] :
% 0.35/0.64        ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 0.35/0.64        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_less_neg_numeral_power_cancel_iff
% 0.35/0.64  thf(fact_5263_Compl__anti__mono,axiom,
% 0.35/0.64      ! [A2: set_int,B2: set_int] :
% 0.35/0.64        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.35/0.64       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B2 ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% 0.35/0.64  
% 0.35/0.64  % Compl_anti_mono
% 0.35/0.64  thf(fact_5264_Compl__subset__Compl__iff,axiom,
% 0.35/0.64      ! [A2: set_int,B2: set_int] :
% 0.35/0.64        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B2 ) )
% 0.35/0.64        = ( ord_less_eq_set_int @ B2 @ A2 ) ) ).
% 0.35/0.64  
% 0.35/0.64  % Compl_subset_Compl_iff
% 0.35/0.64  thf(fact_5265_signed__take__bit__of__0,axiom,
% 0.35/0.64      ! [N: nat] :
% 0.35/0.64        ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
% 0.35/0.64        = zero_zero_int ) ).
% 0.35/0.64  
% 0.35/0.64  % signed_take_bit_of_0
% 0.35/0.64  thf(fact_5266_of__int__0,axiom,
% 0.35/0.64      ( ( ring_17405671764205052669omplex @ zero_zero_int )
% 0.35/0.64      = zero_zero_complex ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_0
% 0.35/0.64  thf(fact_5267_of__int__0,axiom,
% 0.35/0.64      ( ( ring_1_of_int_int @ zero_zero_int )
% 0.35/0.64      = zero_zero_int ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_0
% 0.35/0.64  thf(fact_5268_of__int__0,axiom,
% 0.35/0.64      ( ( ring_1_of_int_real @ zero_zero_int )
% 0.35/0.64      = zero_zero_real ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_0
% 0.35/0.64  thf(fact_5269_of__int__0,axiom,
% 0.35/0.64      ( ( ring_1_of_int_rat @ zero_zero_int )
% 0.35/0.64      = zero_zero_rat ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_0
% 0.35/0.64  thf(fact_5270_of__int__0__eq__iff,axiom,
% 0.35/0.64      ! [Z2: int] :
% 0.35/0.64        ( ( zero_zero_complex
% 0.35/0.64          = ( ring_17405671764205052669omplex @ Z2 ) )
% 0.35/0.64        = ( Z2 = zero_zero_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_0_eq_iff
% 0.35/0.64  thf(fact_5271_of__int__0__eq__iff,axiom,
% 0.35/0.64      ! [Z2: int] :
% 0.35/0.64        ( ( zero_zero_int
% 0.35/0.64          = ( ring_1_of_int_int @ Z2 ) )
% 0.35/0.64        = ( Z2 = zero_zero_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_0_eq_iff
% 0.35/0.64  thf(fact_5272_of__int__0__eq__iff,axiom,
% 0.35/0.64      ! [Z2: int] :
% 0.35/0.64        ( ( zero_zero_real
% 0.35/0.64          = ( ring_1_of_int_real @ Z2 ) )
% 0.35/0.64        = ( Z2 = zero_zero_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_0_eq_iff
% 0.35/0.64  thf(fact_5273_of__int__0__eq__iff,axiom,
% 0.35/0.64      ! [Z2: int] :
% 0.35/0.64        ( ( zero_zero_rat
% 0.35/0.64          = ( ring_1_of_int_rat @ Z2 ) )
% 0.35/0.64        = ( Z2 = zero_zero_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_0_eq_iff
% 0.35/0.64  thf(fact_5274_of__int__eq__0__iff,axiom,
% 0.35/0.64      ! [Z2: int] :
% 0.35/0.64        ( ( ( ring_17405671764205052669omplex @ Z2 )
% 0.35/0.64          = zero_zero_complex )
% 0.35/0.64        = ( Z2 = zero_zero_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_eq_0_iff
% 0.35/0.64  thf(fact_5275_of__int__eq__0__iff,axiom,
% 0.35/0.64      ! [Z2: int] :
% 0.35/0.64        ( ( ( ring_1_of_int_int @ Z2 )
% 0.35/0.64          = zero_zero_int )
% 0.35/0.64        = ( Z2 = zero_zero_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_eq_0_iff
% 0.35/0.64  thf(fact_5276_of__int__eq__0__iff,axiom,
% 0.35/0.64      ! [Z2: int] :
% 0.35/0.64        ( ( ( ring_1_of_int_real @ Z2 )
% 0.35/0.64          = zero_zero_real )
% 0.35/0.64        = ( Z2 = zero_zero_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_eq_0_iff
% 0.35/0.64  thf(fact_5277_of__int__eq__0__iff,axiom,
% 0.35/0.64      ! [Z2: int] :
% 0.35/0.64        ( ( ( ring_1_of_int_rat @ Z2 )
% 0.35/0.64          = zero_zero_rat )
% 0.35/0.64        = ( Z2 = zero_zero_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_eq_0_iff
% 0.35/0.64  thf(fact_5278_of__int__le__iff,axiom,
% 0.35/0.64      ! [W2: int,Z2: int] :
% 0.35/0.64        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) )
% 0.35/0.64        = ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_le_iff
% 0.35/0.64  thf(fact_5279_of__int__le__iff,axiom,
% 0.35/0.64      ! [W2: int,Z2: int] :
% 0.35/0.64        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) )
% 0.35/0.64        = ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_le_iff
% 0.35/0.64  thf(fact_5280_of__int__le__iff,axiom,
% 0.35/0.64      ! [W2: int,Z2: int] :
% 0.35/0.64        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
% 0.35/0.64        = ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_le_iff
% 0.35/0.64  thf(fact_5281_of__int__less__iff,axiom,
% 0.35/0.64      ! [W2: int,Z2: int] :
% 0.35/0.64        ( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) )
% 0.35/0.64        = ( ord_less_int @ W2 @ Z2 ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_less_iff
% 0.35/0.64  thf(fact_5282_of__int__less__iff,axiom,
% 0.35/0.64      ! [W2: int,Z2: int] :
% 0.35/0.64        ( ( ord_less_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) )
% 0.35/0.64        = ( ord_less_int @ W2 @ Z2 ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_less_iff
% 0.35/0.64  thf(fact_5283_of__int__less__iff,axiom,
% 0.35/0.64      ! [W2: int,Z2: int] :
% 0.35/0.64        ( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
% 0.35/0.64        = ( ord_less_int @ W2 @ Z2 ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_less_iff
% 0.35/0.64  thf(fact_5284_of__int__le__0__iff,axiom,
% 0.35/0.64      ! [Z2: int] :
% 0.35/0.64        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
% 0.35/0.64        = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_le_0_iff
% 0.35/0.64  thf(fact_5285_of__int__le__0__iff,axiom,
% 0.35/0.64      ! [Z2: int] :
% 0.35/0.64        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
% 0.35/0.64        = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_le_0_iff
% 0.35/0.64  thf(fact_5286_of__int__le__0__iff,axiom,
% 0.35/0.64      ! [Z2: int] :
% 0.35/0.64        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
% 0.35/0.64        = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_le_0_iff
% 0.35/0.64  thf(fact_5287_of__int__0__le__iff,axiom,
% 0.35/0.64      ! [Z2: int] :
% 0.35/0.64        ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
% 0.35/0.64        = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_0_le_iff
% 0.35/0.64  thf(fact_5288_of__int__0__le__iff,axiom,
% 0.35/0.64      ! [Z2: int] :
% 0.35/0.64        ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 0.35/0.64        = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_0_le_iff
% 0.35/0.64  thf(fact_5289_of__int__0__le__iff,axiom,
% 0.35/0.64      ! [Z2: int] :
% 0.35/0.64        ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
% 0.35/0.64        = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_0_le_iff
% 0.35/0.64  thf(fact_5290_of__int__0__less__iff,axiom,
% 0.35/0.64      ! [Z2: int] :
% 0.35/0.64        ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
% 0.35/0.64        = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_0_less_iff
% 0.35/0.64  thf(fact_5291_of__int__0__less__iff,axiom,
% 0.35/0.64      ! [Z2: int] :
% 0.35/0.64        ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 0.35/0.64        = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_0_less_iff
% 0.35/0.64  thf(fact_5292_of__int__0__less__iff,axiom,
% 0.35/0.64      ! [Z2: int] :
% 0.35/0.64        ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
% 0.35/0.64        = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_0_less_iff
% 0.35/0.64  thf(fact_5293_of__int__less__0__iff,axiom,
% 0.35/0.64      ! [Z2: int] :
% 0.35/0.64        ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
% 0.35/0.64        = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 0.35/0.64  
% 0.35/0.64  % of_int_less_0_iff
% 0.35/0.64  thf(fact_5294_of__int__less__0__iff,axiom,
% 0.35/0.64      ! [Z2: int] :
% 0.35/0.64        ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
% 0.35/0.64        = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_less_0_iff
% 0.35/0.65  thf(fact_5295_of__int__less__0__iff,axiom,
% 0.35/0.65      ! [Z2: int] :
% 0.35/0.65        ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
% 0.35/0.65        = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_less_0_iff
% 0.35/0.65  thf(fact_5296_of__int__le__numeral__iff,axiom,
% 0.35/0.65      ! [Z2: int,N: num] :
% 0.35/0.65        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N ) )
% 0.35/0.65        = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_le_numeral_iff
% 0.35/0.65  thf(fact_5297_of__int__le__numeral__iff,axiom,
% 0.35/0.65      ! [Z2: int,N: num] :
% 0.35/0.65        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N ) )
% 0.35/0.65        = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_le_numeral_iff
% 0.35/0.65  thf(fact_5298_of__int__le__numeral__iff,axiom,
% 0.35/0.65      ! [Z2: int,N: num] :
% 0.35/0.65        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.65        = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_le_numeral_iff
% 0.35/0.65  thf(fact_5299_of__int__numeral__le__iff,axiom,
% 0.35/0.65      ! [N: num,Z2: int] :
% 0.35/0.65        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z2 ) )
% 0.35/0.65        = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_numeral_le_iff
% 0.35/0.65  thf(fact_5300_of__int__numeral__le__iff,axiom,
% 0.35/0.65      ! [N: num,Z2: int] :
% 0.35/0.65        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z2 ) )
% 0.35/0.65        = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_numeral_le_iff
% 0.35/0.65  thf(fact_5301_of__int__numeral__le__iff,axiom,
% 0.35/0.65      ! [N: num,Z2: int] :
% 0.35/0.65        ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z2 ) )
% 0.35/0.65        = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_numeral_le_iff
% 0.35/0.65  thf(fact_5302_of__int__less__numeral__iff,axiom,
% 0.35/0.65      ! [Z2: int,N: num] :
% 0.35/0.65        ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N ) )
% 0.35/0.65        = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_less_numeral_iff
% 0.35/0.65  thf(fact_5303_of__int__less__numeral__iff,axiom,
% 0.35/0.65      ! [Z2: int,N: num] :
% 0.35/0.65        ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N ) )
% 0.35/0.65        = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_less_numeral_iff
% 0.35/0.65  thf(fact_5304_of__int__less__numeral__iff,axiom,
% 0.35/0.65      ! [Z2: int,N: num] :
% 0.35/0.65        ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.65        = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_less_numeral_iff
% 0.35/0.65  thf(fact_5305_of__int__numeral__less__iff,axiom,
% 0.35/0.65      ! [N: num,Z2: int] :
% 0.35/0.65        ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z2 ) )
% 0.35/0.65        = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_numeral_less_iff
% 0.35/0.65  thf(fact_5306_of__int__numeral__less__iff,axiom,
% 0.35/0.65      ! [N: num,Z2: int] :
% 0.35/0.65        ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z2 ) )
% 0.35/0.65        = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_numeral_less_iff
% 0.35/0.65  thf(fact_5307_of__int__numeral__less__iff,axiom,
% 0.35/0.65      ! [N: num,Z2: int] :
% 0.35/0.65        ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z2 ) )
% 0.35/0.65        = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_numeral_less_iff
% 0.35/0.65  thf(fact_5308_of__int__le__1__iff,axiom,
% 0.35/0.65      ! [Z2: int] :
% 0.35/0.65        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
% 0.35/0.65        = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_le_1_iff
% 0.35/0.65  thf(fact_5309_of__int__le__1__iff,axiom,
% 0.35/0.65      ! [Z2: int] :
% 0.35/0.65        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
% 0.35/0.65        = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_le_1_iff
% 0.35/0.65  thf(fact_5310_of__int__le__1__iff,axiom,
% 0.35/0.65      ! [Z2: int] :
% 0.35/0.65        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
% 0.35/0.65        = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_le_1_iff
% 0.35/0.65  thf(fact_5311_of__int__1__le__iff,axiom,
% 0.35/0.65      ! [Z2: int] :
% 0.35/0.65        ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
% 0.35/0.65        = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_1_le_iff
% 0.35/0.65  thf(fact_5312_of__int__1__le__iff,axiom,
% 0.35/0.65      ! [Z2: int] :
% 0.35/0.65        ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 0.35/0.65        = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_1_le_iff
% 0.35/0.65  thf(fact_5313_of__int__1__le__iff,axiom,
% 0.35/0.65      ! [Z2: int] :
% 0.35/0.65        ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
% 0.35/0.65        = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_1_le_iff
% 0.35/0.65  thf(fact_5314_of__int__less__1__iff,axiom,
% 0.35/0.65      ! [Z2: int] :
% 0.35/0.65        ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
% 0.35/0.65        = ( ord_less_int @ Z2 @ one_one_int ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_less_1_iff
% 0.35/0.65  thf(fact_5315_of__int__less__1__iff,axiom,
% 0.35/0.65      ! [Z2: int] :
% 0.35/0.65        ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
% 0.35/0.65        = ( ord_less_int @ Z2 @ one_one_int ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_less_1_iff
% 0.35/0.65  thf(fact_5316_of__int__less__1__iff,axiom,
% 0.35/0.65      ! [Z2: int] :
% 0.35/0.65        ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
% 0.35/0.65        = ( ord_less_int @ Z2 @ one_one_int ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_less_1_iff
% 0.35/0.65  thf(fact_5317_of__int__1__less__iff,axiom,
% 0.35/0.65      ! [Z2: int] :
% 0.35/0.65        ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
% 0.35/0.65        = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_1_less_iff
% 0.35/0.65  thf(fact_5318_of__int__1__less__iff,axiom,
% 0.35/0.65      ! [Z2: int] :
% 0.35/0.65        ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 0.35/0.65        = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_1_less_iff
% 0.35/0.65  thf(fact_5319_of__int__1__less__iff,axiom,
% 0.35/0.65      ! [Z2: int] :
% 0.35/0.65        ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
% 0.35/0.65        = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_1_less_iff
% 0.35/0.65  thf(fact_5320_of__int__le__of__int__power__cancel__iff,axiom,
% 0.35/0.65      ! [B: int,W2: nat,X: int] :
% 0.35/0.65        ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X ) )
% 0.35/0.65        = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_le_of_int_power_cancel_iff
% 0.35/0.65  thf(fact_5321_of__int__le__of__int__power__cancel__iff,axiom,
% 0.35/0.65      ! [B: int,W2: nat,X: int] :
% 0.35/0.65        ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X ) )
% 0.35/0.65        = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_le_of_int_power_cancel_iff
% 0.35/0.65  thf(fact_5322_of__int__le__of__int__power__cancel__iff,axiom,
% 0.35/0.65      ! [B: int,W2: nat,X: int] :
% 0.35/0.65        ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X ) )
% 0.35/0.65        = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_le_of_int_power_cancel_iff
% 0.35/0.65  thf(fact_5323_of__int__power__le__of__int__cancel__iff,axiom,
% 0.35/0.65      ! [X: int,B: int,W2: nat] :
% 0.35/0.65        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
% 0.35/0.65        = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_power_le_of_int_cancel_iff
% 0.35/0.65  thf(fact_5324_of__int__power__le__of__int__cancel__iff,axiom,
% 0.35/0.65      ! [X: int,B: int,W2: nat] :
% 0.35/0.65        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
% 0.35/0.65        = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_power_le_of_int_cancel_iff
% 0.35/0.65  thf(fact_5325_of__int__power__le__of__int__cancel__iff,axiom,
% 0.35/0.65      ! [X: int,B: int,W2: nat] :
% 0.35/0.65        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
% 0.35/0.65        = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_power_le_of_int_cancel_iff
% 0.35/0.65  thf(fact_5326_of__int__less__of__int__power__cancel__iff,axiom,
% 0.35/0.65      ! [B: int,W2: nat,X: int] :
% 0.35/0.65        ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X ) )
% 0.35/0.65        = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_less_of_int_power_cancel_iff
% 0.35/0.65  thf(fact_5327_of__int__less__of__int__power__cancel__iff,axiom,
% 0.35/0.65      ! [B: int,W2: nat,X: int] :
% 0.35/0.65        ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X ) )
% 0.35/0.65        = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_less_of_int_power_cancel_iff
% 0.35/0.65  thf(fact_5328_of__int__less__of__int__power__cancel__iff,axiom,
% 0.35/0.65      ! [B: int,W2: nat,X: int] :
% 0.35/0.65        ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X ) )
% 0.35/0.65        = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_less_of_int_power_cancel_iff
% 0.35/0.65  thf(fact_5329_of__int__power__less__of__int__cancel__iff,axiom,
% 0.35/0.65      ! [X: int,B: int,W2: nat] :
% 0.35/0.65        ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
% 0.35/0.65        = ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_power_less_of_int_cancel_iff
% 0.35/0.65  thf(fact_5330_of__int__power__less__of__int__cancel__iff,axiom,
% 0.35/0.65      ! [X: int,B: int,W2: nat] :
% 0.35/0.65        ( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
% 0.35/0.65        = ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_power_less_of_int_cancel_iff
% 0.35/0.65  thf(fact_5331_of__int__power__less__of__int__cancel__iff,axiom,
% 0.35/0.65      ! [X: int,B: int,W2: nat] :
% 0.35/0.65        ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
% 0.35/0.65        = ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_power_less_of_int_cancel_iff
% 0.35/0.65  thf(fact_5332_numeral__power__le__of__int__cancel__iff,axiom,
% 0.35/0.65      ! [X: num,N: nat,A: int] :
% 0.35/0.65        ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.35/0.65        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % numeral_power_le_of_int_cancel_iff
% 0.35/0.65  thf(fact_5333_numeral__power__le__of__int__cancel__iff,axiom,
% 0.35/0.65      ! [X: num,N: nat,A: int] :
% 0.35/0.65        ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.35/0.65        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % numeral_power_le_of_int_cancel_iff
% 0.35/0.65  thf(fact_5334_numeral__power__le__of__int__cancel__iff,axiom,
% 0.35/0.65      ! [X: num,N: nat,A: int] :
% 0.35/0.65        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.35/0.65        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % numeral_power_le_of_int_cancel_iff
% 0.35/0.65  thf(fact_5335_of__int__le__numeral__power__cancel__iff,axiom,
% 0.35/0.65      ! [A: int,X: num,N: nat] :
% 0.35/0.65        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.35/0.65        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_le_numeral_power_cancel_iff
% 0.35/0.65  thf(fact_5336_of__int__le__numeral__power__cancel__iff,axiom,
% 0.35/0.65      ! [A: int,X: num,N: nat] :
% 0.35/0.65        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.35/0.65        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_le_numeral_power_cancel_iff
% 0.35/0.65  thf(fact_5337_of__int__le__numeral__power__cancel__iff,axiom,
% 0.35/0.65      ! [A: int,X: num,N: nat] :
% 0.35/0.65        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.35/0.65        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_le_numeral_power_cancel_iff
% 0.35/0.65  thf(fact_5338_numeral__power__less__of__int__cancel__iff,axiom,
% 0.35/0.65      ! [X: num,N: nat,A: int] :
% 0.35/0.65        ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.35/0.65        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % numeral_power_less_of_int_cancel_iff
% 0.35/0.65  thf(fact_5339_numeral__power__less__of__int__cancel__iff,axiom,
% 0.35/0.65      ! [X: num,N: nat,A: int] :
% 0.35/0.65        ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.35/0.65        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % numeral_power_less_of_int_cancel_iff
% 0.35/0.65  thf(fact_5340_numeral__power__less__of__int__cancel__iff,axiom,
% 0.35/0.65      ! [X: num,N: nat,A: int] :
% 0.35/0.65        ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.35/0.65        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % numeral_power_less_of_int_cancel_iff
% 0.35/0.65  thf(fact_5341_of__int__less__numeral__power__cancel__iff,axiom,
% 0.35/0.65      ! [A: int,X: num,N: nat] :
% 0.35/0.65        ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.35/0.65        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_less_numeral_power_cancel_iff
% 0.35/0.65  thf(fact_5342_of__int__less__numeral__power__cancel__iff,axiom,
% 0.35/0.65      ! [A: int,X: num,N: nat] :
% 0.35/0.65        ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.35/0.65        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_less_numeral_power_cancel_iff
% 0.35/0.65  thf(fact_5343_of__int__less__numeral__power__cancel__iff,axiom,
% 0.35/0.65      ! [A: int,X: num,N: nat] :
% 0.35/0.65        ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.35/0.65        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_less_numeral_power_cancel_iff
% 0.35/0.65  thf(fact_5344_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.35/0.65      ! [X: num,N: nat,A: int] :
% 0.35/0.65        ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.35/0.65        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % neg_numeral_power_le_of_int_cancel_iff
% 0.35/0.65  thf(fact_5345_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.35/0.65      ! [X: num,N: nat,A: int] :
% 0.35/0.65        ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 0.35/0.65        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % neg_numeral_power_le_of_int_cancel_iff
% 0.35/0.65  thf(fact_5346_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.35/0.65      ! [X: num,N: nat,A: int] :
% 0.35/0.65        ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.35/0.65        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % neg_numeral_power_le_of_int_cancel_iff
% 0.35/0.65  thf(fact_5347_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.35/0.65      ! [X: num,N: nat,A: int] :
% 0.35/0.65        ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.35/0.65        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % neg_numeral_power_le_of_int_cancel_iff
% 0.35/0.65  thf(fact_5348_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.35/0.65      ! [A: int,X: num,N: nat] :
% 0.35/0.65        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 0.35/0.65        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_le_neg_numeral_power_cancel_iff
% 0.35/0.65  thf(fact_5349_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.35/0.65      ! [A: int,X: num,N: nat] :
% 0.35/0.65        ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 0.35/0.65        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_le_neg_numeral_power_cancel_iff
% 0.35/0.65  thf(fact_5350_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.35/0.65      ! [A: int,X: num,N: nat] :
% 0.35/0.65        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 0.35/0.65        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_le_neg_numeral_power_cancel_iff
% 0.35/0.65  thf(fact_5351_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.35/0.65      ! [A: int,X: num,N: nat] :
% 0.35/0.65        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 0.35/0.65        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_le_neg_numeral_power_cancel_iff
% 0.35/0.65  thf(fact_5352_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.35/0.65      ! [X: num,N: nat,A: int] :
% 0.35/0.65        ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.35/0.65        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % neg_numeral_power_less_of_int_cancel_iff
% 0.35/0.65  thf(fact_5353_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.35/0.65      ! [X: num,N: nat,A: int] :
% 0.35/0.65        ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.35/0.65        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % neg_numeral_power_less_of_int_cancel_iff
% 0.35/0.65  thf(fact_5354_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.35/0.65      ! [X: num,N: nat,A: int] :
% 0.35/0.65        ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.35/0.65        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % neg_numeral_power_less_of_int_cancel_iff
% 0.35/0.65  thf(fact_5355_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.35/0.65      ! [X: num,N: nat,A: int] :
% 0.35/0.65        ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 0.35/0.65        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % neg_numeral_power_less_of_int_cancel_iff
% 0.35/0.65  thf(fact_5356_subset__Compl__self__eq,axiom,
% 0.35/0.65      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ( ord_le3146513528884898305at_nat @ A2 @ ( uminus6524753893492686040at_nat @ A2 ) )
% 0.35/0.65        = ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % subset_Compl_self_eq
% 0.35/0.65  thf(fact_5357_subset__Compl__self__eq,axiom,
% 0.35/0.65      ! [A2: set_real] :
% 0.35/0.65        ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ A2 ) )
% 0.35/0.65        = ( A2 = bot_bot_set_real ) ) ).
% 0.35/0.65  
% 0.35/0.65  % subset_Compl_self_eq
% 0.35/0.65  thf(fact_5358_subset__Compl__self__eq,axiom,
% 0.35/0.65      ! [A2: set_nat] :
% 0.35/0.65        ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
% 0.35/0.65        = ( A2 = bot_bot_set_nat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % subset_Compl_self_eq
% 0.35/0.65  thf(fact_5359_subset__Compl__self__eq,axiom,
% 0.35/0.65      ! [A2: set_int] :
% 0.35/0.65        ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
% 0.35/0.65        = ( A2 = bot_bot_set_int ) ) ).
% 0.35/0.65  
% 0.35/0.65  % subset_Compl_self_eq
% 0.35/0.65  thf(fact_5360_real__add__less__0__iff,axiom,
% 0.35/0.65      ! [X: real,Y: real] :
% 0.35/0.65        ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 0.35/0.65        = ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % real_add_less_0_iff
% 0.35/0.65  thf(fact_5361_real__0__less__add__iff,axiom,
% 0.35/0.65      ! [X: real,Y: real] :
% 0.35/0.65        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
% 0.35/0.65        = ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% 0.35/0.65  
% 0.35/0.65  % real_0_less_add_iff
% 0.35/0.65  thf(fact_5362_signed__take__bit__take__bit,axiom,
% 0.35/0.65      ! [M: nat,N: nat,A: int] :
% 0.35/0.65        ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 0.35/0.65        = ( if_int_int @ ( ord_less_eq_nat @ N @ M ) @ ( bit_se2923211474154528505it_int @ N ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % signed_take_bit_take_bit
% 0.35/0.65  thf(fact_5363_int__le__real__less,axiom,
% 0.35/0.65      ( ord_less_eq_int
% 0.35/0.65      = ( ^ [N2: int,M2: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M2 ) @ one_one_real ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % int_le_real_less
% 0.35/0.65  thf(fact_5364_int__less__real__le,axiom,
% 0.35/0.65      ( ord_less_int
% 0.35/0.65      = ( ^ [N2: int,M2: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M2 ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % int_less_real_le
% 0.35/0.65  thf(fact_5365_of__int__nonneg,axiom,
% 0.35/0.65      ! [Z2: int] :
% 0.35/0.65        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.35/0.65       => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_nonneg
% 0.35/0.65  thf(fact_5366_of__int__nonneg,axiom,
% 0.35/0.65      ! [Z2: int] :
% 0.35/0.65        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.35/0.65       => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_nonneg
% 0.35/0.65  thf(fact_5367_of__int__nonneg,axiom,
% 0.35/0.65      ! [Z2: int] :
% 0.35/0.65        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.35/0.65       => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_nonneg
% 0.35/0.65  thf(fact_5368_of__int__pos,axiom,
% 0.35/0.65      ! [Z2: int] :
% 0.35/0.65        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.35/0.65       => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_pos
% 0.35/0.65  thf(fact_5369_of__int__pos,axiom,
% 0.35/0.65      ! [Z2: int] :
% 0.35/0.65        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.35/0.65       => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_pos
% 0.35/0.65  thf(fact_5370_of__int__pos,axiom,
% 0.35/0.65      ! [Z2: int] :
% 0.35/0.65        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.35/0.65       => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % of_int_pos
% 0.35/0.65  thf(fact_5371_take__bit__signed__take__bit,axiom,
% 0.35/0.65      ! [M: nat,N: nat,A: int] :
% 0.35/0.65        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.35/0.65       => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N @ A ) )
% 0.35/0.65          = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % take_bit_signed_take_bit
% 0.35/0.65  thf(fact_5372_signed__take__bit__int__less__exp,axiom,
% 0.35/0.65      ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.65  
% 0.35/0.65  % signed_take_bit_int_less_exp
% 0.35/0.65  thf(fact_5373_signed__take__bit__int__greater__eq__self__iff,axiom,
% 0.35/0.65      ! [K: int,N: nat] :
% 0.35/0.65        ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.35/0.65        = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % signed_take_bit_int_greater_eq_self_iff
% 0.35/0.65  thf(fact_5374_signed__take__bit__int__less__self__iff,axiom,
% 0.35/0.65      ! [N: nat,K: int] :
% 0.35/0.65        ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 0.35/0.65        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 0.35/0.65  
% 0.35/0.65  % signed_take_bit_int_less_self_iff
% 0.35/0.65  thf(fact_5375_signed__take__bit__int__greater__self__iff,axiom,
% 0.35/0.65      ! [K: int,N: nat] :
% 0.35/0.65        ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.35/0.65        = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % signed_take_bit_int_greater_self_iff
% 0.35/0.65  thf(fact_5376_signed__take__bit__int__eq__self__iff,axiom,
% 0.35/0.65      ! [N: nat,K: int] :
% 0.35/0.65        ( ( ( bit_ri631733984087533419it_int @ N @ K )
% 0.35/0.65          = K )
% 0.35/0.65        = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 0.35/0.65          & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % signed_take_bit_int_eq_self_iff
% 0.35/0.65  thf(fact_5377_signed__take__bit__int__eq__self,axiom,
% 0.35/0.65      ! [N: nat,K: int] :
% 0.35/0.65        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 0.35/0.65       => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.65         => ( ( bit_ri631733984087533419it_int @ N @ K )
% 0.35/0.65            = K ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % signed_take_bit_int_eq_self
% 0.35/0.65  thf(fact_5378_signed__take__bit__int__greater__eq,axiom,
% 0.35/0.65      ! [K: int,N: nat] :
% 0.35/0.65        ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.35/0.65       => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % signed_take_bit_int_greater_eq
% 0.35/0.65  thf(fact_5379_floor__exists1,axiom,
% 0.35/0.65      ! [X: real] :
% 0.35/0.65      ? [X4: int] :
% 0.35/0.65        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ X )
% 0.35/0.65        & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X4 @ one_one_int ) ) )
% 0.35/0.65        & ! [Y4: int] :
% 0.35/0.65            ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y4 ) @ X )
% 0.35/0.65              & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
% 0.35/0.65           => ( Y4 = X4 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % floor_exists1
% 0.35/0.65  thf(fact_5380_floor__exists1,axiom,
% 0.35/0.65      ! [X: rat] :
% 0.35/0.65      ? [X4: int] :
% 0.35/0.65        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ X )
% 0.35/0.65        & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X4 @ one_one_int ) ) )
% 0.35/0.65        & ! [Y4: int] :
% 0.35/0.65            ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y4 ) @ X )
% 0.35/0.65              & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
% 0.35/0.65           => ( Y4 = X4 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % floor_exists1
% 0.35/0.65  thf(fact_5381_floor__exists,axiom,
% 0.35/0.65      ! [X: real] :
% 0.35/0.65      ? [Z4: int] :
% 0.35/0.65        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z4 ) @ X )
% 0.35/0.65        & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % floor_exists
% 0.35/0.65  thf(fact_5382_floor__exists,axiom,
% 0.35/0.65      ! [X: rat] :
% 0.35/0.65      ? [Z4: int] :
% 0.35/0.65        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z4 ) @ X )
% 0.35/0.65        & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % floor_exists
% 0.35/0.65  thf(fact_5383_round__unique,axiom,
% 0.35/0.65      ! [X: real,Y: int] :
% 0.35/0.65        ( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y ) )
% 0.35/0.65       => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.35/0.65         => ( ( archim8280529875227126926d_real @ X )
% 0.35/0.65            = Y ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % round_unique
% 0.35/0.65  thf(fact_5384_round__unique,axiom,
% 0.35/0.65      ! [X: rat,Y: int] :
% 0.35/0.65        ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
% 0.35/0.65       => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.65         => ( ( archim7778729529865785530nd_rat @ X )
% 0.35/0.65            = Y ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % round_unique
% 0.35/0.65  thf(fact_5385_set__encode__insert,axiom,
% 0.35/0.65      ! [A2: set_nat,N: nat] :
% 0.35/0.65        ( ( finite_finite_nat @ A2 )
% 0.35/0.65       => ( ~ ( member_nat @ N @ A2 )
% 0.35/0.65         => ( ( nat_set_encode @ ( insert_nat @ N @ A2 ) )
% 0.35/0.65            = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % set_encode_insert
% 0.35/0.65  thf(fact_5386_arith__series__nat,axiom,
% 0.35/0.65      ! [A: nat,D: nat,N: nat] :
% 0.35/0.65        ( ( groups3542108847815614940at_nat
% 0.35/0.65          @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I4 @ D ) )
% 0.35/0.65          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.65        = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % arith_series_nat
% 0.35/0.65  thf(fact_5387_Suc__0__xor__eq,axiom,
% 0.35/0.65      ! [N: nat] :
% 0.35/0.65        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.65        = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.35/0.65          @ ( zero_n2687167440665602831ol_nat
% 0.35/0.65            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Suc_0_xor_eq
% 0.35/0.65  thf(fact_5388_xor__Suc__0__eq,axiom,
% 0.35/0.65      ! [N: nat] :
% 0.35/0.65        ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.35/0.65        = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.35/0.65          @ ( zero_n2687167440665602831ol_nat
% 0.35/0.65            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % xor_Suc_0_eq
% 0.35/0.65  thf(fact_5389_insertCI,axiom,
% 0.35/0.65      ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
% 0.35/0.65        ( ( ~ ( member8757157785044589968at_nat @ A @ B2 )
% 0.35/0.65         => ( A = B ) )
% 0.35/0.65       => ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertCI
% 0.35/0.65  thf(fact_5390_insertCI,axiom,
% 0.35/0.65      ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
% 0.35/0.65        ( ( ~ ( member8440522571783428010at_nat @ A @ B2 )
% 0.35/0.65         => ( A = B ) )
% 0.35/0.65       => ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertCI
% 0.35/0.65  thf(fact_5391_insertCI,axiom,
% 0.35/0.65      ! [A: real,B2: set_real,B: real] :
% 0.35/0.65        ( ( ~ ( member_real @ A @ B2 )
% 0.35/0.65         => ( A = B ) )
% 0.35/0.65       => ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertCI
% 0.35/0.65  thf(fact_5392_insertCI,axiom,
% 0.35/0.65      ! [A: set_nat,B2: set_set_nat,B: set_nat] :
% 0.35/0.65        ( ( ~ ( member_set_nat @ A @ B2 )
% 0.35/0.65         => ( A = B ) )
% 0.35/0.65       => ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertCI
% 0.35/0.65  thf(fact_5393_insertCI,axiom,
% 0.35/0.65      ! [A: nat,B2: set_nat,B: nat] :
% 0.35/0.65        ( ( ~ ( member_nat @ A @ B2 )
% 0.35/0.65         => ( A = B ) )
% 0.35/0.65       => ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertCI
% 0.35/0.65  thf(fact_5394_insertCI,axiom,
% 0.35/0.65      ! [A: int,B2: set_int,B: int] :
% 0.35/0.65        ( ( ~ ( member_int @ A @ B2 )
% 0.35/0.65         => ( A = B ) )
% 0.35/0.65       => ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertCI
% 0.35/0.65  thf(fact_5395_insert__iff,axiom,
% 0.35/0.65      ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65        ( ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ A2 ) )
% 0.35/0.65        = ( ( A = B )
% 0.35/0.65          | ( member8757157785044589968at_nat @ A @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_iff
% 0.35/0.65  thf(fact_5396_insert__iff,axiom,
% 0.35/0.65      ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A2 ) )
% 0.35/0.65        = ( ( A = B )
% 0.35/0.65          | ( member8440522571783428010at_nat @ A @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_iff
% 0.35/0.65  thf(fact_5397_insert__iff,axiom,
% 0.35/0.65      ! [A: real,B: real,A2: set_real] :
% 0.35/0.65        ( ( member_real @ A @ ( insert_real @ B @ A2 ) )
% 0.35/0.65        = ( ( A = B )
% 0.35/0.65          | ( member_real @ A @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_iff
% 0.35/0.65  thf(fact_5398_insert__iff,axiom,
% 0.35/0.65      ! [A: set_nat,B: set_nat,A2: set_set_nat] :
% 0.35/0.65        ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
% 0.35/0.65        = ( ( A = B )
% 0.35/0.65          | ( member_set_nat @ A @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_iff
% 0.35/0.65  thf(fact_5399_insert__iff,axiom,
% 0.35/0.65      ! [A: nat,B: nat,A2: set_nat] :
% 0.35/0.65        ( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
% 0.35/0.65        = ( ( A = B )
% 0.35/0.65          | ( member_nat @ A @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_iff
% 0.35/0.65  thf(fact_5400_insert__iff,axiom,
% 0.35/0.65      ! [A: int,B: int,A2: set_int] :
% 0.35/0.65        ( ( member_int @ A @ ( insert_int @ B @ A2 ) )
% 0.35/0.65        = ( ( A = B )
% 0.35/0.65          | ( member_int @ A @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_iff
% 0.35/0.65  thf(fact_5401_insert__absorb2,axiom,
% 0.35/0.65      ! [X: nat,A2: set_nat] :
% 0.35/0.65        ( ( insert_nat @ X @ ( insert_nat @ X @ A2 ) )
% 0.35/0.65        = ( insert_nat @ X @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_absorb2
% 0.35/0.65  thf(fact_5402_insert__absorb2,axiom,
% 0.35/0.65      ! [X: int,A2: set_int] :
% 0.35/0.65        ( ( insert_int @ X @ ( insert_int @ X @ A2 ) )
% 0.35/0.65        = ( insert_int @ X @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_absorb2
% 0.35/0.65  thf(fact_5403_insert__absorb2,axiom,
% 0.35/0.65      ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ( insert8211810215607154385at_nat @ X @ ( insert8211810215607154385at_nat @ X @ A2 ) )
% 0.35/0.65        = ( insert8211810215607154385at_nat @ X @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_absorb2
% 0.35/0.65  thf(fact_5404_insert__absorb2,axiom,
% 0.35/0.65      ! [X: real,A2: set_real] :
% 0.35/0.65        ( ( insert_real @ X @ ( insert_real @ X @ A2 ) )
% 0.35/0.65        = ( insert_real @ X @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_absorb2
% 0.35/0.65  thf(fact_5405_insert__absorb2,axiom,
% 0.35/0.65      ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65        ( ( insert9069300056098147895at_nat @ X @ ( insert9069300056098147895at_nat @ X @ A2 ) )
% 0.35/0.65        = ( insert9069300056098147895at_nat @ X @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_absorb2
% 0.35/0.65  thf(fact_5406_ComplI,axiom,
% 0.35/0.65      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ~ ( member8440522571783428010at_nat @ C @ A2 )
% 0.35/0.65       => ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % ComplI
% 0.35/0.65  thf(fact_5407_ComplI,axiom,
% 0.35/0.65      ! [C: real,A2: set_real] :
% 0.35/0.65        ( ~ ( member_real @ C @ A2 )
% 0.35/0.65       => ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % ComplI
% 0.35/0.65  thf(fact_5408_ComplI,axiom,
% 0.35/0.65      ! [C: set_nat,A2: set_set_nat] :
% 0.35/0.65        ( ~ ( member_set_nat @ C @ A2 )
% 0.35/0.65       => ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % ComplI
% 0.35/0.65  thf(fact_5409_ComplI,axiom,
% 0.35/0.65      ! [C: nat,A2: set_nat] :
% 0.35/0.65        ( ~ ( member_nat @ C @ A2 )
% 0.35/0.65       => ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % ComplI
% 0.35/0.65  thf(fact_5410_ComplI,axiom,
% 0.35/0.65      ! [C: int,A2: set_int] :
% 0.35/0.65        ( ~ ( member_int @ C @ A2 )
% 0.35/0.65       => ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % ComplI
% 0.35/0.65  thf(fact_5411_Compl__iff,axiom,
% 0.35/0.65      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A2 ) )
% 0.35/0.65        = ( ~ ( member8440522571783428010at_nat @ C @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Compl_iff
% 0.35/0.65  thf(fact_5412_Compl__iff,axiom,
% 0.35/0.65      ! [C: real,A2: set_real] :
% 0.35/0.65        ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
% 0.35/0.65        = ( ~ ( member_real @ C @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Compl_iff
% 0.35/0.65  thf(fact_5413_Compl__iff,axiom,
% 0.35/0.65      ! [C: set_nat,A2: set_set_nat] :
% 0.35/0.65        ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
% 0.35/0.65        = ( ~ ( member_set_nat @ C @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Compl_iff
% 0.35/0.65  thf(fact_5414_Compl__iff,axiom,
% 0.35/0.65      ! [C: nat,A2: set_nat] :
% 0.35/0.65        ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
% 0.35/0.65        = ( ~ ( member_nat @ C @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Compl_iff
% 0.35/0.65  thf(fact_5415_Compl__iff,axiom,
% 0.35/0.65      ! [C: int,A2: set_int] :
% 0.35/0.65        ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
% 0.35/0.65        = ( ~ ( member_int @ C @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Compl_iff
% 0.35/0.65  thf(fact_5416_singletonI,axiom,
% 0.35/0.65      ! [A: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singletonI
% 0.35/0.65  thf(fact_5417_singletonI,axiom,
% 0.35/0.65      ! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singletonI
% 0.35/0.65  thf(fact_5418_singletonI,axiom,
% 0.35/0.65      ! [A: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singletonI
% 0.35/0.65  thf(fact_5419_singletonI,axiom,
% 0.35/0.65      ! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singletonI
% 0.35/0.65  thf(fact_5420_singletonI,axiom,
% 0.35/0.65      ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singletonI
% 0.35/0.65  thf(fact_5421_singletonI,axiom,
% 0.35/0.65      ! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singletonI
% 0.35/0.65  thf(fact_5422_finite__insert,axiom,
% 0.35/0.65      ! [A: real,A2: set_real] :
% 0.35/0.65        ( ( finite_finite_real @ ( insert_real @ A @ A2 ) )
% 0.35/0.65        = ( finite_finite_real @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % finite_insert
% 0.35/0.65  thf(fact_5423_finite__insert,axiom,
% 0.35/0.65      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65        ( ( finite4343798906461161616at_nat @ ( insert9069300056098147895at_nat @ A @ A2 ) )
% 0.35/0.65        = ( finite4343798906461161616at_nat @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % finite_insert
% 0.35/0.65  thf(fact_5424_finite__insert,axiom,
% 0.35/0.65      ! [A: nat,A2: set_nat] :
% 0.35/0.65        ( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
% 0.35/0.65        = ( finite_finite_nat @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % finite_insert
% 0.35/0.65  thf(fact_5425_finite__insert,axiom,
% 0.35/0.65      ! [A: int,A2: set_int] :
% 0.35/0.65        ( ( finite_finite_int @ ( insert_int @ A @ A2 ) )
% 0.35/0.65        = ( finite_finite_int @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % finite_insert
% 0.35/0.65  thf(fact_5426_finite__insert,axiom,
% 0.35/0.65      ! [A: complex,A2: set_complex] :
% 0.35/0.65        ( ( finite3207457112153483333omplex @ ( insert_complex @ A @ A2 ) )
% 0.35/0.65        = ( finite3207457112153483333omplex @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % finite_insert
% 0.35/0.65  thf(fact_5427_finite__insert,axiom,
% 0.35/0.65      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) )
% 0.35/0.65        = ( finite6177210948735845034at_nat @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % finite_insert
% 0.35/0.65  thf(fact_5428_insert__subset,axiom,
% 0.35/0.65      ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65        ( ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X @ A2 ) @ B2 )
% 0.35/0.65        = ( ( member8757157785044589968at_nat @ X @ B2 )
% 0.35/0.65          & ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_subset
% 0.35/0.65  thf(fact_5429_insert__subset,axiom,
% 0.35/0.65      ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) @ B2 )
% 0.35/0.65        = ( ( member8440522571783428010at_nat @ X @ B2 )
% 0.35/0.65          & ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_subset
% 0.35/0.65  thf(fact_5430_insert__subset,axiom,
% 0.35/0.65      ! [X: real,A2: set_real,B2: set_real] :
% 0.35/0.65        ( ( ord_less_eq_set_real @ ( insert_real @ X @ A2 ) @ B2 )
% 0.35/0.65        = ( ( member_real @ X @ B2 )
% 0.35/0.65          & ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_subset
% 0.35/0.65  thf(fact_5431_insert__subset,axiom,
% 0.35/0.65      ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.35/0.65        ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
% 0.35/0.65        = ( ( member_set_nat @ X @ B2 )
% 0.35/0.65          & ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_subset
% 0.35/0.65  thf(fact_5432_insert__subset,axiom,
% 0.35/0.65      ! [X: nat,A2: set_nat,B2: set_nat] :
% 0.35/0.65        ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
% 0.35/0.65        = ( ( member_nat @ X @ B2 )
% 0.35/0.65          & ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_subset
% 0.35/0.65  thf(fact_5433_insert__subset,axiom,
% 0.35/0.65      ! [X: int,A2: set_int,B2: set_int] :
% 0.35/0.65        ( ( ord_less_eq_set_int @ ( insert_int @ X @ A2 ) @ B2 )
% 0.35/0.65        = ( ( member_int @ X @ B2 )
% 0.35/0.65          & ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_subset
% 0.35/0.65  thf(fact_5434_Diff__insert0,axiom,
% 0.35/0.65      ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65        ( ~ ( member8757157785044589968at_nat @ X @ A2 )
% 0.35/0.65       => ( ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ B2 ) )
% 0.35/0.65          = ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Diff_insert0
% 0.35/0.65  thf(fact_5435_Diff__insert0,axiom,
% 0.35/0.65      ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ~ ( member8440522571783428010at_nat @ X @ A2 )
% 0.35/0.65       => ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B2 ) )
% 0.35/0.65          = ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Diff_insert0
% 0.35/0.65  thf(fact_5436_Diff__insert0,axiom,
% 0.35/0.65      ! [X: real,A2: set_real,B2: set_real] :
% 0.35/0.65        ( ~ ( member_real @ X @ A2 )
% 0.35/0.65       => ( ( minus_minus_set_real @ A2 @ ( insert_real @ X @ B2 ) )
% 0.35/0.65          = ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Diff_insert0
% 0.35/0.65  thf(fact_5437_Diff__insert0,axiom,
% 0.35/0.65      ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.35/0.65        ( ~ ( member_set_nat @ X @ A2 )
% 0.35/0.65       => ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
% 0.35/0.65          = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Diff_insert0
% 0.35/0.65  thf(fact_5438_Diff__insert0,axiom,
% 0.35/0.65      ! [X: int,A2: set_int,B2: set_int] :
% 0.35/0.65        ( ~ ( member_int @ X @ A2 )
% 0.35/0.65       => ( ( minus_minus_set_int @ A2 @ ( insert_int @ X @ B2 ) )
% 0.35/0.65          = ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Diff_insert0
% 0.35/0.65  thf(fact_5439_Diff__insert0,axiom,
% 0.35/0.65      ! [X: nat,A2: set_nat,B2: set_nat] :
% 0.35/0.65        ( ~ ( member_nat @ X @ A2 )
% 0.35/0.65       => ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
% 0.35/0.65          = ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Diff_insert0
% 0.35/0.65  thf(fact_5440_insert__Diff1,axiom,
% 0.35/0.65      ! [X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65        ( ( member8757157785044589968at_nat @ X @ B2 )
% 0.35/0.65       => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A2 ) @ B2 )
% 0.35/0.65          = ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_Diff1
% 0.35/0.65  thf(fact_5441_insert__Diff1,axiom,
% 0.35/0.65      ! [X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ( member8440522571783428010at_nat @ X @ B2 )
% 0.35/0.65       => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) @ B2 )
% 0.35/0.65          = ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_Diff1
% 0.35/0.65  thf(fact_5442_insert__Diff1,axiom,
% 0.35/0.65      ! [X: real,B2: set_real,A2: set_real] :
% 0.35/0.65        ( ( member_real @ X @ B2 )
% 0.35/0.65       => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
% 0.35/0.65          = ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_Diff1
% 0.35/0.65  thf(fact_5443_insert__Diff1,axiom,
% 0.35/0.65      ! [X: set_nat,B2: set_set_nat,A2: set_set_nat] :
% 0.35/0.65        ( ( member_set_nat @ X @ B2 )
% 0.35/0.65       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
% 0.35/0.65          = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_Diff1
% 0.35/0.65  thf(fact_5444_insert__Diff1,axiom,
% 0.35/0.65      ! [X: int,B2: set_int,A2: set_int] :
% 0.35/0.65        ( ( member_int @ X @ B2 )
% 0.35/0.65       => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B2 )
% 0.35/0.65          = ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_Diff1
% 0.35/0.65  thf(fact_5445_insert__Diff1,axiom,
% 0.35/0.65      ! [X: nat,B2: set_nat,A2: set_nat] :
% 0.35/0.65        ( ( member_nat @ X @ B2 )
% 0.35/0.65       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
% 0.35/0.65          = ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_Diff1
% 0.35/0.65  thf(fact_5446_xor_Oright__neutral,axiom,
% 0.35/0.65      ! [A: nat] :
% 0.35/0.65        ( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
% 0.35/0.65        = A ) ).
% 0.35/0.65  
% 0.35/0.65  % xor.right_neutral
% 0.35/0.65  thf(fact_5447_xor_Oright__neutral,axiom,
% 0.35/0.65      ! [A: int] :
% 0.35/0.65        ( ( bit_se6526347334894502574or_int @ A @ zero_zero_int )
% 0.35/0.65        = A ) ).
% 0.35/0.65  
% 0.35/0.65  % xor.right_neutral
% 0.35/0.65  thf(fact_5448_xor_Oleft__neutral,axiom,
% 0.35/0.65      ! [A: nat] :
% 0.35/0.65        ( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
% 0.35/0.65        = A ) ).
% 0.35/0.65  
% 0.35/0.65  % xor.left_neutral
% 0.35/0.65  thf(fact_5449_xor_Oleft__neutral,axiom,
% 0.35/0.65      ! [A: int] :
% 0.35/0.65        ( ( bit_se6526347334894502574or_int @ zero_zero_int @ A )
% 0.35/0.65        = A ) ).
% 0.35/0.65  
% 0.35/0.65  % xor.left_neutral
% 0.35/0.65  thf(fact_5450_xor__self__eq,axiom,
% 0.35/0.65      ! [A: nat] :
% 0.35/0.65        ( ( bit_se6528837805403552850or_nat @ A @ A )
% 0.35/0.65        = zero_zero_nat ) ).
% 0.35/0.65  
% 0.35/0.65  % xor_self_eq
% 0.35/0.65  thf(fact_5451_xor__self__eq,axiom,
% 0.35/0.65      ! [A: int] :
% 0.35/0.65        ( ( bit_se6526347334894502574or_int @ A @ A )
% 0.35/0.65        = zero_zero_int ) ).
% 0.35/0.65  
% 0.35/0.65  % xor_self_eq
% 0.35/0.65  thf(fact_5452_bit_Oxor__self,axiom,
% 0.35/0.65      ! [X: int] :
% 0.35/0.65        ( ( bit_se6526347334894502574or_int @ X @ X )
% 0.35/0.65        = zero_zero_int ) ).
% 0.35/0.65  
% 0.35/0.65  % bit.xor_self
% 0.35/0.65  thf(fact_5453_sum_Oneutral__const,axiom,
% 0.35/0.65      ! [A2: set_nat] :
% 0.35/0.65        ( ( groups3542108847815614940at_nat
% 0.35/0.65          @ ^ [Uu3: nat] : zero_zero_nat
% 0.35/0.65          @ A2 )
% 0.35/0.65        = zero_zero_nat ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.neutral_const
% 0.35/0.65  thf(fact_5454_sum_Oneutral__const,axiom,
% 0.35/0.65      ! [A2: set_complex] :
% 0.35/0.65        ( ( groups7754918857620584856omplex
% 0.35/0.65          @ ^ [Uu3: complex] : zero_zero_complex
% 0.35/0.65          @ A2 )
% 0.35/0.65        = zero_zero_complex ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.neutral_const
% 0.35/0.65  thf(fact_5455_sum_Oneutral__const,axiom,
% 0.35/0.65      ! [A2: set_nat] :
% 0.35/0.65        ( ( groups6591440286371151544t_real
% 0.35/0.65          @ ^ [Uu3: nat] : zero_zero_real
% 0.35/0.65          @ A2 )
% 0.35/0.65        = zero_zero_real ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.neutral_const
% 0.35/0.65  thf(fact_5456_singleton__conv,axiom,
% 0.35/0.65      ! [A: produc3843707927480180839at_nat] :
% 0.35/0.65        ( ( collec6321179662152712658at_nat
% 0.35/0.65          @ ^ [X3: produc3843707927480180839at_nat] : ( X3 = A ) )
% 0.35/0.65        = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_conv
% 0.35/0.65  thf(fact_5457_singleton__conv,axiom,
% 0.35/0.65      ! [A: list_nat] :
% 0.35/0.65        ( ( collect_list_nat
% 0.35/0.65          @ ^ [X3: list_nat] : ( X3 = A ) )
% 0.35/0.65        = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_conv
% 0.35/0.65  thf(fact_5458_singleton__conv,axiom,
% 0.35/0.65      ! [A: set_nat] :
% 0.35/0.65        ( ( collect_set_nat
% 0.35/0.65          @ ^ [X3: set_nat] : ( X3 = A ) )
% 0.35/0.65        = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_conv
% 0.35/0.65  thf(fact_5459_singleton__conv,axiom,
% 0.35/0.65      ! [A: product_prod_nat_nat] :
% 0.35/0.65        ( ( collec3392354462482085612at_nat
% 0.35/0.65          @ ^ [X3: product_prod_nat_nat] : ( X3 = A ) )
% 0.35/0.65        = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_conv
% 0.35/0.65  thf(fact_5460_singleton__conv,axiom,
% 0.35/0.65      ! [A: real] :
% 0.35/0.65        ( ( collect_real
% 0.35/0.65          @ ^ [X3: real] : ( X3 = A ) )
% 0.35/0.65        = ( insert_real @ A @ bot_bot_set_real ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_conv
% 0.35/0.65  thf(fact_5461_singleton__conv,axiom,
% 0.35/0.65      ! [A: nat] :
% 0.35/0.65        ( ( collect_nat
% 0.35/0.65          @ ^ [X3: nat] : ( X3 = A ) )
% 0.35/0.65        = ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_conv
% 0.35/0.65  thf(fact_5462_singleton__conv,axiom,
% 0.35/0.65      ! [A: int] :
% 0.35/0.65        ( ( collect_int
% 0.35/0.65          @ ^ [X3: int] : ( X3 = A ) )
% 0.35/0.65        = ( insert_int @ A @ bot_bot_set_int ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_conv
% 0.35/0.65  thf(fact_5463_singleton__conv2,axiom,
% 0.35/0.65      ! [A: produc3843707927480180839at_nat] :
% 0.35/0.65        ( ( collec6321179662152712658at_nat
% 0.35/0.65          @ ( ^ [Y5: produc3843707927480180839at_nat,Z: produc3843707927480180839at_nat] : ( Y5 = Z )
% 0.35/0.65            @ A ) )
% 0.35/0.65        = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_conv2
% 0.35/0.65  thf(fact_5464_singleton__conv2,axiom,
% 0.35/0.65      ! [A: list_nat] :
% 0.35/0.65        ( ( collect_list_nat
% 0.35/0.65          @ ( ^ [Y5: list_nat,Z: list_nat] : ( Y5 = Z )
% 0.35/0.65            @ A ) )
% 0.35/0.65        = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_conv2
% 0.35/0.65  thf(fact_5465_singleton__conv2,axiom,
% 0.35/0.65      ! [A: set_nat] :
% 0.35/0.65        ( ( collect_set_nat
% 0.35/0.65          @ ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z )
% 0.35/0.65            @ A ) )
% 0.35/0.65        = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_conv2
% 0.35/0.65  thf(fact_5466_singleton__conv2,axiom,
% 0.35/0.65      ! [A: product_prod_nat_nat] :
% 0.35/0.65        ( ( collec3392354462482085612at_nat
% 0.35/0.65          @ ( ^ [Y5: product_prod_nat_nat,Z: product_prod_nat_nat] : ( Y5 = Z )
% 0.35/0.65            @ A ) )
% 0.35/0.65        = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_conv2
% 0.35/0.65  thf(fact_5467_singleton__conv2,axiom,
% 0.35/0.65      ! [A: real] :
% 0.35/0.65        ( ( collect_real
% 0.35/0.65          @ ( ^ [Y5: real,Z: real] : ( Y5 = Z )
% 0.35/0.65            @ A ) )
% 0.35/0.65        = ( insert_real @ A @ bot_bot_set_real ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_conv2
% 0.35/0.65  thf(fact_5468_singleton__conv2,axiom,
% 0.35/0.65      ! [A: nat] :
% 0.35/0.65        ( ( collect_nat
% 0.35/0.65          @ ( ^ [Y5: nat,Z: nat] : ( Y5 = Z )
% 0.35/0.65            @ A ) )
% 0.35/0.65        = ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_conv2
% 0.35/0.65  thf(fact_5469_singleton__conv2,axiom,
% 0.35/0.65      ! [A: int] :
% 0.35/0.65        ( ( collect_int
% 0.35/0.65          @ ( ^ [Y5: int,Z: int] : ( Y5 = Z )
% 0.35/0.65            @ A ) )
% 0.35/0.65        = ( insert_int @ A @ bot_bot_set_int ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_conv2
% 0.35/0.65  thf(fact_5470_sum_Oempty,axiom,
% 0.35/0.65      ! [G: real > complex] :
% 0.35/0.65        ( ( groups5754745047067104278omplex @ G @ bot_bot_set_real )
% 0.35/0.65        = zero_zero_complex ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.empty
% 0.35/0.65  thf(fact_5471_sum_Oempty,axiom,
% 0.35/0.65      ! [G: real > real] :
% 0.35/0.65        ( ( groups8097168146408367636l_real @ G @ bot_bot_set_real )
% 0.35/0.65        = zero_zero_real ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.empty
% 0.35/0.65  thf(fact_5472_sum_Oempty,axiom,
% 0.35/0.65      ! [G: real > rat] :
% 0.35/0.65        ( ( groups1300246762558778688al_rat @ G @ bot_bot_set_real )
% 0.35/0.65        = zero_zero_rat ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.empty
% 0.35/0.65  thf(fact_5473_sum_Oempty,axiom,
% 0.35/0.65      ! [G: real > nat] :
% 0.35/0.65        ( ( groups1935376822645274424al_nat @ G @ bot_bot_set_real )
% 0.35/0.65        = zero_zero_nat ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.empty
% 0.35/0.65  thf(fact_5474_sum_Oempty,axiom,
% 0.35/0.65      ! [G: real > int] :
% 0.35/0.65        ( ( groups1932886352136224148al_int @ G @ bot_bot_set_real )
% 0.35/0.65        = zero_zero_int ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.empty
% 0.35/0.65  thf(fact_5475_sum_Oempty,axiom,
% 0.35/0.65      ! [G: nat > complex] :
% 0.35/0.65        ( ( groups2073611262835488442omplex @ G @ bot_bot_set_nat )
% 0.35/0.65        = zero_zero_complex ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.empty
% 0.35/0.65  thf(fact_5476_sum_Oempty,axiom,
% 0.35/0.65      ! [G: nat > rat] :
% 0.35/0.65        ( ( groups2906978787729119204at_rat @ G @ bot_bot_set_nat )
% 0.35/0.65        = zero_zero_rat ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.empty
% 0.35/0.65  thf(fact_5477_sum_Oempty,axiom,
% 0.35/0.65      ! [G: nat > int] :
% 0.35/0.65        ( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
% 0.35/0.65        = zero_zero_int ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.empty
% 0.35/0.65  thf(fact_5478_sum_Oempty,axiom,
% 0.35/0.65      ! [G: int > complex] :
% 0.35/0.65        ( ( groups3049146728041665814omplex @ G @ bot_bot_set_int )
% 0.35/0.65        = zero_zero_complex ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.empty
% 0.35/0.65  thf(fact_5479_sum_Oempty,axiom,
% 0.35/0.65      ! [G: int > real] :
% 0.35/0.65        ( ( groups8778361861064173332t_real @ G @ bot_bot_set_int )
% 0.35/0.65        = zero_zero_real ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.empty
% 0.35/0.65  thf(fact_5480_sum__eq__0__iff,axiom,
% 0.35/0.65      ! [F3: set_int,F: int > nat] :
% 0.35/0.65        ( ( finite_finite_int @ F3 )
% 0.35/0.65       => ( ( ( groups4541462559716669496nt_nat @ F @ F3 )
% 0.35/0.65            = zero_zero_nat )
% 0.35/0.65          = ( ! [X3: int] :
% 0.35/0.65                ( ( member_int @ X3 @ F3 )
% 0.35/0.65               => ( ( F @ X3 )
% 0.35/0.65                  = zero_zero_nat ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum_eq_0_iff
% 0.35/0.65  thf(fact_5481_sum__eq__0__iff,axiom,
% 0.35/0.65      ! [F3: set_complex,F: complex > nat] :
% 0.35/0.65        ( ( finite3207457112153483333omplex @ F3 )
% 0.35/0.65       => ( ( ( groups5693394587270226106ex_nat @ F @ F3 )
% 0.35/0.65            = zero_zero_nat )
% 0.35/0.65          = ( ! [X3: complex] :
% 0.35/0.65                ( ( member_complex @ X3 @ F3 )
% 0.35/0.65               => ( ( F @ X3 )
% 0.35/0.65                  = zero_zero_nat ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum_eq_0_iff
% 0.35/0.65  thf(fact_5482_sum__eq__0__iff,axiom,
% 0.35/0.65      ! [F3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 0.35/0.65        ( ( finite6177210948735845034at_nat @ F3 )
% 0.35/0.65       => ( ( ( groups977919841031483927at_nat @ F @ F3 )
% 0.35/0.65            = zero_zero_nat )
% 0.35/0.65          = ( ! [X3: product_prod_nat_nat] :
% 0.35/0.65                ( ( member8440522571783428010at_nat @ X3 @ F3 )
% 0.35/0.65               => ( ( F @ X3 )
% 0.35/0.65                  = zero_zero_nat ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum_eq_0_iff
% 0.35/0.65  thf(fact_5483_sum__eq__0__iff,axiom,
% 0.35/0.65      ! [F3: set_nat,F: nat > nat] :
% 0.35/0.65        ( ( finite_finite_nat @ F3 )
% 0.35/0.65       => ( ( ( groups3542108847815614940at_nat @ F @ F3 )
% 0.35/0.65            = zero_zero_nat )
% 0.35/0.65          = ( ! [X3: nat] :
% 0.35/0.65                ( ( member_nat @ X3 @ F3 )
% 0.35/0.65               => ( ( F @ X3 )
% 0.35/0.65                  = zero_zero_nat ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum_eq_0_iff
% 0.35/0.65  thf(fact_5484_sum_Oinfinite,axiom,
% 0.35/0.65      ! [A2: set_nat,G: nat > complex] :
% 0.35/0.65        ( ~ ( finite_finite_nat @ A2 )
% 0.35/0.65       => ( ( groups2073611262835488442omplex @ G @ A2 )
% 0.35/0.65          = zero_zero_complex ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.infinite
% 0.35/0.65  thf(fact_5485_sum_Oinfinite,axiom,
% 0.35/0.65      ! [A2: set_int,G: int > complex] :
% 0.35/0.65        ( ~ ( finite_finite_int @ A2 )
% 0.35/0.65       => ( ( groups3049146728041665814omplex @ G @ A2 )
% 0.35/0.65          = zero_zero_complex ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.infinite
% 0.35/0.65  thf(fact_5486_sum_Oinfinite,axiom,
% 0.35/0.65      ! [A2: set_int,G: int > real] :
% 0.35/0.65        ( ~ ( finite_finite_int @ A2 )
% 0.35/0.65       => ( ( groups8778361861064173332t_real @ G @ A2 )
% 0.35/0.65          = zero_zero_real ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.infinite
% 0.35/0.65  thf(fact_5487_sum_Oinfinite,axiom,
% 0.35/0.65      ! [A2: set_complex,G: complex > real] :
% 0.35/0.65        ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.35/0.65       => ( ( groups5808333547571424918x_real @ G @ A2 )
% 0.35/0.65          = zero_zero_real ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.infinite
% 0.35/0.65  thf(fact_5488_sum_Oinfinite,axiom,
% 0.35/0.65      ! [A2: set_nat,G: nat > rat] :
% 0.35/0.65        ( ~ ( finite_finite_nat @ A2 )
% 0.35/0.65       => ( ( groups2906978787729119204at_rat @ G @ A2 )
% 0.35/0.65          = zero_zero_rat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.infinite
% 0.35/0.65  thf(fact_5489_sum_Oinfinite,axiom,
% 0.35/0.65      ! [A2: set_int,G: int > rat] :
% 0.35/0.65        ( ~ ( finite_finite_int @ A2 )
% 0.35/0.65       => ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 0.35/0.65          = zero_zero_rat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.infinite
% 0.35/0.65  thf(fact_5490_sum_Oinfinite,axiom,
% 0.35/0.65      ! [A2: set_complex,G: complex > rat] :
% 0.35/0.65        ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.35/0.65       => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 0.35/0.65          = zero_zero_rat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.infinite
% 0.35/0.65  thf(fact_5491_sum_Oinfinite,axiom,
% 0.35/0.65      ! [A2: set_int,G: int > nat] :
% 0.35/0.65        ( ~ ( finite_finite_int @ A2 )
% 0.35/0.65       => ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 0.35/0.65          = zero_zero_nat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.infinite
% 0.35/0.65  thf(fact_5492_sum_Oinfinite,axiom,
% 0.35/0.65      ! [A2: set_complex,G: complex > nat] :
% 0.35/0.65        ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.35/0.65       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 0.35/0.65          = zero_zero_nat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.infinite
% 0.35/0.65  thf(fact_5493_sum_Oinfinite,axiom,
% 0.35/0.65      ! [A2: set_nat,G: nat > int] :
% 0.35/0.65        ( ~ ( finite_finite_nat @ A2 )
% 0.35/0.65       => ( ( groups3539618377306564664at_int @ G @ A2 )
% 0.35/0.65          = zero_zero_int ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.infinite
% 0.35/0.65  thf(fact_5494_singleton__insert__inj__eq,axiom,
% 0.35/0.65      ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65        ( ( ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat )
% 0.35/0.65          = ( insert9069300056098147895at_nat @ A @ A2 ) )
% 0.35/0.65        = ( ( A = B )
% 0.35/0.65          & ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_insert_inj_eq
% 0.35/0.65  thf(fact_5495_singleton__insert__inj__eq,axiom,
% 0.35/0.65      ! [B: product_prod_nat_nat,A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ( ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat )
% 0.35/0.65          = ( insert8211810215607154385at_nat @ A @ A2 ) )
% 0.35/0.65        = ( ( A = B )
% 0.35/0.65          & ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_insert_inj_eq
% 0.35/0.65  thf(fact_5496_singleton__insert__inj__eq,axiom,
% 0.35/0.65      ! [B: real,A: real,A2: set_real] :
% 0.35/0.65        ( ( ( insert_real @ B @ bot_bot_set_real )
% 0.35/0.65          = ( insert_real @ A @ A2 ) )
% 0.35/0.65        = ( ( A = B )
% 0.35/0.65          & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_insert_inj_eq
% 0.35/0.65  thf(fact_5497_singleton__insert__inj__eq,axiom,
% 0.35/0.65      ! [B: nat,A: nat,A2: set_nat] :
% 0.35/0.65        ( ( ( insert_nat @ B @ bot_bot_set_nat )
% 0.35/0.65          = ( insert_nat @ A @ A2 ) )
% 0.35/0.65        = ( ( A = B )
% 0.35/0.65          & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_insert_inj_eq
% 0.35/0.65  thf(fact_5498_singleton__insert__inj__eq,axiom,
% 0.35/0.65      ! [B: int,A: int,A2: set_int] :
% 0.35/0.65        ( ( ( insert_int @ B @ bot_bot_set_int )
% 0.35/0.65          = ( insert_int @ A @ A2 ) )
% 0.35/0.65        = ( ( A = B )
% 0.35/0.65          & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_insert_inj_eq
% 0.35/0.65  thf(fact_5499_singleton__insert__inj__eq_H,axiom,
% 0.35/0.65      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
% 0.35/0.65        ( ( ( insert9069300056098147895at_nat @ A @ A2 )
% 0.35/0.65          = ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
% 0.35/0.65        = ( ( A = B )
% 0.35/0.65          & ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_insert_inj_eq'
% 0.35/0.65  thf(fact_5500_singleton__insert__inj__eq_H,axiom,
% 0.35/0.65      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
% 0.35/0.65        ( ( ( insert8211810215607154385at_nat @ A @ A2 )
% 0.35/0.65          = ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
% 0.35/0.65        = ( ( A = B )
% 0.35/0.65          & ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_insert_inj_eq'
% 0.35/0.65  thf(fact_5501_singleton__insert__inj__eq_H,axiom,
% 0.35/0.65      ! [A: real,A2: set_real,B: real] :
% 0.35/0.65        ( ( ( insert_real @ A @ A2 )
% 0.35/0.65          = ( insert_real @ B @ bot_bot_set_real ) )
% 0.35/0.65        = ( ( A = B )
% 0.35/0.65          & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_insert_inj_eq'
% 0.35/0.65  thf(fact_5502_singleton__insert__inj__eq_H,axiom,
% 0.35/0.65      ! [A: nat,A2: set_nat,B: nat] :
% 0.35/0.65        ( ( ( insert_nat @ A @ A2 )
% 0.35/0.65          = ( insert_nat @ B @ bot_bot_set_nat ) )
% 0.35/0.65        = ( ( A = B )
% 0.35/0.65          & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_insert_inj_eq'
% 0.35/0.65  thf(fact_5503_singleton__insert__inj__eq_H,axiom,
% 0.35/0.65      ! [A: int,A2: set_int,B: int] :
% 0.35/0.65        ( ( ( insert_int @ A @ A2 )
% 0.35/0.65          = ( insert_int @ B @ bot_bot_set_int ) )
% 0.35/0.65        = ( ( A = B )
% 0.35/0.65          & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_insert_inj_eq'
% 0.35/0.65  thf(fact_5504_atLeastAtMost__singleton__iff,axiom,
% 0.35/0.65      ! [A: nat,B: nat,C: nat] :
% 0.35/0.65        ( ( ( set_or1269000886237332187st_nat @ A @ B )
% 0.35/0.65          = ( insert_nat @ C @ bot_bot_set_nat ) )
% 0.35/0.65        = ( ( A = B )
% 0.35/0.65          & ( B = C ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % atLeastAtMost_singleton_iff
% 0.35/0.65  thf(fact_5505_atLeastAtMost__singleton__iff,axiom,
% 0.35/0.65      ! [A: int,B: int,C: int] :
% 0.35/0.65        ( ( ( set_or1266510415728281911st_int @ A @ B )
% 0.35/0.65          = ( insert_int @ C @ bot_bot_set_int ) )
% 0.35/0.65        = ( ( A = B )
% 0.35/0.65          & ( B = C ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % atLeastAtMost_singleton_iff
% 0.35/0.65  thf(fact_5506_atLeastAtMost__singleton__iff,axiom,
% 0.35/0.65      ! [A: real,B: real,C: real] :
% 0.35/0.65        ( ( ( set_or1222579329274155063t_real @ A @ B )
% 0.35/0.65          = ( insert_real @ C @ bot_bot_set_real ) )
% 0.35/0.65        = ( ( A = B )
% 0.35/0.65          & ( B = C ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % atLeastAtMost_singleton_iff
% 0.35/0.65  thf(fact_5507_atLeastAtMost__singleton,axiom,
% 0.35/0.65      ! [A: nat] :
% 0.35/0.65        ( ( set_or1269000886237332187st_nat @ A @ A )
% 0.35/0.65        = ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % atLeastAtMost_singleton
% 0.35/0.65  thf(fact_5508_atLeastAtMost__singleton,axiom,
% 0.35/0.65      ! [A: int] :
% 0.35/0.65        ( ( set_or1266510415728281911st_int @ A @ A )
% 0.35/0.65        = ( insert_int @ A @ bot_bot_set_int ) ) ).
% 0.35/0.65  
% 0.35/0.65  % atLeastAtMost_singleton
% 0.35/0.65  thf(fact_5509_atLeastAtMost__singleton,axiom,
% 0.35/0.65      ! [A: real] :
% 0.35/0.65        ( ( set_or1222579329274155063t_real @ A @ A )
% 0.35/0.65        = ( insert_real @ A @ bot_bot_set_real ) ) ).
% 0.35/0.65  
% 0.35/0.65  % atLeastAtMost_singleton
% 0.35/0.65  thf(fact_5510_insert__Diff__single,axiom,
% 0.35/0.65      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65        ( ( insert9069300056098147895at_nat @ A @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 0.35/0.65        = ( insert9069300056098147895at_nat @ A @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_Diff_single
% 0.35/0.65  thf(fact_5511_insert__Diff__single,axiom,
% 0.35/0.65      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 0.35/0.65        = ( insert8211810215607154385at_nat @ A @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_Diff_single
% 0.35/0.65  thf(fact_5512_insert__Diff__single,axiom,
% 0.35/0.65      ! [A: real,A2: set_real] :
% 0.35/0.65        ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.35/0.65        = ( insert_real @ A @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_Diff_single
% 0.35/0.65  thf(fact_5513_insert__Diff__single,axiom,
% 0.35/0.65      ! [A: int,A2: set_int] :
% 0.35/0.65        ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.35/0.65        = ( insert_int @ A @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_Diff_single
% 0.35/0.65  thf(fact_5514_insert__Diff__single,axiom,
% 0.35/0.65      ! [A: nat,A2: set_nat] :
% 0.35/0.65        ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 0.35/0.65        = ( insert_nat @ A @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_Diff_single
% 0.35/0.65  thf(fact_5515_finite__Diff__insert,axiom,
% 0.35/0.65      ! [A2: set_real,A: real,B2: set_real] :
% 0.35/0.65        ( ( finite_finite_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) ) )
% 0.35/0.65        = ( finite_finite_real @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % finite_Diff_insert
% 0.35/0.65  thf(fact_5516_finite__Diff__insert,axiom,
% 0.35/0.65      ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65        ( ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) ) )
% 0.35/0.65        = ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % finite_Diff_insert
% 0.35/0.65  thf(fact_5517_finite__Diff__insert,axiom,
% 0.35/0.65      ! [A2: set_int,A: int,B2: set_int] :
% 0.35/0.65        ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) ) )
% 0.35/0.65        = ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % finite_Diff_insert
% 0.35/0.65  thf(fact_5518_finite__Diff__insert,axiom,
% 0.35/0.65      ! [A2: set_complex,A: complex,B2: set_complex] :
% 0.35/0.65        ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ B2 ) ) )
% 0.35/0.65        = ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % finite_Diff_insert
% 0.35/0.65  thf(fact_5519_finite__Diff__insert,axiom,
% 0.35/0.65      ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) ) )
% 0.35/0.65        = ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % finite_Diff_insert
% 0.35/0.65  thf(fact_5520_finite__Diff__insert,axiom,
% 0.35/0.65      ! [A2: set_nat,A: nat,B2: set_nat] :
% 0.35/0.65        ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
% 0.35/0.65        = ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % finite_Diff_insert
% 0.35/0.65  thf(fact_5521_round__0,axiom,
% 0.35/0.65      ( ( archim8280529875227126926d_real @ zero_zero_real )
% 0.35/0.65      = zero_zero_int ) ).
% 0.35/0.65  
% 0.35/0.65  % round_0
% 0.35/0.65  thf(fact_5522_round__0,axiom,
% 0.35/0.65      ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
% 0.35/0.65      = zero_zero_int ) ).
% 0.35/0.65  
% 0.35/0.65  % round_0
% 0.35/0.65  thf(fact_5523_sum_Odelta,axiom,
% 0.35/0.65      ! [S2: set_real,A: real,B: real > complex] :
% 0.35/0.65        ( ( finite_finite_real @ S2 )
% 0.35/0.65       => ( ( ( member_real @ A @ S2 )
% 0.35/0.65           => ( ( groups5754745047067104278omplex
% 0.35/0.65                @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = ( B @ A ) ) )
% 0.35/0.65          & ( ~ ( member_real @ A @ S2 )
% 0.35/0.65           => ( ( groups5754745047067104278omplex
% 0.35/0.65                @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = zero_zero_complex ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.delta
% 0.35/0.65  thf(fact_5524_sum_Odelta,axiom,
% 0.35/0.65      ! [S2: set_nat,A: nat,B: nat > complex] :
% 0.35/0.65        ( ( finite_finite_nat @ S2 )
% 0.35/0.65       => ( ( ( member_nat @ A @ S2 )
% 0.35/0.65           => ( ( groups2073611262835488442omplex
% 0.35/0.65                @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = ( B @ A ) ) )
% 0.35/0.65          & ( ~ ( member_nat @ A @ S2 )
% 0.35/0.65           => ( ( groups2073611262835488442omplex
% 0.35/0.65                @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = zero_zero_complex ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.delta
% 0.35/0.65  thf(fact_5525_sum_Odelta,axiom,
% 0.35/0.65      ! [S2: set_int,A: int,B: int > complex] :
% 0.35/0.65        ( ( finite_finite_int @ S2 )
% 0.35/0.65       => ( ( ( member_int @ A @ S2 )
% 0.35/0.65           => ( ( groups3049146728041665814omplex
% 0.35/0.65                @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = ( B @ A ) ) )
% 0.35/0.65          & ( ~ ( member_int @ A @ S2 )
% 0.35/0.65           => ( ( groups3049146728041665814omplex
% 0.35/0.65                @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = zero_zero_complex ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.delta
% 0.35/0.65  thf(fact_5526_sum_Odelta,axiom,
% 0.35/0.65      ! [S2: set_real,A: real,B: real > real] :
% 0.35/0.65        ( ( finite_finite_real @ S2 )
% 0.35/0.65       => ( ( ( member_real @ A @ S2 )
% 0.35/0.65           => ( ( groups8097168146408367636l_real
% 0.35/0.65                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = ( B @ A ) ) )
% 0.35/0.65          & ( ~ ( member_real @ A @ S2 )
% 0.35/0.65           => ( ( groups8097168146408367636l_real
% 0.35/0.65                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = zero_zero_real ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.delta
% 0.35/0.65  thf(fact_5527_sum_Odelta,axiom,
% 0.35/0.65      ! [S2: set_int,A: int,B: int > real] :
% 0.35/0.65        ( ( finite_finite_int @ S2 )
% 0.35/0.65       => ( ( ( member_int @ A @ S2 )
% 0.35/0.65           => ( ( groups8778361861064173332t_real
% 0.35/0.65                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = ( B @ A ) ) )
% 0.35/0.65          & ( ~ ( member_int @ A @ S2 )
% 0.35/0.65           => ( ( groups8778361861064173332t_real
% 0.35/0.65                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = zero_zero_real ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.delta
% 0.35/0.65  thf(fact_5528_sum_Odelta,axiom,
% 0.35/0.65      ! [S2: set_complex,A: complex,B: complex > real] :
% 0.35/0.65        ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.65       => ( ( ( member_complex @ A @ S2 )
% 0.35/0.65           => ( ( groups5808333547571424918x_real
% 0.35/0.65                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = ( B @ A ) ) )
% 0.35/0.65          & ( ~ ( member_complex @ A @ S2 )
% 0.35/0.65           => ( ( groups5808333547571424918x_real
% 0.35/0.65                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = zero_zero_real ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.delta
% 0.35/0.65  thf(fact_5529_sum_Odelta,axiom,
% 0.35/0.65      ! [S2: set_real,A: real,B: real > rat] :
% 0.35/0.65        ( ( finite_finite_real @ S2 )
% 0.35/0.65       => ( ( ( member_real @ A @ S2 )
% 0.35/0.65           => ( ( groups1300246762558778688al_rat
% 0.35/0.65                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = ( B @ A ) ) )
% 0.35/0.65          & ( ~ ( member_real @ A @ S2 )
% 0.35/0.65           => ( ( groups1300246762558778688al_rat
% 0.35/0.65                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = zero_zero_rat ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.delta
% 0.35/0.65  thf(fact_5530_sum_Odelta,axiom,
% 0.35/0.65      ! [S2: set_nat,A: nat,B: nat > rat] :
% 0.35/0.65        ( ( finite_finite_nat @ S2 )
% 0.35/0.65       => ( ( ( member_nat @ A @ S2 )
% 0.35/0.65           => ( ( groups2906978787729119204at_rat
% 0.35/0.65                @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = ( B @ A ) ) )
% 0.35/0.65          & ( ~ ( member_nat @ A @ S2 )
% 0.35/0.65           => ( ( groups2906978787729119204at_rat
% 0.35/0.65                @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = zero_zero_rat ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.delta
% 0.35/0.65  thf(fact_5531_sum_Odelta,axiom,
% 0.35/0.65      ! [S2: set_int,A: int,B: int > rat] :
% 0.35/0.65        ( ( finite_finite_int @ S2 )
% 0.35/0.65       => ( ( ( member_int @ A @ S2 )
% 0.35/0.65           => ( ( groups3906332499630173760nt_rat
% 0.35/0.65                @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = ( B @ A ) ) )
% 0.35/0.65          & ( ~ ( member_int @ A @ S2 )
% 0.35/0.65           => ( ( groups3906332499630173760nt_rat
% 0.35/0.65                @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = zero_zero_rat ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.delta
% 0.35/0.65  thf(fact_5532_sum_Odelta,axiom,
% 0.35/0.65      ! [S2: set_complex,A: complex,B: complex > rat] :
% 0.35/0.65        ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.65       => ( ( ( member_complex @ A @ S2 )
% 0.35/0.65           => ( ( groups5058264527183730370ex_rat
% 0.35/0.65                @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = ( B @ A ) ) )
% 0.35/0.65          & ( ~ ( member_complex @ A @ S2 )
% 0.35/0.65           => ( ( groups5058264527183730370ex_rat
% 0.35/0.65                @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = zero_zero_rat ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.delta
% 0.35/0.65  thf(fact_5533_sum_Odelta_H,axiom,
% 0.35/0.65      ! [S2: set_real,A: real,B: real > complex] :
% 0.35/0.65        ( ( finite_finite_real @ S2 )
% 0.35/0.65       => ( ( ( member_real @ A @ S2 )
% 0.35/0.65           => ( ( groups5754745047067104278omplex
% 0.35/0.65                @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = ( B @ A ) ) )
% 0.35/0.65          & ( ~ ( member_real @ A @ S2 )
% 0.35/0.65           => ( ( groups5754745047067104278omplex
% 0.35/0.65                @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = zero_zero_complex ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.delta'
% 0.35/0.65  thf(fact_5534_sum_Odelta_H,axiom,
% 0.35/0.65      ! [S2: set_nat,A: nat,B: nat > complex] :
% 0.35/0.65        ( ( finite_finite_nat @ S2 )
% 0.35/0.65       => ( ( ( member_nat @ A @ S2 )
% 0.35/0.65           => ( ( groups2073611262835488442omplex
% 0.35/0.65                @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = ( B @ A ) ) )
% 0.35/0.65          & ( ~ ( member_nat @ A @ S2 )
% 0.35/0.65           => ( ( groups2073611262835488442omplex
% 0.35/0.65                @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = zero_zero_complex ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.delta'
% 0.35/0.65  thf(fact_5535_sum_Odelta_H,axiom,
% 0.35/0.65      ! [S2: set_int,A: int,B: int > complex] :
% 0.35/0.65        ( ( finite_finite_int @ S2 )
% 0.35/0.65       => ( ( ( member_int @ A @ S2 )
% 0.35/0.65           => ( ( groups3049146728041665814omplex
% 0.35/0.65                @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = ( B @ A ) ) )
% 0.35/0.65          & ( ~ ( member_int @ A @ S2 )
% 0.35/0.65           => ( ( groups3049146728041665814omplex
% 0.35/0.65                @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = zero_zero_complex ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.delta'
% 0.35/0.65  thf(fact_5536_sum_Odelta_H,axiom,
% 0.35/0.65      ! [S2: set_real,A: real,B: real > real] :
% 0.35/0.65        ( ( finite_finite_real @ S2 )
% 0.35/0.65       => ( ( ( member_real @ A @ S2 )
% 0.35/0.65           => ( ( groups8097168146408367636l_real
% 0.35/0.65                @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = ( B @ A ) ) )
% 0.35/0.65          & ( ~ ( member_real @ A @ S2 )
% 0.35/0.65           => ( ( groups8097168146408367636l_real
% 0.35/0.65                @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = zero_zero_real ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.delta'
% 0.35/0.65  thf(fact_5537_sum_Odelta_H,axiom,
% 0.35/0.65      ! [S2: set_int,A: int,B: int > real] :
% 0.35/0.65        ( ( finite_finite_int @ S2 )
% 0.35/0.65       => ( ( ( member_int @ A @ S2 )
% 0.35/0.65           => ( ( groups8778361861064173332t_real
% 0.35/0.65                @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = ( B @ A ) ) )
% 0.35/0.65          & ( ~ ( member_int @ A @ S2 )
% 0.35/0.65           => ( ( groups8778361861064173332t_real
% 0.35/0.65                @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = zero_zero_real ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.delta'
% 0.35/0.65  thf(fact_5538_sum_Odelta_H,axiom,
% 0.35/0.65      ! [S2: set_complex,A: complex,B: complex > real] :
% 0.35/0.65        ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.65       => ( ( ( member_complex @ A @ S2 )
% 0.35/0.65           => ( ( groups5808333547571424918x_real
% 0.35/0.65                @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = ( B @ A ) ) )
% 0.35/0.65          & ( ~ ( member_complex @ A @ S2 )
% 0.35/0.65           => ( ( groups5808333547571424918x_real
% 0.35/0.65                @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = zero_zero_real ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.delta'
% 0.35/0.65  thf(fact_5539_sum_Odelta_H,axiom,
% 0.35/0.65      ! [S2: set_real,A: real,B: real > rat] :
% 0.35/0.65        ( ( finite_finite_real @ S2 )
% 0.35/0.65       => ( ( ( member_real @ A @ S2 )
% 0.35/0.65           => ( ( groups1300246762558778688al_rat
% 0.35/0.65                @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = ( B @ A ) ) )
% 0.35/0.65          & ( ~ ( member_real @ A @ S2 )
% 0.35/0.65           => ( ( groups1300246762558778688al_rat
% 0.35/0.65                @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = zero_zero_rat ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.delta'
% 0.35/0.65  thf(fact_5540_sum_Odelta_H,axiom,
% 0.35/0.65      ! [S2: set_nat,A: nat,B: nat > rat] :
% 0.35/0.65        ( ( finite_finite_nat @ S2 )
% 0.35/0.65       => ( ( ( member_nat @ A @ S2 )
% 0.35/0.65           => ( ( groups2906978787729119204at_rat
% 0.35/0.65                @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = ( B @ A ) ) )
% 0.35/0.65          & ( ~ ( member_nat @ A @ S2 )
% 0.35/0.65           => ( ( groups2906978787729119204at_rat
% 0.35/0.65                @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = zero_zero_rat ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.delta'
% 0.35/0.65  thf(fact_5541_sum_Odelta_H,axiom,
% 0.35/0.65      ! [S2: set_int,A: int,B: int > rat] :
% 0.35/0.65        ( ( finite_finite_int @ S2 )
% 0.35/0.65       => ( ( ( member_int @ A @ S2 )
% 0.35/0.65           => ( ( groups3906332499630173760nt_rat
% 0.35/0.65                @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = ( B @ A ) ) )
% 0.35/0.65          & ( ~ ( member_int @ A @ S2 )
% 0.35/0.65           => ( ( groups3906332499630173760nt_rat
% 0.35/0.65                @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = zero_zero_rat ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.delta'
% 0.35/0.65  thf(fact_5542_sum_Odelta_H,axiom,
% 0.35/0.65      ! [S2: set_complex,A: complex,B: complex > rat] :
% 0.35/0.65        ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.65       => ( ( ( member_complex @ A @ S2 )
% 0.35/0.65           => ( ( groups5058264527183730370ex_rat
% 0.35/0.65                @ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = ( B @ A ) ) )
% 0.35/0.65          & ( ~ ( member_complex @ A @ S2 )
% 0.35/0.65           => ( ( groups5058264527183730370ex_rat
% 0.35/0.65                @ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.35/0.65                @ S2 )
% 0.35/0.65              = zero_zero_rat ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.delta'
% 0.35/0.65  thf(fact_5543_sum_Oinsert,axiom,
% 0.35/0.65      ! [A2: set_real,X: real,G: real > real] :
% 0.35/0.65        ( ( finite_finite_real @ A2 )
% 0.35/0.65       => ( ~ ( member_real @ X @ A2 )
% 0.35/0.65         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.65            = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.insert
% 0.35/0.65  thf(fact_5544_sum_Oinsert,axiom,
% 0.35/0.65      ! [A2: set_int,X: int,G: int > real] :
% 0.35/0.65        ( ( finite_finite_int @ A2 )
% 0.35/0.65       => ( ~ ( member_int @ X @ A2 )
% 0.35/0.65         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
% 0.35/0.65            = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.insert
% 0.35/0.65  thf(fact_5545_sum_Oinsert,axiom,
% 0.35/0.65      ! [A2: set_complex,X: complex,G: complex > real] :
% 0.35/0.65        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.65       => ( ~ ( member_complex @ X @ A2 )
% 0.35/0.65         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
% 0.35/0.65            = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.insert
% 0.35/0.65  thf(fact_5546_sum_Oinsert,axiom,
% 0.35/0.65      ! [A2: set_real,X: real,G: real > rat] :
% 0.35/0.65        ( ( finite_finite_real @ A2 )
% 0.35/0.65       => ( ~ ( member_real @ X @ A2 )
% 0.35/0.65         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.65            = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.insert
% 0.35/0.65  thf(fact_5547_sum_Oinsert,axiom,
% 0.35/0.65      ! [A2: set_nat,X: nat,G: nat > rat] :
% 0.35/0.65        ( ( finite_finite_nat @ A2 )
% 0.35/0.65       => ( ~ ( member_nat @ X @ A2 )
% 0.35/0.65         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
% 0.35/0.65            = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.insert
% 0.35/0.65  thf(fact_5548_sum_Oinsert,axiom,
% 0.35/0.65      ! [A2: set_int,X: int,G: int > rat] :
% 0.35/0.65        ( ( finite_finite_int @ A2 )
% 0.35/0.65       => ( ~ ( member_int @ X @ A2 )
% 0.35/0.65         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
% 0.35/0.65            = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.insert
% 0.35/0.65  thf(fact_5549_sum_Oinsert,axiom,
% 0.35/0.65      ! [A2: set_complex,X: complex,G: complex > rat] :
% 0.35/0.65        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.65       => ( ~ ( member_complex @ X @ A2 )
% 0.35/0.65         => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
% 0.35/0.65            = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.insert
% 0.35/0.65  thf(fact_5550_sum_Oinsert,axiom,
% 0.35/0.65      ! [A2: set_real,X: real,G: real > nat] :
% 0.35/0.65        ( ( finite_finite_real @ A2 )
% 0.35/0.65       => ( ~ ( member_real @ X @ A2 )
% 0.35/0.65         => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.65            = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.insert
% 0.35/0.65  thf(fact_5551_sum_Oinsert,axiom,
% 0.35/0.65      ! [A2: set_int,X: int,G: int > nat] :
% 0.35/0.65        ( ( finite_finite_int @ A2 )
% 0.35/0.65       => ( ~ ( member_int @ X @ A2 )
% 0.35/0.65         => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A2 ) )
% 0.35/0.65            = ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.insert
% 0.35/0.65  thf(fact_5552_sum_Oinsert,axiom,
% 0.35/0.65      ! [A2: set_complex,X: complex,G: complex > nat] :
% 0.35/0.65        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.65       => ( ~ ( member_complex @ X @ A2 )
% 0.35/0.65         => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
% 0.35/0.65            = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.insert
% 0.35/0.65  thf(fact_5553_subset__Compl__singleton,axiom,
% 0.35/0.65      ! [A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
% 0.35/0.65        ( ( ord_le1268244103169919719at_nat @ A2 @ ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) )
% 0.35/0.65        = ( ~ ( member8757157785044589968at_nat @ B @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % subset_Compl_singleton
% 0.35/0.65  thf(fact_5554_subset__Compl__singleton,axiom,
% 0.35/0.65      ! [A2: set_set_nat,B: set_nat] :
% 0.35/0.65        ( ( ord_le6893508408891458716et_nat @ A2 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
% 0.35/0.65        = ( ~ ( member_set_nat @ B @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % subset_Compl_singleton
% 0.35/0.65  thf(fact_5555_subset__Compl__singleton,axiom,
% 0.35/0.65      ! [A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
% 0.35/0.65        ( ( ord_le3146513528884898305at_nat @ A2 @ ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) )
% 0.35/0.65        = ( ~ ( member8440522571783428010at_nat @ B @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % subset_Compl_singleton
% 0.35/0.65  thf(fact_5556_subset__Compl__singleton,axiom,
% 0.35/0.65      ! [A2: set_real,B: real] :
% 0.35/0.65        ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ ( insert_real @ B @ bot_bot_set_real ) ) )
% 0.35/0.65        = ( ~ ( member_real @ B @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % subset_Compl_singleton
% 0.35/0.65  thf(fact_5557_subset__Compl__singleton,axiom,
% 0.35/0.65      ! [A2: set_nat,B: nat] :
% 0.35/0.65        ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 0.35/0.65        = ( ~ ( member_nat @ B @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % subset_Compl_singleton
% 0.35/0.65  thf(fact_5558_subset__Compl__singleton,axiom,
% 0.35/0.65      ! [A2: set_int,B: int] :
% 0.35/0.65        ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
% 0.35/0.65        = ( ~ ( member_int @ B @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % subset_Compl_singleton
% 0.35/0.65  thf(fact_5559_set__replicate,axiom,
% 0.35/0.65      ! [N: nat,X: produc3843707927480180839at_nat] :
% 0.35/0.65        ( ( N != zero_zero_nat )
% 0.35/0.65       => ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N @ X ) )
% 0.35/0.65          = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % set_replicate
% 0.35/0.65  thf(fact_5560_set__replicate,axiom,
% 0.35/0.65      ! [N: nat,X: vEBT_VEBT] :
% 0.35/0.65        ( ( N != zero_zero_nat )
% 0.35/0.65       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
% 0.35/0.65          = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % set_replicate
% 0.35/0.65  thf(fact_5561_set__replicate,axiom,
% 0.35/0.65      ! [N: nat,X: product_prod_nat_nat] :
% 0.35/0.65        ( ( N != zero_zero_nat )
% 0.35/0.65       => ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ X ) )
% 0.35/0.65          = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % set_replicate
% 0.35/0.65  thf(fact_5562_set__replicate,axiom,
% 0.35/0.65      ! [N: nat,X: real] :
% 0.35/0.65        ( ( N != zero_zero_nat )
% 0.35/0.65       => ( ( set_real2 @ ( replicate_real @ N @ X ) )
% 0.35/0.65          = ( insert_real @ X @ bot_bot_set_real ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % set_replicate
% 0.35/0.65  thf(fact_5563_set__replicate,axiom,
% 0.35/0.65      ! [N: nat,X: nat] :
% 0.35/0.65        ( ( N != zero_zero_nat )
% 0.35/0.65       => ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
% 0.35/0.65          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % set_replicate
% 0.35/0.65  thf(fact_5564_set__replicate,axiom,
% 0.35/0.65      ! [N: nat,X: int] :
% 0.35/0.65        ( ( N != zero_zero_nat )
% 0.35/0.65       => ( ( set_int2 @ ( replicate_int @ N @ X ) )
% 0.35/0.65          = ( insert_int @ X @ bot_bot_set_int ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % set_replicate
% 0.35/0.65  thf(fact_5565_sum_Ocl__ivl__Suc,axiom,
% 0.35/0.65      ! [N: nat,M: nat,G: nat > complex] :
% 0.35/0.65        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.65         => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.65            = zero_zero_complex ) )
% 0.35/0.65        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.65         => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.65            = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.cl_ivl_Suc
% 0.35/0.65  thf(fact_5566_sum_Ocl__ivl__Suc,axiom,
% 0.35/0.65      ! [N: nat,M: nat,G: nat > rat] :
% 0.35/0.65        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.65         => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.65            = zero_zero_rat ) )
% 0.35/0.65        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.65         => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.65            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.cl_ivl_Suc
% 0.35/0.65  thf(fact_5567_sum_Ocl__ivl__Suc,axiom,
% 0.35/0.65      ! [N: nat,M: nat,G: nat > int] :
% 0.35/0.65        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.65         => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.65            = zero_zero_int ) )
% 0.35/0.65        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.65         => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.65            = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.cl_ivl_Suc
% 0.35/0.65  thf(fact_5568_sum_Ocl__ivl__Suc,axiom,
% 0.35/0.65      ! [N: nat,M: nat,G: nat > nat] :
% 0.35/0.65        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.65         => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.65            = zero_zero_nat ) )
% 0.35/0.65        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.65         => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.65            = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.cl_ivl_Suc
% 0.35/0.65  thf(fact_5569_sum_Ocl__ivl__Suc,axiom,
% 0.35/0.65      ! [N: nat,M: nat,G: nat > real] :
% 0.35/0.65        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.65         => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.65            = zero_zero_real ) )
% 0.35/0.65        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.65         => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.65            = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.cl_ivl_Suc
% 0.35/0.65  thf(fact_5570_xor__nat__numerals_I1_J,axiom,
% 0.35/0.65      ! [Y: num] :
% 0.35/0.65        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.35/0.65        = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % xor_nat_numerals(1)
% 0.35/0.65  thf(fact_5571_xor__nat__numerals_I2_J,axiom,
% 0.35/0.65      ! [Y: num] :
% 0.35/0.65        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.35/0.65        = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % xor_nat_numerals(2)
% 0.35/0.65  thf(fact_5572_xor__nat__numerals_I3_J,axiom,
% 0.35/0.65      ! [X: num] :
% 0.35/0.65        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.35/0.65        = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % xor_nat_numerals(3)
% 0.35/0.65  thf(fact_5573_xor__nat__numerals_I4_J,axiom,
% 0.35/0.65      ! [X: num] :
% 0.35/0.65        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.35/0.65        = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % xor_nat_numerals(4)
% 0.35/0.65  thf(fact_5574_sum__zero__power,axiom,
% 0.35/0.65      ! [A2: set_nat,C: nat > complex] :
% 0.35/0.65        ( ( ( ( finite_finite_nat @ A2 )
% 0.35/0.65            & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.35/0.65         => ( ( groups2073611262835488442omplex
% 0.35/0.65              @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
% 0.35/0.65              @ A2 )
% 0.35/0.65            = ( C @ zero_zero_nat ) ) )
% 0.35/0.65        & ( ~ ( ( finite_finite_nat @ A2 )
% 0.35/0.65              & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.35/0.65         => ( ( groups2073611262835488442omplex
% 0.35/0.65              @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
% 0.35/0.65              @ A2 )
% 0.35/0.65            = zero_zero_complex ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum_zero_power
% 0.35/0.65  thf(fact_5575_sum__zero__power,axiom,
% 0.35/0.65      ! [A2: set_nat,C: nat > rat] :
% 0.35/0.65        ( ( ( ( finite_finite_nat @ A2 )
% 0.35/0.65            & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.35/0.65         => ( ( groups2906978787729119204at_rat
% 0.35/0.65              @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
% 0.35/0.65              @ A2 )
% 0.35/0.65            = ( C @ zero_zero_nat ) ) )
% 0.35/0.65        & ( ~ ( ( finite_finite_nat @ A2 )
% 0.35/0.65              & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.35/0.65         => ( ( groups2906978787729119204at_rat
% 0.35/0.65              @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
% 0.35/0.65              @ A2 )
% 0.35/0.65            = zero_zero_rat ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum_zero_power
% 0.35/0.65  thf(fact_5576_sum__zero__power,axiom,
% 0.35/0.65      ! [A2: set_nat,C: nat > real] :
% 0.35/0.65        ( ( ( ( finite_finite_nat @ A2 )
% 0.35/0.65            & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.35/0.65         => ( ( groups6591440286371151544t_real
% 0.35/0.65              @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
% 0.35/0.65              @ A2 )
% 0.35/0.65            = ( C @ zero_zero_nat ) ) )
% 0.35/0.65        & ( ~ ( ( finite_finite_nat @ A2 )
% 0.35/0.65              & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.35/0.65         => ( ( groups6591440286371151544t_real
% 0.35/0.65              @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
% 0.35/0.65              @ A2 )
% 0.35/0.65            = zero_zero_real ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum_zero_power
% 0.35/0.65  thf(fact_5577_sum__zero__power_H,axiom,
% 0.35/0.65      ! [A2: set_nat,C: nat > complex,D: nat > complex] :
% 0.35/0.65        ( ( ( ( finite_finite_nat @ A2 )
% 0.35/0.65            & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.35/0.65         => ( ( groups2073611262835488442omplex
% 0.35/0.65              @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
% 0.35/0.65              @ A2 )
% 0.35/0.65            = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.35/0.65        & ( ~ ( ( finite_finite_nat @ A2 )
% 0.35/0.65              & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.35/0.65         => ( ( groups2073611262835488442omplex
% 0.35/0.65              @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
% 0.35/0.65              @ A2 )
% 0.35/0.65            = zero_zero_complex ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum_zero_power'
% 0.35/0.65  thf(fact_5578_sum__zero__power_H,axiom,
% 0.35/0.65      ! [A2: set_nat,C: nat > rat,D: nat > rat] :
% 0.35/0.65        ( ( ( ( finite_finite_nat @ A2 )
% 0.35/0.65            & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.35/0.65         => ( ( groups2906978787729119204at_rat
% 0.35/0.65              @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
% 0.35/0.65              @ A2 )
% 0.35/0.65            = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.35/0.65        & ( ~ ( ( finite_finite_nat @ A2 )
% 0.35/0.65              & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.35/0.65         => ( ( groups2906978787729119204at_rat
% 0.35/0.65              @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
% 0.35/0.65              @ A2 )
% 0.35/0.65            = zero_zero_rat ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum_zero_power'
% 0.35/0.65  thf(fact_5579_sum__zero__power_H,axiom,
% 0.35/0.65      ! [A2: set_nat,C: nat > real,D: nat > real] :
% 0.35/0.65        ( ( ( ( finite_finite_nat @ A2 )
% 0.35/0.65            & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.35/0.65         => ( ( groups6591440286371151544t_real
% 0.35/0.65              @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
% 0.35/0.65              @ A2 )
% 0.35/0.65            = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.35/0.65        & ( ~ ( ( finite_finite_nat @ A2 )
% 0.35/0.65              & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.35/0.65         => ( ( groups6591440286371151544t_real
% 0.35/0.65              @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
% 0.35/0.65              @ A2 )
% 0.35/0.65            = zero_zero_real ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum_zero_power'
% 0.35/0.65  thf(fact_5580_uminus__set__def,axiom,
% 0.35/0.65      ( uminus6524753893492686040at_nat
% 0.35/0.65      = ( ^ [A6: set_Pr1261947904930325089at_nat] :
% 0.35/0.65            ( collec3392354462482085612at_nat
% 0.35/0.65            @ ( uminus8676089048583255045_nat_o
% 0.35/0.65              @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A6 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % uminus_set_def
% 0.35/0.65  thf(fact_5581_uminus__set__def,axiom,
% 0.35/0.65      ( uminus612125837232591019t_real
% 0.35/0.65      = ( ^ [A6: set_real] :
% 0.35/0.65            ( collect_real
% 0.35/0.65            @ ( uminus_uminus_real_o
% 0.35/0.65              @ ^ [X3: real] : ( member_real @ X3 @ A6 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % uminus_set_def
% 0.35/0.65  thf(fact_5582_uminus__set__def,axiom,
% 0.35/0.65      ( uminus3195874150345416415st_nat
% 0.35/0.65      = ( ^ [A6: set_list_nat] :
% 0.35/0.65            ( collect_list_nat
% 0.35/0.65            @ ( uminus5770388063884162150_nat_o
% 0.35/0.65              @ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A6 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % uminus_set_def
% 0.35/0.65  thf(fact_5583_uminus__set__def,axiom,
% 0.35/0.65      ( uminus613421341184616069et_nat
% 0.35/0.65      = ( ^ [A6: set_set_nat] :
% 0.35/0.65            ( collect_set_nat
% 0.35/0.65            @ ( uminus6401447641752708672_nat_o
% 0.35/0.65              @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % uminus_set_def
% 0.35/0.65  thf(fact_5584_uminus__set__def,axiom,
% 0.35/0.65      ( uminus5710092332889474511et_nat
% 0.35/0.65      = ( ^ [A6: set_nat] :
% 0.35/0.65            ( collect_nat
% 0.35/0.65            @ ( uminus_uminus_nat_o
% 0.35/0.65              @ ^ [X3: nat] : ( member_nat @ X3 @ A6 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % uminus_set_def
% 0.35/0.65  thf(fact_5585_uminus__set__def,axiom,
% 0.35/0.65      ( uminus1532241313380277803et_int
% 0.35/0.65      = ( ^ [A6: set_int] :
% 0.35/0.65            ( collect_int
% 0.35/0.65            @ ( uminus_uminus_int_o
% 0.35/0.65              @ ^ [X3: int] : ( member_int @ X3 @ A6 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % uminus_set_def
% 0.35/0.65  thf(fact_5586_Collect__neg__eq,axiom,
% 0.35/0.65      ! [P: real > $o] :
% 0.35/0.65        ( ( collect_real
% 0.35/0.65          @ ^ [X3: real] :
% 0.35/0.65              ~ ( P @ X3 ) )
% 0.35/0.65        = ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Collect_neg_eq
% 0.35/0.65  thf(fact_5587_Collect__neg__eq,axiom,
% 0.35/0.65      ! [P: list_nat > $o] :
% 0.35/0.65        ( ( collect_list_nat
% 0.35/0.65          @ ^ [X3: list_nat] :
% 0.35/0.65              ~ ( P @ X3 ) )
% 0.35/0.65        = ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Collect_neg_eq
% 0.35/0.65  thf(fact_5588_Collect__neg__eq,axiom,
% 0.35/0.65      ! [P: set_nat > $o] :
% 0.35/0.65        ( ( collect_set_nat
% 0.35/0.65          @ ^ [X3: set_nat] :
% 0.35/0.65              ~ ( P @ X3 ) )
% 0.35/0.65        = ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Collect_neg_eq
% 0.35/0.65  thf(fact_5589_Collect__neg__eq,axiom,
% 0.35/0.65      ! [P: nat > $o] :
% 0.35/0.65        ( ( collect_nat
% 0.35/0.65          @ ^ [X3: nat] :
% 0.35/0.65              ~ ( P @ X3 ) )
% 0.35/0.65        = ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Collect_neg_eq
% 0.35/0.65  thf(fact_5590_Collect__neg__eq,axiom,
% 0.35/0.65      ! [P: int > $o] :
% 0.35/0.65        ( ( collect_int
% 0.35/0.65          @ ^ [X3: int] :
% 0.35/0.65              ~ ( P @ X3 ) )
% 0.35/0.65        = ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Collect_neg_eq
% 0.35/0.65  thf(fact_5591_Compl__eq,axiom,
% 0.35/0.65      ( uminus6524753893492686040at_nat
% 0.35/0.65      = ( ^ [A6: set_Pr1261947904930325089at_nat] :
% 0.35/0.65            ( collec3392354462482085612at_nat
% 0.35/0.65            @ ^ [X3: product_prod_nat_nat] :
% 0.35/0.65                ~ ( member8440522571783428010at_nat @ X3 @ A6 ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Compl_eq
% 0.35/0.65  thf(fact_5592_Compl__eq,axiom,
% 0.35/0.65      ( uminus612125837232591019t_real
% 0.35/0.65      = ( ^ [A6: set_real] :
% 0.35/0.65            ( collect_real
% 0.35/0.65            @ ^ [X3: real] :
% 0.35/0.65                ~ ( member_real @ X3 @ A6 ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Compl_eq
% 0.35/0.65  thf(fact_5593_Compl__eq,axiom,
% 0.35/0.65      ( uminus3195874150345416415st_nat
% 0.35/0.65      = ( ^ [A6: set_list_nat] :
% 0.35/0.65            ( collect_list_nat
% 0.35/0.65            @ ^ [X3: list_nat] :
% 0.35/0.65                ~ ( member_list_nat @ X3 @ A6 ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Compl_eq
% 0.35/0.65  thf(fact_5594_Compl__eq,axiom,
% 0.35/0.65      ( uminus613421341184616069et_nat
% 0.35/0.65      = ( ^ [A6: set_set_nat] :
% 0.35/0.65            ( collect_set_nat
% 0.35/0.65            @ ^ [X3: set_nat] :
% 0.35/0.65                ~ ( member_set_nat @ X3 @ A6 ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Compl_eq
% 0.35/0.65  thf(fact_5595_Compl__eq,axiom,
% 0.35/0.65      ( uminus5710092332889474511et_nat
% 0.35/0.65      = ( ^ [A6: set_nat] :
% 0.35/0.65            ( collect_nat
% 0.35/0.65            @ ^ [X3: nat] :
% 0.35/0.65                ~ ( member_nat @ X3 @ A6 ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Compl_eq
% 0.35/0.65  thf(fact_5596_Compl__eq,axiom,
% 0.35/0.65      ( uminus1532241313380277803et_int
% 0.35/0.65      = ( ^ [A6: set_int] :
% 0.35/0.65            ( collect_int
% 0.35/0.65            @ ^ [X3: int] :
% 0.35/0.65                ~ ( member_int @ X3 @ A6 ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Compl_eq
% 0.35/0.65  thf(fact_5597_ComplD,axiom,
% 0.35/0.65      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A2 ) )
% 0.35/0.65       => ~ ( member8440522571783428010at_nat @ C @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % ComplD
% 0.35/0.65  thf(fact_5598_ComplD,axiom,
% 0.35/0.65      ! [C: real,A2: set_real] :
% 0.35/0.65        ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
% 0.35/0.65       => ~ ( member_real @ C @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % ComplD
% 0.35/0.65  thf(fact_5599_ComplD,axiom,
% 0.35/0.65      ! [C: set_nat,A2: set_set_nat] :
% 0.35/0.65        ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
% 0.35/0.65       => ~ ( member_set_nat @ C @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % ComplD
% 0.35/0.65  thf(fact_5600_ComplD,axiom,
% 0.35/0.65      ! [C: nat,A2: set_nat] :
% 0.35/0.65        ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
% 0.35/0.65       => ~ ( member_nat @ C @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % ComplD
% 0.35/0.65  thf(fact_5601_ComplD,axiom,
% 0.35/0.65      ! [C: int,A2: set_int] :
% 0.35/0.65        ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
% 0.35/0.65       => ~ ( member_int @ C @ A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % ComplD
% 0.35/0.65  thf(fact_5602_insert__Collect,axiom,
% 0.35/0.65      ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
% 0.35/0.65        ( ( insert8211810215607154385at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
% 0.35/0.65        = ( collec3392354462482085612at_nat
% 0.35/0.65          @ ^ [U2: product_prod_nat_nat] :
% 0.35/0.65              ( ( U2 != A )
% 0.35/0.65             => ( P @ U2 ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_Collect
% 0.35/0.65  thf(fact_5603_insert__Collect,axiom,
% 0.35/0.65      ! [A: produc3843707927480180839at_nat,P: produc3843707927480180839at_nat > $o] :
% 0.35/0.65        ( ( insert9069300056098147895at_nat @ A @ ( collec6321179662152712658at_nat @ P ) )
% 0.35/0.65        = ( collec6321179662152712658at_nat
% 0.35/0.65          @ ^ [U2: produc3843707927480180839at_nat] :
% 0.35/0.65              ( ( U2 != A )
% 0.35/0.65             => ( P @ U2 ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_Collect
% 0.35/0.65  thf(fact_5604_insert__Collect,axiom,
% 0.35/0.65      ! [A: real,P: real > $o] :
% 0.35/0.65        ( ( insert_real @ A @ ( collect_real @ P ) )
% 0.35/0.65        = ( collect_real
% 0.35/0.65          @ ^ [U2: real] :
% 0.35/0.65              ( ( U2 != A )
% 0.35/0.65             => ( P @ U2 ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_Collect
% 0.35/0.65  thf(fact_5605_insert__Collect,axiom,
% 0.35/0.65      ! [A: list_nat,P: list_nat > $o] :
% 0.35/0.65        ( ( insert_list_nat @ A @ ( collect_list_nat @ P ) )
% 0.35/0.65        = ( collect_list_nat
% 0.35/0.65          @ ^ [U2: list_nat] :
% 0.35/0.65              ( ( U2 != A )
% 0.35/0.65             => ( P @ U2 ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_Collect
% 0.35/0.65  thf(fact_5606_insert__Collect,axiom,
% 0.35/0.65      ! [A: set_nat,P: set_nat > $o] :
% 0.35/0.65        ( ( insert_set_nat @ A @ ( collect_set_nat @ P ) )
% 0.35/0.65        = ( collect_set_nat
% 0.35/0.65          @ ^ [U2: set_nat] :
% 0.35/0.65              ( ( U2 != A )
% 0.35/0.65             => ( P @ U2 ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_Collect
% 0.35/0.65  thf(fact_5607_insert__Collect,axiom,
% 0.35/0.65      ! [A: nat,P: nat > $o] :
% 0.35/0.65        ( ( insert_nat @ A @ ( collect_nat @ P ) )
% 0.35/0.65        = ( collect_nat
% 0.35/0.65          @ ^ [U2: nat] :
% 0.35/0.65              ( ( U2 != A )
% 0.35/0.65             => ( P @ U2 ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_Collect
% 0.35/0.65  thf(fact_5608_insert__Collect,axiom,
% 0.35/0.65      ! [A: int,P: int > $o] :
% 0.35/0.65        ( ( insert_int @ A @ ( collect_int @ P ) )
% 0.35/0.65        = ( collect_int
% 0.35/0.65          @ ^ [U2: int] :
% 0.35/0.65              ( ( U2 != A )
% 0.35/0.65             => ( P @ U2 ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_Collect
% 0.35/0.65  thf(fact_5609_insert__compr,axiom,
% 0.35/0.65      ( insert9069300056098147895at_nat
% 0.35/0.65      = ( ^ [A4: produc3843707927480180839at_nat,B6: set_Pr4329608150637261639at_nat] :
% 0.35/0.65            ( collec6321179662152712658at_nat
% 0.35/0.65            @ ^ [X3: produc3843707927480180839at_nat] :
% 0.35/0.65                ( ( X3 = A4 )
% 0.35/0.65                | ( member8757157785044589968at_nat @ X3 @ B6 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_compr
% 0.35/0.65  thf(fact_5610_insert__compr,axiom,
% 0.35/0.65      ( insert8211810215607154385at_nat
% 0.35/0.65      = ( ^ [A4: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
% 0.35/0.65            ( collec3392354462482085612at_nat
% 0.35/0.65            @ ^ [X3: product_prod_nat_nat] :
% 0.35/0.65                ( ( X3 = A4 )
% 0.35/0.65                | ( member8440522571783428010at_nat @ X3 @ B6 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_compr
% 0.35/0.65  thf(fact_5611_insert__compr,axiom,
% 0.35/0.65      ( insert_real
% 0.35/0.65      = ( ^ [A4: real,B6: set_real] :
% 0.35/0.65            ( collect_real
% 0.35/0.65            @ ^ [X3: real] :
% 0.35/0.65                ( ( X3 = A4 )
% 0.35/0.65                | ( member_real @ X3 @ B6 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_compr
% 0.35/0.65  thf(fact_5612_insert__compr,axiom,
% 0.35/0.65      ( insert_list_nat
% 0.35/0.65      = ( ^ [A4: list_nat,B6: set_list_nat] :
% 0.35/0.65            ( collect_list_nat
% 0.35/0.65            @ ^ [X3: list_nat] :
% 0.35/0.65                ( ( X3 = A4 )
% 0.35/0.65                | ( member_list_nat @ X3 @ B6 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_compr
% 0.35/0.65  thf(fact_5613_insert__compr,axiom,
% 0.35/0.65      ( insert_set_nat
% 0.35/0.65      = ( ^ [A4: set_nat,B6: set_set_nat] :
% 0.35/0.65            ( collect_set_nat
% 0.35/0.65            @ ^ [X3: set_nat] :
% 0.35/0.65                ( ( X3 = A4 )
% 0.35/0.65                | ( member_set_nat @ X3 @ B6 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_compr
% 0.35/0.65  thf(fact_5614_insert__compr,axiom,
% 0.35/0.65      ( insert_nat
% 0.35/0.65      = ( ^ [A4: nat,B6: set_nat] :
% 0.35/0.65            ( collect_nat
% 0.35/0.65            @ ^ [X3: nat] :
% 0.35/0.65                ( ( X3 = A4 )
% 0.35/0.65                | ( member_nat @ X3 @ B6 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_compr
% 0.35/0.65  thf(fact_5615_insert__compr,axiom,
% 0.35/0.65      ( insert_int
% 0.35/0.65      = ( ^ [A4: int,B6: set_int] :
% 0.35/0.65            ( collect_int
% 0.35/0.65            @ ^ [X3: int] :
% 0.35/0.65                ( ( X3 = A4 )
% 0.35/0.65                | ( member_int @ X3 @ B6 ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_compr
% 0.35/0.65  thf(fact_5616_insertE,axiom,
% 0.35/0.65      ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65        ( ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ A2 ) )
% 0.35/0.65       => ( ( A != B )
% 0.35/0.65         => ( member8757157785044589968at_nat @ A @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertE
% 0.35/0.65  thf(fact_5617_insertE,axiom,
% 0.35/0.65      ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A2 ) )
% 0.35/0.65       => ( ( A != B )
% 0.35/0.65         => ( member8440522571783428010at_nat @ A @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertE
% 0.35/0.65  thf(fact_5618_insertE,axiom,
% 0.35/0.65      ! [A: real,B: real,A2: set_real] :
% 0.35/0.65        ( ( member_real @ A @ ( insert_real @ B @ A2 ) )
% 0.35/0.65       => ( ( A != B )
% 0.35/0.65         => ( member_real @ A @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertE
% 0.35/0.65  thf(fact_5619_insertE,axiom,
% 0.35/0.65      ! [A: set_nat,B: set_nat,A2: set_set_nat] :
% 0.35/0.65        ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
% 0.35/0.65       => ( ( A != B )
% 0.35/0.65         => ( member_set_nat @ A @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertE
% 0.35/0.65  thf(fact_5620_insertE,axiom,
% 0.35/0.65      ! [A: nat,B: nat,A2: set_nat] :
% 0.35/0.65        ( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
% 0.35/0.65       => ( ( A != B )
% 0.35/0.65         => ( member_nat @ A @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertE
% 0.35/0.65  thf(fact_5621_insertE,axiom,
% 0.35/0.65      ! [A: int,B: int,A2: set_int] :
% 0.35/0.65        ( ( member_int @ A @ ( insert_int @ B @ A2 ) )
% 0.35/0.65       => ( ( A != B )
% 0.35/0.65         => ( member_int @ A @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertE
% 0.35/0.65  thf(fact_5622_insertI1,axiom,
% 0.35/0.65      ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ A @ B2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertI1
% 0.35/0.65  thf(fact_5623_insertI1,axiom,
% 0.35/0.65      ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ B2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertI1
% 0.35/0.65  thf(fact_5624_insertI1,axiom,
% 0.35/0.65      ! [A: real,B2: set_real] : ( member_real @ A @ ( insert_real @ A @ B2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertI1
% 0.35/0.65  thf(fact_5625_insertI1,axiom,
% 0.35/0.65      ! [A: set_nat,B2: set_set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ B2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertI1
% 0.35/0.65  thf(fact_5626_insertI1,axiom,
% 0.35/0.65      ! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertI1
% 0.35/0.65  thf(fact_5627_insertI1,axiom,
% 0.35/0.65      ! [A: int,B2: set_int] : ( member_int @ A @ ( insert_int @ A @ B2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertI1
% 0.35/0.65  thf(fact_5628_insertI2,axiom,
% 0.35/0.65      ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
% 0.35/0.65        ( ( member8757157785044589968at_nat @ A @ B2 )
% 0.35/0.65       => ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertI2
% 0.35/0.65  thf(fact_5629_insertI2,axiom,
% 0.35/0.65      ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
% 0.35/0.65        ( ( member8440522571783428010at_nat @ A @ B2 )
% 0.35/0.65       => ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertI2
% 0.35/0.65  thf(fact_5630_insertI2,axiom,
% 0.35/0.65      ! [A: real,B2: set_real,B: real] :
% 0.35/0.65        ( ( member_real @ A @ B2 )
% 0.35/0.65       => ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertI2
% 0.35/0.65  thf(fact_5631_insertI2,axiom,
% 0.35/0.65      ! [A: set_nat,B2: set_set_nat,B: set_nat] :
% 0.35/0.65        ( ( member_set_nat @ A @ B2 )
% 0.35/0.65       => ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertI2
% 0.35/0.65  thf(fact_5632_insertI2,axiom,
% 0.35/0.65      ! [A: nat,B2: set_nat,B: nat] :
% 0.35/0.65        ( ( member_nat @ A @ B2 )
% 0.35/0.65       => ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertI2
% 0.35/0.65  thf(fact_5633_insertI2,axiom,
% 0.35/0.65      ! [A: int,B2: set_int,B: int] :
% 0.35/0.65        ( ( member_int @ A @ B2 )
% 0.35/0.65       => ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insertI2
% 0.35/0.65  thf(fact_5634_Set_Oset__insert,axiom,
% 0.35/0.65      ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65        ( ( member8757157785044589968at_nat @ X @ A2 )
% 0.35/0.65       => ~ ! [B8: set_Pr4329608150637261639at_nat] :
% 0.35/0.65              ( ( A2
% 0.35/0.65                = ( insert9069300056098147895at_nat @ X @ B8 ) )
% 0.35/0.65             => ( member8757157785044589968at_nat @ X @ B8 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Set.set_insert
% 0.35/0.65  thf(fact_5635_Set_Oset__insert,axiom,
% 0.35/0.65      ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ( member8440522571783428010at_nat @ X @ A2 )
% 0.35/0.65       => ~ ! [B8: set_Pr1261947904930325089at_nat] :
% 0.35/0.65              ( ( A2
% 0.35/0.65                = ( insert8211810215607154385at_nat @ X @ B8 ) )
% 0.35/0.65             => ( member8440522571783428010at_nat @ X @ B8 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Set.set_insert
% 0.35/0.65  thf(fact_5636_Set_Oset__insert,axiom,
% 0.35/0.65      ! [X: real,A2: set_real] :
% 0.35/0.65        ( ( member_real @ X @ A2 )
% 0.35/0.65       => ~ ! [B8: set_real] :
% 0.35/0.65              ( ( A2
% 0.35/0.65                = ( insert_real @ X @ B8 ) )
% 0.35/0.65             => ( member_real @ X @ B8 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Set.set_insert
% 0.35/0.65  thf(fact_5637_Set_Oset__insert,axiom,
% 0.35/0.65      ! [X: set_nat,A2: set_set_nat] :
% 0.35/0.65        ( ( member_set_nat @ X @ A2 )
% 0.35/0.65       => ~ ! [B8: set_set_nat] :
% 0.35/0.65              ( ( A2
% 0.35/0.65                = ( insert_set_nat @ X @ B8 ) )
% 0.35/0.65             => ( member_set_nat @ X @ B8 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Set.set_insert
% 0.35/0.65  thf(fact_5638_Set_Oset__insert,axiom,
% 0.35/0.65      ! [X: nat,A2: set_nat] :
% 0.35/0.65        ( ( member_nat @ X @ A2 )
% 0.35/0.65       => ~ ! [B8: set_nat] :
% 0.35/0.65              ( ( A2
% 0.35/0.65                = ( insert_nat @ X @ B8 ) )
% 0.35/0.65             => ( member_nat @ X @ B8 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Set.set_insert
% 0.35/0.65  thf(fact_5639_Set_Oset__insert,axiom,
% 0.35/0.65      ! [X: int,A2: set_int] :
% 0.35/0.65        ( ( member_int @ X @ A2 )
% 0.35/0.65       => ~ ! [B8: set_int] :
% 0.35/0.65              ( ( A2
% 0.35/0.65                = ( insert_int @ X @ B8 ) )
% 0.35/0.65             => ( member_int @ X @ B8 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % Set.set_insert
% 0.35/0.65  thf(fact_5640_insert__ident,axiom,
% 0.35/0.65      ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65        ( ~ ( member8757157785044589968at_nat @ X @ A2 )
% 0.35/0.65       => ( ~ ( member8757157785044589968at_nat @ X @ B2 )
% 0.35/0.65         => ( ( ( insert9069300056098147895at_nat @ X @ A2 )
% 0.35/0.65              = ( insert9069300056098147895at_nat @ X @ B2 ) )
% 0.35/0.65            = ( A2 = B2 ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_ident
% 0.35/0.65  thf(fact_5641_insert__ident,axiom,
% 0.35/0.65      ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ~ ( member8440522571783428010at_nat @ X @ A2 )
% 0.35/0.65       => ( ~ ( member8440522571783428010at_nat @ X @ B2 )
% 0.35/0.65         => ( ( ( insert8211810215607154385at_nat @ X @ A2 )
% 0.35/0.65              = ( insert8211810215607154385at_nat @ X @ B2 ) )
% 0.35/0.65            = ( A2 = B2 ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_ident
% 0.35/0.65  thf(fact_5642_insert__ident,axiom,
% 0.35/0.65      ! [X: real,A2: set_real,B2: set_real] :
% 0.35/0.65        ( ~ ( member_real @ X @ A2 )
% 0.35/0.65       => ( ~ ( member_real @ X @ B2 )
% 0.35/0.65         => ( ( ( insert_real @ X @ A2 )
% 0.35/0.65              = ( insert_real @ X @ B2 ) )
% 0.35/0.65            = ( A2 = B2 ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_ident
% 0.35/0.65  thf(fact_5643_insert__ident,axiom,
% 0.35/0.65      ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.35/0.65        ( ~ ( member_set_nat @ X @ A2 )
% 0.35/0.65       => ( ~ ( member_set_nat @ X @ B2 )
% 0.35/0.65         => ( ( ( insert_set_nat @ X @ A2 )
% 0.35/0.65              = ( insert_set_nat @ X @ B2 ) )
% 0.35/0.65            = ( A2 = B2 ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_ident
% 0.35/0.65  thf(fact_5644_insert__ident,axiom,
% 0.35/0.65      ! [X: nat,A2: set_nat,B2: set_nat] :
% 0.35/0.65        ( ~ ( member_nat @ X @ A2 )
% 0.35/0.65       => ( ~ ( member_nat @ X @ B2 )
% 0.35/0.65         => ( ( ( insert_nat @ X @ A2 )
% 0.35/0.65              = ( insert_nat @ X @ B2 ) )
% 0.35/0.65            = ( A2 = B2 ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_ident
% 0.35/0.65  thf(fact_5645_insert__ident,axiom,
% 0.35/0.65      ! [X: int,A2: set_int,B2: set_int] :
% 0.35/0.65        ( ~ ( member_int @ X @ A2 )
% 0.35/0.65       => ( ~ ( member_int @ X @ B2 )
% 0.35/0.65         => ( ( ( insert_int @ X @ A2 )
% 0.35/0.65              = ( insert_int @ X @ B2 ) )
% 0.35/0.65            = ( A2 = B2 ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_ident
% 0.35/0.65  thf(fact_5646_insert__absorb,axiom,
% 0.35/0.65      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65        ( ( member8757157785044589968at_nat @ A @ A2 )
% 0.35/0.65       => ( ( insert9069300056098147895at_nat @ A @ A2 )
% 0.35/0.65          = A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_absorb
% 0.35/0.65  thf(fact_5647_insert__absorb,axiom,
% 0.35/0.65      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ( member8440522571783428010at_nat @ A @ A2 )
% 0.35/0.65       => ( ( insert8211810215607154385at_nat @ A @ A2 )
% 0.35/0.65          = A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_absorb
% 0.35/0.65  thf(fact_5648_insert__absorb,axiom,
% 0.35/0.65      ! [A: real,A2: set_real] :
% 0.35/0.65        ( ( member_real @ A @ A2 )
% 0.35/0.65       => ( ( insert_real @ A @ A2 )
% 0.35/0.65          = A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_absorb
% 0.35/0.65  thf(fact_5649_insert__absorb,axiom,
% 0.35/0.65      ! [A: set_nat,A2: set_set_nat] :
% 0.35/0.65        ( ( member_set_nat @ A @ A2 )
% 0.35/0.65       => ( ( insert_set_nat @ A @ A2 )
% 0.35/0.65          = A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_absorb
% 0.35/0.65  thf(fact_5650_insert__absorb,axiom,
% 0.35/0.65      ! [A: nat,A2: set_nat] :
% 0.35/0.65        ( ( member_nat @ A @ A2 )
% 0.35/0.65       => ( ( insert_nat @ A @ A2 )
% 0.35/0.65          = A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_absorb
% 0.35/0.65  thf(fact_5651_insert__absorb,axiom,
% 0.35/0.65      ! [A: int,A2: set_int] :
% 0.35/0.65        ( ( member_int @ A @ A2 )
% 0.35/0.65       => ( ( insert_int @ A @ A2 )
% 0.35/0.65          = A2 ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_absorb
% 0.35/0.65  thf(fact_5652_insert__eq__iff,axiom,
% 0.35/0.65      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65        ( ~ ( member8757157785044589968at_nat @ A @ A2 )
% 0.35/0.65       => ( ~ ( member8757157785044589968at_nat @ B @ B2 )
% 0.35/0.65         => ( ( ( insert9069300056098147895at_nat @ A @ A2 )
% 0.35/0.65              = ( insert9069300056098147895at_nat @ B @ B2 ) )
% 0.35/0.65            = ( ( ( A = B )
% 0.35/0.65               => ( A2 = B2 ) )
% 0.35/0.65              & ( ( A != B )
% 0.35/0.65               => ? [C5: set_Pr4329608150637261639at_nat] :
% 0.35/0.65                    ( ( A2
% 0.35/0.65                      = ( insert9069300056098147895at_nat @ B @ C5 ) )
% 0.35/0.65                    & ~ ( member8757157785044589968at_nat @ B @ C5 )
% 0.35/0.65                    & ( B2
% 0.35/0.65                      = ( insert9069300056098147895at_nat @ A @ C5 ) )
% 0.35/0.65                    & ~ ( member8757157785044589968at_nat @ A @ C5 ) ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_eq_iff
% 0.35/0.65  thf(fact_5653_insert__eq__iff,axiom,
% 0.35/0.65      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ~ ( member8440522571783428010at_nat @ A @ A2 )
% 0.35/0.65       => ( ~ ( member8440522571783428010at_nat @ B @ B2 )
% 0.35/0.65         => ( ( ( insert8211810215607154385at_nat @ A @ A2 )
% 0.35/0.65              = ( insert8211810215607154385at_nat @ B @ B2 ) )
% 0.35/0.65            = ( ( ( A = B )
% 0.35/0.65               => ( A2 = B2 ) )
% 0.35/0.65              & ( ( A != B )
% 0.35/0.65               => ? [C5: set_Pr1261947904930325089at_nat] :
% 0.35/0.65                    ( ( A2
% 0.35/0.65                      = ( insert8211810215607154385at_nat @ B @ C5 ) )
% 0.35/0.65                    & ~ ( member8440522571783428010at_nat @ B @ C5 )
% 0.35/0.65                    & ( B2
% 0.35/0.65                      = ( insert8211810215607154385at_nat @ A @ C5 ) )
% 0.35/0.65                    & ~ ( member8440522571783428010at_nat @ A @ C5 ) ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_eq_iff
% 0.35/0.65  thf(fact_5654_insert__eq__iff,axiom,
% 0.35/0.65      ! [A: real,A2: set_real,B: real,B2: set_real] :
% 0.35/0.65        ( ~ ( member_real @ A @ A2 )
% 0.35/0.65       => ( ~ ( member_real @ B @ B2 )
% 0.35/0.65         => ( ( ( insert_real @ A @ A2 )
% 0.35/0.65              = ( insert_real @ B @ B2 ) )
% 0.35/0.65            = ( ( ( A = B )
% 0.35/0.65               => ( A2 = B2 ) )
% 0.35/0.65              & ( ( A != B )
% 0.35/0.65               => ? [C5: set_real] :
% 0.35/0.65                    ( ( A2
% 0.35/0.65                      = ( insert_real @ B @ C5 ) )
% 0.35/0.65                    & ~ ( member_real @ B @ C5 )
% 0.35/0.65                    & ( B2
% 0.35/0.65                      = ( insert_real @ A @ C5 ) )
% 0.35/0.65                    & ~ ( member_real @ A @ C5 ) ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_eq_iff
% 0.35/0.65  thf(fact_5655_insert__eq__iff,axiom,
% 0.35/0.65      ! [A: set_nat,A2: set_set_nat,B: set_nat,B2: set_set_nat] :
% 0.35/0.65        ( ~ ( member_set_nat @ A @ A2 )
% 0.35/0.65       => ( ~ ( member_set_nat @ B @ B2 )
% 0.35/0.65         => ( ( ( insert_set_nat @ A @ A2 )
% 0.35/0.65              = ( insert_set_nat @ B @ B2 ) )
% 0.35/0.65            = ( ( ( A = B )
% 0.35/0.65               => ( A2 = B2 ) )
% 0.35/0.65              & ( ( A != B )
% 0.35/0.65               => ? [C5: set_set_nat] :
% 0.35/0.65                    ( ( A2
% 0.35/0.65                      = ( insert_set_nat @ B @ C5 ) )
% 0.35/0.65                    & ~ ( member_set_nat @ B @ C5 )
% 0.35/0.65                    & ( B2
% 0.35/0.65                      = ( insert_set_nat @ A @ C5 ) )
% 0.35/0.65                    & ~ ( member_set_nat @ A @ C5 ) ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_eq_iff
% 0.35/0.65  thf(fact_5656_insert__eq__iff,axiom,
% 0.35/0.65      ! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
% 0.35/0.65        ( ~ ( member_nat @ A @ A2 )
% 0.35/0.65       => ( ~ ( member_nat @ B @ B2 )
% 0.35/0.65         => ( ( ( insert_nat @ A @ A2 )
% 0.35/0.65              = ( insert_nat @ B @ B2 ) )
% 0.35/0.65            = ( ( ( A = B )
% 0.35/0.65               => ( A2 = B2 ) )
% 0.35/0.65              & ( ( A != B )
% 0.35/0.65               => ? [C5: set_nat] :
% 0.35/0.65                    ( ( A2
% 0.35/0.65                      = ( insert_nat @ B @ C5 ) )
% 0.35/0.65                    & ~ ( member_nat @ B @ C5 )
% 0.35/0.65                    & ( B2
% 0.35/0.65                      = ( insert_nat @ A @ C5 ) )
% 0.35/0.65                    & ~ ( member_nat @ A @ C5 ) ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_eq_iff
% 0.35/0.65  thf(fact_5657_insert__eq__iff,axiom,
% 0.35/0.65      ! [A: int,A2: set_int,B: int,B2: set_int] :
% 0.35/0.65        ( ~ ( member_int @ A @ A2 )
% 0.35/0.65       => ( ~ ( member_int @ B @ B2 )
% 0.35/0.65         => ( ( ( insert_int @ A @ A2 )
% 0.35/0.65              = ( insert_int @ B @ B2 ) )
% 0.35/0.65            = ( ( ( A = B )
% 0.35/0.65               => ( A2 = B2 ) )
% 0.35/0.65              & ( ( A != B )
% 0.35/0.65               => ? [C5: set_int] :
% 0.35/0.65                    ( ( A2
% 0.35/0.65                      = ( insert_int @ B @ C5 ) )
% 0.35/0.65                    & ~ ( member_int @ B @ C5 )
% 0.35/0.65                    & ( B2
% 0.35/0.65                      = ( insert_int @ A @ C5 ) )
% 0.35/0.65                    & ~ ( member_int @ A @ C5 ) ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_eq_iff
% 0.35/0.65  thf(fact_5658_insert__commute,axiom,
% 0.35/0.65      ! [X: nat,Y: nat,A2: set_nat] :
% 0.35/0.65        ( ( insert_nat @ X @ ( insert_nat @ Y @ A2 ) )
% 0.35/0.65        = ( insert_nat @ Y @ ( insert_nat @ X @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_commute
% 0.35/0.65  thf(fact_5659_insert__commute,axiom,
% 0.35/0.65      ! [X: int,Y: int,A2: set_int] :
% 0.35/0.65        ( ( insert_int @ X @ ( insert_int @ Y @ A2 ) )
% 0.35/0.65        = ( insert_int @ Y @ ( insert_int @ X @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_commute
% 0.35/0.65  thf(fact_5660_insert__commute,axiom,
% 0.35/0.65      ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ( insert8211810215607154385at_nat @ X @ ( insert8211810215607154385at_nat @ Y @ A2 ) )
% 0.35/0.65        = ( insert8211810215607154385at_nat @ Y @ ( insert8211810215607154385at_nat @ X @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_commute
% 0.35/0.65  thf(fact_5661_insert__commute,axiom,
% 0.35/0.65      ! [X: real,Y: real,A2: set_real] :
% 0.35/0.65        ( ( insert_real @ X @ ( insert_real @ Y @ A2 ) )
% 0.35/0.65        = ( insert_real @ Y @ ( insert_real @ X @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_commute
% 0.35/0.65  thf(fact_5662_insert__commute,axiom,
% 0.35/0.65      ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65        ( ( insert9069300056098147895at_nat @ X @ ( insert9069300056098147895at_nat @ Y @ A2 ) )
% 0.35/0.65        = ( insert9069300056098147895at_nat @ Y @ ( insert9069300056098147895at_nat @ X @ A2 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_commute
% 0.35/0.65  thf(fact_5663_mk__disjoint__insert,axiom,
% 0.35/0.65      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65        ( ( member8757157785044589968at_nat @ A @ A2 )
% 0.35/0.65       => ? [B8: set_Pr4329608150637261639at_nat] :
% 0.35/0.65            ( ( A2
% 0.35/0.65              = ( insert9069300056098147895at_nat @ A @ B8 ) )
% 0.35/0.65            & ~ ( member8757157785044589968at_nat @ A @ B8 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % mk_disjoint_insert
% 0.35/0.65  thf(fact_5664_mk__disjoint__insert,axiom,
% 0.35/0.65      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ( member8440522571783428010at_nat @ A @ A2 )
% 0.35/0.65       => ? [B8: set_Pr1261947904930325089at_nat] :
% 0.35/0.65            ( ( A2
% 0.35/0.65              = ( insert8211810215607154385at_nat @ A @ B8 ) )
% 0.35/0.65            & ~ ( member8440522571783428010at_nat @ A @ B8 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % mk_disjoint_insert
% 0.35/0.65  thf(fact_5665_mk__disjoint__insert,axiom,
% 0.35/0.65      ! [A: real,A2: set_real] :
% 0.35/0.65        ( ( member_real @ A @ A2 )
% 0.35/0.65       => ? [B8: set_real] :
% 0.35/0.65            ( ( A2
% 0.35/0.65              = ( insert_real @ A @ B8 ) )
% 0.35/0.65            & ~ ( member_real @ A @ B8 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % mk_disjoint_insert
% 0.35/0.65  thf(fact_5666_mk__disjoint__insert,axiom,
% 0.35/0.65      ! [A: set_nat,A2: set_set_nat] :
% 0.35/0.65        ( ( member_set_nat @ A @ A2 )
% 0.35/0.65       => ? [B8: set_set_nat] :
% 0.35/0.65            ( ( A2
% 0.35/0.65              = ( insert_set_nat @ A @ B8 ) )
% 0.35/0.65            & ~ ( member_set_nat @ A @ B8 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % mk_disjoint_insert
% 0.35/0.65  thf(fact_5667_mk__disjoint__insert,axiom,
% 0.35/0.65      ! [A: nat,A2: set_nat] :
% 0.35/0.65        ( ( member_nat @ A @ A2 )
% 0.35/0.65       => ? [B8: set_nat] :
% 0.35/0.65            ( ( A2
% 0.35/0.65              = ( insert_nat @ A @ B8 ) )
% 0.35/0.65            & ~ ( member_nat @ A @ B8 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % mk_disjoint_insert
% 0.35/0.65  thf(fact_5668_mk__disjoint__insert,axiom,
% 0.35/0.65      ! [A: int,A2: set_int] :
% 0.35/0.65        ( ( member_int @ A @ A2 )
% 0.35/0.65       => ? [B8: set_int] :
% 0.35/0.65            ( ( A2
% 0.35/0.65              = ( insert_int @ A @ B8 ) )
% 0.35/0.65            & ~ ( member_int @ A @ B8 ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % mk_disjoint_insert
% 0.35/0.65  thf(fact_5669_sum_Oinsert__if,axiom,
% 0.35/0.65      ! [A2: set_real,X: real,G: real > real] :
% 0.35/0.65        ( ( finite_finite_real @ A2 )
% 0.35/0.65       => ( ( ( member_real @ X @ A2 )
% 0.35/0.65           => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.65              = ( groups8097168146408367636l_real @ G @ A2 ) ) )
% 0.35/0.65          & ( ~ ( member_real @ X @ A2 )
% 0.35/0.65           => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.65              = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.insert_if
% 0.35/0.65  thf(fact_5670_sum_Oinsert__if,axiom,
% 0.35/0.65      ! [A2: set_int,X: int,G: int > real] :
% 0.35/0.65        ( ( finite_finite_int @ A2 )
% 0.35/0.65       => ( ( ( member_int @ X @ A2 )
% 0.35/0.65           => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
% 0.35/0.65              = ( groups8778361861064173332t_real @ G @ A2 ) ) )
% 0.35/0.65          & ( ~ ( member_int @ X @ A2 )
% 0.35/0.65           => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
% 0.35/0.65              = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.insert_if
% 0.35/0.65  thf(fact_5671_sum_Oinsert__if,axiom,
% 0.35/0.65      ! [A2: set_complex,X: complex,G: complex > real] :
% 0.35/0.65        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.65       => ( ( ( member_complex @ X @ A2 )
% 0.35/0.65           => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
% 0.35/0.65              = ( groups5808333547571424918x_real @ G @ A2 ) ) )
% 0.35/0.65          & ( ~ ( member_complex @ X @ A2 )
% 0.35/0.65           => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
% 0.35/0.65              = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.insert_if
% 0.35/0.65  thf(fact_5672_sum_Oinsert__if,axiom,
% 0.35/0.65      ! [A2: set_real,X: real,G: real > rat] :
% 0.35/0.65        ( ( finite_finite_real @ A2 )
% 0.35/0.65       => ( ( ( member_real @ X @ A2 )
% 0.35/0.65           => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.65              = ( groups1300246762558778688al_rat @ G @ A2 ) ) )
% 0.35/0.65          & ( ~ ( member_real @ X @ A2 )
% 0.35/0.65           => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.65              = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.insert_if
% 0.35/0.65  thf(fact_5673_sum_Oinsert__if,axiom,
% 0.35/0.65      ! [A2: set_nat,X: nat,G: nat > rat] :
% 0.35/0.65        ( ( finite_finite_nat @ A2 )
% 0.35/0.65       => ( ( ( member_nat @ X @ A2 )
% 0.35/0.65           => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
% 0.35/0.65              = ( groups2906978787729119204at_rat @ G @ A2 ) ) )
% 0.35/0.65          & ( ~ ( member_nat @ X @ A2 )
% 0.35/0.65           => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
% 0.35/0.65              = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.insert_if
% 0.35/0.65  thf(fact_5674_sum_Oinsert__if,axiom,
% 0.35/0.65      ! [A2: set_int,X: int,G: int > rat] :
% 0.35/0.65        ( ( finite_finite_int @ A2 )
% 0.35/0.65       => ( ( ( member_int @ X @ A2 )
% 0.35/0.65           => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
% 0.35/0.65              = ( groups3906332499630173760nt_rat @ G @ A2 ) ) )
% 0.35/0.65          & ( ~ ( member_int @ X @ A2 )
% 0.35/0.65           => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
% 0.35/0.65              = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.insert_if
% 0.35/0.65  thf(fact_5675_sum_Oinsert__if,axiom,
% 0.35/0.65      ! [A2: set_complex,X: complex,G: complex > rat] :
% 0.35/0.65        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.65       => ( ( ( member_complex @ X @ A2 )
% 0.35/0.65           => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
% 0.35/0.65              = ( groups5058264527183730370ex_rat @ G @ A2 ) ) )
% 0.35/0.65          & ( ~ ( member_complex @ X @ A2 )
% 0.35/0.65           => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
% 0.35/0.65              = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.insert_if
% 0.35/0.65  thf(fact_5676_sum_Oinsert__if,axiom,
% 0.35/0.65      ! [A2: set_real,X: real,G: real > nat] :
% 0.35/0.65        ( ( finite_finite_real @ A2 )
% 0.35/0.65       => ( ( ( member_real @ X @ A2 )
% 0.35/0.65           => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.65              = ( groups1935376822645274424al_nat @ G @ A2 ) ) )
% 0.35/0.65          & ( ~ ( member_real @ X @ A2 )
% 0.35/0.65           => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.65              = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.insert_if
% 0.35/0.65  thf(fact_5677_sum_Oinsert__if,axiom,
% 0.35/0.65      ! [A2: set_int,X: int,G: int > nat] :
% 0.35/0.65        ( ( finite_finite_int @ A2 )
% 0.35/0.65       => ( ( ( member_int @ X @ A2 )
% 0.35/0.65           => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A2 ) )
% 0.35/0.65              = ( groups4541462559716669496nt_nat @ G @ A2 ) ) )
% 0.35/0.65          & ( ~ ( member_int @ X @ A2 )
% 0.35/0.65           => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A2 ) )
% 0.35/0.65              = ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.insert_if
% 0.35/0.65  thf(fact_5678_sum_Oinsert__if,axiom,
% 0.35/0.65      ! [A2: set_complex,X: complex,G: complex > nat] :
% 0.35/0.65        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.65       => ( ( ( member_complex @ X @ A2 )
% 0.35/0.65           => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
% 0.35/0.65              = ( groups5693394587270226106ex_nat @ G @ A2 ) ) )
% 0.35/0.65          & ( ~ ( member_complex @ X @ A2 )
% 0.35/0.65           => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
% 0.35/0.65              = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.insert_if
% 0.35/0.65  thf(fact_5679_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.35/0.65      ! [G: real > complex,A2: set_real] :
% 0.35/0.65        ( ( ( groups5754745047067104278omplex @ G @ A2 )
% 0.35/0.65         != zero_zero_complex )
% 0.35/0.65       => ~ ! [A3: real] :
% 0.35/0.65              ( ( member_real @ A3 @ A2 )
% 0.35/0.65             => ( ( G @ A3 )
% 0.35/0.65                = zero_zero_complex ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.not_neutral_contains_not_neutral
% 0.35/0.65  thf(fact_5680_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.35/0.65      ! [G: nat > complex,A2: set_nat] :
% 0.35/0.65        ( ( ( groups2073611262835488442omplex @ G @ A2 )
% 0.35/0.65         != zero_zero_complex )
% 0.35/0.65       => ~ ! [A3: nat] :
% 0.35/0.65              ( ( member_nat @ A3 @ A2 )
% 0.35/0.65             => ( ( G @ A3 )
% 0.35/0.65                = zero_zero_complex ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.not_neutral_contains_not_neutral
% 0.35/0.65  thf(fact_5681_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.35/0.65      ! [G: int > complex,A2: set_int] :
% 0.35/0.65        ( ( ( groups3049146728041665814omplex @ G @ A2 )
% 0.35/0.65         != zero_zero_complex )
% 0.35/0.65       => ~ ! [A3: int] :
% 0.35/0.65              ( ( member_int @ A3 @ A2 )
% 0.35/0.65             => ( ( G @ A3 )
% 0.35/0.65                = zero_zero_complex ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.not_neutral_contains_not_neutral
% 0.35/0.65  thf(fact_5682_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.35/0.65      ! [G: real > real,A2: set_real] :
% 0.35/0.65        ( ( ( groups8097168146408367636l_real @ G @ A2 )
% 0.35/0.65         != zero_zero_real )
% 0.35/0.65       => ~ ! [A3: real] :
% 0.35/0.65              ( ( member_real @ A3 @ A2 )
% 0.35/0.65             => ( ( G @ A3 )
% 0.35/0.65                = zero_zero_real ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.not_neutral_contains_not_neutral
% 0.35/0.65  thf(fact_5683_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.35/0.65      ! [G: int > real,A2: set_int] :
% 0.35/0.65        ( ( ( groups8778361861064173332t_real @ G @ A2 )
% 0.35/0.65         != zero_zero_real )
% 0.35/0.65       => ~ ! [A3: int] :
% 0.35/0.65              ( ( member_int @ A3 @ A2 )
% 0.35/0.65             => ( ( G @ A3 )
% 0.35/0.65                = zero_zero_real ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.not_neutral_contains_not_neutral
% 0.35/0.65  thf(fact_5684_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.35/0.65      ! [G: real > rat,A2: set_real] :
% 0.35/0.65        ( ( ( groups1300246762558778688al_rat @ G @ A2 )
% 0.35/0.65         != zero_zero_rat )
% 0.35/0.65       => ~ ! [A3: real] :
% 0.35/0.65              ( ( member_real @ A3 @ A2 )
% 0.35/0.65             => ( ( G @ A3 )
% 0.35/0.65                = zero_zero_rat ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.not_neutral_contains_not_neutral
% 0.35/0.65  thf(fact_5685_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.35/0.65      ! [G: nat > rat,A2: set_nat] :
% 0.35/0.65        ( ( ( groups2906978787729119204at_rat @ G @ A2 )
% 0.35/0.65         != zero_zero_rat )
% 0.35/0.65       => ~ ! [A3: nat] :
% 0.35/0.65              ( ( member_nat @ A3 @ A2 )
% 0.35/0.65             => ( ( G @ A3 )
% 0.35/0.65                = zero_zero_rat ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.not_neutral_contains_not_neutral
% 0.35/0.65  thf(fact_5686_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.35/0.65      ! [G: int > rat,A2: set_int] :
% 0.35/0.65        ( ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 0.35/0.65         != zero_zero_rat )
% 0.35/0.65       => ~ ! [A3: int] :
% 0.35/0.65              ( ( member_int @ A3 @ A2 )
% 0.35/0.65             => ( ( G @ A3 )
% 0.35/0.65                = zero_zero_rat ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.not_neutral_contains_not_neutral
% 0.35/0.65  thf(fact_5687_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.35/0.65      ! [G: real > nat,A2: set_real] :
% 0.35/0.65        ( ( ( groups1935376822645274424al_nat @ G @ A2 )
% 0.35/0.65         != zero_zero_nat )
% 0.35/0.65       => ~ ! [A3: real] :
% 0.35/0.65              ( ( member_real @ A3 @ A2 )
% 0.35/0.65             => ( ( G @ A3 )
% 0.35/0.65                = zero_zero_nat ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.not_neutral_contains_not_neutral
% 0.35/0.65  thf(fact_5688_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.35/0.65      ! [G: int > nat,A2: set_int] :
% 0.35/0.65        ( ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 0.35/0.65         != zero_zero_nat )
% 0.35/0.65       => ~ ! [A3: int] :
% 0.35/0.65              ( ( member_int @ A3 @ A2 )
% 0.35/0.65             => ( ( G @ A3 )
% 0.35/0.65                = zero_zero_nat ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.not_neutral_contains_not_neutral
% 0.35/0.65  thf(fact_5689_sum_Oneutral,axiom,
% 0.35/0.65      ! [A2: set_nat,G: nat > nat] :
% 0.35/0.65        ( ! [X4: nat] :
% 0.35/0.65            ( ( member_nat @ X4 @ A2 )
% 0.35/0.65           => ( ( G @ X4 )
% 0.35/0.65              = zero_zero_nat ) )
% 0.35/0.65       => ( ( groups3542108847815614940at_nat @ G @ A2 )
% 0.35/0.65          = zero_zero_nat ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.neutral
% 0.35/0.65  thf(fact_5690_sum_Oneutral,axiom,
% 0.35/0.65      ! [A2: set_complex,G: complex > complex] :
% 0.35/0.65        ( ! [X4: complex] :
% 0.35/0.65            ( ( member_complex @ X4 @ A2 )
% 0.35/0.65           => ( ( G @ X4 )
% 0.35/0.65              = zero_zero_complex ) )
% 0.35/0.65       => ( ( groups7754918857620584856omplex @ G @ A2 )
% 0.35/0.65          = zero_zero_complex ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.neutral
% 0.35/0.65  thf(fact_5691_sum_Oneutral,axiom,
% 0.35/0.65      ! [A2: set_nat,G: nat > real] :
% 0.35/0.65        ( ! [X4: nat] :
% 0.35/0.65            ( ( member_nat @ X4 @ A2 )
% 0.35/0.65           => ( ( G @ X4 )
% 0.35/0.65              = zero_zero_real ) )
% 0.35/0.65       => ( ( groups6591440286371151544t_real @ G @ A2 )
% 0.35/0.65          = zero_zero_real ) ) ).
% 0.35/0.65  
% 0.35/0.65  % sum.neutral
% 0.35/0.65  thf(fact_5692_singleton__inject,axiom,
% 0.35/0.65      ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
% 0.35/0.65        ( ( ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat )
% 0.35/0.65          = ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
% 0.35/0.65       => ( A = B ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_inject
% 0.35/0.65  thf(fact_5693_singleton__inject,axiom,
% 0.35/0.65      ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
% 0.35/0.65        ( ( ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat )
% 0.35/0.65          = ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
% 0.35/0.65       => ( A = B ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_inject
% 0.35/0.65  thf(fact_5694_singleton__inject,axiom,
% 0.35/0.65      ! [A: real,B: real] :
% 0.35/0.65        ( ( ( insert_real @ A @ bot_bot_set_real )
% 0.35/0.65          = ( insert_real @ B @ bot_bot_set_real ) )
% 0.35/0.65       => ( A = B ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_inject
% 0.35/0.65  thf(fact_5695_singleton__inject,axiom,
% 0.35/0.65      ! [A: nat,B: nat] :
% 0.35/0.65        ( ( ( insert_nat @ A @ bot_bot_set_nat )
% 0.35/0.65          = ( insert_nat @ B @ bot_bot_set_nat ) )
% 0.35/0.65       => ( A = B ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_inject
% 0.35/0.65  thf(fact_5696_singleton__inject,axiom,
% 0.35/0.65      ! [A: int,B: int] :
% 0.35/0.65        ( ( ( insert_int @ A @ bot_bot_set_int )
% 0.35/0.65          = ( insert_int @ B @ bot_bot_set_int ) )
% 0.35/0.65       => ( A = B ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_inject
% 0.35/0.65  thf(fact_5697_insert__not__empty,axiom,
% 0.35/0.65      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65        ( ( insert9069300056098147895at_nat @ A @ A2 )
% 0.35/0.65       != bot_bo228742789529271731at_nat ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_not_empty
% 0.35/0.65  thf(fact_5698_insert__not__empty,axiom,
% 0.35/0.65      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65        ( ( insert8211810215607154385at_nat @ A @ A2 )
% 0.35/0.65       != bot_bo2099793752762293965at_nat ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_not_empty
% 0.35/0.65  thf(fact_5699_insert__not__empty,axiom,
% 0.35/0.65      ! [A: real,A2: set_real] :
% 0.35/0.65        ( ( insert_real @ A @ A2 )
% 0.35/0.65       != bot_bot_set_real ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_not_empty
% 0.35/0.65  thf(fact_5700_insert__not__empty,axiom,
% 0.35/0.65      ! [A: nat,A2: set_nat] :
% 0.35/0.65        ( ( insert_nat @ A @ A2 )
% 0.35/0.65       != bot_bot_set_nat ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_not_empty
% 0.35/0.65  thf(fact_5701_insert__not__empty,axiom,
% 0.35/0.65      ! [A: int,A2: set_int] :
% 0.35/0.65        ( ( insert_int @ A @ A2 )
% 0.35/0.65       != bot_bot_set_int ) ).
% 0.35/0.65  
% 0.35/0.65  % insert_not_empty
% 0.35/0.65  thf(fact_5702_doubleton__eq__iff,axiom,
% 0.35/0.65      ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,C: produc3843707927480180839at_nat,D: produc3843707927480180839at_nat] :
% 0.35/0.65        ( ( ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
% 0.35/0.65          = ( insert9069300056098147895at_nat @ C @ ( insert9069300056098147895at_nat @ D @ bot_bo228742789529271731at_nat ) ) )
% 0.35/0.65        = ( ( ( A = C )
% 0.35/0.65            & ( B = D ) )
% 0.35/0.65          | ( ( A = D )
% 0.35/0.65            & ( B = C ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % doubleton_eq_iff
% 0.35/0.65  thf(fact_5703_doubleton__eq__iff,axiom,
% 0.35/0.65      ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat,D: product_prod_nat_nat] :
% 0.35/0.65        ( ( ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
% 0.35/0.65          = ( insert8211810215607154385at_nat @ C @ ( insert8211810215607154385at_nat @ D @ bot_bo2099793752762293965at_nat ) ) )
% 0.35/0.65        = ( ( ( A = C )
% 0.35/0.65            & ( B = D ) )
% 0.35/0.65          | ( ( A = D )
% 0.35/0.65            & ( B = C ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % doubleton_eq_iff
% 0.35/0.65  thf(fact_5704_doubleton__eq__iff,axiom,
% 0.35/0.65      ! [A: real,B: real,C: real,D: real] :
% 0.35/0.65        ( ( ( insert_real @ A @ ( insert_real @ B @ bot_bot_set_real ) )
% 0.35/0.65          = ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
% 0.35/0.65        = ( ( ( A = C )
% 0.35/0.65            & ( B = D ) )
% 0.35/0.65          | ( ( A = D )
% 0.35/0.65            & ( B = C ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % doubleton_eq_iff
% 0.35/0.65  thf(fact_5705_doubleton__eq__iff,axiom,
% 0.35/0.65      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.35/0.65        ( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
% 0.35/0.65          = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
% 0.35/0.65        = ( ( ( A = C )
% 0.35/0.65            & ( B = D ) )
% 0.35/0.65          | ( ( A = D )
% 0.35/0.65            & ( B = C ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % doubleton_eq_iff
% 0.35/0.65  thf(fact_5706_doubleton__eq__iff,axiom,
% 0.35/0.65      ! [A: int,B: int,C: int,D: int] :
% 0.35/0.65        ( ( ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) )
% 0.35/0.65          = ( insert_int @ C @ ( insert_int @ D @ bot_bot_set_int ) ) )
% 0.35/0.65        = ( ( ( A = C )
% 0.35/0.65            & ( B = D ) )
% 0.35/0.65          | ( ( A = D )
% 0.35/0.65            & ( B = C ) ) ) ) ).
% 0.35/0.65  
% 0.35/0.65  % doubleton_eq_iff
% 0.35/0.65  thf(fact_5707_singleton__iff,axiom,
% 0.35/0.65      ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
% 0.35/0.65        ( ( member8757157785044589968at_nat @ B @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
% 0.35/0.65        = ( B = A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_iff
% 0.35/0.65  thf(fact_5708_singleton__iff,axiom,
% 0.35/0.65      ! [B: set_nat,A: set_nat] :
% 0.35/0.65        ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
% 0.35/0.65        = ( B = A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_iff
% 0.35/0.65  thf(fact_5709_singleton__iff,axiom,
% 0.35/0.65      ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
% 0.35/0.65        ( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
% 0.35/0.65        = ( B = A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_iff
% 0.35/0.65  thf(fact_5710_singleton__iff,axiom,
% 0.35/0.65      ! [B: real,A: real] :
% 0.35/0.65        ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
% 0.35/0.65        = ( B = A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_iff
% 0.35/0.65  thf(fact_5711_singleton__iff,axiom,
% 0.35/0.65      ! [B: nat,A: nat] :
% 0.35/0.65        ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
% 0.35/0.65        = ( B = A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_iff
% 0.35/0.65  thf(fact_5712_singleton__iff,axiom,
% 0.35/0.65      ! [B: int,A: int] :
% 0.35/0.65        ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
% 0.35/0.65        = ( B = A ) ) ).
% 0.35/0.65  
% 0.35/0.65  % singleton_iff
% 0.35/0.65  thf(fact_5713_singletonD,axiom,
% 0.35/0.65      ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
% 0.35/0.66        ( ( member8757157785044589968at_nat @ B @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
% 0.35/0.66       => ( B = A ) ) ).
% 0.35/0.66  
% 0.35/0.66  % singletonD
% 0.35/0.66  thf(fact_5714_singletonD,axiom,
% 0.35/0.66      ! [B: set_nat,A: set_nat] :
% 0.35/0.66        ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
% 0.35/0.66       => ( B = A ) ) ).
% 0.35/0.66  
% 0.35/0.66  % singletonD
% 0.35/0.66  thf(fact_5715_singletonD,axiom,
% 0.35/0.66      ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
% 0.35/0.66        ( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
% 0.35/0.66       => ( B = A ) ) ).
% 0.35/0.66  
% 0.35/0.66  % singletonD
% 0.35/0.66  thf(fact_5716_singletonD,axiom,
% 0.35/0.66      ! [B: real,A: real] :
% 0.35/0.66        ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
% 0.35/0.66       => ( B = A ) ) ).
% 0.35/0.66  
% 0.35/0.66  % singletonD
% 0.35/0.66  thf(fact_5717_singletonD,axiom,
% 0.35/0.66      ! [B: nat,A: nat] :
% 0.35/0.66        ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
% 0.35/0.66       => ( B = A ) ) ).
% 0.35/0.66  
% 0.35/0.66  % singletonD
% 0.35/0.66  thf(fact_5718_singletonD,axiom,
% 0.35/0.66      ! [B: int,A: int] :
% 0.35/0.66        ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
% 0.35/0.66       => ( B = A ) ) ).
% 0.35/0.66  
% 0.35/0.66  % singletonD
% 0.35/0.66  thf(fact_5719_finite_OinsertI,axiom,
% 0.35/0.66      ! [A2: set_real,A: real] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( finite_finite_real @ ( insert_real @ A @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite.insertI
% 0.35/0.66  thf(fact_5720_finite_OinsertI,axiom,
% 0.35/0.66      ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
% 0.35/0.66        ( ( finite4343798906461161616at_nat @ A2 )
% 0.35/0.66       => ( finite4343798906461161616at_nat @ ( insert9069300056098147895at_nat @ A @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite.insertI
% 0.35/0.66  thf(fact_5721_finite_OinsertI,axiom,
% 0.35/0.66      ! [A2: set_nat,A: nat] :
% 0.35/0.66        ( ( finite_finite_nat @ A2 )
% 0.35/0.66       => ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite.insertI
% 0.35/0.66  thf(fact_5722_finite_OinsertI,axiom,
% 0.35/0.66      ! [A2: set_int,A: int] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( finite_finite_int @ ( insert_int @ A @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite.insertI
% 0.35/0.66  thf(fact_5723_finite_OinsertI,axiom,
% 0.35/0.66      ! [A2: set_complex,A: complex] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( finite3207457112153483333omplex @ ( insert_complex @ A @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite.insertI
% 0.35/0.66  thf(fact_5724_finite_OinsertI,axiom,
% 0.35/0.66      ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
% 0.35/0.66        ( ( finite6177210948735845034at_nat @ A2 )
% 0.35/0.66       => ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite.insertI
% 0.35/0.66  thf(fact_5725_subset__insertI2,axiom,
% 0.35/0.66      ! [A2: set_nat,B2: set_nat,B: nat] :
% 0.35/0.66        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.35/0.66       => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_insertI2
% 0.35/0.66  thf(fact_5726_subset__insertI2,axiom,
% 0.35/0.66      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
% 0.35/0.66        ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 0.35/0.66       => ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ B2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_insertI2
% 0.35/0.66  thf(fact_5727_subset__insertI2,axiom,
% 0.35/0.66      ! [A2: set_real,B2: set_real,B: real] :
% 0.35/0.66        ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.35/0.66       => ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ B2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_insertI2
% 0.35/0.66  thf(fact_5728_subset__insertI2,axiom,
% 0.35/0.66      ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
% 0.35/0.66        ( ( ord_le1268244103169919719at_nat @ A2 @ B2 )
% 0.35/0.66       => ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ B @ B2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_insertI2
% 0.35/0.66  thf(fact_5729_subset__insertI2,axiom,
% 0.35/0.66      ! [A2: set_int,B2: set_int,B: int] :
% 0.35/0.66        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.35/0.66       => ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ B2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_insertI2
% 0.35/0.66  thf(fact_5730_subset__insertI,axiom,
% 0.35/0.66      ! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A @ B2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_insertI
% 0.35/0.66  thf(fact_5731_subset__insertI,axiom,
% 0.35/0.66      ! [B2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ B2 @ ( insert8211810215607154385at_nat @ A @ B2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_insertI
% 0.35/0.66  thf(fact_5732_subset__insertI,axiom,
% 0.35/0.66      ! [B2: set_real,A: real] : ( ord_less_eq_set_real @ B2 @ ( insert_real @ A @ B2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_insertI
% 0.35/0.66  thf(fact_5733_subset__insertI,axiom,
% 0.35/0.66      ! [B2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] : ( ord_le1268244103169919719at_nat @ B2 @ ( insert9069300056098147895at_nat @ A @ B2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_insertI
% 0.35/0.66  thf(fact_5734_subset__insertI,axiom,
% 0.35/0.66      ! [B2: set_int,A: int] : ( ord_less_eq_set_int @ B2 @ ( insert_int @ A @ B2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_insertI
% 0.35/0.66  thf(fact_5735_subset__insert,axiom,
% 0.35/0.66      ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.66        ( ~ ( member8757157785044589968at_nat @ X @ A2 )
% 0.35/0.66       => ( ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ B2 ) )
% 0.35/0.66          = ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_insert
% 0.35/0.66  thf(fact_5736_subset__insert,axiom,
% 0.35/0.66      ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.66        ( ~ ( member8440522571783428010at_nat @ X @ A2 )
% 0.35/0.66       => ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B2 ) )
% 0.35/0.66          = ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_insert
% 0.35/0.66  thf(fact_5737_subset__insert,axiom,
% 0.35/0.66      ! [X: real,A2: set_real,B2: set_real] :
% 0.35/0.66        ( ~ ( member_real @ X @ A2 )
% 0.35/0.66       => ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) )
% 0.35/0.66          = ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_insert
% 0.35/0.66  thf(fact_5738_subset__insert,axiom,
% 0.35/0.66      ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.35/0.66        ( ~ ( member_set_nat @ X @ A2 )
% 0.35/0.66       => ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
% 0.35/0.66          = ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_insert
% 0.35/0.66  thf(fact_5739_subset__insert,axiom,
% 0.35/0.66      ! [X: nat,A2: set_nat,B2: set_nat] :
% 0.35/0.66        ( ~ ( member_nat @ X @ A2 )
% 0.35/0.66       => ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
% 0.35/0.66          = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_insert
% 0.35/0.66  thf(fact_5740_subset__insert,axiom,
% 0.35/0.66      ! [X: int,A2: set_int,B2: set_int] :
% 0.35/0.66        ( ~ ( member_int @ X @ A2 )
% 0.35/0.66       => ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) )
% 0.35/0.66          = ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_insert
% 0.35/0.66  thf(fact_5741_insert__mono,axiom,
% 0.35/0.66      ! [C2: set_nat,D6: set_nat,A: nat] :
% 0.35/0.66        ( ( ord_less_eq_set_nat @ C2 @ D6 )
% 0.35/0.66       => ( ord_less_eq_set_nat @ ( insert_nat @ A @ C2 ) @ ( insert_nat @ A @ D6 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_mono
% 0.35/0.66  thf(fact_5742_insert__mono,axiom,
% 0.35/0.66      ! [C2: set_Pr1261947904930325089at_nat,D6: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
% 0.35/0.66        ( ( ord_le3146513528884898305at_nat @ C2 @ D6 )
% 0.35/0.66       => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ A @ C2 ) @ ( insert8211810215607154385at_nat @ A @ D6 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_mono
% 0.35/0.66  thf(fact_5743_insert__mono,axiom,
% 0.35/0.66      ! [C2: set_real,D6: set_real,A: real] :
% 0.35/0.66        ( ( ord_less_eq_set_real @ C2 @ D6 )
% 0.35/0.66       => ( ord_less_eq_set_real @ ( insert_real @ A @ C2 ) @ ( insert_real @ A @ D6 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_mono
% 0.35/0.66  thf(fact_5744_insert__mono,axiom,
% 0.35/0.66      ! [C2: set_Pr4329608150637261639at_nat,D6: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
% 0.35/0.66        ( ( ord_le1268244103169919719at_nat @ C2 @ D6 )
% 0.35/0.66       => ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ A @ C2 ) @ ( insert9069300056098147895at_nat @ A @ D6 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_mono
% 0.35/0.66  thf(fact_5745_insert__mono,axiom,
% 0.35/0.66      ! [C2: set_int,D6: set_int,A: int] :
% 0.35/0.66        ( ( ord_less_eq_set_int @ C2 @ D6 )
% 0.35/0.66       => ( ord_less_eq_set_int @ ( insert_int @ A @ C2 ) @ ( insert_int @ A @ D6 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_mono
% 0.35/0.66  thf(fact_5746_insert__subsetI,axiom,
% 0.35/0.66      ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,X8: set_Pr4329608150637261639at_nat] :
% 0.35/0.66        ( ( member8757157785044589968at_nat @ X @ A2 )
% 0.35/0.66       => ( ( ord_le1268244103169919719at_nat @ X8 @ A2 )
% 0.35/0.66         => ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X @ X8 ) @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_subsetI
% 0.35/0.66  thf(fact_5747_insert__subsetI,axiom,
% 0.35/0.66      ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,X8: set_Pr1261947904930325089at_nat] :
% 0.35/0.66        ( ( member8440522571783428010at_nat @ X @ A2 )
% 0.35/0.66       => ( ( ord_le3146513528884898305at_nat @ X8 @ A2 )
% 0.35/0.66         => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X @ X8 ) @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_subsetI
% 0.35/0.66  thf(fact_5748_insert__subsetI,axiom,
% 0.35/0.66      ! [X: real,A2: set_real,X8: set_real] :
% 0.35/0.66        ( ( member_real @ X @ A2 )
% 0.35/0.66       => ( ( ord_less_eq_set_real @ X8 @ A2 )
% 0.35/0.66         => ( ord_less_eq_set_real @ ( insert_real @ X @ X8 ) @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_subsetI
% 0.35/0.66  thf(fact_5749_insert__subsetI,axiom,
% 0.35/0.66      ! [X: set_nat,A2: set_set_nat,X8: set_set_nat] :
% 0.35/0.66        ( ( member_set_nat @ X @ A2 )
% 0.35/0.66       => ( ( ord_le6893508408891458716et_nat @ X8 @ A2 )
% 0.35/0.66         => ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ X8 ) @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_subsetI
% 0.35/0.66  thf(fact_5750_insert__subsetI,axiom,
% 0.35/0.66      ! [X: nat,A2: set_nat,X8: set_nat] :
% 0.35/0.66        ( ( member_nat @ X @ A2 )
% 0.35/0.66       => ( ( ord_less_eq_set_nat @ X8 @ A2 )
% 0.35/0.66         => ( ord_less_eq_set_nat @ ( insert_nat @ X @ X8 ) @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_subsetI
% 0.35/0.66  thf(fact_5751_insert__subsetI,axiom,
% 0.35/0.66      ! [X: int,A2: set_int,X8: set_int] :
% 0.35/0.66        ( ( member_int @ X @ A2 )
% 0.35/0.66       => ( ( ord_less_eq_set_int @ X8 @ A2 )
% 0.35/0.66         => ( ord_less_eq_set_int @ ( insert_int @ X @ X8 ) @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_subsetI
% 0.35/0.66  thf(fact_5752_insert__Diff__if,axiom,
% 0.35/0.66      ! [X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.66        ( ( ( member8757157785044589968at_nat @ X @ B2 )
% 0.35/0.66         => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A2 ) @ B2 )
% 0.35/0.66            = ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) )
% 0.35/0.66        & ( ~ ( member8757157785044589968at_nat @ X @ B2 )
% 0.35/0.66         => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A2 ) @ B2 )
% 0.35/0.66            = ( insert9069300056098147895at_nat @ X @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_Diff_if
% 0.35/0.66  thf(fact_5753_insert__Diff__if,axiom,
% 0.35/0.66      ! [X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.66        ( ( ( member8440522571783428010at_nat @ X @ B2 )
% 0.35/0.66         => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) @ B2 )
% 0.35/0.66            = ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) )
% 0.35/0.66        & ( ~ ( member8440522571783428010at_nat @ X @ B2 )
% 0.35/0.66         => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) @ B2 )
% 0.35/0.66            = ( insert8211810215607154385at_nat @ X @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_Diff_if
% 0.35/0.66  thf(fact_5754_insert__Diff__if,axiom,
% 0.35/0.66      ! [X: real,B2: set_real,A2: set_real] :
% 0.35/0.66        ( ( ( member_real @ X @ B2 )
% 0.35/0.66         => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
% 0.35/0.66            = ( minus_minus_set_real @ A2 @ B2 ) ) )
% 0.35/0.66        & ( ~ ( member_real @ X @ B2 )
% 0.35/0.66         => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
% 0.35/0.66            = ( insert_real @ X @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_Diff_if
% 0.35/0.66  thf(fact_5755_insert__Diff__if,axiom,
% 0.35/0.66      ! [X: set_nat,B2: set_set_nat,A2: set_set_nat] :
% 0.35/0.66        ( ( ( member_set_nat @ X @ B2 )
% 0.35/0.66         => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
% 0.35/0.66            = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) )
% 0.35/0.66        & ( ~ ( member_set_nat @ X @ B2 )
% 0.35/0.66         => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
% 0.35/0.66            = ( insert_set_nat @ X @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_Diff_if
% 0.35/0.66  thf(fact_5756_insert__Diff__if,axiom,
% 0.35/0.66      ! [X: int,B2: set_int,A2: set_int] :
% 0.35/0.66        ( ( ( member_int @ X @ B2 )
% 0.35/0.66         => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B2 )
% 0.35/0.66            = ( minus_minus_set_int @ A2 @ B2 ) ) )
% 0.35/0.66        & ( ~ ( member_int @ X @ B2 )
% 0.35/0.66         => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B2 )
% 0.35/0.66            = ( insert_int @ X @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_Diff_if
% 0.35/0.66  thf(fact_5757_insert__Diff__if,axiom,
% 0.35/0.66      ! [X: nat,B2: set_nat,A2: set_nat] :
% 0.35/0.66        ( ( ( member_nat @ X @ B2 )
% 0.35/0.66         => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
% 0.35/0.66            = ( minus_minus_set_nat @ A2 @ B2 ) ) )
% 0.35/0.66        & ( ~ ( member_nat @ X @ B2 )
% 0.35/0.66         => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
% 0.35/0.66            = ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_Diff_if
% 0.35/0.66  thf(fact_5758_sum__mono,axiom,
% 0.35/0.66      ! [K4: set_real,F: real > rat,G: real > rat] :
% 0.35/0.66        ( ! [I2: real] :
% 0.35/0.66            ( ( member_real @ I2 @ K4 )
% 0.35/0.66           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.35/0.66       => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K4 ) @ ( groups1300246762558778688al_rat @ G @ K4 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_mono
% 0.35/0.66  thf(fact_5759_sum__mono,axiom,
% 0.35/0.66      ! [K4: set_nat,F: nat > rat,G: nat > rat] :
% 0.35/0.66        ( ! [I2: nat] :
% 0.35/0.66            ( ( member_nat @ I2 @ K4 )
% 0.35/0.66           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.35/0.66       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K4 ) @ ( groups2906978787729119204at_rat @ G @ K4 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_mono
% 0.35/0.66  thf(fact_5760_sum__mono,axiom,
% 0.35/0.66      ! [K4: set_int,F: int > rat,G: int > rat] :
% 0.35/0.66        ( ! [I2: int] :
% 0.35/0.66            ( ( member_int @ I2 @ K4 )
% 0.35/0.66           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.35/0.66       => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K4 ) @ ( groups3906332499630173760nt_rat @ G @ K4 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_mono
% 0.35/0.66  thf(fact_5761_sum__mono,axiom,
% 0.35/0.66      ! [K4: set_real,F: real > nat,G: real > nat] :
% 0.35/0.66        ( ! [I2: real] :
% 0.35/0.66            ( ( member_real @ I2 @ K4 )
% 0.35/0.66           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.35/0.66       => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K4 ) @ ( groups1935376822645274424al_nat @ G @ K4 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_mono
% 0.35/0.66  thf(fact_5762_sum__mono,axiom,
% 0.35/0.66      ! [K4: set_int,F: int > nat,G: int > nat] :
% 0.35/0.66        ( ! [I2: int] :
% 0.35/0.66            ( ( member_int @ I2 @ K4 )
% 0.35/0.66           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.35/0.66       => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K4 ) @ ( groups4541462559716669496nt_nat @ G @ K4 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_mono
% 0.35/0.66  thf(fact_5763_sum__mono,axiom,
% 0.35/0.66      ! [K4: set_real,F: real > int,G: real > int] :
% 0.35/0.66        ( ! [I2: real] :
% 0.35/0.66            ( ( member_real @ I2 @ K4 )
% 0.35/0.66           => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.35/0.66       => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K4 ) @ ( groups1932886352136224148al_int @ G @ K4 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_mono
% 0.35/0.66  thf(fact_5764_sum__mono,axiom,
% 0.35/0.66      ! [K4: set_nat,F: nat > int,G: nat > int] :
% 0.35/0.66        ( ! [I2: nat] :
% 0.35/0.66            ( ( member_nat @ I2 @ K4 )
% 0.35/0.66           => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.35/0.66       => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K4 ) @ ( groups3539618377306564664at_int @ G @ K4 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_mono
% 0.35/0.66  thf(fact_5765_sum__mono,axiom,
% 0.35/0.66      ! [K4: set_int,F: int > int,G: int > int] :
% 0.35/0.66        ( ! [I2: int] :
% 0.35/0.66            ( ( member_int @ I2 @ K4 )
% 0.35/0.66           => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.35/0.66       => ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ K4 ) @ ( groups4538972089207619220nt_int @ G @ K4 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_mono
% 0.35/0.66  thf(fact_5766_sum__mono,axiom,
% 0.35/0.66      ! [K4: set_nat,F: nat > nat,G: nat > nat] :
% 0.35/0.66        ( ! [I2: nat] :
% 0.35/0.66            ( ( member_nat @ I2 @ K4 )
% 0.35/0.66           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.35/0.66       => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K4 ) @ ( groups3542108847815614940at_nat @ G @ K4 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_mono
% 0.35/0.66  thf(fact_5767_sum__mono,axiom,
% 0.35/0.66      ! [K4: set_nat,F: nat > real,G: nat > real] :
% 0.35/0.66        ( ! [I2: nat] :
% 0.35/0.66            ( ( member_nat @ I2 @ K4 )
% 0.35/0.66           => ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.35/0.66       => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K4 ) @ ( groups6591440286371151544t_real @ G @ K4 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_mono
% 0.35/0.66  thf(fact_5768_sum_Oswap__restrict,axiom,
% 0.35/0.66      ! [A2: set_real,B2: set_nat,G: real > nat > nat,R: real > nat > $o] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ( finite_finite_nat @ B2 )
% 0.35/0.66         => ( ( groups1935376822645274424al_nat
% 0.35/0.66              @ ^ [X3: real] :
% 0.35/0.66                  ( groups3542108847815614940at_nat @ ( G @ X3 )
% 0.35/0.66                  @ ( collect_nat
% 0.35/0.66                    @ ^ [Y2: nat] :
% 0.35/0.66                        ( ( member_nat @ Y2 @ B2 )
% 0.35/0.66                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.66              @ A2 )
% 0.35/0.66            = ( groups3542108847815614940at_nat
% 0.35/0.66              @ ^ [Y2: nat] :
% 0.35/0.66                  ( groups1935376822645274424al_nat
% 0.35/0.66                  @ ^ [X3: real] : ( G @ X3 @ Y2 )
% 0.35/0.66                  @ ( collect_real
% 0.35/0.66                    @ ^ [X3: real] :
% 0.35/0.66                        ( ( member_real @ X3 @ A2 )
% 0.35/0.66                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.66              @ B2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.swap_restrict
% 0.35/0.66  thf(fact_5769_sum_Oswap__restrict,axiom,
% 0.35/0.66      ! [A2: set_int,B2: set_nat,G: int > nat > nat,R: int > nat > $o] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ( finite_finite_nat @ B2 )
% 0.35/0.66         => ( ( groups4541462559716669496nt_nat
% 0.35/0.66              @ ^ [X3: int] :
% 0.35/0.66                  ( groups3542108847815614940at_nat @ ( G @ X3 )
% 0.35/0.66                  @ ( collect_nat
% 0.35/0.66                    @ ^ [Y2: nat] :
% 0.35/0.66                        ( ( member_nat @ Y2 @ B2 )
% 0.35/0.66                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.66              @ A2 )
% 0.35/0.66            = ( groups3542108847815614940at_nat
% 0.35/0.66              @ ^ [Y2: nat] :
% 0.35/0.66                  ( groups4541462559716669496nt_nat
% 0.35/0.66                  @ ^ [X3: int] : ( G @ X3 @ Y2 )
% 0.35/0.66                  @ ( collect_int
% 0.35/0.66                    @ ^ [X3: int] :
% 0.35/0.66                        ( ( member_int @ X3 @ A2 )
% 0.35/0.66                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.66              @ B2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.swap_restrict
% 0.35/0.66  thf(fact_5770_sum_Oswap__restrict,axiom,
% 0.35/0.66      ! [A2: set_complex,B2: set_nat,G: complex > nat > nat,R: complex > nat > $o] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ( finite_finite_nat @ B2 )
% 0.35/0.66         => ( ( groups5693394587270226106ex_nat
% 0.35/0.66              @ ^ [X3: complex] :
% 0.35/0.66                  ( groups3542108847815614940at_nat @ ( G @ X3 )
% 0.35/0.66                  @ ( collect_nat
% 0.35/0.66                    @ ^ [Y2: nat] :
% 0.35/0.66                        ( ( member_nat @ Y2 @ B2 )
% 0.35/0.66                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.66              @ A2 )
% 0.35/0.66            = ( groups3542108847815614940at_nat
% 0.35/0.66              @ ^ [Y2: nat] :
% 0.35/0.66                  ( groups5693394587270226106ex_nat
% 0.35/0.66                  @ ^ [X3: complex] : ( G @ X3 @ Y2 )
% 0.35/0.66                  @ ( collect_complex
% 0.35/0.66                    @ ^ [X3: complex] :
% 0.35/0.66                        ( ( member_complex @ X3 @ A2 )
% 0.35/0.66                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.66              @ B2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.swap_restrict
% 0.35/0.66  thf(fact_5771_sum_Oswap__restrict,axiom,
% 0.35/0.66      ! [A2: set_real,B2: set_complex,G: real > complex > complex,R: real > complex > $o] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.66         => ( ( groups5754745047067104278omplex
% 0.35/0.66              @ ^ [X3: real] :
% 0.35/0.66                  ( groups7754918857620584856omplex @ ( G @ X3 )
% 0.35/0.66                  @ ( collect_complex
% 0.35/0.66                    @ ^ [Y2: complex] :
% 0.35/0.66                        ( ( member_complex @ Y2 @ B2 )
% 0.35/0.66                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.66              @ A2 )
% 0.35/0.66            = ( groups7754918857620584856omplex
% 0.35/0.66              @ ^ [Y2: complex] :
% 0.35/0.66                  ( groups5754745047067104278omplex
% 0.35/0.66                  @ ^ [X3: real] : ( G @ X3 @ Y2 )
% 0.35/0.66                  @ ( collect_real
% 0.35/0.66                    @ ^ [X3: real] :
% 0.35/0.66                        ( ( member_real @ X3 @ A2 )
% 0.35/0.66                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.66              @ B2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.swap_restrict
% 0.35/0.66  thf(fact_5772_sum_Oswap__restrict,axiom,
% 0.35/0.66      ! [A2: set_nat,B2: set_complex,G: nat > complex > complex,R: nat > complex > $o] :
% 0.35/0.66        ( ( finite_finite_nat @ A2 )
% 0.35/0.66       => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.66         => ( ( groups2073611262835488442omplex
% 0.35/0.66              @ ^ [X3: nat] :
% 0.35/0.66                  ( groups7754918857620584856omplex @ ( G @ X3 )
% 0.35/0.66                  @ ( collect_complex
% 0.35/0.66                    @ ^ [Y2: complex] :
% 0.35/0.66                        ( ( member_complex @ Y2 @ B2 )
% 0.35/0.66                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.66              @ A2 )
% 0.35/0.66            = ( groups7754918857620584856omplex
% 0.35/0.66              @ ^ [Y2: complex] :
% 0.35/0.66                  ( groups2073611262835488442omplex
% 0.35/0.66                  @ ^ [X3: nat] : ( G @ X3 @ Y2 )
% 0.35/0.66                  @ ( collect_nat
% 0.35/0.66                    @ ^ [X3: nat] :
% 0.35/0.66                        ( ( member_nat @ X3 @ A2 )
% 0.35/0.66                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.66              @ B2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.swap_restrict
% 0.35/0.66  thf(fact_5773_sum_Oswap__restrict,axiom,
% 0.35/0.66      ! [A2: set_int,B2: set_complex,G: int > complex > complex,R: int > complex > $o] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.66         => ( ( groups3049146728041665814omplex
% 0.35/0.66              @ ^ [X3: int] :
% 0.35/0.66                  ( groups7754918857620584856omplex @ ( G @ X3 )
% 0.35/0.66                  @ ( collect_complex
% 0.35/0.66                    @ ^ [Y2: complex] :
% 0.35/0.66                        ( ( member_complex @ Y2 @ B2 )
% 0.35/0.66                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.66              @ A2 )
% 0.35/0.66            = ( groups7754918857620584856omplex
% 0.35/0.66              @ ^ [Y2: complex] :
% 0.35/0.66                  ( groups3049146728041665814omplex
% 0.35/0.66                  @ ^ [X3: int] : ( G @ X3 @ Y2 )
% 0.35/0.66                  @ ( collect_int
% 0.35/0.66                    @ ^ [X3: int] :
% 0.35/0.66                        ( ( member_int @ X3 @ A2 )
% 0.35/0.66                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.66              @ B2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.swap_restrict
% 0.35/0.66  thf(fact_5774_sum_Oswap__restrict,axiom,
% 0.35/0.66      ! [A2: set_real,B2: set_nat,G: real > nat > real,R: real > nat > $o] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ( finite_finite_nat @ B2 )
% 0.35/0.66         => ( ( groups8097168146408367636l_real
% 0.35/0.66              @ ^ [X3: real] :
% 0.35/0.66                  ( groups6591440286371151544t_real @ ( G @ X3 )
% 0.35/0.66                  @ ( collect_nat
% 0.35/0.66                    @ ^ [Y2: nat] :
% 0.35/0.66                        ( ( member_nat @ Y2 @ B2 )
% 0.35/0.66                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.66              @ A2 )
% 0.35/0.66            = ( groups6591440286371151544t_real
% 0.35/0.66              @ ^ [Y2: nat] :
% 0.35/0.66                  ( groups8097168146408367636l_real
% 0.35/0.66                  @ ^ [X3: real] : ( G @ X3 @ Y2 )
% 0.35/0.66                  @ ( collect_real
% 0.35/0.66                    @ ^ [X3: real] :
% 0.35/0.66                        ( ( member_real @ X3 @ A2 )
% 0.35/0.66                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.66              @ B2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.swap_restrict
% 0.35/0.66  thf(fact_5775_sum_Oswap__restrict,axiom,
% 0.35/0.66      ! [A2: set_int,B2: set_nat,G: int > nat > real,R: int > nat > $o] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ( finite_finite_nat @ B2 )
% 0.35/0.66         => ( ( groups8778361861064173332t_real
% 0.35/0.66              @ ^ [X3: int] :
% 0.35/0.66                  ( groups6591440286371151544t_real @ ( G @ X3 )
% 0.35/0.66                  @ ( collect_nat
% 0.35/0.66                    @ ^ [Y2: nat] :
% 0.35/0.66                        ( ( member_nat @ Y2 @ B2 )
% 0.35/0.66                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.66              @ A2 )
% 0.35/0.66            = ( groups6591440286371151544t_real
% 0.35/0.66              @ ^ [Y2: nat] :
% 0.35/0.66                  ( groups8778361861064173332t_real
% 0.35/0.66                  @ ^ [X3: int] : ( G @ X3 @ Y2 )
% 0.35/0.66                  @ ( collect_int
% 0.35/0.66                    @ ^ [X3: int] :
% 0.35/0.66                        ( ( member_int @ X3 @ A2 )
% 0.35/0.66                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.66              @ B2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.swap_restrict
% 0.35/0.66  thf(fact_5776_sum_Oswap__restrict,axiom,
% 0.35/0.66      ! [A2: set_complex,B2: set_nat,G: complex > nat > real,R: complex > nat > $o] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ( finite_finite_nat @ B2 )
% 0.35/0.66         => ( ( groups5808333547571424918x_real
% 0.35/0.66              @ ^ [X3: complex] :
% 0.35/0.66                  ( groups6591440286371151544t_real @ ( G @ X3 )
% 0.35/0.66                  @ ( collect_nat
% 0.35/0.66                    @ ^ [Y2: nat] :
% 0.35/0.66                        ( ( member_nat @ Y2 @ B2 )
% 0.35/0.66                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.66              @ A2 )
% 0.35/0.66            = ( groups6591440286371151544t_real
% 0.35/0.66              @ ^ [Y2: nat] :
% 0.35/0.66                  ( groups5808333547571424918x_real
% 0.35/0.66                  @ ^ [X3: complex] : ( G @ X3 @ Y2 )
% 0.35/0.66                  @ ( collect_complex
% 0.35/0.66                    @ ^ [X3: complex] :
% 0.35/0.66                        ( ( member_complex @ X3 @ A2 )
% 0.35/0.66                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.66              @ B2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.swap_restrict
% 0.35/0.66  thf(fact_5777_sum_Oswap__restrict,axiom,
% 0.35/0.66      ! [A2: set_nat,B2: set_real,G: nat > real > nat,R: nat > real > $o] :
% 0.35/0.66        ( ( finite_finite_nat @ A2 )
% 0.35/0.66       => ( ( finite_finite_real @ B2 )
% 0.35/0.66         => ( ( groups3542108847815614940at_nat
% 0.35/0.66              @ ^ [X3: nat] :
% 0.35/0.66                  ( groups1935376822645274424al_nat @ ( G @ X3 )
% 0.35/0.66                  @ ( collect_real
% 0.35/0.66                    @ ^ [Y2: real] :
% 0.35/0.66                        ( ( member_real @ Y2 @ B2 )
% 0.35/0.66                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.66              @ A2 )
% 0.35/0.66            = ( groups1935376822645274424al_nat
% 0.35/0.66              @ ^ [Y2: real] :
% 0.35/0.66                  ( groups3542108847815614940at_nat
% 0.35/0.66                  @ ^ [X3: nat] : ( G @ X3 @ Y2 )
% 0.35/0.66                  @ ( collect_nat
% 0.35/0.66                    @ ^ [X3: nat] :
% 0.35/0.66                        ( ( member_nat @ X3 @ A2 )
% 0.35/0.66                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.66              @ B2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.swap_restrict
% 0.35/0.66  thf(fact_5778_sum__diff1__nat,axiom,
% 0.35/0.66      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > nat] :
% 0.35/0.66        ( ( ( member8757157785044589968at_nat @ A @ A2 )
% 0.35/0.66         => ( ( groups3860910324918113789at_nat @ F @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 0.35/0.66            = ( minus_minus_nat @ ( groups3860910324918113789at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.35/0.66        & ( ~ ( member8757157785044589968at_nat @ A @ A2 )
% 0.35/0.66         => ( ( groups3860910324918113789at_nat @ F @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 0.35/0.66            = ( groups3860910324918113789at_nat @ F @ A2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_diff1_nat
% 0.35/0.66  thf(fact_5779_sum__diff1__nat,axiom,
% 0.35/0.66      ! [A: set_nat,A2: set_set_nat,F: set_nat > nat] :
% 0.35/0.66        ( ( ( member_set_nat @ A @ A2 )
% 0.35/0.66         => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 0.35/0.66            = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.35/0.66        & ( ~ ( member_set_nat @ A @ A2 )
% 0.35/0.66         => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 0.35/0.66            = ( groups8294997508430121362at_nat @ F @ A2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_diff1_nat
% 0.35/0.66  thf(fact_5780_sum__diff1__nat,axiom,
% 0.35/0.66      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 0.35/0.66        ( ( ( member8440522571783428010at_nat @ A @ A2 )
% 0.35/0.66         => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 0.35/0.66            = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.35/0.66        & ( ~ ( member8440522571783428010at_nat @ A @ A2 )
% 0.35/0.66         => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 0.35/0.66            = ( groups977919841031483927at_nat @ F @ A2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_diff1_nat
% 0.35/0.66  thf(fact_5781_sum__diff1__nat,axiom,
% 0.35/0.66      ! [A: real,A2: set_real,F: real > nat] :
% 0.35/0.66        ( ( ( member_real @ A @ A2 )
% 0.35/0.66         => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.35/0.66            = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.35/0.66        & ( ~ ( member_real @ A @ A2 )
% 0.35/0.66         => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.35/0.66            = ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_diff1_nat
% 0.35/0.66  thf(fact_5782_sum__diff1__nat,axiom,
% 0.35/0.66      ! [A: int,A2: set_int,F: int > nat] :
% 0.35/0.66        ( ( ( member_int @ A @ A2 )
% 0.35/0.66         => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.35/0.66            = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.35/0.66        & ( ~ ( member_int @ A @ A2 )
% 0.35/0.66         => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.35/0.66            = ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_diff1_nat
% 0.35/0.66  thf(fact_5783_sum__diff1__nat,axiom,
% 0.35/0.66      ! [A: nat,A2: set_nat,F: nat > nat] :
% 0.35/0.66        ( ( ( member_nat @ A @ A2 )
% 0.35/0.66         => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 0.35/0.66            = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.35/0.66        & ( ~ ( member_nat @ A @ A2 )
% 0.35/0.66         => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 0.35/0.66            = ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_diff1_nat
% 0.35/0.66  thf(fact_5784_Collect__conv__if,axiom,
% 0.35/0.66      ! [P: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
% 0.35/0.66        ( ( ( P @ A )
% 0.35/0.66         => ( ( collec6321179662152712658at_nat
% 0.35/0.66              @ ^ [X3: produc3843707927480180839at_nat] :
% 0.35/0.66                  ( ( X3 = A )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 0.35/0.66        & ( ~ ( P @ A )
% 0.35/0.66         => ( ( collec6321179662152712658at_nat
% 0.35/0.66              @ ^ [X3: produc3843707927480180839at_nat] :
% 0.35/0.66                  ( ( X3 = A )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = bot_bo228742789529271731at_nat ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Collect_conv_if
% 0.35/0.66  thf(fact_5785_Collect__conv__if,axiom,
% 0.35/0.66      ! [P: list_nat > $o,A: list_nat] :
% 0.35/0.66        ( ( ( P @ A )
% 0.35/0.66         => ( ( collect_list_nat
% 0.35/0.66              @ ^ [X3: list_nat] :
% 0.35/0.66                  ( ( X3 = A )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
% 0.35/0.66        & ( ~ ( P @ A )
% 0.35/0.66         => ( ( collect_list_nat
% 0.35/0.66              @ ^ [X3: list_nat] :
% 0.35/0.66                  ( ( X3 = A )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = bot_bot_set_list_nat ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Collect_conv_if
% 0.35/0.66  thf(fact_5786_Collect__conv__if,axiom,
% 0.35/0.66      ! [P: set_nat > $o,A: set_nat] :
% 0.35/0.66        ( ( ( P @ A )
% 0.35/0.66         => ( ( collect_set_nat
% 0.35/0.66              @ ^ [X3: set_nat] :
% 0.35/0.66                  ( ( X3 = A )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 0.35/0.66        & ( ~ ( P @ A )
% 0.35/0.66         => ( ( collect_set_nat
% 0.35/0.66              @ ^ [X3: set_nat] :
% 0.35/0.66                  ( ( X3 = A )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = bot_bot_set_set_nat ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Collect_conv_if
% 0.35/0.66  thf(fact_5787_Collect__conv__if,axiom,
% 0.35/0.66      ! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
% 0.35/0.66        ( ( ( P @ A )
% 0.35/0.66         => ( ( collec3392354462482085612at_nat
% 0.35/0.66              @ ^ [X3: product_prod_nat_nat] :
% 0.35/0.66                  ( ( X3 = A )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 0.35/0.66        & ( ~ ( P @ A )
% 0.35/0.66         => ( ( collec3392354462482085612at_nat
% 0.35/0.66              @ ^ [X3: product_prod_nat_nat] :
% 0.35/0.66                  ( ( X3 = A )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = bot_bo2099793752762293965at_nat ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Collect_conv_if
% 0.35/0.66  thf(fact_5788_Collect__conv__if,axiom,
% 0.35/0.66      ! [P: real > $o,A: real] :
% 0.35/0.66        ( ( ( P @ A )
% 0.35/0.66         => ( ( collect_real
% 0.35/0.66              @ ^ [X3: real] :
% 0.35/0.66                  ( ( X3 = A )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.35/0.66        & ( ~ ( P @ A )
% 0.35/0.66         => ( ( collect_real
% 0.35/0.66              @ ^ [X3: real] :
% 0.35/0.66                  ( ( X3 = A )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = bot_bot_set_real ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Collect_conv_if
% 0.35/0.66  thf(fact_5789_Collect__conv__if,axiom,
% 0.35/0.66      ! [P: nat > $o,A: nat] :
% 0.35/0.66        ( ( ( P @ A )
% 0.35/0.66         => ( ( collect_nat
% 0.35/0.66              @ ^ [X3: nat] :
% 0.35/0.66                  ( ( X3 = A )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 0.35/0.66        & ( ~ ( P @ A )
% 0.35/0.66         => ( ( collect_nat
% 0.35/0.66              @ ^ [X3: nat] :
% 0.35/0.66                  ( ( X3 = A )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = bot_bot_set_nat ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Collect_conv_if
% 0.35/0.66  thf(fact_5790_Collect__conv__if,axiom,
% 0.35/0.66      ! [P: int > $o,A: int] :
% 0.35/0.66        ( ( ( P @ A )
% 0.35/0.66         => ( ( collect_int
% 0.35/0.66              @ ^ [X3: int] :
% 0.35/0.66                  ( ( X3 = A )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.35/0.66        & ( ~ ( P @ A )
% 0.35/0.66         => ( ( collect_int
% 0.35/0.66              @ ^ [X3: int] :
% 0.35/0.66                  ( ( X3 = A )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = bot_bot_set_int ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Collect_conv_if
% 0.35/0.66  thf(fact_5791_Collect__conv__if2,axiom,
% 0.35/0.66      ! [P: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
% 0.35/0.66        ( ( ( P @ A )
% 0.35/0.66         => ( ( collec6321179662152712658at_nat
% 0.35/0.66              @ ^ [X3: produc3843707927480180839at_nat] :
% 0.35/0.66                  ( ( A = X3 )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 0.35/0.66        & ( ~ ( P @ A )
% 0.35/0.66         => ( ( collec6321179662152712658at_nat
% 0.35/0.66              @ ^ [X3: produc3843707927480180839at_nat] :
% 0.35/0.66                  ( ( A = X3 )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = bot_bo228742789529271731at_nat ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Collect_conv_if2
% 0.35/0.66  thf(fact_5792_Collect__conv__if2,axiom,
% 0.35/0.66      ! [P: list_nat > $o,A: list_nat] :
% 0.35/0.66        ( ( ( P @ A )
% 0.35/0.66         => ( ( collect_list_nat
% 0.35/0.66              @ ^ [X3: list_nat] :
% 0.35/0.66                  ( ( A = X3 )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
% 0.35/0.66        & ( ~ ( P @ A )
% 0.35/0.66         => ( ( collect_list_nat
% 0.35/0.66              @ ^ [X3: list_nat] :
% 0.35/0.66                  ( ( A = X3 )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = bot_bot_set_list_nat ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Collect_conv_if2
% 0.35/0.66  thf(fact_5793_Collect__conv__if2,axiom,
% 0.35/0.66      ! [P: set_nat > $o,A: set_nat] :
% 0.35/0.66        ( ( ( P @ A )
% 0.35/0.66         => ( ( collect_set_nat
% 0.35/0.66              @ ^ [X3: set_nat] :
% 0.35/0.66                  ( ( A = X3 )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 0.35/0.66        & ( ~ ( P @ A )
% 0.35/0.66         => ( ( collect_set_nat
% 0.35/0.66              @ ^ [X3: set_nat] :
% 0.35/0.66                  ( ( A = X3 )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = bot_bot_set_set_nat ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Collect_conv_if2
% 0.35/0.66  thf(fact_5794_Collect__conv__if2,axiom,
% 0.35/0.66      ! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
% 0.35/0.66        ( ( ( P @ A )
% 0.35/0.66         => ( ( collec3392354462482085612at_nat
% 0.35/0.66              @ ^ [X3: product_prod_nat_nat] :
% 0.35/0.66                  ( ( A = X3 )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 0.35/0.66        & ( ~ ( P @ A )
% 0.35/0.66         => ( ( collec3392354462482085612at_nat
% 0.35/0.66              @ ^ [X3: product_prod_nat_nat] :
% 0.35/0.66                  ( ( A = X3 )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = bot_bo2099793752762293965at_nat ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Collect_conv_if2
% 0.35/0.66  thf(fact_5795_Collect__conv__if2,axiom,
% 0.35/0.66      ! [P: real > $o,A: real] :
% 0.35/0.66        ( ( ( P @ A )
% 0.35/0.66         => ( ( collect_real
% 0.35/0.66              @ ^ [X3: real] :
% 0.35/0.66                  ( ( A = X3 )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.35/0.66        & ( ~ ( P @ A )
% 0.35/0.66         => ( ( collect_real
% 0.35/0.66              @ ^ [X3: real] :
% 0.35/0.66                  ( ( A = X3 )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = bot_bot_set_real ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Collect_conv_if2
% 0.35/0.66  thf(fact_5796_Collect__conv__if2,axiom,
% 0.35/0.66      ! [P: nat > $o,A: nat] :
% 0.35/0.66        ( ( ( P @ A )
% 0.35/0.66         => ( ( collect_nat
% 0.35/0.66              @ ^ [X3: nat] :
% 0.35/0.66                  ( ( A = X3 )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 0.35/0.66        & ( ~ ( P @ A )
% 0.35/0.66         => ( ( collect_nat
% 0.35/0.66              @ ^ [X3: nat] :
% 0.35/0.66                  ( ( A = X3 )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = bot_bot_set_nat ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Collect_conv_if2
% 0.35/0.66  thf(fact_5797_Collect__conv__if2,axiom,
% 0.35/0.66      ! [P: int > $o,A: int] :
% 0.35/0.66        ( ( ( P @ A )
% 0.35/0.66         => ( ( collect_int
% 0.35/0.66              @ ^ [X3: int] :
% 0.35/0.66                  ( ( A = X3 )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.35/0.66        & ( ~ ( P @ A )
% 0.35/0.66         => ( ( collect_int
% 0.35/0.66              @ ^ [X3: int] :
% 0.35/0.66                  ( ( A = X3 )
% 0.35/0.66                  & ( P @ X3 ) ) )
% 0.35/0.66            = bot_bot_set_int ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Collect_conv_if2
% 0.35/0.66  thf(fact_5798_round__mono,axiom,
% 0.35/0.66      ! [X: rat,Y: rat] :
% 0.35/0.66        ( ( ord_less_eq_rat @ X @ Y )
% 0.35/0.66       => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % round_mono
% 0.35/0.66  thf(fact_5799_sum_Oinsert__remove,axiom,
% 0.35/0.66      ! [A2: set_complex,G: complex > real,X: complex] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
% 0.35/0.66          = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.insert_remove
% 0.35/0.66  thf(fact_5800_sum_Oinsert__remove,axiom,
% 0.35/0.66      ! [A2: set_complex,G: complex > rat,X: complex] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
% 0.35/0.66          = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.insert_remove
% 0.35/0.66  thf(fact_5801_sum_Oinsert__remove,axiom,
% 0.35/0.66      ! [A2: set_complex,G: complex > nat,X: complex] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
% 0.35/0.66          = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.insert_remove
% 0.35/0.66  thf(fact_5802_sum_Oinsert__remove,axiom,
% 0.35/0.66      ! [A2: set_complex,G: complex > int,X: complex] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X @ A2 ) )
% 0.35/0.66          = ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.insert_remove
% 0.35/0.66  thf(fact_5803_sum_Oinsert__remove,axiom,
% 0.35/0.66      ! [A2: set_real,G: real > real,X: real] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.66          = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.insert_remove
% 0.35/0.66  thf(fact_5804_sum_Oinsert__remove,axiom,
% 0.35/0.66      ! [A2: set_real,G: real > rat,X: real] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.66          = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.insert_remove
% 0.35/0.66  thf(fact_5805_sum_Oinsert__remove,axiom,
% 0.35/0.66      ! [A2: set_real,G: real > nat,X: real] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.66          = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.insert_remove
% 0.35/0.66  thf(fact_5806_sum_Oinsert__remove,axiom,
% 0.35/0.66      ! [A2: set_real,G: real > int,X: real] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ( groups1932886352136224148al_int @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.66          = ( plus_plus_int @ ( G @ X ) @ ( groups1932886352136224148al_int @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.insert_remove
% 0.35/0.66  thf(fact_5807_sum_Oinsert__remove,axiom,
% 0.35/0.66      ! [A2: set_int,G: int > real,X: int] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
% 0.35/0.66          = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.insert_remove
% 0.35/0.66  thf(fact_5808_sum_Oinsert__remove,axiom,
% 0.35/0.66      ! [A2: set_int,G: int > rat,X: int] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
% 0.35/0.66          = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.insert_remove
% 0.35/0.66  thf(fact_5809_sum_Oremove,axiom,
% 0.35/0.66      ! [A2: set_complex,X: complex,G: complex > real] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ( member_complex @ X @ A2 )
% 0.35/0.66         => ( ( groups5808333547571424918x_real @ G @ A2 )
% 0.35/0.66            = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.remove
% 0.35/0.66  thf(fact_5810_sum_Oremove,axiom,
% 0.35/0.66      ! [A2: set_complex,X: complex,G: complex > rat] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ( member_complex @ X @ A2 )
% 0.35/0.66         => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 0.35/0.66            = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.remove
% 0.35/0.66  thf(fact_5811_sum_Oremove,axiom,
% 0.35/0.66      ! [A2: set_complex,X: complex,G: complex > nat] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ( member_complex @ X @ A2 )
% 0.35/0.66         => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 0.35/0.66            = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.remove
% 0.35/0.66  thf(fact_5812_sum_Oremove,axiom,
% 0.35/0.66      ! [A2: set_complex,X: complex,G: complex > int] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ( member_complex @ X @ A2 )
% 0.35/0.66         => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 0.35/0.66            = ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.remove
% 0.35/0.66  thf(fact_5813_sum_Oremove,axiom,
% 0.35/0.66      ! [A2: set_real,X: real,G: real > real] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ( member_real @ X @ A2 )
% 0.35/0.66         => ( ( groups8097168146408367636l_real @ G @ A2 )
% 0.35/0.66            = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.remove
% 0.35/0.66  thf(fact_5814_sum_Oremove,axiom,
% 0.35/0.66      ! [A2: set_real,X: real,G: real > rat] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ( member_real @ X @ A2 )
% 0.35/0.66         => ( ( groups1300246762558778688al_rat @ G @ A2 )
% 0.35/0.66            = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.remove
% 0.35/0.66  thf(fact_5815_sum_Oremove,axiom,
% 0.35/0.66      ! [A2: set_real,X: real,G: real > nat] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ( member_real @ X @ A2 )
% 0.35/0.66         => ( ( groups1935376822645274424al_nat @ G @ A2 )
% 0.35/0.66            = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.remove
% 0.35/0.66  thf(fact_5816_sum_Oremove,axiom,
% 0.35/0.66      ! [A2: set_real,X: real,G: real > int] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ( member_real @ X @ A2 )
% 0.35/0.66         => ( ( groups1932886352136224148al_int @ G @ A2 )
% 0.35/0.66            = ( plus_plus_int @ ( G @ X ) @ ( groups1932886352136224148al_int @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.remove
% 0.35/0.66  thf(fact_5817_sum_Oremove,axiom,
% 0.35/0.66      ! [A2: set_int,X: int,G: int > real] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ( member_int @ X @ A2 )
% 0.35/0.66         => ( ( groups8778361861064173332t_real @ G @ A2 )
% 0.35/0.66            = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.remove
% 0.35/0.66  thf(fact_5818_sum_Oremove,axiom,
% 0.35/0.66      ! [A2: set_int,X: int,G: int > rat] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ( member_int @ X @ A2 )
% 0.35/0.66         => ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 0.35/0.66            = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.remove
% 0.35/0.66  thf(fact_5819_sum__diff1,axiom,
% 0.35/0.66      ! [A2: set_complex,A: complex,F: complex > real] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ( ( member_complex @ A @ A2 )
% 0.35/0.66           => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 0.35/0.66              = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.35/0.66          & ( ~ ( member_complex @ A @ A2 )
% 0.35/0.66           => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 0.35/0.66              = ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_diff1
% 0.35/0.66  thf(fact_5820_sum__diff1,axiom,
% 0.35/0.66      ! [A2: set_real,A: real,F: real > real] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ( ( member_real @ A @ A2 )
% 0.35/0.66           => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.35/0.66              = ( minus_minus_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.35/0.66          & ( ~ ( member_real @ A @ A2 )
% 0.35/0.66           => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.35/0.66              = ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_diff1
% 0.35/0.66  thf(fact_5821_sum__diff1,axiom,
% 0.35/0.66      ! [A2: set_int,A: int,F: int > real] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ( ( member_int @ A @ A2 )
% 0.35/0.66           => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.35/0.66              = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.35/0.66          & ( ~ ( member_int @ A @ A2 )
% 0.35/0.66           => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.35/0.66              = ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_diff1
% 0.35/0.66  thf(fact_5822_sum__diff1,axiom,
% 0.35/0.66      ! [A2: set_complex,A: complex,F: complex > rat] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ( ( member_complex @ A @ A2 )
% 0.35/0.66           => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 0.35/0.66              = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.35/0.66          & ( ~ ( member_complex @ A @ A2 )
% 0.35/0.66           => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 0.35/0.66              = ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_diff1
% 0.35/0.66  thf(fact_5823_sum__diff1,axiom,
% 0.35/0.66      ! [A2: set_real,A: real,F: real > rat] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ( ( member_real @ A @ A2 )
% 0.35/0.66           => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.35/0.66              = ( minus_minus_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.35/0.66          & ( ~ ( member_real @ A @ A2 )
% 0.35/0.66           => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.35/0.66              = ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_diff1
% 0.35/0.66  thf(fact_5824_sum__diff1,axiom,
% 0.35/0.66      ! [A2: set_int,A: int,F: int > rat] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ( ( member_int @ A @ A2 )
% 0.35/0.66           => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.35/0.66              = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.35/0.66          & ( ~ ( member_int @ A @ A2 )
% 0.35/0.66           => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.35/0.66              = ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_diff1
% 0.35/0.66  thf(fact_5825_sum__diff1,axiom,
% 0.35/0.66      ! [A2: set_complex,A: complex,F: complex > int] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ( ( member_complex @ A @ A2 )
% 0.35/0.66           => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 0.35/0.66              = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.35/0.66          & ( ~ ( member_complex @ A @ A2 )
% 0.35/0.66           => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 0.35/0.66              = ( groups5690904116761175830ex_int @ F @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_diff1
% 0.35/0.66  thf(fact_5826_sum__diff1,axiom,
% 0.35/0.66      ! [A2: set_real,A: real,F: real > int] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ( ( member_real @ A @ A2 )
% 0.35/0.66           => ( ( groups1932886352136224148al_int @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.35/0.66              = ( minus_minus_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.35/0.66          & ( ~ ( member_real @ A @ A2 )
% 0.35/0.66           => ( ( groups1932886352136224148al_int @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.35/0.66              = ( groups1932886352136224148al_int @ F @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_diff1
% 0.35/0.66  thf(fact_5827_sum__diff1,axiom,
% 0.35/0.66      ! [A2: set_int,A: int,F: int > int] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ( ( member_int @ A @ A2 )
% 0.35/0.66           => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.35/0.66              = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.35/0.66          & ( ~ ( member_int @ A @ A2 )
% 0.35/0.66           => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.35/0.66              = ( groups4538972089207619220nt_int @ F @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_diff1
% 0.35/0.66  thf(fact_5828_sum__diff1,axiom,
% 0.35/0.66      ! [A2: set_nat,A: nat,F: nat > rat] :
% 0.35/0.66        ( ( finite_finite_nat @ A2 )
% 0.35/0.66       => ( ( ( member_nat @ A @ A2 )
% 0.35/0.66           => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 0.35/0.66              = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.35/0.66          & ( ~ ( member_nat @ A @ A2 )
% 0.35/0.66           => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 0.35/0.66              = ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_diff1
% 0.35/0.66  thf(fact_5829_sum__nonneg,axiom,
% 0.35/0.66      ! [A2: set_real,F: real > real] :
% 0.35/0.66        ( ! [X4: real] :
% 0.35/0.66            ( ( member_real @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.35/0.66       => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonneg
% 0.35/0.66  thf(fact_5830_sum__nonneg,axiom,
% 0.35/0.66      ! [A2: set_int,F: int > real] :
% 0.35/0.66        ( ! [X4: int] :
% 0.35/0.66            ( ( member_int @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.35/0.66       => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonneg
% 0.35/0.66  thf(fact_5831_sum__nonneg,axiom,
% 0.35/0.66      ! [A2: set_real,F: real > rat] :
% 0.35/0.66        ( ! [X4: real] :
% 0.35/0.66            ( ( member_real @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.35/0.66       => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonneg
% 0.35/0.66  thf(fact_5832_sum__nonneg,axiom,
% 0.35/0.66      ! [A2: set_nat,F: nat > rat] :
% 0.35/0.66        ( ! [X4: nat] :
% 0.35/0.66            ( ( member_nat @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.35/0.66       => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonneg
% 0.35/0.66  thf(fact_5833_sum__nonneg,axiom,
% 0.35/0.66      ! [A2: set_int,F: int > rat] :
% 0.35/0.66        ( ! [X4: int] :
% 0.35/0.66            ( ( member_int @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.35/0.66       => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonneg
% 0.35/0.66  thf(fact_5834_sum__nonneg,axiom,
% 0.35/0.66      ! [A2: set_real,F: real > nat] :
% 0.35/0.66        ( ! [X4: real] :
% 0.35/0.66            ( ( member_real @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.35/0.66       => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonneg
% 0.35/0.66  thf(fact_5835_sum__nonneg,axiom,
% 0.35/0.66      ! [A2: set_int,F: int > nat] :
% 0.35/0.66        ( ! [X4: int] :
% 0.35/0.66            ( ( member_int @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.35/0.66       => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonneg
% 0.35/0.66  thf(fact_5836_sum__nonneg,axiom,
% 0.35/0.66      ! [A2: set_real,F: real > int] :
% 0.35/0.66        ( ! [X4: real] :
% 0.35/0.66            ( ( member_real @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 0.35/0.66       => ( ord_less_eq_int @ zero_zero_int @ ( groups1932886352136224148al_int @ F @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonneg
% 0.35/0.66  thf(fact_5837_sum__nonneg,axiom,
% 0.35/0.66      ! [A2: set_nat,F: nat > int] :
% 0.35/0.66        ( ! [X4: nat] :
% 0.35/0.66            ( ( member_nat @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 0.35/0.66       => ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonneg
% 0.35/0.66  thf(fact_5838_sum__nonneg,axiom,
% 0.35/0.66      ! [A2: set_int,F: int > int] :
% 0.35/0.66        ( ! [X4: int] :
% 0.35/0.66            ( ( member_int @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 0.35/0.66       => ( ord_less_eq_int @ zero_zero_int @ ( groups4538972089207619220nt_int @ F @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonneg
% 0.35/0.66  thf(fact_5839_sum__nonpos,axiom,
% 0.35/0.66      ! [A2: set_real,F: real > real] :
% 0.35/0.66        ( ! [X4: real] :
% 0.35/0.66            ( ( member_real @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
% 0.35/0.66       => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonpos
% 0.35/0.66  thf(fact_5840_sum__nonpos,axiom,
% 0.35/0.66      ! [A2: set_int,F: int > real] :
% 0.35/0.66        ( ! [X4: int] :
% 0.35/0.66            ( ( member_int @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
% 0.35/0.66       => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonpos
% 0.35/0.66  thf(fact_5841_sum__nonpos,axiom,
% 0.35/0.66      ! [A2: set_real,F: real > rat] :
% 0.35/0.66        ( ! [X4: real] :
% 0.35/0.66            ( ( member_real @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
% 0.35/0.66       => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonpos
% 0.35/0.66  thf(fact_5842_sum__nonpos,axiom,
% 0.35/0.66      ! [A2: set_nat,F: nat > rat] :
% 0.35/0.66        ( ! [X4: nat] :
% 0.35/0.66            ( ( member_nat @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
% 0.35/0.66       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonpos
% 0.35/0.66  thf(fact_5843_sum__nonpos,axiom,
% 0.35/0.66      ! [A2: set_int,F: int > rat] :
% 0.35/0.66        ( ! [X4: int] :
% 0.35/0.66            ( ( member_int @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
% 0.35/0.66       => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonpos
% 0.35/0.66  thf(fact_5844_sum__nonpos,axiom,
% 0.35/0.66      ! [A2: set_real,F: real > nat] :
% 0.35/0.66        ( ! [X4: real] :
% 0.35/0.66            ( ( member_real @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
% 0.35/0.66       => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonpos
% 0.35/0.66  thf(fact_5845_sum__nonpos,axiom,
% 0.35/0.66      ! [A2: set_int,F: int > nat] :
% 0.35/0.66        ( ! [X4: int] :
% 0.35/0.66            ( ( member_int @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
% 0.35/0.66       => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonpos
% 0.35/0.66  thf(fact_5846_sum__nonpos,axiom,
% 0.35/0.66      ! [A2: set_real,F: real > int] :
% 0.35/0.66        ( ! [X4: real] :
% 0.35/0.66            ( ( member_real @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
% 0.35/0.66       => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ zero_zero_int ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonpos
% 0.35/0.66  thf(fact_5847_sum__nonpos,axiom,
% 0.35/0.66      ! [A2: set_nat,F: nat > int] :
% 0.35/0.66        ( ! [X4: nat] :
% 0.35/0.66            ( ( member_nat @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
% 0.35/0.66       => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ zero_zero_int ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonpos
% 0.35/0.66  thf(fact_5848_sum__nonpos,axiom,
% 0.35/0.66      ! [A2: set_int,F: int > int] :
% 0.35/0.66        ( ! [X4: int] :
% 0.35/0.66            ( ( member_int @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
% 0.35/0.66       => ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ zero_zero_int ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonpos
% 0.35/0.66  thf(fact_5849_sum__mono__inv,axiom,
% 0.35/0.66      ! [F: real > rat,I6: set_real,G: real > rat,I: real] :
% 0.35/0.66        ( ( ( groups1300246762558778688al_rat @ F @ I6 )
% 0.35/0.66          = ( groups1300246762558778688al_rat @ G @ I6 ) )
% 0.35/0.66       => ( ! [I2: real] :
% 0.35/0.66              ( ( member_real @ I2 @ I6 )
% 0.35/0.66             => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.35/0.66         => ( ( member_real @ I @ I6 )
% 0.35/0.66           => ( ( finite_finite_real @ I6 )
% 0.35/0.66             => ( ( F @ I )
% 0.35/0.66                = ( G @ I ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_mono_inv
% 0.35/0.66  thf(fact_5850_sum__mono__inv,axiom,
% 0.35/0.66      ! [F: nat > rat,I6: set_nat,G: nat > rat,I: nat] :
% 0.35/0.66        ( ( ( groups2906978787729119204at_rat @ F @ I6 )
% 0.35/0.66          = ( groups2906978787729119204at_rat @ G @ I6 ) )
% 0.35/0.66       => ( ! [I2: nat] :
% 0.35/0.66              ( ( member_nat @ I2 @ I6 )
% 0.35/0.66             => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.35/0.66         => ( ( member_nat @ I @ I6 )
% 0.35/0.66           => ( ( finite_finite_nat @ I6 )
% 0.35/0.66             => ( ( F @ I )
% 0.35/0.66                = ( G @ I ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_mono_inv
% 0.35/0.66  thf(fact_5851_sum__mono__inv,axiom,
% 0.35/0.66      ! [F: int > rat,I6: set_int,G: int > rat,I: int] :
% 0.35/0.66        ( ( ( groups3906332499630173760nt_rat @ F @ I6 )
% 0.35/0.66          = ( groups3906332499630173760nt_rat @ G @ I6 ) )
% 0.35/0.66       => ( ! [I2: int] :
% 0.35/0.66              ( ( member_int @ I2 @ I6 )
% 0.35/0.66             => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.35/0.66         => ( ( member_int @ I @ I6 )
% 0.35/0.66           => ( ( finite_finite_int @ I6 )
% 0.35/0.66             => ( ( F @ I )
% 0.35/0.66                = ( G @ I ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_mono_inv
% 0.35/0.66  thf(fact_5852_sum__mono__inv,axiom,
% 0.35/0.66      ! [F: complex > rat,I6: set_complex,G: complex > rat,I: complex] :
% 0.35/0.66        ( ( ( groups5058264527183730370ex_rat @ F @ I6 )
% 0.35/0.66          = ( groups5058264527183730370ex_rat @ G @ I6 ) )
% 0.35/0.66       => ( ! [I2: complex] :
% 0.35/0.66              ( ( member_complex @ I2 @ I6 )
% 0.35/0.66             => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.35/0.66         => ( ( member_complex @ I @ I6 )
% 0.35/0.66           => ( ( finite3207457112153483333omplex @ I6 )
% 0.35/0.66             => ( ( F @ I )
% 0.35/0.66                = ( G @ I ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_mono_inv
% 0.35/0.66  thf(fact_5853_sum__mono__inv,axiom,
% 0.35/0.66      ! [F: real > nat,I6: set_real,G: real > nat,I: real] :
% 0.35/0.66        ( ( ( groups1935376822645274424al_nat @ F @ I6 )
% 0.35/0.66          = ( groups1935376822645274424al_nat @ G @ I6 ) )
% 0.35/0.66       => ( ! [I2: real] :
% 0.35/0.66              ( ( member_real @ I2 @ I6 )
% 0.35/0.66             => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.35/0.66         => ( ( member_real @ I @ I6 )
% 0.35/0.66           => ( ( finite_finite_real @ I6 )
% 0.35/0.66             => ( ( F @ I )
% 0.35/0.66                = ( G @ I ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_mono_inv
% 0.35/0.66  thf(fact_5854_sum__mono__inv,axiom,
% 0.35/0.66      ! [F: int > nat,I6: set_int,G: int > nat,I: int] :
% 0.35/0.66        ( ( ( groups4541462559716669496nt_nat @ F @ I6 )
% 0.35/0.66          = ( groups4541462559716669496nt_nat @ G @ I6 ) )
% 0.35/0.66       => ( ! [I2: int] :
% 0.35/0.66              ( ( member_int @ I2 @ I6 )
% 0.35/0.66             => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.35/0.66         => ( ( member_int @ I @ I6 )
% 0.35/0.66           => ( ( finite_finite_int @ I6 )
% 0.35/0.66             => ( ( F @ I )
% 0.35/0.66                = ( G @ I ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_mono_inv
% 0.35/0.66  thf(fact_5855_sum__mono__inv,axiom,
% 0.35/0.66      ! [F: complex > nat,I6: set_complex,G: complex > nat,I: complex] :
% 0.35/0.66        ( ( ( groups5693394587270226106ex_nat @ F @ I6 )
% 0.35/0.66          = ( groups5693394587270226106ex_nat @ G @ I6 ) )
% 0.35/0.66       => ( ! [I2: complex] :
% 0.35/0.66              ( ( member_complex @ I2 @ I6 )
% 0.35/0.66             => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.35/0.66         => ( ( member_complex @ I @ I6 )
% 0.35/0.66           => ( ( finite3207457112153483333omplex @ I6 )
% 0.35/0.66             => ( ( F @ I )
% 0.35/0.66                = ( G @ I ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_mono_inv
% 0.35/0.66  thf(fact_5856_sum__mono__inv,axiom,
% 0.35/0.66      ! [F: real > int,I6: set_real,G: real > int,I: real] :
% 0.35/0.66        ( ( ( groups1932886352136224148al_int @ F @ I6 )
% 0.35/0.66          = ( groups1932886352136224148al_int @ G @ I6 ) )
% 0.35/0.66       => ( ! [I2: real] :
% 0.35/0.66              ( ( member_real @ I2 @ I6 )
% 0.35/0.66             => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.35/0.66         => ( ( member_real @ I @ I6 )
% 0.35/0.66           => ( ( finite_finite_real @ I6 )
% 0.35/0.66             => ( ( F @ I )
% 0.35/0.66                = ( G @ I ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_mono_inv
% 0.35/0.66  thf(fact_5857_sum__mono__inv,axiom,
% 0.35/0.66      ! [F: nat > int,I6: set_nat,G: nat > int,I: nat] :
% 0.35/0.66        ( ( ( groups3539618377306564664at_int @ F @ I6 )
% 0.35/0.66          = ( groups3539618377306564664at_int @ G @ I6 ) )
% 0.35/0.66       => ( ! [I2: nat] :
% 0.35/0.66              ( ( member_nat @ I2 @ I6 )
% 0.35/0.66             => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.35/0.66         => ( ( member_nat @ I @ I6 )
% 0.35/0.66           => ( ( finite_finite_nat @ I6 )
% 0.35/0.66             => ( ( F @ I )
% 0.35/0.66                = ( G @ I ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_mono_inv
% 0.35/0.66  thf(fact_5858_sum__mono__inv,axiom,
% 0.35/0.66      ! [F: int > int,I6: set_int,G: int > int,I: int] :
% 0.35/0.66        ( ( ( groups4538972089207619220nt_int @ F @ I6 )
% 0.35/0.66          = ( groups4538972089207619220nt_int @ G @ I6 ) )
% 0.35/0.66       => ( ! [I2: int] :
% 0.35/0.66              ( ( member_int @ I2 @ I6 )
% 0.35/0.66             => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.35/0.66         => ( ( member_int @ I @ I6 )
% 0.35/0.66           => ( ( finite_finite_int @ I6 )
% 0.35/0.66             => ( ( F @ I )
% 0.35/0.66                = ( G @ I ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_mono_inv
% 0.35/0.66  thf(fact_5859_sum__cong__Suc,axiom,
% 0.35/0.66      ! [A2: set_nat,F: nat > nat,G: nat > nat] :
% 0.35/0.66        ( ~ ( member_nat @ zero_zero_nat @ A2 )
% 0.35/0.66       => ( ! [X4: nat] :
% 0.35/0.66              ( ( member_nat @ ( suc @ X4 ) @ A2 )
% 0.35/0.66             => ( ( F @ ( suc @ X4 ) )
% 0.35/0.66                = ( G @ ( suc @ X4 ) ) ) )
% 0.35/0.66         => ( ( groups3542108847815614940at_nat @ F @ A2 )
% 0.35/0.66            = ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_cong_Suc
% 0.35/0.66  thf(fact_5860_sum__cong__Suc,axiom,
% 0.35/0.66      ! [A2: set_nat,F: nat > real,G: nat > real] :
% 0.35/0.66        ( ~ ( member_nat @ zero_zero_nat @ A2 )
% 0.35/0.66       => ( ! [X4: nat] :
% 0.35/0.66              ( ( member_nat @ ( suc @ X4 ) @ A2 )
% 0.35/0.66             => ( ( F @ ( suc @ X4 ) )
% 0.35/0.66                = ( G @ ( suc @ X4 ) ) ) )
% 0.35/0.66         => ( ( groups6591440286371151544t_real @ F @ A2 )
% 0.35/0.66            = ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_cong_Suc
% 0.35/0.66  thf(fact_5861_sum_Odelta__remove,axiom,
% 0.35/0.66      ! [S2: set_complex,A: complex,B: complex > real,C: complex > real] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.66       => ( ( ( member_complex @ A @ S2 )
% 0.35/0.66           => ( ( groups5808333547571424918x_real
% 0.35/0.66                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.35/0.66                @ S2 )
% 0.35/0.66              = ( plus_plus_real @ ( B @ A ) @ ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
% 0.35/0.66          & ( ~ ( member_complex @ A @ S2 )
% 0.35/0.66           => ( ( groups5808333547571424918x_real
% 0.35/0.66                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.35/0.66                @ S2 )
% 0.35/0.66              = ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.delta_remove
% 0.35/0.66  thf(fact_5862_sum_Odelta__remove,axiom,
% 0.35/0.66      ! [S2: set_complex,A: complex,B: complex > rat,C: complex > rat] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.66       => ( ( ( member_complex @ A @ S2 )
% 0.35/0.66           => ( ( groups5058264527183730370ex_rat
% 0.35/0.66                @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.35/0.66                @ S2 )
% 0.35/0.66              = ( plus_plus_rat @ ( B @ A ) @ ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
% 0.35/0.66          & ( ~ ( member_complex @ A @ S2 )
% 0.35/0.66           => ( ( groups5058264527183730370ex_rat
% 0.35/0.66                @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.35/0.66                @ S2 )
% 0.35/0.66              = ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.delta_remove
% 0.35/0.66  thf(fact_5863_sum_Odelta__remove,axiom,
% 0.35/0.66      ! [S2: set_complex,A: complex,B: complex > nat,C: complex > nat] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.66       => ( ( ( member_complex @ A @ S2 )
% 0.35/0.66           => ( ( groups5693394587270226106ex_nat
% 0.35/0.66                @ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.35/0.66                @ S2 )
% 0.35/0.66              = ( plus_plus_nat @ ( B @ A ) @ ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
% 0.35/0.66          & ( ~ ( member_complex @ A @ S2 )
% 0.35/0.66           => ( ( groups5693394587270226106ex_nat
% 0.35/0.66                @ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.35/0.66                @ S2 )
% 0.35/0.66              = ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.delta_remove
% 0.35/0.66  thf(fact_5864_sum_Odelta__remove,axiom,
% 0.35/0.66      ! [S2: set_complex,A: complex,B: complex > int,C: complex > int] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.66       => ( ( ( member_complex @ A @ S2 )
% 0.35/0.66           => ( ( groups5690904116761175830ex_int
% 0.35/0.66                @ ^ [K3: complex] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.35/0.66                @ S2 )
% 0.35/0.66              = ( plus_plus_int @ ( B @ A ) @ ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
% 0.35/0.66          & ( ~ ( member_complex @ A @ S2 )
% 0.35/0.66           => ( ( groups5690904116761175830ex_int
% 0.35/0.66                @ ^ [K3: complex] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.35/0.66                @ S2 )
% 0.35/0.66              = ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.delta_remove
% 0.35/0.66  thf(fact_5865_sum_Odelta__remove,axiom,
% 0.35/0.66      ! [S2: set_real,A: real,B: real > real,C: real > real] :
% 0.35/0.66        ( ( finite_finite_real @ S2 )
% 0.35/0.66       => ( ( ( member_real @ A @ S2 )
% 0.35/0.66           => ( ( groups8097168146408367636l_real
% 0.35/0.66                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.35/0.66                @ S2 )
% 0.35/0.66              = ( plus_plus_real @ ( B @ A ) @ ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
% 0.35/0.66          & ( ~ ( member_real @ A @ S2 )
% 0.35/0.66           => ( ( groups8097168146408367636l_real
% 0.35/0.66                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.35/0.66                @ S2 )
% 0.35/0.66              = ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.delta_remove
% 0.35/0.66  thf(fact_5866_sum_Odelta__remove,axiom,
% 0.35/0.66      ! [S2: set_real,A: real,B: real > rat,C: real > rat] :
% 0.35/0.66        ( ( finite_finite_real @ S2 )
% 0.35/0.66       => ( ( ( member_real @ A @ S2 )
% 0.35/0.66           => ( ( groups1300246762558778688al_rat
% 0.35/0.66                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.35/0.66                @ S2 )
% 0.35/0.66              = ( plus_plus_rat @ ( B @ A ) @ ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
% 0.35/0.66          & ( ~ ( member_real @ A @ S2 )
% 0.35/0.66           => ( ( groups1300246762558778688al_rat
% 0.35/0.66                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.35/0.66                @ S2 )
% 0.35/0.66              = ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.delta_remove
% 0.35/0.66  thf(fact_5867_sum_Odelta__remove,axiom,
% 0.35/0.66      ! [S2: set_real,A: real,B: real > nat,C: real > nat] :
% 0.35/0.66        ( ( finite_finite_real @ S2 )
% 0.35/0.66       => ( ( ( member_real @ A @ S2 )
% 0.35/0.66           => ( ( groups1935376822645274424al_nat
% 0.35/0.66                @ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.35/0.66                @ S2 )
% 0.35/0.66              = ( plus_plus_nat @ ( B @ A ) @ ( groups1935376822645274424al_nat @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
% 0.35/0.66          & ( ~ ( member_real @ A @ S2 )
% 0.35/0.66           => ( ( groups1935376822645274424al_nat
% 0.35/0.66                @ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.35/0.66                @ S2 )
% 0.35/0.66              = ( groups1935376822645274424al_nat @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.delta_remove
% 0.35/0.66  thf(fact_5868_sum_Odelta__remove,axiom,
% 0.35/0.66      ! [S2: set_real,A: real,B: real > int,C: real > int] :
% 0.35/0.66        ( ( finite_finite_real @ S2 )
% 0.35/0.66       => ( ( ( member_real @ A @ S2 )
% 0.35/0.66           => ( ( groups1932886352136224148al_int
% 0.35/0.66                @ ^ [K3: real] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.35/0.66                @ S2 )
% 0.35/0.66              = ( plus_plus_int @ ( B @ A ) @ ( groups1932886352136224148al_int @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
% 0.35/0.66          & ( ~ ( member_real @ A @ S2 )
% 0.35/0.66           => ( ( groups1932886352136224148al_int
% 0.35/0.66                @ ^ [K3: real] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.35/0.66                @ S2 )
% 0.35/0.66              = ( groups1932886352136224148al_int @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.delta_remove
% 0.35/0.66  thf(fact_5869_sum_Odelta__remove,axiom,
% 0.35/0.66      ! [S2: set_int,A: int,B: int > real,C: int > real] :
% 0.35/0.66        ( ( finite_finite_int @ S2 )
% 0.35/0.66       => ( ( ( member_int @ A @ S2 )
% 0.35/0.66           => ( ( groups8778361861064173332t_real
% 0.35/0.66                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.35/0.66                @ S2 )
% 0.35/0.66              = ( plus_plus_real @ ( B @ A ) @ ( groups8778361861064173332t_real @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
% 0.35/0.66          & ( ~ ( member_int @ A @ S2 )
% 0.35/0.66           => ( ( groups8778361861064173332t_real
% 0.35/0.66                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.35/0.66                @ S2 )
% 0.35/0.66              = ( groups8778361861064173332t_real @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.delta_remove
% 0.35/0.66  thf(fact_5870_sum_Odelta__remove,axiom,
% 0.35/0.66      ! [S2: set_int,A: int,B: int > rat,C: int > rat] :
% 0.35/0.66        ( ( finite_finite_int @ S2 )
% 0.35/0.66       => ( ( ( member_int @ A @ S2 )
% 0.35/0.66           => ( ( groups3906332499630173760nt_rat
% 0.35/0.66                @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.35/0.66                @ S2 )
% 0.35/0.66              = ( plus_plus_rat @ ( B @ A ) @ ( groups3906332499630173760nt_rat @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
% 0.35/0.66          & ( ~ ( member_int @ A @ S2 )
% 0.35/0.66           => ( ( groups3906332499630173760nt_rat
% 0.35/0.66                @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.35/0.66                @ S2 )
% 0.35/0.66              = ( groups3906332499630173760nt_rat @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.delta_remove
% 0.35/0.66  thf(fact_5871_infinite__finite__induct,axiom,
% 0.35/0.66      ! [P: set_Pr4329608150637261639at_nat > $o,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.66        ( ! [A5: set_Pr4329608150637261639at_nat] :
% 0.35/0.66            ( ~ ( finite4343798906461161616at_nat @ A5 )
% 0.35/0.66           => ( P @ A5 ) )
% 0.35/0.66       => ( ( P @ bot_bo228742789529271731at_nat )
% 0.35/0.66         => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
% 0.35/0.66                ( ( finite4343798906461161616at_nat @ F4 )
% 0.35/0.66               => ( ~ ( member8757157785044589968at_nat @ X4 @ F4 )
% 0.35/0.66                 => ( ( P @ F4 )
% 0.35/0.66                   => ( P @ ( insert9069300056098147895at_nat @ X4 @ F4 ) ) ) ) )
% 0.35/0.66           => ( P @ A2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % infinite_finite_induct
% 0.35/0.66  thf(fact_5872_infinite__finite__induct,axiom,
% 0.35/0.66      ! [P: set_set_nat > $o,A2: set_set_nat] :
% 0.35/0.66        ( ! [A5: set_set_nat] :
% 0.35/0.66            ( ~ ( finite1152437895449049373et_nat @ A5 )
% 0.35/0.66           => ( P @ A5 ) )
% 0.35/0.66       => ( ( P @ bot_bot_set_set_nat )
% 0.35/0.66         => ( ! [X4: set_nat,F4: set_set_nat] :
% 0.35/0.66                ( ( finite1152437895449049373et_nat @ F4 )
% 0.35/0.66               => ( ~ ( member_set_nat @ X4 @ F4 )
% 0.35/0.66                 => ( ( P @ F4 )
% 0.35/0.66                   => ( P @ ( insert_set_nat @ X4 @ F4 ) ) ) ) )
% 0.35/0.66           => ( P @ A2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % infinite_finite_induct
% 0.35/0.66  thf(fact_5873_infinite__finite__induct,axiom,
% 0.35/0.66      ! [P: set_complex > $o,A2: set_complex] :
% 0.35/0.66        ( ! [A5: set_complex] :
% 0.35/0.66            ( ~ ( finite3207457112153483333omplex @ A5 )
% 0.35/0.66           => ( P @ A5 ) )
% 0.35/0.66       => ( ( P @ bot_bot_set_complex )
% 0.35/0.66         => ( ! [X4: complex,F4: set_complex] :
% 0.35/0.66                ( ( finite3207457112153483333omplex @ F4 )
% 0.35/0.66               => ( ~ ( member_complex @ X4 @ F4 )
% 0.35/0.66                 => ( ( P @ F4 )
% 0.35/0.66                   => ( P @ ( insert_complex @ X4 @ F4 ) ) ) ) )
% 0.35/0.66           => ( P @ A2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % infinite_finite_induct
% 0.35/0.66  thf(fact_5874_infinite__finite__induct,axiom,
% 0.35/0.66      ! [P: set_Pr1261947904930325089at_nat > $o,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.66        ( ! [A5: set_Pr1261947904930325089at_nat] :
% 0.35/0.66            ( ~ ( finite6177210948735845034at_nat @ A5 )
% 0.35/0.66           => ( P @ A5 ) )
% 0.35/0.66       => ( ( P @ bot_bo2099793752762293965at_nat )
% 0.35/0.66         => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 0.35/0.66                ( ( finite6177210948735845034at_nat @ F4 )
% 0.35/0.66               => ( ~ ( member8440522571783428010at_nat @ X4 @ F4 )
% 0.35/0.66                 => ( ( P @ F4 )
% 0.35/0.66                   => ( P @ ( insert8211810215607154385at_nat @ X4 @ F4 ) ) ) ) )
% 0.35/0.66           => ( P @ A2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % infinite_finite_induct
% 0.35/0.66  thf(fact_5875_infinite__finite__induct,axiom,
% 0.35/0.66      ! [P: set_real > $o,A2: set_real] :
% 0.35/0.66        ( ! [A5: set_real] :
% 0.35/0.66            ( ~ ( finite_finite_real @ A5 )
% 0.35/0.66           => ( P @ A5 ) )
% 0.35/0.66       => ( ( P @ bot_bot_set_real )
% 0.35/0.66         => ( ! [X4: real,F4: set_real] :
% 0.35/0.66                ( ( finite_finite_real @ F4 )
% 0.35/0.66               => ( ~ ( member_real @ X4 @ F4 )
% 0.35/0.66                 => ( ( P @ F4 )
% 0.35/0.66                   => ( P @ ( insert_real @ X4 @ F4 ) ) ) ) )
% 0.35/0.66           => ( P @ A2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % infinite_finite_induct
% 0.35/0.66  thf(fact_5876_infinite__finite__induct,axiom,
% 0.35/0.66      ! [P: set_nat > $o,A2: set_nat] :
% 0.35/0.66        ( ! [A5: set_nat] :
% 0.35/0.66            ( ~ ( finite_finite_nat @ A5 )
% 0.35/0.66           => ( P @ A5 ) )
% 0.35/0.66       => ( ( P @ bot_bot_set_nat )
% 0.35/0.66         => ( ! [X4: nat,F4: set_nat] :
% 0.35/0.66                ( ( finite_finite_nat @ F4 )
% 0.35/0.66               => ( ~ ( member_nat @ X4 @ F4 )
% 0.35/0.66                 => ( ( P @ F4 )
% 0.35/0.66                   => ( P @ ( insert_nat @ X4 @ F4 ) ) ) ) )
% 0.35/0.66           => ( P @ A2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % infinite_finite_induct
% 0.35/0.66  thf(fact_5877_infinite__finite__induct,axiom,
% 0.35/0.66      ! [P: set_int > $o,A2: set_int] :
% 0.35/0.66        ( ! [A5: set_int] :
% 0.35/0.66            ( ~ ( finite_finite_int @ A5 )
% 0.35/0.66           => ( P @ A5 ) )
% 0.35/0.66       => ( ( P @ bot_bot_set_int )
% 0.35/0.66         => ( ! [X4: int,F4: set_int] :
% 0.35/0.66                ( ( finite_finite_int @ F4 )
% 0.35/0.66               => ( ~ ( member_int @ X4 @ F4 )
% 0.35/0.66                 => ( ( P @ F4 )
% 0.35/0.66                   => ( P @ ( insert_int @ X4 @ F4 ) ) ) ) )
% 0.35/0.66           => ( P @ A2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % infinite_finite_induct
% 0.35/0.66  thf(fact_5878_finite__ne__induct,axiom,
% 0.35/0.66      ! [F3: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 0.35/0.66        ( ( finite4343798906461161616at_nat @ F3 )
% 0.35/0.66       => ( ( F3 != bot_bo228742789529271731at_nat )
% 0.35/0.66         => ( ! [X4: produc3843707927480180839at_nat] : ( P @ ( insert9069300056098147895at_nat @ X4 @ bot_bo228742789529271731at_nat ) )
% 0.35/0.66           => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
% 0.35/0.66                  ( ( finite4343798906461161616at_nat @ F4 )
% 0.35/0.66                 => ( ( F4 != bot_bo228742789529271731at_nat )
% 0.35/0.66                   => ( ~ ( member8757157785044589968at_nat @ X4 @ F4 )
% 0.35/0.66                     => ( ( P @ F4 )
% 0.35/0.66                       => ( P @ ( insert9069300056098147895at_nat @ X4 @ F4 ) ) ) ) ) )
% 0.35/0.66             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite_ne_induct
% 0.35/0.66  thf(fact_5879_finite__ne__induct,axiom,
% 0.35/0.66      ! [F3: set_set_nat,P: set_set_nat > $o] :
% 0.35/0.66        ( ( finite1152437895449049373et_nat @ F3 )
% 0.35/0.66       => ( ( F3 != bot_bot_set_set_nat )
% 0.35/0.66         => ( ! [X4: set_nat] : ( P @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
% 0.35/0.66           => ( ! [X4: set_nat,F4: set_set_nat] :
% 0.35/0.66                  ( ( finite1152437895449049373et_nat @ F4 )
% 0.35/0.66                 => ( ( F4 != bot_bot_set_set_nat )
% 0.35/0.66                   => ( ~ ( member_set_nat @ X4 @ F4 )
% 0.35/0.66                     => ( ( P @ F4 )
% 0.35/0.66                       => ( P @ ( insert_set_nat @ X4 @ F4 ) ) ) ) ) )
% 0.35/0.66             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite_ne_induct
% 0.35/0.66  thf(fact_5880_finite__ne__induct,axiom,
% 0.35/0.66      ! [F3: set_complex,P: set_complex > $o] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ F3 )
% 0.35/0.66       => ( ( F3 != bot_bot_set_complex )
% 0.35/0.66         => ( ! [X4: complex] : ( P @ ( insert_complex @ X4 @ bot_bot_set_complex ) )
% 0.35/0.66           => ( ! [X4: complex,F4: set_complex] :
% 0.35/0.66                  ( ( finite3207457112153483333omplex @ F4 )
% 0.35/0.66                 => ( ( F4 != bot_bot_set_complex )
% 0.35/0.66                   => ( ~ ( member_complex @ X4 @ F4 )
% 0.35/0.66                     => ( ( P @ F4 )
% 0.35/0.66                       => ( P @ ( insert_complex @ X4 @ F4 ) ) ) ) ) )
% 0.35/0.66             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite_ne_induct
% 0.35/0.66  thf(fact_5881_finite__ne__induct,axiom,
% 0.35/0.66      ! [F3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.35/0.66        ( ( finite6177210948735845034at_nat @ F3 )
% 0.35/0.66       => ( ( F3 != bot_bo2099793752762293965at_nat )
% 0.35/0.66         => ( ! [X4: product_prod_nat_nat] : ( P @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) )
% 0.35/0.66           => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 0.35/0.66                  ( ( finite6177210948735845034at_nat @ F4 )
% 0.35/0.66                 => ( ( F4 != bot_bo2099793752762293965at_nat )
% 0.35/0.66                   => ( ~ ( member8440522571783428010at_nat @ X4 @ F4 )
% 0.35/0.66                     => ( ( P @ F4 )
% 0.35/0.66                       => ( P @ ( insert8211810215607154385at_nat @ X4 @ F4 ) ) ) ) ) )
% 0.35/0.66             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite_ne_induct
% 0.35/0.66  thf(fact_5882_finite__ne__induct,axiom,
% 0.35/0.66      ! [F3: set_real,P: set_real > $o] :
% 0.35/0.66        ( ( finite_finite_real @ F3 )
% 0.35/0.66       => ( ( F3 != bot_bot_set_real )
% 0.35/0.66         => ( ! [X4: real] : ( P @ ( insert_real @ X4 @ bot_bot_set_real ) )
% 0.35/0.66           => ( ! [X4: real,F4: set_real] :
% 0.35/0.66                  ( ( finite_finite_real @ F4 )
% 0.35/0.66                 => ( ( F4 != bot_bot_set_real )
% 0.35/0.66                   => ( ~ ( member_real @ X4 @ F4 )
% 0.35/0.66                     => ( ( P @ F4 )
% 0.35/0.66                       => ( P @ ( insert_real @ X4 @ F4 ) ) ) ) ) )
% 0.35/0.66             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite_ne_induct
% 0.35/0.66  thf(fact_5883_finite__ne__induct,axiom,
% 0.35/0.66      ! [F3: set_nat,P: set_nat > $o] :
% 0.35/0.66        ( ( finite_finite_nat @ F3 )
% 0.35/0.66       => ( ( F3 != bot_bot_set_nat )
% 0.35/0.66         => ( ! [X4: nat] : ( P @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
% 0.35/0.66           => ( ! [X4: nat,F4: set_nat] :
% 0.35/0.66                  ( ( finite_finite_nat @ F4 )
% 0.35/0.66                 => ( ( F4 != bot_bot_set_nat )
% 0.35/0.66                   => ( ~ ( member_nat @ X4 @ F4 )
% 0.35/0.66                     => ( ( P @ F4 )
% 0.35/0.66                       => ( P @ ( insert_nat @ X4 @ F4 ) ) ) ) ) )
% 0.35/0.66             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite_ne_induct
% 0.35/0.66  thf(fact_5884_finite__ne__induct,axiom,
% 0.35/0.66      ! [F3: set_int,P: set_int > $o] :
% 0.35/0.66        ( ( finite_finite_int @ F3 )
% 0.35/0.66       => ( ( F3 != bot_bot_set_int )
% 0.35/0.66         => ( ! [X4: int] : ( P @ ( insert_int @ X4 @ bot_bot_set_int ) )
% 0.35/0.66           => ( ! [X4: int,F4: set_int] :
% 0.35/0.66                  ( ( finite_finite_int @ F4 )
% 0.35/0.66                 => ( ( F4 != bot_bot_set_int )
% 0.35/0.66                   => ( ~ ( member_int @ X4 @ F4 )
% 0.35/0.66                     => ( ( P @ F4 )
% 0.35/0.66                       => ( P @ ( insert_int @ X4 @ F4 ) ) ) ) ) )
% 0.35/0.66             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite_ne_induct
% 0.35/0.66  thf(fact_5885_finite__induct,axiom,
% 0.35/0.66      ! [F3: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 0.35/0.66        ( ( finite4343798906461161616at_nat @ F3 )
% 0.35/0.66       => ( ( P @ bot_bo228742789529271731at_nat )
% 0.35/0.66         => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
% 0.35/0.66                ( ( finite4343798906461161616at_nat @ F4 )
% 0.35/0.66               => ( ~ ( member8757157785044589968at_nat @ X4 @ F4 )
% 0.35/0.66                 => ( ( P @ F4 )
% 0.35/0.66                   => ( P @ ( insert9069300056098147895at_nat @ X4 @ F4 ) ) ) ) )
% 0.35/0.66           => ( P @ F3 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite_induct
% 0.35/0.66  thf(fact_5886_finite__induct,axiom,
% 0.35/0.66      ! [F3: set_set_nat,P: set_set_nat > $o] :
% 0.35/0.66        ( ( finite1152437895449049373et_nat @ F3 )
% 0.35/0.66       => ( ( P @ bot_bot_set_set_nat )
% 0.35/0.66         => ( ! [X4: set_nat,F4: set_set_nat] :
% 0.35/0.66                ( ( finite1152437895449049373et_nat @ F4 )
% 0.35/0.66               => ( ~ ( member_set_nat @ X4 @ F4 )
% 0.35/0.66                 => ( ( P @ F4 )
% 0.35/0.66                   => ( P @ ( insert_set_nat @ X4 @ F4 ) ) ) ) )
% 0.35/0.66           => ( P @ F3 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite_induct
% 0.35/0.66  thf(fact_5887_finite__induct,axiom,
% 0.35/0.66      ! [F3: set_complex,P: set_complex > $o] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ F3 )
% 0.35/0.66       => ( ( P @ bot_bot_set_complex )
% 0.35/0.66         => ( ! [X4: complex,F4: set_complex] :
% 0.35/0.66                ( ( finite3207457112153483333omplex @ F4 )
% 0.35/0.66               => ( ~ ( member_complex @ X4 @ F4 )
% 0.35/0.66                 => ( ( P @ F4 )
% 0.35/0.66                   => ( P @ ( insert_complex @ X4 @ F4 ) ) ) ) )
% 0.35/0.66           => ( P @ F3 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite_induct
% 0.35/0.66  thf(fact_5888_finite__induct,axiom,
% 0.35/0.66      ! [F3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.35/0.66        ( ( finite6177210948735845034at_nat @ F3 )
% 0.35/0.66       => ( ( P @ bot_bo2099793752762293965at_nat )
% 0.35/0.66         => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 0.35/0.66                ( ( finite6177210948735845034at_nat @ F4 )
% 0.35/0.66               => ( ~ ( member8440522571783428010at_nat @ X4 @ F4 )
% 0.35/0.66                 => ( ( P @ F4 )
% 0.35/0.66                   => ( P @ ( insert8211810215607154385at_nat @ X4 @ F4 ) ) ) ) )
% 0.35/0.66           => ( P @ F3 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite_induct
% 0.35/0.66  thf(fact_5889_finite__induct,axiom,
% 0.35/0.66      ! [F3: set_real,P: set_real > $o] :
% 0.35/0.66        ( ( finite_finite_real @ F3 )
% 0.35/0.66       => ( ( P @ bot_bot_set_real )
% 0.35/0.66         => ( ! [X4: real,F4: set_real] :
% 0.35/0.66                ( ( finite_finite_real @ F4 )
% 0.35/0.66               => ( ~ ( member_real @ X4 @ F4 )
% 0.35/0.66                 => ( ( P @ F4 )
% 0.35/0.66                   => ( P @ ( insert_real @ X4 @ F4 ) ) ) ) )
% 0.35/0.66           => ( P @ F3 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite_induct
% 0.35/0.66  thf(fact_5890_finite__induct,axiom,
% 0.35/0.66      ! [F3: set_nat,P: set_nat > $o] :
% 0.35/0.66        ( ( finite_finite_nat @ F3 )
% 0.35/0.66       => ( ( P @ bot_bot_set_nat )
% 0.35/0.66         => ( ! [X4: nat,F4: set_nat] :
% 0.35/0.66                ( ( finite_finite_nat @ F4 )
% 0.35/0.66               => ( ~ ( member_nat @ X4 @ F4 )
% 0.35/0.66                 => ( ( P @ F4 )
% 0.35/0.66                   => ( P @ ( insert_nat @ X4 @ F4 ) ) ) ) )
% 0.35/0.66           => ( P @ F3 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite_induct
% 0.35/0.66  thf(fact_5891_finite__induct,axiom,
% 0.35/0.66      ! [F3: set_int,P: set_int > $o] :
% 0.35/0.66        ( ( finite_finite_int @ F3 )
% 0.35/0.66       => ( ( P @ bot_bot_set_int )
% 0.35/0.66         => ( ! [X4: int,F4: set_int] :
% 0.35/0.66                ( ( finite_finite_int @ F4 )
% 0.35/0.66               => ( ~ ( member_int @ X4 @ F4 )
% 0.35/0.66                 => ( ( P @ F4 )
% 0.35/0.66                   => ( P @ ( insert_int @ X4 @ F4 ) ) ) ) )
% 0.35/0.66           => ( P @ F3 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite_induct
% 0.35/0.66  thf(fact_5892_finite_Osimps,axiom,
% 0.35/0.66      ( finite4343798906461161616at_nat
% 0.35/0.66      = ( ^ [A4: set_Pr4329608150637261639at_nat] :
% 0.35/0.66            ( ( A4 = bot_bo228742789529271731at_nat )
% 0.35/0.66            | ? [A6: set_Pr4329608150637261639at_nat,B4: produc3843707927480180839at_nat] :
% 0.35/0.66                ( ( A4
% 0.35/0.66                  = ( insert9069300056098147895at_nat @ B4 @ A6 ) )
% 0.35/0.66                & ( finite4343798906461161616at_nat @ A6 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite.simps
% 0.35/0.66  thf(fact_5893_finite_Osimps,axiom,
% 0.35/0.66      ( finite3207457112153483333omplex
% 0.35/0.66      = ( ^ [A4: set_complex] :
% 0.35/0.66            ( ( A4 = bot_bot_set_complex )
% 0.35/0.66            | ? [A6: set_complex,B4: complex] :
% 0.35/0.66                ( ( A4
% 0.35/0.66                  = ( insert_complex @ B4 @ A6 ) )
% 0.35/0.66                & ( finite3207457112153483333omplex @ A6 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite.simps
% 0.35/0.66  thf(fact_5894_finite_Osimps,axiom,
% 0.35/0.66      ( finite6177210948735845034at_nat
% 0.35/0.66      = ( ^ [A4: set_Pr1261947904930325089at_nat] :
% 0.35/0.66            ( ( A4 = bot_bo2099793752762293965at_nat )
% 0.35/0.66            | ? [A6: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat] :
% 0.35/0.66                ( ( A4
% 0.35/0.66                  = ( insert8211810215607154385at_nat @ B4 @ A6 ) )
% 0.35/0.66                & ( finite6177210948735845034at_nat @ A6 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite.simps
% 0.35/0.66  thf(fact_5895_finite_Osimps,axiom,
% 0.35/0.66      ( finite_finite_real
% 0.35/0.66      = ( ^ [A4: set_real] :
% 0.35/0.66            ( ( A4 = bot_bot_set_real )
% 0.35/0.66            | ? [A6: set_real,B4: real] :
% 0.35/0.66                ( ( A4
% 0.35/0.66                  = ( insert_real @ B4 @ A6 ) )
% 0.35/0.66                & ( finite_finite_real @ A6 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite.simps
% 0.35/0.66  thf(fact_5896_finite_Osimps,axiom,
% 0.35/0.66      ( finite_finite_nat
% 0.35/0.66      = ( ^ [A4: set_nat] :
% 0.35/0.66            ( ( A4 = bot_bot_set_nat )
% 0.35/0.66            | ? [A6: set_nat,B4: nat] :
% 0.35/0.66                ( ( A4
% 0.35/0.66                  = ( insert_nat @ B4 @ A6 ) )
% 0.35/0.66                & ( finite_finite_nat @ A6 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite.simps
% 0.35/0.66  thf(fact_5897_finite_Osimps,axiom,
% 0.35/0.66      ( finite_finite_int
% 0.35/0.66      = ( ^ [A4: set_int] :
% 0.35/0.66            ( ( A4 = bot_bot_set_int )
% 0.35/0.66            | ? [A6: set_int,B4: int] :
% 0.35/0.66                ( ( A4
% 0.35/0.66                  = ( insert_int @ B4 @ A6 ) )
% 0.35/0.66                & ( finite_finite_int @ A6 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite.simps
% 0.35/0.66  thf(fact_5898_finite_Ocases,axiom,
% 0.35/0.66      ! [A: set_Pr4329608150637261639at_nat] :
% 0.35/0.66        ( ( finite4343798906461161616at_nat @ A )
% 0.35/0.66       => ( ( A != bot_bo228742789529271731at_nat )
% 0.35/0.66         => ~ ! [A5: set_Pr4329608150637261639at_nat] :
% 0.35/0.66                ( ? [A3: produc3843707927480180839at_nat] :
% 0.35/0.66                    ( A
% 0.35/0.66                    = ( insert9069300056098147895at_nat @ A3 @ A5 ) )
% 0.35/0.66               => ~ ( finite4343798906461161616at_nat @ A5 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite.cases
% 0.35/0.66  thf(fact_5899_finite_Ocases,axiom,
% 0.35/0.66      ! [A: set_complex] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A )
% 0.35/0.66       => ( ( A != bot_bot_set_complex )
% 0.35/0.66         => ~ ! [A5: set_complex] :
% 0.35/0.66                ( ? [A3: complex] :
% 0.35/0.66                    ( A
% 0.35/0.66                    = ( insert_complex @ A3 @ A5 ) )
% 0.35/0.66               => ~ ( finite3207457112153483333omplex @ A5 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite.cases
% 0.35/0.66  thf(fact_5900_finite_Ocases,axiom,
% 0.35/0.66      ! [A: set_Pr1261947904930325089at_nat] :
% 0.35/0.66        ( ( finite6177210948735845034at_nat @ A )
% 0.35/0.66       => ( ( A != bot_bo2099793752762293965at_nat )
% 0.35/0.66         => ~ ! [A5: set_Pr1261947904930325089at_nat] :
% 0.35/0.66                ( ? [A3: product_prod_nat_nat] :
% 0.35/0.66                    ( A
% 0.35/0.66                    = ( insert8211810215607154385at_nat @ A3 @ A5 ) )
% 0.35/0.66               => ~ ( finite6177210948735845034at_nat @ A5 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite.cases
% 0.35/0.66  thf(fact_5901_finite_Ocases,axiom,
% 0.35/0.66      ! [A: set_real] :
% 0.35/0.66        ( ( finite_finite_real @ A )
% 0.35/0.66       => ( ( A != bot_bot_set_real )
% 0.35/0.66         => ~ ! [A5: set_real] :
% 0.35/0.66                ( ? [A3: real] :
% 0.35/0.66                    ( A
% 0.35/0.66                    = ( insert_real @ A3 @ A5 ) )
% 0.35/0.66               => ~ ( finite_finite_real @ A5 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite.cases
% 0.35/0.66  thf(fact_5902_finite_Ocases,axiom,
% 0.35/0.66      ! [A: set_nat] :
% 0.35/0.66        ( ( finite_finite_nat @ A )
% 0.35/0.66       => ( ( A != bot_bot_set_nat )
% 0.35/0.66         => ~ ! [A5: set_nat] :
% 0.35/0.66                ( ? [A3: nat] :
% 0.35/0.66                    ( A
% 0.35/0.66                    = ( insert_nat @ A3 @ A5 ) )
% 0.35/0.66               => ~ ( finite_finite_nat @ A5 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite.cases
% 0.35/0.66  thf(fact_5903_finite_Ocases,axiom,
% 0.35/0.66      ! [A: set_int] :
% 0.35/0.66        ( ( finite_finite_int @ A )
% 0.35/0.66       => ( ( A != bot_bot_set_int )
% 0.35/0.66         => ~ ! [A5: set_int] :
% 0.35/0.66                ( ? [A3: int] :
% 0.35/0.66                    ( A
% 0.35/0.66                    = ( insert_int @ A3 @ A5 ) )
% 0.35/0.66               => ~ ( finite_finite_int @ A5 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % finite.cases
% 0.35/0.66  thf(fact_5904_subset__singletonD,axiom,
% 0.35/0.66      ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
% 0.35/0.66        ( ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
% 0.35/0.66       => ( ( A2 = bot_bo228742789529271731at_nat )
% 0.35/0.66          | ( A2
% 0.35/0.66            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_singletonD
% 0.35/0.66  thf(fact_5905_subset__singletonD,axiom,
% 0.35/0.66      ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
% 0.35/0.66        ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
% 0.35/0.66       => ( ( A2 = bot_bo2099793752762293965at_nat )
% 0.35/0.66          | ( A2
% 0.35/0.66            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_singletonD
% 0.35/0.66  thf(fact_5906_subset__singletonD,axiom,
% 0.35/0.66      ! [A2: set_real,X: real] :
% 0.35/0.66        ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) )
% 0.35/0.66       => ( ( A2 = bot_bot_set_real )
% 0.35/0.66          | ( A2
% 0.35/0.66            = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_singletonD
% 0.35/0.66  thf(fact_5907_subset__singletonD,axiom,
% 0.35/0.66      ! [A2: set_nat,X: nat] :
% 0.35/0.66        ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.35/0.66       => ( ( A2 = bot_bot_set_nat )
% 0.35/0.66          | ( A2
% 0.35/0.66            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_singletonD
% 0.35/0.66  thf(fact_5908_subset__singletonD,axiom,
% 0.35/0.66      ! [A2: set_int,X: int] :
% 0.35/0.66        ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.35/0.66       => ( ( A2 = bot_bot_set_int )
% 0.35/0.66          | ( A2
% 0.35/0.66            = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_singletonD
% 0.35/0.66  thf(fact_5909_subset__singleton__iff,axiom,
% 0.35/0.66      ! [X8: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
% 0.35/0.66        ( ( ord_le1268244103169919719at_nat @ X8 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
% 0.35/0.66        = ( ( X8 = bot_bo228742789529271731at_nat )
% 0.35/0.66          | ( X8
% 0.35/0.66            = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_singleton_iff
% 0.35/0.66  thf(fact_5910_subset__singleton__iff,axiom,
% 0.35/0.66      ! [X8: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
% 0.35/0.66        ( ( ord_le3146513528884898305at_nat @ X8 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
% 0.35/0.66        = ( ( X8 = bot_bo2099793752762293965at_nat )
% 0.35/0.66          | ( X8
% 0.35/0.66            = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_singleton_iff
% 0.35/0.66  thf(fact_5911_subset__singleton__iff,axiom,
% 0.35/0.66      ! [X8: set_real,A: real] :
% 0.35/0.66        ( ( ord_less_eq_set_real @ X8 @ ( insert_real @ A @ bot_bot_set_real ) )
% 0.35/0.66        = ( ( X8 = bot_bot_set_real )
% 0.35/0.66          | ( X8
% 0.35/0.66            = ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_singleton_iff
% 0.35/0.66  thf(fact_5912_subset__singleton__iff,axiom,
% 0.35/0.66      ! [X8: set_nat,A: nat] :
% 0.35/0.66        ( ( ord_less_eq_set_nat @ X8 @ ( insert_nat @ A @ bot_bot_set_nat ) )
% 0.35/0.66        = ( ( X8 = bot_bot_set_nat )
% 0.35/0.66          | ( X8
% 0.35/0.66            = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_singleton_iff
% 0.35/0.66  thf(fact_5913_subset__singleton__iff,axiom,
% 0.35/0.66      ! [X8: set_int,A: int] :
% 0.35/0.66        ( ( ord_less_eq_set_int @ X8 @ ( insert_int @ A @ bot_bot_set_int ) )
% 0.35/0.66        = ( ( X8 = bot_bot_set_int )
% 0.35/0.66          | ( X8
% 0.35/0.66            = ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_singleton_iff
% 0.35/0.66  thf(fact_5914_atLeastAtMost__singleton_H,axiom,
% 0.35/0.66      ! [A: nat,B: nat] :
% 0.35/0.66        ( ( A = B )
% 0.35/0.66       => ( ( set_or1269000886237332187st_nat @ A @ B )
% 0.35/0.66          = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % atLeastAtMost_singleton'
% 0.35/0.66  thf(fact_5915_atLeastAtMost__singleton_H,axiom,
% 0.35/0.66      ! [A: int,B: int] :
% 0.35/0.66        ( ( A = B )
% 0.35/0.66       => ( ( set_or1266510415728281911st_int @ A @ B )
% 0.35/0.66          = ( insert_int @ A @ bot_bot_set_int ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % atLeastAtMost_singleton'
% 0.35/0.66  thf(fact_5916_atLeastAtMost__singleton_H,axiom,
% 0.35/0.66      ! [A: real,B: real] :
% 0.35/0.66        ( ( A = B )
% 0.35/0.66       => ( ( set_or1222579329274155063t_real @ A @ B )
% 0.35/0.66          = ( insert_real @ A @ bot_bot_set_real ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % atLeastAtMost_singleton'
% 0.35/0.66  thf(fact_5917_Diff__insert__absorb,axiom,
% 0.35/0.66      ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.66        ( ~ ( member8757157785044589968at_nat @ X @ A2 )
% 0.35/0.66       => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A2 ) @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
% 0.35/0.66          = A2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Diff_insert_absorb
% 0.35/0.66  thf(fact_5918_Diff__insert__absorb,axiom,
% 0.35/0.66      ! [X: set_nat,A2: set_set_nat] :
% 0.35/0.66        ( ~ ( member_set_nat @ X @ A2 )
% 0.35/0.66       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 0.35/0.66          = A2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Diff_insert_absorb
% 0.35/0.66  thf(fact_5919_Diff__insert__absorb,axiom,
% 0.35/0.66      ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.66        ( ~ ( member8440522571783428010at_nat @ X @ A2 )
% 0.35/0.66       => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
% 0.35/0.66          = A2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Diff_insert_absorb
% 0.35/0.66  thf(fact_5920_Diff__insert__absorb,axiom,
% 0.35/0.66      ! [X: real,A2: set_real] :
% 0.35/0.66        ( ~ ( member_real @ X @ A2 )
% 0.35/0.66       => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ ( insert_real @ X @ bot_bot_set_real ) )
% 0.35/0.66          = A2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Diff_insert_absorb
% 0.35/0.66  thf(fact_5921_Diff__insert__absorb,axiom,
% 0.35/0.66      ! [X: int,A2: set_int] :
% 0.35/0.66        ( ~ ( member_int @ X @ A2 )
% 0.35/0.66       => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.35/0.66          = A2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Diff_insert_absorb
% 0.35/0.66  thf(fact_5922_Diff__insert__absorb,axiom,
% 0.35/0.66      ! [X: nat,A2: set_nat] :
% 0.35/0.66        ( ~ ( member_nat @ X @ A2 )
% 0.35/0.66       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.35/0.66          = A2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Diff_insert_absorb
% 0.35/0.66  thf(fact_5923_Diff__insert2,axiom,
% 0.35/0.66      ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.66        ( ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) )
% 0.35/0.66        = ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ B2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Diff_insert2
% 0.35/0.66  thf(fact_5924_Diff__insert2,axiom,
% 0.35/0.66      ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.66        ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) )
% 0.35/0.66        = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ B2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Diff_insert2
% 0.35/0.66  thf(fact_5925_Diff__insert2,axiom,
% 0.35/0.66      ! [A2: set_real,A: real,B2: set_real] :
% 0.35/0.66        ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
% 0.35/0.66        = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) @ B2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Diff_insert2
% 0.35/0.66  thf(fact_5926_Diff__insert2,axiom,
% 0.35/0.66      ! [A2: set_int,A: int,B2: set_int] :
% 0.35/0.66        ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
% 0.35/0.66        = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Diff_insert2
% 0.35/0.66  thf(fact_5927_Diff__insert2,axiom,
% 0.35/0.66      ! [A2: set_nat,A: nat,B2: set_nat] :
% 0.35/0.66        ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
% 0.35/0.66        = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Diff_insert2
% 0.35/0.66  thf(fact_5928_insert__Diff,axiom,
% 0.35/0.66      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.66        ( ( member8757157785044589968at_nat @ A @ A2 )
% 0.35/0.66       => ( ( insert9069300056098147895at_nat @ A @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 0.35/0.66          = A2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_Diff
% 0.35/0.66  thf(fact_5929_insert__Diff,axiom,
% 0.35/0.66      ! [A: set_nat,A2: set_set_nat] :
% 0.35/0.66        ( ( member_set_nat @ A @ A2 )
% 0.35/0.66       => ( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 0.35/0.66          = A2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_Diff
% 0.35/0.66  thf(fact_5930_insert__Diff,axiom,
% 0.35/0.66      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.66        ( ( member8440522571783428010at_nat @ A @ A2 )
% 0.35/0.66       => ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 0.35/0.66          = A2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_Diff
% 0.35/0.66  thf(fact_5931_insert__Diff,axiom,
% 0.35/0.66      ! [A: real,A2: set_real] :
% 0.35/0.66        ( ( member_real @ A @ A2 )
% 0.35/0.66       => ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.35/0.66          = A2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_Diff
% 0.35/0.66  thf(fact_5932_insert__Diff,axiom,
% 0.35/0.66      ! [A: int,A2: set_int] :
% 0.35/0.66        ( ( member_int @ A @ A2 )
% 0.35/0.66       => ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.35/0.66          = A2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_Diff
% 0.35/0.66  thf(fact_5933_insert__Diff,axiom,
% 0.35/0.66      ! [A: nat,A2: set_nat] :
% 0.35/0.66        ( ( member_nat @ A @ A2 )
% 0.35/0.66       => ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 0.35/0.66          = A2 ) ) ).
% 0.35/0.66  
% 0.35/0.66  % insert_Diff
% 0.35/0.66  thf(fact_5934_Diff__insert,axiom,
% 0.35/0.66      ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.66        ( ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) )
% 0.35/0.66        = ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Diff_insert
% 0.35/0.66  thf(fact_5935_Diff__insert,axiom,
% 0.35/0.66      ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.66        ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) )
% 0.35/0.66        = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Diff_insert
% 0.35/0.66  thf(fact_5936_Diff__insert,axiom,
% 0.35/0.66      ! [A2: set_real,A: real,B2: set_real] :
% 0.35/0.66        ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
% 0.35/0.66        = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ B2 ) @ ( insert_real @ A @ bot_bot_set_real ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Diff_insert
% 0.35/0.66  thf(fact_5937_Diff__insert,axiom,
% 0.35/0.66      ! [A2: set_int,A: int,B2: set_int] :
% 0.35/0.66        ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
% 0.35/0.66        = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Diff_insert
% 0.35/0.66  thf(fact_5938_Diff__insert,axiom,
% 0.35/0.66      ! [A2: set_nat,A: nat,B2: set_nat] :
% 0.35/0.66        ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
% 0.35/0.66        = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Diff_insert
% 0.35/0.66  thf(fact_5939_Compl__insert,axiom,
% 0.35/0.66      ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.66        ( ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ X @ A2 ) )
% 0.35/0.66        = ( minus_3314409938677909166at_nat @ ( uminus935396558254630718at_nat @ A2 ) @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Compl_insert
% 0.35/0.66  thf(fact_5940_Compl__insert,axiom,
% 0.35/0.66      ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.66        ( ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) )
% 0.35/0.66        = ( minus_1356011639430497352at_nat @ ( uminus6524753893492686040at_nat @ A2 ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Compl_insert
% 0.35/0.66  thf(fact_5941_Compl__insert,axiom,
% 0.35/0.66      ! [X: real,A2: set_real] :
% 0.35/0.66        ( ( uminus612125837232591019t_real @ ( insert_real @ X @ A2 ) )
% 0.35/0.66        = ( minus_minus_set_real @ ( uminus612125837232591019t_real @ A2 ) @ ( insert_real @ X @ bot_bot_set_real ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Compl_insert
% 0.35/0.66  thf(fact_5942_Compl__insert,axiom,
% 0.35/0.66      ! [X: int,A2: set_int] :
% 0.35/0.66        ( ( uminus1532241313380277803et_int @ ( insert_int @ X @ A2 ) )
% 0.35/0.66        = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Compl_insert
% 0.35/0.66  thf(fact_5943_Compl__insert,axiom,
% 0.35/0.66      ! [X: nat,A2: set_nat] :
% 0.35/0.66        ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A2 ) )
% 0.35/0.66        = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % Compl_insert
% 0.35/0.66  thf(fact_5944_subset__Diff__insert,axiom,
% 0.35/0.66      ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat] :
% 0.35/0.66        ( ( ord_le1268244103169919719at_nat @ A2 @ ( minus_3314409938677909166at_nat @ B2 @ ( insert9069300056098147895at_nat @ X @ C2 ) ) )
% 0.35/0.66        = ( ( ord_le1268244103169919719at_nat @ A2 @ ( minus_3314409938677909166at_nat @ B2 @ C2 ) )
% 0.35/0.66          & ~ ( member8757157785044589968at_nat @ X @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_Diff_insert
% 0.35/0.66  thf(fact_5945_subset__Diff__insert,axiom,
% 0.35/0.66      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat] :
% 0.35/0.66        ( ( ord_le3146513528884898305at_nat @ A2 @ ( minus_1356011639430497352at_nat @ B2 @ ( insert8211810215607154385at_nat @ X @ C2 ) ) )
% 0.35/0.66        = ( ( ord_le3146513528884898305at_nat @ A2 @ ( minus_1356011639430497352at_nat @ B2 @ C2 ) )
% 0.35/0.66          & ~ ( member8440522571783428010at_nat @ X @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_Diff_insert
% 0.35/0.66  thf(fact_5946_subset__Diff__insert,axiom,
% 0.35/0.66      ! [A2: set_real,B2: set_real,X: real,C2: set_real] :
% 0.35/0.66        ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ ( insert_real @ X @ C2 ) ) )
% 0.35/0.66        = ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ C2 ) )
% 0.35/0.66          & ~ ( member_real @ X @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_Diff_insert
% 0.35/0.66  thf(fact_5947_subset__Diff__insert,axiom,
% 0.35/0.66      ! [A2: set_set_nat,B2: set_set_nat,X: set_nat,C2: set_set_nat] :
% 0.35/0.66        ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ ( insert_set_nat @ X @ C2 ) ) )
% 0.35/0.66        = ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ C2 ) )
% 0.35/0.66          & ~ ( member_set_nat @ X @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_Diff_insert
% 0.35/0.66  thf(fact_5948_subset__Diff__insert,axiom,
% 0.35/0.66      ! [A2: set_nat,B2: set_nat,X: nat,C2: set_nat] :
% 0.35/0.66        ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X @ C2 ) ) )
% 0.35/0.66        = ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C2 ) )
% 0.35/0.66          & ~ ( member_nat @ X @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_Diff_insert
% 0.35/0.66  thf(fact_5949_subset__Diff__insert,axiom,
% 0.35/0.66      ! [A2: set_int,B2: set_int,X: int,C2: set_int] :
% 0.35/0.66        ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ ( insert_int @ X @ C2 ) ) )
% 0.35/0.66        = ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ C2 ) )
% 0.35/0.66          & ~ ( member_int @ X @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % subset_Diff_insert
% 0.35/0.66  thf(fact_5950_sum_Ointer__filter,axiom,
% 0.35/0.66      ! [A2: set_real,G: real > complex,P: real > $o] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ( groups5754745047067104278omplex @ G
% 0.35/0.66            @ ( collect_real
% 0.35/0.66              @ ^ [X3: real] :
% 0.35/0.66                  ( ( member_real @ X3 @ A2 )
% 0.35/0.66                  & ( P @ X3 ) ) ) )
% 0.35/0.66          = ( groups5754745047067104278omplex
% 0.35/0.66            @ ^ [X3: real] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_complex )
% 0.35/0.66            @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.inter_filter
% 0.35/0.66  thf(fact_5951_sum_Ointer__filter,axiom,
% 0.35/0.66      ! [A2: set_nat,G: nat > complex,P: nat > $o] :
% 0.35/0.66        ( ( finite_finite_nat @ A2 )
% 0.35/0.66       => ( ( groups2073611262835488442omplex @ G
% 0.35/0.66            @ ( collect_nat
% 0.35/0.66              @ ^ [X3: nat] :
% 0.35/0.66                  ( ( member_nat @ X3 @ A2 )
% 0.35/0.66                  & ( P @ X3 ) ) ) )
% 0.35/0.66          = ( groups2073611262835488442omplex
% 0.35/0.66            @ ^ [X3: nat] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_complex )
% 0.35/0.66            @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.inter_filter
% 0.35/0.66  thf(fact_5952_sum_Ointer__filter,axiom,
% 0.35/0.66      ! [A2: set_int,G: int > complex,P: int > $o] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ( groups3049146728041665814omplex @ G
% 0.35/0.66            @ ( collect_int
% 0.35/0.66              @ ^ [X3: int] :
% 0.35/0.66                  ( ( member_int @ X3 @ A2 )
% 0.35/0.66                  & ( P @ X3 ) ) ) )
% 0.35/0.66          = ( groups3049146728041665814omplex
% 0.35/0.66            @ ^ [X3: int] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_complex )
% 0.35/0.66            @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.inter_filter
% 0.35/0.66  thf(fact_5953_sum_Ointer__filter,axiom,
% 0.35/0.66      ! [A2: set_real,G: real > real,P: real > $o] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ( groups8097168146408367636l_real @ G
% 0.35/0.66            @ ( collect_real
% 0.35/0.66              @ ^ [X3: real] :
% 0.35/0.66                  ( ( member_real @ X3 @ A2 )
% 0.35/0.66                  & ( P @ X3 ) ) ) )
% 0.35/0.66          = ( groups8097168146408367636l_real
% 0.35/0.66            @ ^ [X3: real] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
% 0.35/0.66            @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.inter_filter
% 0.35/0.66  thf(fact_5954_sum_Ointer__filter,axiom,
% 0.35/0.66      ! [A2: set_int,G: int > real,P: int > $o] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ( groups8778361861064173332t_real @ G
% 0.35/0.66            @ ( collect_int
% 0.35/0.66              @ ^ [X3: int] :
% 0.35/0.66                  ( ( member_int @ X3 @ A2 )
% 0.35/0.66                  & ( P @ X3 ) ) ) )
% 0.35/0.66          = ( groups8778361861064173332t_real
% 0.35/0.66            @ ^ [X3: int] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
% 0.35/0.66            @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.inter_filter
% 0.35/0.66  thf(fact_5955_sum_Ointer__filter,axiom,
% 0.35/0.66      ! [A2: set_complex,G: complex > real,P: complex > $o] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ( groups5808333547571424918x_real @ G
% 0.35/0.66            @ ( collect_complex
% 0.35/0.66              @ ^ [X3: complex] :
% 0.35/0.66                  ( ( member_complex @ X3 @ A2 )
% 0.35/0.66                  & ( P @ X3 ) ) ) )
% 0.35/0.66          = ( groups5808333547571424918x_real
% 0.35/0.66            @ ^ [X3: complex] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
% 0.35/0.66            @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.inter_filter
% 0.35/0.66  thf(fact_5956_sum_Ointer__filter,axiom,
% 0.35/0.66      ! [A2: set_real,G: real > rat,P: real > $o] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ( groups1300246762558778688al_rat @ G
% 0.35/0.66            @ ( collect_real
% 0.35/0.66              @ ^ [X3: real] :
% 0.35/0.66                  ( ( member_real @ X3 @ A2 )
% 0.35/0.66                  & ( P @ X3 ) ) ) )
% 0.35/0.66          = ( groups1300246762558778688al_rat
% 0.35/0.66            @ ^ [X3: real] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
% 0.35/0.66            @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.inter_filter
% 0.35/0.66  thf(fact_5957_sum_Ointer__filter,axiom,
% 0.35/0.66      ! [A2: set_nat,G: nat > rat,P: nat > $o] :
% 0.35/0.66        ( ( finite_finite_nat @ A2 )
% 0.35/0.66       => ( ( groups2906978787729119204at_rat @ G
% 0.35/0.66            @ ( collect_nat
% 0.35/0.66              @ ^ [X3: nat] :
% 0.35/0.66                  ( ( member_nat @ X3 @ A2 )
% 0.35/0.66                  & ( P @ X3 ) ) ) )
% 0.35/0.66          = ( groups2906978787729119204at_rat
% 0.35/0.66            @ ^ [X3: nat] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
% 0.35/0.66            @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.inter_filter
% 0.35/0.66  thf(fact_5958_sum_Ointer__filter,axiom,
% 0.35/0.66      ! [A2: set_int,G: int > rat,P: int > $o] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ( groups3906332499630173760nt_rat @ G
% 0.35/0.66            @ ( collect_int
% 0.35/0.66              @ ^ [X3: int] :
% 0.35/0.66                  ( ( member_int @ X3 @ A2 )
% 0.35/0.66                  & ( P @ X3 ) ) ) )
% 0.35/0.66          = ( groups3906332499630173760nt_rat
% 0.35/0.66            @ ^ [X3: int] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
% 0.35/0.66            @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.inter_filter
% 0.35/0.66  thf(fact_5959_sum_Ointer__filter,axiom,
% 0.35/0.66      ! [A2: set_complex,G: complex > rat,P: complex > $o] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ( groups5058264527183730370ex_rat @ G
% 0.35/0.66            @ ( collect_complex
% 0.35/0.66              @ ^ [X3: complex] :
% 0.35/0.66                  ( ( member_complex @ X3 @ A2 )
% 0.35/0.66                  & ( P @ X3 ) ) ) )
% 0.35/0.66          = ( groups5058264527183730370ex_rat
% 0.35/0.66            @ ^ [X3: complex] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
% 0.35/0.66            @ A2 ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.inter_filter
% 0.35/0.66  thf(fact_5960_member__le__sum,axiom,
% 0.35/0.66      ! [I: complex,A2: set_complex,F: complex > real] :
% 0.35/0.66        ( ( member_complex @ I @ A2 )
% 0.35/0.66       => ( ! [X4: complex] :
% 0.35/0.66              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.66             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.35/0.66         => ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66           => ( ord_less_eq_real @ ( F @ I ) @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % member_le_sum
% 0.35/0.66  thf(fact_5961_member__le__sum,axiom,
% 0.35/0.66      ! [I: real,A2: set_real,F: real > real] :
% 0.35/0.66        ( ( member_real @ I @ A2 )
% 0.35/0.66       => ( ! [X4: real] :
% 0.35/0.66              ( ( member_real @ X4 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.35/0.66             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.35/0.66         => ( ( finite_finite_real @ A2 )
% 0.35/0.66           => ( ord_less_eq_real @ ( F @ I ) @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % member_le_sum
% 0.35/0.66  thf(fact_5962_member__le__sum,axiom,
% 0.35/0.66      ! [I: int,A2: set_int,F: int > real] :
% 0.35/0.66        ( ( member_int @ I @ A2 )
% 0.35/0.66       => ( ! [X4: int] :
% 0.35/0.66              ( ( member_int @ X4 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 0.35/0.66             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.35/0.66         => ( ( finite_finite_int @ A2 )
% 0.35/0.66           => ( ord_less_eq_real @ ( F @ I ) @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % member_le_sum
% 0.35/0.66  thf(fact_5963_member__le__sum,axiom,
% 0.35/0.66      ! [I: complex,A2: set_complex,F: complex > rat] :
% 0.35/0.66        ( ( member_complex @ I @ A2 )
% 0.35/0.66       => ( ! [X4: complex] :
% 0.35/0.66              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.66             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.35/0.66         => ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66           => ( ord_less_eq_rat @ ( F @ I ) @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % member_le_sum
% 0.35/0.66  thf(fact_5964_member__le__sum,axiom,
% 0.35/0.66      ! [I: real,A2: set_real,F: real > rat] :
% 0.35/0.66        ( ( member_real @ I @ A2 )
% 0.35/0.66       => ( ! [X4: real] :
% 0.35/0.66              ( ( member_real @ X4 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.35/0.66             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.35/0.66         => ( ( finite_finite_real @ A2 )
% 0.35/0.66           => ( ord_less_eq_rat @ ( F @ I ) @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % member_le_sum
% 0.35/0.66  thf(fact_5965_member__le__sum,axiom,
% 0.35/0.66      ! [I: int,A2: set_int,F: int > rat] :
% 0.35/0.66        ( ( member_int @ I @ A2 )
% 0.35/0.66       => ( ! [X4: int] :
% 0.35/0.66              ( ( member_int @ X4 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 0.35/0.66             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.35/0.66         => ( ( finite_finite_int @ A2 )
% 0.35/0.66           => ( ord_less_eq_rat @ ( F @ I ) @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % member_le_sum
% 0.35/0.66  thf(fact_5966_member__le__sum,axiom,
% 0.35/0.66      ! [I: nat,A2: set_nat,F: nat > rat] :
% 0.35/0.66        ( ( member_nat @ I @ A2 )
% 0.35/0.66       => ( ! [X4: nat] :
% 0.35/0.66              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
% 0.35/0.66             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.35/0.66         => ( ( finite_finite_nat @ A2 )
% 0.35/0.66           => ( ord_less_eq_rat @ ( F @ I ) @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % member_le_sum
% 0.35/0.66  thf(fact_5967_member__le__sum,axiom,
% 0.35/0.66      ! [I: complex,A2: set_complex,F: complex > nat] :
% 0.35/0.66        ( ( member_complex @ I @ A2 )
% 0.35/0.66       => ( ! [X4: complex] :
% 0.35/0.66              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.66             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.35/0.66         => ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66           => ( ord_less_eq_nat @ ( F @ I ) @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % member_le_sum
% 0.35/0.66  thf(fact_5968_member__le__sum,axiom,
% 0.35/0.66      ! [I: real,A2: set_real,F: real > nat] :
% 0.35/0.66        ( ( member_real @ I @ A2 )
% 0.35/0.66       => ( ! [X4: real] :
% 0.35/0.66              ( ( member_real @ X4 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.35/0.66             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.35/0.66         => ( ( finite_finite_real @ A2 )
% 0.35/0.66           => ( ord_less_eq_nat @ ( F @ I ) @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % member_le_sum
% 0.35/0.66  thf(fact_5969_member__le__sum,axiom,
% 0.35/0.66      ! [I: int,A2: set_int,F: int > nat] :
% 0.35/0.66        ( ( member_int @ I @ A2 )
% 0.35/0.66       => ( ! [X4: int] :
% 0.35/0.66              ( ( member_int @ X4 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 0.35/0.66             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.35/0.66         => ( ( finite_finite_int @ A2 )
% 0.35/0.66           => ( ord_less_eq_nat @ ( F @ I ) @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % member_le_sum
% 0.35/0.66  thf(fact_5970_sum__subtractf__nat,axiom,
% 0.35/0.66      ! [A2: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,F: product_prod_nat_nat > nat] :
% 0.35/0.66        ( ! [X4: product_prod_nat_nat] :
% 0.35/0.66            ( ( member8440522571783428010at_nat @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 0.35/0.66       => ( ( groups977919841031483927at_nat
% 0.35/0.66            @ ^ [X3: product_prod_nat_nat] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.35/0.66            @ A2 )
% 0.35/0.66          = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A2 ) @ ( groups977919841031483927at_nat @ G @ A2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_subtractf_nat
% 0.35/0.66  thf(fact_5971_sum__subtractf__nat,axiom,
% 0.35/0.66      ! [A2: set_real,G: real > nat,F: real > nat] :
% 0.35/0.66        ( ! [X4: real] :
% 0.35/0.66            ( ( member_real @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 0.35/0.66       => ( ( groups1935376822645274424al_nat
% 0.35/0.66            @ ^ [X3: real] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.35/0.66            @ A2 )
% 0.35/0.66          = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_subtractf_nat
% 0.35/0.66  thf(fact_5972_sum__subtractf__nat,axiom,
% 0.35/0.66      ! [A2: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
% 0.35/0.66        ( ! [X4: set_nat] :
% 0.35/0.66            ( ( member_set_nat @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 0.35/0.66       => ( ( groups8294997508430121362at_nat
% 0.35/0.66            @ ^ [X3: set_nat] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.35/0.66            @ A2 )
% 0.35/0.66          = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( groups8294997508430121362at_nat @ G @ A2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_subtractf_nat
% 0.35/0.66  thf(fact_5973_sum__subtractf__nat,axiom,
% 0.35/0.66      ! [A2: set_int,G: int > nat,F: int > nat] :
% 0.35/0.66        ( ! [X4: int] :
% 0.35/0.66            ( ( member_int @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 0.35/0.66       => ( ( groups4541462559716669496nt_nat
% 0.35/0.66            @ ^ [X3: int] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.35/0.66            @ A2 )
% 0.35/0.66          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_subtractf_nat
% 0.35/0.66  thf(fact_5974_sum__subtractf__nat,axiom,
% 0.35/0.66      ! [A2: set_nat,G: nat > nat,F: nat > nat] :
% 0.35/0.66        ( ! [X4: nat] :
% 0.35/0.66            ( ( member_nat @ X4 @ A2 )
% 0.35/0.66           => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 0.35/0.66       => ( ( groups3542108847815614940at_nat
% 0.35/0.66            @ ^ [X3: nat] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.35/0.66            @ A2 )
% 0.35/0.66          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_subtractf_nat
% 0.35/0.66  thf(fact_5975_sum__le__included,axiom,
% 0.35/0.66      ! [S: set_int,T: set_int,G: int > real,I: int > int,F: int > real] :
% 0.35/0.66        ( ( finite_finite_int @ S )
% 0.35/0.66       => ( ( finite_finite_int @ T )
% 0.35/0.66         => ( ! [X4: int] :
% 0.35/0.66                ( ( member_int @ X4 @ T )
% 0.35/0.66               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 0.35/0.66           => ( ! [X4: int] :
% 0.35/0.66                  ( ( member_int @ X4 @ S )
% 0.35/0.66                 => ? [Xa: int] :
% 0.35/0.66                      ( ( member_int @ Xa @ T )
% 0.35/0.66                      & ( ( I @ Xa )
% 0.35/0.66                        = X4 )
% 0.35/0.66                      & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.35/0.66             => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_le_included
% 0.35/0.66  thf(fact_5976_sum__le__included,axiom,
% 0.35/0.66      ! [S: set_int,T: set_complex,G: complex > real,I: complex > int,F: int > real] :
% 0.35/0.66        ( ( finite_finite_int @ S )
% 0.35/0.66       => ( ( finite3207457112153483333omplex @ T )
% 0.35/0.66         => ( ! [X4: complex] :
% 0.35/0.66                ( ( member_complex @ X4 @ T )
% 0.35/0.66               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 0.35/0.66           => ( ! [X4: int] :
% 0.35/0.66                  ( ( member_int @ X4 @ S )
% 0.35/0.66                 => ? [Xa: complex] :
% 0.35/0.66                      ( ( member_complex @ Xa @ T )
% 0.35/0.66                      & ( ( I @ Xa )
% 0.35/0.66                        = X4 )
% 0.35/0.66                      & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.35/0.66             => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_le_included
% 0.35/0.66  thf(fact_5977_sum__le__included,axiom,
% 0.35/0.66      ! [S: set_complex,T: set_int,G: int > real,I: int > complex,F: complex > real] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ S )
% 0.35/0.66       => ( ( finite_finite_int @ T )
% 0.35/0.66         => ( ! [X4: int] :
% 0.35/0.66                ( ( member_int @ X4 @ T )
% 0.35/0.66               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 0.35/0.66           => ( ! [X4: complex] :
% 0.35/0.66                  ( ( member_complex @ X4 @ S )
% 0.35/0.66                 => ? [Xa: int] :
% 0.35/0.66                      ( ( member_int @ Xa @ T )
% 0.35/0.66                      & ( ( I @ Xa )
% 0.35/0.66                        = X4 )
% 0.35/0.66                      & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.35/0.66             => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_le_included
% 0.35/0.66  thf(fact_5978_sum__le__included,axiom,
% 0.35/0.66      ! [S: set_complex,T: set_complex,G: complex > real,I: complex > complex,F: complex > real] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ S )
% 0.35/0.66       => ( ( finite3207457112153483333omplex @ T )
% 0.35/0.66         => ( ! [X4: complex] :
% 0.35/0.66                ( ( member_complex @ X4 @ T )
% 0.35/0.66               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 0.35/0.66           => ( ! [X4: complex] :
% 0.35/0.66                  ( ( member_complex @ X4 @ S )
% 0.35/0.66                 => ? [Xa: complex] :
% 0.35/0.66                      ( ( member_complex @ Xa @ T )
% 0.35/0.66                      & ( ( I @ Xa )
% 0.35/0.66                        = X4 )
% 0.35/0.66                      & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.35/0.66             => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_le_included
% 0.35/0.66  thf(fact_5979_sum__le__included,axiom,
% 0.35/0.66      ! [S: set_nat,T: set_nat,G: nat > rat,I: nat > nat,F: nat > rat] :
% 0.35/0.66        ( ( finite_finite_nat @ S )
% 0.35/0.66       => ( ( finite_finite_nat @ T )
% 0.35/0.66         => ( ! [X4: nat] :
% 0.35/0.66                ( ( member_nat @ X4 @ T )
% 0.35/0.66               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 0.35/0.66           => ( ! [X4: nat] :
% 0.35/0.66                  ( ( member_nat @ X4 @ S )
% 0.35/0.66                 => ? [Xa: nat] :
% 0.35/0.66                      ( ( member_nat @ Xa @ T )
% 0.35/0.66                      & ( ( I @ Xa )
% 0.35/0.66                        = X4 )
% 0.35/0.66                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.35/0.66             => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_le_included
% 0.35/0.66  thf(fact_5980_sum__le__included,axiom,
% 0.35/0.66      ! [S: set_nat,T: set_int,G: int > rat,I: int > nat,F: nat > rat] :
% 0.35/0.66        ( ( finite_finite_nat @ S )
% 0.35/0.66       => ( ( finite_finite_int @ T )
% 0.35/0.66         => ( ! [X4: int] :
% 0.35/0.66                ( ( member_int @ X4 @ T )
% 0.35/0.66               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 0.35/0.66           => ( ! [X4: nat] :
% 0.35/0.66                  ( ( member_nat @ X4 @ S )
% 0.35/0.66                 => ? [Xa: int] :
% 0.35/0.66                      ( ( member_int @ Xa @ T )
% 0.35/0.66                      & ( ( I @ Xa )
% 0.35/0.66                        = X4 )
% 0.35/0.66                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.35/0.66             => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_le_included
% 0.35/0.66  thf(fact_5981_sum__le__included,axiom,
% 0.35/0.66      ! [S: set_nat,T: set_complex,G: complex > rat,I: complex > nat,F: nat > rat] :
% 0.35/0.66        ( ( finite_finite_nat @ S )
% 0.35/0.66       => ( ( finite3207457112153483333omplex @ T )
% 0.35/0.66         => ( ! [X4: complex] :
% 0.35/0.66                ( ( member_complex @ X4 @ T )
% 0.35/0.66               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 0.35/0.66           => ( ! [X4: nat] :
% 0.35/0.66                  ( ( member_nat @ X4 @ S )
% 0.35/0.66                 => ? [Xa: complex] :
% 0.35/0.66                      ( ( member_complex @ Xa @ T )
% 0.35/0.66                      & ( ( I @ Xa )
% 0.35/0.66                        = X4 )
% 0.35/0.66                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.35/0.66             => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_le_included
% 0.35/0.66  thf(fact_5982_sum__le__included,axiom,
% 0.35/0.66      ! [S: set_int,T: set_nat,G: nat > rat,I: nat > int,F: int > rat] :
% 0.35/0.66        ( ( finite_finite_int @ S )
% 0.35/0.66       => ( ( finite_finite_nat @ T )
% 0.35/0.66         => ( ! [X4: nat] :
% 0.35/0.66                ( ( member_nat @ X4 @ T )
% 0.35/0.66               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 0.35/0.66           => ( ! [X4: int] :
% 0.35/0.66                  ( ( member_int @ X4 @ S )
% 0.35/0.66                 => ? [Xa: nat] :
% 0.35/0.66                      ( ( member_nat @ Xa @ T )
% 0.35/0.66                      & ( ( I @ Xa )
% 0.35/0.66                        = X4 )
% 0.35/0.66                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.35/0.66             => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_le_included
% 0.35/0.66  thf(fact_5983_sum__le__included,axiom,
% 0.35/0.66      ! [S: set_int,T: set_int,G: int > rat,I: int > int,F: int > rat] :
% 0.35/0.66        ( ( finite_finite_int @ S )
% 0.35/0.66       => ( ( finite_finite_int @ T )
% 0.35/0.66         => ( ! [X4: int] :
% 0.35/0.66                ( ( member_int @ X4 @ T )
% 0.35/0.66               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 0.35/0.66           => ( ! [X4: int] :
% 0.35/0.66                  ( ( member_int @ X4 @ S )
% 0.35/0.66                 => ? [Xa: int] :
% 0.35/0.66                      ( ( member_int @ Xa @ T )
% 0.35/0.66                      & ( ( I @ Xa )
% 0.35/0.66                        = X4 )
% 0.35/0.66                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.35/0.66             => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_le_included
% 0.35/0.66  thf(fact_5984_sum__le__included,axiom,
% 0.35/0.66      ! [S: set_int,T: set_complex,G: complex > rat,I: complex > int,F: int > rat] :
% 0.35/0.66        ( ( finite_finite_int @ S )
% 0.35/0.66       => ( ( finite3207457112153483333omplex @ T )
% 0.35/0.66         => ( ! [X4: complex] :
% 0.35/0.66                ( ( member_complex @ X4 @ T )
% 0.35/0.66               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 0.35/0.66           => ( ! [X4: int] :
% 0.35/0.66                  ( ( member_int @ X4 @ S )
% 0.35/0.66                 => ? [Xa: complex] :
% 0.35/0.66                      ( ( member_complex @ Xa @ T )
% 0.35/0.66                      & ( ( I @ Xa )
% 0.35/0.66                        = X4 )
% 0.35/0.66                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.35/0.66             => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_le_included
% 0.35/0.66  thf(fact_5985_sum__nonneg__eq__0__iff,axiom,
% 0.35/0.66      ! [A2: set_real,F: real > real] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ! [X4: real] :
% 0.35/0.66              ( ( member_real @ X4 @ A2 )
% 0.35/0.66             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.35/0.66         => ( ( ( groups8097168146408367636l_real @ F @ A2 )
% 0.35/0.66              = zero_zero_real )
% 0.35/0.66            = ( ! [X3: real] :
% 0.35/0.66                  ( ( member_real @ X3 @ A2 )
% 0.35/0.66                 => ( ( F @ X3 )
% 0.35/0.66                    = zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonneg_eq_0_iff
% 0.35/0.66  thf(fact_5986_sum__nonneg__eq__0__iff,axiom,
% 0.35/0.66      ! [A2: set_int,F: int > real] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ! [X4: int] :
% 0.35/0.66              ( ( member_int @ X4 @ A2 )
% 0.35/0.66             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.35/0.66         => ( ( ( groups8778361861064173332t_real @ F @ A2 )
% 0.35/0.66              = zero_zero_real )
% 0.35/0.66            = ( ! [X3: int] :
% 0.35/0.66                  ( ( member_int @ X3 @ A2 )
% 0.35/0.66                 => ( ( F @ X3 )
% 0.35/0.66                    = zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonneg_eq_0_iff
% 0.35/0.66  thf(fact_5987_sum__nonneg__eq__0__iff,axiom,
% 0.35/0.66      ! [A2: set_complex,F: complex > real] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ! [X4: complex] :
% 0.35/0.66              ( ( member_complex @ X4 @ A2 )
% 0.35/0.66             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.35/0.66         => ( ( ( groups5808333547571424918x_real @ F @ A2 )
% 0.35/0.66              = zero_zero_real )
% 0.35/0.66            = ( ! [X3: complex] :
% 0.35/0.66                  ( ( member_complex @ X3 @ A2 )
% 0.35/0.66                 => ( ( F @ X3 )
% 0.35/0.66                    = zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonneg_eq_0_iff
% 0.35/0.66  thf(fact_5988_sum__nonneg__eq__0__iff,axiom,
% 0.35/0.66      ! [A2: set_real,F: real > rat] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ! [X4: real] :
% 0.35/0.66              ( ( member_real @ X4 @ A2 )
% 0.35/0.66             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.35/0.66         => ( ( ( groups1300246762558778688al_rat @ F @ A2 )
% 0.35/0.66              = zero_zero_rat )
% 0.35/0.66            = ( ! [X3: real] :
% 0.35/0.66                  ( ( member_real @ X3 @ A2 )
% 0.35/0.66                 => ( ( F @ X3 )
% 0.35/0.66                    = zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonneg_eq_0_iff
% 0.35/0.66  thf(fact_5989_sum__nonneg__eq__0__iff,axiom,
% 0.35/0.66      ! [A2: set_nat,F: nat > rat] :
% 0.35/0.66        ( ( finite_finite_nat @ A2 )
% 0.35/0.66       => ( ! [X4: nat] :
% 0.35/0.66              ( ( member_nat @ X4 @ A2 )
% 0.35/0.66             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.35/0.66         => ( ( ( groups2906978787729119204at_rat @ F @ A2 )
% 0.35/0.66              = zero_zero_rat )
% 0.35/0.66            = ( ! [X3: nat] :
% 0.35/0.66                  ( ( member_nat @ X3 @ A2 )
% 0.35/0.66                 => ( ( F @ X3 )
% 0.35/0.66                    = zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonneg_eq_0_iff
% 0.35/0.66  thf(fact_5990_sum__nonneg__eq__0__iff,axiom,
% 0.35/0.66      ! [A2: set_int,F: int > rat] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ! [X4: int] :
% 0.35/0.66              ( ( member_int @ X4 @ A2 )
% 0.35/0.66             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.35/0.66         => ( ( ( groups3906332499630173760nt_rat @ F @ A2 )
% 0.35/0.66              = zero_zero_rat )
% 0.35/0.66            = ( ! [X3: int] :
% 0.35/0.66                  ( ( member_int @ X3 @ A2 )
% 0.35/0.66                 => ( ( F @ X3 )
% 0.35/0.66                    = zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonneg_eq_0_iff
% 0.35/0.66  thf(fact_5991_sum__nonneg__eq__0__iff,axiom,
% 0.35/0.66      ! [A2: set_complex,F: complex > rat] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ! [X4: complex] :
% 0.35/0.66              ( ( member_complex @ X4 @ A2 )
% 0.35/0.66             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.35/0.66         => ( ( ( groups5058264527183730370ex_rat @ F @ A2 )
% 0.35/0.66              = zero_zero_rat )
% 0.35/0.66            = ( ! [X3: complex] :
% 0.35/0.66                  ( ( member_complex @ X3 @ A2 )
% 0.35/0.66                 => ( ( F @ X3 )
% 0.35/0.66                    = zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonneg_eq_0_iff
% 0.35/0.66  thf(fact_5992_sum__nonneg__eq__0__iff,axiom,
% 0.35/0.66      ! [A2: set_real,F: real > nat] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ! [X4: real] :
% 0.35/0.66              ( ( member_real @ X4 @ A2 )
% 0.35/0.66             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.35/0.66         => ( ( ( groups1935376822645274424al_nat @ F @ A2 )
% 0.35/0.66              = zero_zero_nat )
% 0.35/0.66            = ( ! [X3: real] :
% 0.35/0.66                  ( ( member_real @ X3 @ A2 )
% 0.35/0.66                 => ( ( F @ X3 )
% 0.35/0.66                    = zero_zero_nat ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonneg_eq_0_iff
% 0.35/0.66  thf(fact_5993_sum__nonneg__eq__0__iff,axiom,
% 0.35/0.66      ! [A2: set_int,F: int > nat] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ! [X4: int] :
% 0.35/0.66              ( ( member_int @ X4 @ A2 )
% 0.35/0.66             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.35/0.66         => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 0.35/0.66              = zero_zero_nat )
% 0.35/0.66            = ( ! [X3: int] :
% 0.35/0.66                  ( ( member_int @ X3 @ A2 )
% 0.35/0.66                 => ( ( F @ X3 )
% 0.35/0.66                    = zero_zero_nat ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonneg_eq_0_iff
% 0.35/0.66  thf(fact_5994_sum__nonneg__eq__0__iff,axiom,
% 0.35/0.66      ! [A2: set_complex,F: complex > nat] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ! [X4: complex] :
% 0.35/0.66              ( ( member_complex @ X4 @ A2 )
% 0.35/0.66             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.35/0.66         => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 0.35/0.66              = zero_zero_nat )
% 0.35/0.66            = ( ! [X3: complex] :
% 0.35/0.66                  ( ( member_complex @ X3 @ A2 )
% 0.35/0.66                 => ( ( F @ X3 )
% 0.35/0.66                    = zero_zero_nat ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_nonneg_eq_0_iff
% 0.35/0.66  thf(fact_5995_sum__strict__mono__ex1,axiom,
% 0.35/0.66      ! [A2: set_int,F: int > real,G: int > real] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ! [X4: int] :
% 0.35/0.66              ( ( member_int @ X4 @ A2 )
% 0.35/0.66             => ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66         => ( ? [X2: int] :
% 0.35/0.66                ( ( member_int @ X2 @ A2 )
% 0.35/0.66                & ( ord_less_real @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.35/0.66           => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_strict_mono_ex1
% 0.35/0.66  thf(fact_5996_sum__strict__mono__ex1,axiom,
% 0.35/0.66      ! [A2: set_complex,F: complex > real,G: complex > real] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ! [X4: complex] :
% 0.35/0.66              ( ( member_complex @ X4 @ A2 )
% 0.35/0.66             => ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66         => ( ? [X2: complex] :
% 0.35/0.66                ( ( member_complex @ X2 @ A2 )
% 0.35/0.66                & ( ord_less_real @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.35/0.66           => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_strict_mono_ex1
% 0.35/0.66  thf(fact_5997_sum__strict__mono__ex1,axiom,
% 0.35/0.66      ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 0.35/0.66        ( ( finite_finite_nat @ A2 )
% 0.35/0.66       => ( ! [X4: nat] :
% 0.35/0.66              ( ( member_nat @ X4 @ A2 )
% 0.35/0.66             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66         => ( ? [X2: nat] :
% 0.35/0.66                ( ( member_nat @ X2 @ A2 )
% 0.35/0.66                & ( ord_less_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.35/0.66           => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_strict_mono_ex1
% 0.35/0.66  thf(fact_5998_sum__strict__mono__ex1,axiom,
% 0.35/0.66      ! [A2: set_int,F: int > rat,G: int > rat] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ! [X4: int] :
% 0.35/0.66              ( ( member_int @ X4 @ A2 )
% 0.35/0.66             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66         => ( ? [X2: int] :
% 0.35/0.66                ( ( member_int @ X2 @ A2 )
% 0.35/0.66                & ( ord_less_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.35/0.66           => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_strict_mono_ex1
% 0.35/0.66  thf(fact_5999_sum__strict__mono__ex1,axiom,
% 0.35/0.66      ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ! [X4: complex] :
% 0.35/0.66              ( ( member_complex @ X4 @ A2 )
% 0.35/0.66             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66         => ( ? [X2: complex] :
% 0.35/0.66                ( ( member_complex @ X2 @ A2 )
% 0.35/0.66                & ( ord_less_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.35/0.66           => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_strict_mono_ex1
% 0.35/0.66  thf(fact_6000_sum__strict__mono__ex1,axiom,
% 0.35/0.66      ! [A2: set_int,F: int > nat,G: int > nat] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ! [X4: int] :
% 0.35/0.66              ( ( member_int @ X4 @ A2 )
% 0.35/0.66             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66         => ( ? [X2: int] :
% 0.35/0.66                ( ( member_int @ X2 @ A2 )
% 0.35/0.66                & ( ord_less_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.35/0.66           => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_strict_mono_ex1
% 0.35/0.66  thf(fact_6001_sum__strict__mono__ex1,axiom,
% 0.35/0.66      ! [A2: set_complex,F: complex > nat,G: complex > nat] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ! [X4: complex] :
% 0.35/0.66              ( ( member_complex @ X4 @ A2 )
% 0.35/0.66             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66         => ( ? [X2: complex] :
% 0.35/0.66                ( ( member_complex @ X2 @ A2 )
% 0.35/0.66                & ( ord_less_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.35/0.66           => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_strict_mono_ex1
% 0.35/0.66  thf(fact_6002_sum__strict__mono__ex1,axiom,
% 0.35/0.66      ! [A2: set_nat,F: nat > int,G: nat > int] :
% 0.35/0.66        ( ( finite_finite_nat @ A2 )
% 0.35/0.66       => ( ! [X4: nat] :
% 0.35/0.66              ( ( member_nat @ X4 @ A2 )
% 0.35/0.66             => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66         => ( ? [X2: nat] :
% 0.35/0.66                ( ( member_nat @ X2 @ A2 )
% 0.35/0.66                & ( ord_less_int @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.35/0.66           => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_strict_mono_ex1
% 0.35/0.66  thf(fact_6003_sum__strict__mono__ex1,axiom,
% 0.35/0.66      ! [A2: set_int,F: int > int,G: int > int] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ! [X4: int] :
% 0.35/0.66              ( ( member_int @ X4 @ A2 )
% 0.35/0.66             => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66         => ( ? [X2: int] :
% 0.35/0.66                ( ( member_int @ X2 @ A2 )
% 0.35/0.66                & ( ord_less_int @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.35/0.66           => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_strict_mono_ex1
% 0.35/0.66  thf(fact_6004_sum__strict__mono__ex1,axiom,
% 0.35/0.66      ! [A2: set_complex,F: complex > int,G: complex > int] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ! [X4: complex] :
% 0.35/0.66              ( ( member_complex @ X4 @ A2 )
% 0.35/0.66             => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66         => ( ? [X2: complex] :
% 0.35/0.66                ( ( member_complex @ X2 @ A2 )
% 0.35/0.66                & ( ord_less_int @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.35/0.66           => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_strict_mono_ex1
% 0.35/0.66  thf(fact_6005_sum_Orelated,axiom,
% 0.35/0.66      ! [R: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
% 0.35/0.66        ( ( R @ zero_zero_complex @ zero_zero_complex )
% 0.35/0.66       => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
% 0.35/0.66              ( ( ( R @ X15 @ X23 )
% 0.35/0.66                & ( R @ Y15 @ Y23 ) )
% 0.35/0.66             => ( R @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
% 0.35/0.66         => ( ( finite_finite_nat @ S2 )
% 0.35/0.66           => ( ! [X4: nat] :
% 0.35/0.66                  ( ( member_nat @ X4 @ S2 )
% 0.35/0.66                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66             => ( R @ ( groups2073611262835488442omplex @ H2 @ S2 ) @ ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.related
% 0.35/0.66  thf(fact_6006_sum_Orelated,axiom,
% 0.35/0.66      ! [R: complex > complex > $o,S2: set_int,H2: int > complex,G: int > complex] :
% 0.35/0.66        ( ( R @ zero_zero_complex @ zero_zero_complex )
% 0.35/0.66       => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
% 0.35/0.66              ( ( ( R @ X15 @ X23 )
% 0.35/0.66                & ( R @ Y15 @ Y23 ) )
% 0.35/0.66             => ( R @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
% 0.35/0.66         => ( ( finite_finite_int @ S2 )
% 0.35/0.66           => ( ! [X4: int] :
% 0.35/0.66                  ( ( member_int @ X4 @ S2 )
% 0.35/0.66                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66             => ( R @ ( groups3049146728041665814omplex @ H2 @ S2 ) @ ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.related
% 0.35/0.66  thf(fact_6007_sum_Orelated,axiom,
% 0.35/0.66      ! [R: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
% 0.35/0.66        ( ( R @ zero_zero_real @ zero_zero_real )
% 0.35/0.66       => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
% 0.35/0.66              ( ( ( R @ X15 @ X23 )
% 0.35/0.66                & ( R @ Y15 @ Y23 ) )
% 0.35/0.66             => ( R @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
% 0.35/0.66         => ( ( finite_finite_int @ S2 )
% 0.35/0.66           => ( ! [X4: int] :
% 0.35/0.66                  ( ( member_int @ X4 @ S2 )
% 0.35/0.66                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66             => ( R @ ( groups8778361861064173332t_real @ H2 @ S2 ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.related
% 0.35/0.66  thf(fact_6008_sum_Orelated,axiom,
% 0.35/0.66      ! [R: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
% 0.35/0.66        ( ( R @ zero_zero_real @ zero_zero_real )
% 0.35/0.66       => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
% 0.35/0.66              ( ( ( R @ X15 @ X23 )
% 0.35/0.66                & ( R @ Y15 @ Y23 ) )
% 0.35/0.66             => ( R @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
% 0.35/0.66         => ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.66           => ( ! [X4: complex] :
% 0.35/0.66                  ( ( member_complex @ X4 @ S2 )
% 0.35/0.66                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66             => ( R @ ( groups5808333547571424918x_real @ H2 @ S2 ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.related
% 0.35/0.66  thf(fact_6009_sum_Orelated,axiom,
% 0.35/0.66      ! [R: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
% 0.35/0.66        ( ( R @ zero_zero_rat @ zero_zero_rat )
% 0.35/0.66       => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
% 0.35/0.66              ( ( ( R @ X15 @ X23 )
% 0.35/0.66                & ( R @ Y15 @ Y23 ) )
% 0.35/0.66             => ( R @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 0.35/0.66         => ( ( finite_finite_nat @ S2 )
% 0.35/0.66           => ( ! [X4: nat] :
% 0.35/0.66                  ( ( member_nat @ X4 @ S2 )
% 0.35/0.66                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66             => ( R @ ( groups2906978787729119204at_rat @ H2 @ S2 ) @ ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.related
% 0.35/0.66  thf(fact_6010_sum_Orelated,axiom,
% 0.35/0.66      ! [R: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
% 0.35/0.66        ( ( R @ zero_zero_rat @ zero_zero_rat )
% 0.35/0.66       => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
% 0.35/0.66              ( ( ( R @ X15 @ X23 )
% 0.35/0.66                & ( R @ Y15 @ Y23 ) )
% 0.35/0.66             => ( R @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 0.35/0.66         => ( ( finite_finite_int @ S2 )
% 0.35/0.66           => ( ! [X4: int] :
% 0.35/0.66                  ( ( member_int @ X4 @ S2 )
% 0.35/0.66                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66             => ( R @ ( groups3906332499630173760nt_rat @ H2 @ S2 ) @ ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.related
% 0.35/0.66  thf(fact_6011_sum_Orelated,axiom,
% 0.35/0.66      ! [R: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
% 0.35/0.66        ( ( R @ zero_zero_rat @ zero_zero_rat )
% 0.35/0.66       => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
% 0.35/0.66              ( ( ( R @ X15 @ X23 )
% 0.35/0.66                & ( R @ Y15 @ Y23 ) )
% 0.35/0.66             => ( R @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 0.35/0.66         => ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.66           => ( ! [X4: complex] :
% 0.35/0.66                  ( ( member_complex @ X4 @ S2 )
% 0.35/0.66                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66             => ( R @ ( groups5058264527183730370ex_rat @ H2 @ S2 ) @ ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.related
% 0.35/0.66  thf(fact_6012_sum_Orelated,axiom,
% 0.35/0.66      ! [R: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
% 0.35/0.66        ( ( R @ zero_zero_nat @ zero_zero_nat )
% 0.35/0.66       => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
% 0.35/0.66              ( ( ( R @ X15 @ X23 )
% 0.35/0.66                & ( R @ Y15 @ Y23 ) )
% 0.35/0.66             => ( R @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
% 0.35/0.66         => ( ( finite_finite_int @ S2 )
% 0.35/0.66           => ( ! [X4: int] :
% 0.35/0.66                  ( ( member_int @ X4 @ S2 )
% 0.35/0.66                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66             => ( R @ ( groups4541462559716669496nt_nat @ H2 @ S2 ) @ ( groups4541462559716669496nt_nat @ G @ S2 ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.related
% 0.35/0.66  thf(fact_6013_sum_Orelated,axiom,
% 0.35/0.66      ! [R: nat > nat > $o,S2: set_complex,H2: complex > nat,G: complex > nat] :
% 0.35/0.66        ( ( R @ zero_zero_nat @ zero_zero_nat )
% 0.35/0.66       => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
% 0.35/0.66              ( ( ( R @ X15 @ X23 )
% 0.35/0.66                & ( R @ Y15 @ Y23 ) )
% 0.35/0.66             => ( R @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
% 0.35/0.66         => ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.66           => ( ! [X4: complex] :
% 0.35/0.66                  ( ( member_complex @ X4 @ S2 )
% 0.35/0.66                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66             => ( R @ ( groups5693394587270226106ex_nat @ H2 @ S2 ) @ ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.related
% 0.35/0.66  thf(fact_6014_sum_Orelated,axiom,
% 0.35/0.66      ! [R: int > int > $o,S2: set_nat,H2: nat > int,G: nat > int] :
% 0.35/0.66        ( ( R @ zero_zero_int @ zero_zero_int )
% 0.35/0.66       => ( ! [X15: int,Y15: int,X23: int,Y23: int] :
% 0.35/0.66              ( ( ( R @ X15 @ X23 )
% 0.35/0.66                & ( R @ Y15 @ Y23 ) )
% 0.35/0.66             => ( R @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
% 0.35/0.66         => ( ( finite_finite_nat @ S2 )
% 0.35/0.66           => ( ! [X4: nat] :
% 0.35/0.66                  ( ( member_nat @ X4 @ S2 )
% 0.35/0.66                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66             => ( R @ ( groups3539618377306564664at_int @ H2 @ S2 ) @ ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum.related
% 0.35/0.66  thf(fact_6015_sum__strict__mono,axiom,
% 0.35/0.66      ! [A2: set_complex,F: complex > real,G: complex > real] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ( A2 != bot_bot_set_complex )
% 0.35/0.66         => ( ! [X4: complex] :
% 0.35/0.66                ( ( member_complex @ X4 @ A2 )
% 0.35/0.66               => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66           => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_strict_mono
% 0.35/0.66  thf(fact_6016_sum__strict__mono,axiom,
% 0.35/0.66      ! [A2: set_real,F: real > real,G: real > real] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ( A2 != bot_bot_set_real )
% 0.35/0.66         => ( ! [X4: real] :
% 0.35/0.66                ( ( member_real @ X4 @ A2 )
% 0.35/0.66               => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66           => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_strict_mono
% 0.35/0.66  thf(fact_6017_sum__strict__mono,axiom,
% 0.35/0.66      ! [A2: set_int,F: int > real,G: int > real] :
% 0.35/0.66        ( ( finite_finite_int @ A2 )
% 0.35/0.66       => ( ( A2 != bot_bot_set_int )
% 0.35/0.66         => ( ! [X4: int] :
% 0.35/0.66                ( ( member_int @ X4 @ A2 )
% 0.35/0.66               => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66           => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_strict_mono
% 0.35/0.66  thf(fact_6018_sum__strict__mono,axiom,
% 0.35/0.66      ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 0.35/0.66        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.66       => ( ( A2 != bot_bot_set_complex )
% 0.35/0.66         => ( ! [X4: complex] :
% 0.35/0.66                ( ( member_complex @ X4 @ A2 )
% 0.35/0.66               => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66           => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_strict_mono
% 0.35/0.66  thf(fact_6019_sum__strict__mono,axiom,
% 0.35/0.66      ! [A2: set_real,F: real > rat,G: real > rat] :
% 0.35/0.66        ( ( finite_finite_real @ A2 )
% 0.35/0.66       => ( ( A2 != bot_bot_set_real )
% 0.35/0.66         => ( ! [X4: real] :
% 0.35/0.66                ( ( member_real @ X4 @ A2 )
% 0.35/0.66               => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.66           => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.66  
% 0.35/0.66  % sum_strict_mono
% 0.35/0.66  thf(fact_6020_sum__strict__mono,axiom,
% 0.35/0.66      ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 0.35/0.66        ( ( finite_finite_nat @ A2 )
% 0.35/0.67       => ( ( A2 != bot_bot_set_nat )
% 0.35/0.67         => ( ! [X4: nat] :
% 0.35/0.67                ( ( member_nat @ X4 @ A2 )
% 0.35/0.67               => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.67           => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_strict_mono
% 0.35/0.67  thf(fact_6021_sum__strict__mono,axiom,
% 0.35/0.67      ! [A2: set_int,F: int > rat,G: int > rat] :
% 0.35/0.67        ( ( finite_finite_int @ A2 )
% 0.35/0.67       => ( ( A2 != bot_bot_set_int )
% 0.35/0.67         => ( ! [X4: int] :
% 0.35/0.67                ( ( member_int @ X4 @ A2 )
% 0.35/0.67               => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.67           => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_strict_mono
% 0.35/0.67  thf(fact_6022_sum__strict__mono,axiom,
% 0.35/0.67      ! [A2: set_complex,F: complex > nat,G: complex > nat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.67       => ( ( A2 != bot_bot_set_complex )
% 0.35/0.67         => ( ! [X4: complex] :
% 0.35/0.67                ( ( member_complex @ X4 @ A2 )
% 0.35/0.67               => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.67           => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_strict_mono
% 0.35/0.67  thf(fact_6023_sum__strict__mono,axiom,
% 0.35/0.67      ! [A2: set_real,F: real > nat,G: real > nat] :
% 0.35/0.67        ( ( finite_finite_real @ A2 )
% 0.35/0.67       => ( ( A2 != bot_bot_set_real )
% 0.35/0.67         => ( ! [X4: real] :
% 0.35/0.67                ( ( member_real @ X4 @ A2 )
% 0.35/0.67               => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.67           => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_strict_mono
% 0.35/0.67  thf(fact_6024_sum__strict__mono,axiom,
% 0.35/0.67      ! [A2: set_int,F: int > nat,G: int > nat] :
% 0.35/0.67        ( ( finite_finite_int @ A2 )
% 0.35/0.67       => ( ( A2 != bot_bot_set_int )
% 0.35/0.67         => ( ! [X4: int] :
% 0.35/0.67                ( ( member_int @ X4 @ A2 )
% 0.35/0.67               => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.67           => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_strict_mono
% 0.35/0.67  thf(fact_6025_sum__eq__Suc0__iff,axiom,
% 0.35/0.67      ! [A2: set_int,F: int > nat] :
% 0.35/0.67        ( ( finite_finite_int @ A2 )
% 0.35/0.67       => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 0.35/0.67            = ( suc @ zero_zero_nat ) )
% 0.35/0.67          = ( ? [X3: int] :
% 0.35/0.67                ( ( member_int @ X3 @ A2 )
% 0.35/0.67                & ( ( F @ X3 )
% 0.35/0.67                  = ( suc @ zero_zero_nat ) )
% 0.35/0.67                & ! [Y2: int] :
% 0.35/0.67                    ( ( member_int @ Y2 @ A2 )
% 0.35/0.67                   => ( ( X3 != Y2 )
% 0.35/0.67                     => ( ( F @ Y2 )
% 0.35/0.67                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_eq_Suc0_iff
% 0.35/0.67  thf(fact_6026_sum__eq__Suc0__iff,axiom,
% 0.35/0.67      ! [A2: set_complex,F: complex > nat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.67       => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 0.35/0.67            = ( suc @ zero_zero_nat ) )
% 0.35/0.67          = ( ? [X3: complex] :
% 0.35/0.67                ( ( member_complex @ X3 @ A2 )
% 0.35/0.67                & ( ( F @ X3 )
% 0.35/0.67                  = ( suc @ zero_zero_nat ) )
% 0.35/0.67                & ! [Y2: complex] :
% 0.35/0.67                    ( ( member_complex @ Y2 @ A2 )
% 0.35/0.67                   => ( ( X3 != Y2 )
% 0.35/0.67                     => ( ( F @ Y2 )
% 0.35/0.67                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_eq_Suc0_iff
% 0.35/0.67  thf(fact_6027_sum__eq__Suc0__iff,axiom,
% 0.35/0.67      ! [A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 0.35/0.67        ( ( finite6177210948735845034at_nat @ A2 )
% 0.35/0.67       => ( ( ( groups977919841031483927at_nat @ F @ A2 )
% 0.35/0.67            = ( suc @ zero_zero_nat ) )
% 0.35/0.67          = ( ? [X3: product_prod_nat_nat] :
% 0.35/0.67                ( ( member8440522571783428010at_nat @ X3 @ A2 )
% 0.35/0.67                & ( ( F @ X3 )
% 0.35/0.67                  = ( suc @ zero_zero_nat ) )
% 0.35/0.67                & ! [Y2: product_prod_nat_nat] :
% 0.35/0.67                    ( ( member8440522571783428010at_nat @ Y2 @ A2 )
% 0.35/0.67                   => ( ( X3 != Y2 )
% 0.35/0.67                     => ( ( F @ Y2 )
% 0.35/0.67                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_eq_Suc0_iff
% 0.35/0.67  thf(fact_6028_sum__eq__Suc0__iff,axiom,
% 0.35/0.67      ! [A2: set_nat,F: nat > nat] :
% 0.35/0.67        ( ( finite_finite_nat @ A2 )
% 0.35/0.67       => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 0.35/0.67            = ( suc @ zero_zero_nat ) )
% 0.35/0.67          = ( ? [X3: nat] :
% 0.35/0.67                ( ( member_nat @ X3 @ A2 )
% 0.35/0.67                & ( ( F @ X3 )
% 0.35/0.67                  = ( suc @ zero_zero_nat ) )
% 0.35/0.67                & ! [Y2: nat] :
% 0.35/0.67                    ( ( member_nat @ Y2 @ A2 )
% 0.35/0.67                   => ( ( X3 != Y2 )
% 0.35/0.67                     => ( ( F @ Y2 )
% 0.35/0.67                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_eq_Suc0_iff
% 0.35/0.67  thf(fact_6029_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.67      ! [S5: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
% 0.35/0.67        ( ( finite_finite_real @ S5 )
% 0.35/0.67       => ( ( finite_finite_real @ T4 )
% 0.35/0.67         => ( ! [A3: real] :
% 0.35/0.67                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 0.35/0.67               => ( ( I @ ( J @ A3 ) )
% 0.35/0.67                  = A3 ) )
% 0.35/0.67           => ( ! [A3: real] :
% 0.35/0.67                  ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 0.35/0.67                 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.35/0.67             => ( ! [B3: real] :
% 0.35/0.67                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.35/0.67                   => ( ( J @ ( I @ B3 ) )
% 0.35/0.67                      = B3 ) )
% 0.35/0.67               => ( ! [B3: real] :
% 0.35/0.67                      ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.35/0.67                     => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
% 0.35/0.67                 => ( ! [A3: real] :
% 0.35/0.67                        ( ( member_real @ A3 @ S5 )
% 0.35/0.67                       => ( ( G @ A3 )
% 0.35/0.67                          = zero_zero_complex ) )
% 0.35/0.67                   => ( ! [B3: real] :
% 0.35/0.67                          ( ( member_real @ B3 @ T4 )
% 0.35/0.67                         => ( ( H2 @ B3 )
% 0.35/0.67                            = zero_zero_complex ) )
% 0.35/0.67                     => ( ! [A3: real] :
% 0.35/0.67                            ( ( member_real @ A3 @ S2 )
% 0.35/0.67                           => ( ( H2 @ ( J @ A3 ) )
% 0.35/0.67                              = ( G @ A3 ) ) )
% 0.35/0.67                       => ( ( groups5754745047067104278omplex @ G @ S2 )
% 0.35/0.67                          = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.reindex_bij_witness_not_neutral
% 0.35/0.67  thf(fact_6030_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.67      ! [S5: set_real,T4: set_int,S2: set_real,I: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
% 0.35/0.67        ( ( finite_finite_real @ S5 )
% 0.35/0.67       => ( ( finite_finite_int @ T4 )
% 0.35/0.67         => ( ! [A3: real] :
% 0.35/0.67                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 0.35/0.67               => ( ( I @ ( J @ A3 ) )
% 0.35/0.67                  = A3 ) )
% 0.35/0.67           => ( ! [A3: real] :
% 0.35/0.67                  ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 0.35/0.67                 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.35/0.67             => ( ! [B3: int] :
% 0.35/0.67                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.35/0.67                   => ( ( J @ ( I @ B3 ) )
% 0.35/0.67                      = B3 ) )
% 0.35/0.67               => ( ! [B3: int] :
% 0.35/0.67                      ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.35/0.67                     => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
% 0.35/0.67                 => ( ! [A3: real] :
% 0.35/0.67                        ( ( member_real @ A3 @ S5 )
% 0.35/0.67                       => ( ( G @ A3 )
% 0.35/0.67                          = zero_zero_complex ) )
% 0.35/0.67                   => ( ! [B3: int] :
% 0.35/0.67                          ( ( member_int @ B3 @ T4 )
% 0.35/0.67                         => ( ( H2 @ B3 )
% 0.35/0.67                            = zero_zero_complex ) )
% 0.35/0.67                     => ( ! [A3: real] :
% 0.35/0.67                            ( ( member_real @ A3 @ S2 )
% 0.35/0.67                           => ( ( H2 @ ( J @ A3 ) )
% 0.35/0.67                              = ( G @ A3 ) ) )
% 0.35/0.67                       => ( ( groups5754745047067104278omplex @ G @ S2 )
% 0.35/0.67                          = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.reindex_bij_witness_not_neutral
% 0.35/0.67  thf(fact_6031_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.67      ! [S5: set_int,T4: set_real,S2: set_int,I: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
% 0.35/0.67        ( ( finite_finite_int @ S5 )
% 0.35/0.67       => ( ( finite_finite_real @ T4 )
% 0.35/0.67         => ( ! [A3: int] :
% 0.35/0.67                ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 0.35/0.67               => ( ( I @ ( J @ A3 ) )
% 0.35/0.67                  = A3 ) )
% 0.35/0.67           => ( ! [A3: int] :
% 0.35/0.67                  ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 0.35/0.67                 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.35/0.67             => ( ! [B3: real] :
% 0.35/0.67                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.35/0.67                   => ( ( J @ ( I @ B3 ) )
% 0.35/0.67                      = B3 ) )
% 0.35/0.67               => ( ! [B3: real] :
% 0.35/0.67                      ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.35/0.67                     => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
% 0.35/0.67                 => ( ! [A3: int] :
% 0.35/0.67                        ( ( member_int @ A3 @ S5 )
% 0.35/0.67                       => ( ( G @ A3 )
% 0.35/0.67                          = zero_zero_complex ) )
% 0.35/0.67                   => ( ! [B3: real] :
% 0.35/0.67                          ( ( member_real @ B3 @ T4 )
% 0.35/0.67                         => ( ( H2 @ B3 )
% 0.35/0.67                            = zero_zero_complex ) )
% 0.35/0.67                     => ( ! [A3: int] :
% 0.35/0.67                            ( ( member_int @ A3 @ S2 )
% 0.35/0.67                           => ( ( H2 @ ( J @ A3 ) )
% 0.35/0.67                              = ( G @ A3 ) ) )
% 0.35/0.67                       => ( ( groups3049146728041665814omplex @ G @ S2 )
% 0.35/0.67                          = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.reindex_bij_witness_not_neutral
% 0.35/0.67  thf(fact_6032_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.67      ! [S5: set_int,T4: set_int,S2: set_int,I: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
% 0.35/0.67        ( ( finite_finite_int @ S5 )
% 0.35/0.67       => ( ( finite_finite_int @ T4 )
% 0.35/0.67         => ( ! [A3: int] :
% 0.35/0.67                ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 0.35/0.67               => ( ( I @ ( J @ A3 ) )
% 0.35/0.67                  = A3 ) )
% 0.35/0.67           => ( ! [A3: int] :
% 0.35/0.67                  ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 0.35/0.67                 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.35/0.67             => ( ! [B3: int] :
% 0.35/0.67                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.35/0.67                   => ( ( J @ ( I @ B3 ) )
% 0.35/0.67                      = B3 ) )
% 0.35/0.67               => ( ! [B3: int] :
% 0.35/0.67                      ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.35/0.67                     => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
% 0.35/0.67                 => ( ! [A3: int] :
% 0.35/0.67                        ( ( member_int @ A3 @ S5 )
% 0.35/0.67                       => ( ( G @ A3 )
% 0.35/0.67                          = zero_zero_complex ) )
% 0.35/0.67                   => ( ! [B3: int] :
% 0.35/0.67                          ( ( member_int @ B3 @ T4 )
% 0.35/0.67                         => ( ( H2 @ B3 )
% 0.35/0.67                            = zero_zero_complex ) )
% 0.35/0.67                     => ( ! [A3: int] :
% 0.35/0.67                            ( ( member_int @ A3 @ S2 )
% 0.35/0.67                           => ( ( H2 @ ( J @ A3 ) )
% 0.35/0.67                              = ( G @ A3 ) ) )
% 0.35/0.67                       => ( ( groups3049146728041665814omplex @ G @ S2 )
% 0.35/0.67                          = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.reindex_bij_witness_not_neutral
% 0.35/0.67  thf(fact_6033_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.67      ! [S5: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T3: set_real,G: real > real,H2: real > real] :
% 0.35/0.67        ( ( finite_finite_real @ S5 )
% 0.35/0.67       => ( ( finite_finite_real @ T4 )
% 0.35/0.67         => ( ! [A3: real] :
% 0.35/0.67                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 0.35/0.67               => ( ( I @ ( J @ A3 ) )
% 0.35/0.67                  = A3 ) )
% 0.35/0.67           => ( ! [A3: real] :
% 0.35/0.67                  ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 0.35/0.67                 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.35/0.67             => ( ! [B3: real] :
% 0.35/0.67                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.35/0.67                   => ( ( J @ ( I @ B3 ) )
% 0.35/0.67                      = B3 ) )
% 0.35/0.67               => ( ! [B3: real] :
% 0.35/0.67                      ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.35/0.67                     => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
% 0.35/0.67                 => ( ! [A3: real] :
% 0.35/0.67                        ( ( member_real @ A3 @ S5 )
% 0.35/0.67                       => ( ( G @ A3 )
% 0.35/0.67                          = zero_zero_real ) )
% 0.35/0.67                   => ( ! [B3: real] :
% 0.35/0.67                          ( ( member_real @ B3 @ T4 )
% 0.35/0.67                         => ( ( H2 @ B3 )
% 0.35/0.67                            = zero_zero_real ) )
% 0.35/0.67                     => ( ! [A3: real] :
% 0.35/0.67                            ( ( member_real @ A3 @ S2 )
% 0.35/0.67                           => ( ( H2 @ ( J @ A3 ) )
% 0.35/0.67                              = ( G @ A3 ) ) )
% 0.35/0.67                       => ( ( groups8097168146408367636l_real @ G @ S2 )
% 0.35/0.67                          = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.reindex_bij_witness_not_neutral
% 0.35/0.67  thf(fact_6034_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.67      ! [S5: set_real,T4: set_int,S2: set_real,I: int > real,J: real > int,T3: set_int,G: real > real,H2: int > real] :
% 0.35/0.67        ( ( finite_finite_real @ S5 )
% 0.35/0.67       => ( ( finite_finite_int @ T4 )
% 0.35/0.67         => ( ! [A3: real] :
% 0.35/0.67                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 0.35/0.67               => ( ( I @ ( J @ A3 ) )
% 0.35/0.67                  = A3 ) )
% 0.35/0.67           => ( ! [A3: real] :
% 0.35/0.67                  ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 0.35/0.67                 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.35/0.67             => ( ! [B3: int] :
% 0.35/0.67                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.35/0.67                   => ( ( J @ ( I @ B3 ) )
% 0.35/0.67                      = B3 ) )
% 0.35/0.67               => ( ! [B3: int] :
% 0.35/0.67                      ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.35/0.67                     => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
% 0.35/0.67                 => ( ! [A3: real] :
% 0.35/0.67                        ( ( member_real @ A3 @ S5 )
% 0.35/0.67                       => ( ( G @ A3 )
% 0.35/0.67                          = zero_zero_real ) )
% 0.35/0.67                   => ( ! [B3: int] :
% 0.35/0.67                          ( ( member_int @ B3 @ T4 )
% 0.35/0.67                         => ( ( H2 @ B3 )
% 0.35/0.67                            = zero_zero_real ) )
% 0.35/0.67                     => ( ! [A3: real] :
% 0.35/0.67                            ( ( member_real @ A3 @ S2 )
% 0.35/0.67                           => ( ( H2 @ ( J @ A3 ) )
% 0.35/0.67                              = ( G @ A3 ) ) )
% 0.35/0.67                       => ( ( groups8097168146408367636l_real @ G @ S2 )
% 0.35/0.67                          = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.reindex_bij_witness_not_neutral
% 0.35/0.67  thf(fact_6035_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.67      ! [S5: set_real,T4: set_complex,S2: set_real,I: complex > real,J: real > complex,T3: set_complex,G: real > real,H2: complex > real] :
% 0.35/0.67        ( ( finite_finite_real @ S5 )
% 0.35/0.67       => ( ( finite3207457112153483333omplex @ T4 )
% 0.35/0.67         => ( ! [A3: real] :
% 0.35/0.67                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 0.35/0.67               => ( ( I @ ( J @ A3 ) )
% 0.35/0.67                  = A3 ) )
% 0.35/0.67           => ( ! [A3: real] :
% 0.35/0.67                  ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 0.35/0.67                 => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 0.35/0.67             => ( ! [B3: complex] :
% 0.35/0.67                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.35/0.67                   => ( ( J @ ( I @ B3 ) )
% 0.35/0.67                      = B3 ) )
% 0.35/0.67               => ( ! [B3: complex] :
% 0.35/0.67                      ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.35/0.67                     => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
% 0.35/0.67                 => ( ! [A3: real] :
% 0.35/0.67                        ( ( member_real @ A3 @ S5 )
% 0.35/0.67                       => ( ( G @ A3 )
% 0.35/0.67                          = zero_zero_real ) )
% 0.35/0.67                   => ( ! [B3: complex] :
% 0.35/0.67                          ( ( member_complex @ B3 @ T4 )
% 0.35/0.67                         => ( ( H2 @ B3 )
% 0.35/0.67                            = zero_zero_real ) )
% 0.35/0.67                     => ( ! [A3: real] :
% 0.35/0.67                            ( ( member_real @ A3 @ S2 )
% 0.35/0.67                           => ( ( H2 @ ( J @ A3 ) )
% 0.35/0.67                              = ( G @ A3 ) ) )
% 0.35/0.67                       => ( ( groups8097168146408367636l_real @ G @ S2 )
% 0.35/0.67                          = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.reindex_bij_witness_not_neutral
% 0.35/0.67  thf(fact_6036_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.67      ! [S5: set_int,T4: set_real,S2: set_int,I: real > int,J: int > real,T3: set_real,G: int > real,H2: real > real] :
% 0.35/0.67        ( ( finite_finite_int @ S5 )
% 0.35/0.67       => ( ( finite_finite_real @ T4 )
% 0.35/0.67         => ( ! [A3: int] :
% 0.35/0.67                ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 0.35/0.67               => ( ( I @ ( J @ A3 ) )
% 0.35/0.67                  = A3 ) )
% 0.35/0.67           => ( ! [A3: int] :
% 0.35/0.67                  ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 0.35/0.67                 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.35/0.67             => ( ! [B3: real] :
% 0.35/0.67                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.35/0.67                   => ( ( J @ ( I @ B3 ) )
% 0.35/0.67                      = B3 ) )
% 0.35/0.67               => ( ! [B3: real] :
% 0.35/0.67                      ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.35/0.67                     => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
% 0.35/0.67                 => ( ! [A3: int] :
% 0.35/0.67                        ( ( member_int @ A3 @ S5 )
% 0.35/0.67                       => ( ( G @ A3 )
% 0.35/0.67                          = zero_zero_real ) )
% 0.35/0.67                   => ( ! [B3: real] :
% 0.35/0.67                          ( ( member_real @ B3 @ T4 )
% 0.35/0.67                         => ( ( H2 @ B3 )
% 0.35/0.67                            = zero_zero_real ) )
% 0.35/0.67                     => ( ! [A3: int] :
% 0.35/0.67                            ( ( member_int @ A3 @ S2 )
% 0.35/0.67                           => ( ( H2 @ ( J @ A3 ) )
% 0.35/0.67                              = ( G @ A3 ) ) )
% 0.35/0.67                       => ( ( groups8778361861064173332t_real @ G @ S2 )
% 0.35/0.67                          = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.reindex_bij_witness_not_neutral
% 0.35/0.67  thf(fact_6037_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.67      ! [S5: set_int,T4: set_int,S2: set_int,I: int > int,J: int > int,T3: set_int,G: int > real,H2: int > real] :
% 0.35/0.67        ( ( finite_finite_int @ S5 )
% 0.35/0.67       => ( ( finite_finite_int @ T4 )
% 0.35/0.67         => ( ! [A3: int] :
% 0.35/0.67                ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 0.35/0.67               => ( ( I @ ( J @ A3 ) )
% 0.35/0.67                  = A3 ) )
% 0.35/0.67           => ( ! [A3: int] :
% 0.35/0.67                  ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 0.35/0.67                 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.35/0.67             => ( ! [B3: int] :
% 0.35/0.67                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.35/0.67                   => ( ( J @ ( I @ B3 ) )
% 0.35/0.67                      = B3 ) )
% 0.35/0.67               => ( ! [B3: int] :
% 0.35/0.67                      ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.35/0.67                     => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
% 0.35/0.67                 => ( ! [A3: int] :
% 0.35/0.67                        ( ( member_int @ A3 @ S5 )
% 0.35/0.67                       => ( ( G @ A3 )
% 0.35/0.67                          = zero_zero_real ) )
% 0.35/0.67                   => ( ! [B3: int] :
% 0.35/0.67                          ( ( member_int @ B3 @ T4 )
% 0.35/0.67                         => ( ( H2 @ B3 )
% 0.35/0.67                            = zero_zero_real ) )
% 0.35/0.67                     => ( ! [A3: int] :
% 0.35/0.67                            ( ( member_int @ A3 @ S2 )
% 0.35/0.67                           => ( ( H2 @ ( J @ A3 ) )
% 0.35/0.67                              = ( G @ A3 ) ) )
% 0.35/0.67                       => ( ( groups8778361861064173332t_real @ G @ S2 )
% 0.35/0.67                          = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.reindex_bij_witness_not_neutral
% 0.35/0.67  thf(fact_6038_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.67      ! [S5: set_int,T4: set_complex,S2: set_int,I: complex > int,J: int > complex,T3: set_complex,G: int > real,H2: complex > real] :
% 0.35/0.67        ( ( finite_finite_int @ S5 )
% 0.35/0.67       => ( ( finite3207457112153483333omplex @ T4 )
% 0.35/0.67         => ( ! [A3: int] :
% 0.35/0.67                ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 0.35/0.67               => ( ( I @ ( J @ A3 ) )
% 0.35/0.67                  = A3 ) )
% 0.35/0.67           => ( ! [A3: int] :
% 0.35/0.67                  ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 0.35/0.67                 => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 0.35/0.67             => ( ! [B3: complex] :
% 0.35/0.67                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.35/0.67                   => ( ( J @ ( I @ B3 ) )
% 0.35/0.67                      = B3 ) )
% 0.35/0.67               => ( ! [B3: complex] :
% 0.35/0.67                      ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.35/0.67                     => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
% 0.35/0.67                 => ( ! [A3: int] :
% 0.35/0.67                        ( ( member_int @ A3 @ S5 )
% 0.35/0.67                       => ( ( G @ A3 )
% 0.35/0.67                          = zero_zero_real ) )
% 0.35/0.67                   => ( ! [B3: complex] :
% 0.35/0.67                          ( ( member_complex @ B3 @ T4 )
% 0.35/0.67                         => ( ( H2 @ B3 )
% 0.35/0.67                            = zero_zero_real ) )
% 0.35/0.67                     => ( ! [A3: int] :
% 0.35/0.67                            ( ( member_int @ A3 @ S2 )
% 0.35/0.67                           => ( ( H2 @ ( J @ A3 ) )
% 0.35/0.67                              = ( G @ A3 ) ) )
% 0.35/0.67                       => ( ( groups8778361861064173332t_real @ G @ S2 )
% 0.35/0.67                          = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.reindex_bij_witness_not_neutral
% 0.35/0.67  thf(fact_6039_sum__SucD,axiom,
% 0.35/0.67      ! [F: nat > nat,A2: set_nat,N: nat] :
% 0.35/0.67        ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 0.35/0.67          = ( suc @ N ) )
% 0.35/0.67       => ? [X4: nat] :
% 0.35/0.67            ( ( member_nat @ X4 @ A2 )
% 0.35/0.67            & ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_SucD
% 0.35/0.67  thf(fact_6040_sum__eq__1__iff,axiom,
% 0.35/0.67      ! [A2: set_int,F: int > nat] :
% 0.35/0.67        ( ( finite_finite_int @ A2 )
% 0.35/0.67       => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 0.35/0.67            = one_one_nat )
% 0.35/0.67          = ( ? [X3: int] :
% 0.35/0.67                ( ( member_int @ X3 @ A2 )
% 0.35/0.67                & ( ( F @ X3 )
% 0.35/0.67                  = one_one_nat )
% 0.35/0.67                & ! [Y2: int] :
% 0.35/0.67                    ( ( member_int @ Y2 @ A2 )
% 0.35/0.67                   => ( ( X3 != Y2 )
% 0.35/0.67                     => ( ( F @ Y2 )
% 0.35/0.67                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_eq_1_iff
% 0.35/0.67  thf(fact_6041_sum__eq__1__iff,axiom,
% 0.35/0.67      ! [A2: set_complex,F: complex > nat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.67       => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 0.35/0.67            = one_one_nat )
% 0.35/0.67          = ( ? [X3: complex] :
% 0.35/0.67                ( ( member_complex @ X3 @ A2 )
% 0.35/0.67                & ( ( F @ X3 )
% 0.35/0.67                  = one_one_nat )
% 0.35/0.67                & ! [Y2: complex] :
% 0.35/0.67                    ( ( member_complex @ Y2 @ A2 )
% 0.35/0.67                   => ( ( X3 != Y2 )
% 0.35/0.67                     => ( ( F @ Y2 )
% 0.35/0.67                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_eq_1_iff
% 0.35/0.67  thf(fact_6042_sum__eq__1__iff,axiom,
% 0.35/0.67      ! [A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 0.35/0.67        ( ( finite6177210948735845034at_nat @ A2 )
% 0.35/0.67       => ( ( ( groups977919841031483927at_nat @ F @ A2 )
% 0.35/0.67            = one_one_nat )
% 0.35/0.67          = ( ? [X3: product_prod_nat_nat] :
% 0.35/0.67                ( ( member8440522571783428010at_nat @ X3 @ A2 )
% 0.35/0.67                & ( ( F @ X3 )
% 0.35/0.67                  = one_one_nat )
% 0.35/0.67                & ! [Y2: product_prod_nat_nat] :
% 0.35/0.67                    ( ( member8440522571783428010at_nat @ Y2 @ A2 )
% 0.35/0.67                   => ( ( X3 != Y2 )
% 0.35/0.67                     => ( ( F @ Y2 )
% 0.35/0.67                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_eq_1_iff
% 0.35/0.67  thf(fact_6043_sum__eq__1__iff,axiom,
% 0.35/0.67      ! [A2: set_nat,F: nat > nat] :
% 0.35/0.67        ( ( finite_finite_nat @ A2 )
% 0.35/0.67       => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 0.35/0.67            = one_one_nat )
% 0.35/0.67          = ( ? [X3: nat] :
% 0.35/0.67                ( ( member_nat @ X3 @ A2 )
% 0.35/0.67                & ( ( F @ X3 )
% 0.35/0.67                  = one_one_nat )
% 0.35/0.67                & ! [Y2: nat] :
% 0.35/0.67                    ( ( member_nat @ Y2 @ A2 )
% 0.35/0.67                   => ( ( X3 != Y2 )
% 0.35/0.67                     => ( ( F @ Y2 )
% 0.35/0.67                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_eq_1_iff
% 0.35/0.67  thf(fact_6044_finite__ranking__induct,axiom,
% 0.35/0.67      ! [S2: set_complex,P: set_complex > $o,F: complex > rat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.67       => ( ( P @ bot_bot_set_complex )
% 0.35/0.67         => ( ! [X4: complex,S6: set_complex] :
% 0.35/0.67                ( ( finite3207457112153483333omplex @ S6 )
% 0.35/0.67               => ( ! [Y4: complex] :
% 0.35/0.67                      ( ( member_complex @ Y4 @ S6 )
% 0.35/0.67                     => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 0.35/0.67                 => ( ( P @ S6 )
% 0.35/0.67                   => ( P @ ( insert_complex @ X4 @ S6 ) ) ) ) )
% 0.35/0.67           => ( P @ S2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_ranking_induct
% 0.35/0.67  thf(fact_6045_finite__ranking__induct,axiom,
% 0.35/0.67      ! [S2: set_real,P: set_real > $o,F: real > rat] :
% 0.35/0.67        ( ( finite_finite_real @ S2 )
% 0.35/0.67       => ( ( P @ bot_bot_set_real )
% 0.35/0.67         => ( ! [X4: real,S6: set_real] :
% 0.35/0.67                ( ( finite_finite_real @ S6 )
% 0.35/0.67               => ( ! [Y4: real] :
% 0.35/0.67                      ( ( member_real @ Y4 @ S6 )
% 0.35/0.67                     => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 0.35/0.67                 => ( ( P @ S6 )
% 0.35/0.67                   => ( P @ ( insert_real @ X4 @ S6 ) ) ) ) )
% 0.35/0.67           => ( P @ S2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_ranking_induct
% 0.35/0.67  thf(fact_6046_finite__ranking__induct,axiom,
% 0.35/0.67      ! [S2: set_nat,P: set_nat > $o,F: nat > rat] :
% 0.35/0.67        ( ( finite_finite_nat @ S2 )
% 0.35/0.67       => ( ( P @ bot_bot_set_nat )
% 0.35/0.67         => ( ! [X4: nat,S6: set_nat] :
% 0.35/0.67                ( ( finite_finite_nat @ S6 )
% 0.35/0.67               => ( ! [Y4: nat] :
% 0.35/0.67                      ( ( member_nat @ Y4 @ S6 )
% 0.35/0.67                     => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 0.35/0.67                 => ( ( P @ S6 )
% 0.35/0.67                   => ( P @ ( insert_nat @ X4 @ S6 ) ) ) ) )
% 0.35/0.67           => ( P @ S2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_ranking_induct
% 0.35/0.67  thf(fact_6047_finite__ranking__induct,axiom,
% 0.35/0.67      ! [S2: set_int,P: set_int > $o,F: int > rat] :
% 0.35/0.67        ( ( finite_finite_int @ S2 )
% 0.35/0.67       => ( ( P @ bot_bot_set_int )
% 0.35/0.67         => ( ! [X4: int,S6: set_int] :
% 0.35/0.67                ( ( finite_finite_int @ S6 )
% 0.35/0.67               => ( ! [Y4: int] :
% 0.35/0.67                      ( ( member_int @ Y4 @ S6 )
% 0.35/0.67                     => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 0.35/0.67                 => ( ( P @ S6 )
% 0.35/0.67                   => ( P @ ( insert_int @ X4 @ S6 ) ) ) ) )
% 0.35/0.67           => ( P @ S2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_ranking_induct
% 0.35/0.67  thf(fact_6048_finite__ranking__induct,axiom,
% 0.35/0.67      ! [S2: set_complex,P: set_complex > $o,F: complex > num] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.67       => ( ( P @ bot_bot_set_complex )
% 0.35/0.67         => ( ! [X4: complex,S6: set_complex] :
% 0.35/0.67                ( ( finite3207457112153483333omplex @ S6 )
% 0.35/0.67               => ( ! [Y4: complex] :
% 0.35/0.67                      ( ( member_complex @ Y4 @ S6 )
% 0.35/0.67                     => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 0.35/0.67                 => ( ( P @ S6 )
% 0.35/0.67                   => ( P @ ( insert_complex @ X4 @ S6 ) ) ) ) )
% 0.35/0.67           => ( P @ S2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_ranking_induct
% 0.35/0.67  thf(fact_6049_finite__ranking__induct,axiom,
% 0.35/0.67      ! [S2: set_real,P: set_real > $o,F: real > num] :
% 0.35/0.67        ( ( finite_finite_real @ S2 )
% 0.35/0.67       => ( ( P @ bot_bot_set_real )
% 0.35/0.67         => ( ! [X4: real,S6: set_real] :
% 0.35/0.67                ( ( finite_finite_real @ S6 )
% 0.35/0.67               => ( ! [Y4: real] :
% 0.35/0.67                      ( ( member_real @ Y4 @ S6 )
% 0.35/0.67                     => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 0.35/0.67                 => ( ( P @ S6 )
% 0.35/0.67                   => ( P @ ( insert_real @ X4 @ S6 ) ) ) ) )
% 0.35/0.67           => ( P @ S2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_ranking_induct
% 0.35/0.67  thf(fact_6050_finite__ranking__induct,axiom,
% 0.35/0.67      ! [S2: set_nat,P: set_nat > $o,F: nat > num] :
% 0.35/0.67        ( ( finite_finite_nat @ S2 )
% 0.35/0.67       => ( ( P @ bot_bot_set_nat )
% 0.35/0.67         => ( ! [X4: nat,S6: set_nat] :
% 0.35/0.67                ( ( finite_finite_nat @ S6 )
% 0.35/0.67               => ( ! [Y4: nat] :
% 0.35/0.67                      ( ( member_nat @ Y4 @ S6 )
% 0.35/0.67                     => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 0.35/0.67                 => ( ( P @ S6 )
% 0.35/0.67                   => ( P @ ( insert_nat @ X4 @ S6 ) ) ) ) )
% 0.35/0.67           => ( P @ S2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_ranking_induct
% 0.35/0.67  thf(fact_6051_finite__ranking__induct,axiom,
% 0.35/0.67      ! [S2: set_int,P: set_int > $o,F: int > num] :
% 0.35/0.67        ( ( finite_finite_int @ S2 )
% 0.35/0.67       => ( ( P @ bot_bot_set_int )
% 0.35/0.67         => ( ! [X4: int,S6: set_int] :
% 0.35/0.67                ( ( finite_finite_int @ S6 )
% 0.35/0.67               => ( ! [Y4: int] :
% 0.35/0.67                      ( ( member_int @ Y4 @ S6 )
% 0.35/0.67                     => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 0.35/0.67                 => ( ( P @ S6 )
% 0.35/0.67                   => ( P @ ( insert_int @ X4 @ S6 ) ) ) ) )
% 0.35/0.67           => ( P @ S2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_ranking_induct
% 0.35/0.67  thf(fact_6052_finite__ranking__induct,axiom,
% 0.35/0.67      ! [S2: set_complex,P: set_complex > $o,F: complex > nat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.67       => ( ( P @ bot_bot_set_complex )
% 0.35/0.67         => ( ! [X4: complex,S6: set_complex] :
% 0.35/0.67                ( ( finite3207457112153483333omplex @ S6 )
% 0.35/0.67               => ( ! [Y4: complex] :
% 0.35/0.67                      ( ( member_complex @ Y4 @ S6 )
% 0.35/0.67                     => ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 0.35/0.67                 => ( ( P @ S6 )
% 0.35/0.67                   => ( P @ ( insert_complex @ X4 @ S6 ) ) ) ) )
% 0.35/0.67           => ( P @ S2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_ranking_induct
% 0.35/0.67  thf(fact_6053_finite__ranking__induct,axiom,
% 0.35/0.67      ! [S2: set_real,P: set_real > $o,F: real > nat] :
% 0.35/0.67        ( ( finite_finite_real @ S2 )
% 0.35/0.67       => ( ( P @ bot_bot_set_real )
% 0.35/0.67         => ( ! [X4: real,S6: set_real] :
% 0.35/0.67                ( ( finite_finite_real @ S6 )
% 0.35/0.67               => ( ! [Y4: real] :
% 0.35/0.67                      ( ( member_real @ Y4 @ S6 )
% 0.35/0.67                     => ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 0.35/0.67                 => ( ( P @ S6 )
% 0.35/0.67                   => ( P @ ( insert_real @ X4 @ S6 ) ) ) ) )
% 0.35/0.67           => ( P @ S2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_ranking_induct
% 0.35/0.67  thf(fact_6054_finite__linorder__min__induct,axiom,
% 0.35/0.67      ! [A2: set_real,P: set_real > $o] :
% 0.35/0.67        ( ( finite_finite_real @ A2 )
% 0.35/0.67       => ( ( P @ bot_bot_set_real )
% 0.35/0.67         => ( ! [B3: real,A5: set_real] :
% 0.35/0.67                ( ( finite_finite_real @ A5 )
% 0.35/0.67               => ( ! [X2: real] :
% 0.35/0.67                      ( ( member_real @ X2 @ A5 )
% 0.35/0.67                     => ( ord_less_real @ B3 @ X2 ) )
% 0.35/0.67                 => ( ( P @ A5 )
% 0.35/0.67                   => ( P @ ( insert_real @ B3 @ A5 ) ) ) ) )
% 0.35/0.67           => ( P @ A2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_linorder_min_induct
% 0.35/0.67  thf(fact_6055_finite__linorder__min__induct,axiom,
% 0.35/0.67      ! [A2: set_rat,P: set_rat > $o] :
% 0.35/0.67        ( ( finite_finite_rat @ A2 )
% 0.35/0.67       => ( ( P @ bot_bot_set_rat )
% 0.35/0.67         => ( ! [B3: rat,A5: set_rat] :
% 0.35/0.67                ( ( finite_finite_rat @ A5 )
% 0.35/0.67               => ( ! [X2: rat] :
% 0.35/0.67                      ( ( member_rat @ X2 @ A5 )
% 0.35/0.67                     => ( ord_less_rat @ B3 @ X2 ) )
% 0.35/0.67                 => ( ( P @ A5 )
% 0.35/0.67                   => ( P @ ( insert_rat @ B3 @ A5 ) ) ) ) )
% 0.35/0.67           => ( P @ A2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_linorder_min_induct
% 0.35/0.67  thf(fact_6056_finite__linorder__min__induct,axiom,
% 0.35/0.67      ! [A2: set_num,P: set_num > $o] :
% 0.35/0.67        ( ( finite_finite_num @ A2 )
% 0.35/0.67       => ( ( P @ bot_bot_set_num )
% 0.35/0.67         => ( ! [B3: num,A5: set_num] :
% 0.35/0.67                ( ( finite_finite_num @ A5 )
% 0.35/0.67               => ( ! [X2: num] :
% 0.35/0.67                      ( ( member_num @ X2 @ A5 )
% 0.35/0.67                     => ( ord_less_num @ B3 @ X2 ) )
% 0.35/0.67                 => ( ( P @ A5 )
% 0.35/0.67                   => ( P @ ( insert_num @ B3 @ A5 ) ) ) ) )
% 0.35/0.67           => ( P @ A2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_linorder_min_induct
% 0.35/0.67  thf(fact_6057_finite__linorder__min__induct,axiom,
% 0.35/0.67      ! [A2: set_nat,P: set_nat > $o] :
% 0.35/0.67        ( ( finite_finite_nat @ A2 )
% 0.35/0.67       => ( ( P @ bot_bot_set_nat )
% 0.35/0.67         => ( ! [B3: nat,A5: set_nat] :
% 0.35/0.67                ( ( finite_finite_nat @ A5 )
% 0.35/0.67               => ( ! [X2: nat] :
% 0.35/0.67                      ( ( member_nat @ X2 @ A5 )
% 0.35/0.67                     => ( ord_less_nat @ B3 @ X2 ) )
% 0.35/0.67                 => ( ( P @ A5 )
% 0.35/0.67                   => ( P @ ( insert_nat @ B3 @ A5 ) ) ) ) )
% 0.35/0.67           => ( P @ A2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_linorder_min_induct
% 0.35/0.67  thf(fact_6058_finite__linorder__min__induct,axiom,
% 0.35/0.67      ! [A2: set_int,P: set_int > $o] :
% 0.35/0.67        ( ( finite_finite_int @ A2 )
% 0.35/0.67       => ( ( P @ bot_bot_set_int )
% 0.35/0.67         => ( ! [B3: int,A5: set_int] :
% 0.35/0.67                ( ( finite_finite_int @ A5 )
% 0.35/0.67               => ( ! [X2: int] :
% 0.35/0.67                      ( ( member_int @ X2 @ A5 )
% 0.35/0.67                     => ( ord_less_int @ B3 @ X2 ) )
% 0.35/0.67                 => ( ( P @ A5 )
% 0.35/0.67                   => ( P @ ( insert_int @ B3 @ A5 ) ) ) ) )
% 0.35/0.67           => ( P @ A2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_linorder_min_induct
% 0.35/0.67  thf(fact_6059_finite__linorder__max__induct,axiom,
% 0.35/0.67      ! [A2: set_real,P: set_real > $o] :
% 0.35/0.67        ( ( finite_finite_real @ A2 )
% 0.35/0.67       => ( ( P @ bot_bot_set_real )
% 0.35/0.67         => ( ! [B3: real,A5: set_real] :
% 0.35/0.67                ( ( finite_finite_real @ A5 )
% 0.35/0.67               => ( ! [X2: real] :
% 0.35/0.67                      ( ( member_real @ X2 @ A5 )
% 0.35/0.67                     => ( ord_less_real @ X2 @ B3 ) )
% 0.35/0.67                 => ( ( P @ A5 )
% 0.35/0.67                   => ( P @ ( insert_real @ B3 @ A5 ) ) ) ) )
% 0.35/0.67           => ( P @ A2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_linorder_max_induct
% 0.35/0.67  thf(fact_6060_finite__linorder__max__induct,axiom,
% 0.35/0.67      ! [A2: set_rat,P: set_rat > $o] :
% 0.35/0.67        ( ( finite_finite_rat @ A2 )
% 0.35/0.67       => ( ( P @ bot_bot_set_rat )
% 0.35/0.67         => ( ! [B3: rat,A5: set_rat] :
% 0.35/0.67                ( ( finite_finite_rat @ A5 )
% 0.35/0.67               => ( ! [X2: rat] :
% 0.35/0.67                      ( ( member_rat @ X2 @ A5 )
% 0.35/0.67                     => ( ord_less_rat @ X2 @ B3 ) )
% 0.35/0.67                 => ( ( P @ A5 )
% 0.35/0.67                   => ( P @ ( insert_rat @ B3 @ A5 ) ) ) ) )
% 0.35/0.67           => ( P @ A2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_linorder_max_induct
% 0.35/0.67  thf(fact_6061_finite__linorder__max__induct,axiom,
% 0.35/0.67      ! [A2: set_num,P: set_num > $o] :
% 0.35/0.67        ( ( finite_finite_num @ A2 )
% 0.35/0.67       => ( ( P @ bot_bot_set_num )
% 0.35/0.67         => ( ! [B3: num,A5: set_num] :
% 0.35/0.67                ( ( finite_finite_num @ A5 )
% 0.35/0.67               => ( ! [X2: num] :
% 0.35/0.67                      ( ( member_num @ X2 @ A5 )
% 0.35/0.67                     => ( ord_less_num @ X2 @ B3 ) )
% 0.35/0.67                 => ( ( P @ A5 )
% 0.35/0.67                   => ( P @ ( insert_num @ B3 @ A5 ) ) ) ) )
% 0.35/0.67           => ( P @ A2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_linorder_max_induct
% 0.35/0.67  thf(fact_6062_finite__linorder__max__induct,axiom,
% 0.35/0.67      ! [A2: set_nat,P: set_nat > $o] :
% 0.35/0.67        ( ( finite_finite_nat @ A2 )
% 0.35/0.67       => ( ( P @ bot_bot_set_nat )
% 0.35/0.67         => ( ! [B3: nat,A5: set_nat] :
% 0.35/0.67                ( ( finite_finite_nat @ A5 )
% 0.35/0.67               => ( ! [X2: nat] :
% 0.35/0.67                      ( ( member_nat @ X2 @ A5 )
% 0.35/0.67                     => ( ord_less_nat @ X2 @ B3 ) )
% 0.35/0.67                 => ( ( P @ A5 )
% 0.35/0.67                   => ( P @ ( insert_nat @ B3 @ A5 ) ) ) ) )
% 0.35/0.67           => ( P @ A2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_linorder_max_induct
% 0.35/0.67  thf(fact_6063_finite__linorder__max__induct,axiom,
% 0.35/0.67      ! [A2: set_int,P: set_int > $o] :
% 0.35/0.67        ( ( finite_finite_int @ A2 )
% 0.35/0.67       => ( ( P @ bot_bot_set_int )
% 0.35/0.67         => ( ! [B3: int,A5: set_int] :
% 0.35/0.67                ( ( finite_finite_int @ A5 )
% 0.35/0.67               => ( ! [X2: int] :
% 0.35/0.67                      ( ( member_int @ X2 @ A5 )
% 0.35/0.67                     => ( ord_less_int @ X2 @ B3 ) )
% 0.35/0.67                 => ( ( P @ A5 )
% 0.35/0.67                   => ( P @ ( insert_int @ B3 @ A5 ) ) ) ) )
% 0.35/0.67           => ( P @ A2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_linorder_max_induct
% 0.35/0.67  thf(fact_6064_finite__subset__induct,axiom,
% 0.35/0.67      ! [F3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 0.35/0.67        ( ( finite4343798906461161616at_nat @ F3 )
% 0.35/0.67       => ( ( ord_le1268244103169919719at_nat @ F3 @ A2 )
% 0.35/0.67         => ( ( P @ bot_bo228742789529271731at_nat )
% 0.35/0.67           => ( ! [A3: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
% 0.35/0.67                  ( ( finite4343798906461161616at_nat @ F4 )
% 0.35/0.67                 => ( ( member8757157785044589968at_nat @ A3 @ A2 )
% 0.35/0.67                   => ( ~ ( member8757157785044589968at_nat @ A3 @ F4 )
% 0.35/0.67                     => ( ( P @ F4 )
% 0.35/0.67                       => ( P @ ( insert9069300056098147895at_nat @ A3 @ F4 ) ) ) ) ) )
% 0.35/0.67             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_subset_induct
% 0.35/0.67  thf(fact_6065_finite__subset__induct,axiom,
% 0.35/0.67      ! [F3: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
% 0.35/0.67        ( ( finite1152437895449049373et_nat @ F3 )
% 0.35/0.67       => ( ( ord_le6893508408891458716et_nat @ F3 @ A2 )
% 0.35/0.67         => ( ( P @ bot_bot_set_set_nat )
% 0.35/0.67           => ( ! [A3: set_nat,F4: set_set_nat] :
% 0.35/0.67                  ( ( finite1152437895449049373et_nat @ F4 )
% 0.35/0.67                 => ( ( member_set_nat @ A3 @ A2 )
% 0.35/0.67                   => ( ~ ( member_set_nat @ A3 @ F4 )
% 0.35/0.67                     => ( ( P @ F4 )
% 0.35/0.67                       => ( P @ ( insert_set_nat @ A3 @ F4 ) ) ) ) ) )
% 0.35/0.67             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_subset_induct
% 0.35/0.67  thf(fact_6066_finite__subset__induct,axiom,
% 0.35/0.67      ! [F3: set_complex,A2: set_complex,P: set_complex > $o] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ F3 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ F3 @ A2 )
% 0.35/0.67         => ( ( P @ bot_bot_set_complex )
% 0.35/0.67           => ( ! [A3: complex,F4: set_complex] :
% 0.35/0.67                  ( ( finite3207457112153483333omplex @ F4 )
% 0.35/0.67                 => ( ( member_complex @ A3 @ A2 )
% 0.35/0.67                   => ( ~ ( member_complex @ A3 @ F4 )
% 0.35/0.67                     => ( ( P @ F4 )
% 0.35/0.67                       => ( P @ ( insert_complex @ A3 @ F4 ) ) ) ) ) )
% 0.35/0.67             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_subset_induct
% 0.35/0.67  thf(fact_6067_finite__subset__induct,axiom,
% 0.35/0.67      ! [F3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.35/0.67        ( ( finite6177210948735845034at_nat @ F3 )
% 0.35/0.67       => ( ( ord_le3146513528884898305at_nat @ F3 @ A2 )
% 0.35/0.67         => ( ( P @ bot_bo2099793752762293965at_nat )
% 0.35/0.67           => ( ! [A3: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 0.35/0.67                  ( ( finite6177210948735845034at_nat @ F4 )
% 0.35/0.67                 => ( ( member8440522571783428010at_nat @ A3 @ A2 )
% 0.35/0.67                   => ( ~ ( member8440522571783428010at_nat @ A3 @ F4 )
% 0.35/0.67                     => ( ( P @ F4 )
% 0.35/0.67                       => ( P @ ( insert8211810215607154385at_nat @ A3 @ F4 ) ) ) ) ) )
% 0.35/0.67             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_subset_induct
% 0.35/0.67  thf(fact_6068_finite__subset__induct,axiom,
% 0.35/0.67      ! [F3: set_real,A2: set_real,P: set_real > $o] :
% 0.35/0.67        ( ( finite_finite_real @ F3 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ F3 @ A2 )
% 0.35/0.67         => ( ( P @ bot_bot_set_real )
% 0.35/0.67           => ( ! [A3: real,F4: set_real] :
% 0.35/0.67                  ( ( finite_finite_real @ F4 )
% 0.35/0.67                 => ( ( member_real @ A3 @ A2 )
% 0.35/0.67                   => ( ~ ( member_real @ A3 @ F4 )
% 0.35/0.67                     => ( ( P @ F4 )
% 0.35/0.67                       => ( P @ ( insert_real @ A3 @ F4 ) ) ) ) ) )
% 0.35/0.67             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_subset_induct
% 0.35/0.67  thf(fact_6069_finite__subset__induct,axiom,
% 0.35/0.67      ! [F3: set_nat,A2: set_nat,P: set_nat > $o] :
% 0.35/0.67        ( ( finite_finite_nat @ F3 )
% 0.35/0.67       => ( ( ord_less_eq_set_nat @ F3 @ A2 )
% 0.35/0.67         => ( ( P @ bot_bot_set_nat )
% 0.35/0.67           => ( ! [A3: nat,F4: set_nat] :
% 0.35/0.67                  ( ( finite_finite_nat @ F4 )
% 0.35/0.67                 => ( ( member_nat @ A3 @ A2 )
% 0.35/0.67                   => ( ~ ( member_nat @ A3 @ F4 )
% 0.35/0.67                     => ( ( P @ F4 )
% 0.35/0.67                       => ( P @ ( insert_nat @ A3 @ F4 ) ) ) ) ) )
% 0.35/0.67             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_subset_induct
% 0.35/0.67  thf(fact_6070_finite__subset__induct,axiom,
% 0.35/0.67      ! [F3: set_int,A2: set_int,P: set_int > $o] :
% 0.35/0.67        ( ( finite_finite_int @ F3 )
% 0.35/0.67       => ( ( ord_less_eq_set_int @ F3 @ A2 )
% 0.35/0.67         => ( ( P @ bot_bot_set_int )
% 0.35/0.67           => ( ! [A3: int,F4: set_int] :
% 0.35/0.67                  ( ( finite_finite_int @ F4 )
% 0.35/0.67                 => ( ( member_int @ A3 @ A2 )
% 0.35/0.67                   => ( ~ ( member_int @ A3 @ F4 )
% 0.35/0.67                     => ( ( P @ F4 )
% 0.35/0.67                       => ( P @ ( insert_int @ A3 @ F4 ) ) ) ) ) )
% 0.35/0.67             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_subset_induct
% 0.35/0.67  thf(fact_6071_finite__subset__induct_H,axiom,
% 0.35/0.67      ! [F3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 0.35/0.67        ( ( finite4343798906461161616at_nat @ F3 )
% 0.35/0.67       => ( ( ord_le1268244103169919719at_nat @ F3 @ A2 )
% 0.35/0.67         => ( ( P @ bot_bo228742789529271731at_nat )
% 0.35/0.67           => ( ! [A3: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
% 0.35/0.67                  ( ( finite4343798906461161616at_nat @ F4 )
% 0.35/0.67                 => ( ( member8757157785044589968at_nat @ A3 @ A2 )
% 0.35/0.67                   => ( ( ord_le1268244103169919719at_nat @ F4 @ A2 )
% 0.35/0.67                     => ( ~ ( member8757157785044589968at_nat @ A3 @ F4 )
% 0.35/0.67                       => ( ( P @ F4 )
% 0.35/0.67                         => ( P @ ( insert9069300056098147895at_nat @ A3 @ F4 ) ) ) ) ) ) )
% 0.35/0.67             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_subset_induct'
% 0.35/0.67  thf(fact_6072_finite__subset__induct_H,axiom,
% 0.35/0.67      ! [F3: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
% 0.35/0.67        ( ( finite1152437895449049373et_nat @ F3 )
% 0.35/0.67       => ( ( ord_le6893508408891458716et_nat @ F3 @ A2 )
% 0.35/0.67         => ( ( P @ bot_bot_set_set_nat )
% 0.35/0.67           => ( ! [A3: set_nat,F4: set_set_nat] :
% 0.35/0.67                  ( ( finite1152437895449049373et_nat @ F4 )
% 0.35/0.67                 => ( ( member_set_nat @ A3 @ A2 )
% 0.35/0.67                   => ( ( ord_le6893508408891458716et_nat @ F4 @ A2 )
% 0.35/0.67                     => ( ~ ( member_set_nat @ A3 @ F4 )
% 0.35/0.67                       => ( ( P @ F4 )
% 0.35/0.67                         => ( P @ ( insert_set_nat @ A3 @ F4 ) ) ) ) ) ) )
% 0.35/0.67             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_subset_induct'
% 0.35/0.67  thf(fact_6073_finite__subset__induct_H,axiom,
% 0.35/0.67      ! [F3: set_complex,A2: set_complex,P: set_complex > $o] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ F3 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ F3 @ A2 )
% 0.35/0.67         => ( ( P @ bot_bot_set_complex )
% 0.35/0.67           => ( ! [A3: complex,F4: set_complex] :
% 0.35/0.67                  ( ( finite3207457112153483333omplex @ F4 )
% 0.35/0.67                 => ( ( member_complex @ A3 @ A2 )
% 0.35/0.67                   => ( ( ord_le211207098394363844omplex @ F4 @ A2 )
% 0.35/0.67                     => ( ~ ( member_complex @ A3 @ F4 )
% 0.35/0.67                       => ( ( P @ F4 )
% 0.35/0.67                         => ( P @ ( insert_complex @ A3 @ F4 ) ) ) ) ) ) )
% 0.35/0.67             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_subset_induct'
% 0.35/0.67  thf(fact_6074_finite__subset__induct_H,axiom,
% 0.35/0.67      ! [F3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.35/0.67        ( ( finite6177210948735845034at_nat @ F3 )
% 0.35/0.67       => ( ( ord_le3146513528884898305at_nat @ F3 @ A2 )
% 0.35/0.67         => ( ( P @ bot_bo2099793752762293965at_nat )
% 0.35/0.67           => ( ! [A3: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 0.35/0.67                  ( ( finite6177210948735845034at_nat @ F4 )
% 0.35/0.67                 => ( ( member8440522571783428010at_nat @ A3 @ A2 )
% 0.35/0.67                   => ( ( ord_le3146513528884898305at_nat @ F4 @ A2 )
% 0.35/0.67                     => ( ~ ( member8440522571783428010at_nat @ A3 @ F4 )
% 0.35/0.67                       => ( ( P @ F4 )
% 0.35/0.67                         => ( P @ ( insert8211810215607154385at_nat @ A3 @ F4 ) ) ) ) ) ) )
% 0.35/0.67             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_subset_induct'
% 0.35/0.67  thf(fact_6075_finite__subset__induct_H,axiom,
% 0.35/0.67      ! [F3: set_real,A2: set_real,P: set_real > $o] :
% 0.35/0.67        ( ( finite_finite_real @ F3 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ F3 @ A2 )
% 0.35/0.67         => ( ( P @ bot_bot_set_real )
% 0.35/0.67           => ( ! [A3: real,F4: set_real] :
% 0.35/0.67                  ( ( finite_finite_real @ F4 )
% 0.35/0.67                 => ( ( member_real @ A3 @ A2 )
% 0.35/0.67                   => ( ( ord_less_eq_set_real @ F4 @ A2 )
% 0.35/0.67                     => ( ~ ( member_real @ A3 @ F4 )
% 0.35/0.67                       => ( ( P @ F4 )
% 0.35/0.67                         => ( P @ ( insert_real @ A3 @ F4 ) ) ) ) ) ) )
% 0.35/0.67             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_subset_induct'
% 0.35/0.67  thf(fact_6076_finite__subset__induct_H,axiom,
% 0.35/0.67      ! [F3: set_nat,A2: set_nat,P: set_nat > $o] :
% 0.35/0.67        ( ( finite_finite_nat @ F3 )
% 0.35/0.67       => ( ( ord_less_eq_set_nat @ F3 @ A2 )
% 0.35/0.67         => ( ( P @ bot_bot_set_nat )
% 0.35/0.67           => ( ! [A3: nat,F4: set_nat] :
% 0.35/0.67                  ( ( finite_finite_nat @ F4 )
% 0.35/0.67                 => ( ( member_nat @ A3 @ A2 )
% 0.35/0.67                   => ( ( ord_less_eq_set_nat @ F4 @ A2 )
% 0.35/0.67                     => ( ~ ( member_nat @ A3 @ F4 )
% 0.35/0.67                       => ( ( P @ F4 )
% 0.35/0.67                         => ( P @ ( insert_nat @ A3 @ F4 ) ) ) ) ) ) )
% 0.35/0.67             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_subset_induct'
% 0.35/0.67  thf(fact_6077_finite__subset__induct_H,axiom,
% 0.35/0.67      ! [F3: set_int,A2: set_int,P: set_int > $o] :
% 0.35/0.67        ( ( finite_finite_int @ F3 )
% 0.35/0.67       => ( ( ord_less_eq_set_int @ F3 @ A2 )
% 0.35/0.67         => ( ( P @ bot_bot_set_int )
% 0.35/0.67           => ( ! [A3: int,F4: set_int] :
% 0.35/0.67                  ( ( finite_finite_int @ F4 )
% 0.35/0.67                 => ( ( member_int @ A3 @ A2 )
% 0.35/0.67                   => ( ( ord_less_eq_set_int @ F4 @ A2 )
% 0.35/0.67                     => ( ~ ( member_int @ A3 @ F4 )
% 0.35/0.67                       => ( ( P @ F4 )
% 0.35/0.67                         => ( P @ ( insert_int @ A3 @ F4 ) ) ) ) ) ) )
% 0.35/0.67             => ( P @ F3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_subset_induct'
% 0.35/0.67  thf(fact_6078_sum__nonneg__leq__bound,axiom,
% 0.35/0.67      ! [S: set_real,F: real > real,B2: real,I: real] :
% 0.35/0.67        ( ( finite_finite_real @ S )
% 0.35/0.67       => ( ! [I2: real] :
% 0.35/0.67              ( ( member_real @ I2 @ S )
% 0.35/0.67             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.67         => ( ( ( groups8097168146408367636l_real @ F @ S )
% 0.35/0.67              = B2 )
% 0.35/0.67           => ( ( member_real @ I @ S )
% 0.35/0.67             => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_nonneg_leq_bound
% 0.35/0.67  thf(fact_6079_sum__nonneg__leq__bound,axiom,
% 0.35/0.67      ! [S: set_int,F: int > real,B2: real,I: int] :
% 0.35/0.67        ( ( finite_finite_int @ S )
% 0.35/0.67       => ( ! [I2: int] :
% 0.35/0.67              ( ( member_int @ I2 @ S )
% 0.35/0.67             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.67         => ( ( ( groups8778361861064173332t_real @ F @ S )
% 0.35/0.67              = B2 )
% 0.35/0.67           => ( ( member_int @ I @ S )
% 0.35/0.67             => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_nonneg_leq_bound
% 0.35/0.67  thf(fact_6080_sum__nonneg__leq__bound,axiom,
% 0.35/0.67      ! [S: set_complex,F: complex > real,B2: real,I: complex] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ S )
% 0.35/0.67       => ( ! [I2: complex] :
% 0.35/0.67              ( ( member_complex @ I2 @ S )
% 0.35/0.67             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.67         => ( ( ( groups5808333547571424918x_real @ F @ S )
% 0.35/0.67              = B2 )
% 0.35/0.67           => ( ( member_complex @ I @ S )
% 0.35/0.67             => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_nonneg_leq_bound
% 0.35/0.67  thf(fact_6081_sum__nonneg__leq__bound,axiom,
% 0.35/0.67      ! [S: set_real,F: real > rat,B2: rat,I: real] :
% 0.35/0.67        ( ( finite_finite_real @ S )
% 0.35/0.67       => ( ! [I2: real] :
% 0.35/0.67              ( ( member_real @ I2 @ S )
% 0.35/0.67             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.67         => ( ( ( groups1300246762558778688al_rat @ F @ S )
% 0.35/0.67              = B2 )
% 0.35/0.67           => ( ( member_real @ I @ S )
% 0.35/0.67             => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_nonneg_leq_bound
% 0.35/0.67  thf(fact_6082_sum__nonneg__leq__bound,axiom,
% 0.35/0.67      ! [S: set_nat,F: nat > rat,B2: rat,I: nat] :
% 0.35/0.67        ( ( finite_finite_nat @ S )
% 0.35/0.67       => ( ! [I2: nat] :
% 0.35/0.67              ( ( member_nat @ I2 @ S )
% 0.35/0.67             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.67         => ( ( ( groups2906978787729119204at_rat @ F @ S )
% 0.35/0.67              = B2 )
% 0.35/0.67           => ( ( member_nat @ I @ S )
% 0.35/0.67             => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_nonneg_leq_bound
% 0.35/0.67  thf(fact_6083_sum__nonneg__leq__bound,axiom,
% 0.35/0.67      ! [S: set_int,F: int > rat,B2: rat,I: int] :
% 0.35/0.67        ( ( finite_finite_int @ S )
% 0.35/0.67       => ( ! [I2: int] :
% 0.35/0.67              ( ( member_int @ I2 @ S )
% 0.35/0.67             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.67         => ( ( ( groups3906332499630173760nt_rat @ F @ S )
% 0.35/0.67              = B2 )
% 0.35/0.67           => ( ( member_int @ I @ S )
% 0.35/0.67             => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_nonneg_leq_bound
% 0.35/0.67  thf(fact_6084_sum__nonneg__leq__bound,axiom,
% 0.35/0.67      ! [S: set_complex,F: complex > rat,B2: rat,I: complex] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ S )
% 0.35/0.67       => ( ! [I2: complex] :
% 0.35/0.67              ( ( member_complex @ I2 @ S )
% 0.35/0.67             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.67         => ( ( ( groups5058264527183730370ex_rat @ F @ S )
% 0.35/0.67              = B2 )
% 0.35/0.67           => ( ( member_complex @ I @ S )
% 0.35/0.67             => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_nonneg_leq_bound
% 0.35/0.67  thf(fact_6085_sum__nonneg__leq__bound,axiom,
% 0.35/0.67      ! [S: set_real,F: real > nat,B2: nat,I: real] :
% 0.35/0.67        ( ( finite_finite_real @ S )
% 0.35/0.67       => ( ! [I2: real] :
% 0.35/0.67              ( ( member_real @ I2 @ S )
% 0.35/0.67             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.35/0.67         => ( ( ( groups1935376822645274424al_nat @ F @ S )
% 0.35/0.67              = B2 )
% 0.35/0.67           => ( ( member_real @ I @ S )
% 0.35/0.67             => ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_nonneg_leq_bound
% 0.35/0.67  thf(fact_6086_sum__nonneg__leq__bound,axiom,
% 0.35/0.67      ! [S: set_int,F: int > nat,B2: nat,I: int] :
% 0.35/0.67        ( ( finite_finite_int @ S )
% 0.35/0.67       => ( ! [I2: int] :
% 0.35/0.67              ( ( member_int @ I2 @ S )
% 0.35/0.67             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.35/0.67         => ( ( ( groups4541462559716669496nt_nat @ F @ S )
% 0.35/0.67              = B2 )
% 0.35/0.67           => ( ( member_int @ I @ S )
% 0.35/0.67             => ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_nonneg_leq_bound
% 0.35/0.67  thf(fact_6087_sum__nonneg__leq__bound,axiom,
% 0.35/0.67      ! [S: set_complex,F: complex > nat,B2: nat,I: complex] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ S )
% 0.35/0.67       => ( ! [I2: complex] :
% 0.35/0.67              ( ( member_complex @ I2 @ S )
% 0.35/0.67             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.35/0.67         => ( ( ( groups5693394587270226106ex_nat @ F @ S )
% 0.35/0.67              = B2 )
% 0.35/0.67           => ( ( member_complex @ I @ S )
% 0.35/0.67             => ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_nonneg_leq_bound
% 0.35/0.67  thf(fact_6088_sum__nonneg__0,axiom,
% 0.35/0.67      ! [S: set_real,F: real > real,I: real] :
% 0.35/0.67        ( ( finite_finite_real @ S )
% 0.35/0.67       => ( ! [I2: real] :
% 0.35/0.67              ( ( member_real @ I2 @ S )
% 0.35/0.67             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.67         => ( ( ( groups8097168146408367636l_real @ F @ S )
% 0.35/0.67              = zero_zero_real )
% 0.35/0.67           => ( ( member_real @ I @ S )
% 0.35/0.67             => ( ( F @ I )
% 0.35/0.67                = zero_zero_real ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_nonneg_0
% 0.35/0.67  thf(fact_6089_sum__nonneg__0,axiom,
% 0.35/0.67      ! [S: set_int,F: int > real,I: int] :
% 0.35/0.67        ( ( finite_finite_int @ S )
% 0.35/0.67       => ( ! [I2: int] :
% 0.35/0.67              ( ( member_int @ I2 @ S )
% 0.35/0.67             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.67         => ( ( ( groups8778361861064173332t_real @ F @ S )
% 0.35/0.67              = zero_zero_real )
% 0.35/0.67           => ( ( member_int @ I @ S )
% 0.35/0.67             => ( ( F @ I )
% 0.35/0.67                = zero_zero_real ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_nonneg_0
% 0.35/0.67  thf(fact_6090_sum__nonneg__0,axiom,
% 0.35/0.67      ! [S: set_complex,F: complex > real,I: complex] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ S )
% 0.35/0.67       => ( ! [I2: complex] :
% 0.35/0.67              ( ( member_complex @ I2 @ S )
% 0.35/0.67             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.67         => ( ( ( groups5808333547571424918x_real @ F @ S )
% 0.35/0.67              = zero_zero_real )
% 0.35/0.67           => ( ( member_complex @ I @ S )
% 0.35/0.67             => ( ( F @ I )
% 0.35/0.67                = zero_zero_real ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_nonneg_0
% 0.35/0.67  thf(fact_6091_sum__nonneg__0,axiom,
% 0.35/0.67      ! [S: set_real,F: real > rat,I: real] :
% 0.35/0.67        ( ( finite_finite_real @ S )
% 0.35/0.67       => ( ! [I2: real] :
% 0.35/0.67              ( ( member_real @ I2 @ S )
% 0.35/0.67             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.67         => ( ( ( groups1300246762558778688al_rat @ F @ S )
% 0.35/0.67              = zero_zero_rat )
% 0.35/0.67           => ( ( member_real @ I @ S )
% 0.35/0.67             => ( ( F @ I )
% 0.35/0.67                = zero_zero_rat ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_nonneg_0
% 0.35/0.67  thf(fact_6092_sum__nonneg__0,axiom,
% 0.35/0.67      ! [S: set_nat,F: nat > rat,I: nat] :
% 0.35/0.67        ( ( finite_finite_nat @ S )
% 0.35/0.67       => ( ! [I2: nat] :
% 0.35/0.67              ( ( member_nat @ I2 @ S )
% 0.35/0.67             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.67         => ( ( ( groups2906978787729119204at_rat @ F @ S )
% 0.35/0.67              = zero_zero_rat )
% 0.35/0.67           => ( ( member_nat @ I @ S )
% 0.35/0.67             => ( ( F @ I )
% 0.35/0.67                = zero_zero_rat ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_nonneg_0
% 0.35/0.67  thf(fact_6093_sum__nonneg__0,axiom,
% 0.35/0.67      ! [S: set_int,F: int > rat,I: int] :
% 0.35/0.67        ( ( finite_finite_int @ S )
% 0.35/0.67       => ( ! [I2: int] :
% 0.35/0.67              ( ( member_int @ I2 @ S )
% 0.35/0.67             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.67         => ( ( ( groups3906332499630173760nt_rat @ F @ S )
% 0.35/0.67              = zero_zero_rat )
% 0.35/0.67           => ( ( member_int @ I @ S )
% 0.35/0.67             => ( ( F @ I )
% 0.35/0.67                = zero_zero_rat ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_nonneg_0
% 0.35/0.67  thf(fact_6094_sum__nonneg__0,axiom,
% 0.35/0.67      ! [S: set_complex,F: complex > rat,I: complex] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ S )
% 0.35/0.67       => ( ! [I2: complex] :
% 0.35/0.67              ( ( member_complex @ I2 @ S )
% 0.35/0.67             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.67         => ( ( ( groups5058264527183730370ex_rat @ F @ S )
% 0.35/0.67              = zero_zero_rat )
% 0.35/0.67           => ( ( member_complex @ I @ S )
% 0.35/0.67             => ( ( F @ I )
% 0.35/0.67                = zero_zero_rat ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_nonneg_0
% 0.35/0.67  thf(fact_6095_sum__nonneg__0,axiom,
% 0.35/0.67      ! [S: set_real,F: real > nat,I: real] :
% 0.35/0.67        ( ( finite_finite_real @ S )
% 0.35/0.67       => ( ! [I2: real] :
% 0.35/0.67              ( ( member_real @ I2 @ S )
% 0.35/0.67             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.35/0.67         => ( ( ( groups1935376822645274424al_nat @ F @ S )
% 0.35/0.67              = zero_zero_nat )
% 0.35/0.67           => ( ( member_real @ I @ S )
% 0.35/0.67             => ( ( F @ I )
% 0.35/0.67                = zero_zero_nat ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_nonneg_0
% 0.35/0.67  thf(fact_6096_sum__nonneg__0,axiom,
% 0.35/0.67      ! [S: set_int,F: int > nat,I: int] :
% 0.35/0.67        ( ( finite_finite_int @ S )
% 0.35/0.67       => ( ! [I2: int] :
% 0.35/0.67              ( ( member_int @ I2 @ S )
% 0.35/0.67             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.35/0.67         => ( ( ( groups4541462559716669496nt_nat @ F @ S )
% 0.35/0.67              = zero_zero_nat )
% 0.35/0.67           => ( ( member_int @ I @ S )
% 0.35/0.67             => ( ( F @ I )
% 0.35/0.67                = zero_zero_nat ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_nonneg_0
% 0.35/0.67  thf(fact_6097_sum__nonneg__0,axiom,
% 0.35/0.67      ! [S: set_complex,F: complex > nat,I: complex] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ S )
% 0.35/0.67       => ( ! [I2: complex] :
% 0.35/0.67              ( ( member_complex @ I2 @ S )
% 0.35/0.67             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.35/0.67         => ( ( ( groups5693394587270226106ex_nat @ F @ S )
% 0.35/0.67              = zero_zero_nat )
% 0.35/0.67           => ( ( member_complex @ I @ S )
% 0.35/0.67             => ( ( F @ I )
% 0.35/0.67                = zero_zero_nat ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_nonneg_0
% 0.35/0.67  thf(fact_6098_finite__empty__induct,axiom,
% 0.35/0.67      ! [A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 0.35/0.67        ( ( finite4343798906461161616at_nat @ A2 )
% 0.35/0.67       => ( ( P @ A2 )
% 0.35/0.67         => ( ! [A3: produc3843707927480180839at_nat,A5: set_Pr4329608150637261639at_nat] :
% 0.35/0.67                ( ( finite4343798906461161616at_nat @ A5 )
% 0.35/0.67               => ( ( member8757157785044589968at_nat @ A3 @ A5 )
% 0.35/0.67                 => ( ( P @ A5 )
% 0.35/0.67                   => ( P @ ( minus_3314409938677909166at_nat @ A5 @ ( insert9069300056098147895at_nat @ A3 @ bot_bo228742789529271731at_nat ) ) ) ) ) )
% 0.35/0.67           => ( P @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_empty_induct
% 0.35/0.67  thf(fact_6099_finite__empty__induct,axiom,
% 0.35/0.67      ! [A2: set_set_nat,P: set_set_nat > $o] :
% 0.35/0.67        ( ( finite1152437895449049373et_nat @ A2 )
% 0.35/0.67       => ( ( P @ A2 )
% 0.35/0.67         => ( ! [A3: set_nat,A5: set_set_nat] :
% 0.35/0.67                ( ( finite1152437895449049373et_nat @ A5 )
% 0.35/0.67               => ( ( member_set_nat @ A3 @ A5 )
% 0.35/0.67                 => ( ( P @ A5 )
% 0.35/0.67                   => ( P @ ( minus_2163939370556025621et_nat @ A5 @ ( insert_set_nat @ A3 @ bot_bot_set_set_nat ) ) ) ) ) )
% 0.35/0.67           => ( P @ bot_bot_set_set_nat ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_empty_induct
% 0.35/0.67  thf(fact_6100_finite__empty__induct,axiom,
% 0.35/0.67      ! [A2: set_complex,P: set_complex > $o] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.67       => ( ( P @ A2 )
% 0.35/0.67         => ( ! [A3: complex,A5: set_complex] :
% 0.35/0.67                ( ( finite3207457112153483333omplex @ A5 )
% 0.35/0.67               => ( ( member_complex @ A3 @ A5 )
% 0.35/0.67                 => ( ( P @ A5 )
% 0.35/0.67                   => ( P @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ A3 @ bot_bot_set_complex ) ) ) ) ) )
% 0.35/0.67           => ( P @ bot_bot_set_complex ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_empty_induct
% 0.35/0.67  thf(fact_6101_finite__empty__induct,axiom,
% 0.35/0.67      ! [A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.35/0.67        ( ( finite6177210948735845034at_nat @ A2 )
% 0.35/0.67       => ( ( P @ A2 )
% 0.35/0.67         => ( ! [A3: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
% 0.35/0.67                ( ( finite6177210948735845034at_nat @ A5 )
% 0.35/0.67               => ( ( member8440522571783428010at_nat @ A3 @ A5 )
% 0.35/0.67                 => ( ( P @ A5 )
% 0.35/0.67                   => ( P @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ A3 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
% 0.35/0.67           => ( P @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_empty_induct
% 0.35/0.67  thf(fact_6102_finite__empty__induct,axiom,
% 0.35/0.67      ! [A2: set_real,P: set_real > $o] :
% 0.35/0.67        ( ( finite_finite_real @ A2 )
% 0.35/0.67       => ( ( P @ A2 )
% 0.35/0.67         => ( ! [A3: real,A5: set_real] :
% 0.35/0.67                ( ( finite_finite_real @ A5 )
% 0.35/0.67               => ( ( member_real @ A3 @ A5 )
% 0.35/0.67                 => ( ( P @ A5 )
% 0.35/0.67                   => ( P @ ( minus_minus_set_real @ A5 @ ( insert_real @ A3 @ bot_bot_set_real ) ) ) ) ) )
% 0.35/0.67           => ( P @ bot_bot_set_real ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_empty_induct
% 0.35/0.67  thf(fact_6103_finite__empty__induct,axiom,
% 0.35/0.67      ! [A2: set_int,P: set_int > $o] :
% 0.35/0.67        ( ( finite_finite_int @ A2 )
% 0.35/0.67       => ( ( P @ A2 )
% 0.35/0.67         => ( ! [A3: int,A5: set_int] :
% 0.35/0.67                ( ( finite_finite_int @ A5 )
% 0.35/0.67               => ( ( member_int @ A3 @ A5 )
% 0.35/0.67                 => ( ( P @ A5 )
% 0.35/0.67                   => ( P @ ( minus_minus_set_int @ A5 @ ( insert_int @ A3 @ bot_bot_set_int ) ) ) ) ) )
% 0.35/0.67           => ( P @ bot_bot_set_int ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_empty_induct
% 0.35/0.67  thf(fact_6104_finite__empty__induct,axiom,
% 0.35/0.67      ! [A2: set_nat,P: set_nat > $o] :
% 0.35/0.67        ( ( finite_finite_nat @ A2 )
% 0.35/0.67       => ( ( P @ A2 )
% 0.35/0.67         => ( ! [A3: nat,A5: set_nat] :
% 0.35/0.67                ( ( finite_finite_nat @ A5 )
% 0.35/0.67               => ( ( member_nat @ A3 @ A5 )
% 0.35/0.67                 => ( ( P @ A5 )
% 0.35/0.67                   => ( P @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ) )
% 0.35/0.67           => ( P @ bot_bot_set_nat ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % finite_empty_induct
% 0.35/0.67  thf(fact_6105_infinite__coinduct,axiom,
% 0.35/0.67      ! [X8: set_Pr4329608150637261639at_nat > $o,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.67        ( ( X8 @ A2 )
% 0.35/0.67       => ( ! [A5: set_Pr4329608150637261639at_nat] :
% 0.35/0.67              ( ( X8 @ A5 )
% 0.35/0.67             => ? [X2: produc3843707927480180839at_nat] :
% 0.35/0.67                  ( ( member8757157785044589968at_nat @ X2 @ A5 )
% 0.35/0.67                  & ( ( X8 @ ( minus_3314409938677909166at_nat @ A5 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) )
% 0.35/0.67                    | ~ ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A5 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) ) ) )
% 0.35/0.67         => ~ ( finite4343798906461161616at_nat @ A2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % infinite_coinduct
% 0.35/0.67  thf(fact_6106_infinite__coinduct,axiom,
% 0.35/0.67      ! [X8: set_complex > $o,A2: set_complex] :
% 0.35/0.67        ( ( X8 @ A2 )
% 0.35/0.67       => ( ! [A5: set_complex] :
% 0.35/0.67              ( ( X8 @ A5 )
% 0.35/0.67             => ? [X2: complex] :
% 0.35/0.67                  ( ( member_complex @ X2 @ A5 )
% 0.35/0.67                  & ( ( X8 @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) )
% 0.35/0.67                    | ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) )
% 0.35/0.67         => ~ ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % infinite_coinduct
% 0.35/0.67  thf(fact_6107_infinite__coinduct,axiom,
% 0.35/0.67      ! [X8: set_Pr1261947904930325089at_nat > $o,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.67        ( ( X8 @ A2 )
% 0.35/0.67       => ( ! [A5: set_Pr1261947904930325089at_nat] :
% 0.35/0.67              ( ( X8 @ A5 )
% 0.35/0.67             => ? [X2: product_prod_nat_nat] :
% 0.35/0.67                  ( ( member8440522571783428010at_nat @ X2 @ A5 )
% 0.35/0.67                  & ( ( X8 @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
% 0.35/0.67                    | ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
% 0.35/0.67         => ~ ( finite6177210948735845034at_nat @ A2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % infinite_coinduct
% 0.35/0.67  thf(fact_6108_infinite__coinduct,axiom,
% 0.35/0.67      ! [X8: set_real > $o,A2: set_real] :
% 0.35/0.67        ( ( X8 @ A2 )
% 0.35/0.67       => ( ! [A5: set_real] :
% 0.35/0.67              ( ( X8 @ A5 )
% 0.35/0.67             => ? [X2: real] :
% 0.35/0.67                  ( ( member_real @ X2 @ A5 )
% 0.35/0.67                  & ( ( X8 @ ( minus_minus_set_real @ A5 @ ( insert_real @ X2 @ bot_bot_set_real ) ) )
% 0.35/0.67                    | ~ ( finite_finite_real @ ( minus_minus_set_real @ A5 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) )
% 0.35/0.67         => ~ ( finite_finite_real @ A2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % infinite_coinduct
% 0.35/0.67  thf(fact_6109_infinite__coinduct,axiom,
% 0.35/0.67      ! [X8: set_int > $o,A2: set_int] :
% 0.35/0.67        ( ( X8 @ A2 )
% 0.35/0.67       => ( ! [A5: set_int] :
% 0.35/0.67              ( ( X8 @ A5 )
% 0.35/0.67             => ? [X2: int] :
% 0.35/0.67                  ( ( member_int @ X2 @ A5 )
% 0.35/0.67                  & ( ( X8 @ ( minus_minus_set_int @ A5 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
% 0.35/0.67                    | ~ ( finite_finite_int @ ( minus_minus_set_int @ A5 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) )
% 0.35/0.67         => ~ ( finite_finite_int @ A2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % infinite_coinduct
% 0.35/0.67  thf(fact_6110_infinite__coinduct,axiom,
% 0.35/0.67      ! [X8: set_nat > $o,A2: set_nat] :
% 0.35/0.67        ( ( X8 @ A2 )
% 0.35/0.67       => ( ! [A5: set_nat] :
% 0.35/0.67              ( ( X8 @ A5 )
% 0.35/0.67             => ? [X2: nat] :
% 0.35/0.67                  ( ( member_nat @ X2 @ A5 )
% 0.35/0.67                  & ( ( X8 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
% 0.35/0.67                    | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) )
% 0.35/0.67         => ~ ( finite_finite_nat @ A2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % infinite_coinduct
% 0.35/0.67  thf(fact_6111_infinite__remove,axiom,
% 0.35/0.67      ! [S2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
% 0.35/0.67        ( ~ ( finite4343798906461161616at_nat @ S2 )
% 0.35/0.67       => ~ ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ S2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % infinite_remove
% 0.35/0.67  thf(fact_6112_infinite__remove,axiom,
% 0.35/0.67      ! [S2: set_complex,A: complex] :
% 0.35/0.67        ( ~ ( finite3207457112153483333omplex @ S2 )
% 0.35/0.67       => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % infinite_remove
% 0.35/0.67  thf(fact_6113_infinite__remove,axiom,
% 0.35/0.67      ! [S2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
% 0.35/0.67        ( ~ ( finite6177210948735845034at_nat @ S2 )
% 0.35/0.67       => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % infinite_remove
% 0.35/0.67  thf(fact_6114_infinite__remove,axiom,
% 0.35/0.67      ! [S2: set_real,A: real] :
% 0.35/0.67        ( ~ ( finite_finite_real @ S2 )
% 0.35/0.67       => ~ ( finite_finite_real @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % infinite_remove
% 0.35/0.67  thf(fact_6115_infinite__remove,axiom,
% 0.35/0.67      ! [S2: set_int,A: int] :
% 0.35/0.67        ( ~ ( finite_finite_int @ S2 )
% 0.35/0.67       => ~ ( finite_finite_int @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % infinite_remove
% 0.35/0.67  thf(fact_6116_infinite__remove,axiom,
% 0.35/0.67      ! [S2: set_nat,A: nat] :
% 0.35/0.67        ( ~ ( finite_finite_nat @ S2 )
% 0.35/0.67       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % infinite_remove
% 0.35/0.67  thf(fact_6117_Diff__single__insert,axiom,
% 0.35/0.67      ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.67        ( ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B2 )
% 0.35/0.67       => ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ B2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % Diff_single_insert
% 0.35/0.67  thf(fact_6118_Diff__single__insert,axiom,
% 0.35/0.67      ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.67        ( ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B2 )
% 0.35/0.67       => ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % Diff_single_insert
% 0.35/0.67  thf(fact_6119_Diff__single__insert,axiom,
% 0.35/0.67      ! [A2: set_real,X: real,B2: set_real] :
% 0.35/0.67        ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 )
% 0.35/0.67       => ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % Diff_single_insert
% 0.35/0.67  thf(fact_6120_Diff__single__insert,axiom,
% 0.35/0.67      ! [A2: set_nat,X: nat,B2: set_nat] :
% 0.35/0.67        ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 )
% 0.35/0.67       => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % Diff_single_insert
% 0.35/0.67  thf(fact_6121_Diff__single__insert,axiom,
% 0.35/0.67      ! [A2: set_int,X: int,B2: set_int] :
% 0.35/0.67        ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 )
% 0.35/0.67       => ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % Diff_single_insert
% 0.35/0.67  thf(fact_6122_subset__insert__iff,axiom,
% 0.35/0.67      ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.67        ( ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ B2 ) )
% 0.35/0.67        = ( ( ( member8757157785044589968at_nat @ X @ A2 )
% 0.35/0.67           => ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B2 ) )
% 0.35/0.67          & ( ~ ( member8757157785044589968at_nat @ X @ A2 )
% 0.35/0.67           => ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % subset_insert_iff
% 0.35/0.67  thf(fact_6123_subset__insert__iff,axiom,
% 0.35/0.67      ! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
% 0.35/0.67        ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
% 0.35/0.67        = ( ( ( member_set_nat @ X @ A2 )
% 0.35/0.67           => ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B2 ) )
% 0.35/0.67          & ( ~ ( member_set_nat @ X @ A2 )
% 0.35/0.67           => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % subset_insert_iff
% 0.35/0.67  thf(fact_6124_subset__insert__iff,axiom,
% 0.35/0.67      ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.67        ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B2 ) )
% 0.35/0.67        = ( ( ( member8440522571783428010at_nat @ X @ A2 )
% 0.35/0.67           => ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B2 ) )
% 0.35/0.67          & ( ~ ( member8440522571783428010at_nat @ X @ A2 )
% 0.35/0.67           => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % subset_insert_iff
% 0.35/0.67  thf(fact_6125_subset__insert__iff,axiom,
% 0.35/0.67      ! [A2: set_real,X: real,B2: set_real] :
% 0.35/0.67        ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) )
% 0.35/0.67        = ( ( ( member_real @ X @ A2 )
% 0.35/0.67           => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 ) )
% 0.35/0.67          & ( ~ ( member_real @ X @ A2 )
% 0.35/0.67           => ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % subset_insert_iff
% 0.35/0.67  thf(fact_6126_subset__insert__iff,axiom,
% 0.35/0.67      ! [A2: set_nat,X: nat,B2: set_nat] :
% 0.35/0.67        ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
% 0.35/0.67        = ( ( ( member_nat @ X @ A2 )
% 0.35/0.67           => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
% 0.35/0.67          & ( ~ ( member_nat @ X @ A2 )
% 0.35/0.67           => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % subset_insert_iff
% 0.35/0.67  thf(fact_6127_subset__insert__iff,axiom,
% 0.35/0.67      ! [A2: set_int,X: int,B2: set_int] :
% 0.35/0.67        ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) )
% 0.35/0.67        = ( ( ( member_int @ X @ A2 )
% 0.35/0.67           => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 ) )
% 0.35/0.67          & ( ~ ( member_int @ X @ A2 )
% 0.35/0.67           => ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % subset_insert_iff
% 0.35/0.67  thf(fact_6128_atLeast0__atMost__Suc,axiom,
% 0.35/0.67      ! [N: nat] :
% 0.35/0.67        ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.35/0.67        = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % atLeast0_atMost_Suc
% 0.35/0.67  thf(fact_6129_Icc__eq__insert__lb__nat,axiom,
% 0.35/0.67      ! [M: nat,N: nat] :
% 0.35/0.67        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.67       => ( ( set_or1269000886237332187st_nat @ M @ N )
% 0.35/0.67          = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % Icc_eq_insert_lb_nat
% 0.35/0.67  thf(fact_6130_atLeastAtMostSuc__conv,axiom,
% 0.35/0.67      ! [M: nat,N: nat] :
% 0.35/0.67        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.35/0.67       => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
% 0.35/0.67          = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % atLeastAtMostSuc_conv
% 0.35/0.67  thf(fact_6131_atLeastAtMost__insertL,axiom,
% 0.35/0.67      ! [M: nat,N: nat] :
% 0.35/0.67        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.67       => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.35/0.67          = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % atLeastAtMost_insertL
% 0.35/0.67  thf(fact_6132_sum_Osetdiff__irrelevant,axiom,
% 0.35/0.67      ! [A2: set_real,G: real > complex] :
% 0.35/0.67        ( ( finite_finite_real @ A2 )
% 0.35/0.67       => ( ( groups5754745047067104278omplex @ G
% 0.35/0.67            @ ( minus_minus_set_real @ A2
% 0.35/0.67              @ ( collect_real
% 0.35/0.67                @ ^ [X3: real] :
% 0.35/0.67                    ( ( G @ X3 )
% 0.35/0.67                    = zero_zero_complex ) ) ) )
% 0.35/0.67          = ( groups5754745047067104278omplex @ G @ A2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.setdiff_irrelevant
% 0.35/0.67  thf(fact_6133_sum_Osetdiff__irrelevant,axiom,
% 0.35/0.67      ! [A2: set_int,G: int > complex] :
% 0.35/0.67        ( ( finite_finite_int @ A2 )
% 0.35/0.67       => ( ( groups3049146728041665814omplex @ G
% 0.35/0.67            @ ( minus_minus_set_int @ A2
% 0.35/0.67              @ ( collect_int
% 0.35/0.67                @ ^ [X3: int] :
% 0.35/0.67                    ( ( G @ X3 )
% 0.35/0.67                    = zero_zero_complex ) ) ) )
% 0.35/0.67          = ( groups3049146728041665814omplex @ G @ A2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.setdiff_irrelevant
% 0.35/0.67  thf(fact_6134_sum_Osetdiff__irrelevant,axiom,
% 0.35/0.67      ! [A2: set_real,G: real > real] :
% 0.35/0.67        ( ( finite_finite_real @ A2 )
% 0.35/0.67       => ( ( groups8097168146408367636l_real @ G
% 0.35/0.67            @ ( minus_minus_set_real @ A2
% 0.35/0.67              @ ( collect_real
% 0.35/0.67                @ ^ [X3: real] :
% 0.35/0.67                    ( ( G @ X3 )
% 0.35/0.67                    = zero_zero_real ) ) ) )
% 0.35/0.67          = ( groups8097168146408367636l_real @ G @ A2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.setdiff_irrelevant
% 0.35/0.67  thf(fact_6135_sum_Osetdiff__irrelevant,axiom,
% 0.35/0.67      ! [A2: set_int,G: int > real] :
% 0.35/0.67        ( ( finite_finite_int @ A2 )
% 0.35/0.67       => ( ( groups8778361861064173332t_real @ G
% 0.35/0.67            @ ( minus_minus_set_int @ A2
% 0.35/0.67              @ ( collect_int
% 0.35/0.67                @ ^ [X3: int] :
% 0.35/0.67                    ( ( G @ X3 )
% 0.35/0.67                    = zero_zero_real ) ) ) )
% 0.35/0.67          = ( groups8778361861064173332t_real @ G @ A2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.setdiff_irrelevant
% 0.35/0.67  thf(fact_6136_sum_Osetdiff__irrelevant,axiom,
% 0.35/0.67      ! [A2: set_complex,G: complex > real] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.67       => ( ( groups5808333547571424918x_real @ G
% 0.35/0.67            @ ( minus_811609699411566653omplex @ A2
% 0.35/0.67              @ ( collect_complex
% 0.35/0.67                @ ^ [X3: complex] :
% 0.35/0.67                    ( ( G @ X3 )
% 0.35/0.67                    = zero_zero_real ) ) ) )
% 0.35/0.67          = ( groups5808333547571424918x_real @ G @ A2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.setdiff_irrelevant
% 0.35/0.67  thf(fact_6137_sum_Osetdiff__irrelevant,axiom,
% 0.35/0.67      ! [A2: set_real,G: real > rat] :
% 0.35/0.67        ( ( finite_finite_real @ A2 )
% 0.35/0.67       => ( ( groups1300246762558778688al_rat @ G
% 0.35/0.67            @ ( minus_minus_set_real @ A2
% 0.35/0.67              @ ( collect_real
% 0.35/0.67                @ ^ [X3: real] :
% 0.35/0.67                    ( ( G @ X3 )
% 0.35/0.67                    = zero_zero_rat ) ) ) )
% 0.35/0.67          = ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.setdiff_irrelevant
% 0.35/0.67  thf(fact_6138_sum_Osetdiff__irrelevant,axiom,
% 0.35/0.67      ! [A2: set_int,G: int > rat] :
% 0.35/0.67        ( ( finite_finite_int @ A2 )
% 0.35/0.67       => ( ( groups3906332499630173760nt_rat @ G
% 0.35/0.67            @ ( minus_minus_set_int @ A2
% 0.35/0.67              @ ( collect_int
% 0.35/0.67                @ ^ [X3: int] :
% 0.35/0.67                    ( ( G @ X3 )
% 0.35/0.67                    = zero_zero_rat ) ) ) )
% 0.35/0.67          = ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.setdiff_irrelevant
% 0.35/0.67  thf(fact_6139_sum_Osetdiff__irrelevant,axiom,
% 0.35/0.67      ! [A2: set_complex,G: complex > rat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.67       => ( ( groups5058264527183730370ex_rat @ G
% 0.35/0.67            @ ( minus_811609699411566653omplex @ A2
% 0.35/0.67              @ ( collect_complex
% 0.35/0.67                @ ^ [X3: complex] :
% 0.35/0.67                    ( ( G @ X3 )
% 0.35/0.67                    = zero_zero_rat ) ) ) )
% 0.35/0.67          = ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.setdiff_irrelevant
% 0.35/0.67  thf(fact_6140_sum_Osetdiff__irrelevant,axiom,
% 0.35/0.67      ! [A2: set_real,G: real > nat] :
% 0.35/0.67        ( ( finite_finite_real @ A2 )
% 0.35/0.67       => ( ( groups1935376822645274424al_nat @ G
% 0.35/0.67            @ ( minus_minus_set_real @ A2
% 0.35/0.67              @ ( collect_real
% 0.35/0.67                @ ^ [X3: real] :
% 0.35/0.67                    ( ( G @ X3 )
% 0.35/0.67                    = zero_zero_nat ) ) ) )
% 0.35/0.67          = ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.setdiff_irrelevant
% 0.35/0.67  thf(fact_6141_sum_Osetdiff__irrelevant,axiom,
% 0.35/0.67      ! [A2: set_int,G: int > nat] :
% 0.35/0.67        ( ( finite_finite_int @ A2 )
% 0.35/0.67       => ( ( groups4541462559716669496nt_nat @ G
% 0.35/0.67            @ ( minus_minus_set_int @ A2
% 0.35/0.67              @ ( collect_int
% 0.35/0.67                @ ^ [X3: int] :
% 0.35/0.67                    ( ( G @ X3 )
% 0.35/0.67                    = zero_zero_nat ) ) ) )
% 0.35/0.67          = ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.setdiff_irrelevant
% 0.35/0.67  thf(fact_6142_set__update__subset__insert,axiom,
% 0.35/0.67      ! [Xs: list_P6011104703257516679at_nat,I: nat,X: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ X ) ) @ ( insert8211810215607154385at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % set_update_subset_insert
% 0.35/0.67  thf(fact_6143_set__update__subset__insert,axiom,
% 0.35/0.67      ! [Xs: list_real,I: nat,X: real] : ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X ) ) @ ( insert_real @ X @ ( set_real2 @ Xs ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % set_update_subset_insert
% 0.35/0.67  thf(fact_6144_set__update__subset__insert,axiom,
% 0.35/0.67      ! [Xs: list_P5464809261938338413at_nat,I: nat,X: produc3843707927480180839at_nat] : ( ord_le1268244103169919719at_nat @ ( set_Pr3765526544606949372at_nat @ ( list_u4696772448584712917at_nat @ Xs @ I @ X ) ) @ ( insert9069300056098147895at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % set_update_subset_insert
% 0.35/0.67  thf(fact_6145_set__update__subset__insert,axiom,
% 0.35/0.67      ! [Xs: list_nat,I: nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ ( insert_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % set_update_subset_insert
% 0.35/0.67  thf(fact_6146_set__update__subset__insert,axiom,
% 0.35/0.67      ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) ) @ ( insert_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % set_update_subset_insert
% 0.35/0.67  thf(fact_6147_set__update__subset__insert,axiom,
% 0.35/0.67      ! [Xs: list_int,I: nat,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X ) ) @ ( insert_int @ X @ ( set_int2 @ Xs ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % set_update_subset_insert
% 0.35/0.67  thf(fact_6148_sum__pos2,axiom,
% 0.35/0.67      ! [I6: set_real,I: real,F: real > real] :
% 0.35/0.67        ( ( finite_finite_real @ I6 )
% 0.35/0.67       => ( ( member_real @ I @ I6 )
% 0.35/0.67         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.35/0.67           => ( ! [I2: real] :
% 0.35/0.67                  ( ( member_real @ I2 @ I6 )
% 0.35/0.67                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.67             => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I6 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_pos2
% 0.35/0.67  thf(fact_6149_sum__pos2,axiom,
% 0.35/0.67      ! [I6: set_int,I: int,F: int > real] :
% 0.35/0.67        ( ( finite_finite_int @ I6 )
% 0.35/0.67       => ( ( member_int @ I @ I6 )
% 0.35/0.67         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.35/0.67           => ( ! [I2: int] :
% 0.35/0.67                  ( ( member_int @ I2 @ I6 )
% 0.35/0.67                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.67             => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I6 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_pos2
% 0.35/0.67  thf(fact_6150_sum__pos2,axiom,
% 0.35/0.67      ! [I6: set_complex,I: complex,F: complex > real] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ I6 )
% 0.35/0.67       => ( ( member_complex @ I @ I6 )
% 0.35/0.67         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.35/0.67           => ( ! [I2: complex] :
% 0.35/0.67                  ( ( member_complex @ I2 @ I6 )
% 0.35/0.67                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.67             => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_pos2
% 0.35/0.67  thf(fact_6151_sum__pos2,axiom,
% 0.35/0.67      ! [I6: set_real,I: real,F: real > rat] :
% 0.35/0.67        ( ( finite_finite_real @ I6 )
% 0.35/0.67       => ( ( member_real @ I @ I6 )
% 0.35/0.67         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 0.35/0.67           => ( ! [I2: real] :
% 0.35/0.67                  ( ( member_real @ I2 @ I6 )
% 0.35/0.67                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.67             => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I6 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_pos2
% 0.35/0.67  thf(fact_6152_sum__pos2,axiom,
% 0.35/0.67      ! [I6: set_nat,I: nat,F: nat > rat] :
% 0.35/0.67        ( ( finite_finite_nat @ I6 )
% 0.35/0.67       => ( ( member_nat @ I @ I6 )
% 0.35/0.67         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 0.35/0.67           => ( ! [I2: nat] :
% 0.35/0.67                  ( ( member_nat @ I2 @ I6 )
% 0.35/0.67                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.67             => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I6 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_pos2
% 0.35/0.67  thf(fact_6153_sum__pos2,axiom,
% 0.35/0.67      ! [I6: set_int,I: int,F: int > rat] :
% 0.35/0.67        ( ( finite_finite_int @ I6 )
% 0.35/0.67       => ( ( member_int @ I @ I6 )
% 0.35/0.67         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 0.35/0.67           => ( ! [I2: int] :
% 0.35/0.67                  ( ( member_int @ I2 @ I6 )
% 0.35/0.67                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.67             => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I6 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_pos2
% 0.35/0.67  thf(fact_6154_sum__pos2,axiom,
% 0.35/0.67      ! [I6: set_complex,I: complex,F: complex > rat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ I6 )
% 0.35/0.67       => ( ( member_complex @ I @ I6 )
% 0.35/0.67         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 0.35/0.67           => ( ! [I2: complex] :
% 0.35/0.67                  ( ( member_complex @ I2 @ I6 )
% 0.35/0.67                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.67             => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I6 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_pos2
% 0.35/0.67  thf(fact_6155_sum__pos2,axiom,
% 0.35/0.67      ! [I6: set_real,I: real,F: real > nat] :
% 0.35/0.67        ( ( finite_finite_real @ I6 )
% 0.35/0.67       => ( ( member_real @ I @ I6 )
% 0.35/0.67         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 0.35/0.67           => ( ! [I2: real] :
% 0.35/0.67                  ( ( member_real @ I2 @ I6 )
% 0.35/0.67                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.35/0.67             => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I6 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_pos2
% 0.35/0.67  thf(fact_6156_sum__pos2,axiom,
% 0.35/0.67      ! [I6: set_int,I: int,F: int > nat] :
% 0.35/0.67        ( ( finite_finite_int @ I6 )
% 0.35/0.67       => ( ( member_int @ I @ I6 )
% 0.35/0.67         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 0.35/0.67           => ( ! [I2: int] :
% 0.35/0.67                  ( ( member_int @ I2 @ I6 )
% 0.35/0.67                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.35/0.67             => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I6 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_pos2
% 0.35/0.67  thf(fact_6157_sum__pos2,axiom,
% 0.35/0.67      ! [I6: set_complex,I: complex,F: complex > nat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ I6 )
% 0.35/0.67       => ( ( member_complex @ I @ I6 )
% 0.35/0.67         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 0.35/0.67           => ( ! [I2: complex] :
% 0.35/0.67                  ( ( member_complex @ I2 @ I6 )
% 0.35/0.67                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.35/0.67             => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I6 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_pos2
% 0.35/0.67  thf(fact_6158_sum__pos,axiom,
% 0.35/0.67      ! [I6: set_complex,F: complex > real] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ I6 )
% 0.35/0.67       => ( ( I6 != bot_bot_set_complex )
% 0.35/0.67         => ( ! [I2: complex] :
% 0.35/0.67                ( ( member_complex @ I2 @ I6 )
% 0.35/0.67               => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.67           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_pos
% 0.35/0.67  thf(fact_6159_sum__pos,axiom,
% 0.35/0.67      ! [I6: set_real,F: real > real] :
% 0.35/0.67        ( ( finite_finite_real @ I6 )
% 0.35/0.67       => ( ( I6 != bot_bot_set_real )
% 0.35/0.67         => ( ! [I2: real] :
% 0.35/0.67                ( ( member_real @ I2 @ I6 )
% 0.35/0.67               => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.67           => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I6 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_pos
% 0.35/0.67  thf(fact_6160_sum__pos,axiom,
% 0.35/0.67      ! [I6: set_int,F: int > real] :
% 0.35/0.67        ( ( finite_finite_int @ I6 )
% 0.35/0.67       => ( ( I6 != bot_bot_set_int )
% 0.35/0.67         => ( ! [I2: int] :
% 0.35/0.67                ( ( member_int @ I2 @ I6 )
% 0.35/0.67               => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.67           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I6 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_pos
% 0.35/0.67  thf(fact_6161_sum__pos,axiom,
% 0.35/0.67      ! [I6: set_complex,F: complex > rat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ I6 )
% 0.35/0.67       => ( ( I6 != bot_bot_set_complex )
% 0.35/0.67         => ( ! [I2: complex] :
% 0.35/0.67                ( ( member_complex @ I2 @ I6 )
% 0.35/0.67               => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.67           => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I6 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_pos
% 0.35/0.67  thf(fact_6162_sum__pos,axiom,
% 0.35/0.67      ! [I6: set_real,F: real > rat] :
% 0.35/0.67        ( ( finite_finite_real @ I6 )
% 0.35/0.67       => ( ( I6 != bot_bot_set_real )
% 0.35/0.67         => ( ! [I2: real] :
% 0.35/0.67                ( ( member_real @ I2 @ I6 )
% 0.35/0.67               => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.67           => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I6 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_pos
% 0.35/0.67  thf(fact_6163_sum__pos,axiom,
% 0.35/0.67      ! [I6: set_nat,F: nat > rat] :
% 0.35/0.67        ( ( finite_finite_nat @ I6 )
% 0.35/0.67       => ( ( I6 != bot_bot_set_nat )
% 0.35/0.67         => ( ! [I2: nat] :
% 0.35/0.67                ( ( member_nat @ I2 @ I6 )
% 0.35/0.67               => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.67           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I6 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_pos
% 0.35/0.67  thf(fact_6164_sum__pos,axiom,
% 0.35/0.67      ! [I6: set_int,F: int > rat] :
% 0.35/0.67        ( ( finite_finite_int @ I6 )
% 0.35/0.67       => ( ( I6 != bot_bot_set_int )
% 0.35/0.67         => ( ! [I2: int] :
% 0.35/0.67                ( ( member_int @ I2 @ I6 )
% 0.35/0.67               => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.67           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I6 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_pos
% 0.35/0.67  thf(fact_6165_sum__pos,axiom,
% 0.35/0.67      ! [I6: set_complex,F: complex > nat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ I6 )
% 0.35/0.67       => ( ( I6 != bot_bot_set_complex )
% 0.35/0.67         => ( ! [I2: complex] :
% 0.35/0.67                ( ( member_complex @ I2 @ I6 )
% 0.35/0.67               => ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.35/0.67           => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I6 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_pos
% 0.35/0.67  thf(fact_6166_sum__pos,axiom,
% 0.35/0.67      ! [I6: set_real,F: real > nat] :
% 0.35/0.67        ( ( finite_finite_real @ I6 )
% 0.35/0.67       => ( ( I6 != bot_bot_set_real )
% 0.35/0.67         => ( ! [I2: real] :
% 0.35/0.67                ( ( member_real @ I2 @ I6 )
% 0.35/0.67               => ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.35/0.67           => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I6 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_pos
% 0.35/0.67  thf(fact_6167_sum__pos,axiom,
% 0.35/0.67      ! [I6: set_int,F: int > nat] :
% 0.35/0.67        ( ( finite_finite_int @ I6 )
% 0.35/0.67       => ( ( I6 != bot_bot_set_int )
% 0.35/0.67         => ( ! [I2: int] :
% 0.35/0.67                ( ( member_int @ I2 @ I6 )
% 0.35/0.67               => ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.35/0.67           => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I6 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_pos
% 0.35/0.67  thf(fact_6168_sum_Omono__neutral__cong__right,axiom,
% 0.35/0.67      ! [T3: set_real,S2: set_real,G: real > complex,H2: real > complex] :
% 0.35/0.67        ( ( finite_finite_real @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: real] :
% 0.35/0.67                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_complex ) )
% 0.35/0.67           => ( ! [X4: real] :
% 0.35/0.67                  ( ( member_real @ X4 @ S2 )
% 0.35/0.67                 => ( ( G @ X4 )
% 0.35/0.67                    = ( H2 @ X4 ) ) )
% 0.35/0.67             => ( ( groups5754745047067104278omplex @ G @ T3 )
% 0.35/0.67                = ( groups5754745047067104278omplex @ H2 @ S2 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_cong_right
% 0.35/0.67  thf(fact_6169_sum_Omono__neutral__cong__right,axiom,
% 0.35/0.67      ! [T3: set_real,S2: set_real,G: real > real,H2: real > real] :
% 0.35/0.67        ( ( finite_finite_real @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: real] :
% 0.35/0.67                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_real ) )
% 0.35/0.67           => ( ! [X4: real] :
% 0.35/0.67                  ( ( member_real @ X4 @ S2 )
% 0.35/0.67                 => ( ( G @ X4 )
% 0.35/0.67                    = ( H2 @ X4 ) ) )
% 0.35/0.67             => ( ( groups8097168146408367636l_real @ G @ T3 )
% 0.35/0.67                = ( groups8097168146408367636l_real @ H2 @ S2 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_cong_right
% 0.35/0.67  thf(fact_6170_sum_Omono__neutral__cong__right,axiom,
% 0.35/0.67      ! [T3: set_complex,S2: set_complex,G: complex > real,H2: complex > real] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ T3 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: complex] :
% 0.35/0.67                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_real ) )
% 0.35/0.67           => ( ! [X4: complex] :
% 0.35/0.67                  ( ( member_complex @ X4 @ S2 )
% 0.35/0.67                 => ( ( G @ X4 )
% 0.35/0.67                    = ( H2 @ X4 ) ) )
% 0.35/0.67             => ( ( groups5808333547571424918x_real @ G @ T3 )
% 0.35/0.67                = ( groups5808333547571424918x_real @ H2 @ S2 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_cong_right
% 0.35/0.67  thf(fact_6171_sum_Omono__neutral__cong__right,axiom,
% 0.35/0.67      ! [T3: set_real,S2: set_real,G: real > rat,H2: real > rat] :
% 0.35/0.67        ( ( finite_finite_real @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: real] :
% 0.35/0.67                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_rat ) )
% 0.35/0.67           => ( ! [X4: real] :
% 0.35/0.67                  ( ( member_real @ X4 @ S2 )
% 0.35/0.67                 => ( ( G @ X4 )
% 0.35/0.67                    = ( H2 @ X4 ) ) )
% 0.35/0.67             => ( ( groups1300246762558778688al_rat @ G @ T3 )
% 0.35/0.67                = ( groups1300246762558778688al_rat @ H2 @ S2 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_cong_right
% 0.35/0.67  thf(fact_6172_sum_Omono__neutral__cong__right,axiom,
% 0.35/0.67      ! [T3: set_complex,S2: set_complex,G: complex > rat,H2: complex > rat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ T3 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: complex] :
% 0.35/0.67                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_rat ) )
% 0.35/0.67           => ( ! [X4: complex] :
% 0.35/0.67                  ( ( member_complex @ X4 @ S2 )
% 0.35/0.67                 => ( ( G @ X4 )
% 0.35/0.67                    = ( H2 @ X4 ) ) )
% 0.35/0.67             => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 0.35/0.67                = ( groups5058264527183730370ex_rat @ H2 @ S2 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_cong_right
% 0.35/0.67  thf(fact_6173_sum_Omono__neutral__cong__right,axiom,
% 0.35/0.67      ! [T3: set_real,S2: set_real,G: real > nat,H2: real > nat] :
% 0.35/0.67        ( ( finite_finite_real @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: real] :
% 0.35/0.67                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_nat ) )
% 0.35/0.67           => ( ! [X4: real] :
% 0.35/0.67                  ( ( member_real @ X4 @ S2 )
% 0.35/0.67                 => ( ( G @ X4 )
% 0.35/0.67                    = ( H2 @ X4 ) ) )
% 0.35/0.67             => ( ( groups1935376822645274424al_nat @ G @ T3 )
% 0.35/0.67                = ( groups1935376822645274424al_nat @ H2 @ S2 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_cong_right
% 0.35/0.67  thf(fact_6174_sum_Omono__neutral__cong__right,axiom,
% 0.35/0.67      ! [T3: set_complex,S2: set_complex,G: complex > nat,H2: complex > nat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ T3 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: complex] :
% 0.35/0.67                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_nat ) )
% 0.35/0.67           => ( ! [X4: complex] :
% 0.35/0.67                  ( ( member_complex @ X4 @ S2 )
% 0.35/0.67                 => ( ( G @ X4 )
% 0.35/0.67                    = ( H2 @ X4 ) ) )
% 0.35/0.67             => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 0.35/0.67                = ( groups5693394587270226106ex_nat @ H2 @ S2 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_cong_right
% 0.35/0.67  thf(fact_6175_sum_Omono__neutral__cong__right,axiom,
% 0.35/0.67      ! [T3: set_real,S2: set_real,G: real > int,H2: real > int] :
% 0.35/0.67        ( ( finite_finite_real @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: real] :
% 0.35/0.67                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_int ) )
% 0.35/0.67           => ( ! [X4: real] :
% 0.35/0.67                  ( ( member_real @ X4 @ S2 )
% 0.35/0.67                 => ( ( G @ X4 )
% 0.35/0.67                    = ( H2 @ X4 ) ) )
% 0.35/0.67             => ( ( groups1932886352136224148al_int @ G @ T3 )
% 0.35/0.67                = ( groups1932886352136224148al_int @ H2 @ S2 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_cong_right
% 0.35/0.67  thf(fact_6176_sum_Omono__neutral__cong__right,axiom,
% 0.35/0.67      ! [T3: set_complex,S2: set_complex,G: complex > int,H2: complex > int] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ T3 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: complex] :
% 0.35/0.67                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_int ) )
% 0.35/0.67           => ( ! [X4: complex] :
% 0.35/0.67                  ( ( member_complex @ X4 @ S2 )
% 0.35/0.67                 => ( ( G @ X4 )
% 0.35/0.67                    = ( H2 @ X4 ) ) )
% 0.35/0.67             => ( ( groups5690904116761175830ex_int @ G @ T3 )
% 0.35/0.67                = ( groups5690904116761175830ex_int @ H2 @ S2 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_cong_right
% 0.35/0.67  thf(fact_6177_sum_Omono__neutral__cong__right,axiom,
% 0.35/0.67      ! [T3: set_nat,S2: set_nat,G: nat > complex,H2: nat > complex] :
% 0.35/0.67        ( ( finite_finite_nat @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: nat] :
% 0.35/0.67                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_complex ) )
% 0.35/0.67           => ( ! [X4: nat] :
% 0.35/0.67                  ( ( member_nat @ X4 @ S2 )
% 0.35/0.67                 => ( ( G @ X4 )
% 0.35/0.67                    = ( H2 @ X4 ) ) )
% 0.35/0.67             => ( ( groups2073611262835488442omplex @ G @ T3 )
% 0.35/0.67                = ( groups2073611262835488442omplex @ H2 @ S2 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_cong_right
% 0.35/0.67  thf(fact_6178_sum_Omono__neutral__cong__left,axiom,
% 0.35/0.67      ! [T3: set_real,S2: set_real,H2: real > complex,G: real > complex] :
% 0.35/0.67        ( ( finite_finite_real @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: real] :
% 0.35/0.67                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.35/0.67               => ( ( H2 @ X4 )
% 0.35/0.67                  = zero_zero_complex ) )
% 0.35/0.67           => ( ! [X4: real] :
% 0.35/0.67                  ( ( member_real @ X4 @ S2 )
% 0.35/0.67                 => ( ( G @ X4 )
% 0.35/0.67                    = ( H2 @ X4 ) ) )
% 0.35/0.67             => ( ( groups5754745047067104278omplex @ G @ S2 )
% 0.35/0.67                = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_cong_left
% 0.35/0.67  thf(fact_6179_sum_Omono__neutral__cong__left,axiom,
% 0.35/0.67      ! [T3: set_real,S2: set_real,H2: real > real,G: real > real] :
% 0.35/0.67        ( ( finite_finite_real @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: real] :
% 0.35/0.67                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.35/0.67               => ( ( H2 @ X4 )
% 0.35/0.67                  = zero_zero_real ) )
% 0.35/0.67           => ( ! [X4: real] :
% 0.35/0.67                  ( ( member_real @ X4 @ S2 )
% 0.35/0.67                 => ( ( G @ X4 )
% 0.35/0.67                    = ( H2 @ X4 ) ) )
% 0.35/0.67             => ( ( groups8097168146408367636l_real @ G @ S2 )
% 0.35/0.67                = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_cong_left
% 0.35/0.67  thf(fact_6180_sum_Omono__neutral__cong__left,axiom,
% 0.35/0.67      ! [T3: set_complex,S2: set_complex,H2: complex > real,G: complex > real] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ T3 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: complex] :
% 0.35/0.67                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.35/0.67               => ( ( H2 @ X4 )
% 0.35/0.67                  = zero_zero_real ) )
% 0.35/0.67           => ( ! [X4: complex] :
% 0.35/0.67                  ( ( member_complex @ X4 @ S2 )
% 0.35/0.67                 => ( ( G @ X4 )
% 0.35/0.67                    = ( H2 @ X4 ) ) )
% 0.35/0.67             => ( ( groups5808333547571424918x_real @ G @ S2 )
% 0.35/0.67                = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_cong_left
% 0.35/0.67  thf(fact_6181_sum_Omono__neutral__cong__left,axiom,
% 0.35/0.67      ! [T3: set_real,S2: set_real,H2: real > rat,G: real > rat] :
% 0.35/0.67        ( ( finite_finite_real @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: real] :
% 0.35/0.67                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.35/0.67               => ( ( H2 @ X4 )
% 0.35/0.67                  = zero_zero_rat ) )
% 0.35/0.67           => ( ! [X4: real] :
% 0.35/0.67                  ( ( member_real @ X4 @ S2 )
% 0.35/0.67                 => ( ( G @ X4 )
% 0.35/0.67                    = ( H2 @ X4 ) ) )
% 0.35/0.67             => ( ( groups1300246762558778688al_rat @ G @ S2 )
% 0.35/0.67                = ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_cong_left
% 0.35/0.67  thf(fact_6182_sum_Omono__neutral__cong__left,axiom,
% 0.35/0.67      ! [T3: set_complex,S2: set_complex,H2: complex > rat,G: complex > rat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ T3 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: complex] :
% 0.35/0.67                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.35/0.67               => ( ( H2 @ X4 )
% 0.35/0.67                  = zero_zero_rat ) )
% 0.35/0.67           => ( ! [X4: complex] :
% 0.35/0.67                  ( ( member_complex @ X4 @ S2 )
% 0.35/0.67                 => ( ( G @ X4 )
% 0.35/0.67                    = ( H2 @ X4 ) ) )
% 0.35/0.67             => ( ( groups5058264527183730370ex_rat @ G @ S2 )
% 0.35/0.67                = ( groups5058264527183730370ex_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_cong_left
% 0.35/0.67  thf(fact_6183_sum_Omono__neutral__cong__left,axiom,
% 0.35/0.67      ! [T3: set_real,S2: set_real,H2: real > nat,G: real > nat] :
% 0.35/0.67        ( ( finite_finite_real @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: real] :
% 0.35/0.67                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.35/0.67               => ( ( H2 @ X4 )
% 0.35/0.67                  = zero_zero_nat ) )
% 0.35/0.67           => ( ! [X4: real] :
% 0.35/0.67                  ( ( member_real @ X4 @ S2 )
% 0.35/0.67                 => ( ( G @ X4 )
% 0.35/0.67                    = ( H2 @ X4 ) ) )
% 0.35/0.67             => ( ( groups1935376822645274424al_nat @ G @ S2 )
% 0.35/0.67                = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_cong_left
% 0.35/0.67  thf(fact_6184_sum_Omono__neutral__cong__left,axiom,
% 0.35/0.67      ! [T3: set_complex,S2: set_complex,H2: complex > nat,G: complex > nat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ T3 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: complex] :
% 0.35/0.67                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.35/0.67               => ( ( H2 @ X4 )
% 0.35/0.67                  = zero_zero_nat ) )
% 0.35/0.67           => ( ! [X4: complex] :
% 0.35/0.67                  ( ( member_complex @ X4 @ S2 )
% 0.35/0.67                 => ( ( G @ X4 )
% 0.35/0.67                    = ( H2 @ X4 ) ) )
% 0.35/0.67             => ( ( groups5693394587270226106ex_nat @ G @ S2 )
% 0.35/0.67                = ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_cong_left
% 0.35/0.67  thf(fact_6185_sum_Omono__neutral__cong__left,axiom,
% 0.35/0.67      ! [T3: set_real,S2: set_real,H2: real > int,G: real > int] :
% 0.35/0.67        ( ( finite_finite_real @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: real] :
% 0.35/0.67                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.35/0.67               => ( ( H2 @ X4 )
% 0.35/0.67                  = zero_zero_int ) )
% 0.35/0.67           => ( ! [X4: real] :
% 0.35/0.67                  ( ( member_real @ X4 @ S2 )
% 0.35/0.67                 => ( ( G @ X4 )
% 0.35/0.67                    = ( H2 @ X4 ) ) )
% 0.35/0.67             => ( ( groups1932886352136224148al_int @ G @ S2 )
% 0.35/0.67                = ( groups1932886352136224148al_int @ H2 @ T3 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_cong_left
% 0.35/0.67  thf(fact_6186_sum_Omono__neutral__cong__left,axiom,
% 0.35/0.67      ! [T3: set_complex,S2: set_complex,H2: complex > int,G: complex > int] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ T3 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: complex] :
% 0.35/0.67                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.35/0.67               => ( ( H2 @ X4 )
% 0.35/0.67                  = zero_zero_int ) )
% 0.35/0.67           => ( ! [X4: complex] :
% 0.35/0.67                  ( ( member_complex @ X4 @ S2 )
% 0.35/0.67                 => ( ( G @ X4 )
% 0.35/0.67                    = ( H2 @ X4 ) ) )
% 0.35/0.67             => ( ( groups5690904116761175830ex_int @ G @ S2 )
% 0.35/0.67                = ( groups5690904116761175830ex_int @ H2 @ T3 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_cong_left
% 0.35/0.67  thf(fact_6187_sum_Omono__neutral__cong__left,axiom,
% 0.35/0.67      ! [T3: set_nat,S2: set_nat,H2: nat > complex,G: nat > complex] :
% 0.35/0.67        ( ( finite_finite_nat @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: nat] :
% 0.35/0.67                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.67               => ( ( H2 @ X4 )
% 0.35/0.67                  = zero_zero_complex ) )
% 0.35/0.67           => ( ! [X4: nat] :
% 0.35/0.67                  ( ( member_nat @ X4 @ S2 )
% 0.35/0.67                 => ( ( G @ X4 )
% 0.35/0.67                    = ( H2 @ X4 ) ) )
% 0.35/0.67             => ( ( groups2073611262835488442omplex @ G @ S2 )
% 0.35/0.67                = ( groups2073611262835488442omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_cong_left
% 0.35/0.67  thf(fact_6188_sum_Omono__neutral__right,axiom,
% 0.35/0.67      ! [T3: set_complex,S2: set_complex,G: complex > real] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ T3 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: complex] :
% 0.35/0.67                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_real ) )
% 0.35/0.67           => ( ( groups5808333547571424918x_real @ G @ T3 )
% 0.35/0.67              = ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_right
% 0.35/0.67  thf(fact_6189_sum_Omono__neutral__right,axiom,
% 0.35/0.67      ! [T3: set_complex,S2: set_complex,G: complex > rat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ T3 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: complex] :
% 0.35/0.67                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_rat ) )
% 0.35/0.67           => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 0.35/0.67              = ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_right
% 0.35/0.67  thf(fact_6190_sum_Omono__neutral__right,axiom,
% 0.35/0.67      ! [T3: set_complex,S2: set_complex,G: complex > nat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ T3 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: complex] :
% 0.35/0.67                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_nat ) )
% 0.35/0.67           => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 0.35/0.67              = ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_right
% 0.35/0.67  thf(fact_6191_sum_Omono__neutral__right,axiom,
% 0.35/0.67      ! [T3: set_complex,S2: set_complex,G: complex > int] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ T3 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: complex] :
% 0.35/0.67                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_int ) )
% 0.35/0.67           => ( ( groups5690904116761175830ex_int @ G @ T3 )
% 0.35/0.67              = ( groups5690904116761175830ex_int @ G @ S2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_right
% 0.35/0.67  thf(fact_6192_sum_Omono__neutral__right,axiom,
% 0.35/0.67      ! [T3: set_nat,S2: set_nat,G: nat > complex] :
% 0.35/0.67        ( ( finite_finite_nat @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: nat] :
% 0.35/0.67                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_complex ) )
% 0.35/0.67           => ( ( groups2073611262835488442omplex @ G @ T3 )
% 0.35/0.67              = ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_right
% 0.35/0.67  thf(fact_6193_sum_Omono__neutral__right,axiom,
% 0.35/0.67      ! [T3: set_nat,S2: set_nat,G: nat > rat] :
% 0.35/0.67        ( ( finite_finite_nat @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: nat] :
% 0.35/0.67                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_rat ) )
% 0.35/0.67           => ( ( groups2906978787729119204at_rat @ G @ T3 )
% 0.35/0.67              = ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_right
% 0.35/0.67  thf(fact_6194_sum_Omono__neutral__right,axiom,
% 0.35/0.67      ! [T3: set_nat,S2: set_nat,G: nat > int] :
% 0.35/0.67        ( ( finite_finite_nat @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: nat] :
% 0.35/0.67                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_int ) )
% 0.35/0.67           => ( ( groups3539618377306564664at_int @ G @ T3 )
% 0.35/0.67              = ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_right
% 0.35/0.67  thf(fact_6195_sum_Omono__neutral__right,axiom,
% 0.35/0.67      ! [T3: set_int,S2: set_int,G: int > complex] :
% 0.35/0.67        ( ( finite_finite_int @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: int] :
% 0.35/0.67                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_complex ) )
% 0.35/0.67           => ( ( groups3049146728041665814omplex @ G @ T3 )
% 0.35/0.67              = ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_right
% 0.35/0.67  thf(fact_6196_sum_Omono__neutral__right,axiom,
% 0.35/0.67      ! [T3: set_int,S2: set_int,G: int > real] :
% 0.35/0.67        ( ( finite_finite_int @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: int] :
% 0.35/0.67                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_real ) )
% 0.35/0.67           => ( ( groups8778361861064173332t_real @ G @ T3 )
% 0.35/0.67              = ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_right
% 0.35/0.67  thf(fact_6197_sum_Omono__neutral__right,axiom,
% 0.35/0.67      ! [T3: set_int,S2: set_int,G: int > rat] :
% 0.35/0.67        ( ( finite_finite_int @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: int] :
% 0.35/0.67                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_rat ) )
% 0.35/0.67           => ( ( groups3906332499630173760nt_rat @ G @ T3 )
% 0.35/0.67              = ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_right
% 0.35/0.67  thf(fact_6198_sum_Omono__neutral__left,axiom,
% 0.35/0.67      ! [T3: set_complex,S2: set_complex,G: complex > real] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ T3 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: complex] :
% 0.35/0.67                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_real ) )
% 0.35/0.67           => ( ( groups5808333547571424918x_real @ G @ S2 )
% 0.35/0.67              = ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_left
% 0.35/0.67  thf(fact_6199_sum_Omono__neutral__left,axiom,
% 0.35/0.67      ! [T3: set_complex,S2: set_complex,G: complex > rat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ T3 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: complex] :
% 0.35/0.67                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_rat ) )
% 0.35/0.67           => ( ( groups5058264527183730370ex_rat @ G @ S2 )
% 0.35/0.67              = ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_left
% 0.35/0.67  thf(fact_6200_sum_Omono__neutral__left,axiom,
% 0.35/0.67      ! [T3: set_complex,S2: set_complex,G: complex > nat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ T3 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: complex] :
% 0.35/0.67                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_nat ) )
% 0.35/0.67           => ( ( groups5693394587270226106ex_nat @ G @ S2 )
% 0.35/0.67              = ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_left
% 0.35/0.67  thf(fact_6201_sum_Omono__neutral__left,axiom,
% 0.35/0.67      ! [T3: set_complex,S2: set_complex,G: complex > int] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ T3 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: complex] :
% 0.35/0.67                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_int ) )
% 0.35/0.67           => ( ( groups5690904116761175830ex_int @ G @ S2 )
% 0.35/0.67              = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_left
% 0.35/0.67  thf(fact_6202_sum_Omono__neutral__left,axiom,
% 0.35/0.67      ! [T3: set_nat,S2: set_nat,G: nat > complex] :
% 0.35/0.67        ( ( finite_finite_nat @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: nat] :
% 0.35/0.67                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_complex ) )
% 0.35/0.67           => ( ( groups2073611262835488442omplex @ G @ S2 )
% 0.35/0.67              = ( groups2073611262835488442omplex @ G @ T3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_left
% 0.35/0.67  thf(fact_6203_sum_Omono__neutral__left,axiom,
% 0.35/0.67      ! [T3: set_nat,S2: set_nat,G: nat > rat] :
% 0.35/0.67        ( ( finite_finite_nat @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: nat] :
% 0.35/0.67                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_rat ) )
% 0.35/0.67           => ( ( groups2906978787729119204at_rat @ G @ S2 )
% 0.35/0.67              = ( groups2906978787729119204at_rat @ G @ T3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_left
% 0.35/0.67  thf(fact_6204_sum_Omono__neutral__left,axiom,
% 0.35/0.67      ! [T3: set_nat,S2: set_nat,G: nat > int] :
% 0.35/0.67        ( ( finite_finite_nat @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: nat] :
% 0.35/0.67                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_int ) )
% 0.35/0.67           => ( ( groups3539618377306564664at_int @ G @ S2 )
% 0.35/0.67              = ( groups3539618377306564664at_int @ G @ T3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_left
% 0.35/0.67  thf(fact_6205_sum_Omono__neutral__left,axiom,
% 0.35/0.67      ! [T3: set_int,S2: set_int,G: int > complex] :
% 0.35/0.67        ( ( finite_finite_int @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: int] :
% 0.35/0.67                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_complex ) )
% 0.35/0.67           => ( ( groups3049146728041665814omplex @ G @ S2 )
% 0.35/0.67              = ( groups3049146728041665814omplex @ G @ T3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_left
% 0.35/0.67  thf(fact_6206_sum_Omono__neutral__left,axiom,
% 0.35/0.67      ! [T3: set_int,S2: set_int,G: int > real] :
% 0.35/0.67        ( ( finite_finite_int @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: int] :
% 0.35/0.67                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_real ) )
% 0.35/0.67           => ( ( groups8778361861064173332t_real @ G @ S2 )
% 0.35/0.67              = ( groups8778361861064173332t_real @ G @ T3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_left
% 0.35/0.67  thf(fact_6207_sum_Omono__neutral__left,axiom,
% 0.35/0.67      ! [T3: set_int,S2: set_int,G: int > rat] :
% 0.35/0.67        ( ( finite_finite_int @ T3 )
% 0.35/0.67       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.35/0.67         => ( ! [X4: int] :
% 0.35/0.67                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.35/0.67               => ( ( G @ X4 )
% 0.35/0.67                  = zero_zero_rat ) )
% 0.35/0.67           => ( ( groups3906332499630173760nt_rat @ G @ S2 )
% 0.35/0.67              = ( groups3906332499630173760nt_rat @ G @ T3 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.mono_neutral_left
% 0.35/0.67  thf(fact_6208_sum_Osame__carrierI,axiom,
% 0.35/0.67      ! [C2: set_real,A2: set_real,B2: set_real,G: real > complex,H2: real > complex] :
% 0.35/0.67        ( ( finite_finite_real @ C2 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.35/0.67         => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 0.35/0.67           => ( ! [A3: real] :
% 0.35/0.67                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.35/0.67                 => ( ( G @ A3 )
% 0.35/0.67                    = zero_zero_complex ) )
% 0.35/0.67             => ( ! [B3: real] :
% 0.35/0.67                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
% 0.35/0.67                   => ( ( H2 @ B3 )
% 0.35/0.67                      = zero_zero_complex ) )
% 0.35/0.67               => ( ( ( groups5754745047067104278omplex @ G @ C2 )
% 0.35/0.67                    = ( groups5754745047067104278omplex @ H2 @ C2 ) )
% 0.35/0.67                 => ( ( groups5754745047067104278omplex @ G @ A2 )
% 0.35/0.67                    = ( groups5754745047067104278omplex @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.same_carrierI
% 0.35/0.67  thf(fact_6209_sum_Osame__carrierI,axiom,
% 0.35/0.67      ! [C2: set_real,A2: set_real,B2: set_real,G: real > real,H2: real > real] :
% 0.35/0.67        ( ( finite_finite_real @ C2 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.35/0.67         => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 0.35/0.67           => ( ! [A3: real] :
% 0.35/0.67                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.35/0.67                 => ( ( G @ A3 )
% 0.35/0.67                    = zero_zero_real ) )
% 0.35/0.67             => ( ! [B3: real] :
% 0.35/0.67                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
% 0.35/0.67                   => ( ( H2 @ B3 )
% 0.35/0.67                      = zero_zero_real ) )
% 0.35/0.67               => ( ( ( groups8097168146408367636l_real @ G @ C2 )
% 0.35/0.67                    = ( groups8097168146408367636l_real @ H2 @ C2 ) )
% 0.35/0.67                 => ( ( groups8097168146408367636l_real @ G @ A2 )
% 0.35/0.67                    = ( groups8097168146408367636l_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.same_carrierI
% 0.35/0.67  thf(fact_6210_sum_Osame__carrierI,axiom,
% 0.35/0.67      ! [C2: set_complex,A2: set_complex,B2: set_complex,G: complex > real,H2: complex > real] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ C2 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 0.35/0.67         => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 0.35/0.67           => ( ! [A3: complex] :
% 0.35/0.67                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 0.35/0.67                 => ( ( G @ A3 )
% 0.35/0.67                    = zero_zero_real ) )
% 0.35/0.67             => ( ! [B3: complex] :
% 0.35/0.67                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 0.35/0.67                   => ( ( H2 @ B3 )
% 0.35/0.67                      = zero_zero_real ) )
% 0.35/0.67               => ( ( ( groups5808333547571424918x_real @ G @ C2 )
% 0.35/0.67                    = ( groups5808333547571424918x_real @ H2 @ C2 ) )
% 0.35/0.67                 => ( ( groups5808333547571424918x_real @ G @ A2 )
% 0.35/0.67                    = ( groups5808333547571424918x_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.same_carrierI
% 0.35/0.67  thf(fact_6211_sum_Osame__carrierI,axiom,
% 0.35/0.67      ! [C2: set_real,A2: set_real,B2: set_real,G: real > rat,H2: real > rat] :
% 0.35/0.67        ( ( finite_finite_real @ C2 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.35/0.67         => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 0.35/0.67           => ( ! [A3: real] :
% 0.35/0.67                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.35/0.67                 => ( ( G @ A3 )
% 0.35/0.67                    = zero_zero_rat ) )
% 0.35/0.67             => ( ! [B3: real] :
% 0.35/0.67                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
% 0.35/0.67                   => ( ( H2 @ B3 )
% 0.35/0.67                      = zero_zero_rat ) )
% 0.35/0.67               => ( ( ( groups1300246762558778688al_rat @ G @ C2 )
% 0.35/0.67                    = ( groups1300246762558778688al_rat @ H2 @ C2 ) )
% 0.35/0.67                 => ( ( groups1300246762558778688al_rat @ G @ A2 )
% 0.35/0.67                    = ( groups1300246762558778688al_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.same_carrierI
% 0.35/0.67  thf(fact_6212_sum_Osame__carrierI,axiom,
% 0.35/0.67      ! [C2: set_complex,A2: set_complex,B2: set_complex,G: complex > rat,H2: complex > rat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ C2 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 0.35/0.67         => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 0.35/0.67           => ( ! [A3: complex] :
% 0.35/0.67                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 0.35/0.67                 => ( ( G @ A3 )
% 0.35/0.67                    = zero_zero_rat ) )
% 0.35/0.67             => ( ! [B3: complex] :
% 0.35/0.67                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 0.35/0.67                   => ( ( H2 @ B3 )
% 0.35/0.67                      = zero_zero_rat ) )
% 0.35/0.67               => ( ( ( groups5058264527183730370ex_rat @ G @ C2 )
% 0.35/0.67                    = ( groups5058264527183730370ex_rat @ H2 @ C2 ) )
% 0.35/0.67                 => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 0.35/0.67                    = ( groups5058264527183730370ex_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.same_carrierI
% 0.35/0.67  thf(fact_6213_sum_Osame__carrierI,axiom,
% 0.35/0.67      ! [C2: set_real,A2: set_real,B2: set_real,G: real > nat,H2: real > nat] :
% 0.35/0.67        ( ( finite_finite_real @ C2 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.35/0.67         => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 0.35/0.67           => ( ! [A3: real] :
% 0.35/0.67                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.35/0.67                 => ( ( G @ A3 )
% 0.35/0.67                    = zero_zero_nat ) )
% 0.35/0.67             => ( ! [B3: real] :
% 0.35/0.67                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
% 0.35/0.67                   => ( ( H2 @ B3 )
% 0.35/0.67                      = zero_zero_nat ) )
% 0.35/0.67               => ( ( ( groups1935376822645274424al_nat @ G @ C2 )
% 0.35/0.67                    = ( groups1935376822645274424al_nat @ H2 @ C2 ) )
% 0.35/0.67                 => ( ( groups1935376822645274424al_nat @ G @ A2 )
% 0.35/0.67                    = ( groups1935376822645274424al_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.same_carrierI
% 0.35/0.67  thf(fact_6214_sum_Osame__carrierI,axiom,
% 0.35/0.67      ! [C2: set_complex,A2: set_complex,B2: set_complex,G: complex > nat,H2: complex > nat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ C2 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 0.35/0.67         => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 0.35/0.67           => ( ! [A3: complex] :
% 0.35/0.67                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 0.35/0.67                 => ( ( G @ A3 )
% 0.35/0.67                    = zero_zero_nat ) )
% 0.35/0.67             => ( ! [B3: complex] :
% 0.35/0.67                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 0.35/0.67                   => ( ( H2 @ B3 )
% 0.35/0.67                      = zero_zero_nat ) )
% 0.35/0.67               => ( ( ( groups5693394587270226106ex_nat @ G @ C2 )
% 0.35/0.67                    = ( groups5693394587270226106ex_nat @ H2 @ C2 ) )
% 0.35/0.67                 => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 0.35/0.67                    = ( groups5693394587270226106ex_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.same_carrierI
% 0.35/0.67  thf(fact_6215_sum_Osame__carrierI,axiom,
% 0.35/0.67      ! [C2: set_real,A2: set_real,B2: set_real,G: real > int,H2: real > int] :
% 0.35/0.67        ( ( finite_finite_real @ C2 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.35/0.67         => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 0.35/0.67           => ( ! [A3: real] :
% 0.35/0.67                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.35/0.67                 => ( ( G @ A3 )
% 0.35/0.67                    = zero_zero_int ) )
% 0.35/0.67             => ( ! [B3: real] :
% 0.35/0.67                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
% 0.35/0.67                   => ( ( H2 @ B3 )
% 0.35/0.67                      = zero_zero_int ) )
% 0.35/0.67               => ( ( ( groups1932886352136224148al_int @ G @ C2 )
% 0.35/0.67                    = ( groups1932886352136224148al_int @ H2 @ C2 ) )
% 0.35/0.67                 => ( ( groups1932886352136224148al_int @ G @ A2 )
% 0.35/0.67                    = ( groups1932886352136224148al_int @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.same_carrierI
% 0.35/0.67  thf(fact_6216_sum_Osame__carrierI,axiom,
% 0.35/0.67      ! [C2: set_complex,A2: set_complex,B2: set_complex,G: complex > int,H2: complex > int] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ C2 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 0.35/0.67         => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 0.35/0.67           => ( ! [A3: complex] :
% 0.35/0.67                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 0.35/0.67                 => ( ( G @ A3 )
% 0.35/0.67                    = zero_zero_int ) )
% 0.35/0.67             => ( ! [B3: complex] :
% 0.35/0.67                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 0.35/0.67                   => ( ( H2 @ B3 )
% 0.35/0.67                      = zero_zero_int ) )
% 0.35/0.67               => ( ( ( groups5690904116761175830ex_int @ G @ C2 )
% 0.35/0.67                    = ( groups5690904116761175830ex_int @ H2 @ C2 ) )
% 0.35/0.67                 => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 0.35/0.67                    = ( groups5690904116761175830ex_int @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.same_carrierI
% 0.35/0.67  thf(fact_6217_sum_Osame__carrierI,axiom,
% 0.35/0.67      ! [C2: set_nat,A2: set_nat,B2: set_nat,G: nat > complex,H2: nat > complex] :
% 0.35/0.67        ( ( finite_finite_nat @ C2 )
% 0.35/0.67       => ( ( ord_less_eq_set_nat @ A2 @ C2 )
% 0.35/0.67         => ( ( ord_less_eq_set_nat @ B2 @ C2 )
% 0.35/0.67           => ( ! [A3: nat] :
% 0.35/0.67                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C2 @ A2 ) )
% 0.35/0.67                 => ( ( G @ A3 )
% 0.35/0.67                    = zero_zero_complex ) )
% 0.35/0.67             => ( ! [B3: nat] :
% 0.35/0.67                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C2 @ B2 ) )
% 0.35/0.67                   => ( ( H2 @ B3 )
% 0.35/0.67                      = zero_zero_complex ) )
% 0.35/0.67               => ( ( ( groups2073611262835488442omplex @ G @ C2 )
% 0.35/0.67                    = ( groups2073611262835488442omplex @ H2 @ C2 ) )
% 0.35/0.67                 => ( ( groups2073611262835488442omplex @ G @ A2 )
% 0.35/0.67                    = ( groups2073611262835488442omplex @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.same_carrierI
% 0.35/0.67  thf(fact_6218_sum_Osame__carrier,axiom,
% 0.35/0.67      ! [C2: set_real,A2: set_real,B2: set_real,G: real > complex,H2: real > complex] :
% 0.35/0.67        ( ( finite_finite_real @ C2 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.35/0.67         => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 0.35/0.67           => ( ! [A3: real] :
% 0.35/0.67                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.35/0.67                 => ( ( G @ A3 )
% 0.35/0.67                    = zero_zero_complex ) )
% 0.35/0.67             => ( ! [B3: real] :
% 0.35/0.67                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
% 0.35/0.67                   => ( ( H2 @ B3 )
% 0.35/0.67                      = zero_zero_complex ) )
% 0.35/0.67               => ( ( ( groups5754745047067104278omplex @ G @ A2 )
% 0.35/0.67                    = ( groups5754745047067104278omplex @ H2 @ B2 ) )
% 0.35/0.67                  = ( ( groups5754745047067104278omplex @ G @ C2 )
% 0.35/0.67                    = ( groups5754745047067104278omplex @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.same_carrier
% 0.35/0.67  thf(fact_6219_sum_Osame__carrier,axiom,
% 0.35/0.67      ! [C2: set_real,A2: set_real,B2: set_real,G: real > real,H2: real > real] :
% 0.35/0.67        ( ( finite_finite_real @ C2 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.35/0.67         => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 0.35/0.67           => ( ! [A3: real] :
% 0.35/0.67                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.35/0.67                 => ( ( G @ A3 )
% 0.35/0.67                    = zero_zero_real ) )
% 0.35/0.67             => ( ! [B3: real] :
% 0.35/0.67                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
% 0.35/0.67                   => ( ( H2 @ B3 )
% 0.35/0.67                      = zero_zero_real ) )
% 0.35/0.67               => ( ( ( groups8097168146408367636l_real @ G @ A2 )
% 0.35/0.67                    = ( groups8097168146408367636l_real @ H2 @ B2 ) )
% 0.35/0.67                  = ( ( groups8097168146408367636l_real @ G @ C2 )
% 0.35/0.67                    = ( groups8097168146408367636l_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.same_carrier
% 0.35/0.67  thf(fact_6220_sum_Osame__carrier,axiom,
% 0.35/0.67      ! [C2: set_complex,A2: set_complex,B2: set_complex,G: complex > real,H2: complex > real] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ C2 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 0.35/0.67         => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 0.35/0.67           => ( ! [A3: complex] :
% 0.35/0.67                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 0.35/0.67                 => ( ( G @ A3 )
% 0.35/0.67                    = zero_zero_real ) )
% 0.35/0.67             => ( ! [B3: complex] :
% 0.35/0.67                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 0.35/0.67                   => ( ( H2 @ B3 )
% 0.35/0.67                      = zero_zero_real ) )
% 0.35/0.67               => ( ( ( groups5808333547571424918x_real @ G @ A2 )
% 0.35/0.67                    = ( groups5808333547571424918x_real @ H2 @ B2 ) )
% 0.35/0.67                  = ( ( groups5808333547571424918x_real @ G @ C2 )
% 0.35/0.67                    = ( groups5808333547571424918x_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.same_carrier
% 0.35/0.67  thf(fact_6221_sum_Osame__carrier,axiom,
% 0.35/0.67      ! [C2: set_real,A2: set_real,B2: set_real,G: real > rat,H2: real > rat] :
% 0.35/0.67        ( ( finite_finite_real @ C2 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.35/0.67         => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 0.35/0.67           => ( ! [A3: real] :
% 0.35/0.67                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.35/0.67                 => ( ( G @ A3 )
% 0.35/0.67                    = zero_zero_rat ) )
% 0.35/0.67             => ( ! [B3: real] :
% 0.35/0.67                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
% 0.35/0.67                   => ( ( H2 @ B3 )
% 0.35/0.67                      = zero_zero_rat ) )
% 0.35/0.67               => ( ( ( groups1300246762558778688al_rat @ G @ A2 )
% 0.35/0.67                    = ( groups1300246762558778688al_rat @ H2 @ B2 ) )
% 0.35/0.67                  = ( ( groups1300246762558778688al_rat @ G @ C2 )
% 0.35/0.67                    = ( groups1300246762558778688al_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.same_carrier
% 0.35/0.67  thf(fact_6222_sum_Osame__carrier,axiom,
% 0.35/0.67      ! [C2: set_complex,A2: set_complex,B2: set_complex,G: complex > rat,H2: complex > rat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ C2 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 0.35/0.67         => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 0.35/0.67           => ( ! [A3: complex] :
% 0.35/0.67                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 0.35/0.67                 => ( ( G @ A3 )
% 0.35/0.67                    = zero_zero_rat ) )
% 0.35/0.67             => ( ! [B3: complex] :
% 0.35/0.67                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 0.35/0.67                   => ( ( H2 @ B3 )
% 0.35/0.67                      = zero_zero_rat ) )
% 0.35/0.67               => ( ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 0.35/0.67                    = ( groups5058264527183730370ex_rat @ H2 @ B2 ) )
% 0.35/0.67                  = ( ( groups5058264527183730370ex_rat @ G @ C2 )
% 0.35/0.67                    = ( groups5058264527183730370ex_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.same_carrier
% 0.35/0.67  thf(fact_6223_sum_Osame__carrier,axiom,
% 0.35/0.67      ! [C2: set_real,A2: set_real,B2: set_real,G: real > nat,H2: real > nat] :
% 0.35/0.67        ( ( finite_finite_real @ C2 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.35/0.67         => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 0.35/0.67           => ( ! [A3: real] :
% 0.35/0.67                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.35/0.67                 => ( ( G @ A3 )
% 0.35/0.67                    = zero_zero_nat ) )
% 0.35/0.67             => ( ! [B3: real] :
% 0.35/0.67                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
% 0.35/0.67                   => ( ( H2 @ B3 )
% 0.35/0.67                      = zero_zero_nat ) )
% 0.35/0.67               => ( ( ( groups1935376822645274424al_nat @ G @ A2 )
% 0.35/0.67                    = ( groups1935376822645274424al_nat @ H2 @ B2 ) )
% 0.35/0.67                  = ( ( groups1935376822645274424al_nat @ G @ C2 )
% 0.35/0.67                    = ( groups1935376822645274424al_nat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.same_carrier
% 0.35/0.67  thf(fact_6224_sum_Osame__carrier,axiom,
% 0.35/0.67      ! [C2: set_complex,A2: set_complex,B2: set_complex,G: complex > nat,H2: complex > nat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ C2 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 0.35/0.67         => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 0.35/0.67           => ( ! [A3: complex] :
% 0.35/0.67                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 0.35/0.67                 => ( ( G @ A3 )
% 0.35/0.67                    = zero_zero_nat ) )
% 0.35/0.67             => ( ! [B3: complex] :
% 0.35/0.67                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 0.35/0.67                   => ( ( H2 @ B3 )
% 0.35/0.67                      = zero_zero_nat ) )
% 0.35/0.67               => ( ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 0.35/0.67                    = ( groups5693394587270226106ex_nat @ H2 @ B2 ) )
% 0.35/0.67                  = ( ( groups5693394587270226106ex_nat @ G @ C2 )
% 0.35/0.67                    = ( groups5693394587270226106ex_nat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.same_carrier
% 0.35/0.67  thf(fact_6225_sum_Osame__carrier,axiom,
% 0.35/0.67      ! [C2: set_real,A2: set_real,B2: set_real,G: real > int,H2: real > int] :
% 0.35/0.67        ( ( finite_finite_real @ C2 )
% 0.35/0.67       => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.35/0.67         => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 0.35/0.67           => ( ! [A3: real] :
% 0.35/0.67                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.35/0.67                 => ( ( G @ A3 )
% 0.35/0.67                    = zero_zero_int ) )
% 0.35/0.67             => ( ! [B3: real] :
% 0.35/0.67                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
% 0.35/0.67                   => ( ( H2 @ B3 )
% 0.35/0.67                      = zero_zero_int ) )
% 0.35/0.67               => ( ( ( groups1932886352136224148al_int @ G @ A2 )
% 0.35/0.67                    = ( groups1932886352136224148al_int @ H2 @ B2 ) )
% 0.35/0.67                  = ( ( groups1932886352136224148al_int @ G @ C2 )
% 0.35/0.67                    = ( groups1932886352136224148al_int @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.same_carrier
% 0.35/0.67  thf(fact_6226_sum_Osame__carrier,axiom,
% 0.35/0.67      ! [C2: set_complex,A2: set_complex,B2: set_complex,G: complex > int,H2: complex > int] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ C2 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 0.35/0.67         => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 0.35/0.67           => ( ! [A3: complex] :
% 0.35/0.67                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 0.35/0.67                 => ( ( G @ A3 )
% 0.35/0.67                    = zero_zero_int ) )
% 0.35/0.67             => ( ! [B3: complex] :
% 0.35/0.67                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 0.35/0.67                   => ( ( H2 @ B3 )
% 0.35/0.67                      = zero_zero_int ) )
% 0.35/0.67               => ( ( ( groups5690904116761175830ex_int @ G @ A2 )
% 0.35/0.67                    = ( groups5690904116761175830ex_int @ H2 @ B2 ) )
% 0.35/0.67                  = ( ( groups5690904116761175830ex_int @ G @ C2 )
% 0.35/0.67                    = ( groups5690904116761175830ex_int @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.same_carrier
% 0.35/0.67  thf(fact_6227_sum_Osame__carrier,axiom,
% 0.35/0.67      ! [C2: set_nat,A2: set_nat,B2: set_nat,G: nat > complex,H2: nat > complex] :
% 0.35/0.67        ( ( finite_finite_nat @ C2 )
% 0.35/0.67       => ( ( ord_less_eq_set_nat @ A2 @ C2 )
% 0.35/0.67         => ( ( ord_less_eq_set_nat @ B2 @ C2 )
% 0.35/0.67           => ( ! [A3: nat] :
% 0.35/0.67                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C2 @ A2 ) )
% 0.35/0.67                 => ( ( G @ A3 )
% 0.35/0.67                    = zero_zero_complex ) )
% 0.35/0.67             => ( ! [B3: nat] :
% 0.35/0.67                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C2 @ B2 ) )
% 0.35/0.67                   => ( ( H2 @ B3 )
% 0.35/0.67                      = zero_zero_complex ) )
% 0.35/0.67               => ( ( ( groups2073611262835488442omplex @ G @ A2 )
% 0.35/0.67                    = ( groups2073611262835488442omplex @ H2 @ B2 ) )
% 0.35/0.67                  = ( ( groups2073611262835488442omplex @ G @ C2 )
% 0.35/0.67                    = ( groups2073611262835488442omplex @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.same_carrier
% 0.35/0.67  thf(fact_6228_sum_Osubset__diff,axiom,
% 0.35/0.67      ! [B2: set_complex,A2: set_complex,G: complex > real] :
% 0.35/0.67        ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 0.35/0.67       => ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.67         => ( ( groups5808333547571424918x_real @ G @ A2 )
% 0.35/0.67            = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5808333547571424918x_real @ G @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.subset_diff
% 0.35/0.67  thf(fact_6229_sum_Osubset__diff,axiom,
% 0.35/0.67      ! [B2: set_complex,A2: set_complex,G: complex > rat] :
% 0.35/0.67        ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 0.35/0.67       => ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.67         => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 0.35/0.67            = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.subset_diff
% 0.35/0.67  thf(fact_6230_sum_Osubset__diff,axiom,
% 0.35/0.67      ! [B2: set_complex,A2: set_complex,G: complex > nat] :
% 0.35/0.67        ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 0.35/0.67       => ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.67         => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 0.35/0.67            = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.subset_diff
% 0.35/0.67  thf(fact_6231_sum_Osubset__diff,axiom,
% 0.35/0.67      ! [B2: set_complex,A2: set_complex,G: complex > int] :
% 0.35/0.67        ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 0.35/0.67       => ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.67         => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 0.35/0.67            = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.subset_diff
% 0.35/0.67  thf(fact_6232_sum_Osubset__diff,axiom,
% 0.35/0.67      ! [B2: set_nat,A2: set_nat,G: nat > rat] :
% 0.35/0.67        ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 0.35/0.67       => ( ( finite_finite_nat @ A2 )
% 0.35/0.67         => ( ( groups2906978787729119204at_rat @ G @ A2 )
% 0.35/0.67            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups2906978787729119204at_rat @ G @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.subset_diff
% 0.35/0.67  thf(fact_6233_sum_Osubset__diff,axiom,
% 0.35/0.67      ! [B2: set_nat,A2: set_nat,G: nat > int] :
% 0.35/0.67        ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 0.35/0.67       => ( ( finite_finite_nat @ A2 )
% 0.35/0.67         => ( ( groups3539618377306564664at_int @ G @ A2 )
% 0.35/0.67            = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups3539618377306564664at_int @ G @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.subset_diff
% 0.35/0.67  thf(fact_6234_sum_Osubset__diff,axiom,
% 0.35/0.67      ! [B2: set_int,A2: set_int,G: int > real] :
% 0.35/0.67        ( ( ord_less_eq_set_int @ B2 @ A2 )
% 0.35/0.67       => ( ( finite_finite_int @ A2 )
% 0.35/0.67         => ( ( groups8778361861064173332t_real @ G @ A2 )
% 0.35/0.67            = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups8778361861064173332t_real @ G @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.subset_diff
% 0.35/0.67  thf(fact_6235_sum_Osubset__diff,axiom,
% 0.35/0.67      ! [B2: set_int,A2: set_int,G: int > rat] :
% 0.35/0.67        ( ( ord_less_eq_set_int @ B2 @ A2 )
% 0.35/0.67       => ( ( finite_finite_int @ A2 )
% 0.35/0.67         => ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 0.35/0.67            = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.subset_diff
% 0.35/0.67  thf(fact_6236_sum_Osubset__diff,axiom,
% 0.35/0.67      ! [B2: set_int,A2: set_int,G: int > nat] :
% 0.35/0.67        ( ( ord_less_eq_set_int @ B2 @ A2 )
% 0.35/0.67       => ( ( finite_finite_int @ A2 )
% 0.35/0.67         => ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 0.35/0.67            = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.subset_diff
% 0.35/0.67  thf(fact_6237_sum_Osubset__diff,axiom,
% 0.35/0.67      ! [B2: set_int,A2: set_int,G: int > int] :
% 0.35/0.67        ( ( ord_less_eq_set_int @ B2 @ A2 )
% 0.35/0.67       => ( ( finite_finite_int @ A2 )
% 0.35/0.67         => ( ( groups4538972089207619220nt_int @ G @ A2 )
% 0.35/0.67            = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4538972089207619220nt_int @ G @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.subset_diff
% 0.35/0.67  thf(fact_6238_sum__diff,axiom,
% 0.35/0.67      ! [A2: set_complex,B2: set_complex,F: complex > real] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 0.35/0.67         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 0.35/0.67            = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_diff
% 0.35/0.67  thf(fact_6239_sum__diff,axiom,
% 0.35/0.67      ! [A2: set_complex,B2: set_complex,F: complex > rat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 0.35/0.67         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 0.35/0.67            = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_diff
% 0.35/0.67  thf(fact_6240_sum__diff,axiom,
% 0.35/0.67      ! [A2: set_complex,B2: set_complex,F: complex > int] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 0.35/0.67         => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 0.35/0.67            = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_diff
% 0.35/0.67  thf(fact_6241_sum__diff,axiom,
% 0.35/0.67      ! [A2: set_nat,B2: set_nat,F: nat > rat] :
% 0.35/0.67        ( ( finite_finite_nat @ A2 )
% 0.35/0.67       => ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 0.35/0.67         => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.35/0.67            = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_diff
% 0.35/0.67  thf(fact_6242_sum__diff,axiom,
% 0.35/0.67      ! [A2: set_nat,B2: set_nat,F: nat > int] :
% 0.35/0.67        ( ( finite_finite_nat @ A2 )
% 0.35/0.67       => ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 0.35/0.67         => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.35/0.67            = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_diff
% 0.35/0.67  thf(fact_6243_sum__diff,axiom,
% 0.35/0.67      ! [A2: set_int,B2: set_int,F: int > real] :
% 0.35/0.67        ( ( finite_finite_int @ A2 )
% 0.35/0.67       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 0.35/0.67         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.35/0.67            = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_diff
% 0.35/0.67  thf(fact_6244_sum__diff,axiom,
% 0.35/0.67      ! [A2: set_int,B2: set_int,F: int > rat] :
% 0.35/0.67        ( ( finite_finite_int @ A2 )
% 0.35/0.67       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 0.35/0.67         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.35/0.67            = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_diff
% 0.35/0.67  thf(fact_6245_sum__diff,axiom,
% 0.35/0.67      ! [A2: set_int,B2: set_int,F: int > int] :
% 0.35/0.67        ( ( finite_finite_int @ A2 )
% 0.35/0.67       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 0.35/0.67         => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.35/0.67            = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ F @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_diff
% 0.35/0.67  thf(fact_6246_sum__diff,axiom,
% 0.35/0.67      ! [A2: set_complex,B2: set_complex,F: complex > complex] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 0.35/0.67         => ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 0.35/0.67            = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ F @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_diff
% 0.35/0.67  thf(fact_6247_sum__diff,axiom,
% 0.35/0.67      ! [A2: set_nat,B2: set_nat,F: nat > real] :
% 0.35/0.67        ( ( finite_finite_nat @ A2 )
% 0.35/0.67       => ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 0.35/0.67         => ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.35/0.67            = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_diff
% 0.35/0.67  thf(fact_6248_sum__diff__nat,axiom,
% 0.35/0.67      ! [B2: set_complex,A2: set_complex,F: complex > nat] :
% 0.35/0.67        ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.67       => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 0.35/0.67         => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 0.35/0.67            = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_diff_nat
% 0.35/0.67  thf(fact_6249_sum__diff__nat,axiom,
% 0.35/0.67      ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 0.35/0.67        ( ( finite6177210948735845034at_nat @ B2 )
% 0.35/0.67       => ( ( ord_le3146513528884898305at_nat @ B2 @ A2 )
% 0.35/0.67         => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 0.35/0.67            = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A2 ) @ ( groups977919841031483927at_nat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_diff_nat
% 0.35/0.67  thf(fact_6250_sum__diff__nat,axiom,
% 0.35/0.67      ! [B2: set_int,A2: set_int,F: int > nat] :
% 0.35/0.67        ( ( finite_finite_int @ B2 )
% 0.35/0.67       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 0.35/0.67         => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.35/0.67            = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_diff_nat
% 0.35/0.67  thf(fact_6251_sum__diff__nat,axiom,
% 0.35/0.67      ! [B2: set_nat,A2: set_nat,F: nat > nat] :
% 0.35/0.67        ( ( finite_finite_nat @ B2 )
% 0.35/0.67       => ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 0.35/0.67         => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.35/0.67            = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_diff_nat
% 0.35/0.67  thf(fact_6252_sum__shift__lb__Suc0__0,axiom,
% 0.35/0.67      ! [F: nat > complex,K: nat] :
% 0.35/0.67        ( ( ( F @ zero_zero_nat )
% 0.35/0.67          = zero_zero_complex )
% 0.35/0.67       => ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.35/0.67          = ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_shift_lb_Suc0_0
% 0.35/0.67  thf(fact_6253_sum__shift__lb__Suc0__0,axiom,
% 0.35/0.67      ! [F: nat > rat,K: nat] :
% 0.35/0.67        ( ( ( F @ zero_zero_nat )
% 0.35/0.67          = zero_zero_rat )
% 0.35/0.67       => ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.35/0.67          = ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_shift_lb_Suc0_0
% 0.35/0.67  thf(fact_6254_sum__shift__lb__Suc0__0,axiom,
% 0.35/0.67      ! [F: nat > int,K: nat] :
% 0.35/0.67        ( ( ( F @ zero_zero_nat )
% 0.35/0.67          = zero_zero_int )
% 0.35/0.67       => ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.35/0.67          = ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_shift_lb_Suc0_0
% 0.35/0.67  thf(fact_6255_sum__shift__lb__Suc0__0,axiom,
% 0.35/0.67      ! [F: nat > nat,K: nat] :
% 0.35/0.67        ( ( ( F @ zero_zero_nat )
% 0.35/0.67          = zero_zero_nat )
% 0.35/0.67       => ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.35/0.67          = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_shift_lb_Suc0_0
% 0.35/0.67  thf(fact_6256_sum__shift__lb__Suc0__0,axiom,
% 0.35/0.67      ! [F: nat > real,K: nat] :
% 0.35/0.67        ( ( ( F @ zero_zero_nat )
% 0.35/0.67          = zero_zero_real )
% 0.35/0.67       => ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.35/0.67          = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum_shift_lb_Suc0_0
% 0.35/0.67  thf(fact_6257_sum_OatLeast0__atMost__Suc,axiom,
% 0.35/0.67      ! [G: nat > rat,N: nat] :
% 0.35/0.67        ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.35/0.67        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.atLeast0_atMost_Suc
% 0.35/0.67  thf(fact_6258_sum_OatLeast0__atMost__Suc,axiom,
% 0.35/0.67      ! [G: nat > int,N: nat] :
% 0.35/0.67        ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.35/0.67        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.atLeast0_atMost_Suc
% 0.35/0.67  thf(fact_6259_sum_OatLeast0__atMost__Suc,axiom,
% 0.35/0.67      ! [G: nat > nat,N: nat] :
% 0.35/0.67        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.35/0.67        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.atLeast0_atMost_Suc
% 0.35/0.67  thf(fact_6260_sum_OatLeast0__atMost__Suc,axiom,
% 0.35/0.67      ! [G: nat > real,N: nat] :
% 0.35/0.67        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.35/0.67        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.atLeast0_atMost_Suc
% 0.35/0.67  thf(fact_6261_sum_OatLeast__Suc__atMost,axiom,
% 0.35/0.67      ! [M: nat,N: nat,G: nat > rat] :
% 0.35/0.67        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.67       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.67          = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.atLeast_Suc_atMost
% 0.35/0.67  thf(fact_6262_sum_OatLeast__Suc__atMost,axiom,
% 0.35/0.67      ! [M: nat,N: nat,G: nat > int] :
% 0.35/0.67        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.67       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.67          = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.atLeast_Suc_atMost
% 0.35/0.67  thf(fact_6263_sum_OatLeast__Suc__atMost,axiom,
% 0.35/0.67      ! [M: nat,N: nat,G: nat > nat] :
% 0.35/0.67        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.67       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.67          = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.atLeast_Suc_atMost
% 0.35/0.67  thf(fact_6264_sum_OatLeast__Suc__atMost,axiom,
% 0.35/0.67      ! [M: nat,N: nat,G: nat > real] :
% 0.35/0.67        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.67       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.67          = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.atLeast_Suc_atMost
% 0.35/0.67  thf(fact_6265_sum_Onat__ivl__Suc_H,axiom,
% 0.35/0.67      ! [M: nat,N: nat,G: nat > rat] :
% 0.35/0.67        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.35/0.67       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.67          = ( plus_plus_rat @ ( G @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.nat_ivl_Suc'
% 0.35/0.67  thf(fact_6266_sum_Onat__ivl__Suc_H,axiom,
% 0.35/0.67      ! [M: nat,N: nat,G: nat > int] :
% 0.35/0.67        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.35/0.67       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.67          = ( plus_plus_int @ ( G @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.nat_ivl_Suc'
% 0.35/0.67  thf(fact_6267_sum_Onat__ivl__Suc_H,axiom,
% 0.35/0.67      ! [M: nat,N: nat,G: nat > nat] :
% 0.35/0.67        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.35/0.67       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.67          = ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.nat_ivl_Suc'
% 0.35/0.67  thf(fact_6268_sum_Onat__ivl__Suc_H,axiom,
% 0.35/0.67      ! [M: nat,N: nat,G: nat > real] :
% 0.35/0.67        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.35/0.67       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.67          = ( plus_plus_real @ ( G @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % sum.nat_ivl_Suc'
% 0.35/0.67  thf(fact_6269_remove__induct,axiom,
% 0.35/0.67      ! [P: set_Pr4329608150637261639at_nat > $o,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.67        ( ( P @ bot_bo228742789529271731at_nat )
% 0.35/0.67       => ( ( ~ ( finite4343798906461161616at_nat @ B2 )
% 0.35/0.67           => ( P @ B2 ) )
% 0.35/0.67         => ( ! [A5: set_Pr4329608150637261639at_nat] :
% 0.35/0.67                ( ( finite4343798906461161616at_nat @ A5 )
% 0.35/0.67               => ( ( A5 != bot_bo228742789529271731at_nat )
% 0.35/0.67                 => ( ( ord_le1268244103169919719at_nat @ A5 @ B2 )
% 0.35/0.67                   => ( ! [X2: produc3843707927480180839at_nat] :
% 0.35/0.67                          ( ( member8757157785044589968at_nat @ X2 @ A5 )
% 0.35/0.67                         => ( P @ ( minus_3314409938677909166at_nat @ A5 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) )
% 0.35/0.67                     => ( P @ A5 ) ) ) ) )
% 0.35/0.67           => ( P @ B2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % remove_induct
% 0.35/0.67  thf(fact_6270_remove__induct,axiom,
% 0.35/0.67      ! [P: set_set_nat > $o,B2: set_set_nat] :
% 0.35/0.67        ( ( P @ bot_bot_set_set_nat )
% 0.35/0.67       => ( ( ~ ( finite1152437895449049373et_nat @ B2 )
% 0.35/0.67           => ( P @ B2 ) )
% 0.35/0.67         => ( ! [A5: set_set_nat] :
% 0.35/0.67                ( ( finite1152437895449049373et_nat @ A5 )
% 0.35/0.67               => ( ( A5 != bot_bot_set_set_nat )
% 0.35/0.67                 => ( ( ord_le6893508408891458716et_nat @ A5 @ B2 )
% 0.35/0.67                   => ( ! [X2: set_nat] :
% 0.35/0.67                          ( ( member_set_nat @ X2 @ A5 )
% 0.35/0.67                         => ( P @ ( minus_2163939370556025621et_nat @ A5 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) )
% 0.35/0.67                     => ( P @ A5 ) ) ) ) )
% 0.35/0.67           => ( P @ B2 ) ) ) ) ).
% 0.35/0.67  
% 0.35/0.67  % remove_induct
% 0.35/0.67  thf(fact_6271_remove__induct,axiom,
% 0.35/0.67      ! [P: set_complex > $o,B2: set_complex] :
% 0.35/0.67        ( ( P @ bot_bot_set_complex )
% 0.35/0.67       => ( ( ~ ( finite3207457112153483333omplex @ B2 )
% 0.35/0.67           => ( P @ B2 ) )
% 0.35/0.67         => ( ! [A5: set_complex] :
% 0.35/0.67                ( ( finite3207457112153483333omplex @ A5 )
% 0.35/0.68               => ( ( A5 != bot_bot_set_complex )
% 0.35/0.68                 => ( ( ord_le211207098394363844omplex @ A5 @ B2 )
% 0.35/0.68                   => ( ! [X2: complex] :
% 0.35/0.68                          ( ( member_complex @ X2 @ A5 )
% 0.35/0.68                         => ( P @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) )
% 0.35/0.68                     => ( P @ A5 ) ) ) ) )
% 0.35/0.68           => ( P @ B2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % remove_induct
% 0.35/0.68  thf(fact_6272_remove__induct,axiom,
% 0.35/0.68      ! [P: set_Pr1261947904930325089at_nat > $o,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.68        ( ( P @ bot_bo2099793752762293965at_nat )
% 0.35/0.68       => ( ( ~ ( finite6177210948735845034at_nat @ B2 )
% 0.35/0.68           => ( P @ B2 ) )
% 0.35/0.68         => ( ! [A5: set_Pr1261947904930325089at_nat] :
% 0.35/0.68                ( ( finite6177210948735845034at_nat @ A5 )
% 0.35/0.68               => ( ( A5 != bot_bo2099793752762293965at_nat )
% 0.35/0.68                 => ( ( ord_le3146513528884898305at_nat @ A5 @ B2 )
% 0.35/0.68                   => ( ! [X2: product_prod_nat_nat] :
% 0.35/0.68                          ( ( member8440522571783428010at_nat @ X2 @ A5 )
% 0.35/0.68                         => ( P @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) )
% 0.35/0.68                     => ( P @ A5 ) ) ) ) )
% 0.35/0.68           => ( P @ B2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % remove_induct
% 0.35/0.68  thf(fact_6273_remove__induct,axiom,
% 0.35/0.68      ! [P: set_real > $o,B2: set_real] :
% 0.35/0.68        ( ( P @ bot_bot_set_real )
% 0.35/0.68       => ( ( ~ ( finite_finite_real @ B2 )
% 0.35/0.68           => ( P @ B2 ) )
% 0.35/0.68         => ( ! [A5: set_real] :
% 0.35/0.68                ( ( finite_finite_real @ A5 )
% 0.35/0.68               => ( ( A5 != bot_bot_set_real )
% 0.35/0.68                 => ( ( ord_less_eq_set_real @ A5 @ B2 )
% 0.35/0.68                   => ( ! [X2: real] :
% 0.35/0.68                          ( ( member_real @ X2 @ A5 )
% 0.35/0.68                         => ( P @ ( minus_minus_set_real @ A5 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) )
% 0.35/0.68                     => ( P @ A5 ) ) ) ) )
% 0.35/0.68           => ( P @ B2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % remove_induct
% 0.35/0.68  thf(fact_6274_remove__induct,axiom,
% 0.35/0.68      ! [P: set_nat > $o,B2: set_nat] :
% 0.35/0.68        ( ( P @ bot_bot_set_nat )
% 0.35/0.68       => ( ( ~ ( finite_finite_nat @ B2 )
% 0.35/0.68           => ( P @ B2 ) )
% 0.35/0.68         => ( ! [A5: set_nat] :
% 0.35/0.68                ( ( finite_finite_nat @ A5 )
% 0.35/0.68               => ( ( A5 != bot_bot_set_nat )
% 0.35/0.68                 => ( ( ord_less_eq_set_nat @ A5 @ B2 )
% 0.35/0.68                   => ( ! [X2: nat] :
% 0.35/0.68                          ( ( member_nat @ X2 @ A5 )
% 0.35/0.68                         => ( P @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
% 0.35/0.68                     => ( P @ A5 ) ) ) ) )
% 0.35/0.68           => ( P @ B2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % remove_induct
% 0.35/0.68  thf(fact_6275_remove__induct,axiom,
% 0.35/0.68      ! [P: set_int > $o,B2: set_int] :
% 0.35/0.68        ( ( P @ bot_bot_set_int )
% 0.35/0.68       => ( ( ~ ( finite_finite_int @ B2 )
% 0.35/0.68           => ( P @ B2 ) )
% 0.35/0.68         => ( ! [A5: set_int] :
% 0.35/0.68                ( ( finite_finite_int @ A5 )
% 0.35/0.68               => ( ( A5 != bot_bot_set_int )
% 0.35/0.68                 => ( ( ord_less_eq_set_int @ A5 @ B2 )
% 0.35/0.68                   => ( ! [X2: int] :
% 0.35/0.68                          ( ( member_int @ X2 @ A5 )
% 0.35/0.68                         => ( P @ ( minus_minus_set_int @ A5 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) )
% 0.35/0.68                     => ( P @ A5 ) ) ) ) )
% 0.35/0.68           => ( P @ B2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % remove_induct
% 0.35/0.68  thf(fact_6276_finite__remove__induct,axiom,
% 0.35/0.68      ! [B2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 0.35/0.68        ( ( finite4343798906461161616at_nat @ B2 )
% 0.35/0.68       => ( ( P @ bot_bo228742789529271731at_nat )
% 0.35/0.68         => ( ! [A5: set_Pr4329608150637261639at_nat] :
% 0.35/0.68                ( ( finite4343798906461161616at_nat @ A5 )
% 0.35/0.68               => ( ( A5 != bot_bo228742789529271731at_nat )
% 0.35/0.68                 => ( ( ord_le1268244103169919719at_nat @ A5 @ B2 )
% 0.35/0.68                   => ( ! [X2: produc3843707927480180839at_nat] :
% 0.35/0.68                          ( ( member8757157785044589968at_nat @ X2 @ A5 )
% 0.35/0.68                         => ( P @ ( minus_3314409938677909166at_nat @ A5 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) )
% 0.35/0.68                     => ( P @ A5 ) ) ) ) )
% 0.35/0.68           => ( P @ B2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % finite_remove_induct
% 0.35/0.68  thf(fact_6277_finite__remove__induct,axiom,
% 0.35/0.68      ! [B2: set_set_nat,P: set_set_nat > $o] :
% 0.35/0.68        ( ( finite1152437895449049373et_nat @ B2 )
% 0.35/0.68       => ( ( P @ bot_bot_set_set_nat )
% 0.35/0.68         => ( ! [A5: set_set_nat] :
% 0.35/0.68                ( ( finite1152437895449049373et_nat @ A5 )
% 0.35/0.68               => ( ( A5 != bot_bot_set_set_nat )
% 0.35/0.68                 => ( ( ord_le6893508408891458716et_nat @ A5 @ B2 )
% 0.35/0.68                   => ( ! [X2: set_nat] :
% 0.35/0.68                          ( ( member_set_nat @ X2 @ A5 )
% 0.35/0.68                         => ( P @ ( minus_2163939370556025621et_nat @ A5 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) )
% 0.35/0.68                     => ( P @ A5 ) ) ) ) )
% 0.35/0.68           => ( P @ B2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % finite_remove_induct
% 0.35/0.68  thf(fact_6278_finite__remove__induct,axiom,
% 0.35/0.68      ! [B2: set_complex,P: set_complex > $o] :
% 0.35/0.68        ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.68       => ( ( P @ bot_bot_set_complex )
% 0.35/0.68         => ( ! [A5: set_complex] :
% 0.35/0.68                ( ( finite3207457112153483333omplex @ A5 )
% 0.35/0.68               => ( ( A5 != bot_bot_set_complex )
% 0.35/0.68                 => ( ( ord_le211207098394363844omplex @ A5 @ B2 )
% 0.35/0.68                   => ( ! [X2: complex] :
% 0.35/0.68                          ( ( member_complex @ X2 @ A5 )
% 0.35/0.68                         => ( P @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) )
% 0.35/0.68                     => ( P @ A5 ) ) ) ) )
% 0.35/0.68           => ( P @ B2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % finite_remove_induct
% 0.35/0.68  thf(fact_6279_finite__remove__induct,axiom,
% 0.35/0.68      ! [B2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.35/0.68        ( ( finite6177210948735845034at_nat @ B2 )
% 0.35/0.68       => ( ( P @ bot_bo2099793752762293965at_nat )
% 0.35/0.68         => ( ! [A5: set_Pr1261947904930325089at_nat] :
% 0.35/0.68                ( ( finite6177210948735845034at_nat @ A5 )
% 0.35/0.68               => ( ( A5 != bot_bo2099793752762293965at_nat )
% 0.35/0.68                 => ( ( ord_le3146513528884898305at_nat @ A5 @ B2 )
% 0.35/0.68                   => ( ! [X2: product_prod_nat_nat] :
% 0.35/0.68                          ( ( member8440522571783428010at_nat @ X2 @ A5 )
% 0.35/0.68                         => ( P @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) )
% 0.35/0.68                     => ( P @ A5 ) ) ) ) )
% 0.35/0.68           => ( P @ B2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % finite_remove_induct
% 0.35/0.68  thf(fact_6280_finite__remove__induct,axiom,
% 0.35/0.68      ! [B2: set_real,P: set_real > $o] :
% 0.35/0.68        ( ( finite_finite_real @ B2 )
% 0.35/0.68       => ( ( P @ bot_bot_set_real )
% 0.35/0.68         => ( ! [A5: set_real] :
% 0.35/0.68                ( ( finite_finite_real @ A5 )
% 0.35/0.68               => ( ( A5 != bot_bot_set_real )
% 0.35/0.68                 => ( ( ord_less_eq_set_real @ A5 @ B2 )
% 0.35/0.68                   => ( ! [X2: real] :
% 0.35/0.68                          ( ( member_real @ X2 @ A5 )
% 0.35/0.68                         => ( P @ ( minus_minus_set_real @ A5 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) )
% 0.35/0.68                     => ( P @ A5 ) ) ) ) )
% 0.35/0.68           => ( P @ B2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % finite_remove_induct
% 0.35/0.68  thf(fact_6281_finite__remove__induct,axiom,
% 0.35/0.68      ! [B2: set_nat,P: set_nat > $o] :
% 0.35/0.68        ( ( finite_finite_nat @ B2 )
% 0.35/0.68       => ( ( P @ bot_bot_set_nat )
% 0.35/0.68         => ( ! [A5: set_nat] :
% 0.35/0.68                ( ( finite_finite_nat @ A5 )
% 0.35/0.68               => ( ( A5 != bot_bot_set_nat )
% 0.35/0.68                 => ( ( ord_less_eq_set_nat @ A5 @ B2 )
% 0.35/0.68                   => ( ! [X2: nat] :
% 0.35/0.68                          ( ( member_nat @ X2 @ A5 )
% 0.35/0.68                         => ( P @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
% 0.35/0.68                     => ( P @ A5 ) ) ) ) )
% 0.35/0.68           => ( P @ B2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % finite_remove_induct
% 0.35/0.68  thf(fact_6282_finite__remove__induct,axiom,
% 0.35/0.68      ! [B2: set_int,P: set_int > $o] :
% 0.35/0.68        ( ( finite_finite_int @ B2 )
% 0.35/0.68       => ( ( P @ bot_bot_set_int )
% 0.35/0.68         => ( ! [A5: set_int] :
% 0.35/0.68                ( ( finite_finite_int @ A5 )
% 0.35/0.68               => ( ( A5 != bot_bot_set_int )
% 0.35/0.68                 => ( ( ord_less_eq_set_int @ A5 @ B2 )
% 0.35/0.68                   => ( ! [X2: int] :
% 0.35/0.68                          ( ( member_int @ X2 @ A5 )
% 0.35/0.68                         => ( P @ ( minus_minus_set_int @ A5 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) )
% 0.35/0.68                     => ( P @ A5 ) ) ) ) )
% 0.35/0.68           => ( P @ B2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % finite_remove_induct
% 0.35/0.68  thf(fact_6283_finite__induct__select,axiom,
% 0.35/0.68      ! [S2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 0.35/0.68        ( ( finite4343798906461161616at_nat @ S2 )
% 0.35/0.68       => ( ( P @ bot_bo228742789529271731at_nat )
% 0.35/0.68         => ( ! [T5: set_Pr4329608150637261639at_nat] :
% 0.35/0.68                ( ( ord_le2604355607129572851at_nat @ T5 @ S2 )
% 0.35/0.68               => ( ( P @ T5 )
% 0.35/0.68                 => ? [X2: produc3843707927480180839at_nat] :
% 0.35/0.68                      ( ( member8757157785044589968at_nat @ X2 @ ( minus_3314409938677909166at_nat @ S2 @ T5 ) )
% 0.35/0.68                      & ( P @ ( insert9069300056098147895at_nat @ X2 @ T5 ) ) ) ) )
% 0.35/0.68           => ( P @ S2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % finite_induct_select
% 0.35/0.68  thf(fact_6284_finite__induct__select,axiom,
% 0.35/0.68      ! [S2: set_complex,P: set_complex > $o] :
% 0.35/0.68        ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.68       => ( ( P @ bot_bot_set_complex )
% 0.35/0.68         => ( ! [T5: set_complex] :
% 0.35/0.68                ( ( ord_less_set_complex @ T5 @ S2 )
% 0.35/0.68               => ( ( P @ T5 )
% 0.35/0.68                 => ? [X2: complex] :
% 0.35/0.68                      ( ( member_complex @ X2 @ ( minus_811609699411566653omplex @ S2 @ T5 ) )
% 0.35/0.68                      & ( P @ ( insert_complex @ X2 @ T5 ) ) ) ) )
% 0.35/0.68           => ( P @ S2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % finite_induct_select
% 0.35/0.68  thf(fact_6285_finite__induct__select,axiom,
% 0.35/0.68      ! [S2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.35/0.68        ( ( finite6177210948735845034at_nat @ S2 )
% 0.35/0.68       => ( ( P @ bot_bo2099793752762293965at_nat )
% 0.35/0.68         => ( ! [T5: set_Pr1261947904930325089at_nat] :
% 0.35/0.68                ( ( ord_le7866589430770878221at_nat @ T5 @ S2 )
% 0.35/0.68               => ( ( P @ T5 )
% 0.35/0.68                 => ? [X2: product_prod_nat_nat] :
% 0.35/0.68                      ( ( member8440522571783428010at_nat @ X2 @ ( minus_1356011639430497352at_nat @ S2 @ T5 ) )
% 0.35/0.68                      & ( P @ ( insert8211810215607154385at_nat @ X2 @ T5 ) ) ) ) )
% 0.35/0.68           => ( P @ S2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % finite_induct_select
% 0.35/0.68  thf(fact_6286_finite__induct__select,axiom,
% 0.35/0.68      ! [S2: set_real,P: set_real > $o] :
% 0.35/0.68        ( ( finite_finite_real @ S2 )
% 0.35/0.68       => ( ( P @ bot_bot_set_real )
% 0.35/0.68         => ( ! [T5: set_real] :
% 0.35/0.68                ( ( ord_less_set_real @ T5 @ S2 )
% 0.35/0.68               => ( ( P @ T5 )
% 0.35/0.68                 => ? [X2: real] :
% 0.35/0.68                      ( ( member_real @ X2 @ ( minus_minus_set_real @ S2 @ T5 ) )
% 0.35/0.68                      & ( P @ ( insert_real @ X2 @ T5 ) ) ) ) )
% 0.35/0.68           => ( P @ S2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % finite_induct_select
% 0.35/0.68  thf(fact_6287_finite__induct__select,axiom,
% 0.35/0.68      ! [S2: set_int,P: set_int > $o] :
% 0.35/0.68        ( ( finite_finite_int @ S2 )
% 0.35/0.68       => ( ( P @ bot_bot_set_int )
% 0.35/0.68         => ( ! [T5: set_int] :
% 0.35/0.68                ( ( ord_less_set_int @ T5 @ S2 )
% 0.35/0.68               => ( ( P @ T5 )
% 0.35/0.68                 => ? [X2: int] :
% 0.35/0.68                      ( ( member_int @ X2 @ ( minus_minus_set_int @ S2 @ T5 ) )
% 0.35/0.68                      & ( P @ ( insert_int @ X2 @ T5 ) ) ) ) )
% 0.35/0.68           => ( P @ S2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % finite_induct_select
% 0.35/0.68  thf(fact_6288_finite__induct__select,axiom,
% 0.35/0.68      ! [S2: set_nat,P: set_nat > $o] :
% 0.35/0.68        ( ( finite_finite_nat @ S2 )
% 0.35/0.68       => ( ( P @ bot_bot_set_nat )
% 0.35/0.68         => ( ! [T5: set_nat] :
% 0.35/0.68                ( ( ord_less_set_nat @ T5 @ S2 )
% 0.35/0.68               => ( ( P @ T5 )
% 0.35/0.68                 => ? [X2: nat] :
% 0.35/0.68                      ( ( member_nat @ X2 @ ( minus_minus_set_nat @ S2 @ T5 ) )
% 0.35/0.68                      & ( P @ ( insert_nat @ X2 @ T5 ) ) ) ) )
% 0.35/0.68           => ( P @ S2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % finite_induct_select
% 0.35/0.68  thf(fact_6289_psubset__insert__iff,axiom,
% 0.35/0.68      ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.68        ( ( ord_le2604355607129572851at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ B2 ) )
% 0.35/0.68        = ( ( ( member8757157785044589968at_nat @ X @ B2 )
% 0.35/0.68           => ( ord_le2604355607129572851at_nat @ A2 @ B2 ) )
% 0.35/0.68          & ( ~ ( member8757157785044589968at_nat @ X @ B2 )
% 0.35/0.68           => ( ( ( member8757157785044589968at_nat @ X @ A2 )
% 0.35/0.68               => ( ord_le2604355607129572851at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B2 ) )
% 0.35/0.68              & ( ~ ( member8757157785044589968at_nat @ X @ A2 )
% 0.35/0.68               => ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % psubset_insert_iff
% 0.35/0.68  thf(fact_6290_psubset__insert__iff,axiom,
% 0.35/0.68      ! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
% 0.35/0.68        ( ( ord_less_set_set_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
% 0.35/0.68        = ( ( ( member_set_nat @ X @ B2 )
% 0.35/0.68           => ( ord_less_set_set_nat @ A2 @ B2 ) )
% 0.35/0.68          & ( ~ ( member_set_nat @ X @ B2 )
% 0.35/0.68           => ( ( ( member_set_nat @ X @ A2 )
% 0.35/0.68               => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B2 ) )
% 0.35/0.68              & ( ~ ( member_set_nat @ X @ A2 )
% 0.35/0.68               => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % psubset_insert_iff
% 0.35/0.68  thf(fact_6291_psubset__insert__iff,axiom,
% 0.35/0.68      ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.68        ( ( ord_le7866589430770878221at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B2 ) )
% 0.35/0.68        = ( ( ( member8440522571783428010at_nat @ X @ B2 )
% 0.35/0.68           => ( ord_le7866589430770878221at_nat @ A2 @ B2 ) )
% 0.35/0.68          & ( ~ ( member8440522571783428010at_nat @ X @ B2 )
% 0.35/0.68           => ( ( ( member8440522571783428010at_nat @ X @ A2 )
% 0.35/0.68               => ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B2 ) )
% 0.35/0.68              & ( ~ ( member8440522571783428010at_nat @ X @ A2 )
% 0.35/0.68               => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % psubset_insert_iff
% 0.35/0.68  thf(fact_6292_psubset__insert__iff,axiom,
% 0.35/0.68      ! [A2: set_real,X: real,B2: set_real] :
% 0.35/0.68        ( ( ord_less_set_real @ A2 @ ( insert_real @ X @ B2 ) )
% 0.35/0.68        = ( ( ( member_real @ X @ B2 )
% 0.35/0.68           => ( ord_less_set_real @ A2 @ B2 ) )
% 0.35/0.68          & ( ~ ( member_real @ X @ B2 )
% 0.35/0.68           => ( ( ( member_real @ X @ A2 )
% 0.35/0.68               => ( ord_less_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 ) )
% 0.35/0.68              & ( ~ ( member_real @ X @ A2 )
% 0.35/0.68               => ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % psubset_insert_iff
% 0.35/0.68  thf(fact_6293_psubset__insert__iff,axiom,
% 0.35/0.68      ! [A2: set_nat,X: nat,B2: set_nat] :
% 0.35/0.68        ( ( ord_less_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
% 0.35/0.68        = ( ( ( member_nat @ X @ B2 )
% 0.35/0.68           => ( ord_less_set_nat @ A2 @ B2 ) )
% 0.35/0.68          & ( ~ ( member_nat @ X @ B2 )
% 0.35/0.68           => ( ( ( member_nat @ X @ A2 )
% 0.35/0.68               => ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
% 0.35/0.68              & ( ~ ( member_nat @ X @ A2 )
% 0.35/0.68               => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % psubset_insert_iff
% 0.35/0.68  thf(fact_6294_psubset__insert__iff,axiom,
% 0.35/0.68      ! [A2: set_int,X: int,B2: set_int] :
% 0.35/0.68        ( ( ord_less_set_int @ A2 @ ( insert_int @ X @ B2 ) )
% 0.35/0.68        = ( ( ( member_int @ X @ B2 )
% 0.35/0.68           => ( ord_less_set_int @ A2 @ B2 ) )
% 0.35/0.68          & ( ~ ( member_int @ X @ B2 )
% 0.35/0.68           => ( ( ( member_int @ X @ A2 )
% 0.35/0.68               => ( ord_less_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 ) )
% 0.35/0.68              & ( ~ ( member_int @ X @ A2 )
% 0.35/0.68               => ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % psubset_insert_iff
% 0.35/0.68  thf(fact_6295_set__replicate__Suc,axiom,
% 0.35/0.68      ! [N: nat,X: produc3843707927480180839at_nat] :
% 0.35/0.68        ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ ( suc @ N ) @ X ) )
% 0.35/0.68        = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % set_replicate_Suc
% 0.35/0.68  thf(fact_6296_set__replicate__Suc,axiom,
% 0.35/0.68      ! [N: nat,X: vEBT_VEBT] :
% 0.35/0.68        ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ ( suc @ N ) @ X ) )
% 0.35/0.68        = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ).
% 0.35/0.68  
% 0.35/0.68  % set_replicate_Suc
% 0.35/0.68  thf(fact_6297_set__replicate__Suc,axiom,
% 0.35/0.68      ! [N: nat,X: product_prod_nat_nat] :
% 0.35/0.68        ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ ( suc @ N ) @ X ) )
% 0.35/0.68        = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % set_replicate_Suc
% 0.35/0.68  thf(fact_6298_set__replicate__Suc,axiom,
% 0.35/0.68      ! [N: nat,X: real] :
% 0.35/0.68        ( ( set_real2 @ ( replicate_real @ ( suc @ N ) @ X ) )
% 0.35/0.68        = ( insert_real @ X @ bot_bot_set_real ) ) ).
% 0.35/0.68  
% 0.35/0.68  % set_replicate_Suc
% 0.35/0.68  thf(fact_6299_set__replicate__Suc,axiom,
% 0.35/0.68      ! [N: nat,X: nat] :
% 0.35/0.68        ( ( set_nat2 @ ( replicate_nat @ ( suc @ N ) @ X ) )
% 0.35/0.68        = ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % set_replicate_Suc
% 0.35/0.68  thf(fact_6300_set__replicate__Suc,axiom,
% 0.35/0.68      ! [N: nat,X: int] :
% 0.35/0.68        ( ( set_int2 @ ( replicate_int @ ( suc @ N ) @ X ) )
% 0.35/0.68        = ( insert_int @ X @ bot_bot_set_int ) ) ).
% 0.35/0.68  
% 0.35/0.68  % set_replicate_Suc
% 0.35/0.68  thf(fact_6301_set__replicate__conv__if,axiom,
% 0.35/0.68      ! [N: nat,X: produc3843707927480180839at_nat] :
% 0.35/0.68        ( ( ( N = zero_zero_nat )
% 0.35/0.68         => ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N @ X ) )
% 0.35/0.68            = bot_bo228742789529271731at_nat ) )
% 0.35/0.68        & ( ( N != zero_zero_nat )
% 0.35/0.68         => ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N @ X ) )
% 0.35/0.68            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % set_replicate_conv_if
% 0.35/0.68  thf(fact_6302_set__replicate__conv__if,axiom,
% 0.35/0.68      ! [N: nat,X: vEBT_VEBT] :
% 0.35/0.68        ( ( ( N = zero_zero_nat )
% 0.35/0.68         => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
% 0.35/0.68            = bot_bo8194388402131092736T_VEBT ) )
% 0.35/0.68        & ( ( N != zero_zero_nat )
% 0.35/0.68         => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
% 0.35/0.68            = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % set_replicate_conv_if
% 0.35/0.68  thf(fact_6303_set__replicate__conv__if,axiom,
% 0.35/0.68      ! [N: nat,X: product_prod_nat_nat] :
% 0.35/0.68        ( ( ( N = zero_zero_nat )
% 0.35/0.68         => ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ X ) )
% 0.35/0.68            = bot_bo2099793752762293965at_nat ) )
% 0.35/0.68        & ( ( N != zero_zero_nat )
% 0.35/0.68         => ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ X ) )
% 0.35/0.68            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % set_replicate_conv_if
% 0.35/0.68  thf(fact_6304_set__replicate__conv__if,axiom,
% 0.35/0.68      ! [N: nat,X: real] :
% 0.35/0.68        ( ( ( N = zero_zero_nat )
% 0.35/0.68         => ( ( set_real2 @ ( replicate_real @ N @ X ) )
% 0.35/0.68            = bot_bot_set_real ) )
% 0.35/0.68        & ( ( N != zero_zero_nat )
% 0.35/0.68         => ( ( set_real2 @ ( replicate_real @ N @ X ) )
% 0.35/0.68            = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % set_replicate_conv_if
% 0.35/0.68  thf(fact_6305_set__replicate__conv__if,axiom,
% 0.35/0.68      ! [N: nat,X: nat] :
% 0.35/0.68        ( ( ( N = zero_zero_nat )
% 0.35/0.68         => ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
% 0.35/0.68            = bot_bot_set_nat ) )
% 0.35/0.68        & ( ( N != zero_zero_nat )
% 0.35/0.68         => ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
% 0.35/0.68            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % set_replicate_conv_if
% 0.35/0.68  thf(fact_6306_set__replicate__conv__if,axiom,
% 0.35/0.68      ! [N: nat,X: int] :
% 0.35/0.68        ( ( ( N = zero_zero_nat )
% 0.35/0.68         => ( ( set_int2 @ ( replicate_int @ N @ X ) )
% 0.35/0.68            = bot_bot_set_int ) )
% 0.35/0.68        & ( ( N != zero_zero_nat )
% 0.35/0.68         => ( ( set_int2 @ ( replicate_int @ N @ X ) )
% 0.35/0.68            = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % set_replicate_conv_if
% 0.35/0.68  thf(fact_6307_sum_OSuc__reindex__ivl,axiom,
% 0.35/0.68      ! [M: nat,N: nat,G: nat > rat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.68       => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.35/0.68          = ( plus_plus_rat @ ( G @ M )
% 0.35/0.68            @ ( groups2906978787729119204at_rat
% 0.35/0.68              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.35/0.68              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum.Suc_reindex_ivl
% 0.35/0.68  thf(fact_6308_sum_OSuc__reindex__ivl,axiom,
% 0.35/0.68      ! [M: nat,N: nat,G: nat > int] :
% 0.35/0.68        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.68       => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.35/0.68          = ( plus_plus_int @ ( G @ M )
% 0.35/0.68            @ ( groups3539618377306564664at_int
% 0.35/0.68              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.35/0.68              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum.Suc_reindex_ivl
% 0.35/0.68  thf(fact_6309_sum_OSuc__reindex__ivl,axiom,
% 0.35/0.68      ! [M: nat,N: nat,G: nat > nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.68       => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.35/0.68          = ( plus_plus_nat @ ( G @ M )
% 0.35/0.68            @ ( groups3542108847815614940at_nat
% 0.35/0.68              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.35/0.68              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum.Suc_reindex_ivl
% 0.35/0.68  thf(fact_6310_sum_OSuc__reindex__ivl,axiom,
% 0.35/0.68      ! [M: nat,N: nat,G: nat > real] :
% 0.35/0.68        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.68       => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.35/0.68          = ( plus_plus_real @ ( G @ M )
% 0.35/0.68            @ ( groups6591440286371151544t_real
% 0.35/0.68              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.35/0.68              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum.Suc_reindex_ivl
% 0.35/0.68  thf(fact_6311_sum__Suc__diff,axiom,
% 0.35/0.68      ! [M: nat,N: nat,F: nat > rat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.35/0.68       => ( ( groups2906978787729119204at_rat
% 0.35/0.68            @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 0.35/0.68            @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68          = ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_Suc_diff
% 0.35/0.68  thf(fact_6312_sum__Suc__diff,axiom,
% 0.35/0.68      ! [M: nat,N: nat,F: nat > int] :
% 0.35/0.68        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.35/0.68       => ( ( groups3539618377306564664at_int
% 0.35/0.68            @ ^ [I4: nat] : ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 0.35/0.68            @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68          = ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_Suc_diff
% 0.35/0.68  thf(fact_6313_sum__Suc__diff,axiom,
% 0.35/0.68      ! [M: nat,N: nat,F: nat > real] :
% 0.35/0.68        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.35/0.68       => ( ( groups6591440286371151544t_real
% 0.35/0.68            @ ^ [I4: nat] : ( minus_minus_real @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 0.35/0.68            @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68          = ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_Suc_diff
% 0.35/0.68  thf(fact_6314_sum__mono2,axiom,
% 0.35/0.68      ! [B2: set_real,A2: set_real,F: real > real] :
% 0.35/0.68        ( ( finite_finite_real @ B2 )
% 0.35/0.68       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.35/0.68         => ( ! [B3: real] :
% 0.35/0.68                ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) )
% 0.35/0.68               => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
% 0.35/0.68           => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_mono2
% 0.35/0.68  thf(fact_6315_sum__mono2,axiom,
% 0.35/0.68      ! [B2: set_complex,A2: set_complex,F: complex > real] :
% 0.35/0.68        ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.68       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.35/0.68         => ( ! [B3: complex] :
% 0.35/0.68                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 0.35/0.68               => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
% 0.35/0.68           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_mono2
% 0.35/0.68  thf(fact_6316_sum__mono2,axiom,
% 0.35/0.68      ! [B2: set_real,A2: set_real,F: real > rat] :
% 0.35/0.68        ( ( finite_finite_real @ B2 )
% 0.35/0.68       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.35/0.68         => ( ! [B3: real] :
% 0.35/0.68                ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) )
% 0.35/0.68               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
% 0.35/0.68           => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_mono2
% 0.35/0.68  thf(fact_6317_sum__mono2,axiom,
% 0.35/0.68      ! [B2: set_complex,A2: set_complex,F: complex > rat] :
% 0.35/0.68        ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.68       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.35/0.68         => ( ! [B3: complex] :
% 0.35/0.68                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 0.35/0.68               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
% 0.35/0.68           => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_mono2
% 0.35/0.68  thf(fact_6318_sum__mono2,axiom,
% 0.35/0.68      ! [B2: set_nat,A2: set_nat,F: nat > rat] :
% 0.35/0.68        ( ( finite_finite_nat @ B2 )
% 0.35/0.68       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.35/0.68         => ( ! [B3: nat] :
% 0.35/0.68                ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B2 @ A2 ) )
% 0.35/0.68               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
% 0.35/0.68           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_mono2
% 0.35/0.68  thf(fact_6319_sum__mono2,axiom,
% 0.35/0.68      ! [B2: set_real,A2: set_real,F: real > nat] :
% 0.35/0.68        ( ( finite_finite_real @ B2 )
% 0.35/0.68       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.35/0.68         => ( ! [B3: real] :
% 0.35/0.68                ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) )
% 0.35/0.68               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
% 0.35/0.68           => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_mono2
% 0.35/0.68  thf(fact_6320_sum__mono2,axiom,
% 0.35/0.68      ! [B2: set_complex,A2: set_complex,F: complex > nat] :
% 0.35/0.68        ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.68       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.35/0.68         => ( ! [B3: complex] :
% 0.35/0.68                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 0.35/0.68               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
% 0.35/0.68           => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_mono2
% 0.35/0.68  thf(fact_6321_sum__mono2,axiom,
% 0.35/0.68      ! [B2: set_real,A2: set_real,F: real > int] :
% 0.35/0.68        ( ( finite_finite_real @ B2 )
% 0.35/0.68       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.35/0.68         => ( ! [B3: real] :
% 0.35/0.68                ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) )
% 0.35/0.68               => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
% 0.35/0.68           => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B2 ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_mono2
% 0.35/0.68  thf(fact_6322_sum__mono2,axiom,
% 0.35/0.68      ! [B2: set_complex,A2: set_complex,F: complex > int] :
% 0.35/0.68        ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.68       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.35/0.68         => ( ! [B3: complex] :
% 0.35/0.68                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 0.35/0.68               => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
% 0.35/0.68           => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_mono2
% 0.35/0.68  thf(fact_6323_sum__mono2,axiom,
% 0.35/0.68      ! [B2: set_nat,A2: set_nat,F: nat > int] :
% 0.35/0.68        ( ( finite_finite_nat @ B2 )
% 0.35/0.68       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.35/0.68         => ( ! [B3: nat] :
% 0.35/0.68                ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B2 @ A2 ) )
% 0.35/0.68               => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
% 0.35/0.68           => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_mono2
% 0.35/0.68  thf(fact_6324_sum_Oub__add__nat,axiom,
% 0.35/0.68      ! [M: nat,N: nat,G: nat > rat,P4: nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.35/0.68       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 0.35/0.68          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum.ub_add_nat
% 0.35/0.68  thf(fact_6325_sum_Oub__add__nat,axiom,
% 0.35/0.68      ! [M: nat,N: nat,G: nat > int,P4: nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.35/0.68       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 0.35/0.68          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum.ub_add_nat
% 0.35/0.68  thf(fact_6326_sum_Oub__add__nat,axiom,
% 0.35/0.68      ! [M: nat,N: nat,G: nat > nat,P4: nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.35/0.68       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 0.35/0.68          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum.ub_add_nat
% 0.35/0.68  thf(fact_6327_sum_Oub__add__nat,axiom,
% 0.35/0.68      ! [M: nat,N: nat,G: nat > real,P4: nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.35/0.68       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 0.35/0.68          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum.ub_add_nat
% 0.35/0.68  thf(fact_6328_sum__count__set,axiom,
% 0.35/0.68      ! [Xs: list_complex,X8: set_complex] :
% 0.35/0.68        ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ X8 )
% 0.35/0.68       => ( ( finite3207457112153483333omplex @ X8 )
% 0.35/0.68         => ( ( groups5693394587270226106ex_nat @ ( count_list_complex @ Xs ) @ X8 )
% 0.35/0.68            = ( size_s3451745648224563538omplex @ Xs ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_count_set
% 0.35/0.68  thf(fact_6329_sum__count__set,axiom,
% 0.35/0.68      ! [Xs: list_P6011104703257516679at_nat,X8: set_Pr1261947904930325089at_nat] :
% 0.35/0.68        ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ X8 )
% 0.35/0.68       => ( ( finite6177210948735845034at_nat @ X8 )
% 0.35/0.68         => ( ( groups977919841031483927at_nat @ ( count_4203492906077236349at_nat @ Xs ) @ X8 )
% 0.35/0.68            = ( size_s5460976970255530739at_nat @ Xs ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_count_set
% 0.35/0.68  thf(fact_6330_sum__count__set,axiom,
% 0.35/0.68      ! [Xs: list_VEBT_VEBT,X8: set_VEBT_VEBT] :
% 0.35/0.68        ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ X8 )
% 0.35/0.68       => ( ( finite5795047828879050333T_VEBT @ X8 )
% 0.35/0.68         => ( ( groups771621172384141258BT_nat @ ( count_list_VEBT_VEBT @ Xs ) @ X8 )
% 0.35/0.68            = ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_count_set
% 0.35/0.68  thf(fact_6331_sum__count__set,axiom,
% 0.35/0.68      ! [Xs: list_o,X8: set_o] :
% 0.35/0.68        ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ X8 )
% 0.35/0.68       => ( ( finite_finite_o @ X8 )
% 0.35/0.68         => ( ( groups8507830703676809646_o_nat @ ( count_list_o @ Xs ) @ X8 )
% 0.35/0.68            = ( size_size_list_o @ Xs ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_count_set
% 0.35/0.68  thf(fact_6332_sum__count__set,axiom,
% 0.35/0.68      ! [Xs: list_int,X8: set_int] :
% 0.35/0.68        ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ X8 )
% 0.35/0.68       => ( ( finite_finite_int @ X8 )
% 0.35/0.68         => ( ( groups4541462559716669496nt_nat @ ( count_list_int @ Xs ) @ X8 )
% 0.35/0.68            = ( size_size_list_int @ Xs ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_count_set
% 0.35/0.68  thf(fact_6333_sum__count__set,axiom,
% 0.35/0.68      ! [Xs: list_nat,X8: set_nat] :
% 0.35/0.68        ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ X8 )
% 0.35/0.68       => ( ( finite_finite_nat @ X8 )
% 0.35/0.68         => ( ( groups3542108847815614940at_nat @ ( count_list_nat @ Xs ) @ X8 )
% 0.35/0.68            = ( size_size_list_nat @ Xs ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_count_set
% 0.35/0.68  thf(fact_6334_sum__strict__mono2,axiom,
% 0.35/0.68      ! [B2: set_real,A2: set_real,B: real,F: real > real] :
% 0.35/0.68        ( ( finite_finite_real @ B2 )
% 0.35/0.68       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.35/0.68         => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
% 0.35/0.68           => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 0.35/0.68             => ( ! [X4: real] :
% 0.35/0.68                    ( ( member_real @ X4 @ B2 )
% 0.35/0.68                   => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.35/0.68               => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_strict_mono2
% 0.35/0.68  thf(fact_6335_sum__strict__mono2,axiom,
% 0.35/0.68      ! [B2: set_complex,A2: set_complex,B: complex,F: complex > real] :
% 0.35/0.68        ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.68       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.35/0.68         => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 0.35/0.68           => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 0.35/0.68             => ( ! [X4: complex] :
% 0.35/0.68                    ( ( member_complex @ X4 @ B2 )
% 0.35/0.68                   => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.35/0.68               => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_strict_mono2
% 0.35/0.68  thf(fact_6336_sum__strict__mono2,axiom,
% 0.35/0.68      ! [B2: set_real,A2: set_real,B: real,F: real > rat] :
% 0.35/0.68        ( ( finite_finite_real @ B2 )
% 0.35/0.68       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.35/0.68         => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
% 0.35/0.68           => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 0.35/0.68             => ( ! [X4: real] :
% 0.35/0.68                    ( ( member_real @ X4 @ B2 )
% 0.35/0.68                   => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.35/0.68               => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_strict_mono2
% 0.35/0.68  thf(fact_6337_sum__strict__mono2,axiom,
% 0.35/0.68      ! [B2: set_complex,A2: set_complex,B: complex,F: complex > rat] :
% 0.35/0.68        ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.68       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.35/0.68         => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 0.35/0.68           => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 0.35/0.68             => ( ! [X4: complex] :
% 0.35/0.68                    ( ( member_complex @ X4 @ B2 )
% 0.35/0.68                   => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.35/0.68               => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_strict_mono2
% 0.35/0.68  thf(fact_6338_sum__strict__mono2,axiom,
% 0.35/0.68      ! [B2: set_nat,A2: set_nat,B: nat,F: nat > rat] :
% 0.35/0.68        ( ( finite_finite_nat @ B2 )
% 0.35/0.68       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.35/0.68         => ( ( member_nat @ B @ ( minus_minus_set_nat @ B2 @ A2 ) )
% 0.35/0.68           => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 0.35/0.68             => ( ! [X4: nat] :
% 0.35/0.68                    ( ( member_nat @ X4 @ B2 )
% 0.35/0.68                   => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.35/0.68               => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_strict_mono2
% 0.35/0.68  thf(fact_6339_sum__strict__mono2,axiom,
% 0.35/0.68      ! [B2: set_real,A2: set_real,B: real,F: real > nat] :
% 0.35/0.68        ( ( finite_finite_real @ B2 )
% 0.35/0.68       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.35/0.68         => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
% 0.35/0.68           => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 0.35/0.68             => ( ! [X4: real] :
% 0.35/0.68                    ( ( member_real @ X4 @ B2 )
% 0.35/0.68                   => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.35/0.68               => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_strict_mono2
% 0.35/0.68  thf(fact_6340_sum__strict__mono2,axiom,
% 0.35/0.68      ! [B2: set_complex,A2: set_complex,B: complex,F: complex > nat] :
% 0.35/0.68        ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.68       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.35/0.68         => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 0.35/0.68           => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 0.35/0.68             => ( ! [X4: complex] :
% 0.35/0.68                    ( ( member_complex @ X4 @ B2 )
% 0.35/0.68                   => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.35/0.68               => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_strict_mono2
% 0.35/0.68  thf(fact_6341_sum__strict__mono2,axiom,
% 0.35/0.68      ! [B2: set_real,A2: set_real,B: real,F: real > int] :
% 0.35/0.68        ( ( finite_finite_real @ B2 )
% 0.35/0.68       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.35/0.68         => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
% 0.35/0.68           => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 0.35/0.68             => ( ! [X4: real] :
% 0.35/0.68                    ( ( member_real @ X4 @ B2 )
% 0.35/0.68                   => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 0.35/0.68               => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B2 ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_strict_mono2
% 0.35/0.68  thf(fact_6342_sum__strict__mono2,axiom,
% 0.35/0.68      ! [B2: set_complex,A2: set_complex,B: complex,F: complex > int] :
% 0.35/0.68        ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.68       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.35/0.68         => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 0.35/0.68           => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 0.35/0.68             => ( ! [X4: complex] :
% 0.35/0.68                    ( ( member_complex @ X4 @ B2 )
% 0.35/0.68                   => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 0.35/0.68               => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_strict_mono2
% 0.35/0.68  thf(fact_6343_sum__strict__mono2,axiom,
% 0.35/0.68      ! [B2: set_nat,A2: set_nat,B: nat,F: nat > int] :
% 0.35/0.68        ( ( finite_finite_nat @ B2 )
% 0.35/0.68       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.35/0.68         => ( ( member_nat @ B @ ( minus_minus_set_nat @ B2 @ A2 ) )
% 0.35/0.68           => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 0.35/0.68             => ( ! [X4: nat] :
% 0.35/0.68                    ( ( member_nat @ X4 @ B2 )
% 0.35/0.68                   => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 0.35/0.68               => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_strict_mono2
% 0.35/0.68  thf(fact_6344_sum__natinterval__diff,axiom,
% 0.35/0.68      ! [M: nat,N: nat,F: nat > complex] :
% 0.35/0.68        ( ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.68         => ( ( groups2073611262835488442omplex
% 0.35/0.68              @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.35/0.68              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68            = ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 0.35/0.68        & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.35/0.68         => ( ( groups2073611262835488442omplex
% 0.35/0.68              @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.35/0.68              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68            = zero_zero_complex ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_natinterval_diff
% 0.35/0.68  thf(fact_6345_sum__natinterval__diff,axiom,
% 0.35/0.68      ! [M: nat,N: nat,F: nat > rat] :
% 0.35/0.68        ( ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.68         => ( ( groups2906978787729119204at_rat
% 0.35/0.68              @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.35/0.68              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68            = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 0.35/0.68        & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.35/0.68         => ( ( groups2906978787729119204at_rat
% 0.35/0.68              @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.35/0.68              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68            = zero_zero_rat ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_natinterval_diff
% 0.35/0.68  thf(fact_6346_sum__natinterval__diff,axiom,
% 0.35/0.68      ! [M: nat,N: nat,F: nat > int] :
% 0.35/0.68        ( ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.68         => ( ( groups3539618377306564664at_int
% 0.35/0.68              @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.35/0.68              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68            = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 0.35/0.68        & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.35/0.68         => ( ( groups3539618377306564664at_int
% 0.35/0.68              @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.35/0.68              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68            = zero_zero_int ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_natinterval_diff
% 0.35/0.68  thf(fact_6347_sum__natinterval__diff,axiom,
% 0.35/0.68      ! [M: nat,N: nat,F: nat > real] :
% 0.35/0.68        ( ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.68         => ( ( groups6591440286371151544t_real
% 0.35/0.68              @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.35/0.68              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68            = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 0.35/0.68        & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.35/0.68         => ( ( groups6591440286371151544t_real
% 0.35/0.68              @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.35/0.68              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68            = zero_zero_real ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_natinterval_diff
% 0.35/0.68  thf(fact_6348_sum__telescope_H_H,axiom,
% 0.35/0.68      ! [M: nat,N: nat,F: nat > rat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.68       => ( ( groups2906978787729119204at_rat
% 0.35/0.68            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 0.35/0.68            @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.35/0.68          = ( minus_minus_rat @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_telescope''
% 0.35/0.68  thf(fact_6349_sum__telescope_H_H,axiom,
% 0.35/0.68      ! [M: nat,N: nat,F: nat > int] :
% 0.35/0.68        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.68       => ( ( groups3539618377306564664at_int
% 0.35/0.68            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 0.35/0.68            @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.35/0.68          = ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_telescope''
% 0.35/0.68  thf(fact_6350_sum__telescope_H_H,axiom,
% 0.35/0.68      ! [M: nat,N: nat,F: nat > real] :
% 0.35/0.68        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.68       => ( ( groups6591440286371151544t_real
% 0.35/0.68            @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 0.35/0.68            @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.35/0.68          = ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_telescope''
% 0.35/0.68  thf(fact_6351_exists__least__lemma,axiom,
% 0.35/0.68      ! [P: nat > $o] :
% 0.35/0.68        ( ~ ( P @ zero_zero_nat )
% 0.35/0.68       => ( ? [X_12: nat] : ( P @ X_12 )
% 0.35/0.68         => ? [N3: nat] :
% 0.35/0.68              ( ~ ( P @ N3 )
% 0.35/0.68              & ( P @ ( suc @ N3 ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % exists_least_lemma
% 0.35/0.68  thf(fact_6352_mask__eq__sum__exp,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int )
% 0.35/0.68        = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.68          @ ( collect_nat
% 0.35/0.68            @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % mask_eq_sum_exp
% 0.35/0.68  thf(fact_6353_mask__eq__sum__exp,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat )
% 0.35/0.68        = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.68          @ ( collect_nat
% 0.35/0.68            @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % mask_eq_sum_exp
% 0.35/0.68  thf(fact_6354_ex__le__of__int,axiom,
% 0.35/0.68      ! [X: real] :
% 0.35/0.68      ? [Z4: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z4 ) ) ).
% 0.35/0.68  
% 0.35/0.68  % ex_le_of_int
% 0.35/0.68  thf(fact_6355_ex__le__of__int,axiom,
% 0.35/0.68      ! [X: rat] :
% 0.35/0.68      ? [Z4: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z4 ) ) ).
% 0.35/0.68  
% 0.35/0.68  % ex_le_of_int
% 0.35/0.68  thf(fact_6356_ex__less__of__int,axiom,
% 0.35/0.68      ! [X: real] :
% 0.35/0.68      ? [Z4: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z4 ) ) ).
% 0.35/0.68  
% 0.35/0.68  % ex_less_of_int
% 0.35/0.68  thf(fact_6357_ex__less__of__int,axiom,
% 0.35/0.68      ! [X: rat] :
% 0.35/0.68      ? [Z4: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z4 ) ) ).
% 0.35/0.68  
% 0.35/0.68  % ex_less_of_int
% 0.35/0.68  thf(fact_6358_ex__of__int__less,axiom,
% 0.35/0.68      ! [X: real] :
% 0.35/0.68      ? [Z4: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z4 ) @ X ) ).
% 0.35/0.68  
% 0.35/0.68  % ex_of_int_less
% 0.35/0.68  thf(fact_6359_ex__of__int__less,axiom,
% 0.35/0.68      ! [X: rat] :
% 0.35/0.68      ? [Z4: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z4 ) @ X ) ).
% 0.35/0.68  
% 0.35/0.68  % ex_of_int_less
% 0.35/0.68  thf(fact_6360_sum__gp__multiplied,axiom,
% 0.35/0.68      ! [M: nat,N: nat,X: complex] :
% 0.35/0.68        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.68       => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 0.35/0.68          = ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_gp_multiplied
% 0.35/0.68  thf(fact_6361_sum__gp__multiplied,axiom,
% 0.35/0.68      ! [M: nat,N: nat,X: rat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.68       => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 0.35/0.68          = ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_gp_multiplied
% 0.35/0.68  thf(fact_6362_sum__gp__multiplied,axiom,
% 0.35/0.68      ! [M: nat,N: nat,X: int] :
% 0.35/0.68        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.68       => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 0.35/0.68          = ( minus_minus_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_gp_multiplied
% 0.35/0.68  thf(fact_6363_sum__gp__multiplied,axiom,
% 0.35/0.68      ! [M: nat,N: nat,X: real] :
% 0.35/0.68        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.68       => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 0.35/0.68          = ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_gp_multiplied
% 0.35/0.68  thf(fact_6364_of__int__round__le,axiom,
% 0.35/0.68      ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_int_round_le
% 0.35/0.68  thf(fact_6365_of__int__round__le,axiom,
% 0.35/0.68      ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_int_round_le
% 0.35/0.68  thf(fact_6366_of__int__round__ge,axiom,
% 0.35/0.68      ! [X: real] : ( ord_less_eq_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_int_round_ge
% 0.35/0.68  thf(fact_6367_of__int__round__ge,axiom,
% 0.35/0.68      ! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_int_round_ge
% 0.35/0.68  thf(fact_6368_of__int__round__gt,axiom,
% 0.35/0.68      ! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_int_round_gt
% 0.35/0.68  thf(fact_6369_of__int__round__gt,axiom,
% 0.35/0.68      ! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_int_round_gt
% 0.35/0.68  thf(fact_6370_xor__nat__unfold,axiom,
% 0.35/0.68      ( bit_se6528837805403552850or_nat
% 0.35/0.68      = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M2 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % xor_nat_unfold
% 0.35/0.68  thf(fact_6371_mask__eq__sum__exp__nat,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
% 0.35/0.68        = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.68          @ ( collect_nat
% 0.35/0.68            @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % mask_eq_sum_exp_nat
% 0.35/0.68  thf(fact_6372_gauss__sum__nat,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( groups3542108847815614940at_nat
% 0.35/0.68          @ ^ [X3: nat] : X3
% 0.35/0.68          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.68        = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % gauss_sum_nat
% 0.35/0.68  thf(fact_6373_sum__gp,axiom,
% 0.35/0.68      ! [N: nat,M: nat,X: complex] :
% 0.35/0.68        ( ( ( ord_less_nat @ N @ M )
% 0.35/0.68         => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68            = zero_zero_complex ) )
% 0.35/0.68        & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.68         => ( ( ( X = one_one_complex )
% 0.35/0.68             => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68                = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 0.35/0.68            & ( ( X != one_one_complex )
% 0.35/0.68             => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68                = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_gp
% 0.35/0.68  thf(fact_6374_sum__gp,axiom,
% 0.35/0.68      ! [N: nat,M: nat,X: rat] :
% 0.35/0.68        ( ( ( ord_less_nat @ N @ M )
% 0.35/0.68         => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68            = zero_zero_rat ) )
% 0.35/0.68        & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.68         => ( ( ( X = one_one_rat )
% 0.35/0.68             => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68                = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 0.35/0.68            & ( ( X != one_one_rat )
% 0.35/0.68             => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68                = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_gp
% 0.35/0.68  thf(fact_6375_sum__gp,axiom,
% 0.35/0.68      ! [N: nat,M: nat,X: real] :
% 0.35/0.68        ( ( ( ord_less_nat @ N @ M )
% 0.35/0.68         => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68            = zero_zero_real ) )
% 0.35/0.68        & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.68         => ( ( ( X = one_one_real )
% 0.35/0.68             => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68                = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 0.35/0.68            & ( ( X != one_one_real )
% 0.35/0.68             => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68                = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_gp
% 0.35/0.68  thf(fact_6376_gauss__sum__from__Suc__0,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.35/0.68        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % gauss_sum_from_Suc_0
% 0.35/0.68  thf(fact_6377_gauss__sum__from__Suc__0,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.35/0.68        = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % gauss_sum_from_Suc_0
% 0.35/0.68  thf(fact_6378_gauss__sum__from__Suc__0,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.35/0.68        = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % gauss_sum_from_Suc_0
% 0.35/0.68  thf(fact_6379_double__gauss__sum__from__Suc__0,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.35/0.68        = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % double_gauss_sum_from_Suc_0
% 0.35/0.68  thf(fact_6380_double__gauss__sum__from__Suc__0,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.35/0.68        = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % double_gauss_sum_from_Suc_0
% 0.35/0.68  thf(fact_6381_double__gauss__sum__from__Suc__0,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.35/0.68        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % double_gauss_sum_from_Suc_0
% 0.35/0.68  thf(fact_6382_double__gauss__sum__from__Suc__0,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.35/0.68        = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % double_gauss_sum_from_Suc_0
% 0.35/0.68  thf(fact_6383_double__gauss__sum__from__Suc__0,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.35/0.68        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % double_gauss_sum_from_Suc_0
% 0.35/0.68  thf(fact_6384_double__gauss__sum__from__Suc__0,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.35/0.68        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % double_gauss_sum_from_Suc_0
% 0.35/0.68  thf(fact_6385_gauss__sum,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.68        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % gauss_sum
% 0.35/0.68  thf(fact_6386_gauss__sum,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.68        = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % gauss_sum
% 0.35/0.68  thf(fact_6387_gauss__sum,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.68        = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % gauss_sum
% 0.35/0.68  thf(fact_6388_arith__series,axiom,
% 0.35/0.68      ! [A: code_integer,D: code_integer,N: nat] :
% 0.35/0.68        ( ( groups7501900531339628137nteger
% 0.35/0.68          @ ^ [I4: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I4 ) @ D ) )
% 0.35/0.68          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.68        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ D ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % arith_series
% 0.35/0.68  thf(fact_6389_arith__series,axiom,
% 0.35/0.68      ! [A: int,D: int,N: nat] :
% 0.35/0.68        ( ( groups3539618377306564664at_int
% 0.35/0.68          @ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
% 0.35/0.68          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.68        = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % arith_series
% 0.35/0.68  thf(fact_6390_arith__series,axiom,
% 0.35/0.68      ! [A: nat,D: nat,N: nat] :
% 0.35/0.68        ( ( groups3542108847815614940at_nat
% 0.35/0.68          @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
% 0.35/0.68          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.68        = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % arith_series
% 0.35/0.68  thf(fact_6391_double__gauss__sum,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.68        = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % double_gauss_sum
% 0.35/0.68  thf(fact_6392_double__gauss__sum,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.68        = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % double_gauss_sum
% 0.35/0.68  thf(fact_6393_double__gauss__sum,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.68        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % double_gauss_sum
% 0.35/0.68  thf(fact_6394_double__gauss__sum,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.68        = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % double_gauss_sum
% 0.35/0.68  thf(fact_6395_double__gauss__sum,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.68        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % double_gauss_sum
% 0.35/0.68  thf(fact_6396_double__gauss__sum,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.68        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % double_gauss_sum
% 0.35/0.68  thf(fact_6397_double__arith__series,axiom,
% 0.35/0.68      ! [A: complex,D: complex,N: nat] :
% 0.35/0.68        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 0.35/0.68          @ ( groups2073611262835488442omplex
% 0.35/0.68            @ ^ [I4: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I4 ) @ D ) )
% 0.35/0.68            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.68        = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ D ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % double_arith_series
% 0.35/0.68  thf(fact_6398_double__arith__series,axiom,
% 0.35/0.68      ! [A: extended_enat,D: extended_enat,N: nat] :
% 0.35/0.68        ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) )
% 0.35/0.68          @ ( groups7108830773950497114d_enat
% 0.35/0.68            @ ^ [I4: nat] : ( plus_p3455044024723400733d_enat @ A @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ I4 ) @ D ) )
% 0.35/0.68            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.68        = ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A ) @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ D ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % double_arith_series
% 0.35/0.68  thf(fact_6399_double__arith__series,axiom,
% 0.35/0.68      ! [A: rat,D: rat,N: nat] :
% 0.35/0.68        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 0.35/0.68          @ ( groups2906978787729119204at_rat
% 0.35/0.68            @ ^ [I4: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I4 ) @ D ) )
% 0.35/0.68            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.68        = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ D ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % double_arith_series
% 0.35/0.68  thf(fact_6400_double__arith__series,axiom,
% 0.35/0.68      ! [A: int,D: int,N: nat] :
% 0.35/0.68        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 0.35/0.68          @ ( groups3539618377306564664at_int
% 0.35/0.68            @ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
% 0.35/0.68            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.68        = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % double_arith_series
% 0.35/0.68  thf(fact_6401_double__arith__series,axiom,
% 0.35/0.68      ! [A: nat,D: nat,N: nat] :
% 0.35/0.68        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
% 0.35/0.68          @ ( groups3542108847815614940at_nat
% 0.35/0.68            @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
% 0.35/0.68            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.68        = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % double_arith_series
% 0.35/0.68  thf(fact_6402_double__arith__series,axiom,
% 0.35/0.68      ! [A: real,D: real,N: nat] :
% 0.35/0.68        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.35/0.68          @ ( groups6591440286371151544t_real
% 0.35/0.68            @ ^ [I4: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I4 ) @ D ) )
% 0.35/0.68            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.68        = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ D ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % double_arith_series
% 0.35/0.68  thf(fact_6403_of__nat__eq__iff,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( ( semiri1314217659103216013at_int @ M )
% 0.35/0.68          = ( semiri1314217659103216013at_int @ N ) )
% 0.35/0.68        = ( M = N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_eq_iff
% 0.35/0.68  thf(fact_6404_of__nat__eq__iff,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( ( semiri5074537144036343181t_real @ M )
% 0.35/0.68          = ( semiri5074537144036343181t_real @ N ) )
% 0.35/0.68        = ( M = N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_eq_iff
% 0.35/0.68  thf(fact_6405_negative__eq__positive,axiom,
% 0.35/0.68      ! [N: nat,M: nat] :
% 0.35/0.68        ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.35/0.68          = ( semiri1314217659103216013at_int @ M ) )
% 0.35/0.68        = ( ( N = zero_zero_nat )
% 0.35/0.68          & ( M = zero_zero_nat ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % negative_eq_positive
% 0.35/0.68  thf(fact_6406_xor__negative__int__iff,axiom,
% 0.35/0.68      ! [K: int,L: int] :
% 0.35/0.68        ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
% 0.35/0.68        = ( ( ord_less_int @ K @ zero_zero_int )
% 0.35/0.68         != ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % xor_negative_int_iff
% 0.35/0.68  thf(fact_6407_of__nat__0,axiom,
% 0.35/0.68      ( ( semiri8010041392384452111omplex @ zero_zero_nat )
% 0.35/0.68      = zero_zero_complex ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_0
% 0.35/0.68  thf(fact_6408_of__nat__0,axiom,
% 0.35/0.68      ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 0.35/0.68      = zero_zero_rat ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_0
% 0.35/0.68  thf(fact_6409_of__nat__0,axiom,
% 0.35/0.68      ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 0.35/0.68      = zero_zero_nat ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_0
% 0.35/0.68  thf(fact_6410_of__nat__0,axiom,
% 0.35/0.68      ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 0.35/0.68      = zero_zero_int ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_0
% 0.35/0.68  thf(fact_6411_of__nat__0,axiom,
% 0.35/0.68      ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 0.35/0.68      = zero_zero_real ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_0
% 0.35/0.68  thf(fact_6412_of__nat__0__eq__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( zero_zero_complex
% 0.35/0.68          = ( semiri8010041392384452111omplex @ N ) )
% 0.35/0.68        = ( zero_zero_nat = N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_0_eq_iff
% 0.35/0.68  thf(fact_6413_of__nat__0__eq__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( zero_zero_rat
% 0.35/0.68          = ( semiri681578069525770553at_rat @ N ) )
% 0.35/0.68        = ( zero_zero_nat = N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_0_eq_iff
% 0.35/0.68  thf(fact_6414_of__nat__0__eq__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( zero_zero_nat
% 0.35/0.68          = ( semiri1316708129612266289at_nat @ N ) )
% 0.35/0.68        = ( zero_zero_nat = N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_0_eq_iff
% 0.35/0.68  thf(fact_6415_of__nat__0__eq__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( zero_zero_int
% 0.35/0.68          = ( semiri1314217659103216013at_int @ N ) )
% 0.35/0.68        = ( zero_zero_nat = N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_0_eq_iff
% 0.35/0.68  thf(fact_6416_of__nat__0__eq__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( zero_zero_real
% 0.35/0.68          = ( semiri5074537144036343181t_real @ N ) )
% 0.35/0.68        = ( zero_zero_nat = N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_0_eq_iff
% 0.35/0.68  thf(fact_6417_of__nat__eq__0__iff,axiom,
% 0.35/0.68      ! [M: nat] :
% 0.35/0.68        ( ( ( semiri8010041392384452111omplex @ M )
% 0.35/0.68          = zero_zero_complex )
% 0.35/0.68        = ( M = zero_zero_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_eq_0_iff
% 0.35/0.68  thf(fact_6418_of__nat__eq__0__iff,axiom,
% 0.35/0.68      ! [M: nat] :
% 0.35/0.68        ( ( ( semiri681578069525770553at_rat @ M )
% 0.35/0.68          = zero_zero_rat )
% 0.35/0.68        = ( M = zero_zero_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_eq_0_iff
% 0.35/0.68  thf(fact_6419_of__nat__eq__0__iff,axiom,
% 0.35/0.68      ! [M: nat] :
% 0.35/0.68        ( ( ( semiri1316708129612266289at_nat @ M )
% 0.35/0.68          = zero_zero_nat )
% 0.35/0.68        = ( M = zero_zero_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_eq_0_iff
% 0.35/0.68  thf(fact_6420_of__nat__eq__0__iff,axiom,
% 0.35/0.68      ! [M: nat] :
% 0.35/0.68        ( ( ( semiri1314217659103216013at_int @ M )
% 0.35/0.68          = zero_zero_int )
% 0.35/0.68        = ( M = zero_zero_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_eq_0_iff
% 0.35/0.68  thf(fact_6421_of__nat__eq__0__iff,axiom,
% 0.35/0.68      ! [M: nat] :
% 0.35/0.68        ( ( ( semiri5074537144036343181t_real @ M )
% 0.35/0.68          = zero_zero_real )
% 0.35/0.68        = ( M = zero_zero_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_eq_0_iff
% 0.35/0.68  thf(fact_6422_of__nat__less__iff,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.35/0.68        = ( ord_less_nat @ M @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_iff
% 0.35/0.68  thf(fact_6423_of__nat__less__iff,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.35/0.68        = ( ord_less_nat @ M @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_iff
% 0.35/0.68  thf(fact_6424_of__nat__less__iff,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.35/0.68        = ( ord_less_nat @ M @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_iff
% 0.35/0.68  thf(fact_6425_of__nat__less__iff,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.35/0.68        = ( ord_less_nat @ M @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_iff
% 0.35/0.68  thf(fact_6426_of__nat__le__iff,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.35/0.68        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_le_iff
% 0.35/0.68  thf(fact_6427_of__nat__le__iff,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.35/0.68        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_le_iff
% 0.35/0.68  thf(fact_6428_of__nat__le__iff,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.35/0.68        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_le_iff
% 0.35/0.68  thf(fact_6429_of__nat__le__iff,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.35/0.68        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_le_iff
% 0.35/0.68  thf(fact_6430_numeral__less__real__of__nat__iff,axiom,
% 0.35/0.68      ! [W2: num,N: nat] :
% 0.35/0.68        ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.35/0.68        = ( ord_less_nat @ ( numeral_numeral_nat @ W2 ) @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % numeral_less_real_of_nat_iff
% 0.35/0.68  thf(fact_6431_real__of__nat__less__numeral__iff,axiom,
% 0.35/0.68      ! [N: nat,W2: num] :
% 0.35/0.68        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W2 ) )
% 0.35/0.68        = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W2 ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % real_of_nat_less_numeral_iff
% 0.35/0.68  thf(fact_6432_of__nat__add,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
% 0.35/0.68        = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_add
% 0.35/0.68  thf(fact_6433_of__nat__add,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
% 0.35/0.68        = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_add
% 0.35/0.68  thf(fact_6434_of__nat__add,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
% 0.35/0.68        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_add
% 0.35/0.68  thf(fact_6435_of__nat__add,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
% 0.35/0.68        = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_add
% 0.35/0.68  thf(fact_6436_numeral__le__real__of__nat__iff,axiom,
% 0.35/0.68      ! [N: num,M: nat] :
% 0.35/0.68        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
% 0.35/0.68        = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% 0.35/0.68  
% 0.35/0.68  % numeral_le_real_of_nat_iff
% 0.35/0.68  thf(fact_6437_of__nat__mult,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
% 0.35/0.68        = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_mult
% 0.35/0.68  thf(fact_6438_of__nat__mult,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
% 0.35/0.68        = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_mult
% 0.35/0.68  thf(fact_6439_of__nat__mult,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
% 0.35/0.68        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_mult
% 0.35/0.68  thf(fact_6440_of__nat__mult,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
% 0.35/0.68        = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_mult
% 0.35/0.68  thf(fact_6441_of__nat__mult,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
% 0.35/0.68        = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_mult
% 0.35/0.68  thf(fact_6442_of__nat__eq__1__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( ( semiri8010041392384452111omplex @ N )
% 0.35/0.68          = one_one_complex )
% 0.35/0.68        = ( N = one_one_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_eq_1_iff
% 0.35/0.68  thf(fact_6443_of__nat__eq__1__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( ( semiri681578069525770553at_rat @ N )
% 0.35/0.68          = one_one_rat )
% 0.35/0.68        = ( N = one_one_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_eq_1_iff
% 0.35/0.68  thf(fact_6444_of__nat__eq__1__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( ( semiri1316708129612266289at_nat @ N )
% 0.35/0.68          = one_one_nat )
% 0.35/0.68        = ( N = one_one_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_eq_1_iff
% 0.35/0.68  thf(fact_6445_of__nat__eq__1__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( ( semiri1314217659103216013at_int @ N )
% 0.35/0.68          = one_one_int )
% 0.35/0.68        = ( N = one_one_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_eq_1_iff
% 0.35/0.68  thf(fact_6446_of__nat__eq__1__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( ( semiri5074537144036343181t_real @ N )
% 0.35/0.68          = one_one_real )
% 0.35/0.68        = ( N = one_one_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_eq_1_iff
% 0.35/0.68  thf(fact_6447_of__nat__1__eq__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( one_one_complex
% 0.35/0.68          = ( semiri8010041392384452111omplex @ N ) )
% 0.35/0.68        = ( N = one_one_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_1_eq_iff
% 0.35/0.68  thf(fact_6448_of__nat__1__eq__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( one_one_rat
% 0.35/0.68          = ( semiri681578069525770553at_rat @ N ) )
% 0.35/0.68        = ( N = one_one_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_1_eq_iff
% 0.35/0.68  thf(fact_6449_of__nat__1__eq__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( one_one_nat
% 0.35/0.68          = ( semiri1316708129612266289at_nat @ N ) )
% 0.35/0.68        = ( N = one_one_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_1_eq_iff
% 0.35/0.68  thf(fact_6450_of__nat__1__eq__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( one_one_int
% 0.35/0.68          = ( semiri1314217659103216013at_int @ N ) )
% 0.35/0.68        = ( N = one_one_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_1_eq_iff
% 0.35/0.68  thf(fact_6451_of__nat__1__eq__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( one_one_real
% 0.35/0.68          = ( semiri5074537144036343181t_real @ N ) )
% 0.35/0.68        = ( N = one_one_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_1_eq_iff
% 0.35/0.68  thf(fact_6452_of__nat__1,axiom,
% 0.35/0.68      ( ( semiri8010041392384452111omplex @ one_one_nat )
% 0.35/0.68      = one_one_complex ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_1
% 0.35/0.68  thf(fact_6453_of__nat__1,axiom,
% 0.35/0.68      ( ( semiri681578069525770553at_rat @ one_one_nat )
% 0.35/0.68      = one_one_rat ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_1
% 0.35/0.68  thf(fact_6454_of__nat__1,axiom,
% 0.35/0.68      ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 0.35/0.68      = one_one_nat ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_1
% 0.35/0.68  thf(fact_6455_of__nat__1,axiom,
% 0.35/0.68      ( ( semiri1314217659103216013at_int @ one_one_nat )
% 0.35/0.68      = one_one_int ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_1
% 0.35/0.68  thf(fact_6456_of__nat__1,axiom,
% 0.35/0.68      ( ( semiri5074537144036343181t_real @ one_one_nat )
% 0.35/0.68      = one_one_real ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_1
% 0.35/0.68  thf(fact_6457_negative__zless,axiom,
% 0.35/0.68      ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 0.35/0.68  
% 0.35/0.68  % negative_zless
% 0.35/0.68  thf(fact_6458_of__nat__of__bool,axiom,
% 0.35/0.68      ! [P: $o] :
% 0.35/0.68        ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.35/0.68        = ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_of_bool
% 0.35/0.68  thf(fact_6459_of__nat__of__bool,axiom,
% 0.35/0.68      ! [P: $o] :
% 0.35/0.68        ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.35/0.68        = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_of_bool
% 0.35/0.68  thf(fact_6460_of__nat__of__bool,axiom,
% 0.35/0.68      ! [P: $o] :
% 0.35/0.68        ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.35/0.68        = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_of_bool
% 0.35/0.68  thf(fact_6461_of__nat__le__0__iff,axiom,
% 0.35/0.68      ! [M: nat] :
% 0.35/0.68        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 0.35/0.68        = ( M = zero_zero_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_le_0_iff
% 0.35/0.68  thf(fact_6462_of__nat__le__0__iff,axiom,
% 0.35/0.68      ! [M: nat] :
% 0.35/0.68        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 0.35/0.68        = ( M = zero_zero_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_le_0_iff
% 0.35/0.68  thf(fact_6463_of__nat__le__0__iff,axiom,
% 0.35/0.68      ! [M: nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 0.35/0.68        = ( M = zero_zero_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_le_0_iff
% 0.35/0.68  thf(fact_6464_of__nat__le__0__iff,axiom,
% 0.35/0.68      ! [M: nat] :
% 0.35/0.68        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 0.35/0.68        = ( M = zero_zero_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_le_0_iff
% 0.35/0.68  thf(fact_6465_of__nat__Suc,axiom,
% 0.35/0.68      ! [M: nat] :
% 0.35/0.68        ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 0.35/0.68        = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_Suc
% 0.35/0.68  thf(fact_6466_of__nat__Suc,axiom,
% 0.35/0.68      ! [M: nat] :
% 0.35/0.68        ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 0.35/0.68        = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_Suc
% 0.35/0.68  thf(fact_6467_of__nat__Suc,axiom,
% 0.35/0.68      ! [M: nat] :
% 0.35/0.68        ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 0.35/0.68        = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_Suc
% 0.35/0.68  thf(fact_6468_of__nat__Suc,axiom,
% 0.35/0.68      ! [M: nat] :
% 0.35/0.68        ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 0.35/0.68        = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_Suc
% 0.35/0.68  thf(fact_6469_of__nat__Suc,axiom,
% 0.35/0.68      ! [M: nat] :
% 0.35/0.68        ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 0.35/0.68        = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_Suc
% 0.35/0.68  thf(fact_6470_of__nat__0__less__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.35/0.68        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_0_less_iff
% 0.35/0.68  thf(fact_6471_of__nat__0__less__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
% 0.35/0.68        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_0_less_iff
% 0.35/0.68  thf(fact_6472_of__nat__0__less__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.35/0.68        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_0_less_iff
% 0.35/0.68  thf(fact_6473_of__nat__0__less__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.35/0.68        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_0_less_iff
% 0.35/0.68  thf(fact_6474_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.35/0.68      ! [B: nat,W2: nat,X: nat] :
% 0.35/0.68        ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.35/0.68        = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_of_nat_power_cancel_iff
% 0.35/0.68  thf(fact_6475_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.35/0.68      ! [B: nat,W2: nat,X: nat] :
% 0.35/0.68        ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.35/0.68        = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_of_nat_power_cancel_iff
% 0.35/0.68  thf(fact_6476_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.35/0.68      ! [B: nat,W2: nat,X: nat] :
% 0.35/0.68        ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.35/0.68        = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_of_nat_power_cancel_iff
% 0.35/0.68  thf(fact_6477_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.35/0.68      ! [B: nat,W2: nat,X: nat] :
% 0.35/0.68        ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.35/0.68        = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_of_nat_power_cancel_iff
% 0.35/0.68  thf(fact_6478_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.35/0.68      ! [X: nat,B: nat,W2: nat] :
% 0.35/0.68        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
% 0.35/0.68        = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_power_less_of_nat_cancel_iff
% 0.35/0.68  thf(fact_6479_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.35/0.68      ! [X: nat,B: nat,W2: nat] :
% 0.35/0.68        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
% 0.35/0.68        = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_power_less_of_nat_cancel_iff
% 0.35/0.68  thf(fact_6480_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.35/0.68      ! [X: nat,B: nat,W2: nat] :
% 0.35/0.68        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
% 0.35/0.68        = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_power_less_of_nat_cancel_iff
% 0.35/0.68  thf(fact_6481_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.35/0.68      ! [X: nat,B: nat,W2: nat] :
% 0.35/0.68        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
% 0.35/0.68        = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_power_less_of_nat_cancel_iff
% 0.35/0.68  thf(fact_6482_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.35/0.68      ! [B: nat,W2: nat,X: nat] :
% 0.35/0.68        ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.35/0.68        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_le_of_nat_power_cancel_iff
% 0.35/0.68  thf(fact_6483_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.35/0.68      ! [B: nat,W2: nat,X: nat] :
% 0.35/0.68        ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.35/0.68        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_le_of_nat_power_cancel_iff
% 0.35/0.68  thf(fact_6484_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.35/0.68      ! [B: nat,W2: nat,X: nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.35/0.68        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_le_of_nat_power_cancel_iff
% 0.35/0.68  thf(fact_6485_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.35/0.68      ! [B: nat,W2: nat,X: nat] :
% 0.35/0.68        ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.35/0.68        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_le_of_nat_power_cancel_iff
% 0.35/0.68  thf(fact_6486_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.35/0.68      ! [X: nat,B: nat,W2: nat] :
% 0.35/0.68        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
% 0.35/0.68        = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_power_le_of_nat_cancel_iff
% 0.35/0.68  thf(fact_6487_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.35/0.68      ! [X: nat,B: nat,W2: nat] :
% 0.35/0.68        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
% 0.35/0.68        = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_power_le_of_nat_cancel_iff
% 0.35/0.68  thf(fact_6488_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.35/0.68      ! [X: nat,B: nat,W2: nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
% 0.35/0.68        = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_power_le_of_nat_cancel_iff
% 0.35/0.68  thf(fact_6489_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.35/0.68      ! [X: nat,B: nat,W2: nat] :
% 0.35/0.68        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
% 0.35/0.68        = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_power_le_of_nat_cancel_iff
% 0.35/0.68  thf(fact_6490_of__nat__zero__less__power__iff,axiom,
% 0.35/0.68      ! [X: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
% 0.35/0.68        = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.35/0.68          | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_zero_less_power_iff
% 0.35/0.68  thf(fact_6491_of__nat__zero__less__power__iff,axiom,
% 0.35/0.68      ! [X: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
% 0.35/0.68        = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.35/0.68          | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_zero_less_power_iff
% 0.35/0.68  thf(fact_6492_of__nat__zero__less__power__iff,axiom,
% 0.35/0.68      ! [X: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
% 0.35/0.68        = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.35/0.68          | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_zero_less_power_iff
% 0.35/0.68  thf(fact_6493_of__nat__zero__less__power__iff,axiom,
% 0.35/0.68      ! [X: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
% 0.35/0.68        = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.35/0.68          | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_zero_less_power_iff
% 0.35/0.68  thf(fact_6494_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.35/0.68      ! [X: nat,I: num,N: nat] :
% 0.35/0.68        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 0.35/0.68        = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_numeral_power_cancel_iff
% 0.35/0.68  thf(fact_6495_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.35/0.68      ! [X: nat,I: num,N: nat] :
% 0.35/0.68        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 0.35/0.68        = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_numeral_power_cancel_iff
% 0.35/0.68  thf(fact_6496_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.35/0.68      ! [X: nat,I: num,N: nat] :
% 0.35/0.68        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 0.35/0.68        = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_numeral_power_cancel_iff
% 0.35/0.68  thf(fact_6497_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.35/0.68      ! [X: nat,I: num,N: nat] :
% 0.35/0.68        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 0.35/0.68        = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_numeral_power_cancel_iff
% 0.35/0.68  thf(fact_6498_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.35/0.68      ! [I: num,N: nat,X: nat] :
% 0.35/0.68        ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.35/0.68        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.35/0.68  
% 0.35/0.68  % numeral_power_less_of_nat_cancel_iff
% 0.35/0.68  thf(fact_6499_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.35/0.68      ! [I: num,N: nat,X: nat] :
% 0.35/0.68        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.35/0.68        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.35/0.68  
% 0.35/0.68  % numeral_power_less_of_nat_cancel_iff
% 0.35/0.68  thf(fact_6500_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.35/0.68      ! [I: num,N: nat,X: nat] :
% 0.35/0.68        ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.35/0.68        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.35/0.68  
% 0.35/0.68  % numeral_power_less_of_nat_cancel_iff
% 0.35/0.68  thf(fact_6501_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.35/0.68      ! [I: num,N: nat,X: nat] :
% 0.35/0.68        ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.35/0.68        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.35/0.68  
% 0.35/0.68  % numeral_power_less_of_nat_cancel_iff
% 0.35/0.68  thf(fact_6502_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.35/0.68      ! [I: num,N: nat,X: nat] :
% 0.35/0.68        ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.35/0.68        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.35/0.68  
% 0.35/0.68  % numeral_power_le_of_nat_cancel_iff
% 0.35/0.68  thf(fact_6503_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.35/0.68      ! [I: num,N: nat,X: nat] :
% 0.35/0.68        ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.35/0.68        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.35/0.68  
% 0.35/0.68  % numeral_power_le_of_nat_cancel_iff
% 0.35/0.68  thf(fact_6504_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.35/0.68      ! [I: num,N: nat,X: nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.35/0.68        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.35/0.68  
% 0.35/0.68  % numeral_power_le_of_nat_cancel_iff
% 0.35/0.68  thf(fact_6505_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.35/0.68      ! [I: num,N: nat,X: nat] :
% 0.35/0.68        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.35/0.68        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.35/0.68  
% 0.35/0.68  % numeral_power_le_of_nat_cancel_iff
% 0.35/0.68  thf(fact_6506_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.35/0.68      ! [X: nat,I: num,N: nat] :
% 0.35/0.68        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 0.35/0.68        = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_le_numeral_power_cancel_iff
% 0.35/0.68  thf(fact_6507_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.35/0.68      ! [X: nat,I: num,N: nat] :
% 0.35/0.68        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 0.35/0.68        = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_le_numeral_power_cancel_iff
% 0.35/0.68  thf(fact_6508_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.35/0.68      ! [X: nat,I: num,N: nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 0.35/0.68        = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_le_numeral_power_cancel_iff
% 0.35/0.68  thf(fact_6509_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.35/0.68      ! [X: nat,I: num,N: nat] :
% 0.35/0.68        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 0.35/0.68        = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_le_numeral_power_cancel_iff
% 0.35/0.68  thf(fact_6510_real__arch__simple,axiom,
% 0.35/0.68      ! [X: real] :
% 0.35/0.68      ? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% 0.35/0.68  
% 0.35/0.68  % real_arch_simple
% 0.35/0.68  thf(fact_6511_real__arch__simple,axiom,
% 0.35/0.68      ! [X: rat] :
% 0.35/0.68      ? [N3: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% 0.35/0.68  
% 0.35/0.68  % real_arch_simple
% 0.35/0.68  thf(fact_6512_reals__Archimedean2,axiom,
% 0.35/0.68      ! [X: rat] :
% 0.35/0.68      ? [N3: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% 0.35/0.68  
% 0.35/0.68  % reals_Archimedean2
% 0.35/0.68  thf(fact_6513_reals__Archimedean2,axiom,
% 0.35/0.68      ! [X: real] :
% 0.35/0.68      ? [N3: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% 0.35/0.68  
% 0.35/0.68  % reals_Archimedean2
% 0.35/0.68  thf(fact_6514_mult__of__nat__commute,axiom,
% 0.35/0.68      ! [X: nat,Y: complex] :
% 0.35/0.68        ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y )
% 0.35/0.68        = ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % mult_of_nat_commute
% 0.35/0.68  thf(fact_6515_mult__of__nat__commute,axiom,
% 0.35/0.68      ! [X: nat,Y: rat] :
% 0.35/0.68        ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
% 0.35/0.68        = ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % mult_of_nat_commute
% 0.35/0.68  thf(fact_6516_mult__of__nat__commute,axiom,
% 0.35/0.68      ! [X: nat,Y: nat] :
% 0.35/0.68        ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
% 0.35/0.68        = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % mult_of_nat_commute
% 0.35/0.68  thf(fact_6517_mult__of__nat__commute,axiom,
% 0.35/0.68      ! [X: nat,Y: int] :
% 0.35/0.68        ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
% 0.35/0.68        = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % mult_of_nat_commute
% 0.35/0.68  thf(fact_6518_mult__of__nat__commute,axiom,
% 0.35/0.68      ! [X: nat,Y: real] :
% 0.35/0.68        ( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
% 0.35/0.68        = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % mult_of_nat_commute
% 0.35/0.68  thf(fact_6519_of__nat__less__of__int__iff,axiom,
% 0.35/0.68      ! [N: nat,X: int] :
% 0.35/0.68        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X ) )
% 0.35/0.68        = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_of_int_iff
% 0.35/0.68  thf(fact_6520_of__nat__less__of__int__iff,axiom,
% 0.35/0.68      ! [N: nat,X: int] :
% 0.35/0.68        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
% 0.35/0.68        = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_of_int_iff
% 0.35/0.68  thf(fact_6521_of__nat__less__of__int__iff,axiom,
% 0.35/0.68      ! [N: nat,X: int] :
% 0.35/0.68        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
% 0.35/0.68        = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_of_int_iff
% 0.35/0.68  thf(fact_6522_of__nat__0__le__iff,axiom,
% 0.35/0.68      ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_0_le_iff
% 0.35/0.68  thf(fact_6523_of__nat__0__le__iff,axiom,
% 0.35/0.68      ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_0_le_iff
% 0.35/0.68  thf(fact_6524_of__nat__0__le__iff,axiom,
% 0.35/0.68      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_0_le_iff
% 0.35/0.68  thf(fact_6525_of__nat__0__le__iff,axiom,
% 0.35/0.68      ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_0_le_iff
% 0.35/0.68  thf(fact_6526_of__nat__less__0__iff,axiom,
% 0.35/0.68      ! [M: nat] :
% 0.35/0.68        ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_0_iff
% 0.35/0.68  thf(fact_6527_of__nat__less__0__iff,axiom,
% 0.35/0.68      ! [M: nat] :
% 0.35/0.68        ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_0_iff
% 0.35/0.68  thf(fact_6528_of__nat__less__0__iff,axiom,
% 0.35/0.68      ! [M: nat] :
% 0.35/0.68        ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_0_iff
% 0.35/0.68  thf(fact_6529_of__nat__less__0__iff,axiom,
% 0.35/0.68      ! [M: nat] :
% 0.35/0.68        ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_0_iff
% 0.35/0.68  thf(fact_6530_of__nat__neq__0,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
% 0.35/0.68       != zero_zero_complex ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_neq_0
% 0.35/0.68  thf(fact_6531_of__nat__neq__0,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
% 0.35/0.68       != zero_zero_rat ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_neq_0
% 0.35/0.68  thf(fact_6532_of__nat__neq__0,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
% 0.35/0.68       != zero_zero_nat ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_neq_0
% 0.35/0.68  thf(fact_6533_of__nat__neq__0,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 0.35/0.68       != zero_zero_int ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_neq_0
% 0.35/0.68  thf(fact_6534_of__nat__neq__0,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
% 0.35/0.68       != zero_zero_real ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_neq_0
% 0.35/0.68  thf(fact_6535_less__imp__of__nat__less,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_nat @ M @ N )
% 0.35/0.68       => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % less_imp_of_nat_less
% 0.35/0.68  thf(fact_6536_less__imp__of__nat__less,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_nat @ M @ N )
% 0.35/0.68       => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % less_imp_of_nat_less
% 0.35/0.68  thf(fact_6537_less__imp__of__nat__less,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_nat @ M @ N )
% 0.35/0.68       => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % less_imp_of_nat_less
% 0.35/0.68  thf(fact_6538_less__imp__of__nat__less,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_nat @ M @ N )
% 0.35/0.68       => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % less_imp_of_nat_less
% 0.35/0.68  thf(fact_6539_of__nat__less__imp__less,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.35/0.68       => ( ord_less_nat @ M @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_imp_less
% 0.35/0.68  thf(fact_6540_of__nat__less__imp__less,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.35/0.68       => ( ord_less_nat @ M @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_imp_less
% 0.35/0.68  thf(fact_6541_of__nat__less__imp__less,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.35/0.68       => ( ord_less_nat @ M @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_imp_less
% 0.35/0.68  thf(fact_6542_of__nat__less__imp__less,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.35/0.68       => ( ord_less_nat @ M @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_imp_less
% 0.35/0.68  thf(fact_6543_of__nat__mono,axiom,
% 0.35/0.68      ! [I: nat,J: nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.68       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_mono
% 0.35/0.68  thf(fact_6544_of__nat__mono,axiom,
% 0.35/0.68      ! [I: nat,J: nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.68       => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_mono
% 0.35/0.68  thf(fact_6545_of__nat__mono,axiom,
% 0.35/0.68      ! [I: nat,J: nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.68       => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_mono
% 0.35/0.68  thf(fact_6546_of__nat__mono,axiom,
% 0.35/0.68      ! [I: nat,J: nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.68       => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_mono
% 0.35/0.68  thf(fact_6547_int__ops_I1_J,axiom,
% 0.35/0.68      ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 0.35/0.68      = zero_zero_int ) ).
% 0.35/0.68  
% 0.35/0.68  % int_ops(1)
% 0.35/0.68  thf(fact_6548_nat__int__comparison_I2_J,axiom,
% 0.35/0.68      ( ord_less_nat
% 0.35/0.68      = ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % nat_int_comparison(2)
% 0.35/0.68  thf(fact_6549_nat__int__comparison_I3_J,axiom,
% 0.35/0.68      ( ord_less_eq_nat
% 0.35/0.68      = ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % nat_int_comparison(3)
% 0.35/0.68  thf(fact_6550_zle__int,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.35/0.68        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % zle_int
% 0.35/0.68  thf(fact_6551_not__int__zless__negative,axiom,
% 0.35/0.68      ! [N: nat,M: nat] :
% 0.35/0.68        ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % not_int_zless_negative
% 0.35/0.68  thf(fact_6552_of__nat__max,axiom,
% 0.35/0.68      ! [X: nat,Y: nat] :
% 0.35/0.68        ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y ) )
% 0.35/0.68        = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_max
% 0.35/0.68  thf(fact_6553_of__nat__max,axiom,
% 0.35/0.68      ! [X: nat,Y: nat] :
% 0.35/0.68        ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y ) )
% 0.35/0.68        = ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_max
% 0.35/0.68  thf(fact_6554_of__nat__max,axiom,
% 0.35/0.68      ! [X: nat,Y: nat] :
% 0.35/0.68        ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y ) )
% 0.35/0.68        = ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_max
% 0.35/0.68  thf(fact_6555_nat__less__as__int,axiom,
% 0.35/0.68      ( ord_less_nat
% 0.35/0.68      = ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % nat_less_as_int
% 0.35/0.68  thf(fact_6556_nat__leq__as__int,axiom,
% 0.35/0.68      ( ord_less_eq_nat
% 0.35/0.68      = ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % nat_leq_as_int
% 0.35/0.68  thf(fact_6557_ex__less__of__nat__mult,axiom,
% 0.35/0.68      ! [X: rat,Y: rat] :
% 0.35/0.68        ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.35/0.68       => ? [N3: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % ex_less_of_nat_mult
% 0.35/0.68  thf(fact_6558_ex__less__of__nat__mult,axiom,
% 0.35/0.68      ! [X: real,Y: real] :
% 0.35/0.68        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.68       => ? [N3: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % ex_less_of_nat_mult
% 0.35/0.68  thf(fact_6559_of__nat__diff,axiom,
% 0.35/0.68      ! [N: nat,M: nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.68       => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.68          = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_diff
% 0.35/0.68  thf(fact_6560_of__nat__diff,axiom,
% 0.35/0.68      ! [N: nat,M: nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.68       => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.68          = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_diff
% 0.35/0.68  thf(fact_6561_of__nat__diff,axiom,
% 0.35/0.68      ! [N: nat,M: nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.68       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.68          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_diff
% 0.35/0.68  thf(fact_6562_of__nat__diff,axiom,
% 0.35/0.68      ! [N: nat,M: nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.68       => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.68          = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_diff
% 0.35/0.68  thf(fact_6563_reals__Archimedean3,axiom,
% 0.35/0.68      ! [X: real] :
% 0.35/0.68        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.68       => ! [Y4: real] :
% 0.35/0.68          ? [N3: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % reals_Archimedean3
% 0.35/0.68  thf(fact_6564_int__cases4,axiom,
% 0.35/0.68      ! [M: int] :
% 0.35/0.68        ( ! [N3: nat] :
% 0.35/0.68            ( M
% 0.35/0.68           != ( semiri1314217659103216013at_int @ N3 ) )
% 0.35/0.68       => ~ ! [N3: nat] :
% 0.35/0.68              ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.35/0.68             => ( M
% 0.35/0.68               != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % int_cases4
% 0.35/0.68  thf(fact_6565_zless__iff__Suc__zadd,axiom,
% 0.35/0.68      ( ord_less_int
% 0.35/0.68      = ( ^ [W3: int,Z3: int] :
% 0.35/0.68          ? [N2: nat] :
% 0.35/0.68            ( Z3
% 0.35/0.68            = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % zless_iff_Suc_zadd
% 0.35/0.68  thf(fact_6566_int__zle__neg,axiom,
% 0.35/0.68      ! [N: nat,M: nat] :
% 0.35/0.68        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
% 0.35/0.68        = ( ( N = zero_zero_nat )
% 0.35/0.68          & ( M = zero_zero_nat ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % int_zle_neg
% 0.35/0.68  thf(fact_6567_sum__roots__unity,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( ord_less_nat @ one_one_nat @ N )
% 0.35/0.68       => ( ( groups7754918857620584856omplex
% 0.35/0.68            @ ^ [X3: complex] : X3
% 0.35/0.68            @ ( collect_complex
% 0.35/0.68              @ ^ [Z3: complex] :
% 0.35/0.68                  ( ( power_power_complex @ Z3 @ N )
% 0.35/0.68                  = one_one_complex ) ) )
% 0.35/0.68          = zero_zero_complex ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_roots_unity
% 0.35/0.68  thf(fact_6568_sum__nth__roots,axiom,
% 0.35/0.68      ! [N: nat,C: complex] :
% 0.35/0.68        ( ( ord_less_nat @ one_one_nat @ N )
% 0.35/0.68       => ( ( groups7754918857620584856omplex
% 0.35/0.68            @ ^ [X3: complex] : X3
% 0.35/0.68            @ ( collect_complex
% 0.35/0.68              @ ^ [Z3: complex] :
% 0.35/0.68                  ( ( power_power_complex @ Z3 @ N )
% 0.35/0.68                  = C ) ) )
% 0.35/0.68          = zero_zero_complex ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_nth_roots
% 0.35/0.68  thf(fact_6569_pos__int__cases,axiom,
% 0.35/0.68      ! [K: int] :
% 0.35/0.68        ( ( ord_less_int @ zero_zero_int @ K )
% 0.35/0.68       => ~ ! [N3: nat] :
% 0.35/0.68              ( ( K
% 0.35/0.68                = ( semiri1314217659103216013at_int @ N3 ) )
% 0.35/0.68             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % pos_int_cases
% 0.35/0.68  thf(fact_6570_zero__less__imp__eq__int,axiom,
% 0.35/0.68      ! [K: int] :
% 0.35/0.68        ( ( ord_less_int @ zero_zero_int @ K )
% 0.35/0.68       => ? [N3: nat] :
% 0.35/0.68            ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.35/0.68            & ( K
% 0.35/0.68              = ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % zero_less_imp_eq_int
% 0.35/0.68  thf(fact_6571_int__cases3,axiom,
% 0.35/0.68      ! [K: int] :
% 0.35/0.68        ( ( K != zero_zero_int )
% 0.35/0.68       => ( ! [N3: nat] :
% 0.35/0.68              ( ( K
% 0.35/0.68                = ( semiri1314217659103216013at_int @ N3 ) )
% 0.35/0.68             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
% 0.35/0.68         => ~ ! [N3: nat] :
% 0.35/0.68                ( ( K
% 0.35/0.68                  = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
% 0.35/0.68               => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % int_cases3
% 0.35/0.68  thf(fact_6572_nat__less__real__le,axiom,
% 0.35/0.68      ( ord_less_nat
% 0.35/0.68      = ( ^ [N2: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % nat_less_real_le
% 0.35/0.68  thf(fact_6573_nat__le__real__less,axiom,
% 0.35/0.68      ( ord_less_eq_nat
% 0.35/0.68      = ( ^ [N2: nat,M2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % nat_le_real_less
% 0.35/0.68  thf(fact_6574_zmult__zless__mono2__lemma,axiom,
% 0.35/0.68      ! [I: int,J: int,K: nat] :
% 0.35/0.68        ( ( ord_less_int @ I @ J )
% 0.35/0.68       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.68         => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % zmult_zless_mono2_lemma
% 0.35/0.68  thf(fact_6575_negD,axiom,
% 0.35/0.68      ! [X: int] :
% 0.35/0.68        ( ( ord_less_int @ X @ zero_zero_int )
% 0.35/0.68       => ? [N3: nat] :
% 0.35/0.68            ( X
% 0.35/0.68            = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % negD
% 0.35/0.68  thf(fact_6576_negative__zless__0,axiom,
% 0.35/0.68      ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% 0.35/0.68  
% 0.35/0.68  % negative_zless_0
% 0.35/0.68  thf(fact_6577_int__ops_I6_J,axiom,
% 0.35/0.68      ! [A: nat,B: nat] :
% 0.35/0.68        ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 0.35/0.68         => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 0.35/0.68            = zero_zero_int ) )
% 0.35/0.68        & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 0.35/0.68         => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 0.35/0.68            = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % int_ops(6)
% 0.35/0.68  thf(fact_6578_simp__from__to,axiom,
% 0.35/0.68      ( set_or1266510415728281911st_int
% 0.35/0.68      = ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % simp_from_to
% 0.35/0.68  thf(fact_6579_nat__approx__posE,axiom,
% 0.35/0.68      ! [E2: rat] :
% 0.35/0.68        ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.35/0.68       => ~ ! [N3: nat] :
% 0.35/0.68              ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ E2 ) ) ).
% 0.35/0.68  
% 0.35/0.68  % nat_approx_posE
% 0.35/0.68  thf(fact_6580_nat__approx__posE,axiom,
% 0.35/0.68      ! [E2: real] :
% 0.35/0.68        ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.35/0.68       => ~ ! [N3: nat] :
% 0.35/0.68              ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ E2 ) ) ).
% 0.35/0.68  
% 0.35/0.68  % nat_approx_posE
% 0.35/0.68  thf(fact_6581_of__nat__less__two__power,axiom,
% 0.35/0.68      ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_two_power
% 0.35/0.68  thf(fact_6582_of__nat__less__two__power,axiom,
% 0.35/0.68      ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_two_power
% 0.35/0.68  thf(fact_6583_of__nat__less__two__power,axiom,
% 0.35/0.68      ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_less_two_power
% 0.35/0.68  thf(fact_6584_inverse__of__nat__le,axiom,
% 0.35/0.68      ! [N: nat,M: nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.68       => ( ( N != zero_zero_nat )
% 0.35/0.68         => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % inverse_of_nat_le
% 0.35/0.68  thf(fact_6585_inverse__of__nat__le,axiom,
% 0.35/0.68      ! [N: nat,M: nat] :
% 0.35/0.68        ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.68       => ( ( N != zero_zero_nat )
% 0.35/0.68         => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % inverse_of_nat_le
% 0.35/0.68  thf(fact_6586_real__archimedian__rdiv__eq__0,axiom,
% 0.35/0.68      ! [X: real,C: real] :
% 0.35/0.68        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.68       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.68         => ( ! [M5: nat] :
% 0.35/0.68                ( ( ord_less_nat @ zero_zero_nat @ M5 )
% 0.35/0.68               => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ X ) @ C ) )
% 0.35/0.68           => ( X = zero_zero_real ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % real_archimedian_rdiv_eq_0
% 0.35/0.68  thf(fact_6587_neg__int__cases,axiom,
% 0.35/0.68      ! [K: int] :
% 0.35/0.68        ( ( ord_less_int @ K @ zero_zero_int )
% 0.35/0.68       => ~ ! [N3: nat] :
% 0.35/0.68              ( ( K
% 0.35/0.68                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
% 0.35/0.68             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % neg_int_cases
% 0.35/0.68  thf(fact_6588_zdiff__int__split,axiom,
% 0.35/0.68      ! [P: int > $o,X: nat,Y: nat] :
% 0.35/0.68        ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
% 0.35/0.68        = ( ( ( ord_less_eq_nat @ Y @ X )
% 0.35/0.68           => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
% 0.35/0.68          & ( ( ord_less_nat @ X @ Y )
% 0.35/0.68           => ( P @ zero_zero_int ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % zdiff_int_split
% 0.35/0.68  thf(fact_6589_XOR__upper,axiom,
% 0.35/0.68      ! [X: int,N: nat,Y: int] :
% 0.35/0.68        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.35/0.68       => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.68         => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.68           => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % XOR_upper
% 0.35/0.68  thf(fact_6590_of__nat__code__if,axiom,
% 0.35/0.68      ( semiri8010041392384452111omplex
% 0.35/0.68      = ( ^ [N2: nat] :
% 0.35/0.68            ( if_complex @ ( N2 = zero_zero_nat ) @ zero_zero_complex
% 0.35/0.68            @ ( produc1917071388513777916omplex
% 0.35/0.68              @ ^ [M2: nat,Q5: nat] : ( if_complex @ ( Q5 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M2 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M2 ) ) @ one_one_complex ) )
% 0.35/0.68              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_code_if
% 0.35/0.68  thf(fact_6591_of__nat__code__if,axiom,
% 0.35/0.68      ( semiri4216267220026989637d_enat
% 0.35/0.68      = ( ^ [N2: nat] :
% 0.35/0.68            ( if_Extended_enat @ ( N2 = zero_zero_nat ) @ zero_z5237406670263579293d_enat
% 0.35/0.68            @ ( produc2676513652042109336d_enat
% 0.35/0.68              @ ^ [M2: nat,Q5: nat] : ( if_Extended_enat @ ( Q5 = zero_zero_nat ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( semiri4216267220026989637d_enat @ M2 ) ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( semiri4216267220026989637d_enat @ M2 ) ) @ one_on7984719198319812577d_enat ) )
% 0.35/0.68              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_code_if
% 0.35/0.68  thf(fact_6592_of__nat__code__if,axiom,
% 0.35/0.68      ( semiri681578069525770553at_rat
% 0.35/0.68      = ( ^ [N2: nat] :
% 0.35/0.68            ( if_rat @ ( N2 = zero_zero_nat ) @ zero_zero_rat
% 0.35/0.68            @ ( produc6207742614233964070at_rat
% 0.35/0.68              @ ^ [M2: nat,Q5: nat] : ( if_rat @ ( Q5 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M2 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M2 ) ) @ one_one_rat ) )
% 0.35/0.68              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_code_if
% 0.35/0.68  thf(fact_6593_of__nat__code__if,axiom,
% 0.35/0.68      ( semiri1316708129612266289at_nat
% 0.35/0.68      = ( ^ [N2: nat] :
% 0.35/0.68            ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
% 0.35/0.68            @ ( produc6842872674320459806at_nat
% 0.35/0.68              @ ^ [M2: nat,Q5: nat] : ( if_nat @ ( Q5 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M2 ) ) @ one_one_nat ) )
% 0.35/0.68              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_code_if
% 0.35/0.68  thf(fact_6594_of__nat__code__if,axiom,
% 0.35/0.68      ( semiri1314217659103216013at_int
% 0.35/0.68      = ( ^ [N2: nat] :
% 0.35/0.68            ( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int
% 0.35/0.68            @ ( produc6840382203811409530at_int
% 0.35/0.68              @ ^ [M2: nat,Q5: nat] : ( if_int @ ( Q5 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ one_one_int ) )
% 0.35/0.68              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_code_if
% 0.35/0.68  thf(fact_6595_of__nat__code__if,axiom,
% 0.35/0.68      ( semiri5074537144036343181t_real
% 0.35/0.68      = ( ^ [N2: nat] :
% 0.35/0.68            ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
% 0.35/0.68            @ ( produc1703576794950452218t_real
% 0.35/0.68              @ ^ [M2: nat,Q5: nat] : ( if_real @ ( Q5 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ one_one_real ) )
% 0.35/0.68              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_code_if
% 0.35/0.68  thf(fact_6596_lemma__termdiff3,axiom,
% 0.35/0.68      ! [H2: real,Z2: real,K4: real,N: nat] :
% 0.35/0.68        ( ( H2 != zero_zero_real )
% 0.35/0.68       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ K4 )
% 0.35/0.68         => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z2 @ H2 ) ) @ K4 )
% 0.35/0.68           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ N ) @ ( power_power_real @ Z2 @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K4 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lemma_termdiff3
% 0.35/0.68  thf(fact_6597_lemma__termdiff3,axiom,
% 0.35/0.68      ! [H2: complex,Z2: complex,K4: real,N: nat] :
% 0.35/0.68        ( ( H2 != zero_zero_complex )
% 0.35/0.68       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ K4 )
% 0.35/0.68         => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z2 @ H2 ) ) @ K4 )
% 0.35/0.68           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ N ) @ ( power_power_complex @ Z2 @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K4 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lemma_termdiff3
% 0.35/0.68  thf(fact_6598_lemma__termdiff2,axiom,
% 0.35/0.68      ! [H2: complex,Z2: complex,N: nat] :
% 0.35/0.68        ( ( H2 != zero_zero_complex )
% 0.35/0.68       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ N ) @ ( power_power_complex @ Z2 @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.35/0.68          = ( times_times_complex @ H2
% 0.35/0.68            @ ( groups2073611262835488442omplex
% 0.35/0.68              @ ^ [P6: nat] :
% 0.35/0.68                  ( groups2073611262835488442omplex
% 0.35/0.68                  @ ^ [Q5: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ Q5 ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
% 0.35/0.68                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
% 0.35/0.68              @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lemma_termdiff2
% 0.35/0.68  thf(fact_6599_lemma__termdiff2,axiom,
% 0.35/0.68      ! [H2: rat,Z2: rat,N: nat] :
% 0.35/0.68        ( ( H2 != zero_zero_rat )
% 0.35/0.68       => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ N ) @ ( power_power_rat @ Z2 @ N ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.35/0.68          = ( times_times_rat @ H2
% 0.35/0.68            @ ( groups2906978787729119204at_rat
% 0.35/0.68              @ ^ [P6: nat] :
% 0.35/0.68                  ( groups2906978787729119204at_rat
% 0.35/0.68                  @ ^ [Q5: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ Q5 ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
% 0.35/0.68                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
% 0.35/0.68              @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lemma_termdiff2
% 0.35/0.68  thf(fact_6600_lemma__termdiff2,axiom,
% 0.35/0.68      ! [H2: real,Z2: real,N: nat] :
% 0.35/0.68        ( ( H2 != zero_zero_real )
% 0.35/0.68       => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ N ) @ ( power_power_real @ Z2 @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.35/0.68          = ( times_times_real @ H2
% 0.35/0.68            @ ( groups6591440286371151544t_real
% 0.35/0.68              @ ^ [P6: nat] :
% 0.35/0.68                  ( groups6591440286371151544t_real
% 0.35/0.68                  @ ^ [Q5: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ Q5 ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
% 0.35/0.68                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
% 0.35/0.68              @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lemma_termdiff2
% 0.35/0.68  thf(fact_6601_of__nat__code,axiom,
% 0.35/0.68      ( semiri8010041392384452111omplex
% 0.35/0.68      = ( ^ [N2: nat] :
% 0.35/0.68            ( semiri2816024913162550771omplex
% 0.35/0.68            @ ^ [I4: complex] : ( plus_plus_complex @ I4 @ one_one_complex )
% 0.35/0.68            @ N2
% 0.35/0.68            @ zero_zero_complex ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_code
% 0.35/0.68  thf(fact_6602_of__nat__code,axiom,
% 0.35/0.68      ( semiri681578069525770553at_rat
% 0.35/0.68      = ( ^ [N2: nat] :
% 0.35/0.68            ( semiri7787848453975740701ux_rat
% 0.35/0.68            @ ^ [I4: rat] : ( plus_plus_rat @ I4 @ one_one_rat )
% 0.35/0.68            @ N2
% 0.35/0.68            @ zero_zero_rat ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_code
% 0.35/0.68  thf(fact_6603_of__nat__code,axiom,
% 0.35/0.68      ( semiri1316708129612266289at_nat
% 0.35/0.68      = ( ^ [N2: nat] :
% 0.35/0.68            ( semiri8422978514062236437ux_nat
% 0.35/0.68            @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ one_one_nat )
% 0.35/0.68            @ N2
% 0.35/0.68            @ zero_zero_nat ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_code
% 0.35/0.68  thf(fact_6604_of__nat__code,axiom,
% 0.35/0.68      ( semiri1314217659103216013at_int
% 0.35/0.68      = ( ^ [N2: nat] :
% 0.35/0.68            ( semiri8420488043553186161ux_int
% 0.35/0.68            @ ^ [I4: int] : ( plus_plus_int @ I4 @ one_one_int )
% 0.35/0.68            @ N2
% 0.35/0.68            @ zero_zero_int ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_code
% 0.35/0.68  thf(fact_6605_of__nat__code,axiom,
% 0.35/0.68      ( semiri5074537144036343181t_real
% 0.35/0.68      = ( ^ [N2: nat] :
% 0.35/0.68            ( semiri7260567687927622513x_real
% 0.35/0.68            @ ^ [I4: real] : ( plus_plus_real @ I4 @ one_one_real )
% 0.35/0.68            @ N2
% 0.35/0.68            @ zero_zero_real ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % of_nat_code
% 0.35/0.68  thf(fact_6606_and__int_Oelims,axiom,
% 0.35/0.68      ! [X: int,Xa2: int,Y: int] :
% 0.35/0.68        ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
% 0.35/0.68          = Y )
% 0.35/0.68       => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.35/0.68              & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.35/0.68           => ( Y
% 0.35/0.68              = ( uminus_uminus_int
% 0.35/0.68                @ ( zero_n2684676970156552555ol_int
% 0.35/0.68                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.35/0.68                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 0.35/0.68          & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.35/0.68                & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.35/0.68           => ( Y
% 0.35/0.68              = ( plus_plus_int
% 0.35/0.68                @ ( zero_n2684676970156552555ol_int
% 0.35/0.68                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.35/0.68                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 0.35/0.68                @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % and_int.elims
% 0.35/0.68  thf(fact_6607_and__int_Osimps,axiom,
% 0.35/0.68      ( bit_se725231765392027082nd_int
% 0.35/0.68      = ( ^ [K3: int,L2: int] :
% 0.35/0.68            ( if_int
% 0.35/0.68            @ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.35/0.68              & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.35/0.68            @ ( uminus_uminus_int
% 0.35/0.68              @ ( zero_n2684676970156552555ol_int
% 0.35/0.68                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 0.35/0.68                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 0.35/0.68            @ ( plus_plus_int
% 0.35/0.68              @ ( zero_n2684676970156552555ol_int
% 0.35/0.68                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 0.35/0.68                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 0.35/0.68              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % and_int.simps
% 0.35/0.68  thf(fact_6608_lessThan__iff,axiom,
% 0.35/0.68      ! [I: set_nat,K: set_nat] :
% 0.35/0.68        ( ( member_set_nat @ I @ ( set_or890127255671739683et_nat @ K ) )
% 0.35/0.68        = ( ord_less_set_nat @ I @ K ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_iff
% 0.35/0.68  thf(fact_6609_lessThan__iff,axiom,
% 0.35/0.68      ! [I: rat,K: rat] :
% 0.35/0.68        ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
% 0.35/0.68        = ( ord_less_rat @ I @ K ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_iff
% 0.35/0.68  thf(fact_6610_lessThan__iff,axiom,
% 0.35/0.68      ! [I: num,K: num] :
% 0.35/0.68        ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
% 0.35/0.68        = ( ord_less_num @ I @ K ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_iff
% 0.35/0.68  thf(fact_6611_lessThan__iff,axiom,
% 0.35/0.68      ! [I: nat,K: nat] :
% 0.35/0.68        ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
% 0.35/0.68        = ( ord_less_nat @ I @ K ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_iff
% 0.35/0.68  thf(fact_6612_lessThan__iff,axiom,
% 0.35/0.68      ! [I: int,K: int] :
% 0.35/0.68        ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
% 0.35/0.68        = ( ord_less_int @ I @ K ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_iff
% 0.35/0.68  thf(fact_6613_lessThan__iff,axiom,
% 0.35/0.68      ! [I: real,K: real] :
% 0.35/0.68        ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
% 0.35/0.68        = ( ord_less_real @ I @ K ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_iff
% 0.35/0.68  thf(fact_6614_bit_Oconj__zero__right,axiom,
% 0.35/0.68      ! [X: int] :
% 0.35/0.68        ( ( bit_se725231765392027082nd_int @ X @ zero_zero_int )
% 0.35/0.68        = zero_zero_int ) ).
% 0.35/0.68  
% 0.35/0.68  % bit.conj_zero_right
% 0.35/0.68  thf(fact_6615_bit_Oconj__zero__left,axiom,
% 0.35/0.68      ! [X: int] :
% 0.35/0.68        ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X )
% 0.35/0.68        = zero_zero_int ) ).
% 0.35/0.68  
% 0.35/0.68  % bit.conj_zero_left
% 0.35/0.68  thf(fact_6616_zero__and__eq,axiom,
% 0.35/0.68      ! [A: int] :
% 0.35/0.68        ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
% 0.35/0.68        = zero_zero_int ) ).
% 0.35/0.68  
% 0.35/0.68  % zero_and_eq
% 0.35/0.68  thf(fact_6617_zero__and__eq,axiom,
% 0.35/0.68      ! [A: nat] :
% 0.35/0.68        ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
% 0.35/0.68        = zero_zero_nat ) ).
% 0.35/0.68  
% 0.35/0.68  % zero_and_eq
% 0.35/0.68  thf(fact_6618_and__zero__eq,axiom,
% 0.35/0.68      ! [A: int] :
% 0.35/0.68        ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
% 0.35/0.68        = zero_zero_int ) ).
% 0.35/0.68  
% 0.35/0.68  % and_zero_eq
% 0.35/0.68  thf(fact_6619_and__zero__eq,axiom,
% 0.35/0.68      ! [A: nat] :
% 0.35/0.68        ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
% 0.35/0.68        = zero_zero_nat ) ).
% 0.35/0.68  
% 0.35/0.68  % and_zero_eq
% 0.35/0.68  thf(fact_6620_finite__lessThan,axiom,
% 0.35/0.68      ! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% 0.35/0.68  
% 0.35/0.68  % finite_lessThan
% 0.35/0.68  thf(fact_6621_lessThan__subset__iff,axiom,
% 0.35/0.68      ! [X: rat,Y: rat] :
% 0.35/0.68        ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X ) @ ( set_ord_lessThan_rat @ Y ) )
% 0.35/0.68        = ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_subset_iff
% 0.35/0.68  thf(fact_6622_lessThan__subset__iff,axiom,
% 0.35/0.68      ! [X: num,Y: num] :
% 0.35/0.68        ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y ) )
% 0.35/0.68        = ( ord_less_eq_num @ X @ Y ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_subset_iff
% 0.35/0.68  thf(fact_6623_lessThan__subset__iff,axiom,
% 0.35/0.68      ! [X: nat,Y: nat] :
% 0.35/0.68        ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
% 0.35/0.68        = ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_subset_iff
% 0.35/0.68  thf(fact_6624_lessThan__subset__iff,axiom,
% 0.35/0.68      ! [X: int,Y: int] :
% 0.35/0.68        ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
% 0.35/0.68        = ( ord_less_eq_int @ X @ Y ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_subset_iff
% 0.35/0.68  thf(fact_6625_lessThan__subset__iff,axiom,
% 0.35/0.68      ! [X: real,Y: real] :
% 0.35/0.68        ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y ) )
% 0.35/0.68        = ( ord_less_eq_real @ X @ Y ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_subset_iff
% 0.35/0.68  thf(fact_6626_and__negative__int__iff,axiom,
% 0.35/0.68      ! [K: int,L: int] :
% 0.35/0.68        ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
% 0.35/0.68        = ( ( ord_less_int @ K @ zero_zero_int )
% 0.35/0.68          & ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % and_negative_int_iff
% 0.35/0.68  thf(fact_6627_lessThan__0,axiom,
% 0.35/0.68      ( ( set_ord_lessThan_nat @ zero_zero_nat )
% 0.35/0.68      = bot_bot_set_nat ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_0
% 0.35/0.68  thf(fact_6628_single__Diff__lessThan,axiom,
% 0.35/0.68      ! [K: nat] :
% 0.35/0.68        ( ( minus_minus_set_nat @ ( insert_nat @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
% 0.35/0.68        = ( insert_nat @ K @ bot_bot_set_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % single_Diff_lessThan
% 0.35/0.68  thf(fact_6629_single__Diff__lessThan,axiom,
% 0.35/0.68      ! [K: int] :
% 0.35/0.68        ( ( minus_minus_set_int @ ( insert_int @ K @ bot_bot_set_int ) @ ( set_ord_lessThan_int @ K ) )
% 0.35/0.68        = ( insert_int @ K @ bot_bot_set_int ) ) ).
% 0.35/0.68  
% 0.35/0.68  % single_Diff_lessThan
% 0.35/0.68  thf(fact_6630_single__Diff__lessThan,axiom,
% 0.35/0.68      ! [K: real] :
% 0.35/0.68        ( ( minus_minus_set_real @ ( insert_real @ K @ bot_bot_set_real ) @ ( set_or5984915006950818249n_real @ K ) )
% 0.35/0.68        = ( insert_real @ K @ bot_bot_set_real ) ) ).
% 0.35/0.68  
% 0.35/0.68  % single_Diff_lessThan
% 0.35/0.68  thf(fact_6631_and__numerals_I1_J,axiom,
% 0.35/0.68      ! [Y: num] :
% 0.35/0.68        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 0.35/0.68        = zero_zero_int ) ).
% 0.35/0.68  
% 0.35/0.68  % and_numerals(1)
% 0.35/0.68  thf(fact_6632_and__numerals_I1_J,axiom,
% 0.35/0.68      ! [Y: num] :
% 0.35/0.68        ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.35/0.68        = zero_zero_nat ) ).
% 0.35/0.68  
% 0.35/0.68  % and_numerals(1)
% 0.35/0.68  thf(fact_6633_and__numerals_I5_J,axiom,
% 0.35/0.68      ! [X: num] :
% 0.35/0.68        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
% 0.35/0.68        = zero_zero_int ) ).
% 0.35/0.68  
% 0.35/0.68  % and_numerals(5)
% 0.35/0.68  thf(fact_6634_and__numerals_I5_J,axiom,
% 0.35/0.68      ! [X: num] :
% 0.35/0.68        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
% 0.35/0.68        = zero_zero_nat ) ).
% 0.35/0.68  
% 0.35/0.68  % and_numerals(5)
% 0.35/0.68  thf(fact_6635_lessThan__non__empty,axiom,
% 0.35/0.68      ! [X: int] :
% 0.35/0.68        ( ( set_ord_lessThan_int @ X )
% 0.35/0.68       != bot_bot_set_int ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_non_empty
% 0.35/0.68  thf(fact_6636_lessThan__non__empty,axiom,
% 0.35/0.68      ! [X: real] :
% 0.35/0.68        ( ( set_or5984915006950818249n_real @ X )
% 0.35/0.68       != bot_bot_set_real ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_non_empty
% 0.35/0.68  thf(fact_6637_infinite__Iio,axiom,
% 0.35/0.68      ! [A: int] :
% 0.35/0.68        ~ ( finite_finite_int @ ( set_ord_lessThan_int @ A ) ) ).
% 0.35/0.68  
% 0.35/0.68  % infinite_Iio
% 0.35/0.68  thf(fact_6638_infinite__Iio,axiom,
% 0.35/0.68      ! [A: real] :
% 0.35/0.68        ~ ( finite_finite_real @ ( set_or5984915006950818249n_real @ A ) ) ).
% 0.35/0.68  
% 0.35/0.68  % infinite_Iio
% 0.35/0.68  thf(fact_6639_lessThan__def,axiom,
% 0.35/0.68      ( set_or890127255671739683et_nat
% 0.35/0.68      = ( ^ [U2: set_nat] :
% 0.35/0.68            ( collect_set_nat
% 0.35/0.68            @ ^ [X3: set_nat] : ( ord_less_set_nat @ X3 @ U2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_def
% 0.35/0.68  thf(fact_6640_lessThan__def,axiom,
% 0.35/0.68      ( set_ord_lessThan_rat
% 0.35/0.68      = ( ^ [U2: rat] :
% 0.35/0.68            ( collect_rat
% 0.35/0.68            @ ^ [X3: rat] : ( ord_less_rat @ X3 @ U2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_def
% 0.35/0.68  thf(fact_6641_lessThan__def,axiom,
% 0.35/0.68      ( set_ord_lessThan_num
% 0.35/0.68      = ( ^ [U2: num] :
% 0.35/0.68            ( collect_num
% 0.35/0.68            @ ^ [X3: num] : ( ord_less_num @ X3 @ U2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_def
% 0.35/0.68  thf(fact_6642_lessThan__def,axiom,
% 0.35/0.68      ( set_ord_lessThan_nat
% 0.35/0.68      = ( ^ [U2: nat] :
% 0.35/0.68            ( collect_nat
% 0.35/0.68            @ ^ [X3: nat] : ( ord_less_nat @ X3 @ U2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_def
% 0.35/0.68  thf(fact_6643_lessThan__def,axiom,
% 0.35/0.68      ( set_ord_lessThan_int
% 0.35/0.68      = ( ^ [U2: int] :
% 0.35/0.68            ( collect_int
% 0.35/0.68            @ ^ [X3: int] : ( ord_less_int @ X3 @ U2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_def
% 0.35/0.68  thf(fact_6644_lessThan__def,axiom,
% 0.35/0.68      ( set_or5984915006950818249n_real
% 0.35/0.68      = ( ^ [U2: real] :
% 0.35/0.68            ( collect_real
% 0.35/0.68            @ ^ [X3: real] : ( ord_less_real @ X3 @ U2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_def
% 0.35/0.68  thf(fact_6645_Iio__eq__empty__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( ( set_ord_lessThan_nat @ N )
% 0.35/0.68          = bot_bot_set_nat )
% 0.35/0.68        = ( N = bot_bot_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % Iio_eq_empty_iff
% 0.35/0.68  thf(fact_6646_lessThan__strict__subset__iff,axiom,
% 0.35/0.68      ! [M: rat,N: rat] :
% 0.35/0.68        ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N ) )
% 0.35/0.68        = ( ord_less_rat @ M @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_strict_subset_iff
% 0.35/0.68  thf(fact_6647_lessThan__strict__subset__iff,axiom,
% 0.35/0.68      ! [M: num,N: num] :
% 0.35/0.68        ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
% 0.35/0.68        = ( ord_less_num @ M @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_strict_subset_iff
% 0.35/0.68  thf(fact_6648_lessThan__strict__subset__iff,axiom,
% 0.35/0.68      ! [M: nat,N: nat] :
% 0.35/0.68        ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.68        = ( ord_less_nat @ M @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_strict_subset_iff
% 0.35/0.68  thf(fact_6649_lessThan__strict__subset__iff,axiom,
% 0.35/0.68      ! [M: int,N: int] :
% 0.35/0.68        ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
% 0.35/0.68        = ( ord_less_int @ M @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_strict_subset_iff
% 0.35/0.68  thf(fact_6650_lessThan__strict__subset__iff,axiom,
% 0.35/0.68      ! [M: real,N: real] :
% 0.35/0.68        ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N ) )
% 0.35/0.68        = ( ord_less_real @ M @ N ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_strict_subset_iff
% 0.35/0.68  thf(fact_6651_lessThan__empty__iff,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( ( set_ord_lessThan_nat @ N )
% 0.35/0.68          = bot_bot_set_nat )
% 0.35/0.68        = ( N = zero_zero_nat ) ) ).
% 0.35/0.68  
% 0.35/0.68  % lessThan_empty_iff
% 0.35/0.68  thf(fact_6652_finite__nat__bounded,axiom,
% 0.35/0.68      ! [S2: set_nat] :
% 0.35/0.68        ( ( finite_finite_nat @ S2 )
% 0.35/0.68       => ? [K2: nat] : ( ord_less_eq_set_nat @ S2 @ ( set_ord_lessThan_nat @ K2 ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % finite_nat_bounded
% 0.35/0.68  thf(fact_6653_finite__nat__iff__bounded,axiom,
% 0.35/0.68      ( finite_finite_nat
% 0.35/0.68      = ( ^ [S7: set_nat] :
% 0.35/0.68          ? [K3: nat] : ( ord_less_eq_set_nat @ S7 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % finite_nat_iff_bounded
% 0.35/0.68  thf(fact_6654_AND__upper2_H_H,axiom,
% 0.35/0.68      ! [Y: int,Z2: int,X: int] :
% 0.35/0.68        ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.35/0.68       => ( ( ord_less_int @ Y @ Z2 )
% 0.35/0.68         => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z2 ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % AND_upper2''
% 0.35/0.68  thf(fact_6655_AND__upper1_H_H,axiom,
% 0.35/0.68      ! [Y: int,Z2: int,Ya: int] :
% 0.35/0.68        ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.35/0.68       => ( ( ord_less_int @ Y @ Z2 )
% 0.35/0.68         => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z2 ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % AND_upper1''
% 0.35/0.68  thf(fact_6656_and__less__eq,axiom,
% 0.35/0.68      ! [L: int,K: int] :
% 0.35/0.68        ( ( ord_less_int @ L @ zero_zero_int )
% 0.35/0.68       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).
% 0.35/0.68  
% 0.35/0.68  % and_less_eq
% 0.35/0.68  thf(fact_6657_sum__diff__distrib,axiom,
% 0.35/0.68      ! [Q: int > nat,P: int > nat,N: int] :
% 0.35/0.68        ( ! [X4: int] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
% 0.35/0.68       => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N ) ) )
% 0.35/0.68          = ( groups4541462559716669496nt_nat
% 0.35/0.68            @ ^ [X3: int] : ( minus_minus_nat @ ( P @ X3 ) @ ( Q @ X3 ) )
% 0.35/0.68            @ ( set_ord_lessThan_int @ N ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_diff_distrib
% 0.35/0.68  thf(fact_6658_sum__diff__distrib,axiom,
% 0.35/0.68      ! [Q: real > nat,P: real > nat,N: real] :
% 0.35/0.68        ( ! [X4: real] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
% 0.35/0.68       => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N ) ) )
% 0.35/0.68          = ( groups1935376822645274424al_nat
% 0.35/0.68            @ ^ [X3: real] : ( minus_minus_nat @ ( P @ X3 ) @ ( Q @ X3 ) )
% 0.35/0.68            @ ( set_or5984915006950818249n_real @ N ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_diff_distrib
% 0.35/0.68  thf(fact_6659_sum__diff__distrib,axiom,
% 0.35/0.68      ! [Q: nat > nat,P: nat > nat,N: nat] :
% 0.35/0.68        ( ! [X4: nat] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
% 0.35/0.68       => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
% 0.35/0.68          = ( groups3542108847815614940at_nat
% 0.35/0.68            @ ^ [X3: nat] : ( minus_minus_nat @ ( P @ X3 ) @ ( Q @ X3 ) )
% 0.35/0.68            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_diff_distrib
% 0.35/0.68  thf(fact_6660_sum_OlessThan__Suc__shift,axiom,
% 0.35/0.68      ! [G: nat > rat,N: nat] :
% 0.35/0.68        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.35/0.68        = ( plus_plus_rat @ ( G @ zero_zero_nat )
% 0.35/0.68          @ ( groups2906978787729119204at_rat
% 0.35/0.68            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.35/0.68            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum.lessThan_Suc_shift
% 0.35/0.68  thf(fact_6661_sum_OlessThan__Suc__shift,axiom,
% 0.35/0.68      ! [G: nat > int,N: nat] :
% 0.35/0.68        ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.35/0.68        = ( plus_plus_int @ ( G @ zero_zero_nat )
% 0.35/0.68          @ ( groups3539618377306564664at_int
% 0.35/0.68            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.35/0.68            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum.lessThan_Suc_shift
% 0.35/0.68  thf(fact_6662_sum_OlessThan__Suc__shift,axiom,
% 0.35/0.68      ! [G: nat > nat,N: nat] :
% 0.35/0.68        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.35/0.68        = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 0.35/0.68          @ ( groups3542108847815614940at_nat
% 0.35/0.68            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.35/0.68            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum.lessThan_Suc_shift
% 0.35/0.68  thf(fact_6663_sum_OlessThan__Suc__shift,axiom,
% 0.35/0.68      ! [G: nat > real,N: nat] :
% 0.35/0.68        ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.35/0.68        = ( plus_plus_real @ ( G @ zero_zero_nat )
% 0.35/0.68          @ ( groups6591440286371151544t_real
% 0.35/0.68            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.35/0.68            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum.lessThan_Suc_shift
% 0.35/0.68  thf(fact_6664_sum__lessThan__telescope_H,axiom,
% 0.35/0.68      ! [F: nat > rat,M: nat] :
% 0.35/0.68        ( ( groups2906978787729119204at_rat
% 0.35/0.68          @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.35/0.68          @ ( set_ord_lessThan_nat @ M ) )
% 0.35/0.68        = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_lessThan_telescope'
% 0.35/0.68  thf(fact_6665_sum__lessThan__telescope_H,axiom,
% 0.35/0.68      ! [F: nat > int,M: nat] :
% 0.35/0.68        ( ( groups3539618377306564664at_int
% 0.35/0.68          @ ^ [N2: nat] : ( minus_minus_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.35/0.68          @ ( set_ord_lessThan_nat @ M ) )
% 0.35/0.68        = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_lessThan_telescope'
% 0.35/0.68  thf(fact_6666_sum__lessThan__telescope_H,axiom,
% 0.35/0.68      ! [F: nat > real,M: nat] :
% 0.35/0.68        ( ( groups6591440286371151544t_real
% 0.35/0.68          @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.35/0.68          @ ( set_ord_lessThan_nat @ M ) )
% 0.35/0.68        = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_lessThan_telescope'
% 0.35/0.68  thf(fact_6667_sum__lessThan__telescope,axiom,
% 0.35/0.68      ! [F: nat > rat,M: nat] :
% 0.35/0.68        ( ( groups2906978787729119204at_rat
% 0.35/0.68          @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.35/0.68          @ ( set_ord_lessThan_nat @ M ) )
% 0.35/0.68        = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_lessThan_telescope
% 0.35/0.68  thf(fact_6668_sum__lessThan__telescope,axiom,
% 0.35/0.68      ! [F: nat > int,M: nat] :
% 0.35/0.68        ( ( groups3539618377306564664at_int
% 0.35/0.68          @ ^ [N2: nat] : ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.35/0.68          @ ( set_ord_lessThan_nat @ M ) )
% 0.35/0.68        = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_lessThan_telescope
% 0.35/0.68  thf(fact_6669_sum__lessThan__telescope,axiom,
% 0.35/0.68      ! [F: nat > real,M: nat] :
% 0.35/0.68        ( ( groups6591440286371151544t_real
% 0.35/0.68          @ ^ [N2: nat] : ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.35/0.68          @ ( set_ord_lessThan_nat @ M ) )
% 0.35/0.68        = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum_lessThan_telescope
% 0.35/0.68  thf(fact_6670_sum_OatLeast1__atMost__eq,axiom,
% 0.35/0.68      ! [G: nat > nat,N: nat] :
% 0.35/0.68        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.35/0.68        = ( groups3542108847815614940at_nat
% 0.35/0.68          @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 0.35/0.68          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum.atLeast1_atMost_eq
% 0.35/0.68  thf(fact_6671_sum_OatLeast1__atMost__eq,axiom,
% 0.35/0.68      ! [G: nat > real,N: nat] :
% 0.35/0.68        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.35/0.68        = ( groups6591440286371151544t_real
% 0.35/0.68          @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 0.35/0.68          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % sum.atLeast1_atMost_eq
% 0.35/0.68  thf(fact_6672_real__sum__nat__ivl__bounded2,axiom,
% 0.35/0.68      ! [N: nat,F: nat > rat,K4: rat,K: nat] :
% 0.35/0.68        ( ! [P7: nat] :
% 0.35/0.68            ( ( ord_less_nat @ P7 @ N )
% 0.35/0.68           => ( ord_less_eq_rat @ ( F @ P7 ) @ K4 ) )
% 0.35/0.68       => ( ( ord_less_eq_rat @ zero_zero_rat @ K4 )
% 0.35/0.68         => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K4 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % real_sum_nat_ivl_bounded2
% 0.35/0.68  thf(fact_6673_real__sum__nat__ivl__bounded2,axiom,
% 0.35/0.68      ! [N: nat,F: nat > int,K4: int,K: nat] :
% 0.35/0.68        ( ! [P7: nat] :
% 0.35/0.68            ( ( ord_less_nat @ P7 @ N )
% 0.35/0.68           => ( ord_less_eq_int @ ( F @ P7 ) @ K4 ) )
% 0.35/0.68       => ( ( ord_less_eq_int @ zero_zero_int @ K4 )
% 0.35/0.68         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K4 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % real_sum_nat_ivl_bounded2
% 0.35/0.68  thf(fact_6674_real__sum__nat__ivl__bounded2,axiom,
% 0.35/0.68      ! [N: nat,F: nat > nat,K4: nat,K: nat] :
% 0.35/0.68        ( ! [P7: nat] :
% 0.35/0.68            ( ( ord_less_nat @ P7 @ N )
% 0.35/0.68           => ( ord_less_eq_nat @ ( F @ P7 ) @ K4 ) )
% 0.35/0.68       => ( ( ord_less_eq_nat @ zero_zero_nat @ K4 )
% 0.35/0.68         => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K4 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % real_sum_nat_ivl_bounded2
% 0.35/0.68  thf(fact_6675_real__sum__nat__ivl__bounded2,axiom,
% 0.35/0.68      ! [N: nat,F: nat > real,K4: real,K: nat] :
% 0.35/0.68        ( ! [P7: nat] :
% 0.35/0.68            ( ( ord_less_nat @ P7 @ N )
% 0.35/0.68           => ( ord_less_eq_real @ ( F @ P7 ) @ K4 ) )
% 0.35/0.68       => ( ( ord_less_eq_real @ zero_zero_real @ K4 )
% 0.35/0.68         => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K4 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % real_sum_nat_ivl_bounded2
% 0.35/0.68  thf(fact_6676_norm__le__zero__iff,axiom,
% 0.35/0.68      ! [X: real] :
% 0.35/0.68        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
% 0.35/0.68        = ( X = zero_zero_real ) ) ).
% 0.35/0.68  
% 0.35/0.68  % norm_le_zero_iff
% 0.35/0.68  thf(fact_6677_norm__le__zero__iff,axiom,
% 0.35/0.68      ! [X: complex] :
% 0.35/0.68        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
% 0.35/0.68        = ( X = zero_zero_complex ) ) ).
% 0.35/0.68  
% 0.35/0.68  % norm_le_zero_iff
% 0.35/0.68  thf(fact_6678_zero__less__norm__iff,axiom,
% 0.35/0.68      ! [X: real] :
% 0.35/0.68        ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
% 0.35/0.68        = ( X != zero_zero_real ) ) ).
% 0.35/0.68  
% 0.35/0.68  % zero_less_norm_iff
% 0.35/0.68  thf(fact_6679_zero__less__norm__iff,axiom,
% 0.35/0.68      ! [X: complex] :
% 0.35/0.68        ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
% 0.35/0.68        = ( X != zero_zero_complex ) ) ).
% 0.35/0.68  
% 0.35/0.68  % zero_less_norm_iff
% 0.35/0.68  thf(fact_6680_norm__eq__zero,axiom,
% 0.35/0.68      ! [X: real] :
% 0.35/0.68        ( ( ( real_V7735802525324610683m_real @ X )
% 0.35/0.68          = zero_zero_real )
% 0.35/0.68        = ( X = zero_zero_real ) ) ).
% 0.35/0.68  
% 0.35/0.68  % norm_eq_zero
% 0.35/0.68  thf(fact_6681_norm__eq__zero,axiom,
% 0.35/0.68      ! [X: complex] :
% 0.35/0.68        ( ( ( real_V1022390504157884413omplex @ X )
% 0.35/0.68          = zero_zero_real )
% 0.35/0.68        = ( X = zero_zero_complex ) ) ).
% 0.35/0.68  
% 0.35/0.68  % norm_eq_zero
% 0.35/0.68  thf(fact_6682_norm__zero,axiom,
% 0.35/0.68      ( ( real_V7735802525324610683m_real @ zero_zero_real )
% 0.35/0.68      = zero_zero_real ) ).
% 0.35/0.68  
% 0.35/0.68  % norm_zero
% 0.35/0.68  thf(fact_6683_norm__zero,axiom,
% 0.35/0.68      ( ( real_V1022390504157884413omplex @ zero_zero_complex )
% 0.35/0.68      = zero_zero_real ) ).
% 0.35/0.68  
% 0.35/0.68  % norm_zero
% 0.35/0.68  thf(fact_6684_power__eq__1__iff,axiom,
% 0.35/0.68      ! [W2: real,N: nat] :
% 0.35/0.68        ( ( ( power_power_real @ W2 @ N )
% 0.35/0.68          = one_one_real )
% 0.35/0.68       => ( ( ( real_V7735802525324610683m_real @ W2 )
% 0.35/0.68            = one_one_real )
% 0.35/0.68          | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % power_eq_1_iff
% 0.35/0.68  thf(fact_6685_power__eq__1__iff,axiom,
% 0.35/0.68      ! [W2: complex,N: nat] :
% 0.35/0.68        ( ( ( power_power_complex @ W2 @ N )
% 0.35/0.68          = one_one_complex )
% 0.35/0.68       => ( ( ( real_V1022390504157884413omplex @ W2 )
% 0.35/0.68            = one_one_real )
% 0.35/0.68          | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % power_eq_1_iff
% 0.35/0.68  thf(fact_6686_norm__diff__triangle__less,axiom,
% 0.35/0.68      ! [X: real,Y: real,E1: real,Z2: real,E22: real] :
% 0.35/0.68        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
% 0.35/0.68       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z2 ) ) @ E22 )
% 0.35/0.68         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % norm_diff_triangle_less
% 0.35/0.68  thf(fact_6687_norm__diff__triangle__less,axiom,
% 0.35/0.68      ! [X: complex,Y: complex,E1: real,Z2: complex,E22: real] :
% 0.35/0.68        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
% 0.35/0.68       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z2 ) ) @ E22 )
% 0.35/0.68         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % norm_diff_triangle_less
% 0.35/0.68  thf(fact_6688_and__nat__numerals_I3_J,axiom,
% 0.35/0.68      ! [X: num] :
% 0.35/0.68        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.35/0.68        = zero_zero_nat ) ).
% 0.35/0.68  
% 0.35/0.68  % and_nat_numerals(3)
% 0.35/0.68  thf(fact_6689_and__nat__numerals_I1_J,axiom,
% 0.35/0.68      ! [Y: num] :
% 0.35/0.68        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.35/0.68        = zero_zero_nat ) ).
% 0.35/0.68  
% 0.35/0.68  % and_nat_numerals(1)
% 0.35/0.68  thf(fact_6690_and__nat__numerals_I4_J,axiom,
% 0.35/0.68      ! [X: num] :
% 0.35/0.68        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.35/0.68        = one_one_nat ) ).
% 0.35/0.68  
% 0.35/0.68  % and_nat_numerals(4)
% 0.35/0.68  thf(fact_6691_and__nat__numerals_I2_J,axiom,
% 0.35/0.68      ! [Y: num] :
% 0.35/0.68        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.35/0.68        = one_one_nat ) ).
% 0.35/0.68  
% 0.35/0.68  % and_nat_numerals(2)
% 0.35/0.68  thf(fact_6692_Suc__0__and__eq,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.68        = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % Suc_0_and_eq
% 0.35/0.68  thf(fact_6693_and__Suc__0__eq,axiom,
% 0.35/0.68      ! [N: nat] :
% 0.35/0.68        ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.35/0.68        = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % and_Suc_0_eq
% 0.35/0.68  thf(fact_6694_and__nat__unfold,axiom,
% 0.35/0.68      ( bit_se727722235901077358nd_nat
% 0.35/0.68      = ( ^ [M2: nat,N2: nat] :
% 0.35/0.68            ( if_nat
% 0.35/0.68            @ ( ( M2 = zero_zero_nat )
% 0.35/0.68              | ( N2 = zero_zero_nat ) )
% 0.35/0.68            @ zero_zero_nat
% 0.35/0.68            @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % and_nat_unfold
% 0.35/0.68  thf(fact_6695_nonzero__norm__divide,axiom,
% 0.35/0.68      ! [B: real,A: real] :
% 0.35/0.68        ( ( B != zero_zero_real )
% 0.35/0.68       => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 0.35/0.68          = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % nonzero_norm_divide
% 0.35/0.68  thf(fact_6696_nonzero__norm__divide,axiom,
% 0.35/0.68      ! [B: complex,A: complex] :
% 0.35/0.68        ( ( B != zero_zero_complex )
% 0.35/0.68       => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.35/0.68          = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % nonzero_norm_divide
% 0.35/0.68  thf(fact_6697_norm__not__less__zero,axiom,
% 0.35/0.68      ! [X: complex] :
% 0.35/0.68        ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).
% 0.35/0.68  
% 0.35/0.68  % norm_not_less_zero
% 0.35/0.68  thf(fact_6698_power__eq__imp__eq__norm,axiom,
% 0.35/0.68      ! [W2: real,N: nat,Z2: real] :
% 0.35/0.68        ( ( ( power_power_real @ W2 @ N )
% 0.35/0.68          = ( power_power_real @ Z2 @ N ) )
% 0.35/0.68       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.68         => ( ( real_V7735802525324610683m_real @ W2 )
% 0.35/0.68            = ( real_V7735802525324610683m_real @ Z2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % power_eq_imp_eq_norm
% 0.35/0.68  thf(fact_6699_power__eq__imp__eq__norm,axiom,
% 0.35/0.68      ! [W2: complex,N: nat,Z2: complex] :
% 0.35/0.68        ( ( ( power_power_complex @ W2 @ N )
% 0.35/0.68          = ( power_power_complex @ Z2 @ N ) )
% 0.35/0.68       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.68         => ( ( real_V1022390504157884413omplex @ W2 )
% 0.35/0.68            = ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % power_eq_imp_eq_norm
% 0.35/0.68  thf(fact_6700_norm__mult__less,axiom,
% 0.35/0.68      ! [X: real,R2: real,Y: real,S: real] :
% 0.35/0.68        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
% 0.35/0.68       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
% 0.35/0.68         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% 0.35/0.68  
% 0.35/0.68  % norm_mult_less
% 0.35/0.68  thf(fact_6701_norm__mult__less,axiom,
% 0.35/0.68      ! [X: complex,R2: real,Y: complex,S: real] :
% 0.35/0.68        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
% 0.35/0.69       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
% 0.35/0.69         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % norm_mult_less
% 0.35/0.69  thf(fact_6702_norm__triangle__lt,axiom,
% 0.35/0.69      ! [X: real,Y: real,E2: real] :
% 0.35/0.69        ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
% 0.35/0.69       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E2 ) ) ).
% 0.35/0.69  
% 0.35/0.69  % norm_triangle_lt
% 0.35/0.69  thf(fact_6703_norm__triangle__lt,axiom,
% 0.35/0.69      ! [X: complex,Y: complex,E2: real] :
% 0.35/0.69        ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
% 0.35/0.69       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E2 ) ) ).
% 0.35/0.69  
% 0.35/0.69  % norm_triangle_lt
% 0.35/0.69  thf(fact_6704_norm__add__less,axiom,
% 0.35/0.69      ! [X: real,R2: real,Y: real,S: real] :
% 0.35/0.69        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
% 0.35/0.69       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
% 0.35/0.69         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % norm_add_less
% 0.35/0.69  thf(fact_6705_norm__add__less,axiom,
% 0.35/0.69      ! [X: complex,R2: real,Y: complex,S: real] :
% 0.35/0.69        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
% 0.35/0.69       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
% 0.35/0.69         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % norm_add_less
% 0.35/0.69  thf(fact_6706_and__int_Opelims,axiom,
% 0.35/0.69      ! [X: int,Xa2: int,Y: int] :
% 0.35/0.69        ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
% 0.35/0.69          = Y )
% 0.35/0.69       => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
% 0.35/0.69         => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.35/0.69                    & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.35/0.69                 => ( Y
% 0.35/0.69                    = ( uminus_uminus_int
% 0.35/0.69                      @ ( zero_n2684676970156552555ol_int
% 0.35/0.69                        @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.35/0.69                          & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 0.35/0.69                & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.35/0.69                      & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.35/0.69                 => ( Y
% 0.35/0.69                    = ( plus_plus_int
% 0.35/0.69                      @ ( zero_n2684676970156552555ol_int
% 0.35/0.69                        @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.35/0.69                          & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 0.35/0.69                      @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.35/0.69             => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % and_int.pelims
% 0.35/0.69  thf(fact_6707_and__int_Opsimps,axiom,
% 0.35/0.69      ! [K: int,L: int] :
% 0.35/0.69        ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
% 0.35/0.69       => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.35/0.69              & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.35/0.69           => ( ( bit_se725231765392027082nd_int @ K @ L )
% 0.35/0.69              = ( uminus_uminus_int
% 0.35/0.69                @ ( zero_n2684676970156552555ol_int
% 0.35/0.69                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.35/0.69                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
% 0.35/0.69          & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.35/0.69                & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.35/0.69           => ( ( bit_se725231765392027082nd_int @ K @ L )
% 0.35/0.69              = ( plus_plus_int
% 0.35/0.69                @ ( zero_n2684676970156552555ol_int
% 0.35/0.69                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.35/0.69                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 0.35/0.69                @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % and_int.psimps
% 0.35/0.69  thf(fact_6708_geometric__deriv__sums,axiom,
% 0.35/0.69      ! [Z2: real] :
% 0.35/0.69        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ one_one_real )
% 0.35/0.69       => ( sums_real
% 0.35/0.69          @ ^ [N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( power_power_real @ Z2 @ N2 ) )
% 0.35/0.69          @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % geometric_deriv_sums
% 0.35/0.69  thf(fact_6709_geometric__deriv__sums,axiom,
% 0.35/0.69      ! [Z2: complex] :
% 0.35/0.69        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ one_one_real )
% 0.35/0.69       => ( sums_complex
% 0.35/0.69          @ ^ [N2: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( power_power_complex @ Z2 @ N2 ) )
% 0.35/0.69          @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % geometric_deriv_sums
% 0.35/0.69  thf(fact_6710_central__binomial__lower__bound,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.69       => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % central_binomial_lower_bound
% 0.35/0.69  thf(fact_6711_polyfun__diff,axiom,
% 0.35/0.69      ! [N: nat,A: nat > complex,X: complex,Y: complex] :
% 0.35/0.69        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.35/0.69       => ( ( minus_minus_complex
% 0.35/0.69            @ ( groups2073611262835488442omplex
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69            @ ( groups2073611262835488442omplex
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ Y @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69          = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
% 0.35/0.69            @ ( groups2073611262835488442omplex
% 0.35/0.69              @ ^ [J3: nat] :
% 0.35/0.69                  ( times_times_complex
% 0.35/0.69                  @ ( groups2073611262835488442omplex
% 0.35/0.69                    @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
% 0.35/0.69                    @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
% 0.35/0.69                  @ ( power_power_complex @ X @ J3 ) )
% 0.35/0.69              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_diff
% 0.35/0.69  thf(fact_6712_polyfun__diff,axiom,
% 0.35/0.69      ! [N: nat,A: nat > rat,X: rat,Y: rat] :
% 0.35/0.69        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.35/0.69       => ( ( minus_minus_rat
% 0.35/0.69            @ ( groups2906978787729119204at_rat
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ X @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69            @ ( groups2906978787729119204at_rat
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ Y @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69          = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
% 0.35/0.69            @ ( groups2906978787729119204at_rat
% 0.35/0.69              @ ^ [J3: nat] :
% 0.35/0.69                  ( times_times_rat
% 0.35/0.69                  @ ( groups2906978787729119204at_rat
% 0.35/0.69                    @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
% 0.35/0.69                    @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
% 0.35/0.69                  @ ( power_power_rat @ X @ J3 ) )
% 0.35/0.69              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_diff
% 0.35/0.69  thf(fact_6713_polyfun__diff,axiom,
% 0.35/0.69      ! [N: nat,A: nat > int,X: int,Y: int] :
% 0.35/0.69        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.35/0.69       => ( ( minus_minus_int
% 0.35/0.69            @ ( groups3539618377306564664at_int
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ X @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69            @ ( groups3539618377306564664at_int
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ Y @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69          = ( times_times_int @ ( minus_minus_int @ X @ Y )
% 0.35/0.69            @ ( groups3539618377306564664at_int
% 0.35/0.69              @ ^ [J3: nat] :
% 0.35/0.69                  ( times_times_int
% 0.35/0.69                  @ ( groups3539618377306564664at_int
% 0.35/0.69                    @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
% 0.35/0.69                    @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
% 0.35/0.69                  @ ( power_power_int @ X @ J3 ) )
% 0.35/0.69              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_diff
% 0.35/0.69  thf(fact_6714_polyfun__diff,axiom,
% 0.35/0.69      ! [N: nat,A: nat > real,X: real,Y: real] :
% 0.35/0.69        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.35/0.69       => ( ( minus_minus_real
% 0.35/0.69            @ ( groups6591440286371151544t_real
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ X @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69            @ ( groups6591440286371151544t_real
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ Y @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69          = ( times_times_real @ ( minus_minus_real @ X @ Y )
% 0.35/0.69            @ ( groups6591440286371151544t_real
% 0.35/0.69              @ ^ [J3: nat] :
% 0.35/0.69                  ( times_times_real
% 0.35/0.69                  @ ( groups6591440286371151544t_real
% 0.35/0.69                    @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
% 0.35/0.69                    @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
% 0.35/0.69                  @ ( power_power_real @ X @ J3 ) )
% 0.35/0.69              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_diff
% 0.35/0.69  thf(fact_6715_modulo__int__unfold,axiom,
% 0.35/0.69      ! [L: int,K: int,N: nat,M: nat] :
% 0.35/0.69        ( ( ( ( ( sgn_sgn_int @ L )
% 0.35/0.69              = zero_zero_int )
% 0.35/0.69            | ( ( sgn_sgn_int @ K )
% 0.35/0.69              = zero_zero_int )
% 0.35/0.69            | ( N = zero_zero_nat ) )
% 0.35/0.69         => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.35/0.69            = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 0.35/0.69        & ( ~ ( ( ( sgn_sgn_int @ L )
% 0.35/0.69                = zero_zero_int )
% 0.35/0.69              | ( ( sgn_sgn_int @ K )
% 0.35/0.69                = zero_zero_int )
% 0.35/0.69              | ( N = zero_zero_nat ) )
% 0.35/0.69         => ( ( ( ( sgn_sgn_int @ K )
% 0.35/0.69                = ( sgn_sgn_int @ L ) )
% 0.35/0.69             => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.35/0.69                = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
% 0.35/0.69            & ( ( ( sgn_sgn_int @ K )
% 0.35/0.69               != ( sgn_sgn_int @ L ) )
% 0.35/0.69             => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.35/0.69                = ( times_times_int @ ( sgn_sgn_int @ L )
% 0.35/0.69                  @ ( minus_minus_int
% 0.35/0.69                    @ ( semiri1314217659103216013at_int
% 0.35/0.69                      @ ( times_times_nat @ N
% 0.35/0.69                        @ ( zero_n2687167440665602831ol_nat
% 0.35/0.69                          @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
% 0.35/0.69                    @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % modulo_int_unfold
% 0.35/0.69  thf(fact_6716_atMost__iff,axiom,
% 0.35/0.69      ! [I: real,K: real] :
% 0.35/0.69        ( ( member_real @ I @ ( set_ord_atMost_real @ K ) )
% 0.35/0.69        = ( ord_less_eq_real @ I @ K ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_iff
% 0.35/0.69  thf(fact_6717_atMost__iff,axiom,
% 0.35/0.69      ! [I: set_nat,K: set_nat] :
% 0.35/0.69        ( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K ) )
% 0.35/0.69        = ( ord_less_eq_set_nat @ I @ K ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_iff
% 0.35/0.69  thf(fact_6718_atMost__iff,axiom,
% 0.35/0.69      ! [I: set_int,K: set_int] :
% 0.35/0.69        ( ( member_set_int @ I @ ( set_or58775011639299419et_int @ K ) )
% 0.35/0.69        = ( ord_less_eq_set_int @ I @ K ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_iff
% 0.35/0.69  thf(fact_6719_atMost__iff,axiom,
% 0.35/0.69      ! [I: rat,K: rat] :
% 0.35/0.69        ( ( member_rat @ I @ ( set_ord_atMost_rat @ K ) )
% 0.35/0.69        = ( ord_less_eq_rat @ I @ K ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_iff
% 0.35/0.69  thf(fact_6720_atMost__iff,axiom,
% 0.35/0.69      ! [I: num,K: num] :
% 0.35/0.69        ( ( member_num @ I @ ( set_ord_atMost_num @ K ) )
% 0.35/0.69        = ( ord_less_eq_num @ I @ K ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_iff
% 0.35/0.69  thf(fact_6721_atMost__iff,axiom,
% 0.35/0.69      ! [I: nat,K: nat] :
% 0.35/0.69        ( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
% 0.35/0.69        = ( ord_less_eq_nat @ I @ K ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_iff
% 0.35/0.69  thf(fact_6722_atMost__iff,axiom,
% 0.35/0.69      ! [I: int,K: int] :
% 0.35/0.69        ( ( member_int @ I @ ( set_ord_atMost_int @ K ) )
% 0.35/0.69        = ( ord_less_eq_int @ I @ K ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_iff
% 0.35/0.69  thf(fact_6723_sgn__0,axiom,
% 0.35/0.69      ( ( sgn_sgn_Code_integer @ zero_z3403309356797280102nteger )
% 0.35/0.69      = zero_z3403309356797280102nteger ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_0
% 0.35/0.69  thf(fact_6724_sgn__0,axiom,
% 0.35/0.69      ( ( sgn_sgn_complex @ zero_zero_complex )
% 0.35/0.69      = zero_zero_complex ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_0
% 0.35/0.69  thf(fact_6725_sgn__0,axiom,
% 0.35/0.69      ( ( sgn_sgn_real @ zero_zero_real )
% 0.35/0.69      = zero_zero_real ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_0
% 0.35/0.69  thf(fact_6726_sgn__0,axiom,
% 0.35/0.69      ( ( sgn_sgn_rat @ zero_zero_rat )
% 0.35/0.69      = zero_zero_rat ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_0
% 0.35/0.69  thf(fact_6727_sgn__0,axiom,
% 0.35/0.69      ( ( sgn_sgn_int @ zero_zero_int )
% 0.35/0.69      = zero_zero_int ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_0
% 0.35/0.69  thf(fact_6728_sgn__zero,axiom,
% 0.35/0.69      ( ( sgn_sgn_complex @ zero_zero_complex )
% 0.35/0.69      = zero_zero_complex ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_zero
% 0.35/0.69  thf(fact_6729_sgn__zero,axiom,
% 0.35/0.69      ( ( sgn_sgn_real @ zero_zero_real )
% 0.35/0.69      = zero_zero_real ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_zero
% 0.35/0.69  thf(fact_6730_finite__atMost,axiom,
% 0.35/0.69      ! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% 0.35/0.69  
% 0.35/0.69  % finite_atMost
% 0.35/0.69  thf(fact_6731_sgn__greater,axiom,
% 0.35/0.69      ! [A: code_integer] :
% 0.35/0.69        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A ) )
% 0.35/0.69        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_greater
% 0.35/0.69  thf(fact_6732_sgn__greater,axiom,
% 0.35/0.69      ! [A: real] :
% 0.35/0.69        ( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A ) )
% 0.35/0.69        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_greater
% 0.35/0.69  thf(fact_6733_sgn__greater,axiom,
% 0.35/0.69      ! [A: rat] :
% 0.35/0.69        ( ( ord_less_rat @ zero_zero_rat @ ( sgn_sgn_rat @ A ) )
% 0.35/0.69        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_greater
% 0.35/0.69  thf(fact_6734_sgn__greater,axiom,
% 0.35/0.69      ! [A: int] :
% 0.35/0.69        ( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A ) )
% 0.35/0.69        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_greater
% 0.35/0.69  thf(fact_6735_sgn__less,axiom,
% 0.35/0.69      ! [A: code_integer] :
% 0.35/0.69        ( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 0.35/0.69        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_less
% 0.35/0.69  thf(fact_6736_sgn__less,axiom,
% 0.35/0.69      ! [A: real] :
% 0.35/0.69        ( ( ord_less_real @ ( sgn_sgn_real @ A ) @ zero_zero_real )
% 0.35/0.69        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_less
% 0.35/0.69  thf(fact_6737_sgn__less,axiom,
% 0.35/0.69      ! [A: rat] :
% 0.35/0.69        ( ( ord_less_rat @ ( sgn_sgn_rat @ A ) @ zero_zero_rat )
% 0.35/0.69        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_less
% 0.35/0.69  thf(fact_6738_sgn__less,axiom,
% 0.35/0.69      ! [A: int] :
% 0.35/0.69        ( ( ord_less_int @ ( sgn_sgn_int @ A ) @ zero_zero_int )
% 0.35/0.69        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_less
% 0.35/0.69  thf(fact_6739_atMost__subset__iff,axiom,
% 0.35/0.69      ! [X: set_int,Y: set_int] :
% 0.35/0.69        ( ( ord_le4403425263959731960et_int @ ( set_or58775011639299419et_int @ X ) @ ( set_or58775011639299419et_int @ Y ) )
% 0.35/0.69        = ( ord_less_eq_set_int @ X @ Y ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_subset_iff
% 0.35/0.69  thf(fact_6740_atMost__subset__iff,axiom,
% 0.35/0.69      ! [X: rat,Y: rat] :
% 0.35/0.69        ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X ) @ ( set_ord_atMost_rat @ Y ) )
% 0.35/0.69        = ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_subset_iff
% 0.35/0.69  thf(fact_6741_atMost__subset__iff,axiom,
% 0.35/0.69      ! [X: num,Y: num] :
% 0.35/0.69        ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X ) @ ( set_ord_atMost_num @ Y ) )
% 0.35/0.69        = ( ord_less_eq_num @ X @ Y ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_subset_iff
% 0.35/0.69  thf(fact_6742_atMost__subset__iff,axiom,
% 0.35/0.69      ! [X: nat,Y: nat] :
% 0.35/0.69        ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
% 0.35/0.69        = ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_subset_iff
% 0.35/0.69  thf(fact_6743_atMost__subset__iff,axiom,
% 0.35/0.69      ! [X: int,Y: int] :
% 0.35/0.69        ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y ) )
% 0.35/0.69        = ( ord_less_eq_int @ X @ Y ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_subset_iff
% 0.35/0.69  thf(fact_6744_sgn__pos,axiom,
% 0.35/0.69      ! [A: code_integer] :
% 0.35/0.69        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 0.35/0.69       => ( ( sgn_sgn_Code_integer @ A )
% 0.35/0.69          = one_one_Code_integer ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_pos
% 0.35/0.69  thf(fact_6745_sgn__pos,axiom,
% 0.35/0.69      ! [A: real] :
% 0.35/0.69        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.69       => ( ( sgn_sgn_real @ A )
% 0.35/0.69          = one_one_real ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_pos
% 0.35/0.69  thf(fact_6746_sgn__pos,axiom,
% 0.35/0.69      ! [A: rat] :
% 0.35/0.69        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.35/0.69       => ( ( sgn_sgn_rat @ A )
% 0.35/0.69          = one_one_rat ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_pos
% 0.35/0.69  thf(fact_6747_sgn__pos,axiom,
% 0.35/0.69      ! [A: int] :
% 0.35/0.69        ( ( ord_less_int @ zero_zero_int @ A )
% 0.35/0.69       => ( ( sgn_sgn_int @ A )
% 0.35/0.69          = one_one_int ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_pos
% 0.35/0.69  thf(fact_6748_Icc__subset__Iic__iff,axiom,
% 0.35/0.69      ! [L: set_int,H2: set_int,H3: set_int] :
% 0.35/0.69        ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ L @ H2 ) @ ( set_or58775011639299419et_int @ H3 ) )
% 0.35/0.69        = ( ~ ( ord_less_eq_set_int @ L @ H2 )
% 0.35/0.69          | ( ord_less_eq_set_int @ H2 @ H3 ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % Icc_subset_Iic_iff
% 0.35/0.69  thf(fact_6749_Icc__subset__Iic__iff,axiom,
% 0.35/0.69      ! [L: rat,H2: rat,H3: rat] :
% 0.35/0.69        ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H2 ) @ ( set_ord_atMost_rat @ H3 ) )
% 0.35/0.69        = ( ~ ( ord_less_eq_rat @ L @ H2 )
% 0.35/0.69          | ( ord_less_eq_rat @ H2 @ H3 ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % Icc_subset_Iic_iff
% 0.35/0.69  thf(fact_6750_Icc__subset__Iic__iff,axiom,
% 0.35/0.69      ! [L: num,H2: num,H3: num] :
% 0.35/0.69        ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L @ H2 ) @ ( set_ord_atMost_num @ H3 ) )
% 0.35/0.69        = ( ~ ( ord_less_eq_num @ L @ H2 )
% 0.35/0.69          | ( ord_less_eq_num @ H2 @ H3 ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % Icc_subset_Iic_iff
% 0.35/0.69  thf(fact_6751_Icc__subset__Iic__iff,axiom,
% 0.35/0.69      ! [L: nat,H2: nat,H3: nat] :
% 0.35/0.69        ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H2 ) @ ( set_ord_atMost_nat @ H3 ) )
% 0.35/0.69        = ( ~ ( ord_less_eq_nat @ L @ H2 )
% 0.35/0.69          | ( ord_less_eq_nat @ H2 @ H3 ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % Icc_subset_Iic_iff
% 0.35/0.69  thf(fact_6752_Icc__subset__Iic__iff,axiom,
% 0.35/0.69      ! [L: int,H2: int,H3: int] :
% 0.35/0.69        ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H2 ) @ ( set_ord_atMost_int @ H3 ) )
% 0.35/0.69        = ( ~ ( ord_less_eq_int @ L @ H2 )
% 0.35/0.69          | ( ord_less_eq_int @ H2 @ H3 ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % Icc_subset_Iic_iff
% 0.35/0.69  thf(fact_6753_Icc__subset__Iic__iff,axiom,
% 0.35/0.69      ! [L: real,H2: real,H3: real] :
% 0.35/0.69        ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H2 ) @ ( set_ord_atMost_real @ H3 ) )
% 0.35/0.69        = ( ~ ( ord_less_eq_real @ L @ H2 )
% 0.35/0.69          | ( ord_less_eq_real @ H2 @ H3 ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % Icc_subset_Iic_iff
% 0.35/0.69  thf(fact_6754_sgn__mult__self__eq,axiom,
% 0.35/0.69      ! [A: code_integer] :
% 0.35/0.69        ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
% 0.35/0.69        = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_mult_self_eq
% 0.35/0.69  thf(fact_6755_sgn__mult__self__eq,axiom,
% 0.35/0.69      ! [A: real] :
% 0.35/0.69        ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ A ) )
% 0.35/0.69        = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_mult_self_eq
% 0.35/0.69  thf(fact_6756_sgn__mult__self__eq,axiom,
% 0.35/0.69      ! [A: rat] :
% 0.35/0.69        ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ A ) )
% 0.35/0.69        = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_mult_self_eq
% 0.35/0.69  thf(fact_6757_sgn__mult__self__eq,axiom,
% 0.35/0.69      ! [A: int] :
% 0.35/0.69        ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ A ) )
% 0.35/0.69        = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_mult_self_eq
% 0.35/0.69  thf(fact_6758_atMost__0,axiom,
% 0.35/0.69      ( ( set_ord_atMost_nat @ zero_zero_nat )
% 0.35/0.69      = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_0
% 0.35/0.69  thf(fact_6759_sgn__neg,axiom,
% 0.35/0.69      ! [A: int] :
% 0.35/0.69        ( ( ord_less_int @ A @ zero_zero_int )
% 0.35/0.69       => ( ( sgn_sgn_int @ A )
% 0.35/0.69          = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_neg
% 0.35/0.69  thf(fact_6760_sgn__neg,axiom,
% 0.35/0.69      ! [A: real] :
% 0.35/0.69        ( ( ord_less_real @ A @ zero_zero_real )
% 0.35/0.69       => ( ( sgn_sgn_real @ A )
% 0.35/0.69          = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_neg
% 0.35/0.69  thf(fact_6761_sgn__neg,axiom,
% 0.35/0.69      ! [A: rat] :
% 0.35/0.69        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.35/0.69       => ( ( sgn_sgn_rat @ A )
% 0.35/0.69          = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_neg
% 0.35/0.69  thf(fact_6762_sgn__neg,axiom,
% 0.35/0.69      ! [A: code_integer] :
% 0.35/0.69        ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 0.35/0.69       => ( ( sgn_sgn_Code_integer @ A )
% 0.35/0.69          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_neg
% 0.35/0.69  thf(fact_6763_sgn__of__nat,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( sgn_sgn_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.35/0.69        = ( zero_n2052037380579107095ol_rat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_of_nat
% 0.35/0.69  thf(fact_6764_sgn__of__nat,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( sgn_sgn_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 0.35/0.69        = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_of_nat
% 0.35/0.69  thf(fact_6765_sgn__of__nat,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( sgn_sgn_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.35/0.69        = ( zero_n3304061248610475627l_real @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_of_nat
% 0.35/0.69  thf(fact_6766_sgn__of__nat,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( sgn_sgn_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.35/0.69        = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_of_nat
% 0.35/0.69  thf(fact_6767_powser__sums__zero__iff,axiom,
% 0.35/0.69      ! [A: nat > complex,X: complex] :
% 0.35/0.69        ( ( sums_complex
% 0.35/0.69          @ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
% 0.35/0.69          @ X )
% 0.35/0.69        = ( ( A @ zero_zero_nat )
% 0.35/0.69          = X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % powser_sums_zero_iff
% 0.35/0.69  thf(fact_6768_powser__sums__zero__iff,axiom,
% 0.35/0.69      ! [A: nat > real,X: real] :
% 0.35/0.69        ( ( sums_real
% 0.35/0.69          @ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
% 0.35/0.69          @ X )
% 0.35/0.69        = ( ( A @ zero_zero_nat )
% 0.35/0.69          = X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % powser_sums_zero_iff
% 0.35/0.69  thf(fact_6769_sgn__eq__0__iff,axiom,
% 0.35/0.69      ! [A: code_integer] :
% 0.35/0.69        ( ( ( sgn_sgn_Code_integer @ A )
% 0.35/0.69          = zero_z3403309356797280102nteger )
% 0.35/0.69        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_eq_0_iff
% 0.35/0.69  thf(fact_6770_sgn__eq__0__iff,axiom,
% 0.35/0.69      ! [A: complex] :
% 0.35/0.69        ( ( ( sgn_sgn_complex @ A )
% 0.35/0.69          = zero_zero_complex )
% 0.35/0.69        = ( A = zero_zero_complex ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_eq_0_iff
% 0.35/0.69  thf(fact_6771_sgn__eq__0__iff,axiom,
% 0.35/0.69      ! [A: real] :
% 0.35/0.69        ( ( ( sgn_sgn_real @ A )
% 0.35/0.69          = zero_zero_real )
% 0.35/0.69        = ( A = zero_zero_real ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_eq_0_iff
% 0.35/0.69  thf(fact_6772_sgn__eq__0__iff,axiom,
% 0.35/0.69      ! [A: rat] :
% 0.35/0.69        ( ( ( sgn_sgn_rat @ A )
% 0.35/0.69          = zero_zero_rat )
% 0.35/0.69        = ( A = zero_zero_rat ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_eq_0_iff
% 0.35/0.69  thf(fact_6773_sgn__eq__0__iff,axiom,
% 0.35/0.69      ! [A: int] :
% 0.35/0.69        ( ( ( sgn_sgn_int @ A )
% 0.35/0.69          = zero_zero_int )
% 0.35/0.69        = ( A = zero_zero_int ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_eq_0_iff
% 0.35/0.69  thf(fact_6774_sgn__0__0,axiom,
% 0.35/0.69      ! [A: code_integer] :
% 0.35/0.69        ( ( ( sgn_sgn_Code_integer @ A )
% 0.35/0.69          = zero_z3403309356797280102nteger )
% 0.35/0.69        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_0_0
% 0.35/0.69  thf(fact_6775_sgn__0__0,axiom,
% 0.35/0.69      ! [A: real] :
% 0.35/0.69        ( ( ( sgn_sgn_real @ A )
% 0.35/0.69          = zero_zero_real )
% 0.35/0.69        = ( A = zero_zero_real ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_0_0
% 0.35/0.69  thf(fact_6776_sgn__0__0,axiom,
% 0.35/0.69      ! [A: rat] :
% 0.35/0.69        ( ( ( sgn_sgn_rat @ A )
% 0.35/0.69          = zero_zero_rat )
% 0.35/0.69        = ( A = zero_zero_rat ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_0_0
% 0.35/0.69  thf(fact_6777_sgn__0__0,axiom,
% 0.35/0.69      ! [A: int] :
% 0.35/0.69        ( ( ( sgn_sgn_int @ A )
% 0.35/0.69          = zero_zero_int )
% 0.35/0.69        = ( A = zero_zero_int ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_0_0
% 0.35/0.69  thf(fact_6778_sgn__zero__iff,axiom,
% 0.35/0.69      ! [X: complex] :
% 0.35/0.69        ( ( ( sgn_sgn_complex @ X )
% 0.35/0.69          = zero_zero_complex )
% 0.35/0.69        = ( X = zero_zero_complex ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_zero_iff
% 0.35/0.69  thf(fact_6779_sgn__zero__iff,axiom,
% 0.35/0.69      ! [X: real] :
% 0.35/0.69        ( ( ( sgn_sgn_real @ X )
% 0.35/0.69          = zero_zero_real )
% 0.35/0.69        = ( X = zero_zero_real ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_zero_iff
% 0.35/0.69  thf(fact_6780_not__empty__eq__Iic__eq__empty,axiom,
% 0.35/0.69      ! [H2: real] :
% 0.35/0.69        ( bot_bot_set_real
% 0.35/0.69       != ( set_ord_atMost_real @ H2 ) ) ).
% 0.35/0.69  
% 0.35/0.69  % not_empty_eq_Iic_eq_empty
% 0.35/0.69  thf(fact_6781_not__empty__eq__Iic__eq__empty,axiom,
% 0.35/0.69      ! [H2: nat] :
% 0.35/0.69        ( bot_bot_set_nat
% 0.35/0.69       != ( set_ord_atMost_nat @ H2 ) ) ).
% 0.35/0.69  
% 0.35/0.69  % not_empty_eq_Iic_eq_empty
% 0.35/0.69  thf(fact_6782_not__empty__eq__Iic__eq__empty,axiom,
% 0.35/0.69      ! [H2: int] :
% 0.35/0.69        ( bot_bot_set_int
% 0.35/0.69       != ( set_ord_atMost_int @ H2 ) ) ).
% 0.35/0.69  
% 0.35/0.69  % not_empty_eq_Iic_eq_empty
% 0.35/0.69  thf(fact_6783_infinite__Iic,axiom,
% 0.35/0.69      ! [A: int] :
% 0.35/0.69        ~ ( finite_finite_int @ ( set_ord_atMost_int @ A ) ) ).
% 0.35/0.69  
% 0.35/0.69  % infinite_Iic
% 0.35/0.69  thf(fact_6784_atMost__def,axiom,
% 0.35/0.69      ( set_ord_atMost_real
% 0.35/0.69      = ( ^ [U2: real] :
% 0.35/0.69            ( collect_real
% 0.35/0.69            @ ^ [X3: real] : ( ord_less_eq_real @ X3 @ U2 ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_def
% 0.35/0.69  thf(fact_6785_atMost__def,axiom,
% 0.35/0.69      ( set_or4236626031148496127et_nat
% 0.35/0.69      = ( ^ [U2: set_nat] :
% 0.35/0.69            ( collect_set_nat
% 0.35/0.69            @ ^ [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ U2 ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_def
% 0.35/0.69  thf(fact_6786_atMost__def,axiom,
% 0.35/0.69      ( set_or58775011639299419et_int
% 0.35/0.69      = ( ^ [U2: set_int] :
% 0.35/0.69            ( collect_set_int
% 0.35/0.69            @ ^ [X3: set_int] : ( ord_less_eq_set_int @ X3 @ U2 ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_def
% 0.35/0.69  thf(fact_6787_atMost__def,axiom,
% 0.35/0.69      ( set_ord_atMost_rat
% 0.35/0.69      = ( ^ [U2: rat] :
% 0.35/0.69            ( collect_rat
% 0.35/0.69            @ ^ [X3: rat] : ( ord_less_eq_rat @ X3 @ U2 ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_def
% 0.35/0.69  thf(fact_6788_atMost__def,axiom,
% 0.35/0.69      ( set_ord_atMost_num
% 0.35/0.69      = ( ^ [U2: num] :
% 0.35/0.69            ( collect_num
% 0.35/0.69            @ ^ [X3: num] : ( ord_less_eq_num @ X3 @ U2 ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_def
% 0.35/0.69  thf(fact_6789_atMost__def,axiom,
% 0.35/0.69      ( set_ord_atMost_nat
% 0.35/0.69      = ( ^ [U2: nat] :
% 0.35/0.69            ( collect_nat
% 0.35/0.69            @ ^ [X3: nat] : ( ord_less_eq_nat @ X3 @ U2 ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_def
% 0.35/0.69  thf(fact_6790_atMost__def,axiom,
% 0.35/0.69      ( set_ord_atMost_int
% 0.35/0.69      = ( ^ [U2: int] :
% 0.35/0.69            ( collect_int
% 0.35/0.69            @ ^ [X3: int] : ( ord_less_eq_int @ X3 @ U2 ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_def
% 0.35/0.69  thf(fact_6791_sgn__not__eq__imp,axiom,
% 0.35/0.69      ! [B: int,A: int] :
% 0.35/0.69        ( ( ( sgn_sgn_int @ B )
% 0.35/0.69         != ( sgn_sgn_int @ A ) )
% 0.35/0.69       => ( ( ( sgn_sgn_int @ A )
% 0.35/0.69           != zero_zero_int )
% 0.35/0.69         => ( ( ( sgn_sgn_int @ B )
% 0.35/0.69             != zero_zero_int )
% 0.35/0.69           => ( ( sgn_sgn_int @ A )
% 0.35/0.69              = ( uminus_uminus_int @ ( sgn_sgn_int @ B ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_not_eq_imp
% 0.35/0.69  thf(fact_6792_sgn__not__eq__imp,axiom,
% 0.35/0.69      ! [B: real,A: real] :
% 0.35/0.69        ( ( ( sgn_sgn_real @ B )
% 0.35/0.69         != ( sgn_sgn_real @ A ) )
% 0.35/0.69       => ( ( ( sgn_sgn_real @ A )
% 0.35/0.69           != zero_zero_real )
% 0.35/0.69         => ( ( ( sgn_sgn_real @ B )
% 0.35/0.69             != zero_zero_real )
% 0.35/0.69           => ( ( sgn_sgn_real @ A )
% 0.35/0.69              = ( uminus_uminus_real @ ( sgn_sgn_real @ B ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_not_eq_imp
% 0.35/0.69  thf(fact_6793_sgn__not__eq__imp,axiom,
% 0.35/0.69      ! [B: rat,A: rat] :
% 0.35/0.69        ( ( ( sgn_sgn_rat @ B )
% 0.35/0.69         != ( sgn_sgn_rat @ A ) )
% 0.35/0.69       => ( ( ( sgn_sgn_rat @ A )
% 0.35/0.69           != zero_zero_rat )
% 0.35/0.69         => ( ( ( sgn_sgn_rat @ B )
% 0.35/0.69             != zero_zero_rat )
% 0.35/0.69           => ( ( sgn_sgn_rat @ A )
% 0.35/0.69              = ( uminus_uminus_rat @ ( sgn_sgn_rat @ B ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_not_eq_imp
% 0.35/0.69  thf(fact_6794_sgn__not__eq__imp,axiom,
% 0.35/0.69      ! [B: code_integer,A: code_integer] :
% 0.35/0.69        ( ( ( sgn_sgn_Code_integer @ B )
% 0.35/0.69         != ( sgn_sgn_Code_integer @ A ) )
% 0.35/0.69       => ( ( ( sgn_sgn_Code_integer @ A )
% 0.35/0.69           != zero_z3403309356797280102nteger )
% 0.35/0.69         => ( ( ( sgn_sgn_Code_integer @ B )
% 0.35/0.69             != zero_z3403309356797280102nteger )
% 0.35/0.69           => ( ( sgn_sgn_Code_integer @ A )
% 0.35/0.69              = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ B ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_not_eq_imp
% 0.35/0.69  thf(fact_6795_atMost__atLeast0,axiom,
% 0.35/0.69      ( set_ord_atMost_nat
% 0.35/0.69      = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atMost_atLeast0
% 0.35/0.69  thf(fact_6796_not__Iic__le__Icc,axiom,
% 0.35/0.69      ! [H2: int,L3: int,H3: int] :
% 0.35/0.69        ~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H2 ) @ ( set_or1266510415728281911st_int @ L3 @ H3 ) ) ).
% 0.35/0.69  
% 0.35/0.69  % not_Iic_le_Icc
% 0.35/0.69  thf(fact_6797_not__Iic__le__Icc,axiom,
% 0.35/0.69      ! [H2: real,L3: real,H3: real] :
% 0.35/0.69        ~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H2 ) @ ( set_or1222579329274155063t_real @ L3 @ H3 ) ) ).
% 0.35/0.69  
% 0.35/0.69  % not_Iic_le_Icc
% 0.35/0.69  thf(fact_6798_finite__nat__iff__bounded__le,axiom,
% 0.35/0.69      ( finite_finite_nat
% 0.35/0.69      = ( ^ [S7: set_nat] :
% 0.35/0.69          ? [K3: nat] : ( ord_less_eq_set_nat @ S7 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % finite_nat_iff_bounded_le
% 0.35/0.69  thf(fact_6799_sgn__1__pos,axiom,
% 0.35/0.69      ! [A: code_integer] :
% 0.35/0.69        ( ( ( sgn_sgn_Code_integer @ A )
% 0.35/0.69          = one_one_Code_integer )
% 0.35/0.69        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_1_pos
% 0.35/0.69  thf(fact_6800_sgn__1__pos,axiom,
% 0.35/0.69      ! [A: real] :
% 0.35/0.69        ( ( ( sgn_sgn_real @ A )
% 0.35/0.69          = one_one_real )
% 0.35/0.69        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_1_pos
% 0.35/0.69  thf(fact_6801_sgn__1__pos,axiom,
% 0.35/0.69      ! [A: rat] :
% 0.35/0.69        ( ( ( sgn_sgn_rat @ A )
% 0.35/0.69          = one_one_rat )
% 0.35/0.69        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_1_pos
% 0.35/0.69  thf(fact_6802_sgn__1__pos,axiom,
% 0.35/0.69      ! [A: int] :
% 0.35/0.69        ( ( ( sgn_sgn_int @ A )
% 0.35/0.69          = one_one_int )
% 0.35/0.69        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_1_pos
% 0.35/0.69  thf(fact_6803_Iic__subset__Iio__iff,axiom,
% 0.35/0.69      ! [A: rat,B: rat] :
% 0.35/0.69        ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ A ) @ ( set_ord_lessThan_rat @ B ) )
% 0.35/0.69        = ( ord_less_rat @ A @ B ) ) ).
% 0.35/0.69  
% 0.35/0.69  % Iic_subset_Iio_iff
% 0.35/0.69  thf(fact_6804_Iic__subset__Iio__iff,axiom,
% 0.35/0.69      ! [A: num,B: num] :
% 0.35/0.69        ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ A ) @ ( set_ord_lessThan_num @ B ) )
% 0.35/0.69        = ( ord_less_num @ A @ B ) ) ).
% 0.35/0.69  
% 0.35/0.69  % Iic_subset_Iio_iff
% 0.35/0.69  thf(fact_6805_Iic__subset__Iio__iff,axiom,
% 0.35/0.69      ! [A: nat,B: nat] :
% 0.35/0.69        ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
% 0.35/0.69        = ( ord_less_nat @ A @ B ) ) ).
% 0.35/0.69  
% 0.35/0.69  % Iic_subset_Iio_iff
% 0.35/0.69  thf(fact_6806_Iic__subset__Iio__iff,axiom,
% 0.35/0.69      ! [A: int,B: int] :
% 0.35/0.69        ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B ) )
% 0.35/0.69        = ( ord_less_int @ A @ B ) ) ).
% 0.35/0.69  
% 0.35/0.69  % Iic_subset_Iio_iff
% 0.35/0.69  thf(fact_6807_Iic__subset__Iio__iff,axiom,
% 0.35/0.69      ! [A: real,B: real] :
% 0.35/0.69        ( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ A ) @ ( set_or5984915006950818249n_real @ B ) )
% 0.35/0.69        = ( ord_less_real @ A @ B ) ) ).
% 0.35/0.69  
% 0.35/0.69  % Iic_subset_Iio_iff
% 0.35/0.69  thf(fact_6808_powser__sums__zero,axiom,
% 0.35/0.69      ! [A: nat > complex] :
% 0.35/0.69        ( sums_complex
% 0.35/0.69        @ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
% 0.35/0.69        @ ( A @ zero_zero_nat ) ) ).
% 0.35/0.69  
% 0.35/0.69  % powser_sums_zero
% 0.35/0.69  thf(fact_6809_powser__sums__zero,axiom,
% 0.35/0.69      ! [A: nat > real] :
% 0.35/0.69        ( sums_real
% 0.35/0.69        @ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
% 0.35/0.69        @ ( A @ zero_zero_nat ) ) ).
% 0.35/0.69  
% 0.35/0.69  % powser_sums_zero
% 0.35/0.69  thf(fact_6810_sgn__1__neg,axiom,
% 0.35/0.69      ! [A: int] :
% 0.35/0.69        ( ( ( sgn_sgn_int @ A )
% 0.35/0.69          = ( uminus_uminus_int @ one_one_int ) )
% 0.35/0.69        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_1_neg
% 0.35/0.69  thf(fact_6811_sgn__1__neg,axiom,
% 0.35/0.69      ! [A: real] :
% 0.35/0.69        ( ( ( sgn_sgn_real @ A )
% 0.35/0.69          = ( uminus_uminus_real @ one_one_real ) )
% 0.35/0.69        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_1_neg
% 0.35/0.69  thf(fact_6812_sgn__1__neg,axiom,
% 0.35/0.69      ! [A: rat] :
% 0.35/0.69        ( ( ( sgn_sgn_rat @ A )
% 0.35/0.69          = ( uminus_uminus_rat @ one_one_rat ) )
% 0.35/0.69        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_1_neg
% 0.35/0.69  thf(fact_6813_sgn__1__neg,axiom,
% 0.35/0.69      ! [A: code_integer] :
% 0.35/0.69        ( ( ( sgn_sgn_Code_integer @ A )
% 0.35/0.69          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.35/0.69        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_1_neg
% 0.35/0.69  thf(fact_6814_sgn__if,axiom,
% 0.35/0.69      ( sgn_sgn_int
% 0.35/0.69      = ( ^ [X3: int] : ( if_int @ ( X3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_if
% 0.35/0.69  thf(fact_6815_sgn__if,axiom,
% 0.35/0.69      ( sgn_sgn_real
% 0.35/0.69      = ( ^ [X3: real] : ( if_real @ ( X3 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X3 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_if
% 0.35/0.69  thf(fact_6816_sgn__if,axiom,
% 0.35/0.69      ( sgn_sgn_rat
% 0.35/0.69      = ( ^ [X3: rat] : ( if_rat @ ( X3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_if
% 0.35/0.69  thf(fact_6817_sgn__if,axiom,
% 0.35/0.69      ( sgn_sgn_Code_integer
% 0.35/0.69      = ( ^ [X3: code_integer] : ( if_Code_integer @ ( X3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X3 ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_if
% 0.35/0.69  thf(fact_6818_zsgn__def,axiom,
% 0.35/0.69      ( sgn_sgn_int
% 0.35/0.69      = ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % zsgn_def
% 0.35/0.69  thf(fact_6819_norm__sgn,axiom,
% 0.35/0.69      ! [X: real] :
% 0.35/0.69        ( ( ( X = zero_zero_real )
% 0.35/0.69         => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
% 0.35/0.69            = zero_zero_real ) )
% 0.35/0.69        & ( ( X != zero_zero_real )
% 0.35/0.69         => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
% 0.35/0.69            = one_one_real ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % norm_sgn
% 0.35/0.69  thf(fact_6820_norm__sgn,axiom,
% 0.35/0.69      ! [X: complex] :
% 0.35/0.69        ( ( ( X = zero_zero_complex )
% 0.35/0.69         => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
% 0.35/0.69            = zero_zero_real ) )
% 0.35/0.69        & ( ( X != zero_zero_complex )
% 0.35/0.69         => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
% 0.35/0.69            = one_one_real ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % norm_sgn
% 0.35/0.69  thf(fact_6821_sum_OatMost__Suc__shift,axiom,
% 0.35/0.69      ! [G: nat > rat,N: nat] :
% 0.35/0.69        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.35/0.69        = ( plus_plus_rat @ ( G @ zero_zero_nat )
% 0.35/0.69          @ ( groups2906978787729119204at_rat
% 0.35/0.69            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.35/0.69            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum.atMost_Suc_shift
% 0.35/0.69  thf(fact_6822_sum_OatMost__Suc__shift,axiom,
% 0.35/0.69      ! [G: nat > int,N: nat] :
% 0.35/0.69        ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.35/0.69        = ( plus_plus_int @ ( G @ zero_zero_nat )
% 0.35/0.69          @ ( groups3539618377306564664at_int
% 0.35/0.69            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.35/0.69            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum.atMost_Suc_shift
% 0.35/0.69  thf(fact_6823_sum_OatMost__Suc__shift,axiom,
% 0.35/0.69      ! [G: nat > nat,N: nat] :
% 0.35/0.69        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.35/0.69        = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 0.35/0.69          @ ( groups3542108847815614940at_nat
% 0.35/0.69            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.35/0.69            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum.atMost_Suc_shift
% 0.35/0.69  thf(fact_6824_sum_OatMost__Suc__shift,axiom,
% 0.35/0.69      ! [G: nat > real,N: nat] :
% 0.35/0.69        ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.35/0.69        = ( plus_plus_real @ ( G @ zero_zero_nat )
% 0.35/0.69          @ ( groups6591440286371151544t_real
% 0.35/0.69            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.35/0.69            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum.atMost_Suc_shift
% 0.35/0.69  thf(fact_6825_sum__telescope,axiom,
% 0.35/0.69      ! [F: nat > rat,I: nat] :
% 0.35/0.69        ( ( groups2906978787729119204at_rat
% 0.35/0.69          @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
% 0.35/0.69          @ ( set_ord_atMost_nat @ I ) )
% 0.35/0.69        = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum_telescope
% 0.35/0.69  thf(fact_6826_sum__telescope,axiom,
% 0.35/0.69      ! [F: nat > int,I: nat] :
% 0.35/0.69        ( ( groups3539618377306564664at_int
% 0.35/0.69          @ ^ [I4: nat] : ( minus_minus_int @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
% 0.35/0.69          @ ( set_ord_atMost_nat @ I ) )
% 0.35/0.69        = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum_telescope
% 0.35/0.69  thf(fact_6827_sum__telescope,axiom,
% 0.35/0.69      ! [F: nat > real,I: nat] :
% 0.35/0.69        ( ( groups6591440286371151544t_real
% 0.35/0.69          @ ^ [I4: nat] : ( minus_minus_real @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
% 0.35/0.69          @ ( set_ord_atMost_nat @ I ) )
% 0.35/0.69        = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum_telescope
% 0.35/0.69  thf(fact_6828_polyfun__eq__coeffs,axiom,
% 0.35/0.69      ! [C: nat > complex,N: nat,D: nat > complex] :
% 0.35/0.69        ( ( ! [X3: complex] :
% 0.35/0.69              ( ( groups2073611262835488442omplex
% 0.35/0.69                @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X3 @ I4 ) )
% 0.35/0.69                @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69              = ( groups2073611262835488442omplex
% 0.35/0.69                @ ^ [I4: nat] : ( times_times_complex @ ( D @ I4 ) @ ( power_power_complex @ X3 @ I4 ) )
% 0.35/0.69                @ ( set_ord_atMost_nat @ N ) ) ) )
% 0.35/0.69        = ( ! [I4: nat] :
% 0.35/0.69              ( ( ord_less_eq_nat @ I4 @ N )
% 0.35/0.69             => ( ( C @ I4 )
% 0.35/0.69                = ( D @ I4 ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_eq_coeffs
% 0.35/0.69  thf(fact_6829_polyfun__eq__coeffs,axiom,
% 0.35/0.69      ! [C: nat > real,N: nat,D: nat > real] :
% 0.35/0.69        ( ( ! [X3: real] :
% 0.35/0.69              ( ( groups6591440286371151544t_real
% 0.35/0.69                @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X3 @ I4 ) )
% 0.35/0.69                @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69              = ( groups6591440286371151544t_real
% 0.35/0.69                @ ^ [I4: nat] : ( times_times_real @ ( D @ I4 ) @ ( power_power_real @ X3 @ I4 ) )
% 0.35/0.69                @ ( set_ord_atMost_nat @ N ) ) ) )
% 0.35/0.69        = ( ! [I4: nat] :
% 0.35/0.69              ( ( ord_less_eq_nat @ I4 @ N )
% 0.35/0.69             => ( ( C @ I4 )
% 0.35/0.69                = ( D @ I4 ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_eq_coeffs
% 0.35/0.69  thf(fact_6830_binomial__maximum_H,axiom,
% 0.35/0.69      ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 0.35/0.69  
% 0.35/0.69  % binomial_maximum'
% 0.35/0.69  thf(fact_6831_binomial__mono,axiom,
% 0.35/0.69      ! [K: nat,K5: nat,N: nat] :
% 0.35/0.69        ( ( ord_less_eq_nat @ K @ K5 )
% 0.35/0.69       => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) @ N )
% 0.35/0.69         => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % binomial_mono
% 0.35/0.69  thf(fact_6832_binomial__antimono,axiom,
% 0.35/0.69      ! [K: nat,K5: nat,N: nat] :
% 0.35/0.69        ( ( ord_less_eq_nat @ K @ K5 )
% 0.35/0.69       => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 0.35/0.69         => ( ( ord_less_eq_nat @ K5 @ N )
% 0.35/0.69           => ( ord_less_eq_nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % binomial_antimono
% 0.35/0.69  thf(fact_6833_binomial__maximum,axiom,
% 0.35/0.69      ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % binomial_maximum
% 0.35/0.69  thf(fact_6834_zero__polynom__imp__zero__coeffs,axiom,
% 0.35/0.69      ! [C: nat > complex,N: nat,K: nat] :
% 0.35/0.69        ( ! [W: complex] :
% 0.35/0.69            ( ( groups2073611262835488442omplex
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ W @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69            = zero_zero_complex )
% 0.35/0.69       => ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.69         => ( ( C @ K )
% 0.35/0.69            = zero_zero_complex ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % zero_polynom_imp_zero_coeffs
% 0.35/0.69  thf(fact_6835_zero__polynom__imp__zero__coeffs,axiom,
% 0.35/0.69      ! [C: nat > real,N: nat,K: nat] :
% 0.35/0.69        ( ! [W: real] :
% 0.35/0.69            ( ( groups6591440286371151544t_real
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ W @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69            = zero_zero_real )
% 0.35/0.69       => ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.69         => ( ( C @ K )
% 0.35/0.69            = zero_zero_real ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % zero_polynom_imp_zero_coeffs
% 0.35/0.69  thf(fact_6836_polyfun__eq__0,axiom,
% 0.35/0.69      ! [C: nat > complex,N: nat] :
% 0.35/0.69        ( ( ! [X3: complex] :
% 0.35/0.69              ( ( groups2073611262835488442omplex
% 0.35/0.69                @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X3 @ I4 ) )
% 0.35/0.69                @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69              = zero_zero_complex ) )
% 0.35/0.69        = ( ! [I4: nat] :
% 0.35/0.69              ( ( ord_less_eq_nat @ I4 @ N )
% 0.35/0.69             => ( ( C @ I4 )
% 0.35/0.69                = zero_zero_complex ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_eq_0
% 0.35/0.69  thf(fact_6837_polyfun__eq__0,axiom,
% 0.35/0.69      ! [C: nat > real,N: nat] :
% 0.35/0.69        ( ( ! [X3: real] :
% 0.35/0.69              ( ( groups6591440286371151544t_real
% 0.35/0.69                @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X3 @ I4 ) )
% 0.35/0.69                @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69              = zero_zero_real ) )
% 0.35/0.69        = ( ! [I4: nat] :
% 0.35/0.69              ( ( ord_less_eq_nat @ I4 @ N )
% 0.35/0.69             => ( ( C @ I4 )
% 0.35/0.69                = zero_zero_real ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_eq_0
% 0.35/0.69  thf(fact_6838_sum_OatMost__shift,axiom,
% 0.35/0.69      ! [G: nat > rat,N: nat] :
% 0.35/0.69        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69        = ( plus_plus_rat @ ( G @ zero_zero_nat )
% 0.35/0.69          @ ( groups2906978787729119204at_rat
% 0.35/0.69            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.35/0.69            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum.atMost_shift
% 0.35/0.69  thf(fact_6839_sum_OatMost__shift,axiom,
% 0.35/0.69      ! [G: nat > int,N: nat] :
% 0.35/0.69        ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69        = ( plus_plus_int @ ( G @ zero_zero_nat )
% 0.35/0.69          @ ( groups3539618377306564664at_int
% 0.35/0.69            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.35/0.69            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum.atMost_shift
% 0.35/0.69  thf(fact_6840_sum_OatMost__shift,axiom,
% 0.35/0.69      ! [G: nat > nat,N: nat] :
% 0.35/0.69        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69        = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 0.35/0.69          @ ( groups3542108847815614940at_nat
% 0.35/0.69            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.35/0.69            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum.atMost_shift
% 0.35/0.69  thf(fact_6841_sum_OatMost__shift,axiom,
% 0.35/0.69      ! [G: nat > real,N: nat] :
% 0.35/0.69        ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69        = ( plus_plus_real @ ( G @ zero_zero_nat )
% 0.35/0.69          @ ( groups6591440286371151544t_real
% 0.35/0.69            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.35/0.69            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum.atMost_shift
% 0.35/0.69  thf(fact_6842_atLeast1__atMost__eq__remove0,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.69        = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % atLeast1_atMost_eq_remove0
% 0.35/0.69  thf(fact_6843_sum_Otriangle__reindex__eq,axiom,
% 0.35/0.69      ! [G: nat > nat > nat,N: nat] :
% 0.35/0.69        ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
% 0.35/0.69          @ ( collec3392354462482085612at_nat
% 0.35/0.69            @ ( produc6081775807080527818_nat_o
% 0.35/0.69              @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 0.35/0.69        = ( groups3542108847815614940at_nat
% 0.35/0.69          @ ^ [K3: nat] :
% 0.35/0.69              ( groups3542108847815614940at_nat
% 0.35/0.69              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ K3 ) )
% 0.35/0.69          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum.triangle_reindex_eq
% 0.35/0.69  thf(fact_6844_sum_Otriangle__reindex__eq,axiom,
% 0.35/0.69      ! [G: nat > nat > real,N: nat] :
% 0.35/0.69        ( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
% 0.35/0.69          @ ( collec3392354462482085612at_nat
% 0.35/0.69            @ ( produc6081775807080527818_nat_o
% 0.35/0.69              @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 0.35/0.69        = ( groups6591440286371151544t_real
% 0.35/0.69          @ ^ [K3: nat] :
% 0.35/0.69              ( groups6591440286371151544t_real
% 0.35/0.69              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ K3 ) )
% 0.35/0.69          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum.triangle_reindex_eq
% 0.35/0.69  thf(fact_6845_binomial__less__binomial__Suc,axiom,
% 0.35/0.69      ! [K: nat,N: nat] :
% 0.35/0.69        ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.69       => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % binomial_less_binomial_Suc
% 0.35/0.69  thf(fact_6846_binomial__strict__antimono,axiom,
% 0.35/0.69      ! [K: nat,K5: nat,N: nat] :
% 0.35/0.69        ( ( ord_less_nat @ K @ K5 )
% 0.35/0.69       => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 0.35/0.69         => ( ( ord_less_eq_nat @ K5 @ N )
% 0.35/0.69           => ( ord_less_nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % binomial_strict_antimono
% 0.35/0.69  thf(fact_6847_binomial__strict__mono,axiom,
% 0.35/0.69      ! [K: nat,K5: nat,N: nat] :
% 0.35/0.69        ( ( ord_less_nat @ K @ K5 )
% 0.35/0.69       => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) @ N )
% 0.35/0.69         => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % binomial_strict_mono
% 0.35/0.69  thf(fact_6848_polyfun__finite__roots,axiom,
% 0.35/0.69      ! [C: nat > complex,N: nat] :
% 0.35/0.69        ( ( finite3207457112153483333omplex
% 0.35/0.69          @ ( collect_complex
% 0.35/0.69            @ ^ [X3: complex] :
% 0.35/0.69                ( ( groups2073611262835488442omplex
% 0.35/0.69                  @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X3 @ I4 ) )
% 0.35/0.69                  @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69                = zero_zero_complex ) ) )
% 0.35/0.69        = ( ? [I4: nat] :
% 0.35/0.69              ( ( ord_less_eq_nat @ I4 @ N )
% 0.35/0.69              & ( ( C @ I4 )
% 0.35/0.69               != zero_zero_complex ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_finite_roots
% 0.35/0.69  thf(fact_6849_polyfun__finite__roots,axiom,
% 0.35/0.69      ! [C: nat > real,N: nat] :
% 0.35/0.69        ( ( finite_finite_real
% 0.35/0.69          @ ( collect_real
% 0.35/0.69            @ ^ [X3: real] :
% 0.35/0.69                ( ( groups6591440286371151544t_real
% 0.35/0.69                  @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X3 @ I4 ) )
% 0.35/0.69                  @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69                = zero_zero_real ) ) )
% 0.35/0.69        = ( ? [I4: nat] :
% 0.35/0.69              ( ( ord_less_eq_nat @ I4 @ N )
% 0.35/0.69              & ( ( C @ I4 )
% 0.35/0.69               != zero_zero_real ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_finite_roots
% 0.35/0.69  thf(fact_6850_polyfun__roots__finite,axiom,
% 0.35/0.69      ! [C: nat > complex,K: nat,N: nat] :
% 0.35/0.69        ( ( ( C @ K )
% 0.35/0.69         != zero_zero_complex )
% 0.35/0.69       => ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.69         => ( finite3207457112153483333omplex
% 0.35/0.69            @ ( collect_complex
% 0.35/0.69              @ ^ [Z3: complex] :
% 0.35/0.69                  ( ( groups2073611262835488442omplex
% 0.35/0.69                    @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z3 @ I4 ) )
% 0.35/0.69                    @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69                  = zero_zero_complex ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_roots_finite
% 0.35/0.69  thf(fact_6851_polyfun__roots__finite,axiom,
% 0.35/0.69      ! [C: nat > real,K: nat,N: nat] :
% 0.35/0.69        ( ( ( C @ K )
% 0.35/0.69         != zero_zero_real )
% 0.35/0.69       => ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.69         => ( finite_finite_real
% 0.35/0.69            @ ( collect_real
% 0.35/0.69              @ ^ [Z3: real] :
% 0.35/0.69                  ( ( groups6591440286371151544t_real
% 0.35/0.69                    @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z3 @ I4 ) )
% 0.35/0.69                    @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69                  = zero_zero_real ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_roots_finite
% 0.35/0.69  thf(fact_6852_polyfun__linear__factor__root,axiom,
% 0.35/0.69      ! [C: nat > complex,A: complex,N: nat] :
% 0.35/0.69        ( ( ( groups2073611262835488442omplex
% 0.35/0.69            @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ A @ I4 ) )
% 0.35/0.69            @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69          = zero_zero_complex )
% 0.35/0.69       => ~ ! [B3: nat > complex] :
% 0.35/0.69              ~ ! [Z5: complex] :
% 0.35/0.69                  ( ( groups2073611262835488442omplex
% 0.35/0.69                    @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z5 @ I4 ) )
% 0.35/0.69                    @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69                  = ( times_times_complex @ ( minus_minus_complex @ Z5 @ A )
% 0.35/0.69                    @ ( groups2073611262835488442omplex
% 0.35/0.69                      @ ^ [I4: nat] : ( times_times_complex @ ( B3 @ I4 ) @ ( power_power_complex @ Z5 @ I4 ) )
% 0.35/0.69                      @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_linear_factor_root
% 0.35/0.69  thf(fact_6853_polyfun__linear__factor__root,axiom,
% 0.35/0.69      ! [C: nat > rat,A: rat,N: nat] :
% 0.35/0.69        ( ( ( groups2906978787729119204at_rat
% 0.35/0.69            @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ A @ I4 ) )
% 0.35/0.69            @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69          = zero_zero_rat )
% 0.35/0.69       => ~ ! [B3: nat > rat] :
% 0.35/0.69              ~ ! [Z5: rat] :
% 0.35/0.69                  ( ( groups2906978787729119204at_rat
% 0.35/0.69                    @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ Z5 @ I4 ) )
% 0.35/0.69                    @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69                  = ( times_times_rat @ ( minus_minus_rat @ Z5 @ A )
% 0.35/0.69                    @ ( groups2906978787729119204at_rat
% 0.35/0.69                      @ ^ [I4: nat] : ( times_times_rat @ ( B3 @ I4 ) @ ( power_power_rat @ Z5 @ I4 ) )
% 0.35/0.69                      @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_linear_factor_root
% 0.35/0.69  thf(fact_6854_polyfun__linear__factor__root,axiom,
% 0.35/0.69      ! [C: nat > int,A: int,N: nat] :
% 0.35/0.69        ( ( ( groups3539618377306564664at_int
% 0.35/0.69            @ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ A @ I4 ) )
% 0.35/0.69            @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69          = zero_zero_int )
% 0.35/0.69       => ~ ! [B3: nat > int] :
% 0.35/0.69              ~ ! [Z5: int] :
% 0.35/0.69                  ( ( groups3539618377306564664at_int
% 0.35/0.69                    @ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ Z5 @ I4 ) )
% 0.35/0.69                    @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69                  = ( times_times_int @ ( minus_minus_int @ Z5 @ A )
% 0.35/0.69                    @ ( groups3539618377306564664at_int
% 0.35/0.69                      @ ^ [I4: nat] : ( times_times_int @ ( B3 @ I4 ) @ ( power_power_int @ Z5 @ I4 ) )
% 0.35/0.69                      @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_linear_factor_root
% 0.35/0.69  thf(fact_6855_polyfun__linear__factor__root,axiom,
% 0.35/0.69      ! [C: nat > real,A: real,N: nat] :
% 0.35/0.69        ( ( ( groups6591440286371151544t_real
% 0.35/0.69            @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ A @ I4 ) )
% 0.35/0.69            @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69          = zero_zero_real )
% 0.35/0.69       => ~ ! [B3: nat > real] :
% 0.35/0.69              ~ ! [Z5: real] :
% 0.35/0.69                  ( ( groups6591440286371151544t_real
% 0.35/0.69                    @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z5 @ I4 ) )
% 0.35/0.69                    @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69                  = ( times_times_real @ ( minus_minus_real @ Z5 @ A )
% 0.35/0.69                    @ ( groups6591440286371151544t_real
% 0.35/0.69                      @ ^ [I4: nat] : ( times_times_real @ ( B3 @ I4 ) @ ( power_power_real @ Z5 @ I4 ) )
% 0.35/0.69                      @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_linear_factor_root
% 0.35/0.69  thf(fact_6856_sum__power__shift,axiom,
% 0.35/0.69      ! [M: nat,N: nat,X: complex] :
% 0.35/0.69        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.69       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.69          = ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum_power_shift
% 0.35/0.69  thf(fact_6857_sum__power__shift,axiom,
% 0.35/0.69      ! [M: nat,N: nat,X: rat] :
% 0.35/0.69        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.69       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.69          = ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum_power_shift
% 0.35/0.69  thf(fact_6858_sum__power__shift,axiom,
% 0.35/0.69      ! [M: nat,N: nat,X: int] :
% 0.35/0.69        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.69       => ( ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.69          = ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum_power_shift
% 0.35/0.69  thf(fact_6859_sum__power__shift,axiom,
% 0.35/0.69      ! [M: nat,N: nat,X: real] :
% 0.35/0.69        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.69       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.69          = ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum_power_shift
% 0.35/0.69  thf(fact_6860_sum_Otriangle__reindex,axiom,
% 0.35/0.69      ! [G: nat > nat > nat,N: nat] :
% 0.35/0.69        ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
% 0.35/0.69          @ ( collec3392354462482085612at_nat
% 0.35/0.69            @ ( produc6081775807080527818_nat_o
% 0.35/0.69              @ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 0.35/0.69        = ( groups3542108847815614940at_nat
% 0.35/0.69          @ ^ [K3: nat] :
% 0.35/0.69              ( groups3542108847815614940at_nat
% 0.35/0.69              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ K3 ) )
% 0.35/0.69          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum.triangle_reindex
% 0.35/0.69  thf(fact_6861_sum_Otriangle__reindex,axiom,
% 0.35/0.69      ! [G: nat > nat > real,N: nat] :
% 0.35/0.69        ( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
% 0.35/0.69          @ ( collec3392354462482085612at_nat
% 0.35/0.69            @ ( produc6081775807080527818_nat_o
% 0.35/0.69              @ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 0.35/0.69        = ( groups6591440286371151544t_real
% 0.35/0.69          @ ^ [K3: nat] :
% 0.35/0.69              ( groups6591440286371151544t_real
% 0.35/0.69              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ K3 ) )
% 0.35/0.69          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum.triangle_reindex
% 0.35/0.69  thf(fact_6862_polynomial__product,axiom,
% 0.35/0.69      ! [M: nat,A: nat > complex,N: nat,B: nat > complex,X: complex] :
% 0.35/0.69        ( ! [I2: nat] :
% 0.35/0.69            ( ( ord_less_nat @ M @ I2 )
% 0.35/0.69           => ( ( A @ I2 )
% 0.35/0.69              = zero_zero_complex ) )
% 0.35/0.69       => ( ! [J2: nat] :
% 0.35/0.69              ( ( ord_less_nat @ N @ J2 )
% 0.35/0.69             => ( ( B @ J2 )
% 0.35/0.69                = zero_zero_complex ) )
% 0.35/0.69         => ( ( times_times_complex
% 0.35/0.69              @ ( groups2073611262835488442omplex
% 0.35/0.69                @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X @ I4 ) )
% 0.35/0.69                @ ( set_ord_atMost_nat @ M ) )
% 0.35/0.69              @ ( groups2073611262835488442omplex
% 0.35/0.69                @ ^ [J3: nat] : ( times_times_complex @ ( B @ J3 ) @ ( power_power_complex @ X @ J3 ) )
% 0.35/0.69                @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69            = ( groups2073611262835488442omplex
% 0.35/0.69              @ ^ [R5: nat] :
% 0.35/0.69                  ( times_times_complex
% 0.35/0.69                  @ ( groups2073611262835488442omplex
% 0.35/0.69                    @ ^ [K3: nat] : ( times_times_complex @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
% 0.35/0.69                    @ ( set_ord_atMost_nat @ R5 ) )
% 0.35/0.69                  @ ( power_power_complex @ X @ R5 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polynomial_product
% 0.35/0.69  thf(fact_6863_polynomial__product,axiom,
% 0.35/0.69      ! [M: nat,A: nat > rat,N: nat,B: nat > rat,X: rat] :
% 0.35/0.69        ( ! [I2: nat] :
% 0.35/0.69            ( ( ord_less_nat @ M @ I2 )
% 0.35/0.69           => ( ( A @ I2 )
% 0.35/0.69              = zero_zero_rat ) )
% 0.35/0.69       => ( ! [J2: nat] :
% 0.35/0.69              ( ( ord_less_nat @ N @ J2 )
% 0.35/0.69             => ( ( B @ J2 )
% 0.35/0.69                = zero_zero_rat ) )
% 0.35/0.69         => ( ( times_times_rat
% 0.35/0.69              @ ( groups2906978787729119204at_rat
% 0.35/0.69                @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ X @ I4 ) )
% 0.35/0.69                @ ( set_ord_atMost_nat @ M ) )
% 0.35/0.69              @ ( groups2906978787729119204at_rat
% 0.35/0.69                @ ^ [J3: nat] : ( times_times_rat @ ( B @ J3 ) @ ( power_power_rat @ X @ J3 ) )
% 0.35/0.69                @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69            = ( groups2906978787729119204at_rat
% 0.35/0.69              @ ^ [R5: nat] :
% 0.35/0.69                  ( times_times_rat
% 0.35/0.69                  @ ( groups2906978787729119204at_rat
% 0.35/0.69                    @ ^ [K3: nat] : ( times_times_rat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
% 0.35/0.69                    @ ( set_ord_atMost_nat @ R5 ) )
% 0.35/0.69                  @ ( power_power_rat @ X @ R5 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polynomial_product
% 0.35/0.69  thf(fact_6864_polynomial__product,axiom,
% 0.35/0.69      ! [M: nat,A: nat > int,N: nat,B: nat > int,X: int] :
% 0.35/0.69        ( ! [I2: nat] :
% 0.35/0.69            ( ( ord_less_nat @ M @ I2 )
% 0.35/0.69           => ( ( A @ I2 )
% 0.35/0.69              = zero_zero_int ) )
% 0.35/0.69       => ( ! [J2: nat] :
% 0.35/0.69              ( ( ord_less_nat @ N @ J2 )
% 0.35/0.69             => ( ( B @ J2 )
% 0.35/0.69                = zero_zero_int ) )
% 0.35/0.69         => ( ( times_times_int
% 0.35/0.69              @ ( groups3539618377306564664at_int
% 0.35/0.69                @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ X @ I4 ) )
% 0.35/0.69                @ ( set_ord_atMost_nat @ M ) )
% 0.35/0.69              @ ( groups3539618377306564664at_int
% 0.35/0.69                @ ^ [J3: nat] : ( times_times_int @ ( B @ J3 ) @ ( power_power_int @ X @ J3 ) )
% 0.35/0.69                @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69            = ( groups3539618377306564664at_int
% 0.35/0.69              @ ^ [R5: nat] :
% 0.35/0.69                  ( times_times_int
% 0.35/0.69                  @ ( groups3539618377306564664at_int
% 0.35/0.69                    @ ^ [K3: nat] : ( times_times_int @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
% 0.35/0.69                    @ ( set_ord_atMost_nat @ R5 ) )
% 0.35/0.69                  @ ( power_power_int @ X @ R5 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polynomial_product
% 0.35/0.69  thf(fact_6865_polynomial__product,axiom,
% 0.35/0.69      ! [M: nat,A: nat > real,N: nat,B: nat > real,X: real] :
% 0.35/0.69        ( ! [I2: nat] :
% 0.35/0.69            ( ( ord_less_nat @ M @ I2 )
% 0.35/0.69           => ( ( A @ I2 )
% 0.35/0.69              = zero_zero_real ) )
% 0.35/0.69       => ( ! [J2: nat] :
% 0.35/0.69              ( ( ord_less_nat @ N @ J2 )
% 0.35/0.69             => ( ( B @ J2 )
% 0.35/0.69                = zero_zero_real ) )
% 0.35/0.69         => ( ( times_times_real
% 0.35/0.69              @ ( groups6591440286371151544t_real
% 0.35/0.69                @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ X @ I4 ) )
% 0.35/0.69                @ ( set_ord_atMost_nat @ M ) )
% 0.35/0.69              @ ( groups6591440286371151544t_real
% 0.35/0.69                @ ^ [J3: nat] : ( times_times_real @ ( B @ J3 ) @ ( power_power_real @ X @ J3 ) )
% 0.35/0.69                @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69            = ( groups6591440286371151544t_real
% 0.35/0.69              @ ^ [R5: nat] :
% 0.35/0.69                  ( times_times_real
% 0.35/0.69                  @ ( groups6591440286371151544t_real
% 0.35/0.69                    @ ^ [K3: nat] : ( times_times_real @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
% 0.35/0.69                    @ ( set_ord_atMost_nat @ R5 ) )
% 0.35/0.69                  @ ( power_power_real @ X @ R5 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polynomial_product
% 0.35/0.69  thf(fact_6866_polyfun__eq__const,axiom,
% 0.35/0.69      ! [C: nat > complex,N: nat,K: complex] :
% 0.35/0.69        ( ( ! [X3: complex] :
% 0.35/0.69              ( ( groups2073611262835488442omplex
% 0.35/0.69                @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X3 @ I4 ) )
% 0.35/0.69                @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69              = K ) )
% 0.35/0.69        = ( ( ( C @ zero_zero_nat )
% 0.35/0.69            = K )
% 0.35/0.69          & ! [X3: nat] :
% 0.35/0.69              ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
% 0.35/0.69             => ( ( C @ X3 )
% 0.35/0.69                = zero_zero_complex ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_eq_const
% 0.35/0.69  thf(fact_6867_polyfun__eq__const,axiom,
% 0.35/0.69      ! [C: nat > real,N: nat,K: real] :
% 0.35/0.69        ( ( ! [X3: real] :
% 0.35/0.69              ( ( groups6591440286371151544t_real
% 0.35/0.69                @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X3 @ I4 ) )
% 0.35/0.69                @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69              = K ) )
% 0.35/0.69        = ( ( ( C @ zero_zero_nat )
% 0.35/0.69            = K )
% 0.35/0.69          & ! [X3: nat] :
% 0.35/0.69              ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
% 0.35/0.69             => ( ( C @ X3 )
% 0.35/0.69                = zero_zero_real ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_eq_const
% 0.35/0.69  thf(fact_6868_polynomial__product__nat,axiom,
% 0.35/0.69      ! [M: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
% 0.35/0.69        ( ! [I2: nat] :
% 0.35/0.69            ( ( ord_less_nat @ M @ I2 )
% 0.35/0.69           => ( ( A @ I2 )
% 0.35/0.69              = zero_zero_nat ) )
% 0.35/0.69       => ( ! [J2: nat] :
% 0.35/0.69              ( ( ord_less_nat @ N @ J2 )
% 0.35/0.69             => ( ( B @ J2 )
% 0.35/0.69                = zero_zero_nat ) )
% 0.35/0.69         => ( ( times_times_nat
% 0.35/0.69              @ ( groups3542108847815614940at_nat
% 0.35/0.69                @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( power_power_nat @ X @ I4 ) )
% 0.35/0.69                @ ( set_ord_atMost_nat @ M ) )
% 0.35/0.69              @ ( groups3542108847815614940at_nat
% 0.35/0.69                @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
% 0.35/0.69                @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69            = ( groups3542108847815614940at_nat
% 0.35/0.69              @ ^ [R5: nat] :
% 0.35/0.69                  ( times_times_nat
% 0.35/0.69                  @ ( groups3542108847815614940at_nat
% 0.35/0.69                    @ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
% 0.35/0.69                    @ ( set_ord_atMost_nat @ R5 ) )
% 0.35/0.69                  @ ( power_power_nat @ X @ R5 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polynomial_product_nat
% 0.35/0.69  thf(fact_6869_and__int_Opinduct,axiom,
% 0.35/0.69      ! [A0: int,A12: int,P: int > int > $o] :
% 0.35/0.69        ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 0.35/0.69       => ( ! [K2: int,L4: int] :
% 0.35/0.69              ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L4 ) )
% 0.35/0.69             => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.35/0.69                      & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.35/0.69                 => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.35/0.69               => ( P @ K2 @ L4 ) ) )
% 0.35/0.69         => ( P @ A0 @ A12 ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % and_int.pinduct
% 0.35/0.69  thf(fact_6870_sum_Ozero__middle,axiom,
% 0.35/0.69      ! [P4: nat,K: nat,G: nat > complex,H2: nat > complex] :
% 0.35/0.69        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 0.35/0.69       => ( ( ord_less_eq_nat @ K @ P4 )
% 0.35/0.69         => ( ( groups2073611262835488442omplex
% 0.35/0.69              @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K ) @ zero_zero_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ P4 ) )
% 0.35/0.69            = ( groups2073611262835488442omplex
% 0.35/0.69              @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum.zero_middle
% 0.35/0.69  thf(fact_6871_sum_Ozero__middle,axiom,
% 0.35/0.69      ! [P4: nat,K: nat,G: nat > rat,H2: nat > rat] :
% 0.35/0.69        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 0.35/0.69       => ( ( ord_less_eq_nat @ K @ P4 )
% 0.35/0.69         => ( ( groups2906978787729119204at_rat
% 0.35/0.69              @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ zero_zero_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ P4 ) )
% 0.35/0.69            = ( groups2906978787729119204at_rat
% 0.35/0.69              @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum.zero_middle
% 0.35/0.69  thf(fact_6872_sum_Ozero__middle,axiom,
% 0.35/0.69      ! [P4: nat,K: nat,G: nat > int,H2: nat > int] :
% 0.35/0.69        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 0.35/0.69       => ( ( ord_less_eq_nat @ K @ P4 )
% 0.35/0.69         => ( ( groups3539618377306564664at_int
% 0.35/0.69              @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ zero_zero_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ P4 ) )
% 0.35/0.69            = ( groups3539618377306564664at_int
% 0.35/0.69              @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum.zero_middle
% 0.35/0.69  thf(fact_6873_sum_Ozero__middle,axiom,
% 0.35/0.69      ! [P4: nat,K: nat,G: nat > nat,H2: nat > nat] :
% 0.35/0.69        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 0.35/0.69       => ( ( ord_less_eq_nat @ K @ P4 )
% 0.35/0.69         => ( ( groups3542108847815614940at_nat
% 0.35/0.69              @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ zero_zero_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ P4 ) )
% 0.35/0.69            = ( groups3542108847815614940at_nat
% 0.35/0.69              @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum.zero_middle
% 0.35/0.69  thf(fact_6874_sum_Ozero__middle,axiom,
% 0.35/0.69      ! [P4: nat,K: nat,G: nat > real,H2: nat > real] :
% 0.35/0.69        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 0.35/0.69       => ( ( ord_less_eq_nat @ K @ P4 )
% 0.35/0.69         => ( ( groups6591440286371151544t_real
% 0.35/0.69              @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ zero_zero_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ P4 ) )
% 0.35/0.69            = ( groups6591440286371151544t_real
% 0.35/0.69              @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum.zero_middle
% 0.35/0.69  thf(fact_6875_root__polyfun,axiom,
% 0.35/0.69      ! [N: nat,Z2: complex,A: complex] :
% 0.35/0.69        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.35/0.69       => ( ( ( power_power_complex @ Z2 @ N )
% 0.35/0.69            = A )
% 0.35/0.69          = ( ( groups2073611262835488442omplex
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_complex @ ( if_complex @ ( I4 = zero_zero_nat ) @ ( uminus1482373934393186551omplex @ A ) @ ( if_complex @ ( I4 = N ) @ one_one_complex @ zero_zero_complex ) ) @ ( power_power_complex @ Z2 @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69            = zero_zero_complex ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % root_polyfun
% 0.35/0.69  thf(fact_6876_root__polyfun,axiom,
% 0.35/0.69      ! [N: nat,Z2: int,A: int] :
% 0.35/0.69        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.35/0.69       => ( ( ( power_power_int @ Z2 @ N )
% 0.35/0.69            = A )
% 0.35/0.69          = ( ( groups3539618377306564664at_int
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_int @ ( if_int @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_int @ A ) @ ( if_int @ ( I4 = N ) @ one_one_int @ zero_zero_int ) ) @ ( power_power_int @ Z2 @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69            = zero_zero_int ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % root_polyfun
% 0.35/0.69  thf(fact_6877_root__polyfun,axiom,
% 0.35/0.69      ! [N: nat,Z2: rat,A: rat] :
% 0.35/0.69        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.35/0.69       => ( ( ( power_power_rat @ Z2 @ N )
% 0.35/0.69            = A )
% 0.35/0.69          = ( ( groups2906978787729119204at_rat
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_rat @ ( if_rat @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_rat @ A ) @ ( if_rat @ ( I4 = N ) @ one_one_rat @ zero_zero_rat ) ) @ ( power_power_rat @ Z2 @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69            = zero_zero_rat ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % root_polyfun
% 0.35/0.69  thf(fact_6878_root__polyfun,axiom,
% 0.35/0.69      ! [N: nat,Z2: code_integer,A: code_integer] :
% 0.35/0.69        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.35/0.69       => ( ( ( power_8256067586552552935nteger @ Z2 @ N )
% 0.35/0.69            = A )
% 0.35/0.69          = ( ( groups7501900531339628137nteger
% 0.35/0.69              @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( if_Code_integer @ ( I4 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ A ) @ ( if_Code_integer @ ( I4 = N ) @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) @ ( power_8256067586552552935nteger @ Z2 @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69            = zero_z3403309356797280102nteger ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % root_polyfun
% 0.35/0.69  thf(fact_6879_root__polyfun,axiom,
% 0.35/0.69      ! [N: nat,Z2: real,A: real] :
% 0.35/0.69        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.35/0.69       => ( ( ( power_power_real @ Z2 @ N )
% 0.35/0.69            = A )
% 0.35/0.69          = ( ( groups6591440286371151544t_real
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_real @ ( if_real @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_real @ A ) @ ( if_real @ ( I4 = N ) @ one_one_real @ zero_zero_real ) ) @ ( power_power_real @ Z2 @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69            = zero_zero_real ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % root_polyfun
% 0.35/0.69  thf(fact_6880_polyfun__diff__alt,axiom,
% 0.35/0.69      ! [N: nat,A: nat > complex,X: complex,Y: complex] :
% 0.35/0.69        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.35/0.69       => ( ( minus_minus_complex
% 0.35/0.69            @ ( groups2073611262835488442omplex
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69            @ ( groups2073611262835488442omplex
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ Y @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69          = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
% 0.35/0.69            @ ( groups2073611262835488442omplex
% 0.35/0.69              @ ^ [J3: nat] :
% 0.35/0.69                  ( groups2073611262835488442omplex
% 0.35/0.69                  @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_complex @ Y @ K3 ) ) @ ( power_power_complex @ X @ J3 ) )
% 0.35/0.69                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
% 0.35/0.69              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_diff_alt
% 0.35/0.69  thf(fact_6881_polyfun__diff__alt,axiom,
% 0.35/0.69      ! [N: nat,A: nat > rat,X: rat,Y: rat] :
% 0.35/0.69        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.35/0.69       => ( ( minus_minus_rat
% 0.35/0.69            @ ( groups2906978787729119204at_rat
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ X @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69            @ ( groups2906978787729119204at_rat
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ Y @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69          = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
% 0.35/0.69            @ ( groups2906978787729119204at_rat
% 0.35/0.69              @ ^ [J3: nat] :
% 0.35/0.69                  ( groups2906978787729119204at_rat
% 0.35/0.69                  @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_rat @ Y @ K3 ) ) @ ( power_power_rat @ X @ J3 ) )
% 0.35/0.69                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
% 0.35/0.69              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_diff_alt
% 0.35/0.69  thf(fact_6882_polyfun__diff__alt,axiom,
% 0.35/0.69      ! [N: nat,A: nat > int,X: int,Y: int] :
% 0.35/0.69        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.35/0.69       => ( ( minus_minus_int
% 0.35/0.69            @ ( groups3539618377306564664at_int
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ X @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69            @ ( groups3539618377306564664at_int
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ Y @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69          = ( times_times_int @ ( minus_minus_int @ X @ Y )
% 0.35/0.69            @ ( groups3539618377306564664at_int
% 0.35/0.69              @ ^ [J3: nat] :
% 0.35/0.69                  ( groups3539618377306564664at_int
% 0.35/0.69                  @ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_int @ Y @ K3 ) ) @ ( power_power_int @ X @ J3 ) )
% 0.35/0.69                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
% 0.35/0.69              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_diff_alt
% 0.35/0.69  thf(fact_6883_polyfun__diff__alt,axiom,
% 0.35/0.69      ! [N: nat,A: nat > real,X: real,Y: real] :
% 0.35/0.69        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.35/0.69       => ( ( minus_minus_real
% 0.35/0.69            @ ( groups6591440286371151544t_real
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ X @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69            @ ( groups6591440286371151544t_real
% 0.35/0.69              @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ Y @ I4 ) )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69          = ( times_times_real @ ( minus_minus_real @ X @ Y )
% 0.35/0.69            @ ( groups6591440286371151544t_real
% 0.35/0.69              @ ^ [J3: nat] :
% 0.35/0.69                  ( groups6591440286371151544t_real
% 0.35/0.69                  @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_real @ Y @ K3 ) ) @ ( power_power_real @ X @ J3 ) )
% 0.35/0.69                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
% 0.35/0.69              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_diff_alt
% 0.35/0.69  thf(fact_6884_divide__int__unfold,axiom,
% 0.35/0.69      ! [L: int,K: int,N: nat,M: nat] :
% 0.35/0.69        ( ( ( ( ( sgn_sgn_int @ L )
% 0.35/0.69              = zero_zero_int )
% 0.35/0.69            | ( ( sgn_sgn_int @ K )
% 0.35/0.69              = zero_zero_int )
% 0.35/0.69            | ( N = zero_zero_nat ) )
% 0.35/0.69         => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.35/0.69            = zero_zero_int ) )
% 0.35/0.69        & ( ~ ( ( ( sgn_sgn_int @ L )
% 0.35/0.69                = zero_zero_int )
% 0.35/0.69              | ( ( sgn_sgn_int @ K )
% 0.35/0.69                = zero_zero_int )
% 0.35/0.69              | ( N = zero_zero_nat ) )
% 0.35/0.69         => ( ( ( ( sgn_sgn_int @ K )
% 0.35/0.69                = ( sgn_sgn_int @ L ) )
% 0.35/0.69             => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.35/0.69                = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
% 0.35/0.69            & ( ( ( sgn_sgn_int @ K )
% 0.35/0.69               != ( sgn_sgn_int @ L ) )
% 0.35/0.69             => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.35/0.69                = ( uminus_uminus_int
% 0.35/0.69                  @ ( semiri1314217659103216013at_int
% 0.35/0.69                    @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
% 0.35/0.69                      @ ( zero_n2687167440665602831ol_nat
% 0.35/0.69                        @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % divide_int_unfold
% 0.35/0.69  thf(fact_6885_polyfun__extremal__lemma,axiom,
% 0.35/0.69      ! [E2: real,C: nat > complex,N: nat] :
% 0.35/0.69        ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.35/0.69       => ? [M8: real] :
% 0.35/0.69          ! [Z5: complex] :
% 0.35/0.69            ( ( ord_less_eq_real @ M8 @ ( real_V1022390504157884413omplex @ Z5 ) )
% 0.35/0.69           => ( ord_less_eq_real
% 0.35/0.69              @ ( real_V1022390504157884413omplex
% 0.35/0.69                @ ( groups2073611262835488442omplex
% 0.35/0.69                  @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z5 @ I4 ) )
% 0.35/0.69                  @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69              @ ( times_times_real @ E2 @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( suc @ N ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_extremal_lemma
% 0.35/0.69  thf(fact_6886_polyfun__extremal__lemma,axiom,
% 0.35/0.69      ! [E2: real,C: nat > real,N: nat] :
% 0.35/0.69        ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.35/0.69       => ? [M8: real] :
% 0.35/0.69          ! [Z5: real] :
% 0.35/0.69            ( ( ord_less_eq_real @ M8 @ ( real_V7735802525324610683m_real @ Z5 ) )
% 0.35/0.69           => ( ord_less_eq_real
% 0.35/0.69              @ ( real_V7735802525324610683m_real
% 0.35/0.69                @ ( groups6591440286371151544t_real
% 0.35/0.69                  @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z5 @ I4 ) )
% 0.35/0.69                  @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69              @ ( times_times_real @ E2 @ ( power_power_real @ ( real_V7735802525324610683m_real @ Z5 ) @ ( suc @ N ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % polyfun_extremal_lemma
% 0.35/0.69  thf(fact_6887_choose__odd__sum,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.69       => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 0.35/0.69            @ ( groups2073611262835488442omplex
% 0.35/0.69              @ ^ [I4: nat] :
% 0.35/0.69                  ( if_complex
% 0.35/0.69                  @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 0.35/0.69                  @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) )
% 0.35/0.69                  @ zero_zero_complex )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69          = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % choose_odd_sum
% 0.35/0.69  thf(fact_6888_choose__odd__sum,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.69       => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 0.35/0.69            @ ( groups2906978787729119204at_rat
% 0.35/0.69              @ ^ [I4: nat] :
% 0.35/0.69                  ( if_rat
% 0.35/0.69                  @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 0.35/0.69                  @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) )
% 0.35/0.69                  @ zero_zero_rat )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69          = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % choose_odd_sum
% 0.35/0.69  thf(fact_6889_choose__odd__sum,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.69       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 0.35/0.69            @ ( groups3539618377306564664at_int
% 0.35/0.69              @ ^ [I4: nat] :
% 0.35/0.69                  ( if_int
% 0.35/0.69                  @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 0.35/0.69                  @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) )
% 0.35/0.69                  @ zero_zero_int )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69          = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % choose_odd_sum
% 0.35/0.69  thf(fact_6890_choose__odd__sum,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.69       => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.35/0.69            @ ( groups6591440286371151544t_real
% 0.35/0.69              @ ^ [I4: nat] :
% 0.35/0.69                  ( if_real
% 0.35/0.69                  @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 0.35/0.69                  @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) )
% 0.35/0.69                  @ zero_zero_real )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % choose_odd_sum
% 0.35/0.69  thf(fact_6891_choose__even__sum,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.69       => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 0.35/0.69            @ ( groups2073611262835488442omplex
% 0.35/0.69              @ ^ [I4: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) @ zero_zero_complex )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69          = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % choose_even_sum
% 0.35/0.69  thf(fact_6892_choose__even__sum,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.69       => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 0.35/0.69            @ ( groups2906978787729119204at_rat
% 0.35/0.69              @ ^ [I4: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) @ zero_zero_rat )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69          = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % choose_even_sum
% 0.35/0.69  thf(fact_6893_choose__even__sum,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.69       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 0.35/0.69            @ ( groups3539618377306564664at_int
% 0.35/0.69              @ ^ [I4: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) @ zero_zero_int )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69          = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % choose_even_sum
% 0.35/0.69  thf(fact_6894_choose__even__sum,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.69       => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.35/0.69            @ ( groups6591440286371151544t_real
% 0.35/0.69              @ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) @ zero_zero_real )
% 0.35/0.69              @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.69          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % choose_even_sum
% 0.35/0.69  thf(fact_6895_choose__alternating__sum,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.69       => ( ( groups2073611262835488442omplex
% 0.35/0.69            @ ^ [I4: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I4 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) )
% 0.35/0.69            @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69          = zero_zero_complex ) ) ).
% 0.35/0.69  
% 0.35/0.69  % choose_alternating_sum
% 0.35/0.69  thf(fact_6896_choose__alternating__sum,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.69       => ( ( groups2906978787729119204at_rat
% 0.35/0.69            @ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I4 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) )
% 0.35/0.69            @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69          = zero_zero_rat ) ) ).
% 0.35/0.69  
% 0.35/0.69  % choose_alternating_sum
% 0.35/0.69  thf(fact_6897_choose__alternating__sum,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.69       => ( ( groups7501900531339628137nteger
% 0.35/0.69            @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I4 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I4 ) ) )
% 0.35/0.69            @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69          = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.69  
% 0.35/0.69  % choose_alternating_sum
% 0.35/0.69  thf(fact_6898_choose__alternating__sum,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.69       => ( ( groups3539618377306564664at_int
% 0.35/0.69            @ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I4 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) )
% 0.35/0.69            @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69          = zero_zero_int ) ) ).
% 0.35/0.69  
% 0.35/0.69  % choose_alternating_sum
% 0.35/0.69  thf(fact_6899_choose__alternating__sum,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.69       => ( ( groups6591440286371151544t_real
% 0.35/0.69            @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) )
% 0.35/0.69            @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69          = zero_zero_real ) ) ).
% 0.35/0.69  
% 0.35/0.69  % choose_alternating_sum
% 0.35/0.69  thf(fact_6900_choose__alternating__linear__sum,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( N != one_one_nat )
% 0.35/0.69       => ( ( groups2073611262835488442omplex
% 0.35/0.69            @ ^ [I4: nat] : ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I4 ) @ ( semiri8010041392384452111omplex @ I4 ) ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) )
% 0.35/0.69            @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69          = zero_zero_complex ) ) ).
% 0.35/0.69  
% 0.35/0.69  % choose_alternating_linear_sum
% 0.35/0.69  thf(fact_6901_choose__alternating__linear__sum,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( N != one_one_nat )
% 0.35/0.69       => ( ( groups2906978787729119204at_rat
% 0.35/0.69            @ ^ [I4: nat] : ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I4 ) @ ( semiri681578069525770553at_rat @ I4 ) ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) )
% 0.35/0.69            @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69          = zero_zero_rat ) ) ).
% 0.35/0.69  
% 0.35/0.69  % choose_alternating_linear_sum
% 0.35/0.69  thf(fact_6902_choose__alternating__linear__sum,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( N != one_one_nat )
% 0.35/0.69       => ( ( groups7501900531339628137nteger
% 0.35/0.69            @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I4 ) @ ( semiri4939895301339042750nteger @ I4 ) ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I4 ) ) )
% 0.35/0.69            @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69          = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.69  
% 0.35/0.69  % choose_alternating_linear_sum
% 0.35/0.69  thf(fact_6903_choose__alternating__linear__sum,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( N != one_one_nat )
% 0.35/0.69       => ( ( groups3539618377306564664at_int
% 0.35/0.69            @ ^ [I4: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I4 ) @ ( semiri1314217659103216013at_int @ I4 ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) )
% 0.35/0.69            @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69          = zero_zero_int ) ) ).
% 0.35/0.69  
% 0.35/0.69  % choose_alternating_linear_sum
% 0.35/0.69  thf(fact_6904_choose__alternating__linear__sum,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( N != one_one_nat )
% 0.35/0.69       => ( ( groups6591440286371151544t_real
% 0.35/0.69            @ ^ [I4: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( semiri5074537144036343181t_real @ I4 ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) )
% 0.35/0.69            @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.69          = zero_zero_real ) ) ).
% 0.35/0.69  
% 0.35/0.69  % choose_alternating_linear_sum
% 0.35/0.69  thf(fact_6905_zero__less__binomial__iff,axiom,
% 0.35/0.69      ! [N: nat,K: nat] :
% 0.35/0.69        ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
% 0.35/0.69        = ( ord_less_eq_nat @ K @ N ) ) ).
% 0.35/0.69  
% 0.35/0.69  % zero_less_binomial_iff
% 0.35/0.69  thf(fact_6906_geometric__sums,axiom,
% 0.35/0.69      ! [C: real] :
% 0.35/0.69        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 0.35/0.69       => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % geometric_sums
% 0.35/0.69  thf(fact_6907_geometric__sums,axiom,
% 0.35/0.69      ! [C: complex] :
% 0.35/0.69        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 0.35/0.69       => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % geometric_sums
% 0.35/0.69  thf(fact_6908_sums__zero,axiom,
% 0.35/0.69      ( sums_complex
% 0.35/0.69      @ ^ [N2: nat] : zero_zero_complex
% 0.35/0.69      @ zero_zero_complex ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_zero
% 0.35/0.69  thf(fact_6909_sums__zero,axiom,
% 0.35/0.69      ( sums_real
% 0.35/0.69      @ ^ [N2: nat] : zero_zero_real
% 0.35/0.69      @ zero_zero_real ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_zero
% 0.35/0.69  thf(fact_6910_sums__zero,axiom,
% 0.35/0.69      ( sums_nat
% 0.35/0.69      @ ^ [N2: nat] : zero_zero_nat
% 0.35/0.69      @ zero_zero_nat ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_zero
% 0.35/0.69  thf(fact_6911_sums__zero,axiom,
% 0.35/0.69      ( sums_int
% 0.35/0.69      @ ^ [N2: nat] : zero_zero_int
% 0.35/0.69      @ zero_zero_int ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_zero
% 0.35/0.69  thf(fact_6912_binomial__1,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
% 0.35/0.69        = N ) ).
% 0.35/0.69  
% 0.35/0.69  % binomial_1
% 0.35/0.69  thf(fact_6913_binomial__0__Suc,axiom,
% 0.35/0.69      ! [K: nat] :
% 0.35/0.69        ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
% 0.35/0.69        = zero_zero_nat ) ).
% 0.35/0.69  
% 0.35/0.69  % binomial_0_Suc
% 0.35/0.69  thf(fact_6914_binomial__eq__0__iff,axiom,
% 0.35/0.69      ! [N: nat,K: nat] :
% 0.35/0.69        ( ( ( binomial @ N @ K )
% 0.35/0.69          = zero_zero_nat )
% 0.35/0.69        = ( ord_less_nat @ N @ K ) ) ).
% 0.35/0.69  
% 0.35/0.69  % binomial_eq_0_iff
% 0.35/0.69  thf(fact_6915_binomial__n__0,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( binomial @ N @ zero_zero_nat )
% 0.35/0.69        = one_one_nat ) ).
% 0.35/0.69  
% 0.35/0.69  % binomial_n_0
% 0.35/0.69  thf(fact_6916_sgn__real__def,axiom,
% 0.35/0.69      ( sgn_sgn_real
% 0.35/0.69      = ( ^ [A4: real] : ( if_real @ ( A4 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A4 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sgn_real_def
% 0.35/0.69  thf(fact_6917_sums__le,axiom,
% 0.35/0.69      ! [F: nat > nat,G: nat > nat,S: nat,T: nat] :
% 0.35/0.69        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.35/0.69       => ( ( sums_nat @ F @ S )
% 0.35/0.69         => ( ( sums_nat @ G @ T )
% 0.35/0.69           => ( ord_less_eq_nat @ S @ T ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_le
% 0.35/0.69  thf(fact_6918_sums__le,axiom,
% 0.35/0.69      ! [F: nat > int,G: nat > int,S: int,T: int] :
% 0.35/0.69        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.35/0.69       => ( ( sums_int @ F @ S )
% 0.35/0.69         => ( ( sums_int @ G @ T )
% 0.35/0.69           => ( ord_less_eq_int @ S @ T ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_le
% 0.35/0.69  thf(fact_6919_sums__0,axiom,
% 0.35/0.69      ! [F: nat > complex] :
% 0.35/0.69        ( ! [N3: nat] :
% 0.35/0.69            ( ( F @ N3 )
% 0.35/0.69            = zero_zero_complex )
% 0.35/0.69       => ( sums_complex @ F @ zero_zero_complex ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_0
% 0.35/0.69  thf(fact_6920_sums__0,axiom,
% 0.35/0.69      ! [F: nat > real] :
% 0.35/0.69        ( ! [N3: nat] :
% 0.35/0.69            ( ( F @ N3 )
% 0.35/0.69            = zero_zero_real )
% 0.35/0.69       => ( sums_real @ F @ zero_zero_real ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_0
% 0.35/0.69  thf(fact_6921_sums__0,axiom,
% 0.35/0.69      ! [F: nat > nat] :
% 0.35/0.69        ( ! [N3: nat] :
% 0.35/0.69            ( ( F @ N3 )
% 0.35/0.69            = zero_zero_nat )
% 0.35/0.69       => ( sums_nat @ F @ zero_zero_nat ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_0
% 0.35/0.69  thf(fact_6922_sums__0,axiom,
% 0.35/0.69      ! [F: nat > int] :
% 0.35/0.69        ( ! [N3: nat] :
% 0.35/0.69            ( ( F @ N3 )
% 0.35/0.69            = zero_zero_int )
% 0.35/0.69       => ( sums_int @ F @ zero_zero_int ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_0
% 0.35/0.69  thf(fact_6923_sums__single,axiom,
% 0.35/0.69      ! [I: nat,F: nat > complex] :
% 0.35/0.69        ( sums_complex
% 0.35/0.69        @ ^ [R5: nat] : ( if_complex @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_complex )
% 0.35/0.69        @ ( F @ I ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_single
% 0.35/0.69  thf(fact_6924_sums__single,axiom,
% 0.35/0.69      ! [I: nat,F: nat > real] :
% 0.35/0.69        ( sums_real
% 0.35/0.69        @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real )
% 0.35/0.69        @ ( F @ I ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_single
% 0.35/0.69  thf(fact_6925_sums__single,axiom,
% 0.35/0.69      ! [I: nat,F: nat > nat] :
% 0.35/0.69        ( sums_nat
% 0.35/0.69        @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat )
% 0.35/0.69        @ ( F @ I ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_single
% 0.35/0.69  thf(fact_6926_sums__single,axiom,
% 0.35/0.69      ! [I: nat,F: nat > int] :
% 0.35/0.69        ( sums_int
% 0.35/0.69        @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int )
% 0.35/0.69        @ ( F @ I ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_single
% 0.35/0.69  thf(fact_6927_binomial__eq__0,axiom,
% 0.35/0.69      ! [N: nat,K: nat] :
% 0.35/0.69        ( ( ord_less_nat @ N @ K )
% 0.35/0.69       => ( ( binomial @ N @ K )
% 0.35/0.69          = zero_zero_nat ) ) ).
% 0.35/0.69  
% 0.35/0.69  % binomial_eq_0
% 0.35/0.69  thf(fact_6928_binomial__symmetric,axiom,
% 0.35/0.69      ! [K: nat,N: nat] :
% 0.35/0.69        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.69       => ( ( binomial @ N @ K )
% 0.35/0.69          = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % binomial_symmetric
% 0.35/0.69  thf(fact_6929_binomial__le__pow,axiom,
% 0.35/0.69      ! [R2: nat,N: nat] :
% 0.35/0.69        ( ( ord_less_eq_nat @ R2 @ N )
% 0.35/0.69       => ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % binomial_le_pow
% 0.35/0.69  thf(fact_6930_sums__mult__iff,axiom,
% 0.35/0.69      ! [C: complex,F: nat > complex,D: complex] :
% 0.35/0.69        ( ( C != zero_zero_complex )
% 0.35/0.69       => ( ( sums_complex
% 0.35/0.69            @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
% 0.35/0.69            @ ( times_times_complex @ C @ D ) )
% 0.35/0.69          = ( sums_complex @ F @ D ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_mult_iff
% 0.35/0.69  thf(fact_6931_sums__mult__iff,axiom,
% 0.35/0.69      ! [C: real,F: nat > real,D: real] :
% 0.35/0.69        ( ( C != zero_zero_real )
% 0.35/0.69       => ( ( sums_real
% 0.35/0.69            @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
% 0.35/0.69            @ ( times_times_real @ C @ D ) )
% 0.35/0.69          = ( sums_real @ F @ D ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_mult_iff
% 0.35/0.69  thf(fact_6932_sums__mult2__iff,axiom,
% 0.35/0.69      ! [C: complex,F: nat > complex,D: complex] :
% 0.35/0.69        ( ( C != zero_zero_complex )
% 0.35/0.69       => ( ( sums_complex
% 0.35/0.69            @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ C )
% 0.35/0.69            @ ( times_times_complex @ D @ C ) )
% 0.35/0.69          = ( sums_complex @ F @ D ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_mult2_iff
% 0.35/0.69  thf(fact_6933_sums__mult2__iff,axiom,
% 0.35/0.69      ! [C: real,F: nat > real,D: real] :
% 0.35/0.69        ( ( C != zero_zero_real )
% 0.35/0.69       => ( ( sums_real
% 0.35/0.69            @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C )
% 0.35/0.69            @ ( times_times_real @ D @ C ) )
% 0.35/0.69          = ( sums_real @ F @ D ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_mult2_iff
% 0.35/0.69  thf(fact_6934_zero__less__binomial,axiom,
% 0.35/0.69      ! [K: nat,N: nat] :
% 0.35/0.69        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.69       => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % zero_less_binomial
% 0.35/0.69  thf(fact_6935_choose__mult,axiom,
% 0.35/0.69      ! [K: nat,M: nat,N: nat] :
% 0.35/0.69        ( ( ord_less_eq_nat @ K @ M )
% 0.35/0.69       => ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.69         => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
% 0.35/0.69            = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % choose_mult
% 0.35/0.69  thf(fact_6936_sums__mult__D,axiom,
% 0.35/0.69      ! [C: complex,F: nat > complex,A: complex] :
% 0.35/0.69        ( ( sums_complex
% 0.35/0.69          @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
% 0.35/0.69          @ A )
% 0.35/0.69       => ( ( C != zero_zero_complex )
% 0.35/0.69         => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_mult_D
% 0.35/0.69  thf(fact_6937_sums__mult__D,axiom,
% 0.35/0.69      ! [C: real,F: nat > real,A: real] :
% 0.35/0.69        ( ( sums_real
% 0.35/0.69          @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
% 0.35/0.69          @ A )
% 0.35/0.69       => ( ( C != zero_zero_real )
% 0.35/0.69         => ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_mult_D
% 0.35/0.69  thf(fact_6938_sums__Suc__imp,axiom,
% 0.35/0.69      ! [F: nat > complex,S: complex] :
% 0.35/0.69        ( ( ( F @ zero_zero_nat )
% 0.35/0.69          = zero_zero_complex )
% 0.35/0.69       => ( ( sums_complex
% 0.35/0.69            @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 0.35/0.69            @ S )
% 0.35/0.69         => ( sums_complex @ F @ S ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_Suc_imp
% 0.35/0.69  thf(fact_6939_sums__Suc__imp,axiom,
% 0.35/0.69      ! [F: nat > real,S: real] :
% 0.35/0.69        ( ( ( F @ zero_zero_nat )
% 0.35/0.69          = zero_zero_real )
% 0.35/0.69       => ( ( sums_real
% 0.35/0.69            @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 0.35/0.69            @ S )
% 0.35/0.69         => ( sums_real @ F @ S ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_Suc_imp
% 0.35/0.69  thf(fact_6940_sums__Suc,axiom,
% 0.35/0.69      ! [F: nat > real,L: real] :
% 0.35/0.69        ( ( sums_real
% 0.35/0.69          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 0.35/0.69          @ L )
% 0.35/0.69       => ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_Suc
% 0.35/0.69  thf(fact_6941_sums__Suc,axiom,
% 0.35/0.69      ! [F: nat > nat,L: nat] :
% 0.35/0.69        ( ( sums_nat
% 0.35/0.69          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 0.35/0.69          @ L )
% 0.35/0.69       => ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_Suc
% 0.35/0.69  thf(fact_6942_sums__Suc,axiom,
% 0.35/0.69      ! [F: nat > int,L: int] :
% 0.35/0.69        ( ( sums_int
% 0.35/0.69          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 0.35/0.69          @ L )
% 0.35/0.69       => ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_Suc
% 0.35/0.69  thf(fact_6943_sums__Suc__iff,axiom,
% 0.35/0.69      ! [F: nat > real,S: real] :
% 0.35/0.69        ( ( sums_real
% 0.35/0.69          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 0.35/0.69          @ S )
% 0.35/0.69        = ( sums_real @ F @ ( plus_plus_real @ S @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_Suc_iff
% 0.35/0.69  thf(fact_6944_sums__zero__iff__shift,axiom,
% 0.35/0.69      ! [N: nat,F: nat > complex,S: complex] :
% 0.35/0.69        ( ! [I2: nat] :
% 0.35/0.69            ( ( ord_less_nat @ I2 @ N )
% 0.35/0.69           => ( ( F @ I2 )
% 0.35/0.69              = zero_zero_complex ) )
% 0.35/0.69       => ( ( sums_complex
% 0.35/0.69            @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 0.35/0.69            @ S )
% 0.35/0.69          = ( sums_complex @ F @ S ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_zero_iff_shift
% 0.35/0.69  thf(fact_6945_sums__zero__iff__shift,axiom,
% 0.35/0.69      ! [N: nat,F: nat > real,S: real] :
% 0.35/0.69        ( ! [I2: nat] :
% 0.35/0.69            ( ( ord_less_nat @ I2 @ N )
% 0.35/0.69           => ( ( F @ I2 )
% 0.35/0.69              = zero_zero_real ) )
% 0.35/0.69       => ( ( sums_real
% 0.35/0.69            @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 0.35/0.69            @ S )
% 0.35/0.69          = ( sums_real @ F @ S ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_zero_iff_shift
% 0.35/0.69  thf(fact_6946_sums__finite,axiom,
% 0.35/0.69      ! [N6: set_nat,F: nat > complex] :
% 0.35/0.69        ( ( finite_finite_nat @ N6 )
% 0.35/0.69       => ( ! [N3: nat] :
% 0.35/0.69              ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.69             => ( ( F @ N3 )
% 0.35/0.69                = zero_zero_complex ) )
% 0.35/0.69         => ( sums_complex @ F @ ( groups2073611262835488442omplex @ F @ N6 ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_finite
% 0.35/0.69  thf(fact_6947_sums__finite,axiom,
% 0.35/0.69      ! [N6: set_nat,F: nat > int] :
% 0.35/0.69        ( ( finite_finite_nat @ N6 )
% 0.35/0.69       => ( ! [N3: nat] :
% 0.35/0.69              ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.69             => ( ( F @ N3 )
% 0.35/0.69                = zero_zero_int ) )
% 0.35/0.69         => ( sums_int @ F @ ( groups3539618377306564664at_int @ F @ N6 ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_finite
% 0.35/0.69  thf(fact_6948_sums__finite,axiom,
% 0.35/0.69      ! [N6: set_nat,F: nat > nat] :
% 0.35/0.69        ( ( finite_finite_nat @ N6 )
% 0.35/0.69       => ( ! [N3: nat] :
% 0.35/0.69              ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.69             => ( ( F @ N3 )
% 0.35/0.69                = zero_zero_nat ) )
% 0.35/0.69         => ( sums_nat @ F @ ( groups3542108847815614940at_nat @ F @ N6 ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_finite
% 0.35/0.69  thf(fact_6949_sums__finite,axiom,
% 0.35/0.69      ! [N6: set_nat,F: nat > real] :
% 0.35/0.69        ( ( finite_finite_nat @ N6 )
% 0.35/0.69       => ( ! [N3: nat] :
% 0.35/0.69              ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.69             => ( ( F @ N3 )
% 0.35/0.69                = zero_zero_real ) )
% 0.35/0.69         => ( sums_real @ F @ ( groups6591440286371151544t_real @ F @ N6 ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_finite
% 0.35/0.69  thf(fact_6950_sums__If__finite,axiom,
% 0.35/0.69      ! [P: nat > $o,F: nat > complex] :
% 0.35/0.69        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.35/0.69       => ( sums_complex
% 0.35/0.69          @ ^ [R5: nat] : ( if_complex @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_complex )
% 0.35/0.69          @ ( groups2073611262835488442omplex @ F @ ( collect_nat @ P ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_If_finite
% 0.35/0.69  thf(fact_6951_sums__If__finite,axiom,
% 0.35/0.69      ! [P: nat > $o,F: nat > int] :
% 0.35/0.69        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.35/0.69       => ( sums_int
% 0.35/0.69          @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int )
% 0.35/0.69          @ ( groups3539618377306564664at_int @ F @ ( collect_nat @ P ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_If_finite
% 0.35/0.69  thf(fact_6952_sums__If__finite,axiom,
% 0.35/0.69      ! [P: nat > $o,F: nat > nat] :
% 0.35/0.69        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.35/0.69       => ( sums_nat
% 0.35/0.69          @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat )
% 0.35/0.69          @ ( groups3542108847815614940at_nat @ F @ ( collect_nat @ P ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_If_finite
% 0.35/0.69  thf(fact_6953_sums__If__finite,axiom,
% 0.35/0.69      ! [P: nat > $o,F: nat > real] :
% 0.35/0.69        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.35/0.69       => ( sums_real
% 0.35/0.69          @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real )
% 0.35/0.69          @ ( groups6591440286371151544t_real @ F @ ( collect_nat @ P ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_If_finite
% 0.35/0.69  thf(fact_6954_sums__If__finite__set,axiom,
% 0.35/0.69      ! [A2: set_nat,F: nat > complex] :
% 0.35/0.69        ( ( finite_finite_nat @ A2 )
% 0.35/0.69       => ( sums_complex
% 0.35/0.69          @ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_complex )
% 0.35/0.69          @ ( groups2073611262835488442omplex @ F @ A2 ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_If_finite_set
% 0.35/0.69  thf(fact_6955_sums__If__finite__set,axiom,
% 0.35/0.69      ! [A2: set_nat,F: nat > int] :
% 0.35/0.69        ( ( finite_finite_nat @ A2 )
% 0.35/0.69       => ( sums_int
% 0.35/0.69          @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_int )
% 0.35/0.69          @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_If_finite_set
% 0.35/0.69  thf(fact_6956_sums__If__finite__set,axiom,
% 0.35/0.69      ! [A2: set_nat,F: nat > nat] :
% 0.35/0.69        ( ( finite_finite_nat @ A2 )
% 0.35/0.69       => ( sums_nat
% 0.35/0.69          @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_nat )
% 0.35/0.69          @ ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_If_finite_set
% 0.35/0.69  thf(fact_6957_sums__If__finite__set,axiom,
% 0.35/0.69      ! [A2: set_nat,F: nat > real] :
% 0.35/0.69        ( ( finite_finite_nat @ A2 )
% 0.35/0.69       => ( sums_real
% 0.35/0.69          @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_real )
% 0.35/0.69          @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_If_finite_set
% 0.35/0.69  thf(fact_6958_powser__sums__if,axiom,
% 0.35/0.69      ! [M: nat,Z2: complex] :
% 0.35/0.69        ( sums_complex
% 0.35/0.69        @ ^ [N2: nat] : ( times_times_complex @ ( if_complex @ ( N2 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z2 @ N2 ) )
% 0.35/0.69        @ ( power_power_complex @ Z2 @ M ) ) ).
% 0.35/0.69  
% 0.35/0.69  % powser_sums_if
% 0.35/0.69  thf(fact_6959_powser__sums__if,axiom,
% 0.35/0.69      ! [M: nat,Z2: real] :
% 0.35/0.69        ( sums_real
% 0.35/0.69        @ ^ [N2: nat] : ( times_times_real @ ( if_real @ ( N2 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z2 @ N2 ) )
% 0.35/0.69        @ ( power_power_real @ Z2 @ M ) ) ).
% 0.35/0.69  
% 0.35/0.69  % powser_sums_if
% 0.35/0.69  thf(fact_6960_powser__sums__if,axiom,
% 0.35/0.69      ! [M: nat,Z2: int] :
% 0.35/0.69        ( sums_int
% 0.35/0.69        @ ^ [N2: nat] : ( times_times_int @ ( if_int @ ( N2 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z2 @ N2 ) )
% 0.35/0.69        @ ( power_power_int @ Z2 @ M ) ) ).
% 0.35/0.69  
% 0.35/0.69  % powser_sums_if
% 0.35/0.69  thf(fact_6961_sums__If__finite__set_H,axiom,
% 0.35/0.69      ! [G: nat > real,S2: real,A2: set_nat,S5: real,F: nat > real] :
% 0.35/0.69        ( ( sums_real @ G @ S2 )
% 0.35/0.69       => ( ( finite_finite_nat @ A2 )
% 0.35/0.69         => ( ( S5
% 0.35/0.69              = ( plus_plus_real @ S2
% 0.35/0.69                @ ( groups6591440286371151544t_real
% 0.35/0.69                  @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.35/0.69                  @ A2 ) ) )
% 0.35/0.69           => ( sums_real
% 0.35/0.69              @ ^ [N2: nat] : ( if_real @ ( member_nat @ N2 @ A2 ) @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.35/0.69              @ S5 ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sums_If_finite_set'
% 0.35/0.69  thf(fact_6962_binomial__ge__n__over__k__pow__k,axiom,
% 0.35/0.69      ! [K: nat,N: nat] :
% 0.35/0.69        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.69       => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % binomial_ge_n_over_k_pow_k
% 0.35/0.69  thf(fact_6963_binomial__ge__n__over__k__pow__k,axiom,
% 0.35/0.69      ! [K: nat,N: nat] :
% 0.35/0.69        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.69       => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % binomial_ge_n_over_k_pow_k
% 0.35/0.69  thf(fact_6964_binomial__le__pow2,axiom,
% 0.35/0.69      ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.69  
% 0.35/0.69  % binomial_le_pow2
% 0.35/0.69  thf(fact_6965_choose__reduce__nat,axiom,
% 0.35/0.69      ! [N: nat,K: nat] :
% 0.35/0.69        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.69       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.69         => ( ( binomial @ N @ K )
% 0.35/0.69            = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % choose_reduce_nat
% 0.35/0.69  thf(fact_6966_times__binomial__minus1__eq,axiom,
% 0.35/0.69      ! [K: nat,N: nat] :
% 0.35/0.69        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.69       => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
% 0.35/0.69          = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % times_binomial_minus1_eq
% 0.35/0.69  thf(fact_6967_sum__choose__diagonal,axiom,
% 0.35/0.69      ! [M: nat,N: nat] :
% 0.35/0.69        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.69       => ( ( groups3542108847815614940at_nat
% 0.35/0.69            @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
% 0.35/0.69            @ ( set_ord_atMost_nat @ M ) )
% 0.35/0.69          = ( binomial @ ( suc @ N ) @ M ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sum_choose_diagonal
% 0.35/0.69  thf(fact_6968_binomial__addition__formula,axiom,
% 0.35/0.69      ! [N: nat,K: nat] :
% 0.35/0.69        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.69       => ( ( binomial @ N @ ( suc @ K ) )
% 0.35/0.69          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % binomial_addition_formula
% 0.35/0.69  thf(fact_6969_upto_Opinduct,axiom,
% 0.35/0.69      ! [A0: int,A12: int,P: int > int > $o] :
% 0.35/0.69        ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 0.35/0.69       => ( ! [I2: int,J2: int] :
% 0.35/0.69              ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
% 0.35/0.69             => ( ( ( ord_less_eq_int @ I2 @ J2 )
% 0.35/0.69                 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
% 0.35/0.69               => ( P @ I2 @ J2 ) ) )
% 0.35/0.69         => ( P @ A0 @ A12 ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % upto.pinduct
% 0.35/0.69  thf(fact_6970_gchoose__row__sum__weighted,axiom,
% 0.35/0.69      ! [R2: complex,M: nat] :
% 0.35/0.69        ( ( groups2073611262835488442omplex
% 0.35/0.69          @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ R2 @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
% 0.35/0.69          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 0.35/0.69        = ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R2 @ ( suc @ M ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % gchoose_row_sum_weighted
% 0.35/0.69  thf(fact_6971_gchoose__row__sum__weighted,axiom,
% 0.35/0.69      ! [R2: rat,M: nat] :
% 0.35/0.69        ( ( groups2906978787729119204at_rat
% 0.35/0.69          @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ R2 @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
% 0.35/0.69          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 0.35/0.69        = ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R2 @ ( suc @ M ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % gchoose_row_sum_weighted
% 0.35/0.69  thf(fact_6972_gchoose__row__sum__weighted,axiom,
% 0.35/0.69      ! [R2: real,M: nat] :
% 0.35/0.69        ( ( groups6591440286371151544t_real
% 0.35/0.69          @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ R2 @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ R2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
% 0.35/0.69          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 0.35/0.69        = ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R2 @ ( suc @ M ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % gchoose_row_sum_weighted
% 0.35/0.69  thf(fact_6973_sqrt__sum__squares__half__less,axiom,
% 0.35/0.69      ! [X: real,U: real,Y: real] :
% 0.35/0.69        ( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.69       => ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.69         => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.69           => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.35/0.69             => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sqrt_sum_squares_half_less
% 0.35/0.69  thf(fact_6974_log2__of__power__le,axiom,
% 0.35/0.69      ! [M: nat,N: nat] :
% 0.35/0.69        ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.69       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.69         => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % log2_of_power_le
% 0.35/0.69  thf(fact_6975_neg__numeral__le__ceiling,axiom,
% 0.35/0.69      ! [V: num,X: real] :
% 0.35/0.69        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
% 0.35/0.69        = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % neg_numeral_le_ceiling
% 0.35/0.69  thf(fact_6976_neg__numeral__le__ceiling,axiom,
% 0.35/0.69      ! [V: num,X: rat] :
% 0.35/0.69        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
% 0.35/0.69        = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % neg_numeral_le_ceiling
% 0.35/0.69  thf(fact_6977_ceiling__less__neg__numeral,axiom,
% 0.35/0.69      ! [X: real,V: num] :
% 0.35/0.69        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.69        = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_less_neg_numeral
% 0.35/0.69  thf(fact_6978_ceiling__less__neg__numeral,axiom,
% 0.35/0.69      ! [X: rat,V: num] :
% 0.35/0.69        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.69        = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_less_neg_numeral
% 0.35/0.69  thf(fact_6979_real__sqrt__less__iff,axiom,
% 0.35/0.69      ! [X: real,Y: real] :
% 0.35/0.69        ( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
% 0.35/0.69        = ( ord_less_real @ X @ Y ) ) ).
% 0.35/0.69  
% 0.35/0.69  % real_sqrt_less_iff
% 0.35/0.69  thf(fact_6980_gbinomial__0_I2_J,axiom,
% 0.35/0.69      ! [K: nat] :
% 0.35/0.69        ( ( gbinomial_complex @ zero_zero_complex @ ( suc @ K ) )
% 0.35/0.69        = zero_zero_complex ) ).
% 0.35/0.69  
% 0.35/0.69  % gbinomial_0(2)
% 0.35/0.69  thf(fact_6981_gbinomial__0_I2_J,axiom,
% 0.35/0.69      ! [K: nat] :
% 0.35/0.69        ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
% 0.35/0.69        = zero_zero_real ) ).
% 0.35/0.69  
% 0.35/0.69  % gbinomial_0(2)
% 0.35/0.69  thf(fact_6982_gbinomial__0_I2_J,axiom,
% 0.35/0.69      ! [K: nat] :
% 0.35/0.69        ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
% 0.35/0.69        = zero_zero_rat ) ).
% 0.35/0.69  
% 0.35/0.69  % gbinomial_0(2)
% 0.35/0.69  thf(fact_6983_gbinomial__0_I2_J,axiom,
% 0.35/0.69      ! [K: nat] :
% 0.35/0.69        ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
% 0.35/0.69        = zero_zero_nat ) ).
% 0.35/0.69  
% 0.35/0.69  % gbinomial_0(2)
% 0.35/0.69  thf(fact_6984_gbinomial__0_I2_J,axiom,
% 0.35/0.69      ! [K: nat] :
% 0.35/0.69        ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
% 0.35/0.69        = zero_zero_int ) ).
% 0.35/0.69  
% 0.35/0.69  % gbinomial_0(2)
% 0.35/0.69  thf(fact_6985_real__sqrt__gt__0__iff,axiom,
% 0.35/0.69      ! [Y: real] :
% 0.35/0.69        ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y ) )
% 0.35/0.69        = ( ord_less_real @ zero_zero_real @ Y ) ) ).
% 0.35/0.69  
% 0.35/0.69  % real_sqrt_gt_0_iff
% 0.35/0.69  thf(fact_6986_real__sqrt__lt__0__iff,axiom,
% 0.35/0.69      ! [X: real] :
% 0.35/0.69        ( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
% 0.35/0.69        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.35/0.69  
% 0.35/0.69  % real_sqrt_lt_0_iff
% 0.35/0.69  thf(fact_6987_ceiling__zero,axiom,
% 0.35/0.69      ( ( archim2889992004027027881ng_rat @ zero_zero_rat )
% 0.35/0.69      = zero_zero_int ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_zero
% 0.35/0.69  thf(fact_6988_ceiling__zero,axiom,
% 0.35/0.69      ( ( archim7802044766580827645g_real @ zero_zero_real )
% 0.35/0.69      = zero_zero_int ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_zero
% 0.35/0.69  thf(fact_6989_gbinomial__0_I1_J,axiom,
% 0.35/0.69      ! [A: complex] :
% 0.35/0.69        ( ( gbinomial_complex @ A @ zero_zero_nat )
% 0.35/0.69        = one_one_complex ) ).
% 0.35/0.69  
% 0.35/0.69  % gbinomial_0(1)
% 0.35/0.69  thf(fact_6990_gbinomial__0_I1_J,axiom,
% 0.35/0.69      ! [A: real] :
% 0.35/0.69        ( ( gbinomial_real @ A @ zero_zero_nat )
% 0.35/0.69        = one_one_real ) ).
% 0.35/0.69  
% 0.35/0.69  % gbinomial_0(1)
% 0.35/0.69  thf(fact_6991_gbinomial__0_I1_J,axiom,
% 0.35/0.69      ! [A: rat] :
% 0.35/0.69        ( ( gbinomial_rat @ A @ zero_zero_nat )
% 0.35/0.69        = one_one_rat ) ).
% 0.35/0.69  
% 0.35/0.69  % gbinomial_0(1)
% 0.35/0.69  thf(fact_6992_gbinomial__0_I1_J,axiom,
% 0.35/0.69      ! [A: nat] :
% 0.35/0.69        ( ( gbinomial_nat @ A @ zero_zero_nat )
% 0.35/0.69        = one_one_nat ) ).
% 0.35/0.69  
% 0.35/0.69  % gbinomial_0(1)
% 0.35/0.69  thf(fact_6993_gbinomial__0_I1_J,axiom,
% 0.35/0.69      ! [A: int] :
% 0.35/0.69        ( ( gbinomial_int @ A @ zero_zero_nat )
% 0.35/0.69        = one_one_int ) ).
% 0.35/0.69  
% 0.35/0.69  % gbinomial_0(1)
% 0.35/0.69  thf(fact_6994_real__sqrt__gt__1__iff,axiom,
% 0.35/0.69      ! [Y: real] :
% 0.35/0.69        ( ( ord_less_real @ one_one_real @ ( sqrt @ Y ) )
% 0.35/0.69        = ( ord_less_real @ one_one_real @ Y ) ) ).
% 0.35/0.69  
% 0.35/0.69  % real_sqrt_gt_1_iff
% 0.35/0.69  thf(fact_6995_real__sqrt__lt__1__iff,axiom,
% 0.35/0.69      ! [X: real] :
% 0.35/0.69        ( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
% 0.35/0.69        = ( ord_less_real @ X @ one_one_real ) ) ).
% 0.35/0.69  
% 0.35/0.69  % real_sqrt_lt_1_iff
% 0.35/0.69  thf(fact_6996_log__eq__one,axiom,
% 0.35/0.69      ! [A: real] :
% 0.35/0.69        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.69       => ( ( A != one_one_real )
% 0.35/0.69         => ( ( log @ A @ A )
% 0.35/0.69            = one_one_real ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % log_eq_one
% 0.35/0.69  thf(fact_6997_log__less__cancel__iff,axiom,
% 0.35/0.69      ! [A: real,X: real,Y: real] :
% 0.35/0.69        ( ( ord_less_real @ one_one_real @ A )
% 0.35/0.69       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.69         => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.35/0.69           => ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
% 0.35/0.69              = ( ord_less_real @ X @ Y ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % log_less_cancel_iff
% 0.35/0.69  thf(fact_6998_log__less__one__cancel__iff,axiom,
% 0.35/0.69      ! [A: real,X: real] :
% 0.35/0.69        ( ( ord_less_real @ one_one_real @ A )
% 0.35/0.69       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.69         => ( ( ord_less_real @ ( log @ A @ X ) @ one_one_real )
% 0.35/0.69            = ( ord_less_real @ X @ A ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % log_less_one_cancel_iff
% 0.35/0.69  thf(fact_6999_one__less__log__cancel__iff,axiom,
% 0.35/0.69      ! [A: real,X: real] :
% 0.35/0.69        ( ( ord_less_real @ one_one_real @ A )
% 0.35/0.69       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.69         => ( ( ord_less_real @ one_one_real @ ( log @ A @ X ) )
% 0.35/0.69            = ( ord_less_real @ A @ X ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % one_less_log_cancel_iff
% 0.35/0.69  thf(fact_7000_log__less__zero__cancel__iff,axiom,
% 0.35/0.69      ! [A: real,X: real] :
% 0.35/0.69        ( ( ord_less_real @ one_one_real @ A )
% 0.35/0.69       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.69         => ( ( ord_less_real @ ( log @ A @ X ) @ zero_zero_real )
% 0.35/0.69            = ( ord_less_real @ X @ one_one_real ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % log_less_zero_cancel_iff
% 0.35/0.69  thf(fact_7001_zero__less__log__cancel__iff,axiom,
% 0.35/0.69      ! [A: real,X: real] :
% 0.35/0.69        ( ( ord_less_real @ one_one_real @ A )
% 0.35/0.69       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.69         => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X ) )
% 0.35/0.69            = ( ord_less_real @ one_one_real @ X ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % zero_less_log_cancel_iff
% 0.35/0.69  thf(fact_7002_ceiling__le__zero,axiom,
% 0.35/0.69      ! [X: real] :
% 0.35/0.69        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
% 0.35/0.69        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_le_zero
% 0.35/0.69  thf(fact_7003_ceiling__le__zero,axiom,
% 0.35/0.69      ! [X: rat] :
% 0.35/0.69        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
% 0.35/0.69        = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_le_zero
% 0.35/0.69  thf(fact_7004_zero__less__ceiling,axiom,
% 0.35/0.69      ! [X: rat] :
% 0.35/0.69        ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
% 0.35/0.69        = ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % zero_less_ceiling
% 0.35/0.69  thf(fact_7005_zero__less__ceiling,axiom,
% 0.35/0.69      ! [X: real] :
% 0.35/0.69        ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
% 0.35/0.69        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % zero_less_ceiling
% 0.35/0.69  thf(fact_7006_ceiling__le__numeral,axiom,
% 0.35/0.69      ! [X: real,V: num] :
% 0.35/0.69        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.69        = ( ord_less_eq_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_le_numeral
% 0.35/0.69  thf(fact_7007_ceiling__le__numeral,axiom,
% 0.35/0.69      ! [X: rat,V: num] :
% 0.35/0.69        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.69        = ( ord_less_eq_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_le_numeral
% 0.35/0.69  thf(fact_7008_ceiling__less__one,axiom,
% 0.35/0.69      ! [X: real] :
% 0.35/0.69        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
% 0.35/0.69        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_less_one
% 0.35/0.69  thf(fact_7009_ceiling__less__one,axiom,
% 0.35/0.69      ! [X: rat] :
% 0.35/0.69        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
% 0.35/0.69        = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_less_one
% 0.35/0.69  thf(fact_7010_numeral__less__ceiling,axiom,
% 0.35/0.69      ! [V: num,X: real] :
% 0.35/0.69        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
% 0.35/0.69        = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % numeral_less_ceiling
% 0.35/0.69  thf(fact_7011_numeral__less__ceiling,axiom,
% 0.35/0.69      ! [V: num,X: rat] :
% 0.35/0.69        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
% 0.35/0.69        = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % numeral_less_ceiling
% 0.35/0.69  thf(fact_7012_one__le__ceiling,axiom,
% 0.35/0.69      ! [X: rat] :
% 0.35/0.69        ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
% 0.35/0.69        = ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % one_le_ceiling
% 0.35/0.69  thf(fact_7013_one__le__ceiling,axiom,
% 0.35/0.69      ! [X: real] :
% 0.35/0.69        ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
% 0.35/0.69        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % one_le_ceiling
% 0.35/0.69  thf(fact_7014_ceiling__le__one,axiom,
% 0.35/0.69      ! [X: real] :
% 0.35/0.69        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
% 0.35/0.69        = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_le_one
% 0.35/0.69  thf(fact_7015_ceiling__le__one,axiom,
% 0.35/0.69      ! [X: rat] :
% 0.35/0.69        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
% 0.35/0.69        = ( ord_less_eq_rat @ X @ one_one_rat ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_le_one
% 0.35/0.69  thf(fact_7016_one__less__ceiling,axiom,
% 0.35/0.69      ! [X: rat] :
% 0.35/0.69        ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
% 0.35/0.69        = ( ord_less_rat @ one_one_rat @ X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % one_less_ceiling
% 0.35/0.69  thf(fact_7017_one__less__ceiling,axiom,
% 0.35/0.69      ! [X: real] :
% 0.35/0.69        ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
% 0.35/0.69        = ( ord_less_real @ one_one_real @ X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % one_less_ceiling
% 0.35/0.69  thf(fact_7018_zero__le__log__cancel__iff,axiom,
% 0.35/0.69      ! [A: real,X: real] :
% 0.35/0.69        ( ( ord_less_real @ one_one_real @ A )
% 0.35/0.69       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.69         => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
% 0.35/0.69            = ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % zero_le_log_cancel_iff
% 0.35/0.69  thf(fact_7019_log__le__zero__cancel__iff,axiom,
% 0.35/0.69      ! [A: real,X: real] :
% 0.35/0.69        ( ( ord_less_real @ one_one_real @ A )
% 0.35/0.69       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.69         => ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
% 0.35/0.69            = ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % log_le_zero_cancel_iff
% 0.35/0.69  thf(fact_7020_one__le__log__cancel__iff,axiom,
% 0.35/0.69      ! [A: real,X: real] :
% 0.35/0.69        ( ( ord_less_real @ one_one_real @ A )
% 0.35/0.69       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.69         => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
% 0.35/0.69            = ( ord_less_eq_real @ A @ X ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % one_le_log_cancel_iff
% 0.35/0.69  thf(fact_7021_log__le__one__cancel__iff,axiom,
% 0.35/0.69      ! [A: real,X: real] :
% 0.35/0.69        ( ( ord_less_real @ one_one_real @ A )
% 0.35/0.69       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.69         => ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
% 0.35/0.69            = ( ord_less_eq_real @ X @ A ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % log_le_one_cancel_iff
% 0.35/0.69  thf(fact_7022_log__le__cancel__iff,axiom,
% 0.35/0.69      ! [A: real,X: real,Y: real] :
% 0.35/0.69        ( ( ord_less_real @ one_one_real @ A )
% 0.35/0.69       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.69         => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.35/0.69           => ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
% 0.35/0.69              = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % log_le_cancel_iff
% 0.35/0.69  thf(fact_7023_ceiling__less__zero,axiom,
% 0.35/0.69      ! [X: real] :
% 0.35/0.69        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
% 0.35/0.69        = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_less_zero
% 0.35/0.69  thf(fact_7024_ceiling__less__zero,axiom,
% 0.35/0.69      ! [X: rat] :
% 0.35/0.69        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
% 0.35/0.69        = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_less_zero
% 0.35/0.69  thf(fact_7025_zero__le__ceiling,axiom,
% 0.35/0.69      ! [X: real] :
% 0.35/0.69        ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
% 0.35/0.69        = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % zero_le_ceiling
% 0.35/0.69  thf(fact_7026_zero__le__ceiling,axiom,
% 0.35/0.69      ! [X: rat] :
% 0.35/0.69        ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
% 0.35/0.69        = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % zero_le_ceiling
% 0.35/0.69  thf(fact_7027_log__pow__cancel,axiom,
% 0.35/0.69      ! [A: real,B: nat] :
% 0.35/0.69        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.69       => ( ( A != one_one_real )
% 0.35/0.69         => ( ( log @ A @ ( power_power_real @ A @ B ) )
% 0.35/0.69            = ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % log_pow_cancel
% 0.35/0.69  thf(fact_7028_ceiling__less__numeral,axiom,
% 0.35/0.69      ! [X: real,V: num] :
% 0.35/0.69        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.69        = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_less_numeral
% 0.35/0.69  thf(fact_7029_ceiling__less__numeral,axiom,
% 0.35/0.69      ! [X: rat,V: num] :
% 0.35/0.69        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.69        = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_less_numeral
% 0.35/0.69  thf(fact_7030_numeral__le__ceiling,axiom,
% 0.35/0.69      ! [V: num,X: real] :
% 0.35/0.69        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
% 0.35/0.69        = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % numeral_le_ceiling
% 0.35/0.69  thf(fact_7031_numeral__le__ceiling,axiom,
% 0.35/0.69      ! [V: num,X: rat] :
% 0.35/0.69        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
% 0.35/0.69        = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % numeral_le_ceiling
% 0.35/0.69  thf(fact_7032_ceiling__le__neg__numeral,axiom,
% 0.35/0.69      ! [X: real,V: num] :
% 0.35/0.69        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.69        = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_le_neg_numeral
% 0.35/0.69  thf(fact_7033_ceiling__le__neg__numeral,axiom,
% 0.35/0.69      ! [X: rat,V: num] :
% 0.35/0.69        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.69        = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_le_neg_numeral
% 0.35/0.69  thf(fact_7034_neg__numeral__less__ceiling,axiom,
% 0.35/0.69      ! [V: num,X: real] :
% 0.35/0.69        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
% 0.35/0.69        = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % neg_numeral_less_ceiling
% 0.35/0.69  thf(fact_7035_neg__numeral__less__ceiling,axiom,
% 0.35/0.69      ! [V: num,X: rat] :
% 0.35/0.69        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
% 0.35/0.69        = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % neg_numeral_less_ceiling
% 0.35/0.69  thf(fact_7036_real__sqrt__less__mono,axiom,
% 0.35/0.69      ! [X: real,Y: real] :
% 0.35/0.69        ( ( ord_less_real @ X @ Y )
% 0.35/0.69       => ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % real_sqrt_less_mono
% 0.35/0.69  thf(fact_7037_real__sqrt__gt__zero,axiom,
% 0.35/0.69      ! [X: real] :
% 0.35/0.69        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.69       => ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % real_sqrt_gt_zero
% 0.35/0.69  thf(fact_7038_ceiling__mono,axiom,
% 0.35/0.69      ! [Y: real,X: real] :
% 0.35/0.69        ( ( ord_less_eq_real @ Y @ X )
% 0.35/0.69       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y ) @ ( archim7802044766580827645g_real @ X ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_mono
% 0.35/0.69  thf(fact_7039_ceiling__mono,axiom,
% 0.35/0.69      ! [Y: rat,X: rat] :
% 0.35/0.69        ( ( ord_less_eq_rat @ Y @ X )
% 0.35/0.69       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y ) @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_mono
% 0.35/0.69  thf(fact_7040_le__of__int__ceiling,axiom,
% 0.35/0.69      ! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % le_of_int_ceiling
% 0.35/0.69  thf(fact_7041_le__of__int__ceiling,axiom,
% 0.35/0.69      ! [X: rat] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % le_of_int_ceiling
% 0.35/0.69  thf(fact_7042_ceiling__less__cancel,axiom,
% 0.35/0.69      ! [X: rat,Y: rat] :
% 0.35/0.69        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) )
% 0.35/0.69       => ( ord_less_rat @ X @ Y ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_less_cancel
% 0.35/0.69  thf(fact_7043_ceiling__less__cancel,axiom,
% 0.35/0.69      ! [X: real,Y: real] :
% 0.35/0.69        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) )
% 0.35/0.69       => ( ord_less_real @ X @ Y ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_less_cancel
% 0.35/0.69  thf(fact_7044_ceiling__le__iff,axiom,
% 0.35/0.69      ! [X: real,Z2: int] :
% 0.35/0.69        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z2 )
% 0.35/0.69        = ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_le_iff
% 0.35/0.69  thf(fact_7045_ceiling__le__iff,axiom,
% 0.35/0.69      ! [X: rat,Z2: int] :
% 0.35/0.69        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 )
% 0.35/0.69        = ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_le_iff
% 0.35/0.69  thf(fact_7046_ceiling__le,axiom,
% 0.35/0.69      ! [X: real,A: int] :
% 0.35/0.69        ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) )
% 0.35/0.69       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ A ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_le
% 0.35/0.69  thf(fact_7047_ceiling__le,axiom,
% 0.35/0.69      ! [X: rat,A: int] :
% 0.35/0.69        ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) )
% 0.35/0.69       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ A ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_le
% 0.35/0.69  thf(fact_7048_less__ceiling__iff,axiom,
% 0.35/0.69      ! [Z2: int,X: rat] :
% 0.35/0.69        ( ( ord_less_int @ Z2 @ ( archim2889992004027027881ng_rat @ X ) )
% 0.35/0.69        = ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % less_ceiling_iff
% 0.35/0.69  thf(fact_7049_less__ceiling__iff,axiom,
% 0.35/0.69      ! [Z2: int,X: real] :
% 0.35/0.69        ( ( ord_less_int @ Z2 @ ( archim7802044766580827645g_real @ X ) )
% 0.35/0.69        = ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % less_ceiling_iff
% 0.35/0.69  thf(fact_7050_gbinomial__of__nat__symmetric,axiom,
% 0.35/0.69      ! [K: nat,N: nat] :
% 0.35/0.69        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.69       => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
% 0.35/0.69          = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % gbinomial_of_nat_symmetric
% 0.35/0.69  thf(fact_7051_sqrt2__less__2,axiom,
% 0.35/0.69      ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.35/0.69  
% 0.35/0.69  % sqrt2_less_2
% 0.35/0.69  thf(fact_7052_of__int__ceiling__le__add__one,axiom,
% 0.35/0.69      ! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).
% 0.35/0.69  
% 0.35/0.69  % of_int_ceiling_le_add_one
% 0.35/0.69  thf(fact_7053_of__int__ceiling__le__add__one,axiom,
% 0.35/0.69      ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).
% 0.35/0.69  
% 0.35/0.69  % of_int_ceiling_le_add_one
% 0.35/0.69  thf(fact_7054_log__base__change,axiom,
% 0.35/0.69      ! [A: real,B: real,X: real] :
% 0.35/0.69        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.69       => ( ( A != one_one_real )
% 0.35/0.69         => ( ( log @ B @ X )
% 0.35/0.69            = ( divide_divide_real @ ( log @ A @ X ) @ ( log @ A @ B ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % log_base_change
% 0.35/0.69  thf(fact_7055_of__int__ceiling__diff__one__le,axiom,
% 0.35/0.69      ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).
% 0.35/0.69  
% 0.35/0.69  % of_int_ceiling_diff_one_le
% 0.35/0.69  thf(fact_7056_of__int__ceiling__diff__one__le,axiom,
% 0.35/0.69      ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).
% 0.35/0.69  
% 0.35/0.69  % of_int_ceiling_diff_one_le
% 0.35/0.69  thf(fact_7057_log__of__power__eq,axiom,
% 0.35/0.69      ! [M: nat,B: real,N: nat] :
% 0.35/0.69        ( ( ( semiri5074537144036343181t_real @ M )
% 0.35/0.69          = ( power_power_real @ B @ N ) )
% 0.35/0.69       => ( ( ord_less_real @ one_one_real @ B )
% 0.35/0.69         => ( ( semiri5074537144036343181t_real @ N )
% 0.35/0.69            = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % log_of_power_eq
% 0.35/0.69  thf(fact_7058_less__log__of__power,axiom,
% 0.35/0.69      ! [B: real,N: nat,M: real] :
% 0.35/0.69        ( ( ord_less_real @ ( power_power_real @ B @ N ) @ M )
% 0.35/0.69       => ( ( ord_less_real @ one_one_real @ B )
% 0.35/0.69         => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % less_log_of_power
% 0.35/0.69  thf(fact_7059_gbinomial__ge__n__over__k__pow__k,axiom,
% 0.35/0.69      ! [K: nat,A: real] :
% 0.35/0.69        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
% 0.35/0.69       => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % gbinomial_ge_n_over_k_pow_k
% 0.35/0.69  thf(fact_7060_gbinomial__ge__n__over__k__pow__k,axiom,
% 0.35/0.69      ! [K: nat,A: rat] :
% 0.35/0.69        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
% 0.35/0.69       => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % gbinomial_ge_n_over_k_pow_k
% 0.35/0.69  thf(fact_7061_real__less__rsqrt,axiom,
% 0.35/0.69      ! [X: real,Y: real] :
% 0.35/0.69        ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
% 0.35/0.69       => ( ord_less_real @ X @ ( sqrt @ Y ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % real_less_rsqrt
% 0.35/0.69  thf(fact_7062_ceiling__split,axiom,
% 0.35/0.69      ! [P: int > $o,T: real] :
% 0.35/0.69        ( ( P @ ( archim7802044766580827645g_real @ T ) )
% 0.35/0.69        = ( ! [I4: int] :
% 0.35/0.69              ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) @ T )
% 0.35/0.69                & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I4 ) ) )
% 0.35/0.69             => ( P @ I4 ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_split
% 0.35/0.69  thf(fact_7063_ceiling__split,axiom,
% 0.35/0.69      ! [P: int > $o,T: rat] :
% 0.35/0.69        ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
% 0.35/0.69        = ( ! [I4: int] :
% 0.35/0.69              ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) @ T )
% 0.35/0.69                & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I4 ) ) )
% 0.35/0.69             => ( P @ I4 ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_split
% 0.35/0.69  thf(fact_7064_ceiling__eq__iff,axiom,
% 0.35/0.69      ! [X: real,A: int] :
% 0.35/0.69        ( ( ( archim7802044766580827645g_real @ X )
% 0.35/0.69          = A )
% 0.35/0.69        = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X )
% 0.35/0.69          & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_eq_iff
% 0.35/0.69  thf(fact_7065_ceiling__eq__iff,axiom,
% 0.35/0.69      ! [X: rat,A: int] :
% 0.35/0.69        ( ( ( archim2889992004027027881ng_rat @ X )
% 0.35/0.69          = A )
% 0.35/0.69        = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X )
% 0.35/0.69          & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_eq_iff
% 0.35/0.69  thf(fact_7066_ceiling__unique,axiom,
% 0.35/0.69      ! [Z2: int,X: real] :
% 0.35/0.69        ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X )
% 0.35/0.69       => ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) )
% 0.35/0.69         => ( ( archim7802044766580827645g_real @ X )
% 0.35/0.69            = Z2 ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_unique
% 0.35/0.69  thf(fact_7067_ceiling__unique,axiom,
% 0.35/0.69      ! [Z2: int,X: rat] :
% 0.35/0.69        ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X )
% 0.35/0.69       => ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z2 ) )
% 0.35/0.69         => ( ( archim2889992004027027881ng_rat @ X )
% 0.35/0.69            = Z2 ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_unique
% 0.35/0.69  thf(fact_7068_ceiling__correct,axiom,
% 0.35/0.69      ! [X: real] :
% 0.35/0.69        ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) @ one_one_real ) @ X )
% 0.35/0.69        & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_correct
% 0.35/0.69  thf(fact_7069_ceiling__correct,axiom,
% 0.35/0.69      ! [X: rat] :
% 0.35/0.69        ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) @ one_one_rat ) @ X )
% 0.35/0.69        & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_correct
% 0.35/0.69  thf(fact_7070_mult__ceiling__le,axiom,
% 0.35/0.69      ! [A: real,B: real] :
% 0.35/0.69        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.35/0.69       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.35/0.69         => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % mult_ceiling_le
% 0.35/0.69  thf(fact_7071_mult__ceiling__le,axiom,
% 0.35/0.69      ! [A: rat,B: rat] :
% 0.35/0.69        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.35/0.69       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.35/0.69         => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % mult_ceiling_le
% 0.35/0.69  thf(fact_7072_log__mult,axiom,
% 0.35/0.69      ! [A: real,X: real,Y: real] :
% 0.35/0.69        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.69       => ( ( A != one_one_real )
% 0.35/0.69         => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.69           => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.35/0.69             => ( ( log @ A @ ( times_times_real @ X @ Y ) )
% 0.35/0.69                = ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % log_mult
% 0.35/0.69  thf(fact_7073_ceiling__less__iff,axiom,
% 0.35/0.69      ! [X: real,Z2: int] :
% 0.35/0.69        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ Z2 )
% 0.35/0.69        = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_less_iff
% 0.35/0.69  thf(fact_7074_ceiling__less__iff,axiom,
% 0.35/0.69      ! [X: rat,Z2: int] :
% 0.35/0.69        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 )
% 0.35/0.69        = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_less_iff
% 0.35/0.69  thf(fact_7075_le__ceiling__iff,axiom,
% 0.35/0.69      ! [Z2: int,X: rat] :
% 0.35/0.69        ( ( ord_less_eq_int @ Z2 @ ( archim2889992004027027881ng_rat @ X ) )
% 0.35/0.69        = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % le_ceiling_iff
% 0.35/0.69  thf(fact_7076_le__ceiling__iff,axiom,
% 0.35/0.69      ! [Z2: int,X: real] :
% 0.35/0.69        ( ( ord_less_eq_int @ Z2 @ ( archim7802044766580827645g_real @ X ) )
% 0.35/0.69        = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X ) ) ).
% 0.35/0.69  
% 0.35/0.69  % le_ceiling_iff
% 0.35/0.69  thf(fact_7077_log__divide,axiom,
% 0.35/0.69      ! [A: real,X: real,Y: real] :
% 0.35/0.69        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.69       => ( ( A != one_one_real )
% 0.35/0.69         => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.69           => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.35/0.69             => ( ( log @ A @ ( divide_divide_real @ X @ Y ) )
% 0.35/0.69                = ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % log_divide
% 0.35/0.69  thf(fact_7078_le__log__of__power,axiom,
% 0.35/0.69      ! [B: real,N: nat,M: real] :
% 0.35/0.69        ( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M )
% 0.35/0.69       => ( ( ord_less_real @ one_one_real @ B )
% 0.35/0.69         => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % le_log_of_power
% 0.35/0.69  thf(fact_7079_log__base__pow,axiom,
% 0.35/0.69      ! [A: real,N: nat,X: real] :
% 0.35/0.69        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.69       => ( ( log @ ( power_power_real @ A @ N ) @ X )
% 0.35/0.69          = ( divide_divide_real @ ( log @ A @ X ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % log_base_pow
% 0.35/0.69  thf(fact_7080_log__nat__power,axiom,
% 0.35/0.69      ! [X: real,B: real,N: nat] :
% 0.35/0.69        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.69       => ( ( log @ B @ ( power_power_real @ X @ N ) )
% 0.35/0.69          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % log_nat_power
% 0.35/0.69  thf(fact_7081_gbinomial__trinomial__revision,axiom,
% 0.35/0.69      ! [K: nat,M: nat,A: complex] :
% 0.35/0.69        ( ( ord_less_eq_nat @ K @ M )
% 0.35/0.69       => ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
% 0.35/0.69          = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % gbinomial_trinomial_revision
% 0.35/0.69  thf(fact_7082_gbinomial__trinomial__revision,axiom,
% 0.35/0.69      ! [K: nat,M: nat,A: rat] :
% 0.35/0.69        ( ( ord_less_eq_nat @ K @ M )
% 0.35/0.69       => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
% 0.35/0.69          = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % gbinomial_trinomial_revision
% 0.35/0.69  thf(fact_7083_gbinomial__trinomial__revision,axiom,
% 0.35/0.69      ! [K: nat,M: nat,A: real] :
% 0.35/0.69        ( ( ord_less_eq_nat @ K @ M )
% 0.35/0.69       => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
% 0.35/0.69          = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % gbinomial_trinomial_revision
% 0.35/0.69  thf(fact_7084_ceiling__log2__div2,axiom,
% 0.35/0.69      ! [N: nat] :
% 0.35/0.69        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.69       => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.35/0.69          = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_log2_div2
% 0.35/0.69  thf(fact_7085_ceiling__log__nat__eq__if,axiom,
% 0.35/0.69      ! [B: nat,N: nat,K: nat] :
% 0.35/0.69        ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.35/0.69       => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 0.35/0.69         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.35/0.69           => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.35/0.69              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_log_nat_eq_if
% 0.35/0.69  thf(fact_7086_ceiling__divide__upper,axiom,
% 0.35/0.69      ! [Q4: real,P4: real] :
% 0.35/0.69        ( ( ord_less_real @ zero_zero_real @ Q4 )
% 0.35/0.69       => ( ord_less_eq_real @ P4 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P4 @ Q4 ) ) ) @ Q4 ) ) ) ).
% 0.35/0.69  
% 0.35/0.69  % ceiling_divide_upper
% 0.35/0.69  thf(fact_7087_ceiling__divide__upper,axiom,
% 0.35/0.69      ! [Q4: rat,P4: rat] :
% 0.35/0.69        ( ( ord_less_rat @ zero_zero_rat @ Q4 )
% 0.35/0.70       => ( ord_less_eq_rat @ P4 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P4 @ Q4 ) ) ) @ Q4 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % ceiling_divide_upper
% 0.35/0.70  thf(fact_7088_lemma__real__divide__sqrt__less,axiom,
% 0.35/0.70      ! [U: real] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ U )
% 0.35/0.70       => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 0.35/0.70  
% 0.35/0.70  % lemma_real_divide_sqrt_less
% 0.35/0.70  thf(fact_7089_log__of__power__less,axiom,
% 0.35/0.70      ! [M: nat,B: real,N: nat] :
% 0.35/0.70        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 0.35/0.70       => ( ( ord_less_real @ one_one_real @ B )
% 0.35/0.70         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.70           => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % log_of_power_less
% 0.35/0.70  thf(fact_7090_ceiling__log__nat__eq__powr__iff,axiom,
% 0.35/0.70      ! [B: nat,K: nat,N: nat] :
% 0.35/0.70        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.35/0.70       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.70         => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.35/0.70              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
% 0.35/0.70            = ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.35/0.70              & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % ceiling_log_nat_eq_powr_iff
% 0.35/0.70  thf(fact_7091_real__less__lsqrt,axiom,
% 0.35/0.70      ! [X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.70       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.35/0.70         => ( ( ord_less_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.70           => ( ord_less_real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % real_less_lsqrt
% 0.35/0.70  thf(fact_7092_ceiling__divide__lower,axiom,
% 0.35/0.70      ! [Q4: real,P4: real] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ Q4 )
% 0.35/0.70       => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P4 @ Q4 ) ) ) @ one_one_real ) @ Q4 ) @ P4 ) ) ).
% 0.35/0.70  
% 0.35/0.70  % ceiling_divide_lower
% 0.35/0.70  thf(fact_7093_ceiling__divide__lower,axiom,
% 0.35/0.70      ! [Q4: rat,P4: rat] :
% 0.35/0.70        ( ( ord_less_rat @ zero_zero_rat @ Q4 )
% 0.35/0.70       => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P4 @ Q4 ) ) ) @ one_one_rat ) @ Q4 ) @ P4 ) ) ).
% 0.35/0.70  
% 0.35/0.70  % ceiling_divide_lower
% 0.35/0.70  thf(fact_7094_log__of__power__le,axiom,
% 0.35/0.70      ! [M: nat,B: real,N: nat] :
% 0.35/0.70        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 0.35/0.70       => ( ( ord_less_real @ one_one_real @ B )
% 0.35/0.70         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.70           => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % log_of_power_le
% 0.35/0.70  thf(fact_7095_ceiling__eq,axiom,
% 0.35/0.70      ! [N: int,X: real] :
% 0.35/0.70        ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
% 0.35/0.70       => ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.35/0.70         => ( ( archim7802044766580827645g_real @ X )
% 0.35/0.70            = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % ceiling_eq
% 0.35/0.70  thf(fact_7096_ceiling__eq,axiom,
% 0.35/0.70      ! [N: int,X: rat] :
% 0.35/0.70        ( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X )
% 0.35/0.70       => ( ( ord_less_eq_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
% 0.35/0.70         => ( ( archim2889992004027027881ng_rat @ X )
% 0.35/0.70            = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % ceiling_eq
% 0.35/0.70  thf(fact_7097_gbinomial__reduce__nat,axiom,
% 0.35/0.70      ! [K: nat,A: complex] :
% 0.35/0.70        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.70       => ( ( gbinomial_complex @ A @ K )
% 0.35/0.70          = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % gbinomial_reduce_nat
% 0.35/0.70  thf(fact_7098_gbinomial__reduce__nat,axiom,
% 0.35/0.70      ! [K: nat,A: real] :
% 0.35/0.70        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.70       => ( ( gbinomial_real @ A @ K )
% 0.35/0.70          = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % gbinomial_reduce_nat
% 0.35/0.70  thf(fact_7099_gbinomial__reduce__nat,axiom,
% 0.35/0.70      ! [K: nat,A: rat] :
% 0.35/0.70        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.70       => ( ( gbinomial_rat @ A @ K )
% 0.35/0.70          = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % gbinomial_reduce_nat
% 0.35/0.70  thf(fact_7100_arsinh__real__aux,axiom,
% 0.35/0.70      ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % arsinh_real_aux
% 0.35/0.70  thf(fact_7101_less__log2__of__power,axiom,
% 0.35/0.70      ! [N: nat,M: nat] :
% 0.35/0.70        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 0.35/0.70       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % less_log2_of_power
% 0.35/0.70  thf(fact_7102_le__log2__of__power,axiom,
% 0.35/0.70      ! [N: nat,M: nat] :
% 0.35/0.70        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 0.35/0.70       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % le_log2_of_power
% 0.35/0.70  thf(fact_7103_gbinomial__sum__up__index,axiom,
% 0.35/0.70      ! [K: nat,N: nat] :
% 0.35/0.70        ( ( groups2073611262835488442omplex
% 0.35/0.70          @ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ K )
% 0.35/0.70          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.70        = ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % gbinomial_sum_up_index
% 0.35/0.70  thf(fact_7104_gbinomial__sum__up__index,axiom,
% 0.35/0.70      ! [K: nat,N: nat] :
% 0.35/0.70        ( ( groups2906978787729119204at_rat
% 0.35/0.70          @ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K )
% 0.35/0.70          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.70        = ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % gbinomial_sum_up_index
% 0.35/0.70  thf(fact_7105_gbinomial__sum__up__index,axiom,
% 0.35/0.70      ! [K: nat,N: nat] :
% 0.35/0.70        ( ( groups6591440286371151544t_real
% 0.35/0.70          @ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K )
% 0.35/0.70          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.70        = ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % gbinomial_sum_up_index
% 0.35/0.70  thf(fact_7106_log2__of__power__less,axiom,
% 0.35/0.70      ! [M: nat,N: nat] :
% 0.35/0.70        ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.70       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.70         => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % log2_of_power_less
% 0.35/0.70  thf(fact_7107_gbinomial__absorption_H,axiom,
% 0.35/0.70      ! [K: nat,A: complex] :
% 0.35/0.70        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.70       => ( ( gbinomial_complex @ A @ K )
% 0.35/0.70          = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % gbinomial_absorption'
% 0.35/0.70  thf(fact_7108_gbinomial__absorption_H,axiom,
% 0.35/0.70      ! [K: nat,A: rat] :
% 0.35/0.70        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.70       => ( ( gbinomial_rat @ A @ K )
% 0.35/0.70          = ( times_times_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % gbinomial_absorption'
% 0.35/0.70  thf(fact_7109_gbinomial__absorption_H,axiom,
% 0.35/0.70      ! [K: nat,A: real] :
% 0.35/0.70        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.70       => ( ( gbinomial_real @ A @ K )
% 0.35/0.70          = ( times_times_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % gbinomial_absorption'
% 0.35/0.70  thf(fact_7110_floor__log__nat__eq__powr__iff,axiom,
% 0.35/0.70      ! [B: nat,K: nat,N: nat] :
% 0.35/0.70        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.35/0.70       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.70         => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.35/0.70              = ( semiri1314217659103216013at_int @ N ) )
% 0.35/0.70            = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.35/0.70              & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_log_nat_eq_powr_iff
% 0.35/0.70  thf(fact_7111_ceiling__log__eq__powr__iff,axiom,
% 0.35/0.70      ! [X: real,B: real,K: nat] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70       => ( ( ord_less_real @ one_one_real @ B )
% 0.35/0.70         => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
% 0.35/0.70              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 0.35/0.70            = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
% 0.35/0.70              & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % ceiling_log_eq_powr_iff
% 0.35/0.70  thf(fact_7112_floor__log__nat__eq__if,axiom,
% 0.35/0.70      ! [B: nat,N: nat,K: nat] :
% 0.35/0.70        ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.35/0.70       => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 0.35/0.70         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.35/0.70           => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.35/0.70              = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_log_nat_eq_if
% 0.35/0.70  thf(fact_7113_floor__log2__div2,axiom,
% 0.35/0.70      ! [N: nat] :
% 0.35/0.70        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.70       => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.35/0.70          = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_log2_div2
% 0.35/0.70  thf(fact_7114_real__sqrt__sum__squares__less,axiom,
% 0.35/0.70      ! [X: real,U: real,Y: real] :
% 0.35/0.70        ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.35/0.70       => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.35/0.70         => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % real_sqrt_sum_squares_less
% 0.35/0.70  thf(fact_7115_dbl__dec__simps_I2_J,axiom,
% 0.35/0.70      ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 0.35/0.70      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.35/0.70  
% 0.35/0.70  % dbl_dec_simps(2)
% 0.35/0.70  thf(fact_7116_dbl__dec__simps_I2_J,axiom,
% 0.35/0.70      ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 0.35/0.70      = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.35/0.70  
% 0.35/0.70  % dbl_dec_simps(2)
% 0.35/0.70  thf(fact_7117_dbl__dec__simps_I2_J,axiom,
% 0.35/0.70      ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 0.35/0.70      = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.35/0.70  
% 0.35/0.70  % dbl_dec_simps(2)
% 0.35/0.70  thf(fact_7118_dbl__dec__simps_I2_J,axiom,
% 0.35/0.70      ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 0.35/0.70      = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.35/0.70  
% 0.35/0.70  % dbl_dec_simps(2)
% 0.35/0.70  thf(fact_7119_dbl__dec__simps_I2_J,axiom,
% 0.35/0.70      ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 0.35/0.70      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.35/0.70  
% 0.35/0.70  % dbl_dec_simps(2)
% 0.35/0.70  thf(fact_7120_abs__idempotent,axiom,
% 0.35/0.70      ! [A: real] :
% 0.35/0.70        ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.35/0.70        = ( abs_abs_real @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_idempotent
% 0.35/0.70  thf(fact_7121_abs__idempotent,axiom,
% 0.35/0.70      ! [A: int] :
% 0.35/0.70        ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 0.35/0.70        = ( abs_abs_int @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_idempotent
% 0.35/0.70  thf(fact_7122_abs__idempotent,axiom,
% 0.35/0.70      ! [A: rat] :
% 0.35/0.70        ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 0.35/0.70        = ( abs_abs_rat @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_idempotent
% 0.35/0.70  thf(fact_7123_abs__idempotent,axiom,
% 0.35/0.70      ! [A: code_integer] :
% 0.35/0.70        ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.35/0.70        = ( abs_abs_Code_integer @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_idempotent
% 0.35/0.70  thf(fact_7124_abs__0__eq,axiom,
% 0.35/0.70      ! [A: code_integer] :
% 0.35/0.70        ( ( zero_z3403309356797280102nteger
% 0.35/0.70          = ( abs_abs_Code_integer @ A ) )
% 0.35/0.70        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_0_eq
% 0.35/0.70  thf(fact_7125_abs__0__eq,axiom,
% 0.35/0.70      ! [A: real] :
% 0.35/0.70        ( ( zero_zero_real
% 0.35/0.70          = ( abs_abs_real @ A ) )
% 0.35/0.70        = ( A = zero_zero_real ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_0_eq
% 0.35/0.70  thf(fact_7126_abs__0__eq,axiom,
% 0.35/0.70      ! [A: rat] :
% 0.35/0.70        ( ( zero_zero_rat
% 0.35/0.70          = ( abs_abs_rat @ A ) )
% 0.35/0.70        = ( A = zero_zero_rat ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_0_eq
% 0.35/0.70  thf(fact_7127_abs__0__eq,axiom,
% 0.35/0.70      ! [A: int] :
% 0.35/0.70        ( ( zero_zero_int
% 0.35/0.70          = ( abs_abs_int @ A ) )
% 0.35/0.70        = ( A = zero_zero_int ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_0_eq
% 0.35/0.70  thf(fact_7128_abs__eq__0,axiom,
% 0.35/0.70      ! [A: code_integer] :
% 0.35/0.70        ( ( ( abs_abs_Code_integer @ A )
% 0.35/0.70          = zero_z3403309356797280102nteger )
% 0.35/0.70        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_eq_0
% 0.35/0.70  thf(fact_7129_abs__eq__0,axiom,
% 0.35/0.70      ! [A: real] :
% 0.35/0.70        ( ( ( abs_abs_real @ A )
% 0.35/0.70          = zero_zero_real )
% 0.35/0.70        = ( A = zero_zero_real ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_eq_0
% 0.35/0.70  thf(fact_7130_abs__eq__0,axiom,
% 0.35/0.70      ! [A: rat] :
% 0.35/0.70        ( ( ( abs_abs_rat @ A )
% 0.35/0.70          = zero_zero_rat )
% 0.35/0.70        = ( A = zero_zero_rat ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_eq_0
% 0.35/0.70  thf(fact_7131_abs__eq__0,axiom,
% 0.35/0.70      ! [A: int] :
% 0.35/0.70        ( ( ( abs_abs_int @ A )
% 0.35/0.70          = zero_zero_int )
% 0.35/0.70        = ( A = zero_zero_int ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_eq_0
% 0.35/0.70  thf(fact_7132_abs__zero,axiom,
% 0.35/0.70      ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 0.35/0.70      = zero_z3403309356797280102nteger ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_zero
% 0.35/0.70  thf(fact_7133_abs__zero,axiom,
% 0.35/0.70      ( ( abs_abs_real @ zero_zero_real )
% 0.35/0.70      = zero_zero_real ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_zero
% 0.35/0.70  thf(fact_7134_abs__zero,axiom,
% 0.35/0.70      ( ( abs_abs_rat @ zero_zero_rat )
% 0.35/0.70      = zero_zero_rat ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_zero
% 0.35/0.70  thf(fact_7135_abs__zero,axiom,
% 0.35/0.70      ( ( abs_abs_int @ zero_zero_int )
% 0.35/0.70      = zero_zero_int ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_zero
% 0.35/0.70  thf(fact_7136_abs__0,axiom,
% 0.35/0.70      ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 0.35/0.70      = zero_z3403309356797280102nteger ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_0
% 0.35/0.70  thf(fact_7137_abs__0,axiom,
% 0.35/0.70      ( ( abs_abs_complex @ zero_zero_complex )
% 0.35/0.70      = zero_zero_complex ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_0
% 0.35/0.70  thf(fact_7138_abs__0,axiom,
% 0.35/0.70      ( ( abs_abs_real @ zero_zero_real )
% 0.35/0.70      = zero_zero_real ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_0
% 0.35/0.70  thf(fact_7139_abs__0,axiom,
% 0.35/0.70      ( ( abs_abs_rat @ zero_zero_rat )
% 0.35/0.70      = zero_zero_rat ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_0
% 0.35/0.70  thf(fact_7140_abs__0,axiom,
% 0.35/0.70      ( ( abs_abs_int @ zero_zero_int )
% 0.35/0.70      = zero_zero_int ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_0
% 0.35/0.70  thf(fact_7141_abs__add__abs,axiom,
% 0.35/0.70      ! [A: code_integer,B: code_integer] :
% 0.35/0.70        ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
% 0.35/0.70        = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_add_abs
% 0.35/0.70  thf(fact_7142_abs__add__abs,axiom,
% 0.35/0.70      ! [A: real,B: real] :
% 0.35/0.70        ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 0.35/0.70        = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_add_abs
% 0.35/0.70  thf(fact_7143_abs__add__abs,axiom,
% 0.35/0.70      ! [A: rat,B: rat] :
% 0.35/0.70        ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
% 0.35/0.70        = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_add_abs
% 0.35/0.70  thf(fact_7144_abs__add__abs,axiom,
% 0.35/0.70      ! [A: int,B: int] :
% 0.35/0.70        ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
% 0.35/0.70        = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_add_abs
% 0.35/0.70  thf(fact_7145_abs__minus__cancel,axiom,
% 0.35/0.70      ! [A: int] :
% 0.35/0.70        ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 0.35/0.70        = ( abs_abs_int @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_minus_cancel
% 0.35/0.70  thf(fact_7146_abs__minus__cancel,axiom,
% 0.35/0.70      ! [A: real] :
% 0.35/0.70        ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 0.35/0.70        = ( abs_abs_real @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_minus_cancel
% 0.35/0.70  thf(fact_7147_abs__minus__cancel,axiom,
% 0.35/0.70      ! [A: rat] :
% 0.35/0.70        ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 0.35/0.70        = ( abs_abs_rat @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_minus_cancel
% 0.35/0.70  thf(fact_7148_abs__minus__cancel,axiom,
% 0.35/0.70      ! [A: code_integer] :
% 0.35/0.70        ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 0.35/0.70        = ( abs_abs_Code_integer @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_minus_cancel
% 0.35/0.70  thf(fact_7149_abs__of__nat,axiom,
% 0.35/0.70      ! [N: nat] :
% 0.35/0.70        ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.35/0.70        = ( semiri681578069525770553at_rat @ N ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_of_nat
% 0.35/0.70  thf(fact_7150_abs__of__nat,axiom,
% 0.35/0.70      ! [N: nat] :
% 0.35/0.70        ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 0.35/0.70        = ( semiri4939895301339042750nteger @ N ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_of_nat
% 0.35/0.70  thf(fact_7151_abs__of__nat,axiom,
% 0.35/0.70      ! [N: nat] :
% 0.35/0.70        ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.35/0.70        = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_of_nat
% 0.35/0.70  thf(fact_7152_abs__of__nat,axiom,
% 0.35/0.70      ! [N: nat] :
% 0.35/0.70        ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.35/0.70        = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_of_nat
% 0.35/0.70  thf(fact_7153_powr__0,axiom,
% 0.35/0.70      ! [Z2: real] :
% 0.35/0.70        ( ( powr_real @ zero_zero_real @ Z2 )
% 0.35/0.70        = zero_zero_real ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_0
% 0.35/0.70  thf(fact_7154_powr__eq__0__iff,axiom,
% 0.35/0.70      ! [W2: real,Z2: real] :
% 0.35/0.70        ( ( ( powr_real @ W2 @ Z2 )
% 0.35/0.70          = zero_zero_real )
% 0.35/0.70        = ( W2 = zero_zero_real ) ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_eq_0_iff
% 0.35/0.70  thf(fact_7155_abs__le__zero__iff,axiom,
% 0.35/0.70      ! [A: code_integer] :
% 0.35/0.70        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 0.35/0.70        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_zero_iff
% 0.35/0.70  thf(fact_7156_abs__le__zero__iff,axiom,
% 0.35/0.70      ! [A: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 0.35/0.70        = ( A = zero_zero_real ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_zero_iff
% 0.35/0.70  thf(fact_7157_abs__le__zero__iff,axiom,
% 0.35/0.70      ! [A: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
% 0.35/0.70        = ( A = zero_zero_rat ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_zero_iff
% 0.35/0.70  thf(fact_7158_abs__le__zero__iff,axiom,
% 0.35/0.70      ! [A: int] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
% 0.35/0.70        = ( A = zero_zero_int ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_zero_iff
% 0.35/0.70  thf(fact_7159_abs__le__self__iff,axiom,
% 0.35/0.70      ! [A: code_integer] :
% 0.35/0.70        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
% 0.35/0.70        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_self_iff
% 0.35/0.70  thf(fact_7160_abs__le__self__iff,axiom,
% 0.35/0.70      ! [A: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 0.35/0.70        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_self_iff
% 0.35/0.70  thf(fact_7161_abs__le__self__iff,axiom,
% 0.35/0.70      ! [A: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
% 0.35/0.70        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_self_iff
% 0.35/0.70  thf(fact_7162_abs__le__self__iff,axiom,
% 0.35/0.70      ! [A: int] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
% 0.35/0.70        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_self_iff
% 0.35/0.70  thf(fact_7163_abs__of__nonneg,axiom,
% 0.35/0.70      ! [A: code_integer] :
% 0.35/0.70        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.35/0.70       => ( ( abs_abs_Code_integer @ A )
% 0.35/0.70          = A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_of_nonneg
% 0.35/0.70  thf(fact_7164_abs__of__nonneg,axiom,
% 0.35/0.70      ! [A: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.35/0.70       => ( ( abs_abs_real @ A )
% 0.35/0.70          = A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_of_nonneg
% 0.35/0.70  thf(fact_7165_abs__of__nonneg,axiom,
% 0.35/0.70      ! [A: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.35/0.70       => ( ( abs_abs_rat @ A )
% 0.35/0.70          = A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_of_nonneg
% 0.35/0.70  thf(fact_7166_abs__of__nonneg,axiom,
% 0.35/0.70      ! [A: int] :
% 0.35/0.70        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.35/0.70       => ( ( abs_abs_int @ A )
% 0.35/0.70          = A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_of_nonneg
% 0.35/0.70  thf(fact_7167_zero__less__abs__iff,axiom,
% 0.35/0.70      ! [A: code_integer] :
% 0.35/0.70        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
% 0.35/0.70        = ( A != zero_z3403309356797280102nteger ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zero_less_abs_iff
% 0.35/0.70  thf(fact_7168_zero__less__abs__iff,axiom,
% 0.35/0.70      ! [A: real] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 0.35/0.70        = ( A != zero_zero_real ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zero_less_abs_iff
% 0.35/0.70  thf(fact_7169_zero__less__abs__iff,axiom,
% 0.35/0.70      ! [A: rat] :
% 0.35/0.70        ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
% 0.35/0.70        = ( A != zero_zero_rat ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zero_less_abs_iff
% 0.35/0.70  thf(fact_7170_zero__less__abs__iff,axiom,
% 0.35/0.70      ! [A: int] :
% 0.35/0.70        ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
% 0.35/0.70        = ( A != zero_zero_int ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zero_less_abs_iff
% 0.35/0.70  thf(fact_7171_powr__zero__eq__one,axiom,
% 0.35/0.70      ! [X: real] :
% 0.35/0.70        ( ( ( X = zero_zero_real )
% 0.35/0.70         => ( ( powr_real @ X @ zero_zero_real )
% 0.35/0.70            = zero_zero_real ) )
% 0.35/0.70        & ( ( X != zero_zero_real )
% 0.35/0.70         => ( ( powr_real @ X @ zero_zero_real )
% 0.35/0.70            = one_one_real ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_zero_eq_one
% 0.35/0.70  thf(fact_7172_floor__zero,axiom,
% 0.35/0.70      ( ( archim6058952711729229775r_real @ zero_zero_real )
% 0.35/0.70      = zero_zero_int ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_zero
% 0.35/0.70  thf(fact_7173_floor__zero,axiom,
% 0.35/0.70      ( ( archim3151403230148437115or_rat @ zero_zero_rat )
% 0.35/0.70      = zero_zero_int ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_zero
% 0.35/0.70  thf(fact_7174_powr__gt__zero,axiom,
% 0.35/0.70      ! [X: real,A: real] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A ) )
% 0.35/0.70        = ( X != zero_zero_real ) ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_gt_zero
% 0.35/0.70  thf(fact_7175_powr__less__cancel__iff,axiom,
% 0.35/0.70      ! [X: real,A: real,B: real] :
% 0.35/0.70        ( ( ord_less_real @ one_one_real @ X )
% 0.35/0.70       => ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 0.35/0.70          = ( ord_less_real @ A @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_less_cancel_iff
% 0.35/0.70  thf(fact_7176_sum__abs,axiom,
% 0.35/0.70      ! [F: nat > real,A2: set_nat] :
% 0.35/0.70        ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 0.35/0.70        @ ( groups6591440286371151544t_real
% 0.35/0.70          @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
% 0.35/0.70          @ A2 ) ) ).
% 0.35/0.70  
% 0.35/0.70  % sum_abs
% 0.35/0.70  thf(fact_7177_zero__le__divide__abs__iff,axiom,
% 0.35/0.70      ! [A: real,B: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
% 0.35/0.70        = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.35/0.70          | ( B = zero_zero_real ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zero_le_divide_abs_iff
% 0.35/0.70  thf(fact_7178_zero__le__divide__abs__iff,axiom,
% 0.35/0.70      ! [A: rat,B: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
% 0.35/0.70        = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.35/0.70          | ( B = zero_zero_rat ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zero_le_divide_abs_iff
% 0.35/0.70  thf(fact_7179_divide__le__0__abs__iff,axiom,
% 0.35/0.70      ! [A: real,B: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
% 0.35/0.70        = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.35/0.70          | ( B = zero_zero_real ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % divide_le_0_abs_iff
% 0.35/0.70  thf(fact_7180_divide__le__0__abs__iff,axiom,
% 0.35/0.70      ! [A: rat,B: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
% 0.35/0.70        = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.35/0.70          | ( B = zero_zero_rat ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % divide_le_0_abs_iff
% 0.35/0.70  thf(fact_7181_abs__of__nonpos,axiom,
% 0.35/0.70      ! [A: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.35/0.70       => ( ( abs_abs_real @ A )
% 0.35/0.70          = ( uminus_uminus_real @ A ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_of_nonpos
% 0.35/0.70  thf(fact_7182_abs__of__nonpos,axiom,
% 0.35/0.70      ! [A: code_integer] :
% 0.35/0.70        ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
% 0.35/0.70       => ( ( abs_abs_Code_integer @ A )
% 0.35/0.70          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_of_nonpos
% 0.35/0.70  thf(fact_7183_abs__of__nonpos,axiom,
% 0.35/0.70      ! [A: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.35/0.70       => ( ( abs_abs_rat @ A )
% 0.35/0.70          = ( uminus_uminus_rat @ A ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_of_nonpos
% 0.35/0.70  thf(fact_7184_abs__of__nonpos,axiom,
% 0.35/0.70      ! [A: int] :
% 0.35/0.70        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.35/0.70       => ( ( abs_abs_int @ A )
% 0.35/0.70          = ( uminus_uminus_int @ A ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_of_nonpos
% 0.35/0.70  thf(fact_7185_abs__sgn__eq__1,axiom,
% 0.35/0.70      ! [A: code_integer] :
% 0.35/0.70        ( ( A != zero_z3403309356797280102nteger )
% 0.35/0.70       => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.35/0.70          = one_one_Code_integer ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_sgn_eq_1
% 0.35/0.70  thf(fact_7186_abs__sgn__eq__1,axiom,
% 0.35/0.70      ! [A: real] :
% 0.35/0.70        ( ( A != zero_zero_real )
% 0.35/0.70       => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 0.35/0.70          = one_one_real ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_sgn_eq_1
% 0.35/0.70  thf(fact_7187_abs__sgn__eq__1,axiom,
% 0.35/0.70      ! [A: rat] :
% 0.35/0.70        ( ( A != zero_zero_rat )
% 0.35/0.70       => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 0.35/0.70          = one_one_rat ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_sgn_eq_1
% 0.35/0.70  thf(fact_7188_abs__sgn__eq__1,axiom,
% 0.35/0.70      ! [A: int] :
% 0.35/0.70        ( ( A != zero_zero_int )
% 0.35/0.70       => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 0.35/0.70          = one_one_int ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_sgn_eq_1
% 0.35/0.70  thf(fact_7189_powr__eq__one__iff,axiom,
% 0.35/0.70      ! [A: real,X: real] :
% 0.35/0.70        ( ( ord_less_real @ one_one_real @ A )
% 0.35/0.70       => ( ( ( powr_real @ A @ X )
% 0.35/0.70            = one_one_real )
% 0.35/0.70          = ( X = zero_zero_real ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_eq_one_iff
% 0.35/0.70  thf(fact_7190_powr__le__cancel__iff,axiom,
% 0.35/0.70      ! [X: real,A: real,B: real] :
% 0.35/0.70        ( ( ord_less_real @ one_one_real @ X )
% 0.35/0.70       => ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 0.35/0.70          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_le_cancel_iff
% 0.35/0.70  thf(fact_7191_artanh__minus__real,axiom,
% 0.35/0.70      ! [X: real] :
% 0.35/0.70        ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.35/0.70       => ( ( artanh_real @ ( uminus_uminus_real @ X ) )
% 0.35/0.70          = ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % artanh_minus_real
% 0.35/0.70  thf(fact_7192_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.35/0.70      ! [A: code_integer] :
% 0.35/0.70        ( ( sgn_sgn_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.35/0.70        = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % idom_abs_sgn_class.abs_sgn
% 0.35/0.70  thf(fact_7193_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.35/0.70      ! [A: complex] :
% 0.35/0.70        ( ( sgn_sgn_complex @ ( abs_abs_complex @ A ) )
% 0.35/0.70        = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % idom_abs_sgn_class.abs_sgn
% 0.35/0.70  thf(fact_7194_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.35/0.70      ! [A: real] :
% 0.35/0.70        ( ( sgn_sgn_real @ ( abs_abs_real @ A ) )
% 0.35/0.70        = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % idom_abs_sgn_class.abs_sgn
% 0.35/0.70  thf(fact_7195_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.35/0.70      ! [A: rat] :
% 0.35/0.70        ( ( sgn_sgn_rat @ ( abs_abs_rat @ A ) )
% 0.35/0.70        = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % idom_abs_sgn_class.abs_sgn
% 0.35/0.70  thf(fact_7196_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.35/0.70      ! [A: int] :
% 0.35/0.70        ( ( sgn_sgn_int @ ( abs_abs_int @ A ) )
% 0.35/0.70        = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % idom_abs_sgn_class.abs_sgn
% 0.35/0.70  thf(fact_7197_sgn__abs,axiom,
% 0.35/0.70      ! [A: code_integer] :
% 0.35/0.70        ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.35/0.70        = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % sgn_abs
% 0.35/0.70  thf(fact_7198_sgn__abs,axiom,
% 0.35/0.70      ! [A: complex] :
% 0.35/0.70        ( ( abs_abs_complex @ ( sgn_sgn_complex @ A ) )
% 0.35/0.70        = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % sgn_abs
% 0.35/0.70  thf(fact_7199_sgn__abs,axiom,
% 0.35/0.70      ! [A: real] :
% 0.35/0.70        ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 0.35/0.70        = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % sgn_abs
% 0.35/0.70  thf(fact_7200_sgn__abs,axiom,
% 0.35/0.70      ! [A: rat] :
% 0.35/0.70        ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 0.35/0.70        = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % sgn_abs
% 0.35/0.70  thf(fact_7201_sgn__abs,axiom,
% 0.35/0.70      ! [A: int] :
% 0.35/0.70        ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 0.35/0.70        = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % sgn_abs
% 0.35/0.70  thf(fact_7202_sum__abs__ge__zero,axiom,
% 0.35/0.70      ! [F: nat > real,A2: set_nat] :
% 0.35/0.70        ( ord_less_eq_real @ zero_zero_real
% 0.35/0.70        @ ( groups6591440286371151544t_real
% 0.35/0.70          @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
% 0.35/0.70          @ A2 ) ) ).
% 0.35/0.70  
% 0.35/0.70  % sum_abs_ge_zero
% 0.35/0.70  thf(fact_7203_zero__less__power__abs__iff,axiom,
% 0.35/0.70      ! [A: code_integer,N: nat] :
% 0.35/0.70        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
% 0.35/0.70        = ( ( A != zero_z3403309356797280102nteger )
% 0.35/0.70          | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zero_less_power_abs_iff
% 0.35/0.70  thf(fact_7204_zero__less__power__abs__iff,axiom,
% 0.35/0.70      ! [A: real,N: nat] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 0.35/0.70        = ( ( A != zero_zero_real )
% 0.35/0.70          | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zero_less_power_abs_iff
% 0.35/0.70  thf(fact_7205_zero__less__power__abs__iff,axiom,
% 0.35/0.70      ! [A: rat,N: nat] :
% 0.35/0.70        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
% 0.35/0.70        = ( ( A != zero_zero_rat )
% 0.35/0.70          | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zero_less_power_abs_iff
% 0.35/0.70  thf(fact_7206_zero__less__power__abs__iff,axiom,
% 0.35/0.70      ! [A: int,N: nat] :
% 0.35/0.70        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
% 0.35/0.70        = ( ( A != zero_zero_int )
% 0.35/0.70          | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zero_less_power_abs_iff
% 0.35/0.70  thf(fact_7207_zero__le__floor,axiom,
% 0.35/0.70      ! [X: real] :
% 0.35/0.70        ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
% 0.35/0.70        = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zero_le_floor
% 0.35/0.70  thf(fact_7208_zero__le__floor,axiom,
% 0.35/0.70      ! [X: rat] :
% 0.35/0.70        ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
% 0.35/0.70        = ( ord_less_eq_rat @ zero_zero_rat @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zero_le_floor
% 0.35/0.70  thf(fact_7209_floor__less__zero,axiom,
% 0.35/0.70      ! [X: real] :
% 0.35/0.70        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
% 0.35/0.70        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_less_zero
% 0.35/0.70  thf(fact_7210_floor__less__zero,axiom,
% 0.35/0.70      ! [X: rat] :
% 0.35/0.70        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
% 0.35/0.70        = ( ord_less_rat @ X @ zero_zero_rat ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_less_zero
% 0.35/0.70  thf(fact_7211_numeral__le__floor,axiom,
% 0.35/0.70      ! [V: num,X: real] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
% 0.35/0.70        = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % numeral_le_floor
% 0.35/0.70  thf(fact_7212_numeral__le__floor,axiom,
% 0.35/0.70      ! [V: num,X: rat] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
% 0.35/0.70        = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % numeral_le_floor
% 0.35/0.70  thf(fact_7213_zero__less__floor,axiom,
% 0.35/0.70      ! [X: real] :
% 0.35/0.70        ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
% 0.35/0.70        = ( ord_less_eq_real @ one_one_real @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zero_less_floor
% 0.35/0.70  thf(fact_7214_zero__less__floor,axiom,
% 0.35/0.70      ! [X: rat] :
% 0.35/0.70        ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
% 0.35/0.70        = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zero_less_floor
% 0.35/0.70  thf(fact_7215_floor__le__zero,axiom,
% 0.35/0.70      ! [X: real] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
% 0.35/0.70        = ( ord_less_real @ X @ one_one_real ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_le_zero
% 0.35/0.70  thf(fact_7216_floor__le__zero,axiom,
% 0.35/0.70      ! [X: rat] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
% 0.35/0.70        = ( ord_less_rat @ X @ one_one_rat ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_le_zero
% 0.35/0.70  thf(fact_7217_floor__less__numeral,axiom,
% 0.35/0.70      ! [X: real,V: num] :
% 0.35/0.70        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.70        = ( ord_less_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_less_numeral
% 0.35/0.70  thf(fact_7218_floor__less__numeral,axiom,
% 0.35/0.70      ! [X: rat,V: num] :
% 0.35/0.70        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.70        = ( ord_less_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_less_numeral
% 0.35/0.70  thf(fact_7219_one__le__floor,axiom,
% 0.35/0.70      ! [X: real] :
% 0.35/0.70        ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
% 0.35/0.70        = ( ord_less_eq_real @ one_one_real @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % one_le_floor
% 0.35/0.70  thf(fact_7220_one__le__floor,axiom,
% 0.35/0.70      ! [X: rat] :
% 0.35/0.70        ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
% 0.35/0.70        = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % one_le_floor
% 0.35/0.70  thf(fact_7221_floor__less__one,axiom,
% 0.35/0.70      ! [X: real] :
% 0.35/0.70        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
% 0.35/0.70        = ( ord_less_real @ X @ one_one_real ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_less_one
% 0.35/0.70  thf(fact_7222_floor__less__one,axiom,
% 0.35/0.70      ! [X: rat] :
% 0.35/0.70        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
% 0.35/0.70        = ( ord_less_rat @ X @ one_one_rat ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_less_one
% 0.35/0.70  thf(fact_7223_log__powr__cancel,axiom,
% 0.35/0.70      ! [A: real,Y: real] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.70       => ( ( A != one_one_real )
% 0.35/0.70         => ( ( log @ A @ ( powr_real @ A @ Y ) )
% 0.35/0.70            = Y ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % log_powr_cancel
% 0.35/0.70  thf(fact_7224_powr__log__cancel,axiom,
% 0.35/0.70      ! [A: real,X: real] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.70       => ( ( A != one_one_real )
% 0.35/0.70         => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70           => ( ( powr_real @ A @ ( log @ A @ X ) )
% 0.35/0.70              = X ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_log_cancel
% 0.35/0.70  thf(fact_7225_numeral__less__floor,axiom,
% 0.35/0.70      ! [V: num,X: real] :
% 0.35/0.70        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
% 0.35/0.70        = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % numeral_less_floor
% 0.35/0.70  thf(fact_7226_numeral__less__floor,axiom,
% 0.35/0.70      ! [V: num,X: rat] :
% 0.35/0.70        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
% 0.35/0.70        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % numeral_less_floor
% 0.35/0.70  thf(fact_7227_floor__le__numeral,axiom,
% 0.35/0.70      ! [X: real,V: num] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.70        = ( ord_less_real @ X @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_le_numeral
% 0.35/0.70  thf(fact_7228_floor__le__numeral,axiom,
% 0.35/0.70      ! [X: rat,V: num] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.70        = ( ord_less_rat @ X @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_le_numeral
% 0.35/0.70  thf(fact_7229_one__less__floor,axiom,
% 0.35/0.70      ! [X: real] :
% 0.35/0.70        ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
% 0.35/0.70        = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % one_less_floor
% 0.35/0.70  thf(fact_7230_one__less__floor,axiom,
% 0.35/0.70      ! [X: rat] :
% 0.35/0.70        ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
% 0.35/0.70        = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % one_less_floor
% 0.35/0.70  thf(fact_7231_floor__le__one,axiom,
% 0.35/0.70      ! [X: real] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
% 0.35/0.70        = ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_le_one
% 0.35/0.70  thf(fact_7232_floor__le__one,axiom,
% 0.35/0.70      ! [X: rat] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
% 0.35/0.70        = ( ord_less_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_le_one
% 0.35/0.70  thf(fact_7233_neg__numeral__le__floor,axiom,
% 0.35/0.70      ! [V: num,X: real] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
% 0.35/0.70        = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % neg_numeral_le_floor
% 0.35/0.70  thf(fact_7234_neg__numeral__le__floor,axiom,
% 0.35/0.70      ! [V: num,X: rat] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
% 0.35/0.70        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % neg_numeral_le_floor
% 0.35/0.70  thf(fact_7235_floor__less__neg__numeral,axiom,
% 0.35/0.70      ! [X: real,V: num] :
% 0.35/0.70        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.70        = ( ord_less_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_less_neg_numeral
% 0.35/0.70  thf(fact_7236_floor__less__neg__numeral,axiom,
% 0.35/0.70      ! [X: rat,V: num] :
% 0.35/0.70        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.70        = ( ord_less_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_less_neg_numeral
% 0.35/0.70  thf(fact_7237_neg__numeral__less__floor,axiom,
% 0.35/0.70      ! [V: num,X: real] :
% 0.35/0.70        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
% 0.35/0.70        = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % neg_numeral_less_floor
% 0.35/0.70  thf(fact_7238_neg__numeral__less__floor,axiom,
% 0.35/0.70      ! [V: num,X: rat] :
% 0.35/0.70        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
% 0.35/0.70        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % neg_numeral_less_floor
% 0.35/0.70  thf(fact_7239_floor__le__neg__numeral,axiom,
% 0.35/0.70      ! [X: real,V: num] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.70        = ( ord_less_real @ X @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_le_neg_numeral
% 0.35/0.70  thf(fact_7240_floor__le__neg__numeral,axiom,
% 0.35/0.70      ! [X: rat,V: num] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.70        = ( ord_less_rat @ X @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_le_neg_numeral
% 0.35/0.70  thf(fact_7241_abs__le__D1,axiom,
% 0.35/0.70      ! [A: real,B: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.35/0.70       => ( ord_less_eq_real @ A @ B ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_D1
% 0.35/0.70  thf(fact_7242_abs__le__D1,axiom,
% 0.35/0.70      ! [A: code_integer,B: code_integer] :
% 0.35/0.70        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.35/0.70       => ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_D1
% 0.35/0.70  thf(fact_7243_abs__le__D1,axiom,
% 0.35/0.70      ! [A: rat,B: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.35/0.70       => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_D1
% 0.35/0.70  thf(fact_7244_abs__le__D1,axiom,
% 0.35/0.70      ! [A: int,B: int] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.35/0.70       => ( ord_less_eq_int @ A @ B ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_D1
% 0.35/0.70  thf(fact_7245_abs__ge__self,axiom,
% 0.35/0.70      ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_ge_self
% 0.35/0.70  thf(fact_7246_abs__ge__self,axiom,
% 0.35/0.70      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_ge_self
% 0.35/0.70  thf(fact_7247_abs__ge__self,axiom,
% 0.35/0.70      ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_ge_self
% 0.35/0.70  thf(fact_7248_abs__ge__self,axiom,
% 0.35/0.70      ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_ge_self
% 0.35/0.70  thf(fact_7249_abs__eq__0__iff,axiom,
% 0.35/0.70      ! [A: code_integer] :
% 0.35/0.70        ( ( ( abs_abs_Code_integer @ A )
% 0.35/0.70          = zero_z3403309356797280102nteger )
% 0.35/0.70        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_eq_0_iff
% 0.35/0.70  thf(fact_7250_abs__eq__0__iff,axiom,
% 0.35/0.70      ! [A: complex] :
% 0.35/0.70        ( ( ( abs_abs_complex @ A )
% 0.35/0.70          = zero_zero_complex )
% 0.35/0.70        = ( A = zero_zero_complex ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_eq_0_iff
% 0.35/0.70  thf(fact_7251_abs__eq__0__iff,axiom,
% 0.35/0.70      ! [A: real] :
% 0.35/0.70        ( ( ( abs_abs_real @ A )
% 0.35/0.70          = zero_zero_real )
% 0.35/0.70        = ( A = zero_zero_real ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_eq_0_iff
% 0.35/0.70  thf(fact_7252_abs__eq__0__iff,axiom,
% 0.35/0.70      ! [A: rat] :
% 0.35/0.70        ( ( ( abs_abs_rat @ A )
% 0.35/0.70          = zero_zero_rat )
% 0.35/0.70        = ( A = zero_zero_rat ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_eq_0_iff
% 0.35/0.70  thf(fact_7253_abs__eq__0__iff,axiom,
% 0.35/0.70      ! [A: int] :
% 0.35/0.70        ( ( ( abs_abs_int @ A )
% 0.35/0.70          = zero_zero_int )
% 0.35/0.70        = ( A = zero_zero_int ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_eq_0_iff
% 0.35/0.70  thf(fact_7254_abs__minus__commute,axiom,
% 0.35/0.70      ! [A: code_integer,B: code_integer] :
% 0.35/0.70        ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.35/0.70        = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_minus_commute
% 0.35/0.70  thf(fact_7255_abs__minus__commute,axiom,
% 0.35/0.70      ! [A: real,B: real] :
% 0.35/0.70        ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 0.35/0.70        = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_minus_commute
% 0.35/0.70  thf(fact_7256_abs__minus__commute,axiom,
% 0.35/0.70      ! [A: rat,B: rat] :
% 0.35/0.70        ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
% 0.35/0.70        = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_minus_commute
% 0.35/0.70  thf(fact_7257_abs__minus__commute,axiom,
% 0.35/0.70      ! [A: int,B: int] :
% 0.35/0.70        ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
% 0.35/0.70        = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_minus_commute
% 0.35/0.70  thf(fact_7258_abs__ge__zero,axiom,
% 0.35/0.70      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_ge_zero
% 0.35/0.70  thf(fact_7259_abs__ge__zero,axiom,
% 0.35/0.70      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_ge_zero
% 0.35/0.70  thf(fact_7260_abs__ge__zero,axiom,
% 0.35/0.70      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_ge_zero
% 0.35/0.70  thf(fact_7261_abs__ge__zero,axiom,
% 0.35/0.70      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_ge_zero
% 0.35/0.70  thf(fact_7262_floor__mono,axiom,
% 0.35/0.70      ! [X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ X @ Y )
% 0.35/0.70       => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_mono
% 0.35/0.70  thf(fact_7263_floor__mono,axiom,
% 0.35/0.70      ! [X: rat,Y: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ X @ Y )
% 0.35/0.70       => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_mono
% 0.35/0.70  thf(fact_7264_abs__of__pos,axiom,
% 0.35/0.70      ! [A: code_integer] :
% 0.35/0.70        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 0.35/0.70       => ( ( abs_abs_Code_integer @ A )
% 0.35/0.70          = A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_of_pos
% 0.35/0.70  thf(fact_7265_abs__of__pos,axiom,
% 0.35/0.70      ! [A: real] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.70       => ( ( abs_abs_real @ A )
% 0.35/0.70          = A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_of_pos
% 0.35/0.70  thf(fact_7266_abs__of__pos,axiom,
% 0.35/0.70      ! [A: rat] :
% 0.35/0.70        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.35/0.70       => ( ( abs_abs_rat @ A )
% 0.35/0.70          = A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_of_pos
% 0.35/0.70  thf(fact_7267_abs__of__pos,axiom,
% 0.35/0.70      ! [A: int] :
% 0.35/0.70        ( ( ord_less_int @ zero_zero_int @ A )
% 0.35/0.70       => ( ( abs_abs_int @ A )
% 0.35/0.70          = A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_of_pos
% 0.35/0.70  thf(fact_7268_abs__not__less__zero,axiom,
% 0.35/0.70      ! [A: code_integer] :
% 0.35/0.70        ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_not_less_zero
% 0.35/0.70  thf(fact_7269_abs__not__less__zero,axiom,
% 0.35/0.70      ! [A: real] :
% 0.35/0.70        ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_not_less_zero
% 0.35/0.70  thf(fact_7270_abs__not__less__zero,axiom,
% 0.35/0.70      ! [A: rat] :
% 0.35/0.70        ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_not_less_zero
% 0.35/0.70  thf(fact_7271_abs__not__less__zero,axiom,
% 0.35/0.70      ! [A: int] :
% 0.35/0.70        ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_not_less_zero
% 0.35/0.70  thf(fact_7272_of__int__floor__le,axiom,
% 0.35/0.70      ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X ) ).
% 0.35/0.70  
% 0.35/0.70  % of_int_floor_le
% 0.35/0.70  thf(fact_7273_of__int__floor__le,axiom,
% 0.35/0.70      ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X ) ).
% 0.35/0.70  
% 0.35/0.70  % of_int_floor_le
% 0.35/0.70  thf(fact_7274_floor__less__cancel,axiom,
% 0.35/0.70      ! [X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) )
% 0.35/0.70       => ( ord_less_real @ X @ Y ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_less_cancel
% 0.35/0.70  thf(fact_7275_floor__less__cancel,axiom,
% 0.35/0.70      ! [X: rat,Y: rat] :
% 0.35/0.70        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) )
% 0.35/0.70       => ( ord_less_rat @ X @ Y ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_less_cancel
% 0.35/0.70  thf(fact_7276_abs__triangle__ineq,axiom,
% 0.35/0.70      ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_triangle_ineq
% 0.35/0.70  thf(fact_7277_abs__triangle__ineq,axiom,
% 0.35/0.70      ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_triangle_ineq
% 0.35/0.70  thf(fact_7278_abs__triangle__ineq,axiom,
% 0.35/0.70      ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_triangle_ineq
% 0.35/0.70  thf(fact_7279_abs__triangle__ineq,axiom,
% 0.35/0.70      ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_triangle_ineq
% 0.35/0.70  thf(fact_7280_abs__mult__less,axiom,
% 0.35/0.70      ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 0.35/0.70        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
% 0.35/0.70       => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
% 0.35/0.70         => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_mult_less
% 0.35/0.70  thf(fact_7281_abs__mult__less,axiom,
% 0.35/0.70      ! [A: real,C: real,B: real,D: real] :
% 0.35/0.70        ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
% 0.35/0.70       => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
% 0.35/0.70         => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_mult_less
% 0.35/0.70  thf(fact_7282_abs__mult__less,axiom,
% 0.35/0.70      ! [A: rat,C: rat,B: rat,D: rat] :
% 0.35/0.70        ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
% 0.35/0.70       => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
% 0.35/0.70         => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_mult_less
% 0.35/0.70  thf(fact_7283_abs__mult__less,axiom,
% 0.35/0.70      ! [A: int,C: int,B: int,D: int] :
% 0.35/0.70        ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
% 0.35/0.70       => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
% 0.35/0.70         => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_mult_less
% 0.35/0.70  thf(fact_7284_abs__triangle__ineq2__sym,axiom,
% 0.35/0.70      ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_triangle_ineq2_sym
% 0.35/0.70  thf(fact_7285_abs__triangle__ineq2__sym,axiom,
% 0.35/0.70      ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_triangle_ineq2_sym
% 0.35/0.70  thf(fact_7286_abs__triangle__ineq2__sym,axiom,
% 0.35/0.70      ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_triangle_ineq2_sym
% 0.35/0.70  thf(fact_7287_abs__triangle__ineq2__sym,axiom,
% 0.35/0.70      ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_triangle_ineq2_sym
% 0.35/0.70  thf(fact_7288_abs__triangle__ineq3,axiom,
% 0.35/0.70      ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_triangle_ineq3
% 0.35/0.70  thf(fact_7289_abs__triangle__ineq3,axiom,
% 0.35/0.70      ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_triangle_ineq3
% 0.35/0.70  thf(fact_7290_abs__triangle__ineq3,axiom,
% 0.35/0.70      ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_triangle_ineq3
% 0.35/0.70  thf(fact_7291_abs__triangle__ineq3,axiom,
% 0.35/0.70      ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_triangle_ineq3
% 0.35/0.70  thf(fact_7292_abs__triangle__ineq2,axiom,
% 0.35/0.70      ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_triangle_ineq2
% 0.35/0.70  thf(fact_7293_abs__triangle__ineq2,axiom,
% 0.35/0.70      ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_triangle_ineq2
% 0.35/0.70  thf(fact_7294_abs__triangle__ineq2,axiom,
% 0.35/0.70      ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_triangle_ineq2
% 0.35/0.70  thf(fact_7295_abs__triangle__ineq2,axiom,
% 0.35/0.70      ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_triangle_ineq2
% 0.35/0.70  thf(fact_7296_nonzero__abs__divide,axiom,
% 0.35/0.70      ! [B: real,A: real] :
% 0.35/0.70        ( ( B != zero_zero_real )
% 0.35/0.70       => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 0.35/0.70          = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nonzero_abs_divide
% 0.35/0.70  thf(fact_7297_nonzero__abs__divide,axiom,
% 0.35/0.70      ! [B: rat,A: rat] :
% 0.35/0.70        ( ( B != zero_zero_rat )
% 0.35/0.70       => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 0.35/0.70          = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nonzero_abs_divide
% 0.35/0.70  thf(fact_7298_abs__leI,axiom,
% 0.35/0.70      ! [A: real,B: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ A @ B )
% 0.35/0.70       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 0.35/0.70         => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_leI
% 0.35/0.70  thf(fact_7299_abs__leI,axiom,
% 0.35/0.70      ! [A: code_integer,B: code_integer] :
% 0.35/0.70        ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.35/0.70       => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.35/0.70         => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_leI
% 0.35/0.70  thf(fact_7300_abs__leI,axiom,
% 0.35/0.70      ! [A: rat,B: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ A @ B )
% 0.35/0.70       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.35/0.70         => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_leI
% 0.35/0.70  thf(fact_7301_abs__leI,axiom,
% 0.35/0.70      ! [A: int,B: int] :
% 0.35/0.70        ( ( ord_less_eq_int @ A @ B )
% 0.35/0.70       => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 0.35/0.70         => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_leI
% 0.35/0.70  thf(fact_7302_abs__le__D2,axiom,
% 0.35/0.70      ! [A: real,B: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.35/0.70       => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_D2
% 0.35/0.70  thf(fact_7303_abs__le__D2,axiom,
% 0.35/0.70      ! [A: code_integer,B: code_integer] :
% 0.35/0.70        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.35/0.70       => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_D2
% 0.35/0.70  thf(fact_7304_abs__le__D2,axiom,
% 0.35/0.70      ! [A: rat,B: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.35/0.70       => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_D2
% 0.35/0.70  thf(fact_7305_abs__le__D2,axiom,
% 0.35/0.70      ! [A: int,B: int] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.35/0.70       => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_D2
% 0.35/0.70  thf(fact_7306_abs__le__iff,axiom,
% 0.35/0.70      ! [A: real,B: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.35/0.70        = ( ( ord_less_eq_real @ A @ B )
% 0.35/0.70          & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_iff
% 0.35/0.70  thf(fact_7307_abs__le__iff,axiom,
% 0.35/0.70      ! [A: code_integer,B: code_integer] :
% 0.35/0.70        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.35/0.70        = ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.35/0.70          & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_iff
% 0.35/0.70  thf(fact_7308_abs__le__iff,axiom,
% 0.35/0.70      ! [A: rat,B: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.35/0.70        = ( ( ord_less_eq_rat @ A @ B )
% 0.35/0.70          & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_iff
% 0.35/0.70  thf(fact_7309_abs__le__iff,axiom,
% 0.35/0.70      ! [A: int,B: int] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.35/0.70        = ( ( ord_less_eq_int @ A @ B )
% 0.35/0.70          & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_iff
% 0.35/0.70  thf(fact_7310_abs__ge__minus__self,axiom,
% 0.35/0.70      ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_ge_minus_self
% 0.35/0.70  thf(fact_7311_abs__ge__minus__self,axiom,
% 0.35/0.70      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_ge_minus_self
% 0.35/0.70  thf(fact_7312_abs__ge__minus__self,axiom,
% 0.35/0.70      ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_ge_minus_self
% 0.35/0.70  thf(fact_7313_abs__ge__minus__self,axiom,
% 0.35/0.70      ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_ge_minus_self
% 0.35/0.70  thf(fact_7314_abs__less__iff,axiom,
% 0.35/0.70      ! [A: int,B: int] :
% 0.35/0.70        ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
% 0.35/0.70        = ( ( ord_less_int @ A @ B )
% 0.35/0.70          & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_less_iff
% 0.35/0.70  thf(fact_7315_abs__less__iff,axiom,
% 0.35/0.70      ! [A: real,B: real] :
% 0.35/0.70        ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
% 0.35/0.70        = ( ( ord_less_real @ A @ B )
% 0.35/0.70          & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_less_iff
% 0.35/0.70  thf(fact_7316_abs__less__iff,axiom,
% 0.35/0.70      ! [A: rat,B: rat] :
% 0.35/0.70        ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
% 0.35/0.70        = ( ( ord_less_rat @ A @ B )
% 0.35/0.70          & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_less_iff
% 0.35/0.70  thf(fact_7317_abs__less__iff,axiom,
% 0.35/0.70      ! [A: code_integer,B: code_integer] :
% 0.35/0.70        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.35/0.70        = ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.35/0.70          & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_less_iff
% 0.35/0.70  thf(fact_7318_powr__less__mono2__neg,axiom,
% 0.35/0.70      ! [A: real,X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_real @ A @ zero_zero_real )
% 0.35/0.70       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70         => ( ( ord_less_real @ X @ Y )
% 0.35/0.70           => ( ord_less_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_less_mono2_neg
% 0.35/0.70  thf(fact_7319_powr__non__neg,axiom,
% 0.35/0.70      ! [A: real,X: real] :
% 0.35/0.70        ~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_non_neg
% 0.35/0.70  thf(fact_7320_powr__less__mono,axiom,
% 0.35/0.70      ! [A: real,B: real,X: real] :
% 0.35/0.70        ( ( ord_less_real @ A @ B )
% 0.35/0.70       => ( ( ord_less_real @ one_one_real @ X )
% 0.35/0.70         => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_less_mono
% 0.35/0.70  thf(fact_7321_powr__less__cancel,axiom,
% 0.35/0.70      ! [X: real,A: real,B: real] :
% 0.35/0.70        ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 0.35/0.70       => ( ( ord_less_real @ one_one_real @ X )
% 0.35/0.70         => ( ord_less_real @ A @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_less_cancel
% 0.35/0.70  thf(fact_7322_dense__eq0__I,axiom,
% 0.35/0.70      ! [X: real] :
% 0.35/0.70        ( ! [E: real] :
% 0.35/0.70            ( ( ord_less_real @ zero_zero_real @ E )
% 0.35/0.70           => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E ) )
% 0.35/0.70       => ( X = zero_zero_real ) ) ).
% 0.35/0.70  
% 0.35/0.70  % dense_eq0_I
% 0.35/0.70  thf(fact_7323_dense__eq0__I,axiom,
% 0.35/0.70      ! [X: rat] :
% 0.35/0.70        ( ! [E: rat] :
% 0.35/0.70            ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.35/0.70           => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E ) )
% 0.35/0.70       => ( X = zero_zero_rat ) ) ).
% 0.35/0.70  
% 0.35/0.70  % dense_eq0_I
% 0.35/0.70  thf(fact_7324_abs__eq__mult,axiom,
% 0.35/0.70      ! [A: code_integer,B: code_integer] :
% 0.35/0.70        ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.35/0.70            | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
% 0.35/0.70          & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.35/0.70            | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
% 0.35/0.70       => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.35/0.70          = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_eq_mult
% 0.35/0.70  thf(fact_7325_abs__eq__mult,axiom,
% 0.35/0.70      ! [A: real,B: real] :
% 0.35/0.70        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.35/0.70            | ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.35/0.70          & ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.35/0.70            | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.35/0.70       => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 0.35/0.70          = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_eq_mult
% 0.35/0.70  thf(fact_7326_abs__eq__mult,axiom,
% 0.35/0.70      ! [A: rat,B: rat] :
% 0.35/0.70        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.35/0.70            | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.35/0.70          & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.35/0.70            | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 0.35/0.70       => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 0.35/0.70          = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_eq_mult
% 0.35/0.70  thf(fact_7327_abs__eq__mult,axiom,
% 0.35/0.70      ! [A: int,B: int] :
% 0.35/0.70        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.35/0.70            | ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.35/0.70          & ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.35/0.70            | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 0.35/0.70       => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 0.35/0.70          = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_eq_mult
% 0.35/0.70  thf(fact_7328_abs__mult__pos,axiom,
% 0.35/0.70      ! [X: code_integer,Y: code_integer] :
% 0.35/0.70        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 0.35/0.70       => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X )
% 0.35/0.70          = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_mult_pos
% 0.35/0.70  thf(fact_7329_abs__mult__pos,axiom,
% 0.35/0.70      ! [X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.70       => ( ( times_times_real @ ( abs_abs_real @ Y ) @ X )
% 0.35/0.70          = ( abs_abs_real @ ( times_times_real @ Y @ X ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_mult_pos
% 0.35/0.70  thf(fact_7330_abs__mult__pos,axiom,
% 0.35/0.70      ! [X: rat,Y: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.35/0.70       => ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X )
% 0.35/0.70          = ( abs_abs_rat @ ( times_times_rat @ Y @ X ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_mult_pos
% 0.35/0.70  thf(fact_7331_abs__mult__pos,axiom,
% 0.35/0.70      ! [X: int,Y: int] :
% 0.35/0.70        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.35/0.70       => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
% 0.35/0.70          = ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_mult_pos
% 0.35/0.70  thf(fact_7332_abs__minus__le__zero,axiom,
% 0.35/0.70      ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_minus_le_zero
% 0.35/0.70  thf(fact_7333_abs__minus__le__zero,axiom,
% 0.35/0.70      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_minus_le_zero
% 0.35/0.70  thf(fact_7334_abs__minus__le__zero,axiom,
% 0.35/0.70      ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_minus_le_zero
% 0.35/0.70  thf(fact_7335_abs__minus__le__zero,axiom,
% 0.35/0.70      ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_minus_le_zero
% 0.35/0.70  thf(fact_7336_eq__abs__iff_H,axiom,
% 0.35/0.70      ! [A: real,B: real] :
% 0.35/0.70        ( ( A
% 0.35/0.70          = ( abs_abs_real @ B ) )
% 0.35/0.70        = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.35/0.70          & ( ( B = A )
% 0.35/0.70            | ( B
% 0.35/0.70              = ( uminus_uminus_real @ A ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % eq_abs_iff'
% 0.35/0.70  thf(fact_7337_eq__abs__iff_H,axiom,
% 0.35/0.70      ! [A: code_integer,B: code_integer] :
% 0.35/0.70        ( ( A
% 0.35/0.70          = ( abs_abs_Code_integer @ B ) )
% 0.35/0.70        = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.35/0.70          & ( ( B = A )
% 0.35/0.70            | ( B
% 0.35/0.70              = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % eq_abs_iff'
% 0.35/0.70  thf(fact_7338_eq__abs__iff_H,axiom,
% 0.35/0.70      ! [A: rat,B: rat] :
% 0.35/0.70        ( ( A
% 0.35/0.70          = ( abs_abs_rat @ B ) )
% 0.35/0.70        = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.35/0.70          & ( ( B = A )
% 0.35/0.70            | ( B
% 0.35/0.70              = ( uminus_uminus_rat @ A ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % eq_abs_iff'
% 0.35/0.70  thf(fact_7339_eq__abs__iff_H,axiom,
% 0.35/0.70      ! [A: int,B: int] :
% 0.35/0.70        ( ( A
% 0.35/0.70          = ( abs_abs_int @ B ) )
% 0.35/0.70        = ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.35/0.70          & ( ( B = A )
% 0.35/0.70            | ( B
% 0.35/0.70              = ( uminus_uminus_int @ A ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % eq_abs_iff'
% 0.35/0.70  thf(fact_7340_abs__eq__iff_H,axiom,
% 0.35/0.70      ! [A: real,B: real] :
% 0.35/0.70        ( ( ( abs_abs_real @ A )
% 0.35/0.70          = B )
% 0.35/0.70        = ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.35/0.70          & ( ( A = B )
% 0.35/0.70            | ( A
% 0.35/0.70              = ( uminus_uminus_real @ B ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_eq_iff'
% 0.35/0.70  thf(fact_7341_abs__eq__iff_H,axiom,
% 0.35/0.70      ! [A: code_integer,B: code_integer] :
% 0.35/0.70        ( ( ( abs_abs_Code_integer @ A )
% 0.35/0.70          = B )
% 0.35/0.70        = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.35/0.70          & ( ( A = B )
% 0.35/0.70            | ( A
% 0.35/0.70              = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_eq_iff'
% 0.35/0.70  thf(fact_7342_abs__eq__iff_H,axiom,
% 0.35/0.70      ! [A: rat,B: rat] :
% 0.35/0.70        ( ( ( abs_abs_rat @ A )
% 0.35/0.70          = B )
% 0.35/0.70        = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.35/0.70          & ( ( A = B )
% 0.35/0.70            | ( A
% 0.35/0.70              = ( uminus_uminus_rat @ B ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_eq_iff'
% 0.35/0.70  thf(fact_7343_abs__eq__iff_H,axiom,
% 0.35/0.70      ! [A: int,B: int] :
% 0.35/0.70        ( ( ( abs_abs_int @ A )
% 0.35/0.70          = B )
% 0.35/0.70        = ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.35/0.70          & ( ( A = B )
% 0.35/0.70            | ( A
% 0.35/0.70              = ( uminus_uminus_int @ B ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_eq_iff'
% 0.35/0.70  thf(fact_7344_abs__div__pos,axiom,
% 0.35/0.70      ! [Y: real,X: real] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ Y )
% 0.35/0.70       => ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y )
% 0.35/0.70          = ( abs_abs_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_div_pos
% 0.35/0.70  thf(fact_7345_abs__div__pos,axiom,
% 0.35/0.70      ! [Y: rat,X: rat] :
% 0.35/0.70        ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.35/0.70       => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y )
% 0.35/0.70          = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_div_pos
% 0.35/0.70  thf(fact_7346_zero__le__power__abs,axiom,
% 0.35/0.70      ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zero_le_power_abs
% 0.35/0.70  thf(fact_7347_zero__le__power__abs,axiom,
% 0.35/0.70      ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zero_le_power_abs
% 0.35/0.70  thf(fact_7348_zero__le__power__abs,axiom,
% 0.35/0.70      ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zero_le_power_abs
% 0.35/0.70  thf(fact_7349_zero__le__power__abs,axiom,
% 0.35/0.70      ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zero_le_power_abs
% 0.35/0.70  thf(fact_7350_abs__if__raw,axiom,
% 0.35/0.70      ( abs_abs_int
% 0.35/0.70      = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_if_raw
% 0.35/0.70  thf(fact_7351_abs__if__raw,axiom,
% 0.35/0.70      ( abs_abs_real
% 0.35/0.70      = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_if_raw
% 0.35/0.70  thf(fact_7352_abs__if__raw,axiom,
% 0.35/0.70      ( abs_abs_rat
% 0.35/0.70      = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_if_raw
% 0.35/0.70  thf(fact_7353_abs__if__raw,axiom,
% 0.35/0.70      ( abs_abs_Code_integer
% 0.35/0.70      = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_if_raw
% 0.35/0.70  thf(fact_7354_abs__of__neg,axiom,
% 0.35/0.70      ! [A: int] :
% 0.35/0.70        ( ( ord_less_int @ A @ zero_zero_int )
% 0.35/0.70       => ( ( abs_abs_int @ A )
% 0.35/0.70          = ( uminus_uminus_int @ A ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_of_neg
% 0.35/0.70  thf(fact_7355_abs__of__neg,axiom,
% 0.35/0.70      ! [A: real] :
% 0.35/0.70        ( ( ord_less_real @ A @ zero_zero_real )
% 0.35/0.70       => ( ( abs_abs_real @ A )
% 0.35/0.70          = ( uminus_uminus_real @ A ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_of_neg
% 0.35/0.70  thf(fact_7356_abs__of__neg,axiom,
% 0.35/0.70      ! [A: rat] :
% 0.35/0.70        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.35/0.70       => ( ( abs_abs_rat @ A )
% 0.35/0.70          = ( uminus_uminus_rat @ A ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_of_neg
% 0.35/0.70  thf(fact_7357_abs__of__neg,axiom,
% 0.35/0.70      ! [A: code_integer] :
% 0.35/0.70        ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 0.35/0.70       => ( ( abs_abs_Code_integer @ A )
% 0.35/0.70          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_of_neg
% 0.35/0.70  thf(fact_7358_abs__if,axiom,
% 0.35/0.70      ( abs_abs_int
% 0.35/0.70      = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_if
% 0.35/0.70  thf(fact_7359_abs__if,axiom,
% 0.35/0.70      ( abs_abs_real
% 0.35/0.70      = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_if
% 0.35/0.70  thf(fact_7360_abs__if,axiom,
% 0.35/0.70      ( abs_abs_rat
% 0.35/0.70      = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_if
% 0.35/0.70  thf(fact_7361_abs__if,axiom,
% 0.35/0.70      ( abs_abs_Code_integer
% 0.35/0.70      = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_if
% 0.35/0.70  thf(fact_7362_abs__diff__triangle__ineq,axiom,
% 0.35/0.70      ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_diff_triangle_ineq
% 0.35/0.70  thf(fact_7363_abs__diff__triangle__ineq,axiom,
% 0.35/0.70      ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_diff_triangle_ineq
% 0.35/0.70  thf(fact_7364_abs__diff__triangle__ineq,axiom,
% 0.35/0.70      ! [A: rat,B: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_diff_triangle_ineq
% 0.35/0.70  thf(fact_7365_abs__diff__triangle__ineq,axiom,
% 0.35/0.70      ! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_diff_triangle_ineq
% 0.35/0.70  thf(fact_7366_abs__triangle__ineq4,axiom,
% 0.35/0.70      ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_triangle_ineq4
% 0.35/0.70  thf(fact_7367_abs__triangle__ineq4,axiom,
% 0.35/0.70      ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_triangle_ineq4
% 0.35/0.70  thf(fact_7368_abs__triangle__ineq4,axiom,
% 0.35/0.70      ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_triangle_ineq4
% 0.35/0.70  thf(fact_7369_abs__triangle__ineq4,axiom,
% 0.35/0.70      ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_triangle_ineq4
% 0.35/0.70  thf(fact_7370_abs__diff__le__iff,axiom,
% 0.35/0.70      ! [X: code_integer,A: code_integer,R2: code_integer] :
% 0.35/0.70        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
% 0.35/0.70        = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
% 0.35/0.70          & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_diff_le_iff
% 0.35/0.70  thf(fact_7371_abs__diff__le__iff,axiom,
% 0.35/0.70      ! [X: real,A: real,R2: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
% 0.35/0.70        = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X )
% 0.35/0.70          & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_diff_le_iff
% 0.35/0.70  thf(fact_7372_abs__diff__le__iff,axiom,
% 0.35/0.70      ! [X: rat,A: rat,R2: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
% 0.35/0.70        = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
% 0.35/0.70          & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_diff_le_iff
% 0.35/0.70  thf(fact_7373_abs__diff__le__iff,axiom,
% 0.35/0.70      ! [X: int,A: int,R2: int] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
% 0.35/0.70        = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X )
% 0.35/0.70          & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_diff_le_iff
% 0.35/0.70  thf(fact_7374_abs__diff__less__iff,axiom,
% 0.35/0.70      ! [X: code_integer,A: code_integer,R2: code_integer] :
% 0.35/0.70        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
% 0.35/0.70        = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
% 0.35/0.70          & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_diff_less_iff
% 0.35/0.70  thf(fact_7375_abs__diff__less__iff,axiom,
% 0.35/0.70      ! [X: real,A: real,R2: real] :
% 0.35/0.70        ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
% 0.35/0.70        = ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X )
% 0.35/0.70          & ( ord_less_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_diff_less_iff
% 0.35/0.70  thf(fact_7376_abs__diff__less__iff,axiom,
% 0.35/0.70      ! [X: rat,A: rat,R2: rat] :
% 0.35/0.70        ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
% 0.35/0.70        = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
% 0.35/0.70          & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_diff_less_iff
% 0.35/0.70  thf(fact_7377_abs__diff__less__iff,axiom,
% 0.35/0.70      ! [X: int,A: int,R2: int] :
% 0.35/0.70        ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
% 0.35/0.70        = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X )
% 0.35/0.70          & ( ord_less_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_diff_less_iff
% 0.35/0.70  thf(fact_7378_le__floor__iff,axiom,
% 0.35/0.70      ! [Z2: int,X: real] :
% 0.35/0.70        ( ( ord_less_eq_int @ Z2 @ ( archim6058952711729229775r_real @ X ) )
% 0.35/0.70        = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % le_floor_iff
% 0.35/0.70  thf(fact_7379_le__floor__iff,axiom,
% 0.35/0.70      ! [Z2: int,X: rat] :
% 0.35/0.70        ( ( ord_less_eq_int @ Z2 @ ( archim3151403230148437115or_rat @ X ) )
% 0.35/0.70        = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % le_floor_iff
% 0.35/0.70  thf(fact_7380_floor__less__iff,axiom,
% 0.35/0.70      ! [X: real,Z2: int] :
% 0.35/0.70        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ Z2 )
% 0.35/0.70        = ( ord_less_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_less_iff
% 0.35/0.70  thf(fact_7381_floor__less__iff,axiom,
% 0.35/0.70      ! [X: rat,Z2: int] :
% 0.35/0.70        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ Z2 )
% 0.35/0.70        = ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_less_iff
% 0.35/0.70  thf(fact_7382_powr__mono2_H,axiom,
% 0.35/0.70      ! [A: real,X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.35/0.70       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70         => ( ( ord_less_eq_real @ X @ Y )
% 0.35/0.70           => ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_mono2'
% 0.35/0.70  thf(fact_7383_powr__less__mono2,axiom,
% 0.35/0.70      ! [A: real,X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.70       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.70         => ( ( ord_less_real @ X @ Y )
% 0.35/0.70           => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_less_mono2
% 0.35/0.70  thf(fact_7384_gr__one__powr,axiom,
% 0.35/0.70      ! [X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_real @ one_one_real @ X )
% 0.35/0.70       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.35/0.70         => ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % gr_one_powr
% 0.35/0.70  thf(fact_7385_powr__inj,axiom,
% 0.35/0.70      ! [A: real,X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.70       => ( ( A != one_one_real )
% 0.35/0.70         => ( ( ( powr_real @ A @ X )
% 0.35/0.70              = ( powr_real @ A @ Y ) )
% 0.35/0.70            = ( X = Y ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_inj
% 0.35/0.70  thf(fact_7386_abs__sgn__eq,axiom,
% 0.35/0.70      ! [A: code_integer] :
% 0.35/0.70        ( ( ( A = zero_z3403309356797280102nteger )
% 0.35/0.70         => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.35/0.70            = zero_z3403309356797280102nteger ) )
% 0.35/0.70        & ( ( A != zero_z3403309356797280102nteger )
% 0.35/0.70         => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 0.35/0.70            = one_one_Code_integer ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_sgn_eq
% 0.35/0.70  thf(fact_7387_abs__sgn__eq,axiom,
% 0.35/0.70      ! [A: real] :
% 0.35/0.70        ( ( ( A = zero_zero_real )
% 0.35/0.70         => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 0.35/0.70            = zero_zero_real ) )
% 0.35/0.70        & ( ( A != zero_zero_real )
% 0.35/0.70         => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 0.35/0.70            = one_one_real ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_sgn_eq
% 0.35/0.70  thf(fact_7388_abs__sgn__eq,axiom,
% 0.35/0.70      ! [A: rat] :
% 0.35/0.70        ( ( ( A = zero_zero_rat )
% 0.35/0.70         => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 0.35/0.70            = zero_zero_rat ) )
% 0.35/0.70        & ( ( A != zero_zero_rat )
% 0.35/0.70         => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 0.35/0.70            = one_one_rat ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_sgn_eq
% 0.35/0.70  thf(fact_7389_abs__sgn__eq,axiom,
% 0.35/0.70      ! [A: int] :
% 0.35/0.70        ( ( ( A = zero_zero_int )
% 0.35/0.70         => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 0.35/0.70            = zero_zero_int ) )
% 0.35/0.70        & ( ( A != zero_zero_int )
% 0.35/0.70         => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 0.35/0.70            = one_one_int ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_sgn_eq
% 0.35/0.70  thf(fact_7390_abs__real__def,axiom,
% 0.35/0.70      ( abs_abs_real
% 0.35/0.70      = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_real_def
% 0.35/0.70  thf(fact_7391_abs__add__one__gt__zero,axiom,
% 0.35/0.70      ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_add_one_gt_zero
% 0.35/0.70  thf(fact_7392_abs__add__one__gt__zero,axiom,
% 0.35/0.70      ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_add_one_gt_zero
% 0.35/0.70  thf(fact_7393_abs__add__one__gt__zero,axiom,
% 0.35/0.70      ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_add_one_gt_zero
% 0.35/0.70  thf(fact_7394_abs__add__one__gt__zero,axiom,
% 0.35/0.70      ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_add_one_gt_zero
% 0.35/0.70  thf(fact_7395_floor__log__eq__powr__iff,axiom,
% 0.35/0.70      ! [X: real,B: real,K: int] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70       => ( ( ord_less_real @ one_one_real @ B )
% 0.35/0.70         => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
% 0.35/0.70              = K )
% 0.35/0.70            = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
% 0.35/0.70              & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_log_eq_powr_iff
% 0.35/0.70  thf(fact_7396_of__int__leD,axiom,
% 0.35/0.70      ! [N: int,X: code_integer] :
% 0.35/0.70        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
% 0.35/0.70       => ( ( N = zero_zero_int )
% 0.35/0.70          | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % of_int_leD
% 0.35/0.70  thf(fact_7397_of__int__leD,axiom,
% 0.35/0.70      ! [N: int,X: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 0.35/0.70       => ( ( N = zero_zero_int )
% 0.35/0.70          | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % of_int_leD
% 0.35/0.70  thf(fact_7398_of__int__leD,axiom,
% 0.35/0.70      ! [N: int,X: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
% 0.35/0.70       => ( ( N = zero_zero_int )
% 0.35/0.70          | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % of_int_leD
% 0.35/0.70  thf(fact_7399_of__int__leD,axiom,
% 0.35/0.70      ! [N: int,X: int] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 0.35/0.70       => ( ( N = zero_zero_int )
% 0.35/0.70          | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % of_int_leD
% 0.35/0.70  thf(fact_7400_of__int__lessD,axiom,
% 0.35/0.70      ! [N: int,X: code_integer] :
% 0.35/0.70        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
% 0.35/0.70       => ( ( N = zero_zero_int )
% 0.35/0.70          | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % of_int_lessD
% 0.35/0.70  thf(fact_7401_of__int__lessD,axiom,
% 0.35/0.70      ! [N: int,X: real] :
% 0.35/0.70        ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 0.35/0.70       => ( ( N = zero_zero_int )
% 0.35/0.70          | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % of_int_lessD
% 0.35/0.70  thf(fact_7402_of__int__lessD,axiom,
% 0.35/0.70      ! [N: int,X: rat] :
% 0.35/0.70        ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
% 0.35/0.70       => ( ( N = zero_zero_int )
% 0.35/0.70          | ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % of_int_lessD
% 0.35/0.70  thf(fact_7403_of__int__lessD,axiom,
% 0.35/0.70      ! [N: int,X: int] :
% 0.35/0.70        ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 0.35/0.70       => ( ( N = zero_zero_int )
% 0.35/0.70          | ( ord_less_int @ one_one_int @ X ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % of_int_lessD
% 0.35/0.70  thf(fact_7404_powr__realpow,axiom,
% 0.35/0.70      ! [X: real,N: nat] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70       => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
% 0.35/0.70          = ( power_power_real @ X @ N ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_realpow
% 0.35/0.70  thf(fact_7405_less__log__iff,axiom,
% 0.35/0.70      ! [B: real,X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_real @ one_one_real @ B )
% 0.35/0.70       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70         => ( ( ord_less_real @ Y @ ( log @ B @ X ) )
% 0.35/0.70            = ( ord_less_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % less_log_iff
% 0.35/0.70  thf(fact_7406_log__less__iff,axiom,
% 0.35/0.70      ! [B: real,X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_real @ one_one_real @ B )
% 0.35/0.70       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70         => ( ( ord_less_real @ ( log @ B @ X ) @ Y )
% 0.35/0.70            = ( ord_less_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % log_less_iff
% 0.35/0.70  thf(fact_7407_less__powr__iff,axiom,
% 0.35/0.70      ! [B: real,X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_real @ one_one_real @ B )
% 0.35/0.70       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70         => ( ( ord_less_real @ X @ ( powr_real @ B @ Y ) )
% 0.35/0.70            = ( ord_less_real @ ( log @ B @ X ) @ Y ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % less_powr_iff
% 0.35/0.70  thf(fact_7408_powr__less__iff,axiom,
% 0.35/0.70      ! [B: real,X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_real @ one_one_real @ B )
% 0.35/0.70       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70         => ( ( ord_less_real @ ( powr_real @ B @ Y ) @ X )
% 0.35/0.70            = ( ord_less_real @ Y @ ( log @ B @ X ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_less_iff
% 0.35/0.70  thf(fact_7409_floor__eq,axiom,
% 0.35/0.70      ! [N: int,X: real] :
% 0.35/0.70        ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
% 0.35/0.70       => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.35/0.70         => ( ( archim6058952711729229775r_real @ X )
% 0.35/0.70            = N ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_eq
% 0.35/0.70  thf(fact_7410_real__of__int__floor__add__one__gt,axiom,
% 0.35/0.70      ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 0.35/0.70  
% 0.35/0.70  % real_of_int_floor_add_one_gt
% 0.35/0.70  thf(fact_7411_real__of__int__floor__gt__diff__one,axiom,
% 0.35/0.70      ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % real_of_int_floor_gt_diff_one
% 0.35/0.70  thf(fact_7412_round__diff__minimal,axiom,
% 0.35/0.70      ! [Z2: real,M: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z2 ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ ( ring_1_of_int_real @ M ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % round_diff_minimal
% 0.35/0.70  thf(fact_7413_round__diff__minimal,axiom,
% 0.35/0.70      ! [Z2: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z2 @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z2 ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z2 @ ( ring_1_of_int_rat @ M ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % round_diff_minimal
% 0.35/0.70  thf(fact_7414_floor__unique,axiom,
% 0.35/0.70      ! [Z2: int,X: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X )
% 0.35/0.70       => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) )
% 0.35/0.70         => ( ( archim6058952711729229775r_real @ X )
% 0.35/0.70            = Z2 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_unique
% 0.35/0.70  thf(fact_7415_floor__unique,axiom,
% 0.35/0.70      ! [Z2: int,X: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
% 0.35/0.70       => ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) )
% 0.35/0.70         => ( ( archim3151403230148437115or_rat @ X )
% 0.35/0.70            = Z2 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_unique
% 0.35/0.70  thf(fact_7416_floor__eq__iff,axiom,
% 0.35/0.70      ! [X: real,A: int] :
% 0.35/0.70        ( ( ( archim6058952711729229775r_real @ X )
% 0.35/0.70          = A )
% 0.35/0.70        = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X )
% 0.35/0.70          & ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_eq_iff
% 0.35/0.70  thf(fact_7417_floor__eq__iff,axiom,
% 0.35/0.70      ! [X: rat,A: int] :
% 0.35/0.70        ( ( ( archim3151403230148437115or_rat @ X )
% 0.35/0.70          = A )
% 0.35/0.70        = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X )
% 0.35/0.70          & ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_eq_iff
% 0.35/0.70  thf(fact_7418_floor__split,axiom,
% 0.35/0.70      ! [P: int > $o,T: real] :
% 0.35/0.70        ( ( P @ ( archim6058952711729229775r_real @ T ) )
% 0.35/0.70        = ( ! [I4: int] :
% 0.35/0.70              ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I4 ) @ T )
% 0.35/0.70                & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) ) )
% 0.35/0.70             => ( P @ I4 ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_split
% 0.35/0.70  thf(fact_7419_floor__split,axiom,
% 0.35/0.70      ! [P: int > $o,T: rat] :
% 0.35/0.70        ( ( P @ ( archim3151403230148437115or_rat @ T ) )
% 0.35/0.70        = ( ! [I4: int] :
% 0.35/0.70              ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I4 ) @ T )
% 0.35/0.70                & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) ) )
% 0.35/0.70             => ( P @ I4 ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_split
% 0.35/0.70  thf(fact_7420_le__mult__floor,axiom,
% 0.35/0.70      ! [A: real,B: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.35/0.70       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.35/0.70         => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % le_mult_floor
% 0.35/0.70  thf(fact_7421_le__mult__floor,axiom,
% 0.35/0.70      ! [A: rat,B: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.35/0.70       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.35/0.70         => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % le_mult_floor
% 0.35/0.70  thf(fact_7422_less__floor__iff,axiom,
% 0.35/0.70      ! [Z2: int,X: real] :
% 0.35/0.70        ( ( ord_less_int @ Z2 @ ( archim6058952711729229775r_real @ X ) )
% 0.35/0.70        = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % less_floor_iff
% 0.35/0.70  thf(fact_7423_less__floor__iff,axiom,
% 0.35/0.70      ! [Z2: int,X: rat] :
% 0.35/0.70        ( ( ord_less_int @ Z2 @ ( archim3151403230148437115or_rat @ X ) )
% 0.35/0.70        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % less_floor_iff
% 0.35/0.70  thf(fact_7424_floor__le__iff,axiom,
% 0.35/0.70      ! [X: real,Z2: int] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ Z2 )
% 0.35/0.70        = ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_le_iff
% 0.35/0.70  thf(fact_7425_floor__le__iff,axiom,
% 0.35/0.70      ! [X: rat,Z2: int] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z2 )
% 0.35/0.70        = ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_le_iff
% 0.35/0.70  thf(fact_7426_floor__correct,axiom,
% 0.35/0.70      ! [X: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X )
% 0.35/0.70        & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_correct
% 0.35/0.70  thf(fact_7427_floor__correct,axiom,
% 0.35/0.70      ! [X: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X )
% 0.35/0.70        & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_correct
% 0.35/0.70  thf(fact_7428_abs__le__square__iff,axiom,
% 0.35/0.70      ! [X: code_integer,Y: code_integer] :
% 0.35/0.70        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y ) )
% 0.35/0.70        = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_square_iff
% 0.35/0.70  thf(fact_7429_abs__le__square__iff,axiom,
% 0.35/0.70      ! [X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) )
% 0.35/0.70        = ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_square_iff
% 0.35/0.70  thf(fact_7430_abs__le__square__iff,axiom,
% 0.35/0.70      ! [X: rat,Y: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y ) )
% 0.35/0.70        = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_square_iff
% 0.35/0.70  thf(fact_7431_abs__le__square__iff,axiom,
% 0.35/0.70      ! [X: int,Y: int] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) )
% 0.35/0.70        = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_le_square_iff
% 0.35/0.70  thf(fact_7432_powr__neg__one,axiom,
% 0.35/0.70      ! [X: real] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70       => ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 0.35/0.70          = ( divide_divide_real @ one_one_real @ X ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_neg_one
% 0.35/0.70  thf(fact_7433_powr__le__iff,axiom,
% 0.35/0.70      ! [B: real,X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_real @ one_one_real @ B )
% 0.35/0.70       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70         => ( ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X )
% 0.35/0.70            = ( ord_less_eq_real @ Y @ ( log @ B @ X ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_le_iff
% 0.35/0.70  thf(fact_7434_le__powr__iff,axiom,
% 0.35/0.70      ! [B: real,X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_real @ one_one_real @ B )
% 0.35/0.70       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70         => ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) )
% 0.35/0.70            = ( ord_less_eq_real @ ( log @ B @ X ) @ Y ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % le_powr_iff
% 0.35/0.70  thf(fact_7435_log__le__iff,axiom,
% 0.35/0.70      ! [B: real,X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_real @ one_one_real @ B )
% 0.35/0.70       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70         => ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y )
% 0.35/0.70            = ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % log_le_iff
% 0.35/0.70  thf(fact_7436_le__log__iff,axiom,
% 0.35/0.70      ! [B: real,X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_real @ one_one_real @ B )
% 0.35/0.70       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70         => ( ( ord_less_eq_real @ Y @ ( log @ B @ X ) )
% 0.35/0.70            = ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % le_log_iff
% 0.35/0.70  thf(fact_7437_floor__eq2,axiom,
% 0.35/0.70      ! [N: int,X: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
% 0.35/0.70       => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.35/0.70         => ( ( archim6058952711729229775r_real @ X )
% 0.35/0.70            = N ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_eq2
% 0.35/0.70  thf(fact_7438_sgn__power__injE,axiom,
% 0.35/0.70      ! [A: real,N: nat,X: real,B: real] :
% 0.35/0.70        ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 0.35/0.70          = X )
% 0.35/0.70       => ( ( X
% 0.35/0.70            = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
% 0.35/0.70         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.70           => ( A = B ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % sgn_power_injE
% 0.35/0.70  thf(fact_7439_floor__divide__lower,axiom,
% 0.35/0.70      ! [Q4: real,P4: real] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ Q4 )
% 0.35/0.70       => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P4 @ Q4 ) ) ) @ Q4 ) @ P4 ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_divide_lower
% 0.35/0.70  thf(fact_7440_floor__divide__lower,axiom,
% 0.35/0.70      ! [Q4: rat,P4: rat] :
% 0.35/0.70        ( ( ord_less_rat @ zero_zero_rat @ Q4 )
% 0.35/0.70       => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P4 @ Q4 ) ) ) @ Q4 ) @ P4 ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_divide_lower
% 0.35/0.70  thf(fact_7441_power2__le__iff__abs__le,axiom,
% 0.35/0.70      ! [Y: code_integer,X: code_integer] :
% 0.35/0.70        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
% 0.35/0.70       => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.70          = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % power2_le_iff_abs_le
% 0.35/0.70  thf(fact_7442_power2__le__iff__abs__le,axiom,
% 0.35/0.70      ! [Y: real,X: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.35/0.70       => ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.70          = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % power2_le_iff_abs_le
% 0.35/0.70  thf(fact_7443_power2__le__iff__abs__le,axiom,
% 0.35/0.70      ! [Y: rat,X: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.35/0.70       => ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.70          = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % power2_le_iff_abs_le
% 0.35/0.70  thf(fact_7444_power2__le__iff__abs__le,axiom,
% 0.35/0.70      ! [Y: int,X: int] :
% 0.35/0.70        ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.35/0.70       => ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.70          = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % power2_le_iff_abs_le
% 0.35/0.70  thf(fact_7445_abs__sqrt__wlog,axiom,
% 0.35/0.70      ! [P: code_integer > code_integer > $o,X: code_integer] :
% 0.35/0.70        ( ! [X4: code_integer] :
% 0.35/0.70            ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
% 0.35/0.70           => ( P @ X4 @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.70       => ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_sqrt_wlog
% 0.35/0.70  thf(fact_7446_abs__sqrt__wlog,axiom,
% 0.35/0.70      ! [P: real > real > $o,X: real] :
% 0.35/0.70        ( ! [X4: real] :
% 0.35/0.70            ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.35/0.70           => ( P @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.70       => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_sqrt_wlog
% 0.35/0.70  thf(fact_7447_abs__sqrt__wlog,axiom,
% 0.35/0.70      ! [P: rat > rat > $o,X: rat] :
% 0.35/0.70        ( ! [X4: rat] :
% 0.35/0.70            ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 0.35/0.70           => ( P @ X4 @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.70       => ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_sqrt_wlog
% 0.35/0.70  thf(fact_7448_abs__sqrt__wlog,axiom,
% 0.35/0.70      ! [P: int > int > $o,X: int] :
% 0.35/0.70        ( ! [X4: int] :
% 0.35/0.70            ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.35/0.70           => ( P @ X4 @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.70       => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_sqrt_wlog
% 0.35/0.70  thf(fact_7449_abs__square__le__1,axiom,
% 0.35/0.70      ! [X: code_integer] :
% 0.35/0.70        ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 0.35/0.70        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_square_le_1
% 0.35/0.70  thf(fact_7450_abs__square__le__1,axiom,
% 0.35/0.70      ! [X: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 0.35/0.70        = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_square_le_1
% 0.35/0.70  thf(fact_7451_abs__square__le__1,axiom,
% 0.35/0.70      ! [X: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 0.35/0.70        = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_square_le_1
% 0.35/0.70  thf(fact_7452_abs__square__le__1,axiom,
% 0.35/0.70      ! [X: int] :
% 0.35/0.70        ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 0.35/0.70        = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_square_le_1
% 0.35/0.70  thf(fact_7453_abs__square__less__1,axiom,
% 0.35/0.70      ! [X: code_integer] :
% 0.35/0.70        ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 0.35/0.70        = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_square_less_1
% 0.35/0.70  thf(fact_7454_abs__square__less__1,axiom,
% 0.35/0.70      ! [X: real] :
% 0.35/0.70        ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 0.35/0.70        = ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_square_less_1
% 0.35/0.70  thf(fact_7455_abs__square__less__1,axiom,
% 0.35/0.70      ! [X: rat] :
% 0.35/0.70        ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 0.35/0.70        = ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_square_less_1
% 0.35/0.70  thf(fact_7456_abs__square__less__1,axiom,
% 0.35/0.70      ! [X: int] :
% 0.35/0.70        ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 0.35/0.70        = ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_square_less_1
% 0.35/0.70  thf(fact_7457_power__mono__even,axiom,
% 0.35/0.70      ! [N: nat,A: code_integer,B: code_integer] :
% 0.35/0.70        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.70       => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
% 0.35/0.70         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % power_mono_even
% 0.35/0.70  thf(fact_7458_power__mono__even,axiom,
% 0.35/0.70      ! [N: nat,A: real,B: real] :
% 0.35/0.70        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.70       => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
% 0.35/0.70         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % power_mono_even
% 0.35/0.70  thf(fact_7459_power__mono__even,axiom,
% 0.35/0.70      ! [N: nat,A: rat,B: rat] :
% 0.35/0.70        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.70       => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
% 0.35/0.70         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % power_mono_even
% 0.35/0.70  thf(fact_7460_power__mono__even,axiom,
% 0.35/0.70      ! [N: nat,A: int,B: int] :
% 0.35/0.70        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.70       => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
% 0.35/0.70         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % power_mono_even
% 0.35/0.70  thf(fact_7461_add__log__eq__powr,axiom,
% 0.35/0.70      ! [B: real,X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ B )
% 0.35/0.70       => ( ( B != one_one_real )
% 0.35/0.70         => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70           => ( ( plus_plus_real @ Y @ ( log @ B @ X ) )
% 0.35/0.70              = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % add_log_eq_powr
% 0.35/0.70  thf(fact_7462_log__add__eq__powr,axiom,
% 0.35/0.70      ! [B: real,X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ B )
% 0.35/0.70       => ( ( B != one_one_real )
% 0.35/0.70         => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70           => ( ( plus_plus_real @ ( log @ B @ X ) @ Y )
% 0.35/0.70              = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % log_add_eq_powr
% 0.35/0.70  thf(fact_7463_minus__log__eq__powr,axiom,
% 0.35/0.70      ! [B: real,X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ B )
% 0.35/0.70       => ( ( B != one_one_real )
% 0.35/0.70         => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70           => ( ( minus_minus_real @ Y @ ( log @ B @ X ) )
% 0.35/0.70              = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % minus_log_eq_powr
% 0.35/0.70  thf(fact_7464_convex__sum__bound__le,axiom,
% 0.35/0.70      ! [I6: set_real,X: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
% 0.35/0.70        ( ! [I2: real] :
% 0.35/0.70            ( ( member_real @ I2 @ I6 )
% 0.35/0.70           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
% 0.35/0.70       => ( ( ( groups7713935264441627589nteger @ X @ I6 )
% 0.35/0.70            = one_one_Code_integer )
% 0.35/0.70         => ( ! [I2: real] :
% 0.35/0.70                ( ( member_real @ I2 @ I6 )
% 0.35/0.70               => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 0.35/0.70           => ( ord_le3102999989581377725nteger
% 0.35/0.70              @ ( abs_abs_Code_integer
% 0.35/0.70                @ ( minus_8373710615458151222nteger
% 0.35/0.70                  @ ( groups7713935264441627589nteger
% 0.35/0.70                    @ ^ [I4: real] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
% 0.35/0.70                    @ I6 )
% 0.35/0.70                  @ B ) )
% 0.35/0.70              @ Delta ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % convex_sum_bound_le
% 0.35/0.70  thf(fact_7465_convex__sum__bound__le,axiom,
% 0.35/0.70      ! [I6: set_nat,X: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
% 0.35/0.70        ( ! [I2: nat] :
% 0.35/0.70            ( ( member_nat @ I2 @ I6 )
% 0.35/0.70           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
% 0.35/0.70       => ( ( ( groups7501900531339628137nteger @ X @ I6 )
% 0.35/0.70            = one_one_Code_integer )
% 0.35/0.70         => ( ! [I2: nat] :
% 0.35/0.70                ( ( member_nat @ I2 @ I6 )
% 0.35/0.70               => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 0.35/0.70           => ( ord_le3102999989581377725nteger
% 0.35/0.70              @ ( abs_abs_Code_integer
% 0.35/0.70                @ ( minus_8373710615458151222nteger
% 0.35/0.70                  @ ( groups7501900531339628137nteger
% 0.35/0.70                    @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
% 0.35/0.70                    @ I6 )
% 0.35/0.70                  @ B ) )
% 0.35/0.70              @ Delta ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % convex_sum_bound_le
% 0.35/0.70  thf(fact_7466_convex__sum__bound__le,axiom,
% 0.35/0.70      ! [I6: set_int,X: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
% 0.35/0.70        ( ! [I2: int] :
% 0.35/0.70            ( ( member_int @ I2 @ I6 )
% 0.35/0.70           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
% 0.35/0.70       => ( ( ( groups7873554091576472773nteger @ X @ I6 )
% 0.35/0.70            = one_one_Code_integer )
% 0.35/0.70         => ( ! [I2: int] :
% 0.35/0.70                ( ( member_int @ I2 @ I6 )
% 0.35/0.70               => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 0.35/0.70           => ( ord_le3102999989581377725nteger
% 0.35/0.70              @ ( abs_abs_Code_integer
% 0.35/0.70                @ ( minus_8373710615458151222nteger
% 0.35/0.70                  @ ( groups7873554091576472773nteger
% 0.35/0.70                    @ ^ [I4: int] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
% 0.35/0.70                    @ I6 )
% 0.35/0.70                  @ B ) )
% 0.35/0.70              @ Delta ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % convex_sum_bound_le
% 0.35/0.70  thf(fact_7467_convex__sum__bound__le,axiom,
% 0.35/0.70      ! [I6: set_real,X: real > real,A: real > real,B: real,Delta: real] :
% 0.35/0.70        ( ! [I2: real] :
% 0.35/0.70            ( ( member_real @ I2 @ I6 )
% 0.35/0.70           => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
% 0.35/0.70       => ( ( ( groups8097168146408367636l_real @ X @ I6 )
% 0.35/0.70            = one_one_real )
% 0.35/0.70         => ( ! [I2: real] :
% 0.35/0.70                ( ( member_real @ I2 @ I6 )
% 0.35/0.70               => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 0.35/0.70           => ( ord_less_eq_real
% 0.35/0.70              @ ( abs_abs_real
% 0.35/0.70                @ ( minus_minus_real
% 0.35/0.70                  @ ( groups8097168146408367636l_real
% 0.35/0.70                    @ ^ [I4: real] : ( times_times_real @ ( A @ I4 ) @ ( X @ I4 ) )
% 0.35/0.70                    @ I6 )
% 0.35/0.70                  @ B ) )
% 0.35/0.70              @ Delta ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % convex_sum_bound_le
% 0.35/0.70  thf(fact_7468_convex__sum__bound__le,axiom,
% 0.35/0.70      ! [I6: set_int,X: int > real,A: int > real,B: real,Delta: real] :
% 0.35/0.70        ( ! [I2: int] :
% 0.35/0.70            ( ( member_int @ I2 @ I6 )
% 0.35/0.70           => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
% 0.35/0.70       => ( ( ( groups8778361861064173332t_real @ X @ I6 )
% 0.35/0.70            = one_one_real )
% 0.35/0.70         => ( ! [I2: int] :
% 0.35/0.70                ( ( member_int @ I2 @ I6 )
% 0.35/0.70               => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 0.35/0.70           => ( ord_less_eq_real
% 0.35/0.70              @ ( abs_abs_real
% 0.35/0.70                @ ( minus_minus_real
% 0.35/0.70                  @ ( groups8778361861064173332t_real
% 0.35/0.70                    @ ^ [I4: int] : ( times_times_real @ ( A @ I4 ) @ ( X @ I4 ) )
% 0.35/0.70                    @ I6 )
% 0.35/0.70                  @ B ) )
% 0.35/0.70              @ Delta ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % convex_sum_bound_le
% 0.35/0.70  thf(fact_7469_convex__sum__bound__le,axiom,
% 0.35/0.70      ! [I6: set_real,X: real > rat,A: real > rat,B: rat,Delta: rat] :
% 0.35/0.70        ( ! [I2: real] :
% 0.35/0.70            ( ( member_real @ I2 @ I6 )
% 0.35/0.70           => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
% 0.35/0.70       => ( ( ( groups1300246762558778688al_rat @ X @ I6 )
% 0.35/0.70            = one_one_rat )
% 0.35/0.70         => ( ! [I2: real] :
% 0.35/0.70                ( ( member_real @ I2 @ I6 )
% 0.35/0.70               => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 0.35/0.70           => ( ord_less_eq_rat
% 0.35/0.70              @ ( abs_abs_rat
% 0.35/0.70                @ ( minus_minus_rat
% 0.35/0.70                  @ ( groups1300246762558778688al_rat
% 0.35/0.70                    @ ^ [I4: real] : ( times_times_rat @ ( A @ I4 ) @ ( X @ I4 ) )
% 0.35/0.70                    @ I6 )
% 0.35/0.70                  @ B ) )
% 0.35/0.70              @ Delta ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % convex_sum_bound_le
% 0.35/0.70  thf(fact_7470_convex__sum__bound__le,axiom,
% 0.35/0.70      ! [I6: set_nat,X: nat > rat,A: nat > rat,B: rat,Delta: rat] :
% 0.35/0.70        ( ! [I2: nat] :
% 0.35/0.70            ( ( member_nat @ I2 @ I6 )
% 0.35/0.70           => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
% 0.35/0.70       => ( ( ( groups2906978787729119204at_rat @ X @ I6 )
% 0.35/0.70            = one_one_rat )
% 0.35/0.70         => ( ! [I2: nat] :
% 0.35/0.70                ( ( member_nat @ I2 @ I6 )
% 0.35/0.70               => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 0.35/0.70           => ( ord_less_eq_rat
% 0.35/0.70              @ ( abs_abs_rat
% 0.35/0.70                @ ( minus_minus_rat
% 0.35/0.70                  @ ( groups2906978787729119204at_rat
% 0.35/0.70                    @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( X @ I4 ) )
% 0.35/0.70                    @ I6 )
% 0.35/0.70                  @ B ) )
% 0.35/0.70              @ Delta ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % convex_sum_bound_le
% 0.35/0.70  thf(fact_7471_convex__sum__bound__le,axiom,
% 0.35/0.70      ! [I6: set_int,X: int > rat,A: int > rat,B: rat,Delta: rat] :
% 0.35/0.70        ( ! [I2: int] :
% 0.35/0.70            ( ( member_int @ I2 @ I6 )
% 0.35/0.70           => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
% 0.35/0.70       => ( ( ( groups3906332499630173760nt_rat @ X @ I6 )
% 0.35/0.70            = one_one_rat )
% 0.35/0.70         => ( ! [I2: int] :
% 0.35/0.70                ( ( member_int @ I2 @ I6 )
% 0.35/0.70               => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 0.35/0.70           => ( ord_less_eq_rat
% 0.35/0.70              @ ( abs_abs_rat
% 0.35/0.70                @ ( minus_minus_rat
% 0.35/0.70                  @ ( groups3906332499630173760nt_rat
% 0.35/0.70                    @ ^ [I4: int] : ( times_times_rat @ ( A @ I4 ) @ ( X @ I4 ) )
% 0.35/0.70                    @ I6 )
% 0.35/0.70                  @ B ) )
% 0.35/0.70              @ Delta ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % convex_sum_bound_le
% 0.35/0.70  thf(fact_7472_convex__sum__bound__le,axiom,
% 0.35/0.70      ! [I6: set_real,X: real > int,A: real > int,B: int,Delta: int] :
% 0.35/0.70        ( ! [I2: real] :
% 0.35/0.70            ( ( member_real @ I2 @ I6 )
% 0.35/0.70           => ( ord_less_eq_int @ zero_zero_int @ ( X @ I2 ) ) )
% 0.35/0.70       => ( ( ( groups1932886352136224148al_int @ X @ I6 )
% 0.35/0.70            = one_one_int )
% 0.35/0.70         => ( ! [I2: real] :
% 0.35/0.70                ( ( member_real @ I2 @ I6 )
% 0.35/0.70               => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 0.35/0.70           => ( ord_less_eq_int
% 0.35/0.70              @ ( abs_abs_int
% 0.35/0.70                @ ( minus_minus_int
% 0.35/0.70                  @ ( groups1932886352136224148al_int
% 0.35/0.70                    @ ^ [I4: real] : ( times_times_int @ ( A @ I4 ) @ ( X @ I4 ) )
% 0.35/0.70                    @ I6 )
% 0.35/0.70                  @ B ) )
% 0.35/0.70              @ Delta ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % convex_sum_bound_le
% 0.35/0.70  thf(fact_7473_convex__sum__bound__le,axiom,
% 0.35/0.70      ! [I6: set_nat,X: nat > int,A: nat > int,B: int,Delta: int] :
% 0.35/0.70        ( ! [I2: nat] :
% 0.35/0.70            ( ( member_nat @ I2 @ I6 )
% 0.35/0.70           => ( ord_less_eq_int @ zero_zero_int @ ( X @ I2 ) ) )
% 0.35/0.70       => ( ( ( groups3539618377306564664at_int @ X @ I6 )
% 0.35/0.70            = one_one_int )
% 0.35/0.70         => ( ! [I2: nat] :
% 0.35/0.70                ( ( member_nat @ I2 @ I6 )
% 0.35/0.70               => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 0.35/0.70           => ( ord_less_eq_int
% 0.35/0.70              @ ( abs_abs_int
% 0.35/0.70                @ ( minus_minus_int
% 0.35/0.70                  @ ( groups3539618377306564664at_int
% 0.35/0.70                    @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( X @ I4 ) )
% 0.35/0.70                    @ I6 )
% 0.35/0.70                  @ B ) )
% 0.35/0.70              @ Delta ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % convex_sum_bound_le
% 0.35/0.70  thf(fact_7474_floor__divide__upper,axiom,
% 0.35/0.70      ! [Q4: real,P4: real] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ Q4 )
% 0.35/0.70       => ( ord_less_real @ P4 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P4 @ Q4 ) ) ) @ one_one_real ) @ Q4 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_divide_upper
% 0.35/0.70  thf(fact_7475_floor__divide__upper,axiom,
% 0.35/0.70      ! [Q4: rat,P4: rat] :
% 0.35/0.70        ( ( ord_less_rat @ zero_zero_rat @ Q4 )
% 0.35/0.70       => ( ord_less_rat @ P4 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P4 @ Q4 ) ) ) @ one_one_rat ) @ Q4 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_divide_upper
% 0.35/0.70  thf(fact_7476_log__minus__eq__powr,axiom,
% 0.35/0.70      ! [B: real,X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ B )
% 0.35/0.70       => ( ( B != one_one_real )
% 0.35/0.70         => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70           => ( ( minus_minus_real @ ( log @ B @ X ) @ Y )
% 0.35/0.70              = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % log_minus_eq_powr
% 0.35/0.70  thf(fact_7477_powr__neg__numeral,axiom,
% 0.35/0.70      ! [X: real,N: num] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70       => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.35/0.70          = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_neg_numeral
% 0.35/0.70  thf(fact_7478_of__int__round__abs__le,axiom,
% 0.35/0.70      ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ X ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % of_int_round_abs_le
% 0.35/0.70  thf(fact_7479_of__int__round__abs__le,axiom,
% 0.35/0.70      ! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % of_int_round_abs_le
% 0.35/0.70  thf(fact_7480_round__unique_H,axiom,
% 0.35/0.70      ! [X: real,N: int] :
% 0.35/0.70        ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.70       => ( ( archim8280529875227126926d_real @ X )
% 0.35/0.70          = N ) ) ).
% 0.35/0.70  
% 0.35/0.70  % round_unique'
% 0.35/0.70  thf(fact_7481_round__unique_H,axiom,
% 0.35/0.70      ! [X: rat,N: int] :
% 0.35/0.70        ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.35/0.70       => ( ( archim7778729529865785530nd_rat @ X )
% 0.35/0.70          = N ) ) ).
% 0.35/0.70  
% 0.35/0.70  % round_unique'
% 0.35/0.70  thf(fact_7482_lemma__interval,axiom,
% 0.35/0.70      ! [A: real,X: real,B: real] :
% 0.35/0.70        ( ( ord_less_real @ A @ X )
% 0.35/0.70       => ( ( ord_less_real @ X @ B )
% 0.35/0.70         => ? [D5: real] :
% 0.35/0.70              ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.35/0.70              & ! [Y4: real] :
% 0.35/0.70                  ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D5 )
% 0.35/0.70                 => ( ( ord_less_eq_real @ A @ Y4 )
% 0.35/0.70                    & ( ord_less_eq_real @ Y4 @ B ) ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % lemma_interval
% 0.35/0.70  thf(fact_7483_lemma__interval__lt,axiom,
% 0.35/0.70      ! [A: real,X: real,B: real] :
% 0.35/0.70        ( ( ord_less_real @ A @ X )
% 0.35/0.70       => ( ( ord_less_real @ X @ B )
% 0.35/0.70         => ? [D5: real] :
% 0.35/0.70              ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.35/0.70              & ! [Y4: real] :
% 0.35/0.70                  ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D5 )
% 0.35/0.70                 => ( ( ord_less_real @ A @ Y4 )
% 0.35/0.70                    & ( ord_less_real @ Y4 @ B ) ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % lemma_interval_lt
% 0.35/0.70  thf(fact_7484_round__altdef,axiom,
% 0.35/0.70      ( archim8280529875227126926d_real
% 0.35/0.70      = ( ^ [X3: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X3 ) ) @ ( archim7802044766580827645g_real @ X3 ) @ ( archim6058952711729229775r_real @ X3 ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % round_altdef
% 0.35/0.70  thf(fact_7485_round__altdef,axiom,
% 0.35/0.70      ( archim7778729529865785530nd_rat
% 0.35/0.70      = ( ^ [X3: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X3 ) ) @ ( archim2889992004027027881ng_rat @ X3 ) @ ( archim3151403230148437115or_rat @ X3 ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % round_altdef
% 0.35/0.70  thf(fact_7486_arctan__double,axiom,
% 0.35/0.70      ! [X: real] :
% 0.35/0.70        ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.35/0.70       => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
% 0.35/0.70          = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % arctan_double
% 0.35/0.70  thf(fact_7487_powr__int,axiom,
% 0.35/0.70      ! [X: real,I: int] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.70       => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.35/0.70           => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
% 0.35/0.70              = ( power_power_real @ X @ ( nat2 @ I ) ) ) )
% 0.35/0.70          & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
% 0.35/0.70           => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
% 0.35/0.70              = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % powr_int
% 0.35/0.70  thf(fact_7488_horner__sum__of__bool__2__less,axiom,
% 0.35/0.70      ! [Bs: list_o] : ( ord_less_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_size_list_o @ Bs ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % horner_sum_of_bool_2_less
% 0.35/0.70  thf(fact_7489_zabs__less__one__iff,axiom,
% 0.35/0.70      ! [Z2: int] :
% 0.35/0.70        ( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
% 0.35/0.70        = ( Z2 = zero_zero_int ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zabs_less_one_iff
% 0.35/0.70  thf(fact_7490_arctan__less__zero__iff,axiom,
% 0.35/0.70      ! [X: real] :
% 0.35/0.70        ( ( ord_less_real @ ( arctan @ X ) @ zero_zero_real )
% 0.35/0.70        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.35/0.70  
% 0.35/0.70  % arctan_less_zero_iff
% 0.35/0.70  thf(fact_7491_zero__less__arctan__iff,axiom,
% 0.35/0.70      ! [X: real] :
% 0.35/0.70        ( ( ord_less_real @ zero_zero_real @ ( arctan @ X ) )
% 0.35/0.70        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zero_less_arctan_iff
% 0.35/0.70  thf(fact_7492_frac__of__int,axiom,
% 0.35/0.70      ! [Z2: int] :
% 0.35/0.70        ( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z2 ) )
% 0.35/0.70        = zero_zero_real ) ).
% 0.35/0.70  
% 0.35/0.70  % frac_of_int
% 0.35/0.70  thf(fact_7493_frac__of__int,axiom,
% 0.35/0.70      ! [Z2: int] :
% 0.35/0.70        ( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 0.35/0.70        = zero_zero_rat ) ).
% 0.35/0.70  
% 0.35/0.70  % frac_of_int
% 0.35/0.70  thf(fact_7494_nat__1,axiom,
% 0.35/0.70      ( ( nat2 @ one_one_int )
% 0.35/0.70      = ( suc @ zero_zero_nat ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_1
% 0.35/0.70  thf(fact_7495_nat__0__iff,axiom,
% 0.35/0.70      ! [I: int] :
% 0.35/0.70        ( ( ( nat2 @ I )
% 0.35/0.70          = zero_zero_nat )
% 0.35/0.70        = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_0_iff
% 0.35/0.70  thf(fact_7496_nat__le__0,axiom,
% 0.35/0.70      ! [Z2: int] :
% 0.35/0.70        ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
% 0.35/0.70       => ( ( nat2 @ Z2 )
% 0.35/0.70          = zero_zero_nat ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_le_0
% 0.35/0.70  thf(fact_7497_zless__nat__conj,axiom,
% 0.35/0.70      ! [W2: int,Z2: int] :
% 0.35/0.70        ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
% 0.35/0.70        = ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.35/0.70          & ( ord_less_int @ W2 @ Z2 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zless_nat_conj
% 0.35/0.70  thf(fact_7498_nat__neg__numeral,axiom,
% 0.35/0.70      ! [K: num] :
% 0.35/0.70        ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.35/0.70        = zero_zero_nat ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_neg_numeral
% 0.35/0.70  thf(fact_7499_nat__zminus__int,axiom,
% 0.35/0.70      ! [N: nat] :
% 0.35/0.70        ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.35/0.70        = zero_zero_nat ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_zminus_int
% 0.35/0.70  thf(fact_7500_zero__less__nat__eq,axiom,
% 0.35/0.70      ! [Z2: int] :
% 0.35/0.70        ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
% 0.35/0.70        = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zero_less_nat_eq
% 0.35/0.70  thf(fact_7501_nat__ceiling__le__eq,axiom,
% 0.35/0.70      ! [X: real,A: nat] :
% 0.35/0.70        ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
% 0.35/0.70        = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_ceiling_le_eq
% 0.35/0.70  thf(fact_7502_one__less__nat__eq,axiom,
% 0.35/0.70      ! [Z2: int] :
% 0.35/0.70        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
% 0.35/0.70        = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 0.35/0.70  
% 0.35/0.70  % one_less_nat_eq
% 0.35/0.70  thf(fact_7503_numeral__power__less__nat__cancel__iff,axiom,
% 0.35/0.70      ! [X: num,N: nat,A: int] :
% 0.35/0.70        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 0.35/0.70        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % numeral_power_less_nat_cancel_iff
% 0.35/0.70  thf(fact_7504_nat__less__numeral__power__cancel__iff,axiom,
% 0.35/0.70      ! [A: int,X: num,N: nat] :
% 0.35/0.70        ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.35/0.70        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_less_numeral_power_cancel_iff
% 0.35/0.70  thf(fact_7505_numeral__power__le__nat__cancel__iff,axiom,
% 0.35/0.70      ! [X: num,N: nat,A: int] :
% 0.35/0.70        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 0.35/0.70        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.35/0.70  
% 0.35/0.70  % numeral_power_le_nat_cancel_iff
% 0.35/0.70  thf(fact_7506_nat__le__numeral__power__cancel__iff,axiom,
% 0.35/0.70      ! [A: int,X: num,N: nat] :
% 0.35/0.70        ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.35/0.70        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_le_numeral_power_cancel_iff
% 0.35/0.70  thf(fact_7507_arctan__less__iff,axiom,
% 0.35/0.70      ! [X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) )
% 0.35/0.70        = ( ord_less_real @ X @ Y ) ) ).
% 0.35/0.70  
% 0.35/0.70  % arctan_less_iff
% 0.35/0.70  thf(fact_7508_arctan__monotone,axiom,
% 0.35/0.70      ! [X: real,Y: real] :
% 0.35/0.70        ( ( ord_less_real @ X @ Y )
% 0.35/0.70       => ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % arctan_monotone
% 0.35/0.70  thf(fact_7509_nat__abs__triangle__ineq,axiom,
% 0.35/0.70      ! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_abs_triangle_ineq
% 0.35/0.70  thf(fact_7510_nat__abs__int__diff,axiom,
% 0.35/0.70      ! [A: nat,B: nat] :
% 0.35/0.70        ( ( ( ord_less_eq_nat @ A @ B )
% 0.35/0.70         => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 0.35/0.70            = ( minus_minus_nat @ B @ A ) ) )
% 0.35/0.70        & ( ~ ( ord_less_eq_nat @ A @ B )
% 0.35/0.70         => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 0.35/0.70            = ( minus_minus_nat @ A @ B ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_abs_int_diff
% 0.35/0.70  thf(fact_7511_nat__zero__as__int,axiom,
% 0.35/0.70      ( zero_zero_nat
% 0.35/0.70      = ( nat2 @ zero_zero_int ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_zero_as_int
% 0.35/0.70  thf(fact_7512_nat__mono,axiom,
% 0.35/0.70      ! [X: int,Y: int] :
% 0.35/0.70        ( ( ord_less_eq_int @ X @ Y )
% 0.35/0.70       => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_mono
% 0.35/0.70  thf(fact_7513_infinite__int__iff__unbounded__le,axiom,
% 0.35/0.70      ! [S2: set_int] :
% 0.35/0.70        ( ( ~ ( finite_finite_int @ S2 ) )
% 0.35/0.70        = ( ! [M2: int] :
% 0.35/0.70            ? [N2: int] :
% 0.35/0.70              ( ( ord_less_eq_int @ M2 @ ( abs_abs_int @ N2 ) )
% 0.35/0.70              & ( member_int @ N2 @ S2 ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % infinite_int_iff_unbounded_le
% 0.35/0.70  thf(fact_7514_infinite__int__iff__unbounded,axiom,
% 0.35/0.70      ! [S2: set_int] :
% 0.35/0.70        ( ( ~ ( finite_finite_int @ S2 ) )
% 0.35/0.70        = ( ! [M2: int] :
% 0.35/0.70            ? [N2: int] :
% 0.35/0.70              ( ( ord_less_int @ M2 @ ( abs_abs_int @ N2 ) )
% 0.35/0.70              & ( member_int @ N2 @ S2 ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % infinite_int_iff_unbounded
% 0.35/0.70  thf(fact_7515_frac__ge__0,axiom,
% 0.35/0.70      ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % frac_ge_0
% 0.35/0.70  thf(fact_7516_frac__ge__0,axiom,
% 0.35/0.70      ! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).
% 0.35/0.70  
% 0.35/0.70  % frac_ge_0
% 0.35/0.70  thf(fact_7517_frac__lt__1,axiom,
% 0.35/0.70      ! [X: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X ) @ one_one_real ) ).
% 0.35/0.70  
% 0.35/0.70  % frac_lt_1
% 0.35/0.70  thf(fact_7518_frac__lt__1,axiom,
% 0.35/0.70      ! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).
% 0.35/0.70  
% 0.35/0.70  % frac_lt_1
% 0.35/0.70  thf(fact_7519_nat__mono__iff,axiom,
% 0.35/0.70      ! [Z2: int,W2: int] :
% 0.35/0.70        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.35/0.70       => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
% 0.35/0.70          = ( ord_less_int @ W2 @ Z2 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_mono_iff
% 0.35/0.70  thf(fact_7520_of__nat__ceiling,axiom,
% 0.35/0.70      ! [R2: real] : ( ord_less_eq_real @ R2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R2 ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % of_nat_ceiling
% 0.35/0.70  thf(fact_7521_of__nat__ceiling,axiom,
% 0.35/0.70      ! [R2: rat] : ( ord_less_eq_rat @ R2 @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R2 ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % of_nat_ceiling
% 0.35/0.70  thf(fact_7522_zless__nat__eq__int__zless,axiom,
% 0.35/0.70      ! [M: nat,Z2: int] :
% 0.35/0.70        ( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
% 0.35/0.70        = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z2 ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zless_nat_eq_int_zless
% 0.35/0.70  thf(fact_7523_nat__le__iff,axiom,
% 0.35/0.70      ! [X: int,N: nat] :
% 0.35/0.70        ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
% 0.35/0.70        = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_le_iff
% 0.35/0.70  thf(fact_7524_zabs__def,axiom,
% 0.35/0.70      ( abs_abs_int
% 0.35/0.70      = ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % zabs_def
% 0.35/0.70  thf(fact_7525_abs__mod__less,axiom,
% 0.35/0.70      ! [L: int,K: int] :
% 0.35/0.70        ( ( L != zero_zero_int )
% 0.35/0.70       => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % abs_mod_less
% 0.35/0.70  thf(fact_7526_of__nat__floor,axiom,
% 0.35/0.70      ! [R2: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 0.35/0.70       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim6058952711729229775r_real @ R2 ) ) ) @ R2 ) ) ).
% 0.35/0.70  
% 0.35/0.70  % of_nat_floor
% 0.35/0.70  thf(fact_7527_of__nat__floor,axiom,
% 0.35/0.70      ! [R2: rat] :
% 0.35/0.70        ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 0.35/0.70       => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R2 ) ) ) @ R2 ) ) ).
% 0.35/0.70  
% 0.35/0.70  % of_nat_floor
% 0.35/0.70  thf(fact_7528_nat__less__eq__zless,axiom,
% 0.35/0.70      ! [W2: int,Z2: int] :
% 0.35/0.70        ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 0.35/0.70       => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
% 0.35/0.70          = ( ord_less_int @ W2 @ Z2 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_less_eq_zless
% 0.35/0.70  thf(fact_7529_nat__le__eq__zle,axiom,
% 0.35/0.70      ! [W2: int,Z2: int] :
% 0.35/0.70        ( ( ( ord_less_int @ zero_zero_int @ W2 )
% 0.35/0.70          | ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
% 0.35/0.70       => ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
% 0.35/0.70          = ( ord_less_eq_int @ W2 @ Z2 ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_le_eq_zle
% 0.35/0.70  thf(fact_7530_nat__eq__iff2,axiom,
% 0.35/0.70      ! [M: nat,W2: int] :
% 0.35/0.70        ( ( M
% 0.35/0.70          = ( nat2 @ W2 ) )
% 0.35/0.70        = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 0.35/0.70           => ( W2
% 0.35/0.70              = ( semiri1314217659103216013at_int @ M ) ) )
% 0.35/0.70          & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
% 0.35/0.70           => ( M = zero_zero_nat ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_eq_iff2
% 0.35/0.70  thf(fact_7531_nat__eq__iff,axiom,
% 0.35/0.70      ! [W2: int,M: nat] :
% 0.35/0.70        ( ( ( nat2 @ W2 )
% 0.35/0.70          = M )
% 0.35/0.70        = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 0.35/0.70           => ( W2
% 0.35/0.70              = ( semiri1314217659103216013at_int @ M ) ) )
% 0.35/0.70          & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
% 0.35/0.70           => ( M = zero_zero_nat ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_eq_iff
% 0.35/0.70  thf(fact_7532_le__mult__nat__floor,axiom,
% 0.35/0.70      ! [A: real,B: real] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ B ) ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % le_mult_nat_floor
% 0.35/0.70  thf(fact_7533_le__mult__nat__floor,axiom,
% 0.35/0.70      ! [A: rat,B: rat] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim3151403230148437115or_rat @ A ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % le_mult_nat_floor
% 0.35/0.70  thf(fact_7534_split__nat,axiom,
% 0.35/0.70      ! [P: nat > $o,I: int] :
% 0.35/0.70        ( ( P @ ( nat2 @ I ) )
% 0.35/0.70        = ( ! [N2: nat] :
% 0.35/0.70              ( ( I
% 0.35/0.70                = ( semiri1314217659103216013at_int @ N2 ) )
% 0.35/0.70             => ( P @ N2 ) )
% 0.35/0.70          & ( ( ord_less_int @ I @ zero_zero_int )
% 0.35/0.70           => ( P @ zero_zero_nat ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % split_nat
% 0.35/0.70  thf(fact_7535_le__nat__iff,axiom,
% 0.35/0.70      ! [K: int,N: nat] :
% 0.35/0.70        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.70       => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
% 0.35/0.70          = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % le_nat_iff
% 0.35/0.70  thf(fact_7536_nat__floor__neg,axiom,
% 0.35/0.70      ! [X: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.35/0.70       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 0.35/0.70          = zero_zero_nat ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_floor_neg
% 0.35/0.70  thf(fact_7537_floor__eq3,axiom,
% 0.35/0.70      ! [N: nat,X: real] :
% 0.35/0.70        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
% 0.35/0.70       => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 0.35/0.70         => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 0.35/0.70            = N ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_eq3
% 0.35/0.70  thf(fact_7538_le__nat__floor,axiom,
% 0.35/0.70      ! [X: nat,A: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
% 0.35/0.70       => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % le_nat_floor
% 0.35/0.70  thf(fact_7539_nat__2,axiom,
% 0.35/0.70      ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.70      = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_2
% 0.35/0.70  thf(fact_7540_nat__less__iff,axiom,
% 0.35/0.70      ! [W2: int,M: nat] :
% 0.35/0.70        ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 0.35/0.70       => ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
% 0.35/0.70          = ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_less_iff
% 0.35/0.70  thf(fact_7541_frac__eq,axiom,
% 0.35/0.70      ! [X: real] :
% 0.35/0.70        ( ( ( archim2898591450579166408c_real @ X )
% 0.35/0.70          = X )
% 0.35/0.70        = ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.70          & ( ord_less_real @ X @ one_one_real ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % frac_eq
% 0.35/0.70  thf(fact_7542_frac__eq,axiom,
% 0.35/0.70      ! [X: rat] :
% 0.35/0.70        ( ( ( archimedean_frac_rat @ X )
% 0.35/0.70          = X )
% 0.35/0.70        = ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.35/0.70          & ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % frac_eq
% 0.35/0.70  thf(fact_7543_floor__eq4,axiom,
% 0.35/0.70      ! [N: nat,X: real] :
% 0.35/0.70        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
% 0.35/0.70       => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 0.35/0.70         => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 0.35/0.70            = N ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_eq4
% 0.35/0.70  thf(fact_7544_frac__add,axiom,
% 0.35/0.70      ! [X: real,Y: real] :
% 0.35/0.70        ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 0.35/0.70         => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
% 0.35/0.70            = ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) ) )
% 0.35/0.70        & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 0.35/0.70         => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
% 0.35/0.70            = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % frac_add
% 0.35/0.70  thf(fact_7545_frac__add,axiom,
% 0.35/0.70      ! [X: rat,Y: rat] :
% 0.35/0.70        ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 0.35/0.70         => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
% 0.35/0.70            = ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) ) )
% 0.35/0.70        & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 0.35/0.70         => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
% 0.35/0.70            = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % frac_add
% 0.35/0.70  thf(fact_7546_diff__nat__eq__if,axiom,
% 0.35/0.70      ! [Z8: int,Z2: int] :
% 0.35/0.70        ( ( ( ord_less_int @ Z8 @ zero_zero_int )
% 0.35/0.70         => ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) )
% 0.35/0.70            = ( nat2 @ Z2 ) ) )
% 0.35/0.70        & ( ~ ( ord_less_int @ Z8 @ zero_zero_int )
% 0.35/0.70         => ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) )
% 0.35/0.70            = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z2 @ Z8 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z2 @ Z8 ) ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % diff_nat_eq_if
% 0.35/0.70  thf(fact_7547_of__int__of__nat,axiom,
% 0.35/0.70      ( ring_1_of_int_rat
% 0.35/0.70      = ( ^ [K3: int] : ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri681578069525770553at_rat @ ( nat2 @ K3 ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % of_int_of_nat
% 0.35/0.70  thf(fact_7548_of__int__of__nat,axiom,
% 0.35/0.70      ( ring_18347121197199848620nteger
% 0.35/0.70      = ( ^ [K3: int] : ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri4939895301339042750nteger @ ( nat2 @ K3 ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % of_int_of_nat
% 0.35/0.70  thf(fact_7549_of__int__of__nat,axiom,
% 0.35/0.70      ( ring_1_of_int_int
% 0.35/0.70      = ( ^ [K3: int] : ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K3 ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % of_int_of_nat
% 0.35/0.70  thf(fact_7550_of__int__of__nat,axiom,
% 0.35/0.70      ( ring_1_of_int_real
% 0.35/0.70      = ( ^ [K3: int] : ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K3 ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % of_int_of_nat
% 0.35/0.70  thf(fact_7551_nat__dvd__iff,axiom,
% 0.35/0.70      ! [Z2: int,M: nat] :
% 0.35/0.70        ( ( dvd_dvd_nat @ ( nat2 @ Z2 ) @ M )
% 0.35/0.70        = ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.35/0.70           => ( dvd_dvd_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) )
% 0.35/0.70          & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.35/0.70           => ( M = zero_zero_nat ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_dvd_iff
% 0.35/0.70  thf(fact_7552_nat__intermed__int__val,axiom,
% 0.35/0.70      ! [M: nat,N: nat,F: nat > int,K: int] :
% 0.35/0.70        ( ! [I2: nat] :
% 0.35/0.70            ( ( ( ord_less_eq_nat @ M @ I2 )
% 0.35/0.70              & ( ord_less_nat @ I2 @ N ) )
% 0.35/0.70           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 0.35/0.70       => ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.70         => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 0.35/0.70           => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.35/0.70             => ? [I2: nat] :
% 0.35/0.70                  ( ( ord_less_eq_nat @ M @ I2 )
% 0.35/0.70                  & ( ord_less_eq_nat @ I2 @ N )
% 0.35/0.70                  & ( ( F @ I2 )
% 0.35/0.70                    = K ) ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_intermed_int_val
% 0.35/0.70  thf(fact_7553_incr__lemma,axiom,
% 0.35/0.70      ! [D: int,Z2: int,X: int] :
% 0.35/0.70        ( ( ord_less_int @ zero_zero_int @ D )
% 0.35/0.70       => ( ord_less_int @ Z2 @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % incr_lemma
% 0.35/0.70  thf(fact_7554_decr__lemma,axiom,
% 0.35/0.70      ! [D: int,X: int,Z2: int] :
% 0.35/0.70        ( ( ord_less_int @ zero_zero_int @ D )
% 0.35/0.70       => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D ) ) @ Z2 ) ) ).
% 0.35/0.70  
% 0.35/0.70  % decr_lemma
% 0.35/0.70  thf(fact_7555_eucl__rel__int__remainderI,axiom,
% 0.35/0.70      ! [R2: int,L: int,K: int,Q4: int] :
% 0.35/0.70        ( ( ( sgn_sgn_int @ R2 )
% 0.35/0.70          = ( sgn_sgn_int @ L ) )
% 0.35/0.70       => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
% 0.35/0.70         => ( ( K
% 0.35/0.70              = ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R2 ) )
% 0.35/0.70           => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % eucl_rel_int_remainderI
% 0.35/0.70  thf(fact_7556_nat__ivt__aux,axiom,
% 0.35/0.70      ! [N: nat,F: nat > int,K: int] :
% 0.35/0.70        ( ! [I2: nat] :
% 0.35/0.70            ( ( ord_less_nat @ I2 @ N )
% 0.35/0.70           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 0.35/0.70       => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.35/0.70         => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.35/0.70           => ? [I2: nat] :
% 0.35/0.70                ( ( ord_less_eq_nat @ I2 @ N )
% 0.35/0.70                & ( ( F @ I2 )
% 0.35/0.70                  = K ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % nat_ivt_aux
% 0.35/0.70  thf(fact_7557_eucl__rel__int_Ocases,axiom,
% 0.35/0.70      ! [A12: int,A23: int,A32: product_prod_int_int] :
% 0.35/0.70        ( ( eucl_rel_int @ A12 @ A23 @ A32 )
% 0.35/0.70       => ( ( ( A23 = zero_zero_int )
% 0.35/0.70           => ( A32
% 0.35/0.70             != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
% 0.35/0.70         => ( ! [Q3: int] :
% 0.35/0.70                ( ( A32
% 0.35/0.70                  = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
% 0.35/0.70               => ( ( A23 != zero_zero_int )
% 0.35/0.70                 => ( A12
% 0.35/0.70                   != ( times_times_int @ Q3 @ A23 ) ) ) )
% 0.35/0.70           => ~ ! [R3: int,Q3: int] :
% 0.35/0.70                  ( ( A32
% 0.35/0.70                    = ( product_Pair_int_int @ Q3 @ R3 ) )
% 0.35/0.70                 => ( ( ( sgn_sgn_int @ R3 )
% 0.35/0.70                      = ( sgn_sgn_int @ A23 ) )
% 0.35/0.70                   => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A23 ) )
% 0.35/0.70                     => ( A12
% 0.35/0.70                       != ( plus_plus_int @ ( times_times_int @ Q3 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % eucl_rel_int.cases
% 0.35/0.70  thf(fact_7558_eucl__rel__int_Osimps,axiom,
% 0.35/0.70      ( eucl_rel_int
% 0.35/0.70      = ( ^ [A1: int,A22: int,A33: product_prod_int_int] :
% 0.35/0.70            ( ? [K3: int] :
% 0.35/0.70                ( ( A1 = K3 )
% 0.35/0.70                & ( A22 = zero_zero_int )
% 0.35/0.70                & ( A33
% 0.35/0.70                  = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
% 0.35/0.70            | ? [L2: int,K3: int,Q5: int] :
% 0.35/0.70                ( ( A1 = K3 )
% 0.35/0.70                & ( A22 = L2 )
% 0.35/0.70                & ( A33
% 0.35/0.70                  = ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
% 0.35/0.70                & ( L2 != zero_zero_int )
% 0.35/0.70                & ( K3
% 0.35/0.70                  = ( times_times_int @ Q5 @ L2 ) ) )
% 0.35/0.70            | ? [R5: int,L2: int,K3: int,Q5: int] :
% 0.35/0.70                ( ( A1 = K3 )
% 0.35/0.70                & ( A22 = L2 )
% 0.35/0.70                & ( A33
% 0.35/0.70                  = ( product_Pair_int_int @ Q5 @ R5 ) )
% 0.35/0.70                & ( ( sgn_sgn_int @ R5 )
% 0.35/0.70                  = ( sgn_sgn_int @ L2 ) )
% 0.35/0.70                & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
% 0.35/0.70                & ( K3
% 0.35/0.70                  = ( plus_plus_int @ ( times_times_int @ Q5 @ L2 ) @ R5 ) ) ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % eucl_rel_int.simps
% 0.35/0.70  thf(fact_7559_floor__add,axiom,
% 0.35/0.70      ! [X: real,Y: real] :
% 0.35/0.70        ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 0.35/0.70         => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
% 0.35/0.70            = ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) )
% 0.35/0.70        & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 0.35/0.70         => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
% 0.35/0.70            = ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ one_one_int ) ) ) ) ).
% 0.35/0.70  
% 0.35/0.70  % floor_add
% 0.35/0.70  thf(fact_7560_floor__add,axiom,
% 0.35/0.70      ! [X: rat,Y: rat] :
% 0.35/0.70        ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 0.35/0.70         => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
% 0.35/0.70            = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) )
% 0.35/0.70        & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 0.35/0.70         => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
% 0.35/0.70            = ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ one_one_int ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % floor_add
% 0.35/0.71  thf(fact_7561_nat0__intermed__int__val,axiom,
% 0.35/0.71      ! [N: nat,F: nat > int,K: int] :
% 0.35/0.71        ( ! [I2: nat] :
% 0.35/0.71            ( ( ord_less_nat @ I2 @ N )
% 0.35/0.71           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 0.35/0.71       => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.35/0.71         => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.35/0.71           => ? [I2: nat] :
% 0.35/0.71                ( ( ord_less_eq_nat @ I2 @ N )
% 0.35/0.71                & ( ( F @ I2 )
% 0.35/0.71                  = K ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nat0_intermed_int_val
% 0.35/0.71  thf(fact_7562_arctan__add,axiom,
% 0.35/0.71      ! [X: real,Y: real] :
% 0.35/0.71        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.35/0.71       => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.35/0.71         => ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y ) )
% 0.35/0.71            = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % arctan_add
% 0.35/0.71  thf(fact_7563_powr__real__of__int,axiom,
% 0.35/0.71      ! [X: real,N: int] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71       => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
% 0.35/0.71           => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 0.35/0.71              = ( power_power_real @ X @ ( nat2 @ N ) ) ) )
% 0.35/0.71          & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
% 0.35/0.71           => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 0.35/0.71              = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % powr_real_of_int
% 0.35/0.71  thf(fact_7564_case__prod__Pair__iden,axiom,
% 0.35/0.71      ! [P4: produc8763457246119570046nteger] :
% 0.35/0.71        ( ( produc3906647086178084059nteger @ produc6137756002093451184nteger @ P4 )
% 0.35/0.71        = P4 ) ).
% 0.35/0.71  
% 0.35/0.71  % case_prod_Pair_iden
% 0.35/0.71  thf(fact_7565_case__prod__Pair__iden,axiom,
% 0.35/0.71      ! [P4: produc1908205239877642774nteger] :
% 0.35/0.71        ( ( produc6512950862096126219nteger @ produc8603105652947943368nteger @ P4 )
% 0.35/0.71        = P4 ) ).
% 0.35/0.71  
% 0.35/0.71  % case_prod_Pair_iden
% 0.35/0.71  thf(fact_7566_case__prod__Pair__iden,axiom,
% 0.35/0.71      ! [P4: produc2285326912895808259nt_int] :
% 0.35/0.71        ( ( produc8492565224438309093nt_int @ produc5700946648718959541nt_int @ P4 )
% 0.35/0.71        = P4 ) ).
% 0.35/0.71  
% 0.35/0.71  % case_prod_Pair_iden
% 0.35/0.71  thf(fact_7567_case__prod__Pair__iden,axiom,
% 0.35/0.71      ! [P4: produc7773217078559923341nt_int] :
% 0.35/0.71        ( ( produc5122537100556696953nt_int @ produc4305682042979456191nt_int @ P4 )
% 0.35/0.71        = P4 ) ).
% 0.35/0.71  
% 0.35/0.71  % case_prod_Pair_iden
% 0.35/0.71  thf(fact_7568_case__prod__Pair__iden,axiom,
% 0.35/0.71      ! [P4: product_prod_int_int] :
% 0.35/0.71        ( ( produc4245557441103728435nt_int @ product_Pair_int_int @ P4 )
% 0.35/0.71        = P4 ) ).
% 0.35/0.71  
% 0.35/0.71  % case_prod_Pair_iden
% 0.35/0.71  thf(fact_7569_the__elem__eq,axiom,
% 0.35/0.71      ! [X: produc3843707927480180839at_nat] :
% 0.35/0.71        ( ( the_el221668144340439132at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
% 0.35/0.71        = X ) ).
% 0.35/0.71  
% 0.35/0.71  % the_elem_eq
% 0.35/0.71  thf(fact_7570_the__elem__eq,axiom,
% 0.35/0.71      ! [X: product_prod_nat_nat] :
% 0.35/0.71        ( ( the_el2281957884133575798at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
% 0.35/0.71        = X ) ).
% 0.35/0.71  
% 0.35/0.71  % the_elem_eq
% 0.35/0.71  thf(fact_7571_the__elem__eq,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( the_elem_real @ ( insert_real @ X @ bot_bot_set_real ) )
% 0.35/0.71        = X ) ).
% 0.35/0.71  
% 0.35/0.71  % the_elem_eq
% 0.35/0.71  thf(fact_7572_the__elem__eq,axiom,
% 0.35/0.71      ! [X: nat] :
% 0.35/0.71        ( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.35/0.71        = X ) ).
% 0.35/0.71  
% 0.35/0.71  % the_elem_eq
% 0.35/0.71  thf(fact_7573_the__elem__eq,axiom,
% 0.35/0.71      ! [X: int] :
% 0.35/0.71        ( ( the_elem_int @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.35/0.71        = X ) ).
% 0.35/0.71  
% 0.35/0.71  % the_elem_eq
% 0.35/0.71  thf(fact_7574_exp__ge__one__minus__x__over__n__power__n,axiom,
% 0.35/0.71      ! [X: real,N: nat] :
% 0.35/0.71        ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
% 0.35/0.71       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.71         => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % exp_ge_one_minus_x_over_n_power_n
% 0.35/0.71  thf(fact_7575_exp__ge__one__plus__x__over__n__power__n,axiom,
% 0.35/0.71      ! [N: nat,X: real] :
% 0.35/0.71        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
% 0.35/0.71       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.71         => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % exp_ge_one_plus_x_over_n_power_n
% 0.35/0.71  thf(fact_7576_sum__diff1_H__aux,axiom,
% 0.35/0.71      ! [F3: set_complex,I6: set_complex,F: complex > complex,I: complex] :
% 0.35/0.71        ( ( finite3207457112153483333omplex @ F3 )
% 0.35/0.71       => ( ( ord_le211207098394363844omplex
% 0.35/0.71            @ ( collect_complex
% 0.35/0.71              @ ^ [I4: complex] :
% 0.35/0.71                  ( ( member_complex @ I4 @ I6 )
% 0.35/0.71                  & ( ( F @ I4 )
% 0.35/0.71                   != zero_zero_complex ) ) )
% 0.35/0.71            @ F3 )
% 0.35/0.71         => ( ( ( member_complex @ I @ I6 )
% 0.35/0.71             => ( ( groups808145749697022017omplex @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.71                = ( minus_minus_complex @ ( groups808145749697022017omplex @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.35/0.71            & ( ~ ( member_complex @ I @ I6 )
% 0.35/0.71             => ( ( groups808145749697022017omplex @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.71                = ( groups808145749697022017omplex @ F @ I6 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum_diff1'_aux
% 0.35/0.71  thf(fact_7577_sum__diff1_H__aux,axiom,
% 0.35/0.71      ! [F3: set_real,I6: set_real,F: real > complex,I: real] :
% 0.35/0.71        ( ( finite_finite_real @ F3 )
% 0.35/0.71       => ( ( ord_less_eq_set_real
% 0.35/0.71            @ ( collect_real
% 0.35/0.71              @ ^ [I4: real] :
% 0.35/0.71                  ( ( member_real @ I4 @ I6 )
% 0.35/0.71                  & ( ( F @ I4 )
% 0.35/0.71                   != zero_zero_complex ) ) )
% 0.35/0.71            @ F3 )
% 0.35/0.71         => ( ( ( member_real @ I @ I6 )
% 0.35/0.71             => ( ( groups5683813829254066239omplex @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.35/0.71                = ( minus_minus_complex @ ( groups5683813829254066239omplex @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.35/0.71            & ( ~ ( member_real @ I @ I6 )
% 0.35/0.71             => ( ( groups5683813829254066239omplex @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.35/0.71                = ( groups5683813829254066239omplex @ F @ I6 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum_diff1'_aux
% 0.35/0.71  thf(fact_7578_sum__diff1_H__aux,axiom,
% 0.35/0.71      ! [F3: set_complex,I6: set_complex,F: complex > real,I: complex] :
% 0.35/0.71        ( ( finite3207457112153483333omplex @ F3 )
% 0.35/0.71       => ( ( ord_le211207098394363844omplex
% 0.35/0.71            @ ( collect_complex
% 0.35/0.71              @ ^ [I4: complex] :
% 0.35/0.71                  ( ( member_complex @ I4 @ I6 )
% 0.35/0.71                  & ( ( F @ I4 )
% 0.35/0.71                   != zero_zero_real ) ) )
% 0.35/0.71            @ F3 )
% 0.35/0.71         => ( ( ( member_complex @ I @ I6 )
% 0.35/0.71             => ( ( groups5737402329758386879x_real @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.71                = ( minus_minus_real @ ( groups5737402329758386879x_real @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.35/0.71            & ( ~ ( member_complex @ I @ I6 )
% 0.35/0.71             => ( ( groups5737402329758386879x_real @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.71                = ( groups5737402329758386879x_real @ F @ I6 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum_diff1'_aux
% 0.35/0.71  thf(fact_7579_sum__diff1_H__aux,axiom,
% 0.35/0.71      ! [F3: set_real,I6: set_real,F: real > real,I: real] :
% 0.35/0.71        ( ( finite_finite_real @ F3 )
% 0.35/0.71       => ( ( ord_less_eq_set_real
% 0.35/0.71            @ ( collect_real
% 0.35/0.71              @ ^ [I4: real] :
% 0.35/0.71                  ( ( member_real @ I4 @ I6 )
% 0.35/0.71                  & ( ( F @ I4 )
% 0.35/0.71                   != zero_zero_real ) ) )
% 0.35/0.71            @ F3 )
% 0.35/0.71         => ( ( ( member_real @ I @ I6 )
% 0.35/0.71             => ( ( groups97945582718554045l_real @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.35/0.71                = ( minus_minus_real @ ( groups97945582718554045l_real @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.35/0.71            & ( ~ ( member_real @ I @ I6 )
% 0.35/0.71             => ( ( groups97945582718554045l_real @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.35/0.71                = ( groups97945582718554045l_real @ F @ I6 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum_diff1'_aux
% 0.35/0.71  thf(fact_7580_sum__diff1_H__aux,axiom,
% 0.35/0.71      ! [F3: set_complex,I6: set_complex,F: complex > rat,I: complex] :
% 0.35/0.71        ( ( finite3207457112153483333omplex @ F3 )
% 0.35/0.71       => ( ( ord_le211207098394363844omplex
% 0.35/0.71            @ ( collect_complex
% 0.35/0.71              @ ^ [I4: complex] :
% 0.35/0.71                  ( ( member_complex @ I4 @ I6 )
% 0.35/0.71                  & ( ( F @ I4 )
% 0.35/0.71                   != zero_zero_rat ) ) )
% 0.35/0.71            @ F3 )
% 0.35/0.71         => ( ( ( member_complex @ I @ I6 )
% 0.35/0.71             => ( ( groups2276542476275365739ex_rat @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.71                = ( minus_minus_rat @ ( groups2276542476275365739ex_rat @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.35/0.71            & ( ~ ( member_complex @ I @ I6 )
% 0.35/0.71             => ( ( groups2276542476275365739ex_rat @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.71                = ( groups2276542476275365739ex_rat @ F @ I6 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum_diff1'_aux
% 0.35/0.71  thf(fact_7581_sum__diff1_H__aux,axiom,
% 0.35/0.71      ! [F3: set_real,I6: set_real,F: real > rat,I: real] :
% 0.35/0.71        ( ( finite_finite_real @ F3 )
% 0.35/0.71       => ( ( ord_less_eq_set_real
% 0.35/0.71            @ ( collect_real
% 0.35/0.71              @ ^ [I4: real] :
% 0.35/0.71                  ( ( member_real @ I4 @ I6 )
% 0.35/0.71                  & ( ( F @ I4 )
% 0.35/0.71                   != zero_zero_rat ) ) )
% 0.35/0.71            @ F3 )
% 0.35/0.71         => ( ( ( member_real @ I @ I6 )
% 0.35/0.71             => ( ( groups3269169158384524137al_rat @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.35/0.71                = ( minus_minus_rat @ ( groups3269169158384524137al_rat @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.35/0.71            & ( ~ ( member_real @ I @ I6 )
% 0.35/0.71             => ( ( groups3269169158384524137al_rat @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.35/0.71                = ( groups3269169158384524137al_rat @ F @ I6 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum_diff1'_aux
% 0.35/0.71  thf(fact_7582_sum__diff1_H__aux,axiom,
% 0.35/0.71      ! [F3: set_complex,I6: set_complex,F: complex > int,I: complex] :
% 0.35/0.71        ( ( finite3207457112153483333omplex @ F3 )
% 0.35/0.71       => ( ( ord_le211207098394363844omplex
% 0.35/0.71            @ ( collect_complex
% 0.35/0.71              @ ^ [I4: complex] :
% 0.35/0.71                  ( ( member_complex @ I4 @ I6 )
% 0.35/0.71                  & ( ( F @ I4 )
% 0.35/0.71                   != zero_zero_int ) ) )
% 0.35/0.71            @ F3 )
% 0.35/0.71         => ( ( ( member_complex @ I @ I6 )
% 0.35/0.71             => ( ( groups2909182065852811199ex_int @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.71                = ( minus_minus_int @ ( groups2909182065852811199ex_int @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.35/0.71            & ( ~ ( member_complex @ I @ I6 )
% 0.35/0.71             => ( ( groups2909182065852811199ex_int @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.71                = ( groups2909182065852811199ex_int @ F @ I6 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum_diff1'_aux
% 0.35/0.71  thf(fact_7583_sum__diff1_H__aux,axiom,
% 0.35/0.71      ! [F3: set_real,I6: set_real,F: real > int,I: real] :
% 0.35/0.71        ( ( finite_finite_real @ F3 )
% 0.35/0.71       => ( ( ord_less_eq_set_real
% 0.35/0.71            @ ( collect_real
% 0.35/0.71              @ ^ [I4: real] :
% 0.35/0.71                  ( ( member_real @ I4 @ I6 )
% 0.35/0.71                  & ( ( F @ I4 )
% 0.35/0.71                   != zero_zero_int ) ) )
% 0.35/0.71            @ F3 )
% 0.35/0.71         => ( ( ( member_real @ I @ I6 )
% 0.35/0.71             => ( ( groups3901808747961969597al_int @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.35/0.71                = ( minus_minus_int @ ( groups3901808747961969597al_int @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.35/0.71            & ( ~ ( member_real @ I @ I6 )
% 0.35/0.71             => ( ( groups3901808747961969597al_int @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.35/0.71                = ( groups3901808747961969597al_int @ F @ I6 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum_diff1'_aux
% 0.35/0.71  thf(fact_7584_sum__diff1_H__aux,axiom,
% 0.35/0.71      ! [F3: set_nat,I6: set_nat,F: nat > complex,I: nat] :
% 0.35/0.71        ( ( finite_finite_nat @ F3 )
% 0.35/0.71       => ( ( ord_less_eq_set_nat
% 0.35/0.71            @ ( collect_nat
% 0.35/0.71              @ ^ [I4: nat] :
% 0.35/0.71                  ( ( member_nat @ I4 @ I6 )
% 0.35/0.71                  & ( ( F @ I4 )
% 0.35/0.71                   != zero_zero_complex ) ) )
% 0.35/0.71            @ F3 )
% 0.35/0.71         => ( ( ( member_nat @ I @ I6 )
% 0.35/0.71             => ( ( groups8515261248781899619omplex @ F @ ( minus_minus_set_nat @ I6 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
% 0.35/0.71                = ( minus_minus_complex @ ( groups8515261248781899619omplex @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.35/0.71            & ( ~ ( member_nat @ I @ I6 )
% 0.35/0.71             => ( ( groups8515261248781899619omplex @ F @ ( minus_minus_set_nat @ I6 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
% 0.35/0.71                = ( groups8515261248781899619omplex @ F @ I6 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum_diff1'_aux
% 0.35/0.71  thf(fact_7585_sum__diff1_H__aux,axiom,
% 0.35/0.71      ! [F3: set_nat,I6: set_nat,F: nat > real,I: nat] :
% 0.35/0.71        ( ( finite_finite_nat @ F3 )
% 0.35/0.71       => ( ( ord_less_eq_set_nat
% 0.35/0.71            @ ( collect_nat
% 0.35/0.71              @ ^ [I4: nat] :
% 0.35/0.71                  ( ( member_nat @ I4 @ I6 )
% 0.35/0.71                  & ( ( F @ I4 )
% 0.35/0.71                   != zero_zero_real ) ) )
% 0.35/0.71            @ F3 )
% 0.35/0.71         => ( ( ( member_nat @ I @ I6 )
% 0.35/0.71             => ( ( groups8560362682196896993t_real @ F @ ( minus_minus_set_nat @ I6 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
% 0.35/0.71                = ( minus_minus_real @ ( groups8560362682196896993t_real @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.35/0.71            & ( ~ ( member_nat @ I @ I6 )
% 0.35/0.71             => ( ( groups8560362682196896993t_real @ F @ ( minus_minus_set_nat @ I6 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
% 0.35/0.71                = ( groups8560362682196896993t_real @ F @ I6 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum_diff1'_aux
% 0.35/0.71  thf(fact_7586_inverse__zero,axiom,
% 0.35/0.71      ( ( invers8013647133539491842omplex @ zero_zero_complex )
% 0.35/0.71      = zero_zero_complex ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_zero
% 0.35/0.71  thf(fact_7587_inverse__zero,axiom,
% 0.35/0.71      ( ( inverse_inverse_real @ zero_zero_real )
% 0.35/0.71      = zero_zero_real ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_zero
% 0.35/0.71  thf(fact_7588_inverse__zero,axiom,
% 0.35/0.71      ( ( inverse_inverse_rat @ zero_zero_rat )
% 0.35/0.71      = zero_zero_rat ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_zero
% 0.35/0.71  thf(fact_7589_inverse__nonzero__iff__nonzero,axiom,
% 0.35/0.71      ! [A: complex] :
% 0.35/0.71        ( ( ( invers8013647133539491842omplex @ A )
% 0.35/0.71          = zero_zero_complex )
% 0.35/0.71        = ( A = zero_zero_complex ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_nonzero_iff_nonzero
% 0.35/0.71  thf(fact_7590_inverse__nonzero__iff__nonzero,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( ( inverse_inverse_real @ A )
% 0.35/0.71          = zero_zero_real )
% 0.35/0.71        = ( A = zero_zero_real ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_nonzero_iff_nonzero
% 0.35/0.71  thf(fact_7591_inverse__nonzero__iff__nonzero,axiom,
% 0.35/0.71      ! [A: rat] :
% 0.35/0.71        ( ( ( inverse_inverse_rat @ A )
% 0.35/0.71          = zero_zero_rat )
% 0.35/0.71        = ( A = zero_zero_rat ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_nonzero_iff_nonzero
% 0.35/0.71  thf(fact_7592_exp__less__cancel__iff,axiom,
% 0.35/0.71      ! [X: real,Y: real] :
% 0.35/0.71        ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 0.35/0.71        = ( ord_less_real @ X @ Y ) ) ).
% 0.35/0.71  
% 0.35/0.71  % exp_less_cancel_iff
% 0.35/0.71  thf(fact_7593_exp__less__mono,axiom,
% 0.35/0.71      ! [X: real,Y: real] :
% 0.35/0.71        ( ( ord_less_real @ X @ Y )
% 0.35/0.71       => ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % exp_less_mono
% 0.35/0.71  thf(fact_7594_inverse__nonpositive__iff__nonpositive,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 0.35/0.71        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_nonpositive_iff_nonpositive
% 0.35/0.71  thf(fact_7595_inverse__nonpositive__iff__nonpositive,axiom,
% 0.35/0.71      ! [A: rat] :
% 0.35/0.71        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 0.35/0.71        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_nonpositive_iff_nonpositive
% 0.35/0.71  thf(fact_7596_inverse__nonnegative__iff__nonnegative,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 0.35/0.71        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_nonnegative_iff_nonnegative
% 0.35/0.71  thf(fact_7597_inverse__nonnegative__iff__nonnegative,axiom,
% 0.35/0.71      ! [A: rat] :
% 0.35/0.71        ( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 0.35/0.71        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_nonnegative_iff_nonnegative
% 0.35/0.71  thf(fact_7598_inverse__less__iff__less,axiom,
% 0.35/0.71      ! [A: real,B: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.71       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.35/0.71         => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.35/0.71            = ( ord_less_real @ B @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_less_iff_less
% 0.35/0.71  thf(fact_7599_inverse__less__iff__less,axiom,
% 0.35/0.71      ! [A: rat,B: rat] :
% 0.35/0.71        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.35/0.71       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.35/0.71         => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.35/0.71            = ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_less_iff_less
% 0.35/0.71  thf(fact_7600_inverse__less__iff__less__neg,axiom,
% 0.35/0.71      ! [A: real,B: real] :
% 0.35/0.71        ( ( ord_less_real @ A @ zero_zero_real )
% 0.35/0.71       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.35/0.71         => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.35/0.71            = ( ord_less_real @ B @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_less_iff_less_neg
% 0.35/0.71  thf(fact_7601_inverse__less__iff__less__neg,axiom,
% 0.35/0.71      ! [A: rat,B: rat] :
% 0.35/0.71        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.35/0.71       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.35/0.71         => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.35/0.71            = ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_less_iff_less_neg
% 0.35/0.71  thf(fact_7602_inverse__negative__iff__negative,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 0.35/0.71        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_negative_iff_negative
% 0.35/0.71  thf(fact_7603_inverse__negative__iff__negative,axiom,
% 0.35/0.71      ! [A: rat] :
% 0.35/0.71        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 0.35/0.71        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_negative_iff_negative
% 0.35/0.71  thf(fact_7604_inverse__positive__iff__positive,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 0.35/0.71        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_positive_iff_positive
% 0.35/0.71  thf(fact_7605_inverse__positive__iff__positive,axiom,
% 0.35/0.71      ! [A: rat] :
% 0.35/0.71        ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 0.35/0.71        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_positive_iff_positive
% 0.35/0.71  thf(fact_7606_exp__zero,axiom,
% 0.35/0.71      ( ( exp_complex @ zero_zero_complex )
% 0.35/0.71      = one_one_complex ) ).
% 0.35/0.71  
% 0.35/0.71  % exp_zero
% 0.35/0.71  thf(fact_7607_exp__zero,axiom,
% 0.35/0.71      ( ( exp_real @ zero_zero_real )
% 0.35/0.71      = one_one_real ) ).
% 0.35/0.71  
% 0.35/0.71  % exp_zero
% 0.35/0.71  thf(fact_7608_sum_Oempty_H,axiom,
% 0.35/0.71      ! [P4: real > complex] :
% 0.35/0.71        ( ( groups5683813829254066239omplex @ P4 @ bot_bot_set_real )
% 0.35/0.71        = zero_zero_complex ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.empty'
% 0.35/0.71  thf(fact_7609_sum_Oempty_H,axiom,
% 0.35/0.71      ! [P4: real > real] :
% 0.35/0.71        ( ( groups97945582718554045l_real @ P4 @ bot_bot_set_real )
% 0.35/0.71        = zero_zero_real ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.empty'
% 0.35/0.71  thf(fact_7610_sum_Oempty_H,axiom,
% 0.35/0.71      ! [P4: real > rat] :
% 0.35/0.71        ( ( groups3269169158384524137al_rat @ P4 @ bot_bot_set_real )
% 0.35/0.71        = zero_zero_rat ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.empty'
% 0.35/0.71  thf(fact_7611_sum_Oempty_H,axiom,
% 0.35/0.71      ! [P4: real > nat] :
% 0.35/0.71        ( ( groups3904299218471019873al_nat @ P4 @ bot_bot_set_real )
% 0.35/0.71        = zero_zero_nat ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.empty'
% 0.35/0.71  thf(fact_7612_sum_Oempty_H,axiom,
% 0.35/0.71      ! [P4: real > int] :
% 0.35/0.71        ( ( groups3901808747961969597al_int @ P4 @ bot_bot_set_real )
% 0.35/0.71        = zero_zero_int ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.empty'
% 0.35/0.71  thf(fact_7613_sum_Oempty_H,axiom,
% 0.35/0.71      ! [P4: nat > complex] :
% 0.35/0.71        ( ( groups8515261248781899619omplex @ P4 @ bot_bot_set_nat )
% 0.35/0.71        = zero_zero_complex ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.empty'
% 0.35/0.71  thf(fact_7614_sum_Oempty_H,axiom,
% 0.35/0.71      ! [P4: nat > real] :
% 0.35/0.71        ( ( groups8560362682196896993t_real @ P4 @ bot_bot_set_nat )
% 0.35/0.71        = zero_zero_real ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.empty'
% 0.35/0.71  thf(fact_7615_sum_Oempty_H,axiom,
% 0.35/0.71      ! [P4: nat > rat] :
% 0.35/0.71        ( ( groups1351286907653491341at_rat @ P4 @ bot_bot_set_nat )
% 0.35/0.71        = zero_zero_rat ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.empty'
% 0.35/0.71  thf(fact_7616_sum_Oempty_H,axiom,
% 0.35/0.71      ! [P4: nat > nat] :
% 0.35/0.71        ( ( groups1986416967739987077at_nat @ P4 @ bot_bot_set_nat )
% 0.35/0.71        = zero_zero_nat ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.empty'
% 0.35/0.71  thf(fact_7617_sum_Oempty_H,axiom,
% 0.35/0.71      ! [P4: nat > int] :
% 0.35/0.71        ( ( groups1983926497230936801at_int @ P4 @ bot_bot_set_nat )
% 0.35/0.71        = zero_zero_int ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.empty'
% 0.35/0.71  thf(fact_7618_sum_Oeq__sum,axiom,
% 0.35/0.71      ! [I6: set_nat,P4: nat > nat] :
% 0.35/0.71        ( ( finite_finite_nat @ I6 )
% 0.35/0.71       => ( ( groups1986416967739987077at_nat @ P4 @ I6 )
% 0.35/0.71          = ( groups3542108847815614940at_nat @ P4 @ I6 ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.eq_sum
% 0.35/0.71  thf(fact_7619_sum_Oeq__sum,axiom,
% 0.35/0.71      ! [I6: set_complex,P4: complex > complex] :
% 0.35/0.71        ( ( finite3207457112153483333omplex @ I6 )
% 0.35/0.71       => ( ( groups808145749697022017omplex @ P4 @ I6 )
% 0.35/0.71          = ( groups7754918857620584856omplex @ P4 @ I6 ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.eq_sum
% 0.35/0.71  thf(fact_7620_sum_Oeq__sum,axiom,
% 0.35/0.71      ! [I6: set_nat,P4: nat > real] :
% 0.35/0.71        ( ( finite_finite_nat @ I6 )
% 0.35/0.71       => ( ( groups8560362682196896993t_real @ P4 @ I6 )
% 0.35/0.71          = ( groups6591440286371151544t_real @ P4 @ I6 ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.eq_sum
% 0.35/0.71  thf(fact_7621_inverse__le__iff__le,axiom,
% 0.35/0.71      ! [A: real,B: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.71       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.35/0.71         => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.35/0.71            = ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_le_iff_le
% 0.35/0.71  thf(fact_7622_inverse__le__iff__le,axiom,
% 0.35/0.71      ! [A: rat,B: rat] :
% 0.35/0.71        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.35/0.71       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.35/0.71         => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.35/0.71            = ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_le_iff_le
% 0.35/0.71  thf(fact_7623_inverse__le__iff__le__neg,axiom,
% 0.35/0.71      ! [A: real,B: real] :
% 0.35/0.71        ( ( ord_less_real @ A @ zero_zero_real )
% 0.35/0.71       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.35/0.71         => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.35/0.71            = ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_le_iff_le_neg
% 0.35/0.71  thf(fact_7624_inverse__le__iff__le__neg,axiom,
% 0.35/0.71      ! [A: rat,B: rat] :
% 0.35/0.71        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.35/0.71       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.35/0.71         => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.35/0.71            = ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_le_iff_le_neg
% 0.35/0.71  thf(fact_7625_right__inverse,axiom,
% 0.35/0.71      ! [A: complex] :
% 0.35/0.71        ( ( A != zero_zero_complex )
% 0.35/0.71       => ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
% 0.35/0.71          = one_one_complex ) ) ).
% 0.35/0.71  
% 0.35/0.71  % right_inverse
% 0.35/0.71  thf(fact_7626_right__inverse,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( A != zero_zero_real )
% 0.35/0.71       => ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
% 0.35/0.71          = one_one_real ) ) ).
% 0.35/0.71  
% 0.35/0.71  % right_inverse
% 0.35/0.71  thf(fact_7627_right__inverse,axiom,
% 0.35/0.71      ! [A: rat] :
% 0.35/0.71        ( ( A != zero_zero_rat )
% 0.35/0.71       => ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
% 0.35/0.71          = one_one_rat ) ) ).
% 0.35/0.71  
% 0.35/0.71  % right_inverse
% 0.35/0.71  thf(fact_7628_left__inverse,axiom,
% 0.35/0.71      ! [A: complex] :
% 0.35/0.71        ( ( A != zero_zero_complex )
% 0.35/0.71       => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
% 0.35/0.71          = one_one_complex ) ) ).
% 0.35/0.71  
% 0.35/0.71  % left_inverse
% 0.35/0.71  thf(fact_7629_left__inverse,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( A != zero_zero_real )
% 0.35/0.71       => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
% 0.35/0.71          = one_one_real ) ) ).
% 0.35/0.71  
% 0.35/0.71  % left_inverse
% 0.35/0.71  thf(fact_7630_left__inverse,axiom,
% 0.35/0.71      ! [A: rat] :
% 0.35/0.71        ( ( A != zero_zero_rat )
% 0.35/0.71       => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
% 0.35/0.71          = one_one_rat ) ) ).
% 0.35/0.71  
% 0.35/0.71  % left_inverse
% 0.35/0.71  thf(fact_7631_one__less__exp__iff,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
% 0.35/0.71        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.35/0.71  
% 0.35/0.71  % one_less_exp_iff
% 0.35/0.71  thf(fact_7632_exp__less__one__iff,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
% 0.35/0.71        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.35/0.71  
% 0.35/0.71  % exp_less_one_iff
% 0.35/0.71  thf(fact_7633_sum_Oinsert_H,axiom,
% 0.35/0.71      ! [I6: set_real,P4: real > complex,I: real] :
% 0.35/0.71        ( ( finite_finite_real
% 0.35/0.71          @ ( collect_real
% 0.35/0.71            @ ^ [X3: real] :
% 0.35/0.71                ( ( member_real @ X3 @ I6 )
% 0.35/0.71                & ( ( P4 @ X3 )
% 0.35/0.71                 != zero_zero_complex ) ) ) )
% 0.35/0.71       => ( ( ( member_real @ I @ I6 )
% 0.35/0.71           => ( ( groups5683813829254066239omplex @ P4 @ ( insert_real @ I @ I6 ) )
% 0.35/0.71              = ( groups5683813829254066239omplex @ P4 @ I6 ) ) )
% 0.35/0.71          & ( ~ ( member_real @ I @ I6 )
% 0.35/0.71           => ( ( groups5683813829254066239omplex @ P4 @ ( insert_real @ I @ I6 ) )
% 0.35/0.71              = ( plus_plus_complex @ ( P4 @ I ) @ ( groups5683813829254066239omplex @ P4 @ I6 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.insert'
% 0.35/0.71  thf(fact_7634_sum_Oinsert_H,axiom,
% 0.35/0.71      ! [I6: set_nat,P4: nat > complex,I: nat] :
% 0.35/0.71        ( ( finite_finite_nat
% 0.35/0.71          @ ( collect_nat
% 0.35/0.71            @ ^ [X3: nat] :
% 0.35/0.71                ( ( member_nat @ X3 @ I6 )
% 0.35/0.71                & ( ( P4 @ X3 )
% 0.35/0.71                 != zero_zero_complex ) ) ) )
% 0.35/0.71       => ( ( ( member_nat @ I @ I6 )
% 0.35/0.71           => ( ( groups8515261248781899619omplex @ P4 @ ( insert_nat @ I @ I6 ) )
% 0.35/0.71              = ( groups8515261248781899619omplex @ P4 @ I6 ) ) )
% 0.35/0.71          & ( ~ ( member_nat @ I @ I6 )
% 0.35/0.71           => ( ( groups8515261248781899619omplex @ P4 @ ( insert_nat @ I @ I6 ) )
% 0.35/0.71              = ( plus_plus_complex @ ( P4 @ I ) @ ( groups8515261248781899619omplex @ P4 @ I6 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.insert'
% 0.35/0.71  thf(fact_7635_sum_Oinsert_H,axiom,
% 0.35/0.71      ! [I6: set_int,P4: int > complex,I: int] :
% 0.35/0.71        ( ( finite_finite_int
% 0.35/0.71          @ ( collect_int
% 0.35/0.71            @ ^ [X3: int] :
% 0.35/0.71                ( ( member_int @ X3 @ I6 )
% 0.35/0.71                & ( ( P4 @ X3 )
% 0.35/0.71                 != zero_zero_complex ) ) ) )
% 0.35/0.71       => ( ( ( member_int @ I @ I6 )
% 0.35/0.71           => ( ( groups267424677133301183omplex @ P4 @ ( insert_int @ I @ I6 ) )
% 0.35/0.71              = ( groups267424677133301183omplex @ P4 @ I6 ) ) )
% 0.35/0.71          & ( ~ ( member_int @ I @ I6 )
% 0.35/0.71           => ( ( groups267424677133301183omplex @ P4 @ ( insert_int @ I @ I6 ) )
% 0.35/0.71              = ( plus_plus_complex @ ( P4 @ I ) @ ( groups267424677133301183omplex @ P4 @ I6 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.insert'
% 0.35/0.71  thf(fact_7636_sum_Oinsert_H,axiom,
% 0.35/0.71      ! [I6: set_complex,P4: complex > complex,I: complex] :
% 0.35/0.71        ( ( finite3207457112153483333omplex
% 0.35/0.71          @ ( collect_complex
% 0.35/0.71            @ ^ [X3: complex] :
% 0.35/0.71                ( ( member_complex @ X3 @ I6 )
% 0.35/0.71                & ( ( P4 @ X3 )
% 0.35/0.71                 != zero_zero_complex ) ) ) )
% 0.35/0.71       => ( ( ( member_complex @ I @ I6 )
% 0.35/0.71           => ( ( groups808145749697022017omplex @ P4 @ ( insert_complex @ I @ I6 ) )
% 0.35/0.71              = ( groups808145749697022017omplex @ P4 @ I6 ) ) )
% 0.35/0.71          & ( ~ ( member_complex @ I @ I6 )
% 0.35/0.71           => ( ( groups808145749697022017omplex @ P4 @ ( insert_complex @ I @ I6 ) )
% 0.35/0.71              = ( plus_plus_complex @ ( P4 @ I ) @ ( groups808145749697022017omplex @ P4 @ I6 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.insert'
% 0.35/0.71  thf(fact_7637_sum_Oinsert_H,axiom,
% 0.35/0.71      ! [I6: set_real,P4: real > real,I: real] :
% 0.35/0.71        ( ( finite_finite_real
% 0.35/0.71          @ ( collect_real
% 0.35/0.71            @ ^ [X3: real] :
% 0.35/0.71                ( ( member_real @ X3 @ I6 )
% 0.35/0.71                & ( ( P4 @ X3 )
% 0.35/0.71                 != zero_zero_real ) ) ) )
% 0.35/0.71       => ( ( ( member_real @ I @ I6 )
% 0.35/0.71           => ( ( groups97945582718554045l_real @ P4 @ ( insert_real @ I @ I6 ) )
% 0.35/0.71              = ( groups97945582718554045l_real @ P4 @ I6 ) ) )
% 0.35/0.71          & ( ~ ( member_real @ I @ I6 )
% 0.35/0.71           => ( ( groups97945582718554045l_real @ P4 @ ( insert_real @ I @ I6 ) )
% 0.35/0.71              = ( plus_plus_real @ ( P4 @ I ) @ ( groups97945582718554045l_real @ P4 @ I6 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.insert'
% 0.35/0.71  thf(fact_7638_sum_Oinsert_H,axiom,
% 0.35/0.71      ! [I6: set_nat,P4: nat > real,I: nat] :
% 0.35/0.71        ( ( finite_finite_nat
% 0.35/0.71          @ ( collect_nat
% 0.35/0.71            @ ^ [X3: nat] :
% 0.35/0.71                ( ( member_nat @ X3 @ I6 )
% 0.35/0.71                & ( ( P4 @ X3 )
% 0.35/0.71                 != zero_zero_real ) ) ) )
% 0.35/0.71       => ( ( ( member_nat @ I @ I6 )
% 0.35/0.71           => ( ( groups8560362682196896993t_real @ P4 @ ( insert_nat @ I @ I6 ) )
% 0.35/0.71              = ( groups8560362682196896993t_real @ P4 @ I6 ) ) )
% 0.35/0.71          & ( ~ ( member_nat @ I @ I6 )
% 0.35/0.71           => ( ( groups8560362682196896993t_real @ P4 @ ( insert_nat @ I @ I6 ) )
% 0.35/0.71              = ( plus_plus_real @ ( P4 @ I ) @ ( groups8560362682196896993t_real @ P4 @ I6 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.insert'
% 0.35/0.71  thf(fact_7639_sum_Oinsert_H,axiom,
% 0.35/0.71      ! [I6: set_int,P4: int > real,I: int] :
% 0.35/0.71        ( ( finite_finite_int
% 0.35/0.71          @ ( collect_int
% 0.35/0.71            @ ^ [X3: int] :
% 0.35/0.71                ( ( member_int @ X3 @ I6 )
% 0.35/0.71                & ( ( P4 @ X3 )
% 0.35/0.71                 != zero_zero_real ) ) ) )
% 0.35/0.71       => ( ( ( member_int @ I @ I6 )
% 0.35/0.71           => ( ( groups1523912220035142973t_real @ P4 @ ( insert_int @ I @ I6 ) )
% 0.35/0.71              = ( groups1523912220035142973t_real @ P4 @ I6 ) ) )
% 0.35/0.71          & ( ~ ( member_int @ I @ I6 )
% 0.35/0.71           => ( ( groups1523912220035142973t_real @ P4 @ ( insert_int @ I @ I6 ) )
% 0.35/0.71              = ( plus_plus_real @ ( P4 @ I ) @ ( groups1523912220035142973t_real @ P4 @ I6 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.insert'
% 0.35/0.71  thf(fact_7640_sum_Oinsert_H,axiom,
% 0.35/0.71      ! [I6: set_complex,P4: complex > real,I: complex] :
% 0.35/0.71        ( ( finite3207457112153483333omplex
% 0.35/0.71          @ ( collect_complex
% 0.35/0.71            @ ^ [X3: complex] :
% 0.35/0.71                ( ( member_complex @ X3 @ I6 )
% 0.35/0.71                & ( ( P4 @ X3 )
% 0.35/0.71                 != zero_zero_real ) ) ) )
% 0.35/0.71       => ( ( ( member_complex @ I @ I6 )
% 0.35/0.71           => ( ( groups5737402329758386879x_real @ P4 @ ( insert_complex @ I @ I6 ) )
% 0.35/0.71              = ( groups5737402329758386879x_real @ P4 @ I6 ) ) )
% 0.35/0.71          & ( ~ ( member_complex @ I @ I6 )
% 0.35/0.71           => ( ( groups5737402329758386879x_real @ P4 @ ( insert_complex @ I @ I6 ) )
% 0.35/0.71              = ( plus_plus_real @ ( P4 @ I ) @ ( groups5737402329758386879x_real @ P4 @ I6 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.insert'
% 0.35/0.71  thf(fact_7641_sum_Oinsert_H,axiom,
% 0.35/0.71      ! [I6: set_real,P4: real > rat,I: real] :
% 0.35/0.71        ( ( finite_finite_real
% 0.35/0.71          @ ( collect_real
% 0.35/0.71            @ ^ [X3: real] :
% 0.35/0.71                ( ( member_real @ X3 @ I6 )
% 0.35/0.71                & ( ( P4 @ X3 )
% 0.35/0.71                 != zero_zero_rat ) ) ) )
% 0.35/0.71       => ( ( ( member_real @ I @ I6 )
% 0.35/0.71           => ( ( groups3269169158384524137al_rat @ P4 @ ( insert_real @ I @ I6 ) )
% 0.35/0.71              = ( groups3269169158384524137al_rat @ P4 @ I6 ) ) )
% 0.35/0.71          & ( ~ ( member_real @ I @ I6 )
% 0.35/0.71           => ( ( groups3269169158384524137al_rat @ P4 @ ( insert_real @ I @ I6 ) )
% 0.35/0.71              = ( plus_plus_rat @ ( P4 @ I ) @ ( groups3269169158384524137al_rat @ P4 @ I6 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.insert'
% 0.35/0.71  thf(fact_7642_sum_Oinsert_H,axiom,
% 0.35/0.71      ! [I6: set_nat,P4: nat > rat,I: nat] :
% 0.35/0.71        ( ( finite_finite_nat
% 0.35/0.71          @ ( collect_nat
% 0.35/0.71            @ ^ [X3: nat] :
% 0.35/0.71                ( ( member_nat @ X3 @ I6 )
% 0.35/0.71                & ( ( P4 @ X3 )
% 0.35/0.71                 != zero_zero_rat ) ) ) )
% 0.35/0.71       => ( ( ( member_nat @ I @ I6 )
% 0.35/0.71           => ( ( groups1351286907653491341at_rat @ P4 @ ( insert_nat @ I @ I6 ) )
% 0.35/0.71              = ( groups1351286907653491341at_rat @ P4 @ I6 ) ) )
% 0.35/0.71          & ( ~ ( member_nat @ I @ I6 )
% 0.35/0.71           => ( ( groups1351286907653491341at_rat @ P4 @ ( insert_nat @ I @ I6 ) )
% 0.35/0.71              = ( plus_plus_rat @ ( P4 @ I ) @ ( groups1351286907653491341at_rat @ P4 @ I6 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.insert'
% 0.35/0.71  thf(fact_7643_nonzero__norm__inverse,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( A != zero_zero_real )
% 0.35/0.71       => ( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A ) )
% 0.35/0.71          = ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_norm_inverse
% 0.35/0.71  thf(fact_7644_nonzero__norm__inverse,axiom,
% 0.35/0.71      ! [A: complex] :
% 0.35/0.71        ( ( A != zero_zero_complex )
% 0.35/0.71       => ( ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ A ) )
% 0.35/0.71          = ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_norm_inverse
% 0.35/0.71  thf(fact_7645_field__class_Ofield__inverse__zero,axiom,
% 0.35/0.71      ( ( invers8013647133539491842omplex @ zero_zero_complex )
% 0.35/0.71      = zero_zero_complex ) ).
% 0.35/0.71  
% 0.35/0.71  % field_class.field_inverse_zero
% 0.35/0.71  thf(fact_7646_field__class_Ofield__inverse__zero,axiom,
% 0.35/0.71      ( ( inverse_inverse_real @ zero_zero_real )
% 0.35/0.71      = zero_zero_real ) ).
% 0.35/0.71  
% 0.35/0.71  % field_class.field_inverse_zero
% 0.35/0.71  thf(fact_7647_field__class_Ofield__inverse__zero,axiom,
% 0.35/0.71      ( ( inverse_inverse_rat @ zero_zero_rat )
% 0.35/0.71      = zero_zero_rat ) ).
% 0.35/0.71  
% 0.35/0.71  % field_class.field_inverse_zero
% 0.35/0.71  thf(fact_7648_inverse__zero__imp__zero,axiom,
% 0.35/0.71      ! [A: complex] :
% 0.35/0.71        ( ( ( invers8013647133539491842omplex @ A )
% 0.35/0.71          = zero_zero_complex )
% 0.35/0.71       => ( A = zero_zero_complex ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_zero_imp_zero
% 0.35/0.71  thf(fact_7649_inverse__zero__imp__zero,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( ( inverse_inverse_real @ A )
% 0.35/0.71          = zero_zero_real )
% 0.35/0.71       => ( A = zero_zero_real ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_zero_imp_zero
% 0.35/0.71  thf(fact_7650_inverse__zero__imp__zero,axiom,
% 0.35/0.71      ! [A: rat] :
% 0.35/0.71        ( ( ( inverse_inverse_rat @ A )
% 0.35/0.71          = zero_zero_rat )
% 0.35/0.71       => ( A = zero_zero_rat ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_zero_imp_zero
% 0.35/0.71  thf(fact_7651_nonzero__inverse__eq__imp__eq,axiom,
% 0.35/0.71      ! [A: complex,B: complex] :
% 0.35/0.71        ( ( ( invers8013647133539491842omplex @ A )
% 0.35/0.71          = ( invers8013647133539491842omplex @ B ) )
% 0.35/0.71       => ( ( A != zero_zero_complex )
% 0.35/0.71         => ( ( B != zero_zero_complex )
% 0.35/0.71           => ( A = B ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_inverse_eq_imp_eq
% 0.35/0.71  thf(fact_7652_nonzero__inverse__eq__imp__eq,axiom,
% 0.35/0.71      ! [A: real,B: real] :
% 0.35/0.71        ( ( ( inverse_inverse_real @ A )
% 0.35/0.71          = ( inverse_inverse_real @ B ) )
% 0.35/0.71       => ( ( A != zero_zero_real )
% 0.35/0.71         => ( ( B != zero_zero_real )
% 0.35/0.71           => ( A = B ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_inverse_eq_imp_eq
% 0.35/0.71  thf(fact_7653_nonzero__inverse__eq__imp__eq,axiom,
% 0.35/0.71      ! [A: rat,B: rat] :
% 0.35/0.71        ( ( ( inverse_inverse_rat @ A )
% 0.35/0.71          = ( inverse_inverse_rat @ B ) )
% 0.35/0.71       => ( ( A != zero_zero_rat )
% 0.35/0.71         => ( ( B != zero_zero_rat )
% 0.35/0.71           => ( A = B ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_inverse_eq_imp_eq
% 0.35/0.71  thf(fact_7654_nonzero__inverse__inverse__eq,axiom,
% 0.35/0.71      ! [A: complex] :
% 0.35/0.71        ( ( A != zero_zero_complex )
% 0.35/0.71       => ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
% 0.35/0.71          = A ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_inverse_inverse_eq
% 0.35/0.71  thf(fact_7655_nonzero__inverse__inverse__eq,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( A != zero_zero_real )
% 0.35/0.71       => ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
% 0.35/0.71          = A ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_inverse_inverse_eq
% 0.35/0.71  thf(fact_7656_nonzero__inverse__inverse__eq,axiom,
% 0.35/0.71      ! [A: rat] :
% 0.35/0.71        ( ( A != zero_zero_rat )
% 0.35/0.71       => ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
% 0.35/0.71          = A ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_inverse_inverse_eq
% 0.35/0.71  thf(fact_7657_nonzero__imp__inverse__nonzero,axiom,
% 0.35/0.71      ! [A: complex] :
% 0.35/0.71        ( ( A != zero_zero_complex )
% 0.35/0.71       => ( ( invers8013647133539491842omplex @ A )
% 0.35/0.71         != zero_zero_complex ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_imp_inverse_nonzero
% 0.35/0.71  thf(fact_7658_nonzero__imp__inverse__nonzero,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( A != zero_zero_real )
% 0.35/0.71       => ( ( inverse_inverse_real @ A )
% 0.35/0.71         != zero_zero_real ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_imp_inverse_nonzero
% 0.35/0.71  thf(fact_7659_nonzero__imp__inverse__nonzero,axiom,
% 0.35/0.71      ! [A: rat] :
% 0.35/0.71        ( ( A != zero_zero_rat )
% 0.35/0.71       => ( ( inverse_inverse_rat @ A )
% 0.35/0.71         != zero_zero_rat ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_imp_inverse_nonzero
% 0.35/0.71  thf(fact_7660_exp__less__cancel,axiom,
% 0.35/0.71      ! [X: real,Y: real] :
% 0.35/0.71        ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 0.35/0.71       => ( ord_less_real @ X @ Y ) ) ).
% 0.35/0.71  
% 0.35/0.71  % exp_less_cancel
% 0.35/0.71  thf(fact_7661_exp__not__eq__zero,axiom,
% 0.35/0.71      ! [X: complex] :
% 0.35/0.71        ( ( exp_complex @ X )
% 0.35/0.71       != zero_zero_complex ) ).
% 0.35/0.71  
% 0.35/0.71  % exp_not_eq_zero
% 0.35/0.71  thf(fact_7662_exp__not__eq__zero,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( exp_real @ X )
% 0.35/0.71       != zero_zero_real ) ).
% 0.35/0.71  
% 0.35/0.71  % exp_not_eq_zero
% 0.35/0.71  thf(fact_7663_sum_Onon__neutral_H,axiom,
% 0.35/0.71      ! [G: real > complex,I6: set_real] :
% 0.35/0.71        ( ( groups5683813829254066239omplex @ G
% 0.35/0.71          @ ( collect_real
% 0.35/0.71            @ ^ [X3: real] :
% 0.35/0.71                ( ( member_real @ X3 @ I6 )
% 0.35/0.71                & ( ( G @ X3 )
% 0.35/0.71                 != zero_zero_complex ) ) ) )
% 0.35/0.71        = ( groups5683813829254066239omplex @ G @ I6 ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.non_neutral'
% 0.35/0.71  thf(fact_7664_sum_Onon__neutral_H,axiom,
% 0.35/0.71      ! [G: nat > complex,I6: set_nat] :
% 0.35/0.71        ( ( groups8515261248781899619omplex @ G
% 0.35/0.71          @ ( collect_nat
% 0.35/0.71            @ ^ [X3: nat] :
% 0.35/0.71                ( ( member_nat @ X3 @ I6 )
% 0.35/0.71                & ( ( G @ X3 )
% 0.35/0.71                 != zero_zero_complex ) ) ) )
% 0.35/0.71        = ( groups8515261248781899619omplex @ G @ I6 ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.non_neutral'
% 0.35/0.71  thf(fact_7665_sum_Onon__neutral_H,axiom,
% 0.35/0.71      ! [G: int > complex,I6: set_int] :
% 0.35/0.71        ( ( groups267424677133301183omplex @ G
% 0.35/0.71          @ ( collect_int
% 0.35/0.71            @ ^ [X3: int] :
% 0.35/0.71                ( ( member_int @ X3 @ I6 )
% 0.35/0.71                & ( ( G @ X3 )
% 0.35/0.71                 != zero_zero_complex ) ) ) )
% 0.35/0.71        = ( groups267424677133301183omplex @ G @ I6 ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.non_neutral'
% 0.35/0.71  thf(fact_7666_sum_Onon__neutral_H,axiom,
% 0.35/0.71      ! [G: real > real,I6: set_real] :
% 0.35/0.71        ( ( groups97945582718554045l_real @ G
% 0.35/0.71          @ ( collect_real
% 0.35/0.71            @ ^ [X3: real] :
% 0.35/0.71                ( ( member_real @ X3 @ I6 )
% 0.35/0.71                & ( ( G @ X3 )
% 0.35/0.71                 != zero_zero_real ) ) ) )
% 0.35/0.71        = ( groups97945582718554045l_real @ G @ I6 ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.non_neutral'
% 0.35/0.71  thf(fact_7667_sum_Onon__neutral_H,axiom,
% 0.35/0.71      ! [G: nat > real,I6: set_nat] :
% 0.35/0.71        ( ( groups8560362682196896993t_real @ G
% 0.35/0.71          @ ( collect_nat
% 0.35/0.71            @ ^ [X3: nat] :
% 0.35/0.71                ( ( member_nat @ X3 @ I6 )
% 0.35/0.71                & ( ( G @ X3 )
% 0.35/0.71                 != zero_zero_real ) ) ) )
% 0.35/0.71        = ( groups8560362682196896993t_real @ G @ I6 ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.non_neutral'
% 0.35/0.71  thf(fact_7668_sum_Onon__neutral_H,axiom,
% 0.35/0.71      ! [G: int > real,I6: set_int] :
% 0.35/0.71        ( ( groups1523912220035142973t_real @ G
% 0.35/0.71          @ ( collect_int
% 0.35/0.71            @ ^ [X3: int] :
% 0.35/0.71                ( ( member_int @ X3 @ I6 )
% 0.35/0.71                & ( ( G @ X3 )
% 0.35/0.71                 != zero_zero_real ) ) ) )
% 0.35/0.71        = ( groups1523912220035142973t_real @ G @ I6 ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.non_neutral'
% 0.35/0.71  thf(fact_7669_sum_Onon__neutral_H,axiom,
% 0.35/0.71      ! [G: real > rat,I6: set_real] :
% 0.35/0.71        ( ( groups3269169158384524137al_rat @ G
% 0.35/0.71          @ ( collect_real
% 0.35/0.71            @ ^ [X3: real] :
% 0.35/0.71                ( ( member_real @ X3 @ I6 )
% 0.35/0.71                & ( ( G @ X3 )
% 0.35/0.71                 != zero_zero_rat ) ) ) )
% 0.35/0.71        = ( groups3269169158384524137al_rat @ G @ I6 ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.non_neutral'
% 0.35/0.71  thf(fact_7670_sum_Onon__neutral_H,axiom,
% 0.35/0.71      ! [G: nat > rat,I6: set_nat] :
% 0.35/0.71        ( ( groups1351286907653491341at_rat @ G
% 0.35/0.71          @ ( collect_nat
% 0.35/0.71            @ ^ [X3: nat] :
% 0.35/0.71                ( ( member_nat @ X3 @ I6 )
% 0.35/0.71                & ( ( G @ X3 )
% 0.35/0.71                 != zero_zero_rat ) ) ) )
% 0.35/0.71        = ( groups1351286907653491341at_rat @ G @ I6 ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.non_neutral'
% 0.35/0.71  thf(fact_7671_sum_Onon__neutral_H,axiom,
% 0.35/0.71      ! [G: int > rat,I6: set_int] :
% 0.35/0.71        ( ( groups2350640619554545897nt_rat @ G
% 0.35/0.71          @ ( collect_int
% 0.35/0.71            @ ^ [X3: int] :
% 0.35/0.71                ( ( member_int @ X3 @ I6 )
% 0.35/0.71                & ( ( G @ X3 )
% 0.35/0.71                 != zero_zero_rat ) ) ) )
% 0.35/0.71        = ( groups2350640619554545897nt_rat @ G @ I6 ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.non_neutral'
% 0.35/0.71  thf(fact_7672_sum_Onon__neutral_H,axiom,
% 0.35/0.71      ! [G: real > nat,I6: set_real] :
% 0.35/0.71        ( ( groups3904299218471019873al_nat @ G
% 0.35/0.71          @ ( collect_real
% 0.35/0.71            @ ^ [X3: real] :
% 0.35/0.71                ( ( member_real @ X3 @ I6 )
% 0.35/0.71                & ( ( G @ X3 )
% 0.35/0.71                 != zero_zero_nat ) ) ) )
% 0.35/0.71        = ( groups3904299218471019873al_nat @ G @ I6 ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.non_neutral'
% 0.35/0.71  thf(fact_7673_norm__inverse__le__norm,axiom,
% 0.35/0.71      ! [R2: real,X: real] :
% 0.35/0.71        ( ( ord_less_eq_real @ R2 @ ( real_V7735802525324610683m_real @ X ) )
% 0.35/0.71       => ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.35/0.71         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ X ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % norm_inverse_le_norm
% 0.35/0.71  thf(fact_7674_norm__inverse__le__norm,axiom,
% 0.35/0.71      ! [R2: real,X: complex] :
% 0.35/0.71        ( ( ord_less_eq_real @ R2 @ ( real_V1022390504157884413omplex @ X ) )
% 0.35/0.71       => ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.35/0.71         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ X ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % norm_inverse_le_norm
% 0.35/0.71  thf(fact_7675_inverse__less__imp__less,axiom,
% 0.35/0.71      ! [A: real,B: real] :
% 0.35/0.71        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.35/0.71       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.71         => ( ord_less_real @ B @ A ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_less_imp_less
% 0.35/0.71  thf(fact_7676_inverse__less__imp__less,axiom,
% 0.35/0.71      ! [A: rat,B: rat] :
% 0.35/0.71        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.35/0.71       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.35/0.71         => ( ord_less_rat @ B @ A ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_less_imp_less
% 0.35/0.71  thf(fact_7677_less__imp__inverse__less,axiom,
% 0.35/0.71      ! [A: real,B: real] :
% 0.35/0.71        ( ( ord_less_real @ A @ B )
% 0.35/0.71       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.71         => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % less_imp_inverse_less
% 0.35/0.71  thf(fact_7678_less__imp__inverse__less,axiom,
% 0.35/0.71      ! [A: rat,B: rat] :
% 0.35/0.71        ( ( ord_less_rat @ A @ B )
% 0.35/0.71       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.35/0.71         => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % less_imp_inverse_less
% 0.35/0.71  thf(fact_7679_inverse__less__imp__less__neg,axiom,
% 0.35/0.71      ! [A: real,B: real] :
% 0.35/0.71        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.35/0.71       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.35/0.71         => ( ord_less_real @ B @ A ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_less_imp_less_neg
% 0.35/0.71  thf(fact_7680_inverse__less__imp__less__neg,axiom,
% 0.35/0.71      ! [A: rat,B: rat] :
% 0.35/0.71        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.35/0.71       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.35/0.71         => ( ord_less_rat @ B @ A ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_less_imp_less_neg
% 0.35/0.71  thf(fact_7681_less__imp__inverse__less__neg,axiom,
% 0.35/0.71      ! [A: real,B: real] :
% 0.35/0.71        ( ( ord_less_real @ A @ B )
% 0.35/0.71       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.35/0.71         => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % less_imp_inverse_less_neg
% 0.35/0.71  thf(fact_7682_less__imp__inverse__less__neg,axiom,
% 0.35/0.71      ! [A: rat,B: rat] :
% 0.35/0.71        ( ( ord_less_rat @ A @ B )
% 0.35/0.71       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.35/0.71         => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % less_imp_inverse_less_neg
% 0.35/0.71  thf(fact_7683_inverse__negative__imp__negative,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 0.35/0.71       => ( ( A != zero_zero_real )
% 0.35/0.71         => ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_negative_imp_negative
% 0.35/0.71  thf(fact_7684_inverse__negative__imp__negative,axiom,
% 0.35/0.71      ! [A: rat] :
% 0.35/0.71        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 0.35/0.71       => ( ( A != zero_zero_rat )
% 0.35/0.71         => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_negative_imp_negative
% 0.35/0.71  thf(fact_7685_inverse__positive__imp__positive,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 0.35/0.71       => ( ( A != zero_zero_real )
% 0.35/0.71         => ( ord_less_real @ zero_zero_real @ A ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_positive_imp_positive
% 0.35/0.71  thf(fact_7686_inverse__positive__imp__positive,axiom,
% 0.35/0.71      ! [A: rat] :
% 0.35/0.71        ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 0.35/0.71       => ( ( A != zero_zero_rat )
% 0.35/0.71         => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_positive_imp_positive
% 0.35/0.71  thf(fact_7687_negative__imp__inverse__negative,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( ord_less_real @ A @ zero_zero_real )
% 0.35/0.71       => ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).
% 0.35/0.71  
% 0.35/0.71  % negative_imp_inverse_negative
% 0.35/0.71  thf(fact_7688_negative__imp__inverse__negative,axiom,
% 0.35/0.71      ! [A: rat] :
% 0.35/0.71        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.35/0.71       => ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat ) ) ).
% 0.35/0.71  
% 0.35/0.71  % negative_imp_inverse_negative
% 0.35/0.71  thf(fact_7689_positive__imp__inverse__positive,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.71       => ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % positive_imp_inverse_positive
% 0.35/0.71  thf(fact_7690_positive__imp__inverse__positive,axiom,
% 0.35/0.71      ! [A: rat] :
% 0.35/0.71        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.35/0.71       => ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % positive_imp_inverse_positive
% 0.35/0.71  thf(fact_7691_nonzero__inverse__mult__distrib,axiom,
% 0.35/0.71      ! [A: complex,B: complex] :
% 0.35/0.71        ( ( A != zero_zero_complex )
% 0.35/0.71       => ( ( B != zero_zero_complex )
% 0.35/0.71         => ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B ) )
% 0.35/0.71            = ( times_times_complex @ ( invers8013647133539491842omplex @ B ) @ ( invers8013647133539491842omplex @ A ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_inverse_mult_distrib
% 0.35/0.71  thf(fact_7692_nonzero__inverse__mult__distrib,axiom,
% 0.35/0.71      ! [A: real,B: real] :
% 0.35/0.71        ( ( A != zero_zero_real )
% 0.35/0.71       => ( ( B != zero_zero_real )
% 0.35/0.71         => ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
% 0.35/0.71            = ( times_times_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_inverse_mult_distrib
% 0.35/0.71  thf(fact_7693_nonzero__inverse__mult__distrib,axiom,
% 0.35/0.71      ! [A: rat,B: rat] :
% 0.35/0.71        ( ( A != zero_zero_rat )
% 0.35/0.71       => ( ( B != zero_zero_rat )
% 0.35/0.71         => ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
% 0.35/0.71            = ( times_times_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_inverse_mult_distrib
% 0.35/0.71  thf(fact_7694_nonzero__inverse__minus__eq,axiom,
% 0.35/0.71      ! [A: complex] :
% 0.35/0.71        ( ( A != zero_zero_complex )
% 0.35/0.71       => ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
% 0.35/0.71          = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_inverse_minus_eq
% 0.35/0.71  thf(fact_7695_nonzero__inverse__minus__eq,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( A != zero_zero_real )
% 0.35/0.71       => ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
% 0.35/0.71          = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_inverse_minus_eq
% 0.35/0.71  thf(fact_7696_nonzero__inverse__minus__eq,axiom,
% 0.35/0.71      ! [A: rat] :
% 0.35/0.71        ( ( A != zero_zero_rat )
% 0.35/0.71       => ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
% 0.35/0.71          = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_inverse_minus_eq
% 0.35/0.71  thf(fact_7697_not__exp__less__zero,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% 0.35/0.71  
% 0.35/0.71  % not_exp_less_zero
% 0.35/0.71  thf(fact_7698_exp__gt__zero,axiom,
% 0.35/0.71      ! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% 0.35/0.71  
% 0.35/0.71  % exp_gt_zero
% 0.35/0.71  thf(fact_7699_exp__total,axiom,
% 0.35/0.71      ! [Y: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ Y )
% 0.35/0.71       => ? [X4: real] :
% 0.35/0.71            ( ( exp_real @ X4 )
% 0.35/0.71            = Y ) ) ).
% 0.35/0.71  
% 0.35/0.71  % exp_total
% 0.35/0.71  thf(fact_7700_nonzero__abs__inverse,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( A != zero_zero_real )
% 0.35/0.71       => ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
% 0.35/0.71          = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_abs_inverse
% 0.35/0.71  thf(fact_7701_nonzero__abs__inverse,axiom,
% 0.35/0.71      ! [A: rat] :
% 0.35/0.71        ( ( A != zero_zero_rat )
% 0.35/0.71       => ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
% 0.35/0.71          = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_abs_inverse
% 0.35/0.71  thf(fact_7702_inverse__le__imp__le,axiom,
% 0.35/0.71      ! [A: real,B: real] :
% 0.35/0.71        ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.35/0.71       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.71         => ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_le_imp_le
% 0.35/0.71  thf(fact_7703_inverse__le__imp__le,axiom,
% 0.35/0.71      ! [A: rat,B: rat] :
% 0.35/0.71        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.35/0.71       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.35/0.71         => ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_le_imp_le
% 0.35/0.71  thf(fact_7704_le__imp__inverse__le,axiom,
% 0.35/0.71      ! [A: real,B: real] :
% 0.35/0.71        ( ( ord_less_eq_real @ A @ B )
% 0.35/0.71       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.71         => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % le_imp_inverse_le
% 0.35/0.71  thf(fact_7705_le__imp__inverse__le,axiom,
% 0.35/0.71      ! [A: rat,B: rat] :
% 0.35/0.71        ( ( ord_less_eq_rat @ A @ B )
% 0.35/0.71       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.35/0.71         => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % le_imp_inverse_le
% 0.35/0.71  thf(fact_7706_inverse__le__imp__le__neg,axiom,
% 0.35/0.71      ! [A: real,B: real] :
% 0.35/0.71        ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.35/0.71       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.35/0.71         => ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_le_imp_le_neg
% 0.35/0.71  thf(fact_7707_inverse__le__imp__le__neg,axiom,
% 0.35/0.71      ! [A: rat,B: rat] :
% 0.35/0.71        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.35/0.71       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.35/0.71         => ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_le_imp_le_neg
% 0.35/0.71  thf(fact_7708_le__imp__inverse__le__neg,axiom,
% 0.35/0.71      ! [A: real,B: real] :
% 0.35/0.71        ( ( ord_less_eq_real @ A @ B )
% 0.35/0.71       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.35/0.71         => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % le_imp_inverse_le_neg
% 0.35/0.71  thf(fact_7709_le__imp__inverse__le__neg,axiom,
% 0.35/0.71      ! [A: rat,B: rat] :
% 0.35/0.71        ( ( ord_less_eq_rat @ A @ B )
% 0.35/0.71       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.35/0.71         => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % le_imp_inverse_le_neg
% 0.35/0.71  thf(fact_7710_inverse__le__1__iff,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_eq_real @ ( inverse_inverse_real @ X ) @ one_one_real )
% 0.35/0.71        = ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.35/0.71          | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_le_1_iff
% 0.35/0.71  thf(fact_7711_inverse__le__1__iff,axiom,
% 0.35/0.71      ! [X: rat] :
% 0.35/0.71        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
% 0.35/0.71        = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.35/0.71          | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_le_1_iff
% 0.35/0.71  thf(fact_7712_one__less__inverse,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.71       => ( ( ord_less_real @ A @ one_one_real )
% 0.35/0.71         => ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % one_less_inverse
% 0.35/0.71  thf(fact_7713_one__less__inverse,axiom,
% 0.35/0.71      ! [A: rat] :
% 0.35/0.71        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.35/0.71       => ( ( ord_less_rat @ A @ one_one_rat )
% 0.35/0.71         => ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % one_less_inverse
% 0.35/0.71  thf(fact_7714_one__less__inverse__iff,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X ) )
% 0.35/0.71        = ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71          & ( ord_less_real @ X @ one_one_real ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % one_less_inverse_iff
% 0.35/0.71  thf(fact_7715_one__less__inverse__iff,axiom,
% 0.35/0.71      ! [X: rat] :
% 0.35/0.71        ( ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
% 0.35/0.71        = ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.35/0.71          & ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % one_less_inverse_iff
% 0.35/0.71  thf(fact_7716_inverse__add,axiom,
% 0.35/0.71      ! [A: complex,B: complex] :
% 0.35/0.71        ( ( A != zero_zero_complex )
% 0.35/0.71       => ( ( B != zero_zero_complex )
% 0.35/0.71         => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 0.35/0.71            = ( times_times_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( invers8013647133539491842omplex @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_add
% 0.35/0.71  thf(fact_7717_inverse__add,axiom,
% 0.35/0.71      ! [A: real,B: real] :
% 0.35/0.71        ( ( A != zero_zero_real )
% 0.35/0.71       => ( ( B != zero_zero_real )
% 0.35/0.71         => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.35/0.71            = ( times_times_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( inverse_inverse_real @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_add
% 0.35/0.71  thf(fact_7718_inverse__add,axiom,
% 0.35/0.71      ! [A: rat,B: rat] :
% 0.35/0.71        ( ( A != zero_zero_rat )
% 0.35/0.71       => ( ( B != zero_zero_rat )
% 0.35/0.71         => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.35/0.71            = ( times_times_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( inverse_inverse_rat @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_add
% 0.35/0.71  thf(fact_7719_division__ring__inverse__add,axiom,
% 0.35/0.71      ! [A: complex,B: complex] :
% 0.35/0.71        ( ( A != zero_zero_complex )
% 0.35/0.71       => ( ( B != zero_zero_complex )
% 0.35/0.71         => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 0.35/0.71            = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( plus_plus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % division_ring_inverse_add
% 0.35/0.71  thf(fact_7720_division__ring__inverse__add,axiom,
% 0.35/0.71      ! [A: real,B: real] :
% 0.35/0.71        ( ( A != zero_zero_real )
% 0.35/0.71       => ( ( B != zero_zero_real )
% 0.35/0.71         => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.35/0.71            = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( plus_plus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % division_ring_inverse_add
% 0.35/0.71  thf(fact_7721_division__ring__inverse__add,axiom,
% 0.35/0.71      ! [A: rat,B: rat] :
% 0.35/0.71        ( ( A != zero_zero_rat )
% 0.35/0.71       => ( ( B != zero_zero_rat )
% 0.35/0.71         => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.35/0.71            = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( plus_plus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % division_ring_inverse_add
% 0.35/0.71  thf(fact_7722_field__class_Ofield__inverse,axiom,
% 0.35/0.71      ! [A: complex] :
% 0.35/0.71        ( ( A != zero_zero_complex )
% 0.35/0.71       => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
% 0.35/0.71          = one_one_complex ) ) ).
% 0.35/0.71  
% 0.35/0.71  % field_class.field_inverse
% 0.35/0.71  thf(fact_7723_field__class_Ofield__inverse,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( A != zero_zero_real )
% 0.35/0.71       => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
% 0.35/0.71          = one_one_real ) ) ).
% 0.35/0.71  
% 0.35/0.71  % field_class.field_inverse
% 0.35/0.71  thf(fact_7724_field__class_Ofield__inverse,axiom,
% 0.35/0.71      ! [A: rat] :
% 0.35/0.71        ( ( A != zero_zero_rat )
% 0.35/0.71       => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
% 0.35/0.71          = one_one_rat ) ) ).
% 0.35/0.71  
% 0.35/0.71  % field_class.field_inverse
% 0.35/0.71  thf(fact_7725_division__ring__inverse__diff,axiom,
% 0.35/0.71      ! [A: complex,B: complex] :
% 0.35/0.71        ( ( A != zero_zero_complex )
% 0.35/0.71       => ( ( B != zero_zero_complex )
% 0.35/0.71         => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 0.35/0.71            = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ B @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % division_ring_inverse_diff
% 0.35/0.71  thf(fact_7726_division__ring__inverse__diff,axiom,
% 0.35/0.71      ! [A: real,B: real] :
% 0.35/0.71        ( ( A != zero_zero_real )
% 0.35/0.71       => ( ( B != zero_zero_real )
% 0.35/0.71         => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.35/0.71            = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ B @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % division_ring_inverse_diff
% 0.35/0.71  thf(fact_7727_division__ring__inverse__diff,axiom,
% 0.35/0.71      ! [A: rat,B: rat] :
% 0.35/0.71        ( ( A != zero_zero_rat )
% 0.35/0.71       => ( ( B != zero_zero_rat )
% 0.35/0.71         => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.35/0.71            = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ B @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % division_ring_inverse_diff
% 0.35/0.71  thf(fact_7728_nonzero__inverse__eq__divide,axiom,
% 0.35/0.71      ! [A: complex] :
% 0.35/0.71        ( ( A != zero_zero_complex )
% 0.35/0.71       => ( ( invers8013647133539491842omplex @ A )
% 0.35/0.71          = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_inverse_eq_divide
% 0.35/0.71  thf(fact_7729_nonzero__inverse__eq__divide,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( A != zero_zero_real )
% 0.35/0.71       => ( ( inverse_inverse_real @ A )
% 0.35/0.71          = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_inverse_eq_divide
% 0.35/0.71  thf(fact_7730_nonzero__inverse__eq__divide,axiom,
% 0.35/0.71      ! [A: rat] :
% 0.35/0.71        ( ( A != zero_zero_rat )
% 0.35/0.71       => ( ( inverse_inverse_rat @ A )
% 0.35/0.71          = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % nonzero_inverse_eq_divide
% 0.35/0.71  thf(fact_7731_sum_Odistrib__triv_H,axiom,
% 0.35/0.71      ! [I6: set_nat,G: nat > real,H2: nat > real] :
% 0.35/0.71        ( ( finite_finite_nat @ I6 )
% 0.35/0.71       => ( ( groups8560362682196896993t_real
% 0.35/0.71            @ ^ [I4: nat] : ( plus_plus_real @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 0.35/0.71            @ I6 )
% 0.35/0.71          = ( plus_plus_real @ ( groups8560362682196896993t_real @ G @ I6 ) @ ( groups8560362682196896993t_real @ H2 @ I6 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.distrib_triv'
% 0.35/0.71  thf(fact_7732_sum_Odistrib__triv_H,axiom,
% 0.35/0.71      ! [I6: set_int,G: int > real,H2: int > real] :
% 0.35/0.71        ( ( finite_finite_int @ I6 )
% 0.35/0.71       => ( ( groups1523912220035142973t_real
% 0.35/0.71            @ ^ [I4: int] : ( plus_plus_real @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 0.35/0.71            @ I6 )
% 0.35/0.71          = ( plus_plus_real @ ( groups1523912220035142973t_real @ G @ I6 ) @ ( groups1523912220035142973t_real @ H2 @ I6 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.distrib_triv'
% 0.35/0.71  thf(fact_7733_sum_Odistrib__triv_H,axiom,
% 0.35/0.71      ! [I6: set_complex,G: complex > real,H2: complex > real] :
% 0.35/0.71        ( ( finite3207457112153483333omplex @ I6 )
% 0.35/0.71       => ( ( groups5737402329758386879x_real
% 0.35/0.71            @ ^ [I4: complex] : ( plus_plus_real @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 0.35/0.71            @ I6 )
% 0.35/0.71          = ( plus_plus_real @ ( groups5737402329758386879x_real @ G @ I6 ) @ ( groups5737402329758386879x_real @ H2 @ I6 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.distrib_triv'
% 0.35/0.71  thf(fact_7734_sum_Odistrib__triv_H,axiom,
% 0.35/0.71      ! [I6: set_nat,G: nat > rat,H2: nat > rat] :
% 0.35/0.71        ( ( finite_finite_nat @ I6 )
% 0.35/0.71       => ( ( groups1351286907653491341at_rat
% 0.35/0.71            @ ^ [I4: nat] : ( plus_plus_rat @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 0.35/0.71            @ I6 )
% 0.35/0.71          = ( plus_plus_rat @ ( groups1351286907653491341at_rat @ G @ I6 ) @ ( groups1351286907653491341at_rat @ H2 @ I6 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.distrib_triv'
% 0.35/0.71  thf(fact_7735_sum_Odistrib__triv_H,axiom,
% 0.35/0.71      ! [I6: set_int,G: int > rat,H2: int > rat] :
% 0.35/0.71        ( ( finite_finite_int @ I6 )
% 0.35/0.71       => ( ( groups2350640619554545897nt_rat
% 0.35/0.71            @ ^ [I4: int] : ( plus_plus_rat @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 0.35/0.71            @ I6 )
% 0.35/0.71          = ( plus_plus_rat @ ( groups2350640619554545897nt_rat @ G @ I6 ) @ ( groups2350640619554545897nt_rat @ H2 @ I6 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.distrib_triv'
% 0.35/0.71  thf(fact_7736_sum_Odistrib__triv_H,axiom,
% 0.35/0.71      ! [I6: set_complex,G: complex > rat,H2: complex > rat] :
% 0.35/0.71        ( ( finite3207457112153483333omplex @ I6 )
% 0.35/0.71       => ( ( groups2276542476275365739ex_rat
% 0.35/0.71            @ ^ [I4: complex] : ( plus_plus_rat @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 0.35/0.71            @ I6 )
% 0.35/0.71          = ( plus_plus_rat @ ( groups2276542476275365739ex_rat @ G @ I6 ) @ ( groups2276542476275365739ex_rat @ H2 @ I6 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.distrib_triv'
% 0.35/0.71  thf(fact_7737_sum_Odistrib__triv_H,axiom,
% 0.35/0.71      ! [I6: set_nat,G: nat > nat,H2: nat > nat] :
% 0.35/0.71        ( ( finite_finite_nat @ I6 )
% 0.35/0.71       => ( ( groups1986416967739987077at_nat
% 0.35/0.71            @ ^ [I4: nat] : ( plus_plus_nat @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 0.35/0.71            @ I6 )
% 0.35/0.71          = ( plus_plus_nat @ ( groups1986416967739987077at_nat @ G @ I6 ) @ ( groups1986416967739987077at_nat @ H2 @ I6 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.distrib_triv'
% 0.35/0.71  thf(fact_7738_sum_Odistrib__triv_H,axiom,
% 0.35/0.71      ! [I6: set_int,G: int > nat,H2: int > nat] :
% 0.35/0.71        ( ( finite_finite_int @ I6 )
% 0.35/0.71       => ( ( groups2985770679641041633nt_nat
% 0.35/0.71            @ ^ [I4: int] : ( plus_plus_nat @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 0.35/0.71            @ I6 )
% 0.35/0.71          = ( plus_plus_nat @ ( groups2985770679641041633nt_nat @ G @ I6 ) @ ( groups2985770679641041633nt_nat @ H2 @ I6 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.distrib_triv'
% 0.35/0.71  thf(fact_7739_sum_Odistrib__triv_H,axiom,
% 0.35/0.71      ! [I6: set_complex,G: complex > nat,H2: complex > nat] :
% 0.35/0.71        ( ( finite3207457112153483333omplex @ I6 )
% 0.35/0.71       => ( ( groups2911672536361861475ex_nat
% 0.35/0.71            @ ^ [I4: complex] : ( plus_plus_nat @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 0.35/0.71            @ I6 )
% 0.35/0.71          = ( plus_plus_nat @ ( groups2911672536361861475ex_nat @ G @ I6 ) @ ( groups2911672536361861475ex_nat @ H2 @ I6 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.distrib_triv'
% 0.35/0.71  thf(fact_7740_sum_Odistrib__triv_H,axiom,
% 0.35/0.71      ! [I6: set_nat,G: nat > int,H2: nat > int] :
% 0.35/0.71        ( ( finite_finite_nat @ I6 )
% 0.35/0.71       => ( ( groups1983926497230936801at_int
% 0.35/0.71            @ ^ [I4: nat] : ( plus_plus_int @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 0.35/0.71            @ I6 )
% 0.35/0.71          = ( plus_plus_int @ ( groups1983926497230936801at_int @ G @ I6 ) @ ( groups1983926497230936801at_int @ H2 @ I6 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.distrib_triv'
% 0.35/0.71  thf(fact_7741_exp__gt__one,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71       => ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % exp_gt_one
% 0.35/0.71  thf(fact_7742_sum_Omono__neutral__cong__right_H,axiom,
% 0.35/0.71      ! [S2: set_real,T3: set_real,G: real > complex,H2: real > complex] :
% 0.35/0.71        ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: real] :
% 0.35/0.71              ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_complex ) )
% 0.35/0.71         => ( ! [X4: real] :
% 0.35/0.71                ( ( member_real @ X4 @ S2 )
% 0.35/0.71               => ( ( G @ X4 )
% 0.35/0.71                  = ( H2 @ X4 ) ) )
% 0.35/0.71           => ( ( groups5683813829254066239omplex @ G @ T3 )
% 0.35/0.71              = ( groups5683813829254066239omplex @ H2 @ S2 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_cong_right'
% 0.35/0.71  thf(fact_7743_sum_Omono__neutral__cong__right_H,axiom,
% 0.35/0.71      ! [S2: set_real,T3: set_real,G: real > real,H2: real > real] :
% 0.35/0.71        ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: real] :
% 0.35/0.71              ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_real ) )
% 0.35/0.71         => ( ! [X4: real] :
% 0.35/0.71                ( ( member_real @ X4 @ S2 )
% 0.35/0.71               => ( ( G @ X4 )
% 0.35/0.71                  = ( H2 @ X4 ) ) )
% 0.35/0.71           => ( ( groups97945582718554045l_real @ G @ T3 )
% 0.35/0.71              = ( groups97945582718554045l_real @ H2 @ S2 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_cong_right'
% 0.35/0.71  thf(fact_7744_sum_Omono__neutral__cong__right_H,axiom,
% 0.35/0.71      ! [S2: set_real,T3: set_real,G: real > rat,H2: real > rat] :
% 0.35/0.71        ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: real] :
% 0.35/0.71              ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_rat ) )
% 0.35/0.71         => ( ! [X4: real] :
% 0.35/0.71                ( ( member_real @ X4 @ S2 )
% 0.35/0.71               => ( ( G @ X4 )
% 0.35/0.71                  = ( H2 @ X4 ) ) )
% 0.35/0.71           => ( ( groups3269169158384524137al_rat @ G @ T3 )
% 0.35/0.71              = ( groups3269169158384524137al_rat @ H2 @ S2 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_cong_right'
% 0.35/0.71  thf(fact_7745_sum_Omono__neutral__cong__right_H,axiom,
% 0.35/0.71      ! [S2: set_real,T3: set_real,G: real > nat,H2: real > nat] :
% 0.35/0.71        ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: real] :
% 0.35/0.71              ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_nat ) )
% 0.35/0.71         => ( ! [X4: real] :
% 0.35/0.71                ( ( member_real @ X4 @ S2 )
% 0.35/0.71               => ( ( G @ X4 )
% 0.35/0.71                  = ( H2 @ X4 ) ) )
% 0.35/0.71           => ( ( groups3904299218471019873al_nat @ G @ T3 )
% 0.35/0.71              = ( groups3904299218471019873al_nat @ H2 @ S2 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_cong_right'
% 0.35/0.71  thf(fact_7746_sum_Omono__neutral__cong__right_H,axiom,
% 0.35/0.71      ! [S2: set_real,T3: set_real,G: real > int,H2: real > int] :
% 0.35/0.71        ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: real] :
% 0.35/0.71              ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_int ) )
% 0.35/0.71         => ( ! [X4: real] :
% 0.35/0.71                ( ( member_real @ X4 @ S2 )
% 0.35/0.71               => ( ( G @ X4 )
% 0.35/0.71                  = ( H2 @ X4 ) ) )
% 0.35/0.71           => ( ( groups3901808747961969597al_int @ G @ T3 )
% 0.35/0.71              = ( groups3901808747961969597al_int @ H2 @ S2 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_cong_right'
% 0.35/0.71  thf(fact_7747_sum_Omono__neutral__cong__right_H,axiom,
% 0.35/0.71      ! [S2: set_nat,T3: set_nat,G: nat > complex,H2: nat > complex] :
% 0.35/0.71        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: nat] :
% 0.35/0.71              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_complex ) )
% 0.35/0.71         => ( ! [X4: nat] :
% 0.35/0.71                ( ( member_nat @ X4 @ S2 )
% 0.35/0.71               => ( ( G @ X4 )
% 0.35/0.71                  = ( H2 @ X4 ) ) )
% 0.35/0.71           => ( ( groups8515261248781899619omplex @ G @ T3 )
% 0.35/0.71              = ( groups8515261248781899619omplex @ H2 @ S2 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_cong_right'
% 0.35/0.71  thf(fact_7748_sum_Omono__neutral__cong__right_H,axiom,
% 0.35/0.71      ! [S2: set_nat,T3: set_nat,G: nat > real,H2: nat > real] :
% 0.35/0.71        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: nat] :
% 0.35/0.71              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_real ) )
% 0.35/0.71         => ( ! [X4: nat] :
% 0.35/0.71                ( ( member_nat @ X4 @ S2 )
% 0.35/0.71               => ( ( G @ X4 )
% 0.35/0.71                  = ( H2 @ X4 ) ) )
% 0.35/0.71           => ( ( groups8560362682196896993t_real @ G @ T3 )
% 0.35/0.71              = ( groups8560362682196896993t_real @ H2 @ S2 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_cong_right'
% 0.35/0.71  thf(fact_7749_sum_Omono__neutral__cong__right_H,axiom,
% 0.35/0.71      ! [S2: set_nat,T3: set_nat,G: nat > rat,H2: nat > rat] :
% 0.35/0.71        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: nat] :
% 0.35/0.71              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_rat ) )
% 0.35/0.71         => ( ! [X4: nat] :
% 0.35/0.71                ( ( member_nat @ X4 @ S2 )
% 0.35/0.71               => ( ( G @ X4 )
% 0.35/0.71                  = ( H2 @ X4 ) ) )
% 0.35/0.71           => ( ( groups1351286907653491341at_rat @ G @ T3 )
% 0.35/0.71              = ( groups1351286907653491341at_rat @ H2 @ S2 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_cong_right'
% 0.35/0.71  thf(fact_7750_sum_Omono__neutral__cong__right_H,axiom,
% 0.35/0.71      ! [S2: set_nat,T3: set_nat,G: nat > nat,H2: nat > nat] :
% 0.35/0.71        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: nat] :
% 0.35/0.71              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_nat ) )
% 0.35/0.71         => ( ! [X4: nat] :
% 0.35/0.71                ( ( member_nat @ X4 @ S2 )
% 0.35/0.71               => ( ( G @ X4 )
% 0.35/0.71                  = ( H2 @ X4 ) ) )
% 0.35/0.71           => ( ( groups1986416967739987077at_nat @ G @ T3 )
% 0.35/0.71              = ( groups1986416967739987077at_nat @ H2 @ S2 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_cong_right'
% 0.35/0.71  thf(fact_7751_sum_Omono__neutral__cong__right_H,axiom,
% 0.35/0.71      ! [S2: set_nat,T3: set_nat,G: nat > int,H2: nat > int] :
% 0.35/0.71        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: nat] :
% 0.35/0.71              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_int ) )
% 0.35/0.71         => ( ! [X4: nat] :
% 0.35/0.71                ( ( member_nat @ X4 @ S2 )
% 0.35/0.71               => ( ( G @ X4 )
% 0.35/0.71                  = ( H2 @ X4 ) ) )
% 0.35/0.71           => ( ( groups1983926497230936801at_int @ G @ T3 )
% 0.35/0.71              = ( groups1983926497230936801at_int @ H2 @ S2 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_cong_right'
% 0.35/0.71  thf(fact_7752_sum_Omono__neutral__cong__left_H,axiom,
% 0.35/0.71      ! [S2: set_real,T3: set_real,H2: real > complex,G: real > complex] :
% 0.35/0.71        ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.35/0.71       => ( ! [I2: real] :
% 0.35/0.71              ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.35/0.71             => ( ( H2 @ I2 )
% 0.35/0.71                = zero_zero_complex ) )
% 0.35/0.71         => ( ! [X4: real] :
% 0.35/0.71                ( ( member_real @ X4 @ S2 )
% 0.35/0.71               => ( ( G @ X4 )
% 0.35/0.71                  = ( H2 @ X4 ) ) )
% 0.35/0.71           => ( ( groups5683813829254066239omplex @ G @ S2 )
% 0.35/0.71              = ( groups5683813829254066239omplex @ H2 @ T3 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_cong_left'
% 0.35/0.71  thf(fact_7753_sum_Omono__neutral__cong__left_H,axiom,
% 0.35/0.71      ! [S2: set_real,T3: set_real,H2: real > real,G: real > real] :
% 0.35/0.71        ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.35/0.71       => ( ! [I2: real] :
% 0.35/0.71              ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.35/0.71             => ( ( H2 @ I2 )
% 0.35/0.71                = zero_zero_real ) )
% 0.35/0.71         => ( ! [X4: real] :
% 0.35/0.71                ( ( member_real @ X4 @ S2 )
% 0.35/0.71               => ( ( G @ X4 )
% 0.35/0.71                  = ( H2 @ X4 ) ) )
% 0.35/0.71           => ( ( groups97945582718554045l_real @ G @ S2 )
% 0.35/0.71              = ( groups97945582718554045l_real @ H2 @ T3 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_cong_left'
% 0.35/0.71  thf(fact_7754_sum_Omono__neutral__cong__left_H,axiom,
% 0.35/0.71      ! [S2: set_real,T3: set_real,H2: real > rat,G: real > rat] :
% 0.35/0.71        ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.35/0.71       => ( ! [I2: real] :
% 0.35/0.71              ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.35/0.71             => ( ( H2 @ I2 )
% 0.35/0.71                = zero_zero_rat ) )
% 0.35/0.71         => ( ! [X4: real] :
% 0.35/0.71                ( ( member_real @ X4 @ S2 )
% 0.35/0.71               => ( ( G @ X4 )
% 0.35/0.71                  = ( H2 @ X4 ) ) )
% 0.35/0.71           => ( ( groups3269169158384524137al_rat @ G @ S2 )
% 0.35/0.71              = ( groups3269169158384524137al_rat @ H2 @ T3 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_cong_left'
% 0.35/0.71  thf(fact_7755_sum_Omono__neutral__cong__left_H,axiom,
% 0.35/0.71      ! [S2: set_real,T3: set_real,H2: real > nat,G: real > nat] :
% 0.35/0.71        ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.35/0.71       => ( ! [I2: real] :
% 0.35/0.71              ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.35/0.71             => ( ( H2 @ I2 )
% 0.35/0.71                = zero_zero_nat ) )
% 0.35/0.71         => ( ! [X4: real] :
% 0.35/0.71                ( ( member_real @ X4 @ S2 )
% 0.35/0.71               => ( ( G @ X4 )
% 0.35/0.71                  = ( H2 @ X4 ) ) )
% 0.35/0.71           => ( ( groups3904299218471019873al_nat @ G @ S2 )
% 0.35/0.71              = ( groups3904299218471019873al_nat @ H2 @ T3 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_cong_left'
% 0.35/0.71  thf(fact_7756_sum_Omono__neutral__cong__left_H,axiom,
% 0.35/0.71      ! [S2: set_real,T3: set_real,H2: real > int,G: real > int] :
% 0.35/0.71        ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.35/0.71       => ( ! [I2: real] :
% 0.35/0.71              ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.35/0.71             => ( ( H2 @ I2 )
% 0.35/0.71                = zero_zero_int ) )
% 0.35/0.71         => ( ! [X4: real] :
% 0.35/0.71                ( ( member_real @ X4 @ S2 )
% 0.35/0.71               => ( ( G @ X4 )
% 0.35/0.71                  = ( H2 @ X4 ) ) )
% 0.35/0.71           => ( ( groups3901808747961969597al_int @ G @ S2 )
% 0.35/0.71              = ( groups3901808747961969597al_int @ H2 @ T3 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_cong_left'
% 0.35/0.71  thf(fact_7757_sum_Omono__neutral__cong__left_H,axiom,
% 0.35/0.71      ! [S2: set_nat,T3: set_nat,H2: nat > complex,G: nat > complex] :
% 0.35/0.71        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.71       => ( ! [I2: nat] :
% 0.35/0.71              ( ( member_nat @ I2 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.71             => ( ( H2 @ I2 )
% 0.35/0.71                = zero_zero_complex ) )
% 0.35/0.71         => ( ! [X4: nat] :
% 0.35/0.71                ( ( member_nat @ X4 @ S2 )
% 0.35/0.71               => ( ( G @ X4 )
% 0.35/0.71                  = ( H2 @ X4 ) ) )
% 0.35/0.71           => ( ( groups8515261248781899619omplex @ G @ S2 )
% 0.35/0.71              = ( groups8515261248781899619omplex @ H2 @ T3 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_cong_left'
% 0.35/0.71  thf(fact_7758_sum_Omono__neutral__cong__left_H,axiom,
% 0.35/0.71      ! [S2: set_nat,T3: set_nat,H2: nat > real,G: nat > real] :
% 0.35/0.71        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.71       => ( ! [I2: nat] :
% 0.35/0.71              ( ( member_nat @ I2 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.71             => ( ( H2 @ I2 )
% 0.35/0.71                = zero_zero_real ) )
% 0.35/0.71         => ( ! [X4: nat] :
% 0.35/0.71                ( ( member_nat @ X4 @ S2 )
% 0.35/0.71               => ( ( G @ X4 )
% 0.35/0.71                  = ( H2 @ X4 ) ) )
% 0.35/0.71           => ( ( groups8560362682196896993t_real @ G @ S2 )
% 0.35/0.71              = ( groups8560362682196896993t_real @ H2 @ T3 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_cong_left'
% 0.35/0.71  thf(fact_7759_sum_Omono__neutral__cong__left_H,axiom,
% 0.35/0.71      ! [S2: set_nat,T3: set_nat,H2: nat > rat,G: nat > rat] :
% 0.35/0.71        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.71       => ( ! [I2: nat] :
% 0.35/0.71              ( ( member_nat @ I2 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.71             => ( ( H2 @ I2 )
% 0.35/0.71                = zero_zero_rat ) )
% 0.35/0.71         => ( ! [X4: nat] :
% 0.35/0.71                ( ( member_nat @ X4 @ S2 )
% 0.35/0.71               => ( ( G @ X4 )
% 0.35/0.71                  = ( H2 @ X4 ) ) )
% 0.35/0.71           => ( ( groups1351286907653491341at_rat @ G @ S2 )
% 0.35/0.71              = ( groups1351286907653491341at_rat @ H2 @ T3 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_cong_left'
% 0.35/0.71  thf(fact_7760_sum_Omono__neutral__cong__left_H,axiom,
% 0.35/0.71      ! [S2: set_nat,T3: set_nat,H2: nat > nat,G: nat > nat] :
% 0.35/0.71        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.71       => ( ! [I2: nat] :
% 0.35/0.71              ( ( member_nat @ I2 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.71             => ( ( H2 @ I2 )
% 0.35/0.71                = zero_zero_nat ) )
% 0.35/0.71         => ( ! [X4: nat] :
% 0.35/0.71                ( ( member_nat @ X4 @ S2 )
% 0.35/0.71               => ( ( G @ X4 )
% 0.35/0.71                  = ( H2 @ X4 ) ) )
% 0.35/0.71           => ( ( groups1986416967739987077at_nat @ G @ S2 )
% 0.35/0.71              = ( groups1986416967739987077at_nat @ H2 @ T3 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_cong_left'
% 0.35/0.71  thf(fact_7761_sum_Omono__neutral__cong__left_H,axiom,
% 0.35/0.71      ! [S2: set_nat,T3: set_nat,H2: nat > int,G: nat > int] :
% 0.35/0.71        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.71       => ( ! [I2: nat] :
% 0.35/0.71              ( ( member_nat @ I2 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.71             => ( ( H2 @ I2 )
% 0.35/0.71                = zero_zero_int ) )
% 0.35/0.71         => ( ! [X4: nat] :
% 0.35/0.71                ( ( member_nat @ X4 @ S2 )
% 0.35/0.71               => ( ( G @ X4 )
% 0.35/0.71                  = ( H2 @ X4 ) ) )
% 0.35/0.71           => ( ( groups1983926497230936801at_int @ G @ S2 )
% 0.35/0.71              = ( groups1983926497230936801at_int @ H2 @ T3 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_cong_left'
% 0.35/0.71  thf(fact_7762_sum_Omono__neutral__right_H,axiom,
% 0.35/0.71      ! [S2: set_nat,T3: set_nat,G: nat > complex] :
% 0.35/0.71        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: nat] :
% 0.35/0.71              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_complex ) )
% 0.35/0.71         => ( ( groups8515261248781899619omplex @ G @ T3 )
% 0.35/0.71            = ( groups8515261248781899619omplex @ G @ S2 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_right'
% 0.35/0.71  thf(fact_7763_sum_Omono__neutral__right_H,axiom,
% 0.35/0.71      ! [S2: set_nat,T3: set_nat,G: nat > real] :
% 0.35/0.71        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: nat] :
% 0.35/0.71              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_real ) )
% 0.35/0.71         => ( ( groups8560362682196896993t_real @ G @ T3 )
% 0.35/0.71            = ( groups8560362682196896993t_real @ G @ S2 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_right'
% 0.35/0.71  thf(fact_7764_sum_Omono__neutral__right_H,axiom,
% 0.35/0.71      ! [S2: set_nat,T3: set_nat,G: nat > rat] :
% 0.35/0.71        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: nat] :
% 0.35/0.71              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_rat ) )
% 0.35/0.71         => ( ( groups1351286907653491341at_rat @ G @ T3 )
% 0.35/0.71            = ( groups1351286907653491341at_rat @ G @ S2 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_right'
% 0.35/0.71  thf(fact_7765_sum_Omono__neutral__right_H,axiom,
% 0.35/0.71      ! [S2: set_nat,T3: set_nat,G: nat > nat] :
% 0.35/0.71        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: nat] :
% 0.35/0.71              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_nat ) )
% 0.35/0.71         => ( ( groups1986416967739987077at_nat @ G @ T3 )
% 0.35/0.71            = ( groups1986416967739987077at_nat @ G @ S2 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_right'
% 0.35/0.71  thf(fact_7766_sum_Omono__neutral__right_H,axiom,
% 0.35/0.71      ! [S2: set_nat,T3: set_nat,G: nat > int] :
% 0.35/0.71        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: nat] :
% 0.35/0.71              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_int ) )
% 0.35/0.71         => ( ( groups1983926497230936801at_int @ G @ T3 )
% 0.35/0.71            = ( groups1983926497230936801at_int @ G @ S2 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_right'
% 0.35/0.71  thf(fact_7767_sum_Omono__neutral__right_H,axiom,
% 0.35/0.71      ! [S2: set_int,T3: set_int,G: int > complex] :
% 0.35/0.71        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: int] :
% 0.35/0.71              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_complex ) )
% 0.35/0.71         => ( ( groups267424677133301183omplex @ G @ T3 )
% 0.35/0.71            = ( groups267424677133301183omplex @ G @ S2 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_right'
% 0.35/0.71  thf(fact_7768_sum_Omono__neutral__right_H,axiom,
% 0.35/0.71      ! [S2: set_int,T3: set_int,G: int > real] :
% 0.35/0.71        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: int] :
% 0.35/0.71              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_real ) )
% 0.35/0.71         => ( ( groups1523912220035142973t_real @ G @ T3 )
% 0.35/0.71            = ( groups1523912220035142973t_real @ G @ S2 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_right'
% 0.35/0.71  thf(fact_7769_sum_Omono__neutral__right_H,axiom,
% 0.35/0.71      ! [S2: set_int,T3: set_int,G: int > rat] :
% 0.35/0.71        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: int] :
% 0.35/0.71              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_rat ) )
% 0.35/0.71         => ( ( groups2350640619554545897nt_rat @ G @ T3 )
% 0.35/0.71            = ( groups2350640619554545897nt_rat @ G @ S2 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_right'
% 0.35/0.71  thf(fact_7770_sum_Omono__neutral__right_H,axiom,
% 0.35/0.71      ! [S2: set_int,T3: set_int,G: int > nat] :
% 0.35/0.71        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: int] :
% 0.35/0.71              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_nat ) )
% 0.35/0.71         => ( ( groups2985770679641041633nt_nat @ G @ T3 )
% 0.35/0.71            = ( groups2985770679641041633nt_nat @ G @ S2 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_right'
% 0.35/0.71  thf(fact_7771_sum_Omono__neutral__right_H,axiom,
% 0.35/0.71      ! [S2: set_int,T3: set_int,G: int > int] :
% 0.35/0.71        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: int] :
% 0.35/0.71              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_int ) )
% 0.35/0.71         => ( ( groups2983280209131991357nt_int @ G @ T3 )
% 0.35/0.71            = ( groups2983280209131991357nt_int @ G @ S2 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_right'
% 0.35/0.71  thf(fact_7772_sum_Omono__neutral__left_H,axiom,
% 0.35/0.71      ! [S2: set_nat,T3: set_nat,G: nat > complex] :
% 0.35/0.71        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: nat] :
% 0.35/0.71              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_complex ) )
% 0.35/0.71         => ( ( groups8515261248781899619omplex @ G @ S2 )
% 0.35/0.71            = ( groups8515261248781899619omplex @ G @ T3 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_left'
% 0.35/0.71  thf(fact_7773_sum_Omono__neutral__left_H,axiom,
% 0.35/0.71      ! [S2: set_nat,T3: set_nat,G: nat > real] :
% 0.35/0.71        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: nat] :
% 0.35/0.71              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_real ) )
% 0.35/0.71         => ( ( groups8560362682196896993t_real @ G @ S2 )
% 0.35/0.71            = ( groups8560362682196896993t_real @ G @ T3 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_left'
% 0.35/0.71  thf(fact_7774_sum_Omono__neutral__left_H,axiom,
% 0.35/0.71      ! [S2: set_nat,T3: set_nat,G: nat > rat] :
% 0.35/0.71        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: nat] :
% 0.35/0.71              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_rat ) )
% 0.35/0.71         => ( ( groups1351286907653491341at_rat @ G @ S2 )
% 0.35/0.71            = ( groups1351286907653491341at_rat @ G @ T3 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_left'
% 0.35/0.71  thf(fact_7775_sum_Omono__neutral__left_H,axiom,
% 0.35/0.71      ! [S2: set_nat,T3: set_nat,G: nat > nat] :
% 0.35/0.71        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: nat] :
% 0.35/0.71              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_nat ) )
% 0.35/0.71         => ( ( groups1986416967739987077at_nat @ G @ S2 )
% 0.35/0.71            = ( groups1986416967739987077at_nat @ G @ T3 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_left'
% 0.35/0.71  thf(fact_7776_sum_Omono__neutral__left_H,axiom,
% 0.35/0.71      ! [S2: set_nat,T3: set_nat,G: nat > int] :
% 0.35/0.71        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: nat] :
% 0.35/0.71              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_int ) )
% 0.35/0.71         => ( ( groups1983926497230936801at_int @ G @ S2 )
% 0.35/0.71            = ( groups1983926497230936801at_int @ G @ T3 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_left'
% 0.35/0.71  thf(fact_7777_sum_Omono__neutral__left_H,axiom,
% 0.35/0.71      ! [S2: set_int,T3: set_int,G: int > complex] :
% 0.35/0.71        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: int] :
% 0.35/0.71              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_complex ) )
% 0.35/0.71         => ( ( groups267424677133301183omplex @ G @ S2 )
% 0.35/0.71            = ( groups267424677133301183omplex @ G @ T3 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_left'
% 0.35/0.71  thf(fact_7778_sum_Omono__neutral__left_H,axiom,
% 0.35/0.71      ! [S2: set_int,T3: set_int,G: int > real] :
% 0.35/0.71        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: int] :
% 0.35/0.71              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_real ) )
% 0.35/0.71         => ( ( groups1523912220035142973t_real @ G @ S2 )
% 0.35/0.71            = ( groups1523912220035142973t_real @ G @ T3 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_left'
% 0.35/0.71  thf(fact_7779_sum_Omono__neutral__left_H,axiom,
% 0.35/0.71      ! [S2: set_int,T3: set_int,G: int > rat] :
% 0.35/0.71        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: int] :
% 0.35/0.71              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_rat ) )
% 0.35/0.71         => ( ( groups2350640619554545897nt_rat @ G @ S2 )
% 0.35/0.71            = ( groups2350640619554545897nt_rat @ G @ T3 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_left'
% 0.35/0.71  thf(fact_7780_sum_Omono__neutral__left_H,axiom,
% 0.35/0.71      ! [S2: set_int,T3: set_int,G: int > nat] :
% 0.35/0.71        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: int] :
% 0.35/0.71              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_nat ) )
% 0.35/0.71         => ( ( groups2985770679641041633nt_nat @ G @ S2 )
% 0.35/0.71            = ( groups2985770679641041633nt_nat @ G @ T3 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_left'
% 0.35/0.71  thf(fact_7781_sum_Omono__neutral__left_H,axiom,
% 0.35/0.71      ! [S2: set_int,T3: set_int,G: int > int] :
% 0.35/0.71        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.35/0.71       => ( ! [X4: int] :
% 0.35/0.71              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.35/0.71             => ( ( G @ X4 )
% 0.35/0.71                = zero_zero_int ) )
% 0.35/0.71         => ( ( groups2983280209131991357nt_int @ G @ S2 )
% 0.35/0.71            = ( groups2983280209131991357nt_int @ G @ T3 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.mono_neutral_left'
% 0.35/0.71  thf(fact_7782_inverse__le__iff,axiom,
% 0.35/0.71      ! [A: real,B: real] :
% 0.35/0.71        ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.35/0.71        = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.35/0.71           => ( ord_less_eq_real @ B @ A ) )
% 0.35/0.71          & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.35/0.71           => ( ord_less_eq_real @ A @ B ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_le_iff
% 0.35/0.71  thf(fact_7783_inverse__le__iff,axiom,
% 0.35/0.71      ! [A: rat,B: rat] :
% 0.35/0.71        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.35/0.71        = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.35/0.71           => ( ord_less_eq_rat @ B @ A ) )
% 0.35/0.71          & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.35/0.71           => ( ord_less_eq_rat @ A @ B ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_le_iff
% 0.35/0.71  thf(fact_7784_inverse__less__iff,axiom,
% 0.35/0.71      ! [A: real,B: real] :
% 0.35/0.71        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.35/0.71        = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.35/0.71           => ( ord_less_real @ B @ A ) )
% 0.35/0.71          & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.35/0.71           => ( ord_less_real @ A @ B ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_less_iff
% 0.35/0.71  thf(fact_7785_inverse__less__iff,axiom,
% 0.35/0.71      ! [A: rat,B: rat] :
% 0.35/0.71        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.35/0.71        = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.35/0.71           => ( ord_less_rat @ B @ A ) )
% 0.35/0.71          & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.35/0.71           => ( ord_less_rat @ A @ B ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_less_iff
% 0.35/0.71  thf(fact_7786_one__le__inverse,axiom,
% 0.35/0.71      ! [A: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.71       => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.35/0.71         => ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % one_le_inverse
% 0.35/0.71  thf(fact_7787_one__le__inverse,axiom,
% 0.35/0.71      ! [A: rat] :
% 0.35/0.71        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.35/0.71       => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.35/0.71         => ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % one_le_inverse
% 0.35/0.71  thf(fact_7788_inverse__less__1__iff,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ ( inverse_inverse_real @ X ) @ one_one_real )
% 0.35/0.71        = ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.35/0.71          | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_less_1_iff
% 0.35/0.71  thf(fact_7789_inverse__less__1__iff,axiom,
% 0.35/0.71      ! [X: rat] :
% 0.35/0.71        ( ( ord_less_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
% 0.35/0.71        = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.35/0.71          | ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_less_1_iff
% 0.35/0.71  thf(fact_7790_one__le__inverse__iff,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X ) )
% 0.35/0.71        = ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71          & ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % one_le_inverse_iff
% 0.35/0.71  thf(fact_7791_one__le__inverse__iff,axiom,
% 0.35/0.71      ! [X: rat] :
% 0.35/0.71        ( ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
% 0.35/0.71        = ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.35/0.71          & ( ord_less_eq_rat @ X @ one_one_rat ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % one_le_inverse_iff
% 0.35/0.71  thf(fact_7792_inverse__diff__inverse,axiom,
% 0.35/0.71      ! [A: complex,B: complex] :
% 0.35/0.71        ( ( A != zero_zero_complex )
% 0.35/0.71       => ( ( B != zero_zero_complex )
% 0.35/0.71         => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 0.35/0.71            = ( uminus1482373934393186551omplex @ ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_diff_inverse
% 0.35/0.71  thf(fact_7793_inverse__diff__inverse,axiom,
% 0.35/0.71      ! [A: real,B: real] :
% 0.35/0.71        ( ( A != zero_zero_real )
% 0.35/0.71       => ( ( B != zero_zero_real )
% 0.35/0.71         => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 0.35/0.71            = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_diff_inverse
% 0.35/0.71  thf(fact_7794_inverse__diff__inverse,axiom,
% 0.35/0.71      ! [A: rat,B: rat] :
% 0.35/0.71        ( ( A != zero_zero_rat )
% 0.35/0.71       => ( ( B != zero_zero_rat )
% 0.35/0.71         => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 0.35/0.71            = ( uminus_uminus_rat @ ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % inverse_diff_inverse
% 0.35/0.71  thf(fact_7795_reals__Archimedean,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71       => ? [N3: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ X ) ) ).
% 0.35/0.71  
% 0.35/0.71  % reals_Archimedean
% 0.35/0.71  thf(fact_7796_reals__Archimedean,axiom,
% 0.35/0.71      ! [X: rat] :
% 0.35/0.71        ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.35/0.71       => ? [N3: nat] : ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ X ) ) ).
% 0.35/0.71  
% 0.35/0.71  % reals_Archimedean
% 0.35/0.71  thf(fact_7797_sum_Odistrib_H,axiom,
% 0.35/0.71      ! [I6: set_real,G: real > complex,H2: real > complex] :
% 0.35/0.71        ( ( finite_finite_real
% 0.35/0.71          @ ( collect_real
% 0.35/0.71            @ ^ [X3: real] :
% 0.35/0.71                ( ( member_real @ X3 @ I6 )
% 0.35/0.71                & ( ( G @ X3 )
% 0.35/0.71                 != zero_zero_complex ) ) ) )
% 0.35/0.71       => ( ( finite_finite_real
% 0.35/0.71            @ ( collect_real
% 0.35/0.71              @ ^ [X3: real] :
% 0.35/0.71                  ( ( member_real @ X3 @ I6 )
% 0.35/0.71                  & ( ( H2 @ X3 )
% 0.35/0.71                   != zero_zero_complex ) ) ) )
% 0.35/0.71         => ( ( groups5683813829254066239omplex
% 0.35/0.71              @ ^ [I4: real] : ( plus_plus_complex @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 0.35/0.71              @ I6 )
% 0.35/0.71            = ( plus_plus_complex @ ( groups5683813829254066239omplex @ G @ I6 ) @ ( groups5683813829254066239omplex @ H2 @ I6 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.distrib'
% 0.35/0.71  thf(fact_7798_sum_Odistrib_H,axiom,
% 0.35/0.71      ! [I6: set_nat,G: nat > complex,H2: nat > complex] :
% 0.35/0.71        ( ( finite_finite_nat
% 0.35/0.71          @ ( collect_nat
% 0.35/0.71            @ ^ [X3: nat] :
% 0.35/0.71                ( ( member_nat @ X3 @ I6 )
% 0.35/0.71                & ( ( G @ X3 )
% 0.35/0.71                 != zero_zero_complex ) ) ) )
% 0.35/0.71       => ( ( finite_finite_nat
% 0.35/0.71            @ ( collect_nat
% 0.35/0.71              @ ^ [X3: nat] :
% 0.35/0.71                  ( ( member_nat @ X3 @ I6 )
% 0.35/0.71                  & ( ( H2 @ X3 )
% 0.35/0.71                   != zero_zero_complex ) ) ) )
% 0.35/0.71         => ( ( groups8515261248781899619omplex
% 0.35/0.71              @ ^ [I4: nat] : ( plus_plus_complex @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 0.35/0.71              @ I6 )
% 0.35/0.71            = ( plus_plus_complex @ ( groups8515261248781899619omplex @ G @ I6 ) @ ( groups8515261248781899619omplex @ H2 @ I6 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.distrib'
% 0.35/0.71  thf(fact_7799_sum_Odistrib_H,axiom,
% 0.35/0.71      ! [I6: set_int,G: int > complex,H2: int > complex] :
% 0.35/0.71        ( ( finite_finite_int
% 0.35/0.71          @ ( collect_int
% 0.35/0.71            @ ^ [X3: int] :
% 0.35/0.71                ( ( member_int @ X3 @ I6 )
% 0.35/0.71                & ( ( G @ X3 )
% 0.35/0.71                 != zero_zero_complex ) ) ) )
% 0.35/0.71       => ( ( finite_finite_int
% 0.35/0.71            @ ( collect_int
% 0.35/0.71              @ ^ [X3: int] :
% 0.35/0.71                  ( ( member_int @ X3 @ I6 )
% 0.35/0.71                  & ( ( H2 @ X3 )
% 0.35/0.71                   != zero_zero_complex ) ) ) )
% 0.35/0.71         => ( ( groups267424677133301183omplex
% 0.35/0.71              @ ^ [I4: int] : ( plus_plus_complex @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 0.35/0.71              @ I6 )
% 0.35/0.71            = ( plus_plus_complex @ ( groups267424677133301183omplex @ G @ I6 ) @ ( groups267424677133301183omplex @ H2 @ I6 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.distrib'
% 0.35/0.71  thf(fact_7800_sum_Odistrib_H,axiom,
% 0.35/0.71      ! [I6: set_complex,G: complex > complex,H2: complex > complex] :
% 0.35/0.71        ( ( finite3207457112153483333omplex
% 0.35/0.71          @ ( collect_complex
% 0.35/0.71            @ ^ [X3: complex] :
% 0.35/0.71                ( ( member_complex @ X3 @ I6 )
% 0.35/0.71                & ( ( G @ X3 )
% 0.35/0.71                 != zero_zero_complex ) ) ) )
% 0.35/0.71       => ( ( finite3207457112153483333omplex
% 0.35/0.71            @ ( collect_complex
% 0.35/0.71              @ ^ [X3: complex] :
% 0.35/0.71                  ( ( member_complex @ X3 @ I6 )
% 0.35/0.71                  & ( ( H2 @ X3 )
% 0.35/0.71                   != zero_zero_complex ) ) ) )
% 0.35/0.71         => ( ( groups808145749697022017omplex
% 0.35/0.71              @ ^ [I4: complex] : ( plus_plus_complex @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 0.35/0.71              @ I6 )
% 0.35/0.71            = ( plus_plus_complex @ ( groups808145749697022017omplex @ G @ I6 ) @ ( groups808145749697022017omplex @ H2 @ I6 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.distrib'
% 0.35/0.71  thf(fact_7801_sum_Odistrib_H,axiom,
% 0.35/0.71      ! [I6: set_real,G: real > real,H2: real > real] :
% 0.35/0.71        ( ( finite_finite_real
% 0.35/0.71          @ ( collect_real
% 0.35/0.71            @ ^ [X3: real] :
% 0.35/0.71                ( ( member_real @ X3 @ I6 )
% 0.35/0.71                & ( ( G @ X3 )
% 0.35/0.71                 != zero_zero_real ) ) ) )
% 0.35/0.71       => ( ( finite_finite_real
% 0.35/0.71            @ ( collect_real
% 0.35/0.71              @ ^ [X3: real] :
% 0.35/0.71                  ( ( member_real @ X3 @ I6 )
% 0.35/0.71                  & ( ( H2 @ X3 )
% 0.35/0.71                   != zero_zero_real ) ) ) )
% 0.35/0.71         => ( ( groups97945582718554045l_real
% 0.35/0.71              @ ^ [I4: real] : ( plus_plus_real @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 0.35/0.71              @ I6 )
% 0.35/0.71            = ( plus_plus_real @ ( groups97945582718554045l_real @ G @ I6 ) @ ( groups97945582718554045l_real @ H2 @ I6 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.distrib'
% 0.35/0.71  thf(fact_7802_sum_Odistrib_H,axiom,
% 0.35/0.71      ! [I6: set_nat,G: nat > real,H2: nat > real] :
% 0.35/0.71        ( ( finite_finite_nat
% 0.35/0.71          @ ( collect_nat
% 0.35/0.71            @ ^ [X3: nat] :
% 0.35/0.71                ( ( member_nat @ X3 @ I6 )
% 0.35/0.71                & ( ( G @ X3 )
% 0.35/0.71                 != zero_zero_real ) ) ) )
% 0.35/0.71       => ( ( finite_finite_nat
% 0.35/0.71            @ ( collect_nat
% 0.35/0.71              @ ^ [X3: nat] :
% 0.35/0.71                  ( ( member_nat @ X3 @ I6 )
% 0.35/0.71                  & ( ( H2 @ X3 )
% 0.35/0.71                   != zero_zero_real ) ) ) )
% 0.35/0.71         => ( ( groups8560362682196896993t_real
% 0.35/0.71              @ ^ [I4: nat] : ( plus_plus_real @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 0.35/0.71              @ I6 )
% 0.35/0.71            = ( plus_plus_real @ ( groups8560362682196896993t_real @ G @ I6 ) @ ( groups8560362682196896993t_real @ H2 @ I6 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.distrib'
% 0.35/0.71  thf(fact_7803_sum_Odistrib_H,axiom,
% 0.35/0.71      ! [I6: set_int,G: int > real,H2: int > real] :
% 0.35/0.71        ( ( finite_finite_int
% 0.35/0.71          @ ( collect_int
% 0.35/0.71            @ ^ [X3: int] :
% 0.35/0.71                ( ( member_int @ X3 @ I6 )
% 0.35/0.71                & ( ( G @ X3 )
% 0.35/0.71                 != zero_zero_real ) ) ) )
% 0.35/0.71       => ( ( finite_finite_int
% 0.35/0.71            @ ( collect_int
% 0.35/0.71              @ ^ [X3: int] :
% 0.35/0.71                  ( ( member_int @ X3 @ I6 )
% 0.35/0.71                  & ( ( H2 @ X3 )
% 0.35/0.71                   != zero_zero_real ) ) ) )
% 0.35/0.71         => ( ( groups1523912220035142973t_real
% 0.35/0.71              @ ^ [I4: int] : ( plus_plus_real @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 0.35/0.71              @ I6 )
% 0.35/0.71            = ( plus_plus_real @ ( groups1523912220035142973t_real @ G @ I6 ) @ ( groups1523912220035142973t_real @ H2 @ I6 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.distrib'
% 0.35/0.71  thf(fact_7804_sum_Odistrib_H,axiom,
% 0.35/0.71      ! [I6: set_complex,G: complex > real,H2: complex > real] :
% 0.35/0.71        ( ( finite3207457112153483333omplex
% 0.35/0.71          @ ( collect_complex
% 0.35/0.71            @ ^ [X3: complex] :
% 0.35/0.71                ( ( member_complex @ X3 @ I6 )
% 0.35/0.71                & ( ( G @ X3 )
% 0.35/0.71                 != zero_zero_real ) ) ) )
% 0.35/0.71       => ( ( finite3207457112153483333omplex
% 0.35/0.71            @ ( collect_complex
% 0.35/0.71              @ ^ [X3: complex] :
% 0.35/0.71                  ( ( member_complex @ X3 @ I6 )
% 0.35/0.71                  & ( ( H2 @ X3 )
% 0.35/0.71                   != zero_zero_real ) ) ) )
% 0.35/0.71         => ( ( groups5737402329758386879x_real
% 0.35/0.71              @ ^ [I4: complex] : ( plus_plus_real @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 0.35/0.71              @ I6 )
% 0.35/0.71            = ( plus_plus_real @ ( groups5737402329758386879x_real @ G @ I6 ) @ ( groups5737402329758386879x_real @ H2 @ I6 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.distrib'
% 0.35/0.71  thf(fact_7805_sum_Odistrib_H,axiom,
% 0.35/0.71      ! [I6: set_real,G: real > rat,H2: real > rat] :
% 0.35/0.71        ( ( finite_finite_real
% 0.35/0.71          @ ( collect_real
% 0.35/0.71            @ ^ [X3: real] :
% 0.35/0.71                ( ( member_real @ X3 @ I6 )
% 0.35/0.71                & ( ( G @ X3 )
% 0.35/0.71                 != zero_zero_rat ) ) ) )
% 0.35/0.71       => ( ( finite_finite_real
% 0.35/0.71            @ ( collect_real
% 0.35/0.71              @ ^ [X3: real] :
% 0.35/0.71                  ( ( member_real @ X3 @ I6 )
% 0.35/0.71                  & ( ( H2 @ X3 )
% 0.35/0.71                   != zero_zero_rat ) ) ) )
% 0.35/0.71         => ( ( groups3269169158384524137al_rat
% 0.35/0.71              @ ^ [I4: real] : ( plus_plus_rat @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 0.35/0.71              @ I6 )
% 0.35/0.71            = ( plus_plus_rat @ ( groups3269169158384524137al_rat @ G @ I6 ) @ ( groups3269169158384524137al_rat @ H2 @ I6 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.distrib'
% 0.35/0.71  thf(fact_7806_sum_Odistrib_H,axiom,
% 0.35/0.71      ! [I6: set_nat,G: nat > rat,H2: nat > rat] :
% 0.35/0.71        ( ( finite_finite_nat
% 0.35/0.71          @ ( collect_nat
% 0.35/0.71            @ ^ [X3: nat] :
% 0.35/0.71                ( ( member_nat @ X3 @ I6 )
% 0.35/0.71                & ( ( G @ X3 )
% 0.35/0.71                 != zero_zero_rat ) ) ) )
% 0.35/0.71       => ( ( finite_finite_nat
% 0.35/0.71            @ ( collect_nat
% 0.35/0.71              @ ^ [X3: nat] :
% 0.35/0.71                  ( ( member_nat @ X3 @ I6 )
% 0.35/0.71                  & ( ( H2 @ X3 )
% 0.35/0.71                   != zero_zero_rat ) ) ) )
% 0.35/0.71         => ( ( groups1351286907653491341at_rat
% 0.35/0.71              @ ^ [I4: nat] : ( plus_plus_rat @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 0.35/0.71              @ I6 )
% 0.35/0.71            = ( plus_plus_rat @ ( groups1351286907653491341at_rat @ G @ I6 ) @ ( groups1351286907653491341at_rat @ H2 @ I6 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.distrib'
% 0.35/0.71  thf(fact_7807_forall__pos__mono__1,axiom,
% 0.35/0.71      ! [P: real > $o,E2: real] :
% 0.35/0.71        ( ! [D5: real,E: real] :
% 0.35/0.71            ( ( ord_less_real @ D5 @ E )
% 0.35/0.71           => ( ( P @ D5 )
% 0.35/0.71             => ( P @ E ) ) )
% 0.35/0.71       => ( ! [N3: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
% 0.35/0.71         => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.35/0.71           => ( P @ E2 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % forall_pos_mono_1
% 0.35/0.71  thf(fact_7808_forall__pos__mono,axiom,
% 0.35/0.71      ! [P: real > $o,E2: real] :
% 0.35/0.71        ( ! [D5: real,E: real] :
% 0.35/0.71            ( ( ord_less_real @ D5 @ E )
% 0.35/0.71           => ( ( P @ D5 )
% 0.35/0.71             => ( P @ E ) ) )
% 0.35/0.71       => ( ! [N3: nat] :
% 0.35/0.71              ( ( N3 != zero_zero_nat )
% 0.35/0.71             => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
% 0.35/0.71         => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.35/0.71           => ( P @ E2 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % forall_pos_mono
% 0.35/0.71  thf(fact_7809_real__arch__inverse,axiom,
% 0.35/0.71      ! [E2: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.35/0.71        = ( ? [N2: nat] :
% 0.35/0.71              ( ( N2 != zero_zero_nat )
% 0.35/0.71              & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.35/0.71              & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E2 ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % real_arch_inverse
% 0.35/0.71  thf(fact_7810_sum_OG__def,axiom,
% 0.35/0.71      ( groups5683813829254066239omplex
% 0.35/0.71      = ( ^ [P6: real > complex,I7: set_real] :
% 0.35/0.71            ( if_complex
% 0.35/0.71            @ ( finite_finite_real
% 0.35/0.71              @ ( collect_real
% 0.35/0.71                @ ^ [X3: real] :
% 0.35/0.71                    ( ( member_real @ X3 @ I7 )
% 0.35/0.71                    & ( ( P6 @ X3 )
% 0.35/0.71                     != zero_zero_complex ) ) ) )
% 0.35/0.71            @ ( groups5754745047067104278omplex @ P6
% 0.35/0.71              @ ( collect_real
% 0.35/0.71                @ ^ [X3: real] :
% 0.35/0.71                    ( ( member_real @ X3 @ I7 )
% 0.35/0.71                    & ( ( P6 @ X3 )
% 0.35/0.71                     != zero_zero_complex ) ) ) )
% 0.35/0.71            @ zero_zero_complex ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.G_def
% 0.35/0.71  thf(fact_7811_sum_OG__def,axiom,
% 0.35/0.71      ( groups8515261248781899619omplex
% 0.35/0.71      = ( ^ [P6: nat > complex,I7: set_nat] :
% 0.35/0.71            ( if_complex
% 0.35/0.71            @ ( finite_finite_nat
% 0.35/0.71              @ ( collect_nat
% 0.35/0.71                @ ^ [X3: nat] :
% 0.35/0.71                    ( ( member_nat @ X3 @ I7 )
% 0.35/0.71                    & ( ( P6 @ X3 )
% 0.35/0.71                     != zero_zero_complex ) ) ) )
% 0.35/0.71            @ ( groups2073611262835488442omplex @ P6
% 0.35/0.71              @ ( collect_nat
% 0.35/0.71                @ ^ [X3: nat] :
% 0.35/0.71                    ( ( member_nat @ X3 @ I7 )
% 0.35/0.71                    & ( ( P6 @ X3 )
% 0.35/0.71                     != zero_zero_complex ) ) ) )
% 0.35/0.71            @ zero_zero_complex ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.G_def
% 0.35/0.71  thf(fact_7812_sum_OG__def,axiom,
% 0.35/0.71      ( groups267424677133301183omplex
% 0.35/0.71      = ( ^ [P6: int > complex,I7: set_int] :
% 0.35/0.71            ( if_complex
% 0.35/0.71            @ ( finite_finite_int
% 0.35/0.71              @ ( collect_int
% 0.35/0.71                @ ^ [X3: int] :
% 0.35/0.71                    ( ( member_int @ X3 @ I7 )
% 0.35/0.71                    & ( ( P6 @ X3 )
% 0.35/0.71                     != zero_zero_complex ) ) ) )
% 0.35/0.71            @ ( groups3049146728041665814omplex @ P6
% 0.35/0.71              @ ( collect_int
% 0.35/0.71                @ ^ [X3: int] :
% 0.35/0.71                    ( ( member_int @ X3 @ I7 )
% 0.35/0.71                    & ( ( P6 @ X3 )
% 0.35/0.71                     != zero_zero_complex ) ) ) )
% 0.35/0.71            @ zero_zero_complex ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.G_def
% 0.35/0.71  thf(fact_7813_sum_OG__def,axiom,
% 0.35/0.71      ( groups97945582718554045l_real
% 0.35/0.71      = ( ^ [P6: real > real,I7: set_real] :
% 0.35/0.71            ( if_real
% 0.35/0.71            @ ( finite_finite_real
% 0.35/0.71              @ ( collect_real
% 0.35/0.71                @ ^ [X3: real] :
% 0.35/0.71                    ( ( member_real @ X3 @ I7 )
% 0.35/0.71                    & ( ( P6 @ X3 )
% 0.35/0.71                     != zero_zero_real ) ) ) )
% 0.35/0.71            @ ( groups8097168146408367636l_real @ P6
% 0.35/0.71              @ ( collect_real
% 0.35/0.71                @ ^ [X3: real] :
% 0.35/0.71                    ( ( member_real @ X3 @ I7 )
% 0.35/0.71                    & ( ( P6 @ X3 )
% 0.35/0.71                     != zero_zero_real ) ) ) )
% 0.35/0.71            @ zero_zero_real ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.G_def
% 0.35/0.71  thf(fact_7814_sum_OG__def,axiom,
% 0.35/0.71      ( groups1523912220035142973t_real
% 0.35/0.71      = ( ^ [P6: int > real,I7: set_int] :
% 0.35/0.71            ( if_real
% 0.35/0.71            @ ( finite_finite_int
% 0.35/0.71              @ ( collect_int
% 0.35/0.71                @ ^ [X3: int] :
% 0.35/0.71                    ( ( member_int @ X3 @ I7 )
% 0.35/0.71                    & ( ( P6 @ X3 )
% 0.35/0.71                     != zero_zero_real ) ) ) )
% 0.35/0.71            @ ( groups8778361861064173332t_real @ P6
% 0.35/0.71              @ ( collect_int
% 0.35/0.71                @ ^ [X3: int] :
% 0.35/0.71                    ( ( member_int @ X3 @ I7 )
% 0.35/0.71                    & ( ( P6 @ X3 )
% 0.35/0.71                     != zero_zero_real ) ) ) )
% 0.35/0.71            @ zero_zero_real ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.G_def
% 0.35/0.71  thf(fact_7815_sum_OG__def,axiom,
% 0.35/0.71      ( groups5737402329758386879x_real
% 0.35/0.71      = ( ^ [P6: complex > real,I7: set_complex] :
% 0.35/0.71            ( if_real
% 0.35/0.71            @ ( finite3207457112153483333omplex
% 0.35/0.71              @ ( collect_complex
% 0.35/0.71                @ ^ [X3: complex] :
% 0.35/0.71                    ( ( member_complex @ X3 @ I7 )
% 0.35/0.71                    & ( ( P6 @ X3 )
% 0.35/0.71                     != zero_zero_real ) ) ) )
% 0.35/0.71            @ ( groups5808333547571424918x_real @ P6
% 0.35/0.71              @ ( collect_complex
% 0.35/0.71                @ ^ [X3: complex] :
% 0.35/0.71                    ( ( member_complex @ X3 @ I7 )
% 0.35/0.71                    & ( ( P6 @ X3 )
% 0.35/0.71                     != zero_zero_real ) ) ) )
% 0.35/0.71            @ zero_zero_real ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.G_def
% 0.35/0.71  thf(fact_7816_sum_OG__def,axiom,
% 0.35/0.71      ( groups3269169158384524137al_rat
% 0.35/0.71      = ( ^ [P6: real > rat,I7: set_real] :
% 0.35/0.71            ( if_rat
% 0.35/0.71            @ ( finite_finite_real
% 0.35/0.71              @ ( collect_real
% 0.35/0.71                @ ^ [X3: real] :
% 0.35/0.71                    ( ( member_real @ X3 @ I7 )
% 0.35/0.71                    & ( ( P6 @ X3 )
% 0.35/0.71                     != zero_zero_rat ) ) ) )
% 0.35/0.71            @ ( groups1300246762558778688al_rat @ P6
% 0.35/0.71              @ ( collect_real
% 0.35/0.71                @ ^ [X3: real] :
% 0.35/0.71                    ( ( member_real @ X3 @ I7 )
% 0.35/0.71                    & ( ( P6 @ X3 )
% 0.35/0.71                     != zero_zero_rat ) ) ) )
% 0.35/0.71            @ zero_zero_rat ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.G_def
% 0.35/0.71  thf(fact_7817_sum_OG__def,axiom,
% 0.35/0.71      ( groups1351286907653491341at_rat
% 0.35/0.71      = ( ^ [P6: nat > rat,I7: set_nat] :
% 0.35/0.71            ( if_rat
% 0.35/0.71            @ ( finite_finite_nat
% 0.35/0.71              @ ( collect_nat
% 0.35/0.71                @ ^ [X3: nat] :
% 0.35/0.71                    ( ( member_nat @ X3 @ I7 )
% 0.35/0.71                    & ( ( P6 @ X3 )
% 0.35/0.71                     != zero_zero_rat ) ) ) )
% 0.35/0.71            @ ( groups2906978787729119204at_rat @ P6
% 0.35/0.71              @ ( collect_nat
% 0.35/0.71                @ ^ [X3: nat] :
% 0.35/0.71                    ( ( member_nat @ X3 @ I7 )
% 0.35/0.71                    & ( ( P6 @ X3 )
% 0.35/0.71                     != zero_zero_rat ) ) ) )
% 0.35/0.71            @ zero_zero_rat ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.G_def
% 0.35/0.71  thf(fact_7818_sum_OG__def,axiom,
% 0.35/0.71      ( groups2350640619554545897nt_rat
% 0.35/0.71      = ( ^ [P6: int > rat,I7: set_int] :
% 0.35/0.71            ( if_rat
% 0.35/0.71            @ ( finite_finite_int
% 0.35/0.71              @ ( collect_int
% 0.35/0.71                @ ^ [X3: int] :
% 0.35/0.71                    ( ( member_int @ X3 @ I7 )
% 0.35/0.71                    & ( ( P6 @ X3 )
% 0.35/0.71                     != zero_zero_rat ) ) ) )
% 0.35/0.71            @ ( groups3906332499630173760nt_rat @ P6
% 0.35/0.71              @ ( collect_int
% 0.35/0.71                @ ^ [X3: int] :
% 0.35/0.71                    ( ( member_int @ X3 @ I7 )
% 0.35/0.71                    & ( ( P6 @ X3 )
% 0.35/0.71                     != zero_zero_rat ) ) ) )
% 0.35/0.71            @ zero_zero_rat ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.G_def
% 0.35/0.71  thf(fact_7819_sum_OG__def,axiom,
% 0.35/0.71      ( groups2276542476275365739ex_rat
% 0.35/0.71      = ( ^ [P6: complex > rat,I7: set_complex] :
% 0.35/0.71            ( if_rat
% 0.35/0.71            @ ( finite3207457112153483333omplex
% 0.35/0.71              @ ( collect_complex
% 0.35/0.71                @ ^ [X3: complex] :
% 0.35/0.71                    ( ( member_complex @ X3 @ I7 )
% 0.35/0.71                    & ( ( P6 @ X3 )
% 0.35/0.71                     != zero_zero_rat ) ) ) )
% 0.35/0.71            @ ( groups5058264527183730370ex_rat @ P6
% 0.35/0.71              @ ( collect_complex
% 0.35/0.71                @ ^ [X3: complex] :
% 0.35/0.71                    ( ( member_complex @ X3 @ I7 )
% 0.35/0.71                    & ( ( P6 @ X3 )
% 0.35/0.71                     != zero_zero_rat ) ) ) )
% 0.35/0.71            @ zero_zero_rat ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum.G_def
% 0.35/0.71  thf(fact_7820_ex__inverse__of__nat__less,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71       => ? [N3: nat] :
% 0.35/0.71            ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.35/0.71            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ X ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % ex_inverse_of_nat_less
% 0.35/0.71  thf(fact_7821_ex__inverse__of__nat__less,axiom,
% 0.35/0.71      ! [X: rat] :
% 0.35/0.71        ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.35/0.71       => ? [N3: nat] :
% 0.35/0.71            ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.35/0.71            & ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ N3 ) ) @ X ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % ex_inverse_of_nat_less
% 0.35/0.71  thf(fact_7822_power__diff__conv__inverse,axiom,
% 0.35/0.71      ! [X: complex,M: nat,N: nat] :
% 0.35/0.71        ( ( X != zero_zero_complex )
% 0.35/0.71       => ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.71         => ( ( power_power_complex @ X @ ( minus_minus_nat @ N @ M ) )
% 0.35/0.71            = ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ M ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % power_diff_conv_inverse
% 0.35/0.71  thf(fact_7823_power__diff__conv__inverse,axiom,
% 0.35/0.71      ! [X: real,M: nat,N: nat] :
% 0.35/0.71        ( ( X != zero_zero_real )
% 0.35/0.71       => ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.71         => ( ( power_power_real @ X @ ( minus_minus_nat @ N @ M ) )
% 0.35/0.71            = ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ M ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % power_diff_conv_inverse
% 0.35/0.71  thf(fact_7824_power__diff__conv__inverse,axiom,
% 0.35/0.71      ! [X: rat,M: nat,N: nat] :
% 0.35/0.71        ( ( X != zero_zero_rat )
% 0.35/0.71       => ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.71         => ( ( power_power_rat @ X @ ( minus_minus_nat @ N @ M ) )
% 0.35/0.71            = ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ M ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % power_diff_conv_inverse
% 0.35/0.71  thf(fact_7825_exp__divide__power__eq,axiom,
% 0.35/0.71      ! [N: nat,X: complex] :
% 0.35/0.71        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.71       => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
% 0.35/0.71          = ( exp_complex @ X ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % exp_divide_power_eq
% 0.35/0.71  thf(fact_7826_exp__divide__power__eq,axiom,
% 0.35/0.71      ! [N: nat,X: real] :
% 0.35/0.71        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.71       => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
% 0.35/0.71          = ( exp_real @ X ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % exp_divide_power_eq
% 0.35/0.71  thf(fact_7827_log__inverse,axiom,
% 0.35/0.71      ! [A: real,X: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.71       => ( ( A != one_one_real )
% 0.35/0.71         => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71           => ( ( log @ A @ ( inverse_inverse_real @ X ) )
% 0.35/0.71              = ( uminus_uminus_real @ ( log @ A @ X ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % log_inverse
% 0.35/0.71  thf(fact_7828_plus__inverse__ge__2,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71       => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % plus_inverse_ge_2
% 0.35/0.71  thf(fact_7829_real__inv__sqrt__pow2,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71       => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.71          = ( inverse_inverse_real @ X ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % real_inv_sqrt_pow2
% 0.35/0.71  thf(fact_7830_sum__diff1_H,axiom,
% 0.35/0.71      ! [I6: set_complex,F: complex > complex,I: complex] :
% 0.35/0.71        ( ( finite3207457112153483333omplex
% 0.35/0.71          @ ( collect_complex
% 0.35/0.71            @ ^ [I4: complex] :
% 0.35/0.71                ( ( member_complex @ I4 @ I6 )
% 0.35/0.71                & ( ( F @ I4 )
% 0.35/0.71                 != zero_zero_complex ) ) ) )
% 0.35/0.71       => ( ( ( member_complex @ I @ I6 )
% 0.35/0.71           => ( ( groups808145749697022017omplex @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.71              = ( minus_minus_complex @ ( groups808145749697022017omplex @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.35/0.71          & ( ~ ( member_complex @ I @ I6 )
% 0.35/0.71           => ( ( groups808145749697022017omplex @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.71              = ( groups808145749697022017omplex @ F @ I6 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum_diff1'
% 0.35/0.71  thf(fact_7831_sum__diff1_H,axiom,
% 0.35/0.71      ! [I6: set_real,F: real > complex,I: real] :
% 0.35/0.71        ( ( finite_finite_real
% 0.35/0.71          @ ( collect_real
% 0.35/0.71            @ ^ [I4: real] :
% 0.35/0.71                ( ( member_real @ I4 @ I6 )
% 0.35/0.71                & ( ( F @ I4 )
% 0.35/0.71                 != zero_zero_complex ) ) ) )
% 0.35/0.71       => ( ( ( member_real @ I @ I6 )
% 0.35/0.71           => ( ( groups5683813829254066239omplex @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.35/0.71              = ( minus_minus_complex @ ( groups5683813829254066239omplex @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.35/0.71          & ( ~ ( member_real @ I @ I6 )
% 0.35/0.71           => ( ( groups5683813829254066239omplex @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.35/0.71              = ( groups5683813829254066239omplex @ F @ I6 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum_diff1'
% 0.35/0.71  thf(fact_7832_sum__diff1_H,axiom,
% 0.35/0.71      ! [I6: set_int,F: int > complex,I: int] :
% 0.35/0.71        ( ( finite_finite_int
% 0.35/0.71          @ ( collect_int
% 0.35/0.71            @ ^ [I4: int] :
% 0.35/0.71                ( ( member_int @ I4 @ I6 )
% 0.35/0.71                & ( ( F @ I4 )
% 0.35/0.71                 != zero_zero_complex ) ) ) )
% 0.35/0.71       => ( ( ( member_int @ I @ I6 )
% 0.35/0.71           => ( ( groups267424677133301183omplex @ F @ ( minus_minus_set_int @ I6 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 0.35/0.71              = ( minus_minus_complex @ ( groups267424677133301183omplex @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.35/0.71          & ( ~ ( member_int @ I @ I6 )
% 0.35/0.71           => ( ( groups267424677133301183omplex @ F @ ( minus_minus_set_int @ I6 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 0.35/0.71              = ( groups267424677133301183omplex @ F @ I6 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum_diff1'
% 0.35/0.71  thf(fact_7833_sum__diff1_H,axiom,
% 0.35/0.71      ! [I6: set_complex,F: complex > real,I: complex] :
% 0.35/0.71        ( ( finite3207457112153483333omplex
% 0.35/0.71          @ ( collect_complex
% 0.35/0.71            @ ^ [I4: complex] :
% 0.35/0.71                ( ( member_complex @ I4 @ I6 )
% 0.35/0.71                & ( ( F @ I4 )
% 0.35/0.71                 != zero_zero_real ) ) ) )
% 0.35/0.71       => ( ( ( member_complex @ I @ I6 )
% 0.35/0.71           => ( ( groups5737402329758386879x_real @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.71              = ( minus_minus_real @ ( groups5737402329758386879x_real @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.35/0.71          & ( ~ ( member_complex @ I @ I6 )
% 0.35/0.71           => ( ( groups5737402329758386879x_real @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.71              = ( groups5737402329758386879x_real @ F @ I6 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum_diff1'
% 0.35/0.71  thf(fact_7834_sum__diff1_H,axiom,
% 0.35/0.71      ! [I6: set_real,F: real > real,I: real] :
% 0.35/0.71        ( ( finite_finite_real
% 0.35/0.71          @ ( collect_real
% 0.35/0.71            @ ^ [I4: real] :
% 0.35/0.71                ( ( member_real @ I4 @ I6 )
% 0.35/0.71                & ( ( F @ I4 )
% 0.35/0.71                 != zero_zero_real ) ) ) )
% 0.35/0.71       => ( ( ( member_real @ I @ I6 )
% 0.35/0.71           => ( ( groups97945582718554045l_real @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.35/0.71              = ( minus_minus_real @ ( groups97945582718554045l_real @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.35/0.71          & ( ~ ( member_real @ I @ I6 )
% 0.35/0.71           => ( ( groups97945582718554045l_real @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.35/0.71              = ( groups97945582718554045l_real @ F @ I6 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum_diff1'
% 0.35/0.71  thf(fact_7835_sum__diff1_H,axiom,
% 0.35/0.71      ! [I6: set_int,F: int > real,I: int] :
% 0.35/0.71        ( ( finite_finite_int
% 0.35/0.71          @ ( collect_int
% 0.35/0.71            @ ^ [I4: int] :
% 0.35/0.71                ( ( member_int @ I4 @ I6 )
% 0.35/0.71                & ( ( F @ I4 )
% 0.35/0.71                 != zero_zero_real ) ) ) )
% 0.35/0.71       => ( ( ( member_int @ I @ I6 )
% 0.35/0.71           => ( ( groups1523912220035142973t_real @ F @ ( minus_minus_set_int @ I6 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 0.35/0.71              = ( minus_minus_real @ ( groups1523912220035142973t_real @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.35/0.71          & ( ~ ( member_int @ I @ I6 )
% 0.35/0.71           => ( ( groups1523912220035142973t_real @ F @ ( minus_minus_set_int @ I6 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 0.35/0.71              = ( groups1523912220035142973t_real @ F @ I6 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum_diff1'
% 0.35/0.71  thf(fact_7836_sum__diff1_H,axiom,
% 0.35/0.71      ! [I6: set_complex,F: complex > rat,I: complex] :
% 0.35/0.71        ( ( finite3207457112153483333omplex
% 0.35/0.71          @ ( collect_complex
% 0.35/0.71            @ ^ [I4: complex] :
% 0.35/0.71                ( ( member_complex @ I4 @ I6 )
% 0.35/0.71                & ( ( F @ I4 )
% 0.35/0.71                 != zero_zero_rat ) ) ) )
% 0.35/0.71       => ( ( ( member_complex @ I @ I6 )
% 0.35/0.71           => ( ( groups2276542476275365739ex_rat @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.71              = ( minus_minus_rat @ ( groups2276542476275365739ex_rat @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.35/0.71          & ( ~ ( member_complex @ I @ I6 )
% 0.35/0.71           => ( ( groups2276542476275365739ex_rat @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.71              = ( groups2276542476275365739ex_rat @ F @ I6 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum_diff1'
% 0.35/0.71  thf(fact_7837_sum__diff1_H,axiom,
% 0.35/0.71      ! [I6: set_real,F: real > rat,I: real] :
% 0.35/0.71        ( ( finite_finite_real
% 0.35/0.71          @ ( collect_real
% 0.35/0.71            @ ^ [I4: real] :
% 0.35/0.71                ( ( member_real @ I4 @ I6 )
% 0.35/0.71                & ( ( F @ I4 )
% 0.35/0.71                 != zero_zero_rat ) ) ) )
% 0.35/0.71       => ( ( ( member_real @ I @ I6 )
% 0.35/0.71           => ( ( groups3269169158384524137al_rat @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.35/0.71              = ( minus_minus_rat @ ( groups3269169158384524137al_rat @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.35/0.71          & ( ~ ( member_real @ I @ I6 )
% 0.35/0.71           => ( ( groups3269169158384524137al_rat @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.35/0.71              = ( groups3269169158384524137al_rat @ F @ I6 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum_diff1'
% 0.35/0.71  thf(fact_7838_sum__diff1_H,axiom,
% 0.35/0.71      ! [I6: set_int,F: int > rat,I: int] :
% 0.35/0.71        ( ( finite_finite_int
% 0.35/0.71          @ ( collect_int
% 0.35/0.71            @ ^ [I4: int] :
% 0.35/0.71                ( ( member_int @ I4 @ I6 )
% 0.35/0.71                & ( ( F @ I4 )
% 0.35/0.71                 != zero_zero_rat ) ) ) )
% 0.35/0.71       => ( ( ( member_int @ I @ I6 )
% 0.35/0.71           => ( ( groups2350640619554545897nt_rat @ F @ ( minus_minus_set_int @ I6 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 0.35/0.71              = ( minus_minus_rat @ ( groups2350640619554545897nt_rat @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.35/0.71          & ( ~ ( member_int @ I @ I6 )
% 0.35/0.71           => ( ( groups2350640619554545897nt_rat @ F @ ( minus_minus_set_int @ I6 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 0.35/0.71              = ( groups2350640619554545897nt_rat @ F @ I6 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum_diff1'
% 0.35/0.71  thf(fact_7839_sum__diff1_H,axiom,
% 0.35/0.71      ! [I6: set_complex,F: complex > int,I: complex] :
% 0.35/0.71        ( ( finite3207457112153483333omplex
% 0.35/0.71          @ ( collect_complex
% 0.35/0.71            @ ^ [I4: complex] :
% 0.35/0.71                ( ( member_complex @ I4 @ I6 )
% 0.35/0.71                & ( ( F @ I4 )
% 0.35/0.71                 != zero_zero_int ) ) ) )
% 0.35/0.71       => ( ( ( member_complex @ I @ I6 )
% 0.35/0.71           => ( ( groups2909182065852811199ex_int @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.71              = ( minus_minus_int @ ( groups2909182065852811199ex_int @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.35/0.71          & ( ~ ( member_complex @ I @ I6 )
% 0.35/0.71           => ( ( groups2909182065852811199ex_int @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.71              = ( groups2909182065852811199ex_int @ F @ I6 ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % sum_diff1'
% 0.35/0.71  thf(fact_7840_Maclaurin__exp__lt,axiom,
% 0.35/0.71      ! [X: real,N: nat] :
% 0.35/0.71        ( ( X != zero_zero_real )
% 0.35/0.71       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.71         => ? [T6: real] :
% 0.35/0.71              ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
% 0.35/0.71              & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
% 0.35/0.71              & ( ( exp_real @ X )
% 0.35/0.71                = ( plus_plus_real
% 0.35/0.71                  @ ( groups6591440286371151544t_real
% 0.35/0.71                    @ ^ [M2: nat] : ( divide_divide_real @ ( power_power_real @ X @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) )
% 0.35/0.71                    @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.71                  @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % Maclaurin_exp_lt
% 0.35/0.71  thf(fact_7841_log__base__10__eq1,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71       => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 0.35/0.71          = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % log_base_10_eq1
% 0.35/0.71  thf(fact_7842_powr__def,axiom,
% 0.35/0.71      ( powr_real
% 0.35/0.71      = ( ^ [X3: real,A4: real] : ( if_real @ ( X3 = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A4 @ ( ln_ln_real @ X3 ) ) ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % powr_def
% 0.35/0.71  thf(fact_7843_log__base__10__eq2,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71       => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 0.35/0.71          = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % log_base_10_eq2
% 0.35/0.71  thf(fact_7844_cosh__zero__iff,axiom,
% 0.35/0.71      ! [X: complex] :
% 0.35/0.71        ( ( ( cosh_complex @ X )
% 0.35/0.71          = zero_zero_complex )
% 0.35/0.71        = ( ( power_power_complex @ ( exp_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.71          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % cosh_zero_iff
% 0.35/0.71  thf(fact_7845_cosh__zero__iff,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ( cosh_real @ X )
% 0.35/0.71          = zero_zero_real )
% 0.35/0.71        = ( ( power_power_real @ ( exp_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.71          = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % cosh_zero_iff
% 0.35/0.71  thf(fact_7846_ln__one,axiom,
% 0.35/0.71      ( ( ln_ln_real @ one_one_real )
% 0.35/0.71      = zero_zero_real ) ).
% 0.35/0.71  
% 0.35/0.71  % ln_one
% 0.35/0.71  thf(fact_7847_fact__0,axiom,
% 0.35/0.71      ( ( semiri5044797733671781792omplex @ zero_zero_nat )
% 0.35/0.71      = one_one_complex ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_0
% 0.35/0.71  thf(fact_7848_fact__0,axiom,
% 0.35/0.71      ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
% 0.35/0.71      = one_one_rat ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_0
% 0.35/0.71  thf(fact_7849_fact__0,axiom,
% 0.35/0.71      ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
% 0.35/0.71      = one_one_int ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_0
% 0.35/0.71  thf(fact_7850_fact__0,axiom,
% 0.35/0.71      ( ( semiri2265585572941072030t_real @ zero_zero_nat )
% 0.35/0.71      = one_one_real ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_0
% 0.35/0.71  thf(fact_7851_fact__0,axiom,
% 0.35/0.71      ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
% 0.35/0.71      = one_one_nat ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_0
% 0.35/0.71  thf(fact_7852_cosh__0,axiom,
% 0.35/0.71      ( ( cosh_complex @ zero_zero_complex )
% 0.35/0.71      = one_one_complex ) ).
% 0.35/0.71  
% 0.35/0.71  % cosh_0
% 0.35/0.71  thf(fact_7853_cosh__0,axiom,
% 0.35/0.71      ( ( cosh_real @ zero_zero_real )
% 0.35/0.71      = one_one_real ) ).
% 0.35/0.71  
% 0.35/0.71  % cosh_0
% 0.35/0.71  thf(fact_7854_ln__inj__iff,axiom,
% 0.35/0.71      ! [X: real,Y: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.35/0.71         => ( ( ( ln_ln_real @ X )
% 0.35/0.71              = ( ln_ln_real @ Y ) )
% 0.35/0.71            = ( X = Y ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % ln_inj_iff
% 0.35/0.71  thf(fact_7855_ln__less__cancel__iff,axiom,
% 0.35/0.71      ! [X: real,Y: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.35/0.71         => ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
% 0.35/0.71            = ( ord_less_real @ X @ Y ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % ln_less_cancel_iff
% 0.35/0.71  thf(fact_7856_fact__Suc__0,axiom,
% 0.35/0.71      ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
% 0.35/0.71      = one_one_complex ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_Suc_0
% 0.35/0.71  thf(fact_7857_fact__Suc__0,axiom,
% 0.35/0.71      ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
% 0.35/0.71      = one_one_rat ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_Suc_0
% 0.35/0.71  thf(fact_7858_fact__Suc__0,axiom,
% 0.35/0.71      ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
% 0.35/0.71      = one_one_int ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_Suc_0
% 0.35/0.71  thf(fact_7859_fact__Suc__0,axiom,
% 0.35/0.71      ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
% 0.35/0.71      = one_one_real ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_Suc_0
% 0.35/0.71  thf(fact_7860_fact__Suc__0,axiom,
% 0.35/0.71      ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
% 0.35/0.71      = one_one_nat ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_Suc_0
% 0.35/0.71  thf(fact_7861_ln__le__cancel__iff,axiom,
% 0.35/0.71      ! [X: real,Y: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.35/0.71         => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
% 0.35/0.71            = ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % ln_le_cancel_iff
% 0.35/0.71  thf(fact_7862_ln__eq__zero__iff,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71       => ( ( ( ln_ln_real @ X )
% 0.35/0.71            = zero_zero_real )
% 0.35/0.71          = ( X = one_one_real ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % ln_eq_zero_iff
% 0.35/0.71  thf(fact_7863_ln__gt__zero__iff,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71       => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.35/0.71          = ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % ln_gt_zero_iff
% 0.35/0.71  thf(fact_7864_ln__less__zero__iff,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71       => ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 0.35/0.71          = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % ln_less_zero_iff
% 0.35/0.71  thf(fact_7865_exp__ln__iff,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ( exp_real @ ( ln_ln_real @ X ) )
% 0.35/0.71          = X )
% 0.35/0.71        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.35/0.71  
% 0.35/0.71  % exp_ln_iff
% 0.35/0.71  thf(fact_7866_exp__ln,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71       => ( ( exp_real @ ( ln_ln_real @ X ) )
% 0.35/0.71          = X ) ) ).
% 0.35/0.71  
% 0.35/0.71  % exp_ln
% 0.35/0.71  thf(fact_7867_ln__le__zero__iff,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71       => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 0.35/0.71          = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % ln_le_zero_iff
% 0.35/0.71  thf(fact_7868_ln__ge__zero__iff,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71       => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.35/0.71          = ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % ln_ge_zero_iff
% 0.35/0.71  thf(fact_7869_fact__nonzero,axiom,
% 0.35/0.71      ! [N: nat] :
% 0.35/0.71        ( ( semiri5044797733671781792omplex @ N )
% 0.35/0.71       != zero_zero_complex ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_nonzero
% 0.35/0.71  thf(fact_7870_fact__nonzero,axiom,
% 0.35/0.71      ! [N: nat] :
% 0.35/0.71        ( ( semiri773545260158071498ct_rat @ N )
% 0.35/0.71       != zero_zero_rat ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_nonzero
% 0.35/0.71  thf(fact_7871_fact__nonzero,axiom,
% 0.35/0.71      ! [N: nat] :
% 0.35/0.71        ( ( semiri1406184849735516958ct_int @ N )
% 0.35/0.71       != zero_zero_int ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_nonzero
% 0.35/0.71  thf(fact_7872_fact__nonzero,axiom,
% 0.35/0.71      ! [N: nat] :
% 0.35/0.71        ( ( semiri2265585572941072030t_real @ N )
% 0.35/0.71       != zero_zero_real ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_nonzero
% 0.35/0.71  thf(fact_7873_fact__nonzero,axiom,
% 0.35/0.71      ! [N: nat] :
% 0.35/0.71        ( ( semiri1408675320244567234ct_nat @ N )
% 0.35/0.71       != zero_zero_nat ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_nonzero
% 0.35/0.71  thf(fact_7874_ln__less__self,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71       => ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% 0.35/0.71  
% 0.35/0.71  % ln_less_self
% 0.35/0.71  thf(fact_7875_fact__ge__zero,axiom,
% 0.35/0.71      ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_ge_zero
% 0.35/0.71  thf(fact_7876_fact__ge__zero,axiom,
% 0.35/0.71      ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_ge_zero
% 0.35/0.71  thf(fact_7877_fact__ge__zero,axiom,
% 0.35/0.71      ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_ge_zero
% 0.35/0.71  thf(fact_7878_fact__ge__zero,axiom,
% 0.35/0.71      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_ge_zero
% 0.35/0.71  thf(fact_7879_fact__not__neg,axiom,
% 0.35/0.71      ! [N: nat] :
% 0.35/0.71        ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_not_neg
% 0.35/0.71  thf(fact_7880_fact__not__neg,axiom,
% 0.35/0.71      ! [N: nat] :
% 0.35/0.71        ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_not_neg
% 0.35/0.71  thf(fact_7881_fact__not__neg,axiom,
% 0.35/0.71      ! [N: nat] :
% 0.35/0.71        ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_not_neg
% 0.35/0.71  thf(fact_7882_fact__not__neg,axiom,
% 0.35/0.71      ! [N: nat] :
% 0.35/0.71        ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_not_neg
% 0.35/0.71  thf(fact_7883_fact__gt__zero,axiom,
% 0.35/0.71      ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_gt_zero
% 0.35/0.71  thf(fact_7884_fact__gt__zero,axiom,
% 0.35/0.71      ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_gt_zero
% 0.35/0.71  thf(fact_7885_fact__gt__zero,axiom,
% 0.35/0.71      ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_gt_zero
% 0.35/0.71  thf(fact_7886_fact__gt__zero,axiom,
% 0.35/0.71      ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_gt_zero
% 0.35/0.71  thf(fact_7887_fact__ge__1,axiom,
% 0.35/0.71      ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_ge_1
% 0.35/0.71  thf(fact_7888_fact__ge__1,axiom,
% 0.35/0.71      ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_ge_1
% 0.35/0.71  thf(fact_7889_fact__ge__1,axiom,
% 0.35/0.71      ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_ge_1
% 0.35/0.71  thf(fact_7890_fact__ge__1,axiom,
% 0.35/0.71      ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_ge_1
% 0.35/0.71  thf(fact_7891_fact__mono,axiom,
% 0.35/0.71      ! [M: nat,N: nat] :
% 0.35/0.71        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.71       => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_mono
% 0.35/0.71  thf(fact_7892_fact__mono,axiom,
% 0.35/0.71      ! [M: nat,N: nat] :
% 0.35/0.71        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.71       => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_mono
% 0.35/0.71  thf(fact_7893_fact__mono,axiom,
% 0.35/0.71      ! [M: nat,N: nat] :
% 0.35/0.71        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.71       => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_mono
% 0.35/0.71  thf(fact_7894_fact__mono,axiom,
% 0.35/0.71      ! [M: nat,N: nat] :
% 0.35/0.71        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.71       => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_mono
% 0.35/0.71  thf(fact_7895_cosh__real__pos,axiom,
% 0.35/0.71      ! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% 0.35/0.71  
% 0.35/0.71  % cosh_real_pos
% 0.35/0.71  thf(fact_7896_fact__dvd,axiom,
% 0.35/0.71      ! [N: nat,M: nat] :
% 0.35/0.71        ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.71       => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_dvd
% 0.35/0.71  thf(fact_7897_fact__dvd,axiom,
% 0.35/0.71      ! [N: nat,M: nat] :
% 0.35/0.71        ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.71       => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_dvd
% 0.35/0.71  thf(fact_7898_fact__dvd,axiom,
% 0.35/0.71      ! [N: nat,M: nat] :
% 0.35/0.71        ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.71       => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_dvd
% 0.35/0.71  thf(fact_7899_fact__dvd,axiom,
% 0.35/0.71      ! [N: nat,M: nat] :
% 0.35/0.71        ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.71       => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_dvd
% 0.35/0.71  thf(fact_7900_ln__bound,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71       => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% 0.35/0.71  
% 0.35/0.71  % ln_bound
% 0.35/0.71  thf(fact_7901_ln__gt__zero,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ one_one_real @ X )
% 0.35/0.71       => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % ln_gt_zero
% 0.35/0.71  thf(fact_7902_ln__less__zero,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71       => ( ( ord_less_real @ X @ one_one_real )
% 0.35/0.71         => ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % ln_less_zero
% 0.35/0.71  thf(fact_7903_ln__gt__zero__imp__gt__one,axiom,
% 0.35/0.71      ! [X: real] :
% 0.35/0.71        ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.35/0.71       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.71         => ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % ln_gt_zero_imp_gt_one
% 0.35/0.71  thf(fact_7904_fact__less__mono,axiom,
% 0.35/0.71      ! [M: nat,N: nat] :
% 0.35/0.71        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.71       => ( ( ord_less_nat @ M @ N )
% 0.35/0.71         => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_less_mono
% 0.35/0.71  thf(fact_7905_fact__less__mono,axiom,
% 0.35/0.71      ! [M: nat,N: nat] :
% 0.35/0.71        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.71       => ( ( ord_less_nat @ M @ N )
% 0.35/0.71         => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_less_mono
% 0.35/0.71  thf(fact_7906_fact__less__mono,axiom,
% 0.35/0.71      ! [M: nat,N: nat] :
% 0.35/0.71        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.71       => ( ( ord_less_nat @ M @ N )
% 0.35/0.71         => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_less_mono
% 0.35/0.71  thf(fact_7907_fact__less__mono,axiom,
% 0.35/0.71      ! [M: nat,N: nat] :
% 0.35/0.71        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.71       => ( ( ord_less_nat @ M @ N )
% 0.35/0.71         => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_less_mono
% 0.35/0.71  thf(fact_7908_cosh__real__nonpos__less__iff,axiom,
% 0.35/0.71      ! [X: real,Y: real] :
% 0.35/0.71        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.35/0.71       => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.35/0.71         => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
% 0.35/0.71            = ( ord_less_real @ Y @ X ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % cosh_real_nonpos_less_iff
% 0.35/0.71  thf(fact_7909_cosh__real__nonneg__less__iff,axiom,
% 0.35/0.71      ! [X: real,Y: real] :
% 0.35/0.71        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.71       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.35/0.71         => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
% 0.35/0.71            = ( ord_less_real @ X @ Y ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % cosh_real_nonneg_less_iff
% 0.35/0.71  thf(fact_7910_cosh__real__strict__mono,axiom,
% 0.35/0.71      ! [X: real,Y: real] :
% 0.35/0.71        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.71       => ( ( ord_less_real @ X @ Y )
% 0.35/0.71         => ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % cosh_real_strict_mono
% 0.35/0.71  thf(fact_7911_fact__mod,axiom,
% 0.35/0.71      ! [M: nat,N: nat] :
% 0.35/0.71        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.71       => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
% 0.35/0.71          = zero_zero_int ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_mod
% 0.35/0.71  thf(fact_7912_fact__mod,axiom,
% 0.35/0.71      ! [M: nat,N: nat] :
% 0.35/0.71        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.71       => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
% 0.35/0.71          = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_mod
% 0.35/0.71  thf(fact_7913_fact__mod,axiom,
% 0.35/0.71      ! [M: nat,N: nat] :
% 0.35/0.71        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.71       => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
% 0.35/0.71          = zero_zero_nat ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_mod
% 0.35/0.71  thf(fact_7914_fact__le__power,axiom,
% 0.35/0.71      ! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_le_power
% 0.35/0.71  thf(fact_7915_fact__le__power,axiom,
% 0.35/0.71      ! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_le_power
% 0.35/0.71  thf(fact_7916_fact__le__power,axiom,
% 0.35/0.71      ! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).
% 0.35/0.71  
% 0.35/0.71  % fact_le_power
% 0.35/0.72  thf(fact_7917_fact__le__power,axiom,
% 0.35/0.72      ! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fact_le_power
% 0.35/0.72  thf(fact_7918_ln__ge__zero__imp__ge__one,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.35/0.72       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72         => ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % ln_ge_zero_imp_ge_one
% 0.35/0.72  thf(fact_7919_cosh__ln__real,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( cosh_real @ ( ln_ln_real @ X ) )
% 0.35/0.72          = ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % cosh_ln_real
% 0.35/0.72  thf(fact_7920_ln__mult,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.35/0.72         => ( ( ln_ln_real @ ( times_times_real @ X @ Y ) )
% 0.35/0.72            = ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % ln_mult
% 0.35/0.72  thf(fact_7921_ln__eq__minus__one,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ( ln_ln_real @ X )
% 0.35/0.72            = ( minus_minus_real @ X @ one_one_real ) )
% 0.35/0.72         => ( X = one_one_real ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % ln_eq_minus_one
% 0.35/0.72  thf(fact_7922_ln__div,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.35/0.72         => ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
% 0.35/0.72            = ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % ln_div
% 0.35/0.72  thf(fact_7923_ln__ge__iff,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ord_less_eq_real @ Y @ ( ln_ln_real @ X ) )
% 0.35/0.72          = ( ord_less_eq_real @ ( exp_real @ Y ) @ X ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % ln_ge_iff
% 0.35/0.72  thf(fact_7924_ln__inverse,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ln_ln_real @ ( inverse_inverse_real @ X ) )
% 0.35/0.72          = ( uminus_uminus_real @ ( ln_ln_real @ X ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % ln_inverse
% 0.35/0.72  thf(fact_7925_choose__dvd,axiom,
% 0.35/0.72      ! [K: nat,N: nat] :
% 0.35/0.72        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.72       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % choose_dvd
% 0.35/0.72  thf(fact_7926_choose__dvd,axiom,
% 0.35/0.72      ! [K: nat,N: nat] :
% 0.35/0.72        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.72       => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % choose_dvd
% 0.35/0.72  thf(fact_7927_choose__dvd,axiom,
% 0.35/0.72      ! [K: nat,N: nat] :
% 0.35/0.72        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.72       => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % choose_dvd
% 0.35/0.72  thf(fact_7928_choose__dvd,axiom,
% 0.35/0.72      ! [K: nat,N: nat] :
% 0.35/0.72        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.72       => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % choose_dvd
% 0.35/0.72  thf(fact_7929_choose__dvd,axiom,
% 0.35/0.72      ! [K: nat,N: nat] :
% 0.35/0.72        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.72       => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % choose_dvd
% 0.35/0.72  thf(fact_7930_ln__2__less__1,axiom,
% 0.35/0.72      ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 0.35/0.72  
% 0.35/0.72  % ln_2_less_1
% 0.35/0.72  thf(fact_7931_ln__le__minus__one,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % ln_le_minus_one
% 0.35/0.72  thf(fact_7932_ln__diff__le,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.35/0.72         => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y ) @ Y ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % ln_diff_le
% 0.35/0.72  thf(fact_7933_ln__add__one__self__le__self2,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.35/0.72       => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 0.35/0.72  
% 0.35/0.72  % ln_add_one_self_le_self2
% 0.35/0.72  thf(fact_7934_ln__realpow,axiom,
% 0.35/0.72      ! [X: real,N: nat] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
% 0.35/0.72          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % ln_realpow
% 0.35/0.72  thf(fact_7935_ln__one__minus__pos__upper__bound,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ one_one_real )
% 0.35/0.72         => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % ln_one_minus_pos_upper_bound
% 0.35/0.72  thf(fact_7936_ln__powr__bound,axiom,
% 0.35/0.72      ! [X: real,A: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ one_one_real @ X )
% 0.35/0.72       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.72         => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % ln_powr_bound
% 0.35/0.72  thf(fact_7937_ln__powr__bound2,axiom,
% 0.35/0.72      ! [X: real,A: real] :
% 0.35/0.72        ( ( ord_less_real @ one_one_real @ X )
% 0.35/0.72       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.72         => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % ln_powr_bound2
% 0.35/0.72  thf(fact_7938_log__eq__div__ln__mult__log,axiom,
% 0.35/0.72      ! [A: real,B: real,X: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.72       => ( ( A != one_one_real )
% 0.35/0.72         => ( ( ord_less_real @ zero_zero_real @ B )
% 0.35/0.72           => ( ( B != one_one_real )
% 0.35/0.72             => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72               => ( ( log @ A @ X )
% 0.35/0.72                  = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % log_eq_div_ln_mult_log
% 0.35/0.72  thf(fact_7939_fact__num__eq__if,axiom,
% 0.35/0.72      ( semiri5044797733671781792omplex
% 0.35/0.72      = ( ^ [M2: nat] : ( if_complex @ ( M2 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fact_num_eq_if
% 0.35/0.72  thf(fact_7940_fact__num__eq__if,axiom,
% 0.35/0.72      ( semiri773545260158071498ct_rat
% 0.35/0.72      = ( ^ [M2: nat] : ( if_rat @ ( M2 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fact_num_eq_if
% 0.35/0.72  thf(fact_7941_fact__num__eq__if,axiom,
% 0.35/0.72      ( semiri1406184849735516958ct_int
% 0.35/0.72      = ( ^ [M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fact_num_eq_if
% 0.35/0.72  thf(fact_7942_fact__num__eq__if,axiom,
% 0.35/0.72      ( semiri2265585572941072030t_real
% 0.35/0.72      = ( ^ [M2: nat] : ( if_real @ ( M2 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fact_num_eq_if
% 0.35/0.72  thf(fact_7943_fact__num__eq__if,axiom,
% 0.35/0.72      ( semiri1408675320244567234ct_nat
% 0.35/0.72      = ( ^ [M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fact_num_eq_if
% 0.35/0.72  thf(fact_7944_fact__reduce,axiom,
% 0.35/0.72      ! [N: nat] :
% 0.35/0.72        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.72       => ( ( semiri5044797733671781792omplex @ N )
% 0.35/0.72          = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fact_reduce
% 0.35/0.72  thf(fact_7945_fact__reduce,axiom,
% 0.35/0.72      ! [N: nat] :
% 0.35/0.72        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.72       => ( ( semiri773545260158071498ct_rat @ N )
% 0.35/0.72          = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fact_reduce
% 0.35/0.72  thf(fact_7946_fact__reduce,axiom,
% 0.35/0.72      ! [N: nat] :
% 0.35/0.72        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.72       => ( ( semiri1406184849735516958ct_int @ N )
% 0.35/0.72          = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fact_reduce
% 0.35/0.72  thf(fact_7947_fact__reduce,axiom,
% 0.35/0.72      ! [N: nat] :
% 0.35/0.72        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.72       => ( ( semiri2265585572941072030t_real @ N )
% 0.35/0.72          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fact_reduce
% 0.35/0.72  thf(fact_7948_fact__reduce,axiom,
% 0.35/0.72      ! [N: nat] :
% 0.35/0.72        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.72       => ( ( semiri1408675320244567234ct_nat @ N )
% 0.35/0.72          = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fact_reduce
% 0.35/0.72  thf(fact_7949_binomial__fact,axiom,
% 0.35/0.72      ! [K: nat,N: nat] :
% 0.35/0.72        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.72       => ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
% 0.35/0.72          = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % binomial_fact
% 0.35/0.72  thf(fact_7950_binomial__fact,axiom,
% 0.35/0.72      ! [K: nat,N: nat] :
% 0.35/0.72        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.72       => ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
% 0.35/0.72          = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % binomial_fact
% 0.35/0.72  thf(fact_7951_binomial__fact,axiom,
% 0.35/0.72      ! [K: nat,N: nat] :
% 0.35/0.72        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.72       => ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
% 0.35/0.72          = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % binomial_fact
% 0.35/0.72  thf(fact_7952_fact__binomial,axiom,
% 0.35/0.72      ! [K: nat,N: nat] :
% 0.35/0.72        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.72       => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) ) )
% 0.35/0.72          = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fact_binomial
% 0.35/0.72  thf(fact_7953_fact__binomial,axiom,
% 0.35/0.72      ! [K: nat,N: nat] :
% 0.35/0.72        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.72       => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
% 0.35/0.72          = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fact_binomial
% 0.35/0.72  thf(fact_7954_fact__binomial,axiom,
% 0.35/0.72      ! [K: nat,N: nat] :
% 0.35/0.72        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.72       => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) )
% 0.35/0.72          = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fact_binomial
% 0.35/0.72  thf(fact_7955_ln__sqrt,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ln_ln_real @ ( sqrt @ X ) )
% 0.35/0.72          = ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % ln_sqrt
% 0.35/0.72  thf(fact_7956_Maclaurin__zero,axiom,
% 0.35/0.72      ! [X: real,N: nat,Diff: nat > complex > real] :
% 0.35/0.72        ( ( X = zero_zero_real )
% 0.35/0.72       => ( ( N != zero_zero_nat )
% 0.35/0.72         => ( ( groups6591440286371151544t_real
% 0.35/0.72              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
% 0.35/0.72              @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.72            = ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % Maclaurin_zero
% 0.35/0.72  thf(fact_7957_Maclaurin__zero,axiom,
% 0.35/0.72      ! [X: real,N: nat,Diff: nat > real > real] :
% 0.35/0.72        ( ( X = zero_zero_real )
% 0.35/0.72       => ( ( N != zero_zero_nat )
% 0.35/0.72         => ( ( groups6591440286371151544t_real
% 0.35/0.72              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
% 0.35/0.72              @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.72            = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % Maclaurin_zero
% 0.35/0.72  thf(fact_7958_Maclaurin__zero,axiom,
% 0.35/0.72      ! [X: real,N: nat,Diff: nat > rat > real] :
% 0.35/0.72        ( ( X = zero_zero_real )
% 0.35/0.72       => ( ( N != zero_zero_nat )
% 0.35/0.72         => ( ( groups6591440286371151544t_real
% 0.35/0.72              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
% 0.35/0.72              @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.72            = ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % Maclaurin_zero
% 0.35/0.72  thf(fact_7959_Maclaurin__zero,axiom,
% 0.35/0.72      ! [X: real,N: nat,Diff: nat > nat > real] :
% 0.35/0.72        ( ( X = zero_zero_real )
% 0.35/0.72       => ( ( N != zero_zero_nat )
% 0.35/0.72         => ( ( groups6591440286371151544t_real
% 0.35/0.72              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
% 0.35/0.72              @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.72            = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % Maclaurin_zero
% 0.35/0.72  thf(fact_7960_Maclaurin__zero,axiom,
% 0.35/0.72      ! [X: real,N: nat,Diff: nat > int > real] :
% 0.35/0.72        ( ( X = zero_zero_real )
% 0.35/0.72       => ( ( N != zero_zero_nat )
% 0.35/0.72         => ( ( groups6591440286371151544t_real
% 0.35/0.72              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
% 0.35/0.72              @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.72            = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % Maclaurin_zero
% 0.35/0.72  thf(fact_7961_Maclaurin__lemma,axiom,
% 0.35/0.72      ! [H2: real,F: real > real,J: nat > real,N: nat] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.35/0.72       => ? [B8: real] :
% 0.35/0.72            ( ( F @ H2 )
% 0.35/0.72            = ( plus_plus_real
% 0.35/0.72              @ ( groups6591440286371151544t_real
% 0.35/0.72                @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 0.35/0.72                @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.72              @ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % Maclaurin_lemma
% 0.35/0.72  thf(fact_7962_sin__coeff__def,axiom,
% 0.35/0.72      ( sin_coeff
% 0.35/0.72      = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_coeff_def
% 0.35/0.72  thf(fact_7963_gbinomial__code,axiom,
% 0.35/0.72      ( gbinomial_complex
% 0.35/0.72      = ( ^ [A4: complex,K3: nat] :
% 0.35/0.72            ( if_complex @ ( K3 = zero_zero_nat ) @ one_one_complex
% 0.35/0.72            @ ( divide1717551699836669952omplex
% 0.35/0.72              @ ( set_fo1517530859248394432omplex
% 0.35/0.72                @ ^ [L2: nat] : ( times_times_complex @ ( minus_minus_complex @ A4 @ ( semiri8010041392384452111omplex @ L2 ) ) )
% 0.35/0.72                @ zero_zero_nat
% 0.35/0.72                @ ( minus_minus_nat @ K3 @ one_one_nat )
% 0.35/0.72                @ one_one_complex )
% 0.35/0.72              @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % gbinomial_code
% 0.35/0.72  thf(fact_7964_gbinomial__code,axiom,
% 0.35/0.72      ( gbinomial_rat
% 0.35/0.72      = ( ^ [A4: rat,K3: nat] :
% 0.35/0.72            ( if_rat @ ( K3 = zero_zero_nat ) @ one_one_rat
% 0.35/0.72            @ ( divide_divide_rat
% 0.35/0.72              @ ( set_fo1949268297981939178at_rat
% 0.35/0.72                @ ^ [L2: nat] : ( times_times_rat @ ( minus_minus_rat @ A4 @ ( semiri681578069525770553at_rat @ L2 ) ) )
% 0.35/0.72                @ zero_zero_nat
% 0.35/0.72                @ ( minus_minus_nat @ K3 @ one_one_nat )
% 0.35/0.72                @ one_one_rat )
% 0.35/0.72              @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % gbinomial_code
% 0.35/0.72  thf(fact_7965_gbinomial__code,axiom,
% 0.35/0.72      ( gbinomial_real
% 0.35/0.72      = ( ^ [A4: real,K3: nat] :
% 0.35/0.72            ( if_real @ ( K3 = zero_zero_nat ) @ one_one_real
% 0.35/0.72            @ ( divide_divide_real
% 0.35/0.72              @ ( set_fo3111899725591712190t_real
% 0.35/0.72                @ ^ [L2: nat] : ( times_times_real @ ( minus_minus_real @ A4 @ ( semiri5074537144036343181t_real @ L2 ) ) )
% 0.35/0.72                @ zero_zero_nat
% 0.35/0.72                @ ( minus_minus_nat @ K3 @ one_one_nat )
% 0.35/0.72                @ one_one_real )
% 0.35/0.72              @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % gbinomial_code
% 0.35/0.72  thf(fact_7966_ln__series,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.35/0.72         => ( ( ln_ln_real @ X )
% 0.35/0.72            = ( suminf_real
% 0.35/0.72              @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N2 ) ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % ln_series
% 0.35/0.72  thf(fact_7967_tanh__ln__real,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( tanh_real @ ( ln_ln_real @ X ) )
% 0.35/0.72          = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tanh_ln_real
% 0.35/0.72  thf(fact_7968_sin__x__sin__y,axiom,
% 0.35/0.72      ! [X: complex,Y: complex] :
% 0.35/0.72        ( sums_complex
% 0.35/0.72        @ ^ [P6: nat] :
% 0.35/0.72            ( groups2073611262835488442omplex
% 0.35/0.72            @ ^ [N2: nat] :
% 0.35/0.72                ( if_complex
% 0.35/0.72                @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 )
% 0.35/0.72                  & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.35/0.72                @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) ) @ ( power_power_complex @ X @ N2 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P6 @ N2 ) ) )
% 0.35/0.72                @ zero_zero_complex )
% 0.35/0.72            @ ( set_ord_atMost_nat @ P6 ) )
% 0.35/0.72        @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ Y ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_x_sin_y
% 0.35/0.72  thf(fact_7969_sin__x__sin__y,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( sums_real
% 0.35/0.72        @ ^ [P6: nat] :
% 0.35/0.72            ( groups6591440286371151544t_real
% 0.35/0.72            @ ^ [N2: nat] :
% 0.35/0.72                ( if_real
% 0.35/0.72                @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 )
% 0.35/0.72                  & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.35/0.72                @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) ) @ ( power_power_real @ X @ N2 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P6 @ N2 ) ) )
% 0.35/0.72                @ zero_zero_real )
% 0.35/0.72            @ ( set_ord_atMost_nat @ P6 ) )
% 0.35/0.72        @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_x_sin_y
% 0.35/0.72  thf(fact_7970_scaleR__cancel__right,axiom,
% 0.35/0.72      ! [A: real,X: complex,B: real] :
% 0.35/0.72        ( ( ( real_V2046097035970521341omplex @ A @ X )
% 0.35/0.72          = ( real_V2046097035970521341omplex @ B @ X ) )
% 0.35/0.72        = ( ( A = B )
% 0.35/0.72          | ( X = zero_zero_complex ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % scaleR_cancel_right
% 0.35/0.72  thf(fact_7971_scaleR__cancel__right,axiom,
% 0.35/0.72      ! [A: real,X: real,B: real] :
% 0.35/0.72        ( ( ( real_V1485227260804924795R_real @ A @ X )
% 0.35/0.72          = ( real_V1485227260804924795R_real @ B @ X ) )
% 0.35/0.72        = ( ( A = B )
% 0.35/0.72          | ( X = zero_zero_real ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % scaleR_cancel_right
% 0.35/0.72  thf(fact_7972_scaleR__zero__right,axiom,
% 0.35/0.72      ! [A: real] :
% 0.35/0.72        ( ( real_V2046097035970521341omplex @ A @ zero_zero_complex )
% 0.35/0.72        = zero_zero_complex ) ).
% 0.35/0.72  
% 0.35/0.72  % scaleR_zero_right
% 0.35/0.72  thf(fact_7973_scaleR__zero__right,axiom,
% 0.35/0.72      ! [A: real] :
% 0.35/0.72        ( ( real_V1485227260804924795R_real @ A @ zero_zero_real )
% 0.35/0.72        = zero_zero_real ) ).
% 0.35/0.72  
% 0.35/0.72  % scaleR_zero_right
% 0.35/0.72  thf(fact_7974_sin__zero,axiom,
% 0.35/0.72      ( ( sin_complex @ zero_zero_complex )
% 0.35/0.72      = zero_zero_complex ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_zero
% 0.35/0.72  thf(fact_7975_sin__zero,axiom,
% 0.35/0.72      ( ( sin_real @ zero_zero_real )
% 0.35/0.72      = zero_zero_real ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_zero
% 0.35/0.72  thf(fact_7976_tanh__0,axiom,
% 0.35/0.72      ( ( tanh_complex @ zero_zero_complex )
% 0.35/0.72      = zero_zero_complex ) ).
% 0.35/0.72  
% 0.35/0.72  % tanh_0
% 0.35/0.72  thf(fact_7977_tanh__0,axiom,
% 0.35/0.72      ( ( tanh_real @ zero_zero_real )
% 0.35/0.72      = zero_zero_real ) ).
% 0.35/0.72  
% 0.35/0.72  % tanh_0
% 0.35/0.72  thf(fact_7978_tanh__real__less__iff,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( ( ord_less_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
% 0.35/0.72        = ( ord_less_real @ X @ Y ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tanh_real_less_iff
% 0.35/0.72  thf(fact_7979_suminf__zero,axiom,
% 0.35/0.72      ( ( suminf_complex
% 0.35/0.72        @ ^ [N2: nat] : zero_zero_complex )
% 0.35/0.72      = zero_zero_complex ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_zero
% 0.35/0.72  thf(fact_7980_suminf__zero,axiom,
% 0.35/0.72      ( ( suminf_real
% 0.35/0.72        @ ^ [N2: nat] : zero_zero_real )
% 0.35/0.72      = zero_zero_real ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_zero
% 0.35/0.72  thf(fact_7981_suminf__zero,axiom,
% 0.35/0.72      ( ( suminf_nat
% 0.35/0.72        @ ^ [N2: nat] : zero_zero_nat )
% 0.35/0.72      = zero_zero_nat ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_zero
% 0.35/0.72  thf(fact_7982_suminf__zero,axiom,
% 0.35/0.72      ( ( suminf_int
% 0.35/0.72        @ ^ [N2: nat] : zero_zero_int )
% 0.35/0.72      = zero_zero_int ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_zero
% 0.35/0.72  thf(fact_7983_scaleR__eq__0__iff,axiom,
% 0.35/0.72      ! [A: real,X: complex] :
% 0.35/0.72        ( ( ( real_V2046097035970521341omplex @ A @ X )
% 0.35/0.72          = zero_zero_complex )
% 0.35/0.72        = ( ( A = zero_zero_real )
% 0.35/0.72          | ( X = zero_zero_complex ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % scaleR_eq_0_iff
% 0.35/0.72  thf(fact_7984_scaleR__eq__0__iff,axiom,
% 0.35/0.72      ! [A: real,X: real] :
% 0.35/0.72        ( ( ( real_V1485227260804924795R_real @ A @ X )
% 0.35/0.72          = zero_zero_real )
% 0.35/0.72        = ( ( A = zero_zero_real )
% 0.35/0.72          | ( X = zero_zero_real ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % scaleR_eq_0_iff
% 0.35/0.72  thf(fact_7985_scaleR__zero__left,axiom,
% 0.35/0.72      ! [X: complex] :
% 0.35/0.72        ( ( real_V2046097035970521341omplex @ zero_zero_real @ X )
% 0.35/0.72        = zero_zero_complex ) ).
% 0.35/0.72  
% 0.35/0.72  % scaleR_zero_left
% 0.35/0.72  thf(fact_7986_scaleR__zero__left,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( real_V1485227260804924795R_real @ zero_zero_real @ X )
% 0.35/0.72        = zero_zero_real ) ).
% 0.35/0.72  
% 0.35/0.72  % scaleR_zero_left
% 0.35/0.72  thf(fact_7987_tanh__real__pos__iff,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X ) )
% 0.35/0.72        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tanh_real_pos_iff
% 0.35/0.72  thf(fact_7988_tanh__real__neg__iff,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ ( tanh_real @ X ) @ zero_zero_real )
% 0.35/0.72        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tanh_real_neg_iff
% 0.35/0.72  thf(fact_7989_sin__coeff__0,axiom,
% 0.35/0.72      ( ( sin_coeff @ zero_zero_nat )
% 0.35/0.72      = zero_zero_real ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_coeff_0
% 0.35/0.72  thf(fact_7990_powser__zero,axiom,
% 0.35/0.72      ! [F: nat > complex] :
% 0.35/0.72        ( ( suminf_complex
% 0.35/0.72          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) )
% 0.35/0.72        = ( F @ zero_zero_nat ) ) ).
% 0.35/0.72  
% 0.35/0.72  % powser_zero
% 0.35/0.72  thf(fact_7991_powser__zero,axiom,
% 0.35/0.72      ! [F: nat > real] :
% 0.35/0.72        ( ( suminf_real
% 0.35/0.72          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) )
% 0.35/0.72        = ( F @ zero_zero_nat ) ) ).
% 0.35/0.72  
% 0.35/0.72  % powser_zero
% 0.35/0.72  thf(fact_7992_fact__mono__nat,axiom,
% 0.35/0.72      ! [M: nat,N: nat] :
% 0.35/0.72        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.72       => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fact_mono_nat
% 0.35/0.72  thf(fact_7993_fact__ge__self,axiom,
% 0.35/0.72      ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fact_ge_self
% 0.35/0.72  thf(fact_7994_scaleR__right__imp__eq,axiom,
% 0.35/0.72      ! [X: complex,A: real,B: real] :
% 0.35/0.72        ( ( X != zero_zero_complex )
% 0.35/0.72       => ( ( ( real_V2046097035970521341omplex @ A @ X )
% 0.35/0.72            = ( real_V2046097035970521341omplex @ B @ X ) )
% 0.35/0.72         => ( A = B ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % scaleR_right_imp_eq
% 0.35/0.72  thf(fact_7995_scaleR__right__imp__eq,axiom,
% 0.35/0.72      ! [X: real,A: real,B: real] :
% 0.35/0.72        ( ( X != zero_zero_real )
% 0.35/0.72       => ( ( ( real_V1485227260804924795R_real @ A @ X )
% 0.35/0.72            = ( real_V1485227260804924795R_real @ B @ X ) )
% 0.35/0.72         => ( A = B ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % scaleR_right_imp_eq
% 0.35/0.72  thf(fact_7996_fact__less__mono__nat,axiom,
% 0.35/0.72      ! [M: nat,N: nat] :
% 0.35/0.72        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.72       => ( ( ord_less_nat @ M @ N )
% 0.35/0.72         => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fact_less_mono_nat
% 0.35/0.72  thf(fact_7997_tanh__real__lt__1,axiom,
% 0.35/0.72      ! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).
% 0.35/0.72  
% 0.35/0.72  % tanh_real_lt_1
% 0.35/0.72  thf(fact_7998_fact__ge__Suc__0__nat,axiom,
% 0.35/0.72      ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fact_ge_Suc_0_nat
% 0.35/0.72  thf(fact_7999_scaleR__right__mono__neg,axiom,
% 0.35/0.72      ! [B: real,A: real,C: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ B @ A )
% 0.35/0.72       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.35/0.72         => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ C ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % scaleR_right_mono_neg
% 0.35/0.72  thf(fact_8000_scaleR__right__mono,axiom,
% 0.35/0.72      ! [A: real,B: real,X: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ A @ B )
% 0.35/0.72       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.72         => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % scaleR_right_mono
% 0.35/0.72  thf(fact_8001_dvd__fact,axiom,
% 0.35/0.72      ! [M: nat,N: nat] :
% 0.35/0.72        ( ( ord_less_eq_nat @ one_one_nat @ M )
% 0.35/0.72       => ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.72         => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % dvd_fact
% 0.35/0.72  thf(fact_8002_vector__fraction__eq__iff,axiom,
% 0.35/0.72      ! [U: real,V: real,A: complex,X: complex] :
% 0.35/0.72        ( ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ U @ V ) @ A )
% 0.35/0.72          = X )
% 0.35/0.72        = ( ( ( V = zero_zero_real )
% 0.35/0.72           => ( X = zero_zero_complex ) )
% 0.35/0.72          & ( ( V != zero_zero_real )
% 0.35/0.72           => ( ( real_V2046097035970521341omplex @ U @ A )
% 0.35/0.72              = ( real_V2046097035970521341omplex @ V @ X ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % vector_fraction_eq_iff
% 0.35/0.72  thf(fact_8003_vector__fraction__eq__iff,axiom,
% 0.35/0.72      ! [U: real,V: real,A: real,X: real] :
% 0.35/0.72        ( ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A )
% 0.35/0.72          = X )
% 0.35/0.72        = ( ( ( V = zero_zero_real )
% 0.35/0.72           => ( X = zero_zero_real ) )
% 0.35/0.72          & ( ( V != zero_zero_real )
% 0.35/0.72           => ( ( real_V1485227260804924795R_real @ U @ A )
% 0.35/0.72              = ( real_V1485227260804924795R_real @ V @ X ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % vector_fraction_eq_iff
% 0.35/0.72  thf(fact_8004_eq__vector__fraction__iff,axiom,
% 0.35/0.72      ! [X: complex,U: real,V: real,A: complex] :
% 0.35/0.72        ( ( X
% 0.35/0.72          = ( real_V2046097035970521341omplex @ ( divide_divide_real @ U @ V ) @ A ) )
% 0.35/0.72        = ( ( ( V = zero_zero_real )
% 0.35/0.72           => ( X = zero_zero_complex ) )
% 0.35/0.72          & ( ( V != zero_zero_real )
% 0.35/0.72           => ( ( real_V2046097035970521341omplex @ V @ X )
% 0.35/0.72              = ( real_V2046097035970521341omplex @ U @ A ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % eq_vector_fraction_iff
% 0.35/0.72  thf(fact_8005_eq__vector__fraction__iff,axiom,
% 0.35/0.72      ! [X: real,U: real,V: real,A: real] :
% 0.35/0.72        ( ( X
% 0.35/0.72          = ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A ) )
% 0.35/0.72        = ( ( ( V = zero_zero_real )
% 0.35/0.72           => ( X = zero_zero_real ) )
% 0.35/0.72          & ( ( V != zero_zero_real )
% 0.35/0.72           => ( ( real_V1485227260804924795R_real @ V @ X )
% 0.35/0.72              = ( real_V1485227260804924795R_real @ U @ A ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % eq_vector_fraction_iff
% 0.35/0.72  thf(fact_8006_Real__Vector__Spaces_Ole__add__iff2,axiom,
% 0.35/0.72      ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E2 ) @ D ) )
% 0.35/0.72        = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % Real_Vector_Spaces.le_add_iff2
% 0.35/0.72  thf(fact_8007_Real__Vector__Spaces_Ole__add__iff1,axiom,
% 0.35/0.72      ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E2 ) @ D ) )
% 0.35/0.72        = ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.35/0.72  
% 0.35/0.72  % Real_Vector_Spaces.le_add_iff1
% 0.35/0.72  thf(fact_8008_suminf__finite,axiom,
% 0.35/0.72      ! [N6: set_nat,F: nat > complex] :
% 0.35/0.72        ( ( finite_finite_nat @ N6 )
% 0.35/0.72       => ( ! [N3: nat] :
% 0.35/0.72              ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.72             => ( ( F @ N3 )
% 0.35/0.72                = zero_zero_complex ) )
% 0.35/0.72         => ( ( suminf_complex @ F )
% 0.35/0.72            = ( groups2073611262835488442omplex @ F @ N6 ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_finite
% 0.35/0.72  thf(fact_8009_suminf__finite,axiom,
% 0.35/0.72      ! [N6: set_nat,F: nat > int] :
% 0.35/0.72        ( ( finite_finite_nat @ N6 )
% 0.35/0.72       => ( ! [N3: nat] :
% 0.35/0.72              ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.72             => ( ( F @ N3 )
% 0.35/0.72                = zero_zero_int ) )
% 0.35/0.72         => ( ( suminf_int @ F )
% 0.35/0.72            = ( groups3539618377306564664at_int @ F @ N6 ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_finite
% 0.35/0.72  thf(fact_8010_suminf__finite,axiom,
% 0.35/0.72      ! [N6: set_nat,F: nat > nat] :
% 0.35/0.72        ( ( finite_finite_nat @ N6 )
% 0.35/0.72       => ( ! [N3: nat] :
% 0.35/0.72              ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.72             => ( ( F @ N3 )
% 0.35/0.72                = zero_zero_nat ) )
% 0.35/0.72         => ( ( suminf_nat @ F )
% 0.35/0.72            = ( groups3542108847815614940at_nat @ F @ N6 ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_finite
% 0.35/0.72  thf(fact_8011_suminf__finite,axiom,
% 0.35/0.72      ! [N6: set_nat,F: nat > real] :
% 0.35/0.72        ( ( finite_finite_nat @ N6 )
% 0.35/0.72       => ( ! [N3: nat] :
% 0.35/0.72              ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.72             => ( ( F @ N3 )
% 0.35/0.72                = zero_zero_real ) )
% 0.35/0.72         => ( ( suminf_real @ F )
% 0.35/0.72            = ( groups6591440286371151544t_real @ F @ N6 ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_finite
% 0.35/0.72  thf(fact_8012_tanh__real__gt__neg1,axiom,
% 0.35/0.72      ! [X: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tanh_real_gt_neg1
% 0.35/0.72  thf(fact_8013_fact__diff__Suc,axiom,
% 0.35/0.72      ! [N: nat,M: nat] :
% 0.35/0.72        ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.35/0.72       => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
% 0.35/0.72          = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fact_diff_Suc
% 0.35/0.72  thf(fact_8014_zero__le__scaleR__iff,axiom,
% 0.35/0.72      ! [A: real,B: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) )
% 0.35/0.72        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.72            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.35/0.72          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.35/0.72            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.35/0.72          | ( A = zero_zero_real ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % zero_le_scaleR_iff
% 0.35/0.72  thf(fact_8015_scaleR__le__0__iff,axiom,
% 0.35/0.72      ! [A: real,B: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ B ) @ zero_zero_real )
% 0.35/0.72        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.72            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.35/0.72          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.35/0.72            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.35/0.72          | ( A = zero_zero_real ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % scaleR_le_0_iff
% 0.35/0.72  thf(fact_8016_scaleR__nonpos__nonpos,axiom,
% 0.35/0.72      ! [A: real,B: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.35/0.72       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.35/0.72         => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % scaleR_nonpos_nonpos
% 0.35/0.72  thf(fact_8017_scaleR__nonpos__nonneg,axiom,
% 0.35/0.72      ! [A: real,X: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.35/0.72       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.72         => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % scaleR_nonpos_nonneg
% 0.35/0.72  thf(fact_8018_scaleR__nonneg__nonpos,axiom,
% 0.35/0.72      ! [A: real,X: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.35/0.72       => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.35/0.72         => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % scaleR_nonneg_nonpos
% 0.35/0.72  thf(fact_8019_scaleR__nonneg__nonneg,axiom,
% 0.35/0.72      ! [A: real,X: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.35/0.72       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.72         => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % scaleR_nonneg_nonneg
% 0.35/0.72  thf(fact_8020_split__scaleR__pos__le,axiom,
% 0.35/0.72      ! [A: real,B: real] :
% 0.35/0.72        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.35/0.72            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.35/0.72          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.35/0.72            & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.35/0.72       => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % split_scaleR_pos_le
% 0.35/0.72  thf(fact_8021_split__scaleR__neg__le,axiom,
% 0.35/0.72      ! [A: real,X: real] :
% 0.35/0.72        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.35/0.72            & ( ord_less_eq_real @ X @ zero_zero_real ) )
% 0.35/0.72          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.35/0.72            & ( ord_less_eq_real @ zero_zero_real @ X ) ) )
% 0.35/0.72       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ).
% 0.35/0.72  
% 0.35/0.72  % split_scaleR_neg_le
% 0.35/0.72  thf(fact_8022_scaleR__mono_H,axiom,
% 0.35/0.72      ! [A: real,B: real,C: real,D: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ A @ B )
% 0.35/0.72       => ( ( ord_less_eq_real @ C @ D )
% 0.35/0.72         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.35/0.72           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.72             => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ D ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % scaleR_mono'
% 0.35/0.72  thf(fact_8023_scaleR__mono,axiom,
% 0.35/0.72      ! [A: real,B: real,X: real,Y: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ A @ B )
% 0.35/0.72       => ( ( ord_less_eq_real @ X @ Y )
% 0.35/0.72         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.35/0.72           => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.72             => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ Y ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % scaleR_mono
% 0.35/0.72  thf(fact_8024_fact__div__fact__le__pow,axiom,
% 0.35/0.72      ! [R2: nat,N: nat] :
% 0.35/0.72        ( ( ord_less_eq_nat @ R2 @ N )
% 0.35/0.72       => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fact_div_fact_le_pow
% 0.35/0.72  thf(fact_8025_scaleR__left__le__one__le,axiom,
% 0.35/0.72      ! [X: real,A: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.35/0.72         => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ X ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % scaleR_left_le_one_le
% 0.35/0.72  thf(fact_8026_binomial__fact__lemma,axiom,
% 0.35/0.72      ! [K: nat,N: nat] :
% 0.35/0.72        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.72       => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
% 0.35/0.72          = ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % binomial_fact_lemma
% 0.35/0.72  thf(fact_8027_neg__less__divideR__eq,axiom,
% 0.35/0.72      ! [C: real,A: real,B: real] :
% 0.35/0.72        ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.72       => ( ( ord_less_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
% 0.35/0.72          = ( ord_less_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % neg_less_divideR_eq
% 0.35/0.72  thf(fact_8028_neg__divideR__less__eq,axiom,
% 0.35/0.72      ! [C: real,B: real,A: real] :
% 0.35/0.72        ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.72       => ( ( ord_less_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
% 0.35/0.72          = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % neg_divideR_less_eq
% 0.35/0.72  thf(fact_8029_pos__less__divideR__eq,axiom,
% 0.35/0.72      ! [C: real,A: real,B: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.72       => ( ( ord_less_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
% 0.35/0.72          = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pos_less_divideR_eq
% 0.35/0.72  thf(fact_8030_pos__divideR__less__eq,axiom,
% 0.35/0.72      ! [C: real,B: real,A: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.72       => ( ( ord_less_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
% 0.35/0.72          = ( ord_less_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pos_divideR_less_eq
% 0.35/0.72  thf(fact_8031_nonzero__inverse__scaleR__distrib,axiom,
% 0.35/0.72      ! [A: real,X: complex] :
% 0.35/0.72        ( ( A != zero_zero_real )
% 0.35/0.72       => ( ( X != zero_zero_complex )
% 0.35/0.72         => ( ( invers8013647133539491842omplex @ ( real_V2046097035970521341omplex @ A @ X ) )
% 0.35/0.72            = ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ A ) @ ( invers8013647133539491842omplex @ X ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % nonzero_inverse_scaleR_distrib
% 0.35/0.72  thf(fact_8032_nonzero__inverse__scaleR__distrib,axiom,
% 0.35/0.72      ! [A: real,X: real] :
% 0.35/0.72        ( ( A != zero_zero_real )
% 0.35/0.72       => ( ( X != zero_zero_real )
% 0.35/0.72         => ( ( inverse_inverse_real @ ( real_V1485227260804924795R_real @ A @ X ) )
% 0.35/0.72            = ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ X ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % nonzero_inverse_scaleR_distrib
% 0.35/0.72  thf(fact_8033_sin__gt__zero__02,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.35/0.72         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_gt_zero_02
% 0.35/0.72  thf(fact_8034_binomial__altdef__nat,axiom,
% 0.35/0.72      ! [K: nat,N: nat] :
% 0.35/0.72        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.72       => ( ( binomial @ N @ K )
% 0.35/0.72          = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % binomial_altdef_nat
% 0.35/0.72  thf(fact_8035_fold__atLeastAtMost__nat_Oelims,axiom,
% 0.35/0.72      ! [X: nat > nat > nat,Xa2: nat,Xb2: nat,Xc: nat,Y: nat] :
% 0.35/0.72        ( ( ( set_fo2584398358068434914at_nat @ X @ Xa2 @ Xb2 @ Xc )
% 0.35/0.72          = Y )
% 0.35/0.72       => ( ( ( ord_less_nat @ Xb2 @ Xa2 )
% 0.35/0.72           => ( Y = Xc ) )
% 0.35/0.72          & ( ~ ( ord_less_nat @ Xb2 @ Xa2 )
% 0.35/0.72           => ( Y
% 0.35/0.72              = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb2 @ ( X @ Xa2 @ Xc ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fold_atLeastAtMost_nat.elims
% 0.35/0.72  thf(fact_8036_fold__atLeastAtMost__nat_Osimps,axiom,
% 0.35/0.72      ( set_fo2584398358068434914at_nat
% 0.35/0.72      = ( ^ [F5: nat > nat > nat,A4: nat,B4: nat,Acc: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A4 ) @ Acc @ ( set_fo2584398358068434914at_nat @ F5 @ ( plus_plus_nat @ A4 @ one_one_nat ) @ B4 @ ( F5 @ A4 @ Acc ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % fold_atLeastAtMost_nat.simps
% 0.35/0.72  thf(fact_8037_pos__le__minus__divideR__eq,axiom,
% 0.35/0.72      ! [C: real,A: real,B: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.72       => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
% 0.35/0.72          = ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pos_le_minus_divideR_eq
% 0.35/0.72  thf(fact_8038_pos__minus__divideR__le__eq,axiom,
% 0.35/0.72      ! [C: real,B: real,A: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.72       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
% 0.35/0.72          = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pos_minus_divideR_le_eq
% 0.35/0.72  thf(fact_8039_neg__le__minus__divideR__eq,axiom,
% 0.35/0.72      ! [C: real,A: real,B: real] :
% 0.35/0.72        ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.72       => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
% 0.35/0.72          = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % neg_le_minus_divideR_eq
% 0.35/0.72  thf(fact_8040_neg__minus__divideR__le__eq,axiom,
% 0.35/0.72      ! [C: real,B: real,A: real] :
% 0.35/0.72        ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.72       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
% 0.35/0.72          = ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % neg_minus_divideR_le_eq
% 0.35/0.72  thf(fact_8041_neg__minus__divideR__less__eq,axiom,
% 0.35/0.72      ! [C: real,B: real,A: real] :
% 0.35/0.72        ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.72       => ( ( ord_less_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
% 0.35/0.72          = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % neg_minus_divideR_less_eq
% 0.35/0.72  thf(fact_8042_neg__less__minus__divideR__eq,axiom,
% 0.35/0.72      ! [C: real,A: real,B: real] :
% 0.35/0.72        ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.72       => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
% 0.35/0.72          = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % neg_less_minus_divideR_eq
% 0.35/0.72  thf(fact_8043_pos__minus__divideR__less__eq,axiom,
% 0.35/0.72      ! [C: real,B: real,A: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.72       => ( ( ord_less_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
% 0.35/0.72          = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pos_minus_divideR_less_eq
% 0.35/0.72  thf(fact_8044_pos__less__minus__divideR__eq,axiom,
% 0.35/0.72      ! [C: real,A: real,B: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.72       => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
% 0.35/0.72          = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pos_less_minus_divideR_eq
% 0.35/0.72  thf(fact_8045_binomial__code,axiom,
% 0.35/0.72      ( binomial
% 0.35/0.72      = ( ^ [N2: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K3 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % binomial_code
% 0.35/0.72  thf(fact_8046_suminf__geometric,axiom,
% 0.35/0.72      ! [C: real] :
% 0.35/0.72        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 0.35/0.72       => ( ( suminf_real @ ( power_power_real @ C ) )
% 0.35/0.72          = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_geometric
% 0.35/0.72  thf(fact_8047_suminf__geometric,axiom,
% 0.35/0.72      ! [C: complex] :
% 0.35/0.72        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 0.35/0.72       => ( ( suminf_complex @ ( power_power_complex @ C ) )
% 0.35/0.72          = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_geometric
% 0.35/0.72  thf(fact_8048_sum__atLeastAtMost__code,axiom,
% 0.35/0.72      ! [F: nat > complex,A: nat,B: nat] :
% 0.35/0.72        ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.35/0.72        = ( set_fo1517530859248394432omplex
% 0.35/0.72          @ ^ [A4: nat] : ( plus_plus_complex @ ( F @ A4 ) )
% 0.35/0.72          @ A
% 0.35/0.72          @ B
% 0.35/0.72          @ zero_zero_complex ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sum_atLeastAtMost_code
% 0.35/0.72  thf(fact_8049_sum__atLeastAtMost__code,axiom,
% 0.35/0.72      ! [F: nat > rat,A: nat,B: nat] :
% 0.35/0.72        ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.35/0.72        = ( set_fo1949268297981939178at_rat
% 0.35/0.72          @ ^ [A4: nat] : ( plus_plus_rat @ ( F @ A4 ) )
% 0.35/0.72          @ A
% 0.35/0.72          @ B
% 0.35/0.72          @ zero_zero_rat ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sum_atLeastAtMost_code
% 0.35/0.72  thf(fact_8050_sum__atLeastAtMost__code,axiom,
% 0.35/0.72      ! [F: nat > int,A: nat,B: nat] :
% 0.35/0.72        ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.35/0.72        = ( set_fo2581907887559384638at_int
% 0.35/0.72          @ ^ [A4: nat] : ( plus_plus_int @ ( F @ A4 ) )
% 0.35/0.72          @ A
% 0.35/0.72          @ B
% 0.35/0.72          @ zero_zero_int ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sum_atLeastAtMost_code
% 0.35/0.72  thf(fact_8051_sum__atLeastAtMost__code,axiom,
% 0.35/0.72      ! [F: nat > nat,A: nat,B: nat] :
% 0.35/0.72        ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.35/0.72        = ( set_fo2584398358068434914at_nat
% 0.35/0.72          @ ^ [A4: nat] : ( plus_plus_nat @ ( F @ A4 ) )
% 0.35/0.72          @ A
% 0.35/0.72          @ B
% 0.35/0.72          @ zero_zero_nat ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sum_atLeastAtMost_code
% 0.35/0.72  thf(fact_8052_sum__atLeastAtMost__code,axiom,
% 0.35/0.72      ! [F: nat > real,A: nat,B: nat] :
% 0.35/0.72        ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.35/0.72        = ( set_fo3111899725591712190t_real
% 0.35/0.72          @ ^ [A4: nat] : ( plus_plus_real @ ( F @ A4 ) )
% 0.35/0.72          @ A
% 0.35/0.72          @ B
% 0.35/0.72          @ zero_zero_real ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sum_atLeastAtMost_code
% 0.35/0.72  thf(fact_8053_tanh__add,axiom,
% 0.35/0.72      ! [X: complex,Y: complex] :
% 0.35/0.72        ( ( ( cosh_complex @ X )
% 0.35/0.72         != zero_zero_complex )
% 0.35/0.72       => ( ( ( cosh_complex @ Y )
% 0.35/0.72           != zero_zero_complex )
% 0.35/0.72         => ( ( tanh_complex @ ( plus_plus_complex @ X @ Y ) )
% 0.35/0.72            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y ) ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tanh_add
% 0.35/0.72  thf(fact_8054_tanh__add,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( ( ( cosh_real @ X )
% 0.35/0.72         != zero_zero_real )
% 0.35/0.72       => ( ( ( cosh_real @ Y )
% 0.35/0.72           != zero_zero_real )
% 0.35/0.72         => ( ( tanh_real @ ( plus_plus_real @ X @ Y ) )
% 0.35/0.72            = ( divide_divide_real @ ( plus_plus_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tanh_add
% 0.35/0.72  thf(fact_8055_cosh__converges,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( sums_real
% 0.35/0.72        @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) @ zero_zero_real )
% 0.35/0.72        @ ( cosh_real @ X ) ) ).
% 0.35/0.72  
% 0.35/0.72  % cosh_converges
% 0.35/0.72  thf(fact_8056_cosh__converges,axiom,
% 0.35/0.72      ! [X: complex] :
% 0.35/0.72        ( sums_complex
% 0.35/0.72        @ ^ [N2: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_complex @ X @ N2 ) ) @ zero_zero_complex )
% 0.35/0.72        @ ( cosh_complex @ X ) ) ).
% 0.35/0.72  
% 0.35/0.72  % cosh_converges
% 0.35/0.72  thf(fact_8057_sums__cos__x__plus__y,axiom,
% 0.35/0.72      ! [X: complex,Y: complex] :
% 0.35/0.72        ( sums_complex
% 0.35/0.72        @ ^ [P6: nat] :
% 0.35/0.72            ( groups2073611262835488442omplex
% 0.35/0.72            @ ^ [N2: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 ) @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_complex @ X @ N2 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P6 @ N2 ) ) ) @ zero_zero_complex )
% 0.35/0.72            @ ( set_ord_atMost_nat @ P6 ) )
% 0.35/0.72        @ ( cos_complex @ ( plus_plus_complex @ X @ Y ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sums_cos_x_plus_y
% 0.35/0.72  thf(fact_8058_sums__cos__x__plus__y,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( sums_real
% 0.35/0.72        @ ^ [P6: nat] :
% 0.35/0.72            ( groups6591440286371151544t_real
% 0.35/0.72            @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 ) @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ X @ N2 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P6 @ N2 ) ) ) @ zero_zero_real )
% 0.35/0.72            @ ( set_ord_atMost_nat @ P6 ) )
% 0.35/0.72        @ ( cos_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sums_cos_x_plus_y
% 0.35/0.72  thf(fact_8059_cos__x__cos__y,axiom,
% 0.35/0.72      ! [X: complex,Y: complex] :
% 0.35/0.72        ( sums_complex
% 0.35/0.72        @ ^ [P6: nat] :
% 0.35/0.72            ( groups2073611262835488442omplex
% 0.35/0.72            @ ^ [N2: nat] :
% 0.35/0.72                ( if_complex
% 0.35/0.72                @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 )
% 0.35/0.72                  & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.35/0.72                @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_complex @ X @ N2 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P6 @ N2 ) ) )
% 0.35/0.72                @ zero_zero_complex )
% 0.35/0.72            @ ( set_ord_atMost_nat @ P6 ) )
% 0.35/0.72        @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % cos_x_cos_y
% 0.35/0.72  thf(fact_8060_cos__x__cos__y,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( sums_real
% 0.35/0.72        @ ^ [P6: nat] :
% 0.35/0.72            ( groups6591440286371151544t_real
% 0.35/0.72            @ ^ [N2: nat] :
% 0.35/0.72                ( if_real
% 0.35/0.72                @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 )
% 0.35/0.72                  & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.35/0.72                @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ X @ N2 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P6 @ N2 ) ) )
% 0.35/0.72                @ zero_zero_real )
% 0.35/0.72            @ ( set_ord_atMost_nat @ P6 ) )
% 0.35/0.72        @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % cos_x_cos_y
% 0.35/0.72  thf(fact_8061_Maclaurin__sin__expansion3,axiom,
% 0.35/0.72      ! [N: nat,X: real] :
% 0.35/0.72        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.72       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72         => ? [T6: real] :
% 0.35/0.72              ( ( ord_less_real @ zero_zero_real @ T6 )
% 0.35/0.72              & ( ord_less_real @ T6 @ X )
% 0.35/0.72              & ( ( sin_real @ X )
% 0.35/0.72                = ( plus_plus_real
% 0.35/0.72                  @ ( groups6591440286371151544t_real
% 0.35/0.72                    @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X @ M2 ) )
% 0.35/0.72                    @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.72                  @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % Maclaurin_sin_expansion3
% 0.35/0.72  thf(fact_8062_Maclaurin__sin__expansion4,axiom,
% 0.35/0.72      ! [X: real,N: nat] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ? [T6: real] :
% 0.35/0.72            ( ( ord_less_real @ zero_zero_real @ T6 )
% 0.35/0.72            & ( ord_less_eq_real @ T6 @ X )
% 0.35/0.72            & ( ( sin_real @ X )
% 0.35/0.72              = ( plus_plus_real
% 0.35/0.72                @ ( groups6591440286371151544t_real
% 0.35/0.72                  @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X @ M2 ) )
% 0.35/0.72                  @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.72                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % Maclaurin_sin_expansion4
% 0.35/0.72  thf(fact_8063_sinh__converges,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( sums_real
% 0.35/0.72        @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) )
% 0.35/0.72        @ ( sinh_real @ X ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sinh_converges
% 0.35/0.72  thf(fact_8064_sinh__converges,axiom,
% 0.35/0.72      ! [X: complex] :
% 0.35/0.72        ( sums_complex
% 0.35/0.72        @ ^ [N2: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_complex @ X @ N2 ) ) )
% 0.35/0.72        @ ( sinh_complex @ X ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sinh_converges
% 0.35/0.72  thf(fact_8065_sinh__real__less__iff,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( ( ord_less_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
% 0.35/0.72        = ( ord_less_real @ X @ Y ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sinh_real_less_iff
% 0.35/0.72  thf(fact_8066_sinh__real__neg__iff,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ ( sinh_real @ X ) @ zero_zero_real )
% 0.35/0.72        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sinh_real_neg_iff
% 0.35/0.72  thf(fact_8067_sinh__real__pos__iff,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X ) )
% 0.35/0.72        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sinh_real_pos_iff
% 0.35/0.72  thf(fact_8068_sinh__0,axiom,
% 0.35/0.72      ( ( sinh_complex @ zero_zero_complex )
% 0.35/0.72      = zero_zero_complex ) ).
% 0.35/0.72  
% 0.35/0.72  % sinh_0
% 0.35/0.72  thf(fact_8069_sinh__0,axiom,
% 0.35/0.72      ( ( sinh_real @ zero_zero_real )
% 0.35/0.72      = zero_zero_real ) ).
% 0.35/0.72  
% 0.35/0.72  % sinh_0
% 0.35/0.72  thf(fact_8070_cos__zero,axiom,
% 0.35/0.72      ( ( cos_complex @ zero_zero_complex )
% 0.35/0.72      = one_one_complex ) ).
% 0.35/0.72  
% 0.35/0.72  % cos_zero
% 0.35/0.72  thf(fact_8071_cos__zero,axiom,
% 0.35/0.72      ( ( cos_real @ zero_zero_real )
% 0.35/0.72      = one_one_real ) ).
% 0.35/0.72  
% 0.35/0.72  % cos_zero
% 0.35/0.72  thf(fact_8072_cos__mono__less__eq,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ord_less_eq_real @ X @ pi )
% 0.35/0.72         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.35/0.72           => ( ( ord_less_eq_real @ Y @ pi )
% 0.35/0.72             => ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
% 0.35/0.72                = ( ord_less_real @ Y @ X ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % cos_mono_less_eq
% 0.35/0.72  thf(fact_8073_cos__monotone__0__pi,axiom,
% 0.35/0.72      ! [Y: real,X: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.35/0.72       => ( ( ord_less_real @ Y @ X )
% 0.35/0.72         => ( ( ord_less_eq_real @ X @ pi )
% 0.35/0.72           => ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % cos_monotone_0_pi
% 0.35/0.72  thf(fact_8074_sincos__principal__value,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72      ? [Y3: real] :
% 0.35/0.72        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 0.35/0.72        & ( ord_less_eq_real @ Y3 @ pi )
% 0.35/0.72        & ( ( sin_real @ Y3 )
% 0.35/0.72          = ( sin_real @ X ) )
% 0.35/0.72        & ( ( cos_real @ Y3 )
% 0.35/0.72          = ( cos_real @ X ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sincos_principal_value
% 0.35/0.72  thf(fact_8075_cos__monotone__minus__pi__0,axiom,
% 0.35/0.72      ! [Y: real,X: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
% 0.35/0.72       => ( ( ord_less_real @ Y @ X )
% 0.35/0.72         => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.35/0.72           => ( ord_less_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % cos_monotone_minus_pi_0
% 0.35/0.72  thf(fact_8076_pi__gt__zero,axiom,
% 0.35/0.72      ord_less_real @ zero_zero_real @ pi ).
% 0.35/0.72  
% 0.35/0.72  % pi_gt_zero
% 0.35/0.72  thf(fact_8077_pi__not__less__zero,axiom,
% 0.35/0.72      ~ ( ord_less_real @ pi @ zero_zero_real ) ).
% 0.35/0.72  
% 0.35/0.72  % pi_not_less_zero
% 0.35/0.72  thf(fact_8078_sin__gt__zero,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ pi )
% 0.35/0.72         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_gt_zero
% 0.35/0.72  thf(fact_8079_sinh__less__cosh__real,axiom,
% 0.35/0.72      ! [X: real] : ( ord_less_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sinh_less_cosh_real
% 0.35/0.72  thf(fact_8080_cos__one__sin__zero,axiom,
% 0.35/0.72      ! [X: complex] :
% 0.35/0.72        ( ( ( cos_complex @ X )
% 0.35/0.72          = one_one_complex )
% 0.35/0.72       => ( ( sin_complex @ X )
% 0.35/0.72          = zero_zero_complex ) ) ).
% 0.35/0.72  
% 0.35/0.72  % cos_one_sin_zero
% 0.35/0.72  thf(fact_8081_cos__one__sin__zero,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ( cos_real @ X )
% 0.35/0.72          = one_one_real )
% 0.35/0.72       => ( ( sin_real @ X )
% 0.35/0.72          = zero_zero_real ) ) ).
% 0.35/0.72  
% 0.35/0.72  % cos_one_sin_zero
% 0.35/0.72  thf(fact_8082_cos__gt__zero,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72         => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % cos_gt_zero
% 0.35/0.72  thf(fact_8083_sin__eq__0__pi,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ pi )
% 0.35/0.72         => ( ( ( sin_real @ X )
% 0.35/0.72              = zero_zero_real )
% 0.35/0.72           => ( X = zero_zero_real ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_eq_0_pi
% 0.35/0.72  thf(fact_8084_sin__zero__pi__iff,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ ( abs_abs_real @ X ) @ pi )
% 0.35/0.72       => ( ( ( sin_real @ X )
% 0.35/0.72            = zero_zero_real )
% 0.35/0.72          = ( X = zero_zero_real ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_zero_pi_iff
% 0.35/0.72  thf(fact_8085_cos__gt__zero__pi,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72         => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % cos_gt_zero_pi
% 0.35/0.72  thf(fact_8086_sin__zero__norm__cos__one,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ( sin_real @ X )
% 0.35/0.72          = zero_zero_real )
% 0.35/0.72       => ( ( real_V7735802525324610683m_real @ ( cos_real @ X ) )
% 0.35/0.72          = one_one_real ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_zero_norm_cos_one
% 0.35/0.72  thf(fact_8087_sin__zero__norm__cos__one,axiom,
% 0.35/0.72      ! [X: complex] :
% 0.35/0.72        ( ( ( sin_complex @ X )
% 0.35/0.72          = zero_zero_complex )
% 0.35/0.72       => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X ) )
% 0.35/0.72          = one_one_real ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_zero_norm_cos_one
% 0.35/0.72  thf(fact_8088_pi__less__4,axiom,
% 0.35/0.72      ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pi_less_4
% 0.35/0.72  thf(fact_8089_cos__two__less__zero,axiom,
% 0.35/0.72      ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 0.35/0.72  
% 0.35/0.72  % cos_two_less_zero
% 0.35/0.72  thf(fact_8090_pi__half__less__two,axiom,
% 0.35/0.72      ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pi_half_less_two
% 0.35/0.72  thf(fact_8091_sin__pi__divide__n__ge__0,axiom,
% 0.35/0.72      ! [N: nat] :
% 0.35/0.72        ( ( N != zero_zero_nat )
% 0.35/0.72       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_pi_divide_n_ge_0
% 0.35/0.72  thf(fact_8092_sincos__total__2pi,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.72          = one_one_real )
% 0.35/0.72       => ~ ! [T6: real] :
% 0.35/0.72              ( ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.35/0.72             => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.35/0.72               => ( ( X
% 0.35/0.72                    = ( cos_real @ T6 ) )
% 0.35/0.72                 => ( Y
% 0.35/0.72                   != ( sin_real @ T6 ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sincos_total_2pi
% 0.35/0.72  thf(fact_8093_pi__half__gt__zero,axiom,
% 0.35/0.72      ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pi_half_gt_zero
% 0.35/0.72  thf(fact_8094_sin__gt__zero2,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_gt_zero2
% 0.35/0.72  thf(fact_8095_sin__lt__zero,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ pi @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.35/0.72         => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_lt_zero
% 0.35/0.72  thf(fact_8096_m2pi__less__pi,axiom,
% 0.35/0.72      ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 0.35/0.72  
% 0.35/0.72  % m2pi_less_pi
% 0.35/0.72  thf(fact_8097_arctan__ubound,axiom,
% 0.35/0.72      ! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % arctan_ubound
% 0.35/0.72  thf(fact_8098_cos__double__less__one,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.35/0.72         => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % cos_double_less_one
% 0.35/0.72  thf(fact_8099_sinh__zero__iff,axiom,
% 0.35/0.72      ! [X: complex] :
% 0.35/0.72        ( ( ( sinh_complex @ X )
% 0.35/0.72          = zero_zero_complex )
% 0.35/0.72        = ( member_complex @ ( exp_complex @ X ) @ ( insert_complex @ one_one_complex @ ( insert_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ bot_bot_set_complex ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sinh_zero_iff
% 0.35/0.72  thf(fact_8100_sinh__zero__iff,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ( sinh_real @ X )
% 0.35/0.72          = zero_zero_real )
% 0.35/0.72        = ( member_real @ ( exp_real @ X ) @ ( insert_real @ one_one_real @ ( insert_real @ ( uminus_uminus_real @ one_one_real ) @ bot_bot_set_real ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sinh_zero_iff
% 0.35/0.72  thf(fact_8101_sin__le__zero,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ pi @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.35/0.72         => ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_le_zero
% 0.35/0.72  thf(fact_8102_minus__pi__half__less__zero,axiom,
% 0.35/0.72      ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 0.35/0.72  
% 0.35/0.72  % minus_pi_half_less_zero
% 0.35/0.72  thf(fact_8103_sin__less__zero,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ zero_zero_real )
% 0.35/0.72         => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_less_zero
% 0.35/0.72  thf(fact_8104_sin__mono__less__eq,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.35/0.72       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.35/0.72           => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72             => ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
% 0.35/0.72                = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_mono_less_eq
% 0.35/0.72  thf(fact_8105_sin__monotone__2pi,axiom,
% 0.35/0.72      ! [Y: real,X: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.35/0.72       => ( ( ord_less_real @ Y @ X )
% 0.35/0.72         => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72           => ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_monotone_2pi
% 0.35/0.72  thf(fact_8106_arctan__bounded,axiom,
% 0.35/0.72      ! [Y: real] :
% 0.35/0.72        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
% 0.35/0.72        & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % arctan_bounded
% 0.35/0.72  thf(fact_8107_arctan__lbound,axiom,
% 0.35/0.72      ! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) ) ).
% 0.35/0.72  
% 0.35/0.72  % arctan_lbound
% 0.35/0.72  thf(fact_8108_sin__pi__divide__n__gt__0,axiom,
% 0.35/0.72      ! [N: nat] :
% 0.35/0.72        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.72       => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_pi_divide_n_gt_0
% 0.35/0.72  thf(fact_8109_sinh__ln__real,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( sinh_real @ ( ln_ln_real @ X ) )
% 0.35/0.72          = ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sinh_ln_real
% 0.35/0.72  thf(fact_8110_Maclaurin__minus__cos__expansion,axiom,
% 0.35/0.72      ! [N: nat,X: real] :
% 0.35/0.72        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.72       => ( ( ord_less_real @ X @ zero_zero_real )
% 0.35/0.72         => ? [T6: real] :
% 0.35/0.72              ( ( ord_less_real @ X @ T6 )
% 0.35/0.72              & ( ord_less_real @ T6 @ zero_zero_real )
% 0.35/0.72              & ( ( cos_real @ X )
% 0.35/0.72                = ( plus_plus_real
% 0.35/0.72                  @ ( groups6591440286371151544t_real
% 0.35/0.72                    @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X @ M2 ) )
% 0.35/0.72                    @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.72                  @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % Maclaurin_minus_cos_expansion
% 0.35/0.72  thf(fact_8111_Maclaurin__cos__expansion2,axiom,
% 0.35/0.72      ! [X: real,N: nat] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.72         => ? [T6: real] :
% 0.35/0.72              ( ( ord_less_real @ zero_zero_real @ T6 )
% 0.35/0.72              & ( ord_less_real @ T6 @ X )
% 0.35/0.72              & ( ( cos_real @ X )
% 0.35/0.72                = ( plus_plus_real
% 0.35/0.72                  @ ( groups6591440286371151544t_real
% 0.35/0.72                    @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X @ M2 ) )
% 0.35/0.72                    @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.72                  @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % Maclaurin_cos_expansion2
% 0.35/0.72  thf(fact_8112_tan__double,axiom,
% 0.35/0.72      ! [X: complex] :
% 0.35/0.72        ( ( ( cos_complex @ X )
% 0.35/0.72         != zero_zero_complex )
% 0.35/0.72       => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.35/0.72           != zero_zero_complex )
% 0.35/0.72         => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.35/0.72            = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_double
% 0.35/0.72  thf(fact_8113_tan__double,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ( cos_real @ X )
% 0.35/0.72         != zero_zero_real )
% 0.35/0.72       => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.35/0.72           != zero_zero_real )
% 0.35/0.72         => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.35/0.72            = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_double
% 0.35/0.72  thf(fact_8114_sin__tan,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72       => ( ( sin_real @ X )
% 0.35/0.72          = ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sin_tan
% 0.35/0.72  thf(fact_8115_cos__tan,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72       => ( ( cos_real @ X )
% 0.35/0.72          = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % cos_tan
% 0.35/0.72  thf(fact_8116_tan__zero,axiom,
% 0.35/0.72      ( ( tan_complex @ zero_zero_complex )
% 0.35/0.72      = zero_zero_complex ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_zero
% 0.35/0.72  thf(fact_8117_tan__zero,axiom,
% 0.35/0.72      ( ( tan_real @ zero_zero_real )
% 0.35/0.72      = zero_zero_real ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_zero
% 0.35/0.72  thf(fact_8118_cos__coeff__0,axiom,
% 0.35/0.72      ( ( cos_coeff @ zero_zero_nat )
% 0.35/0.72      = one_one_real ) ).
% 0.35/0.72  
% 0.35/0.72  % cos_coeff_0
% 0.35/0.72  thf(fact_8119_tan__gt__zero,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72         => ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_gt_zero
% 0.35/0.72  thf(fact_8120_lemma__tan__total,axiom,
% 0.35/0.72      ! [Y: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ Y )
% 0.35/0.72       => ? [X4: real] :
% 0.35/0.72            ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.35/0.72            & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72            & ( ord_less_real @ Y @ ( tan_real @ X4 ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % lemma_tan_total
% 0.35/0.72  thf(fact_8121_lemma__tan__total1,axiom,
% 0.35/0.72      ! [Y: real] :
% 0.35/0.72      ? [X4: real] :
% 0.35/0.72        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.35/0.72        & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72        & ( ( tan_real @ X4 )
% 0.35/0.72          = Y ) ) ).
% 0.35/0.72  
% 0.35/0.72  % lemma_tan_total1
% 0.35/0.72  thf(fact_8122_tan__mono__lt__eq,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.35/0.72           => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72             => ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 0.35/0.72                = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_mono_lt_eq
% 0.35/0.72  thf(fact_8123_tan__monotone_H,axiom,
% 0.35/0.72      ! [Y: real,X: real] :
% 0.35/0.72        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.35/0.72       => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.35/0.72           => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72             => ( ( ord_less_real @ Y @ X )
% 0.35/0.72                = ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_monotone'
% 0.35/0.72  thf(fact_8124_tan__monotone,axiom,
% 0.35/0.72      ! [Y: real,X: real] :
% 0.35/0.72        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.35/0.72       => ( ( ord_less_real @ Y @ X )
% 0.35/0.72         => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72           => ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_monotone
% 0.35/0.72  thf(fact_8125_tan__total,axiom,
% 0.35/0.72      ! [Y: real] :
% 0.35/0.72      ? [X4: real] :
% 0.35/0.72        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.35/0.72        & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72        & ( ( tan_real @ X4 )
% 0.35/0.72          = Y )
% 0.35/0.72        & ! [Y4: real] :
% 0.35/0.72            ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
% 0.35/0.72              & ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72              & ( ( tan_real @ Y4 )
% 0.35/0.72                = Y ) )
% 0.35/0.72           => ( Y4 = X4 ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_total
% 0.35/0.72  thf(fact_8126_add__tan__eq,axiom,
% 0.35/0.72      ! [X: complex,Y: complex] :
% 0.35/0.72        ( ( ( cos_complex @ X )
% 0.35/0.72         != zero_zero_complex )
% 0.35/0.72       => ( ( ( cos_complex @ Y )
% 0.35/0.72           != zero_zero_complex )
% 0.35/0.72         => ( ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) )
% 0.35/0.72            = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % add_tan_eq
% 0.35/0.72  thf(fact_8127_add__tan__eq,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( ( ( cos_real @ X )
% 0.35/0.72         != zero_zero_real )
% 0.35/0.72       => ( ( ( cos_real @ Y )
% 0.35/0.72           != zero_zero_real )
% 0.35/0.72         => ( ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 0.35/0.72            = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % add_tan_eq
% 0.35/0.72  thf(fact_8128_tan__pos__pi2__le,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72         => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_pos_pi2_le
% 0.35/0.72  thf(fact_8129_tan__total__pos,axiom,
% 0.35/0.72      ! [Y: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.35/0.72       => ? [X4: real] :
% 0.35/0.72            ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.35/0.72            & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72            & ( ( tan_real @ X4 )
% 0.35/0.72              = Y ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_total_pos
% 0.35/0.72  thf(fact_8130_tan__less__zero,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ zero_zero_real )
% 0.35/0.72         => ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_less_zero
% 0.35/0.72  thf(fact_8131_tan__mono__le__eq,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.35/0.72           => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72             => ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 0.35/0.72                = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_mono_le_eq
% 0.35/0.72  thf(fact_8132_tan__mono__le,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.35/0.72       => ( ( ord_less_eq_real @ X @ Y )
% 0.35/0.72         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72           => ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_mono_le
% 0.35/0.72  thf(fact_8133_tan__bound__pi2,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.35/0.72       => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_bound_pi2
% 0.35/0.72  thf(fact_8134_arctan,axiom,
% 0.35/0.72      ! [Y: real] :
% 0.35/0.72        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
% 0.35/0.72        & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72        & ( ( tan_real @ ( arctan @ Y ) )
% 0.35/0.72          = Y ) ) ).
% 0.35/0.72  
% 0.35/0.72  % arctan
% 0.35/0.72  thf(fact_8135_arctan__tan,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72         => ( ( arctan @ ( tan_real @ X ) )
% 0.35/0.72            = X ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % arctan_tan
% 0.35/0.72  thf(fact_8136_arctan__unique,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72         => ( ( ( tan_real @ X )
% 0.35/0.72              = Y )
% 0.35/0.72           => ( ( arctan @ Y )
% 0.35/0.72              = X ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % arctan_unique
% 0.35/0.72  thf(fact_8137_tan__add,axiom,
% 0.35/0.72      ! [X: complex,Y: complex] :
% 0.35/0.72        ( ( ( cos_complex @ X )
% 0.35/0.72         != zero_zero_complex )
% 0.35/0.72       => ( ( ( cos_complex @ Y )
% 0.35/0.72           != zero_zero_complex )
% 0.35/0.72         => ( ( ( cos_complex @ ( plus_plus_complex @ X @ Y ) )
% 0.35/0.72             != zero_zero_complex )
% 0.35/0.72           => ( ( tan_complex @ ( plus_plus_complex @ X @ Y ) )
% 0.35/0.72              = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_add
% 0.35/0.72  thf(fact_8138_tan__add,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( ( ( cos_real @ X )
% 0.35/0.72         != zero_zero_real )
% 0.35/0.72       => ( ( ( cos_real @ Y )
% 0.35/0.72           != zero_zero_real )
% 0.35/0.72         => ( ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
% 0.35/0.72             != zero_zero_real )
% 0.35/0.72           => ( ( tan_real @ ( plus_plus_real @ X @ Y ) )
% 0.35/0.72              = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_add
% 0.35/0.72  thf(fact_8139_tan__diff,axiom,
% 0.35/0.72      ! [X: complex,Y: complex] :
% 0.35/0.72        ( ( ( cos_complex @ X )
% 0.35/0.72         != zero_zero_complex )
% 0.35/0.72       => ( ( ( cos_complex @ Y )
% 0.35/0.72           != zero_zero_complex )
% 0.35/0.72         => ( ( ( cos_complex @ ( minus_minus_complex @ X @ Y ) )
% 0.35/0.72             != zero_zero_complex )
% 0.35/0.72           => ( ( tan_complex @ ( minus_minus_complex @ X @ Y ) )
% 0.35/0.72              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_diff
% 0.35/0.72  thf(fact_8140_tan__diff,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( ( ( cos_real @ X )
% 0.35/0.72         != zero_zero_real )
% 0.35/0.72       => ( ( ( cos_real @ Y )
% 0.35/0.72           != zero_zero_real )
% 0.35/0.72         => ( ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
% 0.35/0.72             != zero_zero_real )
% 0.35/0.72           => ( ( tan_real @ ( minus_minus_real @ X @ Y ) )
% 0.35/0.72              = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_diff
% 0.35/0.72  thf(fact_8141_lemma__tan__add1,axiom,
% 0.35/0.72      ! [X: complex,Y: complex] :
% 0.35/0.72        ( ( ( cos_complex @ X )
% 0.35/0.72         != zero_zero_complex )
% 0.35/0.72       => ( ( ( cos_complex @ Y )
% 0.35/0.72           != zero_zero_complex )
% 0.35/0.72         => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) )
% 0.35/0.72            = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % lemma_tan_add1
% 0.35/0.72  thf(fact_8142_lemma__tan__add1,axiom,
% 0.35/0.72      ! [X: real,Y: real] :
% 0.35/0.72        ( ( ( cos_real @ X )
% 0.35/0.72         != zero_zero_real )
% 0.35/0.72       => ( ( ( cos_real @ Y )
% 0.35/0.72           != zero_zero_real )
% 0.35/0.72         => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) )
% 0.35/0.72            = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % lemma_tan_add1
% 0.35/0.72  thf(fact_8143_tan__total__pi4,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.35/0.72       => ? [Z4: real] :
% 0.35/0.72            ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z4 )
% 0.35/0.72            & ( ord_less_real @ Z4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.35/0.72            & ( ( tan_real @ Z4 )
% 0.35/0.72              = X ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_total_pi4
% 0.35/0.72  thf(fact_8144_tan__sec,axiom,
% 0.35/0.72      ! [X: complex] :
% 0.35/0.72        ( ( ( cos_complex @ X )
% 0.35/0.72         != zero_zero_complex )
% 0.35/0.72       => ( ( plus_plus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.72          = ( power_power_complex @ ( invers8013647133539491842omplex @ ( cos_complex @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_sec
% 0.35/0.72  thf(fact_8145_tan__sec,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ( cos_real @ X )
% 0.35/0.72         != zero_zero_real )
% 0.35/0.72       => ( ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.72          = ( power_power_real @ ( inverse_inverse_real @ ( cos_real @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % tan_sec
% 0.35/0.72  thf(fact_8146_complex__unimodular__polar,axiom,
% 0.35/0.72      ! [Z2: complex] :
% 0.35/0.72        ( ( ( real_V1022390504157884413omplex @ Z2 )
% 0.35/0.72          = one_one_real )
% 0.35/0.72       => ~ ! [T6: real] :
% 0.35/0.72              ( ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.35/0.72             => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.35/0.72               => ( Z2
% 0.35/0.72                 != ( complex2 @ ( cos_real @ T6 ) @ ( sin_real @ T6 ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % complex_unimodular_polar
% 0.35/0.72  thf(fact_8147_cot__less__zero,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ zero_zero_real )
% 0.35/0.72         => ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % cot_less_zero
% 0.35/0.72  thf(fact_8148_sum__pos__lt__pair,axiom,
% 0.35/0.72      ! [F: nat > real,K: nat] :
% 0.35/0.72        ( ( summable_real @ F )
% 0.35/0.72       => ( ! [D5: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D5 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D5 ) @ one_one_nat ) ) ) ) )
% 0.35/0.72         => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sum_pos_lt_pair
% 0.35/0.72  thf(fact_8149_arctan__def,axiom,
% 0.35/0.72      ( arctan
% 0.35/0.72      = ( ^ [Y2: real] :
% 0.35/0.72            ( the_real
% 0.35/0.72            @ ^ [X3: real] :
% 0.35/0.72                ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.35/0.72                & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72                & ( ( tan_real @ X3 )
% 0.35/0.72                  = Y2 ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % arctan_def
% 0.35/0.72  thf(fact_8150_cot__gt__zero,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72         => ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % cot_gt_zero
% 0.35/0.72  thf(fact_8151_cot__zero,axiom,
% 0.35/0.72      ( ( cot_complex @ zero_zero_complex )
% 0.35/0.72      = zero_zero_complex ) ).
% 0.35/0.72  
% 0.35/0.72  % cot_zero
% 0.35/0.72  thf(fact_8152_cot__zero,axiom,
% 0.35/0.72      ( ( cot_real @ zero_zero_real )
% 0.35/0.72      = zero_zero_real ) ).
% 0.35/0.72  
% 0.35/0.72  % cot_zero
% 0.35/0.72  thf(fact_8153_summable__single,axiom,
% 0.35/0.72      ! [I: nat,F: nat > complex] :
% 0.35/0.72        ( summable_complex
% 0.35/0.72        @ ^ [R5: nat] : ( if_complex @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_complex ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_single
% 0.35/0.72  thf(fact_8154_summable__single,axiom,
% 0.35/0.72      ! [I: nat,F: nat > real] :
% 0.35/0.72        ( summable_real
% 0.35/0.72        @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_single
% 0.35/0.72  thf(fact_8155_summable__single,axiom,
% 0.35/0.72      ! [I: nat,F: nat > nat] :
% 0.35/0.72        ( summable_nat
% 0.35/0.72        @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_single
% 0.35/0.72  thf(fact_8156_summable__single,axiom,
% 0.35/0.72      ! [I: nat,F: nat > int] :
% 0.35/0.72        ( summable_int
% 0.35/0.72        @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_single
% 0.35/0.72  thf(fact_8157_summable__zero,axiom,
% 0.35/0.72      ( summable_complex
% 0.35/0.72      @ ^ [N2: nat] : zero_zero_complex ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_zero
% 0.35/0.72  thf(fact_8158_summable__zero,axiom,
% 0.35/0.72      ( summable_real
% 0.35/0.72      @ ^ [N2: nat] : zero_zero_real ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_zero
% 0.35/0.72  thf(fact_8159_summable__zero,axiom,
% 0.35/0.72      ( summable_nat
% 0.35/0.72      @ ^ [N2: nat] : zero_zero_nat ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_zero
% 0.35/0.72  thf(fact_8160_summable__zero,axiom,
% 0.35/0.72      ( summable_int
% 0.35/0.72      @ ^ [N2: nat] : zero_zero_int ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_zero
% 0.35/0.72  thf(fact_8161_summable__cmult__iff,axiom,
% 0.35/0.72      ! [C: complex,F: nat > complex] :
% 0.35/0.72        ( ( summable_complex
% 0.35/0.72          @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
% 0.35/0.72        = ( ( C = zero_zero_complex )
% 0.35/0.72          | ( summable_complex @ F ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_cmult_iff
% 0.35/0.72  thf(fact_8162_summable__cmult__iff,axiom,
% 0.35/0.72      ! [C: real,F: nat > real] :
% 0.35/0.72        ( ( summable_real
% 0.35/0.72          @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 0.35/0.72        = ( ( C = zero_zero_real )
% 0.35/0.72          | ( summable_real @ F ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_cmult_iff
% 0.35/0.72  thf(fact_8163_summable__divide__iff,axiom,
% 0.35/0.72      ! [F: nat > complex,C: complex] :
% 0.35/0.72        ( ( summable_complex
% 0.35/0.72          @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
% 0.35/0.72        = ( ( C = zero_zero_complex )
% 0.35/0.72          | ( summable_complex @ F ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_divide_iff
% 0.35/0.72  thf(fact_8164_summable__divide__iff,axiom,
% 0.35/0.72      ! [F: nat > real,C: real] :
% 0.35/0.72        ( ( summable_real
% 0.35/0.72          @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
% 0.35/0.72        = ( ( C = zero_zero_real )
% 0.35/0.72          | ( summable_real @ F ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_divide_iff
% 0.35/0.72  thf(fact_8165_summable__If__finite__set,axiom,
% 0.35/0.72      ! [A2: set_nat,F: nat > complex] :
% 0.35/0.72        ( ( finite_finite_nat @ A2 )
% 0.35/0.72       => ( summable_complex
% 0.35/0.72          @ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_If_finite_set
% 0.35/0.72  thf(fact_8166_summable__If__finite__set,axiom,
% 0.35/0.72      ! [A2: set_nat,F: nat > real] :
% 0.35/0.72        ( ( finite_finite_nat @ A2 )
% 0.35/0.72       => ( summable_real
% 0.35/0.72          @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_If_finite_set
% 0.35/0.72  thf(fact_8167_summable__If__finite__set,axiom,
% 0.35/0.72      ! [A2: set_nat,F: nat > nat] :
% 0.35/0.72        ( ( finite_finite_nat @ A2 )
% 0.35/0.72       => ( summable_nat
% 0.35/0.72          @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_If_finite_set
% 0.35/0.72  thf(fact_8168_summable__If__finite__set,axiom,
% 0.35/0.72      ! [A2: set_nat,F: nat > int] :
% 0.35/0.72        ( ( finite_finite_nat @ A2 )
% 0.35/0.72       => ( summable_int
% 0.35/0.72          @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_If_finite_set
% 0.35/0.72  thf(fact_8169_summable__If__finite,axiom,
% 0.35/0.72      ! [P: nat > $o,F: nat > complex] :
% 0.35/0.72        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.35/0.72       => ( summable_complex
% 0.35/0.72          @ ^ [R5: nat] : ( if_complex @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_If_finite
% 0.35/0.72  thf(fact_8170_summable__If__finite,axiom,
% 0.35/0.72      ! [P: nat > $o,F: nat > real] :
% 0.35/0.72        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.35/0.72       => ( summable_real
% 0.35/0.72          @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_If_finite
% 0.35/0.72  thf(fact_8171_summable__If__finite,axiom,
% 0.35/0.72      ! [P: nat > $o,F: nat > nat] :
% 0.35/0.72        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.35/0.72       => ( summable_nat
% 0.35/0.72          @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_If_finite
% 0.35/0.72  thf(fact_8172_summable__If__finite,axiom,
% 0.35/0.72      ! [P: nat > $o,F: nat > int] :
% 0.35/0.72        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.35/0.72       => ( summable_int
% 0.35/0.72          @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_If_finite
% 0.35/0.72  thf(fact_8173_summable__geometric__iff,axiom,
% 0.35/0.72      ! [C: real] :
% 0.35/0.72        ( ( summable_real @ ( power_power_real @ C ) )
% 0.35/0.72        = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_geometric_iff
% 0.35/0.72  thf(fact_8174_summable__geometric__iff,axiom,
% 0.35/0.72      ! [C: complex] :
% 0.35/0.72        ( ( summable_complex @ ( power_power_complex @ C ) )
% 0.35/0.72        = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_geometric_iff
% 0.35/0.72  thf(fact_8175_summable__comparison__test_H,axiom,
% 0.35/0.72      ! [G: nat > real,N6: nat,F: nat > real] :
% 0.35/0.72        ( ( summable_real @ G )
% 0.35/0.72       => ( ! [N3: nat] :
% 0.35/0.72              ( ( ord_less_eq_nat @ N6 @ N3 )
% 0.35/0.72             => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 0.35/0.72         => ( summable_real @ F ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_comparison_test'
% 0.35/0.72  thf(fact_8176_summable__comparison__test_H,axiom,
% 0.35/0.72      ! [G: nat > real,N6: nat,F: nat > complex] :
% 0.35/0.72        ( ( summable_real @ G )
% 0.35/0.72       => ( ! [N3: nat] :
% 0.35/0.72              ( ( ord_less_eq_nat @ N6 @ N3 )
% 0.35/0.72             => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 0.35/0.72         => ( summable_complex @ F ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_comparison_test'
% 0.35/0.72  thf(fact_8177_summable__comparison__test,axiom,
% 0.35/0.72      ! [F: nat > real,G: nat > real] :
% 0.35/0.72        ( ? [N8: nat] :
% 0.35/0.72          ! [N3: nat] :
% 0.35/0.72            ( ( ord_less_eq_nat @ N8 @ N3 )
% 0.35/0.72           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 0.35/0.72       => ( ( summable_real @ G )
% 0.35/0.72         => ( summable_real @ F ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_comparison_test
% 0.35/0.72  thf(fact_8178_summable__comparison__test,axiom,
% 0.35/0.72      ! [F: nat > complex,G: nat > real] :
% 0.35/0.72        ( ? [N8: nat] :
% 0.35/0.72          ! [N3: nat] :
% 0.35/0.72            ( ( ord_less_eq_nat @ N8 @ N3 )
% 0.35/0.72           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 0.35/0.72       => ( ( summable_real @ G )
% 0.35/0.72         => ( summable_complex @ F ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_comparison_test
% 0.35/0.72  thf(fact_8179_summable__const__iff,axiom,
% 0.35/0.72      ! [C: complex] :
% 0.35/0.72        ( ( summable_complex
% 0.35/0.72          @ ^ [Uu3: nat] : C )
% 0.35/0.72        = ( C = zero_zero_complex ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_const_iff
% 0.35/0.72  thf(fact_8180_summable__const__iff,axiom,
% 0.35/0.72      ! [C: real] :
% 0.35/0.72        ( ( summable_real
% 0.35/0.72          @ ^ [Uu3: nat] : C )
% 0.35/0.72        = ( C = zero_zero_real ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_const_iff
% 0.35/0.72  thf(fact_8181_powser__insidea,axiom,
% 0.35/0.72      ! [F: nat > real,X: real,Z2: real] :
% 0.35/0.72        ( ( summable_real
% 0.35/0.72          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ N2 ) ) )
% 0.35/0.72       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( real_V7735802525324610683m_real @ X ) )
% 0.35/0.72         => ( summable_real
% 0.35/0.72            @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % powser_insidea
% 0.35/0.72  thf(fact_8182_powser__insidea,axiom,
% 0.35/0.72      ! [F: nat > complex,X: complex,Z2: complex] :
% 0.35/0.72        ( ( summable_complex
% 0.35/0.72          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X @ N2 ) ) )
% 0.35/0.72       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( real_V1022390504157884413omplex @ X ) )
% 0.35/0.72         => ( summable_real
% 0.35/0.72            @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % powser_insidea
% 0.35/0.72  thf(fact_8183_suminf__le,axiom,
% 0.35/0.72      ! [F: nat > real,G: nat > real] :
% 0.35/0.72        ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.35/0.72       => ( ( summable_real @ F )
% 0.35/0.72         => ( ( summable_real @ G )
% 0.35/0.72           => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_le
% 0.35/0.72  thf(fact_8184_suminf__le,axiom,
% 0.35/0.72      ! [F: nat > nat,G: nat > nat] :
% 0.35/0.72        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.35/0.72       => ( ( summable_nat @ F )
% 0.35/0.72         => ( ( summable_nat @ G )
% 0.35/0.72           => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_le
% 0.35/0.72  thf(fact_8185_suminf__le,axiom,
% 0.35/0.72      ! [F: nat > int,G: nat > int] :
% 0.35/0.72        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.35/0.72       => ( ( summable_int @ F )
% 0.35/0.72         => ( ( summable_int @ G )
% 0.35/0.72           => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_le
% 0.35/0.72  thf(fact_8186_summable__finite,axiom,
% 0.35/0.72      ! [N6: set_nat,F: nat > complex] :
% 0.35/0.72        ( ( finite_finite_nat @ N6 )
% 0.35/0.72       => ( ! [N3: nat] :
% 0.35/0.72              ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.72             => ( ( F @ N3 )
% 0.35/0.72                = zero_zero_complex ) )
% 0.35/0.72         => ( summable_complex @ F ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_finite
% 0.35/0.72  thf(fact_8187_summable__finite,axiom,
% 0.35/0.72      ! [N6: set_nat,F: nat > real] :
% 0.35/0.72        ( ( finite_finite_nat @ N6 )
% 0.35/0.72       => ( ! [N3: nat] :
% 0.35/0.72              ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.72             => ( ( F @ N3 )
% 0.35/0.72                = zero_zero_real ) )
% 0.35/0.72         => ( summable_real @ F ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_finite
% 0.35/0.72  thf(fact_8188_summable__finite,axiom,
% 0.35/0.72      ! [N6: set_nat,F: nat > nat] :
% 0.35/0.72        ( ( finite_finite_nat @ N6 )
% 0.35/0.72       => ( ! [N3: nat] :
% 0.35/0.72              ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.72             => ( ( F @ N3 )
% 0.35/0.72                = zero_zero_nat ) )
% 0.35/0.72         => ( summable_nat @ F ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_finite
% 0.35/0.72  thf(fact_8189_summable__finite,axiom,
% 0.35/0.72      ! [N6: set_nat,F: nat > int] :
% 0.35/0.72        ( ( finite_finite_nat @ N6 )
% 0.35/0.72       => ( ! [N3: nat] :
% 0.35/0.72              ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.72             => ( ( F @ N3 )
% 0.35/0.72                = zero_zero_int ) )
% 0.35/0.72         => ( summable_int @ F ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_finite
% 0.35/0.72  thf(fact_8190_summable__mult__D,axiom,
% 0.35/0.72      ! [C: complex,F: nat > complex] :
% 0.35/0.72        ( ( summable_complex
% 0.35/0.72          @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
% 0.35/0.72       => ( ( C != zero_zero_complex )
% 0.35/0.72         => ( summable_complex @ F ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_mult_D
% 0.35/0.72  thf(fact_8191_summable__mult__D,axiom,
% 0.35/0.72      ! [C: real,F: nat > real] :
% 0.35/0.72        ( ( summable_real
% 0.35/0.72          @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 0.35/0.72       => ( ( C != zero_zero_real )
% 0.35/0.72         => ( summable_real @ F ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_mult_D
% 0.35/0.72  thf(fact_8192_summable__zero__power,axiom,
% 0.35/0.72      summable_real @ ( power_power_real @ zero_zero_real ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_zero_power
% 0.35/0.72  thf(fact_8193_summable__zero__power,axiom,
% 0.35/0.72      summable_int @ ( power_power_int @ zero_zero_int ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_zero_power
% 0.35/0.72  thf(fact_8194_summable__zero__power,axiom,
% 0.35/0.72      summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_zero_power
% 0.35/0.72  thf(fact_8195_suminf__eq__zero__iff,axiom,
% 0.35/0.72      ! [F: nat > real] :
% 0.35/0.72        ( ( summable_real @ F )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 0.35/0.72         => ( ( ( suminf_real @ F )
% 0.35/0.72              = zero_zero_real )
% 0.35/0.72            = ( ! [N2: nat] :
% 0.35/0.72                  ( ( F @ N2 )
% 0.35/0.72                  = zero_zero_real ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_eq_zero_iff
% 0.35/0.72  thf(fact_8196_suminf__eq__zero__iff,axiom,
% 0.35/0.72      ! [F: nat > nat] :
% 0.35/0.72        ( ( summable_nat @ F )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 0.35/0.72         => ( ( ( suminf_nat @ F )
% 0.35/0.72              = zero_zero_nat )
% 0.35/0.72            = ( ! [N2: nat] :
% 0.35/0.72                  ( ( F @ N2 )
% 0.35/0.72                  = zero_zero_nat ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_eq_zero_iff
% 0.35/0.72  thf(fact_8197_suminf__eq__zero__iff,axiom,
% 0.35/0.72      ! [F: nat > int] :
% 0.35/0.72        ( ( summable_int @ F )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 0.35/0.72         => ( ( ( suminf_int @ F )
% 0.35/0.72              = zero_zero_int )
% 0.35/0.72            = ( ! [N2: nat] :
% 0.35/0.72                  ( ( F @ N2 )
% 0.35/0.72                  = zero_zero_int ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_eq_zero_iff
% 0.35/0.72  thf(fact_8198_suminf__nonneg,axiom,
% 0.35/0.72      ! [F: nat > real] :
% 0.35/0.72        ( ( summable_real @ F )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 0.35/0.72         => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_nonneg
% 0.35/0.72  thf(fact_8199_suminf__nonneg,axiom,
% 0.35/0.72      ! [F: nat > nat] :
% 0.35/0.72        ( ( summable_nat @ F )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 0.35/0.72         => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_nonneg
% 0.35/0.72  thf(fact_8200_suminf__nonneg,axiom,
% 0.35/0.72      ! [F: nat > int] :
% 0.35/0.72        ( ( summable_int @ F )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 0.35/0.72         => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_nonneg
% 0.35/0.72  thf(fact_8201_suminf__pos,axiom,
% 0.35/0.72      ! [F: nat > real] :
% 0.35/0.72        ( ( summable_real @ F )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N3 ) )
% 0.35/0.72         => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_pos
% 0.35/0.72  thf(fact_8202_suminf__pos,axiom,
% 0.35/0.72      ! [F: nat > nat] :
% 0.35/0.72        ( ( summable_nat @ F )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N3 ) )
% 0.35/0.72         => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_pos
% 0.35/0.72  thf(fact_8203_suminf__pos,axiom,
% 0.35/0.72      ! [F: nat > int] :
% 0.35/0.72        ( ( summable_int @ F )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N3 ) )
% 0.35/0.72         => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_pos
% 0.35/0.72  thf(fact_8204_summable__0__powser,axiom,
% 0.35/0.72      ! [F: nat > complex] :
% 0.35/0.72        ( summable_complex
% 0.35/0.72        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_0_powser
% 0.35/0.72  thf(fact_8205_summable__0__powser,axiom,
% 0.35/0.72      ! [F: nat > real] :
% 0.35/0.72        ( summable_real
% 0.35/0.72        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_0_powser
% 0.35/0.72  thf(fact_8206_summable__zero__power_H,axiom,
% 0.35/0.72      ! [F: nat > complex] :
% 0.35/0.72        ( summable_complex
% 0.35/0.72        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_zero_power'
% 0.35/0.72  thf(fact_8207_summable__zero__power_H,axiom,
% 0.35/0.72      ! [F: nat > real] :
% 0.35/0.72        ( summable_real
% 0.35/0.72        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_zero_power'
% 0.35/0.72  thf(fact_8208_summable__zero__power_H,axiom,
% 0.35/0.72      ! [F: nat > int] :
% 0.35/0.72        ( summable_int
% 0.35/0.72        @ ^ [N2: nat] : ( times_times_int @ ( F @ N2 ) @ ( power_power_int @ zero_zero_int @ N2 ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_zero_power'
% 0.35/0.72  thf(fact_8209_summable__norm__comparison__test,axiom,
% 0.35/0.72      ! [F: nat > complex,G: nat > real] :
% 0.35/0.72        ( ? [N8: nat] :
% 0.35/0.72          ! [N3: nat] :
% 0.35/0.72            ( ( ord_less_eq_nat @ N8 @ N3 )
% 0.35/0.72           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 0.35/0.72       => ( ( summable_real @ G )
% 0.35/0.72         => ( summable_real
% 0.35/0.72            @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_norm_comparison_test
% 0.35/0.72  thf(fact_8210_summable__rabs__comparison__test,axiom,
% 0.35/0.72      ! [F: nat > real,G: nat > real] :
% 0.35/0.72        ( ? [N8: nat] :
% 0.35/0.72          ! [N3: nat] :
% 0.35/0.72            ( ( ord_less_eq_nat @ N8 @ N3 )
% 0.35/0.72           => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 0.35/0.72       => ( ( summable_real @ G )
% 0.35/0.72         => ( summable_real
% 0.35/0.72            @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_rabs_comparison_test
% 0.35/0.72  thf(fact_8211_suminf__pos__iff,axiom,
% 0.35/0.72      ! [F: nat > real] :
% 0.35/0.72        ( ( summable_real @ F )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 0.35/0.72         => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
% 0.35/0.72            = ( ? [I4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_pos_iff
% 0.35/0.72  thf(fact_8212_suminf__pos__iff,axiom,
% 0.35/0.72      ! [F: nat > nat] :
% 0.35/0.72        ( ( summable_nat @ F )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 0.35/0.72         => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
% 0.35/0.72            = ( ? [I4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_pos_iff
% 0.35/0.72  thf(fact_8213_suminf__pos__iff,axiom,
% 0.35/0.72      ! [F: nat > int] :
% 0.35/0.72        ( ( summable_int @ F )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 0.35/0.72         => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
% 0.35/0.72            = ( ? [I4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I4 ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_pos_iff
% 0.35/0.72  thf(fact_8214_suminf__pos2,axiom,
% 0.35/0.72      ! [F: nat > real,I: nat] :
% 0.35/0.72        ( ( summable_real @ F )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 0.35/0.72         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.35/0.72           => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_pos2
% 0.35/0.72  thf(fact_8215_suminf__pos2,axiom,
% 0.35/0.72      ! [F: nat > nat,I: nat] :
% 0.35/0.72        ( ( summable_nat @ F )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 0.35/0.72         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 0.35/0.72           => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_pos2
% 0.35/0.72  thf(fact_8216_suminf__pos2,axiom,
% 0.35/0.72      ! [F: nat > int,I: nat] :
% 0.35/0.72        ( ( summable_int @ F )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 0.35/0.72         => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
% 0.35/0.72           => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_pos2
% 0.35/0.72  thf(fact_8217_suminf__le__const,axiom,
% 0.35/0.72      ! [F: nat > int,X: int] :
% 0.35/0.72        ( ( summable_int @ F )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
% 0.35/0.72         => ( ord_less_eq_int @ ( suminf_int @ F ) @ X ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_le_const
% 0.35/0.72  thf(fact_8218_suminf__le__const,axiom,
% 0.35/0.72      ! [F: nat > nat,X: nat] :
% 0.35/0.72        ( ( summable_nat @ F )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
% 0.35/0.72         => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_le_const
% 0.35/0.72  thf(fact_8219_suminf__le__const,axiom,
% 0.35/0.72      ! [F: nat > real,X: real] :
% 0.35/0.72        ( ( summable_real @ F )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
% 0.35/0.72         => ( ord_less_eq_real @ ( suminf_real @ F ) @ X ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_le_const
% 0.35/0.72  thf(fact_8220_powser__inside,axiom,
% 0.35/0.72      ! [F: nat > real,X: real,Z2: real] :
% 0.35/0.72        ( ( summable_real
% 0.35/0.72          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ N2 ) ) )
% 0.35/0.72       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( real_V7735802525324610683m_real @ X ) )
% 0.35/0.72         => ( summable_real
% 0.35/0.72            @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % powser_inside
% 0.35/0.72  thf(fact_8221_powser__inside,axiom,
% 0.35/0.72      ! [F: nat > complex,X: complex,Z2: complex] :
% 0.35/0.72        ( ( summable_complex
% 0.35/0.72          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X @ N2 ) ) )
% 0.35/0.72       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( real_V1022390504157884413omplex @ X ) )
% 0.35/0.72         => ( summable_complex
% 0.35/0.72            @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % powser_inside
% 0.35/0.72  thf(fact_8222_summableI__nonneg__bounded,axiom,
% 0.35/0.72      ! [F: nat > int,X: int] :
% 0.35/0.72        ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
% 0.35/0.72         => ( summable_int @ F ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summableI_nonneg_bounded
% 0.35/0.72  thf(fact_8223_summableI__nonneg__bounded,axiom,
% 0.35/0.72      ! [F: nat > nat,X: nat] :
% 0.35/0.72        ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
% 0.35/0.72         => ( summable_nat @ F ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summableI_nonneg_bounded
% 0.35/0.72  thf(fact_8224_summableI__nonneg__bounded,axiom,
% 0.35/0.72      ! [F: nat > real,X: real] :
% 0.35/0.72        ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
% 0.35/0.72         => ( summable_real @ F ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summableI_nonneg_bounded
% 0.35/0.72  thf(fact_8225_bounded__imp__summable,axiom,
% 0.35/0.72      ! [A: nat > int,B2: int] :
% 0.35/0.72        ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( A @ N3 ) )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B2 )
% 0.35/0.72         => ( summable_int @ A ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % bounded_imp_summable
% 0.35/0.72  thf(fact_8226_bounded__imp__summable,axiom,
% 0.35/0.72      ! [A: nat > nat,B2: nat] :
% 0.35/0.72        ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( A @ N3 ) )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B2 )
% 0.35/0.72         => ( summable_nat @ A ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % bounded_imp_summable
% 0.35/0.72  thf(fact_8227_bounded__imp__summable,axiom,
% 0.35/0.72      ! [A: nat > real,B2: real] :
% 0.35/0.72        ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 0.35/0.72       => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B2 )
% 0.35/0.72         => ( summable_real @ A ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % bounded_imp_summable
% 0.35/0.72  thf(fact_8228_summable__geometric,axiom,
% 0.35/0.72      ! [C: real] :
% 0.35/0.72        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 0.35/0.72       => ( summable_real @ ( power_power_real @ C ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_geometric
% 0.35/0.72  thf(fact_8229_summable__geometric,axiom,
% 0.35/0.72      ! [C: complex] :
% 0.35/0.72        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 0.35/0.72       => ( summable_complex @ ( power_power_complex @ C ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_geometric
% 0.35/0.72  thf(fact_8230_complete__algebra__summable__geometric,axiom,
% 0.35/0.72      ! [X: real] :
% 0.35/0.72        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ one_one_real )
% 0.35/0.72       => ( summable_real @ ( power_power_real @ X ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % complete_algebra_summable_geometric
% 0.35/0.72  thf(fact_8231_complete__algebra__summable__geometric,axiom,
% 0.35/0.72      ! [X: complex] :
% 0.35/0.72        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ one_one_real )
% 0.35/0.72       => ( summable_complex @ ( power_power_complex @ X ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % complete_algebra_summable_geometric
% 0.35/0.72  thf(fact_8232_suminf__split__head,axiom,
% 0.35/0.72      ! [F: nat > real] :
% 0.35/0.72        ( ( summable_real @ F )
% 0.35/0.72       => ( ( suminf_real
% 0.35/0.72            @ ^ [N2: nat] : ( F @ ( suc @ N2 ) ) )
% 0.35/0.72          = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_split_head
% 0.35/0.72  thf(fact_8233_the__elem__def,axiom,
% 0.35/0.72      ( the_el8326832613380209454nt_int
% 0.35/0.72      = ( ^ [X6: set_Pr958786334691620121nt_int] :
% 0.35/0.72            ( the_Pr4378521158711661632nt_int
% 0.35/0.72            @ ^ [X3: product_prod_int_int] :
% 0.35/0.72                ( X6
% 0.35/0.72                = ( insert5033312907999012233nt_int @ X3 @ bot_bo1796632182523588997nt_int ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % the_elem_def
% 0.35/0.72  thf(fact_8234_the__elem__def,axiom,
% 0.35/0.72      ( the_el221668144340439132at_nat
% 0.35/0.72      = ( ^ [X6: set_Pr4329608150637261639at_nat] :
% 0.35/0.72            ( the_Pr4604535484834969198at_nat
% 0.35/0.72            @ ^ [X3: produc3843707927480180839at_nat] :
% 0.35/0.72                ( X6
% 0.35/0.72                = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % the_elem_def
% 0.35/0.72  thf(fact_8235_the__elem__def,axiom,
% 0.35/0.72      ( the_el2281957884133575798at_nat
% 0.35/0.72      = ( ^ [X6: set_Pr1261947904930325089at_nat] :
% 0.35/0.72            ( the_Pr7557018466319803784at_nat
% 0.35/0.72            @ ^ [X3: product_prod_nat_nat] :
% 0.35/0.72                ( X6
% 0.35/0.72                = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % the_elem_def
% 0.35/0.72  thf(fact_8236_the__elem__def,axiom,
% 0.35/0.72      ( the_elem_real
% 0.35/0.72      = ( ^ [X6: set_real] :
% 0.35/0.72            ( the_real
% 0.35/0.72            @ ^ [X3: real] :
% 0.35/0.72                ( X6
% 0.35/0.72                = ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % the_elem_def
% 0.35/0.72  thf(fact_8237_the__elem__def,axiom,
% 0.35/0.72      ( the_elem_nat
% 0.35/0.72      = ( ^ [X6: set_nat] :
% 0.35/0.72            ( the_nat
% 0.35/0.72            @ ^ [X3: nat] :
% 0.35/0.72                ( X6
% 0.35/0.72                = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % the_elem_def
% 0.35/0.72  thf(fact_8238_the__elem__def,axiom,
% 0.35/0.72      ( the_elem_int
% 0.35/0.72      = ( ^ [X6: set_int] :
% 0.35/0.72            ( the_int
% 0.35/0.72            @ ^ [X3: int] :
% 0.35/0.72                ( X6
% 0.35/0.72                = ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % the_elem_def
% 0.35/0.72  thf(fact_8239_sum__le__suminf,axiom,
% 0.35/0.72      ! [F: nat > int,I6: set_nat] :
% 0.35/0.72        ( ( summable_int @ F )
% 0.35/0.72       => ( ( finite_finite_nat @ I6 )
% 0.35/0.72         => ( ! [N3: nat] :
% 0.35/0.72                ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I6 ) )
% 0.35/0.72               => ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) ) )
% 0.35/0.72           => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I6 ) @ ( suminf_int @ F ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sum_le_suminf
% 0.35/0.72  thf(fact_8240_sum__le__suminf,axiom,
% 0.35/0.72      ! [F: nat > nat,I6: set_nat] :
% 0.35/0.72        ( ( summable_nat @ F )
% 0.35/0.72       => ( ( finite_finite_nat @ I6 )
% 0.35/0.72         => ( ! [N3: nat] :
% 0.35/0.72                ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I6 ) )
% 0.35/0.72               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) ) )
% 0.35/0.72           => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I6 ) @ ( suminf_nat @ F ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sum_le_suminf
% 0.35/0.72  thf(fact_8241_sum__le__suminf,axiom,
% 0.35/0.72      ! [F: nat > real,I6: set_nat] :
% 0.35/0.72        ( ( summable_real @ F )
% 0.35/0.72       => ( ( finite_finite_nat @ I6 )
% 0.35/0.72         => ( ! [N3: nat] :
% 0.35/0.72                ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I6 ) )
% 0.35/0.72               => ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) ) )
% 0.35/0.72           => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I6 ) @ ( suminf_real @ F ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sum_le_suminf
% 0.35/0.72  thf(fact_8242_sum__less__suminf,axiom,
% 0.35/0.72      ! [F: nat > int,N: nat] :
% 0.35/0.72        ( ( summable_int @ F )
% 0.35/0.72       => ( ! [M5: nat] :
% 0.35/0.72              ( ( ord_less_eq_nat @ N @ M5 )
% 0.35/0.72             => ( ord_less_int @ zero_zero_int @ ( F @ M5 ) ) )
% 0.35/0.72         => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sum_less_suminf
% 0.35/0.72  thf(fact_8243_sum__less__suminf,axiom,
% 0.35/0.72      ! [F: nat > nat,N: nat] :
% 0.35/0.72        ( ( summable_nat @ F )
% 0.35/0.72       => ( ! [M5: nat] :
% 0.35/0.72              ( ( ord_less_eq_nat @ N @ M5 )
% 0.35/0.72             => ( ord_less_nat @ zero_zero_nat @ ( F @ M5 ) ) )
% 0.35/0.72         => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sum_less_suminf
% 0.35/0.72  thf(fact_8244_sum__less__suminf,axiom,
% 0.35/0.72      ! [F: nat > real,N: nat] :
% 0.35/0.72        ( ( summable_real @ F )
% 0.35/0.72       => ( ! [M5: nat] :
% 0.35/0.72              ( ( ord_less_eq_nat @ N @ M5 )
% 0.35/0.72             => ( ord_less_real @ zero_zero_real @ ( F @ M5 ) ) )
% 0.35/0.72         => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sum_less_suminf
% 0.35/0.72  thf(fact_8245_powser__split__head_I1_J,axiom,
% 0.35/0.72      ! [F: nat > complex,Z2: complex] :
% 0.35/0.72        ( ( summable_complex
% 0.35/0.72          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) )
% 0.35/0.72       => ( ( suminf_complex
% 0.35/0.72            @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) )
% 0.35/0.72          = ( plus_plus_complex @ ( F @ zero_zero_nat )
% 0.35/0.72            @ ( times_times_complex
% 0.35/0.72              @ ( suminf_complex
% 0.35/0.72                @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z2 @ N2 ) ) )
% 0.35/0.72              @ Z2 ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % powser_split_head(1)
% 0.35/0.72  thf(fact_8246_powser__split__head_I1_J,axiom,
% 0.35/0.72      ! [F: nat > real,Z2: real] :
% 0.35/0.72        ( ( summable_real
% 0.35/0.72          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) )
% 0.35/0.72       => ( ( suminf_real
% 0.35/0.72            @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) )
% 0.35/0.72          = ( plus_plus_real @ ( F @ zero_zero_nat )
% 0.35/0.72            @ ( times_times_real
% 0.35/0.72              @ ( suminf_real
% 0.35/0.72                @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z2 @ N2 ) ) )
% 0.35/0.72              @ Z2 ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % powser_split_head(1)
% 0.35/0.72  thf(fact_8247_powser__split__head_I2_J,axiom,
% 0.35/0.72      ! [F: nat > complex,Z2: complex] :
% 0.35/0.72        ( ( summable_complex
% 0.35/0.72          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) )
% 0.35/0.72       => ( ( times_times_complex
% 0.35/0.72            @ ( suminf_complex
% 0.35/0.72              @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z2 @ N2 ) ) )
% 0.35/0.72            @ Z2 )
% 0.35/0.72          = ( minus_minus_complex
% 0.35/0.72            @ ( suminf_complex
% 0.35/0.72              @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) )
% 0.35/0.72            @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % powser_split_head(2)
% 0.35/0.72  thf(fact_8248_powser__split__head_I2_J,axiom,
% 0.35/0.72      ! [F: nat > real,Z2: real] :
% 0.35/0.72        ( ( summable_real
% 0.35/0.72          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) )
% 0.35/0.72       => ( ( times_times_real
% 0.35/0.72            @ ( suminf_real
% 0.35/0.72              @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z2 @ N2 ) ) )
% 0.35/0.72            @ Z2 )
% 0.35/0.72          = ( minus_minus_real
% 0.35/0.72            @ ( suminf_real
% 0.35/0.72              @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) )
% 0.35/0.72            @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % powser_split_head(2)
% 0.35/0.72  thf(fact_8249_summable__partial__sum__bound,axiom,
% 0.35/0.72      ! [F: nat > complex,E2: real] :
% 0.35/0.72        ( ( summable_complex @ F )
% 0.35/0.72       => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.35/0.72         => ~ ! [N9: nat] :
% 0.35/0.72                ~ ! [M3: nat] :
% 0.35/0.72                    ( ( ord_less_eq_nat @ N9 @ M3 )
% 0.35/0.72                   => ! [N5: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M3 @ N5 ) ) ) @ E2 ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_partial_sum_bound
% 0.35/0.72  thf(fact_8250_summable__partial__sum__bound,axiom,
% 0.35/0.72      ! [F: nat > real,E2: real] :
% 0.35/0.72        ( ( summable_real @ F )
% 0.35/0.72       => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.35/0.72         => ~ ! [N9: nat] :
% 0.35/0.72                ~ ! [M3: nat] :
% 0.35/0.72                    ( ( ord_less_eq_nat @ N9 @ M3 )
% 0.35/0.72                   => ! [N5: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M3 @ N5 ) ) ) @ E2 ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_partial_sum_bound
% 0.35/0.72  thf(fact_8251_suminf__exist__split,axiom,
% 0.35/0.72      ! [R2: real,F: nat > real] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.35/0.72       => ( ( summable_real @ F )
% 0.35/0.72         => ? [N9: nat] :
% 0.35/0.72            ! [N5: nat] :
% 0.35/0.72              ( ( ord_less_eq_nat @ N9 @ N5 )
% 0.35/0.72             => ( ord_less_real
% 0.35/0.72                @ ( real_V7735802525324610683m_real
% 0.35/0.72                  @ ( suminf_real
% 0.35/0.72                    @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N5 ) ) ) )
% 0.35/0.72                @ R2 ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_exist_split
% 0.35/0.72  thf(fact_8252_suminf__exist__split,axiom,
% 0.35/0.72      ! [R2: real,F: nat > complex] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.35/0.72       => ( ( summable_complex @ F )
% 0.35/0.72         => ? [N9: nat] :
% 0.35/0.72            ! [N5: nat] :
% 0.35/0.72              ( ( ord_less_eq_nat @ N9 @ N5 )
% 0.35/0.72             => ( ord_less_real
% 0.35/0.72                @ ( real_V1022390504157884413omplex
% 0.35/0.72                  @ ( suminf_complex
% 0.35/0.72                    @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N5 ) ) ) )
% 0.35/0.72                @ R2 ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % suminf_exist_split
% 0.35/0.72  thf(fact_8253_summable__power__series,axiom,
% 0.35/0.72      ! [F: nat > real,Z2: real] :
% 0.35/0.72        ( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
% 0.35/0.72       => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 0.35/0.72         => ( ( ord_less_eq_real @ zero_zero_real @ Z2 )
% 0.35/0.72           => ( ( ord_less_real @ Z2 @ one_one_real )
% 0.35/0.72             => ( summable_real
% 0.35/0.72                @ ^ [I4: nat] : ( times_times_real @ ( F @ I4 ) @ ( power_power_real @ Z2 @ I4 ) ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_power_series
% 0.35/0.72  thf(fact_8254_Abel__lemma,axiom,
% 0.35/0.72      ! [R2: real,R0: real,A: nat > complex,M4: real] :
% 0.35/0.72        ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 0.35/0.72       => ( ( ord_less_real @ R2 @ R0 )
% 0.35/0.72         => ( ! [N3: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N3 ) ) @ ( power_power_real @ R0 @ N3 ) ) @ M4 )
% 0.35/0.72           => ( summable_real
% 0.35/0.72              @ ^ [N2: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N2 ) ) @ ( power_power_real @ R2 @ N2 ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % Abel_lemma
% 0.35/0.72  thf(fact_8255_summable__ratio__test,axiom,
% 0.35/0.72      ! [C: real,N6: nat,F: nat > real] :
% 0.35/0.72        ( ( ord_less_real @ C @ one_one_real )
% 0.35/0.72       => ( ! [N3: nat] :
% 0.35/0.72              ( ( ord_less_eq_nat @ N6 @ N3 )
% 0.35/0.72             => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) ) ) )
% 0.35/0.72         => ( summable_real @ F ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_ratio_test
% 0.35/0.72  thf(fact_8256_summable__ratio__test,axiom,
% 0.35/0.72      ! [C: real,N6: nat,F: nat > complex] :
% 0.35/0.72        ( ( ord_less_real @ C @ one_one_real )
% 0.35/0.72       => ( ! [N3: nat] :
% 0.35/0.72              ( ( ord_less_eq_nat @ N6 @ N3 )
% 0.35/0.72             => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) )
% 0.35/0.72         => ( summable_complex @ F ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % summable_ratio_test
% 0.35/0.72  thf(fact_8257_sum__less__suminf2,axiom,
% 0.35/0.72      ! [F: nat > int,N: nat,I: nat] :
% 0.35/0.72        ( ( summable_int @ F )
% 0.35/0.72       => ( ! [M5: nat] :
% 0.35/0.72              ( ( ord_less_eq_nat @ N @ M5 )
% 0.35/0.72             => ( ord_less_eq_int @ zero_zero_int @ ( F @ M5 ) ) )
% 0.35/0.72         => ( ( ord_less_eq_nat @ N @ I )
% 0.35/0.72           => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
% 0.35/0.72             => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sum_less_suminf2
% 0.35/0.72  thf(fact_8258_sum__less__suminf2,axiom,
% 0.35/0.72      ! [F: nat > nat,N: nat,I: nat] :
% 0.35/0.72        ( ( summable_nat @ F )
% 0.35/0.72       => ( ! [M5: nat] :
% 0.35/0.72              ( ( ord_less_eq_nat @ N @ M5 )
% 0.35/0.72             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M5 ) ) )
% 0.35/0.72         => ( ( ord_less_eq_nat @ N @ I )
% 0.35/0.72           => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 0.35/0.72             => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sum_less_suminf2
% 0.35/0.72  thf(fact_8259_sum__less__suminf2,axiom,
% 0.35/0.72      ! [F: nat > real,N: nat,I: nat] :
% 0.35/0.72        ( ( summable_real @ F )
% 0.35/0.72       => ( ! [M5: nat] :
% 0.35/0.72              ( ( ord_less_eq_nat @ N @ M5 )
% 0.35/0.72             => ( ord_less_eq_real @ zero_zero_real @ ( F @ M5 ) ) )
% 0.35/0.72         => ( ( ord_less_eq_nat @ N @ I )
% 0.35/0.72           => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.35/0.72             => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % sum_less_suminf2
% 0.35/0.72  thf(fact_8260_diffs__equiv,axiom,
% 0.35/0.72      ! [C: nat > complex,X: complex] :
% 0.35/0.72        ( ( summable_complex
% 0.35/0.72          @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) )
% 0.35/0.72       => ( sums_complex
% 0.35/0.72          @ ^ [N2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( C @ N2 ) ) @ ( power_power_complex @ X @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
% 0.35/0.72          @ ( suminf_complex
% 0.35/0.72            @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % diffs_equiv
% 0.35/0.72  thf(fact_8261_diffs__equiv,axiom,
% 0.35/0.72      ! [C: nat > real,X: real] :
% 0.35/0.72        ( ( summable_real
% 0.35/0.72          @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) )
% 0.35/0.72       => ( sums_real
% 0.35/0.72          @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( C @ N2 ) ) @ ( power_power_real @ X @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
% 0.35/0.72          @ ( suminf_real
% 0.35/0.72            @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % diffs_equiv
% 0.35/0.72  thf(fact_8262_arcsin__lt__bounded,axiom,
% 0.35/0.72      ! [Y: real] :
% 0.35/0.72        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.35/0.72       => ( ( ord_less_real @ Y @ one_one_real )
% 0.35/0.72         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 0.35/0.72            & ( ord_less_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % arcsin_lt_bounded
% 0.35/0.72  thf(fact_8263_pochhammer__code,axiom,
% 0.35/0.72      ( comm_s2602460028002588243omplex
% 0.35/0.72      = ( ^ [A4: complex,N2: nat] :
% 0.35/0.72            ( if_complex @ ( N2 = zero_zero_nat ) @ one_one_complex
% 0.35/0.72            @ ( set_fo1517530859248394432omplex
% 0.35/0.72              @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A4 @ ( semiri8010041392384452111omplex @ O ) ) )
% 0.35/0.72              @ zero_zero_nat
% 0.35/0.72              @ ( minus_minus_nat @ N2 @ one_one_nat )
% 0.35/0.72              @ one_one_complex ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_code
% 0.35/0.72  thf(fact_8264_pochhammer__code,axiom,
% 0.35/0.72      ( comm_s4028243227959126397er_rat
% 0.35/0.72      = ( ^ [A4: rat,N2: nat] :
% 0.35/0.72            ( if_rat @ ( N2 = zero_zero_nat ) @ one_one_rat
% 0.35/0.72            @ ( set_fo1949268297981939178at_rat
% 0.35/0.72              @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A4 @ ( semiri681578069525770553at_rat @ O ) ) )
% 0.35/0.72              @ zero_zero_nat
% 0.35/0.72              @ ( minus_minus_nat @ N2 @ one_one_nat )
% 0.35/0.72              @ one_one_rat ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_code
% 0.35/0.72  thf(fact_8265_pochhammer__code,axiom,
% 0.35/0.72      ( comm_s4660882817536571857er_int
% 0.35/0.72      = ( ^ [A4: int,N2: nat] :
% 0.35/0.72            ( if_int @ ( N2 = zero_zero_nat ) @ one_one_int
% 0.35/0.72            @ ( set_fo2581907887559384638at_int
% 0.35/0.72              @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A4 @ ( semiri1314217659103216013at_int @ O ) ) )
% 0.35/0.72              @ zero_zero_nat
% 0.35/0.72              @ ( minus_minus_nat @ N2 @ one_one_nat )
% 0.35/0.72              @ one_one_int ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_code
% 0.35/0.72  thf(fact_8266_pochhammer__code,axiom,
% 0.35/0.72      ( comm_s7457072308508201937r_real
% 0.35/0.72      = ( ^ [A4: real,N2: nat] :
% 0.35/0.72            ( if_real @ ( N2 = zero_zero_nat ) @ one_one_real
% 0.35/0.72            @ ( set_fo3111899725591712190t_real
% 0.35/0.72              @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A4 @ ( semiri5074537144036343181t_real @ O ) ) )
% 0.35/0.72              @ zero_zero_nat
% 0.35/0.72              @ ( minus_minus_nat @ N2 @ one_one_nat )
% 0.35/0.72              @ one_one_real ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_code
% 0.35/0.72  thf(fact_8267_pochhammer__code,axiom,
% 0.35/0.72      ( comm_s4663373288045622133er_nat
% 0.35/0.72      = ( ^ [A4: nat,N2: nat] :
% 0.35/0.72            ( if_nat @ ( N2 = zero_zero_nat ) @ one_one_nat
% 0.35/0.72            @ ( set_fo2584398358068434914at_nat
% 0.35/0.72              @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A4 @ ( semiri1316708129612266289at_nat @ O ) ) )
% 0.35/0.72              @ zero_zero_nat
% 0.35/0.72              @ ( minus_minus_nat @ N2 @ one_one_nat )
% 0.35/0.72              @ one_one_nat ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_code
% 0.35/0.72  thf(fact_8268_or__nat__unfold,axiom,
% 0.35/0.72      ( bit_se1412395901928357646or_nat
% 0.35/0.72      = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M2 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % or_nat_unfold
% 0.35/0.72  thf(fact_8269_Cauchy__iff2,axiom,
% 0.35/0.72      ( topolo4055970368930404560y_real
% 0.35/0.72      = ( ^ [X6: nat > real] :
% 0.35/0.72          ! [J3: nat] :
% 0.35/0.72          ? [M9: nat] :
% 0.35/0.72          ! [M2: nat] :
% 0.35/0.72            ( ( ord_less_eq_nat @ M9 @ M2 )
% 0.35/0.72           => ! [N2: nat] :
% 0.35/0.72                ( ( ord_less_eq_nat @ M9 @ N2 )
% 0.35/0.72               => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % Cauchy_iff2
% 0.35/0.72  thf(fact_8270_or_Oleft__neutral,axiom,
% 0.35/0.72      ! [A: nat] :
% 0.35/0.72        ( ( bit_se1412395901928357646or_nat @ zero_zero_nat @ A )
% 0.35/0.72        = A ) ).
% 0.35/0.72  
% 0.35/0.72  % or.left_neutral
% 0.35/0.72  thf(fact_8271_or_Oleft__neutral,axiom,
% 0.35/0.72      ! [A: int] :
% 0.35/0.72        ( ( bit_se1409905431419307370or_int @ zero_zero_int @ A )
% 0.35/0.72        = A ) ).
% 0.35/0.72  
% 0.35/0.72  % or.left_neutral
% 0.35/0.72  thf(fact_8272_or_Oright__neutral,axiom,
% 0.35/0.72      ! [A: nat] :
% 0.35/0.72        ( ( bit_se1412395901928357646or_nat @ A @ zero_zero_nat )
% 0.35/0.72        = A ) ).
% 0.35/0.72  
% 0.35/0.72  % or.right_neutral
% 0.35/0.72  thf(fact_8273_or_Oright__neutral,axiom,
% 0.35/0.72      ! [A: int] :
% 0.35/0.72        ( ( bit_se1409905431419307370or_int @ A @ zero_zero_int )
% 0.35/0.72        = A ) ).
% 0.35/0.72  
% 0.35/0.72  % or.right_neutral
% 0.35/0.72  thf(fact_8274_The__split__eq,axiom,
% 0.35/0.72      ! [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
% 0.35/0.72        ( ( the_Pr8210177043389155639nteger
% 0.35/0.72          @ ( produc127349428274296955eger_o
% 0.35/0.72            @ ^ [X9: code_integer > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
% 0.35/0.72                ( ( X = X9 )
% 0.35/0.72                & ( Y = Y6 ) ) ) )
% 0.35/0.72        = ( produc6137756002093451184nteger @ X @ Y ) ) ).
% 0.35/0.72  
% 0.35/0.72  % The_split_eq
% 0.35/0.72  thf(fact_8275_The__split__eq,axiom,
% 0.35/0.72      ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
% 0.35/0.72        ( ( the_Pr6653488032121699663nteger
% 0.35/0.72          @ ( produc6253627499356882019eger_o
% 0.35/0.72            @ ^ [X9: produc6241069584506657477e_term > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
% 0.35/0.72                ( ( X = X9 )
% 0.35/0.72                & ( Y = Y6 ) ) ) )
% 0.35/0.72        = ( produc8603105652947943368nteger @ X @ Y ) ) ).
% 0.35/0.72  
% 0.35/0.72  % The_split_eq
% 0.35/0.72  thf(fact_8276_The__split__eq,axiom,
% 0.35/0.72      ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] :
% 0.35/0.72        ( ( the_Pr5445864913131713084nt_int
% 0.35/0.72          @ ( produc1573362020775583542_int_o
% 0.35/0.72            @ ^ [X9: produc8551481072490612790e_term > option6357759511663192854e_term,Y6: product_prod_int_int] :
% 0.35/0.72                ( ( X = X9 )
% 0.35/0.72                & ( Y = Y6 ) ) ) )
% 0.35/0.72        = ( produc5700946648718959541nt_int @ X @ Y ) ) ).
% 0.35/0.72  
% 0.35/0.72  % The_split_eq
% 0.35/0.72  thf(fact_8277_The__split__eq,axiom,
% 0.35/0.72      ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int] :
% 0.35/0.72        ( ( the_Pr6882841213465913158nt_int
% 0.35/0.72          @ ( produc2558449545302689196_int_o
% 0.35/0.72            @ ^ [X9: int > option6357759511663192854e_term,Y6: product_prod_int_int] :
% 0.35/0.72                ( ( X = X9 )
% 0.35/0.72                & ( Y = Y6 ) ) ) )
% 0.35/0.72        = ( produc4305682042979456191nt_int @ X @ Y ) ) ).
% 0.35/0.72  
% 0.35/0.72  % The_split_eq
% 0.35/0.72  thf(fact_8278_The__split__eq,axiom,
% 0.35/0.72      ! [X: nat,Y: nat] :
% 0.35/0.72        ( ( the_Pr7557018466319803784at_nat
% 0.35/0.72          @ ( produc6081775807080527818_nat_o
% 0.35/0.72            @ ^ [X9: nat,Y6: nat] :
% 0.35/0.72                ( ( X = X9 )
% 0.35/0.72                & ( Y = Y6 ) ) ) )
% 0.35/0.72        = ( product_Pair_nat_nat @ X @ Y ) ) ).
% 0.35/0.72  
% 0.35/0.72  % The_split_eq
% 0.35/0.72  thf(fact_8279_The__split__eq,axiom,
% 0.35/0.72      ! [X: int,Y: int] :
% 0.35/0.72        ( ( the_Pr4378521158711661632nt_int
% 0.35/0.72          @ ( produc4947309494688390418_int_o
% 0.35/0.72            @ ^ [X9: int,Y6: int] :
% 0.35/0.72                ( ( X = X9 )
% 0.35/0.72                & ( Y = Y6 ) ) ) )
% 0.35/0.72        = ( product_Pair_int_int @ X @ Y ) ) ).
% 0.35/0.72  
% 0.35/0.72  % The_split_eq
% 0.35/0.72  thf(fact_8280_pochhammer__0,axiom,
% 0.35/0.72      ! [A: complex] :
% 0.35/0.72        ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
% 0.35/0.72        = one_one_complex ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_0
% 0.35/0.72  thf(fact_8281_pochhammer__0,axiom,
% 0.35/0.72      ! [A: real] :
% 0.35/0.72        ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
% 0.35/0.72        = one_one_real ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_0
% 0.35/0.72  thf(fact_8282_pochhammer__0,axiom,
% 0.35/0.72      ! [A: rat] :
% 0.35/0.72        ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
% 0.35/0.72        = one_one_rat ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_0
% 0.35/0.72  thf(fact_8283_pochhammer__0,axiom,
% 0.35/0.72      ! [A: nat] :
% 0.35/0.72        ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
% 0.35/0.72        = one_one_nat ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_0
% 0.35/0.72  thf(fact_8284_pochhammer__0,axiom,
% 0.35/0.72      ! [A: int] :
% 0.35/0.72        ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
% 0.35/0.72        = one_one_int ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_0
% 0.35/0.72  thf(fact_8285_or__nat__numerals_I4_J,axiom,
% 0.35/0.72      ! [X: num] :
% 0.35/0.72        ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.35/0.72        = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % or_nat_numerals(4)
% 0.35/0.72  thf(fact_8286_or__nat__numerals_I2_J,axiom,
% 0.35/0.72      ! [Y: num] :
% 0.35/0.72        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.35/0.72        = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % or_nat_numerals(2)
% 0.35/0.72  thf(fact_8287_or__nat__numerals_I3_J,axiom,
% 0.35/0.72      ! [X: num] :
% 0.35/0.72        ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.35/0.72        = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % or_nat_numerals(3)
% 0.35/0.72  thf(fact_8288_or__nat__numerals_I1_J,axiom,
% 0.35/0.72      ! [Y: num] :
% 0.35/0.72        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.35/0.72        = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % or_nat_numerals(1)
% 0.35/0.72  thf(fact_8289_or__eq__0__iff,axiom,
% 0.35/0.72      ! [A: nat,B: nat] :
% 0.35/0.72        ( ( ( bit_se1412395901928357646or_nat @ A @ B )
% 0.35/0.72          = zero_zero_nat )
% 0.35/0.72        = ( ( A = zero_zero_nat )
% 0.35/0.72          & ( B = zero_zero_nat ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % or_eq_0_iff
% 0.35/0.72  thf(fact_8290_or__eq__0__iff,axiom,
% 0.35/0.72      ! [A: int,B: int] :
% 0.35/0.72        ( ( ( bit_se1409905431419307370or_int @ A @ B )
% 0.35/0.72          = zero_zero_int )
% 0.35/0.72        = ( ( A = zero_zero_int )
% 0.35/0.72          & ( B = zero_zero_int ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % or_eq_0_iff
% 0.35/0.72  thf(fact_8291_bit_Odisj__zero__right,axiom,
% 0.35/0.72      ! [X: int] :
% 0.35/0.72        ( ( bit_se1409905431419307370or_int @ X @ zero_zero_int )
% 0.35/0.72        = X ) ).
% 0.35/0.72  
% 0.35/0.72  % bit.disj_zero_right
% 0.35/0.72  thf(fact_8292_pochhammer__pos,axiom,
% 0.35/0.72      ! [X: real,N: nat] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_pos
% 0.35/0.72  thf(fact_8293_pochhammer__pos,axiom,
% 0.35/0.72      ! [X: rat,N: nat] :
% 0.35/0.72        ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.35/0.72       => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_pos
% 0.35/0.72  thf(fact_8294_pochhammer__pos,axiom,
% 0.35/0.72      ! [X: nat,N: nat] :
% 0.35/0.72        ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.35/0.72       => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_pos
% 0.35/0.72  thf(fact_8295_pochhammer__pos,axiom,
% 0.35/0.72      ! [X: int,N: nat] :
% 0.35/0.72        ( ( ord_less_int @ zero_zero_int @ X )
% 0.35/0.72       => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_pos
% 0.35/0.72  thf(fact_8296_pochhammer__neq__0__mono,axiom,
% 0.35/0.72      ! [A: complex,M: nat,N: nat] :
% 0.35/0.72        ( ( ( comm_s2602460028002588243omplex @ A @ M )
% 0.35/0.72         != zero_zero_complex )
% 0.35/0.72       => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.72         => ( ( comm_s2602460028002588243omplex @ A @ N )
% 0.35/0.72           != zero_zero_complex ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_neq_0_mono
% 0.35/0.72  thf(fact_8297_pochhammer__neq__0__mono,axiom,
% 0.35/0.72      ! [A: real,M: nat,N: nat] :
% 0.35/0.72        ( ( ( comm_s7457072308508201937r_real @ A @ M )
% 0.35/0.72         != zero_zero_real )
% 0.35/0.72       => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.72         => ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.35/0.72           != zero_zero_real ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_neq_0_mono
% 0.35/0.72  thf(fact_8298_pochhammer__neq__0__mono,axiom,
% 0.35/0.72      ! [A: rat,M: nat,N: nat] :
% 0.35/0.72        ( ( ( comm_s4028243227959126397er_rat @ A @ M )
% 0.35/0.72         != zero_zero_rat )
% 0.35/0.72       => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.72         => ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.35/0.72           != zero_zero_rat ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_neq_0_mono
% 0.35/0.72  thf(fact_8299_pochhammer__eq__0__mono,axiom,
% 0.35/0.72      ! [A: complex,N: nat,M: nat] :
% 0.35/0.72        ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 0.35/0.72          = zero_zero_complex )
% 0.35/0.72       => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.72         => ( ( comm_s2602460028002588243omplex @ A @ M )
% 0.35/0.72            = zero_zero_complex ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_eq_0_mono
% 0.35/0.72  thf(fact_8300_pochhammer__eq__0__mono,axiom,
% 0.35/0.72      ! [A: real,N: nat,M: nat] :
% 0.35/0.72        ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.35/0.72          = zero_zero_real )
% 0.35/0.72       => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.72         => ( ( comm_s7457072308508201937r_real @ A @ M )
% 0.35/0.72            = zero_zero_real ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_eq_0_mono
% 0.35/0.72  thf(fact_8301_pochhammer__eq__0__mono,axiom,
% 0.35/0.72      ! [A: rat,N: nat,M: nat] :
% 0.35/0.72        ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.35/0.72          = zero_zero_rat )
% 0.35/0.72       => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.72         => ( ( comm_s4028243227959126397er_rat @ A @ M )
% 0.35/0.72            = zero_zero_rat ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_eq_0_mono
% 0.35/0.72  thf(fact_8302_pochhammer__nonneg,axiom,
% 0.35/0.72      ! [X: real,N: nat] :
% 0.35/0.72        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.72       => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_nonneg
% 0.35/0.72  thf(fact_8303_pochhammer__nonneg,axiom,
% 0.35/0.72      ! [X: rat,N: nat] :
% 0.35/0.72        ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.35/0.72       => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_nonneg
% 0.35/0.72  thf(fact_8304_pochhammer__nonneg,axiom,
% 0.35/0.72      ! [X: nat,N: nat] :
% 0.35/0.72        ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.35/0.72       => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_nonneg
% 0.35/0.72  thf(fact_8305_pochhammer__nonneg,axiom,
% 0.35/0.72      ! [X: int,N: nat] :
% 0.35/0.72        ( ( ord_less_int @ zero_zero_int @ X )
% 0.35/0.72       => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_nonneg
% 0.35/0.72  thf(fact_8306_pochhammer__0__left,axiom,
% 0.35/0.72      ! [N: nat] :
% 0.35/0.72        ( ( ( N = zero_zero_nat )
% 0.35/0.72         => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 0.35/0.72            = one_one_complex ) )
% 0.35/0.72        & ( ( N != zero_zero_nat )
% 0.35/0.72         => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 0.35/0.72            = zero_zero_complex ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_0_left
% 0.35/0.72  thf(fact_8307_pochhammer__0__left,axiom,
% 0.35/0.72      ! [N: nat] :
% 0.35/0.72        ( ( ( N = zero_zero_nat )
% 0.35/0.72         => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 0.35/0.72            = one_one_real ) )
% 0.35/0.72        & ( ( N != zero_zero_nat )
% 0.35/0.72         => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 0.35/0.72            = zero_zero_real ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_0_left
% 0.35/0.72  thf(fact_8308_pochhammer__0__left,axiom,
% 0.35/0.72      ! [N: nat] :
% 0.35/0.72        ( ( ( N = zero_zero_nat )
% 0.35/0.72         => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 0.35/0.72            = one_one_rat ) )
% 0.35/0.72        & ( ( N != zero_zero_nat )
% 0.35/0.72         => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 0.35/0.72            = zero_zero_rat ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_0_left
% 0.35/0.72  thf(fact_8309_pochhammer__0__left,axiom,
% 0.35/0.72      ! [N: nat] :
% 0.35/0.72        ( ( ( N = zero_zero_nat )
% 0.35/0.72         => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 0.35/0.72            = one_one_nat ) )
% 0.35/0.72        & ( ( N != zero_zero_nat )
% 0.35/0.72         => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 0.35/0.72            = zero_zero_nat ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_0_left
% 0.35/0.72  thf(fact_8310_pochhammer__0__left,axiom,
% 0.35/0.72      ! [N: nat] :
% 0.35/0.72        ( ( ( N = zero_zero_nat )
% 0.35/0.72         => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 0.35/0.72            = one_one_int ) )
% 0.35/0.72        & ( ( N != zero_zero_nat )
% 0.35/0.72         => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 0.35/0.72            = zero_zero_int ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_0_left
% 0.35/0.72  thf(fact_8311_bit_Ocomplement__unique,axiom,
% 0.35/0.72      ! [A: code_integer,X: code_integer,Y: code_integer] :
% 0.35/0.72        ( ( ( bit_se3949692690581998587nteger @ A @ X )
% 0.35/0.72          = zero_z3403309356797280102nteger )
% 0.35/0.72       => ( ( ( bit_se1080825931792720795nteger @ A @ X )
% 0.35/0.72            = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.35/0.72         => ( ( ( bit_se3949692690581998587nteger @ A @ Y )
% 0.35/0.72              = zero_z3403309356797280102nteger )
% 0.35/0.72           => ( ( ( bit_se1080825931792720795nteger @ A @ Y )
% 0.35/0.72                = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.35/0.72             => ( X = Y ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % bit.complement_unique
% 0.35/0.72  thf(fact_8312_bit_Ocomplement__unique,axiom,
% 0.35/0.72      ! [A: int,X: int,Y: int] :
% 0.35/0.72        ( ( ( bit_se725231765392027082nd_int @ A @ X )
% 0.35/0.72          = zero_zero_int )
% 0.35/0.72       => ( ( ( bit_se1409905431419307370or_int @ A @ X )
% 0.35/0.72            = ( uminus_uminus_int @ one_one_int ) )
% 0.35/0.72         => ( ( ( bit_se725231765392027082nd_int @ A @ Y )
% 0.35/0.72              = zero_zero_int )
% 0.35/0.72           => ( ( ( bit_se1409905431419307370or_int @ A @ Y )
% 0.35/0.72                = ( uminus_uminus_int @ one_one_int ) )
% 0.35/0.72             => ( X = Y ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % bit.complement_unique
% 0.35/0.72  thf(fact_8313_pochhammer__eq__0__iff,axiom,
% 0.35/0.72      ! [A: complex,N: nat] :
% 0.35/0.72        ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 0.35/0.72          = zero_zero_complex )
% 0.35/0.72        = ( ? [K3: nat] :
% 0.35/0.72              ( ( ord_less_nat @ K3 @ N )
% 0.35/0.72              & ( A
% 0.35/0.72                = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_eq_0_iff
% 0.35/0.72  thf(fact_8314_pochhammer__eq__0__iff,axiom,
% 0.35/0.72      ! [A: rat,N: nat] :
% 0.35/0.72        ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.35/0.72          = zero_zero_rat )
% 0.35/0.72        = ( ? [K3: nat] :
% 0.35/0.72              ( ( ord_less_nat @ K3 @ N )
% 0.35/0.72              & ( A
% 0.35/0.72                = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_eq_0_iff
% 0.35/0.72  thf(fact_8315_pochhammer__eq__0__iff,axiom,
% 0.35/0.72      ! [A: real,N: nat] :
% 0.35/0.72        ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.35/0.72          = zero_zero_real )
% 0.35/0.72        = ( ? [K3: nat] :
% 0.35/0.72              ( ( ord_less_nat @ K3 @ N )
% 0.35/0.72              & ( A
% 0.35/0.72                = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_eq_0_iff
% 0.35/0.72  thf(fact_8316_pochhammer__of__nat__eq__0__iff,axiom,
% 0.35/0.72      ! [N: nat,K: nat] :
% 0.35/0.72        ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.35/0.72          = zero_zero_complex )
% 0.35/0.72        = ( ord_less_nat @ N @ K ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_of_nat_eq_0_iff
% 0.35/0.72  thf(fact_8317_pochhammer__of__nat__eq__0__iff,axiom,
% 0.35/0.72      ! [N: nat,K: nat] :
% 0.35/0.72        ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.35/0.72          = zero_zero_rat )
% 0.35/0.72        = ( ord_less_nat @ N @ K ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_of_nat_eq_0_iff
% 0.35/0.72  thf(fact_8318_pochhammer__of__nat__eq__0__iff,axiom,
% 0.35/0.72      ! [N: nat,K: nat] :
% 0.35/0.72        ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.35/0.72          = zero_z3403309356797280102nteger )
% 0.35/0.72        = ( ord_less_nat @ N @ K ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_of_nat_eq_0_iff
% 0.35/0.72  thf(fact_8319_pochhammer__of__nat__eq__0__iff,axiom,
% 0.35/0.72      ! [N: nat,K: nat] :
% 0.35/0.72        ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.35/0.72          = zero_zero_int )
% 0.35/0.72        = ( ord_less_nat @ N @ K ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_of_nat_eq_0_iff
% 0.35/0.72  thf(fact_8320_pochhammer__of__nat__eq__0__iff,axiom,
% 0.35/0.72      ! [N: nat,K: nat] :
% 0.35/0.72        ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.35/0.72          = zero_zero_real )
% 0.35/0.72        = ( ord_less_nat @ N @ K ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_of_nat_eq_0_iff
% 0.35/0.72  thf(fact_8321_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.35/0.72      ! [N: nat,K: nat] :
% 0.35/0.72        ( ( ord_less_nat @ N @ K )
% 0.35/0.72       => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.35/0.72          = zero_zero_complex ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_of_nat_eq_0_lemma
% 0.35/0.72  thf(fact_8322_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.35/0.72      ! [N: nat,K: nat] :
% 0.35/0.72        ( ( ord_less_nat @ N @ K )
% 0.35/0.72       => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.35/0.72          = zero_zero_rat ) ) ).
% 0.35/0.72  
% 0.35/0.72  % pochhammer_of_nat_eq_0_lemma
% 0.35/0.73  thf(fact_8323_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.35/0.73      ! [N: nat,K: nat] :
% 0.35/0.73        ( ( ord_less_nat @ N @ K )
% 0.35/0.73       => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.35/0.73          = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.73  
% 0.35/0.73  % pochhammer_of_nat_eq_0_lemma
% 0.35/0.73  thf(fact_8324_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.35/0.73      ! [N: nat,K: nat] :
% 0.35/0.73        ( ( ord_less_nat @ N @ K )
% 0.35/0.73       => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.35/0.73          = zero_zero_int ) ) ).
% 0.35/0.73  
% 0.35/0.73  % pochhammer_of_nat_eq_0_lemma
% 0.35/0.73  thf(fact_8325_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.35/0.73      ! [N: nat,K: nat] :
% 0.35/0.73        ( ( ord_less_nat @ N @ K )
% 0.35/0.73       => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.35/0.73          = zero_zero_real ) ) ).
% 0.35/0.73  
% 0.35/0.73  % pochhammer_of_nat_eq_0_lemma
% 0.35/0.73  thf(fact_8326_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.35/0.73      ! [K: nat,N: nat] :
% 0.35/0.73        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.73       => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.35/0.73         != zero_zero_complex ) ) ).
% 0.35/0.73  
% 0.35/0.73  % pochhammer_of_nat_eq_0_lemma'
% 0.35/0.73  thf(fact_8327_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.35/0.73      ! [K: nat,N: nat] :
% 0.35/0.73        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.73       => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.35/0.73         != zero_zero_rat ) ) ).
% 0.35/0.73  
% 0.35/0.73  % pochhammer_of_nat_eq_0_lemma'
% 0.35/0.73  thf(fact_8328_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.35/0.73      ! [K: nat,N: nat] :
% 0.35/0.73        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.73       => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.35/0.73         != zero_z3403309356797280102nteger ) ) ).
% 0.35/0.73  
% 0.35/0.73  % pochhammer_of_nat_eq_0_lemma'
% 0.35/0.73  thf(fact_8329_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.35/0.73      ! [K: nat,N: nat] :
% 0.35/0.73        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.73       => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.35/0.73         != zero_zero_int ) ) ).
% 0.35/0.73  
% 0.35/0.73  % pochhammer_of_nat_eq_0_lemma'
% 0.35/0.73  thf(fact_8330_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.35/0.73      ! [K: nat,N: nat] :
% 0.35/0.73        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.73       => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.35/0.73         != zero_zero_real ) ) ).
% 0.35/0.73  
% 0.35/0.73  % pochhammer_of_nat_eq_0_lemma'
% 0.35/0.73  thf(fact_8331_arcsin__less__arcsin,axiom,
% 0.35/0.73      ! [X: real,Y: real] :
% 0.35/0.73        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.35/0.73       => ( ( ord_less_real @ X @ Y )
% 0.35/0.73         => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.35/0.73           => ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % arcsin_less_arcsin
% 0.35/0.73  thf(fact_8332_arcsin__less__mono,axiom,
% 0.35/0.73      ! [X: real,Y: real] :
% 0.35/0.73        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.35/0.73       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.35/0.73         => ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
% 0.35/0.73            = ( ord_less_real @ X @ Y ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % arcsin_less_mono
% 0.35/0.73  thf(fact_8333_pochhammer__product,axiom,
% 0.35/0.73      ! [M: nat,N: nat,Z2: complex] :
% 0.35/0.73        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.73       => ( ( comm_s2602460028002588243omplex @ Z2 @ N )
% 0.35/0.73          = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % pochhammer_product
% 0.35/0.73  thf(fact_8334_pochhammer__product,axiom,
% 0.35/0.73      ! [M: nat,N: nat,Z2: rat] :
% 0.35/0.73        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.73       => ( ( comm_s4028243227959126397er_rat @ Z2 @ N )
% 0.35/0.73          = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % pochhammer_product
% 0.35/0.73  thf(fact_8335_pochhammer__product,axiom,
% 0.35/0.73      ! [M: nat,N: nat,Z2: nat] :
% 0.35/0.73        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.73       => ( ( comm_s4663373288045622133er_nat @ Z2 @ N )
% 0.35/0.73          = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % pochhammer_product
% 0.35/0.73  thf(fact_8336_pochhammer__product,axiom,
% 0.35/0.73      ! [M: nat,N: nat,Z2: int] :
% 0.35/0.73        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.73       => ( ( comm_s4660882817536571857er_int @ Z2 @ N )
% 0.35/0.73          = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % pochhammer_product
% 0.35/0.73  thf(fact_8337_pochhammer__product,axiom,
% 0.35/0.73      ! [M: nat,N: nat,Z2: real] :
% 0.35/0.73        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.73       => ( ( comm_s7457072308508201937r_real @ Z2 @ N )
% 0.35/0.73          = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % pochhammer_product
% 0.35/0.73  thf(fact_8338_cos__arcsin__nonzero,axiom,
% 0.35/0.73      ! [X: real] :
% 0.35/0.73        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.35/0.73       => ( ( ord_less_real @ X @ one_one_real )
% 0.35/0.73         => ( ( cos_real @ ( arcsin @ X ) )
% 0.35/0.73           != zero_zero_real ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % cos_arcsin_nonzero
% 0.35/0.73  thf(fact_8339_Cauchy__iff,axiom,
% 0.35/0.73      ( topolo6517432010174082258omplex
% 0.35/0.73      = ( ^ [X6: nat > complex] :
% 0.35/0.73          ! [E3: real] :
% 0.35/0.73            ( ( ord_less_real @ zero_zero_real @ E3 )
% 0.35/0.73           => ? [M9: nat] :
% 0.35/0.73              ! [M2: nat] :
% 0.35/0.73                ( ( ord_less_eq_nat @ M9 @ M2 )
% 0.35/0.73               => ! [N2: nat] :
% 0.35/0.73                    ( ( ord_less_eq_nat @ M9 @ N2 )
% 0.35/0.73                   => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) ) @ E3 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % Cauchy_iff
% 0.35/0.73  thf(fact_8340_Cauchy__iff,axiom,
% 0.35/0.73      ( topolo4055970368930404560y_real
% 0.35/0.73      = ( ^ [X6: nat > real] :
% 0.35/0.73          ! [E3: real] :
% 0.35/0.73            ( ( ord_less_real @ zero_zero_real @ E3 )
% 0.35/0.73           => ? [M9: nat] :
% 0.35/0.73              ! [M2: nat] :
% 0.35/0.73                ( ( ord_less_eq_nat @ M9 @ M2 )
% 0.35/0.73               => ! [N2: nat] :
% 0.35/0.73                    ( ( ord_less_eq_nat @ M9 @ N2 )
% 0.35/0.73                   => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) ) @ E3 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % Cauchy_iff
% 0.35/0.73  thf(fact_8341_CauchyI,axiom,
% 0.35/0.73      ! [X8: nat > complex] :
% 0.35/0.73        ( ! [E: real] :
% 0.35/0.73            ( ( ord_less_real @ zero_zero_real @ E )
% 0.35/0.73           => ? [M10: nat] :
% 0.35/0.73              ! [M5: nat] :
% 0.35/0.73                ( ( ord_less_eq_nat @ M10 @ M5 )
% 0.35/0.73               => ! [N3: nat] :
% 0.35/0.73                    ( ( ord_less_eq_nat @ M10 @ N3 )
% 0.35/0.73                   => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) ) @ E ) ) ) )
% 0.35/0.73       => ( topolo6517432010174082258omplex @ X8 ) ) ).
% 0.35/0.73  
% 0.35/0.73  % CauchyI
% 0.35/0.73  thf(fact_8342_CauchyI,axiom,
% 0.35/0.73      ! [X8: nat > real] :
% 0.35/0.73        ( ! [E: real] :
% 0.35/0.73            ( ( ord_less_real @ zero_zero_real @ E )
% 0.35/0.73           => ? [M10: nat] :
% 0.35/0.73              ! [M5: nat] :
% 0.35/0.73                ( ( ord_less_eq_nat @ M10 @ M5 )
% 0.35/0.73               => ! [N3: nat] :
% 0.35/0.73                    ( ( ord_less_eq_nat @ M10 @ N3 )
% 0.35/0.73                   => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) ) @ E ) ) ) )
% 0.35/0.73       => ( topolo4055970368930404560y_real @ X8 ) ) ).
% 0.35/0.73  
% 0.35/0.73  % CauchyI
% 0.35/0.73  thf(fact_8343_CauchyD,axiom,
% 0.35/0.73      ! [X8: nat > complex,E2: real] :
% 0.35/0.73        ( ( topolo6517432010174082258omplex @ X8 )
% 0.35/0.73       => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.35/0.73         => ? [M8: nat] :
% 0.35/0.73            ! [M3: nat] :
% 0.35/0.73              ( ( ord_less_eq_nat @ M8 @ M3 )
% 0.35/0.73             => ! [N5: nat] :
% 0.35/0.73                  ( ( ord_less_eq_nat @ M8 @ N5 )
% 0.35/0.73                 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X8 @ M3 ) @ ( X8 @ N5 ) ) ) @ E2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % CauchyD
% 0.35/0.73  thf(fact_8344_CauchyD,axiom,
% 0.35/0.73      ! [X8: nat > real,E2: real] :
% 0.35/0.73        ( ( topolo4055970368930404560y_real @ X8 )
% 0.35/0.73       => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.35/0.73         => ? [M8: nat] :
% 0.35/0.73            ! [M3: nat] :
% 0.35/0.73              ( ( ord_less_eq_nat @ M8 @ M3 )
% 0.35/0.73             => ! [N5: nat] :
% 0.35/0.73                  ( ( ord_less_eq_nat @ M8 @ N5 )
% 0.35/0.73                 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X8 @ M3 ) @ ( X8 @ N5 ) ) ) @ E2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % CauchyD
% 0.35/0.73  thf(fact_8345_floor__real__def,axiom,
% 0.35/0.73      ( archim6058952711729229775r_real
% 0.35/0.73      = ( ^ [X3: real] :
% 0.35/0.73            ( the_int
% 0.35/0.73            @ ^ [Z3: int] :
% 0.35/0.73                ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X3 )
% 0.35/0.73                & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % floor_real_def
% 0.35/0.73  thf(fact_8346_termdiff__converges,axiom,
% 0.35/0.73      ! [X: real,K4: real,C: nat > real] :
% 0.35/0.73        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ K4 )
% 0.35/0.73       => ( ! [X4: real] :
% 0.35/0.73              ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ K4 )
% 0.35/0.73             => ( summable_real
% 0.35/0.73                @ ^ [N2: nat] : ( times_times_real @ ( C @ N2 ) @ ( power_power_real @ X4 @ N2 ) ) ) )
% 0.35/0.73         => ( summable_real
% 0.35/0.73            @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % termdiff_converges
% 0.35/0.73  thf(fact_8347_termdiff__converges,axiom,
% 0.35/0.73      ! [X: complex,K4: real,C: nat > complex] :
% 0.35/0.73        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ K4 )
% 0.35/0.73       => ( ! [X4: complex] :
% 0.35/0.73              ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ K4 )
% 0.35/0.73             => ( summable_complex
% 0.35/0.73                @ ^ [N2: nat] : ( times_times_complex @ ( C @ N2 ) @ ( power_power_complex @ X4 @ N2 ) ) ) )
% 0.35/0.73         => ( summable_complex
% 0.35/0.73            @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % termdiff_converges
% 0.35/0.73  thf(fact_8348_Suc__0__or__eq,axiom,
% 0.35/0.73      ! [N: nat] :
% 0.35/0.73        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.73        = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % Suc_0_or_eq
% 0.35/0.73  thf(fact_8349_or__Suc__0__eq,axiom,
% 0.35/0.73      ! [N: nat] :
% 0.35/0.73        ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.35/0.73        = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % or_Suc_0_eq
% 0.35/0.73  thf(fact_8350_pochhammer__times__pochhammer__half,axiom,
% 0.35/0.73      ! [Z2: complex,N: nat] :
% 0.35/0.73        ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ ( suc @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 0.35/0.73        = ( groups6464643781859351333omplex
% 0.35/0.73          @ ^ [K3: nat] : ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 0.35/0.73          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % pochhammer_times_pochhammer_half
% 0.35/0.73  thf(fact_8351_pochhammer__times__pochhammer__half,axiom,
% 0.35/0.73      ! [Z2: rat,N: nat] :
% 0.35/0.73        ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 0.35/0.73        = ( groups73079841787564623at_rat
% 0.35/0.73          @ ^ [K3: nat] : ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.35/0.73          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % pochhammer_times_pochhammer_half
% 0.35/0.73  thf(fact_8352_pochhammer__times__pochhammer__half,axiom,
% 0.35/0.73      ! [Z2: real,N: nat] :
% 0.35/0.73        ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ ( suc @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 0.35/0.73        = ( groups129246275422532515t_real
% 0.35/0.73          @ ^ [K3: nat] : ( plus_plus_real @ Z2 @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.73          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % pochhammer_times_pochhammer_half
% 0.35/0.73  thf(fact_8353_floor__rat__def,axiom,
% 0.35/0.73      ( archim3151403230148437115or_rat
% 0.35/0.73      = ( ^ [X3: rat] :
% 0.35/0.73            ( the_int
% 0.35/0.73            @ ^ [Z3: int] :
% 0.35/0.73                ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X3 )
% 0.35/0.73                & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % floor_rat_def
% 0.35/0.73  thf(fact_8354_sum__power2,axiom,
% 0.35/0.73      ! [K: nat] :
% 0.35/0.73        ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 0.35/0.73        = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 0.35/0.73  
% 0.35/0.73  % sum_power2
% 0.35/0.73  thf(fact_8355_VEBT_Osize_I3_J,axiom,
% 0.35/0.73      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.35/0.73        ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.35/0.73        = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % VEBT.size(3)
% 0.35/0.73  thf(fact_8356_is__singleton__the__elem,axiom,
% 0.35/0.73      ( is_sin2937591304547752795at_nat
% 0.35/0.73      = ( ^ [A6: set_Pr4329608150637261639at_nat] :
% 0.35/0.73            ( A6
% 0.35/0.73            = ( insert9069300056098147895at_nat @ ( the_el221668144340439132at_nat @ A6 ) @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % is_singleton_the_elem
% 0.35/0.73  thf(fact_8357_is__singleton__the__elem,axiom,
% 0.35/0.73      ( is_sin2850979758926227957at_nat
% 0.35/0.73      = ( ^ [A6: set_Pr1261947904930325089at_nat] :
% 0.35/0.73            ( A6
% 0.35/0.73            = ( insert8211810215607154385at_nat @ ( the_el2281957884133575798at_nat @ A6 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % is_singleton_the_elem
% 0.35/0.73  thf(fact_8358_is__singleton__the__elem,axiom,
% 0.35/0.73      ( is_singleton_real
% 0.35/0.73      = ( ^ [A6: set_real] :
% 0.35/0.73            ( A6
% 0.35/0.73            = ( insert_real @ ( the_elem_real @ A6 ) @ bot_bot_set_real ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % is_singleton_the_elem
% 0.35/0.73  thf(fact_8359_is__singleton__the__elem,axiom,
% 0.35/0.73      ( is_singleton_nat
% 0.35/0.73      = ( ^ [A6: set_nat] :
% 0.35/0.73            ( A6
% 0.35/0.73            = ( insert_nat @ ( the_elem_nat @ A6 ) @ bot_bot_set_nat ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % is_singleton_the_elem
% 0.35/0.73  thf(fact_8360_is__singleton__the__elem,axiom,
% 0.35/0.73      ( is_singleton_int
% 0.35/0.73      = ( ^ [A6: set_int] :
% 0.35/0.73            ( A6
% 0.35/0.73            = ( insert_int @ ( the_elem_int @ A6 ) @ bot_bot_set_int ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % is_singleton_the_elem
% 0.35/0.73  thf(fact_8361_or__negative__int__iff,axiom,
% 0.35/0.73      ! [K: int,L: int] :
% 0.35/0.73        ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
% 0.35/0.73        = ( ( ord_less_int @ K @ zero_zero_int )
% 0.35/0.73          | ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % or_negative_int_iff
% 0.35/0.73  thf(fact_8362_finite__atLeastLessThan,axiom,
% 0.35/0.73      ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).
% 0.35/0.73  
% 0.35/0.73  % finite_atLeastLessThan
% 0.35/0.73  thf(fact_8363_atLeastLessThan__iff,axiom,
% 0.35/0.73      ! [I: set_nat,L: set_nat,U: set_nat] :
% 0.35/0.73        ( ( member_set_nat @ I @ ( set_or3540276404033026485et_nat @ L @ U ) )
% 0.35/0.73        = ( ( ord_less_eq_set_nat @ L @ I )
% 0.35/0.73          & ( ord_less_set_nat @ I @ U ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_iff
% 0.35/0.73  thf(fact_8364_atLeastLessThan__iff,axiom,
% 0.35/0.73      ! [I: real,L: real,U: real] :
% 0.35/0.73        ( ( member_real @ I @ ( set_or66887138388493659n_real @ L @ U ) )
% 0.35/0.73        = ( ( ord_less_eq_real @ L @ I )
% 0.35/0.73          & ( ord_less_real @ I @ U ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_iff
% 0.35/0.73  thf(fact_8365_atLeastLessThan__iff,axiom,
% 0.35/0.73      ! [I: set_int,L: set_int,U: set_int] :
% 0.35/0.73        ( ( member_set_int @ I @ ( set_or8585797421378605585et_int @ L @ U ) )
% 0.35/0.73        = ( ( ord_less_eq_set_int @ L @ I )
% 0.35/0.73          & ( ord_less_set_int @ I @ U ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_iff
% 0.35/0.73  thf(fact_8366_atLeastLessThan__iff,axiom,
% 0.35/0.73      ! [I: rat,L: rat,U: rat] :
% 0.35/0.73        ( ( member_rat @ I @ ( set_or4029947393144176647an_rat @ L @ U ) )
% 0.35/0.73        = ( ( ord_less_eq_rat @ L @ I )
% 0.35/0.73          & ( ord_less_rat @ I @ U ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_iff
% 0.35/0.73  thf(fact_8367_atLeastLessThan__iff,axiom,
% 0.35/0.73      ! [I: num,L: num,U: num] :
% 0.35/0.73        ( ( member_num @ I @ ( set_or1222409239386451017an_num @ L @ U ) )
% 0.35/0.73        = ( ( ord_less_eq_num @ L @ I )
% 0.35/0.73          & ( ord_less_num @ I @ U ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_iff
% 0.35/0.73  thf(fact_8368_atLeastLessThan__iff,axiom,
% 0.35/0.73      ! [I: nat,L: nat,U: nat] :
% 0.35/0.73        ( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
% 0.35/0.73        = ( ( ord_less_eq_nat @ L @ I )
% 0.35/0.73          & ( ord_less_nat @ I @ U ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_iff
% 0.35/0.73  thf(fact_8369_atLeastLessThan__iff,axiom,
% 0.35/0.73      ! [I: int,L: int,U: int] :
% 0.35/0.73        ( ( member_int @ I @ ( set_or4662586982721622107an_int @ L @ U ) )
% 0.35/0.73        = ( ( ord_less_eq_int @ L @ I )
% 0.35/0.73          & ( ord_less_int @ I @ U ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_iff
% 0.35/0.73  thf(fact_8370_prod__zero__iff,axiom,
% 0.35/0.73      ! [A2: set_nat,F: nat > complex] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( ( groups6464643781859351333omplex @ F @ A2 )
% 0.35/0.73            = zero_zero_complex )
% 0.35/0.73          = ( ? [X3: nat] :
% 0.35/0.73                ( ( member_nat @ X3 @ A2 )
% 0.35/0.73                & ( ( F @ X3 )
% 0.35/0.73                  = zero_zero_complex ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_zero_iff
% 0.35/0.73  thf(fact_8371_prod__zero__iff,axiom,
% 0.35/0.73      ! [A2: set_int,F: int > complex] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ( ( groups7440179247065528705omplex @ F @ A2 )
% 0.35/0.73            = zero_zero_complex )
% 0.35/0.73          = ( ? [X3: int] :
% 0.35/0.73                ( ( member_int @ X3 @ A2 )
% 0.35/0.73                & ( ( F @ X3 )
% 0.35/0.73                  = zero_zero_complex ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_zero_iff
% 0.35/0.73  thf(fact_8372_prod__zero__iff,axiom,
% 0.35/0.73      ! [A2: set_complex,F: complex > complex] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.73       => ( ( ( groups3708469109370488835omplex @ F @ A2 )
% 0.35/0.73            = zero_zero_complex )
% 0.35/0.73          = ( ? [X3: complex] :
% 0.35/0.73                ( ( member_complex @ X3 @ A2 )
% 0.35/0.73                & ( ( F @ X3 )
% 0.35/0.73                  = zero_zero_complex ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_zero_iff
% 0.35/0.73  thf(fact_8373_prod__zero__iff,axiom,
% 0.35/0.73      ! [A2: set_nat,F: nat > real] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( ( groups129246275422532515t_real @ F @ A2 )
% 0.35/0.73            = zero_zero_real )
% 0.35/0.73          = ( ? [X3: nat] :
% 0.35/0.73                ( ( member_nat @ X3 @ A2 )
% 0.35/0.73                & ( ( F @ X3 )
% 0.35/0.73                  = zero_zero_real ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_zero_iff
% 0.35/0.73  thf(fact_8374_prod__zero__iff,axiom,
% 0.35/0.73      ! [A2: set_int,F: int > real] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ( ( groups2316167850115554303t_real @ F @ A2 )
% 0.35/0.73            = zero_zero_real )
% 0.35/0.73          = ( ? [X3: int] :
% 0.35/0.73                ( ( member_int @ X3 @ A2 )
% 0.35/0.73                & ( ( F @ X3 )
% 0.35/0.73                  = zero_zero_real ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_zero_iff
% 0.35/0.73  thf(fact_8375_prod__zero__iff,axiom,
% 0.35/0.73      ! [A2: set_complex,F: complex > real] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.73       => ( ( ( groups766887009212190081x_real @ F @ A2 )
% 0.35/0.73            = zero_zero_real )
% 0.35/0.73          = ( ? [X3: complex] :
% 0.35/0.73                ( ( member_complex @ X3 @ A2 )
% 0.35/0.73                & ( ( F @ X3 )
% 0.35/0.73                  = zero_zero_real ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_zero_iff
% 0.35/0.73  thf(fact_8376_prod__zero__iff,axiom,
% 0.35/0.73      ! [A2: set_nat,F: nat > rat] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( ( groups73079841787564623at_rat @ F @ A2 )
% 0.35/0.73            = zero_zero_rat )
% 0.35/0.73          = ( ? [X3: nat] :
% 0.35/0.73                ( ( member_nat @ X3 @ A2 )
% 0.35/0.73                & ( ( F @ X3 )
% 0.35/0.73                  = zero_zero_rat ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_zero_iff
% 0.35/0.73  thf(fact_8377_prod__zero__iff,axiom,
% 0.35/0.73      ! [A2: set_int,F: int > rat] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ( ( groups1072433553688619179nt_rat @ F @ A2 )
% 0.35/0.73            = zero_zero_rat )
% 0.35/0.73          = ( ? [X3: int] :
% 0.35/0.73                ( ( member_int @ X3 @ A2 )
% 0.35/0.73                & ( ( F @ X3 )
% 0.35/0.73                  = zero_zero_rat ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_zero_iff
% 0.35/0.73  thf(fact_8378_prod__zero__iff,axiom,
% 0.35/0.73      ! [A2: set_complex,F: complex > rat] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.73       => ( ( ( groups225925009352817453ex_rat @ F @ A2 )
% 0.35/0.73            = zero_zero_rat )
% 0.35/0.73          = ( ? [X3: complex] :
% 0.35/0.73                ( ( member_complex @ X3 @ A2 )
% 0.35/0.73                & ( ( F @ X3 )
% 0.35/0.73                  = zero_zero_rat ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_zero_iff
% 0.35/0.73  thf(fact_8379_prod__zero__iff,axiom,
% 0.35/0.73      ! [A2: set_int,F: int > nat] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ( ( groups1707563613775114915nt_nat @ F @ A2 )
% 0.35/0.73            = zero_zero_nat )
% 0.35/0.73          = ( ? [X3: int] :
% 0.35/0.73                ( ( member_int @ X3 @ A2 )
% 0.35/0.73                & ( ( F @ X3 )
% 0.35/0.73                  = zero_zero_nat ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_zero_iff
% 0.35/0.73  thf(fact_8380_atLeastLessThan__empty,axiom,
% 0.35/0.73      ! [B: real,A: real] :
% 0.35/0.73        ( ( ord_less_eq_real @ B @ A )
% 0.35/0.73       => ( ( set_or66887138388493659n_real @ A @ B )
% 0.35/0.73          = bot_bot_set_real ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_empty
% 0.35/0.73  thf(fact_8381_atLeastLessThan__empty,axiom,
% 0.35/0.73      ! [B: set_int,A: set_int] :
% 0.35/0.73        ( ( ord_less_eq_set_int @ B @ A )
% 0.35/0.73       => ( ( set_or8585797421378605585et_int @ A @ B )
% 0.35/0.73          = bot_bot_set_set_int ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_empty
% 0.35/0.73  thf(fact_8382_atLeastLessThan__empty,axiom,
% 0.35/0.73      ! [B: rat,A: rat] :
% 0.35/0.73        ( ( ord_less_eq_rat @ B @ A )
% 0.35/0.73       => ( ( set_or4029947393144176647an_rat @ A @ B )
% 0.35/0.73          = bot_bot_set_rat ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_empty
% 0.35/0.73  thf(fact_8383_atLeastLessThan__empty,axiom,
% 0.35/0.73      ! [B: num,A: num] :
% 0.35/0.73        ( ( ord_less_eq_num @ B @ A )
% 0.35/0.73       => ( ( set_or1222409239386451017an_num @ A @ B )
% 0.35/0.73          = bot_bot_set_num ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_empty
% 0.35/0.73  thf(fact_8384_atLeastLessThan__empty,axiom,
% 0.35/0.73      ! [B: nat,A: nat] :
% 0.35/0.73        ( ( ord_less_eq_nat @ B @ A )
% 0.35/0.73       => ( ( set_or4665077453230672383an_nat @ A @ B )
% 0.35/0.73          = bot_bot_set_nat ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_empty
% 0.35/0.73  thf(fact_8385_atLeastLessThan__empty,axiom,
% 0.35/0.73      ! [B: int,A: int] :
% 0.35/0.73        ( ( ord_less_eq_int @ B @ A )
% 0.35/0.73       => ( ( set_or4662586982721622107an_int @ A @ B )
% 0.35/0.73          = bot_bot_set_int ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_empty
% 0.35/0.73  thf(fact_8386_ivl__subset,axiom,
% 0.35/0.73      ! [I: rat,J: rat,M: rat,N: rat] :
% 0.35/0.73        ( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ I @ J ) @ ( set_or4029947393144176647an_rat @ M @ N ) )
% 0.35/0.73        = ( ( ord_less_eq_rat @ J @ I )
% 0.35/0.73          | ( ( ord_less_eq_rat @ M @ I )
% 0.35/0.73            & ( ord_less_eq_rat @ J @ N ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % ivl_subset
% 0.35/0.73  thf(fact_8387_ivl__subset,axiom,
% 0.35/0.73      ! [I: num,J: num,M: num,N: num] :
% 0.35/0.73        ( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ I @ J ) @ ( set_or1222409239386451017an_num @ M @ N ) )
% 0.35/0.73        = ( ( ord_less_eq_num @ J @ I )
% 0.35/0.73          | ( ( ord_less_eq_num @ M @ I )
% 0.35/0.73            & ( ord_less_eq_num @ J @ N ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % ivl_subset
% 0.35/0.73  thf(fact_8388_ivl__subset,axiom,
% 0.35/0.73      ! [I: nat,J: nat,M: nat,N: nat] :
% 0.35/0.73        ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 0.35/0.73        = ( ( ord_less_eq_nat @ J @ I )
% 0.35/0.73          | ( ( ord_less_eq_nat @ M @ I )
% 0.35/0.73            & ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % ivl_subset
% 0.35/0.73  thf(fact_8389_ivl__subset,axiom,
% 0.35/0.73      ! [I: int,J: int,M: int,N: int] :
% 0.35/0.73        ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I @ J ) @ ( set_or4662586982721622107an_int @ M @ N ) )
% 0.35/0.73        = ( ( ord_less_eq_int @ J @ I )
% 0.35/0.73          | ( ( ord_less_eq_int @ M @ I )
% 0.35/0.73            & ( ord_less_eq_int @ J @ N ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % ivl_subset
% 0.35/0.73  thf(fact_8390_atLeastLessThan__empty__iff2,axiom,
% 0.35/0.73      ! [A: real,B: real] :
% 0.35/0.73        ( ( bot_bot_set_real
% 0.35/0.73          = ( set_or66887138388493659n_real @ A @ B ) )
% 0.35/0.73        = ( ~ ( ord_less_real @ A @ B ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_empty_iff2
% 0.35/0.73  thf(fact_8391_atLeastLessThan__empty__iff2,axiom,
% 0.35/0.73      ! [A: rat,B: rat] :
% 0.35/0.73        ( ( bot_bot_set_rat
% 0.35/0.73          = ( set_or4029947393144176647an_rat @ A @ B ) )
% 0.35/0.73        = ( ~ ( ord_less_rat @ A @ B ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_empty_iff2
% 0.35/0.73  thf(fact_8392_atLeastLessThan__empty__iff2,axiom,
% 0.35/0.73      ! [A: num,B: num] :
% 0.35/0.73        ( ( bot_bot_set_num
% 0.35/0.73          = ( set_or1222409239386451017an_num @ A @ B ) )
% 0.35/0.73        = ( ~ ( ord_less_num @ A @ B ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_empty_iff2
% 0.35/0.73  thf(fact_8393_atLeastLessThan__empty__iff2,axiom,
% 0.35/0.73      ! [A: nat,B: nat] :
% 0.35/0.73        ( ( bot_bot_set_nat
% 0.35/0.73          = ( set_or4665077453230672383an_nat @ A @ B ) )
% 0.35/0.73        = ( ~ ( ord_less_nat @ A @ B ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_empty_iff2
% 0.35/0.73  thf(fact_8394_atLeastLessThan__empty__iff2,axiom,
% 0.35/0.73      ! [A: int,B: int] :
% 0.35/0.73        ( ( bot_bot_set_int
% 0.35/0.73          = ( set_or4662586982721622107an_int @ A @ B ) )
% 0.35/0.73        = ( ~ ( ord_less_int @ A @ B ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_empty_iff2
% 0.35/0.73  thf(fact_8395_atLeastLessThan__empty__iff,axiom,
% 0.35/0.73      ! [A: real,B: real] :
% 0.35/0.73        ( ( ( set_or66887138388493659n_real @ A @ B )
% 0.35/0.73          = bot_bot_set_real )
% 0.35/0.73        = ( ~ ( ord_less_real @ A @ B ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_empty_iff
% 0.35/0.73  thf(fact_8396_atLeastLessThan__empty__iff,axiom,
% 0.35/0.73      ! [A: rat,B: rat] :
% 0.35/0.73        ( ( ( set_or4029947393144176647an_rat @ A @ B )
% 0.35/0.73          = bot_bot_set_rat )
% 0.35/0.73        = ( ~ ( ord_less_rat @ A @ B ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_empty_iff
% 0.35/0.73  thf(fact_8397_atLeastLessThan__empty__iff,axiom,
% 0.35/0.73      ! [A: num,B: num] :
% 0.35/0.73        ( ( ( set_or1222409239386451017an_num @ A @ B )
% 0.35/0.73          = bot_bot_set_num )
% 0.35/0.73        = ( ~ ( ord_less_num @ A @ B ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_empty_iff
% 0.35/0.73  thf(fact_8398_atLeastLessThan__empty__iff,axiom,
% 0.35/0.73      ! [A: nat,B: nat] :
% 0.35/0.73        ( ( ( set_or4665077453230672383an_nat @ A @ B )
% 0.35/0.73          = bot_bot_set_nat )
% 0.35/0.73        = ( ~ ( ord_less_nat @ A @ B ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_empty_iff
% 0.35/0.73  thf(fact_8399_atLeastLessThan__empty__iff,axiom,
% 0.35/0.73      ! [A: int,B: int] :
% 0.35/0.73        ( ( ( set_or4662586982721622107an_int @ A @ B )
% 0.35/0.73          = bot_bot_set_int )
% 0.35/0.73        = ( ~ ( ord_less_int @ A @ B ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_empty_iff
% 0.35/0.73  thf(fact_8400_infinite__Ico__iff,axiom,
% 0.35/0.73      ! [A: real,B: real] :
% 0.35/0.73        ( ( ~ ( finite_finite_real @ ( set_or66887138388493659n_real @ A @ B ) ) )
% 0.35/0.73        = ( ord_less_real @ A @ B ) ) ).
% 0.35/0.73  
% 0.35/0.73  % infinite_Ico_iff
% 0.35/0.73  thf(fact_8401_infinite__Ico__iff,axiom,
% 0.35/0.73      ! [A: rat,B: rat] :
% 0.35/0.73        ( ( ~ ( finite_finite_rat @ ( set_or4029947393144176647an_rat @ A @ B ) ) )
% 0.35/0.73        = ( ord_less_rat @ A @ B ) ) ).
% 0.35/0.73  
% 0.35/0.73  % infinite_Ico_iff
% 0.35/0.73  thf(fact_8402_prod_Oempty,axiom,
% 0.35/0.73      ! [G: real > complex] :
% 0.35/0.73        ( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
% 0.35/0.73        = one_one_complex ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.empty
% 0.35/0.73  thf(fact_8403_prod_Oempty,axiom,
% 0.35/0.73      ! [G: real > real] :
% 0.35/0.73        ( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
% 0.35/0.73        = one_one_real ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.empty
% 0.35/0.73  thf(fact_8404_prod_Oempty,axiom,
% 0.35/0.73      ! [G: real > rat] :
% 0.35/0.73        ( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
% 0.35/0.73        = one_one_rat ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.empty
% 0.35/0.73  thf(fact_8405_prod_Oempty,axiom,
% 0.35/0.73      ! [G: real > nat] :
% 0.35/0.73        ( ( groups4696554848551431203al_nat @ G @ bot_bot_set_real )
% 0.35/0.73        = one_one_nat ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.empty
% 0.35/0.73  thf(fact_8406_prod_Oempty,axiom,
% 0.35/0.73      ! [G: real > int] :
% 0.35/0.73        ( ( groups4694064378042380927al_int @ G @ bot_bot_set_real )
% 0.35/0.73        = one_one_int ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.empty
% 0.35/0.73  thf(fact_8407_prod_Oempty,axiom,
% 0.35/0.73      ! [G: nat > complex] :
% 0.35/0.73        ( ( groups6464643781859351333omplex @ G @ bot_bot_set_nat )
% 0.35/0.73        = one_one_complex ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.empty
% 0.35/0.73  thf(fact_8408_prod_Oempty,axiom,
% 0.35/0.73      ! [G: nat > real] :
% 0.35/0.73        ( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
% 0.35/0.73        = one_one_real ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.empty
% 0.35/0.73  thf(fact_8409_prod_Oempty,axiom,
% 0.35/0.73      ! [G: nat > rat] :
% 0.35/0.73        ( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
% 0.35/0.73        = one_one_rat ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.empty
% 0.35/0.73  thf(fact_8410_prod_Oempty,axiom,
% 0.35/0.73      ! [G: nat > int] :
% 0.35/0.73        ( ( groups705719431365010083at_int @ G @ bot_bot_set_nat )
% 0.35/0.73        = one_one_int ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.empty
% 0.35/0.73  thf(fact_8411_prod_Oempty,axiom,
% 0.35/0.73      ! [G: int > complex] :
% 0.35/0.73        ( ( groups7440179247065528705omplex @ G @ bot_bot_set_int )
% 0.35/0.73        = one_one_complex ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.empty
% 0.35/0.73  thf(fact_8412_prod_Oinfinite,axiom,
% 0.35/0.73      ! [A2: set_nat,G: nat > complex] :
% 0.35/0.73        ( ~ ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( groups6464643781859351333omplex @ G @ A2 )
% 0.35/0.73          = one_one_complex ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.infinite
% 0.35/0.73  thf(fact_8413_prod_Oinfinite,axiom,
% 0.35/0.73      ! [A2: set_int,G: int > complex] :
% 0.35/0.73        ( ~ ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ( groups7440179247065528705omplex @ G @ A2 )
% 0.35/0.73          = one_one_complex ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.infinite
% 0.35/0.73  thf(fact_8414_prod_Oinfinite,axiom,
% 0.35/0.73      ! [A2: set_complex,G: complex > complex] :
% 0.35/0.73        ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.35/0.73       => ( ( groups3708469109370488835omplex @ G @ A2 )
% 0.35/0.73          = one_one_complex ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.infinite
% 0.35/0.73  thf(fact_8415_prod_Oinfinite,axiom,
% 0.35/0.73      ! [A2: set_nat,G: nat > real] :
% 0.35/0.73        ( ~ ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( groups129246275422532515t_real @ G @ A2 )
% 0.35/0.73          = one_one_real ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.infinite
% 0.35/0.73  thf(fact_8416_prod_Oinfinite,axiom,
% 0.35/0.73      ! [A2: set_int,G: int > real] :
% 0.35/0.73        ( ~ ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ( groups2316167850115554303t_real @ G @ A2 )
% 0.35/0.73          = one_one_real ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.infinite
% 0.35/0.73  thf(fact_8417_prod_Oinfinite,axiom,
% 0.35/0.73      ! [A2: set_complex,G: complex > real] :
% 0.35/0.73        ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.35/0.73       => ( ( groups766887009212190081x_real @ G @ A2 )
% 0.35/0.73          = one_one_real ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.infinite
% 0.35/0.73  thf(fact_8418_prod_Oinfinite,axiom,
% 0.35/0.73      ! [A2: set_nat,G: nat > rat] :
% 0.35/0.73        ( ~ ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( groups73079841787564623at_rat @ G @ A2 )
% 0.35/0.73          = one_one_rat ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.infinite
% 0.35/0.73  thf(fact_8419_prod_Oinfinite,axiom,
% 0.35/0.73      ! [A2: set_int,G: int > rat] :
% 0.35/0.73        ( ~ ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ( groups1072433553688619179nt_rat @ G @ A2 )
% 0.35/0.73          = one_one_rat ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.infinite
% 0.35/0.73  thf(fact_8420_prod_Oinfinite,axiom,
% 0.35/0.73      ! [A2: set_complex,G: complex > rat] :
% 0.35/0.73        ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.35/0.73       => ( ( groups225925009352817453ex_rat @ G @ A2 )
% 0.35/0.73          = one_one_rat ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.infinite
% 0.35/0.73  thf(fact_8421_prod_Oinfinite,axiom,
% 0.35/0.73      ! [A2: set_int,G: int > nat] :
% 0.35/0.73        ( ~ ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ( groups1707563613775114915nt_nat @ G @ A2 )
% 0.35/0.73          = one_one_nat ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.infinite
% 0.35/0.73  thf(fact_8422_dvd__prod__eqI,axiom,
% 0.35/0.73      ! [A2: set_real,A: real,B: nat,F: real > nat] :
% 0.35/0.73        ( ( finite_finite_real @ A2 )
% 0.35/0.73       => ( ( member_real @ A @ A2 )
% 0.35/0.73         => ( ( B
% 0.35/0.73              = ( F @ A ) )
% 0.35/0.73           => ( dvd_dvd_nat @ B @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % dvd_prod_eqI
% 0.35/0.73  thf(fact_8423_dvd__prod__eqI,axiom,
% 0.35/0.73      ! [A2: set_int,A: int,B: nat,F: int > nat] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ( member_int @ A @ A2 )
% 0.35/0.73         => ( ( B
% 0.35/0.73              = ( F @ A ) )
% 0.35/0.73           => ( dvd_dvd_nat @ B @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % dvd_prod_eqI
% 0.35/0.73  thf(fact_8424_dvd__prod__eqI,axiom,
% 0.35/0.73      ! [A2: set_complex,A: complex,B: nat,F: complex > nat] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.73       => ( ( member_complex @ A @ A2 )
% 0.35/0.73         => ( ( B
% 0.35/0.73              = ( F @ A ) )
% 0.35/0.73           => ( dvd_dvd_nat @ B @ ( groups861055069439313189ex_nat @ F @ A2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % dvd_prod_eqI
% 0.35/0.73  thf(fact_8425_dvd__prod__eqI,axiom,
% 0.35/0.73      ! [A2: set_real,A: real,B: int,F: real > int] :
% 0.35/0.73        ( ( finite_finite_real @ A2 )
% 0.35/0.73       => ( ( member_real @ A @ A2 )
% 0.35/0.73         => ( ( B
% 0.35/0.73              = ( F @ A ) )
% 0.35/0.73           => ( dvd_dvd_int @ B @ ( groups4694064378042380927al_int @ F @ A2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % dvd_prod_eqI
% 0.35/0.73  thf(fact_8426_dvd__prod__eqI,axiom,
% 0.35/0.73      ! [A2: set_nat,A: nat,B: int,F: nat > int] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( member_nat @ A @ A2 )
% 0.35/0.73         => ( ( B
% 0.35/0.73              = ( F @ A ) )
% 0.35/0.73           => ( dvd_dvd_int @ B @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % dvd_prod_eqI
% 0.35/0.73  thf(fact_8427_dvd__prod__eqI,axiom,
% 0.35/0.73      ! [A2: set_int,A: int,B: int,F: int > int] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ( member_int @ A @ A2 )
% 0.35/0.73         => ( ( B
% 0.35/0.73              = ( F @ A ) )
% 0.35/0.73           => ( dvd_dvd_int @ B @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % dvd_prod_eqI
% 0.35/0.73  thf(fact_8428_dvd__prod__eqI,axiom,
% 0.35/0.73      ! [A2: set_complex,A: complex,B: int,F: complex > int] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.73       => ( ( member_complex @ A @ A2 )
% 0.35/0.73         => ( ( B
% 0.35/0.73              = ( F @ A ) )
% 0.35/0.73           => ( dvd_dvd_int @ B @ ( groups858564598930262913ex_int @ F @ A2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % dvd_prod_eqI
% 0.35/0.73  thf(fact_8429_dvd__prod__eqI,axiom,
% 0.35/0.73      ! [A2: set_real,A: real,B: code_integer,F: real > code_integer] :
% 0.35/0.73        ( ( finite_finite_real @ A2 )
% 0.35/0.73       => ( ( member_real @ A @ A2 )
% 0.35/0.73         => ( ( B
% 0.35/0.73              = ( F @ A ) )
% 0.35/0.73           => ( dvd_dvd_Code_integer @ B @ ( groups6225526099057966256nteger @ F @ A2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % dvd_prod_eqI
% 0.35/0.73  thf(fact_8430_dvd__prod__eqI,axiom,
% 0.35/0.73      ! [A2: set_nat,A: nat,B: code_integer,F: nat > code_integer] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( member_nat @ A @ A2 )
% 0.35/0.73         => ( ( B
% 0.35/0.73              = ( F @ A ) )
% 0.35/0.73           => ( dvd_dvd_Code_integer @ B @ ( groups3455450783089532116nteger @ F @ A2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % dvd_prod_eqI
% 0.35/0.73  thf(fact_8431_dvd__prod__eqI,axiom,
% 0.35/0.73      ! [A2: set_int,A: int,B: code_integer,F: int > code_integer] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ( member_int @ A @ A2 )
% 0.35/0.73         => ( ( B
% 0.35/0.73              = ( F @ A ) )
% 0.35/0.73           => ( dvd_dvd_Code_integer @ B @ ( groups3827104343326376752nteger @ F @ A2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % dvd_prod_eqI
% 0.35/0.73  thf(fact_8432_dvd__prodI,axiom,
% 0.35/0.73      ! [A2: set_real,A: real,F: real > nat] :
% 0.35/0.73        ( ( finite_finite_real @ A2 )
% 0.35/0.73       => ( ( member_real @ A @ A2 )
% 0.35/0.73         => ( dvd_dvd_nat @ ( F @ A ) @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % dvd_prodI
% 0.35/0.73  thf(fact_8433_dvd__prodI,axiom,
% 0.35/0.73      ! [A2: set_int,A: int,F: int > nat] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ( member_int @ A @ A2 )
% 0.35/0.73         => ( dvd_dvd_nat @ ( F @ A ) @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % dvd_prodI
% 0.35/0.73  thf(fact_8434_dvd__prodI,axiom,
% 0.35/0.73      ! [A2: set_complex,A: complex,F: complex > nat] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.73       => ( ( member_complex @ A @ A2 )
% 0.35/0.73         => ( dvd_dvd_nat @ ( F @ A ) @ ( groups861055069439313189ex_nat @ F @ A2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % dvd_prodI
% 0.35/0.73  thf(fact_8435_dvd__prodI,axiom,
% 0.35/0.73      ! [A2: set_real,A: real,F: real > int] :
% 0.35/0.73        ( ( finite_finite_real @ A2 )
% 0.35/0.73       => ( ( member_real @ A @ A2 )
% 0.35/0.73         => ( dvd_dvd_int @ ( F @ A ) @ ( groups4694064378042380927al_int @ F @ A2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % dvd_prodI
% 0.35/0.73  thf(fact_8436_dvd__prodI,axiom,
% 0.35/0.73      ! [A2: set_nat,A: nat,F: nat > int] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( member_nat @ A @ A2 )
% 0.35/0.73         => ( dvd_dvd_int @ ( F @ A ) @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % dvd_prodI
% 0.35/0.73  thf(fact_8437_dvd__prodI,axiom,
% 0.35/0.73      ! [A2: set_int,A: int,F: int > int] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ( member_int @ A @ A2 )
% 0.35/0.73         => ( dvd_dvd_int @ ( F @ A ) @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % dvd_prodI
% 0.35/0.73  thf(fact_8438_dvd__prodI,axiom,
% 0.35/0.73      ! [A2: set_complex,A: complex,F: complex > int] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.73       => ( ( member_complex @ A @ A2 )
% 0.35/0.73         => ( dvd_dvd_int @ ( F @ A ) @ ( groups858564598930262913ex_int @ F @ A2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % dvd_prodI
% 0.35/0.73  thf(fact_8439_dvd__prodI,axiom,
% 0.35/0.73      ! [A2: set_real,A: real,F: real > code_integer] :
% 0.35/0.73        ( ( finite_finite_real @ A2 )
% 0.35/0.73       => ( ( member_real @ A @ A2 )
% 0.35/0.73         => ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups6225526099057966256nteger @ F @ A2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % dvd_prodI
% 0.35/0.73  thf(fact_8440_dvd__prodI,axiom,
% 0.35/0.73      ! [A2: set_nat,A: nat,F: nat > code_integer] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( member_nat @ A @ A2 )
% 0.35/0.73         => ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups3455450783089532116nteger @ F @ A2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % dvd_prodI
% 0.35/0.73  thf(fact_8441_dvd__prodI,axiom,
% 0.35/0.73      ! [A2: set_int,A: int,F: int > code_integer] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ( member_int @ A @ A2 )
% 0.35/0.73         => ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups3827104343326376752nteger @ F @ A2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % dvd_prodI
% 0.35/0.73  thf(fact_8442_ivl__diff,axiom,
% 0.35/0.73      ! [I: rat,N: rat,M: rat] :
% 0.35/0.73        ( ( ord_less_eq_rat @ I @ N )
% 0.35/0.73       => ( ( minus_minus_set_rat @ ( set_or4029947393144176647an_rat @ I @ M ) @ ( set_or4029947393144176647an_rat @ I @ N ) )
% 0.35/0.73          = ( set_or4029947393144176647an_rat @ N @ M ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % ivl_diff
% 0.35/0.73  thf(fact_8443_ivl__diff,axiom,
% 0.35/0.73      ! [I: num,N: num,M: num] :
% 0.35/0.73        ( ( ord_less_eq_num @ I @ N )
% 0.35/0.73       => ( ( minus_minus_set_num @ ( set_or1222409239386451017an_num @ I @ M ) @ ( set_or1222409239386451017an_num @ I @ N ) )
% 0.35/0.73          = ( set_or1222409239386451017an_num @ N @ M ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % ivl_diff
% 0.35/0.73  thf(fact_8444_ivl__diff,axiom,
% 0.35/0.73      ! [I: nat,N: nat,M: nat] :
% 0.35/0.73        ( ( ord_less_eq_nat @ I @ N )
% 0.35/0.73       => ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M ) @ ( set_or4665077453230672383an_nat @ I @ N ) )
% 0.35/0.73          = ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % ivl_diff
% 0.35/0.73  thf(fact_8445_ivl__diff,axiom,
% 0.35/0.73      ! [I: int,N: int,M: int] :
% 0.35/0.73        ( ( ord_less_eq_int @ I @ N )
% 0.35/0.73       => ( ( minus_minus_set_int @ ( set_or4662586982721622107an_int @ I @ M ) @ ( set_or4662586982721622107an_int @ I @ N ) )
% 0.35/0.73          = ( set_or4662586982721622107an_int @ N @ M ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % ivl_diff
% 0.35/0.73  thf(fact_8446_prod_Odelta_H,axiom,
% 0.35/0.73      ! [S2: set_real,A: real,B: real > complex] :
% 0.35/0.73        ( ( finite_finite_real @ S2 )
% 0.35/0.73       => ( ( ( member_real @ A @ S2 )
% 0.35/0.73           => ( ( groups713298508707869441omplex
% 0.35/0.73                @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = ( B @ A ) ) )
% 0.35/0.73          & ( ~ ( member_real @ A @ S2 )
% 0.35/0.73           => ( ( groups713298508707869441omplex
% 0.35/0.73                @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = one_one_complex ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.delta'
% 0.35/0.73  thf(fact_8447_prod_Odelta_H,axiom,
% 0.35/0.73      ! [S2: set_nat,A: nat,B: nat > complex] :
% 0.35/0.73        ( ( finite_finite_nat @ S2 )
% 0.35/0.73       => ( ( ( member_nat @ A @ S2 )
% 0.35/0.73           => ( ( groups6464643781859351333omplex
% 0.35/0.73                @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = ( B @ A ) ) )
% 0.35/0.73          & ( ~ ( member_nat @ A @ S2 )
% 0.35/0.73           => ( ( groups6464643781859351333omplex
% 0.35/0.73                @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = one_one_complex ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.delta'
% 0.35/0.73  thf(fact_8448_prod_Odelta_H,axiom,
% 0.35/0.73      ! [S2: set_int,A: int,B: int > complex] :
% 0.35/0.73        ( ( finite_finite_int @ S2 )
% 0.35/0.73       => ( ( ( member_int @ A @ S2 )
% 0.35/0.73           => ( ( groups7440179247065528705omplex
% 0.35/0.73                @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = ( B @ A ) ) )
% 0.35/0.73          & ( ~ ( member_int @ A @ S2 )
% 0.35/0.73           => ( ( groups7440179247065528705omplex
% 0.35/0.73                @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = one_one_complex ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.delta'
% 0.35/0.73  thf(fact_8449_prod_Odelta_H,axiom,
% 0.35/0.73      ! [S2: set_complex,A: complex,B: complex > complex] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.73       => ( ( ( member_complex @ A @ S2 )
% 0.35/0.73           => ( ( groups3708469109370488835omplex
% 0.35/0.73                @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = ( B @ A ) ) )
% 0.35/0.73          & ( ~ ( member_complex @ A @ S2 )
% 0.35/0.73           => ( ( groups3708469109370488835omplex
% 0.35/0.73                @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = one_one_complex ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.delta'
% 0.35/0.73  thf(fact_8450_prod_Odelta_H,axiom,
% 0.35/0.73      ! [S2: set_real,A: real,B: real > real] :
% 0.35/0.73        ( ( finite_finite_real @ S2 )
% 0.35/0.73       => ( ( ( member_real @ A @ S2 )
% 0.35/0.73           => ( ( groups1681761925125756287l_real
% 0.35/0.73                @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = ( B @ A ) ) )
% 0.35/0.73          & ( ~ ( member_real @ A @ S2 )
% 0.35/0.73           => ( ( groups1681761925125756287l_real
% 0.35/0.73                @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = one_one_real ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.delta'
% 0.35/0.73  thf(fact_8451_prod_Odelta_H,axiom,
% 0.35/0.73      ! [S2: set_nat,A: nat,B: nat > real] :
% 0.35/0.73        ( ( finite_finite_nat @ S2 )
% 0.35/0.73       => ( ( ( member_nat @ A @ S2 )
% 0.35/0.73           => ( ( groups129246275422532515t_real
% 0.35/0.73                @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = ( B @ A ) ) )
% 0.35/0.73          & ( ~ ( member_nat @ A @ S2 )
% 0.35/0.73           => ( ( groups129246275422532515t_real
% 0.35/0.73                @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = one_one_real ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.delta'
% 0.35/0.73  thf(fact_8452_prod_Odelta_H,axiom,
% 0.35/0.73      ! [S2: set_int,A: int,B: int > real] :
% 0.35/0.73        ( ( finite_finite_int @ S2 )
% 0.35/0.73       => ( ( ( member_int @ A @ S2 )
% 0.35/0.73           => ( ( groups2316167850115554303t_real
% 0.35/0.73                @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = ( B @ A ) ) )
% 0.35/0.73          & ( ~ ( member_int @ A @ S2 )
% 0.35/0.73           => ( ( groups2316167850115554303t_real
% 0.35/0.73                @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = one_one_real ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.delta'
% 0.35/0.73  thf(fact_8453_prod_Odelta_H,axiom,
% 0.35/0.73      ! [S2: set_complex,A: complex,B: complex > real] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.73       => ( ( ( member_complex @ A @ S2 )
% 0.35/0.73           => ( ( groups766887009212190081x_real
% 0.35/0.73                @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = ( B @ A ) ) )
% 0.35/0.73          & ( ~ ( member_complex @ A @ S2 )
% 0.35/0.73           => ( ( groups766887009212190081x_real
% 0.35/0.73                @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = one_one_real ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.delta'
% 0.35/0.73  thf(fact_8454_prod_Odelta_H,axiom,
% 0.35/0.73      ! [S2: set_real,A: real,B: real > rat] :
% 0.35/0.73        ( ( finite_finite_real @ S2 )
% 0.35/0.73       => ( ( ( member_real @ A @ S2 )
% 0.35/0.73           => ( ( groups4061424788464935467al_rat
% 0.35/0.73                @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = ( B @ A ) ) )
% 0.35/0.73          & ( ~ ( member_real @ A @ S2 )
% 0.35/0.73           => ( ( groups4061424788464935467al_rat
% 0.35/0.73                @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = one_one_rat ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.delta'
% 0.35/0.73  thf(fact_8455_prod_Odelta_H,axiom,
% 0.35/0.73      ! [S2: set_nat,A: nat,B: nat > rat] :
% 0.35/0.73        ( ( finite_finite_nat @ S2 )
% 0.35/0.73       => ( ( ( member_nat @ A @ S2 )
% 0.35/0.73           => ( ( groups73079841787564623at_rat
% 0.35/0.73                @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = ( B @ A ) ) )
% 0.35/0.73          & ( ~ ( member_nat @ A @ S2 )
% 0.35/0.73           => ( ( groups73079841787564623at_rat
% 0.35/0.73                @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = one_one_rat ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.delta'
% 0.35/0.73  thf(fact_8456_prod_Odelta,axiom,
% 0.35/0.73      ! [S2: set_real,A: real,B: real > complex] :
% 0.35/0.73        ( ( finite_finite_real @ S2 )
% 0.35/0.73       => ( ( ( member_real @ A @ S2 )
% 0.35/0.73           => ( ( groups713298508707869441omplex
% 0.35/0.73                @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = ( B @ A ) ) )
% 0.35/0.73          & ( ~ ( member_real @ A @ S2 )
% 0.35/0.73           => ( ( groups713298508707869441omplex
% 0.35/0.73                @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = one_one_complex ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.delta
% 0.35/0.73  thf(fact_8457_prod_Odelta,axiom,
% 0.35/0.73      ! [S2: set_nat,A: nat,B: nat > complex] :
% 0.35/0.73        ( ( finite_finite_nat @ S2 )
% 0.35/0.73       => ( ( ( member_nat @ A @ S2 )
% 0.35/0.73           => ( ( groups6464643781859351333omplex
% 0.35/0.73                @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = ( B @ A ) ) )
% 0.35/0.73          & ( ~ ( member_nat @ A @ S2 )
% 0.35/0.73           => ( ( groups6464643781859351333omplex
% 0.35/0.73                @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = one_one_complex ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.delta
% 0.35/0.73  thf(fact_8458_prod_Odelta,axiom,
% 0.35/0.73      ! [S2: set_int,A: int,B: int > complex] :
% 0.35/0.73        ( ( finite_finite_int @ S2 )
% 0.35/0.73       => ( ( ( member_int @ A @ S2 )
% 0.35/0.73           => ( ( groups7440179247065528705omplex
% 0.35/0.73                @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = ( B @ A ) ) )
% 0.35/0.73          & ( ~ ( member_int @ A @ S2 )
% 0.35/0.73           => ( ( groups7440179247065528705omplex
% 0.35/0.73                @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = one_one_complex ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.delta
% 0.35/0.73  thf(fact_8459_prod_Odelta,axiom,
% 0.35/0.73      ! [S2: set_complex,A: complex,B: complex > complex] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.73       => ( ( ( member_complex @ A @ S2 )
% 0.35/0.73           => ( ( groups3708469109370488835omplex
% 0.35/0.73                @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = ( B @ A ) ) )
% 0.35/0.73          & ( ~ ( member_complex @ A @ S2 )
% 0.35/0.73           => ( ( groups3708469109370488835omplex
% 0.35/0.73                @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = one_one_complex ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.delta
% 0.35/0.73  thf(fact_8460_prod_Odelta,axiom,
% 0.35/0.73      ! [S2: set_real,A: real,B: real > real] :
% 0.35/0.73        ( ( finite_finite_real @ S2 )
% 0.35/0.73       => ( ( ( member_real @ A @ S2 )
% 0.35/0.73           => ( ( groups1681761925125756287l_real
% 0.35/0.73                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = ( B @ A ) ) )
% 0.35/0.73          & ( ~ ( member_real @ A @ S2 )
% 0.35/0.73           => ( ( groups1681761925125756287l_real
% 0.35/0.73                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = one_one_real ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.delta
% 0.35/0.73  thf(fact_8461_prod_Odelta,axiom,
% 0.35/0.73      ! [S2: set_nat,A: nat,B: nat > real] :
% 0.35/0.73        ( ( finite_finite_nat @ S2 )
% 0.35/0.73       => ( ( ( member_nat @ A @ S2 )
% 0.35/0.73           => ( ( groups129246275422532515t_real
% 0.35/0.73                @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = ( B @ A ) ) )
% 0.35/0.73          & ( ~ ( member_nat @ A @ S2 )
% 0.35/0.73           => ( ( groups129246275422532515t_real
% 0.35/0.73                @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = one_one_real ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.delta
% 0.35/0.73  thf(fact_8462_prod_Odelta,axiom,
% 0.35/0.73      ! [S2: set_int,A: int,B: int > real] :
% 0.35/0.73        ( ( finite_finite_int @ S2 )
% 0.35/0.73       => ( ( ( member_int @ A @ S2 )
% 0.35/0.73           => ( ( groups2316167850115554303t_real
% 0.35/0.73                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = ( B @ A ) ) )
% 0.35/0.73          & ( ~ ( member_int @ A @ S2 )
% 0.35/0.73           => ( ( groups2316167850115554303t_real
% 0.35/0.73                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = one_one_real ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.delta
% 0.35/0.73  thf(fact_8463_prod_Odelta,axiom,
% 0.35/0.73      ! [S2: set_complex,A: complex,B: complex > real] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.73       => ( ( ( member_complex @ A @ S2 )
% 0.35/0.73           => ( ( groups766887009212190081x_real
% 0.35/0.73                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = ( B @ A ) ) )
% 0.35/0.73          & ( ~ ( member_complex @ A @ S2 )
% 0.35/0.73           => ( ( groups766887009212190081x_real
% 0.35/0.73                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = one_one_real ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.delta
% 0.35/0.73  thf(fact_8464_prod_Odelta,axiom,
% 0.35/0.73      ! [S2: set_real,A: real,B: real > rat] :
% 0.35/0.73        ( ( finite_finite_real @ S2 )
% 0.35/0.73       => ( ( ( member_real @ A @ S2 )
% 0.35/0.73           => ( ( groups4061424788464935467al_rat
% 0.35/0.73                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = ( B @ A ) ) )
% 0.35/0.73          & ( ~ ( member_real @ A @ S2 )
% 0.35/0.73           => ( ( groups4061424788464935467al_rat
% 0.35/0.73                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = one_one_rat ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.delta
% 0.35/0.73  thf(fact_8465_prod_Odelta,axiom,
% 0.35/0.73      ! [S2: set_nat,A: nat,B: nat > rat] :
% 0.35/0.73        ( ( finite_finite_nat @ S2 )
% 0.35/0.73       => ( ( ( member_nat @ A @ S2 )
% 0.35/0.73           => ( ( groups73079841787564623at_rat
% 0.35/0.73                @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = ( B @ A ) ) )
% 0.35/0.73          & ( ~ ( member_nat @ A @ S2 )
% 0.35/0.73           => ( ( groups73079841787564623at_rat
% 0.35/0.73                @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 0.35/0.73                @ S2 )
% 0.35/0.73              = one_one_rat ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.delta
% 0.35/0.73  thf(fact_8466_is__singletonI,axiom,
% 0.35/0.73      ! [X: produc3843707927480180839at_nat] : ( is_sin2937591304547752795at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ).
% 0.35/0.73  
% 0.35/0.73  % is_singletonI
% 0.35/0.73  thf(fact_8467_is__singletonI,axiom,
% 0.35/0.73      ! [X: product_prod_nat_nat] : ( is_sin2850979758926227957at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ).
% 0.35/0.73  
% 0.35/0.73  % is_singletonI
% 0.35/0.73  thf(fact_8468_is__singletonI,axiom,
% 0.35/0.73      ! [X: real] : ( is_singleton_real @ ( insert_real @ X @ bot_bot_set_real ) ) ).
% 0.35/0.73  
% 0.35/0.73  % is_singletonI
% 0.35/0.73  thf(fact_8469_is__singletonI,axiom,
% 0.35/0.73      ! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% 0.35/0.73  
% 0.35/0.73  % is_singletonI
% 0.35/0.73  thf(fact_8470_is__singletonI,axiom,
% 0.35/0.73      ! [X: int] : ( is_singleton_int @ ( insert_int @ X @ bot_bot_set_int ) ) ).
% 0.35/0.73  
% 0.35/0.73  % is_singletonI
% 0.35/0.73  thf(fact_8471_prod_Oinsert,axiom,
% 0.35/0.73      ! [A2: set_real,X: real,G: real > complex] :
% 0.35/0.73        ( ( finite_finite_real @ A2 )
% 0.35/0.73       => ( ~ ( member_real @ X @ A2 )
% 0.35/0.73         => ( ( groups713298508707869441omplex @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.73            = ( times_times_complex @ ( G @ X ) @ ( groups713298508707869441omplex @ G @ A2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.insert
% 0.35/0.73  thf(fact_8472_prod_Oinsert,axiom,
% 0.35/0.73      ! [A2: set_nat,X: nat,G: nat > complex] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ~ ( member_nat @ X @ A2 )
% 0.35/0.73         => ( ( groups6464643781859351333omplex @ G @ ( insert_nat @ X @ A2 ) )
% 0.35/0.73            = ( times_times_complex @ ( G @ X ) @ ( groups6464643781859351333omplex @ G @ A2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.insert
% 0.35/0.73  thf(fact_8473_prod_Oinsert,axiom,
% 0.35/0.73      ! [A2: set_int,X: int,G: int > complex] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ~ ( member_int @ X @ A2 )
% 0.35/0.73         => ( ( groups7440179247065528705omplex @ G @ ( insert_int @ X @ A2 ) )
% 0.35/0.73            = ( times_times_complex @ ( G @ X ) @ ( groups7440179247065528705omplex @ G @ A2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.insert
% 0.35/0.73  thf(fact_8474_prod_Oinsert,axiom,
% 0.35/0.73      ! [A2: set_complex,X: complex,G: complex > complex] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.73       => ( ~ ( member_complex @ X @ A2 )
% 0.35/0.73         => ( ( groups3708469109370488835omplex @ G @ ( insert_complex @ X @ A2 ) )
% 0.35/0.73            = ( times_times_complex @ ( G @ X ) @ ( groups3708469109370488835omplex @ G @ A2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.insert
% 0.35/0.73  thf(fact_8475_prod_Oinsert,axiom,
% 0.35/0.73      ! [A2: set_real,X: real,G: real > real] :
% 0.35/0.73        ( ( finite_finite_real @ A2 )
% 0.35/0.73       => ( ~ ( member_real @ X @ A2 )
% 0.35/0.73         => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.73            = ( times_times_real @ ( G @ X ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.insert
% 0.35/0.73  thf(fact_8476_prod_Oinsert,axiom,
% 0.35/0.73      ! [A2: set_nat,X: nat,G: nat > real] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ~ ( member_nat @ X @ A2 )
% 0.35/0.73         => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X @ A2 ) )
% 0.35/0.73            = ( times_times_real @ ( G @ X ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.insert
% 0.35/0.73  thf(fact_8477_prod_Oinsert,axiom,
% 0.35/0.73      ! [A2: set_int,X: int,G: int > real] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ~ ( member_int @ X @ A2 )
% 0.35/0.73         => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X @ A2 ) )
% 0.35/0.73            = ( times_times_real @ ( G @ X ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.insert
% 0.35/0.73  thf(fact_8478_prod_Oinsert,axiom,
% 0.35/0.73      ! [A2: set_complex,X: complex,G: complex > real] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.73       => ( ~ ( member_complex @ X @ A2 )
% 0.35/0.73         => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X @ A2 ) )
% 0.35/0.73            = ( times_times_real @ ( G @ X ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.insert
% 0.35/0.73  thf(fact_8479_prod_Oinsert,axiom,
% 0.35/0.73      ! [A2: set_real,X: real,G: real > rat] :
% 0.35/0.73        ( ( finite_finite_real @ A2 )
% 0.35/0.73       => ( ~ ( member_real @ X @ A2 )
% 0.35/0.73         => ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.73            = ( times_times_rat @ ( G @ X ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.insert
% 0.35/0.73  thf(fact_8480_prod_Oinsert,axiom,
% 0.35/0.73      ! [A2: set_nat,X: nat,G: nat > rat] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ~ ( member_nat @ X @ A2 )
% 0.35/0.73         => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A2 ) )
% 0.35/0.73            = ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.insert
% 0.35/0.73  thf(fact_8481_atLeastLessThan__singleton,axiom,
% 0.35/0.73      ! [M: nat] :
% 0.35/0.73        ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
% 0.35/0.73        = ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_singleton
% 0.35/0.73  thf(fact_8482_sum_Oop__ivl__Suc,axiom,
% 0.35/0.73      ! [N: nat,M: nat,G: nat > complex] :
% 0.35/0.73        ( ( ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = zero_zero_complex ) )
% 0.35/0.73        & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % sum.op_ivl_Suc
% 0.35/0.73  thf(fact_8483_sum_Oop__ivl__Suc,axiom,
% 0.35/0.73      ! [N: nat,M: nat,G: nat > rat] :
% 0.35/0.73        ( ( ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = zero_zero_rat ) )
% 0.35/0.73        & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % sum.op_ivl_Suc
% 0.35/0.73  thf(fact_8484_sum_Oop__ivl__Suc,axiom,
% 0.35/0.73      ! [N: nat,M: nat,G: nat > int] :
% 0.35/0.73        ( ( ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = zero_zero_int ) )
% 0.35/0.73        & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % sum.op_ivl_Suc
% 0.35/0.73  thf(fact_8485_sum_Oop__ivl__Suc,axiom,
% 0.35/0.73      ! [N: nat,M: nat,G: nat > nat] :
% 0.35/0.73        ( ( ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = zero_zero_nat ) )
% 0.35/0.73        & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % sum.op_ivl_Suc
% 0.35/0.73  thf(fact_8486_sum_Oop__ivl__Suc,axiom,
% 0.35/0.73      ! [N: nat,M: nat,G: nat > real] :
% 0.35/0.73        ( ( ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = zero_zero_real ) )
% 0.35/0.73        & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % sum.op_ivl_Suc
% 0.35/0.73  thf(fact_8487_prod_Ocl__ivl__Suc,axiom,
% 0.35/0.73      ! [N: nat,M: nat,G: nat > complex] :
% 0.35/0.73        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.73         => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = one_one_complex ) )
% 0.35/0.73        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.73         => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.cl_ivl_Suc
% 0.35/0.73  thf(fact_8488_prod_Ocl__ivl__Suc,axiom,
% 0.35/0.73      ! [N: nat,M: nat,G: nat > real] :
% 0.35/0.73        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.73         => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = one_one_real ) )
% 0.35/0.73        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.73         => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.cl_ivl_Suc
% 0.35/0.73  thf(fact_8489_prod_Ocl__ivl__Suc,axiom,
% 0.35/0.73      ! [N: nat,M: nat,G: nat > rat] :
% 0.35/0.73        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.73         => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = one_one_rat ) )
% 0.35/0.73        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.73         => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.cl_ivl_Suc
% 0.35/0.73  thf(fact_8490_prod_Ocl__ivl__Suc,axiom,
% 0.35/0.73      ! [N: nat,M: nat,G: nat > int] :
% 0.35/0.73        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.73         => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = one_one_int ) )
% 0.35/0.73        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.73         => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.cl_ivl_Suc
% 0.35/0.73  thf(fact_8491_prod_Ocl__ivl__Suc,axiom,
% 0.35/0.73      ! [N: nat,M: nat,G: nat > nat] :
% 0.35/0.73        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.73         => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = one_one_nat ) )
% 0.35/0.73        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.73         => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.cl_ivl_Suc
% 0.35/0.73  thf(fact_8492_prod_Oop__ivl__Suc,axiom,
% 0.35/0.73      ! [N: nat,M: nat,G: nat > complex] :
% 0.35/0.73        ( ( ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = one_one_complex ) )
% 0.35/0.73        & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.op_ivl_Suc
% 0.35/0.73  thf(fact_8493_prod_Oop__ivl__Suc,axiom,
% 0.35/0.73      ! [N: nat,M: nat,G: nat > real] :
% 0.35/0.73        ( ( ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = one_one_real ) )
% 0.35/0.73        & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.op_ivl_Suc
% 0.35/0.73  thf(fact_8494_prod_Oop__ivl__Suc,axiom,
% 0.35/0.73      ! [N: nat,M: nat,G: nat > rat] :
% 0.35/0.73        ( ( ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = one_one_rat ) )
% 0.35/0.73        & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.op_ivl_Suc
% 0.35/0.73  thf(fact_8495_prod_Oop__ivl__Suc,axiom,
% 0.35/0.73      ! [N: nat,M: nat,G: nat > int] :
% 0.35/0.73        ( ( ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = one_one_int ) )
% 0.35/0.73        & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.op_ivl_Suc
% 0.35/0.73  thf(fact_8496_prod_Oop__ivl__Suc,axiom,
% 0.35/0.73      ! [N: nat,M: nat,G: nat > nat] :
% 0.35/0.73        ( ( ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = one_one_nat ) )
% 0.35/0.73        & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 0.35/0.73            = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.op_ivl_Suc
% 0.35/0.73  thf(fact_8497_sgn__rat__def,axiom,
% 0.35/0.73      ( sgn_sgn_rat
% 0.35/0.73      = ( ^ [A4: rat] : ( if_rat @ ( A4 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A4 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % sgn_rat_def
% 0.35/0.73  thf(fact_8498_abs__rat__def,axiom,
% 0.35/0.73      ( abs_abs_rat
% 0.35/0.73      = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % abs_rat_def
% 0.35/0.73  thf(fact_8499_prod_OatLeastLessThan__concat,axiom,
% 0.35/0.73      ! [M: nat,N: nat,P4: nat,G: nat > complex] :
% 0.35/0.73        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.73       => ( ( ord_less_eq_nat @ N @ P4 )
% 0.35/0.73         => ( ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) )
% 0.35/0.73            = ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.atLeastLessThan_concat
% 0.35/0.73  thf(fact_8500_prod_OatLeastLessThan__concat,axiom,
% 0.35/0.73      ! [M: nat,N: nat,P4: nat,G: nat > real] :
% 0.35/0.73        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.73       => ( ( ord_less_eq_nat @ N @ P4 )
% 0.35/0.73         => ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) )
% 0.35/0.73            = ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.atLeastLessThan_concat
% 0.35/0.73  thf(fact_8501_prod_OatLeastLessThan__concat,axiom,
% 0.35/0.73      ! [M: nat,N: nat,P4: nat,G: nat > rat] :
% 0.35/0.73        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.73       => ( ( ord_less_eq_nat @ N @ P4 )
% 0.35/0.73         => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) )
% 0.35/0.73            = ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.atLeastLessThan_concat
% 0.35/0.73  thf(fact_8502_prod_OatLeastLessThan__concat,axiom,
% 0.35/0.73      ! [M: nat,N: nat,P4: nat,G: nat > int] :
% 0.35/0.73        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.73       => ( ( ord_less_eq_nat @ N @ P4 )
% 0.35/0.73         => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) )
% 0.35/0.73            = ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.atLeastLessThan_concat
% 0.35/0.73  thf(fact_8503_prod_OatLeastLessThan__concat,axiom,
% 0.35/0.73      ! [M: nat,N: nat,P4: nat,G: nat > nat] :
% 0.35/0.73        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.73       => ( ( ord_less_eq_nat @ N @ P4 )
% 0.35/0.73         => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) )
% 0.35/0.73            = ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.atLeastLessThan_concat
% 0.35/0.73  thf(fact_8504_less__eq__rat__def,axiom,
% 0.35/0.73      ( ord_less_eq_rat
% 0.35/0.73      = ( ^ [X3: rat,Y2: rat] :
% 0.35/0.73            ( ( ord_less_rat @ X3 @ Y2 )
% 0.35/0.73            | ( X3 = Y2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % less_eq_rat_def
% 0.35/0.73  thf(fact_8505_prod_Oivl__cong,axiom,
% 0.35/0.73      ! [A: nat,C: nat,B: nat,D: nat,G: nat > nat,H2: nat > nat] :
% 0.35/0.73        ( ( A = C )
% 0.35/0.73       => ( ( B = D )
% 0.35/0.73         => ( ! [X4: nat] :
% 0.35/0.73                ( ( ord_less_eq_nat @ C @ X4 )
% 0.35/0.73               => ( ( ord_less_nat @ X4 @ D )
% 0.35/0.73                 => ( ( G @ X4 )
% 0.35/0.73                    = ( H2 @ X4 ) ) ) )
% 0.35/0.73           => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
% 0.35/0.73              = ( groups708209901874060359at_nat @ H2 @ ( set_or4665077453230672383an_nat @ C @ D ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.ivl_cong
% 0.35/0.73  thf(fact_8506_obtain__pos__sum,axiom,
% 0.35/0.73      ! [R2: rat] :
% 0.35/0.73        ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 0.35/0.73       => ~ ! [S4: rat] :
% 0.35/0.73              ( ( ord_less_rat @ zero_zero_rat @ S4 )
% 0.35/0.73             => ! [T6: rat] :
% 0.35/0.73                  ( ( ord_less_rat @ zero_zero_rat @ T6 )
% 0.35/0.73                 => ( R2
% 0.35/0.73                   != ( plus_plus_rat @ S4 @ T6 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % obtain_pos_sum
% 0.35/0.73  thf(fact_8507_atLeastLessThan__inj_I2_J,axiom,
% 0.35/0.73      ! [A: real,B: real,C: real,D: real] :
% 0.35/0.73        ( ( ( set_or66887138388493659n_real @ A @ B )
% 0.35/0.73          = ( set_or66887138388493659n_real @ C @ D ) )
% 0.35/0.73       => ( ( ord_less_real @ A @ B )
% 0.35/0.73         => ( ( ord_less_real @ C @ D )
% 0.35/0.73           => ( B = D ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_inj(2)
% 0.35/0.73  thf(fact_8508_atLeastLessThan__inj_I2_J,axiom,
% 0.35/0.73      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.35/0.73        ( ( ( set_or4029947393144176647an_rat @ A @ B )
% 0.35/0.73          = ( set_or4029947393144176647an_rat @ C @ D ) )
% 0.35/0.73       => ( ( ord_less_rat @ A @ B )
% 0.35/0.73         => ( ( ord_less_rat @ C @ D )
% 0.35/0.73           => ( B = D ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_inj(2)
% 0.35/0.73  thf(fact_8509_atLeastLessThan__inj_I2_J,axiom,
% 0.35/0.73      ! [A: num,B: num,C: num,D: num] :
% 0.35/0.73        ( ( ( set_or1222409239386451017an_num @ A @ B )
% 0.35/0.73          = ( set_or1222409239386451017an_num @ C @ D ) )
% 0.35/0.73       => ( ( ord_less_num @ A @ B )
% 0.35/0.73         => ( ( ord_less_num @ C @ D )
% 0.35/0.73           => ( B = D ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_inj(2)
% 0.35/0.73  thf(fact_8510_atLeastLessThan__inj_I2_J,axiom,
% 0.35/0.73      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.35/0.73        ( ( ( set_or4665077453230672383an_nat @ A @ B )
% 0.35/0.73          = ( set_or4665077453230672383an_nat @ C @ D ) )
% 0.35/0.73       => ( ( ord_less_nat @ A @ B )
% 0.35/0.73         => ( ( ord_less_nat @ C @ D )
% 0.35/0.73           => ( B = D ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_inj(2)
% 0.35/0.73  thf(fact_8511_atLeastLessThan__inj_I2_J,axiom,
% 0.35/0.73      ! [A: int,B: int,C: int,D: int] :
% 0.35/0.73        ( ( ( set_or4662586982721622107an_int @ A @ B )
% 0.35/0.73          = ( set_or4662586982721622107an_int @ C @ D ) )
% 0.35/0.73       => ( ( ord_less_int @ A @ B )
% 0.35/0.73         => ( ( ord_less_int @ C @ D )
% 0.35/0.73           => ( B = D ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_inj(2)
% 0.35/0.73  thf(fact_8512_atLeastLessThan__inj_I1_J,axiom,
% 0.35/0.73      ! [A: real,B: real,C: real,D: real] :
% 0.35/0.73        ( ( ( set_or66887138388493659n_real @ A @ B )
% 0.35/0.73          = ( set_or66887138388493659n_real @ C @ D ) )
% 0.35/0.73       => ( ( ord_less_real @ A @ B )
% 0.35/0.73         => ( ( ord_less_real @ C @ D )
% 0.35/0.73           => ( A = C ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_inj(1)
% 0.35/0.73  thf(fact_8513_atLeastLessThan__inj_I1_J,axiom,
% 0.35/0.73      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.35/0.73        ( ( ( set_or4029947393144176647an_rat @ A @ B )
% 0.35/0.73          = ( set_or4029947393144176647an_rat @ C @ D ) )
% 0.35/0.73       => ( ( ord_less_rat @ A @ B )
% 0.35/0.73         => ( ( ord_less_rat @ C @ D )
% 0.35/0.73           => ( A = C ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_inj(1)
% 0.35/0.73  thf(fact_8514_atLeastLessThan__inj_I1_J,axiom,
% 0.35/0.73      ! [A: num,B: num,C: num,D: num] :
% 0.35/0.73        ( ( ( set_or1222409239386451017an_num @ A @ B )
% 0.35/0.73          = ( set_or1222409239386451017an_num @ C @ D ) )
% 0.35/0.73       => ( ( ord_less_num @ A @ B )
% 0.35/0.73         => ( ( ord_less_num @ C @ D )
% 0.35/0.73           => ( A = C ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_inj(1)
% 0.35/0.73  thf(fact_8515_atLeastLessThan__inj_I1_J,axiom,
% 0.35/0.73      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.35/0.73        ( ( ( set_or4665077453230672383an_nat @ A @ B )
% 0.35/0.73          = ( set_or4665077453230672383an_nat @ C @ D ) )
% 0.35/0.73       => ( ( ord_less_nat @ A @ B )
% 0.35/0.73         => ( ( ord_less_nat @ C @ D )
% 0.35/0.73           => ( A = C ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_inj(1)
% 0.35/0.73  thf(fact_8516_atLeastLessThan__inj_I1_J,axiom,
% 0.35/0.73      ! [A: int,B: int,C: int,D: int] :
% 0.35/0.73        ( ( ( set_or4662586982721622107an_int @ A @ B )
% 0.35/0.73          = ( set_or4662586982721622107an_int @ C @ D ) )
% 0.35/0.73       => ( ( ord_less_int @ A @ B )
% 0.35/0.73         => ( ( ord_less_int @ C @ D )
% 0.35/0.73           => ( A = C ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_inj(1)
% 0.35/0.73  thf(fact_8517_atLeastLessThan__eq__iff,axiom,
% 0.35/0.73      ! [A: real,B: real,C: real,D: real] :
% 0.35/0.73        ( ( ord_less_real @ A @ B )
% 0.35/0.73       => ( ( ord_less_real @ C @ D )
% 0.35/0.73         => ( ( ( set_or66887138388493659n_real @ A @ B )
% 0.35/0.73              = ( set_or66887138388493659n_real @ C @ D ) )
% 0.35/0.73            = ( ( A = C )
% 0.35/0.73              & ( B = D ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_eq_iff
% 0.35/0.73  thf(fact_8518_atLeastLessThan__eq__iff,axiom,
% 0.35/0.73      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.35/0.73        ( ( ord_less_rat @ A @ B )
% 0.35/0.73       => ( ( ord_less_rat @ C @ D )
% 0.35/0.73         => ( ( ( set_or4029947393144176647an_rat @ A @ B )
% 0.35/0.73              = ( set_or4029947393144176647an_rat @ C @ D ) )
% 0.35/0.73            = ( ( A = C )
% 0.35/0.73              & ( B = D ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_eq_iff
% 0.35/0.73  thf(fact_8519_atLeastLessThan__eq__iff,axiom,
% 0.35/0.73      ! [A: num,B: num,C: num,D: num] :
% 0.35/0.73        ( ( ord_less_num @ A @ B )
% 0.35/0.73       => ( ( ord_less_num @ C @ D )
% 0.35/0.73         => ( ( ( set_or1222409239386451017an_num @ A @ B )
% 0.35/0.73              = ( set_or1222409239386451017an_num @ C @ D ) )
% 0.35/0.73            = ( ( A = C )
% 0.35/0.73              & ( B = D ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_eq_iff
% 0.35/0.73  thf(fact_8520_atLeastLessThan__eq__iff,axiom,
% 0.35/0.73      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.35/0.73        ( ( ord_less_nat @ A @ B )
% 0.35/0.73       => ( ( ord_less_nat @ C @ D )
% 0.35/0.73         => ( ( ( set_or4665077453230672383an_nat @ A @ B )
% 0.35/0.73              = ( set_or4665077453230672383an_nat @ C @ D ) )
% 0.35/0.73            = ( ( A = C )
% 0.35/0.73              & ( B = D ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_eq_iff
% 0.35/0.73  thf(fact_8521_atLeastLessThan__eq__iff,axiom,
% 0.35/0.73      ! [A: int,B: int,C: int,D: int] :
% 0.35/0.73        ( ( ord_less_int @ A @ B )
% 0.35/0.73       => ( ( ord_less_int @ C @ D )
% 0.35/0.73         => ( ( ( set_or4662586982721622107an_int @ A @ B )
% 0.35/0.73              = ( set_or4662586982721622107an_int @ C @ D ) )
% 0.35/0.73            = ( ( A = C )
% 0.35/0.73              & ( B = D ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_eq_iff
% 0.35/0.73  thf(fact_8522_prod_OatLeast0__lessThan__Suc,axiom,
% 0.35/0.73      ! [G: nat > complex,N: nat] :
% 0.35/0.73        ( ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.35/0.73        = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.atLeast0_lessThan_Suc
% 0.35/0.73  thf(fact_8523_prod_OatLeast0__lessThan__Suc,axiom,
% 0.35/0.73      ! [G: nat > real,N: nat] :
% 0.35/0.73        ( ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.35/0.73        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.atLeast0_lessThan_Suc
% 0.35/0.73  thf(fact_8524_prod_OatLeast0__lessThan__Suc,axiom,
% 0.35/0.73      ! [G: nat > rat,N: nat] :
% 0.35/0.73        ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.35/0.73        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.atLeast0_lessThan_Suc
% 0.35/0.73  thf(fact_8525_prod_OatLeast0__lessThan__Suc,axiom,
% 0.35/0.73      ! [G: nat > int,N: nat] :
% 0.35/0.73        ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.35/0.73        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.atLeast0_lessThan_Suc
% 0.35/0.73  thf(fact_8526_prod_OatLeast0__lessThan__Suc,axiom,
% 0.35/0.73      ! [G: nat > nat,N: nat] :
% 0.35/0.73        ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.35/0.73        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.atLeast0_lessThan_Suc
% 0.35/0.73  thf(fact_8527_prod_OatLeast__Suc__lessThan,axiom,
% 0.35/0.73      ! [M: nat,N: nat,G: nat > complex] :
% 0.35/0.73        ( ( ord_less_nat @ M @ N )
% 0.35/0.73       => ( ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 0.35/0.73          = ( times_times_complex @ ( G @ M ) @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.atLeast_Suc_lessThan
% 0.35/0.73  thf(fact_8528_prod_OatLeast__Suc__lessThan,axiom,
% 0.35/0.73      ! [M: nat,N: nat,G: nat > real] :
% 0.35/0.73        ( ( ord_less_nat @ M @ N )
% 0.35/0.73       => ( ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 0.35/0.73          = ( times_times_real @ ( G @ M ) @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.atLeast_Suc_lessThan
% 0.35/0.73  thf(fact_8529_prod_OatLeast__Suc__lessThan,axiom,
% 0.35/0.73      ! [M: nat,N: nat,G: nat > rat] :
% 0.35/0.73        ( ( ord_less_nat @ M @ N )
% 0.35/0.73       => ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 0.35/0.73          = ( times_times_rat @ ( G @ M ) @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.atLeast_Suc_lessThan
% 0.35/0.73  thf(fact_8530_prod_OatLeast__Suc__lessThan,axiom,
% 0.35/0.73      ! [M: nat,N: nat,G: nat > int] :
% 0.35/0.73        ( ( ord_less_nat @ M @ N )
% 0.35/0.73       => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 0.35/0.73          = ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.atLeast_Suc_lessThan
% 0.35/0.73  thf(fact_8531_prod_OatLeast__Suc__lessThan,axiom,
% 0.35/0.73      ! [M: nat,N: nat,G: nat > nat] :
% 0.35/0.73        ( ( ord_less_nat @ M @ N )
% 0.35/0.73       => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 0.35/0.73          = ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.atLeast_Suc_lessThan
% 0.35/0.73  thf(fact_8532_prod_OatLeastLessThan__Suc,axiom,
% 0.35/0.73      ! [A: nat,B: nat,G: nat > complex] :
% 0.35/0.73        ( ( ord_less_eq_nat @ A @ B )
% 0.35/0.73       => ( ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
% 0.35/0.73          = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.atLeastLessThan_Suc
% 0.35/0.73  thf(fact_8533_prod_OatLeastLessThan__Suc,axiom,
% 0.35/0.73      ! [A: nat,B: nat,G: nat > real] :
% 0.35/0.73        ( ( ord_less_eq_nat @ A @ B )
% 0.35/0.73       => ( ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
% 0.35/0.73          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.atLeastLessThan_Suc
% 0.35/0.73  thf(fact_8534_prod_OatLeastLessThan__Suc,axiom,
% 0.35/0.73      ! [A: nat,B: nat,G: nat > rat] :
% 0.35/0.73        ( ( ord_less_eq_nat @ A @ B )
% 0.35/0.73       => ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
% 0.35/0.73          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.atLeastLessThan_Suc
% 0.35/0.73  thf(fact_8535_prod_OatLeastLessThan__Suc,axiom,
% 0.35/0.73      ! [A: nat,B: nat,G: nat > int] :
% 0.35/0.73        ( ( ord_less_eq_nat @ A @ B )
% 0.35/0.73       => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
% 0.35/0.73          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.atLeastLessThan_Suc
% 0.35/0.73  thf(fact_8536_prod_OatLeastLessThan__Suc,axiom,
% 0.35/0.73      ! [A: nat,B: nat,G: nat > nat] :
% 0.35/0.73        ( ( ord_less_eq_nat @ A @ B )
% 0.35/0.73       => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
% 0.35/0.73          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.atLeastLessThan_Suc
% 0.35/0.73  thf(fact_8537_prod_Oswap__restrict,axiom,
% 0.35/0.73      ! [A2: set_real,B2: set_nat,G: real > nat > nat,R: real > nat > $o] :
% 0.35/0.73        ( ( finite_finite_real @ A2 )
% 0.35/0.73       => ( ( finite_finite_nat @ B2 )
% 0.35/0.73         => ( ( groups4696554848551431203al_nat
% 0.35/0.73              @ ^ [X3: real] :
% 0.35/0.73                  ( groups708209901874060359at_nat @ ( G @ X3 )
% 0.35/0.73                  @ ( collect_nat
% 0.35/0.73                    @ ^ [Y2: nat] :
% 0.35/0.73                        ( ( member_nat @ Y2 @ B2 )
% 0.35/0.73                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.73              @ A2 )
% 0.35/0.73            = ( groups708209901874060359at_nat
% 0.35/0.73              @ ^ [Y2: nat] :
% 0.35/0.73                  ( groups4696554848551431203al_nat
% 0.35/0.73                  @ ^ [X3: real] : ( G @ X3 @ Y2 )
% 0.35/0.73                  @ ( collect_real
% 0.35/0.73                    @ ^ [X3: real] :
% 0.35/0.73                        ( ( member_real @ X3 @ A2 )
% 0.35/0.73                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.73              @ B2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.swap_restrict
% 0.35/0.73  thf(fact_8538_prod_Oswap__restrict,axiom,
% 0.35/0.73      ! [A2: set_int,B2: set_nat,G: int > nat > nat,R: int > nat > $o] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ( finite_finite_nat @ B2 )
% 0.35/0.73         => ( ( groups1707563613775114915nt_nat
% 0.35/0.73              @ ^ [X3: int] :
% 0.35/0.73                  ( groups708209901874060359at_nat @ ( G @ X3 )
% 0.35/0.73                  @ ( collect_nat
% 0.35/0.73                    @ ^ [Y2: nat] :
% 0.35/0.73                        ( ( member_nat @ Y2 @ B2 )
% 0.35/0.73                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.73              @ A2 )
% 0.35/0.73            = ( groups708209901874060359at_nat
% 0.35/0.73              @ ^ [Y2: nat] :
% 0.35/0.73                  ( groups1707563613775114915nt_nat
% 0.35/0.73                  @ ^ [X3: int] : ( G @ X3 @ Y2 )
% 0.35/0.73                  @ ( collect_int
% 0.35/0.73                    @ ^ [X3: int] :
% 0.35/0.73                        ( ( member_int @ X3 @ A2 )
% 0.35/0.73                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.73              @ B2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.swap_restrict
% 0.35/0.73  thf(fact_8539_prod_Oswap__restrict,axiom,
% 0.35/0.73      ! [A2: set_complex,B2: set_nat,G: complex > nat > nat,R: complex > nat > $o] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.73       => ( ( finite_finite_nat @ B2 )
% 0.35/0.73         => ( ( groups861055069439313189ex_nat
% 0.35/0.73              @ ^ [X3: complex] :
% 0.35/0.73                  ( groups708209901874060359at_nat @ ( G @ X3 )
% 0.35/0.73                  @ ( collect_nat
% 0.35/0.73                    @ ^ [Y2: nat] :
% 0.35/0.73                        ( ( member_nat @ Y2 @ B2 )
% 0.35/0.73                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.73              @ A2 )
% 0.35/0.73            = ( groups708209901874060359at_nat
% 0.35/0.73              @ ^ [Y2: nat] :
% 0.35/0.73                  ( groups861055069439313189ex_nat
% 0.35/0.73                  @ ^ [X3: complex] : ( G @ X3 @ Y2 )
% 0.35/0.73                  @ ( collect_complex
% 0.35/0.73                    @ ^ [X3: complex] :
% 0.35/0.73                        ( ( member_complex @ X3 @ A2 )
% 0.35/0.73                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.73              @ B2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.swap_restrict
% 0.35/0.73  thf(fact_8540_prod_Oswap__restrict,axiom,
% 0.35/0.73      ! [A2: set_nat,B2: set_real,G: nat > real > nat,R: nat > real > $o] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( finite_finite_real @ B2 )
% 0.35/0.73         => ( ( groups708209901874060359at_nat
% 0.35/0.73              @ ^ [X3: nat] :
% 0.35/0.73                  ( groups4696554848551431203al_nat @ ( G @ X3 )
% 0.35/0.73                  @ ( collect_real
% 0.35/0.73                    @ ^ [Y2: real] :
% 0.35/0.73                        ( ( member_real @ Y2 @ B2 )
% 0.35/0.73                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.73              @ A2 )
% 0.35/0.73            = ( groups4696554848551431203al_nat
% 0.35/0.73              @ ^ [Y2: real] :
% 0.35/0.73                  ( groups708209901874060359at_nat
% 0.35/0.73                  @ ^ [X3: nat] : ( G @ X3 @ Y2 )
% 0.35/0.73                  @ ( collect_nat
% 0.35/0.73                    @ ^ [X3: nat] :
% 0.35/0.73                        ( ( member_nat @ X3 @ A2 )
% 0.35/0.73                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.73              @ B2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.swap_restrict
% 0.35/0.73  thf(fact_8541_prod_Oswap__restrict,axiom,
% 0.35/0.73      ! [A2: set_nat,B2: set_int,G: nat > int > nat,R: nat > int > $o] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( finite_finite_int @ B2 )
% 0.35/0.73         => ( ( groups708209901874060359at_nat
% 0.35/0.73              @ ^ [X3: nat] :
% 0.35/0.73                  ( groups1707563613775114915nt_nat @ ( G @ X3 )
% 0.35/0.73                  @ ( collect_int
% 0.35/0.73                    @ ^ [Y2: int] :
% 0.35/0.73                        ( ( member_int @ Y2 @ B2 )
% 0.35/0.73                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.73              @ A2 )
% 0.35/0.73            = ( groups1707563613775114915nt_nat
% 0.35/0.73              @ ^ [Y2: int] :
% 0.35/0.73                  ( groups708209901874060359at_nat
% 0.35/0.73                  @ ^ [X3: nat] : ( G @ X3 @ Y2 )
% 0.35/0.73                  @ ( collect_nat
% 0.35/0.73                    @ ^ [X3: nat] :
% 0.35/0.73                        ( ( member_nat @ X3 @ A2 )
% 0.35/0.73                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.73              @ B2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.swap_restrict
% 0.35/0.73  thf(fact_8542_prod_Oswap__restrict,axiom,
% 0.35/0.73      ! [A2: set_nat,B2: set_complex,G: nat > complex > nat,R: nat > complex > $o] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.73         => ( ( groups708209901874060359at_nat
% 0.35/0.73              @ ^ [X3: nat] :
% 0.35/0.73                  ( groups861055069439313189ex_nat @ ( G @ X3 )
% 0.35/0.73                  @ ( collect_complex
% 0.35/0.73                    @ ^ [Y2: complex] :
% 0.35/0.73                        ( ( member_complex @ Y2 @ B2 )
% 0.35/0.73                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.73              @ A2 )
% 0.35/0.73            = ( groups861055069439313189ex_nat
% 0.35/0.73              @ ^ [Y2: complex] :
% 0.35/0.73                  ( groups708209901874060359at_nat
% 0.35/0.73                  @ ^ [X3: nat] : ( G @ X3 @ Y2 )
% 0.35/0.73                  @ ( collect_nat
% 0.35/0.73                    @ ^ [X3: nat] :
% 0.35/0.73                        ( ( member_nat @ X3 @ A2 )
% 0.35/0.73                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.73              @ B2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.swap_restrict
% 0.35/0.73  thf(fact_8543_prod_Oswap__restrict,axiom,
% 0.35/0.73      ! [A2: set_nat,B2: set_nat,G: nat > nat > nat,R: nat > nat > $o] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( finite_finite_nat @ B2 )
% 0.35/0.73         => ( ( groups708209901874060359at_nat
% 0.35/0.73              @ ^ [X3: nat] :
% 0.35/0.73                  ( groups708209901874060359at_nat @ ( G @ X3 )
% 0.35/0.73                  @ ( collect_nat
% 0.35/0.73                    @ ^ [Y2: nat] :
% 0.35/0.73                        ( ( member_nat @ Y2 @ B2 )
% 0.35/0.73                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.73              @ A2 )
% 0.35/0.73            = ( groups708209901874060359at_nat
% 0.35/0.73              @ ^ [Y2: nat] :
% 0.35/0.73                  ( groups708209901874060359at_nat
% 0.35/0.73                  @ ^ [X3: nat] : ( G @ X3 @ Y2 )
% 0.35/0.73                  @ ( collect_nat
% 0.35/0.73                    @ ^ [X3: nat] :
% 0.35/0.73                        ( ( member_nat @ X3 @ A2 )
% 0.35/0.73                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.73              @ B2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.swap_restrict
% 0.35/0.73  thf(fact_8544_prod_Oswap__restrict,axiom,
% 0.35/0.73      ! [A2: set_list_nat,B2: set_nat,G: list_nat > nat > nat,R: list_nat > nat > $o] :
% 0.35/0.73        ( ( finite8100373058378681591st_nat @ A2 )
% 0.35/0.73       => ( ( finite_finite_nat @ B2 )
% 0.35/0.73         => ( ( groups2907647131375434839at_nat
% 0.35/0.73              @ ^ [X3: list_nat] :
% 0.35/0.73                  ( groups708209901874060359at_nat @ ( G @ X3 )
% 0.35/0.73                  @ ( collect_nat
% 0.35/0.73                    @ ^ [Y2: nat] :
% 0.35/0.73                        ( ( member_nat @ Y2 @ B2 )
% 0.35/0.73                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.73              @ A2 )
% 0.35/0.73            = ( groups708209901874060359at_nat
% 0.35/0.73              @ ^ [Y2: nat] :
% 0.35/0.73                  ( groups2907647131375434839at_nat
% 0.35/0.73                  @ ^ [X3: list_nat] : ( G @ X3 @ Y2 )
% 0.35/0.73                  @ ( collect_list_nat
% 0.35/0.73                    @ ^ [X3: list_nat] :
% 0.35/0.73                        ( ( member_list_nat @ X3 @ A2 )
% 0.35/0.73                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.73              @ B2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.swap_restrict
% 0.35/0.73  thf(fact_8545_prod_Oswap__restrict,axiom,
% 0.35/0.73      ! [A2: set_set_nat,B2: set_nat,G: set_nat > nat > nat,R: set_nat > nat > $o] :
% 0.35/0.73        ( ( finite1152437895449049373et_nat @ A2 )
% 0.35/0.73       => ( ( finite_finite_nat @ B2 )
% 0.35/0.73         => ( ( groups4248547760180025341at_nat
% 0.35/0.73              @ ^ [X3: set_nat] :
% 0.35/0.73                  ( groups708209901874060359at_nat @ ( G @ X3 )
% 0.35/0.73                  @ ( collect_nat
% 0.35/0.73                    @ ^ [Y2: nat] :
% 0.35/0.73                        ( ( member_nat @ Y2 @ B2 )
% 0.35/0.73                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.73              @ A2 )
% 0.35/0.73            = ( groups708209901874060359at_nat
% 0.35/0.73              @ ^ [Y2: nat] :
% 0.35/0.73                  ( groups4248547760180025341at_nat
% 0.35/0.73                  @ ^ [X3: set_nat] : ( G @ X3 @ Y2 )
% 0.35/0.73                  @ ( collect_set_nat
% 0.35/0.73                    @ ^ [X3: set_nat] :
% 0.35/0.73                        ( ( member_set_nat @ X3 @ A2 )
% 0.35/0.73                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.73              @ B2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.swap_restrict
% 0.35/0.73  thf(fact_8546_prod_Oswap__restrict,axiom,
% 0.35/0.73      ! [A2: set_nat,B2: set_list_nat,G: nat > list_nat > nat,R: nat > list_nat > $o] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( finite8100373058378681591st_nat @ B2 )
% 0.35/0.73         => ( ( groups708209901874060359at_nat
% 0.35/0.73              @ ^ [X3: nat] :
% 0.35/0.73                  ( groups2907647131375434839at_nat @ ( G @ X3 )
% 0.35/0.73                  @ ( collect_list_nat
% 0.35/0.73                    @ ^ [Y2: list_nat] :
% 0.35/0.73                        ( ( member_list_nat @ Y2 @ B2 )
% 0.35/0.73                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.73              @ A2 )
% 0.35/0.73            = ( groups2907647131375434839at_nat
% 0.35/0.73              @ ^ [Y2: list_nat] :
% 0.35/0.73                  ( groups708209901874060359at_nat
% 0.35/0.73                  @ ^ [X3: nat] : ( G @ X3 @ Y2 )
% 0.35/0.73                  @ ( collect_nat
% 0.35/0.73                    @ ^ [X3: nat] :
% 0.35/0.73                        ( ( member_nat @ X3 @ A2 )
% 0.35/0.73                        & ( R @ X3 @ Y2 ) ) ) )
% 0.35/0.73              @ B2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.swap_restrict
% 0.35/0.73  thf(fact_8547_prod_Olast__plus,axiom,
% 0.35/0.73      ! [M: nat,N: nat,G: nat > complex] :
% 0.35/0.73        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.73       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.73          = ( times_times_complex @ ( G @ N ) @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.last_plus
% 0.35/0.73  thf(fact_8548_prod_Olast__plus,axiom,
% 0.35/0.73      ! [M: nat,N: nat,G: nat > real] :
% 0.35/0.73        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.73       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.73          = ( times_times_real @ ( G @ N ) @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.last_plus
% 0.35/0.73  thf(fact_8549_prod_Olast__plus,axiom,
% 0.35/0.73      ! [M: nat,N: nat,G: nat > rat] :
% 0.35/0.73        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.73       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.73          = ( times_times_rat @ ( G @ N ) @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.last_plus
% 0.35/0.73  thf(fact_8550_prod_Olast__plus,axiom,
% 0.35/0.73      ! [M: nat,N: nat,G: nat > int] :
% 0.35/0.73        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.73       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.73          = ( times_times_int @ ( G @ N ) @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.last_plus
% 0.35/0.73  thf(fact_8551_prod_Olast__plus,axiom,
% 0.35/0.73      ! [M: nat,N: nat,G: nat > nat] :
% 0.35/0.73        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.73       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.73          = ( times_times_nat @ ( G @ N ) @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.last_plus
% 0.35/0.73  thf(fact_8552_prod__Suc__Suc__fact,axiom,
% 0.35/0.73      ! [N: nat] :
% 0.35/0.73        ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.35/0.73        = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_Suc_Suc_fact
% 0.35/0.73  thf(fact_8553_prod__Suc__fact,axiom,
% 0.35/0.73      ! [N: nat] :
% 0.35/0.73        ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.35/0.73        = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_Suc_fact
% 0.35/0.73  thf(fact_8554_prod_Onested__swap,axiom,
% 0.35/0.73      ! [A: nat > nat > nat,N: nat] :
% 0.35/0.73        ( ( groups708209901874060359at_nat
% 0.35/0.73          @ ^ [I4: nat] : ( groups708209901874060359at_nat @ ( A @ I4 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I4 ) )
% 0.35/0.73          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.73        = ( groups708209901874060359at_nat
% 0.35/0.73          @ ^ [J3: nat] :
% 0.35/0.73              ( groups708209901874060359at_nat
% 0.35/0.73              @ ^ [I4: nat] : ( A @ I4 @ J3 )
% 0.35/0.73              @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
% 0.35/0.73          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.nested_swap
% 0.35/0.73  thf(fact_8555_prod_Ohead__if,axiom,
% 0.35/0.73      ! [N: nat,M: nat,G: nat > complex] :
% 0.35/0.73        ( ( ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.73            = one_one_complex ) )
% 0.35/0.73        & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.73            = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.head_if
% 0.35/0.73  thf(fact_8556_prod_Ohead__if,axiom,
% 0.35/0.73      ! [N: nat,M: nat,G: nat > real] :
% 0.35/0.73        ( ( ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.73            = one_one_real ) )
% 0.35/0.73        & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.73            = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.head_if
% 0.35/0.73  thf(fact_8557_prod_Ohead__if,axiom,
% 0.35/0.73      ! [N: nat,M: nat,G: nat > rat] :
% 0.35/0.73        ( ( ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.73            = one_one_rat ) )
% 0.35/0.73        & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.73            = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.head_if
% 0.35/0.73  thf(fact_8558_prod_Ohead__if,axiom,
% 0.35/0.73      ! [N: nat,M: nat,G: nat > int] :
% 0.35/0.73        ( ( ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.73            = one_one_int ) )
% 0.35/0.73        & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.73            = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.head_if
% 0.35/0.73  thf(fact_8559_prod_Ohead__if,axiom,
% 0.35/0.73      ! [N: nat,M: nat,G: nat > nat] :
% 0.35/0.73        ( ( ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.73            = one_one_nat ) )
% 0.35/0.73        & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.73         => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.73            = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.head_if
% 0.35/0.73  thf(fact_8560_prod__mono,axiom,
% 0.35/0.73      ! [A2: set_real,F: real > real,G: real > real] :
% 0.35/0.73        ( ! [I2: real] :
% 0.35/0.73            ( ( member_real @ I2 @ A2 )
% 0.35/0.73           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 0.35/0.73              & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 0.35/0.73       => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_mono
% 0.35/0.73  thf(fact_8561_prod__mono,axiom,
% 0.35/0.73      ! [A2: set_nat,F: nat > real,G: nat > real] :
% 0.35/0.73        ( ! [I2: nat] :
% 0.35/0.73            ( ( member_nat @ I2 @ A2 )
% 0.35/0.73           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 0.35/0.73              & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 0.35/0.73       => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_mono
% 0.35/0.73  thf(fact_8562_prod__mono,axiom,
% 0.35/0.73      ! [A2: set_int,F: int > real,G: int > real] :
% 0.35/0.73        ( ! [I2: int] :
% 0.35/0.73            ( ( member_int @ I2 @ A2 )
% 0.35/0.73           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 0.35/0.73              & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 0.35/0.73       => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_mono
% 0.35/0.73  thf(fact_8563_prod__mono,axiom,
% 0.35/0.73      ! [A2: set_real,F: real > rat,G: real > rat] :
% 0.35/0.73        ( ! [I2: real] :
% 0.35/0.73            ( ( member_real @ I2 @ A2 )
% 0.35/0.73           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
% 0.35/0.73              & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 0.35/0.73       => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_mono
% 0.35/0.73  thf(fact_8564_prod__mono,axiom,
% 0.35/0.73      ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 0.35/0.73        ( ! [I2: nat] :
% 0.35/0.73            ( ( member_nat @ I2 @ A2 )
% 0.35/0.73           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
% 0.35/0.73              & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 0.35/0.73       => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_mono
% 0.35/0.73  thf(fact_8565_prod__mono,axiom,
% 0.35/0.73      ! [A2: set_int,F: int > rat,G: int > rat] :
% 0.35/0.73        ( ! [I2: int] :
% 0.35/0.73            ( ( member_int @ I2 @ A2 )
% 0.35/0.73           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
% 0.35/0.73              & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 0.35/0.73       => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_mono
% 0.35/0.73  thf(fact_8566_prod__mono,axiom,
% 0.35/0.73      ! [A2: set_real,F: real > nat,G: real > nat] :
% 0.35/0.73        ( ! [I2: real] :
% 0.35/0.73            ( ( member_real @ I2 @ A2 )
% 0.35/0.73           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
% 0.35/0.73              & ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 0.35/0.73       => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_mono
% 0.35/0.73  thf(fact_8567_prod__mono,axiom,
% 0.35/0.73      ! [A2: set_int,F: int > nat,G: int > nat] :
% 0.35/0.73        ( ! [I2: int] :
% 0.35/0.73            ( ( member_int @ I2 @ A2 )
% 0.35/0.73           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
% 0.35/0.73              & ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 0.35/0.73       => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_mono
% 0.35/0.73  thf(fact_8568_prod__mono,axiom,
% 0.35/0.73      ! [A2: set_real,F: real > int,G: real > int] :
% 0.35/0.73        ( ! [I2: real] :
% 0.35/0.73            ( ( member_real @ I2 @ A2 )
% 0.35/0.73           => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I2 ) )
% 0.35/0.73              & ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 0.35/0.73       => ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ G @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_mono
% 0.35/0.73  thf(fact_8569_prod__mono,axiom,
% 0.35/0.73      ! [A2: set_nat,F: nat > int,G: nat > int] :
% 0.35/0.73        ( ! [I2: nat] :
% 0.35/0.73            ( ( member_nat @ I2 @ A2 )
% 0.35/0.73           => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I2 ) )
% 0.35/0.73              & ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 0.35/0.73       => ( ord_less_eq_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ G @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_mono
% 0.35/0.73  thf(fact_8570_prod__nonneg,axiom,
% 0.35/0.73      ! [A2: set_nat,F: nat > nat] :
% 0.35/0.73        ( ! [X4: nat] :
% 0.35/0.73            ( ( member_nat @ X4 @ A2 )
% 0.35/0.73           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.35/0.73       => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_nonneg
% 0.35/0.73  thf(fact_8571_prod__pos,axiom,
% 0.35/0.73      ! [A2: set_nat,F: nat > nat] :
% 0.35/0.73        ( ! [X4: nat] :
% 0.35/0.73            ( ( member_nat @ X4 @ A2 )
% 0.35/0.73           => ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.35/0.73       => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_pos
% 0.35/0.73  thf(fact_8572_prod__ge__1,axiom,
% 0.35/0.73      ! [A2: set_real,F: real > real] :
% 0.35/0.73        ( ! [X4: real] :
% 0.35/0.73            ( ( member_real @ X4 @ A2 )
% 0.35/0.73           => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
% 0.35/0.73       => ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_ge_1
% 0.35/0.73  thf(fact_8573_prod__ge__1,axiom,
% 0.35/0.73      ! [A2: set_nat,F: nat > real] :
% 0.35/0.73        ( ! [X4: nat] :
% 0.35/0.73            ( ( member_nat @ X4 @ A2 )
% 0.35/0.73           => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
% 0.35/0.73       => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_ge_1
% 0.35/0.73  thf(fact_8574_prod__ge__1,axiom,
% 0.35/0.73      ! [A2: set_int,F: int > real] :
% 0.35/0.73        ( ! [X4: int] :
% 0.35/0.73            ( ( member_int @ X4 @ A2 )
% 0.35/0.73           => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
% 0.35/0.73       => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_ge_1
% 0.35/0.73  thf(fact_8575_prod__ge__1,axiom,
% 0.35/0.73      ! [A2: set_real,F: real > rat] :
% 0.35/0.73        ( ! [X4: real] :
% 0.35/0.73            ( ( member_real @ X4 @ A2 )
% 0.35/0.73           => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
% 0.35/0.73       => ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_ge_1
% 0.35/0.73  thf(fact_8576_prod__ge__1,axiom,
% 0.35/0.73      ! [A2: set_nat,F: nat > rat] :
% 0.35/0.73        ( ! [X4: nat] :
% 0.35/0.73            ( ( member_nat @ X4 @ A2 )
% 0.35/0.73           => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
% 0.35/0.73       => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_ge_1
% 0.35/0.73  thf(fact_8577_prod__ge__1,axiom,
% 0.35/0.73      ! [A2: set_int,F: int > rat] :
% 0.35/0.73        ( ! [X4: int] :
% 0.35/0.73            ( ( member_int @ X4 @ A2 )
% 0.35/0.73           => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
% 0.35/0.73       => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_ge_1
% 0.35/0.73  thf(fact_8578_prod__ge__1,axiom,
% 0.35/0.73      ! [A2: set_real,F: real > nat] :
% 0.35/0.73        ( ! [X4: real] :
% 0.35/0.73            ( ( member_real @ X4 @ A2 )
% 0.35/0.73           => ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
% 0.35/0.73       => ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_ge_1
% 0.35/0.73  thf(fact_8579_prod__ge__1,axiom,
% 0.35/0.73      ! [A2: set_int,F: int > nat] :
% 0.35/0.73        ( ! [X4: int] :
% 0.35/0.73            ( ( member_int @ X4 @ A2 )
% 0.35/0.73           => ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
% 0.35/0.73       => ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_ge_1
% 0.35/0.73  thf(fact_8580_prod__ge__1,axiom,
% 0.35/0.73      ! [A2: set_real,F: real > int] :
% 0.35/0.73        ( ! [X4: real] :
% 0.35/0.73            ( ( member_real @ X4 @ A2 )
% 0.35/0.73           => ( ord_less_eq_int @ one_one_int @ ( F @ X4 ) ) )
% 0.35/0.73       => ( ord_less_eq_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_ge_1
% 0.35/0.73  thf(fact_8581_prod__ge__1,axiom,
% 0.35/0.73      ! [A2: set_nat,F: nat > int] :
% 0.35/0.73        ( ! [X4: nat] :
% 0.35/0.73            ( ( member_nat @ X4 @ A2 )
% 0.35/0.73           => ( ord_less_eq_int @ one_one_int @ ( F @ X4 ) ) )
% 0.35/0.73       => ( ord_less_eq_int @ one_one_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_ge_1
% 0.35/0.73  thf(fact_8582_fact__prod__Suc,axiom,
% 0.35/0.73      ( semiri1406184849735516958ct_int
% 0.35/0.73      = ( ^ [N2: nat] : ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % fact_prod_Suc
% 0.35/0.73  thf(fact_8583_fact__prod__Suc,axiom,
% 0.35/0.73      ( semiri2265585572941072030t_real
% 0.35/0.73      = ( ^ [N2: nat] : ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % fact_prod_Suc
% 0.35/0.73  thf(fact_8584_fact__prod__Suc,axiom,
% 0.35/0.73      ( semiri1408675320244567234ct_nat
% 0.35/0.73      = ( ^ [N2: nat] : ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % fact_prod_Suc
% 0.35/0.73  thf(fact_8585_prod__zero,axiom,
% 0.35/0.73      ! [A2: set_nat,F: nat > complex] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ? [X2: nat] :
% 0.35/0.73              ( ( member_nat @ X2 @ A2 )
% 0.35/0.73              & ( ( F @ X2 )
% 0.35/0.73                = zero_zero_complex ) )
% 0.35/0.73         => ( ( groups6464643781859351333omplex @ F @ A2 )
% 0.35/0.73            = zero_zero_complex ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_zero
% 0.35/0.73  thf(fact_8586_prod__zero,axiom,
% 0.35/0.73      ! [A2: set_int,F: int > complex] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ? [X2: int] :
% 0.35/0.73              ( ( member_int @ X2 @ A2 )
% 0.35/0.73              & ( ( F @ X2 )
% 0.35/0.73                = zero_zero_complex ) )
% 0.35/0.73         => ( ( groups7440179247065528705omplex @ F @ A2 )
% 0.35/0.73            = zero_zero_complex ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_zero
% 0.35/0.73  thf(fact_8587_prod__zero,axiom,
% 0.35/0.73      ! [A2: set_complex,F: complex > complex] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.73       => ( ? [X2: complex] :
% 0.35/0.73              ( ( member_complex @ X2 @ A2 )
% 0.35/0.73              & ( ( F @ X2 )
% 0.35/0.73                = zero_zero_complex ) )
% 0.35/0.73         => ( ( groups3708469109370488835omplex @ F @ A2 )
% 0.35/0.73            = zero_zero_complex ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_zero
% 0.35/0.73  thf(fact_8588_prod__zero,axiom,
% 0.35/0.73      ! [A2: set_nat,F: nat > real] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ? [X2: nat] :
% 0.35/0.73              ( ( member_nat @ X2 @ A2 )
% 0.35/0.73              & ( ( F @ X2 )
% 0.35/0.73                = zero_zero_real ) )
% 0.35/0.73         => ( ( groups129246275422532515t_real @ F @ A2 )
% 0.35/0.73            = zero_zero_real ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_zero
% 0.35/0.73  thf(fact_8589_prod__zero,axiom,
% 0.35/0.73      ! [A2: set_int,F: int > real] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ? [X2: int] :
% 0.35/0.73              ( ( member_int @ X2 @ A2 )
% 0.35/0.73              & ( ( F @ X2 )
% 0.35/0.73                = zero_zero_real ) )
% 0.35/0.73         => ( ( groups2316167850115554303t_real @ F @ A2 )
% 0.35/0.73            = zero_zero_real ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_zero
% 0.35/0.73  thf(fact_8590_prod__zero,axiom,
% 0.35/0.73      ! [A2: set_complex,F: complex > real] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.73       => ( ? [X2: complex] :
% 0.35/0.73              ( ( member_complex @ X2 @ A2 )
% 0.35/0.73              & ( ( F @ X2 )
% 0.35/0.73                = zero_zero_real ) )
% 0.35/0.73         => ( ( groups766887009212190081x_real @ F @ A2 )
% 0.35/0.73            = zero_zero_real ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_zero
% 0.35/0.73  thf(fact_8591_prod__zero,axiom,
% 0.35/0.73      ! [A2: set_nat,F: nat > rat] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ? [X2: nat] :
% 0.35/0.73              ( ( member_nat @ X2 @ A2 )
% 0.35/0.73              & ( ( F @ X2 )
% 0.35/0.73                = zero_zero_rat ) )
% 0.35/0.73         => ( ( groups73079841787564623at_rat @ F @ A2 )
% 0.35/0.73            = zero_zero_rat ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_zero
% 0.35/0.73  thf(fact_8592_prod__zero,axiom,
% 0.35/0.73      ! [A2: set_int,F: int > rat] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ? [X2: int] :
% 0.35/0.73              ( ( member_int @ X2 @ A2 )
% 0.35/0.73              & ( ( F @ X2 )
% 0.35/0.73                = zero_zero_rat ) )
% 0.35/0.73         => ( ( groups1072433553688619179nt_rat @ F @ A2 )
% 0.35/0.73            = zero_zero_rat ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_zero
% 0.35/0.73  thf(fact_8593_prod__zero,axiom,
% 0.35/0.73      ! [A2: set_complex,F: complex > rat] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.73       => ( ? [X2: complex] :
% 0.35/0.73              ( ( member_complex @ X2 @ A2 )
% 0.35/0.73              & ( ( F @ X2 )
% 0.35/0.73                = zero_zero_rat ) )
% 0.35/0.73         => ( ( groups225925009352817453ex_rat @ F @ A2 )
% 0.35/0.73            = zero_zero_rat ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_zero
% 0.35/0.73  thf(fact_8594_prod__zero,axiom,
% 0.35/0.73      ! [A2: set_int,F: int > nat] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ? [X2: int] :
% 0.35/0.73              ( ( member_int @ X2 @ A2 )
% 0.35/0.73              & ( ( F @ X2 )
% 0.35/0.73                = zero_zero_nat ) )
% 0.35/0.73         => ( ( groups1707563613775114915nt_nat @ F @ A2 )
% 0.35/0.73            = zero_zero_nat ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_zero
% 0.35/0.73  thf(fact_8595_atLeastLessThan__subset__iff,axiom,
% 0.35/0.73      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.35/0.73        ( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ A @ B ) @ ( set_or4029947393144176647an_rat @ C @ D ) )
% 0.35/0.73       => ( ( ord_less_eq_rat @ B @ A )
% 0.35/0.73          | ( ( ord_less_eq_rat @ C @ A )
% 0.35/0.73            & ( ord_less_eq_rat @ B @ D ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_subset_iff
% 0.35/0.73  thf(fact_8596_atLeastLessThan__subset__iff,axiom,
% 0.35/0.73      ! [A: num,B: num,C: num,D: num] :
% 0.35/0.73        ( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ A @ B ) @ ( set_or1222409239386451017an_num @ C @ D ) )
% 0.35/0.73       => ( ( ord_less_eq_num @ B @ A )
% 0.35/0.73          | ( ( ord_less_eq_num @ C @ A )
% 0.35/0.73            & ( ord_less_eq_num @ B @ D ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_subset_iff
% 0.35/0.73  thf(fact_8597_atLeastLessThan__subset__iff,axiom,
% 0.35/0.73      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.35/0.73        ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
% 0.35/0.73       => ( ( ord_less_eq_nat @ B @ A )
% 0.35/0.73          | ( ( ord_less_eq_nat @ C @ A )
% 0.35/0.73            & ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_subset_iff
% 0.35/0.73  thf(fact_8598_atLeastLessThan__subset__iff,axiom,
% 0.35/0.73      ! [A: int,B: int,C: int,D: int] :
% 0.35/0.73        ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D ) )
% 0.35/0.73       => ( ( ord_less_eq_int @ B @ A )
% 0.35/0.73          | ( ( ord_less_eq_int @ C @ A )
% 0.35/0.73            & ( ord_less_eq_int @ B @ D ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan_subset_iff
% 0.35/0.73  thf(fact_8599_infinite__Ico,axiom,
% 0.35/0.73      ! [A: real,B: real] :
% 0.35/0.73        ( ( ord_less_real @ A @ B )
% 0.35/0.73       => ~ ( finite_finite_real @ ( set_or66887138388493659n_real @ A @ B ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % infinite_Ico
% 0.35/0.73  thf(fact_8600_infinite__Ico,axiom,
% 0.35/0.73      ! [A: rat,B: rat] :
% 0.35/0.73        ( ( ord_less_rat @ A @ B )
% 0.35/0.73       => ~ ( finite_finite_rat @ ( set_or4029947393144176647an_rat @ A @ B ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % infinite_Ico
% 0.35/0.73  thf(fact_8601_all__nat__less__eq,axiom,
% 0.35/0.73      ! [N: nat,P: nat > $o] :
% 0.35/0.73        ( ( ! [M2: nat] :
% 0.35/0.73              ( ( ord_less_nat @ M2 @ N )
% 0.35/0.73             => ( P @ M2 ) ) )
% 0.35/0.73        = ( ! [X3: nat] :
% 0.35/0.73              ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.35/0.73             => ( P @ X3 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % all_nat_less_eq
% 0.35/0.73  thf(fact_8602_ex__nat__less__eq,axiom,
% 0.35/0.73      ! [N: nat,P: nat > $o] :
% 0.35/0.73        ( ( ? [M2: nat] :
% 0.35/0.73              ( ( ord_less_nat @ M2 @ N )
% 0.35/0.73              & ( P @ M2 ) ) )
% 0.35/0.73        = ( ? [X3: nat] :
% 0.35/0.73              ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.35/0.73              & ( P @ X3 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % ex_nat_less_eq
% 0.35/0.73  thf(fact_8603_fact__prod__rev,axiom,
% 0.35/0.73      ( semiri1406184849735516958ct_int
% 0.35/0.73      = ( ^ [N2: nat] : ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ ( minus_minus_nat @ N2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % fact_prod_rev
% 0.35/0.73  thf(fact_8604_fact__prod__rev,axiom,
% 0.35/0.73      ( semiri2265585572941072030t_real
% 0.35/0.73      = ( ^ [N2: nat] : ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ ( minus_minus_nat @ N2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % fact_prod_rev
% 0.35/0.73  thf(fact_8605_fact__prod__rev,axiom,
% 0.35/0.73      ( semiri1408675320244567234ct_nat
% 0.35/0.73      = ( ^ [N2: nat] : ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ ( minus_minus_nat @ N2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % fact_prod_rev
% 0.35/0.73  thf(fact_8606_pochhammer__prod,axiom,
% 0.35/0.73      ( comm_s4028243227959126397er_rat
% 0.35/0.73      = ( ^ [A4: rat,N2: nat] :
% 0.35/0.73            ( groups73079841787564623at_rat
% 0.35/0.73            @ ^ [I4: nat] : ( plus_plus_rat @ A4 @ ( semiri681578069525770553at_rat @ I4 ) )
% 0.35/0.73            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % pochhammer_prod
% 0.35/0.73  thf(fact_8607_pochhammer__prod,axiom,
% 0.35/0.73      ( comm_s4660882817536571857er_int
% 0.35/0.73      = ( ^ [A4: int,N2: nat] :
% 0.35/0.73            ( groups705719431365010083at_int
% 0.35/0.73            @ ^ [I4: nat] : ( plus_plus_int @ A4 @ ( semiri1314217659103216013at_int @ I4 ) )
% 0.35/0.73            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % pochhammer_prod
% 0.35/0.73  thf(fact_8608_pochhammer__prod,axiom,
% 0.35/0.73      ( comm_s7457072308508201937r_real
% 0.35/0.73      = ( ^ [A4: real,N2: nat] :
% 0.35/0.73            ( groups129246275422532515t_real
% 0.35/0.73            @ ^ [I4: nat] : ( plus_plus_real @ A4 @ ( semiri5074537144036343181t_real @ I4 ) )
% 0.35/0.73            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % pochhammer_prod
% 0.35/0.73  thf(fact_8609_pochhammer__prod,axiom,
% 0.35/0.73      ( comm_s4663373288045622133er_nat
% 0.35/0.73      = ( ^ [A4: nat,N2: nat] :
% 0.35/0.73            ( groups708209901874060359at_nat
% 0.35/0.73            @ ^ [I4: nat] : ( plus_plus_nat @ A4 @ ( semiri1316708129612266289at_nat @ I4 ) )
% 0.35/0.73            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % pochhammer_prod
% 0.35/0.73  thf(fact_8610_lessThan__atLeast0,axiom,
% 0.35/0.73      ( set_ord_lessThan_nat
% 0.35/0.73      = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% 0.35/0.73  
% 0.35/0.73  % lessThan_atLeast0
% 0.35/0.73  thf(fact_8611_prod_Ointer__filter,axiom,
% 0.35/0.73      ! [A2: set_real,G: real > complex,P: real > $o] :
% 0.35/0.73        ( ( finite_finite_real @ A2 )
% 0.35/0.73       => ( ( groups713298508707869441omplex @ G
% 0.35/0.73            @ ( collect_real
% 0.35/0.73              @ ^ [X3: real] :
% 0.35/0.73                  ( ( member_real @ X3 @ A2 )
% 0.35/0.73                  & ( P @ X3 ) ) ) )
% 0.35/0.73          = ( groups713298508707869441omplex
% 0.35/0.73            @ ^ [X3: real] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_complex )
% 0.35/0.73            @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.inter_filter
% 0.35/0.73  thf(fact_8612_prod_Ointer__filter,axiom,
% 0.35/0.73      ! [A2: set_nat,G: nat > complex,P: nat > $o] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( groups6464643781859351333omplex @ G
% 0.35/0.73            @ ( collect_nat
% 0.35/0.73              @ ^ [X3: nat] :
% 0.35/0.73                  ( ( member_nat @ X3 @ A2 )
% 0.35/0.73                  & ( P @ X3 ) ) ) )
% 0.35/0.73          = ( groups6464643781859351333omplex
% 0.35/0.73            @ ^ [X3: nat] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_complex )
% 0.35/0.73            @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.inter_filter
% 0.35/0.73  thf(fact_8613_prod_Ointer__filter,axiom,
% 0.35/0.73      ! [A2: set_int,G: int > complex,P: int > $o] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ( groups7440179247065528705omplex @ G
% 0.35/0.73            @ ( collect_int
% 0.35/0.73              @ ^ [X3: int] :
% 0.35/0.73                  ( ( member_int @ X3 @ A2 )
% 0.35/0.73                  & ( P @ X3 ) ) ) )
% 0.35/0.73          = ( groups7440179247065528705omplex
% 0.35/0.73            @ ^ [X3: int] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_complex )
% 0.35/0.73            @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.inter_filter
% 0.35/0.73  thf(fact_8614_prod_Ointer__filter,axiom,
% 0.35/0.73      ! [A2: set_complex,G: complex > complex,P: complex > $o] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.73       => ( ( groups3708469109370488835omplex @ G
% 0.35/0.73            @ ( collect_complex
% 0.35/0.73              @ ^ [X3: complex] :
% 0.35/0.73                  ( ( member_complex @ X3 @ A2 )
% 0.35/0.73                  & ( P @ X3 ) ) ) )
% 0.35/0.73          = ( groups3708469109370488835omplex
% 0.35/0.73            @ ^ [X3: complex] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_complex )
% 0.35/0.73            @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.inter_filter
% 0.35/0.73  thf(fact_8615_prod_Ointer__filter,axiom,
% 0.35/0.73      ! [A2: set_real,G: real > real,P: real > $o] :
% 0.35/0.73        ( ( finite_finite_real @ A2 )
% 0.35/0.73       => ( ( groups1681761925125756287l_real @ G
% 0.35/0.73            @ ( collect_real
% 0.35/0.73              @ ^ [X3: real] :
% 0.35/0.73                  ( ( member_real @ X3 @ A2 )
% 0.35/0.73                  & ( P @ X3 ) ) ) )
% 0.35/0.73          = ( groups1681761925125756287l_real
% 0.35/0.73            @ ^ [X3: real] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_real )
% 0.35/0.73            @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.inter_filter
% 0.35/0.73  thf(fact_8616_prod_Ointer__filter,axiom,
% 0.35/0.73      ! [A2: set_nat,G: nat > real,P: nat > $o] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( groups129246275422532515t_real @ G
% 0.35/0.73            @ ( collect_nat
% 0.35/0.73              @ ^ [X3: nat] :
% 0.35/0.73                  ( ( member_nat @ X3 @ A2 )
% 0.35/0.73                  & ( P @ X3 ) ) ) )
% 0.35/0.73          = ( groups129246275422532515t_real
% 0.35/0.73            @ ^ [X3: nat] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_real )
% 0.35/0.73            @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.inter_filter
% 0.35/0.73  thf(fact_8617_prod_Ointer__filter,axiom,
% 0.35/0.73      ! [A2: set_int,G: int > real,P: int > $o] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ( groups2316167850115554303t_real @ G
% 0.35/0.73            @ ( collect_int
% 0.35/0.73              @ ^ [X3: int] :
% 0.35/0.73                  ( ( member_int @ X3 @ A2 )
% 0.35/0.73                  & ( P @ X3 ) ) ) )
% 0.35/0.73          = ( groups2316167850115554303t_real
% 0.35/0.73            @ ^ [X3: int] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_real )
% 0.35/0.73            @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.inter_filter
% 0.35/0.73  thf(fact_8618_prod_Ointer__filter,axiom,
% 0.35/0.73      ! [A2: set_complex,G: complex > real,P: complex > $o] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.73       => ( ( groups766887009212190081x_real @ G
% 0.35/0.73            @ ( collect_complex
% 0.35/0.73              @ ^ [X3: complex] :
% 0.35/0.73                  ( ( member_complex @ X3 @ A2 )
% 0.35/0.73                  & ( P @ X3 ) ) ) )
% 0.35/0.73          = ( groups766887009212190081x_real
% 0.35/0.73            @ ^ [X3: complex] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_real )
% 0.35/0.73            @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.inter_filter
% 0.35/0.73  thf(fact_8619_prod_Ointer__filter,axiom,
% 0.35/0.73      ! [A2: set_real,G: real > rat,P: real > $o] :
% 0.35/0.73        ( ( finite_finite_real @ A2 )
% 0.35/0.73       => ( ( groups4061424788464935467al_rat @ G
% 0.35/0.73            @ ( collect_real
% 0.35/0.73              @ ^ [X3: real] :
% 0.35/0.73                  ( ( member_real @ X3 @ A2 )
% 0.35/0.73                  & ( P @ X3 ) ) ) )
% 0.35/0.73          = ( groups4061424788464935467al_rat
% 0.35/0.73            @ ^ [X3: real] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_rat )
% 0.35/0.73            @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.inter_filter
% 0.35/0.73  thf(fact_8620_prod_Ointer__filter,axiom,
% 0.35/0.73      ! [A2: set_nat,G: nat > rat,P: nat > $o] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( groups73079841787564623at_rat @ G
% 0.35/0.73            @ ( collect_nat
% 0.35/0.73              @ ^ [X3: nat] :
% 0.35/0.73                  ( ( member_nat @ X3 @ A2 )
% 0.35/0.73                  & ( P @ X3 ) ) ) )
% 0.35/0.73          = ( groups73079841787564623at_rat
% 0.35/0.73            @ ^ [X3: nat] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_rat )
% 0.35/0.73            @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.inter_filter
% 0.35/0.73  thf(fact_8621_atLeastLessThan0,axiom,
% 0.35/0.73      ! [M: nat] :
% 0.35/0.73        ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
% 0.35/0.73        = bot_bot_set_nat ) ).
% 0.35/0.73  
% 0.35/0.73  % atLeastLessThan0
% 0.35/0.73  thf(fact_8622_prod__le__1,axiom,
% 0.35/0.73      ! [A2: set_real,F: real > real] :
% 0.35/0.73        ( ! [X4: real] :
% 0.35/0.73            ( ( member_real @ X4 @ A2 )
% 0.35/0.73           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 0.35/0.73              & ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
% 0.35/0.73       => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ one_one_real ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_le_1
% 0.35/0.73  thf(fact_8623_prod__le__1,axiom,
% 0.35/0.73      ! [A2: set_nat,F: nat > real] :
% 0.35/0.73        ( ! [X4: nat] :
% 0.35/0.73            ( ( member_nat @ X4 @ A2 )
% 0.35/0.73           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 0.35/0.73              & ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
% 0.35/0.73       => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ one_one_real ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_le_1
% 0.35/0.73  thf(fact_8624_prod__le__1,axiom,
% 0.35/0.73      ! [A2: set_int,F: int > real] :
% 0.35/0.73        ( ! [X4: int] :
% 0.35/0.73            ( ( member_int @ X4 @ A2 )
% 0.35/0.73           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 0.35/0.73              & ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
% 0.35/0.73       => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ one_one_real ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_le_1
% 0.35/0.73  thf(fact_8625_prod__le__1,axiom,
% 0.35/0.73      ! [A2: set_real,F: real > rat] :
% 0.35/0.73        ( ! [X4: real] :
% 0.35/0.73            ( ( member_real @ X4 @ A2 )
% 0.35/0.73           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
% 0.35/0.73              & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
% 0.35/0.73       => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_le_1
% 0.35/0.73  thf(fact_8626_prod__le__1,axiom,
% 0.35/0.73      ! [A2: set_nat,F: nat > rat] :
% 0.35/0.73        ( ! [X4: nat] :
% 0.35/0.73            ( ( member_nat @ X4 @ A2 )
% 0.35/0.73           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
% 0.35/0.73              & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
% 0.35/0.73       => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_le_1
% 0.35/0.73  thf(fact_8627_prod__le__1,axiom,
% 0.35/0.73      ! [A2: set_int,F: int > rat] :
% 0.35/0.73        ( ! [X4: int] :
% 0.35/0.73            ( ( member_int @ X4 @ A2 )
% 0.35/0.73           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
% 0.35/0.73              & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
% 0.35/0.73       => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_le_1
% 0.35/0.73  thf(fact_8628_prod__le__1,axiom,
% 0.35/0.73      ! [A2: set_real,F: real > nat] :
% 0.35/0.73        ( ! [X4: real] :
% 0.35/0.73            ( ( member_real @ X4 @ A2 )
% 0.35/0.73           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
% 0.35/0.73              & ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
% 0.35/0.73       => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ one_one_nat ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_le_1
% 0.35/0.73  thf(fact_8629_prod__le__1,axiom,
% 0.35/0.73      ! [A2: set_int,F: int > nat] :
% 0.35/0.73        ( ! [X4: int] :
% 0.35/0.73            ( ( member_int @ X4 @ A2 )
% 0.35/0.73           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
% 0.35/0.73              & ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
% 0.35/0.73       => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ one_one_nat ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_le_1
% 0.35/0.73  thf(fact_8630_prod__le__1,axiom,
% 0.35/0.73      ! [A2: set_real,F: real > int] :
% 0.35/0.73        ( ! [X4: real] :
% 0.35/0.73            ( ( member_real @ X4 @ A2 )
% 0.35/0.73           => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) )
% 0.35/0.73              & ( ord_less_eq_int @ ( F @ X4 ) @ one_one_int ) ) )
% 0.35/0.73       => ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ one_one_int ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_le_1
% 0.35/0.73  thf(fact_8631_prod__le__1,axiom,
% 0.35/0.73      ! [A2: set_nat,F: nat > int] :
% 0.35/0.73        ( ! [X4: nat] :
% 0.35/0.73            ( ( member_nat @ X4 @ A2 )
% 0.35/0.73           => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) )
% 0.35/0.73              & ( ord_less_eq_int @ ( F @ X4 ) @ one_one_int ) ) )
% 0.35/0.73       => ( ord_less_eq_int @ ( groups705719431365010083at_int @ F @ A2 ) @ one_one_int ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_le_1
% 0.35/0.73  thf(fact_8632_is__singletonI_H,axiom,
% 0.35/0.73      ! [A2: set_set_nat] :
% 0.35/0.73        ( ( A2 != bot_bot_set_set_nat )
% 0.35/0.73       => ( ! [X4: set_nat,Y3: set_nat] :
% 0.35/0.73              ( ( member_set_nat @ X4 @ A2 )
% 0.35/0.73             => ( ( member_set_nat @ Y3 @ A2 )
% 0.35/0.73               => ( X4 = Y3 ) ) )
% 0.35/0.73         => ( is_singleton_set_nat @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % is_singletonI'
% 0.35/0.73  thf(fact_8633_is__singletonI_H,axiom,
% 0.35/0.73      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.73        ( ( A2 != bot_bo2099793752762293965at_nat )
% 0.35/0.73       => ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 0.35/0.73              ( ( member8440522571783428010at_nat @ X4 @ A2 )
% 0.35/0.73             => ( ( member8440522571783428010at_nat @ Y3 @ A2 )
% 0.35/0.73               => ( X4 = Y3 ) ) )
% 0.35/0.73         => ( is_sin2850979758926227957at_nat @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % is_singletonI'
% 0.35/0.73  thf(fact_8634_is__singletonI_H,axiom,
% 0.35/0.73      ! [A2: set_real] :
% 0.35/0.73        ( ( A2 != bot_bot_set_real )
% 0.35/0.73       => ( ! [X4: real,Y3: real] :
% 0.35/0.73              ( ( member_real @ X4 @ A2 )
% 0.35/0.73             => ( ( member_real @ Y3 @ A2 )
% 0.35/0.73               => ( X4 = Y3 ) ) )
% 0.35/0.73         => ( is_singleton_real @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % is_singletonI'
% 0.35/0.73  thf(fact_8635_is__singletonI_H,axiom,
% 0.35/0.73      ! [A2: set_nat] :
% 0.35/0.73        ( ( A2 != bot_bot_set_nat )
% 0.35/0.73       => ( ! [X4: nat,Y3: nat] :
% 0.35/0.73              ( ( member_nat @ X4 @ A2 )
% 0.35/0.73             => ( ( member_nat @ Y3 @ A2 )
% 0.35/0.73               => ( X4 = Y3 ) ) )
% 0.35/0.73         => ( is_singleton_nat @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % is_singletonI'
% 0.35/0.73  thf(fact_8636_is__singletonI_H,axiom,
% 0.35/0.73      ! [A2: set_int] :
% 0.35/0.73        ( ( A2 != bot_bot_set_int )
% 0.35/0.73       => ( ! [X4: int,Y3: int] :
% 0.35/0.73              ( ( member_int @ X4 @ A2 )
% 0.35/0.73             => ( ( member_int @ Y3 @ A2 )
% 0.35/0.73               => ( X4 = Y3 ) ) )
% 0.35/0.73         => ( is_singleton_int @ A2 ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % is_singletonI'
% 0.35/0.73  thf(fact_8637_prod_Orelated,axiom,
% 0.35/0.73      ! [R: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
% 0.35/0.73        ( ( R @ one_one_complex @ one_one_complex )
% 0.35/0.73       => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
% 0.35/0.73              ( ( ( R @ X15 @ X23 )
% 0.35/0.73                & ( R @ Y15 @ Y23 ) )
% 0.35/0.73             => ( R @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
% 0.35/0.73         => ( ( finite_finite_nat @ S2 )
% 0.35/0.73           => ( ! [X4: nat] :
% 0.35/0.73                  ( ( member_nat @ X4 @ S2 )
% 0.35/0.73                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.73             => ( R @ ( groups6464643781859351333omplex @ H2 @ S2 ) @ ( groups6464643781859351333omplex @ G @ S2 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.related
% 0.35/0.73  thf(fact_8638_prod_Orelated,axiom,
% 0.35/0.73      ! [R: complex > complex > $o,S2: set_int,H2: int > complex,G: int > complex] :
% 0.35/0.73        ( ( R @ one_one_complex @ one_one_complex )
% 0.35/0.73       => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
% 0.35/0.73              ( ( ( R @ X15 @ X23 )
% 0.35/0.73                & ( R @ Y15 @ Y23 ) )
% 0.35/0.73             => ( R @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
% 0.35/0.73         => ( ( finite_finite_int @ S2 )
% 0.35/0.73           => ( ! [X4: int] :
% 0.35/0.73                  ( ( member_int @ X4 @ S2 )
% 0.35/0.73                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.73             => ( R @ ( groups7440179247065528705omplex @ H2 @ S2 ) @ ( groups7440179247065528705omplex @ G @ S2 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.related
% 0.35/0.73  thf(fact_8639_prod_Orelated,axiom,
% 0.35/0.73      ! [R: complex > complex > $o,S2: set_complex,H2: complex > complex,G: complex > complex] :
% 0.35/0.73        ( ( R @ one_one_complex @ one_one_complex )
% 0.35/0.73       => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
% 0.35/0.73              ( ( ( R @ X15 @ X23 )
% 0.35/0.73                & ( R @ Y15 @ Y23 ) )
% 0.35/0.73             => ( R @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
% 0.35/0.73         => ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.73           => ( ! [X4: complex] :
% 0.35/0.73                  ( ( member_complex @ X4 @ S2 )
% 0.35/0.73                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.73             => ( R @ ( groups3708469109370488835omplex @ H2 @ S2 ) @ ( groups3708469109370488835omplex @ G @ S2 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.related
% 0.35/0.73  thf(fact_8640_prod_Orelated,axiom,
% 0.35/0.73      ! [R: real > real > $o,S2: set_nat,H2: nat > real,G: nat > real] :
% 0.35/0.73        ( ( R @ one_one_real @ one_one_real )
% 0.35/0.73       => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
% 0.35/0.73              ( ( ( R @ X15 @ X23 )
% 0.35/0.73                & ( R @ Y15 @ Y23 ) )
% 0.35/0.73             => ( R @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
% 0.35/0.73         => ( ( finite_finite_nat @ S2 )
% 0.35/0.73           => ( ! [X4: nat] :
% 0.35/0.73                  ( ( member_nat @ X4 @ S2 )
% 0.35/0.73                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.73             => ( R @ ( groups129246275422532515t_real @ H2 @ S2 ) @ ( groups129246275422532515t_real @ G @ S2 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.related
% 0.35/0.73  thf(fact_8641_prod_Orelated,axiom,
% 0.35/0.73      ! [R: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
% 0.35/0.73        ( ( R @ one_one_real @ one_one_real )
% 0.35/0.73       => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
% 0.35/0.73              ( ( ( R @ X15 @ X23 )
% 0.35/0.73                & ( R @ Y15 @ Y23 ) )
% 0.35/0.73             => ( R @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
% 0.35/0.73         => ( ( finite_finite_int @ S2 )
% 0.35/0.73           => ( ! [X4: int] :
% 0.35/0.73                  ( ( member_int @ X4 @ S2 )
% 0.35/0.73                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.73             => ( R @ ( groups2316167850115554303t_real @ H2 @ S2 ) @ ( groups2316167850115554303t_real @ G @ S2 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.related
% 0.35/0.73  thf(fact_8642_prod_Orelated,axiom,
% 0.35/0.73      ! [R: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
% 0.35/0.73        ( ( R @ one_one_real @ one_one_real )
% 0.35/0.73       => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
% 0.35/0.73              ( ( ( R @ X15 @ X23 )
% 0.35/0.73                & ( R @ Y15 @ Y23 ) )
% 0.35/0.73             => ( R @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
% 0.35/0.73         => ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.73           => ( ! [X4: complex] :
% 0.35/0.73                  ( ( member_complex @ X4 @ S2 )
% 0.35/0.73                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.73             => ( R @ ( groups766887009212190081x_real @ H2 @ S2 ) @ ( groups766887009212190081x_real @ G @ S2 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.related
% 0.35/0.73  thf(fact_8643_prod_Orelated,axiom,
% 0.35/0.73      ! [R: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
% 0.35/0.73        ( ( R @ one_one_rat @ one_one_rat )
% 0.35/0.73       => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
% 0.35/0.73              ( ( ( R @ X15 @ X23 )
% 0.35/0.73                & ( R @ Y15 @ Y23 ) )
% 0.35/0.73             => ( R @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
% 0.35/0.73         => ( ( finite_finite_nat @ S2 )
% 0.35/0.73           => ( ! [X4: nat] :
% 0.35/0.73                  ( ( member_nat @ X4 @ S2 )
% 0.35/0.73                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.73             => ( R @ ( groups73079841787564623at_rat @ H2 @ S2 ) @ ( groups73079841787564623at_rat @ G @ S2 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.related
% 0.35/0.73  thf(fact_8644_prod_Orelated,axiom,
% 0.35/0.73      ! [R: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
% 0.35/0.73        ( ( R @ one_one_rat @ one_one_rat )
% 0.35/0.73       => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
% 0.35/0.73              ( ( ( R @ X15 @ X23 )
% 0.35/0.73                & ( R @ Y15 @ Y23 ) )
% 0.35/0.73             => ( R @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
% 0.35/0.73         => ( ( finite_finite_int @ S2 )
% 0.35/0.73           => ( ! [X4: int] :
% 0.35/0.73                  ( ( member_int @ X4 @ S2 )
% 0.35/0.73                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.73             => ( R @ ( groups1072433553688619179nt_rat @ H2 @ S2 ) @ ( groups1072433553688619179nt_rat @ G @ S2 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.related
% 0.35/0.73  thf(fact_8645_prod_Orelated,axiom,
% 0.35/0.73      ! [R: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
% 0.35/0.73        ( ( R @ one_one_rat @ one_one_rat )
% 0.35/0.73       => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
% 0.35/0.73              ( ( ( R @ X15 @ X23 )
% 0.35/0.73                & ( R @ Y15 @ Y23 ) )
% 0.35/0.73             => ( R @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
% 0.35/0.73         => ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.73           => ( ! [X4: complex] :
% 0.35/0.73                  ( ( member_complex @ X4 @ S2 )
% 0.35/0.73                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.73             => ( R @ ( groups225925009352817453ex_rat @ H2 @ S2 ) @ ( groups225925009352817453ex_rat @ G @ S2 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.related
% 0.35/0.73  thf(fact_8646_prod_Orelated,axiom,
% 0.35/0.73      ! [R: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
% 0.35/0.73        ( ( R @ one_one_nat @ one_one_nat )
% 0.35/0.73       => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
% 0.35/0.73              ( ( ( R @ X15 @ X23 )
% 0.35/0.73                & ( R @ Y15 @ Y23 ) )
% 0.35/0.73             => ( R @ ( times_times_nat @ X15 @ Y15 ) @ ( times_times_nat @ X23 @ Y23 ) ) )
% 0.35/0.73         => ( ( finite_finite_int @ S2 )
% 0.35/0.73           => ( ! [X4: int] :
% 0.35/0.73                  ( ( member_int @ X4 @ S2 )
% 0.35/0.73                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.35/0.73             => ( R @ ( groups1707563613775114915nt_nat @ H2 @ S2 ) @ ( groups1707563613775114915nt_nat @ G @ S2 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.related
% 0.35/0.73  thf(fact_8647_prod_Oinsert__if,axiom,
% 0.35/0.73      ! [A2: set_real,X: real,G: real > complex] :
% 0.35/0.73        ( ( finite_finite_real @ A2 )
% 0.35/0.73       => ( ( ( member_real @ X @ A2 )
% 0.35/0.73           => ( ( groups713298508707869441omplex @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.73              = ( groups713298508707869441omplex @ G @ A2 ) ) )
% 0.35/0.73          & ( ~ ( member_real @ X @ A2 )
% 0.35/0.73           => ( ( groups713298508707869441omplex @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.73              = ( times_times_complex @ ( G @ X ) @ ( groups713298508707869441omplex @ G @ A2 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.insert_if
% 0.35/0.73  thf(fact_8648_prod_Oinsert__if,axiom,
% 0.35/0.73      ! [A2: set_nat,X: nat,G: nat > complex] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( ( member_nat @ X @ A2 )
% 0.35/0.73           => ( ( groups6464643781859351333omplex @ G @ ( insert_nat @ X @ A2 ) )
% 0.35/0.73              = ( groups6464643781859351333omplex @ G @ A2 ) ) )
% 0.35/0.73          & ( ~ ( member_nat @ X @ A2 )
% 0.35/0.73           => ( ( groups6464643781859351333omplex @ G @ ( insert_nat @ X @ A2 ) )
% 0.35/0.73              = ( times_times_complex @ ( G @ X ) @ ( groups6464643781859351333omplex @ G @ A2 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.insert_if
% 0.35/0.73  thf(fact_8649_prod_Oinsert__if,axiom,
% 0.35/0.73      ! [A2: set_int,X: int,G: int > complex] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ( ( member_int @ X @ A2 )
% 0.35/0.73           => ( ( groups7440179247065528705omplex @ G @ ( insert_int @ X @ A2 ) )
% 0.35/0.73              = ( groups7440179247065528705omplex @ G @ A2 ) ) )
% 0.35/0.73          & ( ~ ( member_int @ X @ A2 )
% 0.35/0.73           => ( ( groups7440179247065528705omplex @ G @ ( insert_int @ X @ A2 ) )
% 0.35/0.73              = ( times_times_complex @ ( G @ X ) @ ( groups7440179247065528705omplex @ G @ A2 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.insert_if
% 0.35/0.73  thf(fact_8650_prod_Oinsert__if,axiom,
% 0.35/0.73      ! [A2: set_complex,X: complex,G: complex > complex] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.73       => ( ( ( member_complex @ X @ A2 )
% 0.35/0.73           => ( ( groups3708469109370488835omplex @ G @ ( insert_complex @ X @ A2 ) )
% 0.35/0.73              = ( groups3708469109370488835omplex @ G @ A2 ) ) )
% 0.35/0.73          & ( ~ ( member_complex @ X @ A2 )
% 0.35/0.73           => ( ( groups3708469109370488835omplex @ G @ ( insert_complex @ X @ A2 ) )
% 0.35/0.73              = ( times_times_complex @ ( G @ X ) @ ( groups3708469109370488835omplex @ G @ A2 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.insert_if
% 0.35/0.73  thf(fact_8651_prod_Oinsert__if,axiom,
% 0.35/0.73      ! [A2: set_real,X: real,G: real > real] :
% 0.35/0.73        ( ( finite_finite_real @ A2 )
% 0.35/0.73       => ( ( ( member_real @ X @ A2 )
% 0.35/0.73           => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.73              = ( groups1681761925125756287l_real @ G @ A2 ) ) )
% 0.35/0.73          & ( ~ ( member_real @ X @ A2 )
% 0.35/0.73           => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.73              = ( times_times_real @ ( G @ X ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.insert_if
% 0.35/0.73  thf(fact_8652_prod_Oinsert__if,axiom,
% 0.35/0.73      ! [A2: set_nat,X: nat,G: nat > real] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( ( member_nat @ X @ A2 )
% 0.35/0.73           => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X @ A2 ) )
% 0.35/0.73              = ( groups129246275422532515t_real @ G @ A2 ) ) )
% 0.35/0.73          & ( ~ ( member_nat @ X @ A2 )
% 0.35/0.73           => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X @ A2 ) )
% 0.35/0.73              = ( times_times_real @ ( G @ X ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.insert_if
% 0.35/0.73  thf(fact_8653_prod_Oinsert__if,axiom,
% 0.35/0.73      ! [A2: set_int,X: int,G: int > real] :
% 0.35/0.73        ( ( finite_finite_int @ A2 )
% 0.35/0.73       => ( ( ( member_int @ X @ A2 )
% 0.35/0.73           => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X @ A2 ) )
% 0.35/0.73              = ( groups2316167850115554303t_real @ G @ A2 ) ) )
% 0.35/0.73          & ( ~ ( member_int @ X @ A2 )
% 0.35/0.73           => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X @ A2 ) )
% 0.35/0.73              = ( times_times_real @ ( G @ X ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.insert_if
% 0.35/0.73  thf(fact_8654_prod_Oinsert__if,axiom,
% 0.35/0.73      ! [A2: set_complex,X: complex,G: complex > real] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ A2 )
% 0.35/0.73       => ( ( ( member_complex @ X @ A2 )
% 0.35/0.73           => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X @ A2 ) )
% 0.35/0.73              = ( groups766887009212190081x_real @ G @ A2 ) ) )
% 0.35/0.73          & ( ~ ( member_complex @ X @ A2 )
% 0.35/0.73           => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X @ A2 ) )
% 0.35/0.73              = ( times_times_real @ ( G @ X ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.insert_if
% 0.35/0.73  thf(fact_8655_prod_Oinsert__if,axiom,
% 0.35/0.73      ! [A2: set_real,X: real,G: real > rat] :
% 0.35/0.73        ( ( finite_finite_real @ A2 )
% 0.35/0.73       => ( ( ( member_real @ X @ A2 )
% 0.35/0.73           => ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.73              = ( groups4061424788464935467al_rat @ G @ A2 ) ) )
% 0.35/0.73          & ( ~ ( member_real @ X @ A2 )
% 0.35/0.73           => ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X @ A2 ) )
% 0.35/0.73              = ( times_times_rat @ ( G @ X ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.insert_if
% 0.35/0.73  thf(fact_8656_prod_Oinsert__if,axiom,
% 0.35/0.73      ! [A2: set_nat,X: nat,G: nat > rat] :
% 0.35/0.73        ( ( finite_finite_nat @ A2 )
% 0.35/0.73       => ( ( ( member_nat @ X @ A2 )
% 0.35/0.73           => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A2 ) )
% 0.35/0.73              = ( groups73079841787564623at_rat @ G @ A2 ) ) )
% 0.35/0.73          & ( ~ ( member_nat @ X @ A2 )
% 0.35/0.73           => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A2 ) )
% 0.35/0.73              = ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod.insert_if
% 0.35/0.73  thf(fact_8657_fact__split,axiom,
% 0.35/0.73      ! [K: nat,N: nat] :
% 0.35/0.73        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.73       => ( ( semiri5044797733671781792omplex @ N )
% 0.35/0.73          = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N @ K ) @ N ) ) ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % fact_split
% 0.35/0.73  thf(fact_8658_fact__split,axiom,
% 0.35/0.73      ! [K: nat,N: nat] :
% 0.35/0.73        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.73       => ( ( semiri773545260158071498ct_rat @ N )
% 0.35/0.73          = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N @ K ) @ N ) ) ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % fact_split
% 0.35/0.73  thf(fact_8659_fact__split,axiom,
% 0.35/0.73      ! [K: nat,N: nat] :
% 0.35/0.73        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.73       => ( ( semiri1406184849735516958ct_int @ N )
% 0.35/0.73          = ( times_times_int @ ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N @ K ) @ N ) ) ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % fact_split
% 0.35/0.73  thf(fact_8660_fact__split,axiom,
% 0.35/0.73      ! [K: nat,N: nat] :
% 0.35/0.73        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.73       => ( ( semiri2265585572941072030t_real @ N )
% 0.35/0.73          = ( times_times_real @ ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N @ K ) @ N ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % fact_split
% 0.35/0.73  thf(fact_8661_fact__split,axiom,
% 0.35/0.73      ! [K: nat,N: nat] :
% 0.35/0.73        ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.73       => ( ( semiri1408675320244567234ct_nat @ N )
% 0.35/0.73          = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N @ K ) @ N ) ) ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % fact_split
% 0.35/0.73  thf(fact_8662_prod__dvd__prod__subset,axiom,
% 0.35/0.73      ! [B2: set_complex,A2: set_complex,F: complex > nat] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.73       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.35/0.73         => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ F @ B2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_dvd_prod_subset
% 0.35/0.73  thf(fact_8663_prod__dvd__prod__subset,axiom,
% 0.35/0.73      ! [B2: set_nat,A2: set_nat,F: nat > int] :
% 0.35/0.73        ( ( finite_finite_nat @ B2 )
% 0.35/0.73       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.35/0.73         => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ F @ B2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_dvd_prod_subset
% 0.35/0.73  thf(fact_8664_prod__dvd__prod__subset,axiom,
% 0.35/0.73      ! [B2: set_complex,A2: set_complex,F: complex > int] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.73       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.35/0.73         => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ F @ B2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_dvd_prod_subset
% 0.35/0.73  thf(fact_8665_prod__dvd__prod__subset,axiom,
% 0.35/0.73      ! [B2: set_nat,A2: set_nat,F: nat > code_integer] :
% 0.35/0.73        ( ( finite_finite_nat @ B2 )
% 0.35/0.73       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.35/0.73         => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A2 ) @ ( groups3455450783089532116nteger @ F @ B2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_dvd_prod_subset
% 0.35/0.73  thf(fact_8666_prod__dvd__prod__subset,axiom,
% 0.35/0.73      ! [B2: set_complex,A2: set_complex,F: complex > code_integer] :
% 0.35/0.73        ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.73       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.35/0.73         => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A2 ) @ ( groups8682486955453173170nteger @ F @ B2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_dvd_prod_subset
% 0.35/0.73  thf(fact_8667_prod__dvd__prod__subset,axiom,
% 0.35/0.73      ! [B2: set_int,A2: set_int,F: int > nat] :
% 0.35/0.73        ( ( finite_finite_int @ B2 )
% 0.35/0.73       => ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.35/0.73         => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ F @ B2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_dvd_prod_subset
% 0.35/0.73  thf(fact_8668_prod__dvd__prod__subset,axiom,
% 0.35/0.73      ! [B2: set_int,A2: set_int,F: int > int] :
% 0.35/0.73        ( ( finite_finite_int @ B2 )
% 0.35/0.73       => ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.35/0.73         => ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ ( groups1705073143266064639nt_int @ F @ B2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_dvd_prod_subset
% 0.35/0.73  thf(fact_8669_prod__dvd__prod__subset,axiom,
% 0.35/0.73      ! [B2: set_int,A2: set_int,F: int > code_integer] :
% 0.35/0.73        ( ( finite_finite_int @ B2 )
% 0.35/0.73       => ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.35/0.73         => ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A2 ) @ ( groups3827104343326376752nteger @ F @ B2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_dvd_prod_subset
% 0.35/0.73  thf(fact_8670_prod__dvd__prod__subset,axiom,
% 0.35/0.73      ! [B2: set_nat,A2: set_nat,F: nat > nat] :
% 0.35/0.73        ( ( finite_finite_nat @ B2 )
% 0.35/0.73       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.35/0.73         => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ F @ B2 ) ) ) ) ).
% 0.35/0.73  
% 0.35/0.73  % prod_dvd_prod_subset
% 0.35/0.73  thf(fact_8671_prod__dvd__prod__subset,axiom,
% 0.35/0.73      ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 0.35/0.73        ( ( finite6177210948735845034at_nat @ B2 )
% 0.35/0.73       => ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 0.35/0.73         => ( dvd_dvd_nat @ ( groups4077766827762148844at_nat @ F @ A2 ) @ ( groups4077766827762148844at_nat @ F @ B2 ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod_dvd_prod_subset
% 0.35/0.74  thf(fact_8672_prod__dvd__prod__subset2,axiom,
% 0.35/0.74      ! [B2: set_real,A2: set_real,F: real > nat,G: real > nat] :
% 0.35/0.74        ( ( finite_finite_real @ B2 )
% 0.35/0.74       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.35/0.74         => ( ! [A3: real] :
% 0.35/0.74                ( ( member_real @ A3 @ A2 )
% 0.35/0.74               => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.35/0.74           => ( dvd_dvd_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ B2 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod_dvd_prod_subset2
% 0.35/0.74  thf(fact_8673_prod__dvd__prod__subset2,axiom,
% 0.35/0.74      ! [B2: set_complex,A2: set_complex,F: complex > nat,G: complex > nat] :
% 0.35/0.74        ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.74       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.35/0.74         => ( ! [A3: complex] :
% 0.35/0.74                ( ( member_complex @ A3 @ A2 )
% 0.35/0.74               => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.35/0.74           => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ G @ B2 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod_dvd_prod_subset2
% 0.35/0.74  thf(fact_8674_prod__dvd__prod__subset2,axiom,
% 0.35/0.74      ! [B2: set_real,A2: set_real,F: real > int,G: real > int] :
% 0.35/0.74        ( ( finite_finite_real @ B2 )
% 0.35/0.74       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.35/0.74         => ( ! [A3: real] :
% 0.35/0.74                ( ( member_real @ A3 @ A2 )
% 0.35/0.74               => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.35/0.74           => ( dvd_dvd_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ G @ B2 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod_dvd_prod_subset2
% 0.35/0.74  thf(fact_8675_prod__dvd__prod__subset2,axiom,
% 0.35/0.74      ! [B2: set_nat,A2: set_nat,F: nat > int,G: nat > int] :
% 0.35/0.74        ( ( finite_finite_nat @ B2 )
% 0.35/0.74       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.35/0.74         => ( ! [A3: nat] :
% 0.35/0.74                ( ( member_nat @ A3 @ A2 )
% 0.35/0.74               => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.35/0.74           => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ G @ B2 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod_dvd_prod_subset2
% 0.35/0.74  thf(fact_8676_prod__dvd__prod__subset2,axiom,
% 0.35/0.74      ! [B2: set_complex,A2: set_complex,F: complex > int,G: complex > int] :
% 0.35/0.74        ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.74       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.35/0.74         => ( ! [A3: complex] :
% 0.35/0.74                ( ( member_complex @ A3 @ A2 )
% 0.35/0.74               => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.35/0.74           => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ G @ B2 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod_dvd_prod_subset2
% 0.35/0.74  thf(fact_8677_prod__dvd__prod__subset2,axiom,
% 0.35/0.74      ! [B2: set_real,A2: set_real,F: real > code_integer,G: real > code_integer] :
% 0.35/0.74        ( ( finite_finite_real @ B2 )
% 0.35/0.74       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.35/0.74         => ( ! [A3: real] :
% 0.35/0.74                ( ( member_real @ A3 @ A2 )
% 0.35/0.74               => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.35/0.74           => ( dvd_dvd_Code_integer @ ( groups6225526099057966256nteger @ F @ A2 ) @ ( groups6225526099057966256nteger @ G @ B2 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod_dvd_prod_subset2
% 0.35/0.74  thf(fact_8678_prod__dvd__prod__subset2,axiom,
% 0.35/0.74      ! [B2: set_nat,A2: set_nat,F: nat > code_integer,G: nat > code_integer] :
% 0.35/0.74        ( ( finite_finite_nat @ B2 )
% 0.35/0.74       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.35/0.74         => ( ! [A3: nat] :
% 0.35/0.74                ( ( member_nat @ A3 @ A2 )
% 0.35/0.74               => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.35/0.74           => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A2 ) @ ( groups3455450783089532116nteger @ G @ B2 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod_dvd_prod_subset2
% 0.35/0.74  thf(fact_8679_prod__dvd__prod__subset2,axiom,
% 0.35/0.74      ! [B2: set_complex,A2: set_complex,F: complex > code_integer,G: complex > code_integer] :
% 0.35/0.74        ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.74       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.35/0.74         => ( ! [A3: complex] :
% 0.35/0.74                ( ( member_complex @ A3 @ A2 )
% 0.35/0.74               => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.35/0.74           => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A2 ) @ ( groups8682486955453173170nteger @ G @ B2 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod_dvd_prod_subset2
% 0.35/0.74  thf(fact_8680_prod__dvd__prod__subset2,axiom,
% 0.35/0.74      ! [B2: set_int,A2: set_int,F: int > nat,G: int > nat] :
% 0.35/0.74        ( ( finite_finite_int @ B2 )
% 0.35/0.74       => ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.35/0.74         => ( ! [A3: int] :
% 0.35/0.74                ( ( member_int @ A3 @ A2 )
% 0.35/0.74               => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.35/0.74           => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ B2 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod_dvd_prod_subset2
% 0.35/0.74  thf(fact_8681_prod__dvd__prod__subset2,axiom,
% 0.35/0.74      ! [B2: set_int,A2: set_int,F: int > int,G: int > int] :
% 0.35/0.74        ( ( finite_finite_int @ B2 )
% 0.35/0.74       => ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.35/0.74         => ( ! [A3: int] :
% 0.35/0.74                ( ( member_int @ A3 @ A2 )
% 0.35/0.74               => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.35/0.74           => ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ ( groups1705073143266064639nt_int @ G @ B2 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod_dvd_prod_subset2
% 0.35/0.74  thf(fact_8682_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.74      ! [S5: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T3: set_real,G: real > int,H2: real > int] :
% 0.35/0.74        ( ( finite_finite_real @ S5 )
% 0.35/0.74       => ( ( finite_finite_real @ T4 )
% 0.35/0.74         => ( ! [A3: real] :
% 0.35/0.74                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 0.35/0.74               => ( ( I @ ( J @ A3 ) )
% 0.35/0.74                  = A3 ) )
% 0.35/0.74           => ( ! [A3: real] :
% 0.35/0.74                  ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 0.35/0.74                 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.35/0.74             => ( ! [B3: real] :
% 0.35/0.74                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.35/0.74                   => ( ( J @ ( I @ B3 ) )
% 0.35/0.74                      = B3 ) )
% 0.35/0.74               => ( ! [B3: real] :
% 0.35/0.74                      ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.35/0.74                     => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
% 0.35/0.74                 => ( ! [A3: real] :
% 0.35/0.74                        ( ( member_real @ A3 @ S5 )
% 0.35/0.74                       => ( ( G @ A3 )
% 0.35/0.74                          = one_one_int ) )
% 0.35/0.74                   => ( ! [B3: real] :
% 0.35/0.74                          ( ( member_real @ B3 @ T4 )
% 0.35/0.74                         => ( ( H2 @ B3 )
% 0.35/0.74                            = one_one_int ) )
% 0.35/0.74                     => ( ! [A3: real] :
% 0.35/0.74                            ( ( member_real @ A3 @ S2 )
% 0.35/0.74                           => ( ( H2 @ ( J @ A3 ) )
% 0.35/0.74                              = ( G @ A3 ) ) )
% 0.35/0.74                       => ( ( groups4694064378042380927al_int @ G @ S2 )
% 0.35/0.74                          = ( groups4694064378042380927al_int @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod.reindex_bij_witness_not_neutral
% 0.35/0.74  thf(fact_8683_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.74      ! [S5: set_real,T4: set_int,S2: set_real,I: int > real,J: real > int,T3: set_int,G: real > int,H2: int > int] :
% 0.35/0.74        ( ( finite_finite_real @ S5 )
% 0.35/0.74       => ( ( finite_finite_int @ T4 )
% 0.35/0.74         => ( ! [A3: real] :
% 0.35/0.74                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 0.35/0.74               => ( ( I @ ( J @ A3 ) )
% 0.35/0.74                  = A3 ) )
% 0.35/0.74           => ( ! [A3: real] :
% 0.35/0.74                  ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 0.35/0.74                 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.35/0.74             => ( ! [B3: int] :
% 0.35/0.74                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.35/0.74                   => ( ( J @ ( I @ B3 ) )
% 0.35/0.74                      = B3 ) )
% 0.35/0.74               => ( ! [B3: int] :
% 0.35/0.74                      ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.35/0.74                     => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
% 0.35/0.74                 => ( ! [A3: real] :
% 0.35/0.74                        ( ( member_real @ A3 @ S5 )
% 0.35/0.74                       => ( ( G @ A3 )
% 0.35/0.74                          = one_one_int ) )
% 0.35/0.74                   => ( ! [B3: int] :
% 0.35/0.74                          ( ( member_int @ B3 @ T4 )
% 0.35/0.74                         => ( ( H2 @ B3 )
% 0.35/0.74                            = one_one_int ) )
% 0.35/0.74                     => ( ! [A3: real] :
% 0.35/0.74                            ( ( member_real @ A3 @ S2 )
% 0.35/0.74                           => ( ( H2 @ ( J @ A3 ) )
% 0.35/0.74                              = ( G @ A3 ) ) )
% 0.35/0.74                       => ( ( groups4694064378042380927al_int @ G @ S2 )
% 0.35/0.74                          = ( groups1705073143266064639nt_int @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod.reindex_bij_witness_not_neutral
% 0.35/0.74  thf(fact_8684_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.74      ! [S5: set_real,T4: set_complex,S2: set_real,I: complex > real,J: real > complex,T3: set_complex,G: real > int,H2: complex > int] :
% 0.35/0.74        ( ( finite_finite_real @ S5 )
% 0.35/0.74       => ( ( finite3207457112153483333omplex @ T4 )
% 0.35/0.74         => ( ! [A3: real] :
% 0.35/0.74                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 0.35/0.74               => ( ( I @ ( J @ A3 ) )
% 0.35/0.74                  = A3 ) )
% 0.35/0.74           => ( ! [A3: real] :
% 0.35/0.74                  ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 0.35/0.74                 => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 0.35/0.74             => ( ! [B3: complex] :
% 0.35/0.74                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.35/0.74                   => ( ( J @ ( I @ B3 ) )
% 0.35/0.74                      = B3 ) )
% 0.35/0.74               => ( ! [B3: complex] :
% 0.35/0.74                      ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.35/0.74                     => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
% 0.35/0.74                 => ( ! [A3: real] :
% 0.35/0.74                        ( ( member_real @ A3 @ S5 )
% 0.35/0.74                       => ( ( G @ A3 )
% 0.35/0.74                          = one_one_int ) )
% 0.35/0.74                   => ( ! [B3: complex] :
% 0.35/0.74                          ( ( member_complex @ B3 @ T4 )
% 0.35/0.74                         => ( ( H2 @ B3 )
% 0.35/0.74                            = one_one_int ) )
% 0.35/0.74                     => ( ! [A3: real] :
% 0.35/0.74                            ( ( member_real @ A3 @ S2 )
% 0.35/0.74                           => ( ( H2 @ ( J @ A3 ) )
% 0.35/0.74                              = ( G @ A3 ) ) )
% 0.35/0.74                       => ( ( groups4694064378042380927al_int @ G @ S2 )
% 0.35/0.74                          = ( groups858564598930262913ex_int @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod.reindex_bij_witness_not_neutral
% 0.35/0.74  thf(fact_8685_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.74      ! [S5: set_int,T4: set_real,S2: set_int,I: real > int,J: int > real,T3: set_real,G: int > int,H2: real > int] :
% 0.35/0.74        ( ( finite_finite_int @ S5 )
% 0.35/0.74       => ( ( finite_finite_real @ T4 )
% 0.35/0.74         => ( ! [A3: int] :
% 0.35/0.74                ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 0.35/0.74               => ( ( I @ ( J @ A3 ) )
% 0.35/0.74                  = A3 ) )
% 0.35/0.74           => ( ! [A3: int] :
% 0.35/0.74                  ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 0.35/0.74                 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.35/0.74             => ( ! [B3: real] :
% 0.35/0.74                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.35/0.74                   => ( ( J @ ( I @ B3 ) )
% 0.35/0.74                      = B3 ) )
% 0.35/0.74               => ( ! [B3: real] :
% 0.35/0.74                      ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.35/0.74                     => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
% 0.35/0.74                 => ( ! [A3: int] :
% 0.35/0.74                        ( ( member_int @ A3 @ S5 )
% 0.35/0.74                       => ( ( G @ A3 )
% 0.35/0.74                          = one_one_int ) )
% 0.35/0.74                   => ( ! [B3: real] :
% 0.35/0.74                          ( ( member_real @ B3 @ T4 )
% 0.35/0.74                         => ( ( H2 @ B3 )
% 0.35/0.74                            = one_one_int ) )
% 0.35/0.74                     => ( ! [A3: int] :
% 0.35/0.74                            ( ( member_int @ A3 @ S2 )
% 0.35/0.74                           => ( ( H2 @ ( J @ A3 ) )
% 0.35/0.74                              = ( G @ A3 ) ) )
% 0.35/0.74                       => ( ( groups1705073143266064639nt_int @ G @ S2 )
% 0.35/0.74                          = ( groups4694064378042380927al_int @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod.reindex_bij_witness_not_neutral
% 0.35/0.74  thf(fact_8686_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.74      ! [S5: set_int,T4: set_int,S2: set_int,I: int > int,J: int > int,T3: set_int,G: int > int,H2: int > int] :
% 0.35/0.74        ( ( finite_finite_int @ S5 )
% 0.35/0.74       => ( ( finite_finite_int @ T4 )
% 0.35/0.74         => ( ! [A3: int] :
% 0.35/0.74                ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 0.35/0.74               => ( ( I @ ( J @ A3 ) )
% 0.35/0.74                  = A3 ) )
% 0.35/0.74           => ( ! [A3: int] :
% 0.35/0.74                  ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 0.35/0.74                 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.35/0.74             => ( ! [B3: int] :
% 0.35/0.74                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.35/0.74                   => ( ( J @ ( I @ B3 ) )
% 0.35/0.74                      = B3 ) )
% 0.35/0.74               => ( ! [B3: int] :
% 0.35/0.74                      ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.35/0.74                     => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
% 0.35/0.74                 => ( ! [A3: int] :
% 0.35/0.74                        ( ( member_int @ A3 @ S5 )
% 0.35/0.74                       => ( ( G @ A3 )
% 0.35/0.74                          = one_one_int ) )
% 0.35/0.74                   => ( ! [B3: int] :
% 0.35/0.74                          ( ( member_int @ B3 @ T4 )
% 0.35/0.74                         => ( ( H2 @ B3 )
% 0.35/0.74                            = one_one_int ) )
% 0.35/0.74                     => ( ! [A3: int] :
% 0.35/0.74                            ( ( member_int @ A3 @ S2 )
% 0.35/0.74                           => ( ( H2 @ ( J @ A3 ) )
% 0.35/0.74                              = ( G @ A3 ) ) )
% 0.35/0.74                       => ( ( groups1705073143266064639nt_int @ G @ S2 )
% 0.35/0.74                          = ( groups1705073143266064639nt_int @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod.reindex_bij_witness_not_neutral
% 0.35/0.74  thf(fact_8687_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.74      ! [S5: set_int,T4: set_complex,S2: set_int,I: complex > int,J: int > complex,T3: set_complex,G: int > int,H2: complex > int] :
% 0.35/0.74        ( ( finite_finite_int @ S5 )
% 0.35/0.74       => ( ( finite3207457112153483333omplex @ T4 )
% 0.35/0.74         => ( ! [A3: int] :
% 0.35/0.74                ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 0.35/0.74               => ( ( I @ ( J @ A3 ) )
% 0.35/0.74                  = A3 ) )
% 0.35/0.74           => ( ! [A3: int] :
% 0.35/0.74                  ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 0.35/0.74                 => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 0.35/0.74             => ( ! [B3: complex] :
% 0.35/0.74                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.35/0.74                   => ( ( J @ ( I @ B3 ) )
% 0.35/0.74                      = B3 ) )
% 0.35/0.74               => ( ! [B3: complex] :
% 0.35/0.74                      ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.35/0.74                     => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
% 0.35/0.74                 => ( ! [A3: int] :
% 0.35/0.74                        ( ( member_int @ A3 @ S5 )
% 0.35/0.74                       => ( ( G @ A3 )
% 0.35/0.74                          = one_one_int ) )
% 0.35/0.74                   => ( ! [B3: complex] :
% 0.35/0.74                          ( ( member_complex @ B3 @ T4 )
% 0.35/0.74                         => ( ( H2 @ B3 )
% 0.35/0.74                            = one_one_int ) )
% 0.35/0.74                     => ( ! [A3: int] :
% 0.35/0.74                            ( ( member_int @ A3 @ S2 )
% 0.35/0.74                           => ( ( H2 @ ( J @ A3 ) )
% 0.35/0.74                              = ( G @ A3 ) ) )
% 0.35/0.74                       => ( ( groups1705073143266064639nt_int @ G @ S2 )
% 0.35/0.74                          = ( groups858564598930262913ex_int @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod.reindex_bij_witness_not_neutral
% 0.35/0.74  thf(fact_8688_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.74      ! [S5: set_complex,T4: set_real,S2: set_complex,I: real > complex,J: complex > real,T3: set_real,G: complex > int,H2: real > int] :
% 0.35/0.74        ( ( finite3207457112153483333omplex @ S5 )
% 0.35/0.74       => ( ( finite_finite_real @ T4 )
% 0.35/0.74         => ( ! [A3: complex] :
% 0.35/0.74                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
% 0.35/0.74               => ( ( I @ ( J @ A3 ) )
% 0.35/0.74                  = A3 ) )
% 0.35/0.74           => ( ! [A3: complex] :
% 0.35/0.74                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
% 0.35/0.74                 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.35/0.74             => ( ! [B3: real] :
% 0.35/0.74                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.35/0.74                   => ( ( J @ ( I @ B3 ) )
% 0.35/0.74                      = B3 ) )
% 0.35/0.74               => ( ! [B3: real] :
% 0.35/0.74                      ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.35/0.74                     => ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
% 0.35/0.74                 => ( ! [A3: complex] :
% 0.35/0.74                        ( ( member_complex @ A3 @ S5 )
% 0.35/0.74                       => ( ( G @ A3 )
% 0.35/0.74                          = one_one_int ) )
% 0.35/0.74                   => ( ! [B3: real] :
% 0.35/0.74                          ( ( member_real @ B3 @ T4 )
% 0.35/0.74                         => ( ( H2 @ B3 )
% 0.35/0.74                            = one_one_int ) )
% 0.35/0.74                     => ( ! [A3: complex] :
% 0.35/0.74                            ( ( member_complex @ A3 @ S2 )
% 0.35/0.74                           => ( ( H2 @ ( J @ A3 ) )
% 0.35/0.74                              = ( G @ A3 ) ) )
% 0.35/0.74                       => ( ( groups858564598930262913ex_int @ G @ S2 )
% 0.35/0.74                          = ( groups4694064378042380927al_int @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod.reindex_bij_witness_not_neutral
% 0.35/0.74  thf(fact_8689_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.74      ! [S5: set_complex,T4: set_int,S2: set_complex,I: int > complex,J: complex > int,T3: set_int,G: complex > int,H2: int > int] :
% 0.35/0.74        ( ( finite3207457112153483333omplex @ S5 )
% 0.35/0.74       => ( ( finite_finite_int @ T4 )
% 0.35/0.74         => ( ! [A3: complex] :
% 0.35/0.74                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
% 0.35/0.74               => ( ( I @ ( J @ A3 ) )
% 0.35/0.74                  = A3 ) )
% 0.35/0.74           => ( ! [A3: complex] :
% 0.35/0.74                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
% 0.35/0.74                 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.35/0.74             => ( ! [B3: int] :
% 0.35/0.74                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.35/0.74                   => ( ( J @ ( I @ B3 ) )
% 0.35/0.74                      = B3 ) )
% 0.35/0.74               => ( ! [B3: int] :
% 0.35/0.74                      ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.35/0.74                     => ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
% 0.35/0.74                 => ( ! [A3: complex] :
% 0.35/0.74                        ( ( member_complex @ A3 @ S5 )
% 0.35/0.74                       => ( ( G @ A3 )
% 0.35/0.74                          = one_one_int ) )
% 0.35/0.74                   => ( ! [B3: int] :
% 0.35/0.74                          ( ( member_int @ B3 @ T4 )
% 0.35/0.74                         => ( ( H2 @ B3 )
% 0.35/0.74                            = one_one_int ) )
% 0.35/0.74                     => ( ! [A3: complex] :
% 0.35/0.74                            ( ( member_complex @ A3 @ S2 )
% 0.35/0.74                           => ( ( H2 @ ( J @ A3 ) )
% 0.35/0.74                              = ( G @ A3 ) ) )
% 0.35/0.74                       => ( ( groups858564598930262913ex_int @ G @ S2 )
% 0.35/0.74                          = ( groups1705073143266064639nt_int @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod.reindex_bij_witness_not_neutral
% 0.35/0.74  thf(fact_8690_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.74      ! [S5: set_complex,T4: set_complex,S2: set_complex,I: complex > complex,J: complex > complex,T3: set_complex,G: complex > int,H2: complex > int] :
% 0.35/0.74        ( ( finite3207457112153483333omplex @ S5 )
% 0.35/0.74       => ( ( finite3207457112153483333omplex @ T4 )
% 0.35/0.74         => ( ! [A3: complex] :
% 0.35/0.74                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
% 0.35/0.74               => ( ( I @ ( J @ A3 ) )
% 0.35/0.74                  = A3 ) )
% 0.35/0.74           => ( ! [A3: complex] :
% 0.35/0.74                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
% 0.35/0.74                 => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 0.35/0.74             => ( ! [B3: complex] :
% 0.35/0.74                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.35/0.74                   => ( ( J @ ( I @ B3 ) )
% 0.35/0.74                      = B3 ) )
% 0.35/0.74               => ( ! [B3: complex] :
% 0.35/0.74                      ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.35/0.74                     => ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
% 0.35/0.74                 => ( ! [A3: complex] :
% 0.35/0.74                        ( ( member_complex @ A3 @ S5 )
% 0.35/0.74                       => ( ( G @ A3 )
% 0.35/0.74                          = one_one_int ) )
% 0.35/0.74                   => ( ! [B3: complex] :
% 0.35/0.74                          ( ( member_complex @ B3 @ T4 )
% 0.35/0.74                         => ( ( H2 @ B3 )
% 0.35/0.74                            = one_one_int ) )
% 0.35/0.74                     => ( ! [A3: complex] :
% 0.35/0.74                            ( ( member_complex @ A3 @ S2 )
% 0.35/0.74                           => ( ( H2 @ ( J @ A3 ) )
% 0.35/0.74                              = ( G @ A3 ) ) )
% 0.35/0.74                       => ( ( groups858564598930262913ex_int @ G @ S2 )
% 0.35/0.74                          = ( groups858564598930262913ex_int @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod.reindex_bij_witness_not_neutral
% 0.35/0.74  thf(fact_8691_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.74      ! [S5: set_real,T4: set_nat,S2: set_real,I: nat > real,J: real > nat,T3: set_nat,G: real > complex,H2: nat > complex] :
% 0.35/0.74        ( ( finite_finite_real @ S5 )
% 0.35/0.74       => ( ( finite_finite_nat @ T4 )
% 0.35/0.74         => ( ! [A3: real] :
% 0.35/0.74                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 0.35/0.74               => ( ( I @ ( J @ A3 ) )
% 0.35/0.74                  = A3 ) )
% 0.35/0.74           => ( ! [A3: real] :
% 0.35/0.74                  ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 0.35/0.74                 => ( member_nat @ ( J @ A3 ) @ ( minus_minus_set_nat @ T3 @ T4 ) ) )
% 0.35/0.74             => ( ! [B3: nat] :
% 0.35/0.74                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ T3 @ T4 ) )
% 0.35/0.74                   => ( ( J @ ( I @ B3 ) )
% 0.35/0.74                      = B3 ) )
% 0.35/0.74               => ( ! [B3: nat] :
% 0.35/0.74                      ( ( member_nat @ B3 @ ( minus_minus_set_nat @ T3 @ T4 ) )
% 0.35/0.74                     => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
% 0.35/0.74                 => ( ! [A3: real] :
% 0.35/0.74                        ( ( member_real @ A3 @ S5 )
% 0.35/0.74                       => ( ( G @ A3 )
% 0.35/0.74                          = one_one_complex ) )
% 0.35/0.74                   => ( ! [B3: nat] :
% 0.35/0.74                          ( ( member_nat @ B3 @ T4 )
% 0.35/0.74                         => ( ( H2 @ B3 )
% 0.35/0.74                            = one_one_complex ) )
% 0.35/0.74                     => ( ! [A3: real] :
% 0.35/0.74                            ( ( member_real @ A3 @ S2 )
% 0.35/0.74                           => ( ( H2 @ ( J @ A3 ) )
% 0.35/0.74                              = ( G @ A3 ) ) )
% 0.35/0.74                       => ( ( groups713298508707869441omplex @ G @ S2 )
% 0.35/0.74                          = ( groups6464643781859351333omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod.reindex_bij_witness_not_neutral
% 0.35/0.74  thf(fact_8692_atLeast0__lessThan__Suc,axiom,
% 0.35/0.74      ! [N: nat] :
% 0.35/0.74        ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.35/0.74        = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % atLeast0_lessThan_Suc
% 0.35/0.74  thf(fact_8693_subset__eq__atLeast0__lessThan__finite,axiom,
% 0.35/0.74      ! [N6: set_nat,N: nat] :
% 0.35/0.74        ( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.35/0.74       => ( finite_finite_nat @ N6 ) ) ).
% 0.35/0.74  
% 0.35/0.74  % subset_eq_atLeast0_lessThan_finite
% 0.35/0.74  thf(fact_8694_atLeastLessThanSuc,axiom,
% 0.35/0.74      ! [M: nat,N: nat] :
% 0.35/0.74        ( ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.74         => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 0.35/0.74            = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
% 0.35/0.74        & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.35/0.74         => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 0.35/0.74            = bot_bot_set_nat ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % atLeastLessThanSuc
% 0.35/0.74  thf(fact_8695_fact__eq__fact__times,axiom,
% 0.35/0.74      ! [N: nat,M: nat] :
% 0.35/0.74        ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.74       => ( ( semiri1408675320244567234ct_nat @ M )
% 0.35/0.74          = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
% 0.35/0.74            @ ( groups708209901874060359at_nat
% 0.35/0.74              @ ^ [X3: nat] : X3
% 0.35/0.74              @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % fact_eq_fact_times
% 0.35/0.74  thf(fact_8696_fact__div__fact,axiom,
% 0.35/0.74      ! [N: nat,M: nat] :
% 0.35/0.74        ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.74       => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.35/0.74          = ( groups708209901874060359at_nat
% 0.35/0.74            @ ^ [X3: nat] : X3
% 0.35/0.74            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % fact_div_fact
% 0.35/0.74  thf(fact_8697_OR__upper,axiom,
% 0.35/0.74      ! [X: int,N: nat,Y: int] :
% 0.35/0.74        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.35/0.74       => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.74         => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.74           => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % OR_upper
% 0.35/0.74  thf(fact_8698_atLeast1__lessThan__eq__remove0,axiom,
% 0.35/0.74      ! [N: nat] :
% 0.35/0.74        ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.74        = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % atLeast1_lessThan_eq_remove0
% 0.35/0.74  thf(fact_8699_Chebyshev__sum__upper__nat,axiom,
% 0.35/0.74      ! [N: nat,A: nat > nat,B: nat > nat] :
% 0.35/0.74        ( ! [I2: nat,J2: nat] :
% 0.35/0.74            ( ( ord_less_eq_nat @ I2 @ J2 )
% 0.35/0.74           => ( ( ord_less_nat @ J2 @ N )
% 0.35/0.74             => ( ord_less_eq_nat @ ( A @ I2 ) @ ( A @ J2 ) ) ) )
% 0.35/0.74       => ( ! [I2: nat,J2: nat] :
% 0.35/0.74              ( ( ord_less_eq_nat @ I2 @ J2 )
% 0.35/0.74             => ( ( ord_less_nat @ J2 @ N )
% 0.35/0.74               => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I2 ) ) ) )
% 0.35/0.74         => ( ord_less_eq_nat
% 0.35/0.74            @ ( times_times_nat @ N
% 0.35/0.74              @ ( groups3542108847815614940at_nat
% 0.35/0.74                @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( B @ I4 ) )
% 0.35/0.74                @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.74            @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Chebyshev_sum_upper_nat
% 0.35/0.74  thf(fact_8700_VEBT_Osize__gen_I1_J,axiom,
% 0.35/0.74      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.35/0.74        ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.35/0.74        = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % VEBT.size_gen(1)
% 0.35/0.74  thf(fact_8701_rat__inverse__code,axiom,
% 0.35/0.74      ! [P4: rat] :
% 0.35/0.74        ( ( quotient_of @ ( inverse_inverse_rat @ P4 ) )
% 0.35/0.74        = ( produc4245557441103728435nt_int
% 0.35/0.74          @ ^ [A4: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( A4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A4 ) @ B4 ) @ ( abs_abs_int @ A4 ) ) )
% 0.35/0.74          @ ( quotient_of @ P4 ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % rat_inverse_code
% 0.35/0.74  thf(fact_8702_finite__atLeastLessThan__int,axiom,
% 0.35/0.74      ! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).
% 0.35/0.74  
% 0.35/0.74  % finite_atLeastLessThan_int
% 0.35/0.74  thf(fact_8703_quotient__of__number_I3_J,axiom,
% 0.35/0.74      ! [K: num] :
% 0.35/0.74        ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
% 0.35/0.74        = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% 0.35/0.74  
% 0.35/0.74  % quotient_of_number(3)
% 0.35/0.74  thf(fact_8704_rat__one__code,axiom,
% 0.35/0.74      ( ( quotient_of @ one_one_rat )
% 0.35/0.74      = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% 0.35/0.74  
% 0.35/0.74  % rat_one_code
% 0.35/0.74  thf(fact_8705_rat__zero__code,axiom,
% 0.35/0.74      ( ( quotient_of @ zero_zero_rat )
% 0.35/0.74      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 0.35/0.74  
% 0.35/0.74  % rat_zero_code
% 0.35/0.74  thf(fact_8706_quotient__of__number_I5_J,axiom,
% 0.35/0.74      ! [K: num] :
% 0.35/0.74        ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.35/0.74        = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 0.35/0.74  
% 0.35/0.74  % quotient_of_number(5)
% 0.35/0.74  thf(fact_8707_quotient__of__number_I4_J,axiom,
% 0.35/0.74      ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.35/0.74      = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).
% 0.35/0.74  
% 0.35/0.74  % quotient_of_number(4)
% 0.35/0.74  thf(fact_8708_finite__atLeastZeroLessThan__int,axiom,
% 0.35/0.74      ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% 0.35/0.74  
% 0.35/0.74  % finite_atLeastZeroLessThan_int
% 0.35/0.74  thf(fact_8709_quotient__of__div,axiom,
% 0.35/0.74      ! [R2: rat,N: int,D: int] :
% 0.35/0.74        ( ( ( quotient_of @ R2 )
% 0.35/0.74          = ( product_Pair_int_int @ N @ D ) )
% 0.35/0.74       => ( R2
% 0.35/0.74          = ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % quotient_of_div
% 0.35/0.74  thf(fact_8710_quotient__of__denom__pos,axiom,
% 0.35/0.74      ! [R2: rat,P4: int,Q4: int] :
% 0.35/0.74        ( ( ( quotient_of @ R2 )
% 0.35/0.74          = ( product_Pair_int_int @ P4 @ Q4 ) )
% 0.35/0.74       => ( ord_less_int @ zero_zero_int @ Q4 ) ) ).
% 0.35/0.74  
% 0.35/0.74  % quotient_of_denom_pos
% 0.35/0.74  thf(fact_8711_rat__uminus__code,axiom,
% 0.35/0.74      ! [P4: rat] :
% 0.35/0.74        ( ( quotient_of @ ( uminus_uminus_rat @ P4 ) )
% 0.35/0.74        = ( produc4245557441103728435nt_int
% 0.35/0.74          @ ^ [A4: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A4 ) )
% 0.35/0.74          @ ( quotient_of @ P4 ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % rat_uminus_code
% 0.35/0.74  thf(fact_8712_rat__abs__code,axiom,
% 0.35/0.74      ! [P4: rat] :
% 0.35/0.74        ( ( quotient_of @ ( abs_abs_rat @ P4 ) )
% 0.35/0.74        = ( produc4245557441103728435nt_int
% 0.35/0.74          @ ^ [A4: int] : ( product_Pair_int_int @ ( abs_abs_int @ A4 ) )
% 0.35/0.74          @ ( quotient_of @ P4 ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % rat_abs_code
% 0.35/0.74  thf(fact_8713_rat__less__code,axiom,
% 0.35/0.74      ( ord_less_rat
% 0.35/0.74      = ( ^ [P6: rat,Q5: rat] :
% 0.35/0.74            ( produc4947309494688390418_int_o
% 0.35/0.74            @ ^ [A4: int,C4: int] :
% 0.35/0.74                ( produc4947309494688390418_int_o
% 0.35/0.74                @ ^ [B4: int,D4: int] : ( ord_less_int @ ( times_times_int @ A4 @ D4 ) @ ( times_times_int @ C4 @ B4 ) )
% 0.35/0.74                @ ( quotient_of @ Q5 ) )
% 0.35/0.74            @ ( quotient_of @ P6 ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % rat_less_code
% 0.35/0.74  thf(fact_8714_VEBT_Osize__gen_I2_J,axiom,
% 0.35/0.74      ! [X21: $o,X222: $o] :
% 0.35/0.74        ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 0.35/0.74        = zero_zero_nat ) ).
% 0.35/0.74  
% 0.35/0.74  % VEBT.size_gen(2)
% 0.35/0.74  thf(fact_8715_quotient__of__int,axiom,
% 0.35/0.74      ! [A: int] :
% 0.35/0.74        ( ( quotient_of @ ( of_int @ A ) )
% 0.35/0.74        = ( product_Pair_int_int @ A @ one_one_int ) ) ).
% 0.35/0.74  
% 0.35/0.74  % quotient_of_int
% 0.35/0.74  thf(fact_8716_rat__plus__code,axiom,
% 0.35/0.74      ! [P4: rat,Q4: rat] :
% 0.35/0.74        ( ( quotient_of @ ( plus_plus_rat @ P4 @ Q4 ) )
% 0.35/0.74        = ( produc4245557441103728435nt_int
% 0.35/0.74          @ ^ [A4: int,C4: int] :
% 0.35/0.74              ( produc4245557441103728435nt_int
% 0.35/0.74              @ ^ [B4: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A4 @ D4 ) @ ( times_times_int @ B4 @ C4 ) ) @ ( times_times_int @ C4 @ D4 ) ) )
% 0.35/0.74              @ ( quotient_of @ Q4 ) )
% 0.35/0.74          @ ( quotient_of @ P4 ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % rat_plus_code
% 0.35/0.74  thf(fact_8717_rat__minus__code,axiom,
% 0.35/0.74      ! [P4: rat,Q4: rat] :
% 0.35/0.74        ( ( quotient_of @ ( minus_minus_rat @ P4 @ Q4 ) )
% 0.35/0.74        = ( produc4245557441103728435nt_int
% 0.35/0.74          @ ^ [A4: int,C4: int] :
% 0.35/0.74              ( produc4245557441103728435nt_int
% 0.35/0.74              @ ^ [B4: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A4 @ D4 ) @ ( times_times_int @ B4 @ C4 ) ) @ ( times_times_int @ C4 @ D4 ) ) )
% 0.35/0.74              @ ( quotient_of @ Q4 ) )
% 0.35/0.74          @ ( quotient_of @ P4 ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % rat_minus_code
% 0.35/0.74  thf(fact_8718_normalize__negative,axiom,
% 0.35/0.74      ! [Q4: int,P4: int] :
% 0.35/0.74        ( ( ord_less_int @ Q4 @ zero_zero_int )
% 0.35/0.74       => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q4 ) )
% 0.35/0.74          = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P4 ) @ ( uminus_uminus_int @ Q4 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % normalize_negative
% 0.35/0.74  thf(fact_8719_normalize__denom__zero,axiom,
% 0.35/0.74      ! [P4: int] :
% 0.35/0.74        ( ( normalize @ ( product_Pair_int_int @ P4 @ zero_zero_int ) )
% 0.35/0.74        = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 0.35/0.74  
% 0.35/0.74  % normalize_denom_zero
% 0.35/0.74  thf(fact_8720_normalize__denom__pos,axiom,
% 0.35/0.74      ! [R2: product_prod_int_int,P4: int,Q4: int] :
% 0.35/0.74        ( ( ( normalize @ R2 )
% 0.35/0.74          = ( product_Pair_int_int @ P4 @ Q4 ) )
% 0.35/0.74       => ( ord_less_int @ zero_zero_int @ Q4 ) ) ).
% 0.35/0.74  
% 0.35/0.74  % normalize_denom_pos
% 0.35/0.74  thf(fact_8721_normalize__crossproduct,axiom,
% 0.35/0.74      ! [Q4: int,S: int,P4: int,R2: int] :
% 0.35/0.74        ( ( Q4 != zero_zero_int )
% 0.35/0.74       => ( ( S != zero_zero_int )
% 0.35/0.74         => ( ( ( normalize @ ( product_Pair_int_int @ P4 @ Q4 ) )
% 0.35/0.74              = ( normalize @ ( product_Pair_int_int @ R2 @ S ) ) )
% 0.35/0.74           => ( ( times_times_int @ P4 @ S )
% 0.35/0.74              = ( times_times_int @ R2 @ Q4 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % normalize_crossproduct
% 0.35/0.74  thf(fact_8722_rat__divide__code,axiom,
% 0.35/0.74      ! [P4: rat,Q4: rat] :
% 0.35/0.74        ( ( quotient_of @ ( divide_divide_rat @ P4 @ Q4 ) )
% 0.35/0.74        = ( produc4245557441103728435nt_int
% 0.35/0.74          @ ^ [A4: int,C4: int] :
% 0.35/0.74              ( produc4245557441103728435nt_int
% 0.35/0.74              @ ^ [B4: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A4 @ D4 ) @ ( times_times_int @ C4 @ B4 ) ) )
% 0.35/0.74              @ ( quotient_of @ Q4 ) )
% 0.35/0.74          @ ( quotient_of @ P4 ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % rat_divide_code
% 0.35/0.74  thf(fact_8723_rat__times__code,axiom,
% 0.35/0.74      ! [P4: rat,Q4: rat] :
% 0.35/0.74        ( ( quotient_of @ ( times_times_rat @ P4 @ Q4 ) )
% 0.35/0.74        = ( produc4245557441103728435nt_int
% 0.35/0.74          @ ^ [A4: int,C4: int] :
% 0.35/0.74              ( produc4245557441103728435nt_int
% 0.35/0.74              @ ^ [B4: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A4 @ B4 ) @ ( times_times_int @ C4 @ D4 ) ) )
% 0.35/0.74              @ ( quotient_of @ Q4 ) )
% 0.35/0.74          @ ( quotient_of @ P4 ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % rat_times_code
% 0.35/0.74  thf(fact_8724_Frct__code__post_I5_J,axiom,
% 0.35/0.74      ! [K: num] :
% 0.35/0.74        ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
% 0.35/0.74        = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Frct_code_post(5)
% 0.35/0.74  thf(fact_8725_divmod__step__integer__def,axiom,
% 0.35/0.74      ( unique4921790084139445826nteger
% 0.35/0.74      = ( ^ [L2: num] :
% 0.35/0.74            ( produc6916734918728496179nteger
% 0.35/0.74            @ ^ [Q5: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % divmod_step_integer_def
% 0.35/0.74  thf(fact_8726_Frct__code__post_I6_J,axiom,
% 0.35/0.74      ! [K: num,L: num] :
% 0.35/0.74        ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
% 0.35/0.74        = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Frct_code_post(6)
% 0.35/0.74  thf(fact_8727_card__Collect__less__nat,axiom,
% 0.35/0.74      ! [N: nat] :
% 0.35/0.74        ( ( finite_card_nat
% 0.35/0.74          @ ( collect_nat
% 0.35/0.74            @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
% 0.35/0.74        = N ) ).
% 0.35/0.74  
% 0.35/0.74  % card_Collect_less_nat
% 0.35/0.74  thf(fact_8728_card__Collect__le__nat,axiom,
% 0.35/0.74      ! [N: nat] :
% 0.35/0.74        ( ( finite_card_nat
% 0.35/0.74          @ ( collect_nat
% 0.35/0.74            @ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
% 0.35/0.74        = ( suc @ N ) ) ).
% 0.35/0.74  
% 0.35/0.74  % card_Collect_le_nat
% 0.35/0.74  thf(fact_8729_divmod__integer_H__def,axiom,
% 0.35/0.74      ( unique3479559517661332726nteger
% 0.35/0.74      = ( ^ [M2: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % divmod_integer'_def
% 0.35/0.74  thf(fact_8730_full__exhaustive__integer_H_Ocases,axiom,
% 0.35/0.74      ! [X: produc1908205239877642774nteger] :
% 0.35/0.74        ~ ! [F2: produc6241069584506657477e_term > option6357759511663192854e_term,D5: code_integer,I2: code_integer] :
% 0.35/0.74            ( X
% 0.35/0.74           != ( produc8603105652947943368nteger @ F2 @ ( produc1086072967326762835nteger @ D5 @ I2 ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % full_exhaustive_integer'.cases
% 0.35/0.74  thf(fact_8731_exhaustive__integer_H_Ocases,axiom,
% 0.35/0.74      ! [X: produc8763457246119570046nteger] :
% 0.35/0.74        ~ ! [F2: code_integer > option6357759511663192854e_term,D5: code_integer,I2: code_integer] :
% 0.35/0.74            ( X
% 0.35/0.74           != ( produc6137756002093451184nteger @ F2 @ ( produc1086072967326762835nteger @ D5 @ I2 ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % exhaustive_integer'.cases
% 0.35/0.74  thf(fact_8732_sgn__integer__code,axiom,
% 0.35/0.74      ( sgn_sgn_Code_integer
% 0.35/0.74      = ( ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % sgn_integer_code
% 0.35/0.74  thf(fact_8733_zero__natural_Orsp,axiom,
% 0.35/0.74      zero_zero_nat = zero_zero_nat ).
% 0.35/0.74  
% 0.35/0.74  % zero_natural.rsp
% 0.35/0.74  thf(fact_8734_card__less,axiom,
% 0.35/0.74      ! [M4: set_nat,I: nat] :
% 0.35/0.74        ( ( member_nat @ zero_zero_nat @ M4 )
% 0.35/0.74       => ( ( finite_card_nat
% 0.35/0.74            @ ( collect_nat
% 0.35/0.74              @ ^ [K3: nat] :
% 0.35/0.74                  ( ( member_nat @ K3 @ M4 )
% 0.35/0.74                  & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
% 0.35/0.74         != zero_zero_nat ) ) ).
% 0.35/0.74  
% 0.35/0.74  % card_less
% 0.35/0.74  thf(fact_8735_card__less__Suc,axiom,
% 0.35/0.74      ! [M4: set_nat,I: nat] :
% 0.35/0.74        ( ( member_nat @ zero_zero_nat @ M4 )
% 0.35/0.74       => ( ( suc
% 0.35/0.74            @ ( finite_card_nat
% 0.35/0.74              @ ( collect_nat
% 0.35/0.74                @ ^ [K3: nat] :
% 0.35/0.74                    ( ( member_nat @ ( suc @ K3 ) @ M4 )
% 0.35/0.74                    & ( ord_less_nat @ K3 @ I ) ) ) ) )
% 0.35/0.74          = ( finite_card_nat
% 0.35/0.74            @ ( collect_nat
% 0.35/0.74              @ ^ [K3: nat] :
% 0.35/0.74                  ( ( member_nat @ K3 @ M4 )
% 0.35/0.74                  & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % card_less_Suc
% 0.35/0.74  thf(fact_8736_card__less__Suc2,axiom,
% 0.35/0.74      ! [M4: set_nat,I: nat] :
% 0.35/0.74        ( ~ ( member_nat @ zero_zero_nat @ M4 )
% 0.35/0.74       => ( ( finite_card_nat
% 0.35/0.74            @ ( collect_nat
% 0.35/0.74              @ ^ [K3: nat] :
% 0.35/0.74                  ( ( member_nat @ ( suc @ K3 ) @ M4 )
% 0.35/0.74                  & ( ord_less_nat @ K3 @ I ) ) ) )
% 0.35/0.74          = ( finite_card_nat
% 0.35/0.74            @ ( collect_nat
% 0.35/0.74              @ ^ [K3: nat] :
% 0.35/0.74                  ( ( member_nat @ K3 @ M4 )
% 0.35/0.74                  & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % card_less_Suc2
% 0.35/0.74  thf(fact_8737_subset__eq__atLeast0__lessThan__card,axiom,
% 0.35/0.74      ! [N6: set_nat,N: nat] :
% 0.35/0.74        ( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.35/0.74       => ( ord_less_eq_nat @ ( finite_card_nat @ N6 ) @ N ) ) ).
% 0.35/0.74  
% 0.35/0.74  % subset_eq_atLeast0_lessThan_card
% 0.35/0.74  thf(fact_8738_card__sum__le__nat__sum,axiom,
% 0.35/0.74      ! [S2: set_nat] :
% 0.35/0.74        ( ord_less_eq_nat
% 0.35/0.74        @ ( groups3542108847815614940at_nat
% 0.35/0.74          @ ^ [X3: nat] : X3
% 0.35/0.74          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
% 0.35/0.74        @ ( groups3542108847815614940at_nat
% 0.35/0.74          @ ^ [X3: nat] : X3
% 0.35/0.74          @ S2 ) ) ).
% 0.35/0.74  
% 0.35/0.74  % card_sum_le_nat_sum
% 0.35/0.74  thf(fact_8739_card__nth__roots,axiom,
% 0.35/0.74      ! [C: complex,N: nat] :
% 0.35/0.74        ( ( C != zero_zero_complex )
% 0.35/0.74       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74         => ( ( finite_card_complex
% 0.35/0.74              @ ( collect_complex
% 0.35/0.74                @ ^ [Z3: complex] :
% 0.35/0.74                    ( ( power_power_complex @ Z3 @ N )
% 0.35/0.74                    = C ) ) )
% 0.35/0.74            = N ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % card_nth_roots
% 0.35/0.74  thf(fact_8740_card__roots__unity__eq,axiom,
% 0.35/0.74      ! [N: nat] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( finite_card_complex
% 0.35/0.74            @ ( collect_complex
% 0.35/0.74              @ ^ [Z3: complex] :
% 0.35/0.74                  ( ( power_power_complex @ Z3 @ N )
% 0.35/0.74                  = one_one_complex ) ) )
% 0.35/0.74          = N ) ) ).
% 0.35/0.74  
% 0.35/0.74  % card_roots_unity_eq
% 0.35/0.74  thf(fact_8741_Frct__code__post_I2_J,axiom,
% 0.35/0.74      ! [A: int] :
% 0.35/0.74        ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
% 0.35/0.74        = zero_zero_rat ) ).
% 0.35/0.74  
% 0.35/0.74  % Frct_code_post(2)
% 0.35/0.74  thf(fact_8742_Frct__code__post_I1_J,axiom,
% 0.35/0.74      ! [A: int] :
% 0.35/0.74        ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
% 0.35/0.74        = zero_zero_rat ) ).
% 0.35/0.74  
% 0.35/0.74  % Frct_code_post(1)
% 0.35/0.74  thf(fact_8743_Frct__code__post_I7_J,axiom,
% 0.35/0.74      ! [A: int,B: int] :
% 0.35/0.74        ( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B ) )
% 0.35/0.74        = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Frct_code_post(7)
% 0.35/0.74  thf(fact_8744_Frct__code__post_I8_J,axiom,
% 0.35/0.74      ! [A: int,B: int] :
% 0.35/0.74        ( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B ) ) )
% 0.35/0.74        = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Frct_code_post(8)
% 0.35/0.74  thf(fact_8745_Frct__code__post_I3_J,axiom,
% 0.35/0.74      ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
% 0.35/0.74      = one_one_rat ) ).
% 0.35/0.74  
% 0.35/0.74  % Frct_code_post(3)
% 0.35/0.74  thf(fact_8746_Frct__code__post_I4_J,axiom,
% 0.35/0.74      ! [K: num] :
% 0.35/0.74        ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
% 0.35/0.74        = ( numeral_numeral_rat @ K ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Frct_code_post(4)
% 0.35/0.74  thf(fact_8747_integer__of__int__code,axiom,
% 0.35/0.74      ( code_integer_of_int
% 0.35/0.74      = ( ^ [K3: int] :
% 0.35/0.74            ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
% 0.35/0.74            @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 0.35/0.74              @ ( if_Code_integer
% 0.35/0.74                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.74                  = zero_zero_int )
% 0.35/0.74                @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.35/0.74                @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % integer_of_int_code
% 0.35/0.74  thf(fact_8748_signed__take__bit__negative__iff,axiom,
% 0.35/0.74      ! [N: nat,K: int] :
% 0.35/0.74        ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
% 0.35/0.74        = ( bit_se1146084159140164899it_int @ K @ N ) ) ).
% 0.35/0.74  
% 0.35/0.74  % signed_take_bit_negative_iff
% 0.35/0.74  thf(fact_8749_less__integer_Oabs__eq,axiom,
% 0.35/0.74      ! [Xa2: int,X: int] :
% 0.35/0.74        ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
% 0.35/0.74        = ( ord_less_int @ Xa2 @ X ) ) ).
% 0.35/0.74  
% 0.35/0.74  % less_integer.abs_eq
% 0.35/0.74  thf(fact_8750_less__integer__code_I1_J,axiom,
% 0.35/0.74      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% 0.35/0.74  
% 0.35/0.74  % less_integer_code(1)
% 0.35/0.74  thf(fact_8751_abs__integer__code,axiom,
% 0.35/0.74      ( abs_abs_Code_integer
% 0.35/0.74      = ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % abs_integer_code
% 0.35/0.74  thf(fact_8752_not__bit__Suc__0__Suc,axiom,
% 0.35/0.74      ! [N: nat] :
% 0.35/0.74        ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% 0.35/0.74  
% 0.35/0.74  % not_bit_Suc_0_Suc
% 0.35/0.74  thf(fact_8753_bit__Suc__0__iff,axiom,
% 0.35/0.74      ! [N: nat] :
% 0.35/0.74        ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.74        = ( N = zero_zero_nat ) ) ).
% 0.35/0.74  
% 0.35/0.74  % bit_Suc_0_iff
% 0.35/0.74  thf(fact_8754_not__bit__Suc__0__numeral,axiom,
% 0.35/0.74      ! [N: num] :
% 0.35/0.74        ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% 0.35/0.74  
% 0.35/0.74  % not_bit_Suc_0_numeral
% 0.35/0.74  thf(fact_8755_bit__imp__take__bit__positive,axiom,
% 0.35/0.74      ! [N: nat,M: nat,K: int] :
% 0.35/0.74        ( ( ord_less_nat @ N @ M )
% 0.35/0.74       => ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.35/0.74         => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % bit_imp_take_bit_positive
% 0.35/0.74  thf(fact_8756_int__bit__bound,axiom,
% 0.35/0.74      ! [K: int] :
% 0.35/0.74        ~ ! [N3: nat] :
% 0.35/0.74            ( ! [M3: nat] :
% 0.35/0.74                ( ( ord_less_eq_nat @ N3 @ M3 )
% 0.35/0.74               => ( ( bit_se1146084159140164899it_int @ K @ M3 )
% 0.35/0.74                  = ( bit_se1146084159140164899it_int @ K @ N3 ) ) )
% 0.35/0.74           => ~ ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.35/0.74               => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N3 @ one_one_nat ) )
% 0.35/0.74                  = ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % int_bit_bound
% 0.35/0.74  thf(fact_8757_int__of__integer__code,axiom,
% 0.35/0.74      ( code_int_of_integer
% 0.35/0.74      = ( ^ [K3: code_integer] :
% 0.35/0.74            ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
% 0.35/0.74            @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 0.35/0.74              @ ( produc1553301316500091796er_int
% 0.35/0.74                @ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
% 0.35/0.74                @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % int_of_integer_code
% 0.35/0.74  thf(fact_8758_bit__cut__integer__def,axiom,
% 0.35/0.74      ( code_bit_cut_integer
% 0.35/0.74      = ( ^ [K3: code_integer] :
% 0.35/0.74            ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.74            @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % bit_cut_integer_def
% 0.35/0.74  thf(fact_8759_divmod__integer__def,axiom,
% 0.35/0.74      ( code_divmod_integer
% 0.35/0.74      = ( ^ [K3: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K3 @ L2 ) @ ( modulo364778990260209775nteger @ K3 @ L2 ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % divmod_integer_def
% 0.35/0.74  thf(fact_8760_less__integer_Orep__eq,axiom,
% 0.35/0.74      ( ord_le6747313008572928689nteger
% 0.35/0.74      = ( ^ [X3: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % less_integer.rep_eq
% 0.35/0.74  thf(fact_8761_integer__less__iff,axiom,
% 0.35/0.74      ( ord_le6747313008572928689nteger
% 0.35/0.74      = ( ^ [K3: code_integer,L2: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % integer_less_iff
% 0.35/0.74  thf(fact_8762_binomial__def,axiom,
% 0.35/0.74      ( binomial
% 0.35/0.74      = ( ^ [N2: nat,K3: nat] :
% 0.35/0.74            ( finite_card_set_nat
% 0.35/0.74            @ ( collect_set_nat
% 0.35/0.74              @ ^ [K6: set_nat] :
% 0.35/0.74                  ( ( member_set_nat @ K6 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
% 0.35/0.74                  & ( ( finite_card_nat @ K6 )
% 0.35/0.74                    = K3 ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % binomial_def
% 0.35/0.74  thf(fact_8763_bit__cut__integer__code,axiom,
% 0.35/0.74      ( code_bit_cut_integer
% 0.35/0.74      = ( ^ [K3: code_integer] :
% 0.35/0.74            ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 0.35/0.74            @ ( produc9125791028180074456eger_o
% 0.35/0.74              @ ^ [R5: code_integer,S8: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S8 ) ) @ ( S8 = one_one_Code_integer ) )
% 0.35/0.74              @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % bit_cut_integer_code
% 0.35/0.74  thf(fact_8764_nat__of__integer__code,axiom,
% 0.35/0.74      ( code_nat_of_integer
% 0.35/0.74      = ( ^ [K3: code_integer] :
% 0.35/0.74            ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 0.35/0.74            @ ( produc1555791787009142072er_nat
% 0.35/0.74              @ ^ [L2: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ one_one_nat ) )
% 0.35/0.74              @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % nat_of_integer_code
% 0.35/0.74  thf(fact_8765_divmod__abs__def,axiom,
% 0.35/0.74      ( code_divmod_abs
% 0.35/0.74      = ( ^ [K3: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L2 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % divmod_abs_def
% 0.35/0.74  thf(fact_8766_pair__leqI2,axiom,
% 0.35/0.74      ! [A: nat,B: nat,S: nat,T: nat] :
% 0.35/0.74        ( ( ord_less_eq_nat @ A @ B )
% 0.35/0.74       => ( ( ord_less_eq_nat @ S @ T )
% 0.35/0.74         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % pair_leqI2
% 0.35/0.74  thf(fact_8767_pair__leqI1,axiom,
% 0.35/0.74      ! [A: nat,B: nat,S: nat,T: nat] :
% 0.35/0.74        ( ( ord_less_nat @ A @ B )
% 0.35/0.74       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).
% 0.35/0.74  
% 0.35/0.74  % pair_leqI1
% 0.35/0.74  thf(fact_8768_nat__of__integer__non__positive,axiom,
% 0.35/0.74      ! [K: code_integer] :
% 0.35/0.74        ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
% 0.35/0.74       => ( ( code_nat_of_integer @ K )
% 0.35/0.74          = zero_zero_nat ) ) ).
% 0.35/0.74  
% 0.35/0.74  % nat_of_integer_non_positive
% 0.35/0.74  thf(fact_8769_nat__of__integer__code__post_I1_J,axiom,
% 0.35/0.74      ( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
% 0.35/0.74      = zero_zero_nat ) ).
% 0.35/0.74  
% 0.35/0.74  % nat_of_integer_code_post(1)
% 0.35/0.74  thf(fact_8770_divmod__abs__code_I6_J,axiom,
% 0.35/0.74      ! [J: code_integer] :
% 0.35/0.74        ( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
% 0.35/0.74        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).
% 0.35/0.74  
% 0.35/0.74  % divmod_abs_code(6)
% 0.35/0.74  thf(fact_8771_divmod__abs__code_I5_J,axiom,
% 0.35/0.74      ! [J: code_integer] :
% 0.35/0.74        ( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
% 0.35/0.74        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % divmod_abs_code(5)
% 0.35/0.74  thf(fact_8772_divmod__integer__code,axiom,
% 0.35/0.74      ( code_divmod_integer
% 0.35/0.74      = ( ^ [K3: code_integer,L2: code_integer] :
% 0.35/0.74            ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.35/0.74            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
% 0.35/0.74              @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L2 )
% 0.35/0.74                @ ( produc6916734918728496179nteger
% 0.35/0.74                  @ ^ [R5: code_integer,S8: code_integer] : ( if_Pro6119634080678213985nteger @ ( S8 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S8 ) ) )
% 0.35/0.74                  @ ( code_divmod_abs @ K3 @ L2 ) ) )
% 0.35/0.74              @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 0.35/0.74                @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 0.35/0.74                  @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L2 )
% 0.35/0.74                    @ ( produc6916734918728496179nteger
% 0.35/0.74                      @ ^ [R5: code_integer,S8: code_integer] : ( if_Pro6119634080678213985nteger @ ( S8 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S8 ) ) )
% 0.35/0.74                      @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % divmod_integer_code
% 0.35/0.74  thf(fact_8773_wmin__insertI,axiom,
% 0.35/0.74      ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
% 0.35/0.74        ( ( member8440522571783428010at_nat @ X @ XS )
% 0.35/0.74       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
% 0.35/0.74         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
% 0.35/0.74           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % wmin_insertI
% 0.35/0.74  thf(fact_8774_wmax__insertI,axiom,
% 0.35/0.74      ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
% 0.35/0.74        ( ( member8440522571783428010at_nat @ Y @ YS )
% 0.35/0.74       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
% 0.35/0.74         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
% 0.35/0.74           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % wmax_insertI
% 0.35/0.74  thf(fact_8775_bezw__0,axiom,
% 0.35/0.74      ! [X: nat] :
% 0.35/0.74        ( ( bezw @ X @ zero_zero_nat )
% 0.35/0.74        = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% 0.35/0.74  
% 0.35/0.74  % bezw_0
% 0.35/0.74  thf(fact_8776_Suc__0__mod__numeral,axiom,
% 0.35/0.74      ! [K: num] :
% 0.35/0.74        ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.35/0.74        = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Suc_0_mod_numeral
% 0.35/0.74  thf(fact_8777_wmax__emptyI,axiom,
% 0.35/0.74      ! [X8: set_Pr1261947904930325089at_nat] :
% 0.35/0.74        ( ( finite6177210948735845034at_nat @ X8 )
% 0.35/0.74       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X8 ) @ fun_max_weak ) ) ).
% 0.35/0.74  
% 0.35/0.74  % wmax_emptyI
% 0.35/0.74  thf(fact_8778_wmin__emptyI,axiom,
% 0.35/0.74      ! [X8: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X8 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).
% 0.35/0.74  
% 0.35/0.74  % wmin_emptyI
% 0.35/0.74  thf(fact_8779_smin__insertI,axiom,
% 0.35/0.74      ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
% 0.35/0.74        ( ( member8440522571783428010at_nat @ X @ XS )
% 0.35/0.74       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
% 0.35/0.74         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
% 0.35/0.74           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % smin_insertI
% 0.35/0.74  thf(fact_8780_smax__insertI,axiom,
% 0.35/0.74      ! [Y: product_prod_nat_nat,Y7: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X8: set_Pr1261947904930325089at_nat] :
% 0.35/0.74        ( ( member8440522571783428010at_nat @ Y @ Y7 )
% 0.35/0.74       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
% 0.35/0.74         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X8 @ Y7 ) @ fun_max_strict )
% 0.35/0.74           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ X8 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % smax_insertI
% 0.35/0.74  thf(fact_8781_prod__decode__aux_Osimps,axiom,
% 0.35/0.74      ( nat_prod_decode_aux
% 0.35/0.74      = ( ^ [K3: nat,M2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M2 @ K3 ) @ ( product_Pair_nat_nat @ M2 @ ( minus_minus_nat @ K3 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M2 @ ( suc @ K3 ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod_decode_aux.simps
% 0.35/0.74  thf(fact_8782_prod__decode__aux_Oelims,axiom,
% 0.35/0.74      ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
% 0.35/0.74        ( ( ( nat_prod_decode_aux @ X @ Xa2 )
% 0.35/0.74          = Y )
% 0.35/0.74       => ( ( ( ord_less_eq_nat @ Xa2 @ X )
% 0.35/0.74           => ( Y
% 0.35/0.74              = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
% 0.35/0.74          & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
% 0.35/0.74           => ( Y
% 0.35/0.74              = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod_decode_aux.elims
% 0.35/0.74  thf(fact_8783_Suc__0__div__numeral,axiom,
% 0.35/0.74      ! [K: num] :
% 0.35/0.74        ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.35/0.74        = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Suc_0_div_numeral
% 0.35/0.74  thf(fact_8784_quotient__of__denom__pos_H,axiom,
% 0.35/0.74      ! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % quotient_of_denom_pos'
% 0.35/0.74  thf(fact_8785_smin__emptyI,axiom,
% 0.35/0.74      ! [X8: set_Pr1261947904930325089at_nat] :
% 0.35/0.74        ( ( X8 != bot_bo2099793752762293965at_nat )
% 0.35/0.74       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X8 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).
% 0.35/0.74  
% 0.35/0.74  % smin_emptyI
% 0.35/0.74  thf(fact_8786_smax__emptyI,axiom,
% 0.35/0.74      ! [Y7: set_Pr1261947904930325089at_nat] :
% 0.35/0.74        ( ( finite6177210948735845034at_nat @ Y7 )
% 0.35/0.74       => ( ( Y7 != bot_bo2099793752762293965at_nat )
% 0.35/0.74         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y7 ) @ fun_max_strict ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % smax_emptyI
% 0.35/0.74  thf(fact_8787_bezw_Osimps,axiom,
% 0.35/0.74      ( bezw
% 0.35/0.74      = ( ^ [X3: nat,Y2: nat] : ( if_Pro3027730157355071871nt_int @ ( Y2 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X3 @ Y2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X3 @ Y2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X3 @ Y2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Y2 ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % bezw.simps
% 0.35/0.74  thf(fact_8788_rat__sgn__code,axiom,
% 0.35/0.74      ! [P4: rat] :
% 0.35/0.74        ( ( quotient_of @ ( sgn_sgn_rat @ P4 ) )
% 0.35/0.74        = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P4 ) ) ) @ one_one_int ) ) ).
% 0.35/0.74  
% 0.35/0.74  % rat_sgn_code
% 0.35/0.74  thf(fact_8789_bezw__non__0,axiom,
% 0.35/0.74      ! [Y: nat,X: nat] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ Y )
% 0.35/0.74       => ( ( bezw @ X @ Y )
% 0.35/0.74          = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % bezw_non_0
% 0.35/0.74  thf(fact_8790_bezw_Oelims,axiom,
% 0.35/0.74      ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
% 0.35/0.74        ( ( ( bezw @ X @ Xa2 )
% 0.35/0.74          = Y )
% 0.35/0.74       => ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.74           => ( Y
% 0.35/0.74              = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.35/0.74          & ( ( Xa2 != zero_zero_nat )
% 0.35/0.74           => ( Y
% 0.35/0.74              = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % bezw.elims
% 0.35/0.74  thf(fact_8791_bezw_Opelims,axiom,
% 0.35/0.74      ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
% 0.35/0.74        ( ( ( bezw @ X @ Xa2 )
% 0.35/0.74          = Y )
% 0.35/0.74       => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
% 0.35/0.74         => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.74                 => ( Y
% 0.35/0.74                    = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.35/0.74                & ( ( Xa2 != zero_zero_nat )
% 0.35/0.74                 => ( Y
% 0.35/0.74                    = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) )
% 0.35/0.74             => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % bezw.pelims
% 0.35/0.74  thf(fact_8792_normalize__def,axiom,
% 0.35/0.74      ( normalize
% 0.35/0.74      = ( ^ [P6: product_prod_int_int] :
% 0.35/0.74            ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P6 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P6 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P6 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) )
% 0.35/0.74            @ ( if_Pro3027730157355071871nt_int
% 0.35/0.74              @ ( ( product_snd_int_int @ P6 )
% 0.35/0.74                = zero_zero_int )
% 0.35/0.74              @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.35/0.74              @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P6 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P6 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % normalize_def
% 0.35/0.74  thf(fact_8793_vebt__mint_Opelims,axiom,
% 0.35/0.74      ! [X: vEBT_VEBT,Y: option_nat] :
% 0.35/0.74        ( ( ( vEBT_vebt_mint @ X )
% 0.35/0.74          = Y )
% 0.35/0.74       => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X )
% 0.35/0.74         => ( ! [A3: $o,B3: $o] :
% 0.35/0.74                ( ( X
% 0.35/0.74                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.74               => ( ( ( A3
% 0.35/0.74                     => ( Y
% 0.35/0.74                        = ( some_nat @ zero_zero_nat ) ) )
% 0.35/0.74                    & ( ~ A3
% 0.35/0.74                     => ( ( B3
% 0.35/0.74                         => ( Y
% 0.35/0.74                            = ( some_nat @ one_one_nat ) ) )
% 0.35/0.74                        & ( ~ B3
% 0.35/0.74                         => ( Y = none_nat ) ) ) ) )
% 0.35/0.74                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
% 0.35/0.74           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.74                  ( ( X
% 0.35/0.74                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.35/0.74                 => ( ( Y = none_nat )
% 0.35/0.74                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 0.35/0.74             => ~ ! [Mi3: nat,Ma3: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.35/0.74                    ( ( X
% 0.35/0.74                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.35/0.74                   => ( ( Y
% 0.35/0.74                        = ( some_nat @ Mi3 ) )
% 0.35/0.74                     => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % vebt_mint.pelims
% 0.35/0.74  thf(fact_8794_vebt__maxt_Opelims,axiom,
% 0.35/0.74      ! [X: vEBT_VEBT,Y: option_nat] :
% 0.35/0.74        ( ( ( vEBT_vebt_maxt @ X )
% 0.35/0.74          = Y )
% 0.35/0.74       => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
% 0.35/0.74         => ( ! [A3: $o,B3: $o] :
% 0.35/0.74                ( ( X
% 0.35/0.74                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.35/0.74               => ( ( ( B3
% 0.35/0.74                     => ( Y
% 0.35/0.74                        = ( some_nat @ one_one_nat ) ) )
% 0.35/0.74                    & ( ~ B3
% 0.35/0.74                     => ( ( A3
% 0.35/0.74                         => ( Y
% 0.35/0.74                            = ( some_nat @ zero_zero_nat ) ) )
% 0.35/0.74                        & ( ~ A3
% 0.35/0.74                         => ( Y = none_nat ) ) ) ) )
% 0.35/0.74                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
% 0.35/0.74           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.74                  ( ( X
% 0.35/0.74                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.35/0.74                 => ( ( Y = none_nat )
% 0.35/0.74                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 0.35/0.74             => ~ ! [Mi3: nat,Ma3: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.35/0.74                    ( ( X
% 0.35/0.74                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.35/0.74                   => ( ( Y
% 0.35/0.74                        = ( some_nat @ Ma3 ) )
% 0.35/0.74                     => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % vebt_maxt.pelims
% 0.35/0.74  thf(fact_8795_gcd__pos__int,axiom,
% 0.35/0.74      ! [M: int,N: int] :
% 0.35/0.74        ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
% 0.35/0.74        = ( ( M != zero_zero_int )
% 0.35/0.74          | ( N != zero_zero_int ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_pos_int
% 0.35/0.74  thf(fact_8796_gcd__le1__int,axiom,
% 0.35/0.74      ! [A: int,B: int] :
% 0.35/0.74        ( ( ord_less_int @ zero_zero_int @ A )
% 0.35/0.74       => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_le1_int
% 0.35/0.74  thf(fact_8797_gcd__le2__int,axiom,
% 0.35/0.74      ! [B: int,A: int] :
% 0.35/0.74        ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.74       => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_le2_int
% 0.35/0.74  thf(fact_8798_gcd__non__0__int,axiom,
% 0.35/0.74      ! [Y: int,X: int] :
% 0.35/0.74        ( ( ord_less_int @ zero_zero_int @ Y )
% 0.35/0.74       => ( ( gcd_gcd_int @ X @ Y )
% 0.35/0.74          = ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_non_0_int
% 0.35/0.74  thf(fact_8799_prod__decode__aux_Opelims,axiom,
% 0.35/0.74      ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
% 0.35/0.74        ( ( ( nat_prod_decode_aux @ X @ Xa2 )
% 0.35/0.74          = Y )
% 0.35/0.74       => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
% 0.35/0.74         => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X )
% 0.35/0.74                 => ( Y
% 0.35/0.74                    = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
% 0.35/0.74                & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
% 0.35/0.74                 => ( Y
% 0.35/0.74                    = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) )
% 0.35/0.74             => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % prod_decode_aux.pelims
% 0.35/0.74  thf(fact_8800_gcd__nat_Oeq__neutr__iff,axiom,
% 0.35/0.74      ! [A: nat,B: nat] :
% 0.35/0.74        ( ( ( gcd_gcd_nat @ A @ B )
% 0.35/0.74          = zero_zero_nat )
% 0.35/0.74        = ( ( A = zero_zero_nat )
% 0.35/0.74          & ( B = zero_zero_nat ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_nat.eq_neutr_iff
% 0.35/0.74  thf(fact_8801_gcd__nat_Oleft__neutral,axiom,
% 0.35/0.74      ! [A: nat] :
% 0.35/0.74        ( ( gcd_gcd_nat @ zero_zero_nat @ A )
% 0.35/0.74        = A ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_nat.left_neutral
% 0.35/0.74  thf(fact_8802_gcd__nat_Oneutr__eq__iff,axiom,
% 0.35/0.74      ! [A: nat,B: nat] :
% 0.35/0.74        ( ( zero_zero_nat
% 0.35/0.74          = ( gcd_gcd_nat @ A @ B ) )
% 0.35/0.74        = ( ( A = zero_zero_nat )
% 0.35/0.74          & ( B = zero_zero_nat ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_nat.neutr_eq_iff
% 0.35/0.74  thf(fact_8803_gcd__nat_Oright__neutral,axiom,
% 0.35/0.74      ! [A: nat] :
% 0.35/0.74        ( ( gcd_gcd_nat @ A @ zero_zero_nat )
% 0.35/0.74        = A ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_nat.right_neutral
% 0.35/0.74  thf(fact_8804_gcd__0__nat,axiom,
% 0.35/0.74      ! [X: nat] :
% 0.35/0.74        ( ( gcd_gcd_nat @ X @ zero_zero_nat )
% 0.35/0.74        = X ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_0_nat
% 0.35/0.74  thf(fact_8805_gcd__0__left__nat,axiom,
% 0.35/0.74      ! [X: nat] :
% 0.35/0.74        ( ( gcd_gcd_nat @ zero_zero_nat @ X )
% 0.35/0.74        = X ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_0_left_nat
% 0.35/0.74  thf(fact_8806_gcd__Suc__0,axiom,
% 0.35/0.74      ! [M: nat] :
% 0.35/0.74        ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.35/0.74        = ( suc @ zero_zero_nat ) ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_Suc_0
% 0.35/0.74  thf(fact_8807_gcd__pos__nat,axiom,
% 0.35/0.74      ! [M: nat,N: nat] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
% 0.35/0.74        = ( ( M != zero_zero_nat )
% 0.35/0.74          | ( N != zero_zero_nat ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_pos_nat
% 0.35/0.74  thf(fact_8808_Sup__nat__empty,axiom,
% 0.35/0.74      ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
% 0.35/0.74      = zero_zero_nat ) ).
% 0.35/0.74  
% 0.35/0.74  % Sup_nat_empty
% 0.35/0.74  thf(fact_8809_Inf__nat__def1,axiom,
% 0.35/0.74      ! [K4: set_nat] :
% 0.35/0.74        ( ( K4 != bot_bot_set_nat )
% 0.35/0.74       => ( member_nat @ ( complete_Inf_Inf_nat @ K4 ) @ K4 ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Inf_nat_def1
% 0.35/0.74  thf(fact_8810_gcd__le2__nat,axiom,
% 0.35/0.74      ! [B: nat,A: nat] :
% 0.35/0.74        ( ( B != zero_zero_nat )
% 0.35/0.74       => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_le2_nat
% 0.35/0.74  thf(fact_8811_gcd__le1__nat,axiom,
% 0.35/0.74      ! [A: nat,B: nat] :
% 0.35/0.74        ( ( A != zero_zero_nat )
% 0.35/0.74       => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_le1_nat
% 0.35/0.74  thf(fact_8812_gcd__diff2__nat,axiom,
% 0.35/0.74      ! [M: nat,N: nat] :
% 0.35/0.74        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.74       => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
% 0.35/0.74          = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_diff2_nat
% 0.35/0.74  thf(fact_8813_gcd__diff1__nat,axiom,
% 0.35/0.74      ! [N: nat,M: nat] :
% 0.35/0.74        ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.74       => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
% 0.35/0.74          = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_diff1_nat
% 0.35/0.74  thf(fact_8814_gcd__nat_Oelims,axiom,
% 0.35/0.74      ! [X: nat,Xa2: nat,Y: nat] :
% 0.35/0.74        ( ( ( gcd_gcd_nat @ X @ Xa2 )
% 0.35/0.74          = Y )
% 0.35/0.74       => ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.74           => ( Y = X ) )
% 0.35/0.74          & ( ( Xa2 != zero_zero_nat )
% 0.35/0.74           => ( Y
% 0.35/0.74              = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_nat.elims
% 0.35/0.74  thf(fact_8815_gcd__nat_Osimps,axiom,
% 0.35/0.74      ( gcd_gcd_nat
% 0.35/0.74      = ( ^ [X3: nat,Y2: nat] : ( if_nat @ ( Y2 = zero_zero_nat ) @ X3 @ ( gcd_gcd_nat @ Y2 @ ( modulo_modulo_nat @ X3 @ Y2 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_nat.simps
% 0.35/0.74  thf(fact_8816_gcd__non__0__nat,axiom,
% 0.35/0.74      ! [Y: nat,X: nat] :
% 0.35/0.74        ( ( Y != zero_zero_nat )
% 0.35/0.74       => ( ( gcd_gcd_nat @ X @ Y )
% 0.35/0.74          = ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_non_0_nat
% 0.35/0.74  thf(fact_8817_bezout__nat,axiom,
% 0.35/0.74      ! [A: nat,B: nat] :
% 0.35/0.74        ( ( A != zero_zero_nat )
% 0.35/0.74       => ? [X4: nat,Y3: nat] :
% 0.35/0.74            ( ( times_times_nat @ A @ X4 )
% 0.35/0.74            = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % bezout_nat
% 0.35/0.74  thf(fact_8818_bezout__gcd__nat_H,axiom,
% 0.35/0.74      ! [B: nat,A: nat] :
% 0.35/0.74      ? [X4: nat,Y3: nat] :
% 0.35/0.74        ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y3 ) @ ( times_times_nat @ A @ X4 ) )
% 0.35/0.74          & ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y3 ) )
% 0.35/0.74            = ( gcd_gcd_nat @ A @ B ) ) )
% 0.35/0.74        | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y3 ) @ ( times_times_nat @ B @ X4 ) )
% 0.35/0.74          & ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y3 ) )
% 0.35/0.74            = ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % bezout_gcd_nat'
% 0.35/0.74  thf(fact_8819_gcd__nat_Opelims,axiom,
% 0.35/0.74      ! [X: nat,Xa2: nat,Y: nat] :
% 0.35/0.74        ( ( ( gcd_gcd_nat @ X @ Xa2 )
% 0.35/0.74          = Y )
% 0.35/0.74       => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
% 0.35/0.74         => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.74                 => ( Y = X ) )
% 0.35/0.74                & ( ( Xa2 != zero_zero_nat )
% 0.35/0.74                 => ( Y
% 0.35/0.74                    = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) )
% 0.35/0.74             => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_nat.pelims
% 0.35/0.74  thf(fact_8820_VEBT__internal_OminNull_Opelims_I1_J,axiom,
% 0.35/0.74      ! [X: vEBT_VEBT,Y: $o] :
% 0.35/0.74        ( ( ( vEBT_VEBT_minNull @ X )
% 0.35/0.74          = Y )
% 0.35/0.74       => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
% 0.35/0.74         => ( ( ( X
% 0.35/0.74                = ( vEBT_Leaf @ $false @ $false ) )
% 0.35/0.74             => ( Y
% 0.35/0.74               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
% 0.35/0.74           => ( ! [Uv2: $o] :
% 0.35/0.74                  ( ( X
% 0.35/0.74                    = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.35/0.74                 => ( ~ Y
% 0.35/0.74                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
% 0.35/0.74             => ( ! [Uu2: $o] :
% 0.35/0.74                    ( ( X
% 0.35/0.74                      = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.35/0.74                   => ( ~ Y
% 0.35/0.74                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
% 0.35/0.74               => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.35/0.74                      ( ( X
% 0.35/0.74                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.35/0.74                     => ( Y
% 0.35/0.74                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
% 0.35/0.74                 => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.74                        ( ( X
% 0.35/0.74                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 0.35/0.74                       => ( ~ Y
% 0.35/0.74                         => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % VEBT_internal.minNull.pelims(1)
% 0.35/0.74  thf(fact_8821_VEBT__internal_OminNull_Opelims_I2_J,axiom,
% 0.35/0.74      ! [X: vEBT_VEBT] :
% 0.35/0.74        ( ( vEBT_VEBT_minNull @ X )
% 0.35/0.74       => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
% 0.35/0.74         => ( ( ( X
% 0.35/0.74                = ( vEBT_Leaf @ $false @ $false ) )
% 0.35/0.74             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
% 0.35/0.74           => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.35/0.74                  ( ( X
% 0.35/0.74                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.35/0.74                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % VEBT_internal.minNull.pelims(2)
% 0.35/0.74  thf(fact_8822_VEBT__internal_OminNull_Opelims_I3_J,axiom,
% 0.35/0.74      ! [X: vEBT_VEBT] :
% 0.35/0.74        ( ~ ( vEBT_VEBT_minNull @ X )
% 0.35/0.74       => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
% 0.35/0.74         => ( ! [Uv2: $o] :
% 0.35/0.74                ( ( X
% 0.35/0.74                  = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.35/0.74               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
% 0.35/0.74           => ( ! [Uu2: $o] :
% 0.35/0.74                  ( ( X
% 0.35/0.74                    = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.35/0.74                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
% 0.35/0.74             => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.74                    ( ( X
% 0.35/0.74                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 0.35/0.74                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % VEBT_internal.minNull.pelims(3)
% 0.35/0.74  thf(fact_8823_finite__enumerate,axiom,
% 0.35/0.74      ! [S2: set_nat] :
% 0.35/0.74        ( ( finite_finite_nat @ S2 )
% 0.35/0.74       => ? [R3: nat > nat] :
% 0.35/0.74            ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S2 ) ) )
% 0.35/0.74            & ! [N5: nat] :
% 0.35/0.74                ( ( ord_less_nat @ N5 @ ( finite_card_nat @ S2 ) )
% 0.35/0.74               => ( member_nat @ ( R3 @ N5 ) @ S2 ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % finite_enumerate
% 0.35/0.74  thf(fact_8824_divmod__integer__eq__cases,axiom,
% 0.35/0.74      ( code_divmod_integer
% 0.35/0.74      = ( ^ [K3: code_integer,L2: code_integer] :
% 0.35/0.74            ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.35/0.74            @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 0.35/0.74              @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
% 0.35/0.74                @ ( if_Pro6119634080678213985nteger
% 0.35/0.74                  @ ( ( sgn_sgn_Code_integer @ K3 )
% 0.35/0.74                    = ( sgn_sgn_Code_integer @ L2 ) )
% 0.35/0.74                  @ ( code_divmod_abs @ K3 @ L2 )
% 0.35/0.74                  @ ( produc6916734918728496179nteger
% 0.35/0.74                    @ ^ [R5: code_integer,S8: code_integer] : ( if_Pro6119634080678213985nteger @ ( S8 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S8 ) ) )
% 0.35/0.74                    @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % divmod_integer_eq_cases
% 0.35/0.74  thf(fact_8825_card_Ocomp__fun__commute__on,axiom,
% 0.35/0.74      ( ( comp_nat_nat_nat @ suc @ suc )
% 0.35/0.74      = ( comp_nat_nat_nat @ suc @ suc ) ) ).
% 0.35/0.74  
% 0.35/0.74  % card.comp_fun_commute_on
% 0.35/0.74  thf(fact_8826_root__powr__inverse,axiom,
% 0.35/0.74      ! [N: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.74         => ( ( root @ N @ X )
% 0.35/0.74            = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % root_powr_inverse
% 0.35/0.74  thf(fact_8827_drop__bit__negative__int__iff,axiom,
% 0.35/0.74      ! [N: nat,K: int] :
% 0.35/0.74        ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
% 0.35/0.74        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.35/0.74  
% 0.35/0.74  % drop_bit_negative_int_iff
% 0.35/0.74  thf(fact_8828_real__root__Suc__0,axiom,
% 0.35/0.74      ! [X: real] :
% 0.35/0.74        ( ( root @ ( suc @ zero_zero_nat ) @ X )
% 0.35/0.74        = X ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_Suc_0
% 0.35/0.74  thf(fact_8829_real__root__eq__iff,axiom,
% 0.35/0.74      ! [N: nat,X: real,Y: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ( root @ N @ X )
% 0.35/0.74            = ( root @ N @ Y ) )
% 0.35/0.74          = ( X = Y ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_eq_iff
% 0.35/0.74  thf(fact_8830_root__0,axiom,
% 0.35/0.74      ! [X: real] :
% 0.35/0.74        ( ( root @ zero_zero_nat @ X )
% 0.35/0.74        = zero_zero_real ) ).
% 0.35/0.74  
% 0.35/0.74  % root_0
% 0.35/0.74  thf(fact_8831_drop__bit__of__Suc__0,axiom,
% 0.35/0.74      ! [N: nat] :
% 0.35/0.74        ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.35/0.74        = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % drop_bit_of_Suc_0
% 0.35/0.74  thf(fact_8832_real__root__eq__0__iff,axiom,
% 0.35/0.74      ! [N: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ( root @ N @ X )
% 0.35/0.74            = zero_zero_real )
% 0.35/0.74          = ( X = zero_zero_real ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_eq_0_iff
% 0.35/0.74  thf(fact_8833_real__root__less__iff,axiom,
% 0.35/0.74      ! [N: nat,X: real,Y: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
% 0.35/0.74          = ( ord_less_real @ X @ Y ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_less_iff
% 0.35/0.74  thf(fact_8834_real__root__le__iff,axiom,
% 0.35/0.74      ! [N: nat,X: real,Y: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
% 0.35/0.74          = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_le_iff
% 0.35/0.74  thf(fact_8835_real__root__eq__1__iff,axiom,
% 0.35/0.74      ! [N: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ( root @ N @ X )
% 0.35/0.74            = one_one_real )
% 0.35/0.74          = ( X = one_one_real ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_eq_1_iff
% 0.35/0.74  thf(fact_8836_real__root__one,axiom,
% 0.35/0.74      ! [N: nat] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( root @ N @ one_one_real )
% 0.35/0.74          = one_one_real ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_one
% 0.35/0.74  thf(fact_8837_real__root__lt__0__iff,axiom,
% 0.35/0.74      ! [N: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
% 0.35/0.74          = ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_lt_0_iff
% 0.35/0.74  thf(fact_8838_real__root__gt__0__iff,axiom,
% 0.35/0.74      ! [N: nat,Y: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y ) )
% 0.35/0.74          = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_gt_0_iff
% 0.35/0.74  thf(fact_8839_real__root__le__0__iff,axiom,
% 0.35/0.74      ! [N: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
% 0.35/0.74          = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_le_0_iff
% 0.35/0.74  thf(fact_8840_real__root__ge__0__iff,axiom,
% 0.35/0.74      ! [N: nat,Y: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y ) )
% 0.35/0.74          = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_ge_0_iff
% 0.35/0.74  thf(fact_8841_real__root__lt__1__iff,axiom,
% 0.35/0.74      ! [N: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
% 0.35/0.74          = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_lt_1_iff
% 0.35/0.74  thf(fact_8842_real__root__gt__1__iff,axiom,
% 0.35/0.74      ! [N: nat,Y: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y ) )
% 0.35/0.74          = ( ord_less_real @ one_one_real @ Y ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_gt_1_iff
% 0.35/0.74  thf(fact_8843_real__root__le__1__iff,axiom,
% 0.35/0.74      ! [N: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
% 0.35/0.74          = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_le_1_iff
% 0.35/0.74  thf(fact_8844_real__root__ge__1__iff,axiom,
% 0.35/0.74      ! [N: nat,Y: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y ) )
% 0.35/0.74          = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_ge_1_iff
% 0.35/0.74  thf(fact_8845_real__root__pow__pos2,axiom,
% 0.35/0.74      ! [N: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.74         => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 0.35/0.74            = X ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_pow_pos2
% 0.35/0.74  thf(fact_8846_real__root__less__mono,axiom,
% 0.35/0.74      ! [N: nat,X: real,Y: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_real @ X @ Y )
% 0.35/0.74         => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_less_mono
% 0.35/0.74  thf(fact_8847_real__root__le__mono,axiom,
% 0.35/0.74      ! [N: nat,X: real,Y: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_eq_real @ X @ Y )
% 0.35/0.74         => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_le_mono
% 0.35/0.74  thf(fact_8848_real__root__power,axiom,
% 0.35/0.74      ! [N: nat,X: real,K: nat] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( root @ N @ ( power_power_real @ X @ K ) )
% 0.35/0.74          = ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_power
% 0.35/0.74  thf(fact_8849_real__root__abs,axiom,
% 0.35/0.74      ! [N: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( root @ N @ ( abs_abs_real @ X ) )
% 0.35/0.74          = ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_abs
% 0.35/0.74  thf(fact_8850_sgn__root,axiom,
% 0.35/0.74      ! [N: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( sgn_sgn_real @ ( root @ N @ X ) )
% 0.35/0.74          = ( sgn_sgn_real @ X ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % sgn_root
% 0.35/0.74  thf(fact_8851_real__root__gt__zero,axiom,
% 0.35/0.74      ! [N: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.74         => ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_gt_zero
% 0.35/0.74  thf(fact_8852_real__root__strict__decreasing,axiom,
% 0.35/0.74      ! [N: nat,N6: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_nat @ N @ N6 )
% 0.35/0.74         => ( ( ord_less_real @ one_one_real @ X )
% 0.35/0.74           => ( ord_less_real @ ( root @ N6 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_strict_decreasing
% 0.35/0.74  thf(fact_8853_root__abs__power,axiom,
% 0.35/0.74      ! [N: nat,Y: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y @ N ) ) )
% 0.35/0.74          = ( abs_abs_real @ Y ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % root_abs_power
% 0.35/0.74  thf(fact_8854_real__root__pos__pos,axiom,
% 0.35/0.74      ! [N: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.74         => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_pos_pos
% 0.35/0.74  thf(fact_8855_real__root__strict__increasing,axiom,
% 0.35/0.74      ! [N: nat,N6: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_nat @ N @ N6 )
% 0.35/0.74         => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.74           => ( ( ord_less_real @ X @ one_one_real )
% 0.35/0.74             => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_strict_increasing
% 0.35/0.74  thf(fact_8856_real__root__decreasing,axiom,
% 0.35/0.74      ! [N: nat,N6: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_eq_nat @ N @ N6 )
% 0.35/0.74         => ( ( ord_less_eq_real @ one_one_real @ X )
% 0.35/0.74           => ( ord_less_eq_real @ ( root @ N6 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_decreasing
% 0.35/0.74  thf(fact_8857_real__root__pow__pos,axiom,
% 0.35/0.74      ! [N: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.74         => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 0.35/0.74            = X ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_pow_pos
% 0.35/0.74  thf(fact_8858_real__root__pos__unique,axiom,
% 0.35/0.74      ! [N: nat,Y: real,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.35/0.74         => ( ( ( power_power_real @ Y @ N )
% 0.35/0.74              = X )
% 0.35/0.74           => ( ( root @ N @ X )
% 0.35/0.74              = Y ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_pos_unique
% 0.35/0.74  thf(fact_8859_real__root__power__cancel,axiom,
% 0.35/0.74      ! [N: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.74         => ( ( root @ N @ ( power_power_real @ X @ N ) )
% 0.35/0.74            = X ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_power_cancel
% 0.35/0.74  thf(fact_8860_real__root__increasing,axiom,
% 0.35/0.74      ! [N: nat,N6: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_eq_nat @ N @ N6 )
% 0.35/0.74         => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.74           => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.35/0.74             => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % real_root_increasing
% 0.35/0.74  thf(fact_8861_sgn__power__root,axiom,
% 0.35/0.74      ! [N: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
% 0.35/0.74          = X ) ) ).
% 0.35/0.74  
% 0.35/0.74  % sgn_power_root
% 0.35/0.74  thf(fact_8862_root__sgn__power,axiom,
% 0.35/0.74      ! [N: nat,Y: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) ) )
% 0.35/0.74          = Y ) ) ).
% 0.35/0.74  
% 0.35/0.74  % root_sgn_power
% 0.35/0.74  thf(fact_8863_ln__root,axiom,
% 0.35/0.74      ! [N: nat,B: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.35/0.74         => ( ( ln_ln_real @ ( root @ N @ B ) )
% 0.35/0.74            = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % ln_root
% 0.35/0.74  thf(fact_8864_log__root,axiom,
% 0.35/0.74      ! [N: nat,A: real,B: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.35/0.74         => ( ( log @ B @ ( root @ N @ A ) )
% 0.35/0.74            = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % log_root
% 0.35/0.74  thf(fact_8865_log__base__root,axiom,
% 0.35/0.74      ! [N: nat,B: real,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.35/0.74         => ( ( log @ ( root @ N @ B ) @ X )
% 0.35/0.74            = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % log_base_root
% 0.35/0.74  thf(fact_8866_split__root,axiom,
% 0.35/0.74      ! [P: real > $o,N: nat,X: real] :
% 0.35/0.74        ( ( P @ ( root @ N @ X ) )
% 0.35/0.74        = ( ( ( N = zero_zero_nat )
% 0.35/0.74           => ( P @ zero_zero_real ) )
% 0.35/0.74          & ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74           => ! [Y2: real] :
% 0.35/0.74                ( ( ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N ) )
% 0.35/0.74                  = X )
% 0.35/0.74               => ( P @ Y2 ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % split_root
% 0.35/0.74  thf(fact_8867_Suc__funpow,axiom,
% 0.35/0.74      ! [N: nat] :
% 0.35/0.74        ( ( compow_nat_nat @ N @ suc )
% 0.35/0.74        = ( plus_plus_nat @ N ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Suc_funpow
% 0.35/0.74  thf(fact_8868_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 0.35/0.74      ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 0.35/0.74      @ ^ [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X3 )
% 0.35/0.74      @ ^ [X3: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X3 ) ) ).
% 0.35/0.74  
% 0.35/0.74  % max_nat.semilattice_neutr_order_axioms
% 0.35/0.74  thf(fact_8869_push__bit__of__Suc__0,axiom,
% 0.35/0.74      ! [N: nat] :
% 0.35/0.74        ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.35/0.74        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.74  
% 0.35/0.74  % push_bit_of_Suc_0
% 0.35/0.74  thf(fact_8870_bit__push__bit__iff__nat,axiom,
% 0.35/0.74      ! [M: nat,Q4: nat,N: nat] :
% 0.35/0.74        ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q4 ) @ N )
% 0.35/0.74        = ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.74          & ( bit_se1148574629649215175it_nat @ Q4 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % bit_push_bit_iff_nat
% 0.35/0.74  thf(fact_8871_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
% 0.35/0.74      ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
% 0.35/0.74      @ ^ [M2: nat,N2: nat] :
% 0.35/0.74          ( ( dvd_dvd_nat @ M2 @ N2 )
% 0.35/0.74          & ( M2 != N2 ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % gcd_nat.semilattice_neutr_order_axioms
% 0.35/0.74  thf(fact_8872_min__rpair__set,axiom,
% 0.35/0.74      fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).
% 0.35/0.74  
% 0.35/0.74  % min_rpair_set
% 0.35/0.74  thf(fact_8873_max__rpair__set,axiom,
% 0.35/0.74      fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).
% 0.35/0.74  
% 0.35/0.74  % max_rpair_set
% 0.35/0.74  thf(fact_8874_push__bit__negative__int__iff,axiom,
% 0.35/0.74      ! [N: nat,K: int] :
% 0.35/0.74        ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
% 0.35/0.74        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.35/0.74  
% 0.35/0.74  % push_bit_negative_int_iff
% 0.35/0.74  thf(fact_8875_bit__push__bit__iff__int,axiom,
% 0.35/0.74      ! [M: nat,K: int,N: nat] :
% 0.35/0.74        ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
% 0.35/0.74        = ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.74          & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % bit_push_bit_iff_int
% 0.35/0.74  thf(fact_8876_Gcd__remove0__nat,axiom,
% 0.35/0.74      ! [M4: set_nat] :
% 0.35/0.74        ( ( finite_finite_nat @ M4 )
% 0.35/0.74       => ( ( gcd_Gcd_nat @ M4 )
% 0.35/0.74          = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M4 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Gcd_remove0_nat
% 0.35/0.74  thf(fact_8877_times__int_Oabs__eq,axiom,
% 0.35/0.74      ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.35/0.74        ( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
% 0.35/0.74        = ( abs_Integ
% 0.35/0.74          @ ( produc27273713700761075at_nat
% 0.35/0.74            @ ^ [X3: nat,Y2: nat] :
% 0.35/0.74                ( produc2626176000494625587at_nat
% 0.35/0.74                @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y2 @ V3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V3 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) )
% 0.35/0.74            @ Xa2
% 0.35/0.74            @ X ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % times_int.abs_eq
% 0.35/0.74  thf(fact_8878_eq__Abs__Integ,axiom,
% 0.35/0.74      ! [Z2: int] :
% 0.35/0.74        ~ ! [X4: nat,Y3: nat] :
% 0.35/0.74            ( Z2
% 0.35/0.74           != ( abs_Integ @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % eq_Abs_Integ
% 0.35/0.74  thf(fact_8879_Gcd__in,axiom,
% 0.35/0.74      ! [A2: set_nat] :
% 0.35/0.74        ( ! [A3: nat,B3: nat] :
% 0.35/0.74            ( ( member_nat @ A3 @ A2 )
% 0.35/0.74           => ( ( member_nat @ B3 @ A2 )
% 0.35/0.74             => ( member_nat @ ( gcd_gcd_nat @ A3 @ B3 ) @ A2 ) ) )
% 0.35/0.74       => ( ( A2 != bot_bot_set_nat )
% 0.35/0.74         => ( member_nat @ ( gcd_Gcd_nat @ A2 ) @ A2 ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Gcd_in
% 0.35/0.74  thf(fact_8880_zero__int__def,axiom,
% 0.35/0.74      ( zero_zero_int
% 0.35/0.74      = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % zero_int_def
% 0.35/0.74  thf(fact_8881_int__def,axiom,
% 0.35/0.74      ( semiri1314217659103216013at_int
% 0.35/0.74      = ( ^ [N2: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N2 @ zero_zero_nat ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % int_def
% 0.35/0.74  thf(fact_8882_uminus__int_Oabs__eq,axiom,
% 0.35/0.74      ! [X: product_prod_nat_nat] :
% 0.35/0.74        ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
% 0.35/0.74        = ( abs_Integ
% 0.35/0.74          @ ( produc2626176000494625587at_nat
% 0.35/0.74            @ ^ [X3: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X3 )
% 0.35/0.74            @ X ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % uminus_int.abs_eq
% 0.35/0.74  thf(fact_8883_one__int__def,axiom,
% 0.35/0.74      ( one_one_int
% 0.35/0.74      = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % one_int_def
% 0.35/0.74  thf(fact_8884_less__int_Oabs__eq,axiom,
% 0.35/0.74      ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.35/0.74        ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
% 0.35/0.74        = ( produc8739625826339149834_nat_o
% 0.35/0.74          @ ^ [X3: nat,Y2: nat] :
% 0.35/0.74              ( produc6081775807080527818_nat_o
% 0.35/0.74              @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) )
% 0.35/0.74          @ Xa2
% 0.35/0.74          @ X ) ) ).
% 0.35/0.74  
% 0.35/0.74  % less_int.abs_eq
% 0.35/0.74  thf(fact_8885_less__eq__int_Oabs__eq,axiom,
% 0.35/0.74      ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.35/0.74        ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
% 0.35/0.74        = ( produc8739625826339149834_nat_o
% 0.35/0.74          @ ^ [X3: nat,Y2: nat] :
% 0.35/0.74              ( produc6081775807080527818_nat_o
% 0.35/0.74              @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) )
% 0.35/0.74          @ Xa2
% 0.35/0.74          @ X ) ) ).
% 0.35/0.74  
% 0.35/0.74  % less_eq_int.abs_eq
% 0.35/0.74  thf(fact_8886_plus__int_Oabs__eq,axiom,
% 0.35/0.74      ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.35/0.74        ( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
% 0.35/0.74        = ( abs_Integ
% 0.35/0.74          @ ( produc27273713700761075at_nat
% 0.35/0.74            @ ^ [X3: nat,Y2: nat] :
% 0.35/0.74                ( produc2626176000494625587at_nat
% 0.35/0.74                @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y2 @ V3 ) ) )
% 0.35/0.74            @ Xa2
% 0.35/0.74            @ X ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % plus_int.abs_eq
% 0.35/0.74  thf(fact_8887_minus__int_Oabs__eq,axiom,
% 0.35/0.74      ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.35/0.74        ( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
% 0.35/0.74        = ( abs_Integ
% 0.35/0.74          @ ( produc27273713700761075at_nat
% 0.35/0.74            @ ^ [X3: nat,Y2: nat] :
% 0.35/0.74                ( produc2626176000494625587at_nat
% 0.35/0.74                @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) )
% 0.35/0.74            @ Xa2
% 0.35/0.74            @ X ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % minus_int.abs_eq
% 0.35/0.74  thf(fact_8888_less__eq__int_Orep__eq,axiom,
% 0.35/0.74      ( ord_less_eq_int
% 0.35/0.74      = ( ^ [X3: int,Xa4: int] :
% 0.35/0.74            ( produc8739625826339149834_nat_o
% 0.35/0.74            @ ^ [Y2: nat,Z3: nat] :
% 0.35/0.74                ( produc6081775807080527818_nat_o
% 0.35/0.74                @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y2 @ V3 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
% 0.35/0.74            @ ( rep_Integ @ X3 )
% 0.35/0.74            @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % less_eq_int.rep_eq
% 0.35/0.74  thf(fact_8889_less__int_Orep__eq,axiom,
% 0.35/0.74      ( ord_less_int
% 0.35/0.74      = ( ^ [X3: int,Xa4: int] :
% 0.35/0.74            ( produc8739625826339149834_nat_o
% 0.35/0.74            @ ^ [Y2: nat,Z3: nat] :
% 0.35/0.74                ( produc6081775807080527818_nat_o
% 0.35/0.74                @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y2 @ V3 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
% 0.35/0.74            @ ( rep_Integ @ X3 )
% 0.35/0.74            @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % less_int.rep_eq
% 0.35/0.74  thf(fact_8890_rat__floor__lemma,axiom,
% 0.35/0.74      ! [A: int,B: int] :
% 0.35/0.74        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
% 0.35/0.74        & ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % rat_floor_lemma
% 0.35/0.74  thf(fact_8891_image__minus__const__atLeastLessThan__nat,axiom,
% 0.35/0.74      ! [C: nat,Y: nat,X: nat] :
% 0.35/0.74        ( ( ( ord_less_nat @ C @ Y )
% 0.35/0.74         => ( ( image_nat_nat
% 0.35/0.74              @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.35/0.74              @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 0.35/0.74            = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
% 0.35/0.74        & ( ~ ( ord_less_nat @ C @ Y )
% 0.35/0.74         => ( ( ( ord_less_nat @ X @ Y )
% 0.35/0.74             => ( ( image_nat_nat
% 0.35/0.74                  @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.35/0.74                  @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 0.35/0.74                = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 0.35/0.74            & ( ~ ( ord_less_nat @ X @ Y )
% 0.35/0.74             => ( ( image_nat_nat
% 0.35/0.74                  @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.35/0.74                  @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 0.35/0.74                = bot_bot_set_nat ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % image_minus_const_atLeastLessThan_nat
% 0.35/0.74  thf(fact_8892_less__rat,axiom,
% 0.35/0.74      ! [B: int,D: int,A: int,C: int] :
% 0.35/0.74        ( ( B != zero_zero_int )
% 0.35/0.74       => ( ( D != zero_zero_int )
% 0.35/0.74         => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.35/0.74            = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % less_rat
% 0.35/0.74  thf(fact_8893_zero__notin__Suc__image,axiom,
% 0.35/0.74      ! [A2: set_nat] :
% 0.35/0.74        ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% 0.35/0.74  
% 0.35/0.74  % zero_notin_Suc_image
% 0.35/0.74  thf(fact_8894_Rat__induct__pos,axiom,
% 0.35/0.74      ! [P: rat > $o,Q4: rat] :
% 0.35/0.74        ( ! [A3: int,B3: int] :
% 0.35/0.74            ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.74           => ( P @ ( fract @ A3 @ B3 ) ) )
% 0.35/0.74       => ( P @ Q4 ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Rat_induct_pos
% 0.35/0.74  thf(fact_8895_quotient__of__eq,axiom,
% 0.35/0.74      ! [A: int,B: int,P4: int,Q4: int] :
% 0.35/0.74        ( ( ( quotient_of @ ( fract @ A @ B ) )
% 0.35/0.74          = ( product_Pair_int_int @ P4 @ Q4 ) )
% 0.35/0.74       => ( ( fract @ P4 @ Q4 )
% 0.35/0.74          = ( fract @ A @ B ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % quotient_of_eq
% 0.35/0.74  thf(fact_8896_normalize__eq,axiom,
% 0.35/0.74      ! [A: int,B: int,P4: int,Q4: int] :
% 0.35/0.74        ( ( ( normalize @ ( product_Pair_int_int @ A @ B ) )
% 0.35/0.74          = ( product_Pair_int_int @ P4 @ Q4 ) )
% 0.35/0.74       => ( ( fract @ P4 @ Q4 )
% 0.35/0.74          = ( fract @ A @ B ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % normalize_eq
% 0.35/0.74  thf(fact_8897_quotient__of__Fract,axiom,
% 0.35/0.74      ! [A: int,B: int] :
% 0.35/0.74        ( ( quotient_of @ ( fract @ A @ B ) )
% 0.35/0.74        = ( normalize @ ( product_Pair_int_int @ A @ B ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % quotient_of_Fract
% 0.35/0.74  thf(fact_8898_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 0.35/0.74      ! [N: nat] :
% 0.35/0.74        ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.35/0.74        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % atLeast0_atMost_Suc_eq_insert_0
% 0.35/0.74  thf(fact_8899_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 0.35/0.74      ! [N: nat] :
% 0.35/0.74        ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.35/0.74        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % atLeast0_lessThan_Suc_eq_insert_0
% 0.35/0.74  thf(fact_8900_lessThan__Suc__eq__insert__0,axiom,
% 0.35/0.74      ! [N: nat] :
% 0.35/0.74        ( ( set_ord_lessThan_nat @ ( suc @ N ) )
% 0.35/0.74        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % lessThan_Suc_eq_insert_0
% 0.35/0.74  thf(fact_8901_atMost__Suc__eq__insert__0,axiom,
% 0.35/0.74      ! [N: nat] :
% 0.35/0.74        ( ( set_ord_atMost_nat @ ( suc @ N ) )
% 0.35/0.74        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % atMost_Suc_eq_insert_0
% 0.35/0.74  thf(fact_8902_Fract__less__zero__iff,axiom,
% 0.35/0.74      ! [B: int,A: int] :
% 0.35/0.74        ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.74       => ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 0.35/0.74          = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Fract_less_zero_iff
% 0.35/0.74  thf(fact_8903_zero__less__Fract__iff,axiom,
% 0.35/0.74      ! [B: int,A: int] :
% 0.35/0.74        ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.74       => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 0.35/0.74          = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % zero_less_Fract_iff
% 0.35/0.74  thf(fact_8904_Fract__less__one__iff,axiom,
% 0.35/0.74      ! [B: int,A: int] :
% 0.35/0.74        ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.74       => ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
% 0.35/0.74          = ( ord_less_int @ A @ B ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Fract_less_one_iff
% 0.35/0.74  thf(fact_8905_one__less__Fract__iff,axiom,
% 0.35/0.74      ! [B: int,A: int] :
% 0.35/0.74        ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.74       => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
% 0.35/0.74          = ( ord_less_int @ B @ A ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % one_less_Fract_iff
% 0.35/0.74  thf(fact_8906_zero__le__Fract__iff,axiom,
% 0.35/0.74      ! [B: int,A: int] :
% 0.35/0.74        ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.74       => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 0.35/0.74          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % zero_le_Fract_iff
% 0.35/0.74  thf(fact_8907_Fract__le__zero__iff,axiom,
% 0.35/0.74      ! [B: int,A: int] :
% 0.35/0.74        ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.74       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 0.35/0.74          = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Fract_le_zero_iff
% 0.35/0.74  thf(fact_8908_one__le__Fract__iff,axiom,
% 0.35/0.74      ! [B: int,A: int] :
% 0.35/0.74        ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.74       => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
% 0.35/0.74          = ( ord_less_eq_int @ B @ A ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % one_le_Fract_iff
% 0.35/0.74  thf(fact_8909_Fract__le__one__iff,axiom,
% 0.35/0.74      ! [B: int,A: int] :
% 0.35/0.74        ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.74       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
% 0.35/0.74          = ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Fract_le_one_iff
% 0.35/0.74  thf(fact_8910_uminus__int__def,axiom,
% 0.35/0.74      ( uminus_uminus_int
% 0.35/0.74      = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
% 0.35/0.74        @ ( produc2626176000494625587at_nat
% 0.35/0.74          @ ^ [X3: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X3 ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % uminus_int_def
% 0.35/0.74  thf(fact_8911_finite__int__iff__bounded,axiom,
% 0.35/0.74      ( finite_finite_int
% 0.35/0.74      = ( ^ [S7: set_int] :
% 0.35/0.74          ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S7 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % finite_int_iff_bounded
% 0.35/0.74  thf(fact_8912_finite__int__iff__bounded__le,axiom,
% 0.35/0.74      ( finite_finite_int
% 0.35/0.74      = ( ^ [S7: set_int] :
% 0.35/0.74          ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S7 ) @ ( set_ord_atMost_int @ K3 ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % finite_int_iff_bounded_le
% 0.35/0.74  thf(fact_8913_infinite__int__iff__infinite__nat__abs,axiom,
% 0.35/0.74      ! [S2: set_int] :
% 0.35/0.74        ( ( ~ ( finite_finite_int @ S2 ) )
% 0.35/0.74        = ( ~ ( finite_finite_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ S2 ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % infinite_int_iff_infinite_nat_abs
% 0.35/0.74  thf(fact_8914_times__int__def,axiom,
% 0.35/0.74      ( times_times_int
% 0.35/0.74      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.35/0.74        @ ( produc27273713700761075at_nat
% 0.35/0.74          @ ^ [X3: nat,Y2: nat] :
% 0.35/0.74              ( produc2626176000494625587at_nat
% 0.35/0.74              @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y2 @ V3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V3 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % times_int_def
% 0.35/0.74  thf(fact_8915_minus__int__def,axiom,
% 0.35/0.74      ( minus_minus_int
% 0.35/0.74      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.35/0.74        @ ( produc27273713700761075at_nat
% 0.35/0.74          @ ^ [X3: nat,Y2: nat] :
% 0.35/0.74              ( produc2626176000494625587at_nat
% 0.35/0.74              @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % minus_int_def
% 0.35/0.74  thf(fact_8916_plus__int__def,axiom,
% 0.35/0.74      ( plus_plus_int
% 0.35/0.74      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.35/0.74        @ ( produc27273713700761075at_nat
% 0.35/0.74          @ ^ [X3: nat,Y2: nat] :
% 0.35/0.74              ( produc2626176000494625587at_nat
% 0.35/0.74              @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y2 @ V3 ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % plus_int_def
% 0.35/0.74  thf(fact_8917_infinite__UNIV__nat,axiom,
% 0.35/0.74      ~ ( finite_finite_nat @ top_top_set_nat ) ).
% 0.35/0.74  
% 0.35/0.74  % infinite_UNIV_nat
% 0.35/0.74  thf(fact_8918_nat__not__finite,axiom,
% 0.35/0.74      ~ ( finite_finite_nat @ top_top_set_nat ) ).
% 0.35/0.74  
% 0.35/0.74  % nat_not_finite
% 0.35/0.74  thf(fact_8919_UNIV__nat__eq,axiom,
% 0.35/0.74      ( top_top_set_nat
% 0.35/0.74      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % UNIV_nat_eq
% 0.35/0.74  thf(fact_8920_range__mod,axiom,
% 0.35/0.74      ! [N: nat] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( image_nat_nat
% 0.35/0.74            @ ^ [M2: nat] : ( modulo_modulo_nat @ M2 @ N )
% 0.35/0.74            @ top_top_set_nat )
% 0.35/0.74          = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % range_mod
% 0.35/0.74  thf(fact_8921_card__UNIV__unit,axiom,
% 0.35/0.74      ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
% 0.35/0.74      = one_one_nat ) ).
% 0.35/0.74  
% 0.35/0.74  % card_UNIV_unit
% 0.35/0.74  thf(fact_8922_range__mult,axiom,
% 0.35/0.74      ! [A: real] :
% 0.35/0.74        ( ( ( A = zero_zero_real )
% 0.35/0.74         => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 0.35/0.74            = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 0.35/0.74        & ( ( A != zero_zero_real )
% 0.35/0.74         => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 0.35/0.74            = top_top_set_real ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % range_mult
% 0.35/0.74  thf(fact_8923_infinite__UNIV__int,axiom,
% 0.35/0.74      ~ ( finite_finite_int @ top_top_set_int ) ).
% 0.35/0.74  
% 0.35/0.74  % infinite_UNIV_int
% 0.35/0.74  thf(fact_8924_root__def,axiom,
% 0.35/0.74      ( root
% 0.35/0.74      = ( ^ [N2: nat,X3: real] :
% 0.35/0.74            ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
% 0.35/0.74            @ ( the_in5290026491893676941l_real @ top_top_set_real
% 0.35/0.74              @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N2 ) )
% 0.35/0.74              @ X3 ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % root_def
% 0.35/0.74  thf(fact_8925_DERIV__even__real__root,axiom,
% 0.35/0.74      ! [N: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.74         => ( ( ord_less_real @ X @ zero_zero_real )
% 0.35/0.74           => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_even_real_root
% 0.35/0.74  thf(fact_8926_has__real__derivative__neg__dec__left,axiom,
% 0.35/0.74      ! [F: real > real,L: real,X: real,S2: set_real] :
% 0.35/0.74        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 0.35/0.74       => ( ( ord_less_real @ L @ zero_zero_real )
% 0.35/0.74         => ? [D5: real] :
% 0.35/0.74              ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.35/0.74              & ! [H4: real] :
% 0.35/0.74                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.35/0.74                 => ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S2 )
% 0.35/0.74                   => ( ( ord_less_real @ H4 @ D5 )
% 0.35/0.74                     => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % has_real_derivative_neg_dec_left
% 0.35/0.74  thf(fact_8927_has__real__derivative__pos__inc__left,axiom,
% 0.35/0.74      ! [F: real > real,L: real,X: real,S2: set_real] :
% 0.35/0.74        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 0.35/0.74       => ( ( ord_less_real @ zero_zero_real @ L )
% 0.35/0.74         => ? [D5: real] :
% 0.35/0.74              ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.35/0.74              & ! [H4: real] :
% 0.35/0.74                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.35/0.74                 => ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S2 )
% 0.35/0.74                   => ( ( ord_less_real @ H4 @ D5 )
% 0.35/0.74                     => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % has_real_derivative_pos_inc_left
% 0.35/0.74  thf(fact_8928_has__real__derivative__neg__dec__right,axiom,
% 0.35/0.74      ! [F: real > real,L: real,X: real,S2: set_real] :
% 0.35/0.74        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 0.35/0.74       => ( ( ord_less_real @ L @ zero_zero_real )
% 0.35/0.74         => ? [D5: real] :
% 0.35/0.74              ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.35/0.74              & ! [H4: real] :
% 0.35/0.74                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.35/0.74                 => ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S2 )
% 0.35/0.74                   => ( ( ord_less_real @ H4 @ D5 )
% 0.35/0.74                     => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % has_real_derivative_neg_dec_right
% 0.35/0.74  thf(fact_8929_has__real__derivative__pos__inc__right,axiom,
% 0.35/0.74      ! [F: real > real,L: real,X: real,S2: set_real] :
% 0.35/0.74        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 0.35/0.74       => ( ( ord_less_real @ zero_zero_real @ L )
% 0.35/0.74         => ? [D5: real] :
% 0.35/0.74              ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.35/0.74              & ! [H4: real] :
% 0.35/0.74                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.35/0.74                 => ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S2 )
% 0.35/0.74                   => ( ( ord_less_real @ H4 @ D5 )
% 0.35/0.74                     => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % has_real_derivative_pos_inc_right
% 0.35/0.74  thf(fact_8930_DERIV__neg__imp__decreasing,axiom,
% 0.35/0.74      ! [A: real,B: real,F: real > real] :
% 0.35/0.74        ( ( ord_less_real @ A @ B )
% 0.35/0.74       => ( ! [X4: real] :
% 0.35/0.74              ( ( ord_less_eq_real @ A @ X4 )
% 0.35/0.74             => ( ( ord_less_eq_real @ X4 @ B )
% 0.35/0.74               => ? [Y4: real] :
% 0.35/0.74                    ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.35/0.74                    & ( ord_less_real @ Y4 @ zero_zero_real ) ) ) )
% 0.35/0.74         => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_neg_imp_decreasing
% 0.35/0.74  thf(fact_8931_DERIV__pos__imp__increasing,axiom,
% 0.35/0.74      ! [A: real,B: real,F: real > real] :
% 0.35/0.74        ( ( ord_less_real @ A @ B )
% 0.35/0.74       => ( ! [X4: real] :
% 0.35/0.74              ( ( ord_less_eq_real @ A @ X4 )
% 0.35/0.74             => ( ( ord_less_eq_real @ X4 @ B )
% 0.35/0.74               => ? [Y4: real] :
% 0.35/0.74                    ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.35/0.74                    & ( ord_less_real @ zero_zero_real @ Y4 ) ) ) )
% 0.35/0.74         => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_pos_imp_increasing
% 0.35/0.74  thf(fact_8932_DERIV__pos__inc__right,axiom,
% 0.35/0.74      ! [F: real > real,L: real,X: real] :
% 0.35/0.74        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.35/0.74       => ( ( ord_less_real @ zero_zero_real @ L )
% 0.35/0.74         => ? [D5: real] :
% 0.35/0.74              ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.35/0.74              & ! [H4: real] :
% 0.35/0.74                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.35/0.74                 => ( ( ord_less_real @ H4 @ D5 )
% 0.35/0.74                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_pos_inc_right
% 0.35/0.74  thf(fact_8933_DERIV__neg__dec__right,axiom,
% 0.35/0.74      ! [F: real > real,L: real,X: real] :
% 0.35/0.74        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.35/0.74       => ( ( ord_less_real @ L @ zero_zero_real )
% 0.35/0.74         => ? [D5: real] :
% 0.35/0.74              ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.35/0.74              & ! [H4: real] :
% 0.35/0.74                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.35/0.74                 => ( ( ord_less_real @ H4 @ D5 )
% 0.35/0.74                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_neg_dec_right
% 0.35/0.74  thf(fact_8934_DERIV__pos__inc__left,axiom,
% 0.35/0.74      ! [F: real > real,L: real,X: real] :
% 0.35/0.74        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.35/0.74       => ( ( ord_less_real @ zero_zero_real @ L )
% 0.35/0.74         => ? [D5: real] :
% 0.35/0.74              ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.35/0.74              & ! [H4: real] :
% 0.35/0.74                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.35/0.74                 => ( ( ord_less_real @ H4 @ D5 )
% 0.35/0.74                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_pos_inc_left
% 0.35/0.74  thf(fact_8935_DERIV__neg__dec__left,axiom,
% 0.35/0.74      ! [F: real > real,L: real,X: real] :
% 0.35/0.74        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.35/0.74       => ( ( ord_less_real @ L @ zero_zero_real )
% 0.35/0.74         => ? [D5: real] :
% 0.35/0.74              ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.35/0.74              & ! [H4: real] :
% 0.35/0.74                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.35/0.74                 => ( ( ord_less_real @ H4 @ D5 )
% 0.35/0.74                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_neg_dec_left
% 0.35/0.74  thf(fact_8936_MVT2,axiom,
% 0.35/0.74      ! [A: real,B: real,F: real > real,F6: real > real] :
% 0.35/0.74        ( ( ord_less_real @ A @ B )
% 0.35/0.74       => ( ! [X4: real] :
% 0.35/0.74              ( ( ord_less_eq_real @ A @ X4 )
% 0.35/0.74             => ( ( ord_less_eq_real @ X4 @ B )
% 0.35/0.74               => ( has_fi5821293074295781190e_real @ F @ ( F6 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.35/0.74         => ? [Z4: real] :
% 0.35/0.74              ( ( ord_less_real @ A @ Z4 )
% 0.35/0.74              & ( ord_less_real @ Z4 @ B )
% 0.35/0.74              & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.35/0.74                = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F6 @ Z4 ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % MVT2
% 0.35/0.74  thf(fact_8937_DERIV__local__const,axiom,
% 0.35/0.74      ! [F: real > real,L: real,X: real,D: real] :
% 0.35/0.74        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.35/0.74       => ( ( ord_less_real @ zero_zero_real @ D )
% 0.35/0.74         => ( ! [Y3: real] :
% 0.35/0.74                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
% 0.35/0.74               => ( ( F @ X )
% 0.35/0.74                  = ( F @ Y3 ) ) )
% 0.35/0.74           => ( L = zero_zero_real ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_local_const
% 0.35/0.74  thf(fact_8938_DERIV__ln,axiom,
% 0.35/0.74      ! [X: real] :
% 0.35/0.74        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.74       => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_ln
% 0.35/0.74  thf(fact_8939_DERIV__local__max,axiom,
% 0.35/0.74      ! [F: real > real,L: real,X: real,D: real] :
% 0.35/0.74        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.35/0.74       => ( ( ord_less_real @ zero_zero_real @ D )
% 0.35/0.74         => ( ! [Y3: real] :
% 0.35/0.74                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
% 0.35/0.74               => ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X ) ) )
% 0.35/0.74           => ( L = zero_zero_real ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_local_max
% 0.35/0.74  thf(fact_8940_DERIV__local__min,axiom,
% 0.35/0.74      ! [F: real > real,L: real,X: real,D: real] :
% 0.35/0.74        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.35/0.74       => ( ( ord_less_real @ zero_zero_real @ D )
% 0.35/0.74         => ( ! [Y3: real] :
% 0.35/0.74                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
% 0.35/0.74               => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.35/0.74           => ( L = zero_zero_real ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_local_min
% 0.35/0.74  thf(fact_8941_DERIV__ln__divide,axiom,
% 0.35/0.74      ! [X: real] :
% 0.35/0.74        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.74       => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_ln_divide
% 0.35/0.74  thf(fact_8942_DERIV__pow,axiom,
% 0.35/0.74      ! [N: nat,X: real,S: set_real] :
% 0.35/0.74        ( has_fi5821293074295781190e_real
% 0.35/0.74        @ ^ [X3: real] : ( power_power_real @ X3 @ N )
% 0.35/0.74        @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 0.35/0.74        @ ( topolo2177554685111907308n_real @ X @ S ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_pow
% 0.35/0.74  thf(fact_8943_has__real__derivative__powr,axiom,
% 0.35/0.74      ! [Z2: real,R2: real] :
% 0.35/0.74        ( ( ord_less_real @ zero_zero_real @ Z2 )
% 0.35/0.74       => ( has_fi5821293074295781190e_real
% 0.35/0.74          @ ^ [Z3: real] : ( powr_real @ Z3 @ R2 )
% 0.35/0.74          @ ( times_times_real @ R2 @ ( powr_real @ Z2 @ ( minus_minus_real @ R2 @ one_one_real ) ) )
% 0.35/0.74          @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % has_real_derivative_powr
% 0.35/0.74  thf(fact_8944_DERIV__log,axiom,
% 0.35/0.74      ! [X: real,B: real] :
% 0.35/0.74        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.74       => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_log
% 0.35/0.74  thf(fact_8945_DERIV__fun__powr,axiom,
% 0.35/0.74      ! [G: real > real,M: real,X: real,R2: real] :
% 0.35/0.74        ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.35/0.74       => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 0.35/0.74         => ( has_fi5821293074295781190e_real
% 0.35/0.74            @ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ R2 )
% 0.35/0.74            @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 0.35/0.74            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_fun_powr
% 0.35/0.74  thf(fact_8946_DERIV__powr,axiom,
% 0.35/0.74      ! [G: real > real,M: real,X: real,F: real > real,R2: real] :
% 0.35/0.74        ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.35/0.74       => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 0.35/0.74         => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.35/0.74           => ( has_fi5821293074295781190e_real
% 0.35/0.74              @ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ ( F @ X3 ) )
% 0.35/0.74              @ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
% 0.35/0.74              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_powr
% 0.35/0.74  thf(fact_8947_arcosh__real__has__field__derivative,axiom,
% 0.35/0.74      ! [X: real,A2: set_real] :
% 0.35/0.74        ( ( ord_less_real @ one_one_real @ X )
% 0.35/0.74       => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % arcosh_real_has_field_derivative
% 0.35/0.74  thf(fact_8948_DERIV__real__sqrt,axiom,
% 0.35/0.74      ! [X: real] :
% 0.35/0.74        ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.74       => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_real_sqrt
% 0.35/0.74  thf(fact_8949_DERIV__real__sqrt__generic,axiom,
% 0.35/0.74      ! [X: real,D6: real] :
% 0.35/0.74        ( ( X != zero_zero_real )
% 0.35/0.74       => ( ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.74           => ( D6
% 0.35/0.74              = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.35/0.74         => ( ( ( ord_less_real @ X @ zero_zero_real )
% 0.35/0.74             => ( D6
% 0.35/0.74                = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.35/0.74           => ( has_fi5821293074295781190e_real @ sqrt @ D6 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_real_sqrt_generic
% 0.35/0.74  thf(fact_8950_artanh__real__has__field__derivative,axiom,
% 0.35/0.74      ! [X: real,A2: set_real] :
% 0.35/0.74        ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.35/0.74       => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % artanh_real_has_field_derivative
% 0.35/0.74  thf(fact_8951_DERIV__real__root,axiom,
% 0.35/0.74      ! [N: nat,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.74         => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_real_root
% 0.35/0.74  thf(fact_8952_DERIV__arcsin,axiom,
% 0.35/0.74      ! [X: real] :
% 0.35/0.74        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.35/0.74       => ( ( ord_less_real @ X @ one_one_real )
% 0.35/0.74         => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_arcsin
% 0.35/0.74  thf(fact_8953_Maclaurin__all__le__objl,axiom,
% 0.35/0.74      ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
% 0.35/0.74        ( ( ( ( Diff @ zero_zero_nat )
% 0.35/0.74            = F )
% 0.35/0.74          & ! [M5: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.35/0.74       => ? [T6: real] :
% 0.35/0.74            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
% 0.35/0.74            & ( ( F @ X )
% 0.35/0.74              = ( plus_plus_real
% 0.35/0.74                @ ( groups6591440286371151544t_real
% 0.35/0.74                  @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
% 0.35/0.74                  @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.74                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Maclaurin_all_le_objl
% 0.35/0.74  thf(fact_8954_Maclaurin__all__le,axiom,
% 0.35/0.74      ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
% 0.35/0.74        ( ( ( Diff @ zero_zero_nat )
% 0.35/0.74          = F )
% 0.35/0.74       => ( ! [M5: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.35/0.74         => ? [T6: real] :
% 0.35/0.74              ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
% 0.35/0.74              & ( ( F @ X )
% 0.35/0.74                = ( plus_plus_real
% 0.35/0.74                  @ ( groups6591440286371151544t_real
% 0.35/0.74                    @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
% 0.35/0.74                    @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.74                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Maclaurin_all_le
% 0.35/0.74  thf(fact_8955_DERIV__odd__real__root,axiom,
% 0.35/0.74      ! [N: nat,X: real] :
% 0.35/0.74        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.74       => ( ( X != zero_zero_real )
% 0.35/0.74         => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_odd_real_root
% 0.35/0.74  thf(fact_8956_Maclaurin__minus,axiom,
% 0.35/0.74      ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 0.35/0.74        ( ( ord_less_real @ H2 @ zero_zero_real )
% 0.35/0.74       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74         => ( ( ( Diff @ zero_zero_nat )
% 0.35/0.74              = F )
% 0.35/0.74           => ( ! [M5: nat,T6: real] :
% 0.35/0.74                  ( ( ( ord_less_nat @ M5 @ N )
% 0.35/0.74                    & ( ord_less_eq_real @ H2 @ T6 )
% 0.35/0.74                    & ( ord_less_eq_real @ T6 @ zero_zero_real ) )
% 0.35/0.74                 => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.35/0.74             => ? [T6: real] :
% 0.35/0.74                  ( ( ord_less_real @ H2 @ T6 )
% 0.35/0.74                  & ( ord_less_real @ T6 @ zero_zero_real )
% 0.35/0.74                  & ( ( F @ H2 )
% 0.35/0.74                    = ( plus_plus_real
% 0.35/0.74                      @ ( groups6591440286371151544t_real
% 0.35/0.74                        @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 0.35/0.74                        @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.74                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Maclaurin_minus
% 0.35/0.74  thf(fact_8957_Maclaurin2,axiom,
% 0.35/0.74      ! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
% 0.35/0.74        ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.35/0.74       => ( ( ( Diff @ zero_zero_nat )
% 0.35/0.74            = F )
% 0.35/0.74         => ( ! [M5: nat,T6: real] :
% 0.35/0.74                ( ( ( ord_less_nat @ M5 @ N )
% 0.35/0.74                  & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.35/0.74                  & ( ord_less_eq_real @ T6 @ H2 ) )
% 0.35/0.74               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.35/0.74           => ? [T6: real] :
% 0.35/0.74                ( ( ord_less_real @ zero_zero_real @ T6 )
% 0.35/0.74                & ( ord_less_eq_real @ T6 @ H2 )
% 0.35/0.74                & ( ( F @ H2 )
% 0.35/0.74                  = ( plus_plus_real
% 0.35/0.74                    @ ( groups6591440286371151544t_real
% 0.35/0.74                      @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 0.35/0.74                      @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.74                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Maclaurin2
% 0.35/0.74  thf(fact_8958_Maclaurin,axiom,
% 0.35/0.74      ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 0.35/0.74        ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.35/0.74       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74         => ( ( ( Diff @ zero_zero_nat )
% 0.35/0.74              = F )
% 0.35/0.74           => ( ! [M5: nat,T6: real] :
% 0.35/0.74                  ( ( ( ord_less_nat @ M5 @ N )
% 0.35/0.74                    & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.35/0.74                    & ( ord_less_eq_real @ T6 @ H2 ) )
% 0.35/0.74                 => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.35/0.74             => ? [T6: real] :
% 0.35/0.74                  ( ( ord_less_real @ zero_zero_real @ T6 )
% 0.35/0.74                  & ( ord_less_real @ T6 @ H2 )
% 0.35/0.74                  & ( ( F @ H2 )
% 0.35/0.74                    = ( plus_plus_real
% 0.35/0.74                      @ ( groups6591440286371151544t_real
% 0.35/0.74                        @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 0.35/0.74                        @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.74                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Maclaurin
% 0.35/0.74  thf(fact_8959_Maclaurin__all__lt,axiom,
% 0.35/0.74      ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
% 0.35/0.74        ( ( ( Diff @ zero_zero_nat )
% 0.35/0.74          = F )
% 0.35/0.74       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74         => ( ( X != zero_zero_real )
% 0.35/0.74           => ( ! [M5: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.35/0.74             => ? [T6: real] :
% 0.35/0.74                  ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
% 0.35/0.74                  & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
% 0.35/0.74                  & ( ( F @ X )
% 0.35/0.74                    = ( plus_plus_real
% 0.35/0.74                      @ ( groups6591440286371151544t_real
% 0.35/0.74                        @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
% 0.35/0.74                        @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.74                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Maclaurin_all_lt
% 0.35/0.74  thf(fact_8960_Maclaurin__bi__le,axiom,
% 0.35/0.74      ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
% 0.35/0.74        ( ( ( Diff @ zero_zero_nat )
% 0.35/0.74          = F )
% 0.35/0.74       => ( ! [M5: nat,T6: real] :
% 0.35/0.74              ( ( ( ord_less_nat @ M5 @ N )
% 0.35/0.74                & ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) ) )
% 0.35/0.74             => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.35/0.74         => ? [T6: real] :
% 0.35/0.74              ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
% 0.35/0.74              & ( ( F @ X )
% 0.35/0.74                = ( plus_plus_real
% 0.35/0.74                  @ ( groups6591440286371151544t_real
% 0.35/0.74                    @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
% 0.35/0.74                    @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.74                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Maclaurin_bi_le
% 0.35/0.74  thf(fact_8961_Taylor__down,axiom,
% 0.35/0.74      ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ( Diff @ zero_zero_nat )
% 0.35/0.74            = F )
% 0.35/0.74         => ( ! [M5: nat,T6: real] :
% 0.35/0.74                ( ( ( ord_less_nat @ M5 @ N )
% 0.35/0.74                  & ( ord_less_eq_real @ A @ T6 )
% 0.35/0.74                  & ( ord_less_eq_real @ T6 @ B ) )
% 0.35/0.74               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.35/0.74           => ( ( ord_less_real @ A @ C )
% 0.35/0.74             => ( ( ord_less_eq_real @ C @ B )
% 0.35/0.74               => ? [T6: real] :
% 0.35/0.74                    ( ( ord_less_real @ A @ T6 )
% 0.35/0.74                    & ( ord_less_real @ T6 @ C )
% 0.35/0.74                    & ( ( F @ A )
% 0.35/0.74                      = ( plus_plus_real
% 0.35/0.74                        @ ( groups6591440286371151544t_real
% 0.35/0.74                          @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M2 ) )
% 0.35/0.74                          @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.74                        @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Taylor_down
% 0.35/0.74  thf(fact_8962_Taylor__up,axiom,
% 0.35/0.74      ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ( Diff @ zero_zero_nat )
% 0.35/0.74            = F )
% 0.35/0.74         => ( ! [M5: nat,T6: real] :
% 0.35/0.74                ( ( ( ord_less_nat @ M5 @ N )
% 0.35/0.74                  & ( ord_less_eq_real @ A @ T6 )
% 0.35/0.74                  & ( ord_less_eq_real @ T6 @ B ) )
% 0.35/0.74               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.35/0.74           => ( ( ord_less_eq_real @ A @ C )
% 0.35/0.74             => ( ( ord_less_real @ C @ B )
% 0.35/0.74               => ? [T6: real] :
% 0.35/0.74                    ( ( ord_less_real @ C @ T6 )
% 0.35/0.74                    & ( ord_less_real @ T6 @ B )
% 0.35/0.74                    & ( ( F @ B )
% 0.35/0.74                      = ( plus_plus_real
% 0.35/0.74                        @ ( groups6591440286371151544t_real
% 0.35/0.74                          @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M2 ) )
% 0.35/0.74                          @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.74                        @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Taylor_up
% 0.35/0.74  thf(fact_8963_Taylor,axiom,
% 0.35/0.74      ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( ( Diff @ zero_zero_nat )
% 0.35/0.74            = F )
% 0.35/0.74         => ( ! [M5: nat,T6: real] :
% 0.35/0.74                ( ( ( ord_less_nat @ M5 @ N )
% 0.35/0.74                  & ( ord_less_eq_real @ A @ T6 )
% 0.35/0.74                  & ( ord_less_eq_real @ T6 @ B ) )
% 0.35/0.74               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.35/0.74           => ( ( ord_less_eq_real @ A @ C )
% 0.35/0.74             => ( ( ord_less_eq_real @ C @ B )
% 0.35/0.74               => ( ( ord_less_eq_real @ A @ X )
% 0.35/0.74                 => ( ( ord_less_eq_real @ X @ B )
% 0.35/0.74                   => ( ( X != C )
% 0.35/0.74                     => ? [T6: real] :
% 0.35/0.74                          ( ( ( ord_less_real @ X @ C )
% 0.35/0.74                           => ( ( ord_less_real @ X @ T6 )
% 0.35/0.74                              & ( ord_less_real @ T6 @ C ) ) )
% 0.35/0.74                          & ( ~ ( ord_less_real @ X @ C )
% 0.35/0.74                           => ( ( ord_less_real @ C @ T6 )
% 0.35/0.74                              & ( ord_less_real @ T6 @ X ) ) )
% 0.35/0.74                          & ( ( F @ X )
% 0.35/0.74                            = ( plus_plus_real
% 0.35/0.74                              @ ( groups6591440286371151544t_real
% 0.35/0.74                                @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M2 ) )
% 0.35/0.74                                @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.74                              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Taylor
% 0.35/0.74  thf(fact_8964_Maclaurin__lemma2,axiom,
% 0.35/0.74      ! [N: nat,H2: real,Diff: nat > real > real,K: nat,B2: real] :
% 0.35/0.74        ( ! [M5: nat,T6: real] :
% 0.35/0.74            ( ( ( ord_less_nat @ M5 @ N )
% 0.35/0.74              & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.35/0.74              & ( ord_less_eq_real @ T6 @ H2 ) )
% 0.35/0.74           => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.35/0.74       => ( ( N
% 0.35/0.74            = ( suc @ K ) )
% 0.35/0.74         => ! [M3: nat,T7: real] :
% 0.35/0.74              ( ( ( ord_less_nat @ M3 @ N )
% 0.35/0.74                & ( ord_less_eq_real @ zero_zero_real @ T7 )
% 0.35/0.74                & ( ord_less_eq_real @ T7 @ H2 ) )
% 0.35/0.74             => ( has_fi5821293074295781190e_real
% 0.35/0.74                @ ^ [U2: real] :
% 0.35/0.74                    ( minus_minus_real @ ( Diff @ M3 @ U2 )
% 0.35/0.74                    @ ( plus_plus_real
% 0.35/0.74                      @ ( groups6591440286371151544t_real
% 0.35/0.74                        @ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M3 @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ U2 @ P6 ) )
% 0.35/0.74                        @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M3 ) ) )
% 0.35/0.74                      @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M3 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M3 ) ) ) ) ) )
% 0.35/0.74                @ ( minus_minus_real @ ( Diff @ ( suc @ M3 ) @ T7 )
% 0.35/0.74                  @ ( plus_plus_real
% 0.35/0.74                    @ ( groups6591440286371151544t_real
% 0.35/0.74                      @ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M3 ) @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ T7 @ P6 ) )
% 0.35/0.74                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) )
% 0.35/0.74                    @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ T7 @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) ) ) ) )
% 0.35/0.74                @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % Maclaurin_lemma2
% 0.35/0.74  thf(fact_8965_DERIV__arctan__series,axiom,
% 0.35/0.74      ! [X: real] :
% 0.35/0.74        ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.35/0.74       => ( has_fi5821293074295781190e_real
% 0.35/0.74          @ ^ [X9: real] :
% 0.35/0.74              ( suminf_real
% 0.35/0.74              @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
% 0.35/0.74          @ ( suminf_real
% 0.35/0.74            @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.74          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_arctan_series
% 0.35/0.74  thf(fact_8966_DERIV__real__root__generic,axiom,
% 0.35/0.74      ! [N: nat,X: real,D6: real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( X != zero_zero_real )
% 0.35/0.74         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.74             => ( ( ord_less_real @ zero_zero_real @ X )
% 0.35/0.74               => ( D6
% 0.35/0.74                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 0.35/0.74           => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.74               => ( ( ord_less_real @ X @ zero_zero_real )
% 0.35/0.74                 => ( D6
% 0.35/0.74                    = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
% 0.35/0.74             => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.74                 => ( D6
% 0.35/0.74                    = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 0.35/0.74               => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D6 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_real_root_generic
% 0.35/0.74  thf(fact_8967_DERIV__arccos,axiom,
% 0.35/0.74      ! [X: real] :
% 0.35/0.74        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.35/0.74       => ( ( ord_less_real @ X @ one_one_real )
% 0.35/0.74         => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_arccos
% 0.35/0.74  thf(fact_8968_DERIV__power__series_H,axiom,
% 0.35/0.74      ! [R: real,F: nat > real,X0: real] :
% 0.35/0.74        ( ! [X4: real] :
% 0.35/0.74            ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 0.35/0.74           => ( summable_real
% 0.35/0.74              @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X4 @ N2 ) ) ) )
% 0.35/0.74       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 0.35/0.74         => ( ( ord_less_real @ zero_zero_real @ R )
% 0.35/0.74           => ( has_fi5821293074295781190e_real
% 0.35/0.74              @ ^ [X3: real] :
% 0.35/0.74                  ( suminf_real
% 0.35/0.74                  @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X3 @ ( suc @ N2 ) ) ) )
% 0.35/0.74              @ ( suminf_real
% 0.35/0.74                @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X0 @ N2 ) ) )
% 0.35/0.74              @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_power_series'
% 0.35/0.74  thf(fact_8969_arccos__less__arccos,axiom,
% 0.35/0.74      ! [X: real,Y: real] :
% 0.35/0.74        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.35/0.74       => ( ( ord_less_real @ X @ Y )
% 0.35/0.74         => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.35/0.74           => ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % arccos_less_arccos
% 0.35/0.74  thf(fact_8970_arccos__less__mono,axiom,
% 0.35/0.74      ! [X: real,Y: real] :
% 0.35/0.74        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.35/0.74       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.35/0.74         => ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y ) )
% 0.35/0.74            = ( ord_less_real @ Y @ X ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % arccos_less_mono
% 0.35/0.74  thf(fact_8971_DERIV__isconst3,axiom,
% 0.35/0.74      ! [A: real,B: real,X: real,Y: real,F: real > real] :
% 0.35/0.74        ( ( ord_less_real @ A @ B )
% 0.35/0.74       => ( ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.35/0.74         => ( ( member_real @ Y @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.35/0.74           => ( ! [X4: real] :
% 0.35/0.74                  ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.35/0.74                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.35/0.74             => ( ( F @ X )
% 0.35/0.74                = ( F @ Y ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_isconst3
% 0.35/0.74  thf(fact_8972_arccos__lt__bounded,axiom,
% 0.35/0.74      ! [Y: real] :
% 0.35/0.74        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.35/0.74       => ( ( ord_less_real @ Y @ one_one_real )
% 0.35/0.74         => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y ) )
% 0.35/0.74            & ( ord_less_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % arccos_lt_bounded
% 0.35/0.74  thf(fact_8973_sin__arccos__nonzero,axiom,
% 0.35/0.74      ! [X: real] :
% 0.35/0.74        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.35/0.74       => ( ( ord_less_real @ X @ one_one_real )
% 0.35/0.74         => ( ( sin_real @ ( arccos @ X ) )
% 0.35/0.74           != zero_zero_real ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % sin_arccos_nonzero
% 0.35/0.74  thf(fact_8974_finite__greaterThanLessThan,axiom,
% 0.35/0.74      ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% 0.35/0.74  
% 0.35/0.74  % finite_greaterThanLessThan
% 0.35/0.74  thf(fact_8975_finite__greaterThanLessThan__int,axiom,
% 0.35/0.74      ! [L: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L @ U ) ) ).
% 0.35/0.74  
% 0.35/0.74  % finite_greaterThanLessThan_int
% 0.35/0.74  thf(fact_8976_LIM__fun__gt__zero,axiom,
% 0.35/0.74      ! [F: real > real,L: real,C: real] :
% 0.35/0.74        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.35/0.74       => ( ( ord_less_real @ zero_zero_real @ L )
% 0.35/0.74         => ? [R3: real] :
% 0.35/0.74              ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.35/0.74              & ! [X2: real] :
% 0.35/0.74                  ( ( ( X2 != C )
% 0.35/0.74                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R3 ) )
% 0.35/0.74                 => ( ord_less_real @ zero_zero_real @ ( F @ X2 ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % LIM_fun_gt_zero
% 0.35/0.74  thf(fact_8977_LIM__fun__not__zero,axiom,
% 0.35/0.74      ! [F: real > real,L: real,C: real] :
% 0.35/0.74        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.35/0.74       => ( ( L != zero_zero_real )
% 0.35/0.74         => ? [R3: real] :
% 0.35/0.74              ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.35/0.74              & ! [X2: real] :
% 0.35/0.74                  ( ( ( X2 != C )
% 0.35/0.74                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R3 ) )
% 0.35/0.74                 => ( ( F @ X2 )
% 0.35/0.74                   != zero_zero_real ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % LIM_fun_not_zero
% 0.35/0.74  thf(fact_8978_LIM__fun__less__zero,axiom,
% 0.35/0.74      ! [F: real > real,L: real,C: real] :
% 0.35/0.74        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.35/0.74       => ( ( ord_less_real @ L @ zero_zero_real )
% 0.35/0.74         => ? [R3: real] :
% 0.35/0.74              ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.35/0.74              & ! [X2: real] :
% 0.35/0.74                  ( ( ( X2 != C )
% 0.35/0.74                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R3 ) )
% 0.35/0.74                 => ( ord_less_real @ ( F @ X2 ) @ zero_zero_real ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % LIM_fun_less_zero
% 0.35/0.74  thf(fact_8979_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 0.35/0.74      ! [N: nat,J: nat,I: nat] :
% 0.35/0.74        ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
% 0.35/0.74       => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
% 0.35/0.74          = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % nth_sorted_list_of_set_greaterThanLessThan
% 0.35/0.74  thf(fact_8980_summable__Leibniz_I2_J,axiom,
% 0.35/0.74      ! [A: nat > real] :
% 0.35/0.74        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.35/0.74       => ( ( topolo6980174941875973593q_real @ A )
% 0.35/0.74         => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
% 0.35/0.74           => ! [N5: nat] :
% 0.35/0.74                ( member_real
% 0.35/0.74                @ ( suminf_real
% 0.35/0.74                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 0.35/0.74                @ ( set_or1222579329274155063t_real
% 0.35/0.74                  @ ( groups6591440286371151544t_real
% 0.35/0.74                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.35/0.74                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) )
% 0.35/0.74                  @ ( groups6591440286371151544t_real
% 0.35/0.74                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.35/0.74                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % summable_Leibniz(2)
% 0.35/0.74  thf(fact_8981_summable__Leibniz_I3_J,axiom,
% 0.35/0.74      ! [A: nat > real] :
% 0.35/0.74        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.35/0.74       => ( ( topolo6980174941875973593q_real @ A )
% 0.35/0.74         => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
% 0.35/0.74           => ! [N5: nat] :
% 0.35/0.74                ( member_real
% 0.35/0.74                @ ( suminf_real
% 0.35/0.74                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 0.35/0.74                @ ( set_or1222579329274155063t_real
% 0.35/0.74                  @ ( groups6591440286371151544t_real
% 0.35/0.74                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.35/0.74                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) @ one_one_nat ) ) )
% 0.35/0.74                  @ ( groups6591440286371151544t_real
% 0.35/0.74                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.35/0.74                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) ) ) ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % summable_Leibniz(3)
% 0.35/0.74  thf(fact_8982_trivial__limit__sequentially,axiom,
% 0.35/0.74      at_top_nat != bot_bot_filter_nat ).
% 0.35/0.74  
% 0.35/0.74  % trivial_limit_sequentially
% 0.35/0.74  thf(fact_8983_mult__nat__left__at__top,axiom,
% 0.35/0.74      ! [C: nat] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.35/0.74       => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 0.35/0.74  
% 0.35/0.74  % mult_nat_left_at_top
% 0.35/0.74  thf(fact_8984_mult__nat__right__at__top,axiom,
% 0.35/0.74      ! [C: nat] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.35/0.74       => ( filterlim_nat_nat
% 0.35/0.74          @ ^ [X3: nat] : ( times_times_nat @ X3 @ C )
% 0.35/0.74          @ at_top_nat
% 0.35/0.74          @ at_top_nat ) ) ).
% 0.35/0.74  
% 0.35/0.74  % mult_nat_right_at_top
% 0.35/0.74  thf(fact_8985_LIMSEQ__inverse__zero,axiom,
% 0.35/0.74      ! [X8: nat > real] :
% 0.35/0.74        ( ! [R3: real] :
% 0.35/0.74          ? [N8: nat] :
% 0.35/0.74          ! [N3: nat] :
% 0.35/0.74            ( ( ord_less_eq_nat @ N8 @ N3 )
% 0.35/0.74           => ( ord_less_real @ R3 @ ( X8 @ N3 ) ) )
% 0.35/0.74       => ( filterlim_nat_real
% 0.35/0.74          @ ^ [N2: nat] : ( inverse_inverse_real @ ( X8 @ N2 ) )
% 0.35/0.74          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.35/0.74          @ at_top_nat ) ) ).
% 0.35/0.74  
% 0.35/0.74  % LIMSEQ_inverse_zero
% 0.35/0.74  thf(fact_8986_LIMSEQ__root__const,axiom,
% 0.35/0.74      ! [C: real] :
% 0.35/0.74        ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.74       => ( filterlim_nat_real
% 0.35/0.74          @ ^ [N2: nat] : ( root @ N2 @ C )
% 0.35/0.74          @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.35/0.74          @ at_top_nat ) ) ).
% 0.35/0.74  
% 0.35/0.74  % LIMSEQ_root_const
% 0.35/0.74  thf(fact_8987_increasing__LIMSEQ,axiom,
% 0.35/0.74      ! [F: nat > real,L: real] :
% 0.35/0.74        ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.35/0.74       => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ L )
% 0.35/0.74         => ( ! [E: real] :
% 0.35/0.74                ( ( ord_less_real @ zero_zero_real @ E )
% 0.35/0.74               => ? [N5: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N5 ) @ E ) ) )
% 0.35/0.74           => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % increasing_LIMSEQ
% 0.35/0.74  thf(fact_8988_LIMSEQ__realpow__zero,axiom,
% 0.35/0.74      ! [X: real] :
% 0.35/0.74        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.74       => ( ( ord_less_real @ X @ one_one_real )
% 0.35/0.74         => ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % LIMSEQ_realpow_zero
% 0.35/0.74  thf(fact_8989_LIMSEQ__divide__realpow__zero,axiom,
% 0.35/0.74      ! [X: real,A: real] :
% 0.35/0.74        ( ( ord_less_real @ one_one_real @ X )
% 0.35/0.74       => ( filterlim_nat_real
% 0.35/0.74          @ ^ [N2: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N2 ) )
% 0.35/0.74          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.35/0.74          @ at_top_nat ) ) ).
% 0.35/0.74  
% 0.35/0.74  % LIMSEQ_divide_realpow_zero
% 0.35/0.74  thf(fact_8990_LIMSEQ__abs__realpow__zero2,axiom,
% 0.35/0.74      ! [C: real] :
% 0.35/0.74        ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.35/0.74       => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.35/0.74  
% 0.35/0.74  % LIMSEQ_abs_realpow_zero2
% 0.35/0.74  thf(fact_8991_LIMSEQ__abs__realpow__zero,axiom,
% 0.35/0.74      ! [C: real] :
% 0.35/0.74        ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.35/0.74       => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.35/0.74  
% 0.35/0.74  % LIMSEQ_abs_realpow_zero
% 0.35/0.74  thf(fact_8992_LIMSEQ__inverse__realpow__zero,axiom,
% 0.35/0.74      ! [X: real] :
% 0.35/0.74        ( ( ord_less_real @ one_one_real @ X )
% 0.35/0.74       => ( filterlim_nat_real
% 0.35/0.74          @ ^ [N2: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N2 ) )
% 0.35/0.74          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.35/0.74          @ at_top_nat ) ) ).
% 0.35/0.74  
% 0.35/0.74  % LIMSEQ_inverse_realpow_zero
% 0.35/0.74  thf(fact_8993_DERIV__neg__imp__decreasing__at__top,axiom,
% 0.35/0.74      ! [B: real,F: real > real,Flim: real] :
% 0.35/0.74        ( ! [X4: real] :
% 0.35/0.74            ( ( ord_less_eq_real @ B @ X4 )
% 0.35/0.74           => ? [Y4: real] :
% 0.35/0.74                ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.35/0.74                & ( ord_less_real @ Y4 @ zero_zero_real ) ) )
% 0.35/0.74       => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
% 0.35/0.74         => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_neg_imp_decreasing_at_top
% 0.35/0.74  thf(fact_8994_filterlim__pow__at__bot__even,axiom,
% 0.35/0.74      ! [N: nat,F: real > real,F3: filter_real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
% 0.35/0.74         => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.74           => ( filterlim_real_real
% 0.35/0.74              @ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N )
% 0.35/0.74              @ at_top_real
% 0.35/0.74              @ F3 ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % filterlim_pow_at_bot_even
% 0.35/0.74  thf(fact_8995_eventually__sequentially,axiom,
% 0.35/0.74      ! [P: nat > $o] :
% 0.35/0.74        ( ( eventually_nat @ P @ at_top_nat )
% 0.35/0.74        = ( ? [N4: nat] :
% 0.35/0.74            ! [N2: nat] :
% 0.35/0.74              ( ( ord_less_eq_nat @ N4 @ N2 )
% 0.35/0.74             => ( P @ N2 ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % eventually_sequentially
% 0.35/0.74  thf(fact_8996_eventually__sequentiallyI,axiom,
% 0.35/0.74      ! [C: nat,P: nat > $o] :
% 0.35/0.74        ( ! [X4: nat] :
% 0.35/0.74            ( ( ord_less_eq_nat @ C @ X4 )
% 0.35/0.74           => ( P @ X4 ) )
% 0.35/0.74       => ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.35/0.74  
% 0.35/0.74  % eventually_sequentiallyI
% 0.35/0.74  thf(fact_8997_le__sequentially,axiom,
% 0.35/0.74      ! [F3: filter_nat] :
% 0.35/0.74        ( ( ord_le2510731241096832064er_nat @ F3 @ at_top_nat )
% 0.35/0.74        = ( ! [N4: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N4 ) @ F3 ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % le_sequentially
% 0.35/0.74  thf(fact_8998_eventually__at__left__real,axiom,
% 0.35/0.74      ! [B: real,A: real] :
% 0.35/0.74        ( ( ord_less_real @ B @ A )
% 0.35/0.74       => ( eventually_real
% 0.35/0.74          @ ^ [X3: real] : ( member_real @ X3 @ ( set_or1633881224788618240n_real @ B @ A ) )
% 0.35/0.74          @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % eventually_at_left_real
% 0.35/0.74  thf(fact_8999_DERIV__pos__imp__increasing__at__bot,axiom,
% 0.35/0.74      ! [B: real,F: real > real,Flim: real] :
% 0.35/0.74        ( ! [X4: real] :
% 0.35/0.74            ( ( ord_less_eq_real @ X4 @ B )
% 0.35/0.74           => ? [Y4: real] :
% 0.35/0.74                ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.35/0.74                & ( ord_less_real @ zero_zero_real @ Y4 ) ) )
% 0.35/0.74       => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
% 0.35/0.74         => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % DERIV_pos_imp_increasing_at_bot
% 0.35/0.74  thf(fact_9000_filterlim__pow__at__bot__odd,axiom,
% 0.35/0.74      ! [N: nat,F: real > real,F3: filter_real] :
% 0.35/0.74        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74       => ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
% 0.35/0.74         => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.74           => ( filterlim_real_real
% 0.35/0.74              @ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N )
% 0.35/0.74              @ at_bot_real
% 0.35/0.74              @ F3 ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % filterlim_pow_at_bot_odd
% 0.35/0.74  thf(fact_9001_finite__greaterThanAtMost,axiom,
% 0.35/0.74      ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% 0.35/0.74  
% 0.35/0.74  % finite_greaterThanAtMost
% 0.35/0.74  thf(fact_9002_GreatestI__ex__nat,axiom,
% 0.35/0.74      ! [P: nat > $o,B: nat] :
% 0.35/0.74        ( ? [X_12: nat] : ( P @ X_12 )
% 0.35/0.74       => ( ! [Y3: nat] :
% 0.35/0.74              ( ( P @ Y3 )
% 0.35/0.74             => ( ord_less_eq_nat @ Y3 @ B ) )
% 0.35/0.74         => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.35/0.74  
% 0.35/0.74  % GreatestI_ex_nat
% 0.35/0.74  thf(fact_9003_Greatest__le__nat,axiom,
% 0.35/0.74      ! [P: nat > $o,K: nat,B: nat] :
% 0.35/0.74        ( ( P @ K )
% 0.35/0.74       => ( ! [Y3: nat] :
% 0.35/0.74              ( ( P @ Y3 )
% 0.35/0.75             => ( ord_less_eq_nat @ Y3 @ B ) )
% 0.35/0.75         => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Greatest_le_nat
% 0.35/0.75  thf(fact_9004_GreatestI__nat,axiom,
% 0.35/0.75      ! [P: nat > $o,K: nat,B: nat] :
% 0.35/0.75        ( ( P @ K )
% 0.35/0.75       => ( ! [Y3: nat] :
% 0.35/0.75              ( ( P @ Y3 )
% 0.35/0.75             => ( ord_less_eq_nat @ Y3 @ B ) )
% 0.35/0.75         => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % GreatestI_nat
% 0.35/0.75  thf(fact_9005_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 0.35/0.75      ! [N: nat,J: nat,I: nat] :
% 0.35/0.75        ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
% 0.35/0.75       => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
% 0.35/0.75          = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % nth_sorted_list_of_set_greaterThanAtMost
% 0.35/0.75  thf(fact_9006_finite__greaterThanAtMost__int,axiom,
% 0.35/0.75      ! [L: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L @ U ) ) ).
% 0.35/0.75  
% 0.35/0.75  % finite_greaterThanAtMost_int
% 0.35/0.75  thf(fact_9007_isCont__inverse__function2,axiom,
% 0.35/0.75      ! [A: real,X: real,B: real,G: real > real,F: real > real] :
% 0.35/0.75        ( ( ord_less_real @ A @ X )
% 0.35/0.75       => ( ( ord_less_real @ X @ B )
% 0.35/0.75         => ( ! [Z4: real] :
% 0.35/0.75                ( ( ord_less_eq_real @ A @ Z4 )
% 0.35/0.75               => ( ( ord_less_eq_real @ Z4 @ B )
% 0.35/0.75                 => ( ( G @ ( F @ Z4 ) )
% 0.35/0.75                    = Z4 ) ) )
% 0.35/0.75           => ( ! [Z4: real] :
% 0.35/0.75                  ( ( ord_less_eq_real @ A @ Z4 )
% 0.35/0.75                 => ( ( ord_less_eq_real @ Z4 @ B )
% 0.35/0.75                   => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
% 0.35/0.75             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % isCont_inverse_function2
% 0.35/0.75  thf(fact_9008_isCont__arcosh,axiom,
% 0.35/0.75      ! [X: real] :
% 0.35/0.75        ( ( ord_less_real @ one_one_real @ X )
% 0.35/0.75       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcosh_real ) ) ).
% 0.35/0.75  
% 0.35/0.75  % isCont_arcosh
% 0.35/0.75  thf(fact_9009_DERIV__inverse__function,axiom,
% 0.35/0.75      ! [F: real > real,D6: real,G: real > real,X: real,A: real,B: real] :
% 0.35/0.75        ( ( has_fi5821293074295781190e_real @ F @ D6 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
% 0.35/0.75       => ( ( D6 != zero_zero_real )
% 0.35/0.75         => ( ( ord_less_real @ A @ X )
% 0.35/0.75           => ( ( ord_less_real @ X @ B )
% 0.35/0.75             => ( ! [Y3: real] :
% 0.35/0.75                    ( ( ord_less_real @ A @ Y3 )
% 0.35/0.75                   => ( ( ord_less_real @ Y3 @ B )
% 0.35/0.75                     => ( ( F @ ( G @ Y3 ) )
% 0.35/0.75                        = Y3 ) ) )
% 0.35/0.75               => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
% 0.35/0.75                 => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D6 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % DERIV_inverse_function
% 0.35/0.75  thf(fact_9010_isCont__arccos,axiom,
% 0.35/0.75      ! [X: real] :
% 0.35/0.75        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.35/0.75       => ( ( ord_less_real @ X @ one_one_real )
% 0.35/0.75         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arccos ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % isCont_arccos
% 0.35/0.75  thf(fact_9011_isCont__arcsin,axiom,
% 0.35/0.75      ! [X: real] :
% 0.35/0.75        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.35/0.75       => ( ( ord_less_real @ X @ one_one_real )
% 0.35/0.75         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcsin ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % isCont_arcsin
% 0.35/0.75  thf(fact_9012_LIM__less__bound,axiom,
% 0.35/0.75      ! [B: real,X: real,F: real > real] :
% 0.35/0.75        ( ( ord_less_real @ B @ X )
% 0.35/0.75       => ( ! [X4: real] :
% 0.35/0.75              ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ B @ X ) )
% 0.35/0.75             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.35/0.75         => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
% 0.35/0.75           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % LIM_less_bound
% 0.35/0.75  thf(fact_9013_isCont__artanh,axiom,
% 0.35/0.75      ! [X: real] :
% 0.35/0.75        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.35/0.75       => ( ( ord_less_real @ X @ one_one_real )
% 0.35/0.75         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ artanh_real ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % isCont_artanh
% 0.35/0.75  thf(fact_9014_isCont__inverse__function,axiom,
% 0.35/0.75      ! [D: real,X: real,G: real > real,F: real > real] :
% 0.35/0.75        ( ( ord_less_real @ zero_zero_real @ D )
% 0.35/0.75       => ( ! [Z4: real] :
% 0.35/0.75              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X ) ) @ D )
% 0.35/0.75             => ( ( G @ ( F @ Z4 ) )
% 0.35/0.75                = Z4 ) )
% 0.35/0.75         => ( ! [Z4: real] :
% 0.35/0.75                ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X ) ) @ D )
% 0.35/0.75               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) )
% 0.35/0.75           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % isCont_inverse_function
% 0.35/0.75  thf(fact_9015_GMVT_H,axiom,
% 0.35/0.75      ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F6: real > real] :
% 0.35/0.75        ( ( ord_less_real @ A @ B )
% 0.35/0.75       => ( ! [Z4: real] :
% 0.35/0.75              ( ( ord_less_eq_real @ A @ Z4 )
% 0.35/0.75             => ( ( ord_less_eq_real @ Z4 @ B )
% 0.35/0.75               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
% 0.35/0.75         => ( ! [Z4: real] :
% 0.35/0.75                ( ( ord_less_eq_real @ A @ Z4 )
% 0.35/0.75               => ( ( ord_less_eq_real @ Z4 @ B )
% 0.35/0.75                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ G ) ) )
% 0.35/0.75           => ( ! [Z4: real] :
% 0.35/0.75                  ( ( ord_less_real @ A @ Z4 )
% 0.35/0.75                 => ( ( ord_less_real @ Z4 @ B )
% 0.35/0.75                   => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
% 0.35/0.75             => ( ! [Z4: real] :
% 0.35/0.75                    ( ( ord_less_real @ A @ Z4 )
% 0.35/0.75                   => ( ( ord_less_real @ Z4 @ B )
% 0.35/0.75                     => ( has_fi5821293074295781190e_real @ F @ ( F6 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
% 0.35/0.75               => ? [C3: real] :
% 0.35/0.75                    ( ( ord_less_real @ A @ C3 )
% 0.35/0.75                    & ( ord_less_real @ C3 @ B )
% 0.35/0.75                    & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C3 ) )
% 0.35/0.75                      = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F6 @ C3 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % GMVT'
% 0.35/0.75  thf(fact_9016_INT__greaterThan__UNIV,axiom,
% 0.35/0.75      ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
% 0.35/0.75      = bot_bot_set_nat ) ).
% 0.35/0.75  
% 0.35/0.75  % INT_greaterThan_UNIV
% 0.35/0.75  thf(fact_9017_greaterThan__0,axiom,
% 0.35/0.75      ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
% 0.35/0.75      = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% 0.35/0.75  
% 0.35/0.75  % greaterThan_0
% 0.35/0.75  thf(fact_9018_eventually__at__right__real,axiom,
% 0.35/0.75      ! [A: real,B: real] :
% 0.35/0.75        ( ( ord_less_real @ A @ B )
% 0.35/0.75       => ( eventually_real
% 0.35/0.75          @ ^ [X3: real] : ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.35/0.75          @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % eventually_at_right_real
% 0.35/0.75  thf(fact_9019_greaterThan__Suc,axiom,
% 0.35/0.75      ! [K: nat] :
% 0.35/0.75        ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
% 0.35/0.75        = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % greaterThan_Suc
% 0.35/0.75  thf(fact_9020_GMVT,axiom,
% 0.35/0.75      ! [A: real,B: real,F: real > real,G: real > real] :
% 0.35/0.75        ( ( ord_less_real @ A @ B )
% 0.35/0.75       => ( ! [X4: real] :
% 0.35/0.75              ( ( ( ord_less_eq_real @ A @ X4 )
% 0.35/0.75                & ( ord_less_eq_real @ X4 @ B ) )
% 0.35/0.75             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
% 0.35/0.75         => ( ! [X4: real] :
% 0.35/0.75                ( ( ( ord_less_real @ A @ X4 )
% 0.35/0.75                  & ( ord_less_real @ X4 @ B ) )
% 0.35/0.75               => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.35/0.75           => ( ! [X4: real] :
% 0.35/0.75                  ( ( ( ord_less_eq_real @ A @ X4 )
% 0.35/0.75                    & ( ord_less_eq_real @ X4 @ B ) )
% 0.35/0.75                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ G ) )
% 0.35/0.75             => ( ! [X4: real] :
% 0.35/0.75                    ( ( ( ord_less_real @ A @ X4 )
% 0.35/0.75                      & ( ord_less_real @ X4 @ B ) )
% 0.35/0.75                   => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.35/0.75               => ? [G_c: real,F_c: real,C3: real] :
% 0.35/0.75                    ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 0.35/0.75                    & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 0.35/0.75                    & ( ord_less_real @ A @ C3 )
% 0.35/0.75                    & ( ord_less_real @ C3 @ B )
% 0.35/0.75                    & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
% 0.35/0.75                      = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % GMVT
% 0.35/0.75  thf(fact_9021_atLeast__0,axiom,
% 0.35/0.75      ( ( set_ord_atLeast_nat @ zero_zero_nat )
% 0.35/0.75      = top_top_set_nat ) ).
% 0.35/0.75  
% 0.35/0.75  % atLeast_0
% 0.35/0.75  thf(fact_9022_atLeast__Suc,axiom,
% 0.35/0.75      ! [K: nat] :
% 0.35/0.75        ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 0.35/0.75        = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % atLeast_Suc
% 0.35/0.75  thf(fact_9023_Gcd__eq__Max,axiom,
% 0.35/0.75      ! [M4: set_nat] :
% 0.35/0.75        ( ( finite_finite_nat @ M4 )
% 0.35/0.75       => ( ( M4 != bot_bot_set_nat )
% 0.35/0.75         => ( ~ ( member_nat @ zero_zero_nat @ M4 )
% 0.35/0.75           => ( ( gcd_Gcd_nat @ M4 )
% 0.35/0.75              = ( lattic8265883725875713057ax_nat
% 0.35/0.75                @ ( comple7806235888213564991et_nat
% 0.35/0.75                  @ ( image_nat_set_nat
% 0.35/0.75                    @ ^ [M2: nat] :
% 0.35/0.75                        ( collect_nat
% 0.35/0.75                        @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M2 ) )
% 0.35/0.75                    @ M4 ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Gcd_eq_Max
% 0.35/0.75  thf(fact_9024_Max__divisors__self__nat,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( ( N != zero_zero_nat )
% 0.35/0.75       => ( ( lattic8265883725875713057ax_nat
% 0.35/0.75            @ ( collect_nat
% 0.35/0.75              @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ N ) ) )
% 0.35/0.75          = N ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Max_divisors_self_nat
% 0.35/0.75  thf(fact_9025_bdd__above__nat,axiom,
% 0.35/0.75      condit2214826472909112428ve_nat = finite_finite_nat ).
% 0.35/0.75  
% 0.35/0.75  % bdd_above_nat
% 0.35/0.75  thf(fact_9026_card__le__Suc__Max,axiom,
% 0.35/0.75      ! [S2: set_nat] :
% 0.35/0.75        ( ( finite_finite_nat @ S2 )
% 0.35/0.75       => ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % card_le_Suc_Max
% 0.35/0.75  thf(fact_9027_Sup__nat__def,axiom,
% 0.35/0.75      ( complete_Sup_Sup_nat
% 0.35/0.75      = ( ^ [X6: set_nat] : ( if_nat @ ( X6 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X6 ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Sup_nat_def
% 0.35/0.75  thf(fact_9028_divide__nat__def,axiom,
% 0.35/0.75      ( divide_divide_nat
% 0.35/0.75      = ( ^ [M2: nat,N2: nat] :
% 0.35/0.75            ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
% 0.35/0.75            @ ( lattic8265883725875713057ax_nat
% 0.35/0.75              @ ( collect_nat
% 0.35/0.75                @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N2 ) @ M2 ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % divide_nat_def
% 0.35/0.75  thf(fact_9029_gcd__is__Max__divisors__nat,axiom,
% 0.35/0.75      ! [N: nat,M: nat] :
% 0.35/0.75        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.75       => ( ( gcd_gcd_nat @ M @ N )
% 0.35/0.75          = ( lattic8265883725875713057ax_nat
% 0.35/0.75            @ ( collect_nat
% 0.35/0.75              @ ^ [D4: nat] :
% 0.35/0.75                  ( ( dvd_dvd_nat @ D4 @ M )
% 0.35/0.75                  & ( dvd_dvd_nat @ D4 @ N ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % gcd_is_Max_divisors_nat
% 0.35/0.75  thf(fact_9030_MVT,axiom,
% 0.35/0.75      ! [A: real,B: real,F: real > real] :
% 0.35/0.75        ( ( ord_less_real @ A @ B )
% 0.35/0.75       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.35/0.75         => ( ! [X4: real] :
% 0.35/0.75                ( ( ord_less_real @ A @ X4 )
% 0.35/0.75               => ( ( ord_less_real @ X4 @ B )
% 0.35/0.75                 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.35/0.75           => ? [L4: real,Z4: real] :
% 0.35/0.75                ( ( ord_less_real @ A @ Z4 )
% 0.35/0.75                & ( ord_less_real @ Z4 @ B )
% 0.35/0.75                & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) )
% 0.35/0.75                & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.35/0.75                  = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % MVT
% 0.35/0.75  thf(fact_9031_uniformity__complex__def,axiom,
% 0.35/0.75      ( topolo896644834953643431omplex
% 0.35/0.75      = ( comple8358262395181532106omplex
% 0.35/0.75        @ ( image_5971271580939081552omplex
% 0.35/0.75          @ ^ [E3: real] :
% 0.35/0.75              ( princi3496590319149328850omplex
% 0.35/0.75              @ ( collec8663557070575231912omplex
% 0.35/0.75                @ ( produc6771430404735790350plex_o
% 0.35/0.75                  @ ^ [X3: complex,Y2: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X3 @ Y2 ) @ E3 ) ) ) )
% 0.35/0.75          @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % uniformity_complex_def
% 0.35/0.75  thf(fact_9032_uniformity__real__def,axiom,
% 0.35/0.75      ( topolo1511823702728130853y_real
% 0.35/0.75      = ( comple2936214249959783750l_real
% 0.35/0.75        @ ( image_2178119161166701260l_real
% 0.35/0.75          @ ^ [E3: real] :
% 0.35/0.75              ( princi6114159922880469582l_real
% 0.35/0.75              @ ( collec3799799289383736868l_real
% 0.35/0.75                @ ( produc5414030515140494994real_o
% 0.35/0.75                  @ ^ [X3: real,Y2: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X3 @ Y2 ) @ E3 ) ) ) )
% 0.35/0.75          @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % uniformity_real_def
% 0.35/0.75  thf(fact_9033_Rolle__deriv,axiom,
% 0.35/0.75      ! [A: real,B: real,F: real > real,F6: real > real > real] :
% 0.35/0.75        ( ( ord_less_real @ A @ B )
% 0.35/0.75       => ( ( ( F @ A )
% 0.35/0.75            = ( F @ B ) )
% 0.35/0.75         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.35/0.75           => ( ! [X4: real] :
% 0.35/0.75                  ( ( ord_less_real @ A @ X4 )
% 0.35/0.75                 => ( ( ord_less_real @ X4 @ B )
% 0.35/0.75                   => ( has_de1759254742604945161l_real @ F @ ( F6 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.35/0.75             => ? [Z4: real] :
% 0.35/0.75                  ( ( ord_less_real @ A @ Z4 )
% 0.35/0.75                  & ( ord_less_real @ Z4 @ B )
% 0.35/0.75                  & ( ( F6 @ Z4 )
% 0.35/0.75                    = ( ^ [V3: real] : zero_zero_real ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Rolle_deriv
% 0.35/0.75  thf(fact_9034_mvt,axiom,
% 0.35/0.75      ! [A: real,B: real,F: real > real,F6: real > real > real] :
% 0.35/0.75        ( ( ord_less_real @ A @ B )
% 0.35/0.75       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.35/0.75         => ( ! [X4: real] :
% 0.35/0.75                ( ( ord_less_real @ A @ X4 )
% 0.35/0.75               => ( ( ord_less_real @ X4 @ B )
% 0.35/0.75                 => ( has_de1759254742604945161l_real @ F @ ( F6 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.35/0.75           => ~ ! [Xi: real] :
% 0.35/0.75                  ( ( ord_less_real @ A @ Xi )
% 0.35/0.75                 => ( ( ord_less_real @ Xi @ B )
% 0.35/0.75                   => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.35/0.75                     != ( F6 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % mvt
% 0.35/0.75  thf(fact_9035_DERIV__pos__imp__increasing__open,axiom,
% 0.35/0.75      ! [A: real,B: real,F: real > real] :
% 0.35/0.75        ( ( ord_less_real @ A @ B )
% 0.35/0.75       => ( ! [X4: real] :
% 0.35/0.75              ( ( ord_less_real @ A @ X4 )
% 0.35/0.75             => ( ( ord_less_real @ X4 @ B )
% 0.35/0.75               => ? [Y4: real] :
% 0.35/0.75                    ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.35/0.75                    & ( ord_less_real @ zero_zero_real @ Y4 ) ) ) )
% 0.35/0.75         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.35/0.75           => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % DERIV_pos_imp_increasing_open
% 0.35/0.75  thf(fact_9036_DERIV__neg__imp__decreasing__open,axiom,
% 0.35/0.75      ! [A: real,B: real,F: real > real] :
% 0.35/0.75        ( ( ord_less_real @ A @ B )
% 0.35/0.75       => ( ! [X4: real] :
% 0.35/0.75              ( ( ord_less_real @ A @ X4 )
% 0.35/0.75             => ( ( ord_less_real @ X4 @ B )
% 0.35/0.75               => ? [Y4: real] :
% 0.35/0.75                    ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.35/0.75                    & ( ord_less_real @ Y4 @ zero_zero_real ) ) ) )
% 0.35/0.75         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.35/0.75           => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % DERIV_neg_imp_decreasing_open
% 0.35/0.75  thf(fact_9037_DERIV__isconst__end,axiom,
% 0.35/0.75      ! [A: real,B: real,F: real > real] :
% 0.35/0.75        ( ( ord_less_real @ A @ B )
% 0.35/0.75       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.35/0.75         => ( ! [X4: real] :
% 0.35/0.75                ( ( ord_less_real @ A @ X4 )
% 0.35/0.75               => ( ( ord_less_real @ X4 @ B )
% 0.35/0.75                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.35/0.75           => ( ( F @ B )
% 0.35/0.75              = ( F @ A ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % DERIV_isconst_end
% 0.35/0.75  thf(fact_9038_DERIV__isconst2,axiom,
% 0.35/0.75      ! [A: real,B: real,F: real > real,X: real] :
% 0.35/0.75        ( ( ord_less_real @ A @ B )
% 0.35/0.75       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.35/0.75         => ( ! [X4: real] :
% 0.35/0.75                ( ( ord_less_real @ A @ X4 )
% 0.35/0.75               => ( ( ord_less_real @ X4 @ B )
% 0.35/0.75                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.35/0.75           => ( ( ord_less_eq_real @ A @ X )
% 0.35/0.75             => ( ( ord_less_eq_real @ X @ B )
% 0.35/0.75               => ( ( F @ X )
% 0.35/0.75                  = ( F @ A ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % DERIV_isconst2
% 0.35/0.75  thf(fact_9039_Rolle,axiom,
% 0.35/0.75      ! [A: real,B: real,F: real > real] :
% 0.35/0.75        ( ( ord_less_real @ A @ B )
% 0.35/0.75       => ( ( ( F @ A )
% 0.35/0.75            = ( F @ B ) )
% 0.35/0.75         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.35/0.75           => ( ! [X4: real] :
% 0.35/0.75                  ( ( ord_less_real @ A @ X4 )
% 0.35/0.75                 => ( ( ord_less_real @ X4 @ B )
% 0.35/0.75                   => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.35/0.75             => ? [Z4: real] :
% 0.35/0.75                  ( ( ord_less_real @ A @ Z4 )
% 0.35/0.75                  & ( ord_less_real @ Z4 @ B )
% 0.35/0.75                  & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Rolle
% 0.35/0.75  thf(fact_9040_eventually__prod__sequentially,axiom,
% 0.35/0.75      ! [P: product_prod_nat_nat > $o] :
% 0.35/0.75        ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
% 0.35/0.75        = ( ? [N4: nat] :
% 0.35/0.75            ! [M2: nat] :
% 0.35/0.75              ( ( ord_less_eq_nat @ N4 @ M2 )
% 0.35/0.75             => ! [N2: nat] :
% 0.35/0.75                  ( ( ord_less_eq_nat @ N4 @ N2 )
% 0.35/0.75                 => ( P @ ( product_Pair_nat_nat @ N2 @ M2 ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % eventually_prod_sequentially
% 0.35/0.75  thf(fact_9041_mono__Suc,axiom,
% 0.35/0.75      order_mono_nat_nat @ suc ).
% 0.35/0.75  
% 0.35/0.75  % mono_Suc
% 0.35/0.75  thf(fact_9042_mono__times__nat,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.75       => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % mono_times_nat
% 0.35/0.75  thf(fact_9043_mono__ge2__power__minus__self,axiom,
% 0.35/0.75      ! [K: nat] :
% 0.35/0.75        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.35/0.75       => ( order_mono_nat_nat
% 0.35/0.75          @ ^ [M2: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M2 ) @ M2 ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % mono_ge2_power_minus_self
% 0.35/0.75  thf(fact_9044_inj__sgn__power,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.75       => ( inj_on_real_real
% 0.35/0.75          @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N ) )
% 0.35/0.75          @ top_top_set_real ) ) ).
% 0.35/0.75  
% 0.35/0.75  % inj_sgn_power
% 0.35/0.75  thf(fact_9045_log__inj,axiom,
% 0.35/0.75      ! [B: real] :
% 0.35/0.75        ( ( ord_less_real @ one_one_real @ B )
% 0.35/0.75       => ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % log_inj
% 0.35/0.75  thf(fact_9046_inj__Suc,axiom,
% 0.35/0.75      ! [N6: set_nat] : ( inj_on_nat_nat @ suc @ N6 ) ).
% 0.35/0.75  
% 0.35/0.75  % inj_Suc
% 0.35/0.75  thf(fact_9047_inj__on__diff__nat,axiom,
% 0.35/0.75      ! [N6: set_nat,K: nat] :
% 0.35/0.75        ( ! [N3: nat] :
% 0.35/0.75            ( ( member_nat @ N3 @ N6 )
% 0.35/0.75           => ( ord_less_eq_nat @ K @ N3 ) )
% 0.35/0.75       => ( inj_on_nat_nat
% 0.35/0.75          @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
% 0.35/0.75          @ N6 ) ) ).
% 0.35/0.75  
% 0.35/0.75  % inj_on_diff_nat
% 0.35/0.75  thf(fact_9048_inj__on__set__encode,axiom,
% 0.35/0.75      inj_on_set_nat_nat @ nat_set_encode @ ( collect_set_nat @ finite_finite_nat ) ).
% 0.35/0.75  
% 0.35/0.75  % inj_on_set_encode
% 0.35/0.75  thf(fact_9049_powr__real__of__int_H,axiom,
% 0.35/0.75      ! [X: real,N: int] :
% 0.35/0.75        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.75       => ( ( ( X != zero_zero_real )
% 0.35/0.75            | ( ord_less_int @ zero_zero_int @ N ) )
% 0.35/0.75         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 0.35/0.75            = ( power_int_real @ X @ N ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % powr_real_of_int'
% 0.35/0.75  thf(fact_9050_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 0.35/0.75      ! [X: vEBT_VEBT,Xa2: nat] :
% 0.35/0.75        ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
% 0.35/0.75       => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.35/0.75                ( X
% 0.35/0.75                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.75           => ( Xa2 = one_one_nat ) )
% 0.35/0.75         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.75                ( ( X
% 0.35/0.75                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) )
% 0.35/0.75               => ( ( Deg2 = Xa2 )
% 0.35/0.75                  & ! [X4: vEBT_VEBT] :
% 0.35/0.75                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.75                     => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.75                  & ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.75                  & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.35/0.75                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.75                  & ( case_o184042715313410164at_nat
% 0.35/0.75                    @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 0.35/0.75                      & ! [X3: vEBT_VEBT] :
% 0.35/0.75                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.75                         => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 0.35/0.75                    @ ( produc6081775807080527818_nat_o
% 0.35/0.75                      @ ^ [Mi2: nat,Ma2: nat] :
% 0.35/0.75                          ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.35/0.75                          & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.75                          & ! [I4: nat] :
% 0.35/0.75                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.75                             => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 0.35/0.75                                = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 0.35/0.75                          & ( ( Mi2 = Ma2 )
% 0.35/0.75                           => ! [X3: vEBT_VEBT] :
% 0.35/0.75                                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.75                               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 0.35/0.75                          & ( ( Mi2 != Ma2 )
% 0.35/0.75                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma2 )
% 0.35/0.75                              & ! [X3: nat] :
% 0.35/0.75                                  ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.75                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 0.35/0.75                                   => ( ( ord_less_nat @ Mi2 @ X3 )
% 0.35/0.75                                      & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) )
% 0.35/0.75                    @ Mima ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % VEBT_internal.valid'.elims(3)
% 0.35/0.75  thf(fact_9051_atLeastLessThan__add__Un,axiom,
% 0.35/0.75      ! [I: nat,J: nat,K: nat] :
% 0.35/0.75        ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.75       => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
% 0.35/0.75          = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % atLeastLessThan_add_Un
% 0.35/0.75  thf(fact_9052_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 0.35/0.75      ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
% 0.35/0.75        ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg4 )
% 0.35/0.75        = ( ( Deg = Deg4 )
% 0.35/0.75          & ! [X3: vEBT_VEBT] :
% 0.35/0.75              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.75             => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.75          & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.75          & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.35/0.75            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.75          & ( case_o184042715313410164at_nat
% 0.35/0.75            @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
% 0.35/0.75              & ! [X3: vEBT_VEBT] :
% 0.35/0.75                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.75                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 0.35/0.75            @ ( produc6081775807080527818_nat_o
% 0.35/0.75              @ ^ [Mi2: nat,Ma2: nat] :
% 0.35/0.75                  ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.35/0.75                  & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.35/0.75                  & ! [I4: nat] :
% 0.35/0.75                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.75                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X6 ) )
% 0.35/0.75                        = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 0.35/0.75                  & ( ( Mi2 = Ma2 )
% 0.35/0.75                   => ! [X3: vEBT_VEBT] :
% 0.35/0.75                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.75                       => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 0.35/0.75                  & ( ( Mi2 != Ma2 )
% 0.35/0.75                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma2 )
% 0.35/0.75                      & ! [X3: nat] :
% 0.35/0.75                          ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.35/0.75                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X3 )
% 0.35/0.75                           => ( ( ord_less_nat @ Mi2 @ X3 )
% 0.35/0.75                              & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) )
% 0.35/0.75            @ Mima2 ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % VEBT_internal.valid'.simps(2)
% 0.35/0.75  thf(fact_9053_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 0.35/0.75      ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 0.35/0.75        ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
% 0.35/0.75          = Y )
% 0.35/0.75       => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.35/0.75                ( X
% 0.35/0.75                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.75           => ( Y
% 0.35/0.75              = ( Xa2 != one_one_nat ) ) )
% 0.35/0.75         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.75                ( ( X
% 0.35/0.75                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) )
% 0.35/0.75               => ( Y
% 0.35/0.75                  = ( ~ ( ( Deg2 = Xa2 )
% 0.35/0.75                        & ! [X3: vEBT_VEBT] :
% 0.35/0.75                            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.75                           => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.75                        & ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.75                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.35/0.75                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.75                        & ( case_o184042715313410164at_nat
% 0.35/0.75                          @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 0.35/0.75                            & ! [X3: vEBT_VEBT] :
% 0.35/0.75                                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.75                               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 0.35/0.75                          @ ( produc6081775807080527818_nat_o
% 0.35/0.75                            @ ^ [Mi2: nat,Ma2: nat] :
% 0.35/0.75                                ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.35/0.75                                & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.75                                & ! [I4: nat] :
% 0.35/0.75                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.75                                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 0.35/0.75                                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 0.35/0.75                                & ( ( Mi2 = Ma2 )
% 0.35/0.75                                 => ! [X3: vEBT_VEBT] :
% 0.35/0.75                                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.75                                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 0.35/0.75                                & ( ( Mi2 != Ma2 )
% 0.35/0.75                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma2 )
% 0.35/0.75                                    & ! [X3: nat] :
% 0.35/0.75                                        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.75                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 0.35/0.75                                         => ( ( ord_less_nat @ Mi2 @ X3 )
% 0.35/0.75                                            & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) )
% 0.35/0.75                          @ Mima ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % VEBT_internal.valid'.elims(1)
% 0.35/0.75  thf(fact_9054_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 0.35/0.75      ! [X: vEBT_VEBT,Xa2: nat] :
% 0.35/0.75        ( ( vEBT_VEBT_valid @ X @ Xa2 )
% 0.35/0.75       => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.35/0.75                ( X
% 0.35/0.75                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.75           => ( Xa2 != one_one_nat ) )
% 0.35/0.75         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.75                ( ( X
% 0.35/0.75                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) )
% 0.35/0.75               => ~ ( ( Deg2 = Xa2 )
% 0.35/0.75                    & ! [X2: vEBT_VEBT] :
% 0.35/0.75                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.75                       => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.75                    & ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.75                    & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.35/0.75                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.75                    & ( case_o184042715313410164at_nat
% 0.35/0.75                      @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 0.35/0.75                        & ! [X3: vEBT_VEBT] :
% 0.35/0.75                            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.75                           => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 0.35/0.75                      @ ( produc6081775807080527818_nat_o
% 0.35/0.75                        @ ^ [Mi2: nat,Ma2: nat] :
% 0.35/0.75                            ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.35/0.75                            & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.75                            & ! [I4: nat] :
% 0.35/0.75                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.75                               => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 0.35/0.75                                  = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 0.35/0.75                            & ( ( Mi2 = Ma2 )
% 0.35/0.75                             => ! [X3: vEBT_VEBT] :
% 0.35/0.75                                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.75                                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 0.35/0.75                            & ( ( Mi2 != Ma2 )
% 0.35/0.75                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma2 )
% 0.35/0.75                                & ! [X3: nat] :
% 0.35/0.75                                    ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.75                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 0.35/0.75                                     => ( ( ord_less_nat @ Mi2 @ X3 )
% 0.35/0.75                                        & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) )
% 0.35/0.75                      @ Mima ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % VEBT_internal.valid'.elims(2)
% 0.35/0.75  thf(fact_9055_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 0.35/0.75      ! [X: vEBT_VEBT,Xa2: nat] :
% 0.35/0.75        ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
% 0.35/0.75       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.35/0.75         => ( ! [Uu2: $o,Uv2: $o] :
% 0.35/0.75                ( ( X
% 0.35/0.75                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.75               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 0.35/0.75                 => ( Xa2 = one_one_nat ) ) )
% 0.35/0.75           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.75                  ( ( X
% 0.35/0.75                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) )
% 0.35/0.75                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) @ Xa2 ) )
% 0.35/0.75                   => ( ( Deg2 = Xa2 )
% 0.35/0.75                      & ! [X4: vEBT_VEBT] :
% 0.35/0.75                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.75                         => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.75                      & ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.75                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.35/0.75                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.75                      & ( case_o184042715313410164at_nat
% 0.35/0.75                        @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 0.35/0.75                          & ! [X3: vEBT_VEBT] :
% 0.35/0.75                              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.75                             => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 0.35/0.75                        @ ( produc6081775807080527818_nat_o
% 0.35/0.75                          @ ^ [Mi2: nat,Ma2: nat] :
% 0.35/0.75                              ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.35/0.75                              & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.75                              & ! [I4: nat] :
% 0.35/0.75                                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.75                                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 0.35/0.75                                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 0.35/0.75                              & ( ( Mi2 = Ma2 )
% 0.35/0.75                               => ! [X3: vEBT_VEBT] :
% 0.35/0.75                                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.75                                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 0.35/0.75                              & ( ( Mi2 != Ma2 )
% 0.35/0.75                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma2 )
% 0.35/0.75                                  & ! [X3: nat] :
% 0.35/0.75                                      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.75                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 0.35/0.75                                       => ( ( ord_less_nat @ Mi2 @ X3 )
% 0.35/0.75                                          & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) )
% 0.35/0.75                        @ Mima ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % VEBT_internal.valid'.pelims(3)
% 0.35/0.75  thf(fact_9056_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 0.35/0.75      ! [X: vEBT_VEBT,Xa2: nat] :
% 0.35/0.75        ( ( vEBT_VEBT_valid @ X @ Xa2 )
% 0.35/0.75       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.35/0.75         => ( ! [Uu2: $o,Uv2: $o] :
% 0.35/0.75                ( ( X
% 0.35/0.75                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.75               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 0.35/0.75                 => ( Xa2 != one_one_nat ) ) )
% 0.35/0.75           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.75                  ( ( X
% 0.35/0.75                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) )
% 0.35/0.75                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) @ Xa2 ) )
% 0.35/0.75                   => ~ ( ( Deg2 = Xa2 )
% 0.35/0.75                        & ! [X2: vEBT_VEBT] :
% 0.35/0.75                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.75                           => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.75                        & ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.75                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.35/0.75                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.75                        & ( case_o184042715313410164at_nat
% 0.35/0.75                          @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 0.35/0.75                            & ! [X3: vEBT_VEBT] :
% 0.35/0.75                                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.75                               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 0.35/0.75                          @ ( produc6081775807080527818_nat_o
% 0.35/0.75                            @ ^ [Mi2: nat,Ma2: nat] :
% 0.35/0.75                                ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.35/0.75                                & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.75                                & ! [I4: nat] :
% 0.35/0.75                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.75                                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 0.35/0.75                                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 0.35/0.75                                & ( ( Mi2 = Ma2 )
% 0.35/0.75                                 => ! [X3: vEBT_VEBT] :
% 0.35/0.75                                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.75                                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 0.35/0.75                                & ( ( Mi2 != Ma2 )
% 0.35/0.75                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma2 )
% 0.35/0.75                                    & ! [X3: nat] :
% 0.35/0.75                                        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.75                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 0.35/0.75                                         => ( ( ord_less_nat @ Mi2 @ X3 )
% 0.35/0.75                                            & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) )
% 0.35/0.75                          @ Mima ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % VEBT_internal.valid'.pelims(2)
% 0.35/0.75  thf(fact_9057_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 0.35/0.75      ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 0.35/0.75        ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
% 0.35/0.75          = Y )
% 0.35/0.75       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.35/0.75         => ( ! [Uu2: $o,Uv2: $o] :
% 0.35/0.75                ( ( X
% 0.35/0.75                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.75               => ( ( Y
% 0.35/0.75                    = ( Xa2 = one_one_nat ) )
% 0.35/0.75                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 0.35/0.75           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 0.35/0.75                  ( ( X
% 0.35/0.75                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) )
% 0.35/0.75                 => ( ( Y
% 0.35/0.75                      = ( ( Deg2 = Xa2 )
% 0.35/0.75                        & ! [X3: vEBT_VEBT] :
% 0.35/0.75                            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.75                           => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.75                        & ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.75                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.35/0.75                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.75                        & ( case_o184042715313410164at_nat
% 0.35/0.75                          @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 0.35/0.75                            & ! [X3: vEBT_VEBT] :
% 0.35/0.75                                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.75                               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 0.35/0.75                          @ ( produc6081775807080527818_nat_o
% 0.35/0.75                            @ ^ [Mi2: nat,Ma2: nat] :
% 0.35/0.75                                ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.35/0.75                                & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.75                                & ! [I4: nat] :
% 0.35/0.75                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.75                                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 0.35/0.75                                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 0.35/0.75                                & ( ( Mi2 = Ma2 )
% 0.35/0.75                                 => ! [X3: vEBT_VEBT] :
% 0.35/0.75                                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.75                                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 0.35/0.75                                & ( ( Mi2 != Ma2 )
% 0.35/0.75                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma2 )
% 0.35/0.75                                    & ! [X3: nat] :
% 0.35/0.75                                        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.75                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 0.35/0.75                                         => ( ( ord_less_nat @ Mi2 @ X3 )
% 0.35/0.75                                            & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) )
% 0.35/0.75                          @ Mima ) ) )
% 0.35/0.75                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % VEBT_internal.valid'.pelims(1)
% 0.35/0.75  thf(fact_9058_sup__nat__def,axiom,
% 0.35/0.75      sup_sup_nat = ord_max_nat ).
% 0.35/0.75  
% 0.35/0.75  % sup_nat_def
% 0.35/0.75  thf(fact_9059_min__weak__def,axiom,
% 0.35/0.75      ( fun_min_weak
% 0.35/0.75      = ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % min_weak_def
% 0.35/0.75  thf(fact_9060_max__weak__def,axiom,
% 0.35/0.75      ( fun_max_weak
% 0.35/0.75      = ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % max_weak_def
% 0.35/0.75  thf(fact_9061_take__bit__num__simps_I1_J,axiom,
% 0.35/0.75      ! [M: num] :
% 0.35/0.75        ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 0.35/0.75        = none_num ) ).
% 0.35/0.75  
% 0.35/0.75  % take_bit_num_simps(1)
% 0.35/0.75  thf(fact_9062_take__bit__num__simps_I2_J,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( ( bit_take_bit_num @ ( suc @ N ) @ one )
% 0.35/0.75        = ( some_num @ one ) ) ).
% 0.35/0.75  
% 0.35/0.75  % take_bit_num_simps(2)
% 0.35/0.75  thf(fact_9063_take__bit__num__simps_I5_J,axiom,
% 0.35/0.75      ! [R2: num] :
% 0.35/0.75        ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ one )
% 0.35/0.75        = ( some_num @ one ) ) ).
% 0.35/0.75  
% 0.35/0.75  % take_bit_num_simps(5)
% 0.35/0.75  thf(fact_9064_take__bit__num__simps_I3_J,axiom,
% 0.35/0.75      ! [N: nat,M: num] :
% 0.35/0.75        ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
% 0.35/0.75        = ( case_o6005452278849405969um_num @ none_num
% 0.35/0.75          @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 0.35/0.75          @ ( bit_take_bit_num @ N @ M ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % take_bit_num_simps(3)
% 0.35/0.75  thf(fact_9065_take__bit__num__simps_I4_J,axiom,
% 0.35/0.75      ! [N: nat,M: num] :
% 0.35/0.75        ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
% 0.35/0.75        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % take_bit_num_simps(4)
% 0.35/0.75  thf(fact_9066_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
% 0.35/0.75      ! [N: nat,M: num] :
% 0.35/0.75        ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
% 0.35/0.75        = ( case_nat_option_num @ none_num
% 0.35/0.75          @ ^ [N2: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) )
% 0.35/0.75          @ N ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Code_Abstract_Nat.take_bit_num_code(3)
% 0.35/0.75  thf(fact_9067_take__bit__num__simps_I6_J,axiom,
% 0.35/0.75      ! [R2: num,M: num] :
% 0.35/0.75        ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M ) )
% 0.35/0.75        = ( case_o6005452278849405969um_num @ none_num
% 0.35/0.75          @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 0.35/0.75          @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % take_bit_num_simps(6)
% 0.35/0.75  thf(fact_9068_take__bit__num__simps_I7_J,axiom,
% 0.35/0.75      ! [R2: num,M: num] :
% 0.35/0.75        ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit1 @ M ) )
% 0.35/0.75        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % take_bit_num_simps(7)
% 0.35/0.75  thf(fact_9069_pred__numeral__simps_I1_J,axiom,
% 0.35/0.75      ( ( pred_numeral @ one )
% 0.35/0.75      = zero_zero_nat ) ).
% 0.35/0.75  
% 0.35/0.75  % pred_numeral_simps(1)
% 0.35/0.75  thf(fact_9070_less__Suc__numeral,axiom,
% 0.35/0.75      ! [N: nat,K: num] :
% 0.35/0.75        ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.35/0.75        = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % less_Suc_numeral
% 0.35/0.75  thf(fact_9071_less__numeral__Suc,axiom,
% 0.35/0.75      ! [K: num,N: nat] :
% 0.35/0.75        ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.35/0.75        = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.35/0.75  
% 0.35/0.75  % less_numeral_Suc
% 0.35/0.75  thf(fact_9072_le__numeral__Suc,axiom,
% 0.35/0.75      ! [K: num,N: nat] :
% 0.35/0.75        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.35/0.75        = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.35/0.75  
% 0.35/0.75  % le_numeral_Suc
% 0.35/0.75  thf(fact_9073_le__Suc__numeral,axiom,
% 0.35/0.75      ! [N: nat,K: num] :
% 0.35/0.75        ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.35/0.75        = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % le_Suc_numeral
% 0.35/0.75  thf(fact_9074_atLeastLessThan__nat__numeral,axiom,
% 0.35/0.75      ! [M: nat,K: num] :
% 0.35/0.75        ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.35/0.75         => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.35/0.75            = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 0.35/0.75        & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.35/0.75         => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.35/0.75            = bot_bot_set_nat ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % atLeastLessThan_nat_numeral
% 0.35/0.75  thf(fact_9075_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( ( bit_take_bit_num @ N @ one )
% 0.35/0.75        = ( case_nat_option_num @ none_num
% 0.35/0.75          @ ^ [N2: nat] : ( some_num @ one )
% 0.35/0.75          @ N ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Code_Abstract_Nat.take_bit_num_code(1)
% 0.35/0.75  thf(fact_9076_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
% 0.35/0.75      ! [N: nat,M: num] :
% 0.35/0.75        ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
% 0.35/0.75        = ( case_nat_option_num @ none_num
% 0.35/0.75          @ ^ [N2: nat] :
% 0.35/0.75              ( case_o6005452278849405969um_num @ none_num
% 0.35/0.75              @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 0.35/0.75              @ ( bit_take_bit_num @ N2 @ M ) )
% 0.35/0.75          @ N ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Code_Abstract_Nat.take_bit_num_code(2)
% 0.35/0.75  thf(fact_9077_Bit__Operations_Otake__bit__num__code,axiom,
% 0.35/0.75      ( bit_take_bit_num
% 0.35/0.75      = ( ^ [N2: nat,M2: num] :
% 0.35/0.75            ( produc478579273971653890on_num
% 0.35/0.75            @ ^ [A4: nat,X3: num] :
% 0.35/0.75                ( case_nat_option_num @ none_num
% 0.35/0.75                @ ^ [O: nat] :
% 0.35/0.75                    ( case_num_option_num @ ( some_num @ one )
% 0.35/0.75                    @ ^ [P6: num] :
% 0.35/0.75                        ( case_o6005452278849405969um_num @ none_num
% 0.35/0.75                        @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 0.35/0.75                        @ ( bit_take_bit_num @ O @ P6 ) )
% 0.35/0.75                    @ ^ [P6: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) )
% 0.35/0.75                    @ X3 )
% 0.35/0.75                @ A4 )
% 0.35/0.75            @ ( product_Pair_nat_num @ N2 @ M2 ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Bit_Operations.take_bit_num_code
% 0.35/0.75  thf(fact_9078_take__bit__num__def,axiom,
% 0.35/0.75      ( bit_take_bit_num
% 0.35/0.75      = ( ^ [N2: nat,M2: num] :
% 0.35/0.75            ( if_option_num
% 0.35/0.75            @ ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M2 ) )
% 0.35/0.75              = zero_zero_nat )
% 0.35/0.75            @ none_num
% 0.35/0.75            @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M2 ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % take_bit_num_def
% 0.35/0.75  thf(fact_9079_nat_Odisc__eq__case_I1_J,axiom,
% 0.35/0.75      ! [Nat: nat] :
% 0.35/0.75        ( ( Nat = zero_zero_nat )
% 0.35/0.75        = ( case_nat_o @ $true
% 0.35/0.75          @ ^ [Uu3: nat] : $false
% 0.35/0.75          @ Nat ) ) ).
% 0.35/0.75  
% 0.35/0.75  % nat.disc_eq_case(1)
% 0.35/0.75  thf(fact_9080_nat_Odisc__eq__case_I2_J,axiom,
% 0.35/0.75      ! [Nat: nat] :
% 0.35/0.75        ( ( Nat != zero_zero_nat )
% 0.35/0.75        = ( case_nat_o @ $false
% 0.35/0.75          @ ^ [Uu3: nat] : $true
% 0.35/0.75          @ Nat ) ) ).
% 0.35/0.75  
% 0.35/0.75  % nat.disc_eq_case(2)
% 0.35/0.75  thf(fact_9081_num__of__nat_Osimps_I1_J,axiom,
% 0.35/0.75      ( ( num_of_nat @ zero_zero_nat )
% 0.35/0.75      = one ) ).
% 0.35/0.75  
% 0.35/0.75  % num_of_nat.simps(1)
% 0.35/0.75  thf(fact_9082_less__eq__nat_Osimps_I2_J,axiom,
% 0.35/0.75      ! [M: nat,N: nat] :
% 0.35/0.75        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.35/0.75        = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% 0.35/0.75  
% 0.35/0.75  % less_eq_nat.simps(2)
% 0.35/0.75  thf(fact_9083_max__Suc1,axiom,
% 0.35/0.75      ! [N: nat,M: nat] :
% 0.35/0.75        ( ( ord_max_nat @ ( suc @ N ) @ M )
% 0.35/0.75        = ( case_nat_nat @ ( suc @ N )
% 0.35/0.75          @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ N @ M6 ) )
% 0.35/0.75          @ M ) ) ).
% 0.35/0.75  
% 0.35/0.75  % max_Suc1
% 0.35/0.75  thf(fact_9084_max__Suc2,axiom,
% 0.35/0.75      ! [M: nat,N: nat] :
% 0.35/0.75        ( ( ord_max_nat @ M @ ( suc @ N ) )
% 0.35/0.75        = ( case_nat_nat @ ( suc @ N )
% 0.35/0.75          @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ M6 @ N ) )
% 0.35/0.75          @ M ) ) ).
% 0.35/0.75  
% 0.35/0.75  % max_Suc2
% 0.35/0.75  thf(fact_9085_numeral__num__of__nat,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.75       => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
% 0.35/0.75          = N ) ) ).
% 0.35/0.75  
% 0.35/0.75  % numeral_num_of_nat
% 0.35/0.75  thf(fact_9086_num__of__nat__plus__distrib,axiom,
% 0.35/0.75      ! [M: nat,N: nat] :
% 0.35/0.75        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.75       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.75         => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
% 0.35/0.75            = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % num_of_nat_plus_distrib
% 0.35/0.75  thf(fact_9087_num__of__nat__One,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( ( ord_less_eq_nat @ N @ one_one_nat )
% 0.35/0.75       => ( ( num_of_nat @ N )
% 0.35/0.75          = one ) ) ).
% 0.35/0.75  
% 0.35/0.75  % num_of_nat_One
% 0.35/0.75  thf(fact_9088_diff__Suc,axiom,
% 0.35/0.75      ! [M: nat,N: nat] :
% 0.35/0.75        ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 0.35/0.75        = ( case_nat_nat @ zero_zero_nat
% 0.35/0.75          @ ^ [K3: nat] : K3
% 0.35/0.75          @ ( minus_minus_nat @ M @ N ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % diff_Suc
% 0.35/0.75  thf(fact_9089_num__of__nat__double,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.75       => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
% 0.35/0.75          = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % num_of_nat_double
% 0.35/0.75  thf(fact_9090_pred__def,axiom,
% 0.35/0.75      ( pred
% 0.35/0.75      = ( case_nat_nat @ zero_zero_nat
% 0.35/0.75        @ ^ [X24: nat] : X24 ) ) ).
% 0.35/0.75  
% 0.35/0.75  % pred_def
% 0.35/0.75  thf(fact_9091_min__Suc__Suc,axiom,
% 0.35/0.75      ! [M: nat,N: nat] :
% 0.35/0.75        ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.35/0.75        = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % min_Suc_Suc
% 0.35/0.75  thf(fact_9092_min__0L,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( ( ord_min_nat @ zero_zero_nat @ N )
% 0.35/0.75        = zero_zero_nat ) ).
% 0.35/0.75  
% 0.35/0.75  % min_0L
% 0.35/0.75  thf(fact_9093_min__0R,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( ( ord_min_nat @ N @ zero_zero_nat )
% 0.35/0.75        = zero_zero_nat ) ).
% 0.35/0.75  
% 0.35/0.75  % min_0R
% 0.35/0.75  thf(fact_9094_min__diff,axiom,
% 0.35/0.75      ! [M: nat,I: nat,N: nat] :
% 0.35/0.75        ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
% 0.35/0.75        = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% 0.35/0.75  
% 0.35/0.75  % min_diff
% 0.35/0.75  thf(fact_9095_nat__mult__min__left,axiom,
% 0.35/0.75      ! [M: nat,N: nat,Q4: nat] :
% 0.35/0.75        ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q4 )
% 0.35/0.75        = ( ord_min_nat @ ( times_times_nat @ M @ Q4 ) @ ( times_times_nat @ N @ Q4 ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % nat_mult_min_left
% 0.35/0.75  thf(fact_9096_nat__mult__min__right,axiom,
% 0.35/0.75      ! [M: nat,N: nat,Q4: nat] :
% 0.35/0.75        ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q4 ) )
% 0.35/0.75        = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q4 ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % nat_mult_min_right
% 0.35/0.75  thf(fact_9097_inf__nat__def,axiom,
% 0.35/0.75      inf_inf_nat = ord_min_nat ).
% 0.35/0.75  
% 0.35/0.75  % inf_nat_def
% 0.35/0.75  thf(fact_9098_min__Suc2,axiom,
% 0.35/0.75      ! [M: nat,N: nat] :
% 0.35/0.75        ( ( ord_min_nat @ M @ ( suc @ N ) )
% 0.35/0.75        = ( case_nat_nat @ zero_zero_nat
% 0.35/0.75          @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ M6 @ N ) )
% 0.35/0.75          @ M ) ) ).
% 0.35/0.75  
% 0.35/0.75  % min_Suc2
% 0.35/0.75  thf(fact_9099_min__Suc1,axiom,
% 0.35/0.75      ! [N: nat,M: nat] :
% 0.35/0.75        ( ( ord_min_nat @ ( suc @ N ) @ M )
% 0.35/0.75        = ( case_nat_nat @ zero_zero_nat
% 0.35/0.75          @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ N @ M6 ) )
% 0.35/0.75          @ M ) ) ).
% 0.35/0.75  
% 0.35/0.75  % min_Suc1
% 0.35/0.75  thf(fact_9100_and__not__num_Osimps_I8_J,axiom,
% 0.35/0.75      ! [M: num,N: num] :
% 0.35/0.75        ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.35/0.75        = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.35/0.75          @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.35/0.75          @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % and_not_num.simps(8)
% 0.35/0.75  thf(fact_9101_and__not__num_Osimps_I1_J,axiom,
% 0.35/0.75      ( ( bit_and_not_num @ one @ one )
% 0.35/0.75      = none_num ) ).
% 0.35/0.75  
% 0.35/0.75  % and_not_num.simps(1)
% 0.35/0.75  thf(fact_9102_and__not__num_Osimps_I4_J,axiom,
% 0.35/0.75      ! [M: num] :
% 0.35/0.75        ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
% 0.35/0.75        = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % and_not_num.simps(4)
% 0.35/0.75  thf(fact_9103_and__not__num_Osimps_I2_J,axiom,
% 0.35/0.75      ! [N: num] :
% 0.35/0.75        ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
% 0.35/0.75        = ( some_num @ one ) ) ).
% 0.35/0.75  
% 0.35/0.75  % and_not_num.simps(2)
% 0.35/0.75  thf(fact_9104_and__not__num_Osimps_I3_J,axiom,
% 0.35/0.75      ! [N: num] :
% 0.35/0.75        ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
% 0.35/0.75        = none_num ) ).
% 0.35/0.75  
% 0.35/0.75  % and_not_num.simps(3)
% 0.35/0.75  thf(fact_9105_and__not__num_Osimps_I7_J,axiom,
% 0.35/0.75      ! [M: num] :
% 0.35/0.75        ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
% 0.35/0.75        = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % and_not_num.simps(7)
% 0.35/0.75  thf(fact_9106_and__not__num_Oelims,axiom,
% 0.35/0.75      ! [X: num,Xa2: num,Y: option_num] :
% 0.35/0.75        ( ( ( bit_and_not_num @ X @ Xa2 )
% 0.35/0.75          = Y )
% 0.35/0.75       => ( ( ( X = one )
% 0.35/0.75           => ( ( Xa2 = one )
% 0.35/0.75             => ( Y != none_num ) ) )
% 0.35/0.75         => ( ( ( X = one )
% 0.35/0.75             => ( ? [N3: num] :
% 0.35/0.75                    ( Xa2
% 0.35/0.75                    = ( bit0 @ N3 ) )
% 0.35/0.75               => ( Y
% 0.35/0.75                 != ( some_num @ one ) ) ) )
% 0.35/0.75           => ( ( ( X = one )
% 0.35/0.75               => ( ? [N3: num] :
% 0.35/0.75                      ( Xa2
% 0.35/0.75                      = ( bit1 @ N3 ) )
% 0.35/0.75                 => ( Y != none_num ) ) )
% 0.35/0.75             => ( ! [M5: num] :
% 0.35/0.75                    ( ( X
% 0.35/0.75                      = ( bit0 @ M5 ) )
% 0.35/0.75                   => ( ( Xa2 = one )
% 0.35/0.75                     => ( Y
% 0.35/0.75                       != ( some_num @ ( bit0 @ M5 ) ) ) ) )
% 0.35/0.75               => ( ! [M5: num] :
% 0.35/0.75                      ( ( X
% 0.35/0.75                        = ( bit0 @ M5 ) )
% 0.35/0.75                     => ! [N3: num] :
% 0.35/0.75                          ( ( Xa2
% 0.35/0.75                            = ( bit0 @ N3 ) )
% 0.35/0.75                         => ( Y
% 0.35/0.75                           != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
% 0.35/0.75                 => ( ! [M5: num] :
% 0.35/0.75                        ( ( X
% 0.35/0.75                          = ( bit0 @ M5 ) )
% 0.35/0.75                       => ! [N3: num] :
% 0.35/0.75                            ( ( Xa2
% 0.35/0.75                              = ( bit1 @ N3 ) )
% 0.35/0.75                           => ( Y
% 0.35/0.75                             != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
% 0.35/0.75                   => ( ! [M5: num] :
% 0.35/0.75                          ( ( X
% 0.35/0.75                            = ( bit1 @ M5 ) )
% 0.35/0.75                         => ( ( Xa2 = one )
% 0.35/0.75                           => ( Y
% 0.35/0.75                             != ( some_num @ ( bit0 @ M5 ) ) ) ) )
% 0.35/0.75                     => ( ! [M5: num] :
% 0.35/0.75                            ( ( X
% 0.35/0.75                              = ( bit1 @ M5 ) )
% 0.35/0.75                           => ! [N3: num] :
% 0.35/0.75                                ( ( Xa2
% 0.35/0.75                                  = ( bit0 @ N3 ) )
% 0.35/0.75                               => ( Y
% 0.35/0.75                                 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.35/0.75                                    @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.35/0.75                                    @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
% 0.35/0.75                       => ~ ! [M5: num] :
% 0.35/0.75                              ( ( X
% 0.35/0.75                                = ( bit1 @ M5 ) )
% 0.35/0.75                             => ! [N3: num] :
% 0.35/0.75                                  ( ( Xa2
% 0.35/0.75                                    = ( bit1 @ N3 ) )
% 0.35/0.75                                 => ( Y
% 0.35/0.75                                   != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % and_not_num.elims
% 0.35/0.75  thf(fact_9107_xor__num_Osimps_I8_J,axiom,
% 0.35/0.75      ! [M: num,N: num] :
% 0.35/0.75        ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.35/0.75        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % xor_num.simps(8)
% 0.35/0.75  thf(fact_9108_xor__num_Osimps_I1_J,axiom,
% 0.35/0.75      ( ( bit_un2480387367778600638or_num @ one @ one )
% 0.35/0.75      = none_num ) ).
% 0.35/0.75  
% 0.35/0.75  % xor_num.simps(1)
% 0.35/0.75  thf(fact_9109_xor__num_Oelims,axiom,
% 0.35/0.75      ! [X: num,Xa2: num,Y: option_num] :
% 0.35/0.75        ( ( ( bit_un2480387367778600638or_num @ X @ Xa2 )
% 0.35/0.75          = Y )
% 0.35/0.75       => ( ( ( X = one )
% 0.35/0.75           => ( ( Xa2 = one )
% 0.35/0.75             => ( Y != none_num ) ) )
% 0.35/0.75         => ( ( ( X = one )
% 0.35/0.75             => ! [N3: num] :
% 0.35/0.75                  ( ( Xa2
% 0.35/0.75                    = ( bit0 @ N3 ) )
% 0.35/0.75                 => ( Y
% 0.35/0.75                   != ( some_num @ ( bit1 @ N3 ) ) ) ) )
% 0.35/0.75           => ( ( ( X = one )
% 0.35/0.75               => ! [N3: num] :
% 0.35/0.75                    ( ( Xa2
% 0.35/0.75                      = ( bit1 @ N3 ) )
% 0.35/0.75                   => ( Y
% 0.35/0.75                     != ( some_num @ ( bit0 @ N3 ) ) ) ) )
% 0.35/0.75             => ( ! [M5: num] :
% 0.35/0.75                    ( ( X
% 0.35/0.75                      = ( bit0 @ M5 ) )
% 0.35/0.75                   => ( ( Xa2 = one )
% 0.35/0.75                     => ( Y
% 0.35/0.75                       != ( some_num @ ( bit1 @ M5 ) ) ) ) )
% 0.35/0.75               => ( ! [M5: num] :
% 0.35/0.75                      ( ( X
% 0.35/0.75                        = ( bit0 @ M5 ) )
% 0.35/0.75                     => ! [N3: num] :
% 0.35/0.75                          ( ( Xa2
% 0.35/0.75                            = ( bit0 @ N3 ) )
% 0.35/0.75                         => ( Y
% 0.35/0.75                           != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) )
% 0.35/0.75                 => ( ! [M5: num] :
% 0.35/0.75                        ( ( X
% 0.35/0.75                          = ( bit0 @ M5 ) )
% 0.35/0.75                       => ! [N3: num] :
% 0.35/0.75                            ( ( Xa2
% 0.35/0.75                              = ( bit1 @ N3 ) )
% 0.35/0.75                           => ( Y
% 0.35/0.75                             != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) )
% 0.35/0.75                   => ( ! [M5: num] :
% 0.35/0.75                          ( ( X
% 0.35/0.75                            = ( bit1 @ M5 ) )
% 0.35/0.75                         => ( ( Xa2 = one )
% 0.35/0.75                           => ( Y
% 0.35/0.75                             != ( some_num @ ( bit0 @ M5 ) ) ) ) )
% 0.35/0.75                     => ( ! [M5: num] :
% 0.35/0.75                            ( ( X
% 0.35/0.75                              = ( bit1 @ M5 ) )
% 0.35/0.75                           => ! [N3: num] :
% 0.35/0.75                                ( ( Xa2
% 0.35/0.75                                  = ( bit0 @ N3 ) )
% 0.35/0.75                               => ( Y
% 0.35/0.75                                 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) )
% 0.35/0.75                       => ~ ! [M5: num] :
% 0.35/0.75                              ( ( X
% 0.35/0.75                                = ( bit1 @ M5 ) )
% 0.35/0.75                             => ! [N3: num] :
% 0.35/0.75                                  ( ( Xa2
% 0.35/0.75                                    = ( bit1 @ N3 ) )
% 0.35/0.75                                 => ( Y
% 0.35/0.75                                   != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % xor_num.elims
% 0.35/0.75  thf(fact_9110_xor__num_Osimps_I2_J,axiom,
% 0.35/0.75      ! [N: num] :
% 0.35/0.75        ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
% 0.35/0.75        = ( some_num @ ( bit1 @ N ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % xor_num.simps(2)
% 0.35/0.75  thf(fact_9111_xor__num_Osimps_I3_J,axiom,
% 0.35/0.75      ! [N: num] :
% 0.35/0.75        ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
% 0.35/0.75        = ( some_num @ ( bit0 @ N ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % xor_num.simps(3)
% 0.35/0.75  thf(fact_9112_xor__num_Osimps_I4_J,axiom,
% 0.35/0.75      ! [M: num] :
% 0.35/0.75        ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
% 0.35/0.75        = ( some_num @ ( bit1 @ M ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % xor_num.simps(4)
% 0.35/0.75  thf(fact_9113_xor__num_Osimps_I7_J,axiom,
% 0.35/0.75      ! [M: num] :
% 0.35/0.75        ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
% 0.35/0.75        = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % xor_num.simps(7)
% 0.35/0.75  thf(fact_9114_xor__num_Osimps_I6_J,axiom,
% 0.35/0.75      ! [M: num,N: num] :
% 0.35/0.75        ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.35/0.75        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % xor_num.simps(6)
% 0.35/0.75  thf(fact_9115_and__num_Oelims,axiom,
% 0.35/0.75      ! [X: num,Xa2: num,Y: option_num] :
% 0.35/0.75        ( ( ( bit_un7362597486090784418nd_num @ X @ Xa2 )
% 0.35/0.75          = Y )
% 0.35/0.75       => ( ( ( X = one )
% 0.35/0.75           => ( ( Xa2 = one )
% 0.35/0.75             => ( Y
% 0.35/0.75               != ( some_num @ one ) ) ) )
% 0.35/0.75         => ( ( ( X = one )
% 0.35/0.75             => ( ? [N3: num] :
% 0.35/0.75                    ( Xa2
% 0.35/0.75                    = ( bit0 @ N3 ) )
% 0.35/0.75               => ( Y != none_num ) ) )
% 0.35/0.75           => ( ( ( X = one )
% 0.35/0.75               => ( ? [N3: num] :
% 0.35/0.75                      ( Xa2
% 0.35/0.75                      = ( bit1 @ N3 ) )
% 0.35/0.75                 => ( Y
% 0.35/0.75                   != ( some_num @ one ) ) ) )
% 0.35/0.75             => ( ( ? [M5: num] :
% 0.35/0.75                      ( X
% 0.35/0.75                      = ( bit0 @ M5 ) )
% 0.35/0.75                 => ( ( Xa2 = one )
% 0.35/0.75                   => ( Y != none_num ) ) )
% 0.35/0.75               => ( ! [M5: num] :
% 0.35/0.75                      ( ( X
% 0.35/0.75                        = ( bit0 @ M5 ) )
% 0.35/0.75                     => ! [N3: num] :
% 0.35/0.75                          ( ( Xa2
% 0.35/0.75                            = ( bit0 @ N3 ) )
% 0.35/0.75                         => ( Y
% 0.35/0.75                           != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
% 0.35/0.75                 => ( ! [M5: num] :
% 0.35/0.75                        ( ( X
% 0.35/0.75                          = ( bit0 @ M5 ) )
% 0.35/0.75                       => ! [N3: num] :
% 0.35/0.75                            ( ( Xa2
% 0.35/0.75                              = ( bit1 @ N3 ) )
% 0.35/0.75                           => ( Y
% 0.35/0.75                             != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
% 0.35/0.75                   => ( ( ? [M5: num] :
% 0.35/0.75                            ( X
% 0.35/0.75                            = ( bit1 @ M5 ) )
% 0.35/0.75                       => ( ( Xa2 = one )
% 0.35/0.75                         => ( Y
% 0.35/0.75                           != ( some_num @ one ) ) ) )
% 0.35/0.75                     => ( ! [M5: num] :
% 0.35/0.75                            ( ( X
% 0.35/0.75                              = ( bit1 @ M5 ) )
% 0.35/0.75                           => ! [N3: num] :
% 0.35/0.75                                ( ( Xa2
% 0.35/0.75                                  = ( bit0 @ N3 ) )
% 0.35/0.75                               => ( Y
% 0.35/0.75                                 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
% 0.35/0.75                       => ~ ! [M5: num] :
% 0.35/0.75                              ( ( X
% 0.35/0.75                                = ( bit1 @ M5 ) )
% 0.35/0.75                             => ! [N3: num] :
% 0.35/0.75                                  ( ( Xa2
% 0.35/0.75                                    = ( bit1 @ N3 ) )
% 0.35/0.75                                 => ( Y
% 0.35/0.75                                   != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.35/0.75                                      @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.35/0.75                                      @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % and_num.elims
% 0.35/0.75  thf(fact_9116_and__num_Osimps_I1_J,axiom,
% 0.35/0.75      ( ( bit_un7362597486090784418nd_num @ one @ one )
% 0.35/0.75      = ( some_num @ one ) ) ).
% 0.35/0.75  
% 0.35/0.75  % and_num.simps(1)
% 0.35/0.75  thf(fact_9117_and__num_Osimps_I7_J,axiom,
% 0.35/0.75      ! [M: num] :
% 0.35/0.75        ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
% 0.35/0.75        = ( some_num @ one ) ) ).
% 0.35/0.75  
% 0.35/0.75  % and_num.simps(7)
% 0.35/0.75  thf(fact_9118_and__num_Osimps_I3_J,axiom,
% 0.35/0.75      ! [N: num] :
% 0.35/0.75        ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
% 0.35/0.75        = ( some_num @ one ) ) ).
% 0.35/0.75  
% 0.35/0.75  % and_num.simps(3)
% 0.35/0.75  thf(fact_9119_and__num_Osimps_I4_J,axiom,
% 0.35/0.75      ! [M: num] :
% 0.35/0.75        ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
% 0.35/0.75        = none_num ) ).
% 0.35/0.75  
% 0.35/0.75  % and_num.simps(4)
% 0.35/0.75  thf(fact_9120_and__num_Osimps_I2_J,axiom,
% 0.35/0.75      ! [N: num] :
% 0.35/0.75        ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
% 0.35/0.75        = none_num ) ).
% 0.35/0.75  
% 0.35/0.75  % and_num.simps(2)
% 0.35/0.75  thf(fact_9121_and__num_Osimps_I9_J,axiom,
% 0.35/0.75      ! [M: num,N: num] :
% 0.35/0.75        ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.35/0.75        = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.35/0.75          @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.35/0.75          @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % and_num.simps(9)
% 0.35/0.75  thf(fact_9122_num__of__nat_Osimps_I2_J,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.75         => ( ( num_of_nat @ ( suc @ N ) )
% 0.35/0.75            = ( inc @ ( num_of_nat @ N ) ) ) )
% 0.35/0.75        & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.75         => ( ( num_of_nat @ ( suc @ N ) )
% 0.35/0.75            = one ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % num_of_nat.simps(2)
% 0.35/0.75  thf(fact_9123_Rats__eq__int__div__nat,axiom,
% 0.35/0.75      ( field_5140801741446780682s_real
% 0.35/0.75      = ( collect_real
% 0.35/0.75        @ ^ [Uu3: real] :
% 0.35/0.75          ? [I4: int,N2: nat] :
% 0.35/0.75            ( ( Uu3
% 0.35/0.75              = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.35/0.75            & ( N2 != zero_zero_nat ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Rats_eq_int_div_nat
% 0.35/0.75  thf(fact_9124_Rats__no__bot__less,axiom,
% 0.35/0.75      ! [X: real] :
% 0.35/0.75      ? [X4: real] :
% 0.35/0.75        ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 0.35/0.75        & ( ord_less_real @ X4 @ X ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Rats_no_bot_less
% 0.35/0.75  thf(fact_9125_Rats__dense__in__real,axiom,
% 0.35/0.75      ! [X: real,Y: real] :
% 0.35/0.75        ( ( ord_less_real @ X @ Y )
% 0.35/0.75       => ? [X4: real] :
% 0.35/0.75            ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 0.35/0.75            & ( ord_less_real @ X @ X4 )
% 0.35/0.75            & ( ord_less_real @ X4 @ Y ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Rats_dense_in_real
% 0.35/0.75  thf(fact_9126_pos__deriv__imp__strict__mono,axiom,
% 0.35/0.75      ! [F: real > real,F6: real > real] :
% 0.35/0.75        ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F6 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.35/0.75       => ( ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( F6 @ X4 ) )
% 0.35/0.75         => ( order_7092887310737990675l_real @ F ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % pos_deriv_imp_strict_mono
% 0.35/0.75  thf(fact_9127_positive__rat,axiom,
% 0.35/0.75      ! [A: int,B: int] :
% 0.35/0.75        ( ( positive @ ( fract @ A @ B ) )
% 0.35/0.75        = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % positive_rat
% 0.35/0.75  thf(fact_9128_strict__mono__imp__increasing,axiom,
% 0.35/0.75      ! [F: nat > nat,N: nat] :
% 0.35/0.75        ( ( order_5726023648592871131at_nat @ F )
% 0.35/0.75       => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % strict_mono_imp_increasing
% 0.35/0.75  thf(fact_9129_infinite__enumerate,axiom,
% 0.35/0.75      ! [S2: set_nat] :
% 0.35/0.75        ( ~ ( finite_finite_nat @ S2 )
% 0.35/0.75       => ? [R3: nat > nat] :
% 0.35/0.75            ( ( order_5726023648592871131at_nat @ R3 )
% 0.35/0.75            & ! [N5: nat] : ( member_nat @ ( R3 @ N5 ) @ S2 ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % infinite_enumerate
% 0.35/0.75  thf(fact_9130_less__rat__def,axiom,
% 0.35/0.75      ( ord_less_rat
% 0.35/0.75      = ( ^ [X3: rat,Y2: rat] : ( positive @ ( minus_minus_rat @ Y2 @ X3 ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % less_rat_def
% 0.35/0.75  thf(fact_9131_Rat_Opositive_Orep__eq,axiom,
% 0.35/0.75      ( positive
% 0.35/0.75      = ( ^ [X3: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X3 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X3 ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Rat.positive.rep_eq
% 0.35/0.75  thf(fact_9132_sorted__list__of__set__greaterThanAtMost,axiom,
% 0.35/0.75      ! [I: nat,J: nat] :
% 0.35/0.75        ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
% 0.35/0.75       => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
% 0.35/0.75          = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % sorted_list_of_set_greaterThanAtMost
% 0.35/0.75  thf(fact_9133_sorted__list__of__set__greaterThanLessThan,axiom,
% 0.35/0.75      ! [I: nat,J: nat] :
% 0.35/0.75        ( ( ord_less_nat @ ( suc @ I ) @ J )
% 0.35/0.75       => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
% 0.35/0.75          = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % sorted_list_of_set_greaterThanLessThan
% 0.35/0.75  thf(fact_9134_sorted__list__of__set__lessThan__Suc,axiom,
% 0.35/0.75      ! [K: nat] :
% 0.35/0.75        ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
% 0.35/0.75        = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % sorted_list_of_set_lessThan_Suc
% 0.35/0.75  thf(fact_9135_sorted__list__of__set__atMost__Suc,axiom,
% 0.35/0.75      ! [K: nat] :
% 0.35/0.75        ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
% 0.35/0.75        = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % sorted_list_of_set_atMost_Suc
% 0.35/0.75  thf(fact_9136_upto__aux__rec,axiom,
% 0.35/0.75      ( upto_aux
% 0.35/0.75      = ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upto_aux_rec
% 0.35/0.75  thf(fact_9137_upto_Opelims,axiom,
% 0.35/0.75      ! [X: int,Xa2: int,Y: list_int] :
% 0.35/0.75        ( ( ( upto @ X @ Xa2 )
% 0.35/0.75          = Y )
% 0.35/0.75       => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
% 0.35/0.75         => ~ ( ( ( ( ord_less_eq_int @ X @ Xa2 )
% 0.35/0.75                 => ( Y
% 0.35/0.75                    = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
% 0.35/0.75                & ( ~ ( ord_less_eq_int @ X @ Xa2 )
% 0.35/0.75                 => ( Y = nil_int ) ) )
% 0.35/0.75             => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upto.pelims
% 0.35/0.75  thf(fact_9138_upto_Opsimps,axiom,
% 0.35/0.75      ! [I: int,J: int] :
% 0.35/0.75        ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
% 0.35/0.75       => ( ( ( ord_less_eq_int @ I @ J )
% 0.35/0.75           => ( ( upto @ I @ J )
% 0.35/0.75              = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
% 0.35/0.75          & ( ~ ( ord_less_eq_int @ I @ J )
% 0.35/0.75           => ( ( upto @ I @ J )
% 0.35/0.75              = nil_int ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upto.psimps
% 0.35/0.75  thf(fact_9139_upto__empty,axiom,
% 0.35/0.75      ! [J: int,I: int] :
% 0.35/0.75        ( ( ord_less_int @ J @ I )
% 0.35/0.75       => ( ( upto @ I @ J )
% 0.35/0.75          = nil_int ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upto_empty
% 0.35/0.75  thf(fact_9140_upto__Nil2,axiom,
% 0.35/0.75      ! [I: int,J: int] :
% 0.35/0.75        ( ( nil_int
% 0.35/0.75          = ( upto @ I @ J ) )
% 0.35/0.75        = ( ord_less_int @ J @ I ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upto_Nil2
% 0.35/0.75  thf(fact_9141_upto__Nil,axiom,
% 0.35/0.75      ! [I: int,J: int] :
% 0.35/0.75        ( ( ( upto @ I @ J )
% 0.35/0.75          = nil_int )
% 0.35/0.75        = ( ord_less_int @ J @ I ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upto_Nil
% 0.35/0.75  thf(fact_9142_upto__single,axiom,
% 0.35/0.75      ! [I: int] :
% 0.35/0.75        ( ( upto @ I @ I )
% 0.35/0.75        = ( cons_int @ I @ nil_int ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upto_single
% 0.35/0.75  thf(fact_9143_nth__upto,axiom,
% 0.35/0.75      ! [I: int,K: nat,J: int] :
% 0.35/0.75        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 0.35/0.75       => ( ( nth_int @ ( upto @ I @ J ) @ K )
% 0.35/0.75          = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % nth_upto
% 0.35/0.75  thf(fact_9144_length__upto,axiom,
% 0.35/0.75      ! [I: int,J: int] :
% 0.35/0.75        ( ( size_size_list_int @ ( upto @ I @ J ) )
% 0.35/0.75        = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % length_upto
% 0.35/0.75  thf(fact_9145_upto__rec__numeral_I1_J,axiom,
% 0.35/0.75      ! [M: num,N: num] :
% 0.35/0.75        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.75         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.75            = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 0.35/0.75        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.75         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.75            = nil_int ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upto_rec_numeral(1)
% 0.35/0.75  thf(fact_9146_upto__rec__numeral_I2_J,axiom,
% 0.35/0.75      ! [M: num,N: num] :
% 0.35/0.75        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.75         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.75            = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 0.35/0.75        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.75         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.75            = nil_int ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upto_rec_numeral(2)
% 0.35/0.75  thf(fact_9147_upto__rec__numeral_I3_J,axiom,
% 0.35/0.75      ! [M: num,N: num] :
% 0.35/0.75        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.75         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.75            = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 0.35/0.75        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.75         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.75            = nil_int ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upto_rec_numeral(3)
% 0.35/0.75  thf(fact_9148_upto__rec__numeral_I4_J,axiom,
% 0.35/0.75      ! [M: num,N: num] :
% 0.35/0.75        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.75         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.75            = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 0.35/0.75        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.75         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.75            = nil_int ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upto_rec_numeral(4)
% 0.35/0.75  thf(fact_9149_upto__code,axiom,
% 0.35/0.75      ( upto
% 0.35/0.75      = ( ^ [I4: int,J3: int] : ( upto_aux @ I4 @ J3 @ nil_int ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upto_code
% 0.35/0.75  thf(fact_9150_upto__aux__def,axiom,
% 0.35/0.75      ( upto_aux
% 0.35/0.75      = ( ^ [I4: int,J3: int] : ( append_int @ ( upto @ I4 @ J3 ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upto_aux_def
% 0.35/0.75  thf(fact_9151_distinct__upto,axiom,
% 0.35/0.75      ! [I: int,J: int] : ( distinct_int @ ( upto @ I @ J ) ) ).
% 0.35/0.75  
% 0.35/0.75  % distinct_upto
% 0.35/0.75  thf(fact_9152_atLeastAtMost__upto,axiom,
% 0.35/0.75      ( set_or1266510415728281911st_int
% 0.35/0.75      = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ J3 ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % atLeastAtMost_upto
% 0.35/0.75  thf(fact_9153_upto__split2,axiom,
% 0.35/0.75      ! [I: int,J: int,K: int] :
% 0.35/0.75        ( ( ord_less_eq_int @ I @ J )
% 0.35/0.75       => ( ( ord_less_eq_int @ J @ K )
% 0.35/0.75         => ( ( upto @ I @ K )
% 0.35/0.75            = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upto_split2
% 0.35/0.75  thf(fact_9154_upto__split1,axiom,
% 0.35/0.75      ! [I: int,J: int,K: int] :
% 0.35/0.75        ( ( ord_less_eq_int @ I @ J )
% 0.35/0.75       => ( ( ord_less_eq_int @ J @ K )
% 0.35/0.75         => ( ( upto @ I @ K )
% 0.35/0.75            = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upto_split1
% 0.35/0.75  thf(fact_9155_atLeastLessThan__upto,axiom,
% 0.35/0.75      ( set_or4662586982721622107an_int
% 0.35/0.75      = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % atLeastLessThan_upto
% 0.35/0.75  thf(fact_9156_greaterThanAtMost__upto,axiom,
% 0.35/0.75      ( set_or6656581121297822940st_int
% 0.35/0.75      = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % greaterThanAtMost_upto
% 0.35/0.75  thf(fact_9157_upto_Oelims,axiom,
% 0.35/0.75      ! [X: int,Xa2: int,Y: list_int] :
% 0.35/0.75        ( ( ( upto @ X @ Xa2 )
% 0.35/0.75          = Y )
% 0.35/0.75       => ( ( ( ord_less_eq_int @ X @ Xa2 )
% 0.35/0.75           => ( Y
% 0.35/0.75              = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
% 0.35/0.75          & ( ~ ( ord_less_eq_int @ X @ Xa2 )
% 0.35/0.75           => ( Y = nil_int ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upto.elims
% 0.35/0.75  thf(fact_9158_upto_Osimps,axiom,
% 0.35/0.75      ( upto
% 0.35/0.75      = ( ^ [I4: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J3 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upto.simps
% 0.35/0.75  thf(fact_9159_upto__rec1,axiom,
% 0.35/0.75      ! [I: int,J: int] :
% 0.35/0.75        ( ( ord_less_eq_int @ I @ J )
% 0.35/0.75       => ( ( upto @ I @ J )
% 0.35/0.75          = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upto_rec1
% 0.35/0.75  thf(fact_9160_upto__rec2,axiom,
% 0.35/0.75      ! [I: int,J: int] :
% 0.35/0.75        ( ( ord_less_eq_int @ I @ J )
% 0.35/0.75       => ( ( upto @ I @ J )
% 0.35/0.75          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upto_rec2
% 0.35/0.75  thf(fact_9161_greaterThanLessThan__upto,axiom,
% 0.35/0.75      ( set_or5832277885323065728an_int
% 0.35/0.75      = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % greaterThanLessThan_upto
% 0.35/0.75  thf(fact_9162_upto__split3,axiom,
% 0.35/0.75      ! [I: int,J: int,K: int] :
% 0.35/0.75        ( ( ord_less_eq_int @ I @ J )
% 0.35/0.75       => ( ( ord_less_eq_int @ J @ K )
% 0.35/0.75         => ( ( upto @ I @ K )
% 0.35/0.75            = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upto_split3
% 0.35/0.75  thf(fact_9163_remdups__upt,axiom,
% 0.35/0.75      ! [M: nat,N: nat] :
% 0.35/0.75        ( ( remdups_nat @ ( upt @ M @ N ) )
% 0.35/0.75        = ( upt @ M @ N ) ) ).
% 0.35/0.75  
% 0.35/0.75  % remdups_upt
% 0.35/0.75  thf(fact_9164_hd__upt,axiom,
% 0.35/0.75      ! [I: nat,J: nat] :
% 0.35/0.75        ( ( ord_less_nat @ I @ J )
% 0.35/0.75       => ( ( hd_nat @ ( upt @ I @ J ) )
% 0.35/0.75          = I ) ) ).
% 0.35/0.75  
% 0.35/0.75  % hd_upt
% 0.35/0.75  thf(fact_9165_drop__upt,axiom,
% 0.35/0.75      ! [M: nat,I: nat,J: nat] :
% 0.35/0.75        ( ( drop_nat @ M @ ( upt @ I @ J ) )
% 0.35/0.75        = ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).
% 0.35/0.75  
% 0.35/0.75  % drop_upt
% 0.35/0.75  thf(fact_9166_length__upt,axiom,
% 0.35/0.75      ! [I: nat,J: nat] :
% 0.35/0.75        ( ( size_size_list_nat @ ( upt @ I @ J ) )
% 0.35/0.75        = ( minus_minus_nat @ J @ I ) ) ).
% 0.35/0.75  
% 0.35/0.75  % length_upt
% 0.35/0.75  thf(fact_9167_take__upt,axiom,
% 0.35/0.75      ! [I: nat,M: nat,N: nat] :
% 0.35/0.75        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
% 0.35/0.75       => ( ( take_nat @ M @ ( upt @ I @ N ) )
% 0.35/0.75          = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % take_upt
% 0.35/0.75  thf(fact_9168_upt__conv__Nil,axiom,
% 0.35/0.75      ! [J: nat,I: nat] :
% 0.35/0.75        ( ( ord_less_eq_nat @ J @ I )
% 0.35/0.75       => ( ( upt @ I @ J )
% 0.35/0.75          = nil_nat ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upt_conv_Nil
% 0.35/0.75  thf(fact_9169_sorted__list__of__set__range,axiom,
% 0.35/0.75      ! [M: nat,N: nat] :
% 0.35/0.75        ( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 0.35/0.75        = ( upt @ M @ N ) ) ).
% 0.35/0.75  
% 0.35/0.75  % sorted_list_of_set_range
% 0.35/0.75  thf(fact_9170_upt__eq__Nil__conv,axiom,
% 0.35/0.75      ! [I: nat,J: nat] :
% 0.35/0.75        ( ( ( upt @ I @ J )
% 0.35/0.75          = nil_nat )
% 0.35/0.75        = ( ( J = zero_zero_nat )
% 0.35/0.75          | ( ord_less_eq_nat @ J @ I ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upt_eq_Nil_conv
% 0.35/0.75  thf(fact_9171_nth__upt,axiom,
% 0.35/0.75      ! [I: nat,K: nat,J: nat] :
% 0.35/0.75        ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
% 0.35/0.75       => ( ( nth_nat @ ( upt @ I @ J ) @ K )
% 0.35/0.75          = ( plus_plus_nat @ I @ K ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % nth_upt
% 0.35/0.75  thf(fact_9172_upt__rec__numeral,axiom,
% 0.35/0.75      ! [M: num,N: num] :
% 0.35/0.75        ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.35/0.75         => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.35/0.75            = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
% 0.35/0.75        & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.35/0.75         => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.35/0.75            = nil_nat ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upt_rec_numeral
% 0.35/0.75  thf(fact_9173_upt__0,axiom,
% 0.35/0.75      ! [I: nat] :
% 0.35/0.75        ( ( upt @ I @ zero_zero_nat )
% 0.35/0.75        = nil_nat ) ).
% 0.35/0.75  
% 0.35/0.75  % upt_0
% 0.35/0.75  thf(fact_9174_greaterThanAtMost__upt,axiom,
% 0.35/0.75      ( set_or6659071591806873216st_nat
% 0.35/0.75      = ( ^ [N2: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ ( suc @ M2 ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % greaterThanAtMost_upt
% 0.35/0.75  thf(fact_9175_atLeast__upt,axiom,
% 0.35/0.75      ( set_ord_lessThan_nat
% 0.35/0.75      = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % atLeast_upt
% 0.35/0.75  thf(fact_9176_atLeastLessThan__upt,axiom,
% 0.35/0.75      ( set_or4665077453230672383an_nat
% 0.35/0.75      = ( ^ [I4: nat,J3: nat] : ( set_nat2 @ ( upt @ I4 @ J3 ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % atLeastLessThan_upt
% 0.35/0.75  thf(fact_9177_atLeastAtMost__upt,axiom,
% 0.35/0.75      ( set_or1269000886237332187st_nat
% 0.35/0.75      = ( ^ [N2: nat,M2: nat] : ( set_nat2 @ ( upt @ N2 @ ( suc @ M2 ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % atLeastAtMost_upt
% 0.35/0.75  thf(fact_9178_map__add__upt,axiom,
% 0.35/0.75      ! [N: nat,M: nat] :
% 0.35/0.75        ( ( map_nat_nat
% 0.35/0.75          @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
% 0.35/0.75          @ ( upt @ zero_zero_nat @ M ) )
% 0.35/0.75        = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % map_add_upt
% 0.35/0.75  thf(fact_9179_distinct__upt,axiom,
% 0.35/0.75      ! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).
% 0.35/0.75  
% 0.35/0.75  % distinct_upt
% 0.35/0.75  thf(fact_9180_map__Suc__upt,axiom,
% 0.35/0.75      ! [M: nat,N: nat] :
% 0.35/0.75        ( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
% 0.35/0.75        = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % map_Suc_upt
% 0.35/0.75  thf(fact_9181_greaterThanLessThan__upt,axiom,
% 0.35/0.75      ( set_or5834768355832116004an_nat
% 0.35/0.75      = ( ^ [N2: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ M2 ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % greaterThanLessThan_upt
% 0.35/0.75  thf(fact_9182_upt__conv__Cons__Cons,axiom,
% 0.35/0.75      ! [M: nat,N: nat,Ns: list_nat,Q4: nat] :
% 0.35/0.75        ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
% 0.35/0.75          = ( upt @ M @ Q4 ) )
% 0.35/0.75        = ( ( cons_nat @ N @ Ns )
% 0.35/0.75          = ( upt @ ( suc @ M ) @ Q4 ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upt_conv_Cons_Cons
% 0.35/0.75  thf(fact_9183_atMost__upto,axiom,
% 0.35/0.75      ( set_ord_atMost_nat
% 0.35/0.75      = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % atMost_upto
% 0.35/0.75  thf(fact_9184_upt__conv__Cons,axiom,
% 0.35/0.75      ! [I: nat,J: nat] :
% 0.35/0.75        ( ( ord_less_nat @ I @ J )
% 0.35/0.75       => ( ( upt @ I @ J )
% 0.35/0.75          = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upt_conv_Cons
% 0.35/0.75  thf(fact_9185_map__decr__upt,axiom,
% 0.35/0.75      ! [M: nat,N: nat] :
% 0.35/0.75        ( ( map_nat_nat
% 0.35/0.75          @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 0.35/0.75          @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.35/0.75        = ( upt @ M @ N ) ) ).
% 0.35/0.75  
% 0.35/0.75  % map_decr_upt
% 0.35/0.75  thf(fact_9186_upt__add__eq__append,axiom,
% 0.35/0.75      ! [I: nat,J: nat,K: nat] :
% 0.35/0.75        ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.75       => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
% 0.35/0.75          = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upt_add_eq_append
% 0.35/0.75  thf(fact_9187_upt__eq__Cons__conv,axiom,
% 0.35/0.75      ! [I: nat,J: nat,X: nat,Xs: list_nat] :
% 0.35/0.75        ( ( ( upt @ I @ J )
% 0.35/0.75          = ( cons_nat @ X @ Xs ) )
% 0.35/0.75        = ( ( ord_less_nat @ I @ J )
% 0.35/0.75          & ( I = X )
% 0.35/0.75          & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
% 0.35/0.75            = Xs ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upt_eq_Cons_conv
% 0.35/0.75  thf(fact_9188_upt__rec,axiom,
% 0.35/0.75      ( upt
% 0.35/0.75      = ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upt_rec
% 0.35/0.75  thf(fact_9189_upt__Suc,axiom,
% 0.35/0.75      ! [I: nat,J: nat] :
% 0.35/0.75        ( ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.75         => ( ( upt @ I @ ( suc @ J ) )
% 0.35/0.75            = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
% 0.35/0.75        & ( ~ ( ord_less_eq_nat @ I @ J )
% 0.35/0.75         => ( ( upt @ I @ ( suc @ J ) )
% 0.35/0.75            = nil_nat ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upt_Suc
% 0.35/0.75  thf(fact_9190_upt__Suc__append,axiom,
% 0.35/0.75      ! [I: nat,J: nat] :
% 0.35/0.75        ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.75       => ( ( upt @ I @ ( suc @ J ) )
% 0.35/0.75          = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % upt_Suc_append
% 0.35/0.75  thf(fact_9191_sum__list__upt,axiom,
% 0.35/0.75      ! [M: nat,N: nat] :
% 0.35/0.75        ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.75       => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
% 0.35/0.75          = ( groups3542108847815614940at_nat
% 0.35/0.75            @ ^ [X3: nat] : X3
% 0.35/0.75            @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % sum_list_upt
% 0.35/0.75  thf(fact_9192_card__length__sum__list__rec,axiom,
% 0.35/0.75      ! [M: nat,N6: nat] :
% 0.35/0.75        ( ( ord_less_eq_nat @ one_one_nat @ M )
% 0.35/0.75       => ( ( finite_card_list_nat
% 0.35/0.75            @ ( collect_list_nat
% 0.35/0.75              @ ^ [L2: list_nat] :
% 0.35/0.75                  ( ( ( size_size_list_nat @ L2 )
% 0.35/0.75                    = M )
% 0.35/0.75                  & ( ( groups4561878855575611511st_nat @ L2 )
% 0.35/0.75                    = N6 ) ) ) )
% 0.35/0.75          = ( plus_plus_nat
% 0.35/0.75            @ ( finite_card_list_nat
% 0.35/0.75              @ ( collect_list_nat
% 0.35/0.75                @ ^ [L2: list_nat] :
% 0.35/0.75                    ( ( ( size_size_list_nat @ L2 )
% 0.35/0.75                      = ( minus_minus_nat @ M @ one_one_nat ) )
% 0.35/0.75                    & ( ( groups4561878855575611511st_nat @ L2 )
% 0.35/0.75                      = N6 ) ) ) )
% 0.35/0.75            @ ( finite_card_list_nat
% 0.35/0.75              @ ( collect_list_nat
% 0.35/0.75                @ ^ [L2: list_nat] :
% 0.35/0.75                    ( ( ( size_size_list_nat @ L2 )
% 0.35/0.75                      = M )
% 0.35/0.75                    & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
% 0.35/0.75                      = N6 ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % card_length_sum_list_rec
% 0.35/0.75  thf(fact_9193_card__length__sum__list,axiom,
% 0.35/0.75      ! [M: nat,N6: nat] :
% 0.35/0.75        ( ( finite_card_list_nat
% 0.35/0.75          @ ( collect_list_nat
% 0.35/0.75            @ ^ [L2: list_nat] :
% 0.35/0.75                ( ( ( size_size_list_nat @ L2 )
% 0.35/0.75                  = M )
% 0.35/0.75                & ( ( groups4561878855575611511st_nat @ L2 )
% 0.35/0.75                  = N6 ) ) ) )
% 0.35/0.75        = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N6 @ M ) @ one_one_nat ) @ N6 ) ) ).
% 0.35/0.75  
% 0.35/0.75  % card_length_sum_list
% 0.35/0.75  thf(fact_9194_tl__upt,axiom,
% 0.35/0.75      ! [M: nat,N: nat] :
% 0.35/0.75        ( ( tl_nat @ ( upt @ M @ N ) )
% 0.35/0.75        = ( upt @ ( suc @ M ) @ N ) ) ).
% 0.35/0.75  
% 0.35/0.75  % tl_upt
% 0.35/0.75  thf(fact_9195_sorted__wrt__upt,axiom,
% 0.35/0.75      ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).
% 0.35/0.75  
% 0.35/0.75  % sorted_wrt_upt
% 0.35/0.75  thf(fact_9196_sorted__upt,axiom,
% 0.35/0.75      ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% 0.35/0.75  
% 0.35/0.75  % sorted_upt
% 0.35/0.75  thf(fact_9197_sorted__wrt__less__idx,axiom,
% 0.35/0.75      ! [Ns: list_nat,I: nat] :
% 0.35/0.75        ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
% 0.35/0.75       => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
% 0.35/0.75         => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % sorted_wrt_less_idx
% 0.35/0.75  thf(fact_9198_sorted__wrt__upto,axiom,
% 0.35/0.75      ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).
% 0.35/0.75  
% 0.35/0.75  % sorted_wrt_upto
% 0.35/0.75  thf(fact_9199_sorted__upto,axiom,
% 0.35/0.75      ! [M: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N ) ) ).
% 0.35/0.75  
% 0.35/0.75  % sorted_upto
% 0.35/0.75  thf(fact_9200_pairs__le__eq__Sigma,axiom,
% 0.35/0.75      ! [M: nat] :
% 0.35/0.75        ( ( collec3392354462482085612at_nat
% 0.35/0.75          @ ( produc6081775807080527818_nat_o
% 0.35/0.75            @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ M ) ) )
% 0.35/0.75        = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M )
% 0.35/0.75          @ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ R5 ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % pairs_le_eq_Sigma
% 0.35/0.75  thf(fact_9201_natLess__def,axiom,
% 0.35/0.75      ( bNF_Ca8459412986667044542atLess
% 0.35/0.75      = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % natLess_def
% 0.35/0.75  thf(fact_9202_finite__vimage__Suc__iff,axiom,
% 0.35/0.75      ! [F3: set_nat] :
% 0.35/0.75        ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F3 ) )
% 0.35/0.75        = ( finite_finite_nat @ F3 ) ) ).
% 0.35/0.75  
% 0.35/0.75  % finite_vimage_Suc_iff
% 0.35/0.75  thf(fact_9203_vimage__Suc__insert__0,axiom,
% 0.35/0.75      ! [A2: set_nat] :
% 0.35/0.75        ( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A2 ) )
% 0.35/0.75        = ( vimage_nat_nat @ suc @ A2 ) ) ).
% 0.35/0.75  
% 0.35/0.75  % vimage_Suc_insert_0
% 0.35/0.75  thf(fact_9204_Restr__natLeq,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
% 0.35/0.75          @ ( produc457027306803732586at_nat
% 0.35/0.75            @ ( collect_nat
% 0.35/0.75              @ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) )
% 0.35/0.75            @ ^ [Uu3: nat] :
% 0.35/0.75                ( collect_nat
% 0.35/0.75                @ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) ) ) )
% 0.35/0.75        = ( collec3392354462482085612at_nat
% 0.35/0.75          @ ( produc6081775807080527818_nat_o
% 0.35/0.75            @ ^ [X3: nat,Y2: nat] :
% 0.35/0.75                ( ( ord_less_nat @ X3 @ N )
% 0.35/0.75                & ( ord_less_nat @ Y2 @ N )
% 0.35/0.75                & ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Restr_natLeq
% 0.35/0.75  thf(fact_9205_Arg__bounded,axiom,
% 0.35/0.75      ! [Z2: complex] :
% 0.35/0.75        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z2 ) )
% 0.35/0.75        & ( ord_less_eq_real @ ( arg @ Z2 ) @ pi ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Arg_bounded
% 0.35/0.75  thf(fact_9206_natLeq__def,axiom,
% 0.35/0.75      ( bNF_Ca8665028551170535155natLeq
% 0.35/0.75      = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % natLeq_def
% 0.35/0.75  thf(fact_9207_Restr__natLeq2,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
% 0.35/0.75          @ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
% 0.35/0.75            @ ^ [Uu3: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
% 0.35/0.75        = ( collec3392354462482085612at_nat
% 0.35/0.75          @ ( produc6081775807080527818_nat_o
% 0.35/0.75            @ ^ [X3: nat,Y2: nat] :
% 0.35/0.75                ( ( ord_less_nat @ X3 @ N )
% 0.35/0.75                & ( ord_less_nat @ Y2 @ N )
% 0.35/0.75                & ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Restr_natLeq2
% 0.35/0.75  thf(fact_9208_Arg__correct,axiom,
% 0.35/0.75      ! [Z2: complex] :
% 0.35/0.75        ( ( Z2 != zero_zero_complex )
% 0.35/0.75       => ( ( ( sgn_sgn_complex @ Z2 )
% 0.35/0.75            = ( cis @ ( arg @ Z2 ) ) )
% 0.35/0.75          & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z2 ) )
% 0.35/0.75          & ( ord_less_eq_real @ ( arg @ Z2 ) @ pi ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Arg_correct
% 0.35/0.75  thf(fact_9209_natLeq__underS__less,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
% 0.35/0.75        = ( collect_nat
% 0.35/0.75          @ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % natLeq_underS_less
% 0.35/0.75  thf(fact_9210_cis__Arg__unique,axiom,
% 0.35/0.75      ! [Z2: complex,X: real] :
% 0.35/0.75        ( ( ( sgn_sgn_complex @ Z2 )
% 0.35/0.75          = ( cis @ X ) )
% 0.35/0.75       => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
% 0.35/0.75         => ( ( ord_less_eq_real @ X @ pi )
% 0.35/0.75           => ( ( arg @ Z2 )
% 0.35/0.75              = X ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % cis_Arg_unique
% 0.35/0.75  thf(fact_9211_bij__betw__roots__unity,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.75       => ( bij_betw_nat_complex
% 0.35/0.75          @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.35/0.75          @ ( set_ord_lessThan_nat @ N )
% 0.35/0.75          @ ( collect_complex
% 0.35/0.75            @ ^ [Z3: complex] :
% 0.35/0.75                ( ( power_power_complex @ Z3 @ N )
% 0.35/0.75                = one_one_complex ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % bij_betw_roots_unity
% 0.35/0.75  thf(fact_9212_bij__betw__nth__root__unity,axiom,
% 0.35/0.75      ! [C: complex,N: nat] :
% 0.35/0.75        ( ( C != zero_zero_complex )
% 0.35/0.75       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.75         => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 0.35/0.75            @ ( collect_complex
% 0.35/0.75              @ ^ [Z3: complex] :
% 0.35/0.75                  ( ( power_power_complex @ Z3 @ N )
% 0.35/0.75                  = one_one_complex ) )
% 0.35/0.75            @ ( collect_complex
% 0.35/0.75              @ ^ [Z3: complex] :
% 0.35/0.75                  ( ( power_power_complex @ Z3 @ N )
% 0.35/0.75                  = C ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % bij_betw_nth_root_unity
% 0.35/0.75  thf(fact_9213_Arg__def,axiom,
% 0.35/0.75      ( arg
% 0.35/0.75      = ( ^ [Z3: complex] :
% 0.35/0.75            ( if_real @ ( Z3 = zero_zero_complex ) @ zero_zero_real
% 0.35/0.75            @ ( fChoice_real
% 0.35/0.75              @ ^ [A4: real] :
% 0.35/0.75                  ( ( ( sgn_sgn_complex @ Z3 )
% 0.35/0.75                    = ( cis @ A4 ) )
% 0.35/0.75                  & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A4 )
% 0.35/0.75                  & ( ord_less_eq_real @ A4 @ pi ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Arg_def
% 0.35/0.75  thf(fact_9214_bij__betw__Suc,axiom,
% 0.35/0.75      ! [M4: set_nat,N6: set_nat] :
% 0.35/0.75        ( ( bij_betw_nat_nat @ suc @ M4 @ N6 )
% 0.35/0.75        = ( ( image_nat_nat @ suc @ M4 )
% 0.35/0.75          = N6 ) ) ).
% 0.35/0.75  
% 0.35/0.75  % bij_betw_Suc
% 0.35/0.75  thf(fact_9215_quotient__of__def,axiom,
% 0.35/0.75      ( quotient_of
% 0.35/0.75      = ( ^ [X3: rat] :
% 0.35/0.75            ( the_Pr4378521158711661632nt_int
% 0.35/0.75            @ ^ [Pair: product_prod_int_int] :
% 0.35/0.75                ( ( X3
% 0.35/0.75                  = ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
% 0.35/0.75                & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
% 0.35/0.75                & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % quotient_of_def
% 0.35/0.75  thf(fact_9216_normalize__stable,axiom,
% 0.35/0.75      ! [Q4: int,P4: int] :
% 0.35/0.75        ( ( ord_less_int @ zero_zero_int @ Q4 )
% 0.35/0.75       => ( ( algebr932160517623751201me_int @ P4 @ Q4 )
% 0.35/0.75         => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q4 ) )
% 0.35/0.75            = ( product_Pair_int_int @ P4 @ Q4 ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % normalize_stable
% 0.35/0.75  thf(fact_9217_normalize__coprime,axiom,
% 0.35/0.75      ! [R2: product_prod_int_int,P4: int,Q4: int] :
% 0.35/0.75        ( ( ( normalize @ R2 )
% 0.35/0.75          = ( product_Pair_int_int @ P4 @ Q4 ) )
% 0.35/0.75       => ( algebr932160517623751201me_int @ P4 @ Q4 ) ) ).
% 0.35/0.75  
% 0.35/0.75  % normalize_coprime
% 0.35/0.75  thf(fact_9218_quotient__of__coprime,axiom,
% 0.35/0.75      ! [R2: rat,P4: int,Q4: int] :
% 0.35/0.75        ( ( ( quotient_of @ R2 )
% 0.35/0.75          = ( product_Pair_int_int @ P4 @ Q4 ) )
% 0.35/0.75       => ( algebr932160517623751201me_int @ P4 @ Q4 ) ) ).
% 0.35/0.75  
% 0.35/0.75  % quotient_of_coprime
% 0.35/0.75  thf(fact_9219_Rat__induct,axiom,
% 0.35/0.75      ! [P: rat > $o,Q4: rat] :
% 0.35/0.75        ( ! [A3: int,B3: int] :
% 0.35/0.75            ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.75           => ( ( algebr932160517623751201me_int @ A3 @ B3 )
% 0.35/0.75             => ( P @ ( fract @ A3 @ B3 ) ) ) )
% 0.35/0.75       => ( P @ Q4 ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Rat_induct
% 0.35/0.75  thf(fact_9220_Rat__cases,axiom,
% 0.35/0.75      ! [Q4: rat] :
% 0.35/0.75        ~ ! [A3: int,B3: int] :
% 0.35/0.75            ( ( Q4
% 0.35/0.75              = ( fract @ A3 @ B3 ) )
% 0.35/0.75           => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.75             => ~ ( algebr932160517623751201me_int @ A3 @ B3 ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Rat_cases
% 0.35/0.75  thf(fact_9221_Rat__cases__nonzero,axiom,
% 0.35/0.75      ! [Q4: rat] :
% 0.35/0.75        ( ! [A3: int,B3: int] :
% 0.35/0.75            ( ( Q4
% 0.35/0.75              = ( fract @ A3 @ B3 ) )
% 0.35/0.75           => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.75             => ( ( A3 != zero_zero_int )
% 0.35/0.75               => ~ ( algebr932160517623751201me_int @ A3 @ B3 ) ) ) )
% 0.35/0.75       => ( Q4 = zero_zero_rat ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Rat_cases_nonzero
% 0.35/0.75  thf(fact_9222_quotient__of__unique,axiom,
% 0.35/0.75      ! [R2: rat] :
% 0.35/0.75      ? [X4: product_prod_int_int] :
% 0.35/0.75        ( ( R2
% 0.35/0.75          = ( fract @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) )
% 0.35/0.75        & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X4 ) )
% 0.35/0.75        & ( algebr932160517623751201me_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) )
% 0.35/0.75        & ! [Y4: product_prod_int_int] :
% 0.35/0.75            ( ( ( R2
% 0.35/0.75                = ( fract @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ Y4 ) ) )
% 0.35/0.75              & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y4 ) )
% 0.35/0.75              & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ Y4 ) ) )
% 0.35/0.75           => ( Y4 = X4 ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % quotient_of_unique
% 0.35/0.75  thf(fact_9223_coprime__Suc__0__right,axiom,
% 0.35/0.75      ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).
% 0.35/0.75  
% 0.35/0.75  % coprime_Suc_0_right
% 0.35/0.75  thf(fact_9224_coprime__Suc__0__left,axiom,
% 0.35/0.75      ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).
% 0.35/0.75  
% 0.35/0.75  % coprime_Suc_0_left
% 0.35/0.75  thf(fact_9225_coprime__diff__one__right__nat,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.75       => ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % coprime_diff_one_right_nat
% 0.35/0.75  thf(fact_9226_coprime__diff__one__left__nat,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.75       => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
% 0.35/0.75  
% 0.35/0.75  % coprime_diff_one_left_nat
% 0.35/0.75  thf(fact_9227_Gcd__nat__set__eq__fold,axiom,
% 0.35/0.75      ! [Xs: list_nat] :
% 0.35/0.75        ( ( gcd_Gcd_nat @ ( set_nat2 @ Xs ) )
% 0.35/0.75        = ( fold_nat_nat @ gcd_gcd_nat @ Xs @ zero_zero_nat ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Gcd_nat_set_eq_fold
% 0.35/0.75  thf(fact_9228_Rats__abs__nat__div__natE,axiom,
% 0.35/0.75      ! [X: real] :
% 0.35/0.75        ( ( member_real @ X @ field_5140801741446780682s_real )
% 0.35/0.75       => ~ ! [M5: nat,N3: nat] :
% 0.35/0.75              ( ( N3 != zero_zero_nat )
% 0.35/0.75             => ( ( ( abs_abs_real @ X )
% 0.35/0.75                  = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M5 ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
% 0.35/0.75               => ~ ( algebr934650988132801477me_nat @ M5 @ N3 ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Rats_abs_nat_div_natE
% 0.35/0.75  thf(fact_9229_sort__upt,axiom,
% 0.35/0.75      ! [M: nat,N: nat] :
% 0.35/0.75        ( ( linord738340561235409698at_nat
% 0.35/0.75          @ ^ [X3: nat] : X3
% 0.35/0.75          @ ( upt @ M @ N ) )
% 0.35/0.75        = ( upt @ M @ N ) ) ).
% 0.35/0.75  
% 0.35/0.75  % sort_upt
% 0.35/0.75  thf(fact_9230_sort__upto,axiom,
% 0.35/0.75      ! [I: int,J: int] :
% 0.35/0.75        ( ( linord1735203802627413978nt_int
% 0.35/0.75          @ ^ [X3: int] : X3
% 0.35/0.75          @ ( upto @ I @ J ) )
% 0.35/0.75        = ( upto @ I @ J ) ) ).
% 0.35/0.75  
% 0.35/0.75  % sort_upto
% 0.35/0.75  thf(fact_9231_vanishes__mult__bounded,axiom,
% 0.35/0.75      ! [X8: nat > rat,Y7: nat > rat] :
% 0.35/0.75        ( ? [A8: rat] :
% 0.35/0.75            ( ( ord_less_rat @ zero_zero_rat @ A8 )
% 0.35/0.75            & ! [N3: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N3 ) ) @ A8 ) )
% 0.35/0.75       => ( ( vanishes @ Y7 )
% 0.35/0.75         => ( vanishes
% 0.35/0.75            @ ^ [N2: nat] : ( times_times_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % vanishes_mult_bounded
% 0.35/0.75  thf(fact_9232_vanishes__def,axiom,
% 0.35/0.75      ( vanishes
% 0.35/0.75      = ( ^ [X6: nat > rat] :
% 0.35/0.75          ! [R5: rat] :
% 0.35/0.75            ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.75           => ? [K3: nat] :
% 0.35/0.75              ! [N2: nat] :
% 0.35/0.75                ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.75               => ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N2 ) ) @ R5 ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % vanishes_def
% 0.35/0.75  thf(fact_9233_vanishesI,axiom,
% 0.35/0.75      ! [X8: nat > rat] :
% 0.35/0.75        ( ! [R3: rat] :
% 0.35/0.75            ( ( ord_less_rat @ zero_zero_rat @ R3 )
% 0.35/0.75           => ? [K7: nat] :
% 0.35/0.75              ! [N3: nat] :
% 0.35/0.75                ( ( ord_less_eq_nat @ K7 @ N3 )
% 0.35/0.75               => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N3 ) ) @ R3 ) ) )
% 0.35/0.75       => ( vanishes @ X8 ) ) ).
% 0.35/0.75  
% 0.35/0.75  % vanishesI
% 0.35/0.75  thf(fact_9234_vanishesD,axiom,
% 0.35/0.75      ! [X8: nat > rat,R2: rat] :
% 0.35/0.75        ( ( vanishes @ X8 )
% 0.35/0.75       => ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 0.35/0.75         => ? [K2: nat] :
% 0.35/0.75            ! [N5: nat] :
% 0.35/0.75              ( ( ord_less_eq_nat @ K2 @ N5 )
% 0.35/0.75             => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N5 ) ) @ R2 ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % vanishesD
% 0.35/0.75  thf(fact_9235_and__not__num_Opelims,axiom,
% 0.35/0.75      ! [X: num,Xa2: num,Y: option_num] :
% 0.35/0.75        ( ( ( bit_and_not_num @ X @ Xa2 )
% 0.35/0.75          = Y )
% 0.35/0.75       => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X @ Xa2 ) )
% 0.35/0.75         => ( ( ( X = one )
% 0.35/0.75             => ( ( Xa2 = one )
% 0.35/0.75               => ( ( Y = none_num )
% 0.35/0.75                 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.35/0.75           => ( ( ( X = one )
% 0.35/0.75               => ! [N3: num] :
% 0.35/0.75                    ( ( Xa2
% 0.35/0.75                      = ( bit0 @ N3 ) )
% 0.35/0.75                   => ( ( Y
% 0.35/0.75                        = ( some_num @ one ) )
% 0.35/0.75                     => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
% 0.35/0.75             => ( ( ( X = one )
% 0.35/0.75                 => ! [N3: num] :
% 0.35/0.75                      ( ( Xa2
% 0.35/0.75                        = ( bit1 @ N3 ) )
% 0.35/0.75                     => ( ( Y = none_num )
% 0.35/0.75                       => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
% 0.35/0.75               => ( ! [M5: num] :
% 0.35/0.75                      ( ( X
% 0.35/0.75                        = ( bit0 @ M5 ) )
% 0.35/0.75                     => ( ( Xa2 = one )
% 0.35/0.75                       => ( ( Y
% 0.35/0.75                            = ( some_num @ ( bit0 @ M5 ) ) )
% 0.35/0.75                         => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
% 0.35/0.75                 => ( ! [M5: num] :
% 0.35/0.75                        ( ( X
% 0.35/0.75                          = ( bit0 @ M5 ) )
% 0.35/0.75                       => ! [N3: num] :
% 0.35/0.75                            ( ( Xa2
% 0.35/0.75                              = ( bit0 @ N3 ) )
% 0.35/0.75                           => ( ( Y
% 0.35/0.75                                = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
% 0.35/0.75                             => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 0.35/0.75                   => ( ! [M5: num] :
% 0.35/0.75                          ( ( X
% 0.35/0.75                            = ( bit0 @ M5 ) )
% 0.35/0.75                         => ! [N3: num] :
% 0.35/0.75                              ( ( Xa2
% 0.35/0.75                                = ( bit1 @ N3 ) )
% 0.35/0.75                             => ( ( Y
% 0.35/0.75                                  = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
% 0.35/0.75                               => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
% 0.35/0.75                     => ( ! [M5: num] :
% 0.35/0.75                            ( ( X
% 0.35/0.75                              = ( bit1 @ M5 ) )
% 0.35/0.75                           => ( ( Xa2 = one )
% 0.35/0.75                             => ( ( Y
% 0.35/0.75                                  = ( some_num @ ( bit0 @ M5 ) ) )
% 0.35/0.75                               => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
% 0.35/0.75                       => ( ! [M5: num] :
% 0.35/0.75                              ( ( X
% 0.35/0.75                                = ( bit1 @ M5 ) )
% 0.35/0.75                             => ! [N3: num] :
% 0.35/0.75                                  ( ( Xa2
% 0.35/0.75                                    = ( bit0 @ N3 ) )
% 0.35/0.75                                 => ( ( Y
% 0.35/0.75                                      = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.35/0.75                                        @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.35/0.75                                        @ ( bit_and_not_num @ M5 @ N3 ) ) )
% 0.35/0.75                                   => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 0.35/0.75                         => ~ ! [M5: num] :
% 0.35/0.75                                ( ( X
% 0.35/0.75                                  = ( bit1 @ M5 ) )
% 0.35/0.75                               => ! [N3: num] :
% 0.35/0.75                                    ( ( Xa2
% 0.35/0.75                                      = ( bit1 @ N3 ) )
% 0.35/0.75                                   => ( ( Y
% 0.35/0.75                                        = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
% 0.35/0.75                                     => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % and_not_num.pelims
% 0.35/0.75  thf(fact_9236_and__num_Opelims,axiom,
% 0.35/0.75      ! [X: num,Xa2: num,Y: option_num] :
% 0.35/0.75        ( ( ( bit_un7362597486090784418nd_num @ X @ Xa2 )
% 0.35/0.75          = Y )
% 0.35/0.75       => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X @ Xa2 ) )
% 0.35/0.75         => ( ( ( X = one )
% 0.35/0.75             => ( ( Xa2 = one )
% 0.35/0.75               => ( ( Y
% 0.35/0.75                    = ( some_num @ one ) )
% 0.35/0.75                 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.35/0.75           => ( ( ( X = one )
% 0.35/0.75               => ! [N3: num] :
% 0.35/0.75                    ( ( Xa2
% 0.35/0.75                      = ( bit0 @ N3 ) )
% 0.35/0.75                   => ( ( Y = none_num )
% 0.35/0.75                     => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
% 0.35/0.75             => ( ( ( X = one )
% 0.35/0.75                 => ! [N3: num] :
% 0.35/0.75                      ( ( Xa2
% 0.35/0.75                        = ( bit1 @ N3 ) )
% 0.35/0.75                     => ( ( Y
% 0.35/0.75                          = ( some_num @ one ) )
% 0.35/0.75                       => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
% 0.35/0.75               => ( ! [M5: num] :
% 0.35/0.75                      ( ( X
% 0.35/0.75                        = ( bit0 @ M5 ) )
% 0.35/0.75                     => ( ( Xa2 = one )
% 0.35/0.75                       => ( ( Y = none_num )
% 0.35/0.75                         => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
% 0.35/0.75                 => ( ! [M5: num] :
% 0.35/0.75                        ( ( X
% 0.35/0.75                          = ( bit0 @ M5 ) )
% 0.35/0.75                       => ! [N3: num] :
% 0.35/0.75                            ( ( Xa2
% 0.35/0.75                              = ( bit0 @ N3 ) )
% 0.35/0.75                           => ( ( Y
% 0.35/0.75                                = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
% 0.35/0.75                             => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 0.35/0.75                   => ( ! [M5: num] :
% 0.35/0.75                          ( ( X
% 0.35/0.75                            = ( bit0 @ M5 ) )
% 0.35/0.75                         => ! [N3: num] :
% 0.35/0.75                              ( ( Xa2
% 0.35/0.75                                = ( bit1 @ N3 ) )
% 0.35/0.75                             => ( ( Y
% 0.35/0.75                                  = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
% 0.35/0.75                               => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
% 0.35/0.75                     => ( ! [M5: num] :
% 0.35/0.75                            ( ( X
% 0.35/0.75                              = ( bit1 @ M5 ) )
% 0.35/0.75                           => ( ( Xa2 = one )
% 0.35/0.75                             => ( ( Y
% 0.35/0.75                                  = ( some_num @ one ) )
% 0.35/0.75                               => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
% 0.35/0.75                       => ( ! [M5: num] :
% 0.35/0.75                              ( ( X
% 0.35/0.75                                = ( bit1 @ M5 ) )
% 0.35/0.75                             => ! [N3: num] :
% 0.35/0.75                                  ( ( Xa2
% 0.35/0.75                                    = ( bit0 @ N3 ) )
% 0.35/0.75                                 => ( ( Y
% 0.35/0.75                                      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
% 0.35/0.75                                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 0.35/0.75                         => ~ ! [M5: num] :
% 0.35/0.75                                ( ( X
% 0.35/0.75                                  = ( bit1 @ M5 ) )
% 0.35/0.75                               => ! [N3: num] :
% 0.35/0.75                                    ( ( Xa2
% 0.35/0.75                                      = ( bit1 @ N3 ) )
% 0.35/0.75                                   => ( ( Y
% 0.35/0.75                                        = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.35/0.75                                          @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.35/0.75                                          @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
% 0.35/0.75                                     => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % and_num.pelims
% 0.35/0.75  thf(fact_9237_xor__num_Opelims,axiom,
% 0.35/0.75      ! [X: num,Xa2: num,Y: option_num] :
% 0.35/0.75        ( ( ( bit_un2480387367778600638or_num @ X @ Xa2 )
% 0.35/0.75          = Y )
% 0.35/0.75       => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X @ Xa2 ) )
% 0.35/0.75         => ( ( ( X = one )
% 0.35/0.75             => ( ( Xa2 = one )
% 0.35/0.75               => ( ( Y = none_num )
% 0.35/0.75                 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.35/0.75           => ( ( ( X = one )
% 0.35/0.75               => ! [N3: num] :
% 0.35/0.75                    ( ( Xa2
% 0.35/0.75                      = ( bit0 @ N3 ) )
% 0.35/0.75                   => ( ( Y
% 0.35/0.75                        = ( some_num @ ( bit1 @ N3 ) ) )
% 0.35/0.75                     => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
% 0.35/0.75             => ( ( ( X = one )
% 0.35/0.75                 => ! [N3: num] :
% 0.35/0.75                      ( ( Xa2
% 0.35/0.75                        = ( bit1 @ N3 ) )
% 0.35/0.75                     => ( ( Y
% 0.35/0.75                          = ( some_num @ ( bit0 @ N3 ) ) )
% 0.35/0.75                       => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
% 0.35/0.75               => ( ! [M5: num] :
% 0.35/0.75                      ( ( X
% 0.35/0.75                        = ( bit0 @ M5 ) )
% 0.35/0.75                     => ( ( Xa2 = one )
% 0.35/0.75                       => ( ( Y
% 0.35/0.75                            = ( some_num @ ( bit1 @ M5 ) ) )
% 0.35/0.75                         => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
% 0.35/0.75                 => ( ! [M5: num] :
% 0.35/0.75                        ( ( X
% 0.35/0.75                          = ( bit0 @ M5 ) )
% 0.35/0.75                       => ! [N3: num] :
% 0.35/0.75                            ( ( Xa2
% 0.35/0.75                              = ( bit0 @ N3 ) )
% 0.35/0.75                           => ( ( Y
% 0.35/0.75                                = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) )
% 0.35/0.75                             => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 0.35/0.75                   => ( ! [M5: num] :
% 0.35/0.75                          ( ( X
% 0.35/0.75                            = ( bit0 @ M5 ) )
% 0.35/0.75                         => ! [N3: num] :
% 0.35/0.75                              ( ( Xa2
% 0.35/0.75                                = ( bit1 @ N3 ) )
% 0.35/0.75                             => ( ( Y
% 0.35/0.75                                  = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) )
% 0.35/0.75                               => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
% 0.35/0.75                     => ( ! [M5: num] :
% 0.35/0.75                            ( ( X
% 0.35/0.75                              = ( bit1 @ M5 ) )
% 0.35/0.75                           => ( ( Xa2 = one )
% 0.35/0.75                             => ( ( Y
% 0.35/0.75                                  = ( some_num @ ( bit0 @ M5 ) ) )
% 0.35/0.75                               => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
% 0.35/0.75                       => ( ! [M5: num] :
% 0.35/0.75                              ( ( X
% 0.35/0.75                                = ( bit1 @ M5 ) )
% 0.35/0.75                             => ! [N3: num] :
% 0.35/0.75                                  ( ( Xa2
% 0.35/0.75                                    = ( bit0 @ N3 ) )
% 0.35/0.75                                 => ( ( Y
% 0.35/0.75                                      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) )
% 0.35/0.75                                   => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 0.35/0.75                         => ~ ! [M5: num] :
% 0.35/0.75                                ( ( X
% 0.35/0.75                                  = ( bit1 @ M5 ) )
% 0.35/0.75                               => ! [N3: num] :
% 0.35/0.75                                    ( ( Xa2
% 0.35/0.75                                      = ( bit1 @ N3 ) )
% 0.35/0.75                                   => ( ( Y
% 0.35/0.75                                        = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) )
% 0.35/0.75                                     => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % xor_num.pelims
% 0.35/0.75  thf(fact_9238_Field__natLeq__on,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( ( field_nat
% 0.35/0.75          @ ( collec3392354462482085612at_nat
% 0.35/0.75            @ ( produc6081775807080527818_nat_o
% 0.35/0.75              @ ^ [X3: nat,Y2: nat] :
% 0.35/0.75                  ( ( ord_less_nat @ X3 @ N )
% 0.35/0.75                  & ( ord_less_nat @ Y2 @ N )
% 0.35/0.75                  & ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) )
% 0.35/0.75        = ( collect_nat
% 0.35/0.75          @ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Field_natLeq_on
% 0.35/0.75  thf(fact_9239_wf__less,axiom,
% 0.35/0.75      wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).
% 0.35/0.75  
% 0.35/0.75  % wf_less
% 0.35/0.75  thf(fact_9240_cauchyD,axiom,
% 0.35/0.75      ! [X8: nat > rat,R2: rat] :
% 0.35/0.75        ( ( cauchy @ X8 )
% 0.35/0.75       => ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 0.35/0.75         => ? [K2: nat] :
% 0.35/0.75            ! [M3: nat] :
% 0.35/0.75              ( ( ord_less_eq_nat @ K2 @ M3 )
% 0.35/0.75             => ! [N5: nat] :
% 0.35/0.75                  ( ( ord_less_eq_nat @ K2 @ N5 )
% 0.35/0.75                 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M3 ) @ ( X8 @ N5 ) ) ) @ R2 ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % cauchyD
% 0.35/0.75  thf(fact_9241_cauchy__imp__bounded,axiom,
% 0.35/0.75      ! [X8: nat > rat] :
% 0.35/0.75        ( ( cauchy @ X8 )
% 0.35/0.75       => ? [B3: rat] :
% 0.35/0.75            ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 0.35/0.75            & ! [N5: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N5 ) ) @ B3 ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % cauchy_imp_bounded
% 0.35/0.75  thf(fact_9242_cauchy__not__vanishes__cases,axiom,
% 0.35/0.75      ! [X8: nat > rat] :
% 0.35/0.75        ( ( cauchy @ X8 )
% 0.35/0.75       => ( ~ ( vanishes @ X8 )
% 0.35/0.75         => ? [B3: rat] :
% 0.35/0.75              ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 0.35/0.75              & ? [K2: nat] :
% 0.35/0.75                  ( ! [N5: nat] :
% 0.35/0.75                      ( ( ord_less_eq_nat @ K2 @ N5 )
% 0.35/0.75                     => ( ord_less_rat @ B3 @ ( uminus_uminus_rat @ ( X8 @ N5 ) ) ) )
% 0.35/0.75                  | ! [N5: nat] :
% 0.35/0.75                      ( ( ord_less_eq_nat @ K2 @ N5 )
% 0.35/0.75                     => ( ord_less_rat @ B3 @ ( X8 @ N5 ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % cauchy_not_vanishes_cases
% 0.35/0.75  thf(fact_9243_cauchy__not__vanishes,axiom,
% 0.35/0.75      ! [X8: nat > rat] :
% 0.35/0.75        ( ( cauchy @ X8 )
% 0.35/0.75       => ( ~ ( vanishes @ X8 )
% 0.35/0.75         => ? [B3: rat] :
% 0.35/0.75              ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 0.35/0.75              & ? [K2: nat] :
% 0.35/0.75                ! [N5: nat] :
% 0.35/0.75                  ( ( ord_less_eq_nat @ K2 @ N5 )
% 0.35/0.75                 => ( ord_less_rat @ B3 @ ( abs_abs_rat @ ( X8 @ N5 ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % cauchy_not_vanishes
% 0.35/0.75  thf(fact_9244_cauchy__def,axiom,
% 0.35/0.75      ( cauchy
% 0.35/0.75      = ( ^ [X6: nat > rat] :
% 0.35/0.75          ! [R5: rat] :
% 0.35/0.75            ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.75           => ? [K3: nat] :
% 0.35/0.75              ! [M2: nat] :
% 0.35/0.75                ( ( ord_less_eq_nat @ K3 @ M2 )
% 0.35/0.75               => ! [N2: nat] :
% 0.35/0.75                    ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.75                   => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) ) @ R5 ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % cauchy_def
% 0.35/0.75  thf(fact_9245_cauchyI,axiom,
% 0.35/0.75      ! [X8: nat > rat] :
% 0.35/0.75        ( ! [R3: rat] :
% 0.35/0.75            ( ( ord_less_rat @ zero_zero_rat @ R3 )
% 0.35/0.75           => ? [K7: nat] :
% 0.35/0.75              ! [M5: nat] :
% 0.35/0.75                ( ( ord_less_eq_nat @ K7 @ M5 )
% 0.35/0.75               => ! [N3: nat] :
% 0.35/0.75                    ( ( ord_less_eq_nat @ K7 @ N3 )
% 0.35/0.75                   => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) ) @ R3 ) ) ) )
% 0.35/0.75       => ( cauchy @ X8 ) ) ).
% 0.35/0.75  
% 0.35/0.75  % cauchyI
% 0.35/0.75  thf(fact_9246_le__Real,axiom,
% 0.35/0.75      ! [X8: nat > rat,Y7: nat > rat] :
% 0.35/0.75        ( ( cauchy @ X8 )
% 0.35/0.75       => ( ( cauchy @ Y7 )
% 0.35/0.75         => ( ( ord_less_eq_real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
% 0.35/0.75            = ( ! [R5: rat] :
% 0.35/0.75                  ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.75                 => ? [K3: nat] :
% 0.35/0.75                    ! [N2: nat] :
% 0.35/0.75                      ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.75                     => ( ord_less_eq_rat @ ( X8 @ N2 ) @ ( plus_plus_rat @ ( Y7 @ N2 ) @ R5 ) ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % le_Real
% 0.35/0.75  thf(fact_9247_not__positive__Real,axiom,
% 0.35/0.75      ! [X8: nat > rat] :
% 0.35/0.75        ( ( cauchy @ X8 )
% 0.35/0.75       => ( ( ~ ( positive2 @ ( real2 @ X8 ) ) )
% 0.35/0.75          = ( ! [R5: rat] :
% 0.35/0.75                ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.75               => ? [K3: nat] :
% 0.35/0.75                  ! [N2: nat] :
% 0.35/0.75                    ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.75                   => ( ord_less_eq_rat @ ( X8 @ N2 ) @ R5 ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % not_positive_Real
% 0.35/0.75  thf(fact_9248_less__real__def,axiom,
% 0.35/0.75      ( ord_less_real
% 0.35/0.75      = ( ^ [X3: real,Y2: real] : ( positive2 @ ( minus_minus_real @ Y2 @ X3 ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % less_real_def
% 0.35/0.75  thf(fact_9249_positive__Real,axiom,
% 0.35/0.75      ! [X8: nat > rat] :
% 0.35/0.75        ( ( cauchy @ X8 )
% 0.35/0.75       => ( ( positive2 @ ( real2 @ X8 ) )
% 0.35/0.75          = ( ? [R5: rat] :
% 0.35/0.75                ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.75                & ? [K3: nat] :
% 0.35/0.75                  ! [N2: nat] :
% 0.35/0.75                    ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.75                   => ( ord_less_rat @ R5 @ ( X8 @ N2 ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % positive_Real
% 0.35/0.75  thf(fact_9250_Real_Opositive_Orep__eq,axiom,
% 0.35/0.75      ( positive2
% 0.35/0.75      = ( ^ [X3: real] :
% 0.35/0.75          ? [R5: rat] :
% 0.35/0.75            ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.75            & ? [K3: nat] :
% 0.35/0.75              ! [N2: nat] :
% 0.35/0.75                ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.75               => ( ord_less_rat @ R5 @ ( rep_real @ X3 @ N2 ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Real.positive.rep_eq
% 0.35/0.75  thf(fact_9251_strict__mono__enumerate,axiom,
% 0.35/0.75      ! [S2: set_nat] :
% 0.35/0.75        ( ~ ( finite_finite_nat @ S2 )
% 0.35/0.75       => ( order_5726023648592871131at_nat @ ( infini8530281810654367211te_nat @ S2 ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % strict_mono_enumerate
% 0.35/0.75  thf(fact_9252_enumerate__Ex,axiom,
% 0.35/0.75      ! [S2: set_nat,S: nat] :
% 0.35/0.75        ( ~ ( finite_finite_nat @ S2 )
% 0.35/0.75       => ( ( member_nat @ S @ S2 )
% 0.35/0.75         => ? [N3: nat] :
% 0.35/0.75              ( ( infini8530281810654367211te_nat @ S2 @ N3 )
% 0.35/0.75              = S ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % enumerate_Ex
% 0.35/0.75  thf(fact_9253_le__enumerate,axiom,
% 0.35/0.75      ! [S2: set_nat,N: nat] :
% 0.35/0.75        ( ~ ( finite_finite_nat @ S2 )
% 0.35/0.75       => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % le_enumerate
% 0.35/0.75  thf(fact_9254_range__enumerate,axiom,
% 0.35/0.75      ! [S2: set_nat] :
% 0.35/0.75        ( ~ ( finite_finite_nat @ S2 )
% 0.35/0.75       => ( ( image_nat_nat @ ( infini8530281810654367211te_nat @ S2 ) @ top_top_set_nat )
% 0.35/0.75          = S2 ) ) ).
% 0.35/0.75  
% 0.35/0.75  % range_enumerate
% 0.35/0.75  thf(fact_9255_finite__le__enumerate,axiom,
% 0.35/0.75      ! [S2: set_nat,N: nat] :
% 0.35/0.75        ( ( finite_finite_nat @ S2 )
% 0.35/0.75       => ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
% 0.35/0.75         => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % finite_le_enumerate
% 0.35/0.75  thf(fact_9256_bij__enumerate,axiom,
% 0.35/0.75      ! [S2: set_nat] :
% 0.35/0.75        ( ~ ( finite_finite_nat @ S2 )
% 0.35/0.75       => ( bij_betw_nat_nat @ ( infini8530281810654367211te_nat @ S2 ) @ top_top_set_nat @ S2 ) ) ).
% 0.35/0.75  
% 0.35/0.75  % bij_enumerate
% 0.35/0.75  thf(fact_9257_Least__eq__0,axiom,
% 0.35/0.75      ! [P: nat > $o] :
% 0.35/0.75        ( ( P @ zero_zero_nat )
% 0.35/0.75       => ( ( ord_Least_nat @ P )
% 0.35/0.75          = zero_zero_nat ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Least_eq_0
% 0.35/0.75  thf(fact_9258_Least__Suc,axiom,
% 0.35/0.75      ! [P: nat > $o,N: nat] :
% 0.35/0.75        ( ( P @ N )
% 0.35/0.75       => ( ~ ( P @ zero_zero_nat )
% 0.35/0.75         => ( ( ord_Least_nat @ P )
% 0.35/0.75            = ( suc
% 0.35/0.75              @ ( ord_Least_nat
% 0.35/0.75                @ ^ [M2: nat] : ( P @ ( suc @ M2 ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Least_Suc
% 0.35/0.75  thf(fact_9259_Least__Suc2,axiom,
% 0.35/0.75      ! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
% 0.35/0.75        ( ( P @ N )
% 0.35/0.75       => ( ( Q @ M )
% 0.35/0.75         => ( ~ ( P @ zero_zero_nat )
% 0.35/0.75           => ( ! [K2: nat] :
% 0.35/0.75                  ( ( P @ ( suc @ K2 ) )
% 0.35/0.75                  = ( Q @ K2 ) )
% 0.35/0.75             => ( ( ord_Least_nat @ P )
% 0.35/0.75                = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Least_Suc2
% 0.35/0.75  thf(fact_9260_mask__nat__positive__iff,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.35/0.75        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.35/0.75  
% 0.35/0.75  % mask_nat_positive_iff
% 0.35/0.75  thf(fact_9261_less__eq__mask,axiom,
% 0.35/0.75      ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 0.35/0.75  
% 0.35/0.75  % less_eq_mask
% 0.35/0.75  thf(fact_9262_not__mask__negative__int,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% 0.35/0.75  
% 0.35/0.75  % not_mask_negative_int
% 0.35/0.75  thf(fact_9263_less__mask,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.75       => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % less_mask
% 0.35/0.75  thf(fact_9264_mask__nat__less__exp,axiom,
% 0.35/0.75      ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.75  
% 0.35/0.75  % mask_nat_less_exp
% 0.35/0.75  thf(fact_9265_divmod__nat__code,axiom,
% 0.35/0.75      ( divmod_nat
% 0.35/0.75      = ( ^ [M2: nat,N2: nat] :
% 0.35/0.75            ( produc8678311845419106900er_nat @ code_nat_of_integer @ code_nat_of_integer
% 0.35/0.75            @ ( if_Pro6119634080678213985nteger
% 0.35/0.75              @ ( ( code_integer_of_nat @ M2 )
% 0.35/0.75                = zero_z3403309356797280102nteger )
% 0.35/0.75              @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.35/0.75              @ ( if_Pro6119634080678213985nteger
% 0.35/0.75                @ ( ( code_integer_of_nat @ N2 )
% 0.35/0.75                  = zero_z3403309356797280102nteger )
% 0.35/0.75                @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( code_integer_of_nat @ M2 ) )
% 0.35/0.75                @ ( code_divmod_abs @ ( code_integer_of_nat @ M2 ) @ ( code_integer_of_nat @ N2 ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % divmod_nat_code
% 0.35/0.75  thf(fact_9266_integer__of__nat__0,axiom,
% 0.35/0.75      ( ( code_integer_of_nat @ zero_zero_nat )
% 0.35/0.75      = zero_z3403309356797280102nteger ) ).
% 0.35/0.75  
% 0.35/0.75  % integer_of_nat_0
% 0.35/0.75  thf(fact_9267_last__upt,axiom,
% 0.35/0.75      ! [I: nat,J: nat] :
% 0.35/0.75        ( ( ord_less_nat @ I @ J )
% 0.35/0.75       => ( ( last_nat @ ( upt @ I @ J ) )
% 0.35/0.75          = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % last_upt
% 0.35/0.75  thf(fact_9268_not__nonnegative__int__iff,axiom,
% 0.35/0.75      ! [K: int] :
% 0.35/0.75        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.35/0.75        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.35/0.75  
% 0.35/0.75  % not_nonnegative_int_iff
% 0.35/0.75  thf(fact_9269_not__negative__int__iff,axiom,
% 0.35/0.75      ! [K: int] :
% 0.35/0.75        ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 0.35/0.75        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.35/0.75  
% 0.35/0.75  % not_negative_int_iff
% 0.35/0.75  thf(fact_9270_and__not__num__eq__Some__iff,axiom,
% 0.35/0.75      ! [M: num,N: num,Q4: num] :
% 0.35/0.75        ( ( ( bit_and_not_num @ M @ N )
% 0.35/0.75          = ( some_num @ Q4 ) )
% 0.35/0.75        = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.75          = ( numeral_numeral_int @ Q4 ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % and_not_num_eq_Some_iff
% 0.35/0.75  thf(fact_9271_and__not__num__eq__None__iff,axiom,
% 0.35/0.75      ! [M: num,N: num] :
% 0.35/0.75        ( ( ( bit_and_not_num @ M @ N )
% 0.35/0.75          = none_num )
% 0.35/0.75        = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.75          = zero_zero_int ) ) ).
% 0.35/0.75  
% 0.35/0.75  % and_not_num_eq_None_iff
% 0.35/0.75  thf(fact_9272_Real_Opositive_Oabs__eq,axiom,
% 0.35/0.75      ! [X: nat > rat] :
% 0.35/0.75        ( ( realrel @ X @ X )
% 0.35/0.75       => ( ( positive2 @ ( real2 @ X ) )
% 0.35/0.75          = ( ? [R5: rat] :
% 0.35/0.75                ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.75                & ? [K3: nat] :
% 0.35/0.75                  ! [N2: nat] :
% 0.35/0.75                    ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.75                   => ( ord_less_rat @ R5 @ ( X @ N2 ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Real.positive.abs_eq
% 0.35/0.75  thf(fact_9273_Real_Opositive_Orsp,axiom,
% 0.35/0.75      ( bNF_re728719798268516973at_o_o @ realrel
% 0.35/0.75      @ ^ [Y5: $o,Z: $o] : ( Y5 = Z )
% 0.35/0.75      @ ^ [X6: nat > rat] :
% 0.35/0.75        ? [R5: rat] :
% 0.35/0.75          ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.75          & ? [K3: nat] :
% 0.35/0.75            ! [N2: nat] :
% 0.35/0.75              ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.75             => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) )
% 0.35/0.75      @ ^ [X6: nat > rat] :
% 0.35/0.75        ? [R5: rat] :
% 0.35/0.75          ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.75          & ? [K3: nat] :
% 0.35/0.75            ! [N2: nat] :
% 0.35/0.75              ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.75             => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Real.positive.rsp
% 0.35/0.75  thf(fact_9274_less__eq__natural_Orsp,axiom,
% 0.35/0.75      ( bNF_re578469030762574527_nat_o
% 0.35/0.75      @ ^ [Y5: nat,Z: nat] : ( Y5 = Z )
% 0.35/0.75      @ ( bNF_re4705727531993890431at_o_o
% 0.35/0.75        @ ^ [Y5: nat,Z: nat] : ( Y5 = Z )
% 0.35/0.75        @ ^ [Y5: $o,Z: $o] : ( Y5 = Z ) )
% 0.35/0.75      @ ord_less_eq_nat
% 0.35/0.75      @ ord_less_eq_nat ) ).
% 0.35/0.75  
% 0.35/0.75  % less_eq_natural.rsp
% 0.35/0.75  thf(fact_9275_less__natural_Orsp,axiom,
% 0.35/0.75      ( bNF_re578469030762574527_nat_o
% 0.35/0.75      @ ^ [Y5: nat,Z: nat] : ( Y5 = Z )
% 0.35/0.75      @ ( bNF_re4705727531993890431at_o_o
% 0.35/0.75        @ ^ [Y5: nat,Z: nat] : ( Y5 = Z )
% 0.35/0.75        @ ^ [Y5: $o,Z: $o] : ( Y5 = Z ) )
% 0.35/0.75      @ ord_less_nat
% 0.35/0.75      @ ord_less_nat ) ).
% 0.35/0.75  
% 0.35/0.75  % less_natural.rsp
% 0.35/0.75  thf(fact_9276_less__integer_Orsp,axiom,
% 0.35/0.75      ( bNF_re3403563459893282935_int_o
% 0.35/0.75      @ ^ [Y5: int,Z: int] : ( Y5 = Z )
% 0.35/0.75      @ ( bNF_re5089333283451836215nt_o_o
% 0.35/0.75        @ ^ [Y5: int,Z: int] : ( Y5 = Z )
% 0.35/0.75        @ ^ [Y5: $o,Z: $o] : ( Y5 = Z ) )
% 0.35/0.75      @ ord_less_int
% 0.35/0.75      @ ord_less_int ) ).
% 0.35/0.75  
% 0.35/0.75  % less_integer.rsp
% 0.35/0.75  thf(fact_9277_Real_Opositive_Otransfer,axiom,
% 0.35/0.75      ( bNF_re4297313714947099218al_o_o @ pcr_real
% 0.35/0.75      @ ^ [Y5: $o,Z: $o] : ( Y5 = Z )
% 0.35/0.75      @ ^ [X6: nat > rat] :
% 0.35/0.75        ? [R5: rat] :
% 0.35/0.75          ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.75          & ? [K3: nat] :
% 0.35/0.75            ! [N2: nat] :
% 0.35/0.75              ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.75             => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) )
% 0.35/0.75      @ positive2 ) ).
% 0.35/0.75  
% 0.35/0.75  % Real.positive.transfer
% 0.35/0.75  thf(fact_9278_plus__rat_Otransfer,axiom,
% 0.35/0.75      ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 0.35/0.75      @ ^ [X3: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y2 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) )
% 0.35/0.75      @ plus_plus_rat ) ).
% 0.35/0.75  
% 0.35/0.75  % plus_rat.transfer
% 0.35/0.75  thf(fact_9279_one__rat_Otransfer,axiom,
% 0.35/0.75      pcr_rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ one_one_rat ).
% 0.35/0.75  
% 0.35/0.75  % one_rat.transfer
% 0.35/0.75  thf(fact_9280_zero__rat_Otransfer,axiom,
% 0.35/0.75      pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).
% 0.35/0.75  
% 0.35/0.75  % zero_rat.transfer
% 0.35/0.75  thf(fact_9281_Fract_Otransfer,axiom,
% 0.35/0.75      ( bNF_re3461391660133120880nt_rat
% 0.35/0.75      @ ^ [Y5: int,Z: int] : ( Y5 = Z )
% 0.35/0.75      @ ( bNF_re2214769303045360666nt_rat
% 0.35/0.75        @ ^ [Y5: int,Z: int] : ( Y5 = Z )
% 0.35/0.75        @ pcr_rat )
% 0.35/0.75      @ ^ [A4: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A4 @ B4 ) )
% 0.35/0.75      @ fract ) ).
% 0.35/0.75  
% 0.35/0.75  % Fract.transfer
% 0.35/0.75  thf(fact_9282_uminus__rat_Otransfer,axiom,
% 0.35/0.75      ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
% 0.35/0.75      @ ^ [X3: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X3 ) ) @ ( product_snd_int_int @ X3 ) )
% 0.35/0.75      @ uminus_uminus_rat ) ).
% 0.35/0.75  
% 0.35/0.75  % uminus_rat.transfer
% 0.35/0.75  thf(fact_9283_times__rat_Otransfer,axiom,
% 0.35/0.75      ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 0.35/0.75      @ ^ [X3: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y2 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) )
% 0.35/0.75      @ times_times_rat ) ).
% 0.35/0.75  
% 0.35/0.75  % times_rat.transfer
% 0.35/0.75  thf(fact_9284_Rat_Opositive_Otransfer,axiom,
% 0.35/0.75      ( bNF_re1494630372529172596at_o_o @ pcr_rat
% 0.35/0.75      @ ^ [Y5: $o,Z: $o] : ( Y5 = Z )
% 0.35/0.75      @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
% 0.35/0.75      @ positive ) ).
% 0.35/0.75  
% 0.35/0.75  % Rat.positive.transfer
% 0.35/0.75  thf(fact_9285_inverse__rat_Otransfer,axiom,
% 0.35/0.75      ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
% 0.35/0.75      @ ^ [X3: product_prod_int_int] :
% 0.35/0.75          ( if_Pro3027730157355071871nt_int
% 0.35/0.75          @ ( ( product_fst_int_int @ X3 )
% 0.35/0.75            = zero_zero_int )
% 0.35/0.75          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.35/0.75          @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) )
% 0.35/0.75      @ inverse_inverse_rat ) ).
% 0.35/0.75  
% 0.35/0.75  % inverse_rat.transfer
% 0.35/0.75  thf(fact_9286_times__int_Otransfer,axiom,
% 0.35/0.75      ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 0.35/0.75      @ ( produc27273713700761075at_nat
% 0.35/0.75        @ ^ [X3: nat,Y2: nat] :
% 0.35/0.75            ( produc2626176000494625587at_nat
% 0.35/0.75            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y2 @ V3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V3 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) )
% 0.35/0.75      @ times_times_int ) ).
% 0.35/0.75  
% 0.35/0.75  % times_int.transfer
% 0.35/0.75  thf(fact_9287_minus__int_Otransfer,axiom,
% 0.35/0.75      ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 0.35/0.75      @ ( produc27273713700761075at_nat
% 0.35/0.75        @ ^ [X3: nat,Y2: nat] :
% 0.35/0.75            ( produc2626176000494625587at_nat
% 0.35/0.75            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) )
% 0.35/0.75      @ minus_minus_int ) ).
% 0.35/0.75  
% 0.35/0.75  % minus_int.transfer
% 0.35/0.75  thf(fact_9288_zero__int_Otransfer,axiom,
% 0.35/0.75      pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).
% 0.35/0.75  
% 0.35/0.75  % zero_int.transfer
% 0.35/0.75  thf(fact_9289_int__transfer,axiom,
% 0.35/0.75      ( bNF_re6830278522597306478at_int
% 0.35/0.75      @ ^ [Y5: nat,Z: nat] : ( Y5 = Z )
% 0.35/0.75      @ pcr_int
% 0.35/0.75      @ ^ [N2: nat] : ( product_Pair_nat_nat @ N2 @ zero_zero_nat )
% 0.35/0.75      @ semiri1314217659103216013at_int ) ).
% 0.35/0.75  
% 0.35/0.75  % int_transfer
% 0.35/0.75  thf(fact_9290_uminus__int_Otransfer,axiom,
% 0.35/0.75      ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
% 0.35/0.75      @ ( produc2626176000494625587at_nat
% 0.35/0.75        @ ^ [X3: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X3 ) )
% 0.35/0.75      @ uminus_uminus_int ) ).
% 0.35/0.75  
% 0.35/0.75  % uminus_int.transfer
% 0.35/0.75  thf(fact_9291_one__int_Otransfer,axiom,
% 0.35/0.75      pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).
% 0.35/0.75  
% 0.35/0.75  % one_int.transfer
% 0.35/0.75  thf(fact_9292_less__int_Otransfer,axiom,
% 0.35/0.75      ( bNF_re717283939379294677_int_o @ pcr_int
% 0.35/0.75      @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 0.35/0.75        @ ^ [Y5: $o,Z: $o] : ( Y5 = Z ) )
% 0.35/0.75      @ ( produc8739625826339149834_nat_o
% 0.35/0.75        @ ^ [X3: nat,Y2: nat] :
% 0.35/0.75            ( produc6081775807080527818_nat_o
% 0.35/0.75            @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) )
% 0.35/0.75      @ ord_less_int ) ).
% 0.35/0.75  
% 0.35/0.75  % less_int.transfer
% 0.35/0.75  thf(fact_9293_less__eq__int_Otransfer,axiom,
% 0.35/0.75      ( bNF_re717283939379294677_int_o @ pcr_int
% 0.35/0.75      @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 0.35/0.75        @ ^ [Y5: $o,Z: $o] : ( Y5 = Z ) )
% 0.35/0.75      @ ( produc8739625826339149834_nat_o
% 0.35/0.75        @ ^ [X3: nat,Y2: nat] :
% 0.35/0.75            ( produc6081775807080527818_nat_o
% 0.35/0.75            @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) )
% 0.35/0.75      @ ord_less_eq_int ) ).
% 0.35/0.75  
% 0.35/0.75  % less_eq_int.transfer
% 0.35/0.75  thf(fact_9294_plus__int_Otransfer,axiom,
% 0.35/0.75      ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 0.35/0.75      @ ( produc27273713700761075at_nat
% 0.35/0.75        @ ^ [X3: nat,Y2: nat] :
% 0.35/0.75            ( produc2626176000494625587at_nat
% 0.35/0.75            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y2 @ V3 ) ) ) )
% 0.35/0.75      @ plus_plus_int ) ).
% 0.35/0.75  
% 0.35/0.75  % plus_int.transfer
% 0.35/0.75  thf(fact_9295_times__int_Orsp,axiom,
% 0.35/0.75      ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 0.35/0.75      @ ( produc27273713700761075at_nat
% 0.35/0.75        @ ^ [X3: nat,Y2: nat] :
% 0.35/0.75            ( produc2626176000494625587at_nat
% 0.35/0.75            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y2 @ V3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V3 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) )
% 0.35/0.75      @ ( produc27273713700761075at_nat
% 0.35/0.75        @ ^ [X3: nat,Y2: nat] :
% 0.35/0.75            ( produc2626176000494625587at_nat
% 0.35/0.75            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y2 @ V3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V3 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % times_int.rsp
% 0.35/0.75  thf(fact_9296_intrel__iff,axiom,
% 0.35/0.75      ! [X: nat,Y: nat,U: nat,V: nat] :
% 0.35/0.75        ( ( intrel @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ U @ V ) )
% 0.35/0.75        = ( ( plus_plus_nat @ X @ V )
% 0.35/0.75          = ( plus_plus_nat @ U @ Y ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % intrel_iff
% 0.35/0.75  thf(fact_9297_uminus__int_Orsp,axiom,
% 0.35/0.75      ( bNF_re2241393799969408733at_nat @ intrel @ intrel
% 0.35/0.75      @ ( produc2626176000494625587at_nat
% 0.35/0.75        @ ^ [X3: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X3 ) )
% 0.35/0.75      @ ( produc2626176000494625587at_nat
% 0.35/0.75        @ ^ [X3: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X3 ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % uminus_int.rsp
% 0.35/0.75  thf(fact_9298_zero__int_Orsp,axiom,
% 0.35/0.75      intrel @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.35/0.75  
% 0.35/0.75  % zero_int.rsp
% 0.35/0.75  thf(fact_9299_one__int_Orsp,axiom,
% 0.35/0.75      intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.35/0.75  
% 0.35/0.75  % one_int.rsp
% 0.35/0.75  thf(fact_9300_less__int_Orsp,axiom,
% 0.35/0.75      ( bNF_re4202695980764964119_nat_o @ intrel
% 0.35/0.75      @ ( bNF_re3666534408544137501at_o_o @ intrel
% 0.35/0.75        @ ^ [Y5: $o,Z: $o] : ( Y5 = Z ) )
% 0.35/0.75      @ ( produc8739625826339149834_nat_o
% 0.35/0.75        @ ^ [X3: nat,Y2: nat] :
% 0.35/0.75            ( produc6081775807080527818_nat_o
% 0.35/0.75            @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) )
% 0.35/0.75      @ ( produc8739625826339149834_nat_o
% 0.35/0.75        @ ^ [X3: nat,Y2: nat] :
% 0.35/0.75            ( produc6081775807080527818_nat_o
% 0.35/0.75            @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % less_int.rsp
% 0.35/0.75  thf(fact_9301_less__eq__int_Orsp,axiom,
% 0.35/0.75      ( bNF_re4202695980764964119_nat_o @ intrel
% 0.35/0.75      @ ( bNF_re3666534408544137501at_o_o @ intrel
% 0.35/0.75        @ ^ [Y5: $o,Z: $o] : ( Y5 = Z ) )
% 0.35/0.75      @ ( produc8739625826339149834_nat_o
% 0.35/0.75        @ ^ [X3: nat,Y2: nat] :
% 0.35/0.75            ( produc6081775807080527818_nat_o
% 0.35/0.75            @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) )
% 0.35/0.75      @ ( produc8739625826339149834_nat_o
% 0.35/0.75        @ ^ [X3: nat,Y2: nat] :
% 0.35/0.75            ( produc6081775807080527818_nat_o
% 0.35/0.75            @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % less_eq_int.rsp
% 0.35/0.75  thf(fact_9302_plus__int_Orsp,axiom,
% 0.35/0.75      ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 0.35/0.75      @ ( produc27273713700761075at_nat
% 0.35/0.75        @ ^ [X3: nat,Y2: nat] :
% 0.35/0.75            ( produc2626176000494625587at_nat
% 0.35/0.75            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y2 @ V3 ) ) ) )
% 0.35/0.75      @ ( produc27273713700761075at_nat
% 0.35/0.75        @ ^ [X3: nat,Y2: nat] :
% 0.35/0.75            ( produc2626176000494625587at_nat
% 0.35/0.75            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y2 @ V3 ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % plus_int.rsp
% 0.35/0.75  thf(fact_9303_minus__int_Orsp,axiom,
% 0.35/0.75      ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 0.35/0.75      @ ( produc27273713700761075at_nat
% 0.35/0.75        @ ^ [X3: nat,Y2: nat] :
% 0.35/0.75            ( produc2626176000494625587at_nat
% 0.35/0.75            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) )
% 0.35/0.75      @ ( produc27273713700761075at_nat
% 0.35/0.75        @ ^ [X3: nat,Y2: nat] :
% 0.35/0.75            ( produc2626176000494625587at_nat
% 0.35/0.75            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % minus_int.rsp
% 0.35/0.75  thf(fact_9304_plus__rat_Orsp,axiom,
% 0.35/0.75      ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 0.35/0.75      @ ^ [X3: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y2 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) )
% 0.35/0.75      @ ^ [X3: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y2 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % plus_rat.rsp
% 0.35/0.75  thf(fact_9305_inverse__rat_Orsp,axiom,
% 0.35/0.75      ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
% 0.35/0.75      @ ^ [X3: product_prod_int_int] :
% 0.35/0.75          ( if_Pro3027730157355071871nt_int
% 0.35/0.75          @ ( ( product_fst_int_int @ X3 )
% 0.35/0.75            = zero_zero_int )
% 0.35/0.75          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.35/0.75          @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) )
% 0.35/0.75      @ ^ [X3: product_prod_int_int] :
% 0.35/0.75          ( if_Pro3027730157355071871nt_int
% 0.35/0.75          @ ( ( product_fst_int_int @ X3 )
% 0.35/0.75            = zero_zero_int )
% 0.35/0.75          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.35/0.75          @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % inverse_rat.rsp
% 0.35/0.75  thf(fact_9306_zero__rat_Orsp,axiom,
% 0.35/0.75      ratrel @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ).
% 0.35/0.75  
% 0.35/0.75  % zero_rat.rsp
% 0.35/0.75  thf(fact_9307_one__rat_Orsp,axiom,
% 0.35/0.75      ratrel @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ).
% 0.35/0.75  
% 0.35/0.75  % one_rat.rsp
% 0.35/0.75  thf(fact_9308_Fract_Orsp,axiom,
% 0.35/0.75      ( bNF_re157797125943740599nt_int
% 0.35/0.75      @ ^ [Y5: int,Z: int] : ( Y5 = Z )
% 0.35/0.75      @ ( bNF_re6250860962936578807nt_int
% 0.35/0.75        @ ^ [Y5: int,Z: int] : ( Y5 = Z )
% 0.35/0.75        @ ratrel )
% 0.35/0.75      @ ^ [A4: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A4 @ B4 ) )
% 0.35/0.75      @ ^ [A4: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A4 @ B4 ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Fract.rsp
% 0.35/0.75  thf(fact_9309_uminus__rat_Orsp,axiom,
% 0.35/0.75      ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
% 0.35/0.75      @ ^ [X3: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X3 ) ) @ ( product_snd_int_int @ X3 ) )
% 0.35/0.75      @ ^ [X3: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X3 ) ) @ ( product_snd_int_int @ X3 ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % uminus_rat.rsp
% 0.35/0.75  thf(fact_9310_times__rat_Orsp,axiom,
% 0.35/0.75      ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 0.35/0.75      @ ^ [X3: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y2 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) )
% 0.35/0.75      @ ^ [X3: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y2 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % times_rat.rsp
% 0.35/0.75  thf(fact_9311_Rat_Opositive_Orsp,axiom,
% 0.35/0.75      ( bNF_re8699439704749558557nt_o_o @ ratrel
% 0.35/0.75      @ ^ [Y5: $o,Z: $o] : ( Y5 = Z )
% 0.35/0.75      @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
% 0.35/0.75      @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Rat.positive.rsp
% 0.35/0.75  thf(fact_9312_plus__rat_Oabs__eq,axiom,
% 0.35/0.75      ! [Xa2: product_prod_int_int,X: product_prod_int_int] :
% 0.35/0.75        ( ( ratrel @ Xa2 @ Xa2 )
% 0.35/0.75       => ( ( ratrel @ X @ X )
% 0.35/0.75         => ( ( plus_plus_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X ) )
% 0.35/0.75            = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_snd_int_int @ X ) ) @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % plus_rat.abs_eq
% 0.35/0.75  thf(fact_9313_inverse__rat_Oabs__eq,axiom,
% 0.35/0.75      ! [X: product_prod_int_int] :
% 0.35/0.75        ( ( ratrel @ X @ X )
% 0.35/0.75       => ( ( inverse_inverse_rat @ ( abs_Rat @ X ) )
% 0.35/0.75          = ( abs_Rat
% 0.35/0.75            @ ( if_Pro3027730157355071871nt_int
% 0.35/0.75              @ ( ( product_fst_int_int @ X )
% 0.35/0.75                = zero_zero_int )
% 0.35/0.75              @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.35/0.75              @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % inverse_rat.abs_eq
% 0.35/0.75  thf(fact_9314_one__rat__def,axiom,
% 0.35/0.75      ( one_one_rat
% 0.35/0.75      = ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % one_rat_def
% 0.35/0.75  thf(fact_9315_Fract_Oabs__eq,axiom,
% 0.35/0.75      ( fract
% 0.35/0.75      = ( ^ [Xa4: int,X3: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X3 ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Fract.abs_eq
% 0.35/0.75  thf(fact_9316_zero__rat__def,axiom,
% 0.35/0.75      ( zero_zero_rat
% 0.35/0.75      = ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % zero_rat_def
% 0.35/0.75  thf(fact_9317_uminus__rat_Oabs__eq,axiom,
% 0.35/0.75      ! [X: product_prod_int_int] :
% 0.35/0.75        ( ( ratrel @ X @ X )
% 0.35/0.75       => ( ( uminus_uminus_rat @ ( abs_Rat @ X ) )
% 0.35/0.75          = ( abs_Rat @ ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % uminus_rat.abs_eq
% 0.35/0.75  thf(fact_9318_times__rat_Oabs__eq,axiom,
% 0.35/0.75      ! [Xa2: product_prod_int_int,X: product_prod_int_int] :
% 0.35/0.75        ( ( ratrel @ Xa2 @ Xa2 )
% 0.35/0.75       => ( ( ratrel @ X @ X )
% 0.35/0.75         => ( ( times_times_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X ) )
% 0.35/0.75            = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_fst_int_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % times_rat.abs_eq
% 0.35/0.75  thf(fact_9319_Rat_Opositive_Oabs__eq,axiom,
% 0.35/0.75      ! [X: product_prod_int_int] :
% 0.35/0.75        ( ( ratrel @ X @ X )
% 0.35/0.75       => ( ( positive @ ( abs_Rat @ X ) )
% 0.35/0.75          = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % Rat.positive.abs_eq
% 0.35/0.75  thf(fact_9320_inverse__rat__def,axiom,
% 0.35/0.75      ( inverse_inverse_rat
% 0.35/0.75      = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
% 0.35/0.75        @ ^ [X3: product_prod_int_int] :
% 0.35/0.75            ( if_Pro3027730157355071871nt_int
% 0.35/0.75            @ ( ( product_fst_int_int @ X3 )
% 0.35/0.75              = zero_zero_int )
% 0.35/0.75            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.35/0.75            @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % inverse_rat_def
% 0.35/0.75  thf(fact_9321_uminus__rat__def,axiom,
% 0.35/0.75      ( uminus_uminus_rat
% 0.35/0.75      = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
% 0.35/0.75        @ ^ [X3: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X3 ) ) @ ( product_snd_int_int @ X3 ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % uminus_rat_def
% 0.35/0.75  thf(fact_9322_plus__rat__def,axiom,
% 0.35/0.75      ( plus_plus_rat
% 0.35/0.75      = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 0.35/0.75        @ ^ [X3: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y2 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % plus_rat_def
% 0.35/0.75  thf(fact_9323_times__rat__def,axiom,
% 0.35/0.75      ( times_times_rat
% 0.35/0.75      = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 0.35/0.75        @ ^ [X3: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y2 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) ) ) ) ).
% 0.35/0.75  
% 0.35/0.75  % times_rat_def
% 0.35/0.75  thf(fact_9324_natLeq__on__wo__rel,axiom,
% 0.35/0.75      ! [N: nat] :
% 0.35/0.75        ( bNF_We3818239936649020644el_nat
% 0.35/0.75        @ ( collec3392354462482085612at_nat
% 0.35/0.75          @ ( produc6081775807080527818_nat_o
% 0.35/0.75            @ ^ [X3: nat,Y2: nat] :
% 0.35/0.75                ( ( ord_less_nat @ X3 @ N )
% 0.35/0.75                & ( ord_less_nat @ Y2 @ N )
% 0.35/0.76                & ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ).
% 0.35/0.76  
% 0.35/0.76  % natLeq_on_wo_rel
% 0.35/0.76  thf(fact_9325_pred__nat__trancl__eq__le,axiom,
% 0.35/0.76      ! [M: nat,N: nat] :
% 0.35/0.76        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
% 0.35/0.76        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.35/0.76  
% 0.35/0.76  % pred_nat_trancl_eq_le
% 0.35/0.76  thf(fact_9326_less__eq,axiom,
% 0.35/0.76      ! [M: nat,N: nat] :
% 0.35/0.76        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
% 0.35/0.76        = ( ord_less_nat @ M @ N ) ) ).
% 0.35/0.76  
% 0.35/0.76  % less_eq
% 0.35/0.76  thf(fact_9327_bit__concat__bit__iff,axiom,
% 0.35/0.76      ! [M: nat,K: int,L: int,N: nat] :
% 0.35/0.76        ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
% 0.35/0.76        = ( ( ( ord_less_nat @ N @ M )
% 0.35/0.76            & ( bit_se1146084159140164899it_int @ K @ N ) )
% 0.35/0.76          | ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.76            & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.35/0.76  
% 0.35/0.76  % bit_concat_bit_iff
% 0.35/0.76  thf(fact_9328_concat__bit__0,axiom,
% 0.35/0.76      ! [K: int,L: int] :
% 0.35/0.76        ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
% 0.35/0.76        = L ) ).
% 0.35/0.76  
% 0.35/0.76  % concat_bit_0
% 0.35/0.76  thf(fact_9329_concat__bit__negative__iff,axiom,
% 0.35/0.76      ! [N: nat,K: int,L: int] :
% 0.35/0.76        ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
% 0.35/0.76        = ( ord_less_int @ L @ zero_zero_int ) ) ).
% 0.35/0.76  
% 0.35/0.76  % concat_bit_negative_iff
% 0.35/0.76  thf(fact_9330_gcd__nat_Oordering__top__axioms,axiom,
% 0.35/0.76      ( ordering_top_nat @ dvd_dvd_nat
% 0.35/0.76      @ ^ [M2: nat,N2: nat] :
% 0.35/0.76          ( ( dvd_dvd_nat @ M2 @ N2 )
% 0.35/0.76          & ( M2 != N2 ) )
% 0.35/0.76      @ zero_zero_nat ) ).
% 0.35/0.76  
% 0.35/0.76  % gcd_nat.ordering_top_axioms
% 0.35/0.76  thf(fact_9331_bot__nat__0_Oordering__top__axioms,axiom,
% 0.35/0.76      ( ordering_top_nat
% 0.35/0.76      @ ^ [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X3 )
% 0.35/0.76      @ ^ [X3: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X3 )
% 0.35/0.76      @ zero_zero_nat ) ).
% 0.35/0.76  
% 0.35/0.76  % bot_nat_0.ordering_top_axioms
% 0.35/0.76  
% 0.35/0.76  % Helper facts (40)
% 0.35/0.76  thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 0.35/0.76      ! [X: int,Y: int] :
% 0.35/0.76        ( ( if_int @ $false @ X @ Y )
% 0.35/0.76        = Y ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 0.35/0.76      ! [X: int,Y: int] :
% 0.35/0.76        ( ( if_int @ $true @ X @ Y )
% 0.35/0.76        = X ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.35/0.76      ! [X: nat,Y: nat] :
% 0.35/0.76        ( ( if_nat @ $false @ X @ Y )
% 0.35/0.76        = Y ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.35/0.76      ! [X: nat,Y: nat] :
% 0.35/0.76        ( ( if_nat @ $true @ X @ Y )
% 0.35/0.76        = X ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
% 0.35/0.76      ! [X: num,Y: num] :
% 0.35/0.76        ( ( if_num @ $false @ X @ Y )
% 0.35/0.76        = Y ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
% 0.35/0.76      ! [X: num,Y: num] :
% 0.35/0.76        ( ( if_num @ $true @ X @ Y )
% 0.35/0.76        = X ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 0.35/0.76      ! [X: rat,Y: rat] :
% 0.35/0.76        ( ( if_rat @ $false @ X @ Y )
% 0.35/0.76        = Y ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 0.35/0.76      ! [X: rat,Y: rat] :
% 0.35/0.76        ( ( if_rat @ $true @ X @ Y )
% 0.35/0.76        = X ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 0.35/0.76      ! [X: real,Y: real] :
% 0.35/0.76        ( ( if_real @ $false @ X @ Y )
% 0.35/0.76        = Y ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 0.35/0.76      ! [X: real,Y: real] :
% 0.35/0.76        ( ( if_real @ $true @ X @ Y )
% 0.35/0.76        = X ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
% 0.35/0.76      ! [P: real > $o] :
% 0.35/0.76        ( ( P @ ( fChoice_real @ P ) )
% 0.35/0.76        = ( ? [X6: real] : ( P @ X6 ) ) ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.35/0.76      ! [X: complex,Y: complex] :
% 0.35/0.76        ( ( if_complex @ $false @ X @ Y )
% 0.35/0.76        = Y ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.35/0.76      ! [X: complex,Y: complex] :
% 0.35/0.76        ( ( if_complex @ $true @ X @ Y )
% 0.35/0.76        = X ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 0.35/0.76      ! [X: extended_enat,Y: extended_enat] :
% 0.35/0.76        ( ( if_Extended_enat @ $false @ X @ Y )
% 0.35/0.76        = Y ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 0.35/0.76      ! [X: extended_enat,Y: extended_enat] :
% 0.35/0.76        ( ( if_Extended_enat @ $true @ X @ Y )
% 0.35/0.76        = X ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.35/0.76      ! [X: code_integer,Y: code_integer] :
% 0.35/0.76        ( ( if_Code_integer @ $false @ X @ Y )
% 0.35/0.76        = Y ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.35/0.76      ! [X: code_integer,Y: code_integer] :
% 0.35/0.76        ( ( if_Code_integer @ $true @ X @ Y )
% 0.35/0.76        = X ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 0.35/0.76      ! [X: set_int,Y: set_int] :
% 0.35/0.76        ( ( if_set_int @ $false @ X @ Y )
% 0.35/0.76        = Y ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 0.35/0.76      ! [X: set_int,Y: set_int] :
% 0.35/0.76        ( ( if_set_int @ $true @ X @ Y )
% 0.35/0.76        = X ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 0.35/0.76      ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
% 0.35/0.76        ( ( if_VEBT_VEBT @ $false @ X @ Y )
% 0.35/0.76        = Y ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 0.35/0.76      ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
% 0.35/0.76        ( ( if_VEBT_VEBT @ $true @ X @ Y )
% 0.35/0.76        = X ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.35/0.76      ! [X: list_int,Y: list_int] :
% 0.35/0.76        ( ( if_list_int @ $false @ X @ Y )
% 0.35/0.76        = Y ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.35/0.76      ! [X: list_int,Y: list_int] :
% 0.35/0.76        ( ( if_list_int @ $true @ X @ Y )
% 0.35/0.76        = X ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 0.35/0.76      ! [X: list_nat,Y: list_nat] :
% 0.35/0.76        ( ( if_list_nat @ $false @ X @ Y )
% 0.35/0.76        = Y ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 0.35/0.76      ! [X: list_nat,Y: list_nat] :
% 0.35/0.76        ( ( if_list_nat @ $true @ X @ Y )
% 0.35/0.76        = X ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.35/0.76      ! [X: int > int,Y: int > int] :
% 0.35/0.76        ( ( if_int_int @ $false @ X @ Y )
% 0.35/0.76        = Y ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.35/0.76      ! [X: int > int,Y: int > int] :
% 0.35/0.76        ( ( if_int_int @ $true @ X @ Y )
% 0.35/0.76        = X ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 0.35/0.76      ! [X: option_nat,Y: option_nat] :
% 0.35/0.76        ( ( if_option_nat @ $false @ X @ Y )
% 0.35/0.76        = Y ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 0.35/0.76      ! [X: option_nat,Y: option_nat] :
% 0.35/0.76        ( ( if_option_nat @ $true @ X @ Y )
% 0.35/0.76        = X ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.35/0.76      ! [X: option_num,Y: option_num] :
% 0.35/0.76        ( ( if_option_num @ $false @ X @ Y )
% 0.35/0.76        = Y ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.35/0.76      ! [X: option_num,Y: option_num] :
% 0.35/0.76        ( ( if_option_num @ $true @ X @ Y )
% 0.35/0.76        = X ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.35/0.76      ! [X: product_prod_int_int,Y: product_prod_int_int] :
% 0.35/0.76        ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
% 0.35/0.76        = Y ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.35/0.76      ! [X: product_prod_int_int,Y: product_prod_int_int] :
% 0.35/0.76        ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
% 0.35/0.76        = X ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.35/0.76      ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 0.35/0.76        ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
% 0.35/0.76        = Y ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.35/0.76      ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 0.35/0.76        ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
% 0.35/0.76        = X ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.35/0.76      ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
% 0.35/0.76        ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
% 0.35/0.76        = Y ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.35/0.76      ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
% 0.35/0.76        ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
% 0.35/0.76        = X ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.35/0.76      ! [P: $o] :
% 0.35/0.76        ( ( P = $true )
% 0.35/0.76        | ( P = $false ) ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.35/0.76      ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
% 0.35/0.76        ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
% 0.35/0.76        = Y ) ).
% 0.35/0.76  
% 0.35/0.76  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.35/0.76      ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
% 0.35/0.76        ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
% 0.35/0.76        = X ) ).
% 0.35/0.76  
% 0.35/0.76  % Conjectures (1)
% 0.35/0.76  thf(conj_0,conjecture,
% 0.35/0.76      vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ na ).
% 0.35/0.76  
% 0.35/0.76  %------------------------------------------------------------------------------
% 0.35/0.76  ------------------------
% 6.99/2.84  ---- Embedded in THF ---
% 6.99/2.84  %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 6.99/2.84  %%% Generated on Fri Feb 27 03:17:56 EST 2026
% 6.99/2.84  %%% using '$$dhol' embedding, version 1.3.0.
% 6.99/2.84  %%% Logic specification used:
% 6.99/2.84  %%% thf(spec, logic, $$dhol).
% 6.99/2.84  
% 6.99/2.84  % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.JxDoxSuNLY/DTF2DTF14050.p
% See solution above
% 6.99/2.87  ------------------------
% 9.49/5.21  ---- Cleaned THF ---
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
% 9.49/5.21      produc1319942482725812455at_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 9.49/5.21      produc5542196010084753463at_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
% 9.49/5.21      list_P7828571989066258726nteger: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
% 9.49/5.21      set_Pr1281608226676607948nteger: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 9.49/5.21      produc1908205239877642774nteger: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 9.49/5.21      produc5491161045314408544at_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
% 9.49/5.21      list_P651320350408439699nt_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
% 9.49/5.21      set_Pr9222295170931077689nt_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.49/5.21      produc2285326912895808259nt_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
% 9.49/5.21      list_P5311841565141990158nteger: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
% 9.49/5.21      set_Pr8056137968301705908nteger: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 9.49/5.21      produc8763457246119570046nteger: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
% 9.49/5.21      list_P1316552470764441098e_term: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
% 9.49/5.21      set_Pr7604974323444597168e_term: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
% 9.49/5.21      list_P1743416141875011707e_term: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
% 9.49/5.21      set_Pr3642885161833720865e_term: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
% 9.49/5.21      list_P8915022641806594461nt_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
% 9.49/5.21      set_Pr1872883991513573699nt_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.49/5.21      produc7773217078559923341nt_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 9.49/5.21      list_P5464809261938338413at_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 9.49/5.21      set_Pr4329608150637261639at_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_I_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
% 9.49/5.21      produc1193250871479095198on_num: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
% 9.49/5.21      produc8306885398267862888on_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.49/5.21      produc6121120109295599847at_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_I_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
% 9.49/5.21      produc7036089656553540234on_num: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
% 9.49/5.21      produc2233624965454879586on_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.49/5.21      produc3843707927480180839at_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.49/5.21      list_P8469869581646625389at_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
% 9.49/5.21      list_C878401137130745250e_term: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.49/5.21      set_Pr8693737435421807431at_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
% 9.49/5.21      set_Co9149898834107579976e_term: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
% 9.49/5.21      list_i8448526496819171953e_term: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__VEBT____Definitions__OVEBT_J_J_J,type,
% 9.49/5.21      set_Pr1916528119006554503T_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.21      produc859450856879609959at_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
% 9.49/5.21      set_in3461395444621081367e_term: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
% 9.49/5.21      produc9211091688327510695T_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J,type,
% 9.49/5.21      produc6241069584506657477e_term: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J_J,type,
% 9.49/5.21      set_fi4554929511873752355omplex: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
% 9.49/5.21      set_Pr3451248702717554689st_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
% 9.49/5.21      set_Pr765067013931698361st_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.49/5.21      list_P7413028617227757229T_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J,type,
% 9.49/5.21      produc8551481072490612790e_term: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.49/5.21      set_Pr6192946355708809607T_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
% 9.49/5.21      list_P6254988961118846195et_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 9.49/5.21      list_P5578671422887162913nteger: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
% 9.49/5.21      set_Pr5488025237498180813et_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 9.49/5.21      set_Pr4811707699266497531nteger: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J,type,
% 9.49/5.21      option6357759511663192854e_term: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
% 9.49/5.21      produc3447558737645232053on_num: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
% 9.49/5.21      produc4953844613479565601on_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J_J,type,
% 9.49/5.21      set_fi7789364187291644575l_real: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 9.49/5.21      filter6041513312241820739omplex: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_J,type,
% 9.49/5.21      list_P2623026923184700063T_real: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.49/5.21      list_P877281246627933069T_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
% 9.49/5.21      produc1828647624359046049st_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
% 9.49/5.21      produc1186641810826059865st_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_J,type,
% 9.49/5.21      set_Pr7765410600122031685T_real: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.49/5.21      set_Pr6019664923565264691T_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
% 9.49/5.21      list_P7037539587688870467BT_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
% 9.49/5.21      list_P4547456442757143711BT_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.49/5.21      list_P5647936690300460905T_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.49/5.21      list_P7524865323317820941T_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__List__Olist_I_Eo_J_J_J,type,
% 9.49/5.21      set_Pr6227168374412355847list_o: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.49/5.21      set_li5450038453877631591at_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.21      produc8243902056947475879T_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
% 9.49/5.21      set_Pr7556676689462069481BT_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
% 9.49/5.21      set_Pr5066593544530342725BT_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.49/5.21      set_Pr8044002425091019955T_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.49/5.21      set_se7855581050983116737at_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 9.49/5.21      set_Pr5085853215250843933omplex: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
% 9.49/5.21      produc7819656566062154093et_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 9.49/5.21      produc8923325533196201883nteger: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
% 9.49/5.21      list_P3126845725202233233VEBT_o: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.49/5.21      list_P7495141550334521929T_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 9.49/5.21      filter2146258269922977983l_real: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
% 9.49/5.21      set_Pr3175402225741728619VEBT_o: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.49/5.21      set_Pr7543698050874017315T_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.21      option4927543243414619207at_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 9.49/5.21      list_P8689742595348180415l_real: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.21      filter1242075044329608583at_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 9.49/5.21      set_Pr6218003697084177305l_real: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
% 9.49/5.21      list_P6834414599653733731al_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J_J,type,
% 9.49/5.21      list_P4344331454722006975al_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J_J,type,
% 9.49/5.21      list_P6863124054624500543t_real: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
% 9.49/5.21      produc5170161368751668367T_real: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.21      produc3757001726724277373T_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
% 9.49/5.21      set_Pr3510011417693777981al_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J_J,type,
% 9.49/5.21      set_Pr1019928272762051225al_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J_J,type,
% 9.49/5.21      set_Pr3538720872664544793t_real: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__List__Olist_I_Eo_J_J,type,
% 9.49/5.21      produc7102631898165422375list_o: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.21      list_P6011104703257516679at_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
% 9.49/5.21      list_P3521021558325789923at_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.49/5.21      list_P5707943133018811711nt_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 9.49/5.21      produc9072475918466114483BT_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 9.49/5.21      produc4894624898956917775BT_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.21      produc8025551001238799321T_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.21      produc1531783533982839933T_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.21      set_Pr1261947904930325089at_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.49/5.21      set_Pr958786334691620121nt_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 9.49/5.21      produc4411394909380815293omplex: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_M_Eo_J_J,type,
% 9.49/5.21      list_P3595434254542482545real_o: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J_J,type,
% 9.49/5.21      list_P5232166724548748803o_real: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_M_Eo_J_J,type,
% 9.49/5.21      set_Pr4936984352647145239real_o: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J_J,type,
% 9.49/5.21      set_Pr6573716822653411497o_real: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
% 9.49/5.21      list_P7333126701944960589_nat_o: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
% 9.49/5.21      list_P6285523579766656935_o_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
% 9.49/5.21      list_P3795440434834930179_o_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
% 9.49/5.21      set_list_VEBT_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 9.49/5.21      produc334124729049499915VEBT_o: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.21      produc2504756804600209347T_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
% 9.49/5.21      set_Pr2101469702781467981_o_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
% 9.49/5.21      set_Pr8834758594704517033_o_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 9.49/5.21      produc6271795597528267376eger_o: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 9.49/5.21      produc2422161461964618553l_real: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
% 9.49/5.21      produc3741383161447143261al_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
% 9.49/5.21      produc8786904178792722361al_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
% 9.49/5.21      produc679980390762269497t_real: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 9.49/5.21      product_prod_num_num: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 9.49/5.21      product_prod_nat_num: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.21      product_prod_nat_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
% 9.49/5.21      product_prod_nat_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.49/5.21      product_prod_int_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 9.49/5.21      list_P4002435161011370285od_o_o: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J,type,
% 9.49/5.21      set_list_complex: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 9.49/5.21      set_Product_prod_o_o: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
% 9.49/5.21      set_set_complex: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Option__Ooption_It__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.21      option_VEBT_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
% 9.49/5.21      option_set_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_M_Eo_J,type,
% 9.49/5.21      product_prod_real_o: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J,type,
% 9.49/5.21      product_prod_o_real: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.21      list_VEBT_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 9.49/5.21      set_list_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
% 9.49/5.21      set_list_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
% 9.49/5.21      product_prod_nat_o: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
% 9.49/5.21      product_prod_o_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 9.49/5.21      product_prod_o_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
% 9.49/5.21      list_set_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.21      set_VEBT_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 9.49/5.21      set_set_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
% 9.49/5.21      set_set_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
% 9.49/5.21      set_Code_integer: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
% 9.49/5.21      set_Product_unit: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
% 9.49/5.21      list_complex: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
% 9.49/5.21      set_list_o: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
% 9.49/5.21      product_prod_o_o: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
% 9.49/5.21      set_complex: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Option__Ooption_It__Real__Oreal_J,type,
% 9.49/5.21      option_real: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
% 9.49/5.21      filter_real: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
% 9.49/5.21      option_num: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
% 9.49/5.21      option_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
% 9.49/5.21      option_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
% 9.49/5.21      filter_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
% 9.49/5.21      list_real: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 9.49/5.21      set_real: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Num__Onum_J,type,
% 9.49/5.21      list_num: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
% 9.49/5.21      list_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
% 9.49/5.21      list_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__VEBT____Definitions__OVEBT,type,
% 9.49/5.21      vEBT_VEBT: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
% 9.49/5.21      set_rat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
% 9.49/5.21      set_num: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.21      set_nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
% 9.49/5.21      set_int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Code____Numeral__Ointeger,type,
% 9.49/5.21      code_integer: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Option__Ooption_I_Eo_J,type,
% 9.49/5.21      option_o: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Extended____Nat__Oenat,type,
% 9.49/5.21      extended_enat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__List__Olist_I_Eo_J,type,
% 9.49/5.21      list_o: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Complex__Ocomplex,type,
% 9.49/5.21      complex: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Set__Oset_I_Eo_J,type,
% 9.49/5.21      set_o: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Real__Oreal,type,
% 9.49/5.21      real: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Rat__Orat,type,
% 9.49/5.21      rat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Num__Onum,type,
% 9.49/5.21      num: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Nat__Onat,type,
% 9.49/5.21      nat: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(ty_n_t__Int__Oint,type,
% 9.49/5.21      int: $tType ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
% 9.49/5.21      archim2889992004027027881ng_rat: rat > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
% 9.49/5.21      archim7802044766580827645g_real: real > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
% 9.49/5.21      archim3151403230148437115or_rat: rat > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
% 9.49/5.21      archim6058952711729229775r_real: real > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
% 9.49/5.21      archimedean_frac_rat: rat > rat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
% 9.49/5.21      archim2898591450579166408c_real: real > real ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
% 9.49/5.21      archim7778729529865785530nd_rat: rat > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
% 9.49/5.21      archim8280529875227126926d_real: real > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
% 9.49/5.21      bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
% 9.49/5.21      bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_Eo_001_Eo,type,
% 9.49/5.21      bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_Eo_001_Eo,type,
% 9.49/5.21      bNF_re4297313714947099218al_o_o: ( ( nat > rat ) > real > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( real > $o ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
% 9.49/5.21      bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.49/5.21      bNF_re157797125943740599nt_int: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > product_prod_int_int ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Rat__Orat_J,type,
% 9.49/5.21      bNF_re3461391660133120880nt_rat: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > rat ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > rat ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_Eo_001_Eo,type,
% 9.49/5.21      bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.49/5.21      bNF_re6250860962936578807nt_int: ( int > int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 9.49/5.21      bNF_re2214769303045360666nt_rat: ( int > int > $o ) > ( product_prod_int_int > rat > $o ) > ( int > product_prod_int_int ) > ( int > rat ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
% 9.49/5.21      bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
% 9.49/5.21      bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 9.49/5.21      bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.49/5.21      bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
% 9.49/5.21      bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.49/5.21      bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
% 9.49/5.21      bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_Eo_001_Eo,type,
% 9.49/5.21      bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 9.49/5.21      bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
% 9.49/5.21      bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.49/5.21      bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_Eo_001_Eo,type,
% 9.49/5.21      bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 9.49/5.21      bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 9.49/5.21      bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.21      bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
% 9.49/5.21      bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.21      bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
% 9.49/5.21      bNF_We3818239936649020644el_nat: set_Pr1261947904930325089at_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Binomial_Obinomial,type,
% 9.49/5.21      binomial: nat > nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
% 9.49/5.21      gbinomial_complex: complex > nat > complex ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
% 9.49/5.21      gbinomial_int: int > nat > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
% 9.49/5.21      gbinomial_nat: nat > nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
% 9.49/5.21      gbinomial_rat: rat > nat > rat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
% 9.49/5.21      gbinomial_real: real > nat > real ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Oand__int__rel,type,
% 9.49/5.21      bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Oand__not__num,type,
% 9.49/5.21      bit_and_not_num: num > num > option_num ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
% 9.49/5.21      bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Oconcat__bit,type,
% 9.49/5.21      bit_concat_bit: nat > int > int > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
% 9.49/5.21      bit_ri7919022796975470100ot_int: int > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
% 9.49/5.21      bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
% 9.49/5.21      bit_ri631733984087533419it_int: nat > int > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
% 9.49/5.21      bit_se3949692690581998587nteger: code_integer > code_integer > code_integer ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
% 9.49/5.21      bit_se725231765392027082nd_int: int > int > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
% 9.49/5.21      bit_se727722235901077358nd_nat: nat > nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
% 9.49/5.21      bit_se8568078237143864401it_int: nat > int > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
% 9.49/5.21      bit_se8570568707652914677it_nat: nat > nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
% 9.49/5.21      bit_se1345352211410354436nteger: nat > code_integer > code_integer ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
% 9.49/5.21      bit_se2159334234014336723it_int: nat > int > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
% 9.49/5.21      bit_se2161824704523386999it_nat: nat > nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
% 9.49/5.21      bit_se2000444600071755411sk_int: nat > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
% 9.49/5.21      bit_se2002935070580805687sk_nat: nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Code____Numeral__Ointeger,type,
% 9.49/5.21      bit_se1080825931792720795nteger: code_integer > code_integer > code_integer ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
% 9.49/5.21      bit_se1409905431419307370or_int: int > int > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
% 9.49/5.21      bit_se1412395901928357646or_nat: nat > nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
% 9.49/5.21      bit_se545348938243370406it_int: nat > int > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
% 9.49/5.21      bit_se547839408752420682it_nat: nat > nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
% 9.49/5.21      bit_se2793503036327961859nteger: nat > code_integer > code_integer ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
% 9.49/5.21      bit_se7879613467334960850it_int: nat > int > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
% 9.49/5.21      bit_se7882103937844011126it_nat: nat > nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Code____Numeral__Ointeger,type,
% 9.49/5.21      bit_se1745604003318907178nteger: nat > code_integer > code_integer ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
% 9.49/5.21      bit_se2923211474154528505it_int: nat > int > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
% 9.49/5.21      bit_se2925701944663578781it_nat: nat > nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
% 9.49/5.21      bit_se8260200283734997820nteger: nat > code_integer > code_integer ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
% 9.49/5.21      bit_se4203085406695923979it_int: nat > int > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
% 9.49/5.21      bit_se4205575877204974255it_nat: nat > nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
% 9.49/5.21      bit_se6526347334894502574or_int: int > int > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
% 9.49/5.21      bit_se6528837805403552850or_nat: nat > nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
% 9.49/5.21      bit_se1146084159140164899it_int: int > nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
% 9.49/5.21      bit_se1148574629649215175it_nat: nat > nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Otake__bit__num,type,
% 9.49/5.21      bit_take_bit_num: nat > num > option_num ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
% 9.49/5.21      bit_un7362597486090784418nd_num: num > num > option_num ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
% 9.49/5.21      bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
% 9.49/5.21      bit_un2480387367778600638or_num: num > num > option_num ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
% 9.49/5.21      bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Code__Numeral_Obit__cut__integer,type,
% 9.49/5.21      code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Code__Numeral_Odivmod__abs,type,
% 9.49/5.21      code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Code__Numeral_Odivmod__integer,type,
% 9.49/5.21      code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
% 9.49/5.21      code_int_of_integer: code_integer > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
% 9.49/5.21      code_integer_of_int: int > code_integer ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Code__Numeral_Ointeger__of__nat,type,
% 9.49/5.21      code_integer_of_nat: nat > code_integer ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Code__Numeral_Onat__of__integer,type,
% 9.49/5.21      code_nat_of_integer: code_integer > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 9.49/5.21      comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 9.49/5.21      comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
% 9.49/5.21      complete_Inf_Inf_nat: set_nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.21      comple7806235888213564991et_nat: set_set_nat > set_nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
% 9.49/5.21      complete_Sup_Sup_nat: set_nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Complex_OArg,type,
% 9.49/5.21      arg: complex > real ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Complex_Ocis,type,
% 9.49/5.21      cis: real > complex ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Complex_Ocomplex_OComplex,type,
% 9.49/5.21      complex2: real > real > complex ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
% 9.49/5.21      condit2214826472909112428ve_nat: set_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.49/5.21      differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Deriv_Ohas__derivative_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.49/5.21      has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
% 9.49/5.21      has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Divides_Oadjust__div,type,
% 9.49/5.21      adjust_div: product_prod_int_int > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Divides_Odivmod__nat,type,
% 9.49/5.21      divmod_nat: nat > nat > product_prod_nat_nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Divides_Oeucl__rel__int,type,
% 9.49/5.21      eucl_rel_int: int > int > product_prod_int_int > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
% 9.49/5.21      unique6319869463603278526ux_int: product_prod_int_int > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
% 9.49/5.21      unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
% 9.49/5.21      unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
% 9.49/5.21      unique5052692396658037445od_int: num > num > product_prod_int_int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
% 9.49/5.21      unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
% 9.49/5.21      unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
% 9.49/5.21      unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
% 9.49/5.21      unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Code____Numeral__Ointeger,type,
% 9.49/5.21      comm_s8582702949713902594nteger: code_integer > nat > code_integer ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Complex__Ocomplex,type,
% 9.49/5.21      comm_s2602460028002588243omplex: complex > nat > complex ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Int__Oint,type,
% 9.49/5.21      comm_s4660882817536571857er_int: int > nat > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Nat__Onat,type,
% 9.49/5.21      comm_s4663373288045622133er_nat: nat > nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Rat__Orat,type,
% 9.49/5.21      comm_s4028243227959126397er_rat: rat > nat > rat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
% 9.49/5.21      comm_s7457072308508201937r_real: real > nat > real ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Code____Numeral__Ointeger,type,
% 9.49/5.21      semiri3624122377584611663nteger: nat > code_integer ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
% 9.49/5.21      semiri5044797733671781792omplex: nat > complex ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
% 9.49/5.21      semiri1406184849735516958ct_int: nat > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
% 9.49/5.21      semiri1408675320244567234ct_nat: nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Rat__Orat,type,
% 9.49/5.21      semiri773545260158071498ct_rat: nat > rat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
% 9.49/5.21      semiri2265585572941072030t_real: nat > real ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
% 9.49/5.21      invers8013647133539491842omplex: complex > complex ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
% 9.49/5.21      inverse_inverse_rat: rat > rat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
% 9.49/5.21      inverse_inverse_real: real > real ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
% 9.49/5.21      at_bot_real: filter_real ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
% 9.49/5.21      at_top_nat: filter_nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
% 9.49/5.21      at_top_real: filter_real ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
% 9.49/5.21      eventually_nat: ( nat > $o ) > filter_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Filter_Oeventually_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.21      eventu1038000079068216329at_nat: ( product_prod_nat_nat > $o ) > filter1242075044329608583at_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
% 9.49/5.21      eventually_real: ( real > $o ) > filter_real > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.21      filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
% 9.49/5.21      filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.49/5.21      filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 9.49/5.21      princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 9.49/5.21      princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Filter_Oprod__filter_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.21      prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
% 9.49/5.21      finite_card_complex: set_complex > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
% 9.49/5.21      finite_card_list_nat: set_list_nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
% 9.49/5.21      finite_card_nat: set_nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
% 9.49/5.21      finite410649719033368117t_unit: set_Product_unit > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.21      finite_card_set_nat: set_set_nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
% 9.49/5.21      finite_finite_o: set_o > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
% 9.49/5.21      finite3207457112153483333omplex: set_complex > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
% 9.49/5.21      finite_finite_int: set_int > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_I_Eo_J,type,
% 9.49/5.21      finite_finite_list_o: set_list_o > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 9.49/5.21      finite8712137658972009173omplex: set_list_complex > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Int__Oint_J,type,
% 9.49/5.21      finite3922522038869484883st_int: set_list_int > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
% 9.49/5.21      finite8100373058378681591st_nat: set_list_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.21      finite500796754983035824at_nat: set_li5450038453877631591at_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.21      finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 9.49/5.21      finite_finite_nat: set_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ofinite_001t__Num__Onum,type,
% 9.49/5.21      finite_finite_num: set_num > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.21      finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.49/5.21      finite4343798906461161616at_nat: set_Pr4329608150637261639at_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ofinite_001t__Rat__Orat,type,
% 9.49/5.21      finite_finite_rat: set_rat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
% 9.49/5.21      finite_finite_real: set_real > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 9.49/5.21      finite6551019134538273531omplex: set_set_complex > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
% 9.49/5.21      finite6197958912794628473et_int: set_set_int > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.21      finite1152437895449049373et_nat: set_set_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.21      finite9047747110432174090at_nat: set_se7855581050983116737at_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Finite__Set_Ofinite_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.21      finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 9.49/5.21      bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 9.49/5.21      bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.21      bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun_Ocomp_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 9.49/5.21      comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 9.49/5.21      comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
% 9.49/5.21      comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.21      comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.21      inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.49/5.21      inj_on_real_real: ( real > real ) > set_real > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
% 9.49/5.21      inj_on_set_nat_nat: ( set_nat > nat ) > set_set_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.49/5.21      map_fu4960017516451851995nt_int: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > int > int > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 9.49/5.21      map_fu3667384564859982768at_int: ( int > product_prod_nat_nat ) > ( product_prod_nat_nat > int ) > ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
% 9.49/5.21      map_fu4333342158222067775at_rat: ( rat > product_prod_int_int ) > ( ( product_prod_int_int > product_prod_int_int ) > rat > rat ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > rat > rat > rat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 9.49/5.21      map_fu5673905371560938248nt_rat: ( rat > product_prod_int_int ) > ( product_prod_int_int > rat ) > ( product_prod_int_int > product_prod_int_int ) > rat > rat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun_Ostrict__mono__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.21      strict1292158309912662752at_nat: ( nat > nat ) > set_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun_Othe__inv__into_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.49/5.21      the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun__Def_Omax__strict,type,
% 9.49/5.21      fun_max_strict: set_Pr4329608150637261639at_nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun__Def_Omax__weak,type,
% 9.49/5.21      fun_max_weak: set_Pr4329608150637261639at_nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun__Def_Omin__strict,type,
% 9.49/5.21      fun_min_strict: set_Pr4329608150637261639at_nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun__Def_Omin__weak,type,
% 9.49/5.21      fun_min_weak: set_Pr4329608150637261639at_nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun__Def_Opair__leq,type,
% 9.49/5.21      fun_pair_leq: set_Pr8693737435421807431at_nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun__Def_Opair__less,type,
% 9.49/5.21      fun_pair_less: set_Pr8693737435421807431at_nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Fun__Def_Oreduction__pair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.21      fun_re2478310338295953701at_nat: produc1319942482725812455at_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
% 9.49/5.21      gcd_Gcd_nat: set_nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_GCD_Obezw,type,
% 9.49/5.21      bezw: nat > nat > product_prod_int_int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_GCD_Obezw__rel,type,
% 9.49/5.21      bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Int__Oint,type,
% 9.49/5.21      gcd_gcd_int: int > int > int ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Nat__Onat,type,
% 9.49/5.21      gcd_gcd_nat: nat > nat > nat ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_GCD_Ogcd__nat__rel,type,
% 9.49/5.21      gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Groups_Oabs__class_Oabs_001t__Code____Numeral__Ointeger,type,
% 9.49/5.21      abs_abs_Code_integer: code_integer > code_integer ).
% 9.49/5.21  
% 9.49/5.21  thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex,type,
% 9.49/5.22      abs_abs_complex: complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
% 9.49/5.22      abs_abs_int: int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
% 9.49/5.22      abs_abs_rat: rat > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
% 9.49/5.22      abs_abs_real: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Int__Oint_M_Eo_J,type,
% 9.49/5.22      minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
% 9.49/5.22      minus_1139252259498527702_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
% 9.49/5.22      minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 9.49/5.22      minus_2270307095948843157_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Real__Oreal_M_Eo_J,type,
% 9.49/5.22      minus_minus_real_o: ( real > $o ) > ( real > $o ) > real > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
% 9.49/5.22      minus_6910147592129066416_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
% 9.49/5.22      minus_minus_complex: complex > complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
% 9.49/5.22      minus_minus_int: int > int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 9.49/5.22      minus_minus_nat: nat > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
% 9.49/5.22      minus_minus_rat: rat > rat > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
% 9.49/5.22      minus_minus_real: real > real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 9.49/5.22      minus_811609699411566653omplex: set_complex > set_complex > set_complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
% 9.49/5.22      minus_minus_set_int: set_int > set_int > set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 9.49/5.22      minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      minus_minus_set_nat: set_nat > set_nat > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Num__Onum_J,type,
% 9.49/5.22      minus_minus_set_num: set_num > set_num > set_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.22      minus_1356011639430497352at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 9.49/5.22      minus_3314409938677909166at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Rat__Orat_J,type,
% 9.49/5.22      minus_minus_set_rat: set_rat > set_rat > set_rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
% 9.49/5.22      minus_minus_set_real: set_real > set_real > set_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 9.49/5.22      minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Oone__class_Oone_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      one_one_Code_integer: code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
% 9.49/5.22      one_one_complex: complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
% 9.49/5.22      one_on7984719198319812577d_enat: extended_enat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
% 9.49/5.22      one_one_int: int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 9.49/5.22      one_one_nat: nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
% 9.49/5.22      one_one_rat: rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
% 9.49/5.22      one_one_real: real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Oplus__class_Oplus_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
% 9.49/5.22      plus_plus_complex: complex > complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
% 9.49/5.22      plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
% 9.49/5.22      plus_plus_int: int > int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 9.49/5.22      plus_plus_nat: nat > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
% 9.49/5.22      plus_plus_num: num > num > num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
% 9.49/5.22      plus_plus_rat: rat > rat > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
% 9.49/5.22      plus_plus_real: real > real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Osgn__class_Osgn_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      sgn_sgn_Code_integer: code_integer > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Osgn__class_Osgn_001t__Complex__Ocomplex,type,
% 9.49/5.22      sgn_sgn_complex: complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
% 9.49/5.22      sgn_sgn_int: int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Osgn__class_Osgn_001t__Rat__Orat,type,
% 9.49/5.22      sgn_sgn_rat: rat > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Osgn__class_Osgn_001t__Real__Oreal,type,
% 9.49/5.22      sgn_sgn_real: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Otimes__class_Otimes_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      times_3573771949741848930nteger: code_integer > code_integer > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
% 9.49/5.22      times_times_complex: complex > complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nat__Oenat,type,
% 9.49/5.22      times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
% 9.49/5.22      times_times_int: int > int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
% 9.49/5.22      times_times_nat: nat > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
% 9.49/5.22      times_times_rat: rat > rat > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
% 9.49/5.22      times_times_real: real > real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Int__Oint_M_Eo_J,type,
% 9.49/5.22      uminus_uminus_int_o: ( int > $o ) > int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
% 9.49/5.22      uminus5770388063884162150_nat_o: ( list_nat > $o ) > list_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Nat__Onat_M_Eo_J,type,
% 9.49/5.22      uminus_uminus_nat_o: ( nat > $o ) > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 9.49/5.22      uminus8676089048583255045_nat_o: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Real__Oreal_M_Eo_J,type,
% 9.49/5.22      uminus_uminus_real_o: ( real > $o ) > real > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
% 9.49/5.22      uminus6401447641752708672_nat_o: ( set_nat > $o ) > set_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      uminus1351360451143612070nteger: code_integer > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
% 9.49/5.22      uminus1482373934393186551omplex: complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
% 9.49/5.22      uminus_uminus_int: int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
% 9.49/5.22      uminus_uminus_rat: rat > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
% 9.49/5.22      uminus_uminus_real: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Int__Oint_J,type,
% 9.49/5.22      uminus1532241313380277803et_int: set_int > set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 9.49/5.22      uminus3195874150345416415st_nat: set_list_nat > set_list_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      uminus5710092332889474511et_nat: set_nat > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.22      uminus6524753893492686040at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 9.49/5.22      uminus935396558254630718at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Real__Oreal_J,type,
% 9.49/5.22      uminus612125837232591019t_real: set_real > set_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 9.49/5.22      uminus613421341184616069et_nat: set_set_nat > set_set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ozero__class_Ozero_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      zero_z3403309356797280102nteger: code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
% 9.49/5.22      zero_zero_complex: complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
% 9.49/5.22      zero_z5237406670263579293d_enat: extended_enat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
% 9.49/5.22      zero_zero_int: int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 9.49/5.22      zero_zero_nat: nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,type,
% 9.49/5.22      zero_zero_rat: rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 9.49/5.22      zero_zero_real: real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_Eo_001t__Nat__Onat,type,
% 9.49/5.22      groups8507830703676809646_o_nat: ( $o > nat ) > set_o > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 9.49/5.22      groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Int__Oint,type,
% 9.49/5.22      groups5690904116761175830ex_int: ( complex > int ) > set_complex > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Nat__Onat,type,
% 9.49/5.22      groups5693394587270226106ex_nat: ( complex > nat ) > set_complex > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Rat__Orat,type,
% 9.49/5.22      groups5058264527183730370ex_rat: ( complex > rat ) > set_complex > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Real__Oreal,type,
% 9.49/5.22      groups5808333547571424918x_real: ( complex > real ) > set_complex > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      groups7873554091576472773nteger: ( int > code_integer ) > set_int > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Complex__Ocomplex,type,
% 9.49/5.22      groups3049146728041665814omplex: ( int > complex ) > set_int > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Int__Oint,type,
% 9.49/5.22      groups4538972089207619220nt_int: ( int > int ) > set_int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Nat__Onat,type,
% 9.49/5.22      groups4541462559716669496nt_nat: ( int > nat ) > set_int > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Rat__Orat,type,
% 9.49/5.22      groups3906332499630173760nt_rat: ( int > rat ) > set_int > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Real__Oreal,type,
% 9.49/5.22      groups8778361861064173332t_real: ( int > real ) > set_int > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      groups7501900531339628137nteger: ( nat > code_integer ) > set_nat > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 9.49/5.22      groups2073611262835488442omplex: ( nat > complex ) > set_nat > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
% 9.49/5.22      groups7108830773950497114d_enat: ( nat > extended_enat ) > set_nat > extended_enat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
% 9.49/5.22      groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.22      groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Rat__Orat,type,
% 9.49/5.22      groups2906978787729119204at_rat: ( nat > rat ) > set_nat > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
% 9.49/5.22      groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 9.49/5.22      groups977919841031483927at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Real__Oreal,type,
% 9.49/5.22      groups4567486121110086003t_real: ( product_prod_nat_nat > real ) > set_Pr1261947904930325089at_nat > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Nat__Onat,type,
% 9.49/5.22      groups3860910324918113789at_nat: ( produc3843707927480180839at_nat > nat ) > set_Pr4329608150637261639at_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      groups7713935264441627589nteger: ( real > code_integer ) > set_real > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Complex__Ocomplex,type,
% 9.49/5.22      groups5754745047067104278omplex: ( real > complex ) > set_real > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Int__Oint,type,
% 9.49/5.22      groups1932886352136224148al_int: ( real > int ) > set_real > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Nat__Onat,type,
% 9.49/5.22      groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Rat__Orat,type,
% 9.49/5.22      groups1300246762558778688al_rat: ( real > rat ) > set_real > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.49/5.22      groups8097168146408367636l_real: ( real > real ) > set_real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
% 9.49/5.22      groups8294997508430121362at_nat: ( set_nat > nat ) > set_set_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 9.49/5.22      groups771621172384141258BT_nat: ( vEBT_VEBT > nat ) > set_VEBT_VEBT > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 9.49/5.22      groups808145749697022017omplex: ( complex > complex ) > set_complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Complex__Ocomplex_001t__Int__Oint,type,
% 9.49/5.22      groups2909182065852811199ex_int: ( complex > int ) > set_complex > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Complex__Ocomplex_001t__Nat__Onat,type,
% 9.49/5.22      groups2911672536361861475ex_nat: ( complex > nat ) > set_complex > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Complex__Ocomplex_001t__Rat__Orat,type,
% 9.49/5.22      groups2276542476275365739ex_rat: ( complex > rat ) > set_complex > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Complex__Ocomplex_001t__Real__Oreal,type,
% 9.49/5.22      groups5737402329758386879x_real: ( complex > real ) > set_complex > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Int__Oint_001t__Complex__Ocomplex,type,
% 9.49/5.22      groups267424677133301183omplex: ( int > complex ) > set_int > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Int__Oint_001t__Int__Oint,type,
% 9.49/5.22      groups2983280209131991357nt_int: ( int > int ) > set_int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Int__Oint_001t__Nat__Onat,type,
% 9.49/5.22      groups2985770679641041633nt_nat: ( int > nat ) > set_int > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Int__Oint_001t__Rat__Orat,type,
% 9.49/5.22      groups2350640619554545897nt_rat: ( int > rat ) > set_int > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Int__Oint_001t__Real__Oreal,type,
% 9.49/5.22      groups1523912220035142973t_real: ( int > real ) > set_int > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 9.49/5.22      groups8515261248781899619omplex: ( nat > complex ) > set_nat > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Nat__Onat_001t__Int__Oint,type,
% 9.49/5.22      groups1983926497230936801at_int: ( nat > int ) > set_nat > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.22      groups1986416967739987077at_nat: ( nat > nat ) > set_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Nat__Onat_001t__Rat__Orat,type,
% 9.49/5.22      groups1351286907653491341at_rat: ( nat > rat ) > set_nat > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Nat__Onat_001t__Real__Oreal,type,
% 9.49/5.22      groups8560362682196896993t_real: ( nat > real ) > set_nat > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Real__Oreal_001t__Complex__Ocomplex,type,
% 9.49/5.22      groups5683813829254066239omplex: ( real > complex ) > set_real > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Real__Oreal_001t__Int__Oint,type,
% 9.49/5.22      groups3901808747961969597al_int: ( real > int ) > set_real > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Real__Oreal_001t__Nat__Onat,type,
% 9.49/5.22      groups3904299218471019873al_nat: ( real > nat ) > set_real > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Real__Oreal_001t__Rat__Orat,type,
% 9.49/5.22      groups3269169158384524137al_rat: ( real > rat ) > set_real > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.49/5.22      groups97945582718554045l_real: ( real > real ) > set_real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      groups8682486955453173170nteger: ( complex > code_integer ) > set_complex > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 9.49/5.22      groups3708469109370488835omplex: ( complex > complex ) > set_complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Int__Oint,type,
% 9.49/5.22      groups858564598930262913ex_int: ( complex > int ) > set_complex > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Nat__Onat,type,
% 9.49/5.22      groups861055069439313189ex_nat: ( complex > nat ) > set_complex > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Rat__Orat,type,
% 9.49/5.22      groups225925009352817453ex_rat: ( complex > rat ) > set_complex > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Real__Oreal,type,
% 9.49/5.22      groups766887009212190081x_real: ( complex > real ) > set_complex > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      groups3827104343326376752nteger: ( int > code_integer ) > set_int > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Complex__Ocomplex,type,
% 9.49/5.22      groups7440179247065528705omplex: ( int > complex ) > set_int > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Int__Oint,type,
% 9.49/5.22      groups1705073143266064639nt_int: ( int > int ) > set_int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Nat__Onat,type,
% 9.49/5.22      groups1707563613775114915nt_nat: ( int > nat ) > set_int > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Rat__Orat,type,
% 9.49/5.22      groups1072433553688619179nt_rat: ( int > rat ) > set_int > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Real__Oreal,type,
% 9.49/5.22      groups2316167850115554303t_real: ( int > real ) > set_int > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
% 9.49/5.22      groups2907647131375434839at_nat: ( list_nat > nat ) > set_list_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      groups3455450783089532116nteger: ( nat > code_integer ) > set_nat > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 9.49/5.22      groups6464643781859351333omplex: ( nat > complex ) > set_nat > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Int__Oint,type,
% 9.49/5.22      groups705719431365010083at_int: ( nat > int ) > set_nat > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.22      groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Rat__Orat,type,
% 9.49/5.22      groups73079841787564623at_rat: ( nat > rat ) > set_nat > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Real__Oreal,type,
% 9.49/5.22      groups129246275422532515t_real: ( nat > real ) > set_nat > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 9.49/5.22      groups4077766827762148844at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      groups6225526099057966256nteger: ( real > code_integer ) > set_real > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Complex__Ocomplex,type,
% 9.49/5.22      groups713298508707869441omplex: ( real > complex ) > set_real > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Int__Oint,type,
% 9.49/5.22      groups4694064378042380927al_int: ( real > int ) > set_real > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Nat__Onat,type,
% 9.49/5.22      groups4696554848551431203al_nat: ( real > nat ) > set_real > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Rat__Orat,type,
% 9.49/5.22      groups4061424788464935467al_rat: ( real > rat ) > set_real > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.49/5.22      groups1681761925125756287l_real: ( real > real ) > set_real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
% 9.49/5.22      groups4248547760180025341at_nat: ( set_nat > nat ) > set_set_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001_Eo_001t__Int__Oint,type,
% 9.49/5.22      groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
% 9.49/5.22      groups4561878855575611511st_nat: list_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_HOL_OThe_001t__Int__Oint,type,
% 9.49/5.22      the_int: ( int > $o ) > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_HOL_OThe_001t__Nat__Onat,type,
% 9.49/5.22      the_nat: ( nat > $o ) > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 9.49/5.22      the_Pr8210177043389155639nteger: ( produc8763457246119570046nteger > $o ) > produc8763457246119570046nteger ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.49/5.22      the_Pr6882841213465913158nt_int: ( produc7773217078559923341nt_int > $o ) > produc7773217078559923341nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 9.49/5.22      the_Pr6653488032121699663nteger: ( produc1908205239877642774nteger > $o ) > produc1908205239877642774nteger ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.49/5.22      the_Pr5445864913131713084nt_int: ( produc2285326912895808259nt_int > $o ) > produc2285326912895808259nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.49/5.22      the_Pr4378521158711661632nt_int: ( product_prod_int_int > $o ) > product_prod_int_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      the_Pr7557018466319803784at_nat: ( product_prod_nat_nat > $o ) > product_prod_nat_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.49/5.22      the_Pr4604535484834969198at_nat: ( produc3843707927480180839at_nat > $o ) > produc3843707927480180839at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_HOL_OThe_001t__Real__Oreal,type,
% 9.49/5.22      the_real: ( real > $o ) > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_If_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.49/5.22      if_int_int: $o > ( int > int ) > ( int > int ) > int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_If_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      if_Code_integer: $o > code_integer > code_integer > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_If_001t__Complex__Ocomplex,type,
% 9.49/5.22      if_complex: $o > complex > complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_If_001t__Extended____Nat__Oenat,type,
% 9.49/5.22      if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_If_001t__Int__Oint,type,
% 9.49/5.22      if_int: $o > int > int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
% 9.49/5.22      if_list_int: $o > list_int > list_int > list_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
% 9.49/5.22      if_list_nat: $o > list_nat > list_nat > list_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_If_001t__Nat__Onat,type,
% 9.49/5.22      if_nat: $o > nat > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_If_001t__Num__Onum,type,
% 9.49/5.22      if_num: $o > num > num > num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
% 9.49/5.22      if_option_nat: $o > option_nat > option_nat > option_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_If_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.49/5.22      if_option_num: $o > option_num > option_num > option_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 9.49/5.22      if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 9.49/5.22      if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.49/5.22      if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_If_001t__Rat__Orat,type,
% 9.49/5.22      if_rat: $o > rat > rat > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_If_001t__Real__Oreal,type,
% 9.49/5.22      if_real: $o > real > real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
% 9.49/5.22      if_set_int: $o > set_int > set_int > set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_If_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
% 9.49/5.22      infini8530281810654367211te_nat: set_nat > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Int_OAbs__Integ,type,
% 9.49/5.22      abs_Integ: product_prod_nat_nat > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Int_ORep__Integ,type,
% 9.49/5.22      rep_Integ: int > product_prod_nat_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Int_Oint__ge__less__than,type,
% 9.49/5.22      int_ge_less_than: int > set_Pr958786334691620121nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Int_Oint__ge__less__than2,type,
% 9.49/5.22      int_ge_less_than2: int > set_Pr958786334691620121nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Int_Ointrel,type,
% 9.49/5.22      intrel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Int_Onat,type,
% 9.49/5.22      nat2: int > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Int_Opcr__int,type,
% 9.49/5.22      pcr_int: product_prod_nat_nat > int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Int_Opower__int_001t__Real__Oreal,type,
% 9.49/5.22      power_int_real: real > int > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      ring_18347121197199848620nteger: int > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
% 9.49/5.22      ring_17405671764205052669omplex: int > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
% 9.49/5.22      ring_1_of_int_int: int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
% 9.49/5.22      ring_1_of_int_rat: int > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
% 9.49/5.22      ring_1_of_int_real: int > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
% 9.49/5.22      inf_inf_nat: nat > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.22      inf_in2572325071724192079at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
% 9.49/5.22      semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
% 9.49/5.22      sup_sup_nat: nat > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      sup_sup_set_nat: set_nat > set_nat > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 9.49/5.22      sup_su5525570899277871387at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
% 9.49/5.22      lattic8265883725875713057ax_nat: set_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Nat__Onat,type,
% 9.49/5.22      lattic5364784637807008409ex_nat: ( complex > nat ) > set_complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Num__Onum,type,
% 9.49/5.22      lattic1922116423962787043ex_num: ( complex > num ) > set_complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Rat__Orat,type,
% 9.49/5.22      lattic4729654577720512673ex_rat: ( complex > rat ) > set_complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Real__Oreal,type,
% 9.49/5.22      lattic8794016678065449205x_real: ( complex > real ) > set_complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Num__Onum,type,
% 9.49/5.22      lattic5003618458639192673nt_num: ( int > num ) > set_int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Rat__Orat,type,
% 9.49/5.22      lattic7811156612396918303nt_rat: ( int > rat ) > set_int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Real__Oreal,type,
% 9.49/5.22      lattic2675449441010098035t_real: ( int > real ) > set_int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Num__Onum,type,
% 9.49/5.22      lattic4004264746738138117at_num: ( nat > num ) > set_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Rat__Orat,type,
% 9.49/5.22      lattic6811802900495863747at_rat: ( nat > rat ) > set_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Real__Oreal,type,
% 9.49/5.22      lattic488527866317076247t_real: ( nat > real ) > set_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Nat__Onat,type,
% 9.49/5.22      lattic5055836439445974935al_nat: ( real > nat ) > set_real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Num__Onum,type,
% 9.49/5.22      lattic1613168225601753569al_num: ( real > num ) > set_real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Rat__Orat,type,
% 9.49/5.22      lattic4420706379359479199al_rat: ( real > rat ) > set_real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.49/5.22      lattic8440615504127631091l_real: ( real > real ) > set_real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oappend_001t__Int__Oint,type,
% 9.49/5.22      append_int: list_int > list_int > list_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oappend_001t__Nat__Onat,type,
% 9.49/5.22      append_nat: list_nat > list_nat > list_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ocount__list_001_Eo,type,
% 9.49/5.22      count_list_o: list_o > $o > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ocount__list_001t__Complex__Ocomplex,type,
% 9.49/5.22      count_list_complex: list_complex > complex > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ocount__list_001t__Int__Oint,type,
% 9.49/5.22      count_list_int: list_int > int > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
% 9.49/5.22      count_list_nat: list_nat > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ocount__list_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      count_4203492906077236349at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ocount__list_001t__Real__Oreal,type,
% 9.49/5.22      count_list_real: list_real > real > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ocount__list_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      count_list_set_nat: list_set_nat > set_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ocount__list_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      count_list_VEBT_VEBT: list_VEBT_VEBT > vEBT_VEBT > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Odistinct_001t__Int__Oint,type,
% 9.49/5.22      distinct_int: list_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
% 9.49/5.22      distinct_nat: list_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Odrop_001t__Nat__Onat,type,
% 9.49/5.22      drop_nat: nat > list_nat > list_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oenumerate_001_Eo,type,
% 9.49/5.22      enumerate_o: nat > list_o > list_P7333126701944960589_nat_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oenumerate_001t__Int__Oint,type,
% 9.49/5.22      enumerate_int: nat > list_int > list_P3521021558325789923at_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oenumerate_001t__Nat__Onat,type,
% 9.49/5.22      enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oenumerate_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      enumerate_VEBT_VEBT: nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ofind_001_Eo,type,
% 9.49/5.22      find_o: ( $o > $o ) > list_o > option_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ofind_001t__Int__Oint,type,
% 9.49/5.22      find_int: ( int > $o ) > list_int > option_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ofind_001t__Nat__Onat,type,
% 9.49/5.22      find_nat: ( nat > $o ) > list_nat > option_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ofind_001t__Num__Onum,type,
% 9.49/5.22      find_num: ( num > $o ) > list_num > option_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ofind_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      find_P8199882355184865565at_nat: ( product_prod_nat_nat > $o ) > list_P6011104703257516679at_nat > option4927543243414619207at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ofind_001t__Real__Oreal,type,
% 9.49/5.22      find_real: ( real > $o ) > list_real > option_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ofind_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      find_set_nat: ( set_nat > $o ) > list_set_nat > option_set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ofind_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      find_VEBT_VEBT: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > option_VEBT_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.22      fold_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olast_001t__Nat__Onat,type,
% 9.49/5.22      last_nat: list_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olinorder__class_Osort__key_001t__Int__Oint_001t__Int__Oint,type,
% 9.49/5.22      linord1735203802627413978nt_int: ( int > int ) > list_int > list_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olinorder__class_Osort__key_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.22      linord738340561235409698at_nat: ( nat > nat ) > list_nat > list_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
% 9.49/5.22      linord2614967742042102400et_nat: set_nat > list_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
% 9.49/5.22      cons_int: int > list_int > list_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
% 9.49/5.22      cons_nat: nat > list_nat > list_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
% 9.49/5.22      nil_int: list_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
% 9.49/5.22      nil_nat: list_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
% 9.49/5.22      hd_nat: list_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.22      map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 9.49/5.22      set_Co8062243466402858685e_term: list_C878401137130745250e_term > set_Co9149898834107579976e_term ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 9.49/5.22      set_in5217446777445088012e_term: list_i8448526496819171953e_term > set_in3461395444621081367e_term ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 9.49/5.22      set_Pr8342322266483756581e_term: list_P1316552470764441098e_term > set_Pr7604974323444597168e_term ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 9.49/5.22      set_Pr16608062948090134e_term: list_P1743416141875011707e_term > set_Pr3642885161833720865e_term ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001_Eo,type,
% 9.49/5.22      set_o2: list_o > set_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
% 9.49/5.22      set_complex2: list_complex > set_complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
% 9.49/5.22      set_int2: list_int > set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
% 9.49/5.22      set_nat2: list_nat > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Num__Onum,type,
% 9.49/5.22      set_num2: list_num > set_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 9.49/5.22      set_Pr2999063419360598313nteger: list_P5311841565141990158nteger > set_Pr8056137968301705908nteger ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.49/5.22      set_Pr1633835384712236856nt_int: list_P8915022641806594461nt_int > set_Pr1872883991513573699nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 9.49/5.22      set_Pr2135590979564877377nteger: list_P7828571989066258726nteger > set_Pr1281608226676607948nteger ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.49/5.22      set_Pr4943052134776177454nt_int: list_P651320350408439699nt_int > set_Pr9222295170931077689nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
% 9.49/5.22      set_Product_prod_o_o2: list_P4002435161011370285od_o_o > set_Product_prod_o_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 9.49/5.22      set_Pr2828948584524939422_o_int: list_P3795440434834930179_o_int > set_Pr8834758594704517033_o_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
% 9.49/5.22      set_Pr7006799604034136130_o_nat: list_P6285523579766656935_o_nat > set_Pr2101469702781467981_o_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J,type,
% 9.49/5.22      set_Pr2600826154070092190o_real: list_P5232166724548748803o_real > set_Pr6573716822653411497o_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.22      set_Pr655345902815428824T_VEBT: list_P7495141550334521929T_VEBT > set_Pr7543698050874017315T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 9.49/5.22      set_Pr920681315882439344nteger: list_P5578671422887162913nteger > set_Pr4811707699266497531nteger ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.49/5.22      set_Pr2470121279949933262nt_int: list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
% 9.49/5.22      set_Pr112895574167722958t_real: list_P6863124054624500543t_real > set_Pr3538720872664544793t_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.22      set_Pr8714266321650254504T_VEBT: list_P7524865323317820941T_VEBT > set_Pr8044002425091019955T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.22      set_Pr5518436109238095868at_nat: list_P8469869581646625389at_nat > set_Pr8693737435421807431at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_M_Eo_J,type,
% 9.49/5.22      set_Pr5196769464307566348real_o: list_P3595434254542482545real_o > set_Pr4936984352647145239real_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
% 9.49/5.22      set_Pr8219819362198175822al_int: list_P4344331454722006975al_int > set_Pr1019928272762051225al_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
% 9.49/5.22      set_Pr3174298344852596722al_nat: list_P6834414599653733731al_nat > set_Pr3510011417693777981al_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 9.49/5.22      set_Pr5999470521830281550l_real: list_P8689742595348180415l_real > set_Pr6218003697084177305l_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.22      set_Pr8897343066327330088T_VEBT: list_P877281246627933069T_VEBT > set_Pr6019664923565264691T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
% 9.49/5.22      set_Pr9040384385603167362et_nat: list_P6254988961118846195et_nat > set_Pr5488025237498180813et_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.49/5.22      set_Pr3765526544606949372at_nat: list_P5464809261938338413at_nat > set_Pr4329608150637261639at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 9.49/5.22      set_Pr7708085864119495200VEBT_o: list_P3126845725202233233VEBT_o > set_Pr3175402225741728619VEBT_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 9.49/5.22      set_Pr2853735649769556538BT_int: list_P4547456442757143711BT_int > set_Pr5066593544530342725BT_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 9.49/5.22      set_Pr7031586669278753246BT_nat: list_P7037539587688870467BT_nat > set_Pr7556676689462069481BT_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
% 9.49/5.22      set_Pr1087130671499945274T_real: list_P2623026923184700063T_real > set_Pr7765410600122031685T_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.22      set_Pr9182192707038809660T_VEBT: list_P7413028617227757229T_VEBT > set_Pr6192946355708809607T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
% 9.49/5.22      set_real2: list_real > set_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      set_set_nat2: list_set_nat > set_set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
% 9.49/5.22      tl_nat: list_nat > list_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist__update_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 9.49/5.22      list_u4743598893156345252e_term: list_C878401137130745250e_term > nat > ( code_integer > option6357759511663192854e_term ) > list_C878401137130745250e_term ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist__update_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 9.49/5.22      list_u8946639151299769843e_term: list_i8448526496819171953e_term > nat > ( int > option6357759511663192854e_term ) > list_i8448526496819171953e_term ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist__update_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 9.49/5.22      list_u877304756163299468e_term: list_P1316552470764441098e_term > nat > ( produc6241069584506657477e_term > option6357759511663192854e_term ) > list_P1316552470764441098e_term ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist__update_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 9.49/5.22      list_u3533491785856317309e_term: list_P1743416141875011707e_term > nat > ( produc8551481072490612790e_term > option6357759511663192854e_term ) > list_P1743416141875011707e_term ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist__update_001_Eo,type,
% 9.49/5.22      list_update_o: list_o > nat > $o > list_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist__update_001t__Int__Oint,type,
% 9.49/5.22      list_update_int: list_int > nat > int > list_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
% 9.49/5.22      list_update_nat: list_nat > nat > nat > list_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 9.49/5.22      list_u1133519416628930960nteger: list_P5311841565141990158nteger > nat > produc8763457246119570046nteger > list_P5311841565141990158nteger ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.49/5.22      list_u4780935413889332127nt_int: list_P8915022641806594461nt_int > nat > produc7773217078559923341nt_int > list_P8915022641806594461nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 9.49/5.22      list_u234853988314817064nteger: list_P7828571989066258726nteger > nat > produc1908205239877642774nteger > list_P7828571989066258726nteger ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.49/5.22      list_u7736365598306452245nt_int: list_P651320350408439699nt_int > nat > produc2285326912895808259nt_int > list_P651320350408439699nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 9.49/5.22      list_u2254550707601501961nteger: list_P5578671422887162913nteger > nat > produc8923325533196201883nteger > list_P5578671422887162913nteger ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.49/5.22      list_u3002344382305578791nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int > list_P5707943133018811711nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.49/5.22      list_u4696772448584712917at_nat: list_P5464809261938338413at_nat > nat > produc3843707927480180839at_nat > list_P5464809261938338413at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.22      list_u6961636818849549845T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT > list_P7413028617227757229T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
% 9.49/5.22      list_update_real: list_real > nat > real > list_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      list_update_set_nat: list_set_nat > nat > set_nat > list_set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olistrel1_001_Eo,type,
% 9.49/5.22      listrel1_o: set_Product_prod_o_o > set_Pr6227168374412355847list_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olistrel1_001t__Int__Oint,type,
% 9.49/5.22      listrel1_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olistrel1_001t__Nat__Onat,type,
% 9.49/5.22      listrel1_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olistrel1_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      listrel1_VEBT_VEBT: set_Pr6192946355708809607T_VEBT > set_Pr1916528119006554503T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olistrel1p_001t__Int__Oint,type,
% 9.49/5.22      listrel1p_int: ( int > int > $o ) > list_int > list_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Olistrel1p_001t__Nat__Onat,type,
% 9.49/5.22      listrel1p_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Onth_001_Eo,type,
% 9.49/5.22      nth_o: list_o > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Onth_001t__Int__Oint,type,
% 9.49/5.22      nth_int: list_int > nat > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 9.49/5.22      nth_nat: list_nat > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Onth_001t__Num__Onum,type,
% 9.49/5.22      nth_num: list_num > nat > num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
% 9.49/5.22      nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 9.49/5.22      nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
% 9.49/5.22      nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.22      nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
% 9.49/5.22      nth_Pr112076138515278198_nat_o: list_P7333126701944960589_nat_o > nat > product_prod_nat_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
% 9.49/5.22      nth_Pr3440142176431000676at_int: list_P3521021558325789923at_int > nat > product_prod_nat_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.22      nth_Pr744662078594809490T_VEBT: list_P5647936690300460905T_VEBT > nat > produc8025551001238799321T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 9.49/5.22      nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 9.49/5.22      nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 9.49/5.22      nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.22      nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Onth_001t__Real__Oreal,type,
% 9.49/5.22      nth_real: list_real > nat > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      nth_set_nat: list_set_nat > nat > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
% 9.49/5.22      product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
% 9.49/5.22      product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
% 9.49/5.22      product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
% 9.49/5.22      product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 9.49/5.22      product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 9.49/5.22      produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 9.49/5.22      produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oremdups_001t__Nat__Onat,type,
% 9.49/5.22      remdups_nat: list_nat > list_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oreplicate_001_Eo,type,
% 9.49/5.22      replicate_o: nat > $o > list_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
% 9.49/5.22      replicate_int: nat > int > list_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
% 9.49/5.22      replicate_nat: nat > nat > list_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      replic4235873036481779905at_nat: nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.49/5.22      replic2264142908078655527at_nat: nat > produc3843707927480180839at_nat > list_P5464809261938338413at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
% 9.49/5.22      replicate_real: nat > real > list_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      replicate_set_nat: nat > set_nat > list_set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
% 9.49/5.22      sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
% 9.49/5.22      sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Otake_001t__Nat__Onat,type,
% 9.49/5.22      take_nat: nat > list_nat > list_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oupt,type,
% 9.49/5.22      upt: nat > nat > list_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oupto,type,
% 9.49/5.22      upto: int > int > list_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oupto__aux,type,
% 9.49/5.22      upto_aux: int > int > list_int > list_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Oupto__rel,type,
% 9.49/5.22      upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 9.49/5.22      zip_Co8729459035503499408nteger: list_C878401137130745250e_term > list_P5578671422887162913nteger > list_P5311841565141990158nteger ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.49/5.22      zip_in8766932505889695135nt_int: list_i8448526496819171953e_term > list_P5707943133018811711nt_int > list_P8915022641806594461nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 9.49/5.22      zip_Pr8292346330294042792nteger: list_P1316552470764441098e_term > list_P5578671422887162913nteger > list_P7828571989066258726nteger ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.49/5.22      zip_Pr4168994715204986005nt_int: list_P1743416141875011707e_term > list_P5707943133018811711nt_int > list_P651320350408439699nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001_Eo_001_Eo,type,
% 9.49/5.22      zip_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001_Eo_001t__Int__Oint,type,
% 9.49/5.22      zip_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001_Eo_001t__Nat__Onat,type,
% 9.49/5.22      zip_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001_Eo_001t__Real__Oreal,type,
% 9.49/5.22      zip_o_real: list_o > list_real > list_P5232166724548748803o_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      zip_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001t__Int__Oint_001t__Int__Oint,type,
% 9.49/5.22      zip_int_int: list_int > list_int > list_P5707943133018811711nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001t__Int__Oint_001t__Real__Oreal,type,
% 9.49/5.22      zip_int_real: list_int > list_real > list_P6863124054624500543t_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      zip_int_VEBT_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001t__Nat__Onat_001_Eo,type,
% 9.49/5.22      zip_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.22      zip_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      zip_nat_VEBT_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      zip_Pr4664179122662387191at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P8469869581646625389at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001t__Real__Oreal_001_Eo,type,
% 9.49/5.22      zip_real_o: list_real > list_o > list_P3595434254542482545real_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Int__Oint,type,
% 9.49/5.22      zip_real_int: list_real > list_int > list_P4344331454722006975al_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Nat__Onat,type,
% 9.49/5.22      zip_real_nat: list_real > list_nat > list_P6834414599653733731al_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.49/5.22      zip_real_real: list_real > list_real > list_P8689742595348180415l_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      zip_real_VEBT_VEBT: list_real > list_VEBT_VEBT > list_P877281246627933069T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      zip_set_nat_set_nat: list_set_nat > list_set_nat > list_P6254988961118846195et_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 9.49/5.22      zip_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 9.49/5.22      zip_VEBT_VEBT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 9.49/5.22      zip_VEBT_VEBT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
% 9.49/5.22      zip_VEBT_VEBT_real: list_VEBT_VEBT > list_real > list_P2623026923184700063T_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      zip_VE537291747668921783T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_OSuc,type,
% 9.49/5.22      suc: nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 9.49/5.22      case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 9.49/5.22      case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.49/5.22      case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Onat_Opred,type,
% 9.49/5.22      pred: nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      semiri4939895301339042750nteger: nat > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
% 9.49/5.22      semiri8010041392384452111omplex: nat > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
% 9.49/5.22      semiri4216267220026989637d_enat: nat > extended_enat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 9.49/5.22      semiri1314217659103216013at_int: nat > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 9.49/5.22      semiri1316708129612266289at_nat: nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
% 9.49/5.22      semiri681578069525770553at_rat: nat > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 9.49/5.22      semiri5074537144036343181t_real: nat > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
% 9.49/5.22      semiri2816024913162550771omplex: ( complex > complex ) > nat > complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
% 9.49/5.22      semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
% 9.49/5.22      semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
% 9.49/5.22      semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
% 9.49/5.22      semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
% 9.49/5.22      size_size_list_o: list_o > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 9.49/5.22      size_s3451745648224563538omplex: list_complex > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
% 9.49/5.22      size_size_list_int: list_int > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 9.49/5.22      size_size_list_nat: list_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
% 9.49/5.22      size_size_list_num: list_num > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 9.49/5.22      size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
% 9.49/5.22      size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
% 9.49/5.22      size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.49/5.22      size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
% 9.49/5.22      size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
% 9.49/5.22      size_s2970893825323803983at_int: list_P3521021558325789923at_int > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.22      size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.49/5.22      size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
% 9.49/5.22      size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
% 9.49/5.22      size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
% 9.49/5.22      size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.49/5.22      size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
% 9.49/5.22      size_size_list_real: list_real > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
% 9.49/5.22      size_s3254054031482475050et_nat: list_set_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.22      size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
% 9.49/5.22      size_size_num: num > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
% 9.49/5.22      size_size_option_nat: option_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.49/5.22      size_size_option_num: option_num > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.22      size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      size_size_VEBT_VEBT: vEBT_VEBT > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 9.49/5.22      nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 9.49/5.22      nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat__Bijection_Oset__decode,type,
% 9.49/5.22      nat_set_decode: nat > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Nat__Bijection_Oset__encode,type,
% 9.49/5.22      nat_set_encode: set_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_NthRoot_Oroot,type,
% 9.49/5.22      root: nat > real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_NthRoot_Osqrt,type,
% 9.49/5.22      sqrt: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_OBitM,type,
% 9.49/5.22      bitM: num > num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_Oinc,type,
% 9.49/5.22      inc: num > num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      neg_nu7757733837767384882nteger: code_integer > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
% 9.49/5.22      neg_nu6511756317524482435omplex: complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
% 9.49/5.22      neg_nu3811975205180677377ec_int: int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
% 9.49/5.22      neg_nu3179335615603231917ec_rat: rat > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
% 9.49/5.22      neg_nu6075765906172075777c_real: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_Onum_OBit0,type,
% 9.49/5.22      bit0: num > num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_Onum_OBit1,type,
% 9.49/5.22      bit1: num > num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_Onum_OOne,type,
% 9.49/5.22      one: num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.49/5.22      case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_Onum_Osize__num,type,
% 9.49/5.22      size_num: num > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_Onum__of__nat,type,
% 9.49/5.22      num_of_nat: nat > num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      numera6620942414471956472nteger: num > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
% 9.49/5.22      numera6690914467698888265omplex: num > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
% 9.49/5.22      numera1916890842035813515d_enat: num > extended_enat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
% 9.49/5.22      numeral_numeral_int: num > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
% 9.49/5.22      numeral_numeral_nat: num > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
% 9.49/5.22      numeral_numeral_rat: num > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
% 9.49/5.22      numeral_numeral_real: num > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Num_Opred__numeral,type,
% 9.49/5.22      pred_numeral: num > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_ONone_001t__Int__Oint,type,
% 9.49/5.22      none_int: option_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
% 9.49/5.22      none_nat: option_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
% 9.49/5.22      none_num: option_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_ONone_001t__Real__Oreal,type,
% 9.49/5.22      none_real: option_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      none_set_nat: option_set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_ONone_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      none_VEBT_VEBT: option_VEBT_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_OSome_001_Eo,type,
% 9.49/5.22      some_o: $o > option_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
% 9.49/5.22      some_int: int > option_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
% 9.49/5.22      some_nat: nat > option_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
% 9.49/5.22      some_num: num > option_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_OSome_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      some_VEBT_VEBT: vEBT_VEBT > option_VEBT_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
% 9.49/5.22      case_option_num_num: num > ( num > num ) > option_num > num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
% 9.49/5.22      case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
% 9.49/5.22      map_option_num_num: ( num > num ) > option_num > option_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_Osize__option_001t__Nat__Onat,type,
% 9.49/5.22      size_option_nat: ( nat > nat ) > option_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
% 9.49/5.22      size_option_num: ( num > nat ) > option_num > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      size_o8335143837870341156at_nat: ( product_prod_nat_nat > nat ) > option4927543243414619207at_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
% 9.49/5.22      the_nat2: option_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_Othe_001t__Num__Onum,type,
% 9.49/5.22      the_num: option_num > num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Option_Ooption_Othe_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Order__Relation_OunderS_001t__Nat__Onat,type,
% 9.49/5.22      order_underS_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
% 9.49/5.22      bot_bo5358457235160185703eger_o: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
% 9.49/5.22      bot_bo1403522918969695512_int_o: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
% 9.49/5.22      bot_bo3000040243691356879eger_o: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
% 9.49/5.22      bot_bo8662317086119403298_int_o: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
% 9.49/5.22      bot_bot_int_int_o: int > int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_Eo_J,type,
% 9.49/5.22      bot_bot_int_o: int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
% 9.49/5.22      bot_bot_list_nat_o: list_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
% 9.49/5.22      bot_bot_nat_nat_o: nat > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
% 9.49/5.22      bot_bot_nat_o: nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 9.49/5.22      bot_bo482883023278783056_nat_o: product_prod_nat_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Real__Oreal_M_Eo_J,type,
% 9.49/5.22      bot_bot_real_o: real > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
% 9.49/5.22      bot_bot_set_nat_o: set_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
% 9.49/5.22      bot_bo4199563552545308370d_enat: extended_enat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Nat__Onat_J,type,
% 9.49/5.22      bot_bot_filter_nat: filter_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
% 9.49/5.22      bot_bot_nat: nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 9.49/5.22      bot_bot_set_complex: set_complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
% 9.49/5.22      bot_bot_set_int: set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 9.49/5.22      bot_bot_set_list_nat: set_list_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      bot_bot_set_nat: set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
% 9.49/5.22      bot_bot_set_num: set_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
% 9.49/5.22      bot_bo3145834390647256904nteger: set_Pr8056137968301705908nteger ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
% 9.49/5.22      bot_bo4508923176915781079nt_int: set_Pr1872883991513573699nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
% 9.49/5.22      bot_bo5443222936135328352nteger: set_Pr1281608226676607948nteger ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
% 9.49/5.22      bot_bo572930865798478029nt_int: set_Pr9222295170931077689nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.49/5.22      bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.22      bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 9.49/5.22      bot_bo228742789529271731at_nat: set_Pr4329608150637261639at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
% 9.49/5.22      bot_bot_set_rat: set_rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
% 9.49/5.22      bot_bot_set_real: set_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
% 9.49/5.22      bot_bot_set_set_int: set_set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 9.49/5.22      bot_bot_set_set_nat: set_set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.22      bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
% 9.49/5.22      ord_Least_nat: ( nat > $o ) > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Int__Oint_M_Eo_J,type,
% 9.49/5.22      ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
% 9.49/5.22      ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 9.49/5.22      ord_le549003669493604880_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Real__Oreal_M_Eo_J,type,
% 9.49/5.22      ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
% 9.49/5.22      ord_less_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      ord_le6747313008572928689nteger: code_integer > code_integer > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
% 9.49/5.22      ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 9.49/5.22      ord_less_int: int > int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 9.49/5.22      ord_less_nat: nat > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
% 9.49/5.22      ord_less_num: num > num > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
% 9.49/5.22      ord_less_rat: rat > rat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 9.49/5.22      ord_less_real: real > real > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
% 9.49/5.22      ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 9.49/5.22      ord_less_set_complex: set_complex > set_complex > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
% 9.49/5.22      ord_less_set_int: set_int > set_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      ord_less_set_nat: set_nat > set_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
% 9.49/5.22      ord_less_set_num: set_num > set_num > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.22      ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 9.49/5.22      ord_le2604355607129572851at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
% 9.49/5.22      ord_less_set_rat: set_rat > set_rat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
% 9.49/5.22      ord_less_set_real: set_real > set_real > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
% 9.49/5.22      ord_less_set_set_int: set_set_int > set_set_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 9.49/5.22      ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
% 9.49/5.22      ord_le3636971675376928563eger_o: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
% 9.49/5.22      ord_le2124322318746777828_int_o: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ) > ( ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
% 9.49/5.22      ord_le4340812435750786203eger_o: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
% 9.49/5.22      ord_le5643404153117327598_int_o: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
% 9.49/5.22      ord_le6741204236512500942_int_o: ( int > int > $o ) > ( int > int > $o ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_Eo_J,type,
% 9.49/5.22      ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
% 9.49/5.22      ord_le2646555220125990790_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
% 9.49/5.22      ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 9.49/5.22      ord_le704812498762024988_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
% 9.49/5.22      ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
% 9.49/5.22      ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      ord_le3102999989581377725nteger: code_integer > code_integer > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
% 9.49/5.22      ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
% 9.49/5.22      ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 9.49/5.22      ord_less_eq_int: int > int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 9.49/5.22      ord_less_eq_nat: nat > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
% 9.49/5.22      ord_less_eq_num: num > num > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
% 9.49/5.22      ord_less_eq_rat: rat > rat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 9.49/5.22      ord_less_eq_real: real > real > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
% 9.49/5.22      ord_less_eq_set_o: set_o > set_o > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
% 9.49/5.22      ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 9.49/5.22      ord_le211207098394363844omplex: set_complex > set_complex > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
% 9.49/5.22      ord_less_eq_set_int: set_int > set_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 9.49/5.22      ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
% 9.49/5.22      ord_less_eq_set_num: set_num > set_num > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
% 9.49/5.22      ord_le3216752416896350996nteger: set_Pr8056137968301705908nteger > set_Pr8056137968301705908nteger > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
% 9.49/5.22      ord_le135402666524580259nt_int: set_Pr1872883991513573699nt_int > set_Pr1872883991513573699nt_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
% 9.49/5.22      ord_le653643898420964396nteger: set_Pr1281608226676607948nteger > set_Pr1281608226676607948nteger > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
% 9.49/5.22      ord_le8725513860283290265nt_int: set_Pr9222295170931077689nt_int > set_Pr9222295170931077689nt_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.49/5.22      ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.22      ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 9.49/5.22      ord_le1268244103169919719at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
% 9.49/5.22      ord_less_eq_set_rat: set_rat > set_rat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
% 9.49/5.22      ord_less_eq_set_real: set_real > set_real > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
% 9.49/5.22      ord_le4403425263959731960et_int: set_set_int > set_set_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 9.49/5.22      ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.22      ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      ord_max_Code_integer: code_integer > code_integer > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
% 9.49/5.22      ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
% 9.49/5.22      ord_max_int: int > int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
% 9.49/5.22      ord_max_nat: nat > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
% 9.49/5.22      ord_max_num: num > num > num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
% 9.49/5.22      ord_max_rat: rat > rat > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
% 9.49/5.22      ord_max_real: real > real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
% 9.49/5.22      ord_max_set_int: set_int > set_int > set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      ord_max_set_nat: set_nat > set_nat > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.22      ord_ma7524802468073614006at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Real__Oreal_J,type,
% 9.49/5.22      ord_max_set_real: set_real > set_real > set_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
% 9.49/5.22      ord_min_nat: nat > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
% 9.49/5.22      order_Greatest_nat: ( nat > $o ) > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.22      order_mono_nat_nat: ( nat > nat ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.22      order_5726023648592871131at_nat: ( nat > nat ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.49/5.22      order_7092887310737990675l_real: ( real > real ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
% 9.49/5.22      ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
% 9.49/5.22      top_top_set_int: set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      top_top_set_nat: set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
% 9.49/5.22      top_to1996260823553986621t_unit: set_Product_unit ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
% 9.49/5.22      top_top_set_real: set_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      power_8256067586552552935nteger: code_integer > nat > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
% 9.49/5.22      power_power_complex: complex > nat > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
% 9.49/5.22      power_power_int: int > nat > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
% 9.49/5.22      power_power_nat: nat > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
% 9.49/5.22      power_power_rat: rat > nat > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
% 9.49/5.22      power_power_real: real > nat > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 9.49/5.22      produc6137756002093451184nteger: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc8763457246119570046nteger ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.49/5.22      produc4305682042979456191nt_int: ( int > option6357759511663192854e_term ) > product_prod_int_int > produc7773217078559923341nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
% 9.49/5.22      produc4035269172776083154on_nat: ( nat > nat > $o ) > produc4953844613479565601on_nat > produc2233624965454879586on_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
% 9.49/5.22      produc8929957630744042906on_nat: ( nat > nat > nat ) > produc4953844613479565601on_nat > produc8306885398267862888on_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
% 9.49/5.22      produc3576312749637752826on_num: ( num > num > $o ) > produc3447558737645232053on_num > produc7036089656553540234on_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
% 9.49/5.22      produc5778274026573060048on_num: ( num > num > num ) > produc3447558737645232053on_num > produc1193250871479095198on_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 9.49/5.22      produc8603105652947943368nteger: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc1908205239877642774nteger ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.49/5.22      produc5700946648718959541nt_int: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > produc2285326912895808259nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.49/5.22      produc3994169339658061776at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc6121120109295599847at_nat > produc5491161045314408544at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.49/5.22      produc2899441246263362727at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > produc6121120109295599847at_nat > produc5542196010084753463at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
% 9.49/5.22      product_Pair_o_o: $o > $o > product_prod_o_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
% 9.49/5.22      product_Pair_o_int: $o > int > product_prod_o_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
% 9.49/5.22      product_Pair_o_nat: $o > nat > product_prod_o_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001_Eo_001t__Real__Oreal,type,
% 9.49/5.22      product_Pair_o_real: $o > real > product_prod_o_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
% 9.49/5.22      produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
% 9.49/5.22      product_Pair_int_int: int > int > product_prod_int_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Real__Oreal,type,
% 9.49/5.22      produc801115645435158769t_real: int > real > produc679980390762269497t_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      produc3329399203697025711T_VEBT: int > vEBT_VEBT > produc1531783533982839933T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__List__Olist_I_Eo_J_001t__List__Olist_I_Eo_J,type,
% 9.49/5.22      produc8435520187683070743list_o: list_o > list_o > produc7102631898165422375list_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
% 9.49/5.22      produc364263696895485585st_int: list_int > list_int > produc1186641810826059865st_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
% 9.49/5.22      produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.22      produc3897820843166775703T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > produc9211091688327510695T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001_Eo,type,
% 9.49/5.22      product_Pair_nat_o: nat > $o > product_prod_nat_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
% 9.49/5.22      product_Pair_nat_int: nat > int > product_prod_nat_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.22      product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
% 9.49/5.22      product_Pair_nat_num: nat > num > product_prod_nat_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      produc599794634098209291T_VEBT: nat > vEBT_VEBT > produc8025551001238799321T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
% 9.49/5.22      product_Pair_num_num: num > num > product_prod_num_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
% 9.49/5.22      produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.49/5.22      produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.22      produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001_Eo,type,
% 9.49/5.22      product_Pair_real_o: real > $o > product_prod_real_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Int__Oint,type,
% 9.49/5.22      produc3179012173361985393al_int: real > int > produc8786904178792722361al_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Nat__Onat,type,
% 9.49/5.22      produc3181502643871035669al_nat: real > nat > produc3741383161447143261al_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.49/5.22      produc4511245868158468465l_real: real > real > produc2422161461964618553l_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      produc6931449550656315951T_VEBT: real > vEBT_VEBT > produc3757001726724277373T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.22      produc2922128104949294807at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > produc3843707927480180839at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 9.49/5.22      produc9060074326276436823at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > produc1319942482725812455at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 9.49/5.22      produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 9.49/5.22      produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 9.49/5.22      produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
% 9.49/5.22      produc8117437818029410057T_real: vEBT_VEBT > real > produc5170161368751668367T_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.22      produc457027306803732586at_nat: set_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Omap__prod_001t__Code____Numeral__Ointeger_001t__Nat__Onat_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
% 9.49/5.22      produc8678311845419106900er_nat: ( code_integer > nat ) > ( code_integer > nat ) > produc8923325533196201883nteger > product_prod_nat_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_Eo,type,
% 9.49/5.22      produc127349428274296955eger_o: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > produc8763457246119570046nteger > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 9.49/5.22      produc3906647086178084059nteger: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc8763457246119570046nteger ) > produc8763457246119570046nteger > produc8763457246119570046nteger ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Int__Oint_J,type,
% 9.49/5.22      produc8604463032469472703et_int: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int ) > produc8763457246119570046nteger > set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      produc3558942015123893603et_nat: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat ) > produc8763457246119570046nteger > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Real__Oreal_J,type,
% 9.49/5.22      produc815715089573277247t_real: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real ) > produc8763457246119570046nteger > set_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo,type,
% 9.49/5.22      produc2558449545302689196_int_o: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc7773217078559923341nt_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.49/5.22      produc5122537100556696953nt_int: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > produc7773217078559923341nt_int ) > produc7773217078559923341nt_int > produc7773217078559923341nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      produc8289552606927098482et_nat: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat ) > produc7773217078559923341nt_int > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Real__Oreal_J,type,
% 9.49/5.22      produc8709739885379107790t_real: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real ) > produc7773217078559923341nt_int > set_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_Eo,type,
% 9.49/5.22      produc6253627499356882019eger_o: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > produc1908205239877642774nteger > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 9.49/5.22      produc6512950862096126219nteger: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc1908205239877642774nteger ) > produc1908205239877642774nteger > produc1908205239877642774nteger ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo,type,
% 9.49/5.22      produc1573362020775583542_int_o: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc2285326912895808259nt_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.49/5.22      produc8492565224438309093nt_int: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > produc2285326912895808259nt_int ) > produc2285326912895808259nt_int > produc2285326912895808259nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
% 9.49/5.22      produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
% 9.49/5.22      produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 9.49/5.22      produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 9.49/5.22      produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
% 9.49/5.22      produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
% 9.49/5.22      produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.49/5.22      produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
% 9.49/5.22      produc73460835934605544et_int: ( int > int > set_int ) > product_prod_int_int > set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      produc4251311855443802252et_nat: ( int > int > set_nat ) > product_prod_int_int > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.22      produc1656060378719767003at_nat: ( int > int > set_Pr1261947904930325089at_nat ) > product_prod_int_int > set_Pr1261947904930325089at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Real__Oreal_J,type,
% 9.49/5.22      produc6452406959799940328t_real: ( int > int > set_real ) > product_prod_int_int > set_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 9.49/5.22      produc5233655623923918146et_nat: ( int > int > set_set_nat ) > product_prod_int_int > set_set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 9.49/5.22      produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.22      produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
% 9.49/5.22      produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 9.49/5.22      produc1917071388513777916omplex: ( nat > nat > complex ) > product_prod_nat_nat > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
% 9.49/5.22      produc2676513652042109336d_enat: ( nat > nat > extended_enat ) > product_prod_nat_nat > extended_enat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
% 9.49/5.22      produc6840382203811409530at_int: ( nat > nat > int ) > product_prod_nat_nat > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.22      produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
% 9.49/5.22      produc6207742614233964070at_rat: ( nat > nat > rat ) > product_prod_nat_nat > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
% 9.49/5.22      produc1703576794950452218t_real: ( nat > nat > real ) > product_prod_nat_nat > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.49/5.22      produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
% 9.49/5.22      produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
% 9.49/5.22      product_fst_int_int: product_prod_int_int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.22      product_fst_nat_nat: product_prod_nat_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
% 9.49/5.22      product_snd_int_int: product_prod_int_int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.22      product_snd_nat_nat: product_prod_nat_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rat_OAbs__Rat,type,
% 9.49/5.22      abs_Rat: product_prod_int_int > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rat_OFract,type,
% 9.49/5.22      fract: int > int > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rat_OFrct,type,
% 9.49/5.22      frct: product_prod_int_int > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rat_ORep__Rat,type,
% 9.49/5.22      rep_Rat: rat > product_prod_int_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
% 9.49/5.22      field_5140801741446780682s_real: set_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rat_Onormalize,type,
% 9.49/5.22      normalize: product_prod_int_int > product_prod_int_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rat_Oof__int,type,
% 9.49/5.22      of_int: int > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rat_Opcr__rat,type,
% 9.49/5.22      pcr_rat: product_prod_int_int > rat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rat_Opositive,type,
% 9.49/5.22      positive: rat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rat_Oquotient__of,type,
% 9.49/5.22      quotient_of: rat > product_prod_int_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rat_Oratrel,type,
% 9.49/5.22      ratrel: product_prod_int_int > product_prod_int_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Real_OReal,type,
% 9.49/5.22      real2: ( nat > rat ) > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Real_Ocauchy,type,
% 9.49/5.22      cauchy: ( nat > rat ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Real_Opcr__real,type,
% 9.49/5.22      pcr_real: ( nat > rat ) > real > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Real_Opositive,type,
% 9.49/5.22      positive2: real > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Real_Orealrel,type,
% 9.49/5.22      realrel: ( nat > rat ) > ( nat > rat ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Real_Orep__real,type,
% 9.49/5.22      rep_real: real > nat > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Real_Ovanishes,type,
% 9.49/5.22      vanishes: ( nat > rat ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
% 9.49/5.22      real_V3694042436643373181omplex: complex > complex > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
% 9.49/5.22      real_V975177566351809787t_real: real > real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
% 9.49/5.22      real_V1022390504157884413omplex: complex > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
% 9.49/5.22      real_V7735802525324610683m_real: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
% 9.49/5.22      real_V4546457046886955230omplex: real > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
% 9.49/5.22      real_V2046097035970521341omplex: real > complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
% 9.49/5.22      real_V1485227260804924795R_real: real > real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Relation_OField_001t__Nat__Onat,type,
% 9.49/5.22      field_nat: set_Pr1261947904930325089at_nat > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
% 9.49/5.22      algebr932160517623751201me_int: int > int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
% 9.49/5.22      algebr934650988132801477me_nat: nat > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
% 9.49/5.22      divide1717551699836669952omplex: complex > complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
% 9.49/5.22      divide_divide_int: int > int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
% 9.49/5.22      divide_divide_nat: nat > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
% 9.49/5.22      divide_divide_rat: rat > rat > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
% 9.49/5.22      divide_divide_real: real > real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      dvd_dvd_Code_integer: code_integer > code_integer > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
% 9.49/5.22      dvd_dvd_complex: complex > complex > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
% 9.49/5.22      dvd_dvd_int: int > int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
% 9.49/5.22      dvd_dvd_nat: nat > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
% 9.49/5.22      dvd_dvd_rat: rat > rat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
% 9.49/5.22      dvd_dvd_real: real > real > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      modulo364778990260209775nteger: code_integer > code_integer > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
% 9.49/5.22      modulo_modulo_int: int > int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
% 9.49/5.22      modulo_modulo_nat: nat > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      zero_n356916108424825756nteger: $o > code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
% 9.49/5.22      zero_n1201886186963655149omplex: $o > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
% 9.49/5.22      zero_n2684676970156552555ol_int: $o > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
% 9.49/5.22      zero_n2687167440665602831ol_nat: $o > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
% 9.49/5.22      zero_n2052037380579107095ol_rat: $o > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
% 9.49/5.22      zero_n3304061248610475627l_real: $o > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
% 9.49/5.22      suminf_complex: ( nat > complex ) > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Series_Osuminf_001t__Int__Oint,type,
% 9.49/5.22      suminf_int: ( nat > int ) > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
% 9.49/5.22      suminf_nat: ( nat > nat ) > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
% 9.49/5.22      suminf_real: ( nat > real ) > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
% 9.49/5.22      summable_complex: ( nat > complex ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Series_Osummable_001t__Int__Oint,type,
% 9.49/5.22      summable_int: ( nat > int ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
% 9.49/5.22      summable_nat: ( nat > nat ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
% 9.49/5.22      summable_real: ( nat > real ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
% 9.49/5.22      sums_complex: ( nat > complex ) > complex > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Series_Osums_001t__Int__Oint,type,
% 9.49/5.22      sums_int: ( nat > int ) > int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Series_Osums_001t__Nat__Onat,type,
% 9.49/5.22      sums_nat: ( nat > nat ) > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Series_Osums_001t__Real__Oreal,type,
% 9.49/5.22      sums_real: ( nat > real ) > real > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
% 9.49/5.22      collect_Code_integer: ( code_integer > $o ) > set_Code_integer ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
% 9.49/5.22      collect_complex: ( complex > $o ) > set_complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__Int__Oint,type,
% 9.49/5.22      collect_int: ( int > $o ) > set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__List__Olist_I_Eo_J,type,
% 9.49/5.22      collect_list_o: ( list_o > $o ) > set_list_o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 9.49/5.22      collect_list_complex: ( list_complex > $o ) > set_list_complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
% 9.49/5.22      collect_list_int: ( list_int > $o ) > set_list_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
% 9.49/5.22      collect_list_nat: ( list_nat > $o ) > set_list_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.22      collec3343600615725829874at_nat: ( list_P6011104703257516679at_nat > $o ) > set_li5450038453877631591at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.22      collec5608196760682091941T_VEBT: ( list_VEBT_VEBT > $o ) > set_list_VEBT_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 9.49/5.22      collect_nat: ( nat > $o ) > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__Num__Onum,type,
% 9.49/5.22      collect_num: ( num > $o ) > set_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 9.49/5.22      collec8663557070575231912omplex: ( produc4411394909380815293omplex > $o ) > set_Pr5085853215250843933omplex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.49/5.22      collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 9.49/5.22      collec3799799289383736868l_real: ( produc2422161461964618553l_real > $o ) > set_Pr6218003697084177305l_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.49/5.22      collec6321179662152712658at_nat: ( produc3843707927480180839at_nat > $o ) > set_Pr4329608150637261639at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
% 9.49/5.22      collect_rat: ( rat > $o ) > set_rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 9.49/5.22      collect_real: ( real > $o ) > set_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 9.49/5.22      collect_set_complex: ( set_complex > $o ) > set_set_complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
% 9.49/5.22      collect_set_int: ( set_int > $o ) > set_set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      collect_set_nat: ( set_nat > $o ) > set_set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.22      collec5514110066124741708at_nat: ( set_Pr1261947904930325089at_nat > $o ) > set_se7855581050983116737at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_OPow_001t__Nat__Onat,type,
% 9.49/5.22      pow_nat: set_nat > set_set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
% 9.49/5.22      image_int_int: ( int > int ) > set_int > set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
% 9.49/5.22      image_int_nat: ( int > nat ) > set_int > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.22      image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 9.49/5.22      image_5971271580939081552omplex: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 9.49/5.22      image_2178119161166701260l_real: ( real > filter2146258269922977983l_real ) > set_real > set_fi7789364187291644575l_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.49/5.22      image_real_real: ( real > real ) > set_real > set_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
% 9.49/5.22      insert_complex: complex > set_complex > set_complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
% 9.49/5.22      insert_int: int > set_int > set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
% 9.49/5.22      insert_list_nat: list_nat > set_list_nat > set_list_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
% 9.49/5.22      insert_nat: nat > set_nat > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Oinsert_001t__Num__Onum,type,
% 9.49/5.22      insert_num: num > set_num > set_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.49/5.22      insert5033312907999012233nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.49/5.22      insert9069300056098147895at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Oinsert_001t__Rat__Orat,type,
% 9.49/5.22      insert_rat: rat > set_rat > set_rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
% 9.49/5.22      insert_real: real > set_real > set_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      insert_set_nat: set_nat > set_set_nat > set_set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Oinsert_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      insert_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Ois__singleton_001t__Int__Oint,type,
% 9.49/5.22      is_singleton_int: set_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
% 9.49/5.22      is_singleton_nat: set_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      is_sin2850979758926227957at_nat: set_Pr1261947904930325089at_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.49/5.22      is_sin2937591304547752795at_nat: set_Pr4329608150637261639at_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Ois__singleton_001t__Real__Oreal,type,
% 9.49/5.22      is_singleton_real: set_real > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      is_singleton_set_nat: set_set_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Othe__elem_001t__Int__Oint,type,
% 9.49/5.22      the_elem_int: set_int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
% 9.49/5.22      the_elem_nat: set_nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.49/5.22      the_el8326832613380209454nt_int: set_Pr958786334691620121nt_int > product_prod_int_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      the_el2281957884133575798at_nat: set_Pr1261947904930325089at_nat > product_prod_nat_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.49/5.22      the_el221668144340439132at_nat: set_Pr4329608150637261639at_nat > produc3843707927480180839at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Othe__elem_001t__Real__Oreal,type,
% 9.49/5.22      the_elem_real: set_real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.49/5.22      vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
% 9.49/5.22      set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
% 9.49/5.22      set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
% 9.49/5.22      set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
% 9.49/5.22      set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
% 9.49/5.22      set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
% 9.49/5.22      set_or1266510415728281911st_int: int > int > set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
% 9.49/5.22      set_or1269000886237332187st_nat: nat > nat > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
% 9.49/5.22      set_or7049704709247886629st_num: num > num > set_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
% 9.49/5.22      set_or633870826150836451st_rat: rat > rat > set_rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
% 9.49/5.22      set_or1222579329274155063t_real: real > real > set_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Int__Oint_J,type,
% 9.49/5.22      set_or370866239135849197et_int: set_int > set_int > set_set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
% 9.49/5.22      set_or4662586982721622107an_int: int > int > set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
% 9.49/5.22      set_or4665077453230672383an_nat: nat > nat > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Num__Onum,type,
% 9.49/5.22      set_or1222409239386451017an_num: num > num > set_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Rat__Orat,type,
% 9.49/5.22      set_or4029947393144176647an_rat: rat > rat > set_rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal,type,
% 9.49/5.22      set_or66887138388493659n_real: real > real > set_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Int__Oint_J,type,
% 9.49/5.22      set_or8585797421378605585et_int: set_int > set_int > set_set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
% 9.49/5.22      set_ord_atLeast_nat: nat > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
% 9.49/5.22      set_ord_atMost_int: int > set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
% 9.49/5.22      set_ord_atMost_nat: nat > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Num__Onum,type,
% 9.49/5.22      set_ord_atMost_num: num > set_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Rat__Orat,type,
% 9.49/5.22      set_ord_atMost_rat: rat > set_rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
% 9.49/5.22      set_ord_atMost_real: real > set_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Int__Oint_J,type,
% 9.49/5.22      set_or58775011639299419et_int: set_int > set_set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      set_or4236626031148496127et_nat: set_nat > set_set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
% 9.49/5.22      set_or6656581121297822940st_int: int > int > set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
% 9.49/5.22      set_or6659071591806873216st_nat: nat > nat > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
% 9.49/5.22      set_or5832277885323065728an_int: int > int > set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
% 9.49/5.22      set_or5834768355832116004an_nat: nat > nat > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
% 9.49/5.22      set_or1633881224788618240n_real: real > real > set_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
% 9.49/5.22      set_or1210151606488870762an_nat: nat > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
% 9.49/5.22      set_or5849166863359141190n_real: real > set_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
% 9.49/5.22      set_ord_lessThan_int: int > set_int ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
% 9.49/5.22      set_ord_lessThan_nat: nat > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
% 9.49/5.22      set_ord_lessThan_num: num > set_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
% 9.49/5.22      set_ord_lessThan_rat: rat > set_rat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
% 9.49/5.22      set_or5984915006950818249n_real: real > set_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      set_or890127255671739683et_nat: set_nat > set_set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.49/5.22      topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.49/5.22      topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
% 9.49/5.22      topolo6980174941875973593q_real: ( nat > real ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
% 9.49/5.22      topolo2177554685111907308n_real: real > set_real > filter_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
% 9.49/5.22      topolo2815343760600316023s_real: real > filter_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex,type,
% 9.49/5.22      topolo6517432010174082258omplex: ( nat > complex ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
% 9.49/5.22      topolo4055970368930404560y_real: ( nat > real ) > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
% 9.49/5.22      topolo896644834953643431omplex: filter6041513312241820739omplex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
% 9.49/5.22      topolo1511823702728130853y_real: filter2146258269922977983l_real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Oarccos,type,
% 9.49/5.22      arccos: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
% 9.49/5.22      arcosh_real: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Oarcsin,type,
% 9.49/5.22      arcsin: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Oarctan,type,
% 9.49/5.22      arctan: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
% 9.49/5.22      arsinh_real: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
% 9.49/5.22      artanh_real: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
% 9.49/5.22      cos_complex: complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
% 9.49/5.22      cos_real: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Ocos__coeff,type,
% 9.49/5.22      cos_coeff: nat > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Ocosh_001t__Complex__Ocomplex,type,
% 9.49/5.22      cosh_complex: complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
% 9.49/5.22      cosh_real: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Ocot_001t__Complex__Ocomplex,type,
% 9.49/5.22      cot_complex: complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
% 9.49/5.22      cot_real: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
% 9.49/5.22      diffs_complex: ( nat > complex ) > nat > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
% 9.49/5.22      diffs_real: ( nat > real ) > nat > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
% 9.49/5.22      exp_complex: complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
% 9.49/5.22      exp_real: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
% 9.49/5.22      ln_ln_real: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Olog,type,
% 9.49/5.22      log: real > real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Opi,type,
% 9.49/5.22      pi: real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
% 9.49/5.22      powr_real: real > real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
% 9.49/5.22      sin_complex: complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
% 9.49/5.22      sin_real: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Osin__coeff,type,
% 9.49/5.22      sin_coeff: nat > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Osinh_001t__Complex__Ocomplex,type,
% 9.49/5.22      sinh_complex: complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
% 9.49/5.22      sinh_real: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
% 9.49/5.22      tan_complex: complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
% 9.49/5.22      tan_real: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
% 9.49/5.22      tanh_complex: complex > complex ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
% 9.49/5.22      tanh_real: real > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
% 9.49/5.22      transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
% 9.49/5.22      transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
% 9.49/5.22      vEBT_Leaf: $o > $o > vEBT_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
% 9.49/5.22      vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
% 9.49/5.22      vEBT_size_VEBT: vEBT_VEBT > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
% 9.49/5.22      vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
% 9.49/5.22      vEBT_VEBT_high: nat > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
% 9.49/5.22      vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
% 9.49/5.22      vEBT_VEBT_low: nat > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
% 9.49/5.22      vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
% 9.49/5.22      vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
% 9.49/5.22      vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
% 9.49/5.22      vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
% 9.49/5.22      vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
% 9.49/5.22      vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
% 9.49/5.22      vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Definitions_Oset__vebt,type,
% 9.49/5.22      vEBT_set_vebt: vEBT_VEBT > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
% 9.49/5.22      vEBT_vebt_buildup: nat > vEBT_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
% 9.49/5.22      vEBT_v4011308405150292612up_rel: nat > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Delete_Ovebt__delete,type,
% 9.49/5.22      vEBT_vebt_delete: vEBT_VEBT > nat > vEBT_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
% 9.49/5.22      vEBT_vebt_delete_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Insert_Ovebt__insert,type,
% 9.49/5.22      vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
% 9.49/5.22      vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
% 9.49/5.22      vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
% 9.49/5.22      vEBT_VEBT_minNull: vEBT_VEBT > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
% 9.49/5.22      vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
% 9.49/5.22      vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Member_Ovebt__member,type,
% 9.49/5.22      vEBT_vebt_member: vEBT_VEBT > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
% 9.49/5.22      vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
% 9.49/5.22      vEBT_VEBT_add: option_nat > option_nat > option_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
% 9.49/5.22      vEBT_VEBT_greater: option_nat > option_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
% 9.49/5.22      vEBT_VEBT_less: option_nat > option_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
% 9.49/5.22      vEBT_VEBT_lesseq: option_nat > option_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
% 9.49/5.22      vEBT_VEBT_max_in_set: set_nat > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
% 9.49/5.22      vEBT_VEBT_min_in_set: set_nat > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
% 9.49/5.22      vEBT_VEBT_mul: option_nat > option_nat > option_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
% 9.49/5.22      vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
% 9.49/5.22      vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      vEBT_V1502963449132264192at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
% 9.49/5.22      vEBT_VEBT_power: option_nat > option_nat > option_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
% 9.49/5.22      vEBT_vebt_maxt: vEBT_VEBT > option_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
% 9.49/5.22      vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
% 9.49/5.22      vEBT_vebt_mint: vEBT_VEBT > option_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
% 9.49/5.22      vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Pred_Ois__pred__in__set,type,
% 9.49/5.22      vEBT_is_pred_in_set: set_nat > nat > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Pred_Ovebt__pred,type,
% 9.49/5.22      vEBT_vebt_pred: vEBT_VEBT > nat > option_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
% 9.49/5.22      vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Succ_Ois__succ__in__set,type,
% 9.49/5.22      vEBT_is_succ_in_set: set_nat > nat > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Succ_Ovebt__succ,type,
% 9.49/5.22      vEBT_vebt_succ: vEBT_VEBT > nat > option_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
% 9.49/5.22      vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
% 9.49/5.22      accp_nat: ( nat > nat > $o ) > nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.49/5.22      accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 9.49/5.22      accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 9.49/5.22      accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Wellfounded_Omax__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      max_ex8135407076693332796at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Wellfounded_Omin__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      min_ex6901939911449802026at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Wellfounded_Opred__nat,type,
% 9.49/5.22      pred_nat: set_Pr1261947904930325089at_nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
% 9.49/5.22      wf_nat: set_Pr1261947904930325089at_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_fChoice_001t__Real__Oreal,type,
% 9.49/5.22      fChoice_real: ( real > $o ) > real ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 9.49/5.22      member1535805642427569193e_term: ( code_integer > option6357759511663192854e_term ) > set_Co9149898834107579976e_term > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 9.49/5.22      member8845023287901829240e_term: ( int > option6357759511663192854e_term ) > set_in3461395444621081367e_term > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 9.49/5.22      member4242434998011752849e_term: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > set_Pr7604974323444597168e_term > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
% 9.49/5.22      member3222579708246209666e_term: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr3642885161833720865e_term > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001_Eo,type,
% 9.49/5.22      member_o: $o > set_o > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Complex__Ocomplex,type,
% 9.49/5.22      member_complex: complex > set_complex > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Int__Oint,type,
% 9.49/5.22      member_int: int > set_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
% 9.49/5.22      member_list_o: list_o > set_list_o > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
% 9.49/5.22      member_list_int: list_int > set_list_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
% 9.49/5.22      member_list_nat: list_nat > set_list_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.22      member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Nat__Onat,type,
% 9.49/5.22      member_nat: nat > set_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Num__Onum,type,
% 9.49/5.22      member_num: num > set_num > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 9.49/5.22      member3068662437193594005nteger: produc8763457246119570046nteger > set_Pr8056137968301705908nteger > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.49/5.22      member7034335876925520548nt_int: produc7773217078559923341nt_int > set_Pr1872883991513573699nt_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 9.49/5.22      member4164122664394876845nteger: produc1908205239877642774nteger > set_Pr1281608226676607948nteger > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.49/5.22      member7618704894036264090nt_int: produc2285326912895808259nt_int > set_Pr9222295170931077689nt_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
% 9.49/5.22      member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 9.49/5.22      member7847949116333733898_o_int: product_prod_o_int > set_Pr8834758594704517033_o_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
% 9.49/5.22      member2802428098988154798_o_nat: product_prod_o_nat > set_Pr2101469702781467981_o_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J,type,
% 9.49/5.22      member7400031367953476362o_real: product_prod_o_real > set_Pr6573716822653411497o_real > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.22      member5477980866518848620T_VEBT: produc2504756804600209347T_VEBT > set_Pr7543698050874017315T_VEBT > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 9.49/5.22      member157494554546826820nteger: produc8923325533196201883nteger > set_Pr4811707699266497531nteger > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.49/5.22      member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
% 9.49/5.22      member2744130022092475746t_real: produc679980390762269497t_real > set_Pr3538720872664544793t_real > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.22      member2056185340421749780T_VEBT: produc1531783533982839933T_VEBT > set_Pr8044002425091019955T_VEBT > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__List__Olist_I_Eo_J_J,type,
% 9.49/5.22      member4159035015898711888list_o: produc7102631898165422375list_o > set_Pr6227168374412355847list_o > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
% 9.49/5.22      member6698963635872716290st_int: produc1186641810826059865st_int > set_Pr765067013931698361st_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
% 9.49/5.22      member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
% 9.49/5.22      member4439316823752958928T_VEBT: produc9211091688327510695T_VEBT > set_Pr1916528119006554503T_VEBT > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.49/5.22      member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.49/5.22      member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_M_Eo_J,type,
% 9.49/5.22      member772602641336174712real_o: product_prod_real_o > set_Pr4936984352647145239real_o > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
% 9.49/5.22      member1627681773268152802al_int: produc8786904178792722361al_int > set_Pr1019928272762051225al_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
% 9.49/5.22      member5805532792777349510al_nat: produc3741383161447143261al_nat > set_Pr3510011417693777981al_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 9.49/5.22      member7849222048561428706l_real: produc2422161461964618553l_real > set_Pr6218003697084177305l_real > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.22      member7262085504369356948T_VEBT: produc3757001726724277373T_VEBT > set_Pr6019664923565264691T_VEBT > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
% 9.49/5.22      member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.49/5.22      member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 9.49/5.22      member3307348790968139188VEBT_o: produc334124729049499915VEBT_o > set_Pr3175402225741728619VEBT_o > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 9.49/5.22      member5419026705395827622BT_int: produc4894624898956917775BT_int > set_Pr5066593544530342725BT_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 9.49/5.22      member373505688050248522BT_nat: produc9072475918466114483BT_nat > set_Pr7556676689462069481BT_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
% 9.49/5.22      member8675245146396747942T_real: produc5170161368751668367T_real > set_Pr7765410600122031685T_real > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.49/5.22      member568628332442017744T_VEBT: produc8243902056947475879T_VEBT > set_Pr6192946355708809607T_VEBT > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Rat__Orat,type,
% 9.49/5.22      member_rat: rat > set_rat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Real__Oreal,type,
% 9.49/5.22      member_real: real > set_real > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
% 9.49/5.22      member_set_int: set_int > set_set_int > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.49/5.22      member_set_nat: set_nat > set_set_nat > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
% 9.49/5.22      member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_v_deg____,type,
% 9.49/5.22      deg: nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_v_m____,type,
% 9.49/5.22      m: nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_v_ma____,type,
% 9.49/5.22      ma: nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_v_maxi____,type,
% 9.49/5.22      maxi: nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_v_mi____,type,
% 9.49/5.22      mi: nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_v_na____,type,
% 9.49/5.22      na: nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_v_summary____,type,
% 9.49/5.22      summary: vEBT_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_v_treeList____,type,
% 9.49/5.22      treeList: list_VEBT_VEBT ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_v_xa____,type,
% 9.49/5.22      xa: nat ).
% 9.49/5.22  
% 9.49/5.22  thf(sy_v_ya____,type,
% 9.49/5.22      ya: nat ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_0__C4_Ohyps_C_I3_J,axiom,
% 9.49/5.22      m = na ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_1_xmi,axiom,
% 9.49/5.22      xa != mi ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_2_True,axiom,
% 9.49/5.22      xa = ma ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_3_valid__eq,axiom,
% 9.49/5.22      vEBT_VEBT_valid = vEBT_invar_vebt ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_4_valid__eq1,axiom,
% 9.49/5.22      ! [T: vEBT_VEBT,D: nat] :
% 9.49/5.22        ( ( vEBT_invar_vebt @ T @ D )
% 9.49/5.22       => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_5_valid__eq2,axiom,
% 9.49/5.22      ! [T: vEBT_VEBT,D: nat] :
% 9.49/5.22        ( ( vEBT_VEBT_valid @ T @ D )
% 9.49/5.22       => ( vEBT_invar_vebt @ T @ D ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_6_valid__0__not,axiom,
% 9.49/5.22      ! [T: vEBT_VEBT] :
% 9.49/5.22        ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_7_valid__tree__deg__neq__0,axiom,
% 9.49/5.22      ! [T: vEBT_VEBT] :
% 9.49/5.22        ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_8__092_060open_062high_Ax_An_A_060_Alength_AtreeList_092_060close_062,axiom,
% 9.49/5.22      ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_9__C4_Ohyps_C_I1_J,axiom,
% 9.49/5.22      vEBT_invar_vebt @ summary @ m ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_10_False,axiom,
% 9.49/5.22      ~ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_11_deg__deg__n,axiom,
% 9.49/5.22      ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 9.49/5.22        ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
% 9.49/5.22       => ( Deg = N ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_12_set__vebt__set__vebt_H__valid,axiom,
% 9.49/5.22      ! [T: vEBT_VEBT,N: nat] :
% 9.49/5.22        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.22       => ( ( vEBT_set_vebt @ T )
% 9.49/5.22          = ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_13_notemp,axiom,
% 9.49/5.22      ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) @ X_1 ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_14_hprolist,axiom,
% 9.49/5.22      ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( vEBT_VEBT_high @ xa @ na ) )
% 9.49/5.22      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_15_not__min__Null__member,axiom,
% 9.49/5.22      ! [T: vEBT_VEBT] :
% 9.49/5.22        ( ~ ( vEBT_VEBT_minNull @ T )
% 9.49/5.22       => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_16_deg__not__0,axiom,
% 9.49/5.22      ! [T: vEBT_VEBT,N: nat] :
% 9.49/5.22        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.22       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_17_bit__split__inv,axiom,
% 9.49/5.22      ! [X: nat,D: nat] :
% 9.49/5.22        ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D ) @ ( vEBT_VEBT_low @ X @ D ) @ D )
% 9.49/5.22        = X ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_18__C4_OIH_C_I2_J,axiom,
% 9.49/5.22      ! [X: nat,Y: nat] :
% 9.49/5.22        ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ summary @ X ) @ Y )
% 9.49/5.22        = ( ( X != Y )
% 9.49/5.22          & ( vEBT_V8194947554948674370ptions @ summary @ Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_19_mimapr,axiom,
% 9.49/5.22      ord_less_nat @ mi @ ma ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_20__092_060open_062length_AtreeList_A_061_Alength_A_ItreeList_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_J_092_060close_062,axiom,
% 9.49/5.22      ( ( size_s6755466524823107622T_VEBT @ treeList )
% 9.49/5.22      = ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_21__C4_Ohyps_C_I7_J,axiom,
% 9.49/5.22      ord_less_eq_nat @ mi @ ma ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_22__C4_Ohyps_C_I6_J,axiom,
% 9.49/5.22      ( ( mi = ma )
% 9.49/5.22     => ! [X2: vEBT_VEBT] :
% 9.49/5.22          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
% 9.49/5.22         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_23__C4_OIH_C_I1_J,axiom,
% 9.49/5.22      ! [X2: vEBT_VEBT] :
% 9.49/5.22        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
% 9.49/5.22       => ( ( vEBT_invar_vebt @ X2 @ na )
% 9.49/5.22          & ! [Xa: nat,Xb: nat] :
% 9.49/5.22              ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ X2 @ Xa ) @ Xb )
% 9.49/5.22              = ( ( Xa != Xb )
% 9.49/5.22                & ( vEBT_V8194947554948674370ptions @ X2 @ Xb ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_24_in__children__def,axiom,
% 9.49/5.22      ( vEBT_V5917875025757280293ildren
% 9.49/5.22      = ( ^ [N2: nat,TreeList2: list_VEBT_VEBT,X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ N2 ) ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_25_nth__list__update__eq,axiom,
% 9.49/5.22      ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 9.49/5.22        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.22       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ I )
% 9.49/5.22          = X ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_26_nth__list__update__eq,axiom,
% 9.49/5.22      ! [I: nat,Xs: list_o,X: $o] :
% 9.49/5.22        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.49/5.22       => ( ( nth_o @ ( list_update_o @ Xs @ I @ X ) @ I )
% 9.49/5.22          = X ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_27_nth__list__update__eq,axiom,
% 9.49/5.22      ! [I: nat,Xs: list_nat,X: nat] :
% 9.49/5.22        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 9.49/5.22       => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ I )
% 9.49/5.22          = X ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_28_nth__list__update__eq,axiom,
% 9.49/5.22      ! [I: nat,Xs: list_int,X: int] :
% 9.49/5.22        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 9.49/5.22       => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ I )
% 9.49/5.22          = X ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_29_buildup__gives__valid,axiom,
% 9.49/5.22      ! [N: nat] :
% 9.49/5.22        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.22       => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_30_nth__list__update__neq,axiom,
% 9.49/5.22      ! [I: nat,J: nat,Xs: list_nat,X: nat] :
% 9.49/5.22        ( ( I != J )
% 9.49/5.22       => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
% 9.49/5.22          = ( nth_nat @ Xs @ J ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_31_nth__list__update__neq,axiom,
% 9.49/5.22      ! [I: nat,J: nat,Xs: list_int,X: int] :
% 9.49/5.22        ( ( I != J )
% 9.49/5.22       => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
% 9.49/5.22          = ( nth_int @ Xs @ J ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_32_nth__list__update__neq,axiom,
% 9.49/5.22      ! [I: nat,J: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 9.49/5.22        ( ( I != J )
% 9.49/5.22       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J )
% 9.49/5.22          = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_33_list__update__id,axiom,
% 9.49/5.22      ! [Xs: list_nat,I: nat] :
% 9.49/5.22        ( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
% 9.49/5.22        = Xs ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_34_list__update__id,axiom,
% 9.49/5.22      ! [Xs: list_int,I: nat] :
% 9.49/5.22        ( ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ I ) )
% 9.49/5.22        = Xs ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_35_list__update__id,axiom,
% 9.49/5.22      ! [Xs: list_VEBT_VEBT,I: nat] :
% 9.49/5.22        ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ I ) )
% 9.49/5.22        = Xs ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_36_length__list__update,axiom,
% 9.49/5.22      ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
% 9.49/5.22        ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) )
% 9.49/5.22        = ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_37_length__list__update,axiom,
% 9.49/5.22      ! [Xs: list_o,I: nat,X: $o] :
% 9.49/5.22        ( ( size_size_list_o @ ( list_update_o @ Xs @ I @ X ) )
% 9.49/5.22        = ( size_size_list_o @ Xs ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_38_length__list__update,axiom,
% 9.49/5.22      ! [Xs: list_nat,I: nat,X: nat] :
% 9.49/5.22        ( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X ) )
% 9.49/5.22        = ( size_size_list_nat @ Xs ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_39_length__list__update,axiom,
% 9.49/5.22      ! [Xs: list_int,I: nat,X: int] :
% 9.49/5.22        ( ( size_size_list_int @ ( list_update_int @ Xs @ I @ X ) )
% 9.49/5.22        = ( size_size_list_int @ Xs ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_40_bot__nat__0_Onot__eq__extremum,axiom,
% 9.49/5.22      ! [A: nat] :
% 9.49/5.22        ( ( A != zero_zero_nat )
% 9.49/5.22        = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_41_neq0__conv,axiom,
% 9.49/5.22      ! [N: nat] :
% 9.49/5.22        ( ( N != zero_zero_nat )
% 9.49/5.22        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_42_less__nat__zero__code,axiom,
% 9.49/5.22      ! [N: nat] :
% 9.49/5.22        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_43_not__gr__zero,axiom,
% 9.49/5.22      ! [N: nat] :
% 9.49/5.22        ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 9.49/5.22        = ( N = zero_zero_nat ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_44__092_060open_062mi_A_092_060le_062_Ax_A_092_060and_062_Ax_A_092_060le_062_Ama_092_060close_062,axiom,
% 9.49/5.22      ( ( ord_less_eq_nat @ mi @ xa )
% 9.49/5.22      & ( ord_less_eq_nat @ xa @ ma ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_45_nth__list__update,axiom,
% 9.49/5.22      ! [I: nat,Xs: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
% 9.49/5.22        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.22       => ( ( ( I = J )
% 9.49/5.22           => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J )
% 9.49/5.22              = X ) )
% 9.49/5.22          & ( ( I != J )
% 9.49/5.22           => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J )
% 9.49/5.22              = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_46_nth__list__update,axiom,
% 9.49/5.22      ! [I: nat,Xs: list_o,X: $o,J: nat] :
% 9.49/5.22        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.49/5.22       => ( ( nth_o @ ( list_update_o @ Xs @ I @ X ) @ J )
% 9.49/5.22          = ( ( ( I = J )
% 9.49/5.22             => X )
% 9.49/5.22            & ( ( I != J )
% 9.49/5.22             => ( nth_o @ Xs @ J ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_47_nth__list__update,axiom,
% 9.49/5.22      ! [I: nat,Xs: list_nat,J: nat,X: nat] :
% 9.49/5.22        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 9.49/5.22       => ( ( ( I = J )
% 9.49/5.22           => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
% 9.49/5.22              = X ) )
% 9.49/5.22          & ( ( I != J )
% 9.49/5.22           => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
% 9.49/5.22              = ( nth_nat @ Xs @ J ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_48_nth__list__update,axiom,
% 9.49/5.22      ! [I: nat,Xs: list_int,J: nat,X: int] :
% 9.49/5.22        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 9.49/5.22       => ( ( ( I = J )
% 9.49/5.22           => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
% 9.49/5.22              = X ) )
% 9.49/5.22          & ( ( I != J )
% 9.49/5.22           => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
% 9.49/5.22              = ( nth_int @ Xs @ J ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_49_min__in__set__def,axiom,
% 9.49/5.22      ( vEBT_VEBT_min_in_set
% 9.49/5.22      = ( ^ [Xs2: set_nat,X3: nat] :
% 9.49/5.22            ( ( member_nat @ X3 @ Xs2 )
% 9.49/5.22            & ! [Y2: nat] :
% 9.49/5.22                ( ( member_nat @ Y2 @ Xs2 )
% 9.49/5.22               => ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_50_max__in__set__def,axiom,
% 9.49/5.22      ( vEBT_VEBT_max_in_set
% 9.49/5.22      = ( ^ [Xs2: set_nat,X3: nat] :
% 9.49/5.22            ( ( member_nat @ X3 @ Xs2 )
% 9.49/5.22            & ! [Y2: nat] :
% 9.49/5.22                ( ( member_nat @ Y2 @ Xs2 )
% 9.49/5.22               => ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_51_inthall,axiom,
% 9.49/5.22      ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,N: nat] :
% 9.49/5.22        ( ! [X4: product_prod_nat_nat] :
% 9.49/5.22            ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.49/5.22           => ( P @ X4 ) )
% 9.49/5.22       => ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
% 9.49/5.22         => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ N ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_52_inthall,axiom,
% 9.49/5.22      ! [Xs: list_real,P: real > $o,N: nat] :
% 9.49/5.22        ( ! [X4: real] :
% 9.49/5.22            ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
% 9.49/5.22           => ( P @ X4 ) )
% 9.49/5.22       => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
% 9.49/5.22         => ( P @ ( nth_real @ Xs @ N ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_53_inthall,axiom,
% 9.49/5.22      ! [Xs: list_set_nat,P: set_nat > $o,N: nat] :
% 9.49/5.22        ( ! [X4: set_nat] :
% 9.49/5.22            ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
% 9.49/5.22           => ( P @ X4 ) )
% 9.49/5.22       => ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 9.49/5.22         => ( P @ ( nth_set_nat @ Xs @ N ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_54_inthall,axiom,
% 9.49/5.22      ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
% 9.49/5.22        ( ! [X4: vEBT_VEBT] :
% 9.49/5.22            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.22           => ( P @ X4 ) )
% 9.49/5.22       => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.22         => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_55_inthall,axiom,
% 9.49/5.22      ! [Xs: list_o,P: $o > $o,N: nat] :
% 9.49/5.22        ( ! [X4: $o] :
% 9.49/5.22            ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 9.49/5.22           => ( P @ X4 ) )
% 9.49/5.22       => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 9.49/5.22         => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_56_inthall,axiom,
% 9.49/5.22      ! [Xs: list_nat,P: nat > $o,N: nat] :
% 9.49/5.22        ( ! [X4: nat] :
% 9.49/5.22            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 9.49/5.22           => ( P @ X4 ) )
% 9.49/5.22       => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 9.49/5.22         => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_57_inthall,axiom,
% 9.49/5.22      ! [Xs: list_int,P: int > $o,N: nat] :
% 9.49/5.22        ( ! [X4: int] :
% 9.49/5.22            ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 9.49/5.22           => ( P @ X4 ) )
% 9.49/5.22       => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 9.49/5.22         => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_58_list__update__overwrite,axiom,
% 9.49/5.22      ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT,Y: vEBT_VEBT] :
% 9.49/5.22        ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ I @ Y )
% 9.49/5.22        = ( list_u1324408373059187874T_VEBT @ Xs @ I @ Y ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_59_le__zero__eq,axiom,
% 9.49/5.22      ! [N: nat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 9.49/5.22        = ( N = zero_zero_nat ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_60_le0,axiom,
% 9.49/5.22      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_61_bot__nat__0_Oextremum,axiom,
% 9.49/5.22      ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_62_list__update__beyond,axiom,
% 9.49/5.22      ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
% 9.49/5.22        ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ I )
% 9.49/5.22       => ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X )
% 9.49/5.22          = Xs ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_63_list__update__beyond,axiom,
% 9.49/5.22      ! [Xs: list_o,I: nat,X: $o] :
% 9.49/5.22        ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ I )
% 9.49/5.22       => ( ( list_update_o @ Xs @ I @ X )
% 9.49/5.22          = Xs ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_64_list__update__beyond,axiom,
% 9.49/5.22      ! [Xs: list_nat,I: nat,X: nat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
% 9.49/5.22       => ( ( list_update_nat @ Xs @ I @ X )
% 9.49/5.22          = Xs ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_65_list__update__beyond,axiom,
% 9.49/5.22      ! [Xs: list_int,I: nat,X: int] :
% 9.49/5.22        ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ I )
% 9.49/5.22       => ( ( list_update_int @ Xs @ I @ X )
% 9.49/5.22          = Xs ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_66_mem__Collect__eq,axiom,
% 9.49/5.22      ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
% 9.49/5.22        ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
% 9.49/5.22        = ( P @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_67_mem__Collect__eq,axiom,
% 9.49/5.22      ! [A: real,P: real > $o] :
% 9.49/5.22        ( ( member_real @ A @ ( collect_real @ P ) )
% 9.49/5.22        = ( P @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_68_mem__Collect__eq,axiom,
% 9.49/5.22      ! [A: list_nat,P: list_nat > $o] :
% 9.49/5.22        ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
% 9.49/5.22        = ( P @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_69_mem__Collect__eq,axiom,
% 9.49/5.22      ! [A: set_nat,P: set_nat > $o] :
% 9.49/5.22        ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
% 9.49/5.22        = ( P @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_70_mem__Collect__eq,axiom,
% 9.49/5.22      ! [A: nat,P: nat > $o] :
% 9.49/5.22        ( ( member_nat @ A @ ( collect_nat @ P ) )
% 9.49/5.22        = ( P @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_71_mem__Collect__eq,axiom,
% 9.49/5.22      ! [A: int,P: int > $o] :
% 9.49/5.22        ( ( member_int @ A @ ( collect_int @ P ) )
% 9.49/5.22        = ( P @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_72_Collect__mem__eq,axiom,
% 9.49/5.22      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.22        ( ( collec3392354462482085612at_nat
% 9.49/5.22          @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A2 ) )
% 9.49/5.22        = A2 ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_73_Collect__mem__eq,axiom,
% 9.49/5.22      ! [A2: set_real] :
% 9.49/5.22        ( ( collect_real
% 9.49/5.22          @ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
% 9.49/5.22        = A2 ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_74_Collect__mem__eq,axiom,
% 9.49/5.22      ! [A2: set_list_nat] :
% 9.49/5.22        ( ( collect_list_nat
% 9.49/5.22          @ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A2 ) )
% 9.49/5.22        = A2 ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_75_Collect__mem__eq,axiom,
% 9.49/5.22      ! [A2: set_set_nat] :
% 9.49/5.22        ( ( collect_set_nat
% 9.49/5.22          @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
% 9.49/5.22        = A2 ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_76_Collect__mem__eq,axiom,
% 9.49/5.22      ! [A2: set_nat] :
% 9.49/5.22        ( ( collect_nat
% 9.49/5.22          @ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
% 9.49/5.22        = A2 ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_77_Collect__mem__eq,axiom,
% 9.49/5.22      ! [A2: set_int] :
% 9.49/5.22        ( ( collect_int
% 9.49/5.22          @ ^ [X3: int] : ( member_int @ X3 @ A2 ) )
% 9.49/5.22        = A2 ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_78_Collect__cong,axiom,
% 9.49/5.22      ! [P: real > $o,Q: real > $o] :
% 9.49/5.22        ( ! [X4: real] :
% 9.49/5.22            ( ( P @ X4 )
% 9.49/5.22            = ( Q @ X4 ) )
% 9.49/5.22       => ( ( collect_real @ P )
% 9.49/5.22          = ( collect_real @ Q ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_79_Collect__cong,axiom,
% 9.49/5.22      ! [P: list_nat > $o,Q: list_nat > $o] :
% 9.49/5.22        ( ! [X4: list_nat] :
% 9.49/5.22            ( ( P @ X4 )
% 9.49/5.22            = ( Q @ X4 ) )
% 9.49/5.22       => ( ( collect_list_nat @ P )
% 9.49/5.22          = ( collect_list_nat @ Q ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_80_Collect__cong,axiom,
% 9.49/5.22      ! [P: set_nat > $o,Q: set_nat > $o] :
% 9.49/5.22        ( ! [X4: set_nat] :
% 9.49/5.22            ( ( P @ X4 )
% 9.49/5.22            = ( Q @ X4 ) )
% 9.49/5.22       => ( ( collect_set_nat @ P )
% 9.49/5.22          = ( collect_set_nat @ Q ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_81_Collect__cong,axiom,
% 9.49/5.22      ! [P: nat > $o,Q: nat > $o] :
% 9.49/5.22        ( ! [X4: nat] :
% 9.49/5.22            ( ( P @ X4 )
% 9.49/5.22            = ( Q @ X4 ) )
% 9.49/5.22       => ( ( collect_nat @ P )
% 9.49/5.22          = ( collect_nat @ Q ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_82_Collect__cong,axiom,
% 9.49/5.22      ! [P: int > $o,Q: int > $o] :
% 9.49/5.22        ( ! [X4: int] :
% 9.49/5.22            ( ( P @ X4 )
% 9.49/5.22            = ( Q @ X4 ) )
% 9.49/5.22       => ( ( collect_int @ P )
% 9.49/5.22          = ( collect_int @ Q ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_83_set__swap,axiom,
% 9.49/5.22      ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
% 9.49/5.22        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.22       => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.22         => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs @ I ) ) )
% 9.49/5.22            = ( set_VEBT_VEBT2 @ Xs ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_84_set__swap,axiom,
% 9.49/5.22      ! [I: nat,Xs: list_o,J: nat] :
% 9.49/5.22        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.49/5.22       => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
% 9.49/5.22         => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs @ I @ ( nth_o @ Xs @ J ) ) @ J @ ( nth_o @ Xs @ I ) ) )
% 9.49/5.22            = ( set_o2 @ Xs ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_85_set__swap,axiom,
% 9.49/5.22      ! [I: nat,Xs: list_nat,J: nat] :
% 9.49/5.22        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 9.49/5.22       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
% 9.49/5.22         => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
% 9.49/5.22            = ( set_nat2 @ Xs ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_86_set__swap,axiom,
% 9.49/5.22      ! [I: nat,Xs: list_int,J: nat] :
% 9.49/5.22        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 9.49/5.22       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
% 9.49/5.22         => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ J ) ) @ J @ ( nth_int @ Xs @ I ) ) )
% 9.49/5.22            = ( set_int2 @ Xs ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_87_Nat_Oex__has__greatest__nat,axiom,
% 9.49/5.22      ! [P: nat > $o,K: nat,B: nat] :
% 9.49/5.22        ( ( P @ K )
% 9.49/5.22       => ( ! [Y3: nat] :
% 9.49/5.22              ( ( P @ Y3 )
% 9.49/5.22             => ( ord_less_eq_nat @ Y3 @ B ) )
% 9.49/5.22         => ? [X4: nat] :
% 9.49/5.22              ( ( P @ X4 )
% 9.49/5.22              & ! [Y4: nat] :
% 9.49/5.22                  ( ( P @ Y4 )
% 9.49/5.22                 => ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_88_nat__le__linear,axiom,
% 9.49/5.22      ! [M: nat,N: nat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.22        | ( ord_less_eq_nat @ N @ M ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_89_le__antisym,axiom,
% 9.49/5.22      ! [M: nat,N: nat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.22       => ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.22         => ( M = N ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_90_eq__imp__le,axiom,
% 9.49/5.22      ! [M: nat,N: nat] :
% 9.49/5.22        ( ( M = N )
% 9.49/5.22       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_91_le__trans,axiom,
% 9.49/5.22      ! [I: nat,J: nat,K: nat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.22       => ( ( ord_less_eq_nat @ J @ K )
% 9.49/5.22         => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_92_le__refl,axiom,
% 9.49/5.22      ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_93_zero__le,axiom,
% 9.49/5.22      ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_94_le__0__eq,axiom,
% 9.49/5.22      ! [N: nat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 9.49/5.22        = ( N = zero_zero_nat ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_95_bot__nat__0_Oextremum__uniqueI,axiom,
% 9.49/5.22      ! [A: nat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.49/5.22       => ( A = zero_zero_nat ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_96_bot__nat__0_Oextremum__unique,axiom,
% 9.49/5.22      ! [A: nat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.49/5.22        = ( A = zero_zero_nat ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_97_less__eq__nat_Osimps_I1_J,axiom,
% 9.49/5.22      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_98_less__mono__imp__le__mono,axiom,
% 9.49/5.22      ! [F: nat > nat,I: nat,J: nat] :
% 9.49/5.22        ( ! [I2: nat,J2: nat] :
% 9.49/5.22            ( ( ord_less_nat @ I2 @ J2 )
% 9.49/5.22           => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 9.49/5.22       => ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.22         => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_99_le__neq__implies__less,axiom,
% 9.49/5.22      ! [M: nat,N: nat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.22       => ( ( M != N )
% 9.49/5.22         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_100_less__or__eq__imp__le,axiom,
% 9.49/5.22      ! [M: nat,N: nat] :
% 9.49/5.22        ( ( ( ord_less_nat @ M @ N )
% 9.49/5.22          | ( M = N ) )
% 9.49/5.22       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_101_le__eq__less__or__eq,axiom,
% 9.49/5.22      ( ord_less_eq_nat
% 9.49/5.22      = ( ^ [M2: nat,N2: nat] :
% 9.49/5.22            ( ( ord_less_nat @ M2 @ N2 )
% 9.49/5.22            | ( M2 = N2 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_102_less__imp__le__nat,axiom,
% 9.49/5.22      ! [M: nat,N: nat] :
% 9.49/5.22        ( ( ord_less_nat @ M @ N )
% 9.49/5.22       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_103_nat__less__le,axiom,
% 9.49/5.22      ( ord_less_nat
% 9.49/5.22      = ( ^ [M2: nat,N2: nat] :
% 9.49/5.22            ( ( ord_less_eq_nat @ M2 @ N2 )
% 9.49/5.22            & ( M2 != N2 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_104_ex__least__nat__le,axiom,
% 9.49/5.22      ! [P: nat > $o,N: nat] :
% 9.49/5.22        ( ( P @ N )
% 9.49/5.22       => ( ~ ( P @ zero_zero_nat )
% 9.49/5.22         => ? [K2: nat] :
% 9.49/5.22              ( ( ord_less_eq_nat @ K2 @ N )
% 9.49/5.22              & ! [I3: nat] :
% 9.49/5.22                  ( ( ord_less_nat @ I3 @ K2 )
% 9.49/5.22                 => ~ ( P @ I3 ) )
% 9.49/5.22              & ( P @ K2 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_105_length__pos__if__in__set,axiom,
% 9.49/5.22      ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
% 9.49/5.22        ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.49/5.22       => ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_106_length__pos__if__in__set,axiom,
% 9.49/5.22      ! [X: real,Xs: list_real] :
% 9.49/5.22        ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.49/5.22       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_107_length__pos__if__in__set,axiom,
% 9.49/5.22      ! [X: set_nat,Xs: list_set_nat] :
% 9.49/5.22        ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 9.49/5.22       => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_108_length__pos__if__in__set,axiom,
% 9.49/5.22      ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 9.49/5.22        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.22       => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_109_length__pos__if__in__set,axiom,
% 9.49/5.22      ! [X: $o,Xs: list_o] :
% 9.49/5.22        ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 9.49/5.22       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_110_length__pos__if__in__set,axiom,
% 9.49/5.22      ! [X: nat,Xs: list_nat] :
% 9.49/5.22        ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 9.49/5.22       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_111_length__pos__if__in__set,axiom,
% 9.49/5.22      ! [X: int,Xs: list_int] :
% 9.49/5.22        ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 9.49/5.22       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_112_all__set__conv__all__nth,axiom,
% 9.49/5.22      ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 9.49/5.22        ( ( ! [X3: vEBT_VEBT] :
% 9.49/5.22              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.22             => ( P @ X3 ) ) )
% 9.49/5.22        = ( ! [I4: nat] :
% 9.49/5.22              ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.22             => ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_113_all__set__conv__all__nth,axiom,
% 9.49/5.22      ! [Xs: list_o,P: $o > $o] :
% 9.49/5.22        ( ( ! [X3: $o] :
% 9.49/5.22              ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
% 9.49/5.22             => ( P @ X3 ) ) )
% 9.49/5.22        = ( ! [I4: nat] :
% 9.49/5.22              ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 9.49/5.22             => ( P @ ( nth_o @ Xs @ I4 ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_114_all__set__conv__all__nth,axiom,
% 9.49/5.22      ! [Xs: list_nat,P: nat > $o] :
% 9.49/5.22        ( ( ! [X3: nat] :
% 9.49/5.22              ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 9.49/5.22             => ( P @ X3 ) ) )
% 9.49/5.22        = ( ! [I4: nat] :
% 9.49/5.22              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 9.49/5.22             => ( P @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_115_all__set__conv__all__nth,axiom,
% 9.49/5.22      ! [Xs: list_int,P: int > $o] :
% 9.49/5.22        ( ( ! [X3: int] :
% 9.49/5.22              ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 9.49/5.22             => ( P @ X3 ) ) )
% 9.49/5.22        = ( ! [I4: nat] :
% 9.49/5.22              ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 9.49/5.22             => ( P @ ( nth_int @ Xs @ I4 ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_116_all__nth__imp__all__set,axiom,
% 9.49/5.22      ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,X: product_prod_nat_nat] :
% 9.49/5.22        ( ! [I2: nat] :
% 9.49/5.22            ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 9.49/5.22           => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) ) )
% 9.49/5.22       => ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.49/5.22         => ( P @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_117_all__nth__imp__all__set,axiom,
% 9.49/5.22      ! [Xs: list_real,P: real > $o,X: real] :
% 9.49/5.22        ( ! [I2: nat] :
% 9.49/5.22            ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
% 9.49/5.22           => ( P @ ( nth_real @ Xs @ I2 ) ) )
% 9.49/5.22       => ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.49/5.22         => ( P @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_118_all__nth__imp__all__set,axiom,
% 9.49/5.22      ! [Xs: list_set_nat,P: set_nat > $o,X: set_nat] :
% 9.49/5.22        ( ! [I2: nat] :
% 9.49/5.22            ( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 9.49/5.22           => ( P @ ( nth_set_nat @ Xs @ I2 ) ) )
% 9.49/5.22       => ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 9.49/5.22         => ( P @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_119_all__nth__imp__all__set,axiom,
% 9.49/5.22      ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
% 9.49/5.22        ( ! [I2: nat] :
% 9.49/5.22            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.22           => ( P @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) )
% 9.49/5.22       => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.22         => ( P @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_120_all__nth__imp__all__set,axiom,
% 9.49/5.22      ! [Xs: list_o,P: $o > $o,X: $o] :
% 9.49/5.22        ( ! [I2: nat] :
% 9.49/5.22            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 9.49/5.22           => ( P @ ( nth_o @ Xs @ I2 ) ) )
% 9.49/5.22       => ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 9.49/5.22         => ( P @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_121_all__nth__imp__all__set,axiom,
% 9.49/5.22      ! [Xs: list_nat,P: nat > $o,X: nat] :
% 9.49/5.22        ( ! [I2: nat] :
% 9.49/5.22            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 9.49/5.22           => ( P @ ( nth_nat @ Xs @ I2 ) ) )
% 9.49/5.22       => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 9.49/5.22         => ( P @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_122_all__nth__imp__all__set,axiom,
% 9.49/5.22      ! [Xs: list_int,P: int > $o,X: int] :
% 9.49/5.22        ( ! [I2: nat] :
% 9.49/5.22            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 9.49/5.22           => ( P @ ( nth_int @ Xs @ I2 ) ) )
% 9.49/5.22       => ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 9.49/5.22         => ( P @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_123_in__set__conv__nth,axiom,
% 9.49/5.22      ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
% 9.49/5.22        ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.49/5.22        = ( ? [I4: nat] :
% 9.49/5.22              ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 9.49/5.22              & ( ( nth_Pr7617993195940197384at_nat @ Xs @ I4 )
% 9.49/5.22                = X ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_124_in__set__conv__nth,axiom,
% 9.49/5.22      ! [X: real,Xs: list_real] :
% 9.49/5.22        ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.49/5.22        = ( ? [I4: nat] :
% 9.49/5.22              ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs ) )
% 9.49/5.22              & ( ( nth_real @ Xs @ I4 )
% 9.49/5.22                = X ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_125_in__set__conv__nth,axiom,
% 9.49/5.22      ! [X: set_nat,Xs: list_set_nat] :
% 9.49/5.22        ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 9.49/5.22        = ( ? [I4: nat] :
% 9.49/5.22              ( ( ord_less_nat @ I4 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 9.49/5.22              & ( ( nth_set_nat @ Xs @ I4 )
% 9.49/5.22                = X ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_126_in__set__conv__nth,axiom,
% 9.49/5.22      ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 9.49/5.22        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.22        = ( ? [I4: nat] :
% 9.49/5.22              ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.22              & ( ( nth_VEBT_VEBT @ Xs @ I4 )
% 9.49/5.22                = X ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_127_in__set__conv__nth,axiom,
% 9.49/5.22      ! [X: $o,Xs: list_o] :
% 9.49/5.22        ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 9.49/5.22        = ( ? [I4: nat] :
% 9.49/5.22              ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 9.49/5.22              & ( ( nth_o @ Xs @ I4 )
% 9.49/5.22                = X ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_128_in__set__conv__nth,axiom,
% 9.49/5.22      ! [X: nat,Xs: list_nat] :
% 9.49/5.22        ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 9.49/5.22        = ( ? [I4: nat] :
% 9.49/5.22              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 9.49/5.22              & ( ( nth_nat @ Xs @ I4 )
% 9.49/5.22                = X ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_129_in__set__conv__nth,axiom,
% 9.49/5.22      ! [X: int,Xs: list_int] :
% 9.49/5.22        ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 9.49/5.22        = ( ? [I4: nat] :
% 9.49/5.22              ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 9.49/5.22              & ( ( nth_int @ Xs @ I4 )
% 9.49/5.22                = X ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_130_list__ball__nth,axiom,
% 9.49/5.22      ! [N: nat,Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 9.49/5.22        ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.22       => ( ! [X4: vEBT_VEBT] :
% 9.49/5.22              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.22             => ( P @ X4 ) )
% 9.49/5.22         => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_131_list__ball__nth,axiom,
% 9.49/5.22      ! [N: nat,Xs: list_o,P: $o > $o] :
% 9.49/5.22        ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 9.49/5.22       => ( ! [X4: $o] :
% 9.49/5.22              ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 9.49/5.22             => ( P @ X4 ) )
% 9.49/5.22         => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_132_list__ball__nth,axiom,
% 9.49/5.22      ! [N: nat,Xs: list_nat,P: nat > $o] :
% 9.49/5.22        ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 9.49/5.22       => ( ! [X4: nat] :
% 9.49/5.22              ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 9.49/5.22             => ( P @ X4 ) )
% 9.49/5.22         => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_133_list__ball__nth,axiom,
% 9.49/5.22      ! [N: nat,Xs: list_int,P: int > $o] :
% 9.49/5.22        ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 9.49/5.22       => ( ! [X4: int] :
% 9.49/5.22              ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 9.49/5.22             => ( P @ X4 ) )
% 9.49/5.22         => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_134_nth__mem,axiom,
% 9.49/5.22      ! [N: nat,Xs: list_P6011104703257516679at_nat] :
% 9.49/5.22        ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
% 9.49/5.22       => ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N ) @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_135_nth__mem,axiom,
% 9.49/5.22      ! [N: nat,Xs: list_real] :
% 9.49/5.22        ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
% 9.49/5.22       => ( member_real @ ( nth_real @ Xs @ N ) @ ( set_real2 @ Xs ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_136_nth__mem,axiom,
% 9.49/5.22      ! [N: nat,Xs: list_set_nat] :
% 9.49/5.22        ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 9.49/5.22       => ( member_set_nat @ ( nth_set_nat @ Xs @ N ) @ ( set_set_nat2 @ Xs ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_137_nth__mem,axiom,
% 9.49/5.22      ! [N: nat,Xs: list_VEBT_VEBT] :
% 9.49/5.22        ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.22       => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_138_nth__mem,axiom,
% 9.49/5.22      ! [N: nat,Xs: list_o] :
% 9.49/5.22        ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 9.49/5.22       => ( member_o @ ( nth_o @ Xs @ N ) @ ( set_o2 @ Xs ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_139_nth__mem,axiom,
% 9.49/5.22      ! [N: nat,Xs: list_nat] :
% 9.49/5.22        ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 9.49/5.22       => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_140_nth__mem,axiom,
% 9.49/5.22      ! [N: nat,Xs: list_int] :
% 9.49/5.22        ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 9.49/5.22       => ( member_int @ ( nth_int @ Xs @ N ) @ ( set_int2 @ Xs ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_141_set__update__memI,axiom,
% 9.49/5.22      ! [N: nat,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
% 9.49/5.22        ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
% 9.49/5.22       => ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ N @ X ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_142_set__update__memI,axiom,
% 9.49/5.22      ! [N: nat,Xs: list_real,X: real] :
% 9.49/5.22        ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
% 9.49/5.22       => ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs @ N @ X ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_143_set__update__memI,axiom,
% 9.49/5.22      ! [N: nat,Xs: list_set_nat,X: set_nat] :
% 9.49/5.22        ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 9.49/5.22       => ( member_set_nat @ X @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ N @ X ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_144_set__update__memI,axiom,
% 9.49/5.22      ! [N: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 9.49/5.22        ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.22       => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N @ X ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_145_set__update__memI,axiom,
% 9.49/5.22      ! [N: nat,Xs: list_o,X: $o] :
% 9.49/5.22        ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 9.49/5.22       => ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs @ N @ X ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_146_set__update__memI,axiom,
% 9.49/5.22      ! [N: nat,Xs: list_nat,X: nat] :
% 9.49/5.22        ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 9.49/5.22       => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_147_set__update__memI,axiom,
% 9.49/5.22      ! [N: nat,Xs: list_int,X: int] :
% 9.49/5.22        ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 9.49/5.22       => ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs @ N @ X ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_148_zero__reorient,axiom,
% 9.49/5.22      ! [X: complex] :
% 9.49/5.22        ( ( zero_zero_complex = X )
% 9.49/5.22        = ( X = zero_zero_complex ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_149_zero__reorient,axiom,
% 9.49/5.22      ! [X: real] :
% 9.49/5.22        ( ( zero_zero_real = X )
% 9.49/5.22        = ( X = zero_zero_real ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_150_zero__reorient,axiom,
% 9.49/5.22      ! [X: rat] :
% 9.49/5.22        ( ( zero_zero_rat = X )
% 9.49/5.22        = ( X = zero_zero_rat ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_151_zero__reorient,axiom,
% 9.49/5.22      ! [X: nat] :
% 9.49/5.22        ( ( zero_zero_nat = X )
% 9.49/5.22        = ( X = zero_zero_nat ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_152_zero__reorient,axiom,
% 9.49/5.22      ! [X: int] :
% 9.49/5.22        ( ( zero_zero_int = X )
% 9.49/5.22        = ( X = zero_zero_int ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_153_linorder__neqE__nat,axiom,
% 9.49/5.22      ! [X: nat,Y: nat] :
% 9.49/5.22        ( ( X != Y )
% 9.49/5.22       => ( ~ ( ord_less_nat @ X @ Y )
% 9.49/5.22         => ( ord_less_nat @ Y @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_154_infinite__descent,axiom,
% 9.49/5.22      ! [P: nat > $o,N: nat] :
% 9.49/5.22        ( ! [N3: nat] :
% 9.49/5.22            ( ~ ( P @ N3 )
% 9.49/5.22           => ? [M3: nat] :
% 9.49/5.22                ( ( ord_less_nat @ M3 @ N3 )
% 9.49/5.22                & ~ ( P @ M3 ) ) )
% 9.49/5.22       => ( P @ N ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_155_nat__less__induct,axiom,
% 9.49/5.22      ! [P: nat > $o,N: nat] :
% 9.49/5.22        ( ! [N3: nat] :
% 9.49/5.22            ( ! [M3: nat] :
% 9.49/5.22                ( ( ord_less_nat @ M3 @ N3 )
% 9.49/5.22               => ( P @ M3 ) )
% 9.49/5.22           => ( P @ N3 ) )
% 9.49/5.22       => ( P @ N ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_156_less__irrefl__nat,axiom,
% 9.49/5.22      ! [N: nat] :
% 9.49/5.22        ~ ( ord_less_nat @ N @ N ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_157_less__not__refl3,axiom,
% 9.49/5.22      ! [S: nat,T: nat] :
% 9.49/5.22        ( ( ord_less_nat @ S @ T )
% 9.49/5.22       => ( S != T ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_158_less__not__refl2,axiom,
% 9.49/5.22      ! [N: nat,M: nat] :
% 9.49/5.22        ( ( ord_less_nat @ N @ M )
% 9.49/5.22       => ( M != N ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_159_less__not__refl,axiom,
% 9.49/5.22      ! [N: nat] :
% 9.49/5.22        ~ ( ord_less_nat @ N @ N ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_160_nat__neq__iff,axiom,
% 9.49/5.22      ! [M: nat,N: nat] :
% 9.49/5.22        ( ( M != N )
% 9.49/5.22        = ( ( ord_less_nat @ M @ N )
% 9.49/5.22          | ( ord_less_nat @ N @ M ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_161_size__neq__size__imp__neq,axiom,
% 9.49/5.22      ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
% 9.49/5.22        ( ( ( size_s6755466524823107622T_VEBT @ X )
% 9.49/5.22         != ( size_s6755466524823107622T_VEBT @ Y ) )
% 9.49/5.22       => ( X != Y ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_162_size__neq__size__imp__neq,axiom,
% 9.49/5.22      ! [X: list_o,Y: list_o] :
% 9.49/5.22        ( ( ( size_size_list_o @ X )
% 9.49/5.22         != ( size_size_list_o @ Y ) )
% 9.49/5.22       => ( X != Y ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_163_size__neq__size__imp__neq,axiom,
% 9.49/5.22      ! [X: list_nat,Y: list_nat] :
% 9.49/5.22        ( ( ( size_size_list_nat @ X )
% 9.49/5.22         != ( size_size_list_nat @ Y ) )
% 9.49/5.22       => ( X != Y ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_164_size__neq__size__imp__neq,axiom,
% 9.49/5.22      ! [X: list_int,Y: list_int] :
% 9.49/5.22        ( ( ( size_size_list_int @ X )
% 9.49/5.22         != ( size_size_list_int @ Y ) )
% 9.49/5.22       => ( X != Y ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_165_size__neq__size__imp__neq,axiom,
% 9.49/5.22      ! [X: num,Y: num] :
% 9.49/5.22        ( ( ( size_size_num @ X )
% 9.49/5.22         != ( size_size_num @ Y ) )
% 9.49/5.22       => ( X != Y ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_166_neq__if__length__neq,axiom,
% 9.49/5.22      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 9.49/5.22        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.49/5.22         != ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.49/5.22       => ( Xs != Ys ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_167_neq__if__length__neq,axiom,
% 9.49/5.22      ! [Xs: list_o,Ys: list_o] :
% 9.49/5.22        ( ( ( size_size_list_o @ Xs )
% 9.49/5.22         != ( size_size_list_o @ Ys ) )
% 9.49/5.22       => ( Xs != Ys ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_168_neq__if__length__neq,axiom,
% 9.49/5.22      ! [Xs: list_nat,Ys: list_nat] :
% 9.49/5.22        ( ( ( size_size_list_nat @ Xs )
% 9.49/5.22         != ( size_size_list_nat @ Ys ) )
% 9.49/5.22       => ( Xs != Ys ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_169_neq__if__length__neq,axiom,
% 9.49/5.22      ! [Xs: list_int,Ys: list_int] :
% 9.49/5.22        ( ( ( size_size_list_int @ Xs )
% 9.49/5.22         != ( size_size_list_int @ Ys ) )
% 9.49/5.22       => ( Xs != Ys ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_170_Ex__list__of__length,axiom,
% 9.49/5.22      ! [N: nat] :
% 9.49/5.22      ? [Xs3: list_VEBT_VEBT] :
% 9.49/5.22        ( ( size_s6755466524823107622T_VEBT @ Xs3 )
% 9.49/5.22        = N ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_171_Ex__list__of__length,axiom,
% 9.49/5.22      ! [N: nat] :
% 9.49/5.22      ? [Xs3: list_o] :
% 9.49/5.22        ( ( size_size_list_o @ Xs3 )
% 9.49/5.22        = N ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_172_Ex__list__of__length,axiom,
% 9.49/5.22      ! [N: nat] :
% 9.49/5.22      ? [Xs3: list_nat] :
% 9.49/5.22        ( ( size_size_list_nat @ Xs3 )
% 9.49/5.22        = N ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_173_Ex__list__of__length,axiom,
% 9.49/5.22      ! [N: nat] :
% 9.49/5.22      ? [Xs3: list_int] :
% 9.49/5.22        ( ( size_size_list_int @ Xs3 )
% 9.49/5.22        = N ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_174_list__update__swap,axiom,
% 9.49/5.22      ! [I: nat,I5: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT,X5: vEBT_VEBT] :
% 9.49/5.22        ( ( I != I5 )
% 9.49/5.22       => ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ I5 @ X5 )
% 9.49/5.22          = ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I5 @ X5 ) @ I @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_175_zero__less__iff__neq__zero,axiom,
% 9.49/5.22      ! [N: nat] :
% 9.49/5.22        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.22        = ( N != zero_zero_nat ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_176_gr__implies__not__zero,axiom,
% 9.49/5.22      ! [M: nat,N: nat] :
% 9.49/5.22        ( ( ord_less_nat @ M @ N )
% 9.49/5.22       => ( N != zero_zero_nat ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_177_not__less__zero,axiom,
% 9.49/5.22      ! [N: nat] :
% 9.49/5.22        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_178_gr__zeroI,axiom,
% 9.49/5.22      ! [N: nat] :
% 9.49/5.22        ( ( N != zero_zero_nat )
% 9.49/5.22       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_179_infinite__descent0,axiom,
% 9.49/5.22      ! [P: nat > $o,N: nat] :
% 9.49/5.22        ( ( P @ zero_zero_nat )
% 9.49/5.22       => ( ! [N3: nat] :
% 9.49/5.22              ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.49/5.22             => ( ~ ( P @ N3 )
% 9.49/5.22               => ? [M3: nat] :
% 9.49/5.22                    ( ( ord_less_nat @ M3 @ N3 )
% 9.49/5.22                    & ~ ( P @ M3 ) ) ) )
% 9.49/5.22         => ( P @ N ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_180_gr__implies__not0,axiom,
% 9.49/5.22      ! [M: nat,N: nat] :
% 9.49/5.22        ( ( ord_less_nat @ M @ N )
% 9.49/5.22       => ( N != zero_zero_nat ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_181_less__zeroE,axiom,
% 9.49/5.22      ! [N: nat] :
% 9.49/5.22        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_182_not__less0,axiom,
% 9.49/5.22      ! [N: nat] :
% 9.49/5.22        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_183_not__gr0,axiom,
% 9.49/5.22      ! [N: nat] :
% 9.49/5.22        ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 9.49/5.22        = ( N = zero_zero_nat ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_184_gr0I,axiom,
% 9.49/5.22      ! [N: nat] :
% 9.49/5.22        ( ( N != zero_zero_nat )
% 9.49/5.22       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_185_bot__nat__0_Oextremum__strict,axiom,
% 9.49/5.22      ! [A: nat] :
% 9.49/5.22        ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_186_length__induct,axiom,
% 9.49/5.22      ! [P: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 9.49/5.22        ( ! [Xs3: list_VEBT_VEBT] :
% 9.49/5.22            ( ! [Ys2: list_VEBT_VEBT] :
% 9.49/5.22                ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
% 9.49/5.22               => ( P @ Ys2 ) )
% 9.49/5.22           => ( P @ Xs3 ) )
% 9.49/5.22       => ( P @ Xs ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_187_length__induct,axiom,
% 9.49/5.22      ! [P: list_o > $o,Xs: list_o] :
% 9.49/5.22        ( ! [Xs3: list_o] :
% 9.49/5.22            ( ! [Ys2: list_o] :
% 9.49/5.22                ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
% 9.49/5.22               => ( P @ Ys2 ) )
% 9.49/5.22           => ( P @ Xs3 ) )
% 9.49/5.22       => ( P @ Xs ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_188_length__induct,axiom,
% 9.49/5.22      ! [P: list_nat > $o,Xs: list_nat] :
% 9.49/5.22        ( ! [Xs3: list_nat] :
% 9.49/5.22            ( ! [Ys2: list_nat] :
% 9.49/5.22                ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
% 9.49/5.22               => ( P @ Ys2 ) )
% 9.49/5.22           => ( P @ Xs3 ) )
% 9.49/5.22       => ( P @ Xs ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_189_length__induct,axiom,
% 9.49/5.22      ! [P: list_int > $o,Xs: list_int] :
% 9.49/5.22        ( ! [Xs3: list_int] :
% 9.49/5.22            ( ! [Ys2: list_int] :
% 9.49/5.22                ( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs3 ) )
% 9.49/5.22               => ( P @ Ys2 ) )
% 9.49/5.22           => ( P @ Xs3 ) )
% 9.49/5.22       => ( P @ Xs ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_190_list__eq__iff__nth__eq,axiom,
% 9.49/5.22      ( ( ^ [Y5: list_VEBT_VEBT,Z: list_VEBT_VEBT] : ( Y5 = Z ) )
% 9.49/5.22      = ( ^ [Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 9.49/5.22            ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 9.49/5.22              = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
% 9.49/5.22            & ! [I4: nat] :
% 9.49/5.22                ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 9.49/5.22               => ( ( nth_VEBT_VEBT @ Xs2 @ I4 )
% 9.49/5.22                  = ( nth_VEBT_VEBT @ Ys3 @ I4 ) ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_191_list__eq__iff__nth__eq,axiom,
% 9.49/5.22      ( ( ^ [Y5: list_o,Z: list_o] : ( Y5 = Z ) )
% 9.49/5.22      = ( ^ [Xs2: list_o,Ys3: list_o] :
% 9.49/5.22            ( ( ( size_size_list_o @ Xs2 )
% 9.49/5.22              = ( size_size_list_o @ Ys3 ) )
% 9.49/5.22            & ! [I4: nat] :
% 9.49/5.22                ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
% 9.49/5.22               => ( ( nth_o @ Xs2 @ I4 )
% 9.49/5.22                  = ( nth_o @ Ys3 @ I4 ) ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_192_list__eq__iff__nth__eq,axiom,
% 9.49/5.22      ( ( ^ [Y5: list_nat,Z: list_nat] : ( Y5 = Z ) )
% 9.49/5.22      = ( ^ [Xs2: list_nat,Ys3: list_nat] :
% 9.49/5.22            ( ( ( size_size_list_nat @ Xs2 )
% 9.49/5.22              = ( size_size_list_nat @ Ys3 ) )
% 9.49/5.22            & ! [I4: nat] :
% 9.49/5.22                ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
% 9.49/5.22               => ( ( nth_nat @ Xs2 @ I4 )
% 9.49/5.22                  = ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_193_list__eq__iff__nth__eq,axiom,
% 9.49/5.22      ( ( ^ [Y5: list_int,Z: list_int] : ( Y5 = Z ) )
% 9.49/5.22      = ( ^ [Xs2: list_int,Ys3: list_int] :
% 9.49/5.22            ( ( ( size_size_list_int @ Xs2 )
% 9.49/5.22              = ( size_size_list_int @ Ys3 ) )
% 9.49/5.22            & ! [I4: nat] :
% 9.49/5.22                ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
% 9.49/5.22               => ( ( nth_int @ Xs2 @ I4 )
% 9.49/5.22                  = ( nth_int @ Ys3 @ I4 ) ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_194_Skolem__list__nth,axiom,
% 9.49/5.22      ! [K: nat,P: nat > vEBT_VEBT > $o] :
% 9.49/5.22        ( ( ! [I4: nat] :
% 9.49/5.22              ( ( ord_less_nat @ I4 @ K )
% 9.49/5.22             => ? [X6: vEBT_VEBT] : ( P @ I4 @ X6 ) ) )
% 9.49/5.22        = ( ? [Xs2: list_VEBT_VEBT] :
% 9.49/5.22              ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 9.49/5.22                = K )
% 9.49/5.22              & ! [I4: nat] :
% 9.49/5.22                  ( ( ord_less_nat @ I4 @ K )
% 9.49/5.22                 => ( P @ I4 @ ( nth_VEBT_VEBT @ Xs2 @ I4 ) ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_195_Skolem__list__nth,axiom,
% 9.49/5.22      ! [K: nat,P: nat > $o > $o] :
% 9.49/5.22        ( ( ! [I4: nat] :
% 9.49/5.22              ( ( ord_less_nat @ I4 @ K )
% 9.49/5.22             => ? [X6: $o] : ( P @ I4 @ X6 ) ) )
% 9.49/5.22        = ( ? [Xs2: list_o] :
% 9.49/5.22              ( ( ( size_size_list_o @ Xs2 )
% 9.49/5.22                = K )
% 9.49/5.22              & ! [I4: nat] :
% 9.49/5.22                  ( ( ord_less_nat @ I4 @ K )
% 9.49/5.22                 => ( P @ I4 @ ( nth_o @ Xs2 @ I4 ) ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_196_Skolem__list__nth,axiom,
% 9.49/5.22      ! [K: nat,P: nat > nat > $o] :
% 9.49/5.22        ( ( ! [I4: nat] :
% 9.49/5.22              ( ( ord_less_nat @ I4 @ K )
% 9.49/5.22             => ? [X6: nat] : ( P @ I4 @ X6 ) ) )
% 9.49/5.22        = ( ? [Xs2: list_nat] :
% 9.49/5.22              ( ( ( size_size_list_nat @ Xs2 )
% 9.49/5.22                = K )
% 9.49/5.22              & ! [I4: nat] :
% 9.49/5.22                  ( ( ord_less_nat @ I4 @ K )
% 9.49/5.22                 => ( P @ I4 @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_197_Skolem__list__nth,axiom,
% 9.49/5.22      ! [K: nat,P: nat > int > $o] :
% 9.49/5.22        ( ( ! [I4: nat] :
% 9.49/5.22              ( ( ord_less_nat @ I4 @ K )
% 9.49/5.22             => ? [X6: int] : ( P @ I4 @ X6 ) ) )
% 9.49/5.22        = ( ? [Xs2: list_int] :
% 9.49/5.22              ( ( ( size_size_list_int @ Xs2 )
% 9.49/5.22                = K )
% 9.49/5.22              & ! [I4: nat] :
% 9.49/5.22                  ( ( ord_less_nat @ I4 @ K )
% 9.49/5.22                 => ( P @ I4 @ ( nth_int @ Xs2 @ I4 ) ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_198_nth__equalityI,axiom,
% 9.49/5.22      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 9.49/5.22        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.49/5.22          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.49/5.22       => ( ! [I2: nat] :
% 9.49/5.22              ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.22             => ( ( nth_VEBT_VEBT @ Xs @ I2 )
% 9.49/5.22                = ( nth_VEBT_VEBT @ Ys @ I2 ) ) )
% 9.49/5.22         => ( Xs = Ys ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_199_nth__equalityI,axiom,
% 9.49/5.22      ! [Xs: list_o,Ys: list_o] :
% 9.49/5.22        ( ( ( size_size_list_o @ Xs )
% 9.49/5.22          = ( size_size_list_o @ Ys ) )
% 9.49/5.22       => ( ! [I2: nat] :
% 9.49/5.22              ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 9.49/5.22             => ( ( nth_o @ Xs @ I2 )
% 9.49/5.22                = ( nth_o @ Ys @ I2 ) ) )
% 9.49/5.22         => ( Xs = Ys ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_200_nth__equalityI,axiom,
% 9.49/5.22      ! [Xs: list_nat,Ys: list_nat] :
% 9.49/5.22        ( ( ( size_size_list_nat @ Xs )
% 9.49/5.22          = ( size_size_list_nat @ Ys ) )
% 9.49/5.22       => ( ! [I2: nat] :
% 9.49/5.22              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 9.49/5.22             => ( ( nth_nat @ Xs @ I2 )
% 9.49/5.22                = ( nth_nat @ Ys @ I2 ) ) )
% 9.49/5.22         => ( Xs = Ys ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_201_nth__equalityI,axiom,
% 9.49/5.22      ! [Xs: list_int,Ys: list_int] :
% 9.49/5.22        ( ( ( size_size_list_int @ Xs )
% 9.49/5.22          = ( size_size_list_int @ Ys ) )
% 9.49/5.22       => ( ! [I2: nat] :
% 9.49/5.22              ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 9.49/5.22             => ( ( nth_int @ Xs @ I2 )
% 9.49/5.22                = ( nth_int @ Ys @ I2 ) ) )
% 9.49/5.22         => ( Xs = Ys ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_202_list__update__same__conv,axiom,
% 9.49/5.22      ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 9.49/5.22        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.22       => ( ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X )
% 9.49/5.22            = Xs )
% 9.49/5.22          = ( ( nth_VEBT_VEBT @ Xs @ I )
% 9.49/5.22            = X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_203_list__update__same__conv,axiom,
% 9.49/5.22      ! [I: nat,Xs: list_o,X: $o] :
% 9.49/5.22        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.49/5.22       => ( ( ( list_update_o @ Xs @ I @ X )
% 9.49/5.22            = Xs )
% 9.49/5.22          = ( ( nth_o @ Xs @ I )
% 9.49/5.22            = X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_204_list__update__same__conv,axiom,
% 9.49/5.22      ! [I: nat,Xs: list_nat,X: nat] :
% 9.49/5.22        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 9.49/5.22       => ( ( ( list_update_nat @ Xs @ I @ X )
% 9.49/5.22            = Xs )
% 9.49/5.22          = ( ( nth_nat @ Xs @ I )
% 9.49/5.22            = X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_205_list__update__same__conv,axiom,
% 9.49/5.22      ! [I: nat,Xs: list_int,X: int] :
% 9.49/5.22        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 9.49/5.22       => ( ( ( list_update_int @ Xs @ I @ X )
% 9.49/5.22            = Xs )
% 9.49/5.22          = ( ( nth_int @ Xs @ I )
% 9.49/5.22            = X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_206_buildup__nothing__in__leaf,axiom,
% 9.49/5.22      ! [N: nat,X: nat] :
% 9.49/5.22        ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_207_member__correct,axiom,
% 9.49/5.22      ! [T: vEBT_VEBT,N: nat,X: nat] :
% 9.49/5.22        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.22       => ( ( vEBT_vebt_member @ T @ X )
% 9.49/5.22          = ( member_nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_208_maxidef,axiom,
% 9.49/5.22      ( ( some_nat @ maxi )
% 9.49/5.22      = ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_209_both__member__options__equiv__member,axiom,
% 9.49/5.22      ! [T: vEBT_VEBT,N: nat,X: nat] :
% 9.49/5.22        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.22       => ( ( vEBT_V8194947554948674370ptions @ T @ X )
% 9.49/5.22          = ( vEBT_vebt_member @ T @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_210_valid__member__both__member__options,axiom,
% 9.49/5.22      ! [T: vEBT_VEBT,N: nat,X: nat] :
% 9.49/5.22        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.22       => ( ( vEBT_V8194947554948674370ptions @ T @ X )
% 9.49/5.22         => ( vEBT_vebt_member @ T @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_211_VEBT_Oinject_I1_J,axiom,
% 9.49/5.22      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
% 9.49/5.22        ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 9.49/5.22          = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
% 9.49/5.22        = ( ( X11 = Y11 )
% 9.49/5.22          & ( X12 = Y12 )
% 9.49/5.22          & ( X13 = Y13 )
% 9.49/5.22          & ( X14 = Y14 ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_212_buildup__gives__empty,axiom,
% 9.49/5.22      ! [N: nat] :
% 9.49/5.22        ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
% 9.49/5.22        = bot_bot_set_nat ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_213_dual__order_Orefl,axiom,
% 9.49/5.22      ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_214_dual__order_Orefl,axiom,
% 9.49/5.22      ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_215_dual__order_Orefl,axiom,
% 9.49/5.22      ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_216_dual__order_Orefl,axiom,
% 9.49/5.22      ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_217_dual__order_Orefl,axiom,
% 9.49/5.22      ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_218_order__refl,axiom,
% 9.49/5.22      ! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_219_order__refl,axiom,
% 9.49/5.22      ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_220_order__refl,axiom,
% 9.49/5.22      ! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_221_order__refl,axiom,
% 9.49/5.22      ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_222_order__refl,axiom,
% 9.49/5.22      ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_223_buildup__nothing__in__min__max,axiom,
% 9.49/5.22      ! [N: nat,X: nat] :
% 9.49/5.22        ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_224__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062maxi_O_ASome_Amaxi_A_061_Avebt__maxt_A_ItreeList_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Ahigh_Ax_An_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 9.49/5.22      ~ ! [Maxi: nat] :
% 9.49/5.22          ( ( some_nat @ Maxi )
% 9.49/5.22         != ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_225_set__vebt__finite,axiom,
% 9.49/5.22      ! [T: vEBT_VEBT,N: nat] :
% 9.49/5.22        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.22       => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_226_min__Null__member,axiom,
% 9.49/5.22      ! [T: vEBT_VEBT,X: nat] :
% 9.49/5.22        ( ( vEBT_VEBT_minNull @ T )
% 9.49/5.22       => ~ ( vEBT_vebt_member @ T @ X ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_227_maxbmo,axiom,
% 9.49/5.22      ! [T: vEBT_VEBT,X: nat] :
% 9.49/5.22        ( ( ( vEBT_vebt_maxt @ T )
% 9.49/5.22          = ( some_nat @ X ) )
% 9.49/5.22       => ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_228_both__member__options__def,axiom,
% 9.49/5.22      ( vEBT_V8194947554948674370ptions
% 9.49/5.22      = ( ^ [T2: vEBT_VEBT,X3: nat] :
% 9.49/5.22            ( ( vEBT_V5719532721284313246member @ T2 @ X3 )
% 9.49/5.22            | ( vEBT_VEBT_membermima @ T2 @ X3 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_229_maxt__member,axiom,
% 9.49/5.22      ! [T: vEBT_VEBT,N: nat,Maxi2: nat] :
% 9.49/5.22        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.22       => ( ( ( vEBT_vebt_maxt @ T )
% 9.49/5.22            = ( some_nat @ Maxi2 ) )
% 9.49/5.22         => ( vEBT_vebt_member @ T @ Maxi2 ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_230_member__valid__both__member__options,axiom,
% 9.49/5.22      ! [Tree: vEBT_VEBT,N: nat,X: nat] :
% 9.49/5.22        ( ( vEBT_invar_vebt @ Tree @ N )
% 9.49/5.22       => ( ( vEBT_vebt_member @ Tree @ X )
% 9.49/5.22         => ( ( vEBT_V5719532721284313246member @ Tree @ X )
% 9.49/5.22            | ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_231_maxt__corr__help,axiom,
% 9.49/5.22      ! [T: vEBT_VEBT,N: nat,Maxi2: nat,X: nat] :
% 9.49/5.22        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.22       => ( ( ( vEBT_vebt_maxt @ T )
% 9.49/5.22            = ( some_nat @ Maxi2 ) )
% 9.49/5.22         => ( ( vEBT_vebt_member @ T @ X )
% 9.49/5.22           => ( ord_less_eq_nat @ X @ Maxi2 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_232_maxt__sound,axiom,
% 9.49/5.22      ! [T: vEBT_VEBT,N: nat,X: nat] :
% 9.49/5.22        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.22       => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
% 9.49/5.22         => ( ( vEBT_vebt_maxt @ T )
% 9.49/5.22            = ( some_nat @ X ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_233_maxt__corr,axiom,
% 9.49/5.22      ! [T: vEBT_VEBT,N: nat,X: nat] :
% 9.49/5.22        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.22       => ( ( ( vEBT_vebt_maxt @ T )
% 9.49/5.22            = ( some_nat @ X ) )
% 9.49/5.22         => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_234_List_Ofinite__set,axiom,
% 9.49/5.22      ! [Xs: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_235_List_Ofinite__set,axiom,
% 9.49/5.22      ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_236_List_Ofinite__set,axiom,
% 9.49/5.22      ! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_237_List_Ofinite__set,axiom,
% 9.49/5.22      ! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_238_List_Ofinite__set,axiom,
% 9.49/5.22      ! [Xs: list_P6011104703257516679at_nat] : ( finite6177210948735845034at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_239_pred__none__empty,axiom,
% 9.49/5.22      ! [Xs: set_nat,A: nat] :
% 9.49/5.22        ( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs @ A @ X_1 )
% 9.49/5.22       => ( ( finite_finite_nat @ Xs )
% 9.49/5.22         => ~ ? [X2: nat] :
% 9.49/5.22                ( ( member_nat @ X2 @ Xs )
% 9.49/5.22                & ( ord_less_nat @ X2 @ A ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_240_succ__none__empty,axiom,
% 9.49/5.22      ! [Xs: set_nat,A: nat] :
% 9.49/5.22        ( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs @ A @ X_1 )
% 9.49/5.22       => ( ( finite_finite_nat @ Xs )
% 9.49/5.22         => ~ ? [X2: nat] :
% 9.49/5.22                ( ( member_nat @ X2 @ Xs )
% 9.49/5.22                & ( ord_less_nat @ A @ X2 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_241_obtain__set__pred,axiom,
% 9.49/5.22      ! [Z2: nat,X: nat,A2: set_nat] :
% 9.49/5.22        ( ( ord_less_nat @ Z2 @ X )
% 9.49/5.22       => ( ( vEBT_VEBT_min_in_set @ A2 @ Z2 )
% 9.49/5.22         => ( ( finite_finite_nat @ A2 )
% 9.49/5.22           => ? [X_1: nat] : ( vEBT_is_pred_in_set @ A2 @ X @ X_1 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_242_obtain__set__succ,axiom,
% 9.49/5.22      ! [X: nat,Z2: nat,A2: set_nat,B2: set_nat] :
% 9.49/5.22        ( ( ord_less_nat @ X @ Z2 )
% 9.49/5.22       => ( ( vEBT_VEBT_max_in_set @ A2 @ Z2 )
% 9.49/5.22         => ( ( finite_finite_nat @ B2 )
% 9.49/5.22           => ( ( A2 = B2 )
% 9.49/5.22             => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A2 @ X @ X_1 ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_243_lesseq__shift,axiom,
% 9.49/5.22      ( ord_less_eq_nat
% 9.49/5.22      = ( ^ [X3: nat,Y2: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X3 ) @ ( some_nat @ Y2 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_244_pred__member,axiom,
% 9.49/5.22      ! [T: vEBT_VEBT,X: nat,Y: nat] :
% 9.49/5.22        ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
% 9.49/5.22        = ( ( vEBT_vebt_member @ T @ Y )
% 9.49/5.22          & ( ord_less_nat @ Y @ X )
% 9.49/5.22          & ! [Z3: nat] :
% 9.49/5.22              ( ( ( vEBT_vebt_member @ T @ Z3 )
% 9.49/5.22                & ( ord_less_nat @ Z3 @ X ) )
% 9.49/5.22             => ( ord_less_eq_nat @ Z3 @ Y ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_245_succ__member,axiom,
% 9.49/5.22      ! [T: vEBT_VEBT,X: nat,Y: nat] :
% 9.49/5.22        ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
% 9.49/5.22        = ( ( vEBT_vebt_member @ T @ Y )
% 9.49/5.22          & ( ord_less_nat @ X @ Y )
% 9.49/5.22          & ! [Z3: nat] :
% 9.49/5.22              ( ( ( vEBT_vebt_member @ T @ Z3 )
% 9.49/5.22                & ( ord_less_nat @ X @ Z3 ) )
% 9.49/5.22             => ( ord_less_eq_nat @ Y @ Z3 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_246_bot_Oextremum,axiom,
% 9.49/5.22      ! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_247_bot_Oextremum,axiom,
% 9.49/5.22      ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_248_bot_Oextremum,axiom,
% 9.49/5.22      ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_249_bot_Oextremum,axiom,
% 9.49/5.22      ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_250_bot_Oextremum,axiom,
% 9.49/5.22      ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_251_bot_Oextremum__unique,axiom,
% 9.49/5.22      ! [A: set_Pr1261947904930325089at_nat] :
% 9.49/5.22        ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
% 9.49/5.22        = ( A = bot_bo2099793752762293965at_nat ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_252_bot_Oextremum__unique,axiom,
% 9.49/5.22      ! [A: set_real] :
% 9.49/5.22        ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 9.49/5.22        = ( A = bot_bot_set_real ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_253_bot_Oextremum__unique,axiom,
% 9.49/5.22      ! [A: set_nat] :
% 9.49/5.22        ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 9.49/5.22        = ( A = bot_bot_set_nat ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_254_bot_Oextremum__unique,axiom,
% 9.49/5.22      ! [A: set_int] :
% 9.49/5.22        ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 9.49/5.22        = ( A = bot_bot_set_int ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_255_bot_Oextremum__unique,axiom,
% 9.49/5.22      ! [A: nat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 9.49/5.22        = ( A = bot_bot_nat ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_256_bot_Oextremum__uniqueI,axiom,
% 9.49/5.22      ! [A: set_Pr1261947904930325089at_nat] :
% 9.49/5.22        ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
% 9.49/5.22       => ( A = bot_bo2099793752762293965at_nat ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_257_bot_Oextremum__uniqueI,axiom,
% 9.49/5.22      ! [A: set_real] :
% 9.49/5.22        ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 9.49/5.22       => ( A = bot_bot_set_real ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_258_bot_Oextremum__uniqueI,axiom,
% 9.49/5.22      ! [A: set_nat] :
% 9.49/5.22        ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 9.49/5.22       => ( A = bot_bot_set_nat ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_259_bot_Oextremum__uniqueI,axiom,
% 9.49/5.22      ! [A: set_int] :
% 9.49/5.22        ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 9.49/5.22       => ( A = bot_bot_set_int ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_260_bot_Oextremum__uniqueI,axiom,
% 9.49/5.22      ! [A: nat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 9.49/5.22       => ( A = bot_bot_nat ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_261_bot_Oextremum__strict,axiom,
% 9.49/5.22      ! [A: set_Pr1261947904930325089at_nat] :
% 9.49/5.22        ~ ( ord_le7866589430770878221at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_262_bot_Oextremum__strict,axiom,
% 9.49/5.22      ! [A: set_real] :
% 9.49/5.22        ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_263_bot_Oextremum__strict,axiom,
% 9.49/5.22      ! [A: set_nat] :
% 9.49/5.22        ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_264_bot_Oextremum__strict,axiom,
% 9.49/5.22      ! [A: set_int] :
% 9.49/5.22        ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_265_bot_Oextremum__strict,axiom,
% 9.49/5.22      ! [A: nat] :
% 9.49/5.22        ~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_266_bot_Onot__eq__extremum,axiom,
% 9.49/5.22      ! [A: set_Pr1261947904930325089at_nat] :
% 9.49/5.22        ( ( A != bot_bo2099793752762293965at_nat )
% 9.49/5.22        = ( ord_le7866589430770878221at_nat @ bot_bo2099793752762293965at_nat @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_267_bot_Onot__eq__extremum,axiom,
% 9.49/5.22      ! [A: set_real] :
% 9.49/5.22        ( ( A != bot_bot_set_real )
% 9.49/5.22        = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_268_bot_Onot__eq__extremum,axiom,
% 9.49/5.22      ! [A: set_nat] :
% 9.49/5.22        ( ( A != bot_bot_set_nat )
% 9.49/5.22        = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_269_bot_Onot__eq__extremum,axiom,
% 9.49/5.22      ! [A: set_int] :
% 9.49/5.22        ( ( A != bot_bot_set_int )
% 9.49/5.22        = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_270_bot_Onot__eq__extremum,axiom,
% 9.49/5.22      ! [A: nat] :
% 9.49/5.22        ( ( A != bot_bot_nat )
% 9.49/5.22        = ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_271_finite__list,axiom,
% 9.49/5.22      ! [A2: set_VEBT_VEBT] :
% 9.49/5.22        ( ( finite5795047828879050333T_VEBT @ A2 )
% 9.49/5.22       => ? [Xs3: list_VEBT_VEBT] :
% 9.49/5.22            ( ( set_VEBT_VEBT2 @ Xs3 )
% 9.49/5.22            = A2 ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_272_finite__list,axiom,
% 9.49/5.22      ! [A2: set_nat] :
% 9.49/5.22        ( ( finite_finite_nat @ A2 )
% 9.49/5.22       => ? [Xs3: list_nat] :
% 9.49/5.22            ( ( set_nat2 @ Xs3 )
% 9.49/5.22            = A2 ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_273_finite__list,axiom,
% 9.49/5.22      ! [A2: set_int] :
% 9.49/5.22        ( ( finite_finite_int @ A2 )
% 9.49/5.22       => ? [Xs3: list_int] :
% 9.49/5.22            ( ( set_int2 @ Xs3 )
% 9.49/5.22            = A2 ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_274_finite__list,axiom,
% 9.49/5.22      ! [A2: set_complex] :
% 9.49/5.22        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.22       => ? [Xs3: list_complex] :
% 9.49/5.22            ( ( set_complex2 @ Xs3 )
% 9.49/5.22            = A2 ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_275_finite__list,axiom,
% 9.49/5.22      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.22        ( ( finite6177210948735845034at_nat @ A2 )
% 9.49/5.22       => ? [Xs3: list_P6011104703257516679at_nat] :
% 9.49/5.22            ( ( set_Pr5648618587558075414at_nat @ Xs3 )
% 9.49/5.22            = A2 ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_276_subset__code_I1_J,axiom,
% 9.49/5.22      ! [Xs: list_P6011104703257516679at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.22        ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ B2 )
% 9.49/5.22        = ( ! [X3: product_prod_nat_nat] :
% 9.49/5.22              ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.49/5.22             => ( member8440522571783428010at_nat @ X3 @ B2 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_277_subset__code_I1_J,axiom,
% 9.49/5.22      ! [Xs: list_real,B2: set_real] :
% 9.49/5.22        ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B2 )
% 9.49/5.22        = ( ! [X3: real] :
% 9.49/5.22              ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 9.49/5.22             => ( member_real @ X3 @ B2 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_278_subset__code_I1_J,axiom,
% 9.49/5.22      ! [Xs: list_set_nat,B2: set_set_nat] :
% 9.49/5.22        ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B2 )
% 9.49/5.22        = ( ! [X3: set_nat] :
% 9.49/5.22              ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
% 9.49/5.22             => ( member_set_nat @ X3 @ B2 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_279_subset__code_I1_J,axiom,
% 9.49/5.22      ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
% 9.49/5.22        ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B2 )
% 9.49/5.22        = ( ! [X3: vEBT_VEBT] :
% 9.49/5.22              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.22             => ( member_VEBT_VEBT @ X3 @ B2 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_280_subset__code_I1_J,axiom,
% 9.49/5.22      ! [Xs: list_nat,B2: set_nat] :
% 9.49/5.22        ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
% 9.49/5.22        = ( ! [X3: nat] :
% 9.49/5.22              ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 9.49/5.22             => ( member_nat @ X3 @ B2 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_281_subset__code_I1_J,axiom,
% 9.49/5.22      ! [Xs: list_int,B2: set_int] :
% 9.49/5.22        ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B2 )
% 9.49/5.22        = ( ! [X3: int] :
% 9.49/5.22              ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 9.49/5.22             => ( member_int @ X3 @ B2 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_282_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
% 9.49/5.22      ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
% 9.49/5.22        ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_283_set__update__subsetI,axiom,
% 9.49/5.22      ! [Xs: list_P6011104703257516679at_nat,A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,I: nat] :
% 9.49/5.22        ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ A2 )
% 9.49/5.22       => ( ( member8440522571783428010at_nat @ X @ A2 )
% 9.49/5.22         => ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_284_set__update__subsetI,axiom,
% 9.49/5.22      ! [Xs: list_real,A2: set_real,X: real,I: nat] :
% 9.49/5.22        ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A2 )
% 9.49/5.22       => ( ( member_real @ X @ A2 )
% 9.49/5.22         => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_285_set__update__subsetI,axiom,
% 9.49/5.22      ! [Xs: list_set_nat,A2: set_set_nat,X: set_nat,I: nat] :
% 9.49/5.22        ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A2 )
% 9.49/5.22       => ( ( member_set_nat @ X @ A2 )
% 9.49/5.22         => ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_286_set__update__subsetI,axiom,
% 9.49/5.22      ! [Xs: list_nat,A2: set_nat,X: nat,I: nat] :
% 9.49/5.22        ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
% 9.49/5.22       => ( ( member_nat @ X @ A2 )
% 9.49/5.22         => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_287_set__update__subsetI,axiom,
% 9.49/5.22      ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
% 9.49/5.22        ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
% 9.49/5.22       => ( ( member_VEBT_VEBT @ X @ A2 )
% 9.49/5.22         => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_288_set__update__subsetI,axiom,
% 9.49/5.22      ! [Xs: list_int,A2: set_int,X: int,I: nat] :
% 9.49/5.22        ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
% 9.49/5.22       => ( ( member_int @ X @ A2 )
% 9.49/5.22         => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_289_nle__le,axiom,
% 9.49/5.22      ! [A: rat,B: rat] :
% 9.49/5.22        ( ( ~ ( ord_less_eq_rat @ A @ B ) )
% 9.49/5.22        = ( ( ord_less_eq_rat @ B @ A )
% 9.49/5.22          & ( B != A ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_290_nle__le,axiom,
% 9.49/5.22      ! [A: num,B: num] :
% 9.49/5.22        ( ( ~ ( ord_less_eq_num @ A @ B ) )
% 9.49/5.22        = ( ( ord_less_eq_num @ B @ A )
% 9.49/5.22          & ( B != A ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_291_nle__le,axiom,
% 9.49/5.22      ! [A: nat,B: nat] :
% 9.49/5.22        ( ( ~ ( ord_less_eq_nat @ A @ B ) )
% 9.49/5.22        = ( ( ord_less_eq_nat @ B @ A )
% 9.49/5.22          & ( B != A ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_292_nle__le,axiom,
% 9.49/5.22      ! [A: int,B: int] :
% 9.49/5.22        ( ( ~ ( ord_less_eq_int @ A @ B ) )
% 9.49/5.22        = ( ( ord_less_eq_int @ B @ A )
% 9.49/5.22          & ( B != A ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_293_le__cases3,axiom,
% 9.49/5.22      ! [X: rat,Y: rat,Z2: rat] :
% 9.49/5.22        ( ( ( ord_less_eq_rat @ X @ Y )
% 9.49/5.22         => ~ ( ord_less_eq_rat @ Y @ Z2 ) )
% 9.49/5.22       => ( ( ( ord_less_eq_rat @ Y @ X )
% 9.49/5.22           => ~ ( ord_less_eq_rat @ X @ Z2 ) )
% 9.49/5.22         => ( ( ( ord_less_eq_rat @ X @ Z2 )
% 9.49/5.22             => ~ ( ord_less_eq_rat @ Z2 @ Y ) )
% 9.49/5.22           => ( ( ( ord_less_eq_rat @ Z2 @ Y )
% 9.49/5.22               => ~ ( ord_less_eq_rat @ Y @ X ) )
% 9.49/5.22             => ( ( ( ord_less_eq_rat @ Y @ Z2 )
% 9.49/5.22                 => ~ ( ord_less_eq_rat @ Z2 @ X ) )
% 9.49/5.22               => ~ ( ( ord_less_eq_rat @ Z2 @ X )
% 9.49/5.22                   => ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_294_le__cases3,axiom,
% 9.49/5.22      ! [X: num,Y: num,Z2: num] :
% 9.49/5.22        ( ( ( ord_less_eq_num @ X @ Y )
% 9.49/5.22         => ~ ( ord_less_eq_num @ Y @ Z2 ) )
% 9.49/5.22       => ( ( ( ord_less_eq_num @ Y @ X )
% 9.49/5.22           => ~ ( ord_less_eq_num @ X @ Z2 ) )
% 9.49/5.22         => ( ( ( ord_less_eq_num @ X @ Z2 )
% 9.49/5.22             => ~ ( ord_less_eq_num @ Z2 @ Y ) )
% 9.49/5.22           => ( ( ( ord_less_eq_num @ Z2 @ Y )
% 9.49/5.22               => ~ ( ord_less_eq_num @ Y @ X ) )
% 9.49/5.22             => ( ( ( ord_less_eq_num @ Y @ Z2 )
% 9.49/5.22                 => ~ ( ord_less_eq_num @ Z2 @ X ) )
% 9.49/5.22               => ~ ( ( ord_less_eq_num @ Z2 @ X )
% 9.49/5.22                   => ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_295_le__cases3,axiom,
% 9.49/5.22      ! [X: nat,Y: nat,Z2: nat] :
% 9.49/5.22        ( ( ( ord_less_eq_nat @ X @ Y )
% 9.49/5.22         => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
% 9.49/5.22       => ( ( ( ord_less_eq_nat @ Y @ X )
% 9.49/5.22           => ~ ( ord_less_eq_nat @ X @ Z2 ) )
% 9.49/5.22         => ( ( ( ord_less_eq_nat @ X @ Z2 )
% 9.49/5.22             => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
% 9.49/5.22           => ( ( ( ord_less_eq_nat @ Z2 @ Y )
% 9.49/5.22               => ~ ( ord_less_eq_nat @ Y @ X ) )
% 9.49/5.22             => ( ( ( ord_less_eq_nat @ Y @ Z2 )
% 9.49/5.22                 => ~ ( ord_less_eq_nat @ Z2 @ X ) )
% 9.49/5.22               => ~ ( ( ord_less_eq_nat @ Z2 @ X )
% 9.49/5.22                   => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_296_le__cases3,axiom,
% 9.49/5.22      ! [X: int,Y: int,Z2: int] :
% 9.49/5.22        ( ( ( ord_less_eq_int @ X @ Y )
% 9.49/5.22         => ~ ( ord_less_eq_int @ Y @ Z2 ) )
% 9.49/5.22       => ( ( ( ord_less_eq_int @ Y @ X )
% 9.49/5.22           => ~ ( ord_less_eq_int @ X @ Z2 ) )
% 9.49/5.22         => ( ( ( ord_less_eq_int @ X @ Z2 )
% 9.49/5.22             => ~ ( ord_less_eq_int @ Z2 @ Y ) )
% 9.49/5.22           => ( ( ( ord_less_eq_int @ Z2 @ Y )
% 9.49/5.22               => ~ ( ord_less_eq_int @ Y @ X ) )
% 9.49/5.22             => ( ( ( ord_less_eq_int @ Y @ Z2 )
% 9.49/5.22                 => ~ ( ord_less_eq_int @ Z2 @ X ) )
% 9.49/5.22               => ~ ( ( ord_less_eq_int @ Z2 @ X )
% 9.49/5.22                   => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_297_order__class_Oorder__eq__iff,axiom,
% 9.49/5.22      ( ( ^ [Y5: set_int,Z: set_int] : ( Y5 = Z ) )
% 9.49/5.22      = ( ^ [X3: set_int,Y2: set_int] :
% 9.49/5.22            ( ( ord_less_eq_set_int @ X3 @ Y2 )
% 9.49/5.22            & ( ord_less_eq_set_int @ Y2 @ X3 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_298_order__class_Oorder__eq__iff,axiom,
% 9.49/5.22      ( ( ^ [Y5: rat,Z: rat] : ( Y5 = Z ) )
% 9.49/5.22      = ( ^ [X3: rat,Y2: rat] :
% 9.49/5.22            ( ( ord_less_eq_rat @ X3 @ Y2 )
% 9.49/5.22            & ( ord_less_eq_rat @ Y2 @ X3 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_299_order__class_Oorder__eq__iff,axiom,
% 9.49/5.22      ( ( ^ [Y5: num,Z: num] : ( Y5 = Z ) )
% 9.49/5.22      = ( ^ [X3: num,Y2: num] :
% 9.49/5.22            ( ( ord_less_eq_num @ X3 @ Y2 )
% 9.49/5.22            & ( ord_less_eq_num @ Y2 @ X3 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_300_order__class_Oorder__eq__iff,axiom,
% 9.49/5.22      ( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
% 9.49/5.22      = ( ^ [X3: nat,Y2: nat] :
% 9.49/5.22            ( ( ord_less_eq_nat @ X3 @ Y2 )
% 9.49/5.22            & ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_301_order__class_Oorder__eq__iff,axiom,
% 9.49/5.22      ( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
% 9.49/5.22      = ( ^ [X3: int,Y2: int] :
% 9.49/5.22            ( ( ord_less_eq_int @ X3 @ Y2 )
% 9.49/5.22            & ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_302_ord__eq__le__trans,axiom,
% 9.49/5.22      ! [A: set_int,B: set_int,C: set_int] :
% 9.49/5.22        ( ( A = B )
% 9.49/5.22       => ( ( ord_less_eq_set_int @ B @ C )
% 9.49/5.22         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_303_ord__eq__le__trans,axiom,
% 9.49/5.22      ! [A: rat,B: rat,C: rat] :
% 9.49/5.22        ( ( A = B )
% 9.49/5.22       => ( ( ord_less_eq_rat @ B @ C )
% 9.49/5.22         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_304_ord__eq__le__trans,axiom,
% 9.49/5.22      ! [A: num,B: num,C: num] :
% 9.49/5.22        ( ( A = B )
% 9.49/5.22       => ( ( ord_less_eq_num @ B @ C )
% 9.49/5.22         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_305_ord__eq__le__trans,axiom,
% 9.49/5.22      ! [A: nat,B: nat,C: nat] :
% 9.49/5.22        ( ( A = B )
% 9.49/5.22       => ( ( ord_less_eq_nat @ B @ C )
% 9.49/5.22         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_306_ord__eq__le__trans,axiom,
% 9.49/5.22      ! [A: int,B: int,C: int] :
% 9.49/5.22        ( ( A = B )
% 9.49/5.22       => ( ( ord_less_eq_int @ B @ C )
% 9.49/5.22         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_307_ord__le__eq__trans,axiom,
% 9.49/5.22      ! [A: set_int,B: set_int,C: set_int] :
% 9.49/5.22        ( ( ord_less_eq_set_int @ A @ B )
% 9.49/5.22       => ( ( B = C )
% 9.49/5.22         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_308_ord__le__eq__trans,axiom,
% 9.49/5.22      ! [A: rat,B: rat,C: rat] :
% 9.49/5.22        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.22       => ( ( B = C )
% 9.49/5.22         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_309_ord__le__eq__trans,axiom,
% 9.49/5.22      ! [A: num,B: num,C: num] :
% 9.49/5.22        ( ( ord_less_eq_num @ A @ B )
% 9.49/5.22       => ( ( B = C )
% 9.49/5.22         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_310_ord__le__eq__trans,axiom,
% 9.49/5.22      ! [A: nat,B: nat,C: nat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.22       => ( ( B = C )
% 9.49/5.22         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_311_ord__le__eq__trans,axiom,
% 9.49/5.22      ! [A: int,B: int,C: int] :
% 9.49/5.22        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.22       => ( ( B = C )
% 9.49/5.22         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_312_order__antisym,axiom,
% 9.49/5.22      ! [X: set_int,Y: set_int] :
% 9.49/5.22        ( ( ord_less_eq_set_int @ X @ Y )
% 9.49/5.22       => ( ( ord_less_eq_set_int @ Y @ X )
% 9.49/5.22         => ( X = Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_313_order__antisym,axiom,
% 9.49/5.22      ! [X: rat,Y: rat] :
% 9.49/5.22        ( ( ord_less_eq_rat @ X @ Y )
% 9.49/5.22       => ( ( ord_less_eq_rat @ Y @ X )
% 9.49/5.22         => ( X = Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_314_order__antisym,axiom,
% 9.49/5.22      ! [X: num,Y: num] :
% 9.49/5.22        ( ( ord_less_eq_num @ X @ Y )
% 9.49/5.22       => ( ( ord_less_eq_num @ Y @ X )
% 9.49/5.22         => ( X = Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_315_order__antisym,axiom,
% 9.49/5.22      ! [X: nat,Y: nat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ X @ Y )
% 9.49/5.22       => ( ( ord_less_eq_nat @ Y @ X )
% 9.49/5.22         => ( X = Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_316_order__antisym,axiom,
% 9.49/5.22      ! [X: int,Y: int] :
% 9.49/5.22        ( ( ord_less_eq_int @ X @ Y )
% 9.49/5.22       => ( ( ord_less_eq_int @ Y @ X )
% 9.49/5.22         => ( X = Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_317_order_Otrans,axiom,
% 9.49/5.22      ! [A: set_int,B: set_int,C: set_int] :
% 9.49/5.22        ( ( ord_less_eq_set_int @ A @ B )
% 9.49/5.22       => ( ( ord_less_eq_set_int @ B @ C )
% 9.49/5.22         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_318_order_Otrans,axiom,
% 9.49/5.22      ! [A: rat,B: rat,C: rat] :
% 9.49/5.22        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.22       => ( ( ord_less_eq_rat @ B @ C )
% 9.49/5.22         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_319_order_Otrans,axiom,
% 9.49/5.22      ! [A: num,B: num,C: num] :
% 9.49/5.22        ( ( ord_less_eq_num @ A @ B )
% 9.49/5.22       => ( ( ord_less_eq_num @ B @ C )
% 9.49/5.22         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_320_order_Otrans,axiom,
% 9.49/5.22      ! [A: nat,B: nat,C: nat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.22       => ( ( ord_less_eq_nat @ B @ C )
% 9.49/5.22         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_321_order_Otrans,axiom,
% 9.49/5.22      ! [A: int,B: int,C: int] :
% 9.49/5.22        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.22       => ( ( ord_less_eq_int @ B @ C )
% 9.49/5.22         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_322_order__trans,axiom,
% 9.49/5.22      ! [X: set_int,Y: set_int,Z2: set_int] :
% 9.49/5.22        ( ( ord_less_eq_set_int @ X @ Y )
% 9.49/5.22       => ( ( ord_less_eq_set_int @ Y @ Z2 )
% 9.49/5.22         => ( ord_less_eq_set_int @ X @ Z2 ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_323_order__trans,axiom,
% 9.49/5.22      ! [X: rat,Y: rat,Z2: rat] :
% 9.49/5.22        ( ( ord_less_eq_rat @ X @ Y )
% 9.49/5.22       => ( ( ord_less_eq_rat @ Y @ Z2 )
% 9.49/5.22         => ( ord_less_eq_rat @ X @ Z2 ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_324_order__trans,axiom,
% 9.49/5.22      ! [X: num,Y: num,Z2: num] :
% 9.49/5.22        ( ( ord_less_eq_num @ X @ Y )
% 9.49/5.22       => ( ( ord_less_eq_num @ Y @ Z2 )
% 9.49/5.22         => ( ord_less_eq_num @ X @ Z2 ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_325_order__trans,axiom,
% 9.49/5.22      ! [X: nat,Y: nat,Z2: nat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ X @ Y )
% 9.49/5.22       => ( ( ord_less_eq_nat @ Y @ Z2 )
% 9.49/5.22         => ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_326_order__trans,axiom,
% 9.49/5.22      ! [X: int,Y: int,Z2: int] :
% 9.49/5.22        ( ( ord_less_eq_int @ X @ Y )
% 9.49/5.22       => ( ( ord_less_eq_int @ Y @ Z2 )
% 9.49/5.22         => ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_327_linorder__wlog,axiom,
% 9.49/5.22      ! [P: rat > rat > $o,A: rat,B: rat] :
% 9.49/5.22        ( ! [A3: rat,B3: rat] :
% 9.49/5.22            ( ( ord_less_eq_rat @ A3 @ B3 )
% 9.49/5.22           => ( P @ A3 @ B3 ) )
% 9.49/5.22       => ( ! [A3: rat,B3: rat] :
% 9.49/5.22              ( ( P @ B3 @ A3 )
% 9.49/5.22             => ( P @ A3 @ B3 ) )
% 9.49/5.22         => ( P @ A @ B ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_328_linorder__wlog,axiom,
% 9.49/5.22      ! [P: num > num > $o,A: num,B: num] :
% 9.49/5.22        ( ! [A3: num,B3: num] :
% 9.49/5.22            ( ( ord_less_eq_num @ A3 @ B3 )
% 9.49/5.22           => ( P @ A3 @ B3 ) )
% 9.49/5.22       => ( ! [A3: num,B3: num] :
% 9.49/5.22              ( ( P @ B3 @ A3 )
% 9.49/5.22             => ( P @ A3 @ B3 ) )
% 9.49/5.22         => ( P @ A @ B ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_329_linorder__wlog,axiom,
% 9.49/5.22      ! [P: nat > nat > $o,A: nat,B: nat] :
% 9.49/5.22        ( ! [A3: nat,B3: nat] :
% 9.49/5.22            ( ( ord_less_eq_nat @ A3 @ B3 )
% 9.49/5.22           => ( P @ A3 @ B3 ) )
% 9.49/5.22       => ( ! [A3: nat,B3: nat] :
% 9.49/5.22              ( ( P @ B3 @ A3 )
% 9.49/5.22             => ( P @ A3 @ B3 ) )
% 9.49/5.22         => ( P @ A @ B ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_330_linorder__wlog,axiom,
% 9.49/5.22      ! [P: int > int > $o,A: int,B: int] :
% 9.49/5.22        ( ! [A3: int,B3: int] :
% 9.49/5.22            ( ( ord_less_eq_int @ A3 @ B3 )
% 9.49/5.22           => ( P @ A3 @ B3 ) )
% 9.49/5.22       => ( ! [A3: int,B3: int] :
% 9.49/5.22              ( ( P @ B3 @ A3 )
% 9.49/5.22             => ( P @ A3 @ B3 ) )
% 9.49/5.22         => ( P @ A @ B ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_331_dual__order_Oeq__iff,axiom,
% 9.49/5.22      ( ( ^ [Y5: set_int,Z: set_int] : ( Y5 = Z ) )
% 9.49/5.22      = ( ^ [A4: set_int,B4: set_int] :
% 9.49/5.22            ( ( ord_less_eq_set_int @ B4 @ A4 )
% 9.49/5.22            & ( ord_less_eq_set_int @ A4 @ B4 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_332_dual__order_Oeq__iff,axiom,
% 9.49/5.22      ( ( ^ [Y5: rat,Z: rat] : ( Y5 = Z ) )
% 9.49/5.22      = ( ^ [A4: rat,B4: rat] :
% 9.49/5.22            ( ( ord_less_eq_rat @ B4 @ A4 )
% 9.49/5.22            & ( ord_less_eq_rat @ A4 @ B4 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_333_dual__order_Oeq__iff,axiom,
% 9.49/5.22      ( ( ^ [Y5: num,Z: num] : ( Y5 = Z ) )
% 9.49/5.22      = ( ^ [A4: num,B4: num] :
% 9.49/5.22            ( ( ord_less_eq_num @ B4 @ A4 )
% 9.49/5.22            & ( ord_less_eq_num @ A4 @ B4 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_334_dual__order_Oeq__iff,axiom,
% 9.49/5.22      ( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
% 9.49/5.22      = ( ^ [A4: nat,B4: nat] :
% 9.49/5.22            ( ( ord_less_eq_nat @ B4 @ A4 )
% 9.49/5.22            & ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_335_dual__order_Oeq__iff,axiom,
% 9.49/5.22      ( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
% 9.49/5.22      = ( ^ [A4: int,B4: int] :
% 9.49/5.22            ( ( ord_less_eq_int @ B4 @ A4 )
% 9.49/5.22            & ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_336_dual__order_Oantisym,axiom,
% 9.49/5.22      ! [B: set_int,A: set_int] :
% 9.49/5.22        ( ( ord_less_eq_set_int @ B @ A )
% 9.49/5.22       => ( ( ord_less_eq_set_int @ A @ B )
% 9.49/5.22         => ( A = B ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_337_dual__order_Oantisym,axiom,
% 9.49/5.22      ! [B: rat,A: rat] :
% 9.49/5.22        ( ( ord_less_eq_rat @ B @ A )
% 9.49/5.22       => ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.22         => ( A = B ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_338_dual__order_Oantisym,axiom,
% 9.49/5.22      ! [B: num,A: num] :
% 9.49/5.22        ( ( ord_less_eq_num @ B @ A )
% 9.49/5.22       => ( ( ord_less_eq_num @ A @ B )
% 9.49/5.22         => ( A = B ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_339_dual__order_Oantisym,axiom,
% 9.49/5.22      ! [B: nat,A: nat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ B @ A )
% 9.49/5.22       => ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.22         => ( A = B ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_340_dual__order_Oantisym,axiom,
% 9.49/5.22      ! [B: int,A: int] :
% 9.49/5.22        ( ( ord_less_eq_int @ B @ A )
% 9.49/5.22       => ( ( ord_less_eq_int @ A @ B )
% 9.49/5.22         => ( A = B ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_341_dual__order_Otrans,axiom,
% 9.49/5.22      ! [B: set_int,A: set_int,C: set_int] :
% 9.49/5.22        ( ( ord_less_eq_set_int @ B @ A )
% 9.49/5.22       => ( ( ord_less_eq_set_int @ C @ B )
% 9.49/5.22         => ( ord_less_eq_set_int @ C @ A ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_342_dual__order_Otrans,axiom,
% 9.49/5.22      ! [B: rat,A: rat,C: rat] :
% 9.49/5.22        ( ( ord_less_eq_rat @ B @ A )
% 9.49/5.22       => ( ( ord_less_eq_rat @ C @ B )
% 9.49/5.22         => ( ord_less_eq_rat @ C @ A ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_343_dual__order_Otrans,axiom,
% 9.49/5.22      ! [B: num,A: num,C: num] :
% 9.49/5.22        ( ( ord_less_eq_num @ B @ A )
% 9.49/5.22       => ( ( ord_less_eq_num @ C @ B )
% 9.49/5.22         => ( ord_less_eq_num @ C @ A ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_344_dual__order_Otrans,axiom,
% 9.49/5.22      ! [B: nat,A: nat,C: nat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ B @ A )
% 9.49/5.22       => ( ( ord_less_eq_nat @ C @ B )
% 9.49/5.22         => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_345_dual__order_Otrans,axiom,
% 9.49/5.22      ! [B: int,A: int,C: int] :
% 9.49/5.22        ( ( ord_less_eq_int @ B @ A )
% 9.49/5.22       => ( ( ord_less_eq_int @ C @ B )
% 9.49/5.22         => ( ord_less_eq_int @ C @ A ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_346_antisym,axiom,
% 9.49/5.22      ! [A: set_int,B: set_int] :
% 9.49/5.22        ( ( ord_less_eq_set_int @ A @ B )
% 9.49/5.22       => ( ( ord_less_eq_set_int @ B @ A )
% 9.49/5.22         => ( A = B ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_347_antisym,axiom,
% 9.49/5.22      ! [A: rat,B: rat] :
% 9.49/5.22        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.22       => ( ( ord_less_eq_rat @ B @ A )
% 9.49/5.22         => ( A = B ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_348_antisym,axiom,
% 9.49/5.22      ! [A: num,B: num] :
% 9.49/5.22        ( ( ord_less_eq_num @ A @ B )
% 9.49/5.22       => ( ( ord_less_eq_num @ B @ A )
% 9.49/5.22         => ( A = B ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_349_antisym,axiom,
% 9.49/5.22      ! [A: nat,B: nat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.22       => ( ( ord_less_eq_nat @ B @ A )
% 9.49/5.22         => ( A = B ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_350_antisym,axiom,
% 9.49/5.22      ! [A: int,B: int] :
% 9.49/5.22        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.22       => ( ( ord_less_eq_int @ B @ A )
% 9.49/5.22         => ( A = B ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_351_Orderings_Oorder__eq__iff,axiom,
% 9.49/5.22      ( ( ^ [Y5: set_int,Z: set_int] : ( Y5 = Z ) )
% 9.49/5.22      = ( ^ [A4: set_int,B4: set_int] :
% 9.49/5.22            ( ( ord_less_eq_set_int @ A4 @ B4 )
% 9.49/5.22            & ( ord_less_eq_set_int @ B4 @ A4 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_352_Orderings_Oorder__eq__iff,axiom,
% 9.49/5.22      ( ( ^ [Y5: rat,Z: rat] : ( Y5 = Z ) )
% 9.49/5.22      = ( ^ [A4: rat,B4: rat] :
% 9.49/5.22            ( ( ord_less_eq_rat @ A4 @ B4 )
% 9.49/5.22            & ( ord_less_eq_rat @ B4 @ A4 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_353_Orderings_Oorder__eq__iff,axiom,
% 9.49/5.22      ( ( ^ [Y5: num,Z: num] : ( Y5 = Z ) )
% 9.49/5.22      = ( ^ [A4: num,B4: num] :
% 9.49/5.22            ( ( ord_less_eq_num @ A4 @ B4 )
% 9.49/5.22            & ( ord_less_eq_num @ B4 @ A4 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_354_Orderings_Oorder__eq__iff,axiom,
% 9.49/5.22      ( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
% 9.49/5.22      = ( ^ [A4: nat,B4: nat] :
% 9.49/5.22            ( ( ord_less_eq_nat @ A4 @ B4 )
% 9.49/5.22            & ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_355_Orderings_Oorder__eq__iff,axiom,
% 9.49/5.22      ( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
% 9.49/5.22      = ( ^ [A4: int,B4: int] :
% 9.49/5.22            ( ( ord_less_eq_int @ A4 @ B4 )
% 9.49/5.22            & ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_356_order__subst1,axiom,
% 9.49/5.22      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 9.49/5.22        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_eq_rat @ B @ C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_357_order__subst1,axiom,
% 9.49/5.22      ! [A: rat,F: num > rat,B: num,C: num] :
% 9.49/5.22        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_eq_num @ B @ C )
% 9.49/5.22         => ( ! [X4: num,Y3: num] :
% 9.49/5.22                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_358_order__subst1,axiom,
% 9.49/5.22      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 9.49/5.22        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_eq_nat @ B @ C )
% 9.49/5.22         => ( ! [X4: nat,Y3: nat] :
% 9.49/5.22                ( ( ord_less_eq_nat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_359_order__subst1,axiom,
% 9.49/5.22      ! [A: rat,F: int > rat,B: int,C: int] :
% 9.49/5.22        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_eq_int @ B @ C )
% 9.49/5.22         => ( ! [X4: int,Y3: int] :
% 9.49/5.22                ( ( ord_less_eq_int @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_360_order__subst1,axiom,
% 9.49/5.22      ! [A: num,F: rat > num,B: rat,C: rat] :
% 9.49/5.22        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_eq_rat @ B @ C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_361_order__subst1,axiom,
% 9.49/5.22      ! [A: num,F: num > num,B: num,C: num] :
% 9.49/5.22        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_eq_num @ B @ C )
% 9.49/5.22         => ( ! [X4: num,Y3: num] :
% 9.49/5.22                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_362_order__subst1,axiom,
% 9.49/5.22      ! [A: num,F: nat > num,B: nat,C: nat] :
% 9.49/5.22        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_eq_nat @ B @ C )
% 9.49/5.22         => ( ! [X4: nat,Y3: nat] :
% 9.49/5.22                ( ( ord_less_eq_nat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_363_order__subst1,axiom,
% 9.49/5.22      ! [A: num,F: int > num,B: int,C: int] :
% 9.49/5.22        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_eq_int @ B @ C )
% 9.49/5.22         => ( ! [X4: int,Y3: int] :
% 9.49/5.22                ( ( ord_less_eq_int @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_364_order__subst1,axiom,
% 9.49/5.22      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_eq_rat @ B @ C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_365_order__subst1,axiom,
% 9.49/5.22      ! [A: nat,F: num > nat,B: num,C: num] :
% 9.49/5.22        ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_eq_num @ B @ C )
% 9.49/5.22         => ( ! [X4: num,Y3: num] :
% 9.49/5.22                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_366_order__subst2,axiom,
% 9.49/5.22      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 9.49/5.22        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.22       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_367_order__subst2,axiom,
% 9.49/5.22      ! [A: rat,B: rat,F: rat > num,C: num] :
% 9.49/5.22        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.22       => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_368_order__subst2,axiom,
% 9.49/5.22      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 9.49/5.22        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.22       => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_369_order__subst2,axiom,
% 9.49/5.22      ! [A: rat,B: rat,F: rat > int,C: int] :
% 9.49/5.22        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.22       => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_370_order__subst2,axiom,
% 9.49/5.22      ! [A: num,B: num,F: num > rat,C: rat] :
% 9.49/5.22        ( ( ord_less_eq_num @ A @ B )
% 9.49/5.22       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.49/5.22         => ( ! [X4: num,Y3: num] :
% 9.49/5.22                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_371_order__subst2,axiom,
% 9.49/5.22      ! [A: num,B: num,F: num > num,C: num] :
% 9.49/5.22        ( ( ord_less_eq_num @ A @ B )
% 9.49/5.22       => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 9.49/5.22         => ( ! [X4: num,Y3: num] :
% 9.49/5.22                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_372_order__subst2,axiom,
% 9.49/5.22      ! [A: num,B: num,F: num > nat,C: nat] :
% 9.49/5.22        ( ( ord_less_eq_num @ A @ B )
% 9.49/5.22       => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 9.49/5.22         => ( ! [X4: num,Y3: num] :
% 9.49/5.22                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_373_order__subst2,axiom,
% 9.49/5.22      ! [A: num,B: num,F: num > int,C: int] :
% 9.49/5.22        ( ( ord_less_eq_num @ A @ B )
% 9.49/5.22       => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 9.49/5.22         => ( ! [X4: num,Y3: num] :
% 9.49/5.22                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_374_order__subst2,axiom,
% 9.49/5.22      ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.22       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.49/5.22         => ( ! [X4: nat,Y3: nat] :
% 9.49/5.22                ( ( ord_less_eq_nat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_375_order__subst2,axiom,
% 9.49/5.22      ! [A: nat,B: nat,F: nat > num,C: num] :
% 9.49/5.22        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.22       => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 9.49/5.22         => ( ! [X4: nat,Y3: nat] :
% 9.49/5.22                ( ( ord_less_eq_nat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_376_order__eq__refl,axiom,
% 9.49/5.22      ! [X: set_int,Y: set_int] :
% 9.49/5.22        ( ( X = Y )
% 9.49/5.22       => ( ord_less_eq_set_int @ X @ Y ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_377_order__eq__refl,axiom,
% 9.49/5.22      ! [X: rat,Y: rat] :
% 9.49/5.22        ( ( X = Y )
% 9.49/5.22       => ( ord_less_eq_rat @ X @ Y ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_378_order__eq__refl,axiom,
% 9.49/5.22      ! [X: num,Y: num] :
% 9.49/5.22        ( ( X = Y )
% 9.49/5.22       => ( ord_less_eq_num @ X @ Y ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_379_order__eq__refl,axiom,
% 9.49/5.22      ! [X: nat,Y: nat] :
% 9.49/5.22        ( ( X = Y )
% 9.49/5.22       => ( ord_less_eq_nat @ X @ Y ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_380_order__eq__refl,axiom,
% 9.49/5.22      ! [X: int,Y: int] :
% 9.49/5.22        ( ( X = Y )
% 9.49/5.22       => ( ord_less_eq_int @ X @ Y ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_381_linorder__linear,axiom,
% 9.49/5.22      ! [X: rat,Y: rat] :
% 9.49/5.22        ( ( ord_less_eq_rat @ X @ Y )
% 9.49/5.22        | ( ord_less_eq_rat @ Y @ X ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_382_linorder__linear,axiom,
% 9.49/5.22      ! [X: num,Y: num] :
% 9.49/5.22        ( ( ord_less_eq_num @ X @ Y )
% 9.49/5.22        | ( ord_less_eq_num @ Y @ X ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_383_linorder__linear,axiom,
% 9.49/5.22      ! [X: nat,Y: nat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ X @ Y )
% 9.49/5.22        | ( ord_less_eq_nat @ Y @ X ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_384_linorder__linear,axiom,
% 9.49/5.22      ! [X: int,Y: int] :
% 9.49/5.22        ( ( ord_less_eq_int @ X @ Y )
% 9.49/5.22        | ( ord_less_eq_int @ Y @ X ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_385_ord__eq__le__subst,axiom,
% 9.49/5.22      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 9.49/5.22        ( ( A
% 9.49/5.22          = ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_eq_rat @ B @ C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_386_ord__eq__le__subst,axiom,
% 9.49/5.22      ! [A: num,F: rat > num,B: rat,C: rat] :
% 9.49/5.22        ( ( A
% 9.49/5.22          = ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_eq_rat @ B @ C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_387_ord__eq__le__subst,axiom,
% 9.49/5.22      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 9.49/5.22        ( ( A
% 9.49/5.22          = ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_eq_rat @ B @ C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_388_ord__eq__le__subst,axiom,
% 9.49/5.22      ! [A: int,F: rat > int,B: rat,C: rat] :
% 9.49/5.22        ( ( A
% 9.49/5.22          = ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_eq_rat @ B @ C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_389_ord__eq__le__subst,axiom,
% 9.49/5.22      ! [A: rat,F: num > rat,B: num,C: num] :
% 9.49/5.22        ( ( A
% 9.49/5.22          = ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_eq_num @ B @ C )
% 9.49/5.22         => ( ! [X4: num,Y3: num] :
% 9.49/5.22                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_390_ord__eq__le__subst,axiom,
% 9.49/5.22      ! [A: num,F: num > num,B: num,C: num] :
% 9.49/5.22        ( ( A
% 9.49/5.22          = ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_eq_num @ B @ C )
% 9.49/5.22         => ( ! [X4: num,Y3: num] :
% 9.49/5.22                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_391_ord__eq__le__subst,axiom,
% 9.49/5.22      ! [A: nat,F: num > nat,B: num,C: num] :
% 9.49/5.22        ( ( A
% 9.49/5.22          = ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_eq_num @ B @ C )
% 9.49/5.22         => ( ! [X4: num,Y3: num] :
% 9.49/5.22                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_392_ord__eq__le__subst,axiom,
% 9.49/5.22      ! [A: int,F: num > int,B: num,C: num] :
% 9.49/5.22        ( ( A
% 9.49/5.22          = ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_eq_num @ B @ C )
% 9.49/5.22         => ( ! [X4: num,Y3: num] :
% 9.49/5.22                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_393_ord__eq__le__subst,axiom,
% 9.49/5.22      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 9.49/5.22        ( ( A
% 9.49/5.22          = ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_eq_nat @ B @ C )
% 9.49/5.22         => ( ! [X4: nat,Y3: nat] :
% 9.49/5.22                ( ( ord_less_eq_nat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_394_ord__eq__le__subst,axiom,
% 9.49/5.22      ! [A: num,F: nat > num,B: nat,C: nat] :
% 9.49/5.22        ( ( A
% 9.49/5.22          = ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_eq_nat @ B @ C )
% 9.49/5.22         => ( ! [X4: nat,Y3: nat] :
% 9.49/5.22                ( ( ord_less_eq_nat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_395_ord__le__eq__subst,axiom,
% 9.49/5.22      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 9.49/5.22        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.22       => ( ( ( F @ B )
% 9.49/5.22            = C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_396_ord__le__eq__subst,axiom,
% 9.49/5.22      ! [A: rat,B: rat,F: rat > num,C: num] :
% 9.49/5.22        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.22       => ( ( ( F @ B )
% 9.49/5.22            = C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_397_ord__le__eq__subst,axiom,
% 9.49/5.22      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 9.49/5.22        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.22       => ( ( ( F @ B )
% 9.49/5.22            = C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_398_ord__le__eq__subst,axiom,
% 9.49/5.22      ! [A: rat,B: rat,F: rat > int,C: int] :
% 9.49/5.22        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.22       => ( ( ( F @ B )
% 9.49/5.22            = C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_399_ord__le__eq__subst,axiom,
% 9.49/5.22      ! [A: num,B: num,F: num > rat,C: rat] :
% 9.49/5.22        ( ( ord_less_eq_num @ A @ B )
% 9.49/5.22       => ( ( ( F @ B )
% 9.49/5.22            = C )
% 9.49/5.22         => ( ! [X4: num,Y3: num] :
% 9.49/5.22                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_400_ord__le__eq__subst,axiom,
% 9.49/5.22      ! [A: num,B: num,F: num > num,C: num] :
% 9.49/5.22        ( ( ord_less_eq_num @ A @ B )
% 9.49/5.22       => ( ( ( F @ B )
% 9.49/5.22            = C )
% 9.49/5.22         => ( ! [X4: num,Y3: num] :
% 9.49/5.22                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_401_ord__le__eq__subst,axiom,
% 9.49/5.22      ! [A: num,B: num,F: num > nat,C: nat] :
% 9.49/5.22        ( ( ord_less_eq_num @ A @ B )
% 9.49/5.22       => ( ( ( F @ B )
% 9.49/5.22            = C )
% 9.49/5.22         => ( ! [X4: num,Y3: num] :
% 9.49/5.22                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_402_ord__le__eq__subst,axiom,
% 9.49/5.22      ! [A: num,B: num,F: num > int,C: int] :
% 9.49/5.22        ( ( ord_less_eq_num @ A @ B )
% 9.49/5.22       => ( ( ( F @ B )
% 9.49/5.22            = C )
% 9.49/5.22         => ( ! [X4: num,Y3: num] :
% 9.49/5.22                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_403_ord__le__eq__subst,axiom,
% 9.49/5.22      ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.22       => ( ( ( F @ B )
% 9.49/5.22            = C )
% 9.49/5.22         => ( ! [X4: nat,Y3: nat] :
% 9.49/5.22                ( ( ord_less_eq_nat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_404_ord__le__eq__subst,axiom,
% 9.49/5.22      ! [A: nat,B: nat,F: nat > num,C: num] :
% 9.49/5.22        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.22       => ( ( ( F @ B )
% 9.49/5.22            = C )
% 9.49/5.22         => ( ! [X4: nat,Y3: nat] :
% 9.49/5.22                ( ( ord_less_eq_nat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_405_linorder__le__cases,axiom,
% 9.49/5.22      ! [X: rat,Y: rat] :
% 9.49/5.22        ( ~ ( ord_less_eq_rat @ X @ Y )
% 9.49/5.22       => ( ord_less_eq_rat @ Y @ X ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_406_linorder__le__cases,axiom,
% 9.49/5.22      ! [X: num,Y: num] :
% 9.49/5.22        ( ~ ( ord_less_eq_num @ X @ Y )
% 9.49/5.22       => ( ord_less_eq_num @ Y @ X ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_407_linorder__le__cases,axiom,
% 9.49/5.22      ! [X: nat,Y: nat] :
% 9.49/5.22        ( ~ ( ord_less_eq_nat @ X @ Y )
% 9.49/5.22       => ( ord_less_eq_nat @ Y @ X ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_408_linorder__le__cases,axiom,
% 9.49/5.22      ! [X: int,Y: int] :
% 9.49/5.22        ( ~ ( ord_less_eq_int @ X @ Y )
% 9.49/5.22       => ( ord_less_eq_int @ Y @ X ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_409_order__antisym__conv,axiom,
% 9.49/5.22      ! [Y: set_int,X: set_int] :
% 9.49/5.22        ( ( ord_less_eq_set_int @ Y @ X )
% 9.49/5.22       => ( ( ord_less_eq_set_int @ X @ Y )
% 9.49/5.22          = ( X = Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_410_order__antisym__conv,axiom,
% 9.49/5.22      ! [Y: rat,X: rat] :
% 9.49/5.22        ( ( ord_less_eq_rat @ Y @ X )
% 9.49/5.22       => ( ( ord_less_eq_rat @ X @ Y )
% 9.49/5.22          = ( X = Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_411_order__antisym__conv,axiom,
% 9.49/5.22      ! [Y: num,X: num] :
% 9.49/5.22        ( ( ord_less_eq_num @ Y @ X )
% 9.49/5.22       => ( ( ord_less_eq_num @ X @ Y )
% 9.49/5.22          = ( X = Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_412_order__antisym__conv,axiom,
% 9.49/5.22      ! [Y: nat,X: nat] :
% 9.49/5.22        ( ( ord_less_eq_nat @ Y @ X )
% 9.49/5.22       => ( ( ord_less_eq_nat @ X @ Y )
% 9.49/5.22          = ( X = Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_413_order__antisym__conv,axiom,
% 9.49/5.22      ! [Y: int,X: int] :
% 9.49/5.22        ( ( ord_less_eq_int @ Y @ X )
% 9.49/5.22       => ( ( ord_less_eq_int @ X @ Y )
% 9.49/5.22          = ( X = Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_414_lt__ex,axiom,
% 9.49/5.22      ! [X: real] :
% 9.49/5.22      ? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_415_lt__ex,axiom,
% 9.49/5.22      ! [X: rat] :
% 9.49/5.22      ? [Y3: rat] : ( ord_less_rat @ Y3 @ X ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_416_lt__ex,axiom,
% 9.49/5.22      ! [X: int] :
% 9.49/5.22      ? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_417_gt__ex,axiom,
% 9.49/5.22      ! [X: real] :
% 9.49/5.22      ? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_418_gt__ex,axiom,
% 9.49/5.22      ! [X: rat] :
% 9.49/5.22      ? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_419_gt__ex,axiom,
% 9.49/5.22      ! [X: nat] :
% 9.49/5.22      ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_420_gt__ex,axiom,
% 9.49/5.22      ! [X: int] :
% 9.49/5.22      ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_421_dense,axiom,
% 9.49/5.22      ! [X: real,Y: real] :
% 9.49/5.22        ( ( ord_less_real @ X @ Y )
% 9.49/5.22       => ? [Z4: real] :
% 9.49/5.22            ( ( ord_less_real @ X @ Z4 )
% 9.49/5.22            & ( ord_less_real @ Z4 @ Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_422_dense,axiom,
% 9.49/5.22      ! [X: rat,Y: rat] :
% 9.49/5.22        ( ( ord_less_rat @ X @ Y )
% 9.49/5.22       => ? [Z4: rat] :
% 9.49/5.22            ( ( ord_less_rat @ X @ Z4 )
% 9.49/5.22            & ( ord_less_rat @ Z4 @ Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_423_less__imp__neq,axiom,
% 9.49/5.22      ! [X: real,Y: real] :
% 9.49/5.22        ( ( ord_less_real @ X @ Y )
% 9.49/5.22       => ( X != Y ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_424_less__imp__neq,axiom,
% 9.49/5.22      ! [X: rat,Y: rat] :
% 9.49/5.22        ( ( ord_less_rat @ X @ Y )
% 9.49/5.22       => ( X != Y ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_425_less__imp__neq,axiom,
% 9.49/5.22      ! [X: num,Y: num] :
% 9.49/5.22        ( ( ord_less_num @ X @ Y )
% 9.49/5.22       => ( X != Y ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_426_less__imp__neq,axiom,
% 9.49/5.22      ! [X: nat,Y: nat] :
% 9.49/5.22        ( ( ord_less_nat @ X @ Y )
% 9.49/5.22       => ( X != Y ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_427_less__imp__neq,axiom,
% 9.49/5.22      ! [X: int,Y: int] :
% 9.49/5.22        ( ( ord_less_int @ X @ Y )
% 9.49/5.22       => ( X != Y ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_428_order_Oasym,axiom,
% 9.49/5.22      ! [A: real,B: real] :
% 9.49/5.22        ( ( ord_less_real @ A @ B )
% 9.49/5.22       => ~ ( ord_less_real @ B @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_429_order_Oasym,axiom,
% 9.49/5.22      ! [A: rat,B: rat] :
% 9.49/5.22        ( ( ord_less_rat @ A @ B )
% 9.49/5.22       => ~ ( ord_less_rat @ B @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_430_order_Oasym,axiom,
% 9.49/5.22      ! [A: num,B: num] :
% 9.49/5.22        ( ( ord_less_num @ A @ B )
% 9.49/5.22       => ~ ( ord_less_num @ B @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_431_order_Oasym,axiom,
% 9.49/5.22      ! [A: nat,B: nat] :
% 9.49/5.22        ( ( ord_less_nat @ A @ B )
% 9.49/5.22       => ~ ( ord_less_nat @ B @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_432_order_Oasym,axiom,
% 9.49/5.22      ! [A: int,B: int] :
% 9.49/5.22        ( ( ord_less_int @ A @ B )
% 9.49/5.22       => ~ ( ord_less_int @ B @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_433_ord__eq__less__trans,axiom,
% 9.49/5.22      ! [A: real,B: real,C: real] :
% 9.49/5.22        ( ( A = B )
% 9.49/5.22       => ( ( ord_less_real @ B @ C )
% 9.49/5.22         => ( ord_less_real @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_434_ord__eq__less__trans,axiom,
% 9.49/5.22      ! [A: rat,B: rat,C: rat] :
% 9.49/5.22        ( ( A = B )
% 9.49/5.22       => ( ( ord_less_rat @ B @ C )
% 9.49/5.22         => ( ord_less_rat @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_435_ord__eq__less__trans,axiom,
% 9.49/5.22      ! [A: num,B: num,C: num] :
% 9.49/5.22        ( ( A = B )
% 9.49/5.22       => ( ( ord_less_num @ B @ C )
% 9.49/5.22         => ( ord_less_num @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_436_ord__eq__less__trans,axiom,
% 9.49/5.22      ! [A: nat,B: nat,C: nat] :
% 9.49/5.22        ( ( A = B )
% 9.49/5.22       => ( ( ord_less_nat @ B @ C )
% 9.49/5.22         => ( ord_less_nat @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_437_ord__eq__less__trans,axiom,
% 9.49/5.22      ! [A: int,B: int,C: int] :
% 9.49/5.22        ( ( A = B )
% 9.49/5.22       => ( ( ord_less_int @ B @ C )
% 9.49/5.22         => ( ord_less_int @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_438_ord__less__eq__trans,axiom,
% 9.49/5.22      ! [A: real,B: real,C: real] :
% 9.49/5.22        ( ( ord_less_real @ A @ B )
% 9.49/5.22       => ( ( B = C )
% 9.49/5.22         => ( ord_less_real @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_439_ord__less__eq__trans,axiom,
% 9.49/5.22      ! [A: rat,B: rat,C: rat] :
% 9.49/5.22        ( ( ord_less_rat @ A @ B )
% 9.49/5.22       => ( ( B = C )
% 9.49/5.22         => ( ord_less_rat @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_440_ord__less__eq__trans,axiom,
% 9.49/5.22      ! [A: num,B: num,C: num] :
% 9.49/5.22        ( ( ord_less_num @ A @ B )
% 9.49/5.22       => ( ( B = C )
% 9.49/5.22         => ( ord_less_num @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_441_ord__less__eq__trans,axiom,
% 9.49/5.22      ! [A: nat,B: nat,C: nat] :
% 9.49/5.22        ( ( ord_less_nat @ A @ B )
% 9.49/5.22       => ( ( B = C )
% 9.49/5.22         => ( ord_less_nat @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_442_ord__less__eq__trans,axiom,
% 9.49/5.22      ! [A: int,B: int,C: int] :
% 9.49/5.22        ( ( ord_less_int @ A @ B )
% 9.49/5.22       => ( ( B = C )
% 9.49/5.22         => ( ord_less_int @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_443_less__induct,axiom,
% 9.49/5.22      ! [P: nat > $o,A: nat] :
% 9.49/5.22        ( ! [X4: nat] :
% 9.49/5.22            ( ! [Y4: nat] :
% 9.49/5.22                ( ( ord_less_nat @ Y4 @ X4 )
% 9.49/5.22               => ( P @ Y4 ) )
% 9.49/5.22           => ( P @ X4 ) )
% 9.49/5.22       => ( P @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_444_antisym__conv3,axiom,
% 9.49/5.22      ! [Y: real,X: real] :
% 9.49/5.22        ( ~ ( ord_less_real @ Y @ X )
% 9.49/5.22       => ( ( ~ ( ord_less_real @ X @ Y ) )
% 9.49/5.22          = ( X = Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_445_antisym__conv3,axiom,
% 9.49/5.22      ! [Y: rat,X: rat] :
% 9.49/5.22        ( ~ ( ord_less_rat @ Y @ X )
% 9.49/5.22       => ( ( ~ ( ord_less_rat @ X @ Y ) )
% 9.49/5.22          = ( X = Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_446_antisym__conv3,axiom,
% 9.49/5.22      ! [Y: num,X: num] :
% 9.49/5.22        ( ~ ( ord_less_num @ Y @ X )
% 9.49/5.22       => ( ( ~ ( ord_less_num @ X @ Y ) )
% 9.49/5.22          = ( X = Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_447_antisym__conv3,axiom,
% 9.49/5.22      ! [Y: nat,X: nat] :
% 9.49/5.22        ( ~ ( ord_less_nat @ Y @ X )
% 9.49/5.22       => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 9.49/5.22          = ( X = Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_448_antisym__conv3,axiom,
% 9.49/5.22      ! [Y: int,X: int] :
% 9.49/5.22        ( ~ ( ord_less_int @ Y @ X )
% 9.49/5.22       => ( ( ~ ( ord_less_int @ X @ Y ) )
% 9.49/5.22          = ( X = Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_449_linorder__cases,axiom,
% 9.49/5.22      ! [X: real,Y: real] :
% 9.49/5.22        ( ~ ( ord_less_real @ X @ Y )
% 9.49/5.22       => ( ( X != Y )
% 9.49/5.22         => ( ord_less_real @ Y @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_450_linorder__cases,axiom,
% 9.49/5.22      ! [X: rat,Y: rat] :
% 9.49/5.22        ( ~ ( ord_less_rat @ X @ Y )
% 9.49/5.22       => ( ( X != Y )
% 9.49/5.22         => ( ord_less_rat @ Y @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_451_linorder__cases,axiom,
% 9.49/5.22      ! [X: num,Y: num] :
% 9.49/5.22        ( ~ ( ord_less_num @ X @ Y )
% 9.49/5.22       => ( ( X != Y )
% 9.49/5.22         => ( ord_less_num @ Y @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_452_linorder__cases,axiom,
% 9.49/5.22      ! [X: nat,Y: nat] :
% 9.49/5.22        ( ~ ( ord_less_nat @ X @ Y )
% 9.49/5.22       => ( ( X != Y )
% 9.49/5.22         => ( ord_less_nat @ Y @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_453_linorder__cases,axiom,
% 9.49/5.22      ! [X: int,Y: int] :
% 9.49/5.22        ( ~ ( ord_less_int @ X @ Y )
% 9.49/5.22       => ( ( X != Y )
% 9.49/5.22         => ( ord_less_int @ Y @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_454_dual__order_Oasym,axiom,
% 9.49/5.22      ! [B: real,A: real] :
% 9.49/5.22        ( ( ord_less_real @ B @ A )
% 9.49/5.22       => ~ ( ord_less_real @ A @ B ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_455_dual__order_Oasym,axiom,
% 9.49/5.22      ! [B: rat,A: rat] :
% 9.49/5.22        ( ( ord_less_rat @ B @ A )
% 9.49/5.22       => ~ ( ord_less_rat @ A @ B ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_456_dual__order_Oasym,axiom,
% 9.49/5.22      ! [B: num,A: num] :
% 9.49/5.22        ( ( ord_less_num @ B @ A )
% 9.49/5.22       => ~ ( ord_less_num @ A @ B ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_457_dual__order_Oasym,axiom,
% 9.49/5.22      ! [B: nat,A: nat] :
% 9.49/5.22        ( ( ord_less_nat @ B @ A )
% 9.49/5.22       => ~ ( ord_less_nat @ A @ B ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_458_dual__order_Oasym,axiom,
% 9.49/5.22      ! [B: int,A: int] :
% 9.49/5.22        ( ( ord_less_int @ B @ A )
% 9.49/5.22       => ~ ( ord_less_int @ A @ B ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_459_dual__order_Oirrefl,axiom,
% 9.49/5.22      ! [A: real] :
% 9.49/5.22        ~ ( ord_less_real @ A @ A ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_460_dual__order_Oirrefl,axiom,
% 9.49/5.22      ! [A: rat] :
% 9.49/5.22        ~ ( ord_less_rat @ A @ A ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_461_dual__order_Oirrefl,axiom,
% 9.49/5.22      ! [A: num] :
% 9.49/5.22        ~ ( ord_less_num @ A @ A ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_462_dual__order_Oirrefl,axiom,
% 9.49/5.22      ! [A: nat] :
% 9.49/5.22        ~ ( ord_less_nat @ A @ A ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_463_dual__order_Oirrefl,axiom,
% 9.49/5.22      ! [A: int] :
% 9.49/5.22        ~ ( ord_less_int @ A @ A ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_464_exists__least__iff,axiom,
% 9.49/5.22      ( ( ^ [P2: nat > $o] :
% 9.49/5.22          ? [X7: nat] : ( P2 @ X7 ) )
% 9.49/5.22      = ( ^ [P3: nat > $o] :
% 9.49/5.22          ? [N2: nat] :
% 9.49/5.22            ( ( P3 @ N2 )
% 9.49/5.22            & ! [M2: nat] :
% 9.49/5.22                ( ( ord_less_nat @ M2 @ N2 )
% 9.49/5.22               => ~ ( P3 @ M2 ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_465_linorder__less__wlog,axiom,
% 9.49/5.22      ! [P: real > real > $o,A: real,B: real] :
% 9.49/5.22        ( ! [A3: real,B3: real] :
% 9.49/5.22            ( ( ord_less_real @ A3 @ B3 )
% 9.49/5.22           => ( P @ A3 @ B3 ) )
% 9.49/5.22       => ( ! [A3: real] : ( P @ A3 @ A3 )
% 9.49/5.22         => ( ! [A3: real,B3: real] :
% 9.49/5.22                ( ( P @ B3 @ A3 )
% 9.49/5.22               => ( P @ A3 @ B3 ) )
% 9.49/5.22           => ( P @ A @ B ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_466_linorder__less__wlog,axiom,
% 9.49/5.22      ! [P: rat > rat > $o,A: rat,B: rat] :
% 9.49/5.22        ( ! [A3: rat,B3: rat] :
% 9.49/5.22            ( ( ord_less_rat @ A3 @ B3 )
% 9.49/5.22           => ( P @ A3 @ B3 ) )
% 9.49/5.22       => ( ! [A3: rat] : ( P @ A3 @ A3 )
% 9.49/5.22         => ( ! [A3: rat,B3: rat] :
% 9.49/5.22                ( ( P @ B3 @ A3 )
% 9.49/5.22               => ( P @ A3 @ B3 ) )
% 9.49/5.22           => ( P @ A @ B ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_467_linorder__less__wlog,axiom,
% 9.49/5.22      ! [P: num > num > $o,A: num,B: num] :
% 9.49/5.22        ( ! [A3: num,B3: num] :
% 9.49/5.22            ( ( ord_less_num @ A3 @ B3 )
% 9.49/5.22           => ( P @ A3 @ B3 ) )
% 9.49/5.22       => ( ! [A3: num] : ( P @ A3 @ A3 )
% 9.49/5.22         => ( ! [A3: num,B3: num] :
% 9.49/5.22                ( ( P @ B3 @ A3 )
% 9.49/5.22               => ( P @ A3 @ B3 ) )
% 9.49/5.22           => ( P @ A @ B ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_468_linorder__less__wlog,axiom,
% 9.49/5.22      ! [P: nat > nat > $o,A: nat,B: nat] :
% 9.49/5.22        ( ! [A3: nat,B3: nat] :
% 9.49/5.22            ( ( ord_less_nat @ A3 @ B3 )
% 9.49/5.22           => ( P @ A3 @ B3 ) )
% 9.49/5.22       => ( ! [A3: nat] : ( P @ A3 @ A3 )
% 9.49/5.22         => ( ! [A3: nat,B3: nat] :
% 9.49/5.22                ( ( P @ B3 @ A3 )
% 9.49/5.22               => ( P @ A3 @ B3 ) )
% 9.49/5.22           => ( P @ A @ B ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_469_linorder__less__wlog,axiom,
% 9.49/5.22      ! [P: int > int > $o,A: int,B: int] :
% 9.49/5.22        ( ! [A3: int,B3: int] :
% 9.49/5.22            ( ( ord_less_int @ A3 @ B3 )
% 9.49/5.22           => ( P @ A3 @ B3 ) )
% 9.49/5.22       => ( ! [A3: int] : ( P @ A3 @ A3 )
% 9.49/5.22         => ( ! [A3: int,B3: int] :
% 9.49/5.22                ( ( P @ B3 @ A3 )
% 9.49/5.22               => ( P @ A3 @ B3 ) )
% 9.49/5.22           => ( P @ A @ B ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_470_order_Ostrict__trans,axiom,
% 9.49/5.22      ! [A: real,B: real,C: real] :
% 9.49/5.22        ( ( ord_less_real @ A @ B )
% 9.49/5.22       => ( ( ord_less_real @ B @ C )
% 9.49/5.22         => ( ord_less_real @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_471_order_Ostrict__trans,axiom,
% 9.49/5.22      ! [A: rat,B: rat,C: rat] :
% 9.49/5.22        ( ( ord_less_rat @ A @ B )
% 9.49/5.22       => ( ( ord_less_rat @ B @ C )
% 9.49/5.22         => ( ord_less_rat @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_472_order_Ostrict__trans,axiom,
% 9.49/5.22      ! [A: num,B: num,C: num] :
% 9.49/5.22        ( ( ord_less_num @ A @ B )
% 9.49/5.22       => ( ( ord_less_num @ B @ C )
% 9.49/5.22         => ( ord_less_num @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_473_order_Ostrict__trans,axiom,
% 9.49/5.22      ! [A: nat,B: nat,C: nat] :
% 9.49/5.22        ( ( ord_less_nat @ A @ B )
% 9.49/5.22       => ( ( ord_less_nat @ B @ C )
% 9.49/5.22         => ( ord_less_nat @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_474_order_Ostrict__trans,axiom,
% 9.49/5.22      ! [A: int,B: int,C: int] :
% 9.49/5.22        ( ( ord_less_int @ A @ B )
% 9.49/5.22       => ( ( ord_less_int @ B @ C )
% 9.49/5.22         => ( ord_less_int @ A @ C ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_475_not__less__iff__gr__or__eq,axiom,
% 9.49/5.22      ! [X: real,Y: real] :
% 9.49/5.22        ( ( ~ ( ord_less_real @ X @ Y ) )
% 9.49/5.22        = ( ( ord_less_real @ Y @ X )
% 9.49/5.22          | ( X = Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_476_not__less__iff__gr__or__eq,axiom,
% 9.49/5.22      ! [X: rat,Y: rat] :
% 9.49/5.22        ( ( ~ ( ord_less_rat @ X @ Y ) )
% 9.49/5.22        = ( ( ord_less_rat @ Y @ X )
% 9.49/5.22          | ( X = Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_477_not__less__iff__gr__or__eq,axiom,
% 9.49/5.22      ! [X: num,Y: num] :
% 9.49/5.22        ( ( ~ ( ord_less_num @ X @ Y ) )
% 9.49/5.22        = ( ( ord_less_num @ Y @ X )
% 9.49/5.22          | ( X = Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_478_not__less__iff__gr__or__eq,axiom,
% 9.49/5.22      ! [X: nat,Y: nat] :
% 9.49/5.22        ( ( ~ ( ord_less_nat @ X @ Y ) )
% 9.49/5.22        = ( ( ord_less_nat @ Y @ X )
% 9.49/5.22          | ( X = Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_479_not__less__iff__gr__or__eq,axiom,
% 9.49/5.22      ! [X: int,Y: int] :
% 9.49/5.22        ( ( ~ ( ord_less_int @ X @ Y ) )
% 9.49/5.22        = ( ( ord_less_int @ Y @ X )
% 9.49/5.22          | ( X = Y ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_480_dual__order_Ostrict__trans,axiom,
% 9.49/5.22      ! [B: real,A: real,C: real] :
% 9.49/5.22        ( ( ord_less_real @ B @ A )
% 9.49/5.22       => ( ( ord_less_real @ C @ B )
% 9.49/5.22         => ( ord_less_real @ C @ A ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_481_dual__order_Ostrict__trans,axiom,
% 9.49/5.22      ! [B: rat,A: rat,C: rat] :
% 9.49/5.22        ( ( ord_less_rat @ B @ A )
% 9.49/5.22       => ( ( ord_less_rat @ C @ B )
% 9.49/5.22         => ( ord_less_rat @ C @ A ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_482_dual__order_Ostrict__trans,axiom,
% 9.49/5.22      ! [B: num,A: num,C: num] :
% 9.49/5.22        ( ( ord_less_num @ B @ A )
% 9.49/5.22       => ( ( ord_less_num @ C @ B )
% 9.49/5.22         => ( ord_less_num @ C @ A ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_483_dual__order_Ostrict__trans,axiom,
% 9.49/5.22      ! [B: nat,A: nat,C: nat] :
% 9.49/5.22        ( ( ord_less_nat @ B @ A )
% 9.49/5.22       => ( ( ord_less_nat @ C @ B )
% 9.49/5.22         => ( ord_less_nat @ C @ A ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_484_dual__order_Ostrict__trans,axiom,
% 9.49/5.22      ! [B: int,A: int,C: int] :
% 9.49/5.22        ( ( ord_less_int @ B @ A )
% 9.49/5.22       => ( ( ord_less_int @ C @ B )
% 9.49/5.22         => ( ord_less_int @ C @ A ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_485_order_Ostrict__implies__not__eq,axiom,
% 9.49/5.22      ! [A: real,B: real] :
% 9.49/5.22        ( ( ord_less_real @ A @ B )
% 9.49/5.22       => ( A != B ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_486_order_Ostrict__implies__not__eq,axiom,
% 9.49/5.22      ! [A: rat,B: rat] :
% 9.49/5.22        ( ( ord_less_rat @ A @ B )
% 9.49/5.22       => ( A != B ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_487_order_Ostrict__implies__not__eq,axiom,
% 9.49/5.22      ! [A: num,B: num] :
% 9.49/5.22        ( ( ord_less_num @ A @ B )
% 9.49/5.22       => ( A != B ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_488_order_Ostrict__implies__not__eq,axiom,
% 9.49/5.22      ! [A: nat,B: nat] :
% 9.49/5.22        ( ( ord_less_nat @ A @ B )
% 9.49/5.22       => ( A != B ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_489_order_Ostrict__implies__not__eq,axiom,
% 9.49/5.22      ! [A: int,B: int] :
% 9.49/5.22        ( ( ord_less_int @ A @ B )
% 9.49/5.22       => ( A != B ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_490_dual__order_Ostrict__implies__not__eq,axiom,
% 9.49/5.22      ! [B: real,A: real] :
% 9.49/5.22        ( ( ord_less_real @ B @ A )
% 9.49/5.22       => ( A != B ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_491_dual__order_Ostrict__implies__not__eq,axiom,
% 9.49/5.22      ! [B: rat,A: rat] :
% 9.49/5.22        ( ( ord_less_rat @ B @ A )
% 9.49/5.22       => ( A != B ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_492_dual__order_Ostrict__implies__not__eq,axiom,
% 9.49/5.22      ! [B: num,A: num] :
% 9.49/5.22        ( ( ord_less_num @ B @ A )
% 9.49/5.22       => ( A != B ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_493_dual__order_Ostrict__implies__not__eq,axiom,
% 9.49/5.22      ! [B: nat,A: nat] :
% 9.49/5.22        ( ( ord_less_nat @ B @ A )
% 9.49/5.22       => ( A != B ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_494_dual__order_Ostrict__implies__not__eq,axiom,
% 9.49/5.22      ! [B: int,A: int] :
% 9.49/5.22        ( ( ord_less_int @ B @ A )
% 9.49/5.22       => ( A != B ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_495_linorder__neqE,axiom,
% 9.49/5.22      ! [X: real,Y: real] :
% 9.49/5.22        ( ( X != Y )
% 9.49/5.22       => ( ~ ( ord_less_real @ X @ Y )
% 9.49/5.22         => ( ord_less_real @ Y @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_496_linorder__neqE,axiom,
% 9.49/5.22      ! [X: rat,Y: rat] :
% 9.49/5.22        ( ( X != Y )
% 9.49/5.22       => ( ~ ( ord_less_rat @ X @ Y )
% 9.49/5.22         => ( ord_less_rat @ Y @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_497_linorder__neqE,axiom,
% 9.49/5.22      ! [X: num,Y: num] :
% 9.49/5.22        ( ( X != Y )
% 9.49/5.22       => ( ~ ( ord_less_num @ X @ Y )
% 9.49/5.22         => ( ord_less_num @ Y @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_498_linorder__neqE,axiom,
% 9.49/5.22      ! [X: nat,Y: nat] :
% 9.49/5.22        ( ( X != Y )
% 9.49/5.22       => ( ~ ( ord_less_nat @ X @ Y )
% 9.49/5.22         => ( ord_less_nat @ Y @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_499_linorder__neqE,axiom,
% 9.49/5.22      ! [X: int,Y: int] :
% 9.49/5.22        ( ( X != Y )
% 9.49/5.22       => ( ~ ( ord_less_int @ X @ Y )
% 9.49/5.22         => ( ord_less_int @ Y @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_500_order__less__asym,axiom,
% 9.49/5.22      ! [X: real,Y: real] :
% 9.49/5.22        ( ( ord_less_real @ X @ Y )
% 9.49/5.22       => ~ ( ord_less_real @ Y @ X ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_501_order__less__asym,axiom,
% 9.49/5.22      ! [X: rat,Y: rat] :
% 9.49/5.22        ( ( ord_less_rat @ X @ Y )
% 9.49/5.22       => ~ ( ord_less_rat @ Y @ X ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_502_order__less__asym,axiom,
% 9.49/5.22      ! [X: num,Y: num] :
% 9.49/5.22        ( ( ord_less_num @ X @ Y )
% 9.49/5.22       => ~ ( ord_less_num @ Y @ X ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_503_order__less__asym,axiom,
% 9.49/5.22      ! [X: nat,Y: nat] :
% 9.49/5.22        ( ( ord_less_nat @ X @ Y )
% 9.49/5.22       => ~ ( ord_less_nat @ Y @ X ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_504_order__less__asym,axiom,
% 9.49/5.22      ! [X: int,Y: int] :
% 9.49/5.22        ( ( ord_less_int @ X @ Y )
% 9.49/5.22       => ~ ( ord_less_int @ Y @ X ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_505_linorder__neq__iff,axiom,
% 9.49/5.22      ! [X: real,Y: real] :
% 9.49/5.22        ( ( X != Y )
% 9.49/5.22        = ( ( ord_less_real @ X @ Y )
% 9.49/5.22          | ( ord_less_real @ Y @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_506_linorder__neq__iff,axiom,
% 9.49/5.22      ! [X: rat,Y: rat] :
% 9.49/5.22        ( ( X != Y )
% 9.49/5.22        = ( ( ord_less_rat @ X @ Y )
% 9.49/5.22          | ( ord_less_rat @ Y @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_507_linorder__neq__iff,axiom,
% 9.49/5.22      ! [X: num,Y: num] :
% 9.49/5.22        ( ( X != Y )
% 9.49/5.22        = ( ( ord_less_num @ X @ Y )
% 9.49/5.22          | ( ord_less_num @ Y @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_508_linorder__neq__iff,axiom,
% 9.49/5.22      ! [X: nat,Y: nat] :
% 9.49/5.22        ( ( X != Y )
% 9.49/5.22        = ( ( ord_less_nat @ X @ Y )
% 9.49/5.22          | ( ord_less_nat @ Y @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_509_linorder__neq__iff,axiom,
% 9.49/5.22      ! [X: int,Y: int] :
% 9.49/5.22        ( ( X != Y )
% 9.49/5.22        = ( ( ord_less_int @ X @ Y )
% 9.49/5.22          | ( ord_less_int @ Y @ X ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_510_order__less__asym_H,axiom,
% 9.49/5.22      ! [A: real,B: real] :
% 9.49/5.22        ( ( ord_less_real @ A @ B )
% 9.49/5.22       => ~ ( ord_less_real @ B @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_511_order__less__asym_H,axiom,
% 9.49/5.22      ! [A: rat,B: rat] :
% 9.49/5.22        ( ( ord_less_rat @ A @ B )
% 9.49/5.22       => ~ ( ord_less_rat @ B @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_512_order__less__asym_H,axiom,
% 9.49/5.22      ! [A: num,B: num] :
% 9.49/5.22        ( ( ord_less_num @ A @ B )
% 9.49/5.22       => ~ ( ord_less_num @ B @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_513_order__less__asym_H,axiom,
% 9.49/5.22      ! [A: nat,B: nat] :
% 9.49/5.22        ( ( ord_less_nat @ A @ B )
% 9.49/5.22       => ~ ( ord_less_nat @ B @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_514_order__less__asym_H,axiom,
% 9.49/5.22      ! [A: int,B: int] :
% 9.49/5.22        ( ( ord_less_int @ A @ B )
% 9.49/5.22       => ~ ( ord_less_int @ B @ A ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_515_order__less__trans,axiom,
% 9.49/5.22      ! [X: real,Y: real,Z2: real] :
% 9.49/5.22        ( ( ord_less_real @ X @ Y )
% 9.49/5.22       => ( ( ord_less_real @ Y @ Z2 )
% 9.49/5.22         => ( ord_less_real @ X @ Z2 ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_516_order__less__trans,axiom,
% 9.49/5.22      ! [X: rat,Y: rat,Z2: rat] :
% 9.49/5.22        ( ( ord_less_rat @ X @ Y )
% 9.49/5.22       => ( ( ord_less_rat @ Y @ Z2 )
% 9.49/5.22         => ( ord_less_rat @ X @ Z2 ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_517_order__less__trans,axiom,
% 9.49/5.22      ! [X: num,Y: num,Z2: num] :
% 9.49/5.22        ( ( ord_less_num @ X @ Y )
% 9.49/5.22       => ( ( ord_less_num @ Y @ Z2 )
% 9.49/5.22         => ( ord_less_num @ X @ Z2 ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_518_order__less__trans,axiom,
% 9.49/5.22      ! [X: nat,Y: nat,Z2: nat] :
% 9.49/5.22        ( ( ord_less_nat @ X @ Y )
% 9.49/5.22       => ( ( ord_less_nat @ Y @ Z2 )
% 9.49/5.22         => ( ord_less_nat @ X @ Z2 ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_519_order__less__trans,axiom,
% 9.49/5.22      ! [X: int,Y: int,Z2: int] :
% 9.49/5.22        ( ( ord_less_int @ X @ Y )
% 9.49/5.22       => ( ( ord_less_int @ Y @ Z2 )
% 9.49/5.22         => ( ord_less_int @ X @ Z2 ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_520_ord__eq__less__subst,axiom,
% 9.49/5.22      ! [A: real,F: real > real,B: real,C: real] :
% 9.49/5.22        ( ( A
% 9.49/5.22          = ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_real @ B @ C )
% 9.49/5.22         => ( ! [X4: real,Y3: real] :
% 9.49/5.22                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_521_ord__eq__less__subst,axiom,
% 9.49/5.22      ! [A: rat,F: real > rat,B: real,C: real] :
% 9.49/5.22        ( ( A
% 9.49/5.22          = ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_real @ B @ C )
% 9.49/5.22         => ( ! [X4: real,Y3: real] :
% 9.49/5.22                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_522_ord__eq__less__subst,axiom,
% 9.49/5.22      ! [A: num,F: real > num,B: real,C: real] :
% 9.49/5.22        ( ( A
% 9.49/5.22          = ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_real @ B @ C )
% 9.49/5.22         => ( ! [X4: real,Y3: real] :
% 9.49/5.22                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_523_ord__eq__less__subst,axiom,
% 9.49/5.22      ! [A: nat,F: real > nat,B: real,C: real] :
% 9.49/5.22        ( ( A
% 9.49/5.22          = ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_real @ B @ C )
% 9.49/5.22         => ( ! [X4: real,Y3: real] :
% 9.49/5.22                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_524_ord__eq__less__subst,axiom,
% 9.49/5.22      ! [A: int,F: real > int,B: real,C: real] :
% 9.49/5.22        ( ( A
% 9.49/5.22          = ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_real @ B @ C )
% 9.49/5.22         => ( ! [X4: real,Y3: real] :
% 9.49/5.22                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_525_ord__eq__less__subst,axiom,
% 9.49/5.22      ! [A: real,F: rat > real,B: rat,C: rat] :
% 9.49/5.22        ( ( A
% 9.49/5.22          = ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_rat @ B @ C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_526_ord__eq__less__subst,axiom,
% 9.49/5.22      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 9.49/5.22        ( ( A
% 9.49/5.22          = ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_rat @ B @ C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_527_ord__eq__less__subst,axiom,
% 9.49/5.22      ! [A: num,F: rat > num,B: rat,C: rat] :
% 9.49/5.22        ( ( A
% 9.49/5.22          = ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_rat @ B @ C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_528_ord__eq__less__subst,axiom,
% 9.49/5.22      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 9.49/5.22        ( ( A
% 9.49/5.22          = ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_rat @ B @ C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_529_ord__eq__less__subst,axiom,
% 9.49/5.22      ! [A: int,F: rat > int,B: rat,C: rat] :
% 9.49/5.22        ( ( A
% 9.49/5.22          = ( F @ B ) )
% 9.49/5.22       => ( ( ord_less_rat @ B @ C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_530_ord__less__eq__subst,axiom,
% 9.49/5.22      ! [A: real,B: real,F: real > real,C: real] :
% 9.49/5.22        ( ( ord_less_real @ A @ B )
% 9.49/5.22       => ( ( ( F @ B )
% 9.49/5.22            = C )
% 9.49/5.22         => ( ! [X4: real,Y3: real] :
% 9.49/5.22                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_531_ord__less__eq__subst,axiom,
% 9.49/5.22      ! [A: real,B: real,F: real > rat,C: rat] :
% 9.49/5.22        ( ( ord_less_real @ A @ B )
% 9.49/5.22       => ( ( ( F @ B )
% 9.49/5.22            = C )
% 9.49/5.22         => ( ! [X4: real,Y3: real] :
% 9.49/5.22                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_532_ord__less__eq__subst,axiom,
% 9.49/5.22      ! [A: real,B: real,F: real > num,C: num] :
% 9.49/5.22        ( ( ord_less_real @ A @ B )
% 9.49/5.22       => ( ( ( F @ B )
% 9.49/5.22            = C )
% 9.49/5.22         => ( ! [X4: real,Y3: real] :
% 9.49/5.22                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_533_ord__less__eq__subst,axiom,
% 9.49/5.22      ! [A: real,B: real,F: real > nat,C: nat] :
% 9.49/5.22        ( ( ord_less_real @ A @ B )
% 9.49/5.22       => ( ( ( F @ B )
% 9.49/5.22            = C )
% 9.49/5.22         => ( ! [X4: real,Y3: real] :
% 9.49/5.22                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_534_ord__less__eq__subst,axiom,
% 9.49/5.22      ! [A: real,B: real,F: real > int,C: int] :
% 9.49/5.22        ( ( ord_less_real @ A @ B )
% 9.49/5.22       => ( ( ( F @ B )
% 9.49/5.22            = C )
% 9.49/5.22         => ( ! [X4: real,Y3: real] :
% 9.49/5.22                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_535_ord__less__eq__subst,axiom,
% 9.49/5.22      ! [A: rat,B: rat,F: rat > real,C: real] :
% 9.49/5.22        ( ( ord_less_rat @ A @ B )
% 9.49/5.22       => ( ( ( F @ B )
% 9.49/5.22            = C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_536_ord__less__eq__subst,axiom,
% 9.49/5.22      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 9.49/5.22        ( ( ord_less_rat @ A @ B )
% 9.49/5.22       => ( ( ( F @ B )
% 9.49/5.22            = C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_537_ord__less__eq__subst,axiom,
% 9.49/5.22      ! [A: rat,B: rat,F: rat > num,C: num] :
% 9.49/5.22        ( ( ord_less_rat @ A @ B )
% 9.49/5.22       => ( ( ( F @ B )
% 9.49/5.22            = C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_538_ord__less__eq__subst,axiom,
% 9.49/5.22      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 9.49/5.22        ( ( ord_less_rat @ A @ B )
% 9.49/5.22       => ( ( ( F @ B )
% 9.49/5.22            = C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_539_ord__less__eq__subst,axiom,
% 9.49/5.22      ! [A: rat,B: rat,F: rat > int,C: int] :
% 9.49/5.22        ( ( ord_less_rat @ A @ B )
% 9.49/5.22       => ( ( ( F @ B )
% 9.49/5.22            = C )
% 9.49/5.22         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.22                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.22               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.22           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_540_order__less__irrefl,axiom,
% 9.49/5.22      ! [X: real] :
% 9.49/5.22        ~ ( ord_less_real @ X @ X ) ).
% 9.49/5.22  
% 9.49/5.22  thf(fact_541_order__less__irrefl,axiom,
% 9.49/5.22      ! [X: rat] :
% 9.49/5.22        ~ ( ord_less_rat @ X @ X ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_542_order__less__irrefl,axiom,
% 9.49/5.23      ! [X: num] :
% 9.49/5.23        ~ ( ord_less_num @ X @ X ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_543_order__less__irrefl,axiom,
% 9.49/5.23      ! [X: nat] :
% 9.49/5.23        ~ ( ord_less_nat @ X @ X ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_544_order__less__irrefl,axiom,
% 9.49/5.23      ! [X: int] :
% 9.49/5.23        ~ ( ord_less_int @ X @ X ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_545_order__less__subst1,axiom,
% 9.49/5.23      ! [A: real,F: real > real,B: real,C: real] :
% 9.49/5.23        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_real @ B @ C )
% 9.49/5.23         => ( ! [X4: real,Y3: real] :
% 9.49/5.23                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_546_order__less__subst1,axiom,
% 9.49/5.23      ! [A: real,F: rat > real,B: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_rat @ B @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_547_order__less__subst1,axiom,
% 9.49/5.23      ! [A: real,F: num > real,B: num,C: num] :
% 9.49/5.23        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_num @ B @ C )
% 9.49/5.23         => ( ! [X4: num,Y3: num] :
% 9.49/5.23                ( ( ord_less_num @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_548_order__less__subst1,axiom,
% 9.49/5.23      ! [A: real,F: nat > real,B: nat,C: nat] :
% 9.49/5.23        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_nat @ B @ C )
% 9.49/5.23         => ( ! [X4: nat,Y3: nat] :
% 9.49/5.23                ( ( ord_less_nat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_549_order__less__subst1,axiom,
% 9.49/5.23      ! [A: real,F: int > real,B: int,C: int] :
% 9.49/5.23        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_int @ B @ C )
% 9.49/5.23         => ( ! [X4: int,Y3: int] :
% 9.49/5.23                ( ( ord_less_int @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_550_order__less__subst1,axiom,
% 9.49/5.23      ! [A: rat,F: real > rat,B: real,C: real] :
% 9.49/5.23        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_real @ B @ C )
% 9.49/5.23         => ( ! [X4: real,Y3: real] :
% 9.49/5.23                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_551_order__less__subst1,axiom,
% 9.49/5.23      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_rat @ B @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_552_order__less__subst1,axiom,
% 9.49/5.23      ! [A: rat,F: num > rat,B: num,C: num] :
% 9.49/5.23        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_num @ B @ C )
% 9.49/5.23         => ( ! [X4: num,Y3: num] :
% 9.49/5.23                ( ( ord_less_num @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_553_order__less__subst1,axiom,
% 9.49/5.23      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 9.49/5.23        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_nat @ B @ C )
% 9.49/5.23         => ( ! [X4: nat,Y3: nat] :
% 9.49/5.23                ( ( ord_less_nat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_554_order__less__subst1,axiom,
% 9.49/5.23      ! [A: rat,F: int > rat,B: int,C: int] :
% 9.49/5.23        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_int @ B @ C )
% 9.49/5.23         => ( ! [X4: int,Y3: int] :
% 9.49/5.23                ( ( ord_less_int @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_555_order__less__subst2,axiom,
% 9.49/5.23      ! [A: real,B: real,F: real > real,C: real] :
% 9.49/5.23        ( ( ord_less_real @ A @ B )
% 9.49/5.23       => ( ( ord_less_real @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: real,Y3: real] :
% 9.49/5.23                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_556_order__less__subst2,axiom,
% 9.49/5.23      ! [A: real,B: real,F: real > rat,C: rat] :
% 9.49/5.23        ( ( ord_less_real @ A @ B )
% 9.49/5.23       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: real,Y3: real] :
% 9.49/5.23                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_557_order__less__subst2,axiom,
% 9.49/5.23      ! [A: real,B: real,F: real > num,C: num] :
% 9.49/5.23        ( ( ord_less_real @ A @ B )
% 9.49/5.23       => ( ( ord_less_num @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: real,Y3: real] :
% 9.49/5.23                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_558_order__less__subst2,axiom,
% 9.49/5.23      ! [A: real,B: real,F: real > nat,C: nat] :
% 9.49/5.23        ( ( ord_less_real @ A @ B )
% 9.49/5.23       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: real,Y3: real] :
% 9.49/5.23                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_559_order__less__subst2,axiom,
% 9.49/5.23      ! [A: real,B: real,F: real > int,C: int] :
% 9.49/5.23        ( ( ord_less_real @ A @ B )
% 9.49/5.23       => ( ( ord_less_int @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: real,Y3: real] :
% 9.49/5.23                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_560_order__less__subst2,axiom,
% 9.49/5.23      ! [A: rat,B: rat,F: rat > real,C: real] :
% 9.49/5.23        ( ( ord_less_rat @ A @ B )
% 9.49/5.23       => ( ( ord_less_real @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_561_order__less__subst2,axiom,
% 9.49/5.23      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 9.49/5.23        ( ( ord_less_rat @ A @ B )
% 9.49/5.23       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_562_order__less__subst2,axiom,
% 9.49/5.23      ! [A: rat,B: rat,F: rat > num,C: num] :
% 9.49/5.23        ( ( ord_less_rat @ A @ B )
% 9.49/5.23       => ( ( ord_less_num @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_563_order__less__subst2,axiom,
% 9.49/5.23      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 9.49/5.23        ( ( ord_less_rat @ A @ B )
% 9.49/5.23       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_564_order__less__subst2,axiom,
% 9.49/5.23      ! [A: rat,B: rat,F: rat > int,C: int] :
% 9.49/5.23        ( ( ord_less_rat @ A @ B )
% 9.49/5.23       => ( ( ord_less_int @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_565_order__less__not__sym,axiom,
% 9.49/5.23      ! [X: real,Y: real] :
% 9.49/5.23        ( ( ord_less_real @ X @ Y )
% 9.49/5.23       => ~ ( ord_less_real @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_566_order__less__not__sym,axiom,
% 9.49/5.23      ! [X: rat,Y: rat] :
% 9.49/5.23        ( ( ord_less_rat @ X @ Y )
% 9.49/5.23       => ~ ( ord_less_rat @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_567_order__less__not__sym,axiom,
% 9.49/5.23      ! [X: num,Y: num] :
% 9.49/5.23        ( ( ord_less_num @ X @ Y )
% 9.49/5.23       => ~ ( ord_less_num @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_568_order__less__not__sym,axiom,
% 9.49/5.23      ! [X: nat,Y: nat] :
% 9.49/5.23        ( ( ord_less_nat @ X @ Y )
% 9.49/5.23       => ~ ( ord_less_nat @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_569_order__less__not__sym,axiom,
% 9.49/5.23      ! [X: int,Y: int] :
% 9.49/5.23        ( ( ord_less_int @ X @ Y )
% 9.49/5.23       => ~ ( ord_less_int @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_570_order__less__imp__triv,axiom,
% 9.49/5.23      ! [X: real,Y: real,P: $o] :
% 9.49/5.23        ( ( ord_less_real @ X @ Y )
% 9.49/5.23       => ( ( ord_less_real @ Y @ X )
% 9.49/5.23         => P ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_571_order__less__imp__triv,axiom,
% 9.49/5.23      ! [X: rat,Y: rat,P: $o] :
% 9.49/5.23        ( ( ord_less_rat @ X @ Y )
% 9.49/5.23       => ( ( ord_less_rat @ Y @ X )
% 9.49/5.23         => P ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_572_order__less__imp__triv,axiom,
% 9.49/5.23      ! [X: num,Y: num,P: $o] :
% 9.49/5.23        ( ( ord_less_num @ X @ Y )
% 9.49/5.23       => ( ( ord_less_num @ Y @ X )
% 9.49/5.23         => P ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_573_order__less__imp__triv,axiom,
% 9.49/5.23      ! [X: nat,Y: nat,P: $o] :
% 9.49/5.23        ( ( ord_less_nat @ X @ Y )
% 9.49/5.23       => ( ( ord_less_nat @ Y @ X )
% 9.49/5.23         => P ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_574_order__less__imp__triv,axiom,
% 9.49/5.23      ! [X: int,Y: int,P: $o] :
% 9.49/5.23        ( ( ord_less_int @ X @ Y )
% 9.49/5.23       => ( ( ord_less_int @ Y @ X )
% 9.49/5.23         => P ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_575_linorder__less__linear,axiom,
% 9.49/5.23      ! [X: real,Y: real] :
% 9.49/5.23        ( ( ord_less_real @ X @ Y )
% 9.49/5.23        | ( X = Y )
% 9.49/5.23        | ( ord_less_real @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_576_linorder__less__linear,axiom,
% 9.49/5.23      ! [X: rat,Y: rat] :
% 9.49/5.23        ( ( ord_less_rat @ X @ Y )
% 9.49/5.23        | ( X = Y )
% 9.49/5.23        | ( ord_less_rat @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_577_linorder__less__linear,axiom,
% 9.49/5.23      ! [X: num,Y: num] :
% 9.49/5.23        ( ( ord_less_num @ X @ Y )
% 9.49/5.23        | ( X = Y )
% 9.49/5.23        | ( ord_less_num @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_578_linorder__less__linear,axiom,
% 9.49/5.23      ! [X: nat,Y: nat] :
% 9.49/5.23        ( ( ord_less_nat @ X @ Y )
% 9.49/5.23        | ( X = Y )
% 9.49/5.23        | ( ord_less_nat @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_579_linorder__less__linear,axiom,
% 9.49/5.23      ! [X: int,Y: int] :
% 9.49/5.23        ( ( ord_less_int @ X @ Y )
% 9.49/5.23        | ( X = Y )
% 9.49/5.23        | ( ord_less_int @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_580_order__less__imp__not__eq,axiom,
% 9.49/5.23      ! [X: real,Y: real] :
% 9.49/5.23        ( ( ord_less_real @ X @ Y )
% 9.49/5.23       => ( X != Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_581_order__less__imp__not__eq,axiom,
% 9.49/5.23      ! [X: rat,Y: rat] :
% 9.49/5.23        ( ( ord_less_rat @ X @ Y )
% 9.49/5.23       => ( X != Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_582_order__less__imp__not__eq,axiom,
% 9.49/5.23      ! [X: num,Y: num] :
% 9.49/5.23        ( ( ord_less_num @ X @ Y )
% 9.49/5.23       => ( X != Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_583_order__less__imp__not__eq,axiom,
% 9.49/5.23      ! [X: nat,Y: nat] :
% 9.49/5.23        ( ( ord_less_nat @ X @ Y )
% 9.49/5.23       => ( X != Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_584_order__less__imp__not__eq,axiom,
% 9.49/5.23      ! [X: int,Y: int] :
% 9.49/5.23        ( ( ord_less_int @ X @ Y )
% 9.49/5.23       => ( X != Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_585_order__less__imp__not__eq2,axiom,
% 9.49/5.23      ! [X: real,Y: real] :
% 9.49/5.23        ( ( ord_less_real @ X @ Y )
% 9.49/5.23       => ( Y != X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_586_order__less__imp__not__eq2,axiom,
% 9.49/5.23      ! [X: rat,Y: rat] :
% 9.49/5.23        ( ( ord_less_rat @ X @ Y )
% 9.49/5.23       => ( Y != X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_587_order__less__imp__not__eq2,axiom,
% 9.49/5.23      ! [X: num,Y: num] :
% 9.49/5.23        ( ( ord_less_num @ X @ Y )
% 9.49/5.23       => ( Y != X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_588_order__less__imp__not__eq2,axiom,
% 9.49/5.23      ! [X: nat,Y: nat] :
% 9.49/5.23        ( ( ord_less_nat @ X @ Y )
% 9.49/5.23       => ( Y != X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_589_order__less__imp__not__eq2,axiom,
% 9.49/5.23      ! [X: int,Y: int] :
% 9.49/5.23        ( ( ord_less_int @ X @ Y )
% 9.49/5.23       => ( Y != X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_590_order__less__imp__not__less,axiom,
% 9.49/5.23      ! [X: real,Y: real] :
% 9.49/5.23        ( ( ord_less_real @ X @ Y )
% 9.49/5.23       => ~ ( ord_less_real @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_591_order__less__imp__not__less,axiom,
% 9.49/5.23      ! [X: rat,Y: rat] :
% 9.49/5.23        ( ( ord_less_rat @ X @ Y )
% 9.49/5.23       => ~ ( ord_less_rat @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_592_order__less__imp__not__less,axiom,
% 9.49/5.23      ! [X: num,Y: num] :
% 9.49/5.23        ( ( ord_less_num @ X @ Y )
% 9.49/5.23       => ~ ( ord_less_num @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_593_order__less__imp__not__less,axiom,
% 9.49/5.23      ! [X: nat,Y: nat] :
% 9.49/5.23        ( ( ord_less_nat @ X @ Y )
% 9.49/5.23       => ~ ( ord_less_nat @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_594_order__less__imp__not__less,axiom,
% 9.49/5.23      ! [X: int,Y: int] :
% 9.49/5.23        ( ( ord_less_int @ X @ Y )
% 9.49/5.23       => ~ ( ord_less_int @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_595_leD,axiom,
% 9.49/5.23      ! [Y: real,X: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ Y @ X )
% 9.49/5.23       => ~ ( ord_less_real @ X @ Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_596_leD,axiom,
% 9.49/5.23      ! [Y: set_int,X: set_int] :
% 9.49/5.23        ( ( ord_less_eq_set_int @ Y @ X )
% 9.49/5.23       => ~ ( ord_less_set_int @ X @ Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_597_leD,axiom,
% 9.49/5.23      ! [Y: rat,X: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ Y @ X )
% 9.49/5.23       => ~ ( ord_less_rat @ X @ Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_598_leD,axiom,
% 9.49/5.23      ! [Y: num,X: num] :
% 9.49/5.23        ( ( ord_less_eq_num @ Y @ X )
% 9.49/5.23       => ~ ( ord_less_num @ X @ Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_599_leD,axiom,
% 9.49/5.23      ! [Y: nat,X: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ Y @ X )
% 9.49/5.23       => ~ ( ord_less_nat @ X @ Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_600_leD,axiom,
% 9.49/5.23      ! [Y: int,X: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ Y @ X )
% 9.49/5.23       => ~ ( ord_less_int @ X @ Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_601_leI,axiom,
% 9.49/5.23      ! [X: real,Y: real] :
% 9.49/5.23        ( ~ ( ord_less_real @ X @ Y )
% 9.49/5.23       => ( ord_less_eq_real @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_602_leI,axiom,
% 9.49/5.23      ! [X: rat,Y: rat] :
% 9.49/5.23        ( ~ ( ord_less_rat @ X @ Y )
% 9.49/5.23       => ( ord_less_eq_rat @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_603_leI,axiom,
% 9.49/5.23      ! [X: num,Y: num] :
% 9.49/5.23        ( ~ ( ord_less_num @ X @ Y )
% 9.49/5.23       => ( ord_less_eq_num @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_604_leI,axiom,
% 9.49/5.23      ! [X: nat,Y: nat] :
% 9.49/5.23        ( ~ ( ord_less_nat @ X @ Y )
% 9.49/5.23       => ( ord_less_eq_nat @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_605_leI,axiom,
% 9.49/5.23      ! [X: int,Y: int] :
% 9.49/5.23        ( ~ ( ord_less_int @ X @ Y )
% 9.49/5.23       => ( ord_less_eq_int @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_606_nless__le,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( ~ ( ord_less_real @ A @ B ) )
% 9.49/5.23        = ( ~ ( ord_less_eq_real @ A @ B )
% 9.49/5.23          | ( A = B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_607_nless__le,axiom,
% 9.49/5.23      ! [A: set_int,B: set_int] :
% 9.49/5.23        ( ( ~ ( ord_less_set_int @ A @ B ) )
% 9.49/5.23        = ( ~ ( ord_less_eq_set_int @ A @ B )
% 9.49/5.23          | ( A = B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_608_nless__le,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( ~ ( ord_less_rat @ A @ B ) )
% 9.49/5.23        = ( ~ ( ord_less_eq_rat @ A @ B )
% 9.49/5.23          | ( A = B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_609_nless__le,axiom,
% 9.49/5.23      ! [A: num,B: num] :
% 9.49/5.23        ( ( ~ ( ord_less_num @ A @ B ) )
% 9.49/5.23        = ( ~ ( ord_less_eq_num @ A @ B )
% 9.49/5.23          | ( A = B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_610_nless__le,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( ~ ( ord_less_nat @ A @ B ) )
% 9.49/5.23        = ( ~ ( ord_less_eq_nat @ A @ B )
% 9.49/5.23          | ( A = B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_611_nless__le,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( ~ ( ord_less_int @ A @ B ) )
% 9.49/5.23        = ( ~ ( ord_less_eq_int @ A @ B )
% 9.49/5.23          | ( A = B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_612_antisym__conv1,axiom,
% 9.49/5.23      ! [X: real,Y: real] :
% 9.49/5.23        ( ~ ( ord_less_real @ X @ Y )
% 9.49/5.23       => ( ( ord_less_eq_real @ X @ Y )
% 9.49/5.23          = ( X = Y ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_613_antisym__conv1,axiom,
% 9.49/5.23      ! [X: set_int,Y: set_int] :
% 9.49/5.23        ( ~ ( ord_less_set_int @ X @ Y )
% 9.49/5.23       => ( ( ord_less_eq_set_int @ X @ Y )
% 9.49/5.23          = ( X = Y ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_614_antisym__conv1,axiom,
% 9.49/5.23      ! [X: rat,Y: rat] :
% 9.49/5.23        ( ~ ( ord_less_rat @ X @ Y )
% 9.49/5.23       => ( ( ord_less_eq_rat @ X @ Y )
% 9.49/5.23          = ( X = Y ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_615_antisym__conv1,axiom,
% 9.49/5.23      ! [X: num,Y: num] :
% 9.49/5.23        ( ~ ( ord_less_num @ X @ Y )
% 9.49/5.23       => ( ( ord_less_eq_num @ X @ Y )
% 9.49/5.23          = ( X = Y ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_616_antisym__conv1,axiom,
% 9.49/5.23      ! [X: nat,Y: nat] :
% 9.49/5.23        ( ~ ( ord_less_nat @ X @ Y )
% 9.49/5.23       => ( ( ord_less_eq_nat @ X @ Y )
% 9.49/5.23          = ( X = Y ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_617_antisym__conv1,axiom,
% 9.49/5.23      ! [X: int,Y: int] :
% 9.49/5.23        ( ~ ( ord_less_int @ X @ Y )
% 9.49/5.23       => ( ( ord_less_eq_int @ X @ Y )
% 9.49/5.23          = ( X = Y ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_618_antisym__conv2,axiom,
% 9.49/5.23      ! [X: real,Y: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ X @ Y )
% 9.49/5.23       => ( ( ~ ( ord_less_real @ X @ Y ) )
% 9.49/5.23          = ( X = Y ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_619_antisym__conv2,axiom,
% 9.49/5.23      ! [X: set_int,Y: set_int] :
% 9.49/5.23        ( ( ord_less_eq_set_int @ X @ Y )
% 9.49/5.23       => ( ( ~ ( ord_less_set_int @ X @ Y ) )
% 9.49/5.23          = ( X = Y ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_620_antisym__conv2,axiom,
% 9.49/5.23      ! [X: rat,Y: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ X @ Y )
% 9.49/5.23       => ( ( ~ ( ord_less_rat @ X @ Y ) )
% 9.49/5.23          = ( X = Y ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_621_antisym__conv2,axiom,
% 9.49/5.23      ! [X: num,Y: num] :
% 9.49/5.23        ( ( ord_less_eq_num @ X @ Y )
% 9.49/5.23       => ( ( ~ ( ord_less_num @ X @ Y ) )
% 9.49/5.23          = ( X = Y ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_622_antisym__conv2,axiom,
% 9.49/5.23      ! [X: nat,Y: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ X @ Y )
% 9.49/5.23       => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 9.49/5.23          = ( X = Y ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_623_antisym__conv2,axiom,
% 9.49/5.23      ! [X: int,Y: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ X @ Y )
% 9.49/5.23       => ( ( ~ ( ord_less_int @ X @ Y ) )
% 9.49/5.23          = ( X = Y ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_624_dense__ge,axiom,
% 9.49/5.23      ! [Z2: real,Y: real] :
% 9.49/5.23        ( ! [X4: real] :
% 9.49/5.23            ( ( ord_less_real @ Z2 @ X4 )
% 9.49/5.23           => ( ord_less_eq_real @ Y @ X4 ) )
% 9.49/5.23       => ( ord_less_eq_real @ Y @ Z2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_625_dense__ge,axiom,
% 9.49/5.23      ! [Z2: rat,Y: rat] :
% 9.49/5.23        ( ! [X4: rat] :
% 9.49/5.23            ( ( ord_less_rat @ Z2 @ X4 )
% 9.49/5.23           => ( ord_less_eq_rat @ Y @ X4 ) )
% 9.49/5.23       => ( ord_less_eq_rat @ Y @ Z2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_626_dense__le,axiom,
% 9.49/5.23      ! [Y: real,Z2: real] :
% 9.49/5.23        ( ! [X4: real] :
% 9.49/5.23            ( ( ord_less_real @ X4 @ Y )
% 9.49/5.23           => ( ord_less_eq_real @ X4 @ Z2 ) )
% 9.49/5.23       => ( ord_less_eq_real @ Y @ Z2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_627_dense__le,axiom,
% 9.49/5.23      ! [Y: rat,Z2: rat] :
% 9.49/5.23        ( ! [X4: rat] :
% 9.49/5.23            ( ( ord_less_rat @ X4 @ Y )
% 9.49/5.23           => ( ord_less_eq_rat @ X4 @ Z2 ) )
% 9.49/5.23       => ( ord_less_eq_rat @ Y @ Z2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_628_less__le__not__le,axiom,
% 9.49/5.23      ( ord_less_real
% 9.49/5.23      = ( ^ [X3: real,Y2: real] :
% 9.49/5.23            ( ( ord_less_eq_real @ X3 @ Y2 )
% 9.49/5.23            & ~ ( ord_less_eq_real @ Y2 @ X3 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_629_less__le__not__le,axiom,
% 9.49/5.23      ( ord_less_set_int
% 9.49/5.23      = ( ^ [X3: set_int,Y2: set_int] :
% 9.49/5.23            ( ( ord_less_eq_set_int @ X3 @ Y2 )
% 9.49/5.23            & ~ ( ord_less_eq_set_int @ Y2 @ X3 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_630_less__le__not__le,axiom,
% 9.49/5.23      ( ord_less_rat
% 9.49/5.23      = ( ^ [X3: rat,Y2: rat] :
% 9.49/5.23            ( ( ord_less_eq_rat @ X3 @ Y2 )
% 9.49/5.23            & ~ ( ord_less_eq_rat @ Y2 @ X3 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_631_less__le__not__le,axiom,
% 9.49/5.23      ( ord_less_num
% 9.49/5.23      = ( ^ [X3: num,Y2: num] :
% 9.49/5.23            ( ( ord_less_eq_num @ X3 @ Y2 )
% 9.49/5.23            & ~ ( ord_less_eq_num @ Y2 @ X3 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_632_less__le__not__le,axiom,
% 9.49/5.23      ( ord_less_nat
% 9.49/5.23      = ( ^ [X3: nat,Y2: nat] :
% 9.49/5.23            ( ( ord_less_eq_nat @ X3 @ Y2 )
% 9.49/5.23            & ~ ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_633_less__le__not__le,axiom,
% 9.49/5.23      ( ord_less_int
% 9.49/5.23      = ( ^ [X3: int,Y2: int] :
% 9.49/5.23            ( ( ord_less_eq_int @ X3 @ Y2 )
% 9.49/5.23            & ~ ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_634_not__le__imp__less,axiom,
% 9.49/5.23      ! [Y: real,X: real] :
% 9.49/5.23        ( ~ ( ord_less_eq_real @ Y @ X )
% 9.49/5.23       => ( ord_less_real @ X @ Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_635_not__le__imp__less,axiom,
% 9.49/5.23      ! [Y: rat,X: rat] :
% 9.49/5.23        ( ~ ( ord_less_eq_rat @ Y @ X )
% 9.49/5.23       => ( ord_less_rat @ X @ Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_636_not__le__imp__less,axiom,
% 9.49/5.23      ! [Y: num,X: num] :
% 9.49/5.23        ( ~ ( ord_less_eq_num @ Y @ X )
% 9.49/5.23       => ( ord_less_num @ X @ Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_637_not__le__imp__less,axiom,
% 9.49/5.23      ! [Y: nat,X: nat] :
% 9.49/5.23        ( ~ ( ord_less_eq_nat @ Y @ X )
% 9.49/5.23       => ( ord_less_nat @ X @ Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_638_not__le__imp__less,axiom,
% 9.49/5.23      ! [Y: int,X: int] :
% 9.49/5.23        ( ~ ( ord_less_eq_int @ Y @ X )
% 9.49/5.23       => ( ord_less_int @ X @ Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_639_order_Oorder__iff__strict,axiom,
% 9.49/5.23      ( ord_less_eq_real
% 9.49/5.23      = ( ^ [A4: real,B4: real] :
% 9.49/5.23            ( ( ord_less_real @ A4 @ B4 )
% 9.49/5.23            | ( A4 = B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_640_order_Oorder__iff__strict,axiom,
% 9.49/5.23      ( ord_less_eq_set_int
% 9.49/5.23      = ( ^ [A4: set_int,B4: set_int] :
% 9.49/5.23            ( ( ord_less_set_int @ A4 @ B4 )
% 9.49/5.23            | ( A4 = B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_641_order_Oorder__iff__strict,axiom,
% 9.49/5.23      ( ord_less_eq_rat
% 9.49/5.23      = ( ^ [A4: rat,B4: rat] :
% 9.49/5.23            ( ( ord_less_rat @ A4 @ B4 )
% 9.49/5.23            | ( A4 = B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_642_order_Oorder__iff__strict,axiom,
% 9.49/5.23      ( ord_less_eq_num
% 9.49/5.23      = ( ^ [A4: num,B4: num] :
% 9.49/5.23            ( ( ord_less_num @ A4 @ B4 )
% 9.49/5.23            | ( A4 = B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_643_order_Oorder__iff__strict,axiom,
% 9.49/5.23      ( ord_less_eq_nat
% 9.49/5.23      = ( ^ [A4: nat,B4: nat] :
% 9.49/5.23            ( ( ord_less_nat @ A4 @ B4 )
% 9.49/5.23            | ( A4 = B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_644_order_Oorder__iff__strict,axiom,
% 9.49/5.23      ( ord_less_eq_int
% 9.49/5.23      = ( ^ [A4: int,B4: int] :
% 9.49/5.23            ( ( ord_less_int @ A4 @ B4 )
% 9.49/5.23            | ( A4 = B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_645_order_Ostrict__iff__order,axiom,
% 9.49/5.23      ( ord_less_real
% 9.49/5.23      = ( ^ [A4: real,B4: real] :
% 9.49/5.23            ( ( ord_less_eq_real @ A4 @ B4 )
% 9.49/5.23            & ( A4 != B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_646_order_Ostrict__iff__order,axiom,
% 9.49/5.23      ( ord_less_set_int
% 9.49/5.23      = ( ^ [A4: set_int,B4: set_int] :
% 9.49/5.23            ( ( ord_less_eq_set_int @ A4 @ B4 )
% 9.49/5.23            & ( A4 != B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_647_order_Ostrict__iff__order,axiom,
% 9.49/5.23      ( ord_less_rat
% 9.49/5.23      = ( ^ [A4: rat,B4: rat] :
% 9.49/5.23            ( ( ord_less_eq_rat @ A4 @ B4 )
% 9.49/5.23            & ( A4 != B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_648_order_Ostrict__iff__order,axiom,
% 9.49/5.23      ( ord_less_num
% 9.49/5.23      = ( ^ [A4: num,B4: num] :
% 9.49/5.23            ( ( ord_less_eq_num @ A4 @ B4 )
% 9.49/5.23            & ( A4 != B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_649_order_Ostrict__iff__order,axiom,
% 9.49/5.23      ( ord_less_nat
% 9.49/5.23      = ( ^ [A4: nat,B4: nat] :
% 9.49/5.23            ( ( ord_less_eq_nat @ A4 @ B4 )
% 9.49/5.23            & ( A4 != B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_650_order_Ostrict__iff__order,axiom,
% 9.49/5.23      ( ord_less_int
% 9.49/5.23      = ( ^ [A4: int,B4: int] :
% 9.49/5.23            ( ( ord_less_eq_int @ A4 @ B4 )
% 9.49/5.23            & ( A4 != B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_651_order_Ostrict__trans1,axiom,
% 9.49/5.23      ! [A: real,B: real,C: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.23       => ( ( ord_less_real @ B @ C )
% 9.49/5.23         => ( ord_less_real @ A @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_652_order_Ostrict__trans1,axiom,
% 9.49/5.23      ! [A: set_int,B: set_int,C: set_int] :
% 9.49/5.23        ( ( ord_less_eq_set_int @ A @ B )
% 9.49/5.23       => ( ( ord_less_set_int @ B @ C )
% 9.49/5.23         => ( ord_less_set_int @ A @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_653_order_Ostrict__trans1,axiom,
% 9.49/5.23      ! [A: rat,B: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.23       => ( ( ord_less_rat @ B @ C )
% 9.49/5.23         => ( ord_less_rat @ A @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_654_order_Ostrict__trans1,axiom,
% 9.49/5.23      ! [A: num,B: num,C: num] :
% 9.49/5.23        ( ( ord_less_eq_num @ A @ B )
% 9.49/5.23       => ( ( ord_less_num @ B @ C )
% 9.49/5.23         => ( ord_less_num @ A @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_655_order_Ostrict__trans1,axiom,
% 9.49/5.23      ! [A: nat,B: nat,C: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.23       => ( ( ord_less_nat @ B @ C )
% 9.49/5.23         => ( ord_less_nat @ A @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_656_order_Ostrict__trans1,axiom,
% 9.49/5.23      ! [A: int,B: int,C: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.23       => ( ( ord_less_int @ B @ C )
% 9.49/5.23         => ( ord_less_int @ A @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_657_order_Ostrict__trans2,axiom,
% 9.49/5.23      ! [A: real,B: real,C: real] :
% 9.49/5.23        ( ( ord_less_real @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_real @ B @ C )
% 9.49/5.23         => ( ord_less_real @ A @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_658_order_Ostrict__trans2,axiom,
% 9.49/5.23      ! [A: set_int,B: set_int,C: set_int] :
% 9.49/5.23        ( ( ord_less_set_int @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_set_int @ B @ C )
% 9.49/5.23         => ( ord_less_set_int @ A @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_659_order_Ostrict__trans2,axiom,
% 9.49/5.23      ! [A: rat,B: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_rat @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_rat @ B @ C )
% 9.49/5.23         => ( ord_less_rat @ A @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_660_order_Ostrict__trans2,axiom,
% 9.49/5.23      ! [A: num,B: num,C: num] :
% 9.49/5.23        ( ( ord_less_num @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_num @ B @ C )
% 9.49/5.23         => ( ord_less_num @ A @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_661_order_Ostrict__trans2,axiom,
% 9.49/5.23      ! [A: nat,B: nat,C: nat] :
% 9.49/5.23        ( ( ord_less_nat @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_nat @ B @ C )
% 9.49/5.23         => ( ord_less_nat @ A @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_662_order_Ostrict__trans2,axiom,
% 9.49/5.23      ! [A: int,B: int,C: int] :
% 9.49/5.23        ( ( ord_less_int @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_int @ B @ C )
% 9.49/5.23         => ( ord_less_int @ A @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_663_order_Ostrict__iff__not,axiom,
% 9.49/5.23      ( ord_less_real
% 9.49/5.23      = ( ^ [A4: real,B4: real] :
% 9.49/5.23            ( ( ord_less_eq_real @ A4 @ B4 )
% 9.49/5.23            & ~ ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_664_order_Ostrict__iff__not,axiom,
% 9.49/5.23      ( ord_less_set_int
% 9.49/5.23      = ( ^ [A4: set_int,B4: set_int] :
% 9.49/5.23            ( ( ord_less_eq_set_int @ A4 @ B4 )
% 9.49/5.23            & ~ ( ord_less_eq_set_int @ B4 @ A4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_665_order_Ostrict__iff__not,axiom,
% 9.49/5.23      ( ord_less_rat
% 9.49/5.23      = ( ^ [A4: rat,B4: rat] :
% 9.49/5.23            ( ( ord_less_eq_rat @ A4 @ B4 )
% 9.49/5.23            & ~ ( ord_less_eq_rat @ B4 @ A4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_666_order_Ostrict__iff__not,axiom,
% 9.49/5.23      ( ord_less_num
% 9.49/5.23      = ( ^ [A4: num,B4: num] :
% 9.49/5.23            ( ( ord_less_eq_num @ A4 @ B4 )
% 9.49/5.23            & ~ ( ord_less_eq_num @ B4 @ A4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_667_order_Ostrict__iff__not,axiom,
% 9.49/5.23      ( ord_less_nat
% 9.49/5.23      = ( ^ [A4: nat,B4: nat] :
% 9.49/5.23            ( ( ord_less_eq_nat @ A4 @ B4 )
% 9.49/5.23            & ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_668_order_Ostrict__iff__not,axiom,
% 9.49/5.23      ( ord_less_int
% 9.49/5.23      = ( ^ [A4: int,B4: int] :
% 9.49/5.23            ( ( ord_less_eq_int @ A4 @ B4 )
% 9.49/5.23            & ~ ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_669_dense__ge__bounded,axiom,
% 9.49/5.23      ! [Z2: real,X: real,Y: real] :
% 9.49/5.23        ( ( ord_less_real @ Z2 @ X )
% 9.49/5.23       => ( ! [W: real] :
% 9.49/5.23              ( ( ord_less_real @ Z2 @ W )
% 9.49/5.23             => ( ( ord_less_real @ W @ X )
% 9.49/5.23               => ( ord_less_eq_real @ Y @ W ) ) )
% 9.49/5.23         => ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_670_dense__ge__bounded,axiom,
% 9.49/5.23      ! [Z2: rat,X: rat,Y: rat] :
% 9.49/5.23        ( ( ord_less_rat @ Z2 @ X )
% 9.49/5.23       => ( ! [W: rat] :
% 9.49/5.23              ( ( ord_less_rat @ Z2 @ W )
% 9.49/5.23             => ( ( ord_less_rat @ W @ X )
% 9.49/5.23               => ( ord_less_eq_rat @ Y @ W ) ) )
% 9.49/5.23         => ( ord_less_eq_rat @ Y @ Z2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_671_dense__le__bounded,axiom,
% 9.49/5.23      ! [X: real,Y: real,Z2: real] :
% 9.49/5.23        ( ( ord_less_real @ X @ Y )
% 9.49/5.23       => ( ! [W: real] :
% 9.49/5.23              ( ( ord_less_real @ X @ W )
% 9.49/5.23             => ( ( ord_less_real @ W @ Y )
% 9.49/5.23               => ( ord_less_eq_real @ W @ Z2 ) ) )
% 9.49/5.23         => ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_672_dense__le__bounded,axiom,
% 9.49/5.23      ! [X: rat,Y: rat,Z2: rat] :
% 9.49/5.23        ( ( ord_less_rat @ X @ Y )
% 9.49/5.23       => ( ! [W: rat] :
% 9.49/5.23              ( ( ord_less_rat @ X @ W )
% 9.49/5.23             => ( ( ord_less_rat @ W @ Y )
% 9.49/5.23               => ( ord_less_eq_rat @ W @ Z2 ) ) )
% 9.49/5.23         => ( ord_less_eq_rat @ Y @ Z2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_673_dual__order_Oorder__iff__strict,axiom,
% 9.49/5.23      ( ord_less_eq_real
% 9.49/5.23      = ( ^ [B4: real,A4: real] :
% 9.49/5.23            ( ( ord_less_real @ B4 @ A4 )
% 9.49/5.23            | ( A4 = B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_674_dual__order_Oorder__iff__strict,axiom,
% 9.49/5.23      ( ord_less_eq_set_int
% 9.49/5.23      = ( ^ [B4: set_int,A4: set_int] :
% 9.49/5.23            ( ( ord_less_set_int @ B4 @ A4 )
% 9.49/5.23            | ( A4 = B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_675_dual__order_Oorder__iff__strict,axiom,
% 9.49/5.23      ( ord_less_eq_rat
% 9.49/5.23      = ( ^ [B4: rat,A4: rat] :
% 9.49/5.23            ( ( ord_less_rat @ B4 @ A4 )
% 9.49/5.23            | ( A4 = B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_676_dual__order_Oorder__iff__strict,axiom,
% 9.49/5.23      ( ord_less_eq_num
% 9.49/5.23      = ( ^ [B4: num,A4: num] :
% 9.49/5.23            ( ( ord_less_num @ B4 @ A4 )
% 9.49/5.23            | ( A4 = B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_677_dual__order_Oorder__iff__strict,axiom,
% 9.49/5.23      ( ord_less_eq_nat
% 9.49/5.23      = ( ^ [B4: nat,A4: nat] :
% 9.49/5.23            ( ( ord_less_nat @ B4 @ A4 )
% 9.49/5.23            | ( A4 = B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_678_dual__order_Oorder__iff__strict,axiom,
% 9.49/5.23      ( ord_less_eq_int
% 9.49/5.23      = ( ^ [B4: int,A4: int] :
% 9.49/5.23            ( ( ord_less_int @ B4 @ A4 )
% 9.49/5.23            | ( A4 = B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_679_dual__order_Ostrict__iff__order,axiom,
% 9.49/5.23      ( ord_less_real
% 9.49/5.23      = ( ^ [B4: real,A4: real] :
% 9.49/5.23            ( ( ord_less_eq_real @ B4 @ A4 )
% 9.49/5.23            & ( A4 != B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_680_dual__order_Ostrict__iff__order,axiom,
% 9.49/5.23      ( ord_less_set_int
% 9.49/5.23      = ( ^ [B4: set_int,A4: set_int] :
% 9.49/5.23            ( ( ord_less_eq_set_int @ B4 @ A4 )
% 9.49/5.23            & ( A4 != B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_681_dual__order_Ostrict__iff__order,axiom,
% 9.49/5.23      ( ord_less_rat
% 9.49/5.23      = ( ^ [B4: rat,A4: rat] :
% 9.49/5.23            ( ( ord_less_eq_rat @ B4 @ A4 )
% 9.49/5.23            & ( A4 != B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_682_dual__order_Ostrict__iff__order,axiom,
% 9.49/5.23      ( ord_less_num
% 9.49/5.23      = ( ^ [B4: num,A4: num] :
% 9.49/5.23            ( ( ord_less_eq_num @ B4 @ A4 )
% 9.49/5.23            & ( A4 != B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_683_dual__order_Ostrict__iff__order,axiom,
% 9.49/5.23      ( ord_less_nat
% 9.49/5.23      = ( ^ [B4: nat,A4: nat] :
% 9.49/5.23            ( ( ord_less_eq_nat @ B4 @ A4 )
% 9.49/5.23            & ( A4 != B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_684_dual__order_Ostrict__iff__order,axiom,
% 9.49/5.23      ( ord_less_int
% 9.49/5.23      = ( ^ [B4: int,A4: int] :
% 9.49/5.23            ( ( ord_less_eq_int @ B4 @ A4 )
% 9.49/5.23            & ( A4 != B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_685_dual__order_Ostrict__trans1,axiom,
% 9.49/5.23      ! [B: real,A: real,C: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ B @ A )
% 9.49/5.23       => ( ( ord_less_real @ C @ B )
% 9.49/5.23         => ( ord_less_real @ C @ A ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_686_dual__order_Ostrict__trans1,axiom,
% 9.49/5.23      ! [B: set_int,A: set_int,C: set_int] :
% 9.49/5.23        ( ( ord_less_eq_set_int @ B @ A )
% 9.49/5.23       => ( ( ord_less_set_int @ C @ B )
% 9.49/5.23         => ( ord_less_set_int @ C @ A ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_687_dual__order_Ostrict__trans1,axiom,
% 9.49/5.23      ! [B: rat,A: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ B @ A )
% 9.49/5.23       => ( ( ord_less_rat @ C @ B )
% 9.49/5.23         => ( ord_less_rat @ C @ A ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_688_dual__order_Ostrict__trans1,axiom,
% 9.49/5.23      ! [B: num,A: num,C: num] :
% 9.49/5.23        ( ( ord_less_eq_num @ B @ A )
% 9.49/5.23       => ( ( ord_less_num @ C @ B )
% 9.49/5.23         => ( ord_less_num @ C @ A ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_689_dual__order_Ostrict__trans1,axiom,
% 9.49/5.23      ! [B: nat,A: nat,C: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ B @ A )
% 9.49/5.23       => ( ( ord_less_nat @ C @ B )
% 9.49/5.23         => ( ord_less_nat @ C @ A ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_690_dual__order_Ostrict__trans1,axiom,
% 9.49/5.23      ! [B: int,A: int,C: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ B @ A )
% 9.49/5.23       => ( ( ord_less_int @ C @ B )
% 9.49/5.23         => ( ord_less_int @ C @ A ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_691_dual__order_Ostrict__trans2,axiom,
% 9.49/5.23      ! [B: real,A: real,C: real] :
% 9.49/5.23        ( ( ord_less_real @ B @ A )
% 9.49/5.23       => ( ( ord_less_eq_real @ C @ B )
% 9.49/5.23         => ( ord_less_real @ C @ A ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_692_dual__order_Ostrict__trans2,axiom,
% 9.49/5.23      ! [B: set_int,A: set_int,C: set_int] :
% 9.49/5.23        ( ( ord_less_set_int @ B @ A )
% 9.49/5.23       => ( ( ord_less_eq_set_int @ C @ B )
% 9.49/5.23         => ( ord_less_set_int @ C @ A ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_693_dual__order_Ostrict__trans2,axiom,
% 9.49/5.23      ! [B: rat,A: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_rat @ B @ A )
% 9.49/5.23       => ( ( ord_less_eq_rat @ C @ B )
% 9.49/5.23         => ( ord_less_rat @ C @ A ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_694_dual__order_Ostrict__trans2,axiom,
% 9.49/5.23      ! [B: num,A: num,C: num] :
% 9.49/5.23        ( ( ord_less_num @ B @ A )
% 9.49/5.23       => ( ( ord_less_eq_num @ C @ B )
% 9.49/5.23         => ( ord_less_num @ C @ A ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_695_dual__order_Ostrict__trans2,axiom,
% 9.49/5.23      ! [B: nat,A: nat,C: nat] :
% 9.49/5.23        ( ( ord_less_nat @ B @ A )
% 9.49/5.23       => ( ( ord_less_eq_nat @ C @ B )
% 9.49/5.23         => ( ord_less_nat @ C @ A ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_696_dual__order_Ostrict__trans2,axiom,
% 9.49/5.23      ! [B: int,A: int,C: int] :
% 9.49/5.23        ( ( ord_less_int @ B @ A )
% 9.49/5.23       => ( ( ord_less_eq_int @ C @ B )
% 9.49/5.23         => ( ord_less_int @ C @ A ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_697_dual__order_Ostrict__iff__not,axiom,
% 9.49/5.23      ( ord_less_real
% 9.49/5.23      = ( ^ [B4: real,A4: real] :
% 9.49/5.23            ( ( ord_less_eq_real @ B4 @ A4 )
% 9.49/5.23            & ~ ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_698_dual__order_Ostrict__iff__not,axiom,
% 9.49/5.23      ( ord_less_set_int
% 9.49/5.23      = ( ^ [B4: set_int,A4: set_int] :
% 9.49/5.23            ( ( ord_less_eq_set_int @ B4 @ A4 )
% 9.49/5.23            & ~ ( ord_less_eq_set_int @ A4 @ B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_699_dual__order_Ostrict__iff__not,axiom,
% 9.49/5.23      ( ord_less_rat
% 9.49/5.23      = ( ^ [B4: rat,A4: rat] :
% 9.49/5.23            ( ( ord_less_eq_rat @ B4 @ A4 )
% 9.49/5.23            & ~ ( ord_less_eq_rat @ A4 @ B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_700_dual__order_Ostrict__iff__not,axiom,
% 9.49/5.23      ( ord_less_num
% 9.49/5.23      = ( ^ [B4: num,A4: num] :
% 9.49/5.23            ( ( ord_less_eq_num @ B4 @ A4 )
% 9.49/5.23            & ~ ( ord_less_eq_num @ A4 @ B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_701_dual__order_Ostrict__iff__not,axiom,
% 9.49/5.23      ( ord_less_nat
% 9.49/5.23      = ( ^ [B4: nat,A4: nat] :
% 9.49/5.23            ( ( ord_less_eq_nat @ B4 @ A4 )
% 9.49/5.23            & ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_702_dual__order_Ostrict__iff__not,axiom,
% 9.49/5.23      ( ord_less_int
% 9.49/5.23      = ( ^ [B4: int,A4: int] :
% 9.49/5.23            ( ( ord_less_eq_int @ B4 @ A4 )
% 9.49/5.23            & ~ ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_703_order_Ostrict__implies__order,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( ord_less_real @ A @ B )
% 9.49/5.23       => ( ord_less_eq_real @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_704_order_Ostrict__implies__order,axiom,
% 9.49/5.23      ! [A: set_int,B: set_int] :
% 9.49/5.23        ( ( ord_less_set_int @ A @ B )
% 9.49/5.23       => ( ord_less_eq_set_int @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_705_order_Ostrict__implies__order,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_rat @ A @ B )
% 9.49/5.23       => ( ord_less_eq_rat @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_706_order_Ostrict__implies__order,axiom,
% 9.49/5.23      ! [A: num,B: num] :
% 9.49/5.23        ( ( ord_less_num @ A @ B )
% 9.49/5.23       => ( ord_less_eq_num @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_707_order_Ostrict__implies__order,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_nat @ A @ B )
% 9.49/5.23       => ( ord_less_eq_nat @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_708_order_Ostrict__implies__order,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( ord_less_int @ A @ B )
% 9.49/5.23       => ( ord_less_eq_int @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_709_dual__order_Ostrict__implies__order,axiom,
% 9.49/5.23      ! [B: real,A: real] :
% 9.49/5.23        ( ( ord_less_real @ B @ A )
% 9.49/5.23       => ( ord_less_eq_real @ B @ A ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_710_dual__order_Ostrict__implies__order,axiom,
% 9.49/5.23      ! [B: set_int,A: set_int] :
% 9.49/5.23        ( ( ord_less_set_int @ B @ A )
% 9.49/5.23       => ( ord_less_eq_set_int @ B @ A ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_711_dual__order_Ostrict__implies__order,axiom,
% 9.49/5.23      ! [B: rat,A: rat] :
% 9.49/5.23        ( ( ord_less_rat @ B @ A )
% 9.49/5.23       => ( ord_less_eq_rat @ B @ A ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_712_dual__order_Ostrict__implies__order,axiom,
% 9.49/5.23      ! [B: num,A: num] :
% 9.49/5.23        ( ( ord_less_num @ B @ A )
% 9.49/5.23       => ( ord_less_eq_num @ B @ A ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_713_dual__order_Ostrict__implies__order,axiom,
% 9.49/5.23      ! [B: nat,A: nat] :
% 9.49/5.23        ( ( ord_less_nat @ B @ A )
% 9.49/5.23       => ( ord_less_eq_nat @ B @ A ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_714_dual__order_Ostrict__implies__order,axiom,
% 9.49/5.23      ! [B: int,A: int] :
% 9.49/5.23        ( ( ord_less_int @ B @ A )
% 9.49/5.23       => ( ord_less_eq_int @ B @ A ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_715_order__le__less,axiom,
% 9.49/5.23      ( ord_less_eq_real
% 9.49/5.23      = ( ^ [X3: real,Y2: real] :
% 9.49/5.23            ( ( ord_less_real @ X3 @ Y2 )
% 9.49/5.23            | ( X3 = Y2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_716_order__le__less,axiom,
% 9.49/5.23      ( ord_less_eq_set_int
% 9.49/5.23      = ( ^ [X3: set_int,Y2: set_int] :
% 9.49/5.23            ( ( ord_less_set_int @ X3 @ Y2 )
% 9.49/5.23            | ( X3 = Y2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_717_order__le__less,axiom,
% 9.49/5.23      ( ord_less_eq_rat
% 9.49/5.23      = ( ^ [X3: rat,Y2: rat] :
% 9.49/5.23            ( ( ord_less_rat @ X3 @ Y2 )
% 9.49/5.23            | ( X3 = Y2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_718_order__le__less,axiom,
% 9.49/5.23      ( ord_less_eq_num
% 9.49/5.23      = ( ^ [X3: num,Y2: num] :
% 9.49/5.23            ( ( ord_less_num @ X3 @ Y2 )
% 9.49/5.23            | ( X3 = Y2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_719_order__le__less,axiom,
% 9.49/5.23      ( ord_less_eq_nat
% 9.49/5.23      = ( ^ [X3: nat,Y2: nat] :
% 9.49/5.23            ( ( ord_less_nat @ X3 @ Y2 )
% 9.49/5.23            | ( X3 = Y2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_720_order__le__less,axiom,
% 9.49/5.23      ( ord_less_eq_int
% 9.49/5.23      = ( ^ [X3: int,Y2: int] :
% 9.49/5.23            ( ( ord_less_int @ X3 @ Y2 )
% 9.49/5.23            | ( X3 = Y2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_721_order__less__le,axiom,
% 9.49/5.23      ( ord_less_real
% 9.49/5.23      = ( ^ [X3: real,Y2: real] :
% 9.49/5.23            ( ( ord_less_eq_real @ X3 @ Y2 )
% 9.49/5.23            & ( X3 != Y2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_722_order__less__le,axiom,
% 9.49/5.23      ( ord_less_set_int
% 9.49/5.23      = ( ^ [X3: set_int,Y2: set_int] :
% 9.49/5.23            ( ( ord_less_eq_set_int @ X3 @ Y2 )
% 9.49/5.23            & ( X3 != Y2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_723_order__less__le,axiom,
% 9.49/5.23      ( ord_less_rat
% 9.49/5.23      = ( ^ [X3: rat,Y2: rat] :
% 9.49/5.23            ( ( ord_less_eq_rat @ X3 @ Y2 )
% 9.49/5.23            & ( X3 != Y2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_724_order__less__le,axiom,
% 9.49/5.23      ( ord_less_num
% 9.49/5.23      = ( ^ [X3: num,Y2: num] :
% 9.49/5.23            ( ( ord_less_eq_num @ X3 @ Y2 )
% 9.49/5.23            & ( X3 != Y2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_725_order__less__le,axiom,
% 9.49/5.23      ( ord_less_nat
% 9.49/5.23      = ( ^ [X3: nat,Y2: nat] :
% 9.49/5.23            ( ( ord_less_eq_nat @ X3 @ Y2 )
% 9.49/5.23            & ( X3 != Y2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_726_order__less__le,axiom,
% 9.49/5.23      ( ord_less_int
% 9.49/5.23      = ( ^ [X3: int,Y2: int] :
% 9.49/5.23            ( ( ord_less_eq_int @ X3 @ Y2 )
% 9.49/5.23            & ( X3 != Y2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_727_linorder__not__le,axiom,
% 9.49/5.23      ! [X: real,Y: real] :
% 9.49/5.23        ( ( ~ ( ord_less_eq_real @ X @ Y ) )
% 9.49/5.23        = ( ord_less_real @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_728_linorder__not__le,axiom,
% 9.49/5.23      ! [X: rat,Y: rat] :
% 9.49/5.23        ( ( ~ ( ord_less_eq_rat @ X @ Y ) )
% 9.49/5.23        = ( ord_less_rat @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_729_linorder__not__le,axiom,
% 9.49/5.23      ! [X: num,Y: num] :
% 9.49/5.23        ( ( ~ ( ord_less_eq_num @ X @ Y ) )
% 9.49/5.23        = ( ord_less_num @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_730_linorder__not__le,axiom,
% 9.49/5.23      ! [X: nat,Y: nat] :
% 9.49/5.23        ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
% 9.49/5.23        = ( ord_less_nat @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_731_linorder__not__le,axiom,
% 9.49/5.23      ! [X: int,Y: int] :
% 9.49/5.23        ( ( ~ ( ord_less_eq_int @ X @ Y ) )
% 9.49/5.23        = ( ord_less_int @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_732_linorder__not__less,axiom,
% 9.49/5.23      ! [X: real,Y: real] :
% 9.49/5.23        ( ( ~ ( ord_less_real @ X @ Y ) )
% 9.49/5.23        = ( ord_less_eq_real @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_733_linorder__not__less,axiom,
% 9.49/5.23      ! [X: rat,Y: rat] :
% 9.49/5.23        ( ( ~ ( ord_less_rat @ X @ Y ) )
% 9.49/5.23        = ( ord_less_eq_rat @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_734_linorder__not__less,axiom,
% 9.49/5.23      ! [X: num,Y: num] :
% 9.49/5.23        ( ( ~ ( ord_less_num @ X @ Y ) )
% 9.49/5.23        = ( ord_less_eq_num @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_735_linorder__not__less,axiom,
% 9.49/5.23      ! [X: nat,Y: nat] :
% 9.49/5.23        ( ( ~ ( ord_less_nat @ X @ Y ) )
% 9.49/5.23        = ( ord_less_eq_nat @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_736_linorder__not__less,axiom,
% 9.49/5.23      ! [X: int,Y: int] :
% 9.49/5.23        ( ( ~ ( ord_less_int @ X @ Y ) )
% 9.49/5.23        = ( ord_less_eq_int @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_737_order__less__imp__le,axiom,
% 9.49/5.23      ! [X: real,Y: real] :
% 9.49/5.23        ( ( ord_less_real @ X @ Y )
% 9.49/5.23       => ( ord_less_eq_real @ X @ Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_738_order__less__imp__le,axiom,
% 9.49/5.23      ! [X: set_int,Y: set_int] :
% 9.49/5.23        ( ( ord_less_set_int @ X @ Y )
% 9.49/5.23       => ( ord_less_eq_set_int @ X @ Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_739_order__less__imp__le,axiom,
% 9.49/5.23      ! [X: rat,Y: rat] :
% 9.49/5.23        ( ( ord_less_rat @ X @ Y )
% 9.49/5.23       => ( ord_less_eq_rat @ X @ Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_740_order__less__imp__le,axiom,
% 9.49/5.23      ! [X: num,Y: num] :
% 9.49/5.23        ( ( ord_less_num @ X @ Y )
% 9.49/5.23       => ( ord_less_eq_num @ X @ Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_741_order__less__imp__le,axiom,
% 9.49/5.23      ! [X: nat,Y: nat] :
% 9.49/5.23        ( ( ord_less_nat @ X @ Y )
% 9.49/5.23       => ( ord_less_eq_nat @ X @ Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_742_order__less__imp__le,axiom,
% 9.49/5.23      ! [X: int,Y: int] :
% 9.49/5.23        ( ( ord_less_int @ X @ Y )
% 9.49/5.23       => ( ord_less_eq_int @ X @ Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_743_order__le__neq__trans,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.23       => ( ( A != B )
% 9.49/5.23         => ( ord_less_real @ A @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_744_order__le__neq__trans,axiom,
% 9.49/5.23      ! [A: set_int,B: set_int] :
% 9.49/5.23        ( ( ord_less_eq_set_int @ A @ B )
% 9.49/5.23       => ( ( A != B )
% 9.49/5.23         => ( ord_less_set_int @ A @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_745_order__le__neq__trans,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.23       => ( ( A != B )
% 9.49/5.23         => ( ord_less_rat @ A @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_746_order__le__neq__trans,axiom,
% 9.49/5.23      ! [A: num,B: num] :
% 9.49/5.23        ( ( ord_less_eq_num @ A @ B )
% 9.49/5.23       => ( ( A != B )
% 9.49/5.23         => ( ord_less_num @ A @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_747_order__le__neq__trans,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.23       => ( ( A != B )
% 9.49/5.23         => ( ord_less_nat @ A @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_748_order__le__neq__trans,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.23       => ( ( A != B )
% 9.49/5.23         => ( ord_less_int @ A @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_749_order__neq__le__trans,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( A != B )
% 9.49/5.23       => ( ( ord_less_eq_real @ A @ B )
% 9.49/5.23         => ( ord_less_real @ A @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_750_order__neq__le__trans,axiom,
% 9.49/5.23      ! [A: set_int,B: set_int] :
% 9.49/5.23        ( ( A != B )
% 9.49/5.23       => ( ( ord_less_eq_set_int @ A @ B )
% 9.49/5.23         => ( ord_less_set_int @ A @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_751_order__neq__le__trans,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( A != B )
% 9.49/5.23       => ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.23         => ( ord_less_rat @ A @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_752_order__neq__le__trans,axiom,
% 9.49/5.23      ! [A: num,B: num] :
% 9.49/5.23        ( ( A != B )
% 9.49/5.23       => ( ( ord_less_eq_num @ A @ B )
% 9.49/5.23         => ( ord_less_num @ A @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_753_order__neq__le__trans,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( A != B )
% 9.49/5.23       => ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.23         => ( ord_less_nat @ A @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_754_order__neq__le__trans,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( A != B )
% 9.49/5.23       => ( ( ord_less_eq_int @ A @ B )
% 9.49/5.23         => ( ord_less_int @ A @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_755_order__le__less__trans,axiom,
% 9.49/5.23      ! [X: real,Y: real,Z2: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ X @ Y )
% 9.49/5.23       => ( ( ord_less_real @ Y @ Z2 )
% 9.49/5.23         => ( ord_less_real @ X @ Z2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_756_order__le__less__trans,axiom,
% 9.49/5.23      ! [X: set_int,Y: set_int,Z2: set_int] :
% 9.49/5.23        ( ( ord_less_eq_set_int @ X @ Y )
% 9.49/5.23       => ( ( ord_less_set_int @ Y @ Z2 )
% 9.49/5.23         => ( ord_less_set_int @ X @ Z2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_757_order__le__less__trans,axiom,
% 9.49/5.23      ! [X: rat,Y: rat,Z2: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ X @ Y )
% 9.49/5.23       => ( ( ord_less_rat @ Y @ Z2 )
% 9.49/5.23         => ( ord_less_rat @ X @ Z2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_758_order__le__less__trans,axiom,
% 9.49/5.23      ! [X: num,Y: num,Z2: num] :
% 9.49/5.23        ( ( ord_less_eq_num @ X @ Y )
% 9.49/5.23       => ( ( ord_less_num @ Y @ Z2 )
% 9.49/5.23         => ( ord_less_num @ X @ Z2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_759_order__le__less__trans,axiom,
% 9.49/5.23      ! [X: nat,Y: nat,Z2: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ X @ Y )
% 9.49/5.23       => ( ( ord_less_nat @ Y @ Z2 )
% 9.49/5.23         => ( ord_less_nat @ X @ Z2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_760_order__le__less__trans,axiom,
% 9.49/5.23      ! [X: int,Y: int,Z2: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ X @ Y )
% 9.49/5.23       => ( ( ord_less_int @ Y @ Z2 )
% 9.49/5.23         => ( ord_less_int @ X @ Z2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_761_order__less__le__trans,axiom,
% 9.49/5.23      ! [X: real,Y: real,Z2: real] :
% 9.49/5.23        ( ( ord_less_real @ X @ Y )
% 9.49/5.23       => ( ( ord_less_eq_real @ Y @ Z2 )
% 9.49/5.23         => ( ord_less_real @ X @ Z2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_762_order__less__le__trans,axiom,
% 9.49/5.23      ! [X: set_int,Y: set_int,Z2: set_int] :
% 9.49/5.23        ( ( ord_less_set_int @ X @ Y )
% 9.49/5.23       => ( ( ord_less_eq_set_int @ Y @ Z2 )
% 9.49/5.23         => ( ord_less_set_int @ X @ Z2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_763_order__less__le__trans,axiom,
% 9.49/5.23      ! [X: rat,Y: rat,Z2: rat] :
% 9.49/5.23        ( ( ord_less_rat @ X @ Y )
% 9.49/5.23       => ( ( ord_less_eq_rat @ Y @ Z2 )
% 9.49/5.23         => ( ord_less_rat @ X @ Z2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_764_order__less__le__trans,axiom,
% 9.49/5.23      ! [X: num,Y: num,Z2: num] :
% 9.49/5.23        ( ( ord_less_num @ X @ Y )
% 9.49/5.23       => ( ( ord_less_eq_num @ Y @ Z2 )
% 9.49/5.23         => ( ord_less_num @ X @ Z2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_765_order__less__le__trans,axiom,
% 9.49/5.23      ! [X: nat,Y: nat,Z2: nat] :
% 9.49/5.23        ( ( ord_less_nat @ X @ Y )
% 9.49/5.23       => ( ( ord_less_eq_nat @ Y @ Z2 )
% 9.49/5.23         => ( ord_less_nat @ X @ Z2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_766_order__less__le__trans,axiom,
% 9.49/5.23      ! [X: int,Y: int,Z2: int] :
% 9.49/5.23        ( ( ord_less_int @ X @ Y )
% 9.49/5.23       => ( ( ord_less_eq_int @ Y @ Z2 )
% 9.49/5.23         => ( ord_less_int @ X @ Z2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_767_order__le__less__subst1,axiom,
% 9.49/5.23      ! [A: real,F: real > real,B: real,C: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_real @ B @ C )
% 9.49/5.23         => ( ! [X4: real,Y3: real] :
% 9.49/5.23                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_768_order__le__less__subst1,axiom,
% 9.49/5.23      ! [A: real,F: rat > real,B: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_rat @ B @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_769_order__le__less__subst1,axiom,
% 9.49/5.23      ! [A: real,F: num > real,B: num,C: num] :
% 9.49/5.23        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_num @ B @ C )
% 9.49/5.23         => ( ! [X4: num,Y3: num] :
% 9.49/5.23                ( ( ord_less_num @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_770_order__le__less__subst1,axiom,
% 9.49/5.23      ! [A: real,F: nat > real,B: nat,C: nat] :
% 9.49/5.23        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_nat @ B @ C )
% 9.49/5.23         => ( ! [X4: nat,Y3: nat] :
% 9.49/5.23                ( ( ord_less_nat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_771_order__le__less__subst1,axiom,
% 9.49/5.23      ! [A: real,F: int > real,B: int,C: int] :
% 9.49/5.23        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_int @ B @ C )
% 9.49/5.23         => ( ! [X4: int,Y3: int] :
% 9.49/5.23                ( ( ord_less_int @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_772_order__le__less__subst1,axiom,
% 9.49/5.23      ! [A: rat,F: real > rat,B: real,C: real] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_real @ B @ C )
% 9.49/5.23         => ( ! [X4: real,Y3: real] :
% 9.49/5.23                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_773_order__le__less__subst1,axiom,
% 9.49/5.23      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_rat @ B @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_774_order__le__less__subst1,axiom,
% 9.49/5.23      ! [A: rat,F: num > rat,B: num,C: num] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_num @ B @ C )
% 9.49/5.23         => ( ! [X4: num,Y3: num] :
% 9.49/5.23                ( ( ord_less_num @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_775_order__le__less__subst1,axiom,
% 9.49/5.23      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_nat @ B @ C )
% 9.49/5.23         => ( ! [X4: nat,Y3: nat] :
% 9.49/5.23                ( ( ord_less_nat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_776_order__le__less__subst1,axiom,
% 9.49/5.23      ! [A: rat,F: int > rat,B: int,C: int] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_int @ B @ C )
% 9.49/5.23         => ( ! [X4: int,Y3: int] :
% 9.49/5.23                ( ( ord_less_int @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_777_order__le__less__subst2,axiom,
% 9.49/5.23      ! [A: rat,B: rat,F: rat > real,C: real] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.23       => ( ( ord_less_real @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_778_order__le__less__subst2,axiom,
% 9.49/5.23      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.23       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_779_order__le__less__subst2,axiom,
% 9.49/5.23      ! [A: rat,B: rat,F: rat > num,C: num] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.23       => ( ( ord_less_num @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_780_order__le__less__subst2,axiom,
% 9.49/5.23      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.23       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_781_order__le__less__subst2,axiom,
% 9.49/5.23      ! [A: rat,B: rat,F: rat > int,C: int] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.23       => ( ( ord_less_int @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_782_order__le__less__subst2,axiom,
% 9.49/5.23      ! [A: num,B: num,F: num > real,C: real] :
% 9.49/5.23        ( ( ord_less_eq_num @ A @ B )
% 9.49/5.23       => ( ( ord_less_real @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: num,Y3: num] :
% 9.49/5.23                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_783_order__le__less__subst2,axiom,
% 9.49/5.23      ! [A: num,B: num,F: num > rat,C: rat] :
% 9.49/5.23        ( ( ord_less_eq_num @ A @ B )
% 9.49/5.23       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: num,Y3: num] :
% 9.49/5.23                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_784_order__le__less__subst2,axiom,
% 9.49/5.23      ! [A: num,B: num,F: num > num,C: num] :
% 9.49/5.23        ( ( ord_less_eq_num @ A @ B )
% 9.49/5.23       => ( ( ord_less_num @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: num,Y3: num] :
% 9.49/5.23                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_785_order__le__less__subst2,axiom,
% 9.49/5.23      ! [A: num,B: num,F: num > nat,C: nat] :
% 9.49/5.23        ( ( ord_less_eq_num @ A @ B )
% 9.49/5.23       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: num,Y3: num] :
% 9.49/5.23                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_786_order__le__less__subst2,axiom,
% 9.49/5.23      ! [A: num,B: num,F: num > int,C: int] :
% 9.49/5.23        ( ( ord_less_eq_num @ A @ B )
% 9.49/5.23       => ( ( ord_less_int @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: num,Y3: num] :
% 9.49/5.23                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_787_order__less__le__subst1,axiom,
% 9.49/5.23      ! [A: real,F: rat > real,B: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_eq_rat @ B @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_788_order__less__le__subst1,axiom,
% 9.49/5.23      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_eq_rat @ B @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_789_order__less__le__subst1,axiom,
% 9.49/5.23      ! [A: num,F: rat > num,B: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_num @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_eq_rat @ B @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_790_order__less__le__subst1,axiom,
% 9.49/5.23      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_nat @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_eq_rat @ B @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_791_order__less__le__subst1,axiom,
% 9.49/5.23      ! [A: int,F: rat > int,B: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_int @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_eq_rat @ B @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_eq_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_792_order__less__le__subst1,axiom,
% 9.49/5.23      ! [A: real,F: num > real,B: num,C: num] :
% 9.49/5.23        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_eq_num @ B @ C )
% 9.49/5.23         => ( ! [X4: num,Y3: num] :
% 9.49/5.23                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_793_order__less__le__subst1,axiom,
% 9.49/5.23      ! [A: rat,F: num > rat,B: num,C: num] :
% 9.49/5.23        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_eq_num @ B @ C )
% 9.49/5.23         => ( ! [X4: num,Y3: num] :
% 9.49/5.23                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_794_order__less__le__subst1,axiom,
% 9.49/5.23      ! [A: num,F: num > num,B: num,C: num] :
% 9.49/5.23        ( ( ord_less_num @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_eq_num @ B @ C )
% 9.49/5.23         => ( ! [X4: num,Y3: num] :
% 9.49/5.23                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_795_order__less__le__subst1,axiom,
% 9.49/5.23      ! [A: nat,F: num > nat,B: num,C: num] :
% 9.49/5.23        ( ( ord_less_nat @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_eq_num @ B @ C )
% 9.49/5.23         => ( ! [X4: num,Y3: num] :
% 9.49/5.23                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_796_order__less__le__subst1,axiom,
% 9.49/5.23      ! [A: int,F: num > int,B: num,C: num] :
% 9.49/5.23        ( ( ord_less_int @ A @ ( F @ B ) )
% 9.49/5.23       => ( ( ord_less_eq_num @ B @ C )
% 9.49/5.23         => ( ! [X4: num,Y3: num] :
% 9.49/5.23                ( ( ord_less_eq_num @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_797_order__less__le__subst2,axiom,
% 9.49/5.23      ! [A: real,B: real,F: real > real,C: real] :
% 9.49/5.23        ( ( ord_less_real @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: real,Y3: real] :
% 9.49/5.23                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_798_order__less__le__subst2,axiom,
% 9.49/5.23      ! [A: rat,B: rat,F: rat > real,C: real] :
% 9.49/5.23        ( ( ord_less_rat @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_799_order__less__le__subst2,axiom,
% 9.49/5.23      ! [A: num,B: num,F: num > real,C: real] :
% 9.49/5.23        ( ( ord_less_num @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: num,Y3: num] :
% 9.49/5.23                ( ( ord_less_num @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_800_order__less__le__subst2,axiom,
% 9.49/5.23      ! [A: nat,B: nat,F: nat > real,C: real] :
% 9.49/5.23        ( ( ord_less_nat @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: nat,Y3: nat] :
% 9.49/5.23                ( ( ord_less_nat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_801_order__less__le__subst2,axiom,
% 9.49/5.23      ! [A: int,B: int,F: int > real,C: real] :
% 9.49/5.23        ( ( ord_less_int @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: int,Y3: int] :
% 9.49/5.23                ( ( ord_less_int @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_802_order__less__le__subst2,axiom,
% 9.49/5.23      ! [A: real,B: real,F: real > rat,C: rat] :
% 9.49/5.23        ( ( ord_less_real @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: real,Y3: real] :
% 9.49/5.23                ( ( ord_less_real @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_803_order__less__le__subst2,axiom,
% 9.49/5.23      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 9.49/5.23        ( ( ord_less_rat @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: rat,Y3: rat] :
% 9.49/5.23                ( ( ord_less_rat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_804_order__less__le__subst2,axiom,
% 9.49/5.23      ! [A: num,B: num,F: num > rat,C: rat] :
% 9.49/5.23        ( ( ord_less_num @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: num,Y3: num] :
% 9.49/5.23                ( ( ord_less_num @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_805_order__less__le__subst2,axiom,
% 9.49/5.23      ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 9.49/5.23        ( ( ord_less_nat @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: nat,Y3: nat] :
% 9.49/5.23                ( ( ord_less_nat @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_806_order__less__le__subst2,axiom,
% 9.49/5.23      ! [A: int,B: int,F: int > rat,C: rat] :
% 9.49/5.23        ( ( ord_less_int @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.49/5.23         => ( ! [X4: int,Y3: int] :
% 9.49/5.23                ( ( ord_less_int @ X4 @ Y3 )
% 9.49/5.23               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
% 9.49/5.23           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_807_linorder__le__less__linear,axiom,
% 9.49/5.23      ! [X: real,Y: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ X @ Y )
% 9.49/5.23        | ( ord_less_real @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_808_linorder__le__less__linear,axiom,
% 9.49/5.23      ! [X: rat,Y: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ X @ Y )
% 9.49/5.23        | ( ord_less_rat @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_809_linorder__le__less__linear,axiom,
% 9.49/5.23      ! [X: num,Y: num] :
% 9.49/5.23        ( ( ord_less_eq_num @ X @ Y )
% 9.49/5.23        | ( ord_less_num @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_810_linorder__le__less__linear,axiom,
% 9.49/5.23      ! [X: nat,Y: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ X @ Y )
% 9.49/5.23        | ( ord_less_nat @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_811_linorder__le__less__linear,axiom,
% 9.49/5.23      ! [X: int,Y: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ X @ Y )
% 9.49/5.23        | ( ord_less_int @ Y @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_812_order__le__imp__less__or__eq,axiom,
% 9.49/5.23      ! [X: real,Y: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ X @ Y )
% 9.49/5.23       => ( ( ord_less_real @ X @ Y )
% 9.49/5.23          | ( X = Y ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_813_order__le__imp__less__or__eq,axiom,
% 9.49/5.23      ! [X: set_int,Y: set_int] :
% 9.49/5.23        ( ( ord_less_eq_set_int @ X @ Y )
% 9.49/5.23       => ( ( ord_less_set_int @ X @ Y )
% 9.49/5.23          | ( X = Y ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_814_order__le__imp__less__or__eq,axiom,
% 9.49/5.23      ! [X: rat,Y: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ X @ Y )
% 9.49/5.23       => ( ( ord_less_rat @ X @ Y )
% 9.49/5.23          | ( X = Y ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_815_order__le__imp__less__or__eq,axiom,
% 9.49/5.23      ! [X: num,Y: num] :
% 9.49/5.23        ( ( ord_less_eq_num @ X @ Y )
% 9.49/5.23       => ( ( ord_less_num @ X @ Y )
% 9.49/5.23          | ( X = Y ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_816_order__le__imp__less__or__eq,axiom,
% 9.49/5.23      ! [X: nat,Y: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ X @ Y )
% 9.49/5.23       => ( ( ord_less_nat @ X @ Y )
% 9.49/5.23          | ( X = Y ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_817_order__le__imp__less__or__eq,axiom,
% 9.49/5.23      ! [X: int,Y: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ X @ Y )
% 9.49/5.23       => ( ( ord_less_int @ X @ Y )
% 9.49/5.23          | ( X = Y ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_818_greater__shift,axiom,
% 9.49/5.23      ( ord_less_nat
% 9.49/5.23      = ( ^ [Y2: nat,X3: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X3 ) @ ( some_nat @ Y2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_819_less__shift,axiom,
% 9.49/5.23      ( ord_less_nat
% 9.49/5.23      = ( ^ [X3: nat,Y2: nat] : ( vEBT_VEBT_less @ ( some_nat @ X3 ) @ ( some_nat @ Y2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_820_mint__sound,axiom,
% 9.49/5.23      ! [T: vEBT_VEBT,N: nat,X: nat] :
% 9.49/5.23        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.23       => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
% 9.49/5.23         => ( ( vEBT_vebt_mint @ T )
% 9.49/5.23            = ( some_nat @ X ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_821_mint__corr,axiom,
% 9.49/5.23      ! [T: vEBT_VEBT,N: nat,X: nat] :
% 9.49/5.23        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.23       => ( ( ( vEBT_vebt_mint @ T )
% 9.49/5.23            = ( some_nat @ X ) )
% 9.49/5.23         => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_822_maxt__corr__help__empty,axiom,
% 9.49/5.23      ! [T: vEBT_VEBT,N: nat] :
% 9.49/5.23        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.23       => ( ( ( vEBT_vebt_maxt @ T )
% 9.49/5.23            = none_nat )
% 9.49/5.23         => ( ( vEBT_VEBT_set_vebt @ T )
% 9.49/5.23            = bot_bot_set_nat ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_823_mint__corr__help,axiom,
% 9.49/5.23      ! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
% 9.49/5.23        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.23       => ( ( ( vEBT_vebt_mint @ T )
% 9.49/5.23            = ( some_nat @ Mini ) )
% 9.49/5.23         => ( ( vEBT_vebt_member @ T @ X )
% 9.49/5.23           => ( ord_less_eq_nat @ Mini @ X ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_824_infinite__growing,axiom,
% 9.49/5.23      ! [X8: set_real] :
% 9.49/5.23        ( ( X8 != bot_bot_set_real )
% 9.49/5.23       => ( ! [X4: real] :
% 9.49/5.23              ( ( member_real @ X4 @ X8 )
% 9.49/5.23             => ? [Xa: real] :
% 9.49/5.23                  ( ( member_real @ Xa @ X8 )
% 9.49/5.23                  & ( ord_less_real @ X4 @ Xa ) ) )
% 9.49/5.23         => ~ ( finite_finite_real @ X8 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_825_infinite__growing,axiom,
% 9.49/5.23      ! [X8: set_rat] :
% 9.49/5.23        ( ( X8 != bot_bot_set_rat )
% 9.49/5.23       => ( ! [X4: rat] :
% 9.49/5.23              ( ( member_rat @ X4 @ X8 )
% 9.49/5.23             => ? [Xa: rat] :
% 9.49/5.23                  ( ( member_rat @ Xa @ X8 )
% 9.49/5.23                  & ( ord_less_rat @ X4 @ Xa ) ) )
% 9.49/5.23         => ~ ( finite_finite_rat @ X8 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_826_infinite__growing,axiom,
% 9.49/5.23      ! [X8: set_num] :
% 9.49/5.23        ( ( X8 != bot_bot_set_num )
% 9.49/5.23       => ( ! [X4: num] :
% 9.49/5.23              ( ( member_num @ X4 @ X8 )
% 9.49/5.23             => ? [Xa: num] :
% 9.49/5.23                  ( ( member_num @ Xa @ X8 )
% 9.49/5.23                  & ( ord_less_num @ X4 @ Xa ) ) )
% 9.49/5.23         => ~ ( finite_finite_num @ X8 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_827_infinite__growing,axiom,
% 9.49/5.23      ! [X8: set_nat] :
% 9.49/5.23        ( ( X8 != bot_bot_set_nat )
% 9.49/5.23       => ( ! [X4: nat] :
% 9.49/5.23              ( ( member_nat @ X4 @ X8 )
% 9.49/5.23             => ? [Xa: nat] :
% 9.49/5.23                  ( ( member_nat @ Xa @ X8 )
% 9.49/5.23                  & ( ord_less_nat @ X4 @ Xa ) ) )
% 9.49/5.23         => ~ ( finite_finite_nat @ X8 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_828_infinite__growing,axiom,
% 9.49/5.23      ! [X8: set_int] :
% 9.49/5.23        ( ( X8 != bot_bot_set_int )
% 9.49/5.23       => ( ! [X4: int] :
% 9.49/5.23              ( ( member_int @ X4 @ X8 )
% 9.49/5.23             => ? [Xa: int] :
% 9.49/5.23                  ( ( member_int @ Xa @ X8 )
% 9.49/5.23                  & ( ord_less_int @ X4 @ Xa ) ) )
% 9.49/5.23         => ~ ( finite_finite_int @ X8 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_829_ex__min__if__finite,axiom,
% 9.49/5.23      ! [S2: set_real] :
% 9.49/5.23        ( ( finite_finite_real @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_real )
% 9.49/5.23         => ? [X4: real] :
% 9.49/5.23              ( ( member_real @ X4 @ S2 )
% 9.49/5.23              & ~ ? [Xa: real] :
% 9.49/5.23                    ( ( member_real @ Xa @ S2 )
% 9.49/5.23                    & ( ord_less_real @ Xa @ X4 ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_830_ex__min__if__finite,axiom,
% 9.49/5.23      ! [S2: set_rat] :
% 9.49/5.23        ( ( finite_finite_rat @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_rat )
% 9.49/5.23         => ? [X4: rat] :
% 9.49/5.23              ( ( member_rat @ X4 @ S2 )
% 9.49/5.23              & ~ ? [Xa: rat] :
% 9.49/5.23                    ( ( member_rat @ Xa @ S2 )
% 9.49/5.23                    & ( ord_less_rat @ Xa @ X4 ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_831_ex__min__if__finite,axiom,
% 9.49/5.23      ! [S2: set_num] :
% 9.49/5.23        ( ( finite_finite_num @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_num )
% 9.49/5.23         => ? [X4: num] :
% 9.49/5.23              ( ( member_num @ X4 @ S2 )
% 9.49/5.23              & ~ ? [Xa: num] :
% 9.49/5.23                    ( ( member_num @ Xa @ S2 )
% 9.49/5.23                    & ( ord_less_num @ Xa @ X4 ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_832_ex__min__if__finite,axiom,
% 9.49/5.23      ! [S2: set_nat] :
% 9.49/5.23        ( ( finite_finite_nat @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_nat )
% 9.49/5.23         => ? [X4: nat] :
% 9.49/5.23              ( ( member_nat @ X4 @ S2 )
% 9.49/5.23              & ~ ? [Xa: nat] :
% 9.49/5.23                    ( ( member_nat @ Xa @ S2 )
% 9.49/5.23                    & ( ord_less_nat @ Xa @ X4 ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_833_ex__min__if__finite,axiom,
% 9.49/5.23      ! [S2: set_int] :
% 9.49/5.23        ( ( finite_finite_int @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_int )
% 9.49/5.23         => ? [X4: int] :
% 9.49/5.23              ( ( member_int @ X4 @ S2 )
% 9.49/5.23              & ~ ? [Xa: int] :
% 9.49/5.23                    ( ( member_int @ Xa @ S2 )
% 9.49/5.23                    & ( ord_less_int @ Xa @ X4 ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_834_finite__has__minimal,axiom,
% 9.49/5.23      ! [A2: set_real] :
% 9.49/5.23        ( ( finite_finite_real @ A2 )
% 9.49/5.23       => ( ( A2 != bot_bot_set_real )
% 9.49/5.23         => ? [X4: real] :
% 9.49/5.23              ( ( member_real @ X4 @ A2 )
% 9.49/5.23              & ! [Xa: real] :
% 9.49/5.23                  ( ( member_real @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_real @ Xa @ X4 )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_835_finite__has__minimal,axiom,
% 9.49/5.23      ! [A2: set_set_int] :
% 9.49/5.23        ( ( finite6197958912794628473et_int @ A2 )
% 9.49/5.23       => ( ( A2 != bot_bot_set_set_int )
% 9.49/5.23         => ? [X4: set_int] :
% 9.49/5.23              ( ( member_set_int @ X4 @ A2 )
% 9.49/5.23              & ! [Xa: set_int] :
% 9.49/5.23                  ( ( member_set_int @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_set_int @ Xa @ X4 )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_836_finite__has__minimal,axiom,
% 9.49/5.23      ! [A2: set_rat] :
% 9.49/5.23        ( ( finite_finite_rat @ A2 )
% 9.49/5.23       => ( ( A2 != bot_bot_set_rat )
% 9.49/5.23         => ? [X4: rat] :
% 9.49/5.23              ( ( member_rat @ X4 @ A2 )
% 9.49/5.23              & ! [Xa: rat] :
% 9.49/5.23                  ( ( member_rat @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_rat @ Xa @ X4 )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_837_finite__has__minimal,axiom,
% 9.49/5.23      ! [A2: set_num] :
% 9.49/5.23        ( ( finite_finite_num @ A2 )
% 9.49/5.23       => ( ( A2 != bot_bot_set_num )
% 9.49/5.23         => ? [X4: num] :
% 9.49/5.23              ( ( member_num @ X4 @ A2 )
% 9.49/5.23              & ! [Xa: num] :
% 9.49/5.23                  ( ( member_num @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_num @ Xa @ X4 )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_838_finite__has__minimal,axiom,
% 9.49/5.23      ! [A2: set_nat] :
% 9.49/5.23        ( ( finite_finite_nat @ A2 )
% 9.49/5.23       => ( ( A2 != bot_bot_set_nat )
% 9.49/5.23         => ? [X4: nat] :
% 9.49/5.23              ( ( member_nat @ X4 @ A2 )
% 9.49/5.23              & ! [Xa: nat] :
% 9.49/5.23                  ( ( member_nat @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_nat @ Xa @ X4 )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_839_finite__has__minimal,axiom,
% 9.49/5.23      ! [A2: set_int] :
% 9.49/5.23        ( ( finite_finite_int @ A2 )
% 9.49/5.23       => ( ( A2 != bot_bot_set_int )
% 9.49/5.23         => ? [X4: int] :
% 9.49/5.23              ( ( member_int @ X4 @ A2 )
% 9.49/5.23              & ! [Xa: int] :
% 9.49/5.23                  ( ( member_int @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_int @ Xa @ X4 )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_840_finite__has__maximal,axiom,
% 9.49/5.23      ! [A2: set_real] :
% 9.49/5.23        ( ( finite_finite_real @ A2 )
% 9.49/5.23       => ( ( A2 != bot_bot_set_real )
% 9.49/5.23         => ? [X4: real] :
% 9.49/5.23              ( ( member_real @ X4 @ A2 )
% 9.49/5.23              & ! [Xa: real] :
% 9.49/5.23                  ( ( member_real @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_real @ X4 @ Xa )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_841_finite__has__maximal,axiom,
% 9.49/5.23      ! [A2: set_set_int] :
% 9.49/5.23        ( ( finite6197958912794628473et_int @ A2 )
% 9.49/5.23       => ( ( A2 != bot_bot_set_set_int )
% 9.49/5.23         => ? [X4: set_int] :
% 9.49/5.23              ( ( member_set_int @ X4 @ A2 )
% 9.49/5.23              & ! [Xa: set_int] :
% 9.49/5.23                  ( ( member_set_int @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_set_int @ X4 @ Xa )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_842_finite__has__maximal,axiom,
% 9.49/5.23      ! [A2: set_rat] :
% 9.49/5.23        ( ( finite_finite_rat @ A2 )
% 9.49/5.23       => ( ( A2 != bot_bot_set_rat )
% 9.49/5.23         => ? [X4: rat] :
% 9.49/5.23              ( ( member_rat @ X4 @ A2 )
% 9.49/5.23              & ! [Xa: rat] :
% 9.49/5.23                  ( ( member_rat @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_rat @ X4 @ Xa )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_843_finite__has__maximal,axiom,
% 9.49/5.23      ! [A2: set_num] :
% 9.49/5.23        ( ( finite_finite_num @ A2 )
% 9.49/5.23       => ( ( A2 != bot_bot_set_num )
% 9.49/5.23         => ? [X4: num] :
% 9.49/5.23              ( ( member_num @ X4 @ A2 )
% 9.49/5.23              & ! [Xa: num] :
% 9.49/5.23                  ( ( member_num @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_num @ X4 @ Xa )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_844_finite__has__maximal,axiom,
% 9.49/5.23      ! [A2: set_nat] :
% 9.49/5.23        ( ( finite_finite_nat @ A2 )
% 9.49/5.23       => ( ( A2 != bot_bot_set_nat )
% 9.49/5.23         => ? [X4: nat] :
% 9.49/5.23              ( ( member_nat @ X4 @ A2 )
% 9.49/5.23              & ! [Xa: nat] :
% 9.49/5.23                  ( ( member_nat @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_nat @ X4 @ Xa )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_845_finite__has__maximal,axiom,
% 9.49/5.23      ! [A2: set_int] :
% 9.49/5.23        ( ( finite_finite_int @ A2 )
% 9.49/5.23       => ( ( A2 != bot_bot_set_int )
% 9.49/5.23         => ? [X4: int] :
% 9.49/5.23              ( ( member_int @ X4 @ A2 )
% 9.49/5.23              & ! [Xa: int] :
% 9.49/5.23                  ( ( member_int @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_int @ X4 @ Xa )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_846_mint__member,axiom,
% 9.49/5.23      ! [T: vEBT_VEBT,N: nat,Maxi2: nat] :
% 9.49/5.23        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.23       => ( ( ( vEBT_vebt_mint @ T )
% 9.49/5.23            = ( some_nat @ Maxi2 ) )
% 9.49/5.23         => ( vEBT_vebt_member @ T @ Maxi2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_847_option_Oinject,axiom,
% 9.49/5.23      ! [X22: nat,Y22: nat] :
% 9.49/5.23        ( ( ( some_nat @ X22 )
% 9.49/5.23          = ( some_nat @ Y22 ) )
% 9.49/5.23        = ( X22 = Y22 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_848_option_Oinject,axiom,
% 9.49/5.23      ! [X22: product_prod_nat_nat,Y22: product_prod_nat_nat] :
% 9.49/5.23        ( ( ( some_P7363390416028606310at_nat @ X22 )
% 9.49/5.23          = ( some_P7363390416028606310at_nat @ Y22 ) )
% 9.49/5.23        = ( X22 = Y22 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_849_option_Oinject,axiom,
% 9.49/5.23      ! [X22: num,Y22: num] :
% 9.49/5.23        ( ( ( some_num @ X22 )
% 9.49/5.23          = ( some_num @ Y22 ) )
% 9.49/5.23        = ( X22 = Y22 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_850_minminNull,axiom,
% 9.49/5.23      ! [T: vEBT_VEBT] :
% 9.49/5.23        ( ( ( vEBT_vebt_mint @ T )
% 9.49/5.23          = none_nat )
% 9.49/5.23       => ( vEBT_VEBT_minNull @ T ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_851_minNullmin,axiom,
% 9.49/5.23      ! [T: vEBT_VEBT] :
% 9.49/5.23        ( ( vEBT_VEBT_minNull @ T )
% 9.49/5.23       => ( ( vEBT_vebt_mint @ T )
% 9.49/5.23          = none_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_852_mint__corr__help__empty,axiom,
% 9.49/5.23      ! [T: vEBT_VEBT,N: nat] :
% 9.49/5.23        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.23       => ( ( ( vEBT_vebt_mint @ T )
% 9.49/5.23            = none_nat )
% 9.49/5.23         => ( ( vEBT_VEBT_set_vebt @ T )
% 9.49/5.23            = bot_bot_set_nat ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_853_not__Some__eq,axiom,
% 9.49/5.23      ! [X: option_nat] :
% 9.49/5.23        ( ( ! [Y2: nat] :
% 9.49/5.23              ( X
% 9.49/5.23             != ( some_nat @ Y2 ) ) )
% 9.49/5.23        = ( X = none_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_854_not__Some__eq,axiom,
% 9.49/5.23      ! [X: option4927543243414619207at_nat] :
% 9.49/5.23        ( ( ! [Y2: product_prod_nat_nat] :
% 9.49/5.23              ( X
% 9.49/5.23             != ( some_P7363390416028606310at_nat @ Y2 ) ) )
% 9.49/5.23        = ( X = none_P5556105721700978146at_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_855_not__Some__eq,axiom,
% 9.49/5.23      ! [X: option_num] :
% 9.49/5.23        ( ( ! [Y2: num] :
% 9.49/5.23              ( X
% 9.49/5.23             != ( some_num @ Y2 ) ) )
% 9.49/5.23        = ( X = none_num ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_856_not__None__eq,axiom,
% 9.49/5.23      ! [X: option_nat] :
% 9.49/5.23        ( ( X != none_nat )
% 9.49/5.23        = ( ? [Y2: nat] :
% 9.49/5.23              ( X
% 9.49/5.23              = ( some_nat @ Y2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_857_not__None__eq,axiom,
% 9.49/5.23      ! [X: option4927543243414619207at_nat] :
% 9.49/5.23        ( ( X != none_P5556105721700978146at_nat )
% 9.49/5.23        = ( ? [Y2: product_prod_nat_nat] :
% 9.49/5.23              ( X
% 9.49/5.23              = ( some_P7363390416028606310at_nat @ Y2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_858_not__None__eq,axiom,
% 9.49/5.23      ! [X: option_num] :
% 9.49/5.23        ( ( X != none_num )
% 9.49/5.23        = ( ? [Y2: num] :
% 9.49/5.23              ( X
% 9.49/5.23              = ( some_num @ Y2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_859_combine__options__cases,axiom,
% 9.49/5.23      ! [X: option_nat,P: option_nat > option_nat > $o,Y: option_nat] :
% 9.49/5.23        ( ( ( X = none_nat )
% 9.49/5.23         => ( P @ X @ Y ) )
% 9.49/5.23       => ( ( ( Y = none_nat )
% 9.49/5.23           => ( P @ X @ Y ) )
% 9.49/5.23         => ( ! [A3: nat,B3: nat] :
% 9.49/5.23                ( ( X
% 9.49/5.23                  = ( some_nat @ A3 ) )
% 9.49/5.23               => ( ( Y
% 9.49/5.23                    = ( some_nat @ B3 ) )
% 9.49/5.23                 => ( P @ X @ Y ) ) )
% 9.49/5.23           => ( P @ X @ Y ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_860_combine__options__cases,axiom,
% 9.49/5.23      ! [X: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
% 9.49/5.23        ( ( ( X = none_nat )
% 9.49/5.23         => ( P @ X @ Y ) )
% 9.49/5.23       => ( ( ( Y = none_P5556105721700978146at_nat )
% 9.49/5.23           => ( P @ X @ Y ) )
% 9.49/5.23         => ( ! [A3: nat,B3: product_prod_nat_nat] :
% 9.49/5.23                ( ( X
% 9.49/5.23                  = ( some_nat @ A3 ) )
% 9.49/5.23               => ( ( Y
% 9.49/5.23                    = ( some_P7363390416028606310at_nat @ B3 ) )
% 9.49/5.23                 => ( P @ X @ Y ) ) )
% 9.49/5.23           => ( P @ X @ Y ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_861_combine__options__cases,axiom,
% 9.49/5.23      ! [X: option_nat,P: option_nat > option_num > $o,Y: option_num] :
% 9.49/5.23        ( ( ( X = none_nat )
% 9.49/5.23         => ( P @ X @ Y ) )
% 9.49/5.23       => ( ( ( Y = none_num )
% 9.49/5.23           => ( P @ X @ Y ) )
% 9.49/5.23         => ( ! [A3: nat,B3: num] :
% 9.49/5.23                ( ( X
% 9.49/5.23                  = ( some_nat @ A3 ) )
% 9.49/5.23               => ( ( Y
% 9.49/5.23                    = ( some_num @ B3 ) )
% 9.49/5.23                 => ( P @ X @ Y ) ) )
% 9.49/5.23           => ( P @ X @ Y ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_862_combine__options__cases,axiom,
% 9.49/5.23      ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y: option_nat] :
% 9.49/5.23        ( ( ( X = none_P5556105721700978146at_nat )
% 9.49/5.23         => ( P @ X @ Y ) )
% 9.49/5.23       => ( ( ( Y = none_nat )
% 9.49/5.23           => ( P @ X @ Y ) )
% 9.49/5.23         => ( ! [A3: product_prod_nat_nat,B3: nat] :
% 9.49/5.23                ( ( X
% 9.49/5.23                  = ( some_P7363390416028606310at_nat @ A3 ) )
% 9.49/5.23               => ( ( Y
% 9.49/5.23                    = ( some_nat @ B3 ) )
% 9.49/5.23                 => ( P @ X @ Y ) ) )
% 9.49/5.23           => ( P @ X @ Y ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_863_combine__options__cases,axiom,
% 9.49/5.23      ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
% 9.49/5.23        ( ( ( X = none_P5556105721700978146at_nat )
% 9.49/5.23         => ( P @ X @ Y ) )
% 9.49/5.23       => ( ( ( Y = none_P5556105721700978146at_nat )
% 9.49/5.23           => ( P @ X @ Y ) )
% 9.49/5.23         => ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
% 9.49/5.23                ( ( X
% 9.49/5.23                  = ( some_P7363390416028606310at_nat @ A3 ) )
% 9.49/5.23               => ( ( Y
% 9.49/5.23                    = ( some_P7363390416028606310at_nat @ B3 ) )
% 9.49/5.23                 => ( P @ X @ Y ) ) )
% 9.49/5.23           => ( P @ X @ Y ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_864_combine__options__cases,axiom,
% 9.49/5.23      ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y: option_num] :
% 9.49/5.23        ( ( ( X = none_P5556105721700978146at_nat )
% 9.49/5.23         => ( P @ X @ Y ) )
% 9.49/5.23       => ( ( ( Y = none_num )
% 9.49/5.23           => ( P @ X @ Y ) )
% 9.49/5.23         => ( ! [A3: product_prod_nat_nat,B3: num] :
% 9.49/5.23                ( ( X
% 9.49/5.23                  = ( some_P7363390416028606310at_nat @ A3 ) )
% 9.49/5.23               => ( ( Y
% 9.49/5.23                    = ( some_num @ B3 ) )
% 9.49/5.23                 => ( P @ X @ Y ) ) )
% 9.49/5.23           => ( P @ X @ Y ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_865_combine__options__cases,axiom,
% 9.49/5.23      ! [X: option_num,P: option_num > option_nat > $o,Y: option_nat] :
% 9.49/5.23        ( ( ( X = none_num )
% 9.49/5.23         => ( P @ X @ Y ) )
% 9.49/5.23       => ( ( ( Y = none_nat )
% 9.49/5.23           => ( P @ X @ Y ) )
% 9.49/5.23         => ( ! [A3: num,B3: nat] :
% 9.49/5.23                ( ( X
% 9.49/5.23                  = ( some_num @ A3 ) )
% 9.49/5.23               => ( ( Y
% 9.49/5.23                    = ( some_nat @ B3 ) )
% 9.49/5.23                 => ( P @ X @ Y ) ) )
% 9.49/5.23           => ( P @ X @ Y ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_866_combine__options__cases,axiom,
% 9.49/5.23      ! [X: option_num,P: option_num > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
% 9.49/5.23        ( ( ( X = none_num )
% 9.49/5.23         => ( P @ X @ Y ) )
% 9.49/5.23       => ( ( ( Y = none_P5556105721700978146at_nat )
% 9.49/5.23           => ( P @ X @ Y ) )
% 9.49/5.23         => ( ! [A3: num,B3: product_prod_nat_nat] :
% 9.49/5.23                ( ( X
% 9.49/5.23                  = ( some_num @ A3 ) )
% 9.49/5.23               => ( ( Y
% 9.49/5.23                    = ( some_P7363390416028606310at_nat @ B3 ) )
% 9.49/5.23                 => ( P @ X @ Y ) ) )
% 9.49/5.23           => ( P @ X @ Y ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_867_combine__options__cases,axiom,
% 9.49/5.23      ! [X: option_num,P: option_num > option_num > $o,Y: option_num] :
% 9.49/5.23        ( ( ( X = none_num )
% 9.49/5.23         => ( P @ X @ Y ) )
% 9.49/5.23       => ( ( ( Y = none_num )
% 9.49/5.23           => ( P @ X @ Y ) )
% 9.49/5.23         => ( ! [A3: num,B3: num] :
% 9.49/5.23                ( ( X
% 9.49/5.23                  = ( some_num @ A3 ) )
% 9.49/5.23               => ( ( Y
% 9.49/5.23                    = ( some_num @ B3 ) )
% 9.49/5.23                 => ( P @ X @ Y ) ) )
% 9.49/5.23           => ( P @ X @ Y ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_868_split__option__all,axiom,
% 9.49/5.23      ( ( ^ [P2: option_nat > $o] :
% 9.49/5.23          ! [X7: option_nat] : ( P2 @ X7 ) )
% 9.49/5.23      = ( ^ [P3: option_nat > $o] :
% 9.49/5.23            ( ( P3 @ none_nat )
% 9.49/5.23            & ! [X3: nat] : ( P3 @ ( some_nat @ X3 ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_869_split__option__all,axiom,
% 9.49/5.23      ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 9.49/5.23          ! [X7: option4927543243414619207at_nat] : ( P2 @ X7 ) )
% 9.49/5.23      = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 9.49/5.23            ( ( P3 @ none_P5556105721700978146at_nat )
% 9.49/5.23            & ! [X3: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X3 ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_870_split__option__all,axiom,
% 9.49/5.23      ( ( ^ [P2: option_num > $o] :
% 9.49/5.23          ! [X7: option_num] : ( P2 @ X7 ) )
% 9.49/5.23      = ( ^ [P3: option_num > $o] :
% 9.49/5.23            ( ( P3 @ none_num )
% 9.49/5.23            & ! [X3: num] : ( P3 @ ( some_num @ X3 ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_871_split__option__ex,axiom,
% 9.49/5.23      ( ( ^ [P2: option_nat > $o] :
% 9.49/5.23          ? [X7: option_nat] : ( P2 @ X7 ) )
% 9.49/5.23      = ( ^ [P3: option_nat > $o] :
% 9.49/5.23            ( ( P3 @ none_nat )
% 9.49/5.23            | ? [X3: nat] : ( P3 @ ( some_nat @ X3 ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_872_split__option__ex,axiom,
% 9.49/5.23      ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 9.49/5.23          ? [X7: option4927543243414619207at_nat] : ( P2 @ X7 ) )
% 9.49/5.23      = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 9.49/5.23            ( ( P3 @ none_P5556105721700978146at_nat )
% 9.49/5.23            | ? [X3: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X3 ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_873_split__option__ex,axiom,
% 9.49/5.23      ( ( ^ [P2: option_num > $o] :
% 9.49/5.23          ? [X7: option_num] : ( P2 @ X7 ) )
% 9.49/5.23      = ( ^ [P3: option_num > $o] :
% 9.49/5.23            ( ( P3 @ none_num )
% 9.49/5.23            | ? [X3: num] : ( P3 @ ( some_num @ X3 ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_874_option_Oexhaust,axiom,
% 9.49/5.23      ! [Y: option_nat] :
% 9.49/5.23        ( ( Y != none_nat )
% 9.49/5.23       => ~ ! [X23: nat] :
% 9.49/5.23              ( Y
% 9.49/5.23             != ( some_nat @ X23 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_875_option_Oexhaust,axiom,
% 9.49/5.23      ! [Y: option4927543243414619207at_nat] :
% 9.49/5.23        ( ( Y != none_P5556105721700978146at_nat )
% 9.49/5.23       => ~ ! [X23: product_prod_nat_nat] :
% 9.49/5.23              ( Y
% 9.49/5.23             != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_876_option_Oexhaust,axiom,
% 9.49/5.23      ! [Y: option_num] :
% 9.49/5.23        ( ( Y != none_num )
% 9.49/5.23       => ~ ! [X23: num] :
% 9.49/5.23              ( Y
% 9.49/5.23             != ( some_num @ X23 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_877_option_OdiscI,axiom,
% 9.49/5.23      ! [Option: option_nat,X22: nat] :
% 9.49/5.23        ( ( Option
% 9.49/5.23          = ( some_nat @ X22 ) )
% 9.49/5.23       => ( Option != none_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_878_option_OdiscI,axiom,
% 9.49/5.23      ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
% 9.49/5.23        ( ( Option
% 9.49/5.23          = ( some_P7363390416028606310at_nat @ X22 ) )
% 9.49/5.23       => ( Option != none_P5556105721700978146at_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_879_option_OdiscI,axiom,
% 9.49/5.23      ! [Option: option_num,X22: num] :
% 9.49/5.23        ( ( Option
% 9.49/5.23          = ( some_num @ X22 ) )
% 9.49/5.23       => ( Option != none_num ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_880_option_Odistinct_I1_J,axiom,
% 9.49/5.23      ! [X22: nat] :
% 9.49/5.23        ( none_nat
% 9.49/5.23       != ( some_nat @ X22 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_881_option_Odistinct_I1_J,axiom,
% 9.49/5.23      ! [X22: product_prod_nat_nat] :
% 9.49/5.23        ( none_P5556105721700978146at_nat
% 9.49/5.23       != ( some_P7363390416028606310at_nat @ X22 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_882_option_Odistinct_I1_J,axiom,
% 9.49/5.23      ! [X22: num] :
% 9.49/5.23        ( none_num
% 9.49/5.23       != ( some_num @ X22 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_883_bot__nat__def,axiom,
% 9.49/5.23      bot_bot_nat = zero_zero_nat ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_884_finite__maxlen,axiom,
% 9.49/5.23      ! [M4: set_list_VEBT_VEBT] :
% 9.49/5.23        ( ( finite3004134309566078307T_VEBT @ M4 )
% 9.49/5.23       => ? [N3: nat] :
% 9.49/5.23          ! [X2: list_VEBT_VEBT] :
% 9.49/5.23            ( ( member2936631157270082147T_VEBT @ X2 @ M4 )
% 9.49/5.23           => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X2 ) @ N3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_885_finite__maxlen,axiom,
% 9.49/5.23      ! [M4: set_list_o] :
% 9.49/5.23        ( ( finite_finite_list_o @ M4 )
% 9.49/5.23       => ? [N3: nat] :
% 9.49/5.23          ! [X2: list_o] :
% 9.49/5.23            ( ( member_list_o @ X2 @ M4 )
% 9.49/5.23           => ( ord_less_nat @ ( size_size_list_o @ X2 ) @ N3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_886_finite__maxlen,axiom,
% 9.49/5.23      ! [M4: set_list_nat] :
% 9.49/5.23        ( ( finite8100373058378681591st_nat @ M4 )
% 9.49/5.23       => ? [N3: nat] :
% 9.49/5.23          ! [X2: list_nat] :
% 9.49/5.23            ( ( member_list_nat @ X2 @ M4 )
% 9.49/5.23           => ( ord_less_nat @ ( size_size_list_nat @ X2 ) @ N3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_887_finite__maxlen,axiom,
% 9.49/5.23      ! [M4: set_list_int] :
% 9.49/5.23        ( ( finite3922522038869484883st_int @ M4 )
% 9.49/5.23       => ? [N3: nat] :
% 9.49/5.23          ! [X2: list_int] :
% 9.49/5.23            ( ( member_list_int @ X2 @ M4 )
% 9.49/5.23           => ( ord_less_nat @ ( size_size_list_int @ X2 ) @ N3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_888_finite__subset,axiom,
% 9.49/5.23      ! [A2: set_nat,B2: set_nat] :
% 9.49/5.23        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.49/5.23       => ( ( finite_finite_nat @ B2 )
% 9.49/5.23         => ( finite_finite_nat @ A2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_889_finite__subset,axiom,
% 9.49/5.23      ! [A2: set_complex,B2: set_complex] :
% 9.49/5.23        ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.49/5.23       => ( ( finite3207457112153483333omplex @ B2 )
% 9.49/5.23         => ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_890_finite__subset,axiom,
% 9.49/5.23      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.23        ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 9.49/5.23       => ( ( finite6177210948735845034at_nat @ B2 )
% 9.49/5.23         => ( finite6177210948735845034at_nat @ A2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_891_finite__subset,axiom,
% 9.49/5.23      ! [A2: set_int,B2: set_int] :
% 9.49/5.23        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.49/5.23       => ( ( finite_finite_int @ B2 )
% 9.49/5.23         => ( finite_finite_int @ A2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_892_infinite__super,axiom,
% 9.49/5.23      ! [S2: set_nat,T3: set_nat] :
% 9.49/5.23        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.49/5.23       => ( ~ ( finite_finite_nat @ S2 )
% 9.49/5.23         => ~ ( finite_finite_nat @ T3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_893_infinite__super,axiom,
% 9.49/5.23      ! [S2: set_complex,T3: set_complex] :
% 9.49/5.23        ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.49/5.23       => ( ~ ( finite3207457112153483333omplex @ S2 )
% 9.49/5.23         => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_894_infinite__super,axiom,
% 9.49/5.23      ! [S2: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
% 9.49/5.23        ( ( ord_le3146513528884898305at_nat @ S2 @ T3 )
% 9.49/5.23       => ( ~ ( finite6177210948735845034at_nat @ S2 )
% 9.49/5.23         => ~ ( finite6177210948735845034at_nat @ T3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_895_infinite__super,axiom,
% 9.49/5.23      ! [S2: set_int,T3: set_int] :
% 9.49/5.23        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.49/5.23       => ( ~ ( finite_finite_int @ S2 )
% 9.49/5.23         => ~ ( finite_finite_int @ T3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_896_rev__finite__subset,axiom,
% 9.49/5.23      ! [B2: set_nat,A2: set_nat] :
% 9.49/5.23        ( ( finite_finite_nat @ B2 )
% 9.49/5.23       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.49/5.23         => ( finite_finite_nat @ A2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_897_rev__finite__subset,axiom,
% 9.49/5.23      ! [B2: set_complex,A2: set_complex] :
% 9.49/5.23        ( ( finite3207457112153483333omplex @ B2 )
% 9.49/5.23       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.49/5.23         => ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_898_rev__finite__subset,axiom,
% 9.49/5.23      ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.23        ( ( finite6177210948735845034at_nat @ B2 )
% 9.49/5.23       => ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 9.49/5.23         => ( finite6177210948735845034at_nat @ A2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_899_rev__finite__subset,axiom,
% 9.49/5.23      ! [B2: set_int,A2: set_int] :
% 9.49/5.23        ( ( finite_finite_int @ B2 )
% 9.49/5.23       => ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.49/5.23         => ( finite_finite_int @ A2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_900_finite__psubset__induct,axiom,
% 9.49/5.23      ! [A2: set_nat,P: set_nat > $o] :
% 9.49/5.23        ( ( finite_finite_nat @ A2 )
% 9.49/5.23       => ( ! [A5: set_nat] :
% 9.49/5.23              ( ( finite_finite_nat @ A5 )
% 9.49/5.23             => ( ! [B5: set_nat] :
% 9.49/5.23                    ( ( ord_less_set_nat @ B5 @ A5 )
% 9.49/5.23                   => ( P @ B5 ) )
% 9.49/5.23               => ( P @ A5 ) ) )
% 9.49/5.23         => ( P @ A2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_901_finite__psubset__induct,axiom,
% 9.49/5.23      ! [A2: set_int,P: set_int > $o] :
% 9.49/5.23        ( ( finite_finite_int @ A2 )
% 9.49/5.23       => ( ! [A5: set_int] :
% 9.49/5.23              ( ( finite_finite_int @ A5 )
% 9.49/5.23             => ( ! [B5: set_int] :
% 9.49/5.23                    ( ( ord_less_set_int @ B5 @ A5 )
% 9.49/5.23                   => ( P @ B5 ) )
% 9.49/5.23               => ( P @ A5 ) ) )
% 9.49/5.23         => ( P @ A2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_902_finite__psubset__induct,axiom,
% 9.49/5.23      ! [A2: set_complex,P: set_complex > $o] :
% 9.49/5.23        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.23       => ( ! [A5: set_complex] :
% 9.49/5.23              ( ( finite3207457112153483333omplex @ A5 )
% 9.49/5.23             => ( ! [B5: set_complex] :
% 9.49/5.23                    ( ( ord_less_set_complex @ B5 @ A5 )
% 9.49/5.23                   => ( P @ B5 ) )
% 9.49/5.23               => ( P @ A5 ) ) )
% 9.49/5.23         => ( P @ A2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_903_finite__psubset__induct,axiom,
% 9.49/5.23      ! [A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.49/5.23        ( ( finite6177210948735845034at_nat @ A2 )
% 9.49/5.23       => ( ! [A5: set_Pr1261947904930325089at_nat] :
% 9.49/5.23              ( ( finite6177210948735845034at_nat @ A5 )
% 9.49/5.23             => ( ! [B5: set_Pr1261947904930325089at_nat] :
% 9.49/5.23                    ( ( ord_le7866589430770878221at_nat @ B5 @ A5 )
% 9.49/5.23                   => ( P @ B5 ) )
% 9.49/5.23               => ( P @ A5 ) ) )
% 9.49/5.23         => ( P @ A2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_904_finite__has__minimal2,axiom,
% 9.49/5.23      ! [A2: set_real,A: real] :
% 9.49/5.23        ( ( finite_finite_real @ A2 )
% 9.49/5.23       => ( ( member_real @ A @ A2 )
% 9.49/5.23         => ? [X4: real] :
% 9.49/5.23              ( ( member_real @ X4 @ A2 )
% 9.49/5.23              & ( ord_less_eq_real @ X4 @ A )
% 9.49/5.23              & ! [Xa: real] :
% 9.49/5.23                  ( ( member_real @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_real @ Xa @ X4 )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_905_finite__has__minimal2,axiom,
% 9.49/5.23      ! [A2: set_set_nat,A: set_nat] :
% 9.49/5.23        ( ( finite1152437895449049373et_nat @ A2 )
% 9.49/5.23       => ( ( member_set_nat @ A @ A2 )
% 9.49/5.23         => ? [X4: set_nat] :
% 9.49/5.23              ( ( member_set_nat @ X4 @ A2 )
% 9.49/5.23              & ( ord_less_eq_set_nat @ X4 @ A )
% 9.49/5.23              & ! [Xa: set_nat] :
% 9.49/5.23                  ( ( member_set_nat @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_set_nat @ Xa @ X4 )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_906_finite__has__minimal2,axiom,
% 9.49/5.23      ! [A2: set_set_int,A: set_int] :
% 9.49/5.23        ( ( finite6197958912794628473et_int @ A2 )
% 9.49/5.23       => ( ( member_set_int @ A @ A2 )
% 9.49/5.23         => ? [X4: set_int] :
% 9.49/5.23              ( ( member_set_int @ X4 @ A2 )
% 9.49/5.23              & ( ord_less_eq_set_int @ X4 @ A )
% 9.49/5.23              & ! [Xa: set_int] :
% 9.49/5.23                  ( ( member_set_int @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_set_int @ Xa @ X4 )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_907_finite__has__minimal2,axiom,
% 9.49/5.23      ! [A2: set_rat,A: rat] :
% 9.49/5.23        ( ( finite_finite_rat @ A2 )
% 9.49/5.23       => ( ( member_rat @ A @ A2 )
% 9.49/5.23         => ? [X4: rat] :
% 9.49/5.23              ( ( member_rat @ X4 @ A2 )
% 9.49/5.23              & ( ord_less_eq_rat @ X4 @ A )
% 9.49/5.23              & ! [Xa: rat] :
% 9.49/5.23                  ( ( member_rat @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_rat @ Xa @ X4 )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_908_finite__has__minimal2,axiom,
% 9.49/5.23      ! [A2: set_num,A: num] :
% 9.49/5.23        ( ( finite_finite_num @ A2 )
% 9.49/5.23       => ( ( member_num @ A @ A2 )
% 9.49/5.23         => ? [X4: num] :
% 9.49/5.23              ( ( member_num @ X4 @ A2 )
% 9.49/5.23              & ( ord_less_eq_num @ X4 @ A )
% 9.49/5.23              & ! [Xa: num] :
% 9.49/5.23                  ( ( member_num @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_num @ Xa @ X4 )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_909_finite__has__minimal2,axiom,
% 9.49/5.23      ! [A2: set_nat,A: nat] :
% 9.49/5.23        ( ( finite_finite_nat @ A2 )
% 9.49/5.23       => ( ( member_nat @ A @ A2 )
% 9.49/5.23         => ? [X4: nat] :
% 9.49/5.23              ( ( member_nat @ X4 @ A2 )
% 9.49/5.23              & ( ord_less_eq_nat @ X4 @ A )
% 9.49/5.23              & ! [Xa: nat] :
% 9.49/5.23                  ( ( member_nat @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_nat @ Xa @ X4 )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_910_finite__has__minimal2,axiom,
% 9.49/5.23      ! [A2: set_int,A: int] :
% 9.49/5.23        ( ( finite_finite_int @ A2 )
% 9.49/5.23       => ( ( member_int @ A @ A2 )
% 9.49/5.23         => ? [X4: int] :
% 9.49/5.23              ( ( member_int @ X4 @ A2 )
% 9.49/5.23              & ( ord_less_eq_int @ X4 @ A )
% 9.49/5.23              & ! [Xa: int] :
% 9.49/5.23                  ( ( member_int @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_int @ Xa @ X4 )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_911_finite__has__maximal2,axiom,
% 9.49/5.23      ! [A2: set_real,A: real] :
% 9.49/5.23        ( ( finite_finite_real @ A2 )
% 9.49/5.23       => ( ( member_real @ A @ A2 )
% 9.49/5.23         => ? [X4: real] :
% 9.49/5.23              ( ( member_real @ X4 @ A2 )
% 9.49/5.23              & ( ord_less_eq_real @ A @ X4 )
% 9.49/5.23              & ! [Xa: real] :
% 9.49/5.23                  ( ( member_real @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_real @ X4 @ Xa )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_912_finite__has__maximal2,axiom,
% 9.49/5.23      ! [A2: set_set_nat,A: set_nat] :
% 9.49/5.23        ( ( finite1152437895449049373et_nat @ A2 )
% 9.49/5.23       => ( ( member_set_nat @ A @ A2 )
% 9.49/5.23         => ? [X4: set_nat] :
% 9.49/5.23              ( ( member_set_nat @ X4 @ A2 )
% 9.49/5.23              & ( ord_less_eq_set_nat @ A @ X4 )
% 9.49/5.23              & ! [Xa: set_nat] :
% 9.49/5.23                  ( ( member_set_nat @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_set_nat @ X4 @ Xa )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_913_finite__has__maximal2,axiom,
% 9.49/5.23      ! [A2: set_set_int,A: set_int] :
% 9.49/5.23        ( ( finite6197958912794628473et_int @ A2 )
% 9.49/5.23       => ( ( member_set_int @ A @ A2 )
% 9.49/5.23         => ? [X4: set_int] :
% 9.49/5.23              ( ( member_set_int @ X4 @ A2 )
% 9.49/5.23              & ( ord_less_eq_set_int @ A @ X4 )
% 9.49/5.23              & ! [Xa: set_int] :
% 9.49/5.23                  ( ( member_set_int @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_set_int @ X4 @ Xa )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_914_finite__has__maximal2,axiom,
% 9.49/5.23      ! [A2: set_rat,A: rat] :
% 9.49/5.23        ( ( finite_finite_rat @ A2 )
% 9.49/5.23       => ( ( member_rat @ A @ A2 )
% 9.49/5.23         => ? [X4: rat] :
% 9.49/5.23              ( ( member_rat @ X4 @ A2 )
% 9.49/5.23              & ( ord_less_eq_rat @ A @ X4 )
% 9.49/5.23              & ! [Xa: rat] :
% 9.49/5.23                  ( ( member_rat @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_rat @ X4 @ Xa )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_915_finite__has__maximal2,axiom,
% 9.49/5.23      ! [A2: set_num,A: num] :
% 9.49/5.23        ( ( finite_finite_num @ A2 )
% 9.49/5.23       => ( ( member_num @ A @ A2 )
% 9.49/5.23         => ? [X4: num] :
% 9.49/5.23              ( ( member_num @ X4 @ A2 )
% 9.49/5.23              & ( ord_less_eq_num @ A @ X4 )
% 9.49/5.23              & ! [Xa: num] :
% 9.49/5.23                  ( ( member_num @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_num @ X4 @ Xa )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_916_finite__has__maximal2,axiom,
% 9.49/5.23      ! [A2: set_nat,A: nat] :
% 9.49/5.23        ( ( finite_finite_nat @ A2 )
% 9.49/5.23       => ( ( member_nat @ A @ A2 )
% 9.49/5.23         => ? [X4: nat] :
% 9.49/5.23              ( ( member_nat @ X4 @ A2 )
% 9.49/5.23              & ( ord_less_eq_nat @ A @ X4 )
% 9.49/5.23              & ! [Xa: nat] :
% 9.49/5.23                  ( ( member_nat @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_nat @ X4 @ Xa )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_917_finite__has__maximal2,axiom,
% 9.49/5.23      ! [A2: set_int,A: int] :
% 9.49/5.23        ( ( finite_finite_int @ A2 )
% 9.49/5.23       => ( ( member_int @ A @ A2 )
% 9.49/5.23         => ? [X4: int] :
% 9.49/5.23              ( ( member_int @ X4 @ A2 )
% 9.49/5.23              & ( ord_less_eq_int @ A @ X4 )
% 9.49/5.23              & ! [Xa: int] :
% 9.49/5.23                  ( ( member_int @ Xa @ A2 )
% 9.49/5.23                 => ( ( ord_less_eq_int @ X4 @ Xa )
% 9.49/5.23                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_918_infinite__imp__nonempty,axiom,
% 9.49/5.23      ! [S2: set_complex] :
% 9.49/5.23        ( ~ ( finite3207457112153483333omplex @ S2 )
% 9.49/5.23       => ( S2 != bot_bot_set_complex ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_919_infinite__imp__nonempty,axiom,
% 9.49/5.23      ! [S2: set_Pr1261947904930325089at_nat] :
% 9.49/5.23        ( ~ ( finite6177210948735845034at_nat @ S2 )
% 9.49/5.23       => ( S2 != bot_bo2099793752762293965at_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_920_infinite__imp__nonempty,axiom,
% 9.49/5.23      ! [S2: set_real] :
% 9.49/5.23        ( ~ ( finite_finite_real @ S2 )
% 9.49/5.23       => ( S2 != bot_bot_set_real ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_921_infinite__imp__nonempty,axiom,
% 9.49/5.23      ! [S2: set_nat] :
% 9.49/5.23        ( ~ ( finite_finite_nat @ S2 )
% 9.49/5.23       => ( S2 != bot_bot_set_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_922_infinite__imp__nonempty,axiom,
% 9.49/5.23      ! [S2: set_int] :
% 9.49/5.23        ( ~ ( finite_finite_int @ S2 )
% 9.49/5.23       => ( S2 != bot_bot_set_int ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_923_finite_OemptyI,axiom,
% 9.49/5.23      finite3207457112153483333omplex @ bot_bot_set_complex ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_924_finite_OemptyI,axiom,
% 9.49/5.23      finite6177210948735845034at_nat @ bot_bo2099793752762293965at_nat ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_925_finite_OemptyI,axiom,
% 9.49/5.23      finite_finite_real @ bot_bot_set_real ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_926_finite_OemptyI,axiom,
% 9.49/5.23      finite_finite_nat @ bot_bot_set_nat ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_927_finite_OemptyI,axiom,
% 9.49/5.23      finite_finite_int @ bot_bot_set_int ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_928_pred__correct,axiom,
% 9.49/5.23      ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
% 9.49/5.23        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.23       => ( ( ( vEBT_vebt_pred @ T @ X )
% 9.49/5.23            = ( some_nat @ Sx ) )
% 9.49/5.23          = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_929_succ__correct,axiom,
% 9.49/5.23      ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
% 9.49/5.23        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.23       => ( ( ( vEBT_vebt_succ @ T @ X )
% 9.49/5.23            = ( some_nat @ Sx ) )
% 9.49/5.23          = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_930_pred__corr,axiom,
% 9.49/5.23      ! [T: vEBT_VEBT,N: nat,X: nat,Px: nat] :
% 9.49/5.23        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.23       => ( ( ( vEBT_vebt_pred @ T @ X )
% 9.49/5.23            = ( some_nat @ Px ) )
% 9.49/5.23          = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Px ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_931_succ__corr,axiom,
% 9.49/5.23      ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
% 9.49/5.23        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.23       => ( ( ( vEBT_vebt_succ @ T @ X )
% 9.49/5.23            = ( some_nat @ Sx ) )
% 9.49/5.23          = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_932_subset__empty,axiom,
% 9.49/5.23      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.23        ( ( ord_le3146513528884898305at_nat @ A2 @ bot_bo2099793752762293965at_nat )
% 9.49/5.23        = ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_933_subset__empty,axiom,
% 9.49/5.23      ! [A2: set_real] :
% 9.49/5.23        ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
% 9.49/5.23        = ( A2 = bot_bot_set_real ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_934_subset__empty,axiom,
% 9.49/5.23      ! [A2: set_nat] :
% 9.49/5.23        ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
% 9.49/5.23        = ( A2 = bot_bot_set_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_935_subset__empty,axiom,
% 9.49/5.23      ! [A2: set_int] :
% 9.49/5.23        ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
% 9.49/5.23        = ( A2 = bot_bot_set_int ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_936_empty__subsetI,axiom,
% 9.49/5.23      ! [A2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A2 ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_937_empty__subsetI,axiom,
% 9.49/5.23      ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_938_empty__subsetI,axiom,
% 9.49/5.23      ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_939_empty__subsetI,axiom,
% 9.49/5.23      ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_940_is__succ__in__set__def,axiom,
% 9.49/5.23      ( vEBT_is_succ_in_set
% 9.49/5.23      = ( ^ [Xs2: set_nat,X3: nat,Y2: nat] :
% 9.49/5.23            ( ( member_nat @ Y2 @ Xs2 )
% 9.49/5.23            & ( ord_less_nat @ X3 @ Y2 )
% 9.49/5.23            & ! [Z3: nat] :
% 9.49/5.23                ( ( member_nat @ Z3 @ Xs2 )
% 9.49/5.23               => ( ( ord_less_nat @ X3 @ Z3 )
% 9.49/5.23                 => ( ord_less_eq_nat @ Y2 @ Z3 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_941_is__pred__in__set__def,axiom,
% 9.49/5.23      ( vEBT_is_pred_in_set
% 9.49/5.23      = ( ^ [Xs2: set_nat,X3: nat,Y2: nat] :
% 9.49/5.23            ( ( member_nat @ Y2 @ Xs2 )
% 9.49/5.23            & ( ord_less_nat @ Y2 @ X3 )
% 9.49/5.23            & ! [Z3: nat] :
% 9.49/5.23                ( ( member_nat @ Z3 @ Xs2 )
% 9.49/5.23               => ( ( ord_less_nat @ Z3 @ X3 )
% 9.49/5.23                 => ( ord_less_eq_nat @ Z3 @ Y2 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_942_mi__eq__ma__no__ch,axiom,
% 9.49/5.23      ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.49/5.23        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
% 9.49/5.23       => ( ( Mi = Ma )
% 9.49/5.23         => ( ! [X2: vEBT_VEBT] :
% 9.49/5.23                ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.49/5.23               => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) )
% 9.49/5.23            & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_943_vebt__member_Osimps_I3_J,axiom,
% 9.49/5.23      ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
% 9.49/5.23        ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_944_psubsetI,axiom,
% 9.49/5.23      ! [A2: set_int,B2: set_int] :
% 9.49/5.23        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.49/5.23       => ( ( A2 != B2 )
% 9.49/5.23         => ( ord_less_set_int @ A2 @ B2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_945_empty__iff,axiom,
% 9.49/5.23      ! [C: set_nat] :
% 9.49/5.23        ~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_946_empty__iff,axiom,
% 9.49/5.23      ! [C: product_prod_nat_nat] :
% 9.49/5.23        ~ ( member8440522571783428010at_nat @ C @ bot_bo2099793752762293965at_nat ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_947_empty__iff,axiom,
% 9.49/5.23      ! [C: real] :
% 9.49/5.23        ~ ( member_real @ C @ bot_bot_set_real ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_948_empty__iff,axiom,
% 9.49/5.23      ! [C: nat] :
% 9.49/5.23        ~ ( member_nat @ C @ bot_bot_set_nat ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_949_empty__iff,axiom,
% 9.49/5.23      ! [C: int] :
% 9.49/5.23        ~ ( member_int @ C @ bot_bot_set_int ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_950_all__not__in__conv,axiom,
% 9.49/5.23      ! [A2: set_set_nat] :
% 9.49/5.23        ( ( ! [X3: set_nat] :
% 9.49/5.23              ~ ( member_set_nat @ X3 @ A2 ) )
% 9.49/5.23        = ( A2 = bot_bot_set_set_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_951_all__not__in__conv,axiom,
% 9.49/5.23      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.23        ( ( ! [X3: product_prod_nat_nat] :
% 9.49/5.23              ~ ( member8440522571783428010at_nat @ X3 @ A2 ) )
% 9.49/5.23        = ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_952_all__not__in__conv,axiom,
% 9.49/5.23      ! [A2: set_real] :
% 9.49/5.23        ( ( ! [X3: real] :
% 9.49/5.23              ~ ( member_real @ X3 @ A2 ) )
% 9.49/5.23        = ( A2 = bot_bot_set_real ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_953_all__not__in__conv,axiom,
% 9.49/5.23      ! [A2: set_nat] :
% 9.49/5.23        ( ( ! [X3: nat] :
% 9.49/5.23              ~ ( member_nat @ X3 @ A2 ) )
% 9.49/5.23        = ( A2 = bot_bot_set_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_954_all__not__in__conv,axiom,
% 9.49/5.23      ! [A2: set_int] :
% 9.49/5.23        ( ( ! [X3: int] :
% 9.49/5.23              ~ ( member_int @ X3 @ A2 ) )
% 9.49/5.23        = ( A2 = bot_bot_set_int ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_955_Collect__empty__eq,axiom,
% 9.49/5.23      ! [P: list_nat > $o] :
% 9.49/5.23        ( ( ( collect_list_nat @ P )
% 9.49/5.23          = bot_bot_set_list_nat )
% 9.49/5.23        = ( ! [X3: list_nat] :
% 9.49/5.23              ~ ( P @ X3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_956_Collect__empty__eq,axiom,
% 9.49/5.23      ! [P: set_nat > $o] :
% 9.49/5.23        ( ( ( collect_set_nat @ P )
% 9.49/5.23          = bot_bot_set_set_nat )
% 9.49/5.23        = ( ! [X3: set_nat] :
% 9.49/5.23              ~ ( P @ X3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_957_Collect__empty__eq,axiom,
% 9.49/5.23      ! [P: product_prod_nat_nat > $o] :
% 9.49/5.23        ( ( ( collec3392354462482085612at_nat @ P )
% 9.49/5.23          = bot_bo2099793752762293965at_nat )
% 9.49/5.23        = ( ! [X3: product_prod_nat_nat] :
% 9.49/5.23              ~ ( P @ X3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_958_Collect__empty__eq,axiom,
% 9.49/5.23      ! [P: real > $o] :
% 9.49/5.23        ( ( ( collect_real @ P )
% 9.49/5.23          = bot_bot_set_real )
% 9.49/5.23        = ( ! [X3: real] :
% 9.49/5.23              ~ ( P @ X3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_959_Collect__empty__eq,axiom,
% 9.49/5.23      ! [P: nat > $o] :
% 9.49/5.23        ( ( ( collect_nat @ P )
% 9.49/5.23          = bot_bot_set_nat )
% 9.49/5.23        = ( ! [X3: nat] :
% 9.49/5.23              ~ ( P @ X3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_960_Collect__empty__eq,axiom,
% 9.49/5.23      ! [P: int > $o] :
% 9.49/5.23        ( ( ( collect_int @ P )
% 9.49/5.23          = bot_bot_set_int )
% 9.49/5.23        = ( ! [X3: int] :
% 9.49/5.23              ~ ( P @ X3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_961_empty__Collect__eq,axiom,
% 9.49/5.23      ! [P: list_nat > $o] :
% 9.49/5.23        ( ( bot_bot_set_list_nat
% 9.49/5.23          = ( collect_list_nat @ P ) )
% 9.49/5.23        = ( ! [X3: list_nat] :
% 9.49/5.23              ~ ( P @ X3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_962_empty__Collect__eq,axiom,
% 9.49/5.23      ! [P: set_nat > $o] :
% 9.49/5.23        ( ( bot_bot_set_set_nat
% 9.49/5.23          = ( collect_set_nat @ P ) )
% 9.49/5.23        = ( ! [X3: set_nat] :
% 9.49/5.23              ~ ( P @ X3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_963_empty__Collect__eq,axiom,
% 9.49/5.23      ! [P: product_prod_nat_nat > $o] :
% 9.49/5.23        ( ( bot_bo2099793752762293965at_nat
% 9.49/5.23          = ( collec3392354462482085612at_nat @ P ) )
% 9.49/5.23        = ( ! [X3: product_prod_nat_nat] :
% 9.49/5.23              ~ ( P @ X3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_964_empty__Collect__eq,axiom,
% 9.49/5.23      ! [P: real > $o] :
% 9.49/5.23        ( ( bot_bot_set_real
% 9.49/5.23          = ( collect_real @ P ) )
% 9.49/5.23        = ( ! [X3: real] :
% 9.49/5.23              ~ ( P @ X3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_965_empty__Collect__eq,axiom,
% 9.49/5.23      ! [P: nat > $o] :
% 9.49/5.23        ( ( bot_bot_set_nat
% 9.49/5.23          = ( collect_nat @ P ) )
% 9.49/5.23        = ( ! [X3: nat] :
% 9.49/5.23              ~ ( P @ X3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_966_empty__Collect__eq,axiom,
% 9.49/5.23      ! [P: int > $o] :
% 9.49/5.23        ( ( bot_bot_set_int
% 9.49/5.23          = ( collect_int @ P ) )
% 9.49/5.23        = ( ! [X3: int] :
% 9.49/5.23              ~ ( P @ X3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_967_subsetI,axiom,
% 9.49/5.23      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.23        ( ! [X4: product_prod_nat_nat] :
% 9.49/5.23            ( ( member8440522571783428010at_nat @ X4 @ A2 )
% 9.49/5.23           => ( member8440522571783428010at_nat @ X4 @ B2 ) )
% 9.49/5.23       => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_968_subsetI,axiom,
% 9.49/5.23      ! [A2: set_real,B2: set_real] :
% 9.49/5.23        ( ! [X4: real] :
% 9.49/5.23            ( ( member_real @ X4 @ A2 )
% 9.49/5.23           => ( member_real @ X4 @ B2 ) )
% 9.49/5.23       => ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_969_subsetI,axiom,
% 9.49/5.23      ! [A2: set_set_nat,B2: set_set_nat] :
% 9.49/5.23        ( ! [X4: set_nat] :
% 9.49/5.23            ( ( member_set_nat @ X4 @ A2 )
% 9.49/5.23           => ( member_set_nat @ X4 @ B2 ) )
% 9.49/5.23       => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_970_subsetI,axiom,
% 9.49/5.23      ! [A2: set_nat,B2: set_nat] :
% 9.49/5.23        ( ! [X4: nat] :
% 9.49/5.23            ( ( member_nat @ X4 @ A2 )
% 9.49/5.23           => ( member_nat @ X4 @ B2 ) )
% 9.49/5.23       => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_971_subsetI,axiom,
% 9.49/5.23      ! [A2: set_int,B2: set_int] :
% 9.49/5.23        ( ! [X4: int] :
% 9.49/5.23            ( ( member_int @ X4 @ A2 )
% 9.49/5.23           => ( member_int @ X4 @ B2 ) )
% 9.49/5.23       => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_972_subset__antisym,axiom,
% 9.49/5.23      ! [A2: set_int,B2: set_int] :
% 9.49/5.23        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.49/5.23       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 9.49/5.23         => ( A2 = B2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_973_geqmaxNone,axiom,
% 9.49/5.23      ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
% 9.49/5.23        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 9.49/5.23       => ( ( ord_less_eq_nat @ Ma @ X )
% 9.49/5.23         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.23            = none_nat ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_974_psubsetD,axiom,
% 9.49/5.23      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
% 9.49/5.23        ( ( ord_le7866589430770878221at_nat @ A2 @ B2 )
% 9.49/5.23       => ( ( member8440522571783428010at_nat @ C @ A2 )
% 9.49/5.23         => ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_975_psubsetD,axiom,
% 9.49/5.23      ! [A2: set_real,B2: set_real,C: real] :
% 9.49/5.23        ( ( ord_less_set_real @ A2 @ B2 )
% 9.49/5.23       => ( ( member_real @ C @ A2 )
% 9.49/5.23         => ( member_real @ C @ B2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_976_psubsetD,axiom,
% 9.49/5.23      ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
% 9.49/5.23        ( ( ord_less_set_set_nat @ A2 @ B2 )
% 9.49/5.23       => ( ( member_set_nat @ C @ A2 )
% 9.49/5.23         => ( member_set_nat @ C @ B2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_977_psubsetD,axiom,
% 9.49/5.23      ! [A2: set_nat,B2: set_nat,C: nat] :
% 9.49/5.23        ( ( ord_less_set_nat @ A2 @ B2 )
% 9.49/5.23       => ( ( member_nat @ C @ A2 )
% 9.49/5.23         => ( member_nat @ C @ B2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_978_psubsetD,axiom,
% 9.49/5.23      ! [A2: set_int,B2: set_int,C: int] :
% 9.49/5.23        ( ( ord_less_set_int @ A2 @ B2 )
% 9.49/5.23       => ( ( member_int @ C @ A2 )
% 9.49/5.23         => ( member_int @ C @ B2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_979_vebt__succ_Osimps_I3_J,axiom,
% 9.49/5.23      ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
% 9.49/5.23        ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va )
% 9.49/5.23        = none_nat ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_980_vebt__pred_Osimps_I4_J,axiom,
% 9.49/5.23      ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
% 9.49/5.23        ( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va ) @ Vb )
% 9.49/5.23        = none_nat ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_981_vebt__member_Osimps_I2_J,axiom,
% 9.49/5.23      ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
% 9.49/5.23        ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_982_VEBT__internal_OminNull_Osimps_I4_J,axiom,
% 9.49/5.23      ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_983_vebt__delete_Osimps_I4_J,axiom,
% 9.49/5.23      ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
% 9.49/5.23        ( ( vEBT_vebt_delete @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Uu )
% 9.49/5.23        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_984_vebt__pred_Osimps_I5_J,axiom,
% 9.49/5.23      ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
% 9.49/5.23        ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
% 9.49/5.23        = none_nat ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_985_vebt__succ_Osimps_I4_J,axiom,
% 9.49/5.23      ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
% 9.49/5.23        ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
% 9.49/5.23        = none_nat ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_986_not__psubset__empty,axiom,
% 9.49/5.23      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.23        ~ ( ord_le7866589430770878221at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_987_not__psubset__empty,axiom,
% 9.49/5.23      ! [A2: set_real] :
% 9.49/5.23        ~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_988_not__psubset__empty,axiom,
% 9.49/5.23      ! [A2: set_nat] :
% 9.49/5.23        ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_989_not__psubset__empty,axiom,
% 9.49/5.23      ! [A2: set_int] :
% 9.49/5.23        ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_990_emptyE,axiom,
% 9.49/5.23      ! [A: set_nat] :
% 9.49/5.23        ~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_991_emptyE,axiom,
% 9.49/5.23      ! [A: product_prod_nat_nat] :
% 9.49/5.23        ~ ( member8440522571783428010at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_992_emptyE,axiom,
% 9.49/5.23      ! [A: real] :
% 9.49/5.23        ~ ( member_real @ A @ bot_bot_set_real ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_993_emptyE,axiom,
% 9.49/5.23      ! [A: nat] :
% 9.49/5.23        ~ ( member_nat @ A @ bot_bot_set_nat ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_994_emptyE,axiom,
% 9.49/5.23      ! [A: int] :
% 9.49/5.23        ~ ( member_int @ A @ bot_bot_set_int ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_995_equals0D,axiom,
% 9.49/5.23      ! [A2: set_set_nat,A: set_nat] :
% 9.49/5.23        ( ( A2 = bot_bot_set_set_nat )
% 9.49/5.23       => ~ ( member_set_nat @ A @ A2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_996_equals0D,axiom,
% 9.49/5.23      ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
% 9.49/5.23        ( ( A2 = bot_bo2099793752762293965at_nat )
% 9.49/5.23       => ~ ( member8440522571783428010at_nat @ A @ A2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_997_equals0D,axiom,
% 9.49/5.23      ! [A2: set_real,A: real] :
% 9.49/5.23        ( ( A2 = bot_bot_set_real )
% 9.49/5.23       => ~ ( member_real @ A @ A2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_998_equals0D,axiom,
% 9.49/5.23      ! [A2: set_nat,A: nat] :
% 9.49/5.23        ( ( A2 = bot_bot_set_nat )
% 9.49/5.23       => ~ ( member_nat @ A @ A2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_999_equals0D,axiom,
% 9.49/5.23      ! [A2: set_int,A: int] :
% 9.49/5.23        ( ( A2 = bot_bot_set_int )
% 9.49/5.23       => ~ ( member_int @ A @ A2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1000_equals0I,axiom,
% 9.49/5.23      ! [A2: set_set_nat] :
% 9.49/5.23        ( ! [Y3: set_nat] :
% 9.49/5.23            ~ ( member_set_nat @ Y3 @ A2 )
% 9.49/5.23       => ( A2 = bot_bot_set_set_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1001_equals0I,axiom,
% 9.49/5.23      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.23        ( ! [Y3: product_prod_nat_nat] :
% 9.49/5.23            ~ ( member8440522571783428010at_nat @ Y3 @ A2 )
% 9.49/5.23       => ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1002_equals0I,axiom,
% 9.49/5.23      ! [A2: set_real] :
% 9.49/5.23        ( ! [Y3: real] :
% 9.49/5.23            ~ ( member_real @ Y3 @ A2 )
% 9.49/5.23       => ( A2 = bot_bot_set_real ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1003_equals0I,axiom,
% 9.49/5.23      ! [A2: set_nat] :
% 9.49/5.23        ( ! [Y3: nat] :
% 9.49/5.23            ~ ( member_nat @ Y3 @ A2 )
% 9.49/5.23       => ( A2 = bot_bot_set_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1004_equals0I,axiom,
% 9.49/5.23      ! [A2: set_int] :
% 9.49/5.23        ( ! [Y3: int] :
% 9.49/5.23            ~ ( member_int @ Y3 @ A2 )
% 9.49/5.23       => ( A2 = bot_bot_set_int ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1005_ex__in__conv,axiom,
% 9.49/5.23      ! [A2: set_set_nat] :
% 9.49/5.23        ( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
% 9.49/5.23        = ( A2 != bot_bot_set_set_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1006_ex__in__conv,axiom,
% 9.49/5.23      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.23        ( ( ? [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A2 ) )
% 9.49/5.23        = ( A2 != bot_bo2099793752762293965at_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1007_ex__in__conv,axiom,
% 9.49/5.23      ! [A2: set_real] :
% 9.49/5.23        ( ( ? [X3: real] : ( member_real @ X3 @ A2 ) )
% 9.49/5.23        = ( A2 != bot_bot_set_real ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1008_ex__in__conv,axiom,
% 9.49/5.23      ! [A2: set_nat] :
% 9.49/5.23        ( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
% 9.49/5.23        = ( A2 != bot_bot_set_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1009_ex__in__conv,axiom,
% 9.49/5.23      ! [A2: set_int] :
% 9.49/5.23        ( ( ? [X3: int] : ( member_int @ X3 @ A2 ) )
% 9.49/5.23        = ( A2 != bot_bot_set_int ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1010_bot__set__def,axiom,
% 9.49/5.23      ( bot_bot_set_list_nat
% 9.49/5.23      = ( collect_list_nat @ bot_bot_list_nat_o ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1011_bot__set__def,axiom,
% 9.49/5.23      ( bot_bot_set_set_nat
% 9.49/5.23      = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1012_bot__set__def,axiom,
% 9.49/5.23      ( bot_bo2099793752762293965at_nat
% 9.49/5.23      = ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1013_bot__set__def,axiom,
% 9.49/5.23      ( bot_bot_set_real
% 9.49/5.23      = ( collect_real @ bot_bot_real_o ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1014_bot__set__def,axiom,
% 9.49/5.23      ( bot_bot_set_nat
% 9.49/5.23      = ( collect_nat @ bot_bot_nat_o ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1015_bot__set__def,axiom,
% 9.49/5.23      ( bot_bot_set_int
% 9.49/5.23      = ( collect_int @ bot_bot_int_o ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1016_vebt__delete_Osimps_I5_J,axiom,
% 9.49/5.23      ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X: nat] :
% 9.49/5.23        ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) @ X )
% 9.49/5.23        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1017_in__mono,axiom,
% 9.49/5.23      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
% 9.49/5.23        ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 9.49/5.23       => ( ( member8440522571783428010at_nat @ X @ A2 )
% 9.49/5.23         => ( member8440522571783428010at_nat @ X @ B2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1018_in__mono,axiom,
% 9.49/5.23      ! [A2: set_real,B2: set_real,X: real] :
% 9.49/5.23        ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.49/5.23       => ( ( member_real @ X @ A2 )
% 9.49/5.23         => ( member_real @ X @ B2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1019_in__mono,axiom,
% 9.49/5.23      ! [A2: set_set_nat,B2: set_set_nat,X: set_nat] :
% 9.49/5.23        ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
% 9.49/5.23       => ( ( member_set_nat @ X @ A2 )
% 9.49/5.23         => ( member_set_nat @ X @ B2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1020_in__mono,axiom,
% 9.49/5.23      ! [A2: set_nat,B2: set_nat,X: nat] :
% 9.49/5.23        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.49/5.23       => ( ( member_nat @ X @ A2 )
% 9.49/5.23         => ( member_nat @ X @ B2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1021_in__mono,axiom,
% 9.49/5.23      ! [A2: set_int,B2: set_int,X: int] :
% 9.49/5.23        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.49/5.23       => ( ( member_int @ X @ A2 )
% 9.49/5.23         => ( member_int @ X @ B2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1022_subsetD,axiom,
% 9.49/5.23      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
% 9.49/5.23        ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 9.49/5.23       => ( ( member8440522571783428010at_nat @ C @ A2 )
% 9.49/5.23         => ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1023_subsetD,axiom,
% 9.49/5.23      ! [A2: set_real,B2: set_real,C: real] :
% 9.49/5.23        ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.49/5.23       => ( ( member_real @ C @ A2 )
% 9.49/5.23         => ( member_real @ C @ B2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1024_subsetD,axiom,
% 9.49/5.23      ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
% 9.49/5.23        ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
% 9.49/5.23       => ( ( member_set_nat @ C @ A2 )
% 9.49/5.23         => ( member_set_nat @ C @ B2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1025_subsetD,axiom,
% 9.49/5.23      ! [A2: set_nat,B2: set_nat,C: nat] :
% 9.49/5.23        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.49/5.23       => ( ( member_nat @ C @ A2 )
% 9.49/5.23         => ( member_nat @ C @ B2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1026_subsetD,axiom,
% 9.49/5.23      ! [A2: set_int,B2: set_int,C: int] :
% 9.49/5.23        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.49/5.23       => ( ( member_int @ C @ A2 )
% 9.49/5.23         => ( member_int @ C @ B2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1027_equalityE,axiom,
% 9.49/5.23      ! [A2: set_int,B2: set_int] :
% 9.49/5.23        ( ( A2 = B2 )
% 9.49/5.23       => ~ ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.49/5.23           => ~ ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1028_subset__eq,axiom,
% 9.49/5.23      ( ord_le3146513528884898305at_nat
% 9.49/5.23      = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 9.49/5.23          ! [X3: product_prod_nat_nat] :
% 9.49/5.23            ( ( member8440522571783428010at_nat @ X3 @ A6 )
% 9.49/5.23           => ( member8440522571783428010at_nat @ X3 @ B6 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1029_subset__eq,axiom,
% 9.49/5.23      ( ord_less_eq_set_real
% 9.49/5.23      = ( ^ [A6: set_real,B6: set_real] :
% 9.49/5.23          ! [X3: real] :
% 9.49/5.23            ( ( member_real @ X3 @ A6 )
% 9.49/5.23           => ( member_real @ X3 @ B6 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1030_subset__eq,axiom,
% 9.49/5.23      ( ord_le6893508408891458716et_nat
% 9.49/5.23      = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 9.49/5.23          ! [X3: set_nat] :
% 9.49/5.23            ( ( member_set_nat @ X3 @ A6 )
% 9.49/5.23           => ( member_set_nat @ X3 @ B6 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1031_subset__eq,axiom,
% 9.49/5.23      ( ord_less_eq_set_nat
% 9.49/5.23      = ( ^ [A6: set_nat,B6: set_nat] :
% 9.49/5.23          ! [X3: nat] :
% 9.49/5.23            ( ( member_nat @ X3 @ A6 )
% 9.49/5.23           => ( member_nat @ X3 @ B6 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1032_subset__eq,axiom,
% 9.49/5.23      ( ord_less_eq_set_int
% 9.49/5.23      = ( ^ [A6: set_int,B6: set_int] :
% 9.49/5.23          ! [X3: int] :
% 9.49/5.23            ( ( member_int @ X3 @ A6 )
% 9.49/5.23           => ( member_int @ X3 @ B6 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1033_equalityD1,axiom,
% 9.49/5.23      ! [A2: set_int,B2: set_int] :
% 9.49/5.23        ( ( A2 = B2 )
% 9.49/5.23       => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1034_equalityD2,axiom,
% 9.49/5.23      ! [A2: set_int,B2: set_int] :
% 9.49/5.23        ( ( A2 = B2 )
% 9.49/5.23       => ( ord_less_eq_set_int @ B2 @ A2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1035_subset__iff,axiom,
% 9.49/5.23      ( ord_le3146513528884898305at_nat
% 9.49/5.23      = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 9.49/5.23          ! [T2: product_prod_nat_nat] :
% 9.49/5.23            ( ( member8440522571783428010at_nat @ T2 @ A6 )
% 9.49/5.23           => ( member8440522571783428010at_nat @ T2 @ B6 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1036_subset__iff,axiom,
% 9.49/5.23      ( ord_less_eq_set_real
% 9.49/5.23      = ( ^ [A6: set_real,B6: set_real] :
% 9.49/5.23          ! [T2: real] :
% 9.49/5.23            ( ( member_real @ T2 @ A6 )
% 9.49/5.23           => ( member_real @ T2 @ B6 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1037_subset__iff,axiom,
% 9.49/5.23      ( ord_le6893508408891458716et_nat
% 9.49/5.23      = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 9.49/5.23          ! [T2: set_nat] :
% 9.49/5.23            ( ( member_set_nat @ T2 @ A6 )
% 9.49/5.23           => ( member_set_nat @ T2 @ B6 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1038_subset__iff,axiom,
% 9.49/5.23      ( ord_less_eq_set_nat
% 9.49/5.23      = ( ^ [A6: set_nat,B6: set_nat] :
% 9.49/5.23          ! [T2: nat] :
% 9.49/5.23            ( ( member_nat @ T2 @ A6 )
% 9.49/5.23           => ( member_nat @ T2 @ B6 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1039_subset__iff,axiom,
% 9.49/5.23      ( ord_less_eq_set_int
% 9.49/5.23      = ( ^ [A6: set_int,B6: set_int] :
% 9.49/5.23          ! [T2: int] :
% 9.49/5.23            ( ( member_int @ T2 @ A6 )
% 9.49/5.23           => ( member_int @ T2 @ B6 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1040_subset__refl,axiom,
% 9.49/5.23      ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1041_Collect__mono,axiom,
% 9.49/5.23      ! [P: real > $o,Q: real > $o] :
% 9.49/5.23        ( ! [X4: real] :
% 9.49/5.23            ( ( P @ X4 )
% 9.49/5.23           => ( Q @ X4 ) )
% 9.49/5.23       => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1042_Collect__mono,axiom,
% 9.49/5.23      ! [P: list_nat > $o,Q: list_nat > $o] :
% 9.49/5.23        ( ! [X4: list_nat] :
% 9.49/5.23            ( ( P @ X4 )
% 9.49/5.23           => ( Q @ X4 ) )
% 9.49/5.23       => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1043_Collect__mono,axiom,
% 9.49/5.23      ! [P: set_nat > $o,Q: set_nat > $o] :
% 9.49/5.23        ( ! [X4: set_nat] :
% 9.49/5.23            ( ( P @ X4 )
% 9.49/5.23           => ( Q @ X4 ) )
% 9.49/5.23       => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1044_Collect__mono,axiom,
% 9.49/5.23      ! [P: nat > $o,Q: nat > $o] :
% 9.49/5.23        ( ! [X4: nat] :
% 9.49/5.23            ( ( P @ X4 )
% 9.49/5.23           => ( Q @ X4 ) )
% 9.49/5.23       => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1045_Collect__mono,axiom,
% 9.49/5.23      ! [P: int > $o,Q: int > $o] :
% 9.49/5.23        ( ! [X4: int] :
% 9.49/5.23            ( ( P @ X4 )
% 9.49/5.23           => ( Q @ X4 ) )
% 9.49/5.23       => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1046_subset__trans,axiom,
% 9.49/5.23      ! [A2: set_int,B2: set_int,C2: set_int] :
% 9.49/5.23        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.49/5.23       => ( ( ord_less_eq_set_int @ B2 @ C2 )
% 9.49/5.23         => ( ord_less_eq_set_int @ A2 @ C2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1047_set__eq__subset,axiom,
% 9.49/5.23      ( ( ^ [Y5: set_int,Z: set_int] : ( Y5 = Z ) )
% 9.49/5.23      = ( ^ [A6: set_int,B6: set_int] :
% 9.49/5.23            ( ( ord_less_eq_set_int @ A6 @ B6 )
% 9.49/5.23            & ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1048_Collect__mono__iff,axiom,
% 9.49/5.23      ! [P: real > $o,Q: real > $o] :
% 9.49/5.23        ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
% 9.49/5.23        = ( ! [X3: real] :
% 9.49/5.23              ( ( P @ X3 )
% 9.49/5.23             => ( Q @ X3 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1049_Collect__mono__iff,axiom,
% 9.49/5.23      ! [P: list_nat > $o,Q: list_nat > $o] :
% 9.49/5.23        ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
% 9.49/5.23        = ( ! [X3: list_nat] :
% 9.49/5.23              ( ( P @ X3 )
% 9.49/5.23             => ( Q @ X3 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1050_Collect__mono__iff,axiom,
% 9.49/5.23      ! [P: set_nat > $o,Q: set_nat > $o] :
% 9.49/5.23        ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
% 9.49/5.23        = ( ! [X3: set_nat] :
% 9.49/5.23              ( ( P @ X3 )
% 9.49/5.23             => ( Q @ X3 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1051_Collect__mono__iff,axiom,
% 9.49/5.23      ! [P: nat > $o,Q: nat > $o] :
% 9.49/5.23        ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
% 9.49/5.23        = ( ! [X3: nat] :
% 9.49/5.23              ( ( P @ X3 )
% 9.49/5.23             => ( Q @ X3 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1052_Collect__mono__iff,axiom,
% 9.49/5.23      ! [P: int > $o,Q: int > $o] :
% 9.49/5.23        ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
% 9.49/5.23        = ( ! [X3: int] :
% 9.49/5.23              ( ( P @ X3 )
% 9.49/5.23             => ( Q @ X3 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1053_subset__iff__psubset__eq,axiom,
% 9.49/5.23      ( ord_less_eq_set_int
% 9.49/5.23      = ( ^ [A6: set_int,B6: set_int] :
% 9.49/5.23            ( ( ord_less_set_int @ A6 @ B6 )
% 9.49/5.23            | ( A6 = B6 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1054_subset__psubset__trans,axiom,
% 9.49/5.23      ! [A2: set_int,B2: set_int,C2: set_int] :
% 9.49/5.23        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.49/5.23       => ( ( ord_less_set_int @ B2 @ C2 )
% 9.49/5.23         => ( ord_less_set_int @ A2 @ C2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1055_subset__not__subset__eq,axiom,
% 9.49/5.23      ( ord_less_set_int
% 9.49/5.23      = ( ^ [A6: set_int,B6: set_int] :
% 9.49/5.23            ( ( ord_less_eq_set_int @ A6 @ B6 )
% 9.49/5.23            & ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1056_psubset__subset__trans,axiom,
% 9.49/5.23      ! [A2: set_int,B2: set_int,C2: set_int] :
% 9.49/5.23        ( ( ord_less_set_int @ A2 @ B2 )
% 9.49/5.23       => ( ( ord_less_eq_set_int @ B2 @ C2 )
% 9.49/5.23         => ( ord_less_set_int @ A2 @ C2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1057_psubset__imp__subset,axiom,
% 9.49/5.23      ! [A2: set_int,B2: set_int] :
% 9.49/5.23        ( ( ord_less_set_int @ A2 @ B2 )
% 9.49/5.23       => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1058_psubset__eq,axiom,
% 9.49/5.23      ( ord_less_set_int
% 9.49/5.23      = ( ^ [A6: set_int,B6: set_int] :
% 9.49/5.23            ( ( ord_less_eq_set_int @ A6 @ B6 )
% 9.49/5.23            & ( A6 != B6 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1059_psubsetE,axiom,
% 9.49/5.23      ! [A2: set_int,B2: set_int] :
% 9.49/5.23        ( ( ord_less_set_int @ A2 @ B2 )
% 9.49/5.23       => ~ ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.49/5.23           => ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1060_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
% 9.49/5.23      ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
% 9.49/5.23        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X )
% 9.49/5.23        = ( ( X = Mi )
% 9.49/5.23          | ( X = Ma ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1061_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
% 9.49/5.23      ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
% 9.49/5.23        ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1062_VEBT__internal_OminNull_Osimps_I5_J,axiom,
% 9.49/5.23      ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 9.49/5.23        ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1063_tvalid,axiom,
% 9.49/5.23      vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ deg ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1064_vebt__mint_Osimps_I3_J,axiom,
% 9.49/5.23      ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 9.49/5.23        ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 9.49/5.23        = ( some_nat @ Mi ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1065_vebt__maxt_Osimps_I3_J,axiom,
% 9.49/5.23      ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 9.49/5.23        ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 9.49/5.23        = ( some_nat @ Ma ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1066_vebt__mint_Osimps_I2_J,axiom,
% 9.49/5.23      ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 9.49/5.23        ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 9.49/5.23        = none_nat ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1067_vebt__maxt_Osimps_I2_J,axiom,
% 9.49/5.23      ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 9.49/5.23        ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 9.49/5.23        = none_nat ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1068_calculation,axiom,
% 9.49/5.23      ( ( ya = mi )
% 9.49/5.23     => ( ( xa != ya )
% 9.49/5.23        & ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ ya ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1069_prod_Oinject,axiom,
% 9.49/5.23      ! [X1: int,X22: int,Y1: int,Y22: int] :
% 9.49/5.23        ( ( ( product_Pair_int_int @ X1 @ X22 )
% 9.49/5.23          = ( product_Pair_int_int @ Y1 @ Y22 ) )
% 9.49/5.23        = ( ( X1 = Y1 )
% 9.49/5.23          & ( X22 = Y22 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1070_prod_Oinject,axiom,
% 9.49/5.23      ! [X1: code_integer > option6357759511663192854e_term,X22: produc8923325533196201883nteger,Y1: code_integer > option6357759511663192854e_term,Y22: produc8923325533196201883nteger] :
% 9.49/5.23        ( ( ( produc6137756002093451184nteger @ X1 @ X22 )
% 9.49/5.23          = ( produc6137756002093451184nteger @ Y1 @ Y22 ) )
% 9.49/5.23        = ( ( X1 = Y1 )
% 9.49/5.23          & ( X22 = Y22 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1071_prod_Oinject,axiom,
% 9.49/5.23      ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X22: produc8923325533196201883nteger,Y1: produc6241069584506657477e_term > option6357759511663192854e_term,Y22: produc8923325533196201883nteger] :
% 9.49/5.23        ( ( ( produc8603105652947943368nteger @ X1 @ X22 )
% 9.49/5.23          = ( produc8603105652947943368nteger @ Y1 @ Y22 ) )
% 9.49/5.23        = ( ( X1 = Y1 )
% 9.49/5.23          & ( X22 = Y22 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1072_prod_Oinject,axiom,
% 9.49/5.23      ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X22: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y22: product_prod_int_int] :
% 9.49/5.23        ( ( ( produc5700946648718959541nt_int @ X1 @ X22 )
% 9.49/5.23          = ( produc5700946648718959541nt_int @ Y1 @ Y22 ) )
% 9.49/5.23        = ( ( X1 = Y1 )
% 9.49/5.23          & ( X22 = Y22 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1073_prod_Oinject,axiom,
% 9.49/5.23      ! [X1: int > option6357759511663192854e_term,X22: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y22: product_prod_int_int] :
% 9.49/5.23        ( ( ( produc4305682042979456191nt_int @ X1 @ X22 )
% 9.49/5.23          = ( produc4305682042979456191nt_int @ Y1 @ Y22 ) )
% 9.49/5.23        = ( ( X1 = Y1 )
% 9.49/5.23          & ( X22 = Y22 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1074_old_Oprod_Oinject,axiom,
% 9.49/5.23      ! [A: int,B: int,A7: int,B7: int] :
% 9.49/5.23        ( ( ( product_Pair_int_int @ A @ B )
% 9.49/5.23          = ( product_Pair_int_int @ A7 @ B7 ) )
% 9.49/5.23        = ( ( A = A7 )
% 9.49/5.23          & ( B = B7 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1075_old_Oprod_Oinject,axiom,
% 9.49/5.23      ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A7: code_integer > option6357759511663192854e_term,B7: produc8923325533196201883nteger] :
% 9.49/5.23        ( ( ( produc6137756002093451184nteger @ A @ B )
% 9.49/5.23          = ( produc6137756002093451184nteger @ A7 @ B7 ) )
% 9.49/5.23        = ( ( A = A7 )
% 9.49/5.23          & ( B = B7 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1076_old_Oprod_Oinject,axiom,
% 9.49/5.23      ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A7: produc6241069584506657477e_term > option6357759511663192854e_term,B7: produc8923325533196201883nteger] :
% 9.49/5.23        ( ( ( produc8603105652947943368nteger @ A @ B )
% 9.49/5.23          = ( produc8603105652947943368nteger @ A7 @ B7 ) )
% 9.49/5.23        = ( ( A = A7 )
% 9.49/5.23          & ( B = B7 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1077_old_Oprod_Oinject,axiom,
% 9.49/5.23      ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A7: produc8551481072490612790e_term > option6357759511663192854e_term,B7: product_prod_int_int] :
% 9.49/5.23        ( ( ( produc5700946648718959541nt_int @ A @ B )
% 9.49/5.23          = ( produc5700946648718959541nt_int @ A7 @ B7 ) )
% 9.49/5.23        = ( ( A = A7 )
% 9.49/5.23          & ( B = B7 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1078_old_Oprod_Oinject,axiom,
% 9.49/5.23      ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A7: int > option6357759511663192854e_term,B7: product_prod_int_int] :
% 9.49/5.23        ( ( ( produc4305682042979456191nt_int @ A @ B )
% 9.49/5.23          = ( produc4305682042979456191nt_int @ A7 @ B7 ) )
% 9.49/5.23        = ( ( A = A7 )
% 9.49/5.23          & ( B = B7 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1079_infinite__nat__iff__unbounded__le,axiom,
% 9.49/5.23      ! [S2: set_nat] :
% 9.49/5.23        ( ( ~ ( finite_finite_nat @ S2 ) )
% 9.49/5.23        = ( ! [M2: nat] :
% 9.49/5.23            ? [N2: nat] :
% 9.49/5.23              ( ( ord_less_eq_nat @ M2 @ N2 )
% 9.49/5.23              & ( member_nat @ N2 @ S2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1080_finite__nat__set__iff__bounded__le,axiom,
% 9.49/5.23      ( finite_finite_nat
% 9.49/5.23      = ( ^ [N4: set_nat] :
% 9.49/5.23          ? [M2: nat] :
% 9.49/5.23          ! [X3: nat] :
% 9.49/5.23            ( ( member_nat @ X3 @ N4 )
% 9.49/5.23           => ( ord_less_eq_nat @ X3 @ M2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1081_unbounded__k__infinite,axiom,
% 9.49/5.23      ! [K: nat,S2: set_nat] :
% 9.49/5.23        ( ! [M5: nat] :
% 9.49/5.23            ( ( ord_less_nat @ K @ M5 )
% 9.49/5.23           => ? [N5: nat] :
% 9.49/5.23                ( ( ord_less_nat @ M5 @ N5 )
% 9.49/5.23                & ( member_nat @ N5 @ S2 ) ) )
% 9.49/5.23       => ~ ( finite_finite_nat @ S2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1082__C4_Ohyps_C_I4_J,axiom,
% 9.49/5.23      ( deg
% 9.49/5.23      = ( plus_plus_nat @ na @ m ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1083_VEBT__internal_Ooption__shift_Ocases,axiom,
% 9.49/5.23      ! [X: produc8306885398267862888on_nat] :
% 9.49/5.23        ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
% 9.49/5.23            ( X
% 9.49/5.23           != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 9.49/5.23       => ( ! [Uw2: nat > nat > nat,V2: nat] :
% 9.49/5.23              ( X
% 9.49/5.23             != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 9.49/5.23         => ~ ! [F2: nat > nat > nat,A3: nat,B3: nat] :
% 9.49/5.23                ( X
% 9.49/5.23               != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A3 ) @ ( some_nat @ B3 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1084_VEBT__internal_Ooption__shift_Ocases,axiom,
% 9.49/5.23      ! [X: produc5542196010084753463at_nat] :
% 9.49/5.23        ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
% 9.49/5.23            ( X
% 9.49/5.23           != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 9.49/5.23       => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
% 9.49/5.23              ( X
% 9.49/5.23             != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 9.49/5.23         => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
% 9.49/5.23                ( X
% 9.49/5.23               != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B3 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1085_VEBT__internal_Ooption__shift_Ocases,axiom,
% 9.49/5.23      ! [X: produc1193250871479095198on_num] :
% 9.49/5.23        ( ! [Uu2: num > num > num,Uv2: option_num] :
% 9.49/5.23            ( X
% 9.49/5.23           != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 9.49/5.23       => ( ! [Uw2: num > num > num,V2: num] :
% 9.49/5.23              ( X
% 9.49/5.23             != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 9.49/5.23         => ~ ! [F2: num > num > num,A3: num,B3: num] :
% 9.49/5.23                ( X
% 9.49/5.23               != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A3 ) @ ( some_num @ B3 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1086_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 9.49/5.23      ! [X: produc2233624965454879586on_nat] :
% 9.49/5.23        ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
% 9.49/5.23            ( X
% 9.49/5.23           != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 9.49/5.23       => ( ! [Uw2: nat > nat > $o,V2: nat] :
% 9.49/5.23              ( X
% 9.49/5.23             != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 9.49/5.23         => ~ ! [F2: nat > nat > $o,X4: nat,Y3: nat] :
% 9.49/5.23                ( X
% 9.49/5.23               != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X4 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1087_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 9.49/5.23      ! [X: produc5491161045314408544at_nat] :
% 9.49/5.23        ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
% 9.49/5.23            ( X
% 9.49/5.23           != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 9.49/5.23       => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
% 9.49/5.23              ( X
% 9.49/5.23             != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 9.49/5.23         => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 9.49/5.23                ( X
% 9.49/5.23               != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X4 ) @ ( some_P7363390416028606310at_nat @ Y3 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1088_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 9.49/5.23      ! [X: produc7036089656553540234on_num] :
% 9.49/5.23        ( ! [Uu2: num > num > $o,Uv2: option_num] :
% 9.49/5.23            ( X
% 9.49/5.23           != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 9.49/5.23       => ( ! [Uw2: num > num > $o,V2: num] :
% 9.49/5.23              ( X
% 9.49/5.23             != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 9.49/5.23         => ~ ! [F2: num > num > $o,X4: num,Y3: num] :
% 9.49/5.23                ( X
% 9.49/5.23               != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X4 ) @ ( some_num @ Y3 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1089_prod__induct3,axiom,
% 9.49/5.23      ! [P: produc8763457246119570046nteger > $o,X: produc8763457246119570046nteger] :
% 9.49/5.23        ( ! [A3: code_integer > option6357759511663192854e_term,B3: code_integer,C3: code_integer] : ( P @ ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C3 ) ) )
% 9.49/5.23       => ( P @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1090_prod__induct3,axiom,
% 9.49/5.23      ! [P: produc1908205239877642774nteger > $o,X: produc1908205239877642774nteger] :
% 9.49/5.23        ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: code_integer,C3: code_integer] : ( P @ ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C3 ) ) )
% 9.49/5.23       => ( P @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1091_prod__induct3,axiom,
% 9.49/5.23      ! [P: produc2285326912895808259nt_int > $o,X: produc2285326912895808259nt_int] :
% 9.49/5.23        ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: int,C3: int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C3 ) ) )
% 9.49/5.23       => ( P @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1092_prod__induct3,axiom,
% 9.49/5.23      ! [P: produc7773217078559923341nt_int > $o,X: produc7773217078559923341nt_int] :
% 9.49/5.23        ( ! [A3: int > option6357759511663192854e_term,B3: int,C3: int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C3 ) ) )
% 9.49/5.23       => ( P @ X ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1093_prod__cases3,axiom,
% 9.49/5.23      ! [Y: produc8763457246119570046nteger] :
% 9.49/5.23        ~ ! [A3: code_integer > option6357759511663192854e_term,B3: code_integer,C3: code_integer] :
% 9.49/5.23            ( Y
% 9.49/5.23           != ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1094_prod__cases3,axiom,
% 9.49/5.23      ! [Y: produc1908205239877642774nteger] :
% 9.49/5.23        ~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: code_integer,C3: code_integer] :
% 9.49/5.23            ( Y
% 9.49/5.23           != ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1095_prod__cases3,axiom,
% 9.49/5.23      ! [Y: produc2285326912895808259nt_int] :
% 9.49/5.23        ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: int,C3: int] :
% 9.49/5.23            ( Y
% 9.49/5.23           != ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1096_prod__cases3,axiom,
% 9.49/5.23      ! [Y: produc7773217078559923341nt_int] :
% 9.49/5.23        ~ ! [A3: int > option6357759511663192854e_term,B3: int,C3: int] :
% 9.49/5.23            ( Y
% 9.49/5.23           != ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1097_Pair__inject,axiom,
% 9.49/5.23      ! [A: int,B: int,A7: int,B7: int] :
% 9.49/5.23        ( ( ( product_Pair_int_int @ A @ B )
% 9.49/5.23          = ( product_Pair_int_int @ A7 @ B7 ) )
% 9.49/5.23       => ~ ( ( A = A7 )
% 9.49/5.23           => ( B != B7 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1098_Pair__inject,axiom,
% 9.49/5.23      ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A7: code_integer > option6357759511663192854e_term,B7: produc8923325533196201883nteger] :
% 9.49/5.23        ( ( ( produc6137756002093451184nteger @ A @ B )
% 9.49/5.23          = ( produc6137756002093451184nteger @ A7 @ B7 ) )
% 9.49/5.23       => ~ ( ( A = A7 )
% 9.49/5.23           => ( B != B7 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1099_Pair__inject,axiom,
% 9.49/5.23      ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A7: produc6241069584506657477e_term > option6357759511663192854e_term,B7: produc8923325533196201883nteger] :
% 9.49/5.23        ( ( ( produc8603105652947943368nteger @ A @ B )
% 9.49/5.23          = ( produc8603105652947943368nteger @ A7 @ B7 ) )
% 9.49/5.23       => ~ ( ( A = A7 )
% 9.49/5.23           => ( B != B7 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1100_Pair__inject,axiom,
% 9.49/5.23      ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A7: produc8551481072490612790e_term > option6357759511663192854e_term,B7: product_prod_int_int] :
% 9.49/5.23        ( ( ( produc5700946648718959541nt_int @ A @ B )
% 9.49/5.23          = ( produc5700946648718959541nt_int @ A7 @ B7 ) )
% 9.49/5.23       => ~ ( ( A = A7 )
% 9.49/5.23           => ( B != B7 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1101_Pair__inject,axiom,
% 9.49/5.23      ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A7: int > option6357759511663192854e_term,B7: product_prod_int_int] :
% 9.49/5.23        ( ( ( produc4305682042979456191nt_int @ A @ B )
% 9.49/5.23          = ( produc4305682042979456191nt_int @ A7 @ B7 ) )
% 9.49/5.23       => ~ ( ( A = A7 )
% 9.49/5.23           => ( B != B7 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1102_prod__cases,axiom,
% 9.49/5.23      ! [P: product_prod_int_int > $o,P4: product_prod_int_int] :
% 9.49/5.23        ( ! [A3: int,B3: int] : ( P @ ( product_Pair_int_int @ A3 @ B3 ) )
% 9.49/5.23       => ( P @ P4 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1103_prod__cases,axiom,
% 9.49/5.23      ! [P: produc8763457246119570046nteger > $o,P4: produc8763457246119570046nteger] :
% 9.49/5.23        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] : ( P @ ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 9.49/5.23       => ( P @ P4 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1104_prod__cases,axiom,
% 9.49/5.23      ! [P: produc1908205239877642774nteger > $o,P4: produc1908205239877642774nteger] :
% 9.49/5.23        ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] : ( P @ ( produc8603105652947943368nteger @ A3 @ B3 ) )
% 9.49/5.23       => ( P @ P4 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1105_prod__cases,axiom,
% 9.49/5.23      ! [P: produc2285326912895808259nt_int > $o,P4: produc2285326912895808259nt_int] :
% 9.49/5.23        ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ B3 ) )
% 9.49/5.23       => ( P @ P4 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1106_prod__cases,axiom,
% 9.49/5.23      ! [P: produc7773217078559923341nt_int > $o,P4: produc7773217078559923341nt_int] :
% 9.49/5.23        ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ B3 ) )
% 9.49/5.23       => ( P @ P4 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1107_surj__pair,axiom,
% 9.49/5.23      ! [P4: product_prod_int_int] :
% 9.49/5.23      ? [X4: int,Y3: int] :
% 9.49/5.23        ( P4
% 9.49/5.23        = ( product_Pair_int_int @ X4 @ Y3 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1108_surj__pair,axiom,
% 9.49/5.23      ! [P4: produc8763457246119570046nteger] :
% 9.49/5.23      ? [X4: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
% 9.49/5.23        ( P4
% 9.49/5.23        = ( produc6137756002093451184nteger @ X4 @ Y3 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1109_surj__pair,axiom,
% 9.49/5.23      ! [P4: produc1908205239877642774nteger] :
% 9.49/5.23      ? [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
% 9.49/5.23        ( P4
% 9.49/5.23        = ( produc8603105652947943368nteger @ X4 @ Y3 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1110_surj__pair,axiom,
% 9.49/5.23      ! [P4: produc2285326912895808259nt_int] :
% 9.49/5.23      ? [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
% 9.49/5.23        ( P4
% 9.49/5.23        = ( produc5700946648718959541nt_int @ X4 @ Y3 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1111_surj__pair,axiom,
% 9.49/5.23      ! [P4: produc7773217078559923341nt_int] :
% 9.49/5.23      ? [X4: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
% 9.49/5.23        ( P4
% 9.49/5.23        = ( produc4305682042979456191nt_int @ X4 @ Y3 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1112_old_Oprod_Oexhaust,axiom,
% 9.49/5.23      ! [Y: product_prod_int_int] :
% 9.49/5.23        ~ ! [A3: int,B3: int] :
% 9.49/5.23            ( Y
% 9.49/5.23           != ( product_Pair_int_int @ A3 @ B3 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1113_old_Oprod_Oexhaust,axiom,
% 9.49/5.23      ! [Y: produc8763457246119570046nteger] :
% 9.49/5.23        ~ ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 9.49/5.23            ( Y
% 9.49/5.23           != ( produc6137756002093451184nteger @ A3 @ B3 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1114_old_Oprod_Oexhaust,axiom,
% 9.49/5.23      ! [Y: produc1908205239877642774nteger] :
% 9.49/5.23        ~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 9.49/5.23            ( Y
% 9.49/5.23           != ( produc8603105652947943368nteger @ A3 @ B3 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1115_old_Oprod_Oexhaust,axiom,
% 9.49/5.23      ! [Y: produc2285326912895808259nt_int] :
% 9.49/5.23        ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
% 9.49/5.23            ( Y
% 9.49/5.23           != ( produc5700946648718959541nt_int @ A3 @ B3 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1116_old_Oprod_Oexhaust,axiom,
% 9.49/5.23      ! [Y: produc7773217078559923341nt_int] :
% 9.49/5.23        ~ ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
% 9.49/5.23            ( Y
% 9.49/5.23           != ( produc4305682042979456191nt_int @ A3 @ B3 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1117_bounded__Max__nat,axiom,
% 9.49/5.23      ! [P: nat > $o,X: nat,M4: nat] :
% 9.49/5.23        ( ( P @ X )
% 9.49/5.23       => ( ! [X4: nat] :
% 9.49/5.23              ( ( P @ X4 )
% 9.49/5.23             => ( ord_less_eq_nat @ X4 @ M4 ) )
% 9.49/5.23         => ~ ! [M5: nat] :
% 9.49/5.23                ( ( P @ M5 )
% 9.49/5.23               => ~ ! [X2: nat] :
% 9.49/5.23                      ( ( P @ X2 )
% 9.49/5.23                     => ( ord_less_eq_nat @ X2 @ M5 ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1118_finite__transitivity__chain,axiom,
% 9.49/5.23      ! [A2: set_set_nat,R: set_nat > set_nat > $o] :
% 9.49/5.23        ( ( finite1152437895449049373et_nat @ A2 )
% 9.49/5.23       => ( ! [X4: set_nat] :
% 9.49/5.23              ~ ( R @ X4 @ X4 )
% 9.49/5.23         => ( ! [X4: set_nat,Y3: set_nat,Z4: set_nat] :
% 9.49/5.23                ( ( R @ X4 @ Y3 )
% 9.49/5.23               => ( ( R @ Y3 @ Z4 )
% 9.49/5.23                 => ( R @ X4 @ Z4 ) ) )
% 9.49/5.23           => ( ! [X4: set_nat] :
% 9.49/5.23                  ( ( member_set_nat @ X4 @ A2 )
% 9.49/5.23                 => ? [Y4: set_nat] :
% 9.49/5.23                      ( ( member_set_nat @ Y4 @ A2 )
% 9.49/5.23                      & ( R @ X4 @ Y4 ) ) )
% 9.49/5.23             => ( A2 = bot_bot_set_set_nat ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1119_finite__transitivity__chain,axiom,
% 9.49/5.23      ! [A2: set_complex,R: complex > complex > $o] :
% 9.49/5.23        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.23       => ( ! [X4: complex] :
% 9.49/5.23              ~ ( R @ X4 @ X4 )
% 9.49/5.23         => ( ! [X4: complex,Y3: complex,Z4: complex] :
% 9.49/5.23                ( ( R @ X4 @ Y3 )
% 9.49/5.23               => ( ( R @ Y3 @ Z4 )
% 9.49/5.23                 => ( R @ X4 @ Z4 ) ) )
% 9.49/5.23           => ( ! [X4: complex] :
% 9.49/5.23                  ( ( member_complex @ X4 @ A2 )
% 9.49/5.23                 => ? [Y4: complex] :
% 9.49/5.23                      ( ( member_complex @ Y4 @ A2 )
% 9.49/5.23                      & ( R @ X4 @ Y4 ) ) )
% 9.49/5.23             => ( A2 = bot_bot_set_complex ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1120_finite__transitivity__chain,axiom,
% 9.49/5.23      ! [A2: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 9.49/5.23        ( ( finite6177210948735845034at_nat @ A2 )
% 9.49/5.23       => ( ! [X4: product_prod_nat_nat] :
% 9.49/5.23              ~ ( R @ X4 @ X4 )
% 9.49/5.23         => ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat,Z4: product_prod_nat_nat] :
% 9.49/5.23                ( ( R @ X4 @ Y3 )
% 9.49/5.23               => ( ( R @ Y3 @ Z4 )
% 9.49/5.23                 => ( R @ X4 @ Z4 ) ) )
% 9.49/5.23           => ( ! [X4: product_prod_nat_nat] :
% 9.49/5.23                  ( ( member8440522571783428010at_nat @ X4 @ A2 )
% 9.49/5.23                 => ? [Y4: product_prod_nat_nat] :
% 9.49/5.23                      ( ( member8440522571783428010at_nat @ Y4 @ A2 )
% 9.49/5.23                      & ( R @ X4 @ Y4 ) ) )
% 9.49/5.23             => ( A2 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1121_finite__transitivity__chain,axiom,
% 9.49/5.23      ! [A2: set_real,R: real > real > $o] :
% 9.49/5.23        ( ( finite_finite_real @ A2 )
% 9.49/5.23       => ( ! [X4: real] :
% 9.49/5.23              ~ ( R @ X4 @ X4 )
% 9.49/5.23         => ( ! [X4: real,Y3: real,Z4: real] :
% 9.49/5.23                ( ( R @ X4 @ Y3 )
% 9.49/5.23               => ( ( R @ Y3 @ Z4 )
% 9.49/5.23                 => ( R @ X4 @ Z4 ) ) )
% 9.49/5.23           => ( ! [X4: real] :
% 9.49/5.23                  ( ( member_real @ X4 @ A2 )
% 9.49/5.23                 => ? [Y4: real] :
% 9.49/5.23                      ( ( member_real @ Y4 @ A2 )
% 9.49/5.23                      & ( R @ X4 @ Y4 ) ) )
% 9.49/5.23             => ( A2 = bot_bot_set_real ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1122_finite__transitivity__chain,axiom,
% 9.49/5.23      ! [A2: set_nat,R: nat > nat > $o] :
% 9.49/5.23        ( ( finite_finite_nat @ A2 )
% 9.49/5.23       => ( ! [X4: nat] :
% 9.49/5.23              ~ ( R @ X4 @ X4 )
% 9.49/5.23         => ( ! [X4: nat,Y3: nat,Z4: nat] :
% 9.49/5.23                ( ( R @ X4 @ Y3 )
% 9.49/5.23               => ( ( R @ Y3 @ Z4 )
% 9.49/5.23                 => ( R @ X4 @ Z4 ) ) )
% 9.49/5.23           => ( ! [X4: nat] :
% 9.49/5.23                  ( ( member_nat @ X4 @ A2 )
% 9.49/5.23                 => ? [Y4: nat] :
% 9.49/5.23                      ( ( member_nat @ Y4 @ A2 )
% 9.49/5.23                      & ( R @ X4 @ Y4 ) ) )
% 9.49/5.23             => ( A2 = bot_bot_set_nat ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1123_finite__transitivity__chain,axiom,
% 9.49/5.23      ! [A2: set_int,R: int > int > $o] :
% 9.49/5.23        ( ( finite_finite_int @ A2 )
% 9.49/5.23       => ( ! [X4: int] :
% 9.49/5.23              ~ ( R @ X4 @ X4 )
% 9.49/5.23         => ( ! [X4: int,Y3: int,Z4: int] :
% 9.49/5.23                ( ( R @ X4 @ Y3 )
% 9.49/5.23               => ( ( R @ Y3 @ Z4 )
% 9.49/5.23                 => ( R @ X4 @ Z4 ) ) )
% 9.49/5.23           => ( ! [X4: int] :
% 9.49/5.23                  ( ( member_int @ X4 @ A2 )
% 9.49/5.23                 => ? [Y4: int] :
% 9.49/5.23                      ( ( member_int @ Y4 @ A2 )
% 9.49/5.23                      & ( R @ X4 @ Y4 ) ) )
% 9.49/5.23             => ( A2 = bot_bot_set_int ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1124_finite__nat__set__iff__bounded,axiom,
% 9.49/5.23      ( finite_finite_nat
% 9.49/5.23      = ( ^ [N4: set_nat] :
% 9.49/5.23          ? [M2: nat] :
% 9.49/5.23          ! [X3: nat] :
% 9.49/5.23            ( ( member_nat @ X3 @ N4 )
% 9.49/5.23           => ( ord_less_nat @ X3 @ M2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1125_infinite__nat__iff__unbounded,axiom,
% 9.49/5.23      ! [S2: set_nat] :
% 9.49/5.23        ( ( ~ ( finite_finite_nat @ S2 ) )
% 9.49/5.23        = ( ! [M2: nat] :
% 9.49/5.23            ? [N2: nat] :
% 9.49/5.23              ( ( ord_less_nat @ M2 @ N2 )
% 9.49/5.23              & ( member_nat @ N2 @ S2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1126_bounded__nat__set__is__finite,axiom,
% 9.49/5.23      ! [N6: set_nat,N: nat] :
% 9.49/5.23        ( ! [X4: nat] :
% 9.49/5.23            ( ( member_nat @ X4 @ N6 )
% 9.49/5.23           => ( ord_less_nat @ X4 @ N ) )
% 9.49/5.23       => ( finite_finite_nat @ N6 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1127_Collect__empty__eq__bot,axiom,
% 9.49/5.23      ! [P: list_nat > $o] :
% 9.49/5.23        ( ( ( collect_list_nat @ P )
% 9.49/5.23          = bot_bot_set_list_nat )
% 9.49/5.23        = ( P = bot_bot_list_nat_o ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1128_Collect__empty__eq__bot,axiom,
% 9.49/5.23      ! [P: set_nat > $o] :
% 9.49/5.23        ( ( ( collect_set_nat @ P )
% 9.49/5.23          = bot_bot_set_set_nat )
% 9.49/5.23        = ( P = bot_bot_set_nat_o ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1129_Collect__empty__eq__bot,axiom,
% 9.49/5.23      ! [P: product_prod_nat_nat > $o] :
% 9.49/5.23        ( ( ( collec3392354462482085612at_nat @ P )
% 9.49/5.23          = bot_bo2099793752762293965at_nat )
% 9.49/5.23        = ( P = bot_bo482883023278783056_nat_o ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1130_Collect__empty__eq__bot,axiom,
% 9.49/5.23      ! [P: real > $o] :
% 9.49/5.23        ( ( ( collect_real @ P )
% 9.49/5.23          = bot_bot_set_real )
% 9.49/5.23        = ( P = bot_bot_real_o ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1131_Collect__empty__eq__bot,axiom,
% 9.49/5.23      ! [P: nat > $o] :
% 9.49/5.23        ( ( ( collect_nat @ P )
% 9.49/5.23          = bot_bot_set_nat )
% 9.49/5.23        = ( P = bot_bot_nat_o ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1132_Collect__empty__eq__bot,axiom,
% 9.49/5.23      ! [P: int > $o] :
% 9.49/5.23        ( ( ( collect_int @ P )
% 9.49/5.23          = bot_bot_set_int )
% 9.49/5.23        = ( P = bot_bot_int_o ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1133_bot__empty__eq,axiom,
% 9.49/5.23      ( bot_bot_set_nat_o
% 9.49/5.23      = ( ^ [X3: set_nat] : ( member_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1134_bot__empty__eq,axiom,
% 9.49/5.23      ( bot_bo482883023278783056_nat_o
% 9.49/5.23      = ( ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1135_bot__empty__eq,axiom,
% 9.49/5.23      ( bot_bot_real_o
% 9.49/5.23      = ( ^ [X3: real] : ( member_real @ X3 @ bot_bot_set_real ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1136_bot__empty__eq,axiom,
% 9.49/5.23      ( bot_bot_nat_o
% 9.49/5.23      = ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1137_bot__empty__eq,axiom,
% 9.49/5.23      ( bot_bot_int_o
% 9.49/5.23      = ( ^ [X3: int] : ( member_int @ X3 @ bot_bot_set_int ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1138_arg__min__if__finite_I2_J,axiom,
% 9.49/5.23      ! [S2: set_complex,F: complex > real] :
% 9.49/5.23        ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_complex )
% 9.49/5.23         => ~ ? [X2: complex] :
% 9.49/5.23                ( ( member_complex @ X2 @ S2 )
% 9.49/5.23                & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic8794016678065449205x_real @ F @ S2 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1139_arg__min__if__finite_I2_J,axiom,
% 9.49/5.23      ! [S2: set_real,F: real > real] :
% 9.49/5.23        ( ( finite_finite_real @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_real )
% 9.49/5.23         => ~ ? [X2: real] :
% 9.49/5.23                ( ( member_real @ X2 @ S2 )
% 9.49/5.23                & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic8440615504127631091l_real @ F @ S2 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1140_arg__min__if__finite_I2_J,axiom,
% 9.49/5.23      ! [S2: set_nat,F: nat > real] :
% 9.49/5.23        ( ( finite_finite_nat @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_nat )
% 9.49/5.23         => ~ ? [X2: nat] :
% 9.49/5.23                ( ( member_nat @ X2 @ S2 )
% 9.49/5.23                & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic488527866317076247t_real @ F @ S2 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1141_arg__min__if__finite_I2_J,axiom,
% 9.49/5.23      ! [S2: set_int,F: int > real] :
% 9.49/5.23        ( ( finite_finite_int @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_int )
% 9.49/5.23         => ~ ? [X2: int] :
% 9.49/5.23                ( ( member_int @ X2 @ S2 )
% 9.49/5.23                & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic2675449441010098035t_real @ F @ S2 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1142_arg__min__if__finite_I2_J,axiom,
% 9.49/5.23      ! [S2: set_complex,F: complex > rat] :
% 9.49/5.23        ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_complex )
% 9.49/5.23         => ~ ? [X2: complex] :
% 9.49/5.23                ( ( member_complex @ X2 @ S2 )
% 9.49/5.23                & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S2 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1143_arg__min__if__finite_I2_J,axiom,
% 9.49/5.23      ! [S2: set_real,F: real > rat] :
% 9.49/5.23        ( ( finite_finite_real @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_real )
% 9.49/5.23         => ~ ? [X2: real] :
% 9.49/5.23                ( ( member_real @ X2 @ S2 )
% 9.49/5.23                & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic4420706379359479199al_rat @ F @ S2 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1144_arg__min__if__finite_I2_J,axiom,
% 9.49/5.23      ! [S2: set_nat,F: nat > rat] :
% 9.49/5.23        ( ( finite_finite_nat @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_nat )
% 9.49/5.23         => ~ ? [X2: nat] :
% 9.49/5.23                ( ( member_nat @ X2 @ S2 )
% 9.49/5.23                & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic6811802900495863747at_rat @ F @ S2 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1145_arg__min__if__finite_I2_J,axiom,
% 9.49/5.23      ! [S2: set_int,F: int > rat] :
% 9.49/5.23        ( ( finite_finite_int @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_int )
% 9.49/5.23         => ~ ? [X2: int] :
% 9.49/5.23                ( ( member_int @ X2 @ S2 )
% 9.49/5.23                & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S2 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1146_arg__min__if__finite_I2_J,axiom,
% 9.49/5.23      ! [S2: set_complex,F: complex > num] :
% 9.49/5.23        ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_complex )
% 9.49/5.23         => ~ ? [X2: complex] :
% 9.49/5.23                ( ( member_complex @ X2 @ S2 )
% 9.49/5.23                & ( ord_less_num @ ( F @ X2 ) @ ( F @ ( lattic1922116423962787043ex_num @ F @ S2 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1147_arg__min__if__finite_I2_J,axiom,
% 9.49/5.23      ! [S2: set_real,F: real > num] :
% 9.49/5.23        ( ( finite_finite_real @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_real )
% 9.49/5.23         => ~ ? [X2: real] :
% 9.49/5.23                ( ( member_real @ X2 @ S2 )
% 9.49/5.23                & ( ord_less_num @ ( F @ X2 ) @ ( F @ ( lattic1613168225601753569al_num @ F @ S2 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1148_arg__min__least,axiom,
% 9.49/5.23      ! [S2: set_complex,Y: complex,F: complex > rat] :
% 9.49/5.23        ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_complex )
% 9.49/5.23         => ( ( member_complex @ Y @ S2 )
% 9.49/5.23           => ( ord_less_eq_rat @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1149_arg__min__least,axiom,
% 9.49/5.23      ! [S2: set_real,Y: real,F: real > rat] :
% 9.49/5.23        ( ( finite_finite_real @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_real )
% 9.49/5.23         => ( ( member_real @ Y @ S2 )
% 9.49/5.23           => ( ord_less_eq_rat @ ( F @ ( lattic4420706379359479199al_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1150_arg__min__least,axiom,
% 9.49/5.23      ! [S2: set_nat,Y: nat,F: nat > rat] :
% 9.49/5.23        ( ( finite_finite_nat @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_nat )
% 9.49/5.23         => ( ( member_nat @ Y @ S2 )
% 9.49/5.23           => ( ord_less_eq_rat @ ( F @ ( lattic6811802900495863747at_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1151_arg__min__least,axiom,
% 9.49/5.23      ! [S2: set_int,Y: int,F: int > rat] :
% 9.49/5.23        ( ( finite_finite_int @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_int )
% 9.49/5.23         => ( ( member_int @ Y @ S2 )
% 9.49/5.23           => ( ord_less_eq_rat @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1152_arg__min__least,axiom,
% 9.49/5.23      ! [S2: set_complex,Y: complex,F: complex > num] :
% 9.49/5.23        ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_complex )
% 9.49/5.23         => ( ( member_complex @ Y @ S2 )
% 9.49/5.23           => ( ord_less_eq_num @ ( F @ ( lattic1922116423962787043ex_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1153_arg__min__least,axiom,
% 9.49/5.23      ! [S2: set_real,Y: real,F: real > num] :
% 9.49/5.23        ( ( finite_finite_real @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_real )
% 9.49/5.23         => ( ( member_real @ Y @ S2 )
% 9.49/5.23           => ( ord_less_eq_num @ ( F @ ( lattic1613168225601753569al_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1154_arg__min__least,axiom,
% 9.49/5.23      ! [S2: set_nat,Y: nat,F: nat > num] :
% 9.49/5.23        ( ( finite_finite_nat @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_nat )
% 9.49/5.23         => ( ( member_nat @ Y @ S2 )
% 9.49/5.23           => ( ord_less_eq_num @ ( F @ ( lattic4004264746738138117at_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1155_arg__min__least,axiom,
% 9.49/5.23      ! [S2: set_int,Y: int,F: int > num] :
% 9.49/5.23        ( ( finite_finite_int @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_int )
% 9.49/5.23         => ( ( member_int @ Y @ S2 )
% 9.49/5.23           => ( ord_less_eq_num @ ( F @ ( lattic5003618458639192673nt_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1156_arg__min__least,axiom,
% 9.49/5.23      ! [S2: set_complex,Y: complex,F: complex > nat] :
% 9.49/5.23        ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_complex )
% 9.49/5.23         => ( ( member_complex @ Y @ S2 )
% 9.49/5.23           => ( ord_less_eq_nat @ ( F @ ( lattic5364784637807008409ex_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1157_arg__min__least,axiom,
% 9.49/5.23      ! [S2: set_real,Y: real,F: real > nat] :
% 9.49/5.23        ( ( finite_finite_real @ S2 )
% 9.49/5.23       => ( ( S2 != bot_bot_set_real )
% 9.49/5.23         => ( ( member_real @ Y @ S2 )
% 9.49/5.23           => ( ord_less_eq_nat @ ( F @ ( lattic5055836439445974935al_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1158_nat__descend__induct,axiom,
% 9.49/5.23      ! [N: nat,P: nat > $o,M: nat] :
% 9.49/5.23        ( ! [K2: nat] :
% 9.49/5.23            ( ( ord_less_nat @ N @ K2 )
% 9.49/5.23           => ( P @ K2 ) )
% 9.49/5.23       => ( ! [K2: nat] :
% 9.49/5.23              ( ( ord_less_eq_nat @ K2 @ N )
% 9.49/5.23             => ( ! [I3: nat] :
% 9.49/5.23                    ( ( ord_less_nat @ K2 @ I3 )
% 9.49/5.23                   => ( P @ I3 ) )
% 9.49/5.23               => ( P @ K2 ) ) )
% 9.49/5.23         => ( P @ M ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1159_subset__emptyI,axiom,
% 9.49/5.23      ! [A2: set_set_nat] :
% 9.49/5.23        ( ! [X4: set_nat] :
% 9.49/5.23            ~ ( member_set_nat @ X4 @ A2 )
% 9.49/5.23       => ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1160_subset__emptyI,axiom,
% 9.49/5.23      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.23        ( ! [X4: product_prod_nat_nat] :
% 9.49/5.23            ~ ( member8440522571783428010at_nat @ X4 @ A2 )
% 9.49/5.23       => ( ord_le3146513528884898305at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1161_subset__emptyI,axiom,
% 9.49/5.23      ! [A2: set_real] :
% 9.49/5.23        ( ! [X4: real] :
% 9.49/5.23            ~ ( member_real @ X4 @ A2 )
% 9.49/5.23       => ( ord_less_eq_set_real @ A2 @ bot_bot_set_real ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1162_subset__emptyI,axiom,
% 9.49/5.23      ! [A2: set_nat] :
% 9.49/5.23        ( ! [X4: nat] :
% 9.49/5.23            ~ ( member_nat @ X4 @ A2 )
% 9.49/5.23       => ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1163_subset__emptyI,axiom,
% 9.49/5.23      ! [A2: set_int] :
% 9.49/5.23        ( ! [X4: int] :
% 9.49/5.23            ~ ( member_int @ X4 @ A2 )
% 9.49/5.23       => ( ord_less_eq_set_int @ A2 @ bot_bot_set_int ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1164_field__lbound__gt__zero,axiom,
% 9.49/5.23      ! [D1: real,D2: real] :
% 9.49/5.23        ( ( ord_less_real @ zero_zero_real @ D1 )
% 9.49/5.23       => ( ( ord_less_real @ zero_zero_real @ D2 )
% 9.49/5.23         => ? [E: real] :
% 9.49/5.23              ( ( ord_less_real @ zero_zero_real @ E )
% 9.49/5.23              & ( ord_less_real @ E @ D1 )
% 9.49/5.23              & ( ord_less_real @ E @ D2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1165_field__lbound__gt__zero,axiom,
% 9.49/5.23      ! [D1: rat,D2: rat] :
% 9.49/5.23        ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 9.49/5.23       => ( ( ord_less_rat @ zero_zero_rat @ D2 )
% 9.49/5.23         => ? [E: rat] :
% 9.49/5.23              ( ( ord_less_rat @ zero_zero_rat @ E )
% 9.49/5.23              & ( ord_less_rat @ E @ D1 )
% 9.49/5.23              & ( ord_less_rat @ E @ D2 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1166_less__numeral__extra_I3_J,axiom,
% 9.49/5.23      ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1167_less__numeral__extra_I3_J,axiom,
% 9.49/5.23      ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1168_less__numeral__extra_I3_J,axiom,
% 9.49/5.23      ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1169_less__numeral__extra_I3_J,axiom,
% 9.49/5.23      ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1170_complete__interval,axiom,
% 9.49/5.23      ! [A: real,B: real,P: real > $o] :
% 9.49/5.23        ( ( ord_less_real @ A @ B )
% 9.49/5.23       => ( ( P @ A )
% 9.49/5.23         => ( ~ ( P @ B )
% 9.49/5.23           => ? [C3: real] :
% 9.49/5.23                ( ( ord_less_eq_real @ A @ C3 )
% 9.49/5.23                & ( ord_less_eq_real @ C3 @ B )
% 9.49/5.23                & ! [X2: real] :
% 9.49/5.23                    ( ( ( ord_less_eq_real @ A @ X2 )
% 9.49/5.23                      & ( ord_less_real @ X2 @ C3 ) )
% 9.49/5.23                   => ( P @ X2 ) )
% 9.49/5.23                & ! [D3: real] :
% 9.49/5.23                    ( ! [X4: real] :
% 9.49/5.23                        ( ( ( ord_less_eq_real @ A @ X4 )
% 9.49/5.23                          & ( ord_less_real @ X4 @ D3 ) )
% 9.49/5.23                       => ( P @ X4 ) )
% 9.49/5.23                   => ( ord_less_eq_real @ D3 @ C3 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1171_complete__interval,axiom,
% 9.49/5.23      ! [A: nat,B: nat,P: nat > $o] :
% 9.49/5.23        ( ( ord_less_nat @ A @ B )
% 9.49/5.23       => ( ( P @ A )
% 9.49/5.23         => ( ~ ( P @ B )
% 9.49/5.23           => ? [C3: nat] :
% 9.49/5.23                ( ( ord_less_eq_nat @ A @ C3 )
% 9.49/5.23                & ( ord_less_eq_nat @ C3 @ B )
% 9.49/5.23                & ! [X2: nat] :
% 9.49/5.23                    ( ( ( ord_less_eq_nat @ A @ X2 )
% 9.49/5.23                      & ( ord_less_nat @ X2 @ C3 ) )
% 9.49/5.23                   => ( P @ X2 ) )
% 9.49/5.23                & ! [D3: nat] :
% 9.49/5.23                    ( ! [X4: nat] :
% 9.49/5.23                        ( ( ( ord_less_eq_nat @ A @ X4 )
% 9.49/5.23                          & ( ord_less_nat @ X4 @ D3 ) )
% 9.49/5.23                       => ( P @ X4 ) )
% 9.49/5.23                   => ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1172_complete__interval,axiom,
% 9.49/5.23      ! [A: int,B: int,P: int > $o] :
% 9.49/5.23        ( ( ord_less_int @ A @ B )
% 9.49/5.23       => ( ( P @ A )
% 9.49/5.23         => ( ~ ( P @ B )
% 9.49/5.23           => ? [C3: int] :
% 9.49/5.23                ( ( ord_less_eq_int @ A @ C3 )
% 9.49/5.23                & ( ord_less_eq_int @ C3 @ B )
% 9.49/5.23                & ! [X2: int] :
% 9.49/5.23                    ( ( ( ord_less_eq_int @ A @ X2 )
% 9.49/5.23                      & ( ord_less_int @ X2 @ C3 ) )
% 9.49/5.23                   => ( P @ X2 ) )
% 9.49/5.23                & ! [D3: int] :
% 9.49/5.23                    ( ! [X4: int] :
% 9.49/5.23                        ( ( ( ord_less_eq_int @ A @ X4 )
% 9.49/5.23                          & ( ord_less_int @ X4 @ D3 ) )
% 9.49/5.23                       => ( P @ X4 ) )
% 9.49/5.23                   => ( ord_less_eq_int @ D3 @ C3 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1173_verit__comp__simplify1_I3_J,axiom,
% 9.49/5.23      ! [B7: real,A7: real] :
% 9.49/5.23        ( ( ~ ( ord_less_eq_real @ B7 @ A7 ) )
% 9.49/5.23        = ( ord_less_real @ A7 @ B7 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1174_verit__comp__simplify1_I3_J,axiom,
% 9.49/5.23      ! [B7: rat,A7: rat] :
% 9.49/5.23        ( ( ~ ( ord_less_eq_rat @ B7 @ A7 ) )
% 9.49/5.23        = ( ord_less_rat @ A7 @ B7 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1175_verit__comp__simplify1_I3_J,axiom,
% 9.49/5.23      ! [B7: num,A7: num] :
% 9.49/5.23        ( ( ~ ( ord_less_eq_num @ B7 @ A7 ) )
% 9.49/5.23        = ( ord_less_num @ A7 @ B7 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1176_verit__comp__simplify1_I3_J,axiom,
% 9.49/5.23      ! [B7: nat,A7: nat] :
% 9.49/5.23        ( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
% 9.49/5.23        = ( ord_less_nat @ A7 @ B7 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1177_verit__comp__simplify1_I3_J,axiom,
% 9.49/5.23      ! [B7: int,A7: int] :
% 9.49/5.23        ( ( ~ ( ord_less_eq_int @ B7 @ A7 ) )
% 9.49/5.23        = ( ord_less_int @ A7 @ B7 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1178_add__right__cancel,axiom,
% 9.49/5.23      ! [B: real,A: real,C: real] :
% 9.49/5.23        ( ( ( plus_plus_real @ B @ A )
% 9.49/5.23          = ( plus_plus_real @ C @ A ) )
% 9.49/5.23        = ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1179_add__right__cancel,axiom,
% 9.49/5.23      ! [B: rat,A: rat,C: rat] :
% 9.49/5.23        ( ( ( plus_plus_rat @ B @ A )
% 9.49/5.23          = ( plus_plus_rat @ C @ A ) )
% 9.49/5.23        = ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1180_add__right__cancel,axiom,
% 9.49/5.23      ! [B: nat,A: nat,C: nat] :
% 9.49/5.23        ( ( ( plus_plus_nat @ B @ A )
% 9.49/5.23          = ( plus_plus_nat @ C @ A ) )
% 9.49/5.23        = ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1181_add__right__cancel,axiom,
% 9.49/5.23      ! [B: int,A: int,C: int] :
% 9.49/5.23        ( ( ( plus_plus_int @ B @ A )
% 9.49/5.23          = ( plus_plus_int @ C @ A ) )
% 9.49/5.23        = ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1182_add__left__cancel,axiom,
% 9.49/5.23      ! [A: real,B: real,C: real] :
% 9.49/5.23        ( ( ( plus_plus_real @ A @ B )
% 9.49/5.23          = ( plus_plus_real @ A @ C ) )
% 9.49/5.23        = ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1183_add__left__cancel,axiom,
% 9.49/5.23      ! [A: rat,B: rat,C: rat] :
% 9.49/5.23        ( ( ( plus_plus_rat @ A @ B )
% 9.49/5.23          = ( plus_plus_rat @ A @ C ) )
% 9.49/5.23        = ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1184_add__left__cancel,axiom,
% 9.49/5.23      ! [A: nat,B: nat,C: nat] :
% 9.49/5.23        ( ( ( plus_plus_nat @ A @ B )
% 9.49/5.23          = ( plus_plus_nat @ A @ C ) )
% 9.49/5.23        = ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1185_add__left__cancel,axiom,
% 9.49/5.23      ! [A: int,B: int,C: int] :
% 9.49/5.23        ( ( ( plus_plus_int @ A @ B )
% 9.49/5.23          = ( plus_plus_int @ A @ C ) )
% 9.49/5.23        = ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1186_add__le__cancel__left,axiom,
% 9.49/5.23      ! [C: real,A: real,B: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 9.49/5.23        = ( ord_less_eq_real @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1187_add__le__cancel__left,axiom,
% 9.49/5.23      ! [C: rat,A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 9.49/5.23        = ( ord_less_eq_rat @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1188_add__le__cancel__left,axiom,
% 9.49/5.23      ! [C: nat,A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 9.49/5.23        = ( ord_less_eq_nat @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1189_add__le__cancel__left,axiom,
% 9.49/5.23      ! [C: int,A: int,B: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 9.49/5.23        = ( ord_less_eq_int @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1190_add__le__cancel__right,axiom,
% 9.49/5.23      ! [A: real,C: real,B: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 9.49/5.23        = ( ord_less_eq_real @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1191_add__le__cancel__right,axiom,
% 9.49/5.23      ! [A: rat,C: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 9.49/5.23        = ( ord_less_eq_rat @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1192_add__le__cancel__right,axiom,
% 9.49/5.23      ! [A: nat,C: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 9.49/5.23        = ( ord_less_eq_nat @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1193_add__le__cancel__right,axiom,
% 9.49/5.23      ! [A: int,C: int,B: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 9.49/5.23        = ( ord_less_eq_int @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1194_add_Oright__neutral,axiom,
% 9.49/5.23      ! [A: complex] :
% 9.49/5.23        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1195_add_Oright__neutral,axiom,
% 9.49/5.23      ! [A: real] :
% 9.49/5.23        ( ( plus_plus_real @ A @ zero_zero_real )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1196_add_Oright__neutral,axiom,
% 9.49/5.23      ! [A: rat] :
% 9.49/5.23        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1197_add_Oright__neutral,axiom,
% 9.49/5.23      ! [A: nat] :
% 9.49/5.23        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1198_add_Oright__neutral,axiom,
% 9.49/5.23      ! [A: int] :
% 9.49/5.23        ( ( plus_plus_int @ A @ zero_zero_int )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1199_double__zero__sym,axiom,
% 9.49/5.23      ! [A: real] :
% 9.49/5.23        ( ( zero_zero_real
% 9.49/5.23          = ( plus_plus_real @ A @ A ) )
% 9.49/5.23        = ( A = zero_zero_real ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1200_double__zero__sym,axiom,
% 9.49/5.23      ! [A: rat] :
% 9.49/5.23        ( ( zero_zero_rat
% 9.49/5.23          = ( plus_plus_rat @ A @ A ) )
% 9.49/5.23        = ( A = zero_zero_rat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1201_double__zero__sym,axiom,
% 9.49/5.23      ! [A: int] :
% 9.49/5.23        ( ( zero_zero_int
% 9.49/5.23          = ( plus_plus_int @ A @ A ) )
% 9.49/5.23        = ( A = zero_zero_int ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1202_add__cancel__left__left,axiom,
% 9.49/5.23      ! [B: complex,A: complex] :
% 9.49/5.23        ( ( ( plus_plus_complex @ B @ A )
% 9.49/5.23          = A )
% 9.49/5.23        = ( B = zero_zero_complex ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1203_add__cancel__left__left,axiom,
% 9.49/5.23      ! [B: real,A: real] :
% 9.49/5.23        ( ( ( plus_plus_real @ B @ A )
% 9.49/5.23          = A )
% 9.49/5.23        = ( B = zero_zero_real ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1204_add__cancel__left__left,axiom,
% 9.49/5.23      ! [B: rat,A: rat] :
% 9.49/5.23        ( ( ( plus_plus_rat @ B @ A )
% 9.49/5.23          = A )
% 9.49/5.23        = ( B = zero_zero_rat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1205_add__cancel__left__left,axiom,
% 9.49/5.23      ! [B: nat,A: nat] :
% 9.49/5.23        ( ( ( plus_plus_nat @ B @ A )
% 9.49/5.23          = A )
% 9.49/5.23        = ( B = zero_zero_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1206_add__cancel__left__left,axiom,
% 9.49/5.23      ! [B: int,A: int] :
% 9.49/5.23        ( ( ( plus_plus_int @ B @ A )
% 9.49/5.23          = A )
% 9.49/5.23        = ( B = zero_zero_int ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1207_add__cancel__left__right,axiom,
% 9.49/5.23      ! [A: complex,B: complex] :
% 9.49/5.23        ( ( ( plus_plus_complex @ A @ B )
% 9.49/5.23          = A )
% 9.49/5.23        = ( B = zero_zero_complex ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1208_add__cancel__left__right,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( ( plus_plus_real @ A @ B )
% 9.49/5.23          = A )
% 9.49/5.23        = ( B = zero_zero_real ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1209_add__cancel__left__right,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( ( plus_plus_rat @ A @ B )
% 9.49/5.23          = A )
% 9.49/5.23        = ( B = zero_zero_rat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1210_add__cancel__left__right,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( ( plus_plus_nat @ A @ B )
% 9.49/5.23          = A )
% 9.49/5.23        = ( B = zero_zero_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1211_add__cancel__left__right,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( ( plus_plus_int @ A @ B )
% 9.49/5.23          = A )
% 9.49/5.23        = ( B = zero_zero_int ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1212_add__cancel__right__left,axiom,
% 9.49/5.23      ! [A: complex,B: complex] :
% 9.49/5.23        ( ( A
% 9.49/5.23          = ( plus_plus_complex @ B @ A ) )
% 9.49/5.23        = ( B = zero_zero_complex ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1213_add__cancel__right__left,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( A
% 9.49/5.23          = ( plus_plus_real @ B @ A ) )
% 9.49/5.23        = ( B = zero_zero_real ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1214_add__cancel__right__left,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( A
% 9.49/5.23          = ( plus_plus_rat @ B @ A ) )
% 9.49/5.23        = ( B = zero_zero_rat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1215_add__cancel__right__left,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( A
% 9.49/5.23          = ( plus_plus_nat @ B @ A ) )
% 9.49/5.23        = ( B = zero_zero_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1216_add__cancel__right__left,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( A
% 9.49/5.23          = ( plus_plus_int @ B @ A ) )
% 9.49/5.23        = ( B = zero_zero_int ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1217_add__cancel__right__right,axiom,
% 9.49/5.23      ! [A: complex,B: complex] :
% 9.49/5.23        ( ( A
% 9.49/5.23          = ( plus_plus_complex @ A @ B ) )
% 9.49/5.23        = ( B = zero_zero_complex ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1218_add__cancel__right__right,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( A
% 9.49/5.23          = ( plus_plus_real @ A @ B ) )
% 9.49/5.23        = ( B = zero_zero_real ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1219_add__cancel__right__right,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( A
% 9.49/5.23          = ( plus_plus_rat @ A @ B ) )
% 9.49/5.23        = ( B = zero_zero_rat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1220_add__cancel__right__right,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( A
% 9.49/5.23          = ( plus_plus_nat @ A @ B ) )
% 9.49/5.23        = ( B = zero_zero_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1221_add__cancel__right__right,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( A
% 9.49/5.23          = ( plus_plus_int @ A @ B ) )
% 9.49/5.23        = ( B = zero_zero_int ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1222_add__eq__0__iff__both__eq__0,axiom,
% 9.49/5.23      ! [X: nat,Y: nat] :
% 9.49/5.23        ( ( ( plus_plus_nat @ X @ Y )
% 9.49/5.23          = zero_zero_nat )
% 9.49/5.23        = ( ( X = zero_zero_nat )
% 9.49/5.23          & ( Y = zero_zero_nat ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1223_zero__eq__add__iff__both__eq__0,axiom,
% 9.49/5.23      ! [X: nat,Y: nat] :
% 9.49/5.23        ( ( zero_zero_nat
% 9.49/5.23          = ( plus_plus_nat @ X @ Y ) )
% 9.49/5.23        = ( ( X = zero_zero_nat )
% 9.49/5.23          & ( Y = zero_zero_nat ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1224_add__0,axiom,
% 9.49/5.23      ! [A: complex] :
% 9.49/5.23        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1225_add__0,axiom,
% 9.49/5.23      ! [A: real] :
% 9.49/5.23        ( ( plus_plus_real @ zero_zero_real @ A )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1226_add__0,axiom,
% 9.49/5.23      ! [A: rat] :
% 9.49/5.23        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1227_add__0,axiom,
% 9.49/5.23      ! [A: nat] :
% 9.49/5.23        ( ( plus_plus_nat @ zero_zero_nat @ A )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1228_add__0,axiom,
% 9.49/5.23      ! [A: int] :
% 9.49/5.23        ( ( plus_plus_int @ zero_zero_int @ A )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1229_add__less__cancel__left,axiom,
% 9.49/5.23      ! [C: real,A: real,B: real] :
% 9.49/5.23        ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 9.49/5.23        = ( ord_less_real @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1230_add__less__cancel__left,axiom,
% 9.49/5.23      ! [C: rat,A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 9.49/5.23        = ( ord_less_rat @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1231_add__less__cancel__left,axiom,
% 9.49/5.23      ! [C: nat,A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 9.49/5.23        = ( ord_less_nat @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1232_add__less__cancel__left,axiom,
% 9.49/5.23      ! [C: int,A: int,B: int] :
% 9.49/5.23        ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 9.49/5.23        = ( ord_less_int @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1233_add__less__cancel__right,axiom,
% 9.49/5.23      ! [A: real,C: real,B: real] :
% 9.49/5.23        ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 9.49/5.23        = ( ord_less_real @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1234_add__less__cancel__right,axiom,
% 9.49/5.23      ! [A: rat,C: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 9.49/5.23        = ( ord_less_rat @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1235_add__less__cancel__right,axiom,
% 9.49/5.23      ! [A: nat,C: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 9.49/5.23        = ( ord_less_nat @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1236_add__less__cancel__right,axiom,
% 9.49/5.23      ! [A: int,C: int,B: int] :
% 9.49/5.23        ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 9.49/5.23        = ( ord_less_int @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1237_add__is__0,axiom,
% 9.49/5.23      ! [M: nat,N: nat] :
% 9.49/5.23        ( ( ( plus_plus_nat @ M @ N )
% 9.49/5.23          = zero_zero_nat )
% 9.49/5.23        = ( ( M = zero_zero_nat )
% 9.49/5.23          & ( N = zero_zero_nat ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1238_Nat_Oadd__0__right,axiom,
% 9.49/5.23      ! [M: nat] :
% 9.49/5.23        ( ( plus_plus_nat @ M @ zero_zero_nat )
% 9.49/5.23        = M ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1239_nat__add__left__cancel__less,axiom,
% 9.49/5.23      ! [K: nat,M: nat,N: nat] :
% 9.49/5.23        ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 9.49/5.23        = ( ord_less_nat @ M @ N ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1240_nat__add__left__cancel__le,axiom,
% 9.49/5.23      ! [K: nat,M: nat,N: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 9.49/5.23        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1241_zero__le__double__add__iff__zero__le__single__add,axiom,
% 9.49/5.23      ! [A: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 9.49/5.23        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1242_zero__le__double__add__iff__zero__le__single__add,axiom,
% 9.49/5.23      ! [A: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 9.49/5.23        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1243_zero__le__double__add__iff__zero__le__single__add,axiom,
% 9.49/5.23      ! [A: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 9.49/5.23        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1244_double__add__le__zero__iff__single__add__le__zero,axiom,
% 9.49/5.23      ! [A: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 9.49/5.23        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1245_double__add__le__zero__iff__single__add__le__zero,axiom,
% 9.49/5.23      ! [A: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 9.49/5.23        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1246_double__add__le__zero__iff__single__add__le__zero,axiom,
% 9.49/5.23      ! [A: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 9.49/5.23        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1247_le__add__same__cancel2,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 9.49/5.23        = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1248_le__add__same__cancel2,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 9.49/5.23        = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1249_le__add__same__cancel2,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 9.49/5.23        = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1250_le__add__same__cancel2,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
% 9.49/5.23        = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1251_le__add__same__cancel1,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 9.49/5.23        = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1252_le__add__same__cancel1,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 9.49/5.23        = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1253_le__add__same__cancel1,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 9.49/5.23        = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1254_le__add__same__cancel1,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
% 9.49/5.23        = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1255_add__le__same__cancel2,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 9.49/5.23        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1256_add__le__same__cancel2,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 9.49/5.23        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1257_add__le__same__cancel2,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 9.49/5.23        = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1258_add__le__same__cancel2,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
% 9.49/5.23        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1259_add__le__same__cancel1,axiom,
% 9.49/5.23      ! [B: real,A: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 9.49/5.23        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1260_add__le__same__cancel1,axiom,
% 9.49/5.23      ! [B: rat,A: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 9.49/5.23        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1261_add__le__same__cancel1,axiom,
% 9.49/5.23      ! [B: nat,A: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 9.49/5.23        = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1262_add__le__same__cancel1,axiom,
% 9.49/5.23      ! [B: int,A: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
% 9.49/5.23        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1263_add__less__same__cancel1,axiom,
% 9.49/5.23      ! [B: real,A: real] :
% 9.49/5.23        ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
% 9.49/5.23        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1264_add__less__same__cancel1,axiom,
% 9.49/5.23      ! [B: rat,A: rat] :
% 9.49/5.23        ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 9.49/5.23        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1265_add__less__same__cancel1,axiom,
% 9.49/5.23      ! [B: nat,A: nat] :
% 9.49/5.23        ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 9.49/5.23        = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1266_add__less__same__cancel1,axiom,
% 9.49/5.23      ! [B: int,A: int] :
% 9.49/5.23        ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
% 9.49/5.23        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1267_add__less__same__cancel2,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
% 9.49/5.23        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1268_add__less__same__cancel2,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 9.49/5.23        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1269_add__less__same__cancel2,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 9.49/5.23        = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1270_add__less__same__cancel2,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
% 9.49/5.23        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1271_less__add__same__cancel1,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
% 9.49/5.23        = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1272_less__add__same__cancel1,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 9.49/5.23        = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1273_less__add__same__cancel1,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 9.49/5.23        = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1274_less__add__same__cancel1,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
% 9.49/5.23        = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1275_less__add__same__cancel2,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
% 9.49/5.23        = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1276_less__add__same__cancel2,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 9.49/5.23        = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1277_less__add__same__cancel2,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 9.49/5.23        = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1278_less__add__same__cancel2,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
% 9.49/5.23        = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1279_double__add__less__zero__iff__single__add__less__zero,axiom,
% 9.49/5.23      ! [A: real] :
% 9.49/5.23        ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 9.49/5.23        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1280_double__add__less__zero__iff__single__add__less__zero,axiom,
% 9.49/5.23      ! [A: rat] :
% 9.49/5.23        ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 9.49/5.23        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1281_double__add__less__zero__iff__single__add__less__zero,axiom,
% 9.49/5.23      ! [A: int] :
% 9.49/5.23        ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 9.49/5.23        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1282_zero__less__double__add__iff__zero__less__single__add,axiom,
% 9.49/5.23      ! [A: real] :
% 9.49/5.23        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 9.49/5.23        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1283_zero__less__double__add__iff__zero__less__single__add,axiom,
% 9.49/5.23      ! [A: rat] :
% 9.49/5.23        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 9.49/5.23        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1284_zero__less__double__add__iff__zero__less__single__add,axiom,
% 9.49/5.23      ! [A: int] :
% 9.49/5.23        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 9.49/5.23        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1285_add__gr__0,axiom,
% 9.49/5.23      ! [M: nat,N: nat] :
% 9.49/5.23        ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 9.49/5.23        = ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.49/5.23          | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1286_add__right__imp__eq,axiom,
% 9.49/5.23      ! [B: real,A: real,C: real] :
% 9.49/5.23        ( ( ( plus_plus_real @ B @ A )
% 9.49/5.23          = ( plus_plus_real @ C @ A ) )
% 9.49/5.23       => ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1287_add__right__imp__eq,axiom,
% 9.49/5.23      ! [B: rat,A: rat,C: rat] :
% 9.49/5.23        ( ( ( plus_plus_rat @ B @ A )
% 9.49/5.23          = ( plus_plus_rat @ C @ A ) )
% 9.49/5.23       => ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1288_add__right__imp__eq,axiom,
% 9.49/5.23      ! [B: nat,A: nat,C: nat] :
% 9.49/5.23        ( ( ( plus_plus_nat @ B @ A )
% 9.49/5.23          = ( plus_plus_nat @ C @ A ) )
% 9.49/5.23       => ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1289_add__right__imp__eq,axiom,
% 9.49/5.23      ! [B: int,A: int,C: int] :
% 9.49/5.23        ( ( ( plus_plus_int @ B @ A )
% 9.49/5.23          = ( plus_plus_int @ C @ A ) )
% 9.49/5.23       => ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1290_add__left__imp__eq,axiom,
% 9.49/5.23      ! [A: real,B: real,C: real] :
% 9.49/5.23        ( ( ( plus_plus_real @ A @ B )
% 9.49/5.23          = ( plus_plus_real @ A @ C ) )
% 9.49/5.23       => ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1291_add__left__imp__eq,axiom,
% 9.49/5.23      ! [A: rat,B: rat,C: rat] :
% 9.49/5.23        ( ( ( plus_plus_rat @ A @ B )
% 9.49/5.23          = ( plus_plus_rat @ A @ C ) )
% 9.49/5.23       => ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1292_add__left__imp__eq,axiom,
% 9.49/5.23      ! [A: nat,B: nat,C: nat] :
% 9.49/5.23        ( ( ( plus_plus_nat @ A @ B )
% 9.49/5.23          = ( plus_plus_nat @ A @ C ) )
% 9.49/5.23       => ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1293_add__left__imp__eq,axiom,
% 9.49/5.23      ! [A: int,B: int,C: int] :
% 9.49/5.23        ( ( ( plus_plus_int @ A @ B )
% 9.49/5.23          = ( plus_plus_int @ A @ C ) )
% 9.49/5.23       => ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1294_add_Oleft__commute,axiom,
% 9.49/5.23      ! [B: real,A: real,C: real] :
% 9.49/5.23        ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 9.49/5.23        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1295_add_Oleft__commute,axiom,
% 9.49/5.23      ! [B: rat,A: rat,C: rat] :
% 9.49/5.23        ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
% 9.49/5.23        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1296_add_Oleft__commute,axiom,
% 9.49/5.23      ! [B: nat,A: nat,C: nat] :
% 9.49/5.23        ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 9.49/5.23        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1297_add_Oleft__commute,axiom,
% 9.49/5.23      ! [B: int,A: int,C: int] :
% 9.49/5.23        ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 9.49/5.23        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1298_add_Ocommute,axiom,
% 9.49/5.23      ( plus_plus_real
% 9.49/5.23      = ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1299_add_Ocommute,axiom,
% 9.49/5.23      ( plus_plus_rat
% 9.49/5.23      = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ B4 @ A4 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1300_add_Ocommute,axiom,
% 9.49/5.23      ( plus_plus_nat
% 9.49/5.23      = ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1301_add_Ocommute,axiom,
% 9.49/5.23      ( plus_plus_int
% 9.49/5.23      = ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1302_add_Oright__cancel,axiom,
% 9.49/5.23      ! [B: real,A: real,C: real] :
% 9.49/5.23        ( ( ( plus_plus_real @ B @ A )
% 9.49/5.23          = ( plus_plus_real @ C @ A ) )
% 9.49/5.23        = ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1303_add_Oright__cancel,axiom,
% 9.49/5.23      ! [B: rat,A: rat,C: rat] :
% 9.49/5.23        ( ( ( plus_plus_rat @ B @ A )
% 9.49/5.23          = ( plus_plus_rat @ C @ A ) )
% 9.49/5.23        = ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1304_add_Oright__cancel,axiom,
% 9.49/5.23      ! [B: int,A: int,C: int] :
% 9.49/5.23        ( ( ( plus_plus_int @ B @ A )
% 9.49/5.23          = ( plus_plus_int @ C @ A ) )
% 9.49/5.23        = ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1305_add_Oleft__cancel,axiom,
% 9.49/5.23      ! [A: real,B: real,C: real] :
% 9.49/5.23        ( ( ( plus_plus_real @ A @ B )
% 9.49/5.23          = ( plus_plus_real @ A @ C ) )
% 9.49/5.23        = ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1306_add_Oleft__cancel,axiom,
% 9.49/5.23      ! [A: rat,B: rat,C: rat] :
% 9.49/5.23        ( ( ( plus_plus_rat @ A @ B )
% 9.49/5.23          = ( plus_plus_rat @ A @ C ) )
% 9.49/5.23        = ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1307_add_Oleft__cancel,axiom,
% 9.49/5.23      ! [A: int,B: int,C: int] :
% 9.49/5.23        ( ( ( plus_plus_int @ A @ B )
% 9.49/5.23          = ( plus_plus_int @ A @ C ) )
% 9.49/5.23        = ( B = C ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1308_add_Oassoc,axiom,
% 9.49/5.23      ! [A: real,B: real,C: real] :
% 9.49/5.23        ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 9.49/5.23        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1309_add_Oassoc,axiom,
% 9.49/5.23      ! [A: rat,B: rat,C: rat] :
% 9.49/5.23        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 9.49/5.23        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1310_add_Oassoc,axiom,
% 9.49/5.23      ! [A: nat,B: nat,C: nat] :
% 9.49/5.23        ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 9.49/5.23        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1311_add_Oassoc,axiom,
% 9.49/5.23      ! [A: int,B: int,C: int] :
% 9.49/5.23        ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.49/5.23        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1312_group__cancel_Oadd2,axiom,
% 9.49/5.23      ! [B2: real,K: real,B: real,A: real] :
% 9.49/5.23        ( ( B2
% 9.49/5.23          = ( plus_plus_real @ K @ B ) )
% 9.49/5.23       => ( ( plus_plus_real @ A @ B2 )
% 9.49/5.23          = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1313_group__cancel_Oadd2,axiom,
% 9.49/5.23      ! [B2: rat,K: rat,B: rat,A: rat] :
% 9.49/5.23        ( ( B2
% 9.49/5.23          = ( plus_plus_rat @ K @ B ) )
% 9.49/5.23       => ( ( plus_plus_rat @ A @ B2 )
% 9.49/5.23          = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1314_group__cancel_Oadd2,axiom,
% 9.49/5.23      ! [B2: nat,K: nat,B: nat,A: nat] :
% 9.49/5.23        ( ( B2
% 9.49/5.23          = ( plus_plus_nat @ K @ B ) )
% 9.49/5.23       => ( ( plus_plus_nat @ A @ B2 )
% 9.49/5.23          = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1315_group__cancel_Oadd2,axiom,
% 9.49/5.23      ! [B2: int,K: int,B: int,A: int] :
% 9.49/5.23        ( ( B2
% 9.49/5.23          = ( plus_plus_int @ K @ B ) )
% 9.49/5.23       => ( ( plus_plus_int @ A @ B2 )
% 9.49/5.23          = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1316_group__cancel_Oadd1,axiom,
% 9.49/5.23      ! [A2: real,K: real,A: real,B: real] :
% 9.49/5.23        ( ( A2
% 9.49/5.23          = ( plus_plus_real @ K @ A ) )
% 9.49/5.23       => ( ( plus_plus_real @ A2 @ B )
% 9.49/5.23          = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1317_group__cancel_Oadd1,axiom,
% 9.49/5.23      ! [A2: rat,K: rat,A: rat,B: rat] :
% 9.49/5.23        ( ( A2
% 9.49/5.23          = ( plus_plus_rat @ K @ A ) )
% 9.49/5.23       => ( ( plus_plus_rat @ A2 @ B )
% 9.49/5.23          = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1318_group__cancel_Oadd1,axiom,
% 9.49/5.23      ! [A2: nat,K: nat,A: nat,B: nat] :
% 9.49/5.23        ( ( A2
% 9.49/5.23          = ( plus_plus_nat @ K @ A ) )
% 9.49/5.23       => ( ( plus_plus_nat @ A2 @ B )
% 9.49/5.23          = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1319_group__cancel_Oadd1,axiom,
% 9.49/5.23      ! [A2: int,K: int,A: int,B: int] :
% 9.49/5.23        ( ( A2
% 9.49/5.23          = ( plus_plus_int @ K @ A ) )
% 9.49/5.23       => ( ( plus_plus_int @ A2 @ B )
% 9.49/5.23          = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1320_add__mono__thms__linordered__semiring_I4_J,axiom,
% 9.49/5.23      ! [I: real,J: real,K: real,L: real] :
% 9.49/5.23        ( ( ( I = J )
% 9.49/5.23          & ( K = L ) )
% 9.49/5.23       => ( ( plus_plus_real @ I @ K )
% 9.49/5.23          = ( plus_plus_real @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1321_add__mono__thms__linordered__semiring_I4_J,axiom,
% 9.49/5.23      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.49/5.23        ( ( ( I = J )
% 9.49/5.23          & ( K = L ) )
% 9.49/5.23       => ( ( plus_plus_rat @ I @ K )
% 9.49/5.23          = ( plus_plus_rat @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1322_add__mono__thms__linordered__semiring_I4_J,axiom,
% 9.49/5.23      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.49/5.23        ( ( ( I = J )
% 9.49/5.23          & ( K = L ) )
% 9.49/5.23       => ( ( plus_plus_nat @ I @ K )
% 9.49/5.23          = ( plus_plus_nat @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1323_add__mono__thms__linordered__semiring_I4_J,axiom,
% 9.49/5.23      ! [I: int,J: int,K: int,L: int] :
% 9.49/5.23        ( ( ( I = J )
% 9.49/5.23          & ( K = L ) )
% 9.49/5.23       => ( ( plus_plus_int @ I @ K )
% 9.49/5.23          = ( plus_plus_int @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1324_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.49/5.23      ! [A: real,B: real,C: real] :
% 9.49/5.23        ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 9.49/5.23        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1325_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.49/5.23      ! [A: rat,B: rat,C: rat] :
% 9.49/5.23        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 9.49/5.23        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1326_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.49/5.23      ! [A: nat,B: nat,C: nat] :
% 9.49/5.23        ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 9.49/5.23        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1327_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.49/5.23      ! [A: int,B: int,C: int] :
% 9.49/5.23        ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.49/5.23        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1328_verit__sum__simplify,axiom,
% 9.49/5.23      ! [A: complex] :
% 9.49/5.23        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1329_verit__sum__simplify,axiom,
% 9.49/5.23      ! [A: real] :
% 9.49/5.23        ( ( plus_plus_real @ A @ zero_zero_real )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1330_verit__sum__simplify,axiom,
% 9.49/5.23      ! [A: rat] :
% 9.49/5.23        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1331_verit__sum__simplify,axiom,
% 9.49/5.23      ! [A: nat] :
% 9.49/5.23        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1332_verit__sum__simplify,axiom,
% 9.49/5.23      ! [A: int] :
% 9.49/5.23        ( ( plus_plus_int @ A @ zero_zero_int )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1333_add__mono__thms__linordered__semiring_I3_J,axiom,
% 9.49/5.23      ! [I: real,J: real,K: real,L: real] :
% 9.49/5.23        ( ( ( ord_less_eq_real @ I @ J )
% 9.49/5.23          & ( K = L ) )
% 9.49/5.23       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1334_add__mono__thms__linordered__semiring_I3_J,axiom,
% 9.49/5.23      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.49/5.23        ( ( ( ord_less_eq_rat @ I @ J )
% 9.49/5.23          & ( K = L ) )
% 9.49/5.23       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1335_add__mono__thms__linordered__semiring_I3_J,axiom,
% 9.49/5.23      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.49/5.23        ( ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.23          & ( K = L ) )
% 9.49/5.23       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1336_add__mono__thms__linordered__semiring_I3_J,axiom,
% 9.49/5.23      ! [I: int,J: int,K: int,L: int] :
% 9.49/5.23        ( ( ( ord_less_eq_int @ I @ J )
% 9.49/5.23          & ( K = L ) )
% 9.49/5.23       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1337_add__mono__thms__linordered__semiring_I2_J,axiom,
% 9.49/5.23      ! [I: real,J: real,K: real,L: real] :
% 9.49/5.23        ( ( ( I = J )
% 9.49/5.23          & ( ord_less_eq_real @ K @ L ) )
% 9.49/5.23       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1338_add__mono__thms__linordered__semiring_I2_J,axiom,
% 9.49/5.23      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.49/5.23        ( ( ( I = J )
% 9.49/5.23          & ( ord_less_eq_rat @ K @ L ) )
% 9.49/5.23       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1339_add__mono__thms__linordered__semiring_I2_J,axiom,
% 9.49/5.23      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.49/5.23        ( ( ( I = J )
% 9.49/5.23          & ( ord_less_eq_nat @ K @ L ) )
% 9.49/5.23       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1340_add__mono__thms__linordered__semiring_I2_J,axiom,
% 9.49/5.23      ! [I: int,J: int,K: int,L: int] :
% 9.49/5.23        ( ( ( I = J )
% 9.49/5.23          & ( ord_less_eq_int @ K @ L ) )
% 9.49/5.23       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1341_add__mono__thms__linordered__semiring_I1_J,axiom,
% 9.49/5.23      ! [I: real,J: real,K: real,L: real] :
% 9.49/5.23        ( ( ( ord_less_eq_real @ I @ J )
% 9.49/5.23          & ( ord_less_eq_real @ K @ L ) )
% 9.49/5.23       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1342_add__mono__thms__linordered__semiring_I1_J,axiom,
% 9.49/5.23      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.49/5.23        ( ( ( ord_less_eq_rat @ I @ J )
% 9.49/5.23          & ( ord_less_eq_rat @ K @ L ) )
% 9.49/5.23       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1343_add__mono__thms__linordered__semiring_I1_J,axiom,
% 9.49/5.23      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.49/5.23        ( ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.23          & ( ord_less_eq_nat @ K @ L ) )
% 9.49/5.23       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1344_add__mono__thms__linordered__semiring_I1_J,axiom,
% 9.49/5.23      ! [I: int,J: int,K: int,L: int] :
% 9.49/5.23        ( ( ( ord_less_eq_int @ I @ J )
% 9.49/5.23          & ( ord_less_eq_int @ K @ L ) )
% 9.49/5.23       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1345_add__mono,axiom,
% 9.49/5.23      ! [A: real,B: real,C: real,D: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_real @ C @ D )
% 9.49/5.23         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1346_add__mono,axiom,
% 9.49/5.23      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_rat @ C @ D )
% 9.49/5.23         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1347_add__mono,axiom,
% 9.49/5.23      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_nat @ C @ D )
% 9.49/5.23         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1348_add__mono,axiom,
% 9.49/5.23      ! [A: int,B: int,C: int,D: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_int @ C @ D )
% 9.49/5.23         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1349_add__left__mono,axiom,
% 9.49/5.23      ! [A: real,B: real,C: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.23       => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1350_add__left__mono,axiom,
% 9.49/5.23      ! [A: rat,B: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.23       => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1351_add__left__mono,axiom,
% 9.49/5.23      ! [A: nat,B: nat,C: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.23       => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1352_add__left__mono,axiom,
% 9.49/5.23      ! [A: int,B: int,C: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.23       => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1353_less__eqE,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.23       => ~ ! [C3: nat] :
% 9.49/5.23              ( B
% 9.49/5.23             != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1354_add__right__mono,axiom,
% 9.49/5.23      ! [A: real,B: real,C: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.23       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1355_add__right__mono,axiom,
% 9.49/5.23      ! [A: rat,B: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.23       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1356_add__right__mono,axiom,
% 9.49/5.23      ! [A: nat,B: nat,C: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.23       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1357_add__right__mono,axiom,
% 9.49/5.23      ! [A: int,B: int,C: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.23       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1358_le__iff__add,axiom,
% 9.49/5.23      ( ord_less_eq_nat
% 9.49/5.23      = ( ^ [A4: nat,B4: nat] :
% 9.49/5.23          ? [C4: nat] :
% 9.49/5.23            ( B4
% 9.49/5.23            = ( plus_plus_nat @ A4 @ C4 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1359_add__le__imp__le__left,axiom,
% 9.49/5.23      ! [C: real,A: real,B: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 9.49/5.23       => ( ord_less_eq_real @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1360_add__le__imp__le__left,axiom,
% 9.49/5.23      ! [C: rat,A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 9.49/5.23       => ( ord_less_eq_rat @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1361_add__le__imp__le__left,axiom,
% 9.49/5.23      ! [C: nat,A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 9.49/5.23       => ( ord_less_eq_nat @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1362_add__le__imp__le__left,axiom,
% 9.49/5.23      ! [C: int,A: int,B: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 9.49/5.23       => ( ord_less_eq_int @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1363_add__le__imp__le__right,axiom,
% 9.49/5.23      ! [A: real,C: real,B: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 9.49/5.23       => ( ord_less_eq_real @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1364_add__le__imp__le__right,axiom,
% 9.49/5.23      ! [A: rat,C: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 9.49/5.23       => ( ord_less_eq_rat @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1365_add__le__imp__le__right,axiom,
% 9.49/5.23      ! [A: nat,C: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 9.49/5.23       => ( ord_less_eq_nat @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1366_add__le__imp__le__right,axiom,
% 9.49/5.23      ! [A: int,C: int,B: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 9.49/5.23       => ( ord_less_eq_int @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1367_comm__monoid__add__class_Oadd__0,axiom,
% 9.49/5.23      ! [A: complex] :
% 9.49/5.23        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1368_comm__monoid__add__class_Oadd__0,axiom,
% 9.49/5.23      ! [A: real] :
% 9.49/5.23        ( ( plus_plus_real @ zero_zero_real @ A )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1369_comm__monoid__add__class_Oadd__0,axiom,
% 9.49/5.23      ! [A: rat] :
% 9.49/5.23        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1370_comm__monoid__add__class_Oadd__0,axiom,
% 9.49/5.23      ! [A: nat] :
% 9.49/5.23        ( ( plus_plus_nat @ zero_zero_nat @ A )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1371_comm__monoid__add__class_Oadd__0,axiom,
% 9.49/5.23      ! [A: int] :
% 9.49/5.23        ( ( plus_plus_int @ zero_zero_int @ A )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1372_add_Ocomm__neutral,axiom,
% 9.49/5.23      ! [A: complex] :
% 9.49/5.23        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1373_add_Ocomm__neutral,axiom,
% 9.49/5.23      ! [A: real] :
% 9.49/5.23        ( ( plus_plus_real @ A @ zero_zero_real )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1374_add_Ocomm__neutral,axiom,
% 9.49/5.23      ! [A: rat] :
% 9.49/5.23        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1375_add_Ocomm__neutral,axiom,
% 9.49/5.23      ! [A: nat] :
% 9.49/5.23        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1376_add_Ocomm__neutral,axiom,
% 9.49/5.23      ! [A: int] :
% 9.49/5.23        ( ( plus_plus_int @ A @ zero_zero_int )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1377_add_Ogroup__left__neutral,axiom,
% 9.49/5.23      ! [A: complex] :
% 9.49/5.23        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1378_add_Ogroup__left__neutral,axiom,
% 9.49/5.23      ! [A: real] :
% 9.49/5.23        ( ( plus_plus_real @ zero_zero_real @ A )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1379_add_Ogroup__left__neutral,axiom,
% 9.49/5.23      ! [A: rat] :
% 9.49/5.23        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1380_add_Ogroup__left__neutral,axiom,
% 9.49/5.23      ! [A: int] :
% 9.49/5.23        ( ( plus_plus_int @ zero_zero_int @ A )
% 9.49/5.23        = A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1381_add__mono__thms__linordered__field_I5_J,axiom,
% 9.49/5.23      ! [I: real,J: real,K: real,L: real] :
% 9.49/5.23        ( ( ( ord_less_real @ I @ J )
% 9.49/5.23          & ( ord_less_real @ K @ L ) )
% 9.49/5.23       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1382_add__mono__thms__linordered__field_I5_J,axiom,
% 9.49/5.23      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.49/5.23        ( ( ( ord_less_rat @ I @ J )
% 9.49/5.23          & ( ord_less_rat @ K @ L ) )
% 9.49/5.23       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1383_add__mono__thms__linordered__field_I5_J,axiom,
% 9.49/5.23      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.49/5.23        ( ( ( ord_less_nat @ I @ J )
% 9.49/5.23          & ( ord_less_nat @ K @ L ) )
% 9.49/5.23       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1384_add__mono__thms__linordered__field_I5_J,axiom,
% 9.49/5.23      ! [I: int,J: int,K: int,L: int] :
% 9.49/5.23        ( ( ( ord_less_int @ I @ J )
% 9.49/5.23          & ( ord_less_int @ K @ L ) )
% 9.49/5.23       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1385_add__mono__thms__linordered__field_I2_J,axiom,
% 9.49/5.23      ! [I: real,J: real,K: real,L: real] :
% 9.49/5.23        ( ( ( I = J )
% 9.49/5.23          & ( ord_less_real @ K @ L ) )
% 9.49/5.23       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1386_add__mono__thms__linordered__field_I2_J,axiom,
% 9.49/5.23      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.49/5.23        ( ( ( I = J )
% 9.49/5.23          & ( ord_less_rat @ K @ L ) )
% 9.49/5.23       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1387_add__mono__thms__linordered__field_I2_J,axiom,
% 9.49/5.23      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.49/5.23        ( ( ( I = J )
% 9.49/5.23          & ( ord_less_nat @ K @ L ) )
% 9.49/5.23       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1388_add__mono__thms__linordered__field_I2_J,axiom,
% 9.49/5.23      ! [I: int,J: int,K: int,L: int] :
% 9.49/5.23        ( ( ( I = J )
% 9.49/5.23          & ( ord_less_int @ K @ L ) )
% 9.49/5.23       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1389_add__mono__thms__linordered__field_I1_J,axiom,
% 9.49/5.23      ! [I: real,J: real,K: real,L: real] :
% 9.49/5.23        ( ( ( ord_less_real @ I @ J )
% 9.49/5.23          & ( K = L ) )
% 9.49/5.23       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1390_add__mono__thms__linordered__field_I1_J,axiom,
% 9.49/5.23      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.49/5.23        ( ( ( ord_less_rat @ I @ J )
% 9.49/5.23          & ( K = L ) )
% 9.49/5.23       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1391_add__mono__thms__linordered__field_I1_J,axiom,
% 9.49/5.23      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.49/5.23        ( ( ( ord_less_nat @ I @ J )
% 9.49/5.23          & ( K = L ) )
% 9.49/5.23       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1392_add__mono__thms__linordered__field_I1_J,axiom,
% 9.49/5.23      ! [I: int,J: int,K: int,L: int] :
% 9.49/5.23        ( ( ( ord_less_int @ I @ J )
% 9.49/5.23          & ( K = L ) )
% 9.49/5.23       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1393_add__strict__mono,axiom,
% 9.49/5.23      ! [A: real,B: real,C: real,D: real] :
% 9.49/5.23        ( ( ord_less_real @ A @ B )
% 9.49/5.23       => ( ( ord_less_real @ C @ D )
% 9.49/5.23         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1394_add__strict__mono,axiom,
% 9.49/5.23      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.49/5.23        ( ( ord_less_rat @ A @ B )
% 9.49/5.23       => ( ( ord_less_rat @ C @ D )
% 9.49/5.23         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1395_add__strict__mono,axiom,
% 9.49/5.23      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.49/5.23        ( ( ord_less_nat @ A @ B )
% 9.49/5.23       => ( ( ord_less_nat @ C @ D )
% 9.49/5.23         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1396_add__strict__mono,axiom,
% 9.49/5.23      ! [A: int,B: int,C: int,D: int] :
% 9.49/5.23        ( ( ord_less_int @ A @ B )
% 9.49/5.23       => ( ( ord_less_int @ C @ D )
% 9.49/5.23         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1397_add__strict__left__mono,axiom,
% 9.49/5.23      ! [A: real,B: real,C: real] :
% 9.49/5.23        ( ( ord_less_real @ A @ B )
% 9.49/5.23       => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1398_add__strict__left__mono,axiom,
% 9.49/5.23      ! [A: rat,B: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_rat @ A @ B )
% 9.49/5.23       => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1399_add__strict__left__mono,axiom,
% 9.49/5.23      ! [A: nat,B: nat,C: nat] :
% 9.49/5.23        ( ( ord_less_nat @ A @ B )
% 9.49/5.23       => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1400_add__strict__left__mono,axiom,
% 9.49/5.23      ! [A: int,B: int,C: int] :
% 9.49/5.23        ( ( ord_less_int @ A @ B )
% 9.49/5.23       => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1401_add__strict__right__mono,axiom,
% 9.49/5.23      ! [A: real,B: real,C: real] :
% 9.49/5.23        ( ( ord_less_real @ A @ B )
% 9.49/5.23       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1402_add__strict__right__mono,axiom,
% 9.49/5.23      ! [A: rat,B: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_rat @ A @ B )
% 9.49/5.23       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1403_add__strict__right__mono,axiom,
% 9.49/5.23      ! [A: nat,B: nat,C: nat] :
% 9.49/5.23        ( ( ord_less_nat @ A @ B )
% 9.49/5.23       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1404_add__strict__right__mono,axiom,
% 9.49/5.23      ! [A: int,B: int,C: int] :
% 9.49/5.23        ( ( ord_less_int @ A @ B )
% 9.49/5.23       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1405_add__less__imp__less__left,axiom,
% 9.49/5.23      ! [C: real,A: real,B: real] :
% 9.49/5.23        ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 9.49/5.23       => ( ord_less_real @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1406_add__less__imp__less__left,axiom,
% 9.49/5.23      ! [C: rat,A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 9.49/5.23       => ( ord_less_rat @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1407_add__less__imp__less__left,axiom,
% 9.49/5.23      ! [C: nat,A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 9.49/5.23       => ( ord_less_nat @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1408_add__less__imp__less__left,axiom,
% 9.49/5.23      ! [C: int,A: int,B: int] :
% 9.49/5.23        ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 9.49/5.23       => ( ord_less_int @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1409_add__less__imp__less__right,axiom,
% 9.49/5.23      ! [A: real,C: real,B: real] :
% 9.49/5.23        ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 9.49/5.23       => ( ord_less_real @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1410_add__less__imp__less__right,axiom,
% 9.49/5.23      ! [A: rat,C: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 9.49/5.23       => ( ord_less_rat @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1411_add__less__imp__less__right,axiom,
% 9.49/5.23      ! [A: nat,C: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 9.49/5.23       => ( ord_less_nat @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1412_add__less__imp__less__right,axiom,
% 9.49/5.23      ! [A: int,C: int,B: int] :
% 9.49/5.23        ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 9.49/5.23       => ( ord_less_int @ A @ B ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1413_plus__nat_Oadd__0,axiom,
% 9.49/5.23      ! [N: nat] :
% 9.49/5.23        ( ( plus_plus_nat @ zero_zero_nat @ N )
% 9.49/5.23        = N ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1414_add__eq__self__zero,axiom,
% 9.49/5.23      ! [M: nat,N: nat] :
% 9.49/5.23        ( ( ( plus_plus_nat @ M @ N )
% 9.49/5.23          = M )
% 9.49/5.23       => ( N = zero_zero_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1415_add__lessD1,axiom,
% 9.49/5.23      ! [I: nat,J: nat,K: nat] :
% 9.49/5.23        ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 9.49/5.23       => ( ord_less_nat @ I @ K ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1416_add__less__mono,axiom,
% 9.49/5.23      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.49/5.23        ( ( ord_less_nat @ I @ J )
% 9.49/5.23       => ( ( ord_less_nat @ K @ L )
% 9.49/5.23         => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1417_not__add__less1,axiom,
% 9.49/5.23      ! [I: nat,J: nat] :
% 9.49/5.23        ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1418_not__add__less2,axiom,
% 9.49/5.23      ! [J: nat,I: nat] :
% 9.49/5.23        ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1419_add__less__mono1,axiom,
% 9.49/5.23      ! [I: nat,J: nat,K: nat] :
% 9.49/5.23        ( ( ord_less_nat @ I @ J )
% 9.49/5.23       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1420_trans__less__add1,axiom,
% 9.49/5.23      ! [I: nat,J: nat,M: nat] :
% 9.49/5.23        ( ( ord_less_nat @ I @ J )
% 9.49/5.23       => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1421_trans__less__add2,axiom,
% 9.49/5.23      ! [I: nat,J: nat,M: nat] :
% 9.49/5.23        ( ( ord_less_nat @ I @ J )
% 9.49/5.23       => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1422_less__add__eq__less,axiom,
% 9.49/5.23      ! [K: nat,L: nat,M: nat,N: nat] :
% 9.49/5.23        ( ( ord_less_nat @ K @ L )
% 9.49/5.23       => ( ( ( plus_plus_nat @ M @ L )
% 9.49/5.23            = ( plus_plus_nat @ K @ N ) )
% 9.49/5.23         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1423_add__leE,axiom,
% 9.49/5.23      ! [M: nat,K: nat,N: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 9.49/5.23       => ~ ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.23           => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1424_le__add1,axiom,
% 9.49/5.23      ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1425_le__add2,axiom,
% 9.49/5.23      ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1426_add__leD1,axiom,
% 9.49/5.23      ! [M: nat,K: nat,N: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 9.49/5.23       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1427_add__leD2,axiom,
% 9.49/5.23      ! [M: nat,K: nat,N: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 9.49/5.23       => ( ord_less_eq_nat @ K @ N ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1428_le__Suc__ex,axiom,
% 9.49/5.23      ! [K: nat,L: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ K @ L )
% 9.49/5.23       => ? [N3: nat] :
% 9.49/5.23            ( L
% 9.49/5.23            = ( plus_plus_nat @ K @ N3 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1429_add__le__mono,axiom,
% 9.49/5.23      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.23       => ( ( ord_less_eq_nat @ K @ L )
% 9.49/5.23         => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1430_add__le__mono1,axiom,
% 9.49/5.23      ! [I: nat,J: nat,K: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.23       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1431_trans__le__add1,axiom,
% 9.49/5.23      ! [I: nat,J: nat,M: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.23       => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1432_trans__le__add2,axiom,
% 9.49/5.23      ! [I: nat,J: nat,M: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.23       => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1433_nat__le__iff__add,axiom,
% 9.49/5.23      ( ord_less_eq_nat
% 9.49/5.23      = ( ^ [M2: nat,N2: nat] :
% 9.49/5.23          ? [K3: nat] :
% 9.49/5.23            ( N2
% 9.49/5.23            = ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1434_add__nonpos__eq__0__iff,axiom,
% 9.49/5.23      ! [X: real,Y: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.49/5.23       => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 9.49/5.23         => ( ( ( plus_plus_real @ X @ Y )
% 9.49/5.23              = zero_zero_real )
% 9.49/5.23            = ( ( X = zero_zero_real )
% 9.49/5.23              & ( Y = zero_zero_real ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1435_add__nonpos__eq__0__iff,axiom,
% 9.49/5.23      ! [X: rat,Y: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 9.49/5.23       => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 9.49/5.23         => ( ( ( plus_plus_rat @ X @ Y )
% 9.49/5.23              = zero_zero_rat )
% 9.49/5.23            = ( ( X = zero_zero_rat )
% 9.49/5.23              & ( Y = zero_zero_rat ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1436_add__nonpos__eq__0__iff,axiom,
% 9.49/5.23      ! [X: nat,Y: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ X @ zero_zero_nat )
% 9.49/5.23       => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
% 9.49/5.23         => ( ( ( plus_plus_nat @ X @ Y )
% 9.49/5.23              = zero_zero_nat )
% 9.49/5.23            = ( ( X = zero_zero_nat )
% 9.49/5.23              & ( Y = zero_zero_nat ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1437_add__nonpos__eq__0__iff,axiom,
% 9.49/5.23      ! [X: int,Y: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ X @ zero_zero_int )
% 9.49/5.23       => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 9.49/5.23         => ( ( ( plus_plus_int @ X @ Y )
% 9.49/5.23              = zero_zero_int )
% 9.49/5.23            = ( ( X = zero_zero_int )
% 9.49/5.23              & ( Y = zero_zero_int ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1438_add__nonneg__eq__0__iff,axiom,
% 9.49/5.23      ! [X: real,Y: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.49/5.23       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.49/5.23         => ( ( ( plus_plus_real @ X @ Y )
% 9.49/5.23              = zero_zero_real )
% 9.49/5.23            = ( ( X = zero_zero_real )
% 9.49/5.23              & ( Y = zero_zero_real ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1439_add__nonneg__eq__0__iff,axiom,
% 9.49/5.23      ! [X: rat,Y: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.49/5.23       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.49/5.23         => ( ( ( plus_plus_rat @ X @ Y )
% 9.49/5.23              = zero_zero_rat )
% 9.49/5.23            = ( ( X = zero_zero_rat )
% 9.49/5.23              & ( Y = zero_zero_rat ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1440_add__nonneg__eq__0__iff,axiom,
% 9.49/5.23      ! [X: nat,Y: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 9.49/5.23       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 9.49/5.23         => ( ( ( plus_plus_nat @ X @ Y )
% 9.49/5.23              = zero_zero_nat )
% 9.49/5.23            = ( ( X = zero_zero_nat )
% 9.49/5.23              & ( Y = zero_zero_nat ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1441_add__nonneg__eq__0__iff,axiom,
% 9.49/5.23      ! [X: int,Y: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.49/5.23       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.49/5.23         => ( ( ( plus_plus_int @ X @ Y )
% 9.49/5.23              = zero_zero_int )
% 9.49/5.23            = ( ( X = zero_zero_int )
% 9.49/5.23              & ( Y = zero_zero_int ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1442_add__nonpos__nonpos,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.49/5.23       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 9.49/5.23         => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1443_add__nonpos__nonpos,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.49/5.23       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 9.49/5.23         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1444_add__nonpos__nonpos,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.49/5.23       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 9.49/5.23         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1445_add__nonpos__nonpos,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.49/5.23       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 9.49/5.23         => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1446_add__nonneg__nonneg,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.23       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.49/5.23         => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1447_add__nonneg__nonneg,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.23       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.49/5.23         => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1448_add__nonneg__nonneg,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.23       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.49/5.23         => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1449_add__nonneg__nonneg,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.23       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.49/5.23         => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1450_add__increasing2,axiom,
% 9.49/5.23      ! [C: real,B: real,A: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.49/5.23       => ( ( ord_less_eq_real @ B @ A )
% 9.49/5.23         => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1451_add__increasing2,axiom,
% 9.49/5.23      ! [C: rat,B: rat,A: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.49/5.23       => ( ( ord_less_eq_rat @ B @ A )
% 9.49/5.23         => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1452_add__increasing2,axiom,
% 9.49/5.23      ! [C: nat,B: nat,A: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.49/5.23       => ( ( ord_less_eq_nat @ B @ A )
% 9.49/5.23         => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1453_add__increasing2,axiom,
% 9.49/5.23      ! [C: int,B: int,A: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.49/5.23       => ( ( ord_less_eq_int @ B @ A )
% 9.49/5.23         => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1454_add__decreasing2,axiom,
% 9.49/5.23      ! [C: real,A: real,B: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.49/5.23       => ( ( ord_less_eq_real @ A @ B )
% 9.49/5.23         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1455_add__decreasing2,axiom,
% 9.49/5.23      ! [C: rat,A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.49/5.23       => ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.23         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1456_add__decreasing2,axiom,
% 9.49/5.23      ! [C: nat,A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 9.49/5.23       => ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.23         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1457_add__decreasing2,axiom,
% 9.49/5.23      ! [C: int,A: int,B: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.49/5.23       => ( ( ord_less_eq_int @ A @ B )
% 9.49/5.23         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1458_add__increasing,axiom,
% 9.49/5.23      ! [A: real,B: real,C: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.23       => ( ( ord_less_eq_real @ B @ C )
% 9.49/5.23         => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1459_add__increasing,axiom,
% 9.49/5.23      ! [A: rat,B: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.23       => ( ( ord_less_eq_rat @ B @ C )
% 9.49/5.23         => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1460_add__increasing,axiom,
% 9.49/5.23      ! [A: nat,B: nat,C: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.23       => ( ( ord_less_eq_nat @ B @ C )
% 9.49/5.23         => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1461_add__increasing,axiom,
% 9.49/5.23      ! [A: int,B: int,C: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.23       => ( ( ord_less_eq_int @ B @ C )
% 9.49/5.23         => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1462_add__decreasing,axiom,
% 9.49/5.23      ! [A: real,C: real,B: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.49/5.23       => ( ( ord_less_eq_real @ C @ B )
% 9.49/5.23         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1463_add__decreasing,axiom,
% 9.49/5.23      ! [A: rat,C: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.49/5.23       => ( ( ord_less_eq_rat @ C @ B )
% 9.49/5.23         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1464_add__decreasing,axiom,
% 9.49/5.23      ! [A: nat,C: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.49/5.23       => ( ( ord_less_eq_nat @ C @ B )
% 9.49/5.23         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1465_add__decreasing,axiom,
% 9.49/5.23      ! [A: int,C: int,B: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.49/5.23       => ( ( ord_less_eq_int @ C @ B )
% 9.49/5.23         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1466_add__less__le__mono,axiom,
% 9.49/5.23      ! [A: real,B: real,C: real,D: real] :
% 9.49/5.23        ( ( ord_less_real @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_real @ C @ D )
% 9.49/5.23         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1467_add__less__le__mono,axiom,
% 9.49/5.23      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.49/5.23        ( ( ord_less_rat @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_rat @ C @ D )
% 9.49/5.23         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1468_add__less__le__mono,axiom,
% 9.49/5.23      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.49/5.23        ( ( ord_less_nat @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_nat @ C @ D )
% 9.49/5.23         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1469_add__less__le__mono,axiom,
% 9.49/5.23      ! [A: int,B: int,C: int,D: int] :
% 9.49/5.23        ( ( ord_less_int @ A @ B )
% 9.49/5.23       => ( ( ord_less_eq_int @ C @ D )
% 9.49/5.23         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1470_add__le__less__mono,axiom,
% 9.49/5.23      ! [A: real,B: real,C: real,D: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.23       => ( ( ord_less_real @ C @ D )
% 9.49/5.23         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1471_add__le__less__mono,axiom,
% 9.49/5.23      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.23       => ( ( ord_less_rat @ C @ D )
% 9.49/5.23         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1472_add__le__less__mono,axiom,
% 9.49/5.23      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.23       => ( ( ord_less_nat @ C @ D )
% 9.49/5.23         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1473_add__le__less__mono,axiom,
% 9.49/5.23      ! [A: int,B: int,C: int,D: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.23       => ( ( ord_less_int @ C @ D )
% 9.49/5.23         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1474_add__mono__thms__linordered__field_I3_J,axiom,
% 9.49/5.23      ! [I: real,J: real,K: real,L: real] :
% 9.49/5.23        ( ( ( ord_less_real @ I @ J )
% 9.49/5.23          & ( ord_less_eq_real @ K @ L ) )
% 9.49/5.23       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1475_add__mono__thms__linordered__field_I3_J,axiom,
% 9.49/5.23      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.49/5.23        ( ( ( ord_less_rat @ I @ J )
% 9.49/5.23          & ( ord_less_eq_rat @ K @ L ) )
% 9.49/5.23       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1476_add__mono__thms__linordered__field_I3_J,axiom,
% 9.49/5.23      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.49/5.23        ( ( ( ord_less_nat @ I @ J )
% 9.49/5.23          & ( ord_less_eq_nat @ K @ L ) )
% 9.49/5.23       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1477_add__mono__thms__linordered__field_I3_J,axiom,
% 9.49/5.23      ! [I: int,J: int,K: int,L: int] :
% 9.49/5.23        ( ( ( ord_less_int @ I @ J )
% 9.49/5.23          & ( ord_less_eq_int @ K @ L ) )
% 9.49/5.23       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1478_add__mono__thms__linordered__field_I4_J,axiom,
% 9.49/5.23      ! [I: real,J: real,K: real,L: real] :
% 9.49/5.23        ( ( ( ord_less_eq_real @ I @ J )
% 9.49/5.23          & ( ord_less_real @ K @ L ) )
% 9.49/5.23       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1479_add__mono__thms__linordered__field_I4_J,axiom,
% 9.49/5.23      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.49/5.23        ( ( ( ord_less_eq_rat @ I @ J )
% 9.49/5.23          & ( ord_less_rat @ K @ L ) )
% 9.49/5.23       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1480_add__mono__thms__linordered__field_I4_J,axiom,
% 9.49/5.23      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.49/5.23        ( ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.23          & ( ord_less_nat @ K @ L ) )
% 9.49/5.23       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1481_add__mono__thms__linordered__field_I4_J,axiom,
% 9.49/5.23      ! [I: int,J: int,K: int,L: int] :
% 9.49/5.23        ( ( ( ord_less_eq_int @ I @ J )
% 9.49/5.23          & ( ord_less_int @ K @ L ) )
% 9.49/5.23       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1482_add__neg__neg,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( ord_less_real @ A @ zero_zero_real )
% 9.49/5.23       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.49/5.23         => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1483_add__neg__neg,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.49/5.23       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.49/5.23         => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1484_add__neg__neg,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_nat @ A @ zero_zero_nat )
% 9.49/5.23       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 9.49/5.23         => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1485_add__neg__neg,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( ord_less_int @ A @ zero_zero_int )
% 9.49/5.23       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.49/5.23         => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1486_add__pos__pos,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.23       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.49/5.23         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1487_add__pos__pos,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.23       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.49/5.23         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1488_add__pos__pos,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.49/5.23       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.49/5.23         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1489_add__pos__pos,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( ord_less_int @ zero_zero_int @ A )
% 9.49/5.23       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.23         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1490_canonically__ordered__monoid__add__class_OlessE,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_nat @ A @ B )
% 9.49/5.23       => ~ ! [C3: nat] :
% 9.49/5.23              ( ( B
% 9.49/5.23                = ( plus_plus_nat @ A @ C3 ) )
% 9.49/5.23             => ( C3 = zero_zero_nat ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1491_pos__add__strict,axiom,
% 9.49/5.23      ! [A: real,B: real,C: real] :
% 9.49/5.23        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.23       => ( ( ord_less_real @ B @ C )
% 9.49/5.23         => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1492_pos__add__strict,axiom,
% 9.49/5.23      ! [A: rat,B: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.23       => ( ( ord_less_rat @ B @ C )
% 9.49/5.23         => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1493_pos__add__strict,axiom,
% 9.49/5.23      ! [A: nat,B: nat,C: nat] :
% 9.49/5.23        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.49/5.23       => ( ( ord_less_nat @ B @ C )
% 9.49/5.23         => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1494_pos__add__strict,axiom,
% 9.49/5.23      ! [A: int,B: int,C: int] :
% 9.49/5.23        ( ( ord_less_int @ zero_zero_int @ A )
% 9.49/5.23       => ( ( ord_less_int @ B @ C )
% 9.49/5.23         => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1495_less__imp__add__positive,axiom,
% 9.49/5.23      ! [I: nat,J: nat] :
% 9.49/5.23        ( ( ord_less_nat @ I @ J )
% 9.49/5.23       => ? [K2: nat] :
% 9.49/5.23            ( ( ord_less_nat @ zero_zero_nat @ K2 )
% 9.49/5.23            & ( ( plus_plus_nat @ I @ K2 )
% 9.49/5.23              = J ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1496_mono__nat__linear__lb,axiom,
% 9.49/5.23      ! [F: nat > nat,M: nat,K: nat] :
% 9.49/5.23        ( ! [M5: nat,N3: nat] :
% 9.49/5.23            ( ( ord_less_nat @ M5 @ N3 )
% 9.49/5.23           => ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
% 9.49/5.23       => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1497_add__neg__nonpos,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( ord_less_real @ A @ zero_zero_real )
% 9.49/5.23       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 9.49/5.23         => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1498_add__neg__nonpos,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.49/5.23       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 9.49/5.23         => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1499_add__neg__nonpos,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_nat @ A @ zero_zero_nat )
% 9.49/5.23       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 9.49/5.23         => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1500_add__neg__nonpos,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( ord_less_int @ A @ zero_zero_int )
% 9.49/5.23       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 9.49/5.23         => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1501_add__nonneg__pos,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.23       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.49/5.23         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1502_add__nonneg__pos,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.23       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.49/5.23         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1503_add__nonneg__pos,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.23       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.49/5.23         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1504_add__nonneg__pos,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.23       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.23         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1505_add__nonpos__neg,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.49/5.23       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.49/5.23         => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1506_add__nonpos__neg,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.49/5.23       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.49/5.23         => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1507_add__nonpos__neg,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.49/5.23       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 9.49/5.23         => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1508_add__nonpos__neg,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.49/5.23       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.49/5.23         => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1509_add__pos__nonneg,axiom,
% 9.49/5.23      ! [A: real,B: real] :
% 9.49/5.23        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.23       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.49/5.23         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1510_add__pos__nonneg,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.23       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.49/5.23         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1511_add__pos__nonneg,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.49/5.23       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.49/5.23         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1512_add__pos__nonneg,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( ord_less_int @ zero_zero_int @ A )
% 9.49/5.23       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.49/5.23         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1513_add__strict__increasing,axiom,
% 9.49/5.23      ! [A: real,B: real,C: real] :
% 9.49/5.23        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.23       => ( ( ord_less_eq_real @ B @ C )
% 9.49/5.23         => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1514_add__strict__increasing,axiom,
% 9.49/5.23      ! [A: rat,B: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.23       => ( ( ord_less_eq_rat @ B @ C )
% 9.49/5.23         => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1515_add__strict__increasing,axiom,
% 9.49/5.23      ! [A: nat,B: nat,C: nat] :
% 9.49/5.23        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.49/5.23       => ( ( ord_less_eq_nat @ B @ C )
% 9.49/5.23         => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1516_add__strict__increasing,axiom,
% 9.49/5.23      ! [A: int,B: int,C: int] :
% 9.49/5.23        ( ( ord_less_int @ zero_zero_int @ A )
% 9.49/5.23       => ( ( ord_less_eq_int @ B @ C )
% 9.49/5.23         => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1517_add__strict__increasing2,axiom,
% 9.49/5.23      ! [A: real,B: real,C: real] :
% 9.49/5.23        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.23       => ( ( ord_less_real @ B @ C )
% 9.49/5.23         => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1518_add__strict__increasing2,axiom,
% 9.49/5.23      ! [A: rat,B: rat,C: rat] :
% 9.49/5.23        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.23       => ( ( ord_less_rat @ B @ C )
% 9.49/5.23         => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1519_add__strict__increasing2,axiom,
% 9.49/5.23      ! [A: nat,B: nat,C: nat] :
% 9.49/5.23        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.23       => ( ( ord_less_nat @ B @ C )
% 9.49/5.23         => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1520_add__strict__increasing2,axiom,
% 9.49/5.23      ! [A: int,B: int,C: int] :
% 9.49/5.23        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.23       => ( ( ord_less_int @ B @ C )
% 9.49/5.23         => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1521_verit__comp__simplify1_I2_J,axiom,
% 9.49/5.23      ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1522_verit__comp__simplify1_I2_J,axiom,
% 9.49/5.23      ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1523_verit__comp__simplify1_I2_J,axiom,
% 9.49/5.23      ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1524_verit__comp__simplify1_I2_J,axiom,
% 9.49/5.23      ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1525_verit__comp__simplify1_I2_J,axiom,
% 9.49/5.23      ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1526_verit__la__disequality,axiom,
% 9.49/5.23      ! [A: rat,B: rat] :
% 9.49/5.23        ( ( A = B )
% 9.49/5.23        | ~ ( ord_less_eq_rat @ A @ B )
% 9.49/5.23        | ~ ( ord_less_eq_rat @ B @ A ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1527_verit__la__disequality,axiom,
% 9.49/5.23      ! [A: num,B: num] :
% 9.49/5.23        ( ( A = B )
% 9.49/5.23        | ~ ( ord_less_eq_num @ A @ B )
% 9.49/5.23        | ~ ( ord_less_eq_num @ B @ A ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1528_verit__la__disequality,axiom,
% 9.49/5.23      ! [A: nat,B: nat] :
% 9.49/5.23        ( ( A = B )
% 9.49/5.23        | ~ ( ord_less_eq_nat @ A @ B )
% 9.49/5.23        | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1529_verit__la__disequality,axiom,
% 9.49/5.23      ! [A: int,B: int] :
% 9.49/5.23        ( ( A = B )
% 9.49/5.23        | ~ ( ord_less_eq_int @ A @ B )
% 9.49/5.23        | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1530_verit__comp__simplify1_I1_J,axiom,
% 9.49/5.23      ! [A: real] :
% 9.49/5.23        ~ ( ord_less_real @ A @ A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1531_verit__comp__simplify1_I1_J,axiom,
% 9.49/5.23      ! [A: rat] :
% 9.49/5.23        ~ ( ord_less_rat @ A @ A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1532_verit__comp__simplify1_I1_J,axiom,
% 9.49/5.23      ! [A: num] :
% 9.49/5.23        ~ ( ord_less_num @ A @ A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1533_verit__comp__simplify1_I1_J,axiom,
% 9.49/5.23      ! [A: nat] :
% 9.49/5.23        ~ ( ord_less_nat @ A @ A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1534_verit__comp__simplify1_I1_J,axiom,
% 9.49/5.23      ! [A: int] :
% 9.49/5.23        ~ ( ord_less_int @ A @ A ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1535_ex__gt__or__lt,axiom,
% 9.49/5.23      ! [A: real] :
% 9.49/5.23      ? [B3: real] :
% 9.49/5.23        ( ( ord_less_real @ A @ B3 )
% 9.49/5.23        | ( ord_less_real @ B3 @ A ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1536_subrelI,axiom,
% 9.49/5.23      ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
% 9.49/5.23        ( ! [X4: nat,Y3: nat] :
% 9.49/5.23            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R2 )
% 9.49/5.23           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ S ) )
% 9.49/5.23       => ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1537_subrelI,axiom,
% 9.49/5.23      ! [R2: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
% 9.49/5.23        ( ! [X4: int,Y3: int] :
% 9.49/5.23            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ R2 )
% 9.49/5.23           => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ S ) )
% 9.49/5.23       => ( ord_le2843351958646193337nt_int @ R2 @ S ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1538_subrelI,axiom,
% 9.49/5.23      ! [R2: set_Pr8056137968301705908nteger,S: set_Pr8056137968301705908nteger] :
% 9.49/5.23        ( ! [X4: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
% 9.49/5.23            ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y3 ) @ R2 )
% 9.49/5.23           => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y3 ) @ S ) )
% 9.49/5.23       => ( ord_le3216752416896350996nteger @ R2 @ S ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1539_subrelI,axiom,
% 9.49/5.23      ! [R2: set_Pr1281608226676607948nteger,S: set_Pr1281608226676607948nteger] :
% 9.49/5.23        ( ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
% 9.49/5.23            ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y3 ) @ R2 )
% 9.49/5.23           => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y3 ) @ S ) )
% 9.49/5.23       => ( ord_le653643898420964396nteger @ R2 @ S ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1540_subrelI,axiom,
% 9.49/5.23      ! [R2: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
% 9.49/5.23        ( ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
% 9.49/5.23            ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y3 ) @ R2 )
% 9.49/5.23           => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y3 ) @ S ) )
% 9.49/5.23       => ( ord_le8725513860283290265nt_int @ R2 @ S ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1541_subrelI,axiom,
% 9.49/5.23      ! [R2: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
% 9.49/5.23        ( ! [X4: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
% 9.49/5.23            ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y3 ) @ R2 )
% 9.49/5.23           => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y3 ) @ S ) )
% 9.49/5.23       => ( ord_le135402666524580259nt_int @ R2 @ S ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1542_ssubst__Pair__rhs,axiom,
% 9.49/5.23      ! [R2: nat,S: nat,R: set_Pr1261947904930325089at_nat,S3: nat] :
% 9.49/5.23        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S ) @ R )
% 9.49/5.23       => ( ( S3 = S )
% 9.49/5.23         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S3 ) @ R ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1543_ssubst__Pair__rhs,axiom,
% 9.49/5.23      ! [R2: int,S: int,R: set_Pr958786334691620121nt_int,S3: int] :
% 9.49/5.23        ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S ) @ R )
% 9.49/5.23       => ( ( S3 = S )
% 9.49/5.23         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S3 ) @ R ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1544_ssubst__Pair__rhs,axiom,
% 9.49/5.23      ! [R2: code_integer > option6357759511663192854e_term,S: produc8923325533196201883nteger,R: set_Pr8056137968301705908nteger,S3: produc8923325533196201883nteger] :
% 9.49/5.23        ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ R2 @ S ) @ R )
% 9.49/5.23       => ( ( S3 = S )
% 9.49/5.23         => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ R2 @ S3 ) @ R ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1545_ssubst__Pair__rhs,axiom,
% 9.49/5.23      ! [R2: produc6241069584506657477e_term > option6357759511663192854e_term,S: produc8923325533196201883nteger,R: set_Pr1281608226676607948nteger,S3: produc8923325533196201883nteger] :
% 9.49/5.23        ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R2 @ S ) @ R )
% 9.49/5.23       => ( ( S3 = S )
% 9.49/5.23         => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R2 @ S3 ) @ R ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1546_ssubst__Pair__rhs,axiom,
% 9.49/5.23      ! [R2: produc8551481072490612790e_term > option6357759511663192854e_term,S: product_prod_int_int,R: set_Pr9222295170931077689nt_int,S3: product_prod_int_int] :
% 9.49/5.23        ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R2 @ S ) @ R )
% 9.49/5.23       => ( ( S3 = S )
% 9.49/5.23         => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R2 @ S3 ) @ R ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1547_ssubst__Pair__rhs,axiom,
% 9.49/5.23      ! [R2: int > option6357759511663192854e_term,S: product_prod_int_int,R: set_Pr1872883991513573699nt_int,S3: product_prod_int_int] :
% 9.49/5.23        ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R2 @ S ) @ R )
% 9.49/5.23       => ( ( S3 = S )
% 9.49/5.23         => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R2 @ S3 ) @ R ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1548_le__numeral__extra_I3_J,axiom,
% 9.49/5.23      ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1549_le__numeral__extra_I3_J,axiom,
% 9.49/5.23      ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1550_le__numeral__extra_I3_J,axiom,
% 9.49/5.23      ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1551_le__numeral__extra_I3_J,axiom,
% 9.49/5.23      ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1552_double__eq__0__iff,axiom,
% 9.49/5.23      ! [A: real] :
% 9.49/5.23        ( ( ( plus_plus_real @ A @ A )
% 9.49/5.23          = zero_zero_real )
% 9.49/5.23        = ( A = zero_zero_real ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1553_double__eq__0__iff,axiom,
% 9.49/5.23      ! [A: rat] :
% 9.49/5.23        ( ( ( plus_plus_rat @ A @ A )
% 9.49/5.23          = zero_zero_rat )
% 9.49/5.23        = ( A = zero_zero_rat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1554_double__eq__0__iff,axiom,
% 9.49/5.23      ! [A: int] :
% 9.49/5.23        ( ( ( plus_plus_int @ A @ A )
% 9.49/5.23          = zero_zero_int )
% 9.49/5.23        = ( A = zero_zero_int ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1555_add__shift,axiom,
% 9.49/5.23      ! [X: nat,Y: nat,Z2: nat] :
% 9.49/5.23        ( ( ( plus_plus_nat @ X @ Y )
% 9.49/5.23          = Z2 )
% 9.49/5.23        = ( ( vEBT_VEBT_add @ ( some_nat @ X ) @ ( some_nat @ Y ) )
% 9.49/5.23          = ( some_nat @ Z2 ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1556_field__le__epsilon,axiom,
% 9.49/5.23      ! [X: real,Y: real] :
% 9.49/5.23        ( ! [E: real] :
% 9.49/5.23            ( ( ord_less_real @ zero_zero_real @ E )
% 9.49/5.23           => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E ) ) )
% 9.49/5.23       => ( ord_less_eq_real @ X @ Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1557_field__le__epsilon,axiom,
% 9.49/5.23      ! [X: rat,Y: rat] :
% 9.49/5.23        ( ! [E: rat] :
% 9.49/5.23            ( ( ord_less_rat @ zero_zero_rat @ E )
% 9.49/5.23           => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E ) ) )
% 9.49/5.23       => ( ord_less_eq_rat @ X @ Y ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1558_add__less__zeroD,axiom,
% 9.49/5.23      ! [X: real,Y: real] :
% 9.49/5.23        ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 9.49/5.23       => ( ( ord_less_real @ X @ zero_zero_real )
% 9.49/5.23          | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1559_add__less__zeroD,axiom,
% 9.49/5.23      ! [X: rat,Y: rat] :
% 9.49/5.23        ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
% 9.49/5.23       => ( ( ord_less_rat @ X @ zero_zero_rat )
% 9.49/5.23          | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1560_add__less__zeroD,axiom,
% 9.49/5.23      ! [X: int,Y: int] :
% 9.49/5.23        ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
% 9.49/5.23       => ( ( ord_less_int @ X @ zero_zero_int )
% 9.49/5.23          | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1561_nth__enumerate__eq,axiom,
% 9.49/5.23      ! [M: nat,Xs: list_VEBT_VEBT,N: nat] :
% 9.49/5.23        ( ( ord_less_nat @ M @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.23       => ( ( nth_Pr744662078594809490T_VEBT @ ( enumerate_VEBT_VEBT @ N @ Xs ) @ M )
% 9.49/5.23          = ( produc599794634098209291T_VEBT @ ( plus_plus_nat @ N @ M ) @ ( nth_VEBT_VEBT @ Xs @ M ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1562_nth__enumerate__eq,axiom,
% 9.49/5.23      ! [M: nat,Xs: list_o,N: nat] :
% 9.49/5.23        ( ( ord_less_nat @ M @ ( size_size_list_o @ Xs ) )
% 9.49/5.23       => ( ( nth_Pr112076138515278198_nat_o @ ( enumerate_o @ N @ Xs ) @ M )
% 9.49/5.23          = ( product_Pair_nat_o @ ( plus_plus_nat @ N @ M ) @ ( nth_o @ Xs @ M ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1563_nth__enumerate__eq,axiom,
% 9.49/5.23      ! [M: nat,Xs: list_nat,N: nat] :
% 9.49/5.23        ( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
% 9.49/5.23       => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M )
% 9.49/5.23          = ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1564_nth__enumerate__eq,axiom,
% 9.49/5.23      ! [M: nat,Xs: list_int,N: nat] :
% 9.49/5.23        ( ( ord_less_nat @ M @ ( size_size_list_int @ Xs ) )
% 9.49/5.23       => ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N @ Xs ) @ M )
% 9.49/5.23          = ( product_Pair_nat_int @ ( plus_plus_nat @ N @ M ) @ ( nth_int @ Xs @ M ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1565_Euclid__induct,axiom,
% 9.49/5.23      ! [P: nat > nat > $o,A: nat,B: nat] :
% 9.49/5.23        ( ! [A3: nat,B3: nat] :
% 9.49/5.23            ( ( P @ A3 @ B3 )
% 9.49/5.23            = ( P @ B3 @ A3 ) )
% 9.49/5.23       => ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
% 9.49/5.23         => ( ! [A3: nat,B3: nat] :
% 9.49/5.23                ( ( P @ A3 @ B3 )
% 9.49/5.23               => ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
% 9.49/5.23           => ( P @ A @ B ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1566_add__0__iff,axiom,
% 9.49/5.23      ! [B: complex,A: complex] :
% 9.49/5.23        ( ( B
% 9.49/5.23          = ( plus_plus_complex @ B @ A ) )
% 9.49/5.23        = ( A = zero_zero_complex ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1567_add__0__iff,axiom,
% 9.49/5.23      ! [B: real,A: real] :
% 9.49/5.23        ( ( B
% 9.49/5.23          = ( plus_plus_real @ B @ A ) )
% 9.49/5.23        = ( A = zero_zero_real ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1568_add__0__iff,axiom,
% 9.49/5.23      ! [B: rat,A: rat] :
% 9.49/5.23        ( ( B
% 9.49/5.23          = ( plus_plus_rat @ B @ A ) )
% 9.49/5.23        = ( A = zero_zero_rat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1569_add__0__iff,axiom,
% 9.49/5.23      ! [B: nat,A: nat] :
% 9.49/5.23        ( ( B
% 9.49/5.23          = ( plus_plus_nat @ B @ A ) )
% 9.49/5.23        = ( A = zero_zero_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1570_add__0__iff,axiom,
% 9.49/5.23      ! [B: int,A: int] :
% 9.49/5.23        ( ( B
% 9.49/5.23          = ( plus_plus_int @ B @ A ) )
% 9.49/5.23        = ( A = zero_zero_int ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1571_minf_I8_J,axiom,
% 9.49/5.23      ! [T: real] :
% 9.49/5.23      ? [Z4: real] :
% 9.49/5.23      ! [X2: real] :
% 9.49/5.23        ( ( ord_less_real @ X2 @ Z4 )
% 9.49/5.23       => ~ ( ord_less_eq_real @ T @ X2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1572_minf_I8_J,axiom,
% 9.49/5.23      ! [T: rat] :
% 9.49/5.23      ? [Z4: rat] :
% 9.49/5.23      ! [X2: rat] :
% 9.49/5.23        ( ( ord_less_rat @ X2 @ Z4 )
% 9.49/5.23       => ~ ( ord_less_eq_rat @ T @ X2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1573_minf_I8_J,axiom,
% 9.49/5.23      ! [T: num] :
% 9.49/5.23      ? [Z4: num] :
% 9.49/5.23      ! [X2: num] :
% 9.49/5.23        ( ( ord_less_num @ X2 @ Z4 )
% 9.49/5.23       => ~ ( ord_less_eq_num @ T @ X2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1574_minf_I8_J,axiom,
% 9.49/5.23      ! [T: nat] :
% 9.49/5.23      ? [Z4: nat] :
% 9.49/5.23      ! [X2: nat] :
% 9.49/5.23        ( ( ord_less_nat @ X2 @ Z4 )
% 9.49/5.23       => ~ ( ord_less_eq_nat @ T @ X2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1575_minf_I8_J,axiom,
% 9.49/5.23      ! [T: int] :
% 9.49/5.23      ? [Z4: int] :
% 9.49/5.23      ! [X2: int] :
% 9.49/5.23        ( ( ord_less_int @ X2 @ Z4 )
% 9.49/5.23       => ~ ( ord_less_eq_int @ T @ X2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1576_minf_I6_J,axiom,
% 9.49/5.23      ! [T: real] :
% 9.49/5.23      ? [Z4: real] :
% 9.49/5.23      ! [X2: real] :
% 9.49/5.23        ( ( ord_less_real @ X2 @ Z4 )
% 9.49/5.23       => ( ord_less_eq_real @ X2 @ T ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1577_minf_I6_J,axiom,
% 9.49/5.23      ! [T: rat] :
% 9.49/5.23      ? [Z4: rat] :
% 9.49/5.23      ! [X2: rat] :
% 9.49/5.23        ( ( ord_less_rat @ X2 @ Z4 )
% 9.49/5.23       => ( ord_less_eq_rat @ X2 @ T ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1578_minf_I6_J,axiom,
% 9.49/5.23      ! [T: num] :
% 9.49/5.23      ? [Z4: num] :
% 9.49/5.23      ! [X2: num] :
% 9.49/5.23        ( ( ord_less_num @ X2 @ Z4 )
% 9.49/5.23       => ( ord_less_eq_num @ X2 @ T ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1579_minf_I6_J,axiom,
% 9.49/5.23      ! [T: nat] :
% 9.49/5.23      ? [Z4: nat] :
% 9.49/5.23      ! [X2: nat] :
% 9.49/5.23        ( ( ord_less_nat @ X2 @ Z4 )
% 9.49/5.23       => ( ord_less_eq_nat @ X2 @ T ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1580_minf_I6_J,axiom,
% 9.49/5.23      ! [T: int] :
% 9.49/5.23      ? [Z4: int] :
% 9.49/5.23      ! [X2: int] :
% 9.49/5.23        ( ( ord_less_int @ X2 @ Z4 )
% 9.49/5.23       => ( ord_less_eq_int @ X2 @ T ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1581_pinf_I8_J,axiom,
% 9.49/5.23      ! [T: real] :
% 9.49/5.23      ? [Z4: real] :
% 9.49/5.23      ! [X2: real] :
% 9.49/5.23        ( ( ord_less_real @ Z4 @ X2 )
% 9.49/5.23       => ( ord_less_eq_real @ T @ X2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1582_pinf_I8_J,axiom,
% 9.49/5.23      ! [T: rat] :
% 9.49/5.23      ? [Z4: rat] :
% 9.49/5.23      ! [X2: rat] :
% 9.49/5.23        ( ( ord_less_rat @ Z4 @ X2 )
% 9.49/5.23       => ( ord_less_eq_rat @ T @ X2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1583_pinf_I8_J,axiom,
% 9.49/5.23      ! [T: num] :
% 9.49/5.23      ? [Z4: num] :
% 9.49/5.23      ! [X2: num] :
% 9.49/5.23        ( ( ord_less_num @ Z4 @ X2 )
% 9.49/5.23       => ( ord_less_eq_num @ T @ X2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1584_pinf_I8_J,axiom,
% 9.49/5.23      ! [T: nat] :
% 9.49/5.23      ? [Z4: nat] :
% 9.49/5.23      ! [X2: nat] :
% 9.49/5.23        ( ( ord_less_nat @ Z4 @ X2 )
% 9.49/5.23       => ( ord_less_eq_nat @ T @ X2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1585_pinf_I8_J,axiom,
% 9.49/5.23      ! [T: int] :
% 9.49/5.23      ? [Z4: int] :
% 9.49/5.23      ! [X2: int] :
% 9.49/5.23        ( ( ord_less_int @ Z4 @ X2 )
% 9.49/5.23       => ( ord_less_eq_int @ T @ X2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1586_pinf_I6_J,axiom,
% 9.49/5.23      ! [T: real] :
% 9.49/5.23      ? [Z4: real] :
% 9.49/5.23      ! [X2: real] :
% 9.49/5.23        ( ( ord_less_real @ Z4 @ X2 )
% 9.49/5.23       => ~ ( ord_less_eq_real @ X2 @ T ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1587_pinf_I6_J,axiom,
% 9.49/5.23      ! [T: rat] :
% 9.49/5.23      ? [Z4: rat] :
% 9.49/5.23      ! [X2: rat] :
% 9.49/5.23        ( ( ord_less_rat @ Z4 @ X2 )
% 9.49/5.23       => ~ ( ord_less_eq_rat @ X2 @ T ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1588_pinf_I6_J,axiom,
% 9.49/5.23      ! [T: num] :
% 9.49/5.23      ? [Z4: num] :
% 9.49/5.23      ! [X2: num] :
% 9.49/5.23        ( ( ord_less_num @ Z4 @ X2 )
% 9.49/5.23       => ~ ( ord_less_eq_num @ X2 @ T ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1589_pinf_I6_J,axiom,
% 9.49/5.23      ! [T: nat] :
% 9.49/5.23      ? [Z4: nat] :
% 9.49/5.23      ! [X2: nat] :
% 9.49/5.23        ( ( ord_less_nat @ Z4 @ X2 )
% 9.49/5.23       => ~ ( ord_less_eq_nat @ X2 @ T ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1590_pinf_I6_J,axiom,
% 9.49/5.23      ! [T: int] :
% 9.49/5.23      ? [Z4: int] :
% 9.49/5.23      ! [X2: int] :
% 9.49/5.23        ( ( ord_less_int @ Z4 @ X2 )
% 9.49/5.23       => ~ ( ord_less_eq_int @ X2 @ T ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1591_length__enumerate,axiom,
% 9.49/5.23      ! [N: nat,Xs: list_VEBT_VEBT] :
% 9.49/5.23        ( ( size_s4762443039079500285T_VEBT @ ( enumerate_VEBT_VEBT @ N @ Xs ) )
% 9.49/5.23        = ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1592_length__enumerate,axiom,
% 9.49/5.23      ! [N: nat,Xs: list_o] :
% 9.49/5.23        ( ( size_s6491369823275344609_nat_o @ ( enumerate_o @ N @ Xs ) )
% 9.49/5.23        = ( size_size_list_o @ Xs ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1593_length__enumerate,axiom,
% 9.49/5.23      ! [N: nat,Xs: list_nat] :
% 9.49/5.23        ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
% 9.49/5.23        = ( size_size_list_nat @ Xs ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1594_length__enumerate,axiom,
% 9.49/5.23      ! [N: nat,Xs: list_int] :
% 9.49/5.23        ( ( size_s2970893825323803983at_int @ ( enumerate_int @ N @ Xs ) )
% 9.49/5.23        = ( size_size_list_int @ Xs ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1595_add__def,axiom,
% 9.49/5.23      ( vEBT_VEBT_add
% 9.49/5.23      = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1596_less__by__empty,axiom,
% 9.49/5.23      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.23        ( ( A2 = bot_bo2099793752762293965at_nat )
% 9.49/5.23       => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1597_linorder__neqE__linordered__idom,axiom,
% 9.49/5.23      ! [X: real,Y: real] :
% 9.49/5.23        ( ( X != Y )
% 9.49/5.23       => ( ~ ( ord_less_real @ X @ Y )
% 9.49/5.23         => ( ord_less_real @ Y @ X ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1598_linorder__neqE__linordered__idom,axiom,
% 9.49/5.23      ! [X: rat,Y: rat] :
% 9.49/5.23        ( ( X != Y )
% 9.49/5.23       => ( ~ ( ord_less_rat @ X @ Y )
% 9.49/5.23         => ( ord_less_rat @ Y @ X ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1599_linorder__neqE__linordered__idom,axiom,
% 9.49/5.23      ! [X: int,Y: int] :
% 9.49/5.23        ( ( X != Y )
% 9.49/5.23       => ( ~ ( ord_less_int @ X @ Y )
% 9.49/5.23         => ( ord_less_int @ Y @ X ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1600_linordered__field__no__ub,axiom,
% 9.49/5.23      ! [X2: real] :
% 9.49/5.23      ? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1601_linordered__field__no__ub,axiom,
% 9.49/5.23      ! [X2: rat] :
% 9.49/5.23      ? [X_1: rat] : ( ord_less_rat @ X2 @ X_1 ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1602_linordered__field__no__lb,axiom,
% 9.49/5.23      ! [X2: real] :
% 9.49/5.23      ? [Y3: real] : ( ord_less_real @ Y3 @ X2 ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1603_linordered__field__no__lb,axiom,
% 9.49/5.23      ! [X2: rat] :
% 9.49/5.23      ? [Y3: rat] : ( ord_less_rat @ Y3 @ X2 ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1604_pinf_I1_J,axiom,
% 9.49/5.23      ! [P: real > $o,P5: real > $o,Q: real > $o,Q2: real > $o] :
% 9.49/5.23        ( ? [Z5: real] :
% 9.49/5.23          ! [X4: real] :
% 9.49/5.23            ( ( ord_less_real @ Z5 @ X4 )
% 9.49/5.23           => ( ( P @ X4 )
% 9.49/5.23              = ( P5 @ X4 ) ) )
% 9.49/5.23       => ( ? [Z5: real] :
% 9.49/5.23            ! [X4: real] :
% 9.49/5.23              ( ( ord_less_real @ Z5 @ X4 )
% 9.49/5.23             => ( ( Q @ X4 )
% 9.49/5.23                = ( Q2 @ X4 ) ) )
% 9.49/5.23         => ? [Z4: real] :
% 9.49/5.23            ! [X2: real] :
% 9.49/5.23              ( ( ord_less_real @ Z4 @ X2 )
% 9.49/5.23             => ( ( ( P @ X2 )
% 9.49/5.23                  & ( Q @ X2 ) )
% 9.49/5.23                = ( ( P5 @ X2 )
% 9.49/5.23                  & ( Q2 @ X2 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1605_pinf_I1_J,axiom,
% 9.49/5.23      ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q2: rat > $o] :
% 9.49/5.23        ( ? [Z5: rat] :
% 9.49/5.23          ! [X4: rat] :
% 9.49/5.23            ( ( ord_less_rat @ Z5 @ X4 )
% 9.49/5.23           => ( ( P @ X4 )
% 9.49/5.23              = ( P5 @ X4 ) ) )
% 9.49/5.23       => ( ? [Z5: rat] :
% 9.49/5.23            ! [X4: rat] :
% 9.49/5.23              ( ( ord_less_rat @ Z5 @ X4 )
% 9.49/5.23             => ( ( Q @ X4 )
% 9.49/5.23                = ( Q2 @ X4 ) ) )
% 9.49/5.23         => ? [Z4: rat] :
% 9.49/5.23            ! [X2: rat] :
% 9.49/5.23              ( ( ord_less_rat @ Z4 @ X2 )
% 9.49/5.23             => ( ( ( P @ X2 )
% 9.49/5.23                  & ( Q @ X2 ) )
% 9.49/5.23                = ( ( P5 @ X2 )
% 9.49/5.23                  & ( Q2 @ X2 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1606_pinf_I1_J,axiom,
% 9.49/5.23      ! [P: num > $o,P5: num > $o,Q: num > $o,Q2: num > $o] :
% 9.49/5.23        ( ? [Z5: num] :
% 9.49/5.23          ! [X4: num] :
% 9.49/5.23            ( ( ord_less_num @ Z5 @ X4 )
% 9.49/5.23           => ( ( P @ X4 )
% 9.49/5.23              = ( P5 @ X4 ) ) )
% 9.49/5.23       => ( ? [Z5: num] :
% 9.49/5.23            ! [X4: num] :
% 9.49/5.23              ( ( ord_less_num @ Z5 @ X4 )
% 9.49/5.23             => ( ( Q @ X4 )
% 9.49/5.23                = ( Q2 @ X4 ) ) )
% 9.49/5.23         => ? [Z4: num] :
% 9.49/5.23            ! [X2: num] :
% 9.49/5.23              ( ( ord_less_num @ Z4 @ X2 )
% 9.49/5.23             => ( ( ( P @ X2 )
% 9.49/5.23                  & ( Q @ X2 ) )
% 9.49/5.23                = ( ( P5 @ X2 )
% 9.49/5.23                  & ( Q2 @ X2 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1607_pinf_I1_J,axiom,
% 9.49/5.23      ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 9.49/5.23        ( ? [Z5: nat] :
% 9.49/5.23          ! [X4: nat] :
% 9.49/5.23            ( ( ord_less_nat @ Z5 @ X4 )
% 9.49/5.23           => ( ( P @ X4 )
% 9.49/5.23              = ( P5 @ X4 ) ) )
% 9.49/5.23       => ( ? [Z5: nat] :
% 9.49/5.23            ! [X4: nat] :
% 9.49/5.23              ( ( ord_less_nat @ Z5 @ X4 )
% 9.49/5.23             => ( ( Q @ X4 )
% 9.49/5.23                = ( Q2 @ X4 ) ) )
% 9.49/5.23         => ? [Z4: nat] :
% 9.49/5.23            ! [X2: nat] :
% 9.49/5.23              ( ( ord_less_nat @ Z4 @ X2 )
% 9.49/5.23             => ( ( ( P @ X2 )
% 9.49/5.23                  & ( Q @ X2 ) )
% 9.49/5.23                = ( ( P5 @ X2 )
% 9.49/5.23                  & ( Q2 @ X2 ) ) ) ) ) ) ).
% 9.49/5.23  
% 9.49/5.23  thf(fact_1608_pinf_I1_J,axiom,
% 9.49/5.23      ! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
% 9.49/5.23        ( ? [Z5: int] :
% 9.49/5.23          ! [X4: int] :
% 9.49/5.23            ( ( ord_less_int @ Z5 @ X4 )
% 9.49/5.23           => ( ( P @ X4 )
% 9.49/5.23              = ( P5 @ X4 ) ) )
% 9.49/5.24       => ( ? [Z5: int] :
% 9.49/5.24            ! [X4: int] :
% 9.49/5.24              ( ( ord_less_int @ Z5 @ X4 )
% 9.49/5.24             => ( ( Q @ X4 )
% 9.49/5.24                = ( Q2 @ X4 ) ) )
% 9.49/5.24         => ? [Z4: int] :
% 9.49/5.24            ! [X2: int] :
% 9.49/5.24              ( ( ord_less_int @ Z4 @ X2 )
% 9.49/5.24             => ( ( ( P @ X2 )
% 9.49/5.24                  & ( Q @ X2 ) )
% 9.49/5.24                = ( ( P5 @ X2 )
% 9.49/5.24                  & ( Q2 @ X2 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1609_pinf_I2_J,axiom,
% 9.49/5.24      ! [P: real > $o,P5: real > $o,Q: real > $o,Q2: real > $o] :
% 9.49/5.24        ( ? [Z5: real] :
% 9.49/5.24          ! [X4: real] :
% 9.49/5.24            ( ( ord_less_real @ Z5 @ X4 )
% 9.49/5.24           => ( ( P @ X4 )
% 9.49/5.24              = ( P5 @ X4 ) ) )
% 9.49/5.24       => ( ? [Z5: real] :
% 9.49/5.24            ! [X4: real] :
% 9.49/5.24              ( ( ord_less_real @ Z5 @ X4 )
% 9.49/5.24             => ( ( Q @ X4 )
% 9.49/5.24                = ( Q2 @ X4 ) ) )
% 9.49/5.24         => ? [Z4: real] :
% 9.49/5.24            ! [X2: real] :
% 9.49/5.24              ( ( ord_less_real @ Z4 @ X2 )
% 9.49/5.24             => ( ( ( P @ X2 )
% 9.49/5.24                  | ( Q @ X2 ) )
% 9.49/5.24                = ( ( P5 @ X2 )
% 9.49/5.24                  | ( Q2 @ X2 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1610_pinf_I2_J,axiom,
% 9.49/5.24      ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q2: rat > $o] :
% 9.49/5.24        ( ? [Z5: rat] :
% 9.49/5.24          ! [X4: rat] :
% 9.49/5.24            ( ( ord_less_rat @ Z5 @ X4 )
% 9.49/5.24           => ( ( P @ X4 )
% 9.49/5.24              = ( P5 @ X4 ) ) )
% 9.49/5.24       => ( ? [Z5: rat] :
% 9.49/5.24            ! [X4: rat] :
% 9.49/5.24              ( ( ord_less_rat @ Z5 @ X4 )
% 9.49/5.24             => ( ( Q @ X4 )
% 9.49/5.24                = ( Q2 @ X4 ) ) )
% 9.49/5.24         => ? [Z4: rat] :
% 9.49/5.24            ! [X2: rat] :
% 9.49/5.24              ( ( ord_less_rat @ Z4 @ X2 )
% 9.49/5.24             => ( ( ( P @ X2 )
% 9.49/5.24                  | ( Q @ X2 ) )
% 9.49/5.24                = ( ( P5 @ X2 )
% 9.49/5.24                  | ( Q2 @ X2 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1611_pinf_I2_J,axiom,
% 9.49/5.24      ! [P: num > $o,P5: num > $o,Q: num > $o,Q2: num > $o] :
% 9.49/5.24        ( ? [Z5: num] :
% 9.49/5.24          ! [X4: num] :
% 9.49/5.24            ( ( ord_less_num @ Z5 @ X4 )
% 9.49/5.24           => ( ( P @ X4 )
% 9.49/5.24              = ( P5 @ X4 ) ) )
% 9.49/5.24       => ( ? [Z5: num] :
% 9.49/5.24            ! [X4: num] :
% 9.49/5.24              ( ( ord_less_num @ Z5 @ X4 )
% 9.49/5.24             => ( ( Q @ X4 )
% 9.49/5.24                = ( Q2 @ X4 ) ) )
% 9.49/5.24         => ? [Z4: num] :
% 9.49/5.24            ! [X2: num] :
% 9.49/5.24              ( ( ord_less_num @ Z4 @ X2 )
% 9.49/5.24             => ( ( ( P @ X2 )
% 9.49/5.24                  | ( Q @ X2 ) )
% 9.49/5.24                = ( ( P5 @ X2 )
% 9.49/5.24                  | ( Q2 @ X2 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1612_pinf_I2_J,axiom,
% 9.49/5.24      ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 9.49/5.24        ( ? [Z5: nat] :
% 9.49/5.24          ! [X4: nat] :
% 9.49/5.24            ( ( ord_less_nat @ Z5 @ X4 )
% 9.49/5.24           => ( ( P @ X4 )
% 9.49/5.24              = ( P5 @ X4 ) ) )
% 9.49/5.24       => ( ? [Z5: nat] :
% 9.49/5.24            ! [X4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ Z5 @ X4 )
% 9.49/5.24             => ( ( Q @ X4 )
% 9.49/5.24                = ( Q2 @ X4 ) ) )
% 9.49/5.24         => ? [Z4: nat] :
% 9.49/5.24            ! [X2: nat] :
% 9.49/5.24              ( ( ord_less_nat @ Z4 @ X2 )
% 9.49/5.24             => ( ( ( P @ X2 )
% 9.49/5.24                  | ( Q @ X2 ) )
% 9.49/5.24                = ( ( P5 @ X2 )
% 9.49/5.24                  | ( Q2 @ X2 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1613_pinf_I2_J,axiom,
% 9.49/5.24      ! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
% 9.49/5.24        ( ? [Z5: int] :
% 9.49/5.24          ! [X4: int] :
% 9.49/5.24            ( ( ord_less_int @ Z5 @ X4 )
% 9.49/5.24           => ( ( P @ X4 )
% 9.49/5.24              = ( P5 @ X4 ) ) )
% 9.49/5.24       => ( ? [Z5: int] :
% 9.49/5.24            ! [X4: int] :
% 9.49/5.24              ( ( ord_less_int @ Z5 @ X4 )
% 9.49/5.24             => ( ( Q @ X4 )
% 9.49/5.24                = ( Q2 @ X4 ) ) )
% 9.49/5.24         => ? [Z4: int] :
% 9.49/5.24            ! [X2: int] :
% 9.49/5.24              ( ( ord_less_int @ Z4 @ X2 )
% 9.49/5.24             => ( ( ( P @ X2 )
% 9.49/5.24                  | ( Q @ X2 ) )
% 9.49/5.24                = ( ( P5 @ X2 )
% 9.49/5.24                  | ( Q2 @ X2 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1614_pinf_I3_J,axiom,
% 9.49/5.24      ! [T: real] :
% 9.49/5.24      ? [Z4: real] :
% 9.49/5.24      ! [X2: real] :
% 9.49/5.24        ( ( ord_less_real @ Z4 @ X2 )
% 9.49/5.24       => ( X2 != T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1615_pinf_I3_J,axiom,
% 9.49/5.24      ! [T: rat] :
% 9.49/5.24      ? [Z4: rat] :
% 9.49/5.24      ! [X2: rat] :
% 9.49/5.24        ( ( ord_less_rat @ Z4 @ X2 )
% 9.49/5.24       => ( X2 != T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1616_pinf_I3_J,axiom,
% 9.49/5.24      ! [T: num] :
% 9.49/5.24      ? [Z4: num] :
% 9.49/5.24      ! [X2: num] :
% 9.49/5.24        ( ( ord_less_num @ Z4 @ X2 )
% 9.49/5.24       => ( X2 != T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1617_pinf_I3_J,axiom,
% 9.49/5.24      ! [T: nat] :
% 9.49/5.24      ? [Z4: nat] :
% 9.49/5.24      ! [X2: nat] :
% 9.49/5.24        ( ( ord_less_nat @ Z4 @ X2 )
% 9.49/5.24       => ( X2 != T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1618_pinf_I3_J,axiom,
% 9.49/5.24      ! [T: int] :
% 9.49/5.24      ? [Z4: int] :
% 9.49/5.24      ! [X2: int] :
% 9.49/5.24        ( ( ord_less_int @ Z4 @ X2 )
% 9.49/5.24       => ( X2 != T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1619_pinf_I4_J,axiom,
% 9.49/5.24      ! [T: real] :
% 9.49/5.24      ? [Z4: real] :
% 9.49/5.24      ! [X2: real] :
% 9.49/5.24        ( ( ord_less_real @ Z4 @ X2 )
% 9.49/5.24       => ( X2 != T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1620_pinf_I4_J,axiom,
% 9.49/5.24      ! [T: rat] :
% 9.49/5.24      ? [Z4: rat] :
% 9.49/5.24      ! [X2: rat] :
% 9.49/5.24        ( ( ord_less_rat @ Z4 @ X2 )
% 9.49/5.24       => ( X2 != T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1621_pinf_I4_J,axiom,
% 9.49/5.24      ! [T: num] :
% 9.49/5.24      ? [Z4: num] :
% 9.49/5.24      ! [X2: num] :
% 9.49/5.24        ( ( ord_less_num @ Z4 @ X2 )
% 9.49/5.24       => ( X2 != T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1622_pinf_I4_J,axiom,
% 9.49/5.24      ! [T: nat] :
% 9.49/5.24      ? [Z4: nat] :
% 9.49/5.24      ! [X2: nat] :
% 9.49/5.24        ( ( ord_less_nat @ Z4 @ X2 )
% 9.49/5.24       => ( X2 != T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1623_pinf_I4_J,axiom,
% 9.49/5.24      ! [T: int] :
% 9.49/5.24      ? [Z4: int] :
% 9.49/5.24      ! [X2: int] :
% 9.49/5.24        ( ( ord_less_int @ Z4 @ X2 )
% 9.49/5.24       => ( X2 != T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1624_pinf_I5_J,axiom,
% 9.49/5.24      ! [T: real] :
% 9.49/5.24      ? [Z4: real] :
% 9.49/5.24      ! [X2: real] :
% 9.49/5.24        ( ( ord_less_real @ Z4 @ X2 )
% 9.49/5.24       => ~ ( ord_less_real @ X2 @ T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1625_pinf_I5_J,axiom,
% 9.49/5.24      ! [T: rat] :
% 9.49/5.24      ? [Z4: rat] :
% 9.49/5.24      ! [X2: rat] :
% 9.49/5.24        ( ( ord_less_rat @ Z4 @ X2 )
% 9.49/5.24       => ~ ( ord_less_rat @ X2 @ T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1626_pinf_I5_J,axiom,
% 9.49/5.24      ! [T: num] :
% 9.49/5.24      ? [Z4: num] :
% 9.49/5.24      ! [X2: num] :
% 9.49/5.24        ( ( ord_less_num @ Z4 @ X2 )
% 9.49/5.24       => ~ ( ord_less_num @ X2 @ T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1627_pinf_I5_J,axiom,
% 9.49/5.24      ! [T: nat] :
% 9.49/5.24      ? [Z4: nat] :
% 9.49/5.24      ! [X2: nat] :
% 9.49/5.24        ( ( ord_less_nat @ Z4 @ X2 )
% 9.49/5.24       => ~ ( ord_less_nat @ X2 @ T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1628_pinf_I5_J,axiom,
% 9.49/5.24      ! [T: int] :
% 9.49/5.24      ? [Z4: int] :
% 9.49/5.24      ! [X2: int] :
% 9.49/5.24        ( ( ord_less_int @ Z4 @ X2 )
% 9.49/5.24       => ~ ( ord_less_int @ X2 @ T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1629_pinf_I7_J,axiom,
% 9.49/5.24      ! [T: real] :
% 9.49/5.24      ? [Z4: real] :
% 9.49/5.24      ! [X2: real] :
% 9.49/5.24        ( ( ord_less_real @ Z4 @ X2 )
% 9.49/5.24       => ( ord_less_real @ T @ X2 ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1630_pinf_I7_J,axiom,
% 9.49/5.24      ! [T: rat] :
% 9.49/5.24      ? [Z4: rat] :
% 9.49/5.24      ! [X2: rat] :
% 9.49/5.24        ( ( ord_less_rat @ Z4 @ X2 )
% 9.49/5.24       => ( ord_less_rat @ T @ X2 ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1631_pinf_I7_J,axiom,
% 9.49/5.24      ! [T: num] :
% 9.49/5.24      ? [Z4: num] :
% 9.49/5.24      ! [X2: num] :
% 9.49/5.24        ( ( ord_less_num @ Z4 @ X2 )
% 9.49/5.24       => ( ord_less_num @ T @ X2 ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1632_pinf_I7_J,axiom,
% 9.49/5.24      ! [T: nat] :
% 9.49/5.24      ? [Z4: nat] :
% 9.49/5.24      ! [X2: nat] :
% 9.49/5.24        ( ( ord_less_nat @ Z4 @ X2 )
% 9.49/5.24       => ( ord_less_nat @ T @ X2 ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1633_pinf_I7_J,axiom,
% 9.49/5.24      ! [T: int] :
% 9.49/5.24      ? [Z4: int] :
% 9.49/5.24      ! [X2: int] :
% 9.49/5.24        ( ( ord_less_int @ Z4 @ X2 )
% 9.49/5.24       => ( ord_less_int @ T @ X2 ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1634_minf_I1_J,axiom,
% 9.49/5.24      ! [P: real > $o,P5: real > $o,Q: real > $o,Q2: real > $o] :
% 9.49/5.24        ( ? [Z5: real] :
% 9.49/5.24          ! [X4: real] :
% 9.49/5.24            ( ( ord_less_real @ X4 @ Z5 )
% 9.49/5.24           => ( ( P @ X4 )
% 9.49/5.24              = ( P5 @ X4 ) ) )
% 9.49/5.24       => ( ? [Z5: real] :
% 9.49/5.24            ! [X4: real] :
% 9.49/5.24              ( ( ord_less_real @ X4 @ Z5 )
% 9.49/5.24             => ( ( Q @ X4 )
% 9.49/5.24                = ( Q2 @ X4 ) ) )
% 9.49/5.24         => ? [Z4: real] :
% 9.49/5.24            ! [X2: real] :
% 9.49/5.24              ( ( ord_less_real @ X2 @ Z4 )
% 9.49/5.24             => ( ( ( P @ X2 )
% 9.49/5.24                  & ( Q @ X2 ) )
% 9.49/5.24                = ( ( P5 @ X2 )
% 9.49/5.24                  & ( Q2 @ X2 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1635_minf_I1_J,axiom,
% 9.49/5.24      ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q2: rat > $o] :
% 9.49/5.24        ( ? [Z5: rat] :
% 9.49/5.24          ! [X4: rat] :
% 9.49/5.24            ( ( ord_less_rat @ X4 @ Z5 )
% 9.49/5.24           => ( ( P @ X4 )
% 9.49/5.24              = ( P5 @ X4 ) ) )
% 9.49/5.24       => ( ? [Z5: rat] :
% 9.49/5.24            ! [X4: rat] :
% 9.49/5.24              ( ( ord_less_rat @ X4 @ Z5 )
% 9.49/5.24             => ( ( Q @ X4 )
% 9.49/5.24                = ( Q2 @ X4 ) ) )
% 9.49/5.24         => ? [Z4: rat] :
% 9.49/5.24            ! [X2: rat] :
% 9.49/5.24              ( ( ord_less_rat @ X2 @ Z4 )
% 9.49/5.24             => ( ( ( P @ X2 )
% 9.49/5.24                  & ( Q @ X2 ) )
% 9.49/5.24                = ( ( P5 @ X2 )
% 9.49/5.24                  & ( Q2 @ X2 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1636_minf_I1_J,axiom,
% 9.49/5.24      ! [P: num > $o,P5: num > $o,Q: num > $o,Q2: num > $o] :
% 9.49/5.24        ( ? [Z5: num] :
% 9.49/5.24          ! [X4: num] :
% 9.49/5.24            ( ( ord_less_num @ X4 @ Z5 )
% 9.49/5.24           => ( ( P @ X4 )
% 9.49/5.24              = ( P5 @ X4 ) ) )
% 9.49/5.24       => ( ? [Z5: num] :
% 9.49/5.24            ! [X4: num] :
% 9.49/5.24              ( ( ord_less_num @ X4 @ Z5 )
% 9.49/5.24             => ( ( Q @ X4 )
% 9.49/5.24                = ( Q2 @ X4 ) ) )
% 9.49/5.24         => ? [Z4: num] :
% 9.49/5.24            ! [X2: num] :
% 9.49/5.24              ( ( ord_less_num @ X2 @ Z4 )
% 9.49/5.24             => ( ( ( P @ X2 )
% 9.49/5.24                  & ( Q @ X2 ) )
% 9.49/5.24                = ( ( P5 @ X2 )
% 9.49/5.24                  & ( Q2 @ X2 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1637_minf_I1_J,axiom,
% 9.49/5.24      ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 9.49/5.24        ( ? [Z5: nat] :
% 9.49/5.24          ! [X4: nat] :
% 9.49/5.24            ( ( ord_less_nat @ X4 @ Z5 )
% 9.49/5.24           => ( ( P @ X4 )
% 9.49/5.24              = ( P5 @ X4 ) ) )
% 9.49/5.24       => ( ? [Z5: nat] :
% 9.49/5.24            ! [X4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ X4 @ Z5 )
% 9.49/5.24             => ( ( Q @ X4 )
% 9.49/5.24                = ( Q2 @ X4 ) ) )
% 9.49/5.24         => ? [Z4: nat] :
% 9.49/5.24            ! [X2: nat] :
% 9.49/5.24              ( ( ord_less_nat @ X2 @ Z4 )
% 9.49/5.24             => ( ( ( P @ X2 )
% 9.49/5.24                  & ( Q @ X2 ) )
% 9.49/5.24                = ( ( P5 @ X2 )
% 9.49/5.24                  & ( Q2 @ X2 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1638_minf_I1_J,axiom,
% 9.49/5.24      ! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
% 9.49/5.24        ( ? [Z5: int] :
% 9.49/5.24          ! [X4: int] :
% 9.49/5.24            ( ( ord_less_int @ X4 @ Z5 )
% 9.49/5.24           => ( ( P @ X4 )
% 9.49/5.24              = ( P5 @ X4 ) ) )
% 9.49/5.24       => ( ? [Z5: int] :
% 9.49/5.24            ! [X4: int] :
% 9.49/5.24              ( ( ord_less_int @ X4 @ Z5 )
% 9.49/5.24             => ( ( Q @ X4 )
% 9.49/5.24                = ( Q2 @ X4 ) ) )
% 9.49/5.24         => ? [Z4: int] :
% 9.49/5.24            ! [X2: int] :
% 9.49/5.24              ( ( ord_less_int @ X2 @ Z4 )
% 9.49/5.24             => ( ( ( P @ X2 )
% 9.49/5.24                  & ( Q @ X2 ) )
% 9.49/5.24                = ( ( P5 @ X2 )
% 9.49/5.24                  & ( Q2 @ X2 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1639_minf_I2_J,axiom,
% 9.49/5.24      ! [P: real > $o,P5: real > $o,Q: real > $o,Q2: real > $o] :
% 9.49/5.24        ( ? [Z5: real] :
% 9.49/5.24          ! [X4: real] :
% 9.49/5.24            ( ( ord_less_real @ X4 @ Z5 )
% 9.49/5.24           => ( ( P @ X4 )
% 9.49/5.24              = ( P5 @ X4 ) ) )
% 9.49/5.24       => ( ? [Z5: real] :
% 9.49/5.24            ! [X4: real] :
% 9.49/5.24              ( ( ord_less_real @ X4 @ Z5 )
% 9.49/5.24             => ( ( Q @ X4 )
% 9.49/5.24                = ( Q2 @ X4 ) ) )
% 9.49/5.24         => ? [Z4: real] :
% 9.49/5.24            ! [X2: real] :
% 9.49/5.24              ( ( ord_less_real @ X2 @ Z4 )
% 9.49/5.24             => ( ( ( P @ X2 )
% 9.49/5.24                  | ( Q @ X2 ) )
% 9.49/5.24                = ( ( P5 @ X2 )
% 9.49/5.24                  | ( Q2 @ X2 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1640_minf_I2_J,axiom,
% 9.49/5.24      ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q2: rat > $o] :
% 9.49/5.24        ( ? [Z5: rat] :
% 9.49/5.24          ! [X4: rat] :
% 9.49/5.24            ( ( ord_less_rat @ X4 @ Z5 )
% 9.49/5.24           => ( ( P @ X4 )
% 9.49/5.24              = ( P5 @ X4 ) ) )
% 9.49/5.24       => ( ? [Z5: rat] :
% 9.49/5.24            ! [X4: rat] :
% 9.49/5.24              ( ( ord_less_rat @ X4 @ Z5 )
% 9.49/5.24             => ( ( Q @ X4 )
% 9.49/5.24                = ( Q2 @ X4 ) ) )
% 9.49/5.24         => ? [Z4: rat] :
% 9.49/5.24            ! [X2: rat] :
% 9.49/5.24              ( ( ord_less_rat @ X2 @ Z4 )
% 9.49/5.24             => ( ( ( P @ X2 )
% 9.49/5.24                  | ( Q @ X2 ) )
% 9.49/5.24                = ( ( P5 @ X2 )
% 9.49/5.24                  | ( Q2 @ X2 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1641_minf_I2_J,axiom,
% 9.49/5.24      ! [P: num > $o,P5: num > $o,Q: num > $o,Q2: num > $o] :
% 9.49/5.24        ( ? [Z5: num] :
% 9.49/5.24          ! [X4: num] :
% 9.49/5.24            ( ( ord_less_num @ X4 @ Z5 )
% 9.49/5.24           => ( ( P @ X4 )
% 9.49/5.24              = ( P5 @ X4 ) ) )
% 9.49/5.24       => ( ? [Z5: num] :
% 9.49/5.24            ! [X4: num] :
% 9.49/5.24              ( ( ord_less_num @ X4 @ Z5 )
% 9.49/5.24             => ( ( Q @ X4 )
% 9.49/5.24                = ( Q2 @ X4 ) ) )
% 9.49/5.24         => ? [Z4: num] :
% 9.49/5.24            ! [X2: num] :
% 9.49/5.24              ( ( ord_less_num @ X2 @ Z4 )
% 9.49/5.24             => ( ( ( P @ X2 )
% 9.49/5.24                  | ( Q @ X2 ) )
% 9.49/5.24                = ( ( P5 @ X2 )
% 9.49/5.24                  | ( Q2 @ X2 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1642_minf_I2_J,axiom,
% 9.49/5.24      ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 9.49/5.24        ( ? [Z5: nat] :
% 9.49/5.24          ! [X4: nat] :
% 9.49/5.24            ( ( ord_less_nat @ X4 @ Z5 )
% 9.49/5.24           => ( ( P @ X4 )
% 9.49/5.24              = ( P5 @ X4 ) ) )
% 9.49/5.24       => ( ? [Z5: nat] :
% 9.49/5.24            ! [X4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ X4 @ Z5 )
% 9.49/5.24             => ( ( Q @ X4 )
% 9.49/5.24                = ( Q2 @ X4 ) ) )
% 9.49/5.24         => ? [Z4: nat] :
% 9.49/5.24            ! [X2: nat] :
% 9.49/5.24              ( ( ord_less_nat @ X2 @ Z4 )
% 9.49/5.24             => ( ( ( P @ X2 )
% 9.49/5.24                  | ( Q @ X2 ) )
% 9.49/5.24                = ( ( P5 @ X2 )
% 9.49/5.24                  | ( Q2 @ X2 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1643_minf_I2_J,axiom,
% 9.49/5.24      ! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
% 9.49/5.24        ( ? [Z5: int] :
% 9.49/5.24          ! [X4: int] :
% 9.49/5.24            ( ( ord_less_int @ X4 @ Z5 )
% 9.49/5.24           => ( ( P @ X4 )
% 9.49/5.24              = ( P5 @ X4 ) ) )
% 9.49/5.24       => ( ? [Z5: int] :
% 9.49/5.24            ! [X4: int] :
% 9.49/5.24              ( ( ord_less_int @ X4 @ Z5 )
% 9.49/5.24             => ( ( Q @ X4 )
% 9.49/5.24                = ( Q2 @ X4 ) ) )
% 9.49/5.24         => ? [Z4: int] :
% 9.49/5.24            ! [X2: int] :
% 9.49/5.24              ( ( ord_less_int @ X2 @ Z4 )
% 9.49/5.24             => ( ( ( P @ X2 )
% 9.49/5.24                  | ( Q @ X2 ) )
% 9.49/5.24                = ( ( P5 @ X2 )
% 9.49/5.24                  | ( Q2 @ X2 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1644_minf_I3_J,axiom,
% 9.49/5.24      ! [T: real] :
% 9.49/5.24      ? [Z4: real] :
% 9.49/5.24      ! [X2: real] :
% 9.49/5.24        ( ( ord_less_real @ X2 @ Z4 )
% 9.49/5.24       => ( X2 != T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1645_minf_I3_J,axiom,
% 9.49/5.24      ! [T: rat] :
% 9.49/5.24      ? [Z4: rat] :
% 9.49/5.24      ! [X2: rat] :
% 9.49/5.24        ( ( ord_less_rat @ X2 @ Z4 )
% 9.49/5.24       => ( X2 != T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1646_minf_I3_J,axiom,
% 9.49/5.24      ! [T: num] :
% 9.49/5.24      ? [Z4: num] :
% 9.49/5.24      ! [X2: num] :
% 9.49/5.24        ( ( ord_less_num @ X2 @ Z4 )
% 9.49/5.24       => ( X2 != T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1647_minf_I3_J,axiom,
% 9.49/5.24      ! [T: nat] :
% 9.49/5.24      ? [Z4: nat] :
% 9.49/5.24      ! [X2: nat] :
% 9.49/5.24        ( ( ord_less_nat @ X2 @ Z4 )
% 9.49/5.24       => ( X2 != T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1648_minf_I3_J,axiom,
% 9.49/5.24      ! [T: int] :
% 9.49/5.24      ? [Z4: int] :
% 9.49/5.24      ! [X2: int] :
% 9.49/5.24        ( ( ord_less_int @ X2 @ Z4 )
% 9.49/5.24       => ( X2 != T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1649_minf_I4_J,axiom,
% 9.49/5.24      ! [T: real] :
% 9.49/5.24      ? [Z4: real] :
% 9.49/5.24      ! [X2: real] :
% 9.49/5.24        ( ( ord_less_real @ X2 @ Z4 )
% 9.49/5.24       => ( X2 != T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1650_minf_I4_J,axiom,
% 9.49/5.24      ! [T: rat] :
% 9.49/5.24      ? [Z4: rat] :
% 9.49/5.24      ! [X2: rat] :
% 9.49/5.24        ( ( ord_less_rat @ X2 @ Z4 )
% 9.49/5.24       => ( X2 != T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1651_minf_I4_J,axiom,
% 9.49/5.24      ! [T: num] :
% 9.49/5.24      ? [Z4: num] :
% 9.49/5.24      ! [X2: num] :
% 9.49/5.24        ( ( ord_less_num @ X2 @ Z4 )
% 9.49/5.24       => ( X2 != T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1652_minf_I4_J,axiom,
% 9.49/5.24      ! [T: nat] :
% 9.49/5.24      ? [Z4: nat] :
% 9.49/5.24      ! [X2: nat] :
% 9.49/5.24        ( ( ord_less_nat @ X2 @ Z4 )
% 9.49/5.24       => ( X2 != T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1653_minf_I4_J,axiom,
% 9.49/5.24      ! [T: int] :
% 9.49/5.24      ? [Z4: int] :
% 9.49/5.24      ! [X2: int] :
% 9.49/5.24        ( ( ord_less_int @ X2 @ Z4 )
% 9.49/5.24       => ( X2 != T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1654_minf_I5_J,axiom,
% 9.49/5.24      ! [T: real] :
% 9.49/5.24      ? [Z4: real] :
% 9.49/5.24      ! [X2: real] :
% 9.49/5.24        ( ( ord_less_real @ X2 @ Z4 )
% 9.49/5.24       => ( ord_less_real @ X2 @ T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1655_minf_I5_J,axiom,
% 9.49/5.24      ! [T: rat] :
% 9.49/5.24      ? [Z4: rat] :
% 9.49/5.24      ! [X2: rat] :
% 9.49/5.24        ( ( ord_less_rat @ X2 @ Z4 )
% 9.49/5.24       => ( ord_less_rat @ X2 @ T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1656_minf_I5_J,axiom,
% 9.49/5.24      ! [T: num] :
% 9.49/5.24      ? [Z4: num] :
% 9.49/5.24      ! [X2: num] :
% 9.49/5.24        ( ( ord_less_num @ X2 @ Z4 )
% 9.49/5.24       => ( ord_less_num @ X2 @ T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1657_minf_I5_J,axiom,
% 9.49/5.24      ! [T: nat] :
% 9.49/5.24      ? [Z4: nat] :
% 9.49/5.24      ! [X2: nat] :
% 9.49/5.24        ( ( ord_less_nat @ X2 @ Z4 )
% 9.49/5.24       => ( ord_less_nat @ X2 @ T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1658_minf_I5_J,axiom,
% 9.49/5.24      ! [T: int] :
% 9.49/5.24      ? [Z4: int] :
% 9.49/5.24      ! [X2: int] :
% 9.49/5.24        ( ( ord_less_int @ X2 @ Z4 )
% 9.49/5.24       => ( ord_less_int @ X2 @ T ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1659_minf_I7_J,axiom,
% 9.49/5.24      ! [T: real] :
% 9.49/5.24      ? [Z4: real] :
% 9.49/5.24      ! [X2: real] :
% 9.49/5.24        ( ( ord_less_real @ X2 @ Z4 )
% 9.49/5.24       => ~ ( ord_less_real @ T @ X2 ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1660_minf_I7_J,axiom,
% 9.49/5.24      ! [T: rat] :
% 9.49/5.24      ? [Z4: rat] :
% 9.49/5.24      ! [X2: rat] :
% 9.49/5.24        ( ( ord_less_rat @ X2 @ Z4 )
% 9.49/5.24       => ~ ( ord_less_rat @ T @ X2 ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1661_minf_I7_J,axiom,
% 9.49/5.24      ! [T: num] :
% 9.49/5.24      ? [Z4: num] :
% 9.49/5.24      ! [X2: num] :
% 9.49/5.24        ( ( ord_less_num @ X2 @ Z4 )
% 9.49/5.24       => ~ ( ord_less_num @ T @ X2 ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1662_minf_I7_J,axiom,
% 9.49/5.24      ! [T: nat] :
% 9.49/5.24      ? [Z4: nat] :
% 9.49/5.24      ! [X2: nat] :
% 9.49/5.24        ( ( ord_less_nat @ X2 @ Z4 )
% 9.49/5.24       => ~ ( ord_less_nat @ T @ X2 ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1663_minf_I7_J,axiom,
% 9.49/5.24      ! [T: int] :
% 9.49/5.24      ? [Z4: int] :
% 9.49/5.24      ! [X2: int] :
% 9.49/5.24        ( ( ord_less_int @ X2 @ Z4 )
% 9.49/5.24       => ~ ( ord_less_int @ T @ X2 ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1664_deg__SUcn__Node,axiom,
% 9.49/5.24      ! [Tree: vEBT_VEBT,N: nat] :
% 9.49/5.24        ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
% 9.49/5.24       => ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S4: vEBT_VEBT] :
% 9.49/5.24            ( Tree
% 9.49/5.24            = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList3 @ S4 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1665_nth__zip,axiom,
% 9.49/5.24      ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 9.49/5.24        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.24       => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.49/5.24         => ( ( nth_Pr4953567300277697838T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) @ I )
% 9.49/5.24            = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1666_nth__zip,axiom,
% 9.49/5.24      ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
% 9.49/5.24        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.24       => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys ) )
% 9.49/5.24         => ( ( nth_Pr4606735188037164562VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys ) @ I )
% 9.49/5.24            = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_o @ Ys @ I ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1667_nth__zip,axiom,
% 9.49/5.24      ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
% 9.49/5.24        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.24       => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
% 9.49/5.24         => ( ( nth_Pr1791586995822124652BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) @ I )
% 9.49/5.24            = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1668_nth__zip,axiom,
% 9.49/5.24      ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
% 9.49/5.24        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.24       => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys ) )
% 9.49/5.24         => ( ( nth_Pr6837108013167703752BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) @ I )
% 9.49/5.24            = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_int @ Ys @ I ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1669_nth__zip,axiom,
% 9.49/5.24      ! [I: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
% 9.49/5.24        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.49/5.24       => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.49/5.24         => ( ( nth_Pr6777367263587873994T_VEBT @ ( zip_o_VEBT_VEBT @ Xs @ Ys ) @ I )
% 9.49/5.24            = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1670_nth__zip,axiom,
% 9.49/5.24      ! [I: nat,Xs: list_o,Ys: list_o] :
% 9.49/5.24        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.49/5.24       => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys ) )
% 9.49/5.24         => ( ( nth_Product_prod_o_o @ ( zip_o_o @ Xs @ Ys ) @ I )
% 9.49/5.24            = ( product_Pair_o_o @ ( nth_o @ Xs @ I ) @ ( nth_o @ Ys @ I ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1671_nth__zip,axiom,
% 9.49/5.24      ! [I: nat,Xs: list_o,Ys: list_nat] :
% 9.49/5.24        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.49/5.24       => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
% 9.49/5.24         => ( ( nth_Pr5826913651314560976_o_nat @ ( zip_o_nat @ Xs @ Ys ) @ I )
% 9.49/5.24            = ( product_Pair_o_nat @ ( nth_o @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1672_nth__zip,axiom,
% 9.49/5.24      ! [I: nat,Xs: list_o,Ys: list_int] :
% 9.49/5.24        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.49/5.24       => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys ) )
% 9.49/5.24         => ( ( nth_Pr1649062631805364268_o_int @ ( zip_o_int @ Xs @ Ys ) @ I )
% 9.49/5.24            = ( product_Pair_o_int @ ( nth_o @ Xs @ I ) @ ( nth_int @ Ys @ I ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1673_nth__zip,axiom,
% 9.49/5.24      ! [I: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
% 9.49/5.24        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 9.49/5.24       => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.49/5.24         => ( ( nth_Pr744662078594809490T_VEBT @ ( zip_nat_VEBT_VEBT @ Xs @ Ys ) @ I )
% 9.49/5.24            = ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1674_nth__zip,axiom,
% 9.49/5.24      ! [I: nat,Xs: list_nat,Ys: list_o] :
% 9.49/5.24        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 9.49/5.24       => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys ) )
% 9.49/5.24         => ( ( nth_Pr112076138515278198_nat_o @ ( zip_nat_o @ Xs @ Ys ) @ I )
% 9.49/5.24            = ( product_Pair_nat_o @ ( nth_nat @ Xs @ I ) @ ( nth_o @ Ys @ I ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1675_find__Some__iff2,axiom,
% 9.49/5.24      ! [X: product_prod_nat_nat,P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
% 9.49/5.24        ( ( ( some_P7363390416028606310at_nat @ X )
% 9.49/5.24          = ( find_P8199882355184865565at_nat @ P @ Xs ) )
% 9.49/5.24        = ( ? [I4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 9.49/5.24              & ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
% 9.49/5.24              & ( X
% 9.49/5.24                = ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
% 9.49/5.24              & ! [J3: nat] :
% 9.49/5.24                  ( ( ord_less_nat @ J3 @ I4 )
% 9.49/5.24                 => ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1676_find__Some__iff2,axiom,
% 9.49/5.24      ! [X: num,P: num > $o,Xs: list_num] :
% 9.49/5.24        ( ( ( some_num @ X )
% 9.49/5.24          = ( find_num @ P @ Xs ) )
% 9.49/5.24        = ( ? [I4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I4 @ ( size_size_list_num @ Xs ) )
% 9.49/5.24              & ( P @ ( nth_num @ Xs @ I4 ) )
% 9.49/5.24              & ( X
% 9.49/5.24                = ( nth_num @ Xs @ I4 ) )
% 9.49/5.24              & ! [J3: nat] :
% 9.49/5.24                  ( ( ord_less_nat @ J3 @ I4 )
% 9.49/5.24                 => ~ ( P @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1677_find__Some__iff2,axiom,
% 9.49/5.24      ! [X: vEBT_VEBT,P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 9.49/5.24        ( ( ( some_VEBT_VEBT @ X )
% 9.49/5.24          = ( find_VEBT_VEBT @ P @ Xs ) )
% 9.49/5.24        = ( ? [I4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.24              & ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) )
% 9.49/5.24              & ( X
% 9.49/5.24                = ( nth_VEBT_VEBT @ Xs @ I4 ) )
% 9.49/5.24              & ! [J3: nat] :
% 9.49/5.24                  ( ( ord_less_nat @ J3 @ I4 )
% 9.49/5.24                 => ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1678_find__Some__iff2,axiom,
% 9.49/5.24      ! [X: $o,P: $o > $o,Xs: list_o] :
% 9.49/5.24        ( ( ( some_o @ X )
% 9.49/5.24          = ( find_o @ P @ Xs ) )
% 9.49/5.24        = ( ? [I4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 9.49/5.24              & ( P @ ( nth_o @ Xs @ I4 ) )
% 9.49/5.24              & ( X
% 9.49/5.24                = ( nth_o @ Xs @ I4 ) )
% 9.49/5.24              & ! [J3: nat] :
% 9.49/5.24                  ( ( ord_less_nat @ J3 @ I4 )
% 9.49/5.24                 => ~ ( P @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1679_find__Some__iff2,axiom,
% 9.49/5.24      ! [X: nat,P: nat > $o,Xs: list_nat] :
% 9.49/5.24        ( ( ( some_nat @ X )
% 9.49/5.24          = ( find_nat @ P @ Xs ) )
% 9.49/5.24        = ( ? [I4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 9.49/5.24              & ( P @ ( nth_nat @ Xs @ I4 ) )
% 9.49/5.24              & ( X
% 9.49/5.24                = ( nth_nat @ Xs @ I4 ) )
% 9.49/5.24              & ! [J3: nat] :
% 9.49/5.24                  ( ( ord_less_nat @ J3 @ I4 )
% 9.49/5.24                 => ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1680_find__Some__iff2,axiom,
% 9.49/5.24      ! [X: int,P: int > $o,Xs: list_int] :
% 9.49/5.24        ( ( ( some_int @ X )
% 9.49/5.24          = ( find_int @ P @ Xs ) )
% 9.49/5.24        = ( ? [I4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 9.49/5.24              & ( P @ ( nth_int @ Xs @ I4 ) )
% 9.49/5.24              & ( X
% 9.49/5.24                = ( nth_int @ Xs @ I4 ) )
% 9.49/5.24              & ! [J3: nat] :
% 9.49/5.24                  ( ( ord_less_nat @ J3 @ I4 )
% 9.49/5.24                 => ~ ( P @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1681_find__Some__iff,axiom,
% 9.49/5.24      ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
% 9.49/5.24        ( ( ( find_P8199882355184865565at_nat @ P @ Xs )
% 9.49/5.24          = ( some_P7363390416028606310at_nat @ X ) )
% 9.49/5.24        = ( ? [I4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 9.49/5.24              & ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
% 9.49/5.24              & ( X
% 9.49/5.24                = ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
% 9.49/5.24              & ! [J3: nat] :
% 9.49/5.24                  ( ( ord_less_nat @ J3 @ I4 )
% 9.49/5.24                 => ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1682_find__Some__iff,axiom,
% 9.49/5.24      ! [P: num > $o,Xs: list_num,X: num] :
% 9.49/5.24        ( ( ( find_num @ P @ Xs )
% 9.49/5.24          = ( some_num @ X ) )
% 9.49/5.24        = ( ? [I4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I4 @ ( size_size_list_num @ Xs ) )
% 9.49/5.24              & ( P @ ( nth_num @ Xs @ I4 ) )
% 9.49/5.24              & ( X
% 9.49/5.24                = ( nth_num @ Xs @ I4 ) )
% 9.49/5.24              & ! [J3: nat] :
% 9.49/5.24                  ( ( ord_less_nat @ J3 @ I4 )
% 9.49/5.24                 => ~ ( P @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1683_find__Some__iff,axiom,
% 9.49/5.24      ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 9.49/5.24        ( ( ( find_VEBT_VEBT @ P @ Xs )
% 9.49/5.24          = ( some_VEBT_VEBT @ X ) )
% 9.49/5.24        = ( ? [I4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.24              & ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) )
% 9.49/5.24              & ( X
% 9.49/5.24                = ( nth_VEBT_VEBT @ Xs @ I4 ) )
% 9.49/5.24              & ! [J3: nat] :
% 9.49/5.24                  ( ( ord_less_nat @ J3 @ I4 )
% 9.49/5.24                 => ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1684_find__Some__iff,axiom,
% 9.49/5.24      ! [P: $o > $o,Xs: list_o,X: $o] :
% 9.49/5.24        ( ( ( find_o @ P @ Xs )
% 9.49/5.24          = ( some_o @ X ) )
% 9.49/5.24        = ( ? [I4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 9.49/5.24              & ( P @ ( nth_o @ Xs @ I4 ) )
% 9.49/5.24              & ( X
% 9.49/5.24                = ( nth_o @ Xs @ I4 ) )
% 9.49/5.24              & ! [J3: nat] :
% 9.49/5.24                  ( ( ord_less_nat @ J3 @ I4 )
% 9.49/5.24                 => ~ ( P @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1685_find__Some__iff,axiom,
% 9.49/5.24      ! [P: nat > $o,Xs: list_nat,X: nat] :
% 9.49/5.24        ( ( ( find_nat @ P @ Xs )
% 9.49/5.24          = ( some_nat @ X ) )
% 9.49/5.24        = ( ? [I4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 9.49/5.24              & ( P @ ( nth_nat @ Xs @ I4 ) )
% 9.49/5.24              & ( X
% 9.49/5.24                = ( nth_nat @ Xs @ I4 ) )
% 9.49/5.24              & ! [J3: nat] :
% 9.49/5.24                  ( ( ord_less_nat @ J3 @ I4 )
% 9.49/5.24                 => ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1686_find__Some__iff,axiom,
% 9.49/5.24      ! [P: int > $o,Xs: list_int,X: int] :
% 9.49/5.24        ( ( ( find_int @ P @ Xs )
% 9.49/5.24          = ( some_int @ X ) )
% 9.49/5.24        = ( ? [I4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 9.49/5.24              & ( P @ ( nth_int @ Xs @ I4 ) )
% 9.49/5.24              & ( X
% 9.49/5.24                = ( nth_int @ Xs @ I4 ) )
% 9.49/5.24              & ! [J3: nat] :
% 9.49/5.24                  ( ( ord_less_nat @ J3 @ I4 )
% 9.49/5.24                 => ~ ( P @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1687_count__notin,axiom,
% 9.49/5.24      ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
% 9.49/5.24        ( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.49/5.24       => ( ( count_4203492906077236349at_nat @ Xs @ X )
% 9.49/5.24          = zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1688_count__notin,axiom,
% 9.49/5.24      ! [X: real,Xs: list_real] :
% 9.49/5.24        ( ~ ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.49/5.24       => ( ( count_list_real @ Xs @ X )
% 9.49/5.24          = zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1689_count__notin,axiom,
% 9.49/5.24      ! [X: set_nat,Xs: list_set_nat] :
% 9.49/5.24        ( ~ ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 9.49/5.24       => ( ( count_list_set_nat @ Xs @ X )
% 9.49/5.24          = zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1690_count__notin,axiom,
% 9.49/5.24      ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 9.49/5.24        ( ~ ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.24       => ( ( count_list_VEBT_VEBT @ Xs @ X )
% 9.49/5.24          = zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1691_count__notin,axiom,
% 9.49/5.24      ! [X: int,Xs: list_int] :
% 9.49/5.24        ( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
% 9.49/5.24       => ( ( count_list_int @ Xs @ X )
% 9.49/5.24          = zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1692_count__notin,axiom,
% 9.49/5.24      ! [X: nat,Xs: list_nat] :
% 9.49/5.24        ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 9.49/5.24       => ( ( count_list_nat @ Xs @ X )
% 9.49/5.24          = zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1693__C4_Ohyps_C_I9_J,axiom,
% 9.49/5.24      ( ( mi != ma )
% 9.49/5.24     => ! [I3: nat] :
% 9.49/5.24          ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 9.49/5.24         => ( ( ( ( vEBT_VEBT_high @ ma @ na )
% 9.49/5.24                = I3 )
% 9.49/5.24             => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I3 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
% 9.49/5.24            & ! [X2: nat] :
% 9.49/5.24                ( ( ( ( vEBT_VEBT_high @ X2 @ na )
% 9.49/5.24                    = I3 )
% 9.49/5.24                  & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I3 ) @ ( vEBT_VEBT_low @ X2 @ na ) ) )
% 9.49/5.24               => ( ( ord_less_nat @ mi @ X2 )
% 9.49/5.24                  & ( ord_less_eq_nat @ X2 @ ma ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1694_listrel1__iff__update,axiom,
% 9.49/5.24      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
% 9.49/5.24        ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Xs @ Ys ) @ ( listrel1_VEBT_VEBT @ R2 ) )
% 9.49/5.24        = ( ? [Y2: vEBT_VEBT,N2: nat] :
% 9.49/5.24              ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ N2 ) @ Y2 ) @ R2 )
% 9.49/5.24              & ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.49/5.24              & ( Ys
% 9.49/5.24                = ( list_u1324408373059187874T_VEBT @ Xs @ N2 @ Y2 ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1695_listrel1__iff__update,axiom,
% 9.49/5.24      ! [Xs: list_o,Ys: list_o,R2: set_Product_prod_o_o] :
% 9.49/5.24        ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( listrel1_o @ R2 ) )
% 9.49/5.24        = ( ? [Y2: $o,N2: nat] :
% 9.49/5.24              ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( nth_o @ Xs @ N2 ) @ Y2 ) @ R2 )
% 9.49/5.24              & ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
% 9.49/5.24              & ( Ys
% 9.49/5.24                = ( list_update_o @ Xs @ N2 @ Y2 ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1696_listrel1__iff__update,axiom,
% 9.49/5.24      ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
% 9.49/5.24        ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
% 9.49/5.24        = ( ? [Y2: nat,N2: nat] :
% 9.49/5.24              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ Y2 ) @ R2 )
% 9.49/5.24              & ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
% 9.49/5.24              & ( Ys
% 9.49/5.24                = ( list_update_nat @ Xs @ N2 @ Y2 ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1697_listrel1__iff__update,axiom,
% 9.49/5.24      ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
% 9.49/5.24        ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
% 9.49/5.24        = ( ? [Y2: int,N2: nat] :
% 9.49/5.24              ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N2 ) @ Y2 ) @ R2 )
% 9.49/5.24              & ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
% 9.49/5.24              & ( Ys
% 9.49/5.24                = ( list_update_int @ Xs @ N2 @ Y2 ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1698_vebt__pred_Osimps_I6_J,axiom,
% 9.49/5.24      ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
% 9.49/5.24        ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
% 9.49/5.24        = none_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1699_vebt__succ_Osimps_I5_J,axiom,
% 9.49/5.24      ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
% 9.49/5.24        ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
% 9.49/5.24        = none_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1700_pair__lessI2,axiom,
% 9.49/5.24      ! [A: nat,B: nat,S: nat,T: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.24       => ( ( ord_less_nat @ S @ T )
% 9.49/5.24         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1701_even__odd__cases,axiom,
% 9.49/5.24      ! [X: nat] :
% 9.49/5.24        ( ! [N3: nat] :
% 9.49/5.24            ( X
% 9.49/5.24           != ( plus_plus_nat @ N3 @ N3 ) )
% 9.49/5.24       => ~ ! [N3: nat] :
% 9.49/5.24              ( X
% 9.49/5.24             != ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1702_power__shift,axiom,
% 9.49/5.24      ! [X: nat,Y: nat,Z2: nat] :
% 9.49/5.24        ( ( ( power_power_nat @ X @ Y )
% 9.49/5.24          = Z2 )
% 9.49/5.24        = ( ( vEBT_VEBT_power @ ( some_nat @ X ) @ ( some_nat @ Y ) )
% 9.49/5.24          = ( some_nat @ Z2 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1703__C4_Ohyps_C_I8_J,axiom,
% 9.49/5.24      ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1704__C4_Ohyps_C_I2_J,axiom,
% 9.49/5.24      ( ( size_s6755466524823107622T_VEBT @ treeList )
% 9.49/5.24      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1705_old_Onat_Oinject,axiom,
% 9.49/5.24      ! [Nat: nat,Nat2: nat] :
% 9.49/5.24        ( ( ( suc @ Nat )
% 9.49/5.24          = ( suc @ Nat2 ) )
% 9.49/5.24        = ( Nat = Nat2 ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1706_nat_Oinject,axiom,
% 9.49/5.24      ! [X22: nat,Y22: nat] :
% 9.49/5.24        ( ( ( suc @ X22 )
% 9.49/5.24          = ( suc @ Y22 ) )
% 9.49/5.24        = ( X22 = Y22 ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1707_local_Opower__def,axiom,
% 9.49/5.24      ( vEBT_VEBT_power
% 9.49/5.24      = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1708_dp,axiom,
% 9.49/5.24      ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1709__C4_Ohyps_C_I5_J,axiom,
% 9.49/5.24      ! [I3: nat] :
% 9.49/5.24        ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 9.49/5.24       => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I3 ) @ X6 ) )
% 9.49/5.24          = ( vEBT_V8194947554948674370ptions @ summary @ I3 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1710_high__bound__aux,axiom,
% 9.49/5.24      ! [Ma: nat,N: nat,M: nat] :
% 9.49/5.24        ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 9.49/5.24       => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1711_member__bound,axiom,
% 9.49/5.24      ! [Tree: vEBT_VEBT,X: nat,N: nat] :
% 9.49/5.24        ( ( vEBT_vebt_member @ Tree @ X )
% 9.49/5.24       => ( ( vEBT_invar_vebt @ Tree @ N )
% 9.49/5.24         => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1712_numeral__le__iff,axiom,
% 9.49/5.24      ! [M: num,N: num] :
% 9.49/5.24        ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 9.49/5.24        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1713_numeral__le__iff,axiom,
% 9.49/5.24      ! [M: num,N: num] :
% 9.49/5.24        ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 9.49/5.24        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1714_numeral__le__iff,axiom,
% 9.49/5.24      ! [M: num,N: num] :
% 9.49/5.24        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 9.49/5.24        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1715_numeral__le__iff,axiom,
% 9.49/5.24      ! [M: num,N: num] :
% 9.49/5.24        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.49/5.24        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1716_numeral__le__iff,axiom,
% 9.49/5.24      ! [M: num,N: num] :
% 9.49/5.24        ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.49/5.24        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1717_numeral__less__iff,axiom,
% 9.49/5.24      ! [M: num,N: num] :
% 9.49/5.24        ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 9.49/5.24        = ( ord_less_num @ M @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1718_numeral__less__iff,axiom,
% 9.49/5.24      ! [M: num,N: num] :
% 9.49/5.24        ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 9.49/5.24        = ( ord_less_num @ M @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1719_numeral__less__iff,axiom,
% 9.49/5.24      ! [M: num,N: num] :
% 9.49/5.24        ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 9.49/5.24        = ( ord_less_num @ M @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1720_numeral__less__iff,axiom,
% 9.49/5.24      ! [M: num,N: num] :
% 9.49/5.24        ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.49/5.24        = ( ord_less_num @ M @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1721_numeral__less__iff,axiom,
% 9.49/5.24      ! [M: num,N: num] :
% 9.49/5.24        ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.49/5.24        = ( ord_less_num @ M @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1722_lessI,axiom,
% 9.49/5.24      ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1723_Suc__mono,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ M @ N )
% 9.49/5.24       => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1724_Suc__less__eq,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 9.49/5.24        = ( ord_less_nat @ M @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1725_Suc__le__mono,axiom,
% 9.49/5.24      ! [N: nat,M: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 9.49/5.24        = ( ord_less_eq_nat @ N @ M ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1726_misiz,axiom,
% 9.49/5.24      ! [T: vEBT_VEBT,N: nat,M: nat] :
% 9.49/5.24        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.24       => ( ( ( some_nat @ M )
% 9.49/5.24            = ( vEBT_vebt_mint @ T ) )
% 9.49/5.24         => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1727_add__Suc__right,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 9.49/5.24        = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1728_helpyd,axiom,
% 9.49/5.24      ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
% 9.49/5.24        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.24       => ( ( ( vEBT_vebt_succ @ T @ X )
% 9.49/5.24            = ( some_nat @ Y ) )
% 9.49/5.24         => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1729_helpypredd,axiom,
% 9.49/5.24      ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
% 9.49/5.24        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.24       => ( ( ( vEBT_vebt_pred @ T @ X )
% 9.49/5.24            = ( some_nat @ Y ) )
% 9.49/5.24         => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1730__092_060open_062x_A_060_A2_A_094_Adeg_092_060close_062,axiom,
% 9.49/5.24      ord_less_nat @ xa @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1731__092_060open_062ma_A_092_060le_062_A2_A_094_Adeg_092_060close_062,axiom,
% 9.49/5.24      ord_less_eq_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1732_delt__out__of__range,axiom,
% 9.49/5.24      ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.49/5.24        ( ( ( ord_less_nat @ X @ Mi )
% 9.49/5.24          | ( ord_less_nat @ Ma @ X ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.24         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.24            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1733_del__single__cont,axiom,
% 9.49/5.24      ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.49/5.24        ( ( ( X = Mi )
% 9.49/5.24          & ( X = Ma ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.24         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.24            = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1734_mi__ma__2__deg,axiom,
% 9.49/5.24      ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 9.49/5.24        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 9.49/5.24       => ( ( ord_less_eq_nat @ Mi @ Ma )
% 9.49/5.24          & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1735_pred__max,axiom,
% 9.49/5.24      ! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.49/5.24        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.24       => ( ( ord_less_nat @ Ma @ X )
% 9.49/5.24         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.24            = ( some_nat @ Ma ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1736_succ__min,axiom,
% 9.49/5.24      ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.49/5.24        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.24       => ( ( ord_less_nat @ X @ Mi )
% 9.49/5.24         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.24            = ( some_nat @ Mi ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1737_less__Suc0,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.49/5.24        = ( N = zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1738_zero__less__Suc,axiom,
% 9.49/5.24      ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1739_nothprolist,axiom,
% 9.49/5.24      ! [I: nat] :
% 9.49/5.24        ( ( ( I
% 9.49/5.24           != ( vEBT_VEBT_high @ xa @ na ) )
% 9.49/5.24          & ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) )
% 9.49/5.24       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ I )
% 9.49/5.24          = ( nth_VEBT_VEBT @ treeList @ I ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1740_pair__less__iff1,axiom,
% 9.49/5.24      ! [X: nat,Y: nat,Z2: nat] :
% 9.49/5.24        ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z2 ) ) @ fun_pair_less )
% 9.49/5.24        = ( ord_less_nat @ Y @ Z2 ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1741_n__not__Suc__n,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( N
% 9.49/5.24       != ( suc @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1742_Suc__inject,axiom,
% 9.49/5.24      ! [X: nat,Y: nat] :
% 9.49/5.24        ( ( ( suc @ X )
% 9.49/5.24          = ( suc @ Y ) )
% 9.49/5.24       => ( X = Y ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1743_less__2__cases,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24       => ( ( N = zero_zero_nat )
% 9.49/5.24          | ( N
% 9.49/5.24            = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1744_less__2__cases__iff,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24        = ( ( N = zero_zero_nat )
% 9.49/5.24          | ( N
% 9.49/5.24            = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1745_numeral__1__eq__Suc__0,axiom,
% 9.49/5.24      ( ( numeral_numeral_nat @ one )
% 9.49/5.24      = ( suc @ zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1746_numeral__2__eq__2,axiom,
% 9.49/5.24      ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 9.49/5.24      = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1747_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
% 9.49/5.24      ! [X: nat,N: nat,M: nat] :
% 9.49/5.24        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 9.49/5.24       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.49/5.24           => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1748_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
% 9.49/5.24      ! [X: nat,N: nat,M: nat] :
% 9.49/5.24        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 9.49/5.24       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.49/5.24           => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1749_num_Osize_I5_J,axiom,
% 9.49/5.24      ! [X22: num] :
% 9.49/5.24        ( ( size_size_num @ ( bit0 @ X22 ) )
% 9.49/5.24        = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1750_zero__neq__numeral,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ( zero_zero_complex
% 9.49/5.24       != ( numera6690914467698888265omplex @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1751_zero__neq__numeral,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ( zero_z5237406670263579293d_enat
% 9.49/5.24       != ( numera1916890842035813515d_enat @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1752_zero__neq__numeral,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ( zero_zero_real
% 9.49/5.24       != ( numeral_numeral_real @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1753_zero__neq__numeral,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ( zero_zero_rat
% 9.49/5.24       != ( numeral_numeral_rat @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1754_zero__neq__numeral,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ( zero_zero_nat
% 9.49/5.24       != ( numeral_numeral_nat @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1755_zero__neq__numeral,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ( zero_zero_int
% 9.49/5.24       != ( numeral_numeral_int @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1756_vebt__buildup_Ocases,axiom,
% 9.49/5.24      ! [X: nat] :
% 9.49/5.24        ( ( X != zero_zero_nat )
% 9.49/5.24       => ( ( X
% 9.49/5.24           != ( suc @ zero_zero_nat ) )
% 9.49/5.24         => ~ ! [Va2: nat] :
% 9.49/5.24                ( X
% 9.49/5.24               != ( suc @ ( suc @ Va2 ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1757_nat_Odistinct_I1_J,axiom,
% 9.49/5.24      ! [X22: nat] :
% 9.49/5.24        ( zero_zero_nat
% 9.49/5.24       != ( suc @ X22 ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1758_old_Onat_Odistinct_I2_J,axiom,
% 9.49/5.24      ! [Nat2: nat] :
% 9.49/5.24        ( ( suc @ Nat2 )
% 9.49/5.24       != zero_zero_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1759_old_Onat_Odistinct_I1_J,axiom,
% 9.49/5.24      ! [Nat2: nat] :
% 9.49/5.24        ( zero_zero_nat
% 9.49/5.24       != ( suc @ Nat2 ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1760_nat_OdiscI,axiom,
% 9.49/5.24      ! [Nat: nat,X22: nat] :
% 9.49/5.24        ( ( Nat
% 9.49/5.24          = ( suc @ X22 ) )
% 9.49/5.24       => ( Nat != zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1761_old_Onat_Oexhaust,axiom,
% 9.49/5.24      ! [Y: nat] :
% 9.49/5.24        ( ( Y != zero_zero_nat )
% 9.49/5.24       => ~ ! [Nat3: nat] :
% 9.49/5.24              ( Y
% 9.49/5.24             != ( suc @ Nat3 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1762_nat__induct,axiom,
% 9.49/5.24      ! [P: nat > $o,N: nat] :
% 9.49/5.24        ( ( P @ zero_zero_nat )
% 9.49/5.24       => ( ! [N3: nat] :
% 9.49/5.24              ( ( P @ N3 )
% 9.49/5.24             => ( P @ ( suc @ N3 ) ) )
% 9.49/5.24         => ( P @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1763_diff__induct,axiom,
% 9.49/5.24      ! [P: nat > nat > $o,M: nat,N: nat] :
% 9.49/5.24        ( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
% 9.49/5.24       => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
% 9.49/5.24         => ( ! [X4: nat,Y3: nat] :
% 9.49/5.24                ( ( P @ X4 @ Y3 )
% 9.49/5.24               => ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
% 9.49/5.24           => ( P @ M @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1764_zero__induct,axiom,
% 9.49/5.24      ! [P: nat > $o,K: nat] :
% 9.49/5.24        ( ( P @ K )
% 9.49/5.24       => ( ! [N3: nat] :
% 9.49/5.24              ( ( P @ ( suc @ N3 ) )
% 9.49/5.24             => ( P @ N3 ) )
% 9.49/5.24         => ( P @ zero_zero_nat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1765_Suc__neq__Zero,axiom,
% 9.49/5.24      ! [M: nat] :
% 9.49/5.24        ( ( suc @ M )
% 9.49/5.24       != zero_zero_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1766_Zero__neq__Suc,axiom,
% 9.49/5.24      ! [M: nat] :
% 9.49/5.24        ( zero_zero_nat
% 9.49/5.24       != ( suc @ M ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1767_Zero__not__Suc,axiom,
% 9.49/5.24      ! [M: nat] :
% 9.49/5.24        ( zero_zero_nat
% 9.49/5.24       != ( suc @ M ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1768_not0__implies__Suc,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( N != zero_zero_nat )
% 9.49/5.24       => ? [M5: nat] :
% 9.49/5.24            ( N
% 9.49/5.24            = ( suc @ M5 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1769_Nat_OlessE,axiom,
% 9.49/5.24      ! [I: nat,K: nat] :
% 9.49/5.24        ( ( ord_less_nat @ I @ K )
% 9.49/5.24       => ( ( K
% 9.49/5.24           != ( suc @ I ) )
% 9.49/5.24         => ~ ! [J2: nat] :
% 9.49/5.24                ( ( ord_less_nat @ I @ J2 )
% 9.49/5.24               => ( K
% 9.49/5.24                 != ( suc @ J2 ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1770_Suc__lessD,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ ( suc @ M ) @ N )
% 9.49/5.24       => ( ord_less_nat @ M @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1771_Suc__lessE,axiom,
% 9.49/5.24      ! [I: nat,K: nat] :
% 9.49/5.24        ( ( ord_less_nat @ ( suc @ I ) @ K )
% 9.49/5.24       => ~ ! [J2: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I @ J2 )
% 9.49/5.24             => ( K
% 9.49/5.24               != ( suc @ J2 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1772_Suc__lessI,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ M @ N )
% 9.49/5.24       => ( ( ( suc @ M )
% 9.49/5.24           != N )
% 9.49/5.24         => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1773_less__SucE,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 9.49/5.24       => ( ~ ( ord_less_nat @ M @ N )
% 9.49/5.24         => ( M = N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1774_less__SucI,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ M @ N )
% 9.49/5.24       => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1775_Ex__less__Suc,axiom,
% 9.49/5.24      ! [N: nat,P: nat > $o] :
% 9.49/5.24        ( ( ? [I4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 9.49/5.24              & ( P @ I4 ) ) )
% 9.49/5.24        = ( ( P @ N )
% 9.49/5.24          | ? [I4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I4 @ N )
% 9.49/5.24              & ( P @ I4 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1776_less__Suc__eq,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 9.49/5.24        = ( ( ord_less_nat @ M @ N )
% 9.49/5.24          | ( M = N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1777_not__less__eq,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ~ ( ord_less_nat @ M @ N ) )
% 9.49/5.24        = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1778_All__less__Suc,axiom,
% 9.49/5.24      ! [N: nat,P: nat > $o] :
% 9.49/5.24        ( ( ! [I4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 9.49/5.24             => ( P @ I4 ) ) )
% 9.49/5.24        = ( ( P @ N )
% 9.49/5.24          & ! [I4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I4 @ N )
% 9.49/5.24             => ( P @ I4 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1779_Suc__less__eq2,axiom,
% 9.49/5.24      ! [N: nat,M: nat] :
% 9.49/5.24        ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.49/5.24        = ( ? [M6: nat] :
% 9.49/5.24              ( ( M
% 9.49/5.24                = ( suc @ M6 ) )
% 9.49/5.24              & ( ord_less_nat @ N @ M6 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1780_less__antisym,axiom,
% 9.49/5.24      ! [N: nat,M: nat] :
% 9.49/5.24        ( ~ ( ord_less_nat @ N @ M )
% 9.49/5.24       => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 9.49/5.24         => ( M = N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1781_Suc__less__SucD,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 9.49/5.24       => ( ord_less_nat @ M @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1782_less__trans__Suc,axiom,
% 9.49/5.24      ! [I: nat,J: nat,K: nat] :
% 9.49/5.24        ( ( ord_less_nat @ I @ J )
% 9.49/5.24       => ( ( ord_less_nat @ J @ K )
% 9.49/5.24         => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1783_less__Suc__induct,axiom,
% 9.49/5.24      ! [I: nat,J: nat,P: nat > nat > $o] :
% 9.49/5.24        ( ( ord_less_nat @ I @ J )
% 9.49/5.24       => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
% 9.49/5.24         => ( ! [I2: nat,J2: nat,K2: nat] :
% 9.49/5.24                ( ( ord_less_nat @ I2 @ J2 )
% 9.49/5.24               => ( ( ord_less_nat @ J2 @ K2 )
% 9.49/5.24                 => ( ( P @ I2 @ J2 )
% 9.49/5.24                   => ( ( P @ J2 @ K2 )
% 9.49/5.24                     => ( P @ I2 @ K2 ) ) ) ) )
% 9.49/5.24           => ( P @ I @ J ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1784_strict__inc__induct,axiom,
% 9.49/5.24      ! [I: nat,J: nat,P: nat > $o] :
% 9.49/5.24        ( ( ord_less_nat @ I @ J )
% 9.49/5.24       => ( ! [I2: nat] :
% 9.49/5.24              ( ( J
% 9.49/5.24                = ( suc @ I2 ) )
% 9.49/5.24             => ( P @ I2 ) )
% 9.49/5.24         => ( ! [I2: nat] :
% 9.49/5.24                ( ( ord_less_nat @ I2 @ J )
% 9.49/5.24               => ( ( P @ ( suc @ I2 ) )
% 9.49/5.24                 => ( P @ I2 ) ) )
% 9.49/5.24           => ( P @ I ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1785_not__less__less__Suc__eq,axiom,
% 9.49/5.24      ! [N: nat,M: nat] :
% 9.49/5.24        ( ~ ( ord_less_nat @ N @ M )
% 9.49/5.24       => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 9.49/5.24          = ( N = M ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1786_Suc__leD,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 9.49/5.24       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1787_le__SucE,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.49/5.24       => ( ~ ( ord_less_eq_nat @ M @ N )
% 9.49/5.24         => ( M
% 9.49/5.24            = ( suc @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1788_le__SucI,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.24       => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1789_Suc__le__D,axiom,
% 9.49/5.24      ! [N: nat,M7: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
% 9.49/5.24       => ? [M5: nat] :
% 9.49/5.24            ( M7
% 9.49/5.24            = ( suc @ M5 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1790_le__Suc__eq,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.49/5.24        = ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.24          | ( M
% 9.49/5.24            = ( suc @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1791_Suc__n__not__le__n,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1792_not__less__eq__eq,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 9.49/5.24        = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1793_full__nat__induct,axiom,
% 9.49/5.24      ! [P: nat > $o,N: nat] :
% 9.49/5.24        ( ! [N3: nat] :
% 9.49/5.24            ( ! [M3: nat] :
% 9.49/5.24                ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
% 9.49/5.24               => ( P @ M3 ) )
% 9.49/5.24           => ( P @ N3 ) )
% 9.49/5.24       => ( P @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1794_nat__induct__at__least,axiom,
% 9.49/5.24      ! [M: nat,N: nat,P: nat > $o] :
% 9.49/5.24        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.24       => ( ( P @ M )
% 9.49/5.24         => ( ! [N3: nat] :
% 9.49/5.24                ( ( ord_less_eq_nat @ M @ N3 )
% 9.49/5.24               => ( ( P @ N3 )
% 9.49/5.24                 => ( P @ ( suc @ N3 ) ) ) )
% 9.49/5.24           => ( P @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1795_transitive__stepwise__le,axiom,
% 9.49/5.24      ! [M: nat,N: nat,R: nat > nat > $o] :
% 9.49/5.24        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.24       => ( ! [X4: nat] : ( R @ X4 @ X4 )
% 9.49/5.24         => ( ! [X4: nat,Y3: nat,Z4: nat] :
% 9.49/5.24                ( ( R @ X4 @ Y3 )
% 9.49/5.24               => ( ( R @ Y3 @ Z4 )
% 9.49/5.24                 => ( R @ X4 @ Z4 ) ) )
% 9.49/5.24           => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
% 9.49/5.24             => ( R @ M @ N ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1796_add__Suc__shift,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 9.49/5.24        = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1797_add__Suc,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 9.49/5.24        = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1798_nat__arith_Osuc1,axiom,
% 9.49/5.24      ! [A2: nat,K: nat,A: nat] :
% 9.49/5.24        ( ( A2
% 9.49/5.24          = ( plus_plus_nat @ K @ A ) )
% 9.49/5.24       => ( ( suc @ A2 )
% 9.49/5.24          = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1799_num_Osize_I4_J,axiom,
% 9.49/5.24      ( ( size_size_num @ one )
% 9.49/5.24      = zero_zero_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1800_listrel1__eq__len,axiom,
% 9.49/5.24      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
% 9.49/5.24        ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Xs @ Ys ) @ ( listrel1_VEBT_VEBT @ R2 ) )
% 9.49/5.24       => ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.49/5.24          = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1801_listrel1__eq__len,axiom,
% 9.49/5.24      ! [Xs: list_o,Ys: list_o,R2: set_Product_prod_o_o] :
% 9.49/5.24        ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( listrel1_o @ R2 ) )
% 9.49/5.24       => ( ( size_size_list_o @ Xs )
% 9.49/5.24          = ( size_size_list_o @ Ys ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1802_listrel1__eq__len,axiom,
% 9.49/5.24      ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
% 9.49/5.24        ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
% 9.49/5.24       => ( ( size_size_list_nat @ Xs )
% 9.49/5.24          = ( size_size_list_nat @ Ys ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1803_listrel1__eq__len,axiom,
% 9.49/5.24      ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
% 9.49/5.24        ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
% 9.49/5.24       => ( ( size_size_list_int @ Xs )
% 9.49/5.24          = ( size_size_list_int @ Ys ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1804_invar__vebt_Ointros_I3_J,axiom,
% 9.49/5.24      ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 9.49/5.24        ( ! [X4: vEBT_VEBT] :
% 9.49/5.24            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.49/5.24           => ( vEBT_invar_vebt @ X4 @ N ) )
% 9.49/5.24       => ( ( vEBT_invar_vebt @ Summary @ M )
% 9.49/5.24         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.49/5.24              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.49/5.24           => ( ( M
% 9.49/5.24                = ( suc @ N ) )
% 9.49/5.24             => ( ( Deg
% 9.49/5.24                  = ( plus_plus_nat @ N @ M ) )
% 9.49/5.24               => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 9.49/5.24                 => ( ! [X4: vEBT_VEBT] :
% 9.49/5.24                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.49/5.24                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
% 9.49/5.24                   => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1805_invar__vebt_Ointros_I2_J,axiom,
% 9.49/5.24      ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 9.49/5.24        ( ! [X4: vEBT_VEBT] :
% 9.49/5.24            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.49/5.24           => ( vEBT_invar_vebt @ X4 @ N ) )
% 9.49/5.24       => ( ( vEBT_invar_vebt @ Summary @ M )
% 9.49/5.24         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.49/5.24              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.49/5.24           => ( ( M = N )
% 9.49/5.24             => ( ( Deg
% 9.49/5.24                  = ( plus_plus_nat @ N @ M ) )
% 9.49/5.24               => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 9.49/5.24                 => ( ! [X4: vEBT_VEBT] :
% 9.49/5.24                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.49/5.24                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
% 9.49/5.24                   => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1806_find__cong,axiom,
% 9.49/5.24      ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 9.49/5.24        ( ( Xs = Ys )
% 9.49/5.24       => ( ! [X4: product_prod_nat_nat] :
% 9.49/5.24              ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Ys ) )
% 9.49/5.24             => ( ( P @ X4 )
% 9.49/5.24                = ( Q @ X4 ) ) )
% 9.49/5.24         => ( ( find_P8199882355184865565at_nat @ P @ Xs )
% 9.49/5.24            = ( find_P8199882355184865565at_nat @ Q @ Ys ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1807_find__cong,axiom,
% 9.49/5.24      ! [Xs: list_real,Ys: list_real,P: real > $o,Q: real > $o] :
% 9.49/5.24        ( ( Xs = Ys )
% 9.49/5.24       => ( ! [X4: real] :
% 9.49/5.24              ( ( member_real @ X4 @ ( set_real2 @ Ys ) )
% 9.49/5.24             => ( ( P @ X4 )
% 9.49/5.24                = ( Q @ X4 ) ) )
% 9.49/5.24         => ( ( find_real @ P @ Xs )
% 9.49/5.24            = ( find_real @ Q @ Ys ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1808_find__cong,axiom,
% 9.49/5.24      ! [Xs: list_set_nat,Ys: list_set_nat,P: set_nat > $o,Q: set_nat > $o] :
% 9.49/5.24        ( ( Xs = Ys )
% 9.49/5.24       => ( ! [X4: set_nat] :
% 9.49/5.24              ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Ys ) )
% 9.49/5.24             => ( ( P @ X4 )
% 9.49/5.24                = ( Q @ X4 ) ) )
% 9.49/5.24         => ( ( find_set_nat @ P @ Xs )
% 9.49/5.24            = ( find_set_nat @ Q @ Ys ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1809_find__cong,axiom,
% 9.49/5.24      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,P: vEBT_VEBT > $o,Q: vEBT_VEBT > $o] :
% 9.49/5.24        ( ( Xs = Ys )
% 9.49/5.24       => ( ! [X4: vEBT_VEBT] :
% 9.49/5.24              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Ys ) )
% 9.49/5.24             => ( ( P @ X4 )
% 9.49/5.24                = ( Q @ X4 ) ) )
% 9.49/5.24         => ( ( find_VEBT_VEBT @ P @ Xs )
% 9.49/5.24            = ( find_VEBT_VEBT @ Q @ Ys ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1810_find__cong,axiom,
% 9.49/5.24      ! [Xs: list_int,Ys: list_int,P: int > $o,Q: int > $o] :
% 9.49/5.24        ( ( Xs = Ys )
% 9.49/5.24       => ( ! [X4: int] :
% 9.49/5.24              ( ( member_int @ X4 @ ( set_int2 @ Ys ) )
% 9.49/5.24             => ( ( P @ X4 )
% 9.49/5.24                = ( Q @ X4 ) ) )
% 9.49/5.24         => ( ( find_int @ P @ Xs )
% 9.49/5.24            = ( find_int @ Q @ Ys ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1811_find__cong,axiom,
% 9.49/5.24      ! [Xs: list_nat,Ys: list_nat,P: nat > $o,Q: nat > $o] :
% 9.49/5.24        ( ( Xs = Ys )
% 9.49/5.24       => ( ! [X4: nat] :
% 9.49/5.24              ( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
% 9.49/5.24             => ( ( P @ X4 )
% 9.49/5.24                = ( Q @ X4 ) ) )
% 9.49/5.24         => ( ( find_nat @ P @ Xs )
% 9.49/5.24            = ( find_nat @ Q @ Ys ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1812_zip__same,axiom,
% 9.49/5.24      ! [A: product_prod_nat_nat,B: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
% 9.49/5.24        ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Xs ) ) )
% 9.49/5.24        = ( ( member8440522571783428010at_nat @ A @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.49/5.24          & ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1813_zip__same,axiom,
% 9.49/5.24      ! [A: real,B: real,Xs: list_real] :
% 9.49/5.24        ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ A @ B ) @ ( set_Pr5999470521830281550l_real @ ( zip_real_real @ Xs @ Xs ) ) )
% 9.49/5.24        = ( ( member_real @ A @ ( set_real2 @ Xs ) )
% 9.49/5.24          & ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1814_zip__same,axiom,
% 9.49/5.24      ! [A: set_nat,B: set_nat,Xs: list_set_nat] :
% 9.49/5.24        ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A @ B ) @ ( set_Pr9040384385603167362et_nat @ ( zip_set_nat_set_nat @ Xs @ Xs ) ) )
% 9.49/5.24        = ( ( member_set_nat @ A @ ( set_set_nat2 @ Xs ) )
% 9.49/5.24          & ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1815_zip__same,axiom,
% 9.49/5.24      ! [A: vEBT_VEBT,B: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 9.49/5.24        ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ A @ B ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Xs ) ) )
% 9.49/5.24        = ( ( member_VEBT_VEBT @ A @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.24          & ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1816_zip__same,axiom,
% 9.49/5.24      ! [A: nat,B: nat,Xs: list_nat] :
% 9.49/5.24        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
% 9.49/5.24        = ( ( member_nat @ A @ ( set_nat2 @ Xs ) )
% 9.49/5.24          & ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1817_zip__same,axiom,
% 9.49/5.24      ! [A: int,B: int,Xs: list_int] :
% 9.49/5.24        ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Xs ) ) )
% 9.49/5.24        = ( ( member_int @ A @ ( set_int2 @ Xs ) )
% 9.49/5.24          & ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1818_in__set__zipE,axiom,
% 9.49/5.24      ! [X: real,Y: real,Xs: list_real,Ys: list_real] :
% 9.49/5.24        ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X @ Y ) @ ( set_Pr5999470521830281550l_real @ ( zip_real_real @ Xs @ Ys ) ) )
% 9.49/5.24       => ~ ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.49/5.24           => ~ ( member_real @ Y @ ( set_real2 @ Ys ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1819_in__set__zipE,axiom,
% 9.49/5.24      ! [X: real,Y: vEBT_VEBT,Xs: list_real,Ys: list_VEBT_VEBT] :
% 9.49/5.24        ( ( member7262085504369356948T_VEBT @ ( produc6931449550656315951T_VEBT @ X @ Y ) @ ( set_Pr8897343066327330088T_VEBT @ ( zip_real_VEBT_VEBT @ Xs @ Ys ) ) )
% 9.49/5.24       => ~ ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.49/5.24           => ~ ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1820_in__set__zipE,axiom,
% 9.49/5.24      ! [X: real,Y: int,Xs: list_real,Ys: list_int] :
% 9.49/5.24        ( ( member1627681773268152802al_int @ ( produc3179012173361985393al_int @ X @ Y ) @ ( set_Pr8219819362198175822al_int @ ( zip_real_int @ Xs @ Ys ) ) )
% 9.49/5.24       => ~ ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.49/5.24           => ~ ( member_int @ Y @ ( set_int2 @ Ys ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1821_in__set__zipE,axiom,
% 9.49/5.24      ! [X: real,Y: nat,Xs: list_real,Ys: list_nat] :
% 9.49/5.24        ( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X @ Y ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs @ Ys ) ) )
% 9.49/5.24       => ~ ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.49/5.24           => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1822_in__set__zipE,axiom,
% 9.49/5.24      ! [X: vEBT_VEBT,Y: real,Xs: list_VEBT_VEBT,Ys: list_real] :
% 9.49/5.24        ( ( member8675245146396747942T_real @ ( produc8117437818029410057T_real @ X @ Y ) @ ( set_Pr1087130671499945274T_real @ ( zip_VEBT_VEBT_real @ Xs @ Ys ) ) )
% 9.49/5.24       => ~ ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.24           => ~ ( member_real @ Y @ ( set_real2 @ Ys ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1823_in__set__zipE,axiom,
% 9.49/5.24      ! [X: vEBT_VEBT,Y: vEBT_VEBT,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 9.49/5.24        ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X @ Y ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) )
% 9.49/5.24       => ~ ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.24           => ~ ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1824_in__set__zipE,axiom,
% 9.49/5.24      ! [X: vEBT_VEBT,Y: int,Xs: list_VEBT_VEBT,Ys: list_int] :
% 9.49/5.24        ( ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X @ Y ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) ) )
% 9.49/5.24       => ~ ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.24           => ~ ( member_int @ Y @ ( set_int2 @ Ys ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1825_in__set__zipE,axiom,
% 9.49/5.24      ! [X: vEBT_VEBT,Y: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
% 9.49/5.24        ( ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X @ Y ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) )
% 9.49/5.24       => ~ ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.24           => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1826_in__set__zipE,axiom,
% 9.49/5.24      ! [X: int,Y: real,Xs: list_int,Ys: list_real] :
% 9.49/5.24        ( ( member2744130022092475746t_real @ ( produc801115645435158769t_real @ X @ Y ) @ ( set_Pr112895574167722958t_real @ ( zip_int_real @ Xs @ Ys ) ) )
% 9.49/5.24       => ~ ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 9.49/5.24           => ~ ( member_real @ Y @ ( set_real2 @ Ys ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1827_in__set__zipE,axiom,
% 9.49/5.24      ! [X: int,Y: vEBT_VEBT,Xs: list_int,Ys: list_VEBT_VEBT] :
% 9.49/5.24        ( ( member2056185340421749780T_VEBT @ ( produc3329399203697025711T_VEBT @ X @ Y ) @ ( set_Pr8714266321650254504T_VEBT @ ( zip_int_VEBT_VEBT @ Xs @ Ys ) ) )
% 9.49/5.24       => ~ ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 9.49/5.24           => ~ ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1828_set__zip__leftD,axiom,
% 9.49/5.24      ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
% 9.49/5.24        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
% 9.49/5.24       => ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1829_set__zip__leftD,axiom,
% 9.49/5.24      ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
% 9.49/5.24        ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
% 9.49/5.24       => ( member_int @ X @ ( set_int2 @ Xs ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1830_set__zip__leftD,axiom,
% 9.49/5.24      ! [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger,Xs: list_C878401137130745250e_term,Ys: list_P5578671422887162913nteger] :
% 9.49/5.24        ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ ( set_Pr2999063419360598313nteger @ ( zip_Co8729459035503499408nteger @ Xs @ Ys ) ) )
% 9.49/5.24       => ( member1535805642427569193e_term @ X @ ( set_Co8062243466402858685e_term @ Xs ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1831_set__zip__leftD,axiom,
% 9.49/5.24      ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger,Xs: list_P1316552470764441098e_term,Ys: list_P5578671422887162913nteger] :
% 9.49/5.24        ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ ( set_Pr2135590979564877377nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys ) ) )
% 9.49/5.24       => ( member4242434998011752849e_term @ X @ ( set_Pr8342322266483756581e_term @ Xs ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1832_set__zip__leftD,axiom,
% 9.49/5.24      ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int] :
% 9.49/5.24        ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ ( set_Pr4943052134776177454nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) ) )
% 9.49/5.24       => ( member3222579708246209666e_term @ X @ ( set_Pr16608062948090134e_term @ Xs ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1833_set__zip__leftD,axiom,
% 9.49/5.24      ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int] :
% 9.49/5.24        ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ ( set_Pr1633835384712236856nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) ) )
% 9.49/5.24       => ( member8845023287901829240e_term @ X @ ( set_in5217446777445088012e_term @ Xs ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1834_set__zip__rightD,axiom,
% 9.49/5.24      ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
% 9.49/5.24        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
% 9.49/5.24       => ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1835_set__zip__rightD,axiom,
% 9.49/5.24      ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
% 9.49/5.24        ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
% 9.49/5.24       => ( member_int @ Y @ ( set_int2 @ Ys ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1836_set__zip__rightD,axiom,
% 9.49/5.24      ! [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger,Xs: list_C878401137130745250e_term,Ys: list_P5578671422887162913nteger] :
% 9.49/5.24        ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ ( set_Pr2999063419360598313nteger @ ( zip_Co8729459035503499408nteger @ Xs @ Ys ) ) )
% 9.49/5.24       => ( member157494554546826820nteger @ Y @ ( set_Pr920681315882439344nteger @ Ys ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1837_set__zip__rightD,axiom,
% 9.49/5.24      ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger,Xs: list_P1316552470764441098e_term,Ys: list_P5578671422887162913nteger] :
% 9.49/5.24        ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ ( set_Pr2135590979564877377nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys ) ) )
% 9.49/5.24       => ( member157494554546826820nteger @ Y @ ( set_Pr920681315882439344nteger @ Ys ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1838_set__zip__rightD,axiom,
% 9.49/5.24      ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int] :
% 9.49/5.24        ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ ( set_Pr4943052134776177454nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) ) )
% 9.49/5.24       => ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1839_set__zip__rightD,axiom,
% 9.49/5.24      ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int] :
% 9.49/5.24        ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ ( set_Pr1633835384712236856nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) ) )
% 9.49/5.24       => ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1840_zip__update,axiom,
% 9.49/5.24      ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
% 9.49/5.24        ( ( zip_VE537291747668921783T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Ys @ I @ Y ) )
% 9.49/5.24        = ( list_u6961636818849549845T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) @ I @ ( produc537772716801021591T_VEBT @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1841_zip__update,axiom,
% 9.49/5.24      ! [Xs: list_int,I: nat,X: int,Ys: list_int,Y: int] :
% 9.49/5.24        ( ( zip_int_int @ ( list_update_int @ Xs @ I @ X ) @ ( list_update_int @ Ys @ I @ Y ) )
% 9.49/5.24        = ( list_u3002344382305578791nt_int @ ( zip_int_int @ Xs @ Ys ) @ I @ ( product_Pair_int_int @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1842_zip__update,axiom,
% 9.49/5.24      ! [Xs: list_C878401137130745250e_term,I: nat,X: code_integer > option6357759511663192854e_term,Ys: list_P5578671422887162913nteger,Y: produc8923325533196201883nteger] :
% 9.49/5.24        ( ( zip_Co8729459035503499408nteger @ ( list_u4743598893156345252e_term @ Xs @ I @ X ) @ ( list_u2254550707601501961nteger @ Ys @ I @ Y ) )
% 9.49/5.24        = ( list_u1133519416628930960nteger @ ( zip_Co8729459035503499408nteger @ Xs @ Ys ) @ I @ ( produc6137756002093451184nteger @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1843_zip__update,axiom,
% 9.49/5.24      ! [Xs: list_P1316552470764441098e_term,I: nat,X: produc6241069584506657477e_term > option6357759511663192854e_term,Ys: list_P5578671422887162913nteger,Y: produc8923325533196201883nteger] :
% 9.49/5.24        ( ( zip_Pr8292346330294042792nteger @ ( list_u877304756163299468e_term @ Xs @ I @ X ) @ ( list_u2254550707601501961nteger @ Ys @ I @ Y ) )
% 9.49/5.24        = ( list_u234853988314817064nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys ) @ I @ ( produc8603105652947943368nteger @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1844_zip__update,axiom,
% 9.49/5.24      ! [Xs: list_P1743416141875011707e_term,I: nat,X: produc8551481072490612790e_term > option6357759511663192854e_term,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
% 9.49/5.24        ( ( zip_Pr4168994715204986005nt_int @ ( list_u3533491785856317309e_term @ Xs @ I @ X ) @ ( list_u3002344382305578791nt_int @ Ys @ I @ Y ) )
% 9.49/5.24        = ( list_u7736365598306452245nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) @ I @ ( produc5700946648718959541nt_int @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1845_zip__update,axiom,
% 9.49/5.24      ! [Xs: list_i8448526496819171953e_term,I: nat,X: int > option6357759511663192854e_term,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
% 9.49/5.24        ( ( zip_in8766932505889695135nt_int @ ( list_u8946639151299769843e_term @ Xs @ I @ X ) @ ( list_u3002344382305578791nt_int @ Ys @ I @ Y ) )
% 9.49/5.24        = ( list_u4780935413889332127nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) @ I @ ( produc4305682042979456191nt_int @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1846_not__numeral__le__zero,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1847_not__numeral__le__zero,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1848_not__numeral__le__zero,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1849_not__numeral__le__zero,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1850_not__numeral__le__zero,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1851_zero__le__numeral,axiom,
% 9.49/5.24      ! [N: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1852_zero__le__numeral,axiom,
% 9.49/5.24      ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1853_zero__le__numeral,axiom,
% 9.49/5.24      ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1854_zero__le__numeral,axiom,
% 9.49/5.24      ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1855_zero__le__numeral,axiom,
% 9.49/5.24      ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1856_zero__less__numeral,axiom,
% 9.49/5.24      ! [N: num] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1857_zero__less__numeral,axiom,
% 9.49/5.24      ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1858_zero__less__numeral,axiom,
% 9.49/5.24      ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1859_zero__less__numeral,axiom,
% 9.49/5.24      ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1860_zero__less__numeral,axiom,
% 9.49/5.24      ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1861_not__numeral__less__zero,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1862_not__numeral__less__zero,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1863_not__numeral__less__zero,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1864_not__numeral__less__zero,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1865_not__numeral__less__zero,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1866_lift__Suc__mono__less,axiom,
% 9.49/5.24      ! [F: nat > real,N: nat,N7: nat] :
% 9.49/5.24        ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.49/5.24       => ( ( ord_less_nat @ N @ N7 )
% 9.49/5.24         => ( ord_less_real @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1867_lift__Suc__mono__less,axiom,
% 9.49/5.24      ! [F: nat > rat,N: nat,N7: nat] :
% 9.49/5.24        ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.49/5.24       => ( ( ord_less_nat @ N @ N7 )
% 9.49/5.24         => ( ord_less_rat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1868_lift__Suc__mono__less,axiom,
% 9.49/5.24      ! [F: nat > num,N: nat,N7: nat] :
% 9.49/5.24        ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.49/5.24       => ( ( ord_less_nat @ N @ N7 )
% 9.49/5.24         => ( ord_less_num @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1869_lift__Suc__mono__less,axiom,
% 9.49/5.24      ! [F: nat > nat,N: nat,N7: nat] :
% 9.49/5.24        ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.49/5.24       => ( ( ord_less_nat @ N @ N7 )
% 9.49/5.24         => ( ord_less_nat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1870_lift__Suc__mono__less,axiom,
% 9.49/5.24      ! [F: nat > int,N: nat,N7: nat] :
% 9.49/5.24        ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.49/5.24       => ( ( ord_less_nat @ N @ N7 )
% 9.49/5.24         => ( ord_less_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1871_lift__Suc__mono__less__iff,axiom,
% 9.49/5.24      ! [F: nat > real,N: nat,M: nat] :
% 9.49/5.24        ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.49/5.24       => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 9.49/5.24          = ( ord_less_nat @ N @ M ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1872_lift__Suc__mono__less__iff,axiom,
% 9.49/5.24      ! [F: nat > rat,N: nat,M: nat] :
% 9.49/5.24        ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.49/5.24       => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
% 9.49/5.24          = ( ord_less_nat @ N @ M ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1873_lift__Suc__mono__less__iff,axiom,
% 9.49/5.24      ! [F: nat > num,N: nat,M: nat] :
% 9.49/5.24        ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.49/5.24       => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
% 9.49/5.24          = ( ord_less_nat @ N @ M ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1874_lift__Suc__mono__less__iff,axiom,
% 9.49/5.24      ! [F: nat > nat,N: nat,M: nat] :
% 9.49/5.24        ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.49/5.24       => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 9.49/5.24          = ( ord_less_nat @ N @ M ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1875_lift__Suc__mono__less__iff,axiom,
% 9.49/5.24      ! [F: nat > int,N: nat,M: nat] :
% 9.49/5.24        ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.49/5.24       => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 9.49/5.24          = ( ord_less_nat @ N @ M ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1876_lift__Suc__mono__le,axiom,
% 9.49/5.24      ! [F: nat > set_int,N: nat,N7: nat] :
% 9.49/5.24        ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ N @ N7 )
% 9.49/5.24         => ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1877_lift__Suc__mono__le,axiom,
% 9.49/5.24      ! [F: nat > rat,N: nat,N7: nat] :
% 9.49/5.24        ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ N @ N7 )
% 9.49/5.24         => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1878_lift__Suc__mono__le,axiom,
% 9.49/5.24      ! [F: nat > num,N: nat,N7: nat] :
% 9.49/5.24        ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ N @ N7 )
% 9.49/5.24         => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1879_lift__Suc__mono__le,axiom,
% 9.49/5.24      ! [F: nat > nat,N: nat,N7: nat] :
% 9.49/5.24        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ N @ N7 )
% 9.49/5.24         => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1880_lift__Suc__mono__le,axiom,
% 9.49/5.24      ! [F: nat > int,N: nat,N7: nat] :
% 9.49/5.24        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ N @ N7 )
% 9.49/5.24         => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1881_lift__Suc__antimono__le,axiom,
% 9.49/5.24      ! [F: nat > set_int,N: nat,N7: nat] :
% 9.49/5.24        ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ N @ N7 )
% 9.49/5.24         => ( ord_less_eq_set_int @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1882_lift__Suc__antimono__le,axiom,
% 9.49/5.24      ! [F: nat > rat,N: nat,N7: nat] :
% 9.49/5.24        ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ N @ N7 )
% 9.49/5.24         => ( ord_less_eq_rat @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1883_lift__Suc__antimono__le,axiom,
% 9.49/5.24      ! [F: nat > num,N: nat,N7: nat] :
% 9.49/5.24        ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ N @ N7 )
% 9.49/5.24         => ( ord_less_eq_num @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1884_lift__Suc__antimono__le,axiom,
% 9.49/5.24      ! [F: nat > nat,N: nat,N7: nat] :
% 9.49/5.24        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ N @ N7 )
% 9.49/5.24         => ( ord_less_eq_nat @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1885_lift__Suc__antimono__le,axiom,
% 9.49/5.24      ! [F: nat > int,N: nat,N7: nat] :
% 9.49/5.24        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ N @ N7 )
% 9.49/5.24         => ( ord_less_eq_int @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1886_Ex__less__Suc2,axiom,
% 9.49/5.24      ! [N: nat,P: nat > $o] :
% 9.49/5.24        ( ( ? [I4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 9.49/5.24              & ( P @ I4 ) ) )
% 9.49/5.24        = ( ( P @ zero_zero_nat )
% 9.49/5.24          | ? [I4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I4 @ N )
% 9.49/5.24              & ( P @ ( suc @ I4 ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1887_gr0__conv__Suc,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24        = ( ? [M2: nat] :
% 9.49/5.24              ( N
% 9.49/5.24              = ( suc @ M2 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1888_All__less__Suc2,axiom,
% 9.49/5.24      ! [N: nat,P: nat > $o] :
% 9.49/5.24        ( ( ! [I4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 9.49/5.24             => ( P @ I4 ) ) )
% 9.49/5.24        = ( ( P @ zero_zero_nat )
% 9.49/5.24          & ! [I4: nat] :
% 9.49/5.24              ( ( ord_less_nat @ I4 @ N )
% 9.49/5.24             => ( P @ ( suc @ I4 ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1889_gr0__implies__Suc,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24       => ? [M5: nat] :
% 9.49/5.24            ( N
% 9.49/5.24            = ( suc @ M5 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1890_less__Suc__eq__0__disj,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 9.49/5.24        = ( ( M = zero_zero_nat )
% 9.49/5.24          | ? [J3: nat] :
% 9.49/5.24              ( ( M
% 9.49/5.24                = ( suc @ J3 ) )
% 9.49/5.24              & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1891_le__imp__less__Suc,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.24       => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1892_less__eq__Suc__le,axiom,
% 9.49/5.24      ( ord_less_nat
% 9.49/5.24      = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1893_less__Suc__eq__le,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 9.49/5.24        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1894_le__less__Suc__eq,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.24       => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 9.49/5.24          = ( N = M ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1895_Suc__le__lessD,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 9.49/5.24       => ( ord_less_nat @ M @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1896_inc__induct,axiom,
% 9.49/5.24      ! [I: nat,J: nat,P: nat > $o] :
% 9.49/5.24        ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.24       => ( ( P @ J )
% 9.49/5.24         => ( ! [N3: nat] :
% 9.49/5.24                ( ( ord_less_eq_nat @ I @ N3 )
% 9.49/5.24               => ( ( ord_less_nat @ N3 @ J )
% 9.49/5.24                 => ( ( P @ ( suc @ N3 ) )
% 9.49/5.24                   => ( P @ N3 ) ) ) )
% 9.49/5.24           => ( P @ I ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1897_dec__induct,axiom,
% 9.49/5.24      ! [I: nat,J: nat,P: nat > $o] :
% 9.49/5.24        ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.24       => ( ( P @ I )
% 9.49/5.24         => ( ! [N3: nat] :
% 9.49/5.24                ( ( ord_less_eq_nat @ I @ N3 )
% 9.49/5.24               => ( ( ord_less_nat @ N3 @ J )
% 9.49/5.24                 => ( ( P @ N3 )
% 9.49/5.24                   => ( P @ ( suc @ N3 ) ) ) ) )
% 9.49/5.24           => ( P @ J ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1898_Suc__le__eq,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 9.49/5.24        = ( ord_less_nat @ M @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1899_Suc__leI,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ M @ N )
% 9.49/5.24       => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1900_one__is__add,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ( suc @ zero_zero_nat )
% 9.49/5.24          = ( plus_plus_nat @ M @ N ) )
% 9.49/5.24        = ( ( ( M
% 9.49/5.24              = ( suc @ zero_zero_nat ) )
% 9.49/5.24            & ( N = zero_zero_nat ) )
% 9.49/5.24          | ( ( M = zero_zero_nat )
% 9.49/5.24            & ( N
% 9.49/5.24              = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1901_add__is__1,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ( plus_plus_nat @ M @ N )
% 9.49/5.24          = ( suc @ zero_zero_nat ) )
% 9.49/5.24        = ( ( ( M
% 9.49/5.24              = ( suc @ zero_zero_nat ) )
% 9.49/5.24            & ( N = zero_zero_nat ) )
% 9.49/5.24          | ( ( M = zero_zero_nat )
% 9.49/5.24            & ( N
% 9.49/5.24              = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1902_less__imp__Suc__add,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ M @ N )
% 9.49/5.24       => ? [K2: nat] :
% 9.49/5.24            ( N
% 9.49/5.24            = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1903_less__iff__Suc__add,axiom,
% 9.49/5.24      ( ord_less_nat
% 9.49/5.24      = ( ^ [M2: nat,N2: nat] :
% 9.49/5.24          ? [K3: nat] :
% 9.49/5.24            ( N2
% 9.49/5.24            = ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1904_less__add__Suc2,axiom,
% 9.49/5.24      ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1905_less__add__Suc1,axiom,
% 9.49/5.24      ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1906_less__natE,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ M @ N )
% 9.49/5.24       => ~ ! [Q3: nat] :
% 9.49/5.24              ( N
% 9.49/5.24             != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1907_invar__vebt_Ointros_I5_J,axiom,
% 9.49/5.24      ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 9.49/5.24        ( ! [X4: vEBT_VEBT] :
% 9.49/5.24            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.49/5.24           => ( vEBT_invar_vebt @ X4 @ N ) )
% 9.49/5.24       => ( ( vEBT_invar_vebt @ Summary @ M )
% 9.49/5.24         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.49/5.24              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.49/5.24           => ( ( M
% 9.49/5.24                = ( suc @ N ) )
% 9.49/5.24             => ( ( Deg
% 9.49/5.24                  = ( plus_plus_nat @ N @ M ) )
% 9.49/5.24               => ( ! [I2: nat] :
% 9.49/5.24                      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.49/5.24                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
% 9.49/5.24                        = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 9.49/5.24                 => ( ( ( Mi = Ma )
% 9.49/5.24                     => ! [X4: vEBT_VEBT] :
% 9.49/5.24                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.49/5.24                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
% 9.49/5.24                   => ( ( ord_less_eq_nat @ Mi @ Ma )
% 9.49/5.24                     => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.49/5.24                       => ( ( ( Mi != Ma )
% 9.49/5.24                           => ! [I2: nat] :
% 9.49/5.24                                ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.49/5.24                               => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 9.49/5.24                                      = I2 )
% 9.49/5.24                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 9.49/5.24                                  & ! [X4: nat] :
% 9.49/5.24                                      ( ( ( ( vEBT_VEBT_high @ X4 @ N )
% 9.49/5.24                                          = I2 )
% 9.49/5.24                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
% 9.49/5.24                                     => ( ( ord_less_nat @ Mi @ X4 )
% 9.49/5.24                                        & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
% 9.49/5.24                         => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1908_in__set__impl__in__set__zip2,axiom,
% 9.49/5.24      ! [Xs: list_VEBT_VEBT,Ys: list_real,Y: real] :
% 9.49/5.24        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.49/5.24          = ( size_size_list_real @ Ys ) )
% 9.49/5.24       => ( ( member_real @ Y @ ( set_real2 @ Ys ) )
% 9.49/5.24         => ~ ! [X4: vEBT_VEBT] :
% 9.49/5.24                ~ ( member8675245146396747942T_real @ ( produc8117437818029410057T_real @ X4 @ Y ) @ ( set_Pr1087130671499945274T_real @ ( zip_VEBT_VEBT_real @ Xs @ Ys ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1909_in__set__impl__in__set__zip2,axiom,
% 9.49/5.24      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
% 9.49/5.24        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.49/5.24          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.49/5.24       => ( ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys ) )
% 9.49/5.24         => ~ ! [X4: vEBT_VEBT] :
% 9.49/5.24                ~ ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X4 @ Y ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1910_in__set__impl__in__set__zip2,axiom,
% 9.49/5.24      ! [Xs: list_VEBT_VEBT,Ys: list_o,Y: $o] :
% 9.49/5.24        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.49/5.24          = ( size_size_list_o @ Ys ) )
% 9.49/5.24       => ( ( member_o @ Y @ ( set_o2 @ Ys ) )
% 9.49/5.24         => ~ ! [X4: vEBT_VEBT] :
% 9.49/5.24                ~ ( member3307348790968139188VEBT_o @ ( produc8721562602347293563VEBT_o @ X4 @ Y ) @ ( set_Pr7708085864119495200VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1911_in__set__impl__in__set__zip2,axiom,
% 9.49/5.24      ! [Xs: list_VEBT_VEBT,Ys: list_nat,Y: nat] :
% 9.49/5.24        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.49/5.24          = ( size_size_list_nat @ Ys ) )
% 9.49/5.24       => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
% 9.49/5.24         => ~ ! [X4: vEBT_VEBT] :
% 9.49/5.24                ~ ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X4 @ Y ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1912_in__set__impl__in__set__zip2,axiom,
% 9.49/5.24      ! [Xs: list_VEBT_VEBT,Ys: list_int,Y: int] :
% 9.49/5.24        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.49/5.24          = ( size_size_list_int @ Ys ) )
% 9.49/5.24       => ( ( member_int @ Y @ ( set_int2 @ Ys ) )
% 9.49/5.24         => ~ ! [X4: vEBT_VEBT] :
% 9.49/5.24                ~ ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X4 @ Y ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1913_in__set__impl__in__set__zip2,axiom,
% 9.49/5.24      ! [Xs: list_o,Ys: list_real,Y: real] :
% 9.49/5.24        ( ( ( size_size_list_o @ Xs )
% 9.49/5.24          = ( size_size_list_real @ Ys ) )
% 9.49/5.24       => ( ( member_real @ Y @ ( set_real2 @ Ys ) )
% 9.49/5.24         => ~ ! [X4: $o] :
% 9.49/5.24                ~ ( member7400031367953476362o_real @ ( product_Pair_o_real @ X4 @ Y ) @ ( set_Pr2600826154070092190o_real @ ( zip_o_real @ Xs @ Ys ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1914_in__set__impl__in__set__zip2,axiom,
% 9.49/5.24      ! [Xs: list_o,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
% 9.49/5.24        ( ( ( size_size_list_o @ Xs )
% 9.49/5.24          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.49/5.24       => ( ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys ) )
% 9.49/5.24         => ~ ! [X4: $o] :
% 9.49/5.24                ~ ( member5477980866518848620T_VEBT @ ( produc2982872950893828659T_VEBT @ X4 @ Y ) @ ( set_Pr655345902815428824T_VEBT @ ( zip_o_VEBT_VEBT @ Xs @ Ys ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1915_in__set__impl__in__set__zip2,axiom,
% 9.49/5.24      ! [Xs: list_o,Ys: list_o,Y: $o] :
% 9.49/5.24        ( ( ( size_size_list_o @ Xs )
% 9.49/5.24          = ( size_size_list_o @ Ys ) )
% 9.49/5.24       => ( ( member_o @ Y @ ( set_o2 @ Ys ) )
% 9.49/5.24         => ~ ! [X4: $o] :
% 9.49/5.24                ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y ) @ ( set_Product_prod_o_o2 @ ( zip_o_o @ Xs @ Ys ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1916_in__set__impl__in__set__zip2,axiom,
% 9.49/5.24      ! [Xs: list_o,Ys: list_nat,Y: nat] :
% 9.49/5.24        ( ( ( size_size_list_o @ Xs )
% 9.49/5.24          = ( size_size_list_nat @ Ys ) )
% 9.49/5.24       => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
% 9.49/5.24         => ~ ! [X4: $o] :
% 9.49/5.24                ~ ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ X4 @ Y ) @ ( set_Pr7006799604034136130_o_nat @ ( zip_o_nat @ Xs @ Ys ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1917_in__set__impl__in__set__zip2,axiom,
% 9.49/5.24      ! [Xs: list_o,Ys: list_int,Y: int] :
% 9.49/5.24        ( ( ( size_size_list_o @ Xs )
% 9.49/5.24          = ( size_size_list_int @ Ys ) )
% 9.49/5.24       => ( ( member_int @ Y @ ( set_int2 @ Ys ) )
% 9.49/5.24         => ~ ! [X4: $o] :
% 9.49/5.24                ~ ( member7847949116333733898_o_int @ ( product_Pair_o_int @ X4 @ Y ) @ ( set_Pr2828948584524939422_o_int @ ( zip_o_int @ Xs @ Ys ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1918_in__set__impl__in__set__zip1,axiom,
% 9.49/5.24      ! [Xs: list_real,Ys: list_VEBT_VEBT,X: real] :
% 9.49/5.24        ( ( ( size_size_list_real @ Xs )
% 9.49/5.24          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.49/5.24       => ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.49/5.24         => ~ ! [Y3: vEBT_VEBT] :
% 9.49/5.24                ~ ( member7262085504369356948T_VEBT @ ( produc6931449550656315951T_VEBT @ X @ Y3 ) @ ( set_Pr8897343066327330088T_VEBT @ ( zip_real_VEBT_VEBT @ Xs @ Ys ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1919_in__set__impl__in__set__zip1,axiom,
% 9.49/5.24      ! [Xs: list_real,Ys: list_o,X: real] :
% 9.49/5.24        ( ( ( size_size_list_real @ Xs )
% 9.49/5.24          = ( size_size_list_o @ Ys ) )
% 9.49/5.24       => ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.49/5.24         => ~ ! [Y3: $o] :
% 9.49/5.24                ~ ( member772602641336174712real_o @ ( product_Pair_real_o @ X @ Y3 ) @ ( set_Pr5196769464307566348real_o @ ( zip_real_o @ Xs @ Ys ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1920_in__set__impl__in__set__zip1,axiom,
% 9.49/5.24      ! [Xs: list_real,Ys: list_nat,X: real] :
% 9.49/5.24        ( ( ( size_size_list_real @ Xs )
% 9.49/5.24          = ( size_size_list_nat @ Ys ) )
% 9.49/5.24       => ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.49/5.24         => ~ ! [Y3: nat] :
% 9.49/5.24                ~ ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X @ Y3 ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs @ Ys ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1921_in__set__impl__in__set__zip1,axiom,
% 9.49/5.24      ! [Xs: list_real,Ys: list_int,X: real] :
% 9.49/5.24        ( ( ( size_size_list_real @ Xs )
% 9.49/5.24          = ( size_size_list_int @ Ys ) )
% 9.49/5.24       => ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.49/5.24         => ~ ! [Y3: int] :
% 9.49/5.24                ~ ( member1627681773268152802al_int @ ( produc3179012173361985393al_int @ X @ Y3 ) @ ( set_Pr8219819362198175822al_int @ ( zip_real_int @ Xs @ Ys ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1922_in__set__impl__in__set__zip1,axiom,
% 9.49/5.24      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,X: vEBT_VEBT] :
% 9.49/5.24        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.49/5.24          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.49/5.24       => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.24         => ~ ! [Y3: vEBT_VEBT] :
% 9.49/5.24                ~ ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X @ Y3 ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1923_in__set__impl__in__set__zip1,axiom,
% 9.49/5.24      ! [Xs: list_VEBT_VEBT,Ys: list_o,X: vEBT_VEBT] :
% 9.49/5.24        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.49/5.24          = ( size_size_list_o @ Ys ) )
% 9.49/5.24       => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.24         => ~ ! [Y3: $o] :
% 9.49/5.24                ~ ( member3307348790968139188VEBT_o @ ( produc8721562602347293563VEBT_o @ X @ Y3 ) @ ( set_Pr7708085864119495200VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1924_in__set__impl__in__set__zip1,axiom,
% 9.49/5.24      ! [Xs: list_VEBT_VEBT,Ys: list_nat,X: vEBT_VEBT] :
% 9.49/5.24        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.49/5.24          = ( size_size_list_nat @ Ys ) )
% 9.49/5.24       => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.24         => ~ ! [Y3: nat] :
% 9.49/5.24                ~ ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X @ Y3 ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1925_in__set__impl__in__set__zip1,axiom,
% 9.49/5.24      ! [Xs: list_VEBT_VEBT,Ys: list_int,X: vEBT_VEBT] :
% 9.49/5.24        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.49/5.24          = ( size_size_list_int @ Ys ) )
% 9.49/5.24       => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.24         => ~ ! [Y3: int] :
% 9.49/5.24                ~ ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X @ Y3 ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1926_in__set__impl__in__set__zip1,axiom,
% 9.49/5.24      ! [Xs: list_o,Ys: list_VEBT_VEBT,X: $o] :
% 9.49/5.24        ( ( ( size_size_list_o @ Xs )
% 9.49/5.24          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.49/5.24       => ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 9.49/5.24         => ~ ! [Y3: vEBT_VEBT] :
% 9.49/5.24                ~ ( member5477980866518848620T_VEBT @ ( produc2982872950893828659T_VEBT @ X @ Y3 ) @ ( set_Pr655345902815428824T_VEBT @ ( zip_o_VEBT_VEBT @ Xs @ Ys ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1927_in__set__impl__in__set__zip1,axiom,
% 9.49/5.24      ! [Xs: list_o,Ys: list_o,X: $o] :
% 9.49/5.24        ( ( ( size_size_list_o @ Xs )
% 9.49/5.24          = ( size_size_list_o @ Ys ) )
% 9.49/5.24       => ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 9.49/5.24         => ~ ! [Y3: $o] :
% 9.49/5.24                ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y3 ) @ ( set_Product_prod_o_o2 @ ( zip_o_o @ Xs @ Ys ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1928_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 9.49/5.24      ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
% 9.49/5.24        ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) )
% 9.49/5.24        = ( some_P7363390416028606310at_nat @ ( F @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1929_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 9.49/5.24      ! [F: num > num > num,A: num,B: num] :
% 9.49/5.24        ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B ) )
% 9.49/5.24        = ( some_num @ ( F @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1930_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 9.49/5.24      ! [F: nat > nat > nat,A: nat,B: nat] :
% 9.49/5.24        ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
% 9.49/5.24        = ( some_nat @ ( F @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1931_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 9.49/5.24      ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
% 9.49/5.24        ( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
% 9.49/5.24        = none_P5556105721700978146at_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1932_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 9.49/5.24      ! [Uu: num > num > num,Uv: option_num] :
% 9.49/5.24        ( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
% 9.49/5.24        = none_num ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1933_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 9.49/5.24      ! [Uu: nat > nat > nat,Uv: option_nat] :
% 9.49/5.24        ( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
% 9.49/5.24        = none_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1934_ex__least__nat__less,axiom,
% 9.49/5.24      ! [P: nat > $o,N: nat] :
% 9.49/5.24        ( ( P @ N )
% 9.49/5.24       => ( ~ ( P @ zero_zero_nat )
% 9.49/5.24         => ? [K2: nat] :
% 9.49/5.24              ( ( ord_less_nat @ K2 @ N )
% 9.49/5.24              & ! [I3: nat] :
% 9.49/5.24                  ( ( ord_less_eq_nat @ I3 @ K2 )
% 9.49/5.24                 => ~ ( P @ I3 ) )
% 9.49/5.24              & ( P @ ( suc @ K2 ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1935_find__None__iff,axiom,
% 9.49/5.24      ! [P: real > $o,Xs: list_real] :
% 9.49/5.24        ( ( ( find_real @ P @ Xs )
% 9.49/5.24          = none_real )
% 9.49/5.24        = ( ~ ? [X3: real] :
% 9.49/5.24                ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 9.49/5.24                & ( P @ X3 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1936_find__None__iff,axiom,
% 9.49/5.24      ! [P: set_nat > $o,Xs: list_set_nat] :
% 9.49/5.24        ( ( ( find_set_nat @ P @ Xs )
% 9.49/5.24          = none_set_nat )
% 9.49/5.24        = ( ~ ? [X3: set_nat] :
% 9.49/5.24                ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
% 9.49/5.24                & ( P @ X3 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1937_find__None__iff,axiom,
% 9.49/5.24      ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 9.49/5.24        ( ( ( find_VEBT_VEBT @ P @ Xs )
% 9.49/5.24          = none_VEBT_VEBT )
% 9.49/5.24        = ( ~ ? [X3: vEBT_VEBT] :
% 9.49/5.24                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.24                & ( P @ X3 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1938_find__None__iff,axiom,
% 9.49/5.24      ! [P: int > $o,Xs: list_int] :
% 9.49/5.24        ( ( ( find_int @ P @ Xs )
% 9.49/5.24          = none_int )
% 9.49/5.24        = ( ~ ? [X3: int] :
% 9.49/5.24                ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 9.49/5.24                & ( P @ X3 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1939_find__None__iff,axiom,
% 9.49/5.24      ! [P: nat > $o,Xs: list_nat] :
% 9.49/5.24        ( ( ( find_nat @ P @ Xs )
% 9.49/5.24          = none_nat )
% 9.49/5.24        = ( ~ ? [X3: nat] :
% 9.49/5.24                ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 9.49/5.24                & ( P @ X3 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1940_find__None__iff,axiom,
% 9.49/5.24      ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
% 9.49/5.24        ( ( ( find_P8199882355184865565at_nat @ P @ Xs )
% 9.49/5.24          = none_P5556105721700978146at_nat )
% 9.49/5.24        = ( ~ ? [X3: product_prod_nat_nat] :
% 9.49/5.24                ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.49/5.24                & ( P @ X3 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1941_find__None__iff,axiom,
% 9.49/5.24      ! [P: num > $o,Xs: list_num] :
% 9.49/5.24        ( ( ( find_num @ P @ Xs )
% 9.49/5.24          = none_num )
% 9.49/5.24        = ( ~ ? [X3: num] :
% 9.49/5.24                ( ( member_num @ X3 @ ( set_num2 @ Xs ) )
% 9.49/5.24                & ( P @ X3 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1942_find__None__iff2,axiom,
% 9.49/5.24      ! [P: real > $o,Xs: list_real] :
% 9.49/5.24        ( ( none_real
% 9.49/5.24          = ( find_real @ P @ Xs ) )
% 9.49/5.24        = ( ~ ? [X3: real] :
% 9.49/5.24                ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 9.49/5.24                & ( P @ X3 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1943_find__None__iff2,axiom,
% 9.49/5.24      ! [P: set_nat > $o,Xs: list_set_nat] :
% 9.49/5.24        ( ( none_set_nat
% 9.49/5.24          = ( find_set_nat @ P @ Xs ) )
% 9.49/5.24        = ( ~ ? [X3: set_nat] :
% 9.49/5.24                ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
% 9.49/5.24                & ( P @ X3 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1944_find__None__iff2,axiom,
% 9.49/5.24      ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 9.49/5.24        ( ( none_VEBT_VEBT
% 9.49/5.24          = ( find_VEBT_VEBT @ P @ Xs ) )
% 9.49/5.24        = ( ~ ? [X3: vEBT_VEBT] :
% 9.49/5.24                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.24                & ( P @ X3 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1945_find__None__iff2,axiom,
% 9.49/5.24      ! [P: int > $o,Xs: list_int] :
% 9.49/5.24        ( ( none_int
% 9.49/5.24          = ( find_int @ P @ Xs ) )
% 9.49/5.24        = ( ~ ? [X3: int] :
% 9.49/5.24                ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 9.49/5.24                & ( P @ X3 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1946_find__None__iff2,axiom,
% 9.49/5.24      ! [P: nat > $o,Xs: list_nat] :
% 9.49/5.24        ( ( none_nat
% 9.49/5.24          = ( find_nat @ P @ Xs ) )
% 9.49/5.24        = ( ~ ? [X3: nat] :
% 9.49/5.24                ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 9.49/5.24                & ( P @ X3 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1947_find__None__iff2,axiom,
% 9.49/5.24      ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
% 9.49/5.24        ( ( none_P5556105721700978146at_nat
% 9.49/5.24          = ( find_P8199882355184865565at_nat @ P @ Xs ) )
% 9.49/5.24        = ( ~ ? [X3: product_prod_nat_nat] :
% 9.49/5.24                ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.49/5.24                & ( P @ X3 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1948_find__None__iff2,axiom,
% 9.49/5.24      ! [P: num > $o,Xs: list_num] :
% 9.49/5.24        ( ( none_num
% 9.49/5.24          = ( find_num @ P @ Xs ) )
% 9.49/5.24        = ( ~ ? [X3: num] :
% 9.49/5.24                ( ( member_num @ X3 @ ( set_num2 @ Xs ) )
% 9.49/5.24                & ( P @ X3 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1949_invar__vebt_Ointros_I4_J,axiom,
% 9.49/5.24      ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 9.49/5.24        ( ! [X4: vEBT_VEBT] :
% 9.49/5.24            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.49/5.24           => ( vEBT_invar_vebt @ X4 @ N ) )
% 9.49/5.24       => ( ( vEBT_invar_vebt @ Summary @ M )
% 9.49/5.24         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.49/5.24              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.49/5.24           => ( ( M = N )
% 9.49/5.24             => ( ( Deg
% 9.49/5.24                  = ( plus_plus_nat @ N @ M ) )
% 9.49/5.24               => ( ! [I2: nat] :
% 9.49/5.24                      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.49/5.24                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
% 9.49/5.24                        = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 9.49/5.24                 => ( ( ( Mi = Ma )
% 9.49/5.24                     => ! [X4: vEBT_VEBT] :
% 9.49/5.24                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.49/5.24                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
% 9.49/5.24                   => ( ( ord_less_eq_nat @ Mi @ Ma )
% 9.49/5.24                     => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.49/5.24                       => ( ( ( Mi != Ma )
% 9.49/5.24                           => ! [I2: nat] :
% 9.49/5.24                                ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.49/5.24                               => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 9.49/5.24                                      = I2 )
% 9.49/5.24                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 9.49/5.24                                  & ! [X4: nat] :
% 9.49/5.24                                      ( ( ( ( vEBT_VEBT_high @ X4 @ N )
% 9.49/5.24                                          = I2 )
% 9.49/5.24                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
% 9.49/5.24                                     => ( ( ord_less_nat @ Mi @ X4 )
% 9.49/5.24                                        & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
% 9.49/5.24                         => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1950_pair__lessI1,axiom,
% 9.49/5.24      ! [A: nat,B: nat,S: nat,T: nat] :
% 9.49/5.24        ( ( ord_less_nat @ A @ B )
% 9.49/5.24       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1951_count__le__length,axiom,
% 9.49/5.24      ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] : ( ord_less_eq_nat @ ( count_list_VEBT_VEBT @ Xs @ X ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1952_count__le__length,axiom,
% 9.49/5.24      ! [Xs: list_o,X: $o] : ( ord_less_eq_nat @ ( count_list_o @ Xs @ X ) @ ( size_size_list_o @ Xs ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1953_count__le__length,axiom,
% 9.49/5.24      ! [Xs: list_nat,X: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X ) @ ( size_size_list_nat @ Xs ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1954_count__le__length,axiom,
% 9.49/5.24      ! [Xs: list_int,X: int] : ( ord_less_eq_nat @ ( count_list_int @ Xs @ X ) @ ( size_size_list_int @ Xs ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1955_option_Osize_I4_J,axiom,
% 9.49/5.24      ! [X22: nat] :
% 9.49/5.24        ( ( size_size_option_nat @ ( some_nat @ X22 ) )
% 9.49/5.24        = ( suc @ zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1956_option_Osize_I4_J,axiom,
% 9.49/5.24      ! [X22: product_prod_nat_nat] :
% 9.49/5.24        ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 9.49/5.24        = ( suc @ zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1957_option_Osize_I4_J,axiom,
% 9.49/5.24      ! [X22: num] :
% 9.49/5.24        ( ( size_size_option_num @ ( some_num @ X22 ) )
% 9.49/5.24        = ( suc @ zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1958_option_Osize_I3_J,axiom,
% 9.49/5.24      ( ( size_size_option_nat @ none_nat )
% 9.49/5.24      = ( suc @ zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1959_option_Osize_I3_J,axiom,
% 9.49/5.24      ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
% 9.49/5.24      = ( suc @ zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1960_option_Osize_I3_J,axiom,
% 9.49/5.24      ( ( size_size_option_num @ none_num )
% 9.49/5.24      = ( suc @ zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1961_vebt__member_Osimps_I4_J,axiom,
% 9.49/5.24      ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
% 9.49/5.24        ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1962_VEBT__internal_Ooption__shift_Oelims,axiom,
% 9.49/5.24      ! [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb2: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
% 9.49/5.24        ( ( ( vEBT_V1502963449132264192at_nat @ X @ Xa2 @ Xb2 )
% 9.49/5.24          = Y )
% 9.49/5.24       => ( ( ( Xa2 = none_P5556105721700978146at_nat )
% 9.49/5.24           => ( Y != none_P5556105721700978146at_nat ) )
% 9.49/5.24         => ( ( ? [V2: product_prod_nat_nat] :
% 9.49/5.24                  ( Xa2
% 9.49/5.24                  = ( some_P7363390416028606310at_nat @ V2 ) )
% 9.49/5.24             => ( ( Xb2 = none_P5556105721700978146at_nat )
% 9.49/5.24               => ( Y != none_P5556105721700978146at_nat ) ) )
% 9.49/5.24           => ~ ! [A3: product_prod_nat_nat] :
% 9.49/5.24                  ( ( Xa2
% 9.49/5.24                    = ( some_P7363390416028606310at_nat @ A3 ) )
% 9.49/5.24                 => ! [B3: product_prod_nat_nat] :
% 9.49/5.24                      ( ( Xb2
% 9.49/5.24                        = ( some_P7363390416028606310at_nat @ B3 ) )
% 9.49/5.24                     => ( Y
% 9.49/5.24                       != ( some_P7363390416028606310at_nat @ ( X @ A3 @ B3 ) ) ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1963_VEBT__internal_Ooption__shift_Oelims,axiom,
% 9.49/5.24      ! [X: num > num > num,Xa2: option_num,Xb2: option_num,Y: option_num] :
% 9.49/5.24        ( ( ( vEBT_V819420779217536731ft_num @ X @ Xa2 @ Xb2 )
% 9.49/5.24          = Y )
% 9.49/5.24       => ( ( ( Xa2 = none_num )
% 9.49/5.24           => ( Y != none_num ) )
% 9.49/5.24         => ( ( ? [V2: num] :
% 9.49/5.24                  ( Xa2
% 9.49/5.24                  = ( some_num @ V2 ) )
% 9.49/5.24             => ( ( Xb2 = none_num )
% 9.49/5.24               => ( Y != none_num ) ) )
% 9.49/5.24           => ~ ! [A3: num] :
% 9.49/5.24                  ( ( Xa2
% 9.49/5.24                    = ( some_num @ A3 ) )
% 9.49/5.24                 => ! [B3: num] :
% 9.49/5.24                      ( ( Xb2
% 9.49/5.24                        = ( some_num @ B3 ) )
% 9.49/5.24                     => ( Y
% 9.49/5.24                       != ( some_num @ ( X @ A3 @ B3 ) ) ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1964_VEBT__internal_Ooption__shift_Oelims,axiom,
% 9.49/5.24      ! [X: nat > nat > nat,Xa2: option_nat,Xb2: option_nat,Y: option_nat] :
% 9.49/5.24        ( ( ( vEBT_V4262088993061758097ft_nat @ X @ Xa2 @ Xb2 )
% 9.49/5.24          = Y )
% 9.49/5.24       => ( ( ( Xa2 = none_nat )
% 9.49/5.24           => ( Y != none_nat ) )
% 9.49/5.24         => ( ( ? [V2: nat] :
% 9.49/5.24                  ( Xa2
% 9.49/5.24                  = ( some_nat @ V2 ) )
% 9.49/5.24             => ( ( Xb2 = none_nat )
% 9.49/5.24               => ( Y != none_nat ) ) )
% 9.49/5.24           => ~ ! [A3: nat] :
% 9.49/5.24                  ( ( Xa2
% 9.49/5.24                    = ( some_nat @ A3 ) )
% 9.49/5.24                 => ! [B3: nat] :
% 9.49/5.24                      ( ( Xb2
% 9.49/5.24                        = ( some_nat @ B3 ) )
% 9.49/5.24                     => ( Y
% 9.49/5.24                       != ( some_nat @ ( X @ A3 @ B3 ) ) ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1965_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 9.49/5.24      ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
% 9.49/5.24        ( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
% 9.49/5.24        = none_P5556105721700978146at_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1966_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 9.49/5.24      ! [Uw: num > num > num,V: num] :
% 9.49/5.24        ( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
% 9.49/5.24        = none_num ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1967_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 9.49/5.24      ! [Uw: nat > nat > nat,V: nat] :
% 9.49/5.24        ( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
% 9.49/5.24        = none_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1968_vebt__delete_Osimps_I6_J,axiom,
% 9.49/5.24      ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X: nat] :
% 9.49/5.24        ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ X )
% 9.49/5.24        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1969_sum__power2__eq__zero__iff,axiom,
% 9.49/5.24      ! [X: rat,Y: rat] :
% 9.49/5.24        ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24          = zero_zero_rat )
% 9.49/5.24        = ( ( X = zero_zero_rat )
% 9.49/5.24          & ( Y = zero_zero_rat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1970_sum__power2__eq__zero__iff,axiom,
% 9.49/5.24      ! [X: real,Y: real] :
% 9.49/5.24        ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24          = zero_zero_real )
% 9.49/5.24        = ( ( X = zero_zero_real )
% 9.49/5.24          & ( Y = zero_zero_real ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1971_sum__power2__eq__zero__iff,axiom,
% 9.49/5.24      ! [X: int,Y: int] :
% 9.49/5.24        ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24          = zero_zero_int )
% 9.49/5.24        = ( ( X = zero_zero_int )
% 9.49/5.24          & ( Y = zero_zero_int ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1972_zero__less__power2,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24        = ( A != zero_zero_real ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1973_zero__less__power2,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24        = ( A != zero_zero_rat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1974_zero__less__power2,axiom,
% 9.49/5.24      ! [A: int] :
% 9.49/5.24        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24        = ( A != zero_zero_int ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1975_power2__eq__iff__nonneg,axiom,
% 9.49/5.24      ! [X: real,Y: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.49/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.49/5.24         => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24              = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24            = ( X = Y ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1976_power2__eq__iff__nonneg,axiom,
% 9.49/5.24      ! [X: rat,Y: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.49/5.24       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.49/5.24         => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24              = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24            = ( X = Y ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1977_power2__eq__iff__nonneg,axiom,
% 9.49/5.24      ! [X: nat,Y: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 9.49/5.24       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 9.49/5.24         => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24              = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24            = ( X = Y ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1978_power2__eq__iff__nonneg,axiom,
% 9.49/5.24      ! [X: int,Y: int] :
% 9.49/5.24        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.49/5.24       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.49/5.24         => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24              = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24            = ( X = Y ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1979_power2__less__eq__zero__iff,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 9.49/5.24        = ( A = zero_zero_real ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1980_power2__less__eq__zero__iff,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 9.49/5.24        = ( A = zero_zero_rat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1981_power2__less__eq__zero__iff,axiom,
% 9.49/5.24      ! [A: int] :
% 9.49/5.24        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 9.49/5.24        = ( A = zero_zero_int ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1982_post__member__pre__member,axiom,
% 9.49/5.24      ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
% 9.49/5.24        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.24       => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.24         => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.24           => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y )
% 9.49/5.24             => ( ( vEBT_vebt_member @ T @ Y )
% 9.49/5.24                | ( X = Y ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1983_valid__insert__both__member__options__add,axiom,
% 9.49/5.24      ! [T: vEBT_VEBT,N: nat,X: nat] :
% 9.49/5.24        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.24       => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.24         => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1984_valid__insert__both__member__options__pres,axiom,
% 9.49/5.24      ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
% 9.49/5.24        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.24       => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.24         => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.24           => ( ( vEBT_V8194947554948674370ptions @ T @ X )
% 9.49/5.24             => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1985_insert__simp__mima,axiom,
% 9.49/5.24      ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.49/5.24        ( ( ( X = Mi )
% 9.49/5.24          | ( X = Ma ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.24         => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.24            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1986_power__mono__iff,axiom,
% 9.49/5.24      ! [A: real,B: real,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.49/5.24         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24           => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 9.49/5.24              = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1987_power__mono__iff,axiom,
% 9.49/5.24      ! [A: rat,B: rat,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.24       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.49/5.24         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24           => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 9.49/5.24              = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1988_power__mono__iff,axiom,
% 9.49/5.24      ! [A: nat,B: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.24       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.49/5.24         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24           => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 9.49/5.24              = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1989_power__mono__iff,axiom,
% 9.49/5.24      ! [A: int,B: int,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.24       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.49/5.24         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24           => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 9.49/5.24              = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1990_zero__eq__power2,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24          = zero_zero_rat )
% 9.49/5.24        = ( A = zero_zero_rat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1991_zero__eq__power2,axiom,
% 9.49/5.24      ! [A: nat] :
% 9.49/5.24        ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24          = zero_zero_nat )
% 9.49/5.24        = ( A = zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1992_zero__eq__power2,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24          = zero_zero_real )
% 9.49/5.24        = ( A = zero_zero_real ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1993_zero__eq__power2,axiom,
% 9.49/5.24      ! [A: int] :
% 9.49/5.24        ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24          = zero_zero_int )
% 9.49/5.24        = ( A = zero_zero_int ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1994_zero__eq__power2,axiom,
% 9.49/5.24      ! [A: complex] :
% 9.49/5.24        ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24          = zero_zero_complex )
% 9.49/5.24        = ( A = zero_zero_complex ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1995_power__0__Suc,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
% 9.49/5.24        = zero_zero_rat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1996_power__0__Suc,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 9.49/5.24        = zero_zero_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1997_power__0__Suc,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 9.49/5.24        = zero_zero_real ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1998_power__0__Suc,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 9.49/5.24        = zero_zero_int ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_1999_power__0__Suc,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
% 9.49/5.24        = zero_zero_complex ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2000_power__zero__numeral,axiom,
% 9.49/5.24      ! [K: num] :
% 9.49/5.24        ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 9.49/5.24        = zero_zero_rat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2001_power__zero__numeral,axiom,
% 9.49/5.24      ! [K: num] :
% 9.49/5.24        ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 9.49/5.24        = zero_zero_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2002_power__zero__numeral,axiom,
% 9.49/5.24      ! [K: num] :
% 9.49/5.24        ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 9.49/5.24        = zero_zero_real ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2003_power__zero__numeral,axiom,
% 9.49/5.24      ! [K: num] :
% 9.49/5.24        ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 9.49/5.24        = zero_zero_int ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2004_power__zero__numeral,axiom,
% 9.49/5.24      ! [K: num] :
% 9.49/5.24        ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 9.49/5.24        = zero_zero_complex ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2005_power__Suc0__right,axiom,
% 9.49/5.24      ! [A: nat] :
% 9.49/5.24        ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2006_power__Suc0__right,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2007_power__Suc0__right,axiom,
% 9.49/5.24      ! [A: int] :
% 9.49/5.24        ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2008_power__Suc0__right,axiom,
% 9.49/5.24      ! [A: complex] :
% 9.49/5.24        ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2009_nat__power__eq__Suc__0__iff,axiom,
% 9.49/5.24      ! [X: nat,M: nat] :
% 9.49/5.24        ( ( ( power_power_nat @ X @ M )
% 9.49/5.24          = ( suc @ zero_zero_nat ) )
% 9.49/5.24        = ( ( M = zero_zero_nat )
% 9.49/5.24          | ( X
% 9.49/5.24            = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2010_power__Suc__0,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.49/5.24        = ( suc @ zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2011_nat__zero__less__power__iff,axiom,
% 9.49/5.24      ! [X: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
% 9.49/5.24        = ( ( ord_less_nat @ zero_zero_nat @ X )
% 9.49/5.24          | ( N = zero_zero_nat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2012_power__eq__0__iff,axiom,
% 9.49/5.24      ! [A: rat,N: nat] :
% 9.49/5.24        ( ( ( power_power_rat @ A @ N )
% 9.49/5.24          = zero_zero_rat )
% 9.49/5.24        = ( ( A = zero_zero_rat )
% 9.49/5.24          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2013_power__eq__0__iff,axiom,
% 9.49/5.24      ! [A: nat,N: nat] :
% 9.49/5.24        ( ( ( power_power_nat @ A @ N )
% 9.49/5.24          = zero_zero_nat )
% 9.49/5.24        = ( ( A = zero_zero_nat )
% 9.49/5.24          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2014_power__eq__0__iff,axiom,
% 9.49/5.24      ! [A: real,N: nat] :
% 9.49/5.24        ( ( ( power_power_real @ A @ N )
% 9.49/5.24          = zero_zero_real )
% 9.49/5.24        = ( ( A = zero_zero_real )
% 9.49/5.24          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2015_power__eq__0__iff,axiom,
% 9.49/5.24      ! [A: int,N: nat] :
% 9.49/5.24        ( ( ( power_power_int @ A @ N )
% 9.49/5.24          = zero_zero_int )
% 9.49/5.24        = ( ( A = zero_zero_int )
% 9.49/5.24          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2016_power__eq__0__iff,axiom,
% 9.49/5.24      ! [A: complex,N: nat] :
% 9.49/5.24        ( ( ( power_power_complex @ A @ N )
% 9.49/5.24          = zero_zero_complex )
% 9.49/5.24        = ( ( A = zero_zero_complex )
% 9.49/5.24          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2017_power__not__zero,axiom,
% 9.49/5.24      ! [A: rat,N: nat] :
% 9.49/5.24        ( ( A != zero_zero_rat )
% 9.49/5.24       => ( ( power_power_rat @ A @ N )
% 9.49/5.24         != zero_zero_rat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2018_power__not__zero,axiom,
% 9.49/5.24      ! [A: nat,N: nat] :
% 9.49/5.24        ( ( A != zero_zero_nat )
% 9.49/5.24       => ( ( power_power_nat @ A @ N )
% 9.49/5.24         != zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2019_power__not__zero,axiom,
% 9.49/5.24      ! [A: real,N: nat] :
% 9.49/5.24        ( ( A != zero_zero_real )
% 9.49/5.24       => ( ( power_power_real @ A @ N )
% 9.49/5.24         != zero_zero_real ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2020_power__not__zero,axiom,
% 9.49/5.24      ! [A: int,N: nat] :
% 9.49/5.24        ( ( A != zero_zero_int )
% 9.49/5.24       => ( ( power_power_int @ A @ N )
% 9.49/5.24         != zero_zero_int ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2021_power__not__zero,axiom,
% 9.49/5.24      ! [A: complex,N: nat] :
% 9.49/5.24        ( ( A != zero_zero_complex )
% 9.49/5.24       => ( ( power_power_complex @ A @ N )
% 9.49/5.24         != zero_zero_complex ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2022_power__mono,axiom,
% 9.49/5.24      ! [A: real,B: real,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.24         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2023_power__mono,axiom,
% 9.49/5.24      ! [A: rat,B: rat,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.24       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.24         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2024_power__mono,axiom,
% 9.49/5.24      ! [A: nat,B: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.24       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.24         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2025_power__mono,axiom,
% 9.49/5.24      ! [A: int,B: int,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.24       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.24         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2026_zero__le__power,axiom,
% 9.49/5.24      ! [A: real,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.24       => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2027_zero__le__power,axiom,
% 9.49/5.24      ! [A: rat,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.24       => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2028_zero__le__power,axiom,
% 9.49/5.24      ! [A: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.24       => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2029_zero__le__power,axiom,
% 9.49/5.24      ! [A: int,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.24       => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2030_zero__less__power,axiom,
% 9.49/5.24      ! [A: real,N: nat] :
% 9.49/5.24        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.24       => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2031_zero__less__power,axiom,
% 9.49/5.24      ! [A: rat,N: nat] :
% 9.49/5.24        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.24       => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2032_zero__less__power,axiom,
% 9.49/5.24      ! [A: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.49/5.24       => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2033_zero__less__power,axiom,
% 9.49/5.24      ! [A: int,N: nat] :
% 9.49/5.24        ( ( ord_less_int @ zero_zero_int @ A )
% 9.49/5.24       => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2034_nat__power__less__imp__less,axiom,
% 9.49/5.24      ! [I: nat,M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ I )
% 9.49/5.24       => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 9.49/5.24         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2035_power__less__imp__less__base,axiom,
% 9.49/5.24      ! [A: real,N: nat,B: real] :
% 9.49/5.24        ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 9.49/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.49/5.24         => ( ord_less_real @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2036_power__less__imp__less__base,axiom,
% 9.49/5.24      ! [A: rat,N: nat,B: rat] :
% 9.49/5.24        ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 9.49/5.24       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.49/5.24         => ( ord_less_rat @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2037_power__less__imp__less__base,axiom,
% 9.49/5.24      ! [A: nat,N: nat,B: nat] :
% 9.49/5.24        ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.49/5.24         => ( ord_less_nat @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2038_power__less__imp__less__base,axiom,
% 9.49/5.24      ! [A: int,N: nat,B: int] :
% 9.49/5.24        ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 9.49/5.24       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.49/5.24         => ( ord_less_int @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2039_power__inject__base,axiom,
% 9.49/5.24      ! [A: real,N: nat,B: real] :
% 9.49/5.24        ( ( ( power_power_real @ A @ ( suc @ N ) )
% 9.49/5.24          = ( power_power_real @ B @ ( suc @ N ) ) )
% 9.49/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.24         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.49/5.24           => ( A = B ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2040_power__inject__base,axiom,
% 9.49/5.24      ! [A: rat,N: nat,B: rat] :
% 9.49/5.24        ( ( ( power_power_rat @ A @ ( suc @ N ) )
% 9.49/5.24          = ( power_power_rat @ B @ ( suc @ N ) ) )
% 9.49/5.24       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.24         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.49/5.24           => ( A = B ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2041_power__inject__base,axiom,
% 9.49/5.24      ! [A: nat,N: nat,B: nat] :
% 9.49/5.24        ( ( ( power_power_nat @ A @ ( suc @ N ) )
% 9.49/5.24          = ( power_power_nat @ B @ ( suc @ N ) ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.24         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.49/5.24           => ( A = B ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2042_power__inject__base,axiom,
% 9.49/5.24      ! [A: int,N: nat,B: int] :
% 9.49/5.24        ( ( ( power_power_int @ A @ ( suc @ N ) )
% 9.49/5.24          = ( power_power_int @ B @ ( suc @ N ) ) )
% 9.49/5.24       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.24         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.49/5.24           => ( A = B ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2043_power__le__imp__le__base,axiom,
% 9.49/5.24      ! [A: real,N: nat,B: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
% 9.49/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.49/5.24         => ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2044_power__le__imp__le__base,axiom,
% 9.49/5.24      ! [A: rat,N: nat,B: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
% 9.49/5.24       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.49/5.24         => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2045_power__le__imp__le__base,axiom,
% 9.49/5.24      ! [A: nat,N: nat,B: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.49/5.24         => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2046_power__le__imp__le__base,axiom,
% 9.49/5.24      ! [A: int,N: nat,B: int] :
% 9.49/5.24        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
% 9.49/5.24       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.49/5.24         => ( ord_less_eq_int @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2047_zero__power,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24       => ( ( power_power_rat @ zero_zero_rat @ N )
% 9.49/5.24          = zero_zero_rat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2048_zero__power,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24       => ( ( power_power_nat @ zero_zero_nat @ N )
% 9.49/5.24          = zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2049_zero__power,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24       => ( ( power_power_real @ zero_zero_real @ N )
% 9.49/5.24          = zero_zero_real ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2050_zero__power,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24       => ( ( power_power_int @ zero_zero_int @ N )
% 9.49/5.24          = zero_zero_int ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2051_zero__power,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24       => ( ( power_power_complex @ zero_zero_complex @ N )
% 9.49/5.24          = zero_zero_complex ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2052_power__gt__expt,axiom,
% 9.49/5.24      ! [N: nat,K: nat] :
% 9.49/5.24        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.49/5.24       => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2053_nat__one__le__power,axiom,
% 9.49/5.24      ! [I: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 9.49/5.24       => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2054_zero__power2,axiom,
% 9.49/5.24      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24      = zero_zero_rat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2055_zero__power2,axiom,
% 9.49/5.24      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24      = zero_zero_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2056_zero__power2,axiom,
% 9.49/5.24      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24      = zero_zero_real ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2057_zero__power2,axiom,
% 9.49/5.24      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24      = zero_zero_int ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2058_zero__power2,axiom,
% 9.49/5.24      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24      = zero_zero_complex ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2059_power__eq__imp__eq__base,axiom,
% 9.49/5.24      ! [A: real,N: nat,B: real] :
% 9.49/5.24        ( ( ( power_power_real @ A @ N )
% 9.49/5.24          = ( power_power_real @ B @ N ) )
% 9.49/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.24         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.49/5.24           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24             => ( A = B ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2060_power__eq__imp__eq__base,axiom,
% 9.49/5.24      ! [A: rat,N: nat,B: rat] :
% 9.49/5.24        ( ( ( power_power_rat @ A @ N )
% 9.49/5.24          = ( power_power_rat @ B @ N ) )
% 9.49/5.24       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.24         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.49/5.24           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24             => ( A = B ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2061_power__eq__imp__eq__base,axiom,
% 9.49/5.24      ! [A: nat,N: nat,B: nat] :
% 9.49/5.24        ( ( ( power_power_nat @ A @ N )
% 9.49/5.24          = ( power_power_nat @ B @ N ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.24         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.49/5.24           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24             => ( A = B ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2062_power__eq__imp__eq__base,axiom,
% 9.49/5.24      ! [A: int,N: nat,B: int] :
% 9.49/5.24        ( ( ( power_power_int @ A @ N )
% 9.49/5.24          = ( power_power_int @ B @ N ) )
% 9.49/5.24       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.24         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.49/5.24           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24             => ( A = B ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2063_power__eq__iff__eq__base,axiom,
% 9.49/5.24      ! [N: nat,A: real,B: real] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.24         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.49/5.24           => ( ( ( power_power_real @ A @ N )
% 9.49/5.24                = ( power_power_real @ B @ N ) )
% 9.49/5.24              = ( A = B ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2064_power__eq__iff__eq__base,axiom,
% 9.49/5.24      ! [N: nat,A: rat,B: rat] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.24         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.49/5.24           => ( ( ( power_power_rat @ A @ N )
% 9.49/5.24                = ( power_power_rat @ B @ N ) )
% 9.49/5.24              = ( A = B ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2065_power__eq__iff__eq__base,axiom,
% 9.49/5.24      ! [N: nat,A: nat,B: nat] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.24         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.49/5.24           => ( ( ( power_power_nat @ A @ N )
% 9.49/5.24                = ( power_power_nat @ B @ N ) )
% 9.49/5.24              = ( A = B ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2066_power__eq__iff__eq__base,axiom,
% 9.49/5.24      ! [N: nat,A: int,B: int] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.24         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.49/5.24           => ( ( ( power_power_int @ A @ N )
% 9.49/5.24                = ( power_power_int @ B @ N ) )
% 9.49/5.24              = ( A = B ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2067_less__exp,axiom,
% 9.49/5.24      ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2068_power2__nat__le__imp__le,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 9.49/5.24       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2069_power2__nat__le__eq__le,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2070_self__le__ge2__pow,axiom,
% 9.49/5.24      ! [K: nat,M: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.49/5.24       => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2071_zero__le__power2,axiom,
% 9.49/5.24      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2072_zero__le__power2,axiom,
% 9.49/5.24      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2073_zero__le__power2,axiom,
% 9.49/5.24      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2074_power2__eq__imp__eq,axiom,
% 9.49/5.24      ! [X: real,Y: real] :
% 9.49/5.24        ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24          = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.49/5.24         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.49/5.24           => ( X = Y ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2075_power2__eq__imp__eq,axiom,
% 9.49/5.24      ! [X: rat,Y: rat] :
% 9.49/5.24        ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24          = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24       => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.49/5.24         => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.49/5.24           => ( X = Y ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2076_power2__eq__imp__eq,axiom,
% 9.49/5.24      ! [X: nat,Y: nat] :
% 9.49/5.24        ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24          = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 9.49/5.24         => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 9.49/5.24           => ( X = Y ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2077_power2__eq__imp__eq,axiom,
% 9.49/5.24      ! [X: int,Y: int] :
% 9.49/5.24        ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24          = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24       => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.49/5.24         => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.49/5.24           => ( X = Y ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2078_power2__le__imp__le,axiom,
% 9.49/5.24      ! [X: real,Y: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.49/5.24         => ( ord_less_eq_real @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2079_power2__le__imp__le,axiom,
% 9.49/5.24      ! [X: rat,Y: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.49/5.24         => ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2080_power2__le__imp__le,axiom,
% 9.49/5.24      ! [X: nat,Y: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 9.49/5.24         => ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2081_power2__le__imp__le,axiom,
% 9.49/5.24      ! [X: int,Y: int] :
% 9.49/5.24        ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.49/5.24         => ( ord_less_eq_int @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2082_power2__less__0,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2083_power2__less__0,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2084_power2__less__0,axiom,
% 9.49/5.24      ! [A: int] :
% 9.49/5.24        ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2085_power__strict__mono,axiom,
% 9.49/5.24      ! [A: real,B: real,N: nat] :
% 9.49/5.24        ( ( ord_less_real @ A @ B )
% 9.49/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.24         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24           => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2086_power__strict__mono,axiom,
% 9.49/5.24      ! [A: rat,B: rat,N: nat] :
% 9.49/5.24        ( ( ord_less_rat @ A @ B )
% 9.49/5.24       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.24         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24           => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2087_power__strict__mono,axiom,
% 9.49/5.24      ! [A: nat,B: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ A @ B )
% 9.49/5.24       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.24         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24           => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2088_power__strict__mono,axiom,
% 9.49/5.24      ! [A: int,B: int,N: nat] :
% 9.49/5.24        ( ( ord_less_int @ A @ B )
% 9.49/5.24       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.24         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.24           => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2089_power2__less__imp__less,axiom,
% 9.49/5.24      ! [X: real,Y: real] :
% 9.49/5.24        ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.49/5.24         => ( ord_less_real @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2090_power2__less__imp__less,axiom,
% 9.49/5.24      ! [X: rat,Y: rat] :
% 9.49/5.24        ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.49/5.24         => ( ord_less_rat @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2091_power2__less__imp__less,axiom,
% 9.49/5.24      ! [X: nat,Y: nat] :
% 9.49/5.24        ( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 9.49/5.24         => ( ord_less_nat @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2092_power2__less__imp__less,axiom,
% 9.49/5.24      ! [X: int,Y: int] :
% 9.49/5.24        ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.24       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.49/5.24         => ( ord_less_int @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2093_sum__power2__le__zero__iff,axiom,
% 9.49/5.24      ! [X: real,Y: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
% 9.49/5.24        = ( ( X = zero_zero_real )
% 9.49/5.24          & ( Y = zero_zero_real ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2094_sum__power2__le__zero__iff,axiom,
% 9.49/5.24      ! [X: rat,Y: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
% 9.49/5.24        = ( ( X = zero_zero_rat )
% 9.49/5.24          & ( Y = zero_zero_rat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2095_sum__power2__le__zero__iff,axiom,
% 9.49/5.24      ! [X: int,Y: int] :
% 9.49/5.24        ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
% 9.49/5.24        = ( ( X = zero_zero_int )
% 9.49/5.24          & ( Y = zero_zero_int ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2096_sum__power2__ge__zero,axiom,
% 9.49/5.24      ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2097_sum__power2__ge__zero,axiom,
% 9.49/5.24      ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2098_sum__power2__ge__zero,axiom,
% 9.49/5.24      ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2099_not__sum__power2__lt__zero,axiom,
% 9.49/5.24      ! [X: real,Y: real] :
% 9.49/5.24        ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2100_not__sum__power2__lt__zero,axiom,
% 9.49/5.24      ! [X: rat,Y: rat] :
% 9.49/5.24        ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2101_not__sum__power2__lt__zero,axiom,
% 9.49/5.24      ! [X: int,Y: int] :
% 9.49/5.24        ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2102_sum__power2__gt__zero__iff,axiom,
% 9.49/5.24      ! [X: real,Y: real] :
% 9.49/5.24        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.49/5.24        = ( ( X != zero_zero_real )
% 9.49/5.24          | ( Y != zero_zero_real ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2103_sum__power2__gt__zero__iff,axiom,
% 9.49/5.24      ! [X: rat,Y: rat] :
% 9.49/5.24        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.49/5.24        = ( ( X != zero_zero_rat )
% 9.49/5.24          | ( Y != zero_zero_rat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2104_sum__power2__gt__zero__iff,axiom,
% 9.49/5.24      ! [X: int,Y: int] :
% 9.49/5.24        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.49/5.24        = ( ( X != zero_zero_int )
% 9.49/5.24          | ( Y != zero_zero_int ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2105_member__inv,axiom,
% 9.49/5.24      ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 9.49/5.24        ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.24       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.24          & ( ( X = Mi )
% 9.49/5.24            | ( X = Ma )
% 9.49/5.24            | ( ( ord_less_nat @ X @ Ma )
% 9.49/5.24              & ( ord_less_nat @ Mi @ X )
% 9.49/5.24              & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.24              & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2106_pred__list__to__short,axiom,
% 9.49/5.24      ! [Deg: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
% 9.49/5.24        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.24       => ( ( ord_less_eq_nat @ X @ Ma )
% 9.49/5.24         => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.49/5.24           => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.24              = none_nat ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2107_succ__list__to__short,axiom,
% 9.49/5.24      ! [Deg: nat,Mi: nat,X: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 9.49/5.24        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.24       => ( ( ord_less_eq_nat @ Mi @ X )
% 9.49/5.24         => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.49/5.24           => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.24              = none_nat ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2108_both__member__options__ding,axiom,
% 9.49/5.24      ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
% 9.49/5.24        ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
% 9.49/5.24       => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.49/5.24         => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.49/5.24           => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2109_semiring__norm_I76_J,axiom,
% 9.49/5.24      ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2110_set__n__deg__not__0,axiom,
% 9.49/5.24      ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
% 9.49/5.24        ( ! [X4: vEBT_VEBT] :
% 9.49/5.24            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.49/5.24           => ( vEBT_invar_vebt @ X4 @ N ) )
% 9.49/5.24       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.49/5.24            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.49/5.24         => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2111_vebt__insert_Osimps_I4_J,axiom,
% 9.49/5.24      ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 9.49/5.24        ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
% 9.49/5.24        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2112_exp__add__not__zero__imp__right,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 9.49/5.24         != zero_zero_nat )
% 9.49/5.24       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.24         != zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2113_exp__add__not__zero__imp__right,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 9.49/5.24         != zero_zero_int )
% 9.49/5.24       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 9.49/5.24         != zero_zero_int ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2114_exp__add__not__zero__imp__left,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 9.49/5.24         != zero_zero_nat )
% 9.49/5.24       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 9.49/5.24         != zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2115_exp__add__not__zero__imp__left,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 9.49/5.24         != zero_zero_int )
% 9.49/5.24       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 9.49/5.24         != zero_zero_int ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2116_bit__concat__def,axiom,
% 9.49/5.24      ( vEBT_VEBT_bit_concat
% 9.49/5.24      = ( ^ [H: nat,L2: nat,D4: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D4 ) ) @ L2 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2117_pow__sum,axiom,
% 9.49/5.24      ! [A: nat,B: nat] :
% 9.49/5.24        ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 9.49/5.24        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2118_high__def,axiom,
% 9.49/5.24      ( vEBT_VEBT_high
% 9.49/5.24      = ( ^ [X3: nat,N2: nat] : ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2119__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
% 9.49/5.24      ( ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24      = na ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2120_mult__zero__left,axiom,
% 9.49/5.24      ! [A: complex] :
% 9.49/5.24        ( ( times_times_complex @ zero_zero_complex @ A )
% 9.49/5.24        = zero_zero_complex ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2121_mult__zero__left,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( times_times_real @ zero_zero_real @ A )
% 9.49/5.24        = zero_zero_real ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2122_mult__zero__left,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( times_times_rat @ zero_zero_rat @ A )
% 9.49/5.24        = zero_zero_rat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2123_mult__zero__left,axiom,
% 9.49/5.24      ! [A: nat] :
% 9.49/5.24        ( ( times_times_nat @ zero_zero_nat @ A )
% 9.49/5.24        = zero_zero_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2124_mult__zero__left,axiom,
% 9.49/5.24      ! [A: int] :
% 9.49/5.24        ( ( times_times_int @ zero_zero_int @ A )
% 9.49/5.24        = zero_zero_int ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2125_mult__zero__right,axiom,
% 9.49/5.24      ! [A: complex] :
% 9.49/5.24        ( ( times_times_complex @ A @ zero_zero_complex )
% 9.49/5.24        = zero_zero_complex ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2126_mult__zero__right,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( times_times_real @ A @ zero_zero_real )
% 9.49/5.24        = zero_zero_real ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2127_mult__zero__right,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( times_times_rat @ A @ zero_zero_rat )
% 9.49/5.24        = zero_zero_rat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2128_mult__zero__right,axiom,
% 9.49/5.24      ! [A: nat] :
% 9.49/5.24        ( ( times_times_nat @ A @ zero_zero_nat )
% 9.49/5.24        = zero_zero_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2129_mult__zero__right,axiom,
% 9.49/5.24      ! [A: int] :
% 9.49/5.24        ( ( times_times_int @ A @ zero_zero_int )
% 9.49/5.24        = zero_zero_int ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2130_mult__eq__0__iff,axiom,
% 9.49/5.24      ! [A: complex,B: complex] :
% 9.49/5.24        ( ( ( times_times_complex @ A @ B )
% 9.49/5.24          = zero_zero_complex )
% 9.49/5.24        = ( ( A = zero_zero_complex )
% 9.49/5.24          | ( B = zero_zero_complex ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2131_mult__eq__0__iff,axiom,
% 9.49/5.24      ! [A: real,B: real] :
% 9.49/5.24        ( ( ( times_times_real @ A @ B )
% 9.49/5.24          = zero_zero_real )
% 9.49/5.24        = ( ( A = zero_zero_real )
% 9.49/5.24          | ( B = zero_zero_real ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2132_mult__eq__0__iff,axiom,
% 9.49/5.24      ! [A: rat,B: rat] :
% 9.49/5.24        ( ( ( times_times_rat @ A @ B )
% 9.49/5.24          = zero_zero_rat )
% 9.49/5.24        = ( ( A = zero_zero_rat )
% 9.49/5.24          | ( B = zero_zero_rat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2133_mult__eq__0__iff,axiom,
% 9.49/5.24      ! [A: nat,B: nat] :
% 9.49/5.24        ( ( ( times_times_nat @ A @ B )
% 9.49/5.24          = zero_zero_nat )
% 9.49/5.24        = ( ( A = zero_zero_nat )
% 9.49/5.24          | ( B = zero_zero_nat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2134_mult__eq__0__iff,axiom,
% 9.49/5.24      ! [A: int,B: int] :
% 9.49/5.24        ( ( ( times_times_int @ A @ B )
% 9.49/5.24          = zero_zero_int )
% 9.49/5.24        = ( ( A = zero_zero_int )
% 9.49/5.24          | ( B = zero_zero_int ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2135_mult__cancel__left,axiom,
% 9.49/5.24      ! [C: complex,A: complex,B: complex] :
% 9.49/5.24        ( ( ( times_times_complex @ C @ A )
% 9.49/5.24          = ( times_times_complex @ C @ B ) )
% 9.49/5.24        = ( ( C = zero_zero_complex )
% 9.49/5.24          | ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2136_mult__cancel__left,axiom,
% 9.49/5.24      ! [C: real,A: real,B: real] :
% 9.49/5.24        ( ( ( times_times_real @ C @ A )
% 9.49/5.24          = ( times_times_real @ C @ B ) )
% 9.49/5.24        = ( ( C = zero_zero_real )
% 9.49/5.24          | ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2137_mult__cancel__left,axiom,
% 9.49/5.24      ! [C: rat,A: rat,B: rat] :
% 9.49/5.24        ( ( ( times_times_rat @ C @ A )
% 9.49/5.24          = ( times_times_rat @ C @ B ) )
% 9.49/5.24        = ( ( C = zero_zero_rat )
% 9.49/5.24          | ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2138_mult__cancel__left,axiom,
% 9.49/5.24      ! [C: nat,A: nat,B: nat] :
% 9.49/5.24        ( ( ( times_times_nat @ C @ A )
% 9.49/5.24          = ( times_times_nat @ C @ B ) )
% 9.49/5.24        = ( ( C = zero_zero_nat )
% 9.49/5.24          | ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2139_mult__cancel__left,axiom,
% 9.49/5.24      ! [C: int,A: int,B: int] :
% 9.49/5.24        ( ( ( times_times_int @ C @ A )
% 9.49/5.24          = ( times_times_int @ C @ B ) )
% 9.49/5.24        = ( ( C = zero_zero_int )
% 9.49/5.24          | ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2140_mult__cancel__right,axiom,
% 9.49/5.24      ! [A: complex,C: complex,B: complex] :
% 9.49/5.24        ( ( ( times_times_complex @ A @ C )
% 9.49/5.24          = ( times_times_complex @ B @ C ) )
% 9.49/5.24        = ( ( C = zero_zero_complex )
% 9.49/5.24          | ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2141_mult__cancel__right,axiom,
% 9.49/5.24      ! [A: real,C: real,B: real] :
% 9.49/5.24        ( ( ( times_times_real @ A @ C )
% 9.49/5.24          = ( times_times_real @ B @ C ) )
% 9.49/5.24        = ( ( C = zero_zero_real )
% 9.49/5.24          | ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2142_mult__cancel__right,axiom,
% 9.49/5.24      ! [A: rat,C: rat,B: rat] :
% 9.49/5.24        ( ( ( times_times_rat @ A @ C )
% 9.49/5.24          = ( times_times_rat @ B @ C ) )
% 9.49/5.24        = ( ( C = zero_zero_rat )
% 9.49/5.24          | ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2143_mult__cancel__right,axiom,
% 9.49/5.24      ! [A: nat,C: nat,B: nat] :
% 9.49/5.24        ( ( ( times_times_nat @ A @ C )
% 9.49/5.24          = ( times_times_nat @ B @ C ) )
% 9.49/5.24        = ( ( C = zero_zero_nat )
% 9.49/5.24          | ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2144_mult__cancel__right,axiom,
% 9.49/5.24      ! [A: int,C: int,B: int] :
% 9.49/5.24        ( ( ( times_times_int @ A @ C )
% 9.49/5.24          = ( times_times_int @ B @ C ) )
% 9.49/5.24        = ( ( C = zero_zero_int )
% 9.49/5.24          | ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2145_high__inv,axiom,
% 9.49/5.24      ! [X: nat,N: nat,Y: nat] :
% 9.49/5.24        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.24       => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
% 9.49/5.24          = Y ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2146_low__inv,axiom,
% 9.49/5.24      ! [X: nat,N: nat,Y: nat] :
% 9.49/5.24        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.24       => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
% 9.49/5.24          = X ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2147_bits__div__0,axiom,
% 9.49/5.24      ! [A: nat] :
% 9.49/5.24        ( ( divide_divide_nat @ zero_zero_nat @ A )
% 9.49/5.24        = zero_zero_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2148_bits__div__0,axiom,
% 9.49/5.24      ! [A: int] :
% 9.49/5.24        ( ( divide_divide_int @ zero_zero_int @ A )
% 9.49/5.24        = zero_zero_int ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2149_bits__div__0,axiom,
% 9.49/5.24      ! [A: code_integer] :
% 9.49/5.24        ( ( divide6298287555418463151nteger @ zero_z3403309356797280102nteger @ A )
% 9.49/5.24        = zero_z3403309356797280102nteger ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2150_bits__div__by__0,axiom,
% 9.49/5.24      ! [A: nat] :
% 9.49/5.24        ( ( divide_divide_nat @ A @ zero_zero_nat )
% 9.49/5.24        = zero_zero_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2151_bits__div__by__0,axiom,
% 9.49/5.24      ! [A: int] :
% 9.49/5.24        ( ( divide_divide_int @ A @ zero_zero_int )
% 9.49/5.24        = zero_zero_int ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2152_bits__div__by__0,axiom,
% 9.49/5.24      ! [A: code_integer] :
% 9.49/5.24        ( ( divide6298287555418463151nteger @ A @ zero_z3403309356797280102nteger )
% 9.49/5.24        = zero_z3403309356797280102nteger ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2153_div__0,axiom,
% 9.49/5.24      ! [A: complex] :
% 9.49/5.24        ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
% 9.49/5.24        = zero_zero_complex ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2154_div__0,axiom,
% 9.49/5.24      ! [A: nat] :
% 9.49/5.24        ( ( divide_divide_nat @ zero_zero_nat @ A )
% 9.49/5.24        = zero_zero_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2155_div__0,axiom,
% 9.49/5.24      ! [A: int] :
% 9.49/5.24        ( ( divide_divide_int @ zero_zero_int @ A )
% 9.49/5.24        = zero_zero_int ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2156_div__0,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( divide_divide_real @ zero_zero_real @ A )
% 9.49/5.24        = zero_zero_real ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2157_div__0,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( divide_divide_rat @ zero_zero_rat @ A )
% 9.49/5.24        = zero_zero_rat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2158_div__0,axiom,
% 9.49/5.24      ! [A: code_integer] :
% 9.49/5.24        ( ( divide6298287555418463151nteger @ zero_z3403309356797280102nteger @ A )
% 9.49/5.24        = zero_z3403309356797280102nteger ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2159_divide__eq__0__iff,axiom,
% 9.49/5.24      ! [A: complex,B: complex] :
% 9.49/5.24        ( ( ( divide1717551699836669952omplex @ A @ B )
% 9.49/5.24          = zero_zero_complex )
% 9.49/5.24        = ( ( A = zero_zero_complex )
% 9.49/5.24          | ( B = zero_zero_complex ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2160_divide__eq__0__iff,axiom,
% 9.49/5.24      ! [A: real,B: real] :
% 9.49/5.24        ( ( ( divide_divide_real @ A @ B )
% 9.49/5.24          = zero_zero_real )
% 9.49/5.24        = ( ( A = zero_zero_real )
% 9.49/5.24          | ( B = zero_zero_real ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2161_divide__eq__0__iff,axiom,
% 9.49/5.24      ! [A: rat,B: rat] :
% 9.49/5.24        ( ( ( divide_divide_rat @ A @ B )
% 9.49/5.24          = zero_zero_rat )
% 9.49/5.24        = ( ( A = zero_zero_rat )
% 9.49/5.24          | ( B = zero_zero_rat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2162_div__by__0,axiom,
% 9.49/5.24      ! [A: complex] :
% 9.49/5.24        ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 9.49/5.24        = zero_zero_complex ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2163_div__by__0,axiom,
% 9.49/5.24      ! [A: nat] :
% 9.49/5.24        ( ( divide_divide_nat @ A @ zero_zero_nat )
% 9.49/5.24        = zero_zero_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2164_div__by__0,axiom,
% 9.49/5.24      ! [A: int] :
% 9.49/5.24        ( ( divide_divide_int @ A @ zero_zero_int )
% 9.49/5.24        = zero_zero_int ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2165_div__by__0,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( divide_divide_real @ A @ zero_zero_real )
% 9.49/5.24        = zero_zero_real ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2166_div__by__0,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( divide_divide_rat @ A @ zero_zero_rat )
% 9.49/5.24        = zero_zero_rat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2167_div__by__0,axiom,
% 9.49/5.24      ! [A: code_integer] :
% 9.49/5.24        ( ( divide6298287555418463151nteger @ A @ zero_z3403309356797280102nteger )
% 9.49/5.24        = zero_z3403309356797280102nteger ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2168_divide__cancel__left,axiom,
% 9.49/5.24      ! [C: complex,A: complex,B: complex] :
% 9.49/5.24        ( ( ( divide1717551699836669952omplex @ C @ A )
% 9.49/5.24          = ( divide1717551699836669952omplex @ C @ B ) )
% 9.49/5.24        = ( ( C = zero_zero_complex )
% 9.49/5.24          | ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2169_divide__cancel__left,axiom,
% 9.49/5.24      ! [C: real,A: real,B: real] :
% 9.49/5.24        ( ( ( divide_divide_real @ C @ A )
% 9.49/5.24          = ( divide_divide_real @ C @ B ) )
% 9.49/5.24        = ( ( C = zero_zero_real )
% 9.49/5.24          | ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2170_divide__cancel__left,axiom,
% 9.49/5.24      ! [C: rat,A: rat,B: rat] :
% 9.49/5.24        ( ( ( divide_divide_rat @ C @ A )
% 9.49/5.24          = ( divide_divide_rat @ C @ B ) )
% 9.49/5.24        = ( ( C = zero_zero_rat )
% 9.49/5.24          | ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2171_divide__cancel__right,axiom,
% 9.49/5.24      ! [A: complex,C: complex,B: complex] :
% 9.49/5.24        ( ( ( divide1717551699836669952omplex @ A @ C )
% 9.49/5.24          = ( divide1717551699836669952omplex @ B @ C ) )
% 9.49/5.24        = ( ( C = zero_zero_complex )
% 9.49/5.24          | ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2172_divide__cancel__right,axiom,
% 9.49/5.24      ! [A: real,C: real,B: real] :
% 9.49/5.24        ( ( ( divide_divide_real @ A @ C )
% 9.49/5.24          = ( divide_divide_real @ B @ C ) )
% 9.49/5.24        = ( ( C = zero_zero_real )
% 9.49/5.24          | ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2173_divide__cancel__right,axiom,
% 9.49/5.24      ! [A: rat,C: rat,B: rat] :
% 9.49/5.24        ( ( ( divide_divide_rat @ A @ C )
% 9.49/5.24          = ( divide_divide_rat @ B @ C ) )
% 9.49/5.24        = ( ( C = zero_zero_rat )
% 9.49/5.24          | ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2174_division__ring__divide__zero,axiom,
% 9.49/5.24      ! [A: complex] :
% 9.49/5.24        ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 9.49/5.24        = zero_zero_complex ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2175_division__ring__divide__zero,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( divide_divide_real @ A @ zero_zero_real )
% 9.49/5.24        = zero_zero_real ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2176_division__ring__divide__zero,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( divide_divide_rat @ A @ zero_zero_rat )
% 9.49/5.24        = zero_zero_rat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2177_mult__1,axiom,
% 9.49/5.24      ! [A: complex] :
% 9.49/5.24        ( ( times_times_complex @ one_one_complex @ A )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2178_mult__1,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( times_times_real @ one_one_real @ A )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2179_mult__1,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( times_times_rat @ one_one_rat @ A )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2180_mult__1,axiom,
% 9.49/5.24      ! [A: nat] :
% 9.49/5.24        ( ( times_times_nat @ one_one_nat @ A )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2181_mult__1,axiom,
% 9.49/5.24      ! [A: int] :
% 9.49/5.24        ( ( times_times_int @ one_one_int @ A )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2182_mult_Oright__neutral,axiom,
% 9.49/5.24      ! [A: complex] :
% 9.49/5.24        ( ( times_times_complex @ A @ one_one_complex )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2183_mult_Oright__neutral,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( times_times_real @ A @ one_one_real )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2184_mult_Oright__neutral,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( times_times_rat @ A @ one_one_rat )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2185_mult_Oright__neutral,axiom,
% 9.49/5.24      ! [A: nat] :
% 9.49/5.24        ( ( times_times_nat @ A @ one_one_nat )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2186_mult_Oright__neutral,axiom,
% 9.49/5.24      ! [A: int] :
% 9.49/5.24        ( ( times_times_int @ A @ one_one_int )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2187_mult__is__0,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ( times_times_nat @ M @ N )
% 9.49/5.24          = zero_zero_nat )
% 9.49/5.24        = ( ( M = zero_zero_nat )
% 9.49/5.24          | ( N = zero_zero_nat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2188_mult__0__right,axiom,
% 9.49/5.24      ! [M: nat] :
% 9.49/5.24        ( ( times_times_nat @ M @ zero_zero_nat )
% 9.49/5.24        = zero_zero_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2189_mult__cancel1,axiom,
% 9.49/5.24      ! [K: nat,M: nat,N: nat] :
% 9.49/5.24        ( ( ( times_times_nat @ K @ M )
% 9.49/5.24          = ( times_times_nat @ K @ N ) )
% 9.49/5.24        = ( ( M = N )
% 9.49/5.24          | ( K = zero_zero_nat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2190_mult__cancel2,axiom,
% 9.49/5.24      ! [M: nat,K: nat,N: nat] :
% 9.49/5.24        ( ( ( times_times_nat @ M @ K )
% 9.49/5.24          = ( times_times_nat @ N @ K ) )
% 9.49/5.24        = ( ( M = N )
% 9.49/5.24          | ( K = zero_zero_nat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2191_nat__mult__eq__1__iff,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ( times_times_nat @ M @ N )
% 9.49/5.24          = one_one_nat )
% 9.49/5.24        = ( ( M = one_one_nat )
% 9.49/5.24          & ( N = one_one_nat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2192_nat__1__eq__mult__iff,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( one_one_nat
% 9.49/5.24          = ( times_times_nat @ M @ N ) )
% 9.49/5.24        = ( ( M = one_one_nat )
% 9.49/5.24          & ( N = one_one_nat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2193_semiring__norm_I78_J,axiom,
% 9.49/5.24      ! [M: num,N: num] :
% 9.49/5.24        ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.49/5.24        = ( ord_less_num @ M @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2194_semiring__norm_I75_J,axiom,
% 9.49/5.24      ! [M: num] :
% 9.49/5.24        ~ ( ord_less_num @ M @ one ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2195_sum__squares__eq__zero__iff,axiom,
% 9.49/5.24      ! [X: real,Y: real] :
% 9.49/5.24        ( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
% 9.49/5.24          = zero_zero_real )
% 9.49/5.24        = ( ( X = zero_zero_real )
% 9.49/5.24          & ( Y = zero_zero_real ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2196_sum__squares__eq__zero__iff,axiom,
% 9.49/5.24      ! [X: rat,Y: rat] :
% 9.49/5.24        ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
% 9.49/5.24          = zero_zero_rat )
% 9.49/5.24        = ( ( X = zero_zero_rat )
% 9.49/5.24          & ( Y = zero_zero_rat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2197_sum__squares__eq__zero__iff,axiom,
% 9.49/5.24      ! [X: int,Y: int] :
% 9.49/5.24        ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
% 9.49/5.24          = zero_zero_int )
% 9.49/5.24        = ( ( X = zero_zero_int )
% 9.49/5.24          & ( Y = zero_zero_int ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2198_mult__cancel__left1,axiom,
% 9.49/5.24      ! [C: complex,B: complex] :
% 9.49/5.24        ( ( C
% 9.49/5.24          = ( times_times_complex @ C @ B ) )
% 9.49/5.24        = ( ( C = zero_zero_complex )
% 9.49/5.24          | ( B = one_one_complex ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2199_mult__cancel__left1,axiom,
% 9.49/5.24      ! [C: real,B: real] :
% 9.49/5.24        ( ( C
% 9.49/5.24          = ( times_times_real @ C @ B ) )
% 9.49/5.24        = ( ( C = zero_zero_real )
% 9.49/5.24          | ( B = one_one_real ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2200_mult__cancel__left1,axiom,
% 9.49/5.24      ! [C: rat,B: rat] :
% 9.49/5.24        ( ( C
% 9.49/5.24          = ( times_times_rat @ C @ B ) )
% 9.49/5.24        = ( ( C = zero_zero_rat )
% 9.49/5.24          | ( B = one_one_rat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2201_mult__cancel__left1,axiom,
% 9.49/5.24      ! [C: int,B: int] :
% 9.49/5.24        ( ( C
% 9.49/5.24          = ( times_times_int @ C @ B ) )
% 9.49/5.24        = ( ( C = zero_zero_int )
% 9.49/5.24          | ( B = one_one_int ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2202_mult__cancel__left2,axiom,
% 9.49/5.24      ! [C: complex,A: complex] :
% 9.49/5.24        ( ( ( times_times_complex @ C @ A )
% 9.49/5.24          = C )
% 9.49/5.24        = ( ( C = zero_zero_complex )
% 9.49/5.24          | ( A = one_one_complex ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2203_mult__cancel__left2,axiom,
% 9.49/5.24      ! [C: real,A: real] :
% 9.49/5.24        ( ( ( times_times_real @ C @ A )
% 9.49/5.24          = C )
% 9.49/5.24        = ( ( C = zero_zero_real )
% 9.49/5.24          | ( A = one_one_real ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2204_mult__cancel__left2,axiom,
% 9.49/5.24      ! [C: rat,A: rat] :
% 9.49/5.24        ( ( ( times_times_rat @ C @ A )
% 9.49/5.24          = C )
% 9.49/5.24        = ( ( C = zero_zero_rat )
% 9.49/5.24          | ( A = one_one_rat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2205_mult__cancel__left2,axiom,
% 9.49/5.24      ! [C: int,A: int] :
% 9.49/5.24        ( ( ( times_times_int @ C @ A )
% 9.49/5.24          = C )
% 9.49/5.24        = ( ( C = zero_zero_int )
% 9.49/5.24          | ( A = one_one_int ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2206_mult__cancel__right1,axiom,
% 9.49/5.24      ! [C: complex,B: complex] :
% 9.49/5.24        ( ( C
% 9.49/5.24          = ( times_times_complex @ B @ C ) )
% 9.49/5.24        = ( ( C = zero_zero_complex )
% 9.49/5.24          | ( B = one_one_complex ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2207_mult__cancel__right1,axiom,
% 9.49/5.24      ! [C: real,B: real] :
% 9.49/5.24        ( ( C
% 9.49/5.24          = ( times_times_real @ B @ C ) )
% 9.49/5.24        = ( ( C = zero_zero_real )
% 9.49/5.24          | ( B = one_one_real ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2208_mult__cancel__right1,axiom,
% 9.49/5.24      ! [C: rat,B: rat] :
% 9.49/5.24        ( ( C
% 9.49/5.24          = ( times_times_rat @ B @ C ) )
% 9.49/5.24        = ( ( C = zero_zero_rat )
% 9.49/5.24          | ( B = one_one_rat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2209_mult__cancel__right1,axiom,
% 9.49/5.24      ! [C: int,B: int] :
% 9.49/5.24        ( ( C
% 9.49/5.24          = ( times_times_int @ B @ C ) )
% 9.49/5.24        = ( ( C = zero_zero_int )
% 9.49/5.24          | ( B = one_one_int ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2210_mult__cancel__right2,axiom,
% 9.49/5.24      ! [A: complex,C: complex] :
% 9.49/5.24        ( ( ( times_times_complex @ A @ C )
% 9.49/5.24          = C )
% 9.49/5.24        = ( ( C = zero_zero_complex )
% 9.49/5.24          | ( A = one_one_complex ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2211_mult__cancel__right2,axiom,
% 9.49/5.24      ! [A: real,C: real] :
% 9.49/5.24        ( ( ( times_times_real @ A @ C )
% 9.49/5.24          = C )
% 9.49/5.24        = ( ( C = zero_zero_real )
% 9.49/5.24          | ( A = one_one_real ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2212_mult__cancel__right2,axiom,
% 9.49/5.24      ! [A: rat,C: rat] :
% 9.49/5.24        ( ( ( times_times_rat @ A @ C )
% 9.49/5.24          = C )
% 9.49/5.24        = ( ( C = zero_zero_rat )
% 9.49/5.24          | ( A = one_one_rat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2213_mult__cancel__right2,axiom,
% 9.49/5.24      ! [A: int,C: int] :
% 9.49/5.24        ( ( ( times_times_int @ A @ C )
% 9.49/5.24          = C )
% 9.49/5.24        = ( ( C = zero_zero_int )
% 9.49/5.24          | ( A = one_one_int ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2214_mult__divide__mult__cancel__left__if,axiom,
% 9.49/5.24      ! [C: complex,A: complex,B: complex] :
% 9.49/5.24        ( ( ( C = zero_zero_complex )
% 9.49/5.24         => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 9.49/5.24            = zero_zero_complex ) )
% 9.49/5.24        & ( ( C != zero_zero_complex )
% 9.49/5.24         => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 9.49/5.24            = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2215_mult__divide__mult__cancel__left__if,axiom,
% 9.49/5.24      ! [C: real,A: real,B: real] :
% 9.49/5.24        ( ( ( C = zero_zero_real )
% 9.49/5.24         => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.49/5.24            = zero_zero_real ) )
% 9.49/5.24        & ( ( C != zero_zero_real )
% 9.49/5.24         => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.49/5.24            = ( divide_divide_real @ A @ B ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2216_mult__divide__mult__cancel__left__if,axiom,
% 9.49/5.24      ! [C: rat,A: rat,B: rat] :
% 9.49/5.24        ( ( ( C = zero_zero_rat )
% 9.49/5.24         => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.49/5.24            = zero_zero_rat ) )
% 9.49/5.24        & ( ( C != zero_zero_rat )
% 9.49/5.24         => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.49/5.24            = ( divide_divide_rat @ A @ B ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2217_nonzero__mult__divide__mult__cancel__left,axiom,
% 9.49/5.24      ! [C: complex,A: complex,B: complex] :
% 9.49/5.24        ( ( C != zero_zero_complex )
% 9.49/5.24       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 9.49/5.24          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2218_nonzero__mult__divide__mult__cancel__left,axiom,
% 9.49/5.24      ! [C: real,A: real,B: real] :
% 9.49/5.24        ( ( C != zero_zero_real )
% 9.49/5.24       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.49/5.24          = ( divide_divide_real @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2219_nonzero__mult__divide__mult__cancel__left,axiom,
% 9.49/5.24      ! [C: rat,A: rat,B: rat] :
% 9.49/5.24        ( ( C != zero_zero_rat )
% 9.49/5.24       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.49/5.24          = ( divide_divide_rat @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2220_nonzero__mult__div__cancel__left,axiom,
% 9.49/5.24      ! [A: complex,B: complex] :
% 9.49/5.24        ( ( A != zero_zero_complex )
% 9.49/5.24       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
% 9.49/5.24          = B ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2221_nonzero__mult__div__cancel__left,axiom,
% 9.49/5.24      ! [A: nat,B: nat] :
% 9.49/5.24        ( ( A != zero_zero_nat )
% 9.49/5.24       => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
% 9.49/5.24          = B ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2222_nonzero__mult__div__cancel__left,axiom,
% 9.49/5.24      ! [A: int,B: int] :
% 9.49/5.24        ( ( A != zero_zero_int )
% 9.49/5.24       => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
% 9.49/5.24          = B ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2223_nonzero__mult__div__cancel__left,axiom,
% 9.49/5.24      ! [A: real,B: real] :
% 9.49/5.24        ( ( A != zero_zero_real )
% 9.49/5.24       => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
% 9.49/5.24          = B ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2224_nonzero__mult__div__cancel__left,axiom,
% 9.49/5.24      ! [A: rat,B: rat] :
% 9.49/5.24        ( ( A != zero_zero_rat )
% 9.49/5.24       => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
% 9.49/5.24          = B ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2225_nonzero__mult__div__cancel__left,axiom,
% 9.49/5.24      ! [A: code_integer,B: code_integer] :
% 9.49/5.24        ( ( A != zero_z3403309356797280102nteger )
% 9.49/5.24       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ A )
% 9.49/5.24          = B ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2226_nonzero__mult__divide__mult__cancel__left2,axiom,
% 9.49/5.24      ! [C: complex,A: complex,B: complex] :
% 9.49/5.24        ( ( C != zero_zero_complex )
% 9.49/5.24       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
% 9.49/5.24          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2227_nonzero__mult__divide__mult__cancel__left2,axiom,
% 9.49/5.24      ! [C: real,A: real,B: real] :
% 9.49/5.24        ( ( C != zero_zero_real )
% 9.49/5.24       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
% 9.49/5.24          = ( divide_divide_real @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2228_nonzero__mult__divide__mult__cancel__left2,axiom,
% 9.49/5.24      ! [C: rat,A: rat,B: rat] :
% 9.49/5.24        ( ( C != zero_zero_rat )
% 9.49/5.24       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
% 9.49/5.24          = ( divide_divide_rat @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2229_nonzero__mult__divide__mult__cancel__right,axiom,
% 9.49/5.24      ! [C: complex,A: complex,B: complex] :
% 9.49/5.24        ( ( C != zero_zero_complex )
% 9.49/5.24       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 9.49/5.24          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2230_nonzero__mult__divide__mult__cancel__right,axiom,
% 9.49/5.24      ! [C: real,A: real,B: real] :
% 9.49/5.24        ( ( C != zero_zero_real )
% 9.49/5.24       => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.49/5.24          = ( divide_divide_real @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2231_nonzero__mult__divide__mult__cancel__right,axiom,
% 9.49/5.24      ! [C: rat,A: rat,B: rat] :
% 9.49/5.24        ( ( C != zero_zero_rat )
% 9.49/5.24       => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.49/5.24          = ( divide_divide_rat @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2232_nonzero__mult__div__cancel__right,axiom,
% 9.49/5.24      ! [B: complex,A: complex] :
% 9.49/5.24        ( ( B != zero_zero_complex )
% 9.49/5.24       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
% 9.49/5.24          = A ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2233_nonzero__mult__div__cancel__right,axiom,
% 9.49/5.24      ! [B: nat,A: nat] :
% 9.49/5.24        ( ( B != zero_zero_nat )
% 9.49/5.24       => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
% 9.49/5.24          = A ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2234_nonzero__mult__div__cancel__right,axiom,
% 9.49/5.24      ! [B: int,A: int] :
% 9.49/5.24        ( ( B != zero_zero_int )
% 9.49/5.24       => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
% 9.49/5.24          = A ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2235_nonzero__mult__div__cancel__right,axiom,
% 9.49/5.24      ! [B: real,A: real] :
% 9.49/5.24        ( ( B != zero_zero_real )
% 9.49/5.24       => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
% 9.49/5.24          = A ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2236_nonzero__mult__div__cancel__right,axiom,
% 9.49/5.24      ! [B: rat,A: rat] :
% 9.49/5.24        ( ( B != zero_zero_rat )
% 9.49/5.24       => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
% 9.49/5.24          = A ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2237_nonzero__mult__div__cancel__right,axiom,
% 9.49/5.24      ! [B: code_integer,A: code_integer] :
% 9.49/5.24        ( ( B != zero_z3403309356797280102nteger )
% 9.49/5.24       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
% 9.49/5.24          = A ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2238_nonzero__mult__divide__mult__cancel__right2,axiom,
% 9.49/5.24      ! [C: complex,A: complex,B: complex] :
% 9.49/5.24        ( ( C != zero_zero_complex )
% 9.49/5.24       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
% 9.49/5.24          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2239_nonzero__mult__divide__mult__cancel__right2,axiom,
% 9.49/5.24      ! [C: real,A: real,B: real] :
% 9.49/5.24        ( ( C != zero_zero_real )
% 9.49/5.24       => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
% 9.49/5.24          = ( divide_divide_real @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2240_nonzero__mult__divide__mult__cancel__right2,axiom,
% 9.49/5.24      ! [C: rat,A: rat,B: rat] :
% 9.49/5.24        ( ( C != zero_zero_rat )
% 9.49/5.24       => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
% 9.49/5.24          = ( divide_divide_rat @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2241_divide__eq__1__iff,axiom,
% 9.49/5.24      ! [A: complex,B: complex] :
% 9.49/5.24        ( ( ( divide1717551699836669952omplex @ A @ B )
% 9.49/5.24          = one_one_complex )
% 9.49/5.24        = ( ( B != zero_zero_complex )
% 9.49/5.24          & ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2242_divide__eq__1__iff,axiom,
% 9.49/5.24      ! [A: real,B: real] :
% 9.49/5.24        ( ( ( divide_divide_real @ A @ B )
% 9.49/5.24          = one_one_real )
% 9.49/5.24        = ( ( B != zero_zero_real )
% 9.49/5.24          & ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2243_divide__eq__1__iff,axiom,
% 9.49/5.24      ! [A: rat,B: rat] :
% 9.49/5.24        ( ( ( divide_divide_rat @ A @ B )
% 9.49/5.24          = one_one_rat )
% 9.49/5.24        = ( ( B != zero_zero_rat )
% 9.49/5.24          & ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2244_div__self,axiom,
% 9.49/5.24      ! [A: complex] :
% 9.49/5.24        ( ( A != zero_zero_complex )
% 9.49/5.24       => ( ( divide1717551699836669952omplex @ A @ A )
% 9.49/5.24          = one_one_complex ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2245_div__self,axiom,
% 9.49/5.24      ! [A: nat] :
% 9.49/5.24        ( ( A != zero_zero_nat )
% 9.49/5.24       => ( ( divide_divide_nat @ A @ A )
% 9.49/5.24          = one_one_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2246_div__self,axiom,
% 9.49/5.24      ! [A: int] :
% 9.49/5.24        ( ( A != zero_zero_int )
% 9.49/5.24       => ( ( divide_divide_int @ A @ A )
% 9.49/5.24          = one_one_int ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2247_div__self,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( A != zero_zero_real )
% 9.49/5.24       => ( ( divide_divide_real @ A @ A )
% 9.49/5.24          = one_one_real ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2248_div__self,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( A != zero_zero_rat )
% 9.49/5.24       => ( ( divide_divide_rat @ A @ A )
% 9.49/5.24          = one_one_rat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2249_div__self,axiom,
% 9.49/5.24      ! [A: code_integer] :
% 9.49/5.24        ( ( A != zero_z3403309356797280102nteger )
% 9.49/5.24       => ( ( divide6298287555418463151nteger @ A @ A )
% 9.49/5.24          = one_one_Code_integer ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2250_one__eq__divide__iff,axiom,
% 9.49/5.24      ! [A: complex,B: complex] :
% 9.49/5.24        ( ( one_one_complex
% 9.49/5.24          = ( divide1717551699836669952omplex @ A @ B ) )
% 9.49/5.24        = ( ( B != zero_zero_complex )
% 9.49/5.24          & ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2251_one__eq__divide__iff,axiom,
% 9.49/5.24      ! [A: real,B: real] :
% 9.49/5.24        ( ( one_one_real
% 9.49/5.24          = ( divide_divide_real @ A @ B ) )
% 9.49/5.24        = ( ( B != zero_zero_real )
% 9.49/5.24          & ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2252_one__eq__divide__iff,axiom,
% 9.49/5.24      ! [A: rat,B: rat] :
% 9.49/5.24        ( ( one_one_rat
% 9.49/5.24          = ( divide_divide_rat @ A @ B ) )
% 9.49/5.24        = ( ( B != zero_zero_rat )
% 9.49/5.24          & ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2253_divide__self,axiom,
% 9.49/5.24      ! [A: complex] :
% 9.49/5.24        ( ( A != zero_zero_complex )
% 9.49/5.24       => ( ( divide1717551699836669952omplex @ A @ A )
% 9.49/5.24          = one_one_complex ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2254_divide__self,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( A != zero_zero_real )
% 9.49/5.24       => ( ( divide_divide_real @ A @ A )
% 9.49/5.24          = one_one_real ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2255_divide__self,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( A != zero_zero_rat )
% 9.49/5.24       => ( ( divide_divide_rat @ A @ A )
% 9.49/5.24          = one_one_rat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2256_divide__self__if,axiom,
% 9.49/5.24      ! [A: complex] :
% 9.49/5.24        ( ( ( A = zero_zero_complex )
% 9.49/5.24         => ( ( divide1717551699836669952omplex @ A @ A )
% 9.49/5.24            = zero_zero_complex ) )
% 9.49/5.24        & ( ( A != zero_zero_complex )
% 9.49/5.24         => ( ( divide1717551699836669952omplex @ A @ A )
% 9.49/5.24            = one_one_complex ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2257_divide__self__if,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( ( A = zero_zero_real )
% 9.49/5.24         => ( ( divide_divide_real @ A @ A )
% 9.49/5.24            = zero_zero_real ) )
% 9.49/5.24        & ( ( A != zero_zero_real )
% 9.49/5.24         => ( ( divide_divide_real @ A @ A )
% 9.49/5.24            = one_one_real ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2258_divide__self__if,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( ( A = zero_zero_rat )
% 9.49/5.24         => ( ( divide_divide_rat @ A @ A )
% 9.49/5.24            = zero_zero_rat ) )
% 9.49/5.24        & ( ( A != zero_zero_rat )
% 9.49/5.24         => ( ( divide_divide_rat @ A @ A )
% 9.49/5.24            = one_one_rat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2259_divide__eq__eq__1,axiom,
% 9.49/5.24      ! [B: real,A: real] :
% 9.49/5.24        ( ( ( divide_divide_real @ B @ A )
% 9.49/5.24          = one_one_real )
% 9.49/5.24        = ( ( A != zero_zero_real )
% 9.49/5.24          & ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2260_divide__eq__eq__1,axiom,
% 9.49/5.24      ! [B: rat,A: rat] :
% 9.49/5.24        ( ( ( divide_divide_rat @ B @ A )
% 9.49/5.24          = one_one_rat )
% 9.49/5.24        = ( ( A != zero_zero_rat )
% 9.49/5.24          & ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2261_eq__divide__eq__1,axiom,
% 9.49/5.24      ! [B: real,A: real] :
% 9.49/5.24        ( ( one_one_real
% 9.49/5.24          = ( divide_divide_real @ B @ A ) )
% 9.49/5.24        = ( ( A != zero_zero_real )
% 9.49/5.24          & ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2262_eq__divide__eq__1,axiom,
% 9.49/5.24      ! [B: rat,A: rat] :
% 9.49/5.24        ( ( one_one_rat
% 9.49/5.24          = ( divide_divide_rat @ B @ A ) )
% 9.49/5.24        = ( ( A != zero_zero_rat )
% 9.49/5.24          & ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2263_one__divide__eq__0__iff,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( ( divide_divide_real @ one_one_real @ A )
% 9.49/5.24          = zero_zero_real )
% 9.49/5.24        = ( A = zero_zero_real ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2264_one__divide__eq__0__iff,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( ( divide_divide_rat @ one_one_rat @ A )
% 9.49/5.24          = zero_zero_rat )
% 9.49/5.24        = ( A = zero_zero_rat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2265_zero__eq__1__divide__iff,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( zero_zero_real
% 9.49/5.24          = ( divide_divide_real @ one_one_real @ A ) )
% 9.49/5.24        = ( A = zero_zero_real ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2266_zero__eq__1__divide__iff,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( zero_zero_rat
% 9.49/5.24          = ( divide_divide_rat @ one_one_rat @ A ) )
% 9.49/5.24        = ( A = zero_zero_rat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2267_power__inject__exp,axiom,
% 9.49/5.24      ! [A: real,M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_real @ one_one_real @ A )
% 9.49/5.24       => ( ( ( power_power_real @ A @ M )
% 9.49/5.24            = ( power_power_real @ A @ N ) )
% 9.49/5.24          = ( M = N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2268_power__inject__exp,axiom,
% 9.49/5.24      ! [A: rat,M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_rat @ one_one_rat @ A )
% 9.49/5.24       => ( ( ( power_power_rat @ A @ M )
% 9.49/5.24            = ( power_power_rat @ A @ N ) )
% 9.49/5.24          = ( M = N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2269_power__inject__exp,axiom,
% 9.49/5.24      ! [A: nat,M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ one_one_nat @ A )
% 9.49/5.24       => ( ( ( power_power_nat @ A @ M )
% 9.49/5.24            = ( power_power_nat @ A @ N ) )
% 9.49/5.24          = ( M = N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2270_power__inject__exp,axiom,
% 9.49/5.24      ! [A: int,M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_int @ one_one_int @ A )
% 9.49/5.24       => ( ( ( power_power_int @ A @ M )
% 9.49/5.24            = ( power_power_int @ A @ N ) )
% 9.49/5.24          = ( M = N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2271_one__eq__mult__iff,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ( suc @ zero_zero_nat )
% 9.49/5.24          = ( times_times_nat @ M @ N ) )
% 9.49/5.24        = ( ( M
% 9.49/5.24            = ( suc @ zero_zero_nat ) )
% 9.49/5.24          & ( N
% 9.49/5.24            = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2272_mult__eq__1__iff,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ( times_times_nat @ M @ N )
% 9.49/5.24          = ( suc @ zero_zero_nat ) )
% 9.49/5.24        = ( ( M
% 9.49/5.24            = ( suc @ zero_zero_nat ) )
% 9.49/5.24          & ( N
% 9.49/5.24            = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2273_nat__mult__less__cancel__disj,axiom,
% 9.49/5.24      ! [K: nat,M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.49/5.24        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.24          & ( ord_less_nat @ M @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2274_mult__less__cancel2,axiom,
% 9.49/5.24      ! [M: nat,K: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 9.49/5.24        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.24          & ( ord_less_nat @ M @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2275_nat__0__less__mult__iff,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 9.49/5.24        = ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.49/5.24          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2276_mult__Suc__right,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( times_times_nat @ M @ ( suc @ N ) )
% 9.49/5.24        = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2277_less__one,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ N @ one_one_nat )
% 9.49/5.24        = ( N = zero_zero_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2278_nat__mult__div__cancel__disj,axiom,
% 9.49/5.24      ! [K: nat,M: nat,N: nat] :
% 9.49/5.24        ( ( ( K = zero_zero_nat )
% 9.49/5.24         => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.49/5.24            = zero_zero_nat ) )
% 9.49/5.24        & ( ( K != zero_zero_nat )
% 9.49/5.24         => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.49/5.24            = ( divide_divide_nat @ M @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2279_both__member__options__from__complete__tree__to__child,axiom,
% 9.49/5.24      ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 9.49/5.24       => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.24         => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.49/5.24            | ( X = Mi )
% 9.49/5.24            | ( X = Ma ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2280_zero__le__divide__1__iff,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 9.49/5.24        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2281_zero__le__divide__1__iff,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 9.49/5.24        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2282_divide__le__0__1__iff,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 9.49/5.24        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2283_divide__le__0__1__iff,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 9.49/5.24        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2284_divide__less__0__1__iff,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 9.49/5.24        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2285_divide__less__0__1__iff,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 9.49/5.24        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2286_divide__less__eq__1__neg,axiom,
% 9.49/5.24      ! [A: real,B: real] :
% 9.49/5.24        ( ( ord_less_real @ A @ zero_zero_real )
% 9.49/5.24       => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 9.49/5.24          = ( ord_less_real @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2287_divide__less__eq__1__neg,axiom,
% 9.49/5.24      ! [A: rat,B: rat] :
% 9.49/5.24        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.49/5.24       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 9.49/5.24          = ( ord_less_rat @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2288_divide__less__eq__1__pos,axiom,
% 9.49/5.24      ! [A: real,B: real] :
% 9.49/5.24        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.24       => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 9.49/5.24          = ( ord_less_real @ B @ A ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2289_divide__less__eq__1__pos,axiom,
% 9.49/5.24      ! [A: rat,B: rat] :
% 9.49/5.24        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.24       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 9.49/5.24          = ( ord_less_rat @ B @ A ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2290_less__divide__eq__1__neg,axiom,
% 9.49/5.24      ! [A: real,B: real] :
% 9.49/5.24        ( ( ord_less_real @ A @ zero_zero_real )
% 9.49/5.24       => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 9.49/5.24          = ( ord_less_real @ B @ A ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2291_less__divide__eq__1__neg,axiom,
% 9.49/5.24      ! [A: rat,B: rat] :
% 9.49/5.24        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.49/5.24       => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 9.49/5.24          = ( ord_less_rat @ B @ A ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2292_less__divide__eq__1__pos,axiom,
% 9.49/5.24      ! [A: real,B: real] :
% 9.49/5.24        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.24       => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 9.49/5.24          = ( ord_less_real @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2293_less__divide__eq__1__pos,axiom,
% 9.49/5.24      ! [A: rat,B: rat] :
% 9.49/5.24        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.24       => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 9.49/5.24          = ( ord_less_rat @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2294_zero__less__divide__1__iff,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 9.49/5.24        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2295_zero__less__divide__1__iff,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 9.49/5.24        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2296_le__divide__eq__numeral1_I1_J,axiom,
% 9.49/5.24      ! [A: real,B: real,W2: num] :
% 9.49/5.24        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
% 9.49/5.24        = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2297_le__divide__eq__numeral1_I1_J,axiom,
% 9.49/5.24      ! [A: rat,B: rat,W2: num] :
% 9.49/5.24        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
% 9.49/5.24        = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2298_divide__le__eq__numeral1_I1_J,axiom,
% 9.49/5.24      ! [B: real,W2: num,A: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
% 9.49/5.24        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2299_divide__le__eq__numeral1_I1_J,axiom,
% 9.49/5.24      ! [B: rat,W2: num,A: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
% 9.49/5.24        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2300_eq__divide__eq__numeral1_I1_J,axiom,
% 9.49/5.24      ! [A: complex,B: complex,W2: num] :
% 9.49/5.24        ( ( A
% 9.49/5.24          = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) ) )
% 9.49/5.24        = ( ( ( ( numera6690914467698888265omplex @ W2 )
% 9.49/5.24             != zero_zero_complex )
% 9.49/5.24           => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) )
% 9.49/5.24              = B ) )
% 9.49/5.24          & ( ( ( numera6690914467698888265omplex @ W2 )
% 9.49/5.24              = zero_zero_complex )
% 9.49/5.24           => ( A = zero_zero_complex ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2301_eq__divide__eq__numeral1_I1_J,axiom,
% 9.49/5.24      ! [A: real,B: real,W2: num] :
% 9.49/5.24        ( ( A
% 9.49/5.24          = ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
% 9.49/5.24        = ( ( ( ( numeral_numeral_real @ W2 )
% 9.49/5.24             != zero_zero_real )
% 9.49/5.24           => ( ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) )
% 9.49/5.24              = B ) )
% 9.49/5.24          & ( ( ( numeral_numeral_real @ W2 )
% 9.49/5.24              = zero_zero_real )
% 9.49/5.24           => ( A = zero_zero_real ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2302_eq__divide__eq__numeral1_I1_J,axiom,
% 9.49/5.24      ! [A: rat,B: rat,W2: num] :
% 9.49/5.24        ( ( A
% 9.49/5.24          = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
% 9.49/5.24        = ( ( ( ( numeral_numeral_rat @ W2 )
% 9.49/5.24             != zero_zero_rat )
% 9.49/5.24           => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) )
% 9.49/5.24              = B ) )
% 9.49/5.24          & ( ( ( numeral_numeral_rat @ W2 )
% 9.49/5.24              = zero_zero_rat )
% 9.49/5.24           => ( A = zero_zero_rat ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2303_divide__eq__eq__numeral1_I1_J,axiom,
% 9.49/5.24      ! [B: complex,W2: num,A: complex] :
% 9.49/5.24        ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) )
% 9.49/5.24          = A )
% 9.49/5.24        = ( ( ( ( numera6690914467698888265omplex @ W2 )
% 9.49/5.24             != zero_zero_complex )
% 9.49/5.24           => ( B
% 9.49/5.24              = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) ) ) )
% 9.49/5.24          & ( ( ( numera6690914467698888265omplex @ W2 )
% 9.49/5.24              = zero_zero_complex )
% 9.49/5.24           => ( A = zero_zero_complex ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2304_divide__eq__eq__numeral1_I1_J,axiom,
% 9.49/5.24      ! [B: real,W2: num,A: real] :
% 9.49/5.24        ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) )
% 9.49/5.24          = A )
% 9.49/5.24        = ( ( ( ( numeral_numeral_real @ W2 )
% 9.49/5.24             != zero_zero_real )
% 9.49/5.24           => ( B
% 9.49/5.24              = ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) )
% 9.49/5.24          & ( ( ( numeral_numeral_real @ W2 )
% 9.49/5.24              = zero_zero_real )
% 9.49/5.24           => ( A = zero_zero_real ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2305_divide__eq__eq__numeral1_I1_J,axiom,
% 9.49/5.24      ! [B: rat,W2: num,A: rat] :
% 9.49/5.24        ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) )
% 9.49/5.24          = A )
% 9.49/5.24        = ( ( ( ( numeral_numeral_rat @ W2 )
% 9.49/5.24             != zero_zero_rat )
% 9.49/5.24           => ( B
% 9.49/5.24              = ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) )
% 9.49/5.24          & ( ( ( numeral_numeral_rat @ W2 )
% 9.49/5.24              = zero_zero_rat )
% 9.49/5.24           => ( A = zero_zero_rat ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2306_divide__less__eq__numeral1_I1_J,axiom,
% 9.49/5.24      ! [B: real,W2: num,A: real] :
% 9.49/5.24        ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
% 9.49/5.24        = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2307_divide__less__eq__numeral1_I1_J,axiom,
% 9.49/5.24      ! [B: rat,W2: num,A: rat] :
% 9.49/5.24        ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
% 9.49/5.24        = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2308_less__divide__eq__numeral1_I1_J,axiom,
% 9.49/5.24      ! [A: real,B: real,W2: num] :
% 9.49/5.24        ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
% 9.49/5.24        = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2309_less__divide__eq__numeral1_I1_J,axiom,
% 9.49/5.24      ! [A: rat,B: rat,W2: num] :
% 9.49/5.24        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
% 9.49/5.24        = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2310_nonzero__divide__mult__cancel__right,axiom,
% 9.49/5.24      ! [B: complex,A: complex] :
% 9.49/5.24        ( ( B != zero_zero_complex )
% 9.49/5.24       => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
% 9.49/5.24          = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2311_nonzero__divide__mult__cancel__right,axiom,
% 9.49/5.24      ! [B: real,A: real] :
% 9.49/5.24        ( ( B != zero_zero_real )
% 9.49/5.24       => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
% 9.49/5.24          = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2312_nonzero__divide__mult__cancel__right,axiom,
% 9.49/5.24      ! [B: rat,A: rat] :
% 9.49/5.24        ( ( B != zero_zero_rat )
% 9.49/5.24       => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
% 9.49/5.24          = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2313_nonzero__divide__mult__cancel__left,axiom,
% 9.49/5.24      ! [A: complex,B: complex] :
% 9.49/5.24        ( ( A != zero_zero_complex )
% 9.49/5.24       => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
% 9.49/5.24          = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2314_nonzero__divide__mult__cancel__left,axiom,
% 9.49/5.24      ! [A: real,B: real] :
% 9.49/5.24        ( ( A != zero_zero_real )
% 9.49/5.24       => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
% 9.49/5.24          = ( divide_divide_real @ one_one_real @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2315_nonzero__divide__mult__cancel__left,axiom,
% 9.49/5.24      ! [A: rat,B: rat] :
% 9.49/5.24        ( ( A != zero_zero_rat )
% 9.49/5.24       => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
% 9.49/5.24          = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2316_power__strict__increasing__iff,axiom,
% 9.49/5.24      ! [B: real,X: nat,Y: nat] :
% 9.49/5.24        ( ( ord_less_real @ one_one_real @ B )
% 9.49/5.24       => ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
% 9.49/5.24          = ( ord_less_nat @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2317_power__strict__increasing__iff,axiom,
% 9.49/5.24      ! [B: rat,X: nat,Y: nat] :
% 9.49/5.24        ( ( ord_less_rat @ one_one_rat @ B )
% 9.49/5.24       => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
% 9.49/5.24          = ( ord_less_nat @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2318_power__strict__increasing__iff,axiom,
% 9.49/5.24      ! [B: nat,X: nat,Y: nat] :
% 9.49/5.24        ( ( ord_less_nat @ one_one_nat @ B )
% 9.49/5.24       => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
% 9.49/5.24          = ( ord_less_nat @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2319_power__strict__increasing__iff,axiom,
% 9.49/5.24      ! [B: int,X: nat,Y: nat] :
% 9.49/5.24        ( ( ord_less_int @ one_one_int @ B )
% 9.49/5.24       => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
% 9.49/5.24          = ( ord_less_nat @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2320_both__member__options__from__chilf__to__complete__tree,axiom,
% 9.49/5.24      ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 9.49/5.24        ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.24       => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 9.49/5.24         => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.49/5.24           => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2321_one__le__mult__iff,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 9.49/5.24        = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.49/5.24          & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2322_nat__mult__le__cancel__disj,axiom,
% 9.49/5.24      ! [K: nat,M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.49/5.24        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.24         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2323_mult__le__cancel2,axiom,
% 9.49/5.24      ! [M: nat,K: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 9.49/5.24        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.24         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2324_divide__le__eq__1__neg,axiom,
% 9.49/5.24      ! [A: real,B: real] :
% 9.49/5.24        ( ( ord_less_real @ A @ zero_zero_real )
% 9.49/5.24       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 9.49/5.24          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2325_divide__le__eq__1__neg,axiom,
% 9.49/5.24      ! [A: rat,B: rat] :
% 9.49/5.24        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.49/5.24       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 9.49/5.24          = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2326_divide__le__eq__1__pos,axiom,
% 9.49/5.24      ! [A: real,B: real] :
% 9.49/5.24        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.24       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 9.49/5.24          = ( ord_less_eq_real @ B @ A ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2327_divide__le__eq__1__pos,axiom,
% 9.49/5.24      ! [A: rat,B: rat] :
% 9.49/5.24        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.24       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 9.49/5.24          = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2328_le__divide__eq__1__neg,axiom,
% 9.49/5.24      ! [A: real,B: real] :
% 9.49/5.24        ( ( ord_less_real @ A @ zero_zero_real )
% 9.49/5.24       => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 9.49/5.24          = ( ord_less_eq_real @ B @ A ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2329_le__divide__eq__1__neg,axiom,
% 9.49/5.24      ! [A: rat,B: rat] :
% 9.49/5.24        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.49/5.24       => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 9.49/5.24          = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2330_le__divide__eq__1__pos,axiom,
% 9.49/5.24      ! [A: real,B: real] :
% 9.49/5.24        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.24       => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 9.49/5.24          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2331_le__divide__eq__1__pos,axiom,
% 9.49/5.24      ! [A: rat,B: rat] :
% 9.49/5.24        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.24       => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 9.49/5.24          = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2332_power__strict__decreasing__iff,axiom,
% 9.49/5.24      ! [B: real,M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_real @ zero_zero_real @ B )
% 9.49/5.24       => ( ( ord_less_real @ B @ one_one_real )
% 9.49/5.24         => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 9.49/5.24            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2333_power__strict__decreasing__iff,axiom,
% 9.49/5.24      ! [B: rat,M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.49/5.24       => ( ( ord_less_rat @ B @ one_one_rat )
% 9.49/5.24         => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 9.49/5.24            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2334_power__strict__decreasing__iff,axiom,
% 9.49/5.24      ! [B: nat,M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.49/5.24       => ( ( ord_less_nat @ B @ one_one_nat )
% 9.49/5.24         => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 9.49/5.24            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2335_power__strict__decreasing__iff,axiom,
% 9.49/5.24      ! [B: int,M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.24       => ( ( ord_less_int @ B @ one_one_int )
% 9.49/5.24         => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 9.49/5.24            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2336_power__increasing__iff,axiom,
% 9.49/5.24      ! [B: real,X: nat,Y: nat] :
% 9.49/5.24        ( ( ord_less_real @ one_one_real @ B )
% 9.49/5.24       => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
% 9.49/5.24          = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2337_power__increasing__iff,axiom,
% 9.49/5.24      ! [B: rat,X: nat,Y: nat] :
% 9.49/5.24        ( ( ord_less_rat @ one_one_rat @ B )
% 9.49/5.24       => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
% 9.49/5.24          = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2338_power__increasing__iff,axiom,
% 9.49/5.24      ! [B: nat,X: nat,Y: nat] :
% 9.49/5.24        ( ( ord_less_nat @ one_one_nat @ B )
% 9.49/5.24       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
% 9.49/5.24          = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2339_power__increasing__iff,axiom,
% 9.49/5.24      ! [B: int,X: nat,Y: nat] :
% 9.49/5.24        ( ( ord_less_int @ one_one_int @ B )
% 9.49/5.24       => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
% 9.49/5.24          = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2340_numeral__le__one__iff,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
% 9.49/5.24        = ( ord_less_eq_num @ N @ one ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2341_numeral__le__one__iff,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 9.49/5.24        = ( ord_less_eq_num @ N @ one ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2342_numeral__le__one__iff,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 9.49/5.24        = ( ord_less_eq_num @ N @ one ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2343_numeral__le__one__iff,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 9.49/5.24        = ( ord_less_eq_num @ N @ one ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2344_numeral__le__one__iff,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 9.49/5.24        = ( ord_less_eq_num @ N @ one ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2345_one__less__numeral__iff,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
% 9.49/5.24        = ( ord_less_num @ one @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2346_one__less__numeral__iff,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 9.49/5.24        = ( ord_less_num @ one @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2347_one__less__numeral__iff,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 9.49/5.24        = ( ord_less_num @ one @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2348_one__less__numeral__iff,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 9.49/5.24        = ( ord_less_num @ one @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2349_one__less__numeral__iff,axiom,
% 9.49/5.24      ! [N: num] :
% 9.49/5.24        ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 9.49/5.24        = ( ord_less_num @ one @ N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2350_bits__1__div__2,axiom,
% 9.49/5.24      ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.24      = zero_zero_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2351_bits__1__div__2,axiom,
% 9.49/5.24      ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.24      = zero_zero_int ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2352_bits__1__div__2,axiom,
% 9.49/5.24      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.49/5.24      = zero_z3403309356797280102nteger ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2353_power__decreasing__iff,axiom,
% 9.49/5.24      ! [B: real,M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_real @ zero_zero_real @ B )
% 9.49/5.24       => ( ( ord_less_real @ B @ one_one_real )
% 9.49/5.24         => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 9.49/5.24            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2354_power__decreasing__iff,axiom,
% 9.49/5.24      ! [B: rat,M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.49/5.24       => ( ( ord_less_rat @ B @ one_one_rat )
% 9.49/5.24         => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 9.49/5.24            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2355_power__decreasing__iff,axiom,
% 9.49/5.24      ! [B: nat,M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.49/5.24       => ( ( ord_less_nat @ B @ one_one_nat )
% 9.49/5.24         => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 9.49/5.24            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2356_power__decreasing__iff,axiom,
% 9.49/5.24      ! [B: int,M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.24       => ( ( ord_less_int @ B @ one_one_int )
% 9.49/5.24         => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 9.49/5.24            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2357__092_060open_062y_A_061_A_Iif_Ax_A_061_Ama_Athen_Ahigh_Ax_An_A_K_A2_A_094_A_Ideg_Adiv_A2_J_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Ahigh_Ax_An_J_J_Aelse_Ama_J_092_060close_062,axiom,
% 9.49/5.24      ( ( ( xa = ma )
% 9.49/5.24       => ( ya
% 9.49/5.24          = ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ) ) )
% 9.49/5.24      & ( ( xa != ma )
% 9.49/5.24       => ( ya = ma ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2358_ssms,axiom,
% 9.49/5.24      vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ( if_nat @ ( xa = ma ) @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ) @ ma ) ) ) @ deg @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ summary ) @ ya ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2359_right__inverse__eq,axiom,
% 9.49/5.24      ! [B: complex,A: complex] :
% 9.49/5.24        ( ( B != zero_zero_complex )
% 9.49/5.24       => ( ( ( divide1717551699836669952omplex @ A @ B )
% 9.49/5.24            = one_one_complex )
% 9.49/5.24          = ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2360_right__inverse__eq,axiom,
% 9.49/5.24      ! [B: real,A: real] :
% 9.49/5.24        ( ( B != zero_zero_real )
% 9.49/5.24       => ( ( ( divide_divide_real @ A @ B )
% 9.49/5.24            = one_one_real )
% 9.49/5.24          = ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2361_right__inverse__eq,axiom,
% 9.49/5.24      ! [B: rat,A: rat] :
% 9.49/5.24        ( ( B != zero_zero_rat )
% 9.49/5.24       => ( ( ( divide_divide_rat @ A @ B )
% 9.49/5.24            = one_one_rat )
% 9.49/5.24          = ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2362_less__1__mult,axiom,
% 9.49/5.24      ! [M: real,N: real] :
% 9.49/5.24        ( ( ord_less_real @ one_one_real @ M )
% 9.49/5.24       => ( ( ord_less_real @ one_one_real @ N )
% 9.49/5.24         => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2363_less__1__mult,axiom,
% 9.49/5.24      ! [M: rat,N: rat] :
% 9.49/5.24        ( ( ord_less_rat @ one_one_rat @ M )
% 9.49/5.24       => ( ( ord_less_rat @ one_one_rat @ N )
% 9.49/5.24         => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2364_less__1__mult,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ one_one_nat @ M )
% 9.49/5.24       => ( ( ord_less_nat @ one_one_nat @ N )
% 9.49/5.24         => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2365_less__1__mult,axiom,
% 9.49/5.24      ! [M: int,N: int] :
% 9.49/5.24        ( ( ord_less_int @ one_one_int @ M )
% 9.49/5.24       => ( ( ord_less_int @ one_one_int @ N )
% 9.49/5.24         => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2366_mult__eq__self__implies__10,axiom,
% 9.49/5.24      ! [M: nat,N: nat] :
% 9.49/5.24        ( ( M
% 9.49/5.24          = ( times_times_nat @ M @ N ) )
% 9.49/5.24       => ( ( N = one_one_nat )
% 9.49/5.24          | ( M = zero_zero_nat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2367_nonzero__eq__divide__eq,axiom,
% 9.49/5.24      ! [C: complex,A: complex,B: complex] :
% 9.49/5.24        ( ( C != zero_zero_complex )
% 9.49/5.24       => ( ( A
% 9.49/5.24            = ( divide1717551699836669952omplex @ B @ C ) )
% 9.49/5.24          = ( ( times_times_complex @ A @ C )
% 9.49/5.24            = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2368_nonzero__eq__divide__eq,axiom,
% 9.49/5.24      ! [C: real,A: real,B: real] :
% 9.49/5.24        ( ( C != zero_zero_real )
% 9.49/5.24       => ( ( A
% 9.49/5.24            = ( divide_divide_real @ B @ C ) )
% 9.49/5.24          = ( ( times_times_real @ A @ C )
% 9.49/5.24            = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2369_nonzero__eq__divide__eq,axiom,
% 9.49/5.24      ! [C: rat,A: rat,B: rat] :
% 9.49/5.24        ( ( C != zero_zero_rat )
% 9.49/5.24       => ( ( A
% 9.49/5.24            = ( divide_divide_rat @ B @ C ) )
% 9.49/5.24          = ( ( times_times_rat @ A @ C )
% 9.49/5.24            = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2370_nonzero__divide__eq__eq,axiom,
% 9.49/5.24      ! [C: complex,B: complex,A: complex] :
% 9.49/5.24        ( ( C != zero_zero_complex )
% 9.49/5.24       => ( ( ( divide1717551699836669952omplex @ B @ C )
% 9.49/5.24            = A )
% 9.49/5.24          = ( B
% 9.49/5.24            = ( times_times_complex @ A @ C ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2371_nonzero__divide__eq__eq,axiom,
% 9.49/5.24      ! [C: real,B: real,A: real] :
% 9.49/5.24        ( ( C != zero_zero_real )
% 9.49/5.24       => ( ( ( divide_divide_real @ B @ C )
% 9.49/5.24            = A )
% 9.49/5.24          = ( B
% 9.49/5.24            = ( times_times_real @ A @ C ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2372_nonzero__divide__eq__eq,axiom,
% 9.49/5.24      ! [C: rat,B: rat,A: rat] :
% 9.49/5.24        ( ( C != zero_zero_rat )
% 9.49/5.24       => ( ( ( divide_divide_rat @ B @ C )
% 9.49/5.24            = A )
% 9.49/5.24          = ( B
% 9.49/5.24            = ( times_times_rat @ A @ C ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2373_eq__divide__imp,axiom,
% 9.49/5.24      ! [C: complex,A: complex,B: complex] :
% 9.49/5.24        ( ( C != zero_zero_complex )
% 9.49/5.24       => ( ( ( times_times_complex @ A @ C )
% 9.49/5.24            = B )
% 9.49/5.24         => ( A
% 9.49/5.24            = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2374_eq__divide__imp,axiom,
% 9.49/5.24      ! [C: real,A: real,B: real] :
% 9.49/5.24        ( ( C != zero_zero_real )
% 9.49/5.24       => ( ( ( times_times_real @ A @ C )
% 9.49/5.24            = B )
% 9.49/5.24         => ( A
% 9.49/5.24            = ( divide_divide_real @ B @ C ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2375_eq__divide__imp,axiom,
% 9.49/5.24      ! [C: rat,A: rat,B: rat] :
% 9.49/5.24        ( ( C != zero_zero_rat )
% 9.49/5.24       => ( ( ( times_times_rat @ A @ C )
% 9.49/5.24            = B )
% 9.49/5.24         => ( A
% 9.49/5.24            = ( divide_divide_rat @ B @ C ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2376_divide__eq__imp,axiom,
% 9.49/5.24      ! [C: complex,B: complex,A: complex] :
% 9.49/5.24        ( ( C != zero_zero_complex )
% 9.49/5.24       => ( ( B
% 9.49/5.24            = ( times_times_complex @ A @ C ) )
% 9.49/5.24         => ( ( divide1717551699836669952omplex @ B @ C )
% 9.49/5.24            = A ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2377_divide__eq__imp,axiom,
% 9.49/5.24      ! [C: real,B: real,A: real] :
% 9.49/5.24        ( ( C != zero_zero_real )
% 9.49/5.24       => ( ( B
% 9.49/5.24            = ( times_times_real @ A @ C ) )
% 9.49/5.24         => ( ( divide_divide_real @ B @ C )
% 9.49/5.24            = A ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2378_divide__eq__imp,axiom,
% 9.49/5.24      ! [C: rat,B: rat,A: rat] :
% 9.49/5.24        ( ( C != zero_zero_rat )
% 9.49/5.24       => ( ( B
% 9.49/5.24            = ( times_times_rat @ A @ C ) )
% 9.49/5.24         => ( ( divide_divide_rat @ B @ C )
% 9.49/5.24            = A ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2379_eq__divide__eq,axiom,
% 9.49/5.24      ! [A: complex,B: complex,C: complex] :
% 9.49/5.24        ( ( A
% 9.49/5.24          = ( divide1717551699836669952omplex @ B @ C ) )
% 9.49/5.24        = ( ( ( C != zero_zero_complex )
% 9.49/5.24           => ( ( times_times_complex @ A @ C )
% 9.49/5.24              = B ) )
% 9.49/5.24          & ( ( C = zero_zero_complex )
% 9.49/5.24           => ( A = zero_zero_complex ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2380_eq__divide__eq,axiom,
% 9.49/5.24      ! [A: real,B: real,C: real] :
% 9.49/5.24        ( ( A
% 9.49/5.24          = ( divide_divide_real @ B @ C ) )
% 9.49/5.24        = ( ( ( C != zero_zero_real )
% 9.49/5.24           => ( ( times_times_real @ A @ C )
% 9.49/5.24              = B ) )
% 9.49/5.24          & ( ( C = zero_zero_real )
% 9.49/5.24           => ( A = zero_zero_real ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2381_eq__divide__eq,axiom,
% 9.49/5.24      ! [A: rat,B: rat,C: rat] :
% 9.49/5.24        ( ( A
% 9.49/5.24          = ( divide_divide_rat @ B @ C ) )
% 9.49/5.24        = ( ( ( C != zero_zero_rat )
% 9.49/5.24           => ( ( times_times_rat @ A @ C )
% 9.49/5.24              = B ) )
% 9.49/5.24          & ( ( C = zero_zero_rat )
% 9.49/5.24           => ( A = zero_zero_rat ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2382_divide__eq__eq,axiom,
% 9.49/5.24      ! [B: complex,C: complex,A: complex] :
% 9.49/5.24        ( ( ( divide1717551699836669952omplex @ B @ C )
% 9.49/5.24          = A )
% 9.49/5.24        = ( ( ( C != zero_zero_complex )
% 9.49/5.24           => ( B
% 9.49/5.24              = ( times_times_complex @ A @ C ) ) )
% 9.49/5.24          & ( ( C = zero_zero_complex )
% 9.49/5.24           => ( A = zero_zero_complex ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2383_divide__eq__eq,axiom,
% 9.49/5.24      ! [B: real,C: real,A: real] :
% 9.49/5.24        ( ( ( divide_divide_real @ B @ C )
% 9.49/5.24          = A )
% 9.49/5.24        = ( ( ( C != zero_zero_real )
% 9.49/5.24           => ( B
% 9.49/5.24              = ( times_times_real @ A @ C ) ) )
% 9.49/5.24          & ( ( C = zero_zero_real )
% 9.49/5.24           => ( A = zero_zero_real ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2384_divide__eq__eq,axiom,
% 9.49/5.24      ! [B: rat,C: rat,A: rat] :
% 9.49/5.24        ( ( ( divide_divide_rat @ B @ C )
% 9.49/5.24          = A )
% 9.49/5.24        = ( ( ( C != zero_zero_rat )
% 9.49/5.24           => ( B
% 9.49/5.24              = ( times_times_rat @ A @ C ) ) )
% 9.49/5.24          & ( ( C = zero_zero_rat )
% 9.49/5.24           => ( A = zero_zero_rat ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2385_frac__eq__eq,axiom,
% 9.49/5.24      ! [Y: complex,Z2: complex,X: complex,W2: complex] :
% 9.49/5.24        ( ( Y != zero_zero_complex )
% 9.49/5.24       => ( ( Z2 != zero_zero_complex )
% 9.49/5.24         => ( ( ( divide1717551699836669952omplex @ X @ Y )
% 9.49/5.24              = ( divide1717551699836669952omplex @ W2 @ Z2 ) )
% 9.49/5.24            = ( ( times_times_complex @ X @ Z2 )
% 9.49/5.24              = ( times_times_complex @ W2 @ Y ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2386_frac__eq__eq,axiom,
% 9.49/5.24      ! [Y: real,Z2: real,X: real,W2: real] :
% 9.49/5.24        ( ( Y != zero_zero_real )
% 9.49/5.24       => ( ( Z2 != zero_zero_real )
% 9.49/5.24         => ( ( ( divide_divide_real @ X @ Y )
% 9.49/5.24              = ( divide_divide_real @ W2 @ Z2 ) )
% 9.49/5.24            = ( ( times_times_real @ X @ Z2 )
% 9.49/5.24              = ( times_times_real @ W2 @ Y ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2387_frac__eq__eq,axiom,
% 9.49/5.24      ! [Y: rat,Z2: rat,X: rat,W2: rat] :
% 9.49/5.24        ( ( Y != zero_zero_rat )
% 9.49/5.24       => ( ( Z2 != zero_zero_rat )
% 9.49/5.24         => ( ( ( divide_divide_rat @ X @ Y )
% 9.49/5.24              = ( divide_divide_rat @ W2 @ Z2 ) )
% 9.49/5.24            = ( ( times_times_rat @ X @ Z2 )
% 9.49/5.24              = ( times_times_rat @ W2 @ Y ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2388_nat__mult__1__right,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( times_times_nat @ N @ one_one_nat )
% 9.49/5.24        = N ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2389_one__reorient,axiom,
% 9.49/5.24      ! [X: complex] :
% 9.49/5.24        ( ( one_one_complex = X )
% 9.49/5.24        = ( X = one_one_complex ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2390_one__reorient,axiom,
% 9.49/5.24      ! [X: real] :
% 9.49/5.24        ( ( one_one_real = X )
% 9.49/5.24        = ( X = one_one_real ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2391_one__reorient,axiom,
% 9.49/5.24      ! [X: rat] :
% 9.49/5.24        ( ( one_one_rat = X )
% 9.49/5.24        = ( X = one_one_rat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2392_one__reorient,axiom,
% 9.49/5.24      ! [X: nat] :
% 9.49/5.24        ( ( one_one_nat = X )
% 9.49/5.24        = ( X = one_one_nat ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2393_one__reorient,axiom,
% 9.49/5.24      ! [X: int] :
% 9.49/5.24        ( ( one_one_int = X )
% 9.49/5.24        = ( X = one_one_int ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2394_nat__mult__1,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( times_times_nat @ one_one_nat @ N )
% 9.49/5.24        = N ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2395_mult_Oleft__commute,axiom,
% 9.49/5.24      ! [B: complex,A: complex,C: complex] :
% 9.49/5.24        ( ( times_times_complex @ B @ ( times_times_complex @ A @ C ) )
% 9.49/5.24        = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2396_mult_Oleft__commute,axiom,
% 9.49/5.24      ! [B: real,A: real,C: real] :
% 9.49/5.24        ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
% 9.49/5.24        = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2397_mult_Oleft__commute,axiom,
% 9.49/5.24      ! [B: rat,A: rat,C: rat] :
% 9.49/5.24        ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
% 9.49/5.24        = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2398_mult_Oleft__commute,axiom,
% 9.49/5.24      ! [B: nat,A: nat,C: nat] :
% 9.49/5.24        ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 9.49/5.24        = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2399_mult_Oleft__commute,axiom,
% 9.49/5.24      ! [B: int,A: int,C: int] :
% 9.49/5.24        ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
% 9.49/5.24        = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2400_mult_Ocomm__neutral,axiom,
% 9.49/5.24      ! [A: complex] :
% 9.49/5.24        ( ( times_times_complex @ A @ one_one_complex )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2401_mult_Ocomm__neutral,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( times_times_real @ A @ one_one_real )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2402_mult_Ocomm__neutral,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( times_times_rat @ A @ one_one_rat )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2403_mult_Ocomm__neutral,axiom,
% 9.49/5.24      ! [A: nat] :
% 9.49/5.24        ( ( times_times_nat @ A @ one_one_nat )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2404_mult_Ocomm__neutral,axiom,
% 9.49/5.24      ! [A: int] :
% 9.49/5.24        ( ( times_times_int @ A @ one_one_int )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2405_mult_Ocommute,axiom,
% 9.49/5.24      ( times_times_complex
% 9.49/5.24      = ( ^ [A4: complex,B4: complex] : ( times_times_complex @ B4 @ A4 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2406_mult_Ocommute,axiom,
% 9.49/5.24      ( times_times_real
% 9.49/5.24      = ( ^ [A4: real,B4: real] : ( times_times_real @ B4 @ A4 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2407_mult_Ocommute,axiom,
% 9.49/5.24      ( times_times_rat
% 9.49/5.24      = ( ^ [A4: rat,B4: rat] : ( times_times_rat @ B4 @ A4 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2408_mult_Ocommute,axiom,
% 9.49/5.24      ( times_times_nat
% 9.49/5.24      = ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2409_mult_Ocommute,axiom,
% 9.49/5.24      ( times_times_int
% 9.49/5.24      = ( ^ [A4: int,B4: int] : ( times_times_int @ B4 @ A4 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2410_mult_Oassoc,axiom,
% 9.49/5.24      ! [A: complex,B: complex,C: complex] :
% 9.49/5.24        ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
% 9.49/5.24        = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2411_mult_Oassoc,axiom,
% 9.49/5.24      ! [A: real,B: real,C: real] :
% 9.49/5.24        ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 9.49/5.24        = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2412_mult_Oassoc,axiom,
% 9.49/5.24      ! [A: rat,B: rat,C: rat] :
% 9.49/5.24        ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 9.49/5.24        = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2413_mult_Oassoc,axiom,
% 9.49/5.24      ! [A: nat,B: nat,C: nat] :
% 9.49/5.24        ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 9.49/5.24        = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2414_mult_Oassoc,axiom,
% 9.49/5.24      ! [A: int,B: int,C: int] :
% 9.49/5.24        ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 9.49/5.24        = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2415_comm__monoid__mult__class_Omult__1,axiom,
% 9.49/5.24      ! [A: complex] :
% 9.49/5.24        ( ( times_times_complex @ one_one_complex @ A )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2416_comm__monoid__mult__class_Omult__1,axiom,
% 9.49/5.24      ! [A: real] :
% 9.49/5.24        ( ( times_times_real @ one_one_real @ A )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2417_comm__monoid__mult__class_Omult__1,axiom,
% 9.49/5.24      ! [A: rat] :
% 9.49/5.24        ( ( times_times_rat @ one_one_rat @ A )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2418_comm__monoid__mult__class_Omult__1,axiom,
% 9.49/5.24      ! [A: nat] :
% 9.49/5.24        ( ( times_times_nat @ one_one_nat @ A )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2419_comm__monoid__mult__class_Omult__1,axiom,
% 9.49/5.24      ! [A: int] :
% 9.49/5.24        ( ( times_times_int @ one_one_int @ A )
% 9.49/5.24        = A ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2420_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.49/5.24      ! [A: complex,B: complex,C: complex] :
% 9.49/5.24        ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
% 9.49/5.24        = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2421_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.49/5.24      ! [A: real,B: real,C: real] :
% 9.49/5.24        ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 9.49/5.24        = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2422_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.49/5.24      ! [A: rat,B: rat,C: rat] :
% 9.49/5.24        ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 9.49/5.24        = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2423_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.49/5.24      ! [A: nat,B: nat,C: nat] :
% 9.49/5.24        ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 9.49/5.24        = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2424_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.49/5.24      ! [A: int,B: int,C: int] :
% 9.49/5.24        ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 9.49/5.24        = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2425_nat__mult__eq__cancel__disj,axiom,
% 9.49/5.24      ! [K: nat,M: nat,N: nat] :
% 9.49/5.24        ( ( ( times_times_nat @ K @ M )
% 9.49/5.24          = ( times_times_nat @ K @ N ) )
% 9.49/5.24        = ( ( K = zero_zero_nat )
% 9.49/5.24          | ( M = N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2426_nat__mult__div__cancel1,axiom,
% 9.49/5.24      ! [K: nat,M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.24       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.49/5.24          = ( divide_divide_nat @ M @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2427_divide__less__eq,axiom,
% 9.49/5.24      ! [B: real,C: real,A: real] :
% 9.49/5.24        ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 9.49/5.24        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 9.49/5.24          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24               => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 9.49/5.24              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24               => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2428_divide__less__eq,axiom,
% 9.49/5.24      ! [B: rat,C: rat,A: rat] :
% 9.49/5.24        ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 9.49/5.24        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 9.49/5.24          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24               => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 9.49/5.24              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24               => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2429_less__divide__eq,axiom,
% 9.49/5.24      ! [A: real,B: real,C: real] :
% 9.49/5.24        ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.49/5.24        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 9.49/5.24          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24               => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 9.49/5.24              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24               => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2430_less__divide__eq,axiom,
% 9.49/5.24      ! [A: rat,B: rat,C: rat] :
% 9.49/5.24        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.49/5.24        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 9.49/5.24          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24               => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 9.49/5.24              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24               => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2431_neg__divide__less__eq,axiom,
% 9.49/5.24      ! [C: real,B: real,A: real] :
% 9.49/5.24        ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24       => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 9.49/5.24          = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2432_neg__divide__less__eq,axiom,
% 9.49/5.24      ! [C: rat,B: rat,A: rat] :
% 9.49/5.24        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 9.49/5.24          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2433_neg__less__divide__eq,axiom,
% 9.49/5.24      ! [C: real,A: real,B: real] :
% 9.49/5.24        ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24       => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.49/5.24          = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2434_neg__less__divide__eq,axiom,
% 9.49/5.24      ! [C: rat,A: rat,B: rat] :
% 9.49/5.24        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24       => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.49/5.24          = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2435_pos__divide__less__eq,axiom,
% 9.49/5.24      ! [C: real,B: real,A: real] :
% 9.49/5.24        ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24       => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 9.49/5.24          = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2436_pos__divide__less__eq,axiom,
% 9.49/5.24      ! [C: rat,B: rat,A: rat] :
% 9.49/5.24        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 9.49/5.24          = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2437_pos__less__divide__eq,axiom,
% 9.49/5.24      ! [C: real,A: real,B: real] :
% 9.49/5.24        ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24       => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.49/5.24          = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2438_pos__less__divide__eq,axiom,
% 9.49/5.24      ! [C: rat,A: rat,B: rat] :
% 9.49/5.24        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24       => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.49/5.24          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2439_mult__imp__div__pos__less,axiom,
% 9.49/5.24      ! [Y: real,X: real,Z2: real] :
% 9.49/5.24        ( ( ord_less_real @ zero_zero_real @ Y )
% 9.49/5.24       => ( ( ord_less_real @ X @ ( times_times_real @ Z2 @ Y ) )
% 9.49/5.24         => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z2 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2440_mult__imp__div__pos__less,axiom,
% 9.49/5.24      ! [Y: rat,X: rat,Z2: rat] :
% 9.49/5.24        ( ( ord_less_rat @ zero_zero_rat @ Y )
% 9.49/5.24       => ( ( ord_less_rat @ X @ ( times_times_rat @ Z2 @ Y ) )
% 9.49/5.24         => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2441_mult__imp__less__div__pos,axiom,
% 9.49/5.24      ! [Y: real,Z2: real,X: real] :
% 9.49/5.24        ( ( ord_less_real @ zero_zero_real @ Y )
% 9.49/5.24       => ( ( ord_less_real @ ( times_times_real @ Z2 @ Y ) @ X )
% 9.49/5.24         => ( ord_less_real @ Z2 @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2442_mult__imp__less__div__pos,axiom,
% 9.49/5.24      ! [Y: rat,Z2: rat,X: rat] :
% 9.49/5.24        ( ( ord_less_rat @ zero_zero_rat @ Y )
% 9.49/5.24       => ( ( ord_less_rat @ ( times_times_rat @ Z2 @ Y ) @ X )
% 9.49/5.24         => ( ord_less_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2443_divide__strict__left__mono,axiom,
% 9.49/5.24      ! [B: real,A: real,C: real] :
% 9.49/5.24        ( ( ord_less_real @ B @ A )
% 9.49/5.24       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.49/5.24           => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2444_divide__strict__left__mono,axiom,
% 9.49/5.24      ! [B: rat,A: rat,C: rat] :
% 9.49/5.24        ( ( ord_less_rat @ B @ A )
% 9.49/5.24       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.49/5.24           => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2445_divide__strict__left__mono__neg,axiom,
% 9.49/5.24      ! [A: real,B: real,C: real] :
% 9.49/5.24        ( ( ord_less_real @ A @ B )
% 9.49/5.24       => ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.49/5.24           => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2446_divide__strict__left__mono__neg,axiom,
% 9.49/5.24      ! [A: rat,B: rat,C: rat] :
% 9.49/5.24        ( ( ord_less_rat @ A @ B )
% 9.49/5.24       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.49/5.24           => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2447_eq__divide__eq__numeral_I1_J,axiom,
% 9.49/5.24      ! [W2: num,B: complex,C: complex] :
% 9.49/5.24        ( ( ( numera6690914467698888265omplex @ W2 )
% 9.49/5.24          = ( divide1717551699836669952omplex @ B @ C ) )
% 9.49/5.24        = ( ( ( C != zero_zero_complex )
% 9.49/5.24           => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C )
% 9.49/5.24              = B ) )
% 9.49/5.24          & ( ( C = zero_zero_complex )
% 9.49/5.24           => ( ( numera6690914467698888265omplex @ W2 )
% 9.49/5.24              = zero_zero_complex ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2448_eq__divide__eq__numeral_I1_J,axiom,
% 9.49/5.24      ! [W2: num,B: real,C: real] :
% 9.49/5.24        ( ( ( numeral_numeral_real @ W2 )
% 9.49/5.24          = ( divide_divide_real @ B @ C ) )
% 9.49/5.24        = ( ( ( C != zero_zero_real )
% 9.49/5.24           => ( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C )
% 9.49/5.24              = B ) )
% 9.49/5.24          & ( ( C = zero_zero_real )
% 9.49/5.24           => ( ( numeral_numeral_real @ W2 )
% 9.49/5.24              = zero_zero_real ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2449_eq__divide__eq__numeral_I1_J,axiom,
% 9.49/5.24      ! [W2: num,B: rat,C: rat] :
% 9.49/5.24        ( ( ( numeral_numeral_rat @ W2 )
% 9.49/5.24          = ( divide_divide_rat @ B @ C ) )
% 9.49/5.24        = ( ( ( C != zero_zero_rat )
% 9.49/5.24           => ( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C )
% 9.49/5.24              = B ) )
% 9.49/5.24          & ( ( C = zero_zero_rat )
% 9.49/5.24           => ( ( numeral_numeral_rat @ W2 )
% 9.49/5.24              = zero_zero_rat ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2450_divide__eq__eq__numeral_I1_J,axiom,
% 9.49/5.24      ! [B: complex,C: complex,W2: num] :
% 9.49/5.24        ( ( ( divide1717551699836669952omplex @ B @ C )
% 9.49/5.24          = ( numera6690914467698888265omplex @ W2 ) )
% 9.49/5.24        = ( ( ( C != zero_zero_complex )
% 9.49/5.24           => ( B
% 9.49/5.24              = ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C ) ) )
% 9.49/5.24          & ( ( C = zero_zero_complex )
% 9.49/5.24           => ( ( numera6690914467698888265omplex @ W2 )
% 9.49/5.24              = zero_zero_complex ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2451_divide__eq__eq__numeral_I1_J,axiom,
% 9.49/5.24      ! [B: real,C: real,W2: num] :
% 9.49/5.24        ( ( ( divide_divide_real @ B @ C )
% 9.49/5.24          = ( numeral_numeral_real @ W2 ) )
% 9.49/5.24        = ( ( ( C != zero_zero_real )
% 9.49/5.24           => ( B
% 9.49/5.24              = ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 9.49/5.24          & ( ( C = zero_zero_real )
% 9.49/5.24           => ( ( numeral_numeral_real @ W2 )
% 9.49/5.24              = zero_zero_real ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2452_divide__eq__eq__numeral_I1_J,axiom,
% 9.49/5.24      ! [B: rat,C: rat,W2: num] :
% 9.49/5.24        ( ( ( divide_divide_rat @ B @ C )
% 9.49/5.24          = ( numeral_numeral_rat @ W2 ) )
% 9.49/5.24        = ( ( ( C != zero_zero_rat )
% 9.49/5.24           => ( B
% 9.49/5.24              = ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 9.49/5.24          & ( ( C = zero_zero_rat )
% 9.49/5.24           => ( ( numeral_numeral_rat @ W2 )
% 9.49/5.24              = zero_zero_rat ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2453_add__divide__eq__if__simps_I2_J,axiom,
% 9.49/5.24      ! [Z2: complex,A: complex,B: complex] :
% 9.49/5.24        ( ( ( Z2 = zero_zero_complex )
% 9.49/5.24         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 9.49/5.24            = B ) )
% 9.49/5.24        & ( ( Z2 != zero_zero_complex )
% 9.49/5.24         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 9.49/5.24            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2454_add__divide__eq__if__simps_I2_J,axiom,
% 9.49/5.24      ! [Z2: real,A: real,B: real] :
% 9.49/5.24        ( ( ( Z2 = zero_zero_real )
% 9.49/5.24         => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 9.49/5.24            = B ) )
% 9.49/5.24        & ( ( Z2 != zero_zero_real )
% 9.49/5.24         => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 9.49/5.24            = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2455_add__divide__eq__if__simps_I2_J,axiom,
% 9.49/5.24      ! [Z2: rat,A: rat,B: rat] :
% 9.49/5.24        ( ( ( Z2 = zero_zero_rat )
% 9.49/5.24         => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 9.49/5.24            = B ) )
% 9.49/5.24        & ( ( Z2 != zero_zero_rat )
% 9.49/5.24         => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 9.49/5.24            = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2456_add__divide__eq__if__simps_I1_J,axiom,
% 9.49/5.24      ! [Z2: complex,A: complex,B: complex] :
% 9.49/5.24        ( ( ( Z2 = zero_zero_complex )
% 9.49/5.24         => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 9.49/5.24            = A ) )
% 9.49/5.24        & ( ( Z2 != zero_zero_complex )
% 9.49/5.24         => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 9.49/5.24            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2457_add__divide__eq__if__simps_I1_J,axiom,
% 9.49/5.24      ! [Z2: real,A: real,B: real] :
% 9.49/5.24        ( ( ( Z2 = zero_zero_real )
% 9.49/5.24         => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 9.49/5.24            = A ) )
% 9.49/5.24        & ( ( Z2 != zero_zero_real )
% 9.49/5.24         => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 9.49/5.24            = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2458_add__divide__eq__if__simps_I1_J,axiom,
% 9.49/5.24      ! [Z2: rat,A: rat,B: rat] :
% 9.49/5.24        ( ( ( Z2 = zero_zero_rat )
% 9.49/5.24         => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 9.49/5.24            = A ) )
% 9.49/5.24        & ( ( Z2 != zero_zero_rat )
% 9.49/5.24         => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 9.49/5.24            = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2459_add__frac__eq,axiom,
% 9.49/5.24      ! [Y: complex,Z2: complex,X: complex,W2: complex] :
% 9.49/5.24        ( ( Y != zero_zero_complex )
% 9.49/5.24       => ( ( Z2 != zero_zero_complex )
% 9.49/5.24         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W2 @ Z2 ) )
% 9.49/5.24            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ W2 @ Y ) ) @ ( times_times_complex @ Y @ Z2 ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2460_add__frac__eq,axiom,
% 9.49/5.24      ! [Y: real,Z2: real,X: real,W2: real] :
% 9.49/5.24        ( ( Y != zero_zero_real )
% 9.49/5.24       => ( ( Z2 != zero_zero_real )
% 9.49/5.24         => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z2 ) )
% 9.49/5.24            = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2461_add__frac__eq,axiom,
% 9.49/5.24      ! [Y: rat,Z2: rat,X: rat,W2: rat] :
% 9.49/5.24        ( ( Y != zero_zero_rat )
% 9.49/5.24       => ( ( Z2 != zero_zero_rat )
% 9.49/5.24         => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z2 ) )
% 9.49/5.24            = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2462_add__frac__num,axiom,
% 9.49/5.24      ! [Y: complex,X: complex,Z2: complex] :
% 9.49/5.24        ( ( Y != zero_zero_complex )
% 9.49/5.24       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z2 )
% 9.49/5.24          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z2 @ Y ) ) @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2463_add__frac__num,axiom,
% 9.49/5.24      ! [Y: real,X: real,Z2: real] :
% 9.49/5.24        ( ( Y != zero_zero_real )
% 9.49/5.24       => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z2 )
% 9.49/5.24          = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z2 @ Y ) ) @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2464_add__frac__num,axiom,
% 9.49/5.24      ! [Y: rat,X: rat,Z2: rat] :
% 9.49/5.24        ( ( Y != zero_zero_rat )
% 9.49/5.24       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 )
% 9.49/5.24          = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z2 @ Y ) ) @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2465_add__num__frac,axiom,
% 9.49/5.24      ! [Y: complex,Z2: complex,X: complex] :
% 9.49/5.24        ( ( Y != zero_zero_complex )
% 9.49/5.24       => ( ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ X @ Y ) )
% 9.49/5.24          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z2 @ Y ) ) @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2466_add__num__frac,axiom,
% 9.49/5.24      ! [Y: real,Z2: real,X: real] :
% 9.49/5.24        ( ( Y != zero_zero_real )
% 9.49/5.24       => ( ( plus_plus_real @ Z2 @ ( divide_divide_real @ X @ Y ) )
% 9.49/5.24          = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z2 @ Y ) ) @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2467_add__num__frac,axiom,
% 9.49/5.24      ! [Y: rat,Z2: rat,X: rat] :
% 9.49/5.24        ( ( Y != zero_zero_rat )
% 9.49/5.24       => ( ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) )
% 9.49/5.24          = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z2 @ Y ) ) @ Y ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2468_add__divide__eq__iff,axiom,
% 9.49/5.24      ! [Z2: complex,X: complex,Y: complex] :
% 9.49/5.24        ( ( Z2 != zero_zero_complex )
% 9.49/5.24       => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z2 ) )
% 9.49/5.24          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2469_add__divide__eq__iff,axiom,
% 9.49/5.24      ! [Z2: real,X: real,Y: real] :
% 9.49/5.24        ( ( Z2 != zero_zero_real )
% 9.49/5.24       => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z2 ) )
% 9.49/5.24          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2470_add__divide__eq__iff,axiom,
% 9.49/5.24      ! [Z2: rat,X: rat,Y: rat] :
% 9.49/5.24        ( ( Z2 != zero_zero_rat )
% 9.49/5.24       => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z2 ) )
% 9.49/5.24          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2471_divide__add__eq__iff,axiom,
% 9.49/5.24      ! [Z2: complex,X: complex,Y: complex] :
% 9.49/5.24        ( ( Z2 != zero_zero_complex )
% 9.49/5.24       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z2 ) @ Y )
% 9.49/5.24          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2472_divide__add__eq__iff,axiom,
% 9.49/5.24      ! [Z2: real,X: real,Y: real] :
% 9.49/5.24        ( ( Z2 != zero_zero_real )
% 9.49/5.24       => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z2 ) @ Y )
% 9.49/5.24          = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2473_divide__add__eq__iff,axiom,
% 9.49/5.24      ! [Z2: rat,X: rat,Y: rat] :
% 9.49/5.24        ( ( Z2 != zero_zero_rat )
% 9.49/5.24       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z2 ) @ Y )
% 9.49/5.24          = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2474_divide__less__eq__1,axiom,
% 9.49/5.24      ! [B: real,A: real] :
% 9.49/5.24        ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 9.49/5.24        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.24            & ( ord_less_real @ B @ A ) )
% 9.49/5.24          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.49/5.24            & ( ord_less_real @ A @ B ) )
% 9.49/5.24          | ( A = zero_zero_real ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2475_divide__less__eq__1,axiom,
% 9.49/5.24      ! [B: rat,A: rat] :
% 9.49/5.24        ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 9.49/5.24        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.24            & ( ord_less_rat @ B @ A ) )
% 9.49/5.24          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.49/5.24            & ( ord_less_rat @ A @ B ) )
% 9.49/5.24          | ( A = zero_zero_rat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2476_less__divide__eq__1,axiom,
% 9.49/5.24      ! [B: real,A: real] :
% 9.49/5.24        ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 9.49/5.24        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.24            & ( ord_less_real @ A @ B ) )
% 9.49/5.24          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.49/5.24            & ( ord_less_real @ B @ A ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2477_less__divide__eq__1,axiom,
% 9.49/5.24      ! [B: rat,A: rat] :
% 9.49/5.24        ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 9.49/5.24        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.24            & ( ord_less_rat @ A @ B ) )
% 9.49/5.24          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.49/5.24            & ( ord_less_rat @ B @ A ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2478_less__half__sum,axiom,
% 9.49/5.24      ! [A: real,B: real] :
% 9.49/5.24        ( ( ord_less_real @ A @ B )
% 9.49/5.24       => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2479_less__half__sum,axiom,
% 9.49/5.24      ! [A: rat,B: rat] :
% 9.49/5.24        ( ( ord_less_rat @ A @ B )
% 9.49/5.24       => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2480_gt__half__sum,axiom,
% 9.49/5.24      ! [A: real,B: real] :
% 9.49/5.24        ( ( ord_less_real @ A @ B )
% 9.49/5.24       => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2481_gt__half__sum,axiom,
% 9.49/5.24      ! [A: rat,B: rat] :
% 9.49/5.24        ( ( ord_less_rat @ A @ B )
% 9.49/5.24       => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2482_mult__left__le,axiom,
% 9.49/5.24      ! [C: real,A: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ C @ one_one_real )
% 9.49/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.24         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2483_mult__left__le,axiom,
% 9.49/5.24      ! [C: rat,A: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ C @ one_one_rat )
% 9.49/5.24       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.24         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2484_mult__left__le,axiom,
% 9.49/5.24      ! [C: nat,A: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ C @ one_one_nat )
% 9.49/5.24       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.24         => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2485_mult__left__le,axiom,
% 9.49/5.24      ! [C: int,A: int] :
% 9.49/5.24        ( ( ord_less_eq_int @ C @ one_one_int )
% 9.49/5.24       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.24         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2486_mult__le__one,axiom,
% 9.49/5.24      ! [A: real,B: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ A @ one_one_real )
% 9.49/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.49/5.24         => ( ( ord_less_eq_real @ B @ one_one_real )
% 9.49/5.24           => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2487_mult__le__one,axiom,
% 9.49/5.24      ! [A: rat,B: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ A @ one_one_rat )
% 9.49/5.24       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.49/5.24         => ( ( ord_less_eq_rat @ B @ one_one_rat )
% 9.49/5.24           => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2488_mult__le__one,axiom,
% 9.49/5.24      ! [A: nat,B: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ A @ one_one_nat )
% 9.49/5.24       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.49/5.24         => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 9.49/5.24           => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2489_mult__le__one,axiom,
% 9.49/5.24      ! [A: int,B: int] :
% 9.49/5.24        ( ( ord_less_eq_int @ A @ one_one_int )
% 9.49/5.24       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.49/5.24         => ( ( ord_less_eq_int @ B @ one_one_int )
% 9.49/5.24           => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2490_mult__right__le__one__le,axiom,
% 9.49/5.24      ! [X: real,Y: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.49/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.49/5.24         => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.49/5.24           => ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2491_mult__right__le__one__le,axiom,
% 9.49/5.24      ! [X: rat,Y: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.49/5.24       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.49/5.24         => ( ( ord_less_eq_rat @ Y @ one_one_rat )
% 9.49/5.24           => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2492_mult__right__le__one__le,axiom,
% 9.49/5.24      ! [X: int,Y: int] :
% 9.49/5.24        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.49/5.24       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.49/5.24         => ( ( ord_less_eq_int @ Y @ one_one_int )
% 9.49/5.24           => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2493_mult__left__le__one__le,axiom,
% 9.49/5.24      ! [X: real,Y: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.49/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.49/5.24         => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.49/5.24           => ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2494_mult__left__le__one__le,axiom,
% 9.49/5.24      ! [X: rat,Y: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.49/5.24       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.49/5.24         => ( ( ord_less_eq_rat @ Y @ one_one_rat )
% 9.49/5.24           => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2495_mult__left__le__one__le,axiom,
% 9.49/5.24      ! [X: int,Y: int] :
% 9.49/5.24        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.49/5.24       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.49/5.24         => ( ( ord_less_eq_int @ Y @ one_one_int )
% 9.49/5.24           => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2496_power__less__power__Suc,axiom,
% 9.49/5.24      ! [A: real,N: nat] :
% 9.49/5.24        ( ( ord_less_real @ one_one_real @ A )
% 9.49/5.24       => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2497_power__less__power__Suc,axiom,
% 9.49/5.24      ! [A: rat,N: nat] :
% 9.49/5.24        ( ( ord_less_rat @ one_one_rat @ A )
% 9.49/5.24       => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2498_power__less__power__Suc,axiom,
% 9.49/5.24      ! [A: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ one_one_nat @ A )
% 9.49/5.24       => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2499_power__less__power__Suc,axiom,
% 9.49/5.24      ! [A: int,N: nat] :
% 9.49/5.24        ( ( ord_less_int @ one_one_int @ A )
% 9.49/5.24       => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2500_power__gt1__lemma,axiom,
% 9.49/5.24      ! [A: real,N: nat] :
% 9.49/5.24        ( ( ord_less_real @ one_one_real @ A )
% 9.49/5.24       => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2501_power__gt1__lemma,axiom,
% 9.49/5.24      ! [A: rat,N: nat] :
% 9.49/5.24        ( ( ord_less_rat @ one_one_rat @ A )
% 9.49/5.24       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2502_power__gt1__lemma,axiom,
% 9.49/5.24      ! [A: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ one_one_nat @ A )
% 9.49/5.24       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2503_power__gt1__lemma,axiom,
% 9.49/5.24      ! [A: int,N: nat] :
% 9.49/5.24        ( ( ord_less_int @ one_one_int @ A )
% 9.49/5.24       => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2504_mult__not__zero,axiom,
% 9.49/5.24      ! [A: complex,B: complex] :
% 9.49/5.24        ( ( ( times_times_complex @ A @ B )
% 9.49/5.24         != zero_zero_complex )
% 9.49/5.24       => ( ( A != zero_zero_complex )
% 9.49/5.24          & ( B != zero_zero_complex ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2505_mult__not__zero,axiom,
% 9.49/5.24      ! [A: real,B: real] :
% 9.49/5.24        ( ( ( times_times_real @ A @ B )
% 9.49/5.24         != zero_zero_real )
% 9.49/5.24       => ( ( A != zero_zero_real )
% 9.49/5.24          & ( B != zero_zero_real ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2506_mult__not__zero,axiom,
% 9.49/5.24      ! [A: rat,B: rat] :
% 9.49/5.24        ( ( ( times_times_rat @ A @ B )
% 9.49/5.24         != zero_zero_rat )
% 9.49/5.24       => ( ( A != zero_zero_rat )
% 9.49/5.24          & ( B != zero_zero_rat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2507_mult__not__zero,axiom,
% 9.49/5.24      ! [A: nat,B: nat] :
% 9.49/5.24        ( ( ( times_times_nat @ A @ B )
% 9.49/5.24         != zero_zero_nat )
% 9.49/5.24       => ( ( A != zero_zero_nat )
% 9.49/5.24          & ( B != zero_zero_nat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2508_mult__not__zero,axiom,
% 9.49/5.24      ! [A: int,B: int] :
% 9.49/5.24        ( ( ( times_times_int @ A @ B )
% 9.49/5.24         != zero_zero_int )
% 9.49/5.24       => ( ( A != zero_zero_int )
% 9.49/5.24          & ( B != zero_zero_int ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2509_divisors__zero,axiom,
% 9.49/5.24      ! [A: complex,B: complex] :
% 9.49/5.24        ( ( ( times_times_complex @ A @ B )
% 9.49/5.24          = zero_zero_complex )
% 9.49/5.24       => ( ( A = zero_zero_complex )
% 9.49/5.24          | ( B = zero_zero_complex ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2510_divisors__zero,axiom,
% 9.49/5.24      ! [A: real,B: real] :
% 9.49/5.24        ( ( ( times_times_real @ A @ B )
% 9.49/5.24          = zero_zero_real )
% 9.49/5.24       => ( ( A = zero_zero_real )
% 9.49/5.24          | ( B = zero_zero_real ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2511_divisors__zero,axiom,
% 9.49/5.24      ! [A: rat,B: rat] :
% 9.49/5.24        ( ( ( times_times_rat @ A @ B )
% 9.49/5.24          = zero_zero_rat )
% 9.49/5.24       => ( ( A = zero_zero_rat )
% 9.49/5.24          | ( B = zero_zero_rat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2512_divisors__zero,axiom,
% 9.49/5.24      ! [A: nat,B: nat] :
% 9.49/5.24        ( ( ( times_times_nat @ A @ B )
% 9.49/5.24          = zero_zero_nat )
% 9.49/5.24       => ( ( A = zero_zero_nat )
% 9.49/5.24          | ( B = zero_zero_nat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2513_divisors__zero,axiom,
% 9.49/5.24      ! [A: int,B: int] :
% 9.49/5.24        ( ( ( times_times_int @ A @ B )
% 9.49/5.24          = zero_zero_int )
% 9.49/5.24       => ( ( A = zero_zero_int )
% 9.49/5.24          | ( B = zero_zero_int ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2514_no__zero__divisors,axiom,
% 9.49/5.24      ! [A: complex,B: complex] :
% 9.49/5.24        ( ( A != zero_zero_complex )
% 9.49/5.24       => ( ( B != zero_zero_complex )
% 9.49/5.24         => ( ( times_times_complex @ A @ B )
% 9.49/5.24           != zero_zero_complex ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2515_no__zero__divisors,axiom,
% 9.49/5.24      ! [A: real,B: real] :
% 9.49/5.24        ( ( A != zero_zero_real )
% 9.49/5.24       => ( ( B != zero_zero_real )
% 9.49/5.24         => ( ( times_times_real @ A @ B )
% 9.49/5.24           != zero_zero_real ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2516_no__zero__divisors,axiom,
% 9.49/5.24      ! [A: rat,B: rat] :
% 9.49/5.24        ( ( A != zero_zero_rat )
% 9.49/5.24       => ( ( B != zero_zero_rat )
% 9.49/5.24         => ( ( times_times_rat @ A @ B )
% 9.49/5.24           != zero_zero_rat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2517_no__zero__divisors,axiom,
% 9.49/5.24      ! [A: nat,B: nat] :
% 9.49/5.24        ( ( A != zero_zero_nat )
% 9.49/5.24       => ( ( B != zero_zero_nat )
% 9.49/5.24         => ( ( times_times_nat @ A @ B )
% 9.49/5.24           != zero_zero_nat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2518_no__zero__divisors,axiom,
% 9.49/5.24      ! [A: int,B: int] :
% 9.49/5.24        ( ( A != zero_zero_int )
% 9.49/5.24       => ( ( B != zero_zero_int )
% 9.49/5.24         => ( ( times_times_int @ A @ B )
% 9.49/5.24           != zero_zero_int ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2519_mult__left__cancel,axiom,
% 9.49/5.24      ! [C: complex,A: complex,B: complex] :
% 9.49/5.24        ( ( C != zero_zero_complex )
% 9.49/5.24       => ( ( ( times_times_complex @ C @ A )
% 9.49/5.24            = ( times_times_complex @ C @ B ) )
% 9.49/5.24          = ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2520_mult__left__cancel,axiom,
% 9.49/5.24      ! [C: real,A: real,B: real] :
% 9.49/5.24        ( ( C != zero_zero_real )
% 9.49/5.24       => ( ( ( times_times_real @ C @ A )
% 9.49/5.24            = ( times_times_real @ C @ B ) )
% 9.49/5.24          = ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2521_mult__left__cancel,axiom,
% 9.49/5.24      ! [C: rat,A: rat,B: rat] :
% 9.49/5.24        ( ( C != zero_zero_rat )
% 9.49/5.24       => ( ( ( times_times_rat @ C @ A )
% 9.49/5.24            = ( times_times_rat @ C @ B ) )
% 9.49/5.24          = ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2522_mult__left__cancel,axiom,
% 9.49/5.24      ! [C: nat,A: nat,B: nat] :
% 9.49/5.24        ( ( C != zero_zero_nat )
% 9.49/5.24       => ( ( ( times_times_nat @ C @ A )
% 9.49/5.24            = ( times_times_nat @ C @ B ) )
% 9.49/5.24          = ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2523_mult__left__cancel,axiom,
% 9.49/5.24      ! [C: int,A: int,B: int] :
% 9.49/5.24        ( ( C != zero_zero_int )
% 9.49/5.24       => ( ( ( times_times_int @ C @ A )
% 9.49/5.24            = ( times_times_int @ C @ B ) )
% 9.49/5.24          = ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2524_mult__right__cancel,axiom,
% 9.49/5.24      ! [C: complex,A: complex,B: complex] :
% 9.49/5.24        ( ( C != zero_zero_complex )
% 9.49/5.24       => ( ( ( times_times_complex @ A @ C )
% 9.49/5.24            = ( times_times_complex @ B @ C ) )
% 9.49/5.24          = ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2525_mult__right__cancel,axiom,
% 9.49/5.24      ! [C: real,A: real,B: real] :
% 9.49/5.24        ( ( C != zero_zero_real )
% 9.49/5.24       => ( ( ( times_times_real @ A @ C )
% 9.49/5.24            = ( times_times_real @ B @ C ) )
% 9.49/5.24          = ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2526_mult__right__cancel,axiom,
% 9.49/5.24      ! [C: rat,A: rat,B: rat] :
% 9.49/5.24        ( ( C != zero_zero_rat )
% 9.49/5.24       => ( ( ( times_times_rat @ A @ C )
% 9.49/5.24            = ( times_times_rat @ B @ C ) )
% 9.49/5.24          = ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2527_mult__right__cancel,axiom,
% 9.49/5.24      ! [C: nat,A: nat,B: nat] :
% 9.49/5.24        ( ( C != zero_zero_nat )
% 9.49/5.24       => ( ( ( times_times_nat @ A @ C )
% 9.49/5.24            = ( times_times_nat @ B @ C ) )
% 9.49/5.24          = ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2528_mult__right__cancel,axiom,
% 9.49/5.24      ! [C: int,A: int,B: int] :
% 9.49/5.24        ( ( C != zero_zero_int )
% 9.49/5.24       => ( ( ( times_times_int @ A @ C )
% 9.49/5.24            = ( times_times_int @ B @ C ) )
% 9.49/5.24          = ( A = B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2529_Suc__mult__cancel1,axiom,
% 9.49/5.24      ! [K: nat,M: nat,N: nat] :
% 9.49/5.24        ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 9.49/5.24          = ( times_times_nat @ ( suc @ K ) @ N ) )
% 9.49/5.24        = ( M = N ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2530_mult__0,axiom,
% 9.49/5.24      ! [N: nat] :
% 9.49/5.24        ( ( times_times_nat @ zero_zero_nat @ N )
% 9.49/5.24        = zero_zero_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2531_le__numeral__extra_I4_J,axiom,
% 9.49/5.24      ord_less_eq_real @ one_one_real @ one_one_real ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2532_le__numeral__extra_I4_J,axiom,
% 9.49/5.24      ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2533_le__numeral__extra_I4_J,axiom,
% 9.49/5.24      ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2534_le__numeral__extra_I4_J,axiom,
% 9.49/5.24      ord_less_eq_int @ one_one_int @ one_one_int ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2535_zero__neq__one,axiom,
% 9.49/5.24      zero_zero_complex != one_one_complex ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2536_zero__neq__one,axiom,
% 9.49/5.24      zero_zero_real != one_one_real ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2537_zero__neq__one,axiom,
% 9.49/5.24      zero_zero_rat != one_one_rat ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2538_zero__neq__one,axiom,
% 9.49/5.24      zero_zero_nat != one_one_nat ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2539_zero__neq__one,axiom,
% 9.49/5.24      zero_zero_int != one_one_int ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2540_less__numeral__extra_I4_J,axiom,
% 9.49/5.24      ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2541_less__numeral__extra_I4_J,axiom,
% 9.49/5.24      ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2542_less__numeral__extra_I4_J,axiom,
% 9.49/5.24      ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2543_less__numeral__extra_I4_J,axiom,
% 9.49/5.24      ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2544_le__cube,axiom,
% 9.49/5.24      ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2545_le__square,axiom,
% 9.49/5.24      ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2546_mult__le__mono,axiom,
% 9.49/5.24      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.24       => ( ( ord_less_eq_nat @ K @ L )
% 9.49/5.24         => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2547_mult__le__mono1,axiom,
% 9.49/5.24      ! [I: nat,J: nat,K: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.24       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2548_mult__le__mono2,axiom,
% 9.49/5.24      ! [I: nat,J: nat,K: nat] :
% 9.49/5.24        ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.24       => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2549_add__mult__distrib2,axiom,
% 9.49/5.24      ! [K: nat,M: nat,N: nat] :
% 9.49/5.24        ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
% 9.49/5.24        = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2550_add__mult__distrib,axiom,
% 9.49/5.24      ! [M: nat,N: nat,K: nat] :
% 9.49/5.24        ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 9.49/5.24        = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2551_nat__mult__eq__cancel1,axiom,
% 9.49/5.24      ! [K: nat,M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.24       => ( ( ( times_times_nat @ K @ M )
% 9.49/5.24            = ( times_times_nat @ K @ N ) )
% 9.49/5.24          = ( M = N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2552_nat__mult__less__cancel1,axiom,
% 9.49/5.24      ! [K: nat,M: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.24       => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.49/5.24          = ( ord_less_nat @ M @ N ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2553_divide__le__eq,axiom,
% 9.49/5.24      ! [B: real,C: real,A: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 9.49/5.24        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 9.49/5.24          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24               => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 9.49/5.24              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24               => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2554_divide__le__eq,axiom,
% 9.49/5.24      ! [B: rat,C: rat,A: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 9.49/5.24        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 9.49/5.24          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24               => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 9.49/5.24              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24               => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2555_le__divide__eq,axiom,
% 9.49/5.24      ! [A: real,B: real,C: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.49/5.24        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 9.49/5.24          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24               => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 9.49/5.24              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24               => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2556_le__divide__eq,axiom,
% 9.49/5.24      ! [A: rat,B: rat,C: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.49/5.24        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 9.49/5.24          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24               => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 9.49/5.24              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24               => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2557_divide__left__mono,axiom,
% 9.49/5.24      ! [B: real,A: real,C: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ B @ A )
% 9.49/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.49/5.24         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.49/5.24           => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2558_divide__left__mono,axiom,
% 9.49/5.24      ! [B: rat,A: rat,C: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ B @ A )
% 9.49/5.24       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.49/5.24         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.49/5.24           => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2559_neg__divide__le__eq,axiom,
% 9.49/5.24      ! [C: real,B: real,A: real] :
% 9.49/5.24        ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 9.49/5.24          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2560_neg__divide__le__eq,axiom,
% 9.49/5.24      ! [C: rat,B: rat,A: rat] :
% 9.49/5.24        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 9.49/5.24          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2561_neg__le__divide__eq,axiom,
% 9.49/5.24      ! [C: real,A: real,B: real] :
% 9.49/5.24        ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24       => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.49/5.24          = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2562_neg__le__divide__eq,axiom,
% 9.49/5.24      ! [C: rat,A: rat,B: rat] :
% 9.49/5.24        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24       => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.49/5.24          = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2563_pos__divide__le__eq,axiom,
% 9.49/5.24      ! [C: real,B: real,A: real] :
% 9.49/5.24        ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 9.49/5.24          = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2564_pos__divide__le__eq,axiom,
% 9.49/5.24      ! [C: rat,B: rat,A: rat] :
% 9.49/5.24        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 9.49/5.24          = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2565_pos__le__divide__eq,axiom,
% 9.49/5.24      ! [C: real,A: real,B: real] :
% 9.49/5.24        ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24       => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.49/5.24          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2566_pos__le__divide__eq,axiom,
% 9.49/5.24      ! [C: rat,A: rat,B: rat] :
% 9.49/5.24        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24       => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.49/5.24          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2567_mult__imp__div__pos__le,axiom,
% 9.49/5.24      ! [Y: real,X: real,Z2: real] :
% 9.49/5.24        ( ( ord_less_real @ zero_zero_real @ Y )
% 9.49/5.24       => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z2 @ Y ) )
% 9.49/5.24         => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z2 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2568_mult__imp__div__pos__le,axiom,
% 9.49/5.24      ! [Y: rat,X: rat,Z2: rat] :
% 9.49/5.24        ( ( ord_less_rat @ zero_zero_rat @ Y )
% 9.49/5.24       => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z2 @ Y ) )
% 9.49/5.24         => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2569_mult__imp__le__div__pos,axiom,
% 9.49/5.24      ! [Y: real,Z2: real,X: real] :
% 9.49/5.24        ( ( ord_less_real @ zero_zero_real @ Y )
% 9.49/5.24       => ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ Y ) @ X )
% 9.49/5.24         => ( ord_less_eq_real @ Z2 @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2570_mult__imp__le__div__pos,axiom,
% 9.49/5.24      ! [Y: rat,Z2: rat,X: rat] :
% 9.49/5.24        ( ( ord_less_rat @ zero_zero_rat @ Y )
% 9.49/5.24       => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ Y ) @ X )
% 9.49/5.24         => ( ord_less_eq_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2571_divide__left__mono__neg,axiom,
% 9.49/5.24      ! [A: real,B: real,C: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.24       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.49/5.24         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.49/5.24           => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2572_divide__left__mono__neg,axiom,
% 9.49/5.24      ! [A: rat,B: rat,C: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.24       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.49/5.24         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.49/5.24           => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2573_divide__less__eq__numeral_I1_J,axiom,
% 9.49/5.24      ! [B: real,C: real,W2: num] :
% 9.49/5.24        ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
% 9.49/5.24        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 9.49/5.24          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24               => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
% 9.49/5.24              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24               => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2574_divide__less__eq__numeral_I1_J,axiom,
% 9.49/5.24      ! [B: rat,C: rat,W2: num] :
% 9.49/5.24        ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W2 ) )
% 9.49/5.24        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 9.49/5.24          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24               => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
% 9.49/5.24              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24               => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2575_less__divide__eq__numeral_I1_J,axiom,
% 9.49/5.24      ! [W2: num,B: real,C: real] :
% 9.49/5.24        ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
% 9.49/5.24        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
% 9.49/5.24          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24               => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 9.49/5.24              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24               => ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2576_less__divide__eq__numeral_I1_J,axiom,
% 9.49/5.24      ! [W2: num,B: rat,C: rat] :
% 9.49/5.24        ( ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C ) )
% 9.49/5.24        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
% 9.49/5.24          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24               => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 9.49/5.24              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24               => ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2577_divide__le__eq__1,axiom,
% 9.49/5.24      ! [B: real,A: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 9.49/5.24        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.24            & ( ord_less_eq_real @ B @ A ) )
% 9.49/5.24          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.49/5.24            & ( ord_less_eq_real @ A @ B ) )
% 9.49/5.24          | ( A = zero_zero_real ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2578_divide__le__eq__1,axiom,
% 9.49/5.24      ! [B: rat,A: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 9.49/5.24        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.24            & ( ord_less_eq_rat @ B @ A ) )
% 9.49/5.24          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.49/5.24            & ( ord_less_eq_rat @ A @ B ) )
% 9.49/5.24          | ( A = zero_zero_rat ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2579_le__divide__eq__1,axiom,
% 9.49/5.24      ! [B: real,A: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 9.49/5.24        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.24            & ( ord_less_eq_real @ A @ B ) )
% 9.49/5.24          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.49/5.24            & ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2580_le__divide__eq__1,axiom,
% 9.49/5.24      ! [B: rat,A: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 9.49/5.24        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.24            & ( ord_less_eq_rat @ A @ B ) )
% 9.49/5.24          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.49/5.24            & ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2581_mult__le__cancel__left1,axiom,
% 9.49/5.24      ! [C: real,B: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
% 9.49/5.24        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ord_less_eq_real @ one_one_real @ B ) )
% 9.49/5.24          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24           => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2582_mult__le__cancel__left1,axiom,
% 9.49/5.24      ! [C: rat,B: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
% 9.49/5.24        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 9.49/5.24          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24           => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2583_mult__le__cancel__left1,axiom,
% 9.49/5.24      ! [C: int,B: int] :
% 9.49/5.24        ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
% 9.49/5.24        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.49/5.24           => ( ord_less_eq_int @ one_one_int @ B ) )
% 9.49/5.24          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.49/5.24           => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2584_mult__le__cancel__left2,axiom,
% 9.49/5.24      ! [C: real,A: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
% 9.49/5.24        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ord_less_eq_real @ A @ one_one_real ) )
% 9.49/5.24          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24           => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2585_mult__le__cancel__left2,axiom,
% 9.49/5.24      ! [C: rat,A: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
% 9.49/5.24        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 9.49/5.24          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24           => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2586_mult__le__cancel__left2,axiom,
% 9.49/5.24      ! [C: int,A: int] :
% 9.49/5.24        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
% 9.49/5.24        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.49/5.24           => ( ord_less_eq_int @ A @ one_one_int ) )
% 9.49/5.24          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.49/5.24           => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2587_mult__le__cancel__right1,axiom,
% 9.49/5.24      ! [C: real,B: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
% 9.49/5.24        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ord_less_eq_real @ one_one_real @ B ) )
% 9.49/5.24          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24           => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2588_mult__le__cancel__right1,axiom,
% 9.49/5.24      ! [C: rat,B: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
% 9.49/5.24        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 9.49/5.24          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24           => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2589_mult__le__cancel__right1,axiom,
% 9.49/5.24      ! [C: int,B: int] :
% 9.49/5.24        ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
% 9.49/5.24        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.49/5.24           => ( ord_less_eq_int @ one_one_int @ B ) )
% 9.49/5.24          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.49/5.24           => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2590_mult__le__cancel__right2,axiom,
% 9.49/5.24      ! [A: real,C: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
% 9.49/5.24        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ord_less_eq_real @ A @ one_one_real ) )
% 9.49/5.24          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24           => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2591_mult__le__cancel__right2,axiom,
% 9.49/5.24      ! [A: rat,C: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
% 9.49/5.24        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 9.49/5.24          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24           => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2592_mult__le__cancel__right2,axiom,
% 9.49/5.24      ! [A: int,C: int] :
% 9.49/5.24        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
% 9.49/5.24        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.49/5.24           => ( ord_less_eq_int @ A @ one_one_int ) )
% 9.49/5.24          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.49/5.24           => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2593_mult__less__cancel__left1,axiom,
% 9.49/5.24      ! [C: real,B: real] :
% 9.49/5.24        ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
% 9.49/5.24        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ord_less_real @ one_one_real @ B ) )
% 9.49/5.24          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.49/5.24           => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2594_mult__less__cancel__left1,axiom,
% 9.49/5.24      ! [C: rat,B: rat] :
% 9.49/5.24        ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
% 9.49/5.24        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ord_less_rat @ one_one_rat @ B ) )
% 9.49/5.24          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.49/5.24           => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2595_mult__less__cancel__left1,axiom,
% 9.49/5.24      ! [C: int,B: int] :
% 9.49/5.24        ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
% 9.49/5.24        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.49/5.24           => ( ord_less_int @ one_one_int @ B ) )
% 9.49/5.24          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.49/5.24           => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2596_mult__less__cancel__left2,axiom,
% 9.49/5.24      ! [C: real,A: real] :
% 9.49/5.24        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
% 9.49/5.24        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ord_less_real @ A @ one_one_real ) )
% 9.49/5.24          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.49/5.24           => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2597_mult__less__cancel__left2,axiom,
% 9.49/5.24      ! [C: rat,A: rat] :
% 9.49/5.24        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
% 9.49/5.24        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ord_less_rat @ A @ one_one_rat ) )
% 9.49/5.24          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.49/5.24           => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2598_mult__less__cancel__left2,axiom,
% 9.49/5.24      ! [C: int,A: int] :
% 9.49/5.24        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
% 9.49/5.24        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.49/5.24           => ( ord_less_int @ A @ one_one_int ) )
% 9.49/5.24          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.49/5.24           => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2599_mult__less__cancel__right1,axiom,
% 9.49/5.24      ! [C: real,B: real] :
% 9.49/5.24        ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
% 9.49/5.24        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ord_less_real @ one_one_real @ B ) )
% 9.49/5.24          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.49/5.24           => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2600_mult__less__cancel__right1,axiom,
% 9.49/5.24      ! [C: rat,B: rat] :
% 9.49/5.24        ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
% 9.49/5.24        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ord_less_rat @ one_one_rat @ B ) )
% 9.49/5.24          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.49/5.24           => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2601_mult__less__cancel__right1,axiom,
% 9.49/5.24      ! [C: int,B: int] :
% 9.49/5.24        ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
% 9.49/5.24        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.49/5.24           => ( ord_less_int @ one_one_int @ B ) )
% 9.49/5.24          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.49/5.24           => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2602_mult__less__cancel__right2,axiom,
% 9.49/5.24      ! [A: real,C: real] :
% 9.49/5.24        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
% 9.49/5.24        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ord_less_real @ A @ one_one_real ) )
% 9.49/5.24          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.49/5.24           => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2603_mult__less__cancel__right2,axiom,
% 9.49/5.24      ! [A: rat,C: rat] :
% 9.49/5.24        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
% 9.49/5.24        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ord_less_rat @ A @ one_one_rat ) )
% 9.49/5.24          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.49/5.24           => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2604_mult__less__cancel__right2,axiom,
% 9.49/5.24      ! [A: int,C: int] :
% 9.49/5.24        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
% 9.49/5.24        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.49/5.24           => ( ord_less_int @ A @ one_one_int ) )
% 9.49/5.24          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.49/5.24           => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2605_field__le__mult__one__interval,axiom,
% 9.49/5.24      ! [X: real,Y: real] :
% 9.49/5.24        ( ! [Z4: real] :
% 9.49/5.24            ( ( ord_less_real @ zero_zero_real @ Z4 )
% 9.49/5.24           => ( ( ord_less_real @ Z4 @ one_one_real )
% 9.49/5.24             => ( ord_less_eq_real @ ( times_times_real @ Z4 @ X ) @ Y ) ) )
% 9.49/5.24       => ( ord_less_eq_real @ X @ Y ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2606_field__le__mult__one__interval,axiom,
% 9.49/5.24      ! [X: rat,Y: rat] :
% 9.49/5.24        ( ! [Z4: rat] :
% 9.49/5.24            ( ( ord_less_rat @ zero_zero_rat @ Z4 )
% 9.49/5.24           => ( ( ord_less_rat @ Z4 @ one_one_rat )
% 9.49/5.24             => ( ord_less_eq_rat @ ( times_times_rat @ Z4 @ X ) @ Y ) ) )
% 9.49/5.24       => ( ord_less_eq_rat @ X @ Y ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2607_convex__bound__le,axiom,
% 9.49/5.24      ! [X: real,A: real,Y: real,U: real,V: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ X @ A )
% 9.49/5.24       => ( ( ord_less_eq_real @ Y @ A )
% 9.49/5.24         => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 9.49/5.24           => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 9.49/5.24             => ( ( ( plus_plus_real @ U @ V )
% 9.49/5.24                  = one_one_real )
% 9.49/5.24               => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2608_convex__bound__le,axiom,
% 9.49/5.24      ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ X @ A )
% 9.49/5.24       => ( ( ord_less_eq_rat @ Y @ A )
% 9.49/5.24         => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 9.49/5.24           => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 9.49/5.24             => ( ( ( plus_plus_rat @ U @ V )
% 9.49/5.24                  = one_one_rat )
% 9.49/5.24               => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2609_convex__bound__le,axiom,
% 9.49/5.24      ! [X: int,A: int,Y: int,U: int,V: int] :
% 9.49/5.24        ( ( ord_less_eq_int @ X @ A )
% 9.49/5.24       => ( ( ord_less_eq_int @ Y @ A )
% 9.49/5.24         => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 9.49/5.24           => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 9.49/5.24             => ( ( ( plus_plus_int @ U @ V )
% 9.49/5.24                  = one_one_int )
% 9.49/5.24               => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2610_power__Suc__less,axiom,
% 9.49/5.24      ! [A: real,N: nat] :
% 9.49/5.24        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.24       => ( ( ord_less_real @ A @ one_one_real )
% 9.49/5.24         => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2611_power__Suc__less,axiom,
% 9.49/5.24      ! [A: rat,N: nat] :
% 9.49/5.24        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.24       => ( ( ord_less_rat @ A @ one_one_rat )
% 9.49/5.24         => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2612_power__Suc__less,axiom,
% 9.49/5.24      ! [A: nat,N: nat] :
% 9.49/5.24        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.49/5.24       => ( ( ord_less_nat @ A @ one_one_nat )
% 9.49/5.24         => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2613_power__Suc__less,axiom,
% 9.49/5.24      ! [A: int,N: nat] :
% 9.49/5.24        ( ( ord_less_int @ zero_zero_int @ A )
% 9.49/5.24       => ( ( ord_less_int @ A @ one_one_int )
% 9.49/5.24         => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2614_le__divide__eq__numeral_I1_J,axiom,
% 9.49/5.24      ! [W2: num,B: real,C: real] :
% 9.49/5.24        ( ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
% 9.49/5.24        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
% 9.49/5.24          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24               => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 9.49/5.24              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24               => ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2615_le__divide__eq__numeral_I1_J,axiom,
% 9.49/5.24      ! [W2: num,B: rat,C: rat] :
% 9.49/5.24        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C ) )
% 9.49/5.24        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
% 9.49/5.24          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.24           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24               => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 9.49/5.24              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.24               => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% 9.49/5.24  
% 9.49/5.24  thf(fact_2616_divide__le__eq__numeral_I1_J,axiom,
% 9.49/5.24      ! [B: real,C: real,W2: num] :
% 9.49/5.24        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
% 9.49/5.24        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 9.49/5.24          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.24           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.24               => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
% 9.49/5.25              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.25               => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2617_divide__le__eq__numeral_I1_J,axiom,
% 9.49/5.25      ! [B: rat,C: rat,W2: num] :
% 9.49/5.25        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W2 ) )
% 9.49/5.25        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.25           => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 9.49/5.25          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.25           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.25               => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
% 9.49/5.25              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.25               => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2618_convex__bound__lt,axiom,
% 9.49/5.25      ! [X: real,A: real,Y: real,U: real,V: real] :
% 9.49/5.25        ( ( ord_less_real @ X @ A )
% 9.49/5.25       => ( ( ord_less_real @ Y @ A )
% 9.49/5.25         => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 9.49/5.25           => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 9.49/5.25             => ( ( ( plus_plus_real @ U @ V )
% 9.49/5.25                  = one_one_real )
% 9.49/5.25               => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2619_convex__bound__lt,axiom,
% 9.49/5.25      ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
% 9.49/5.25        ( ( ord_less_rat @ X @ A )
% 9.49/5.25       => ( ( ord_less_rat @ Y @ A )
% 9.49/5.25         => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 9.49/5.25           => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 9.49/5.25             => ( ( ( plus_plus_rat @ U @ V )
% 9.49/5.25                  = one_one_rat )
% 9.49/5.25               => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2620_convex__bound__lt,axiom,
% 9.49/5.25      ! [X: int,A: int,Y: int,U: int,V: int] :
% 9.49/5.25        ( ( ord_less_int @ X @ A )
% 9.49/5.25       => ( ( ord_less_int @ Y @ A )
% 9.49/5.25         => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 9.49/5.25           => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 9.49/5.25             => ( ( ( plus_plus_int @ U @ V )
% 9.49/5.25                  = one_one_int )
% 9.49/5.25               => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2621_divide__right__mono__neg,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.49/5.25         => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2622_divide__right__mono__neg,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.49/5.25         => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2623_divide__nonpos__nonpos,axiom,
% 9.49/5.25      ! [X: real,Y: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.49/5.25       => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 9.49/5.25         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2624_divide__nonpos__nonpos,axiom,
% 9.49/5.25      ! [X: rat,Y: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 9.49/5.25       => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 9.49/5.25         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2625_divide__nonpos__nonneg,axiom,
% 9.49/5.25      ! [X: real,Y: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.49/5.25       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.49/5.25         => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2626_divide__nonpos__nonneg,axiom,
% 9.49/5.25      ! [X: rat,Y: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 9.49/5.25       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.49/5.25         => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2627_divide__nonneg__nonpos,axiom,
% 9.49/5.25      ! [X: real,Y: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.49/5.25       => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 9.49/5.25         => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2628_divide__nonneg__nonpos,axiom,
% 9.49/5.25      ! [X: rat,Y: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.49/5.25       => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 9.49/5.25         => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2629_divide__nonneg__nonneg,axiom,
% 9.49/5.25      ! [X: real,Y: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.49/5.25       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.49/5.25         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2630_divide__nonneg__nonneg,axiom,
% 9.49/5.25      ! [X: rat,Y: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.49/5.25       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.49/5.25         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2631_zero__le__divide__iff,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 9.49/5.25        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.25            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 9.49/5.25          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.49/5.25            & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2632_zero__le__divide__iff,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 9.49/5.25        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.25            & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 9.49/5.25          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.49/5.25            & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2633_divide__right__mono,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.49/5.25         => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2634_divide__right__mono,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.49/5.25         => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2635_divide__le__0__iff,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 9.49/5.25        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.25            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 9.49/5.25          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.49/5.25            & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2636_divide__le__0__iff,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 9.49/5.25        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.25            & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 9.49/5.25          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.49/5.25            & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2637_divide__neg__neg,axiom,
% 9.49/5.25      ! [X: real,Y: real] :
% 9.49/5.25        ( ( ord_less_real @ X @ zero_zero_real )
% 9.49/5.25       => ( ( ord_less_real @ Y @ zero_zero_real )
% 9.49/5.25         => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2638_divide__neg__neg,axiom,
% 9.49/5.25      ! [X: rat,Y: rat] :
% 9.49/5.25        ( ( ord_less_rat @ X @ zero_zero_rat )
% 9.49/5.25       => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 9.49/5.25         => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2639_divide__neg__pos,axiom,
% 9.49/5.25      ! [X: real,Y: real] :
% 9.49/5.25        ( ( ord_less_real @ X @ zero_zero_real )
% 9.49/5.25       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.49/5.25         => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2640_divide__neg__pos,axiom,
% 9.49/5.25      ! [X: rat,Y: rat] :
% 9.49/5.25        ( ( ord_less_rat @ X @ zero_zero_rat )
% 9.49/5.25       => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 9.49/5.25         => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2641_divide__pos__neg,axiom,
% 9.49/5.25      ! [X: real,Y: real] :
% 9.49/5.25        ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.25       => ( ( ord_less_real @ Y @ zero_zero_real )
% 9.49/5.25         => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2642_divide__pos__neg,axiom,
% 9.49/5.25      ! [X: rat,Y: rat] :
% 9.49/5.25        ( ( ord_less_rat @ zero_zero_rat @ X )
% 9.49/5.25       => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 9.49/5.25         => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2643_divide__pos__pos,axiom,
% 9.49/5.25      ! [X: real,Y: real] :
% 9.49/5.25        ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.25       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.49/5.25         => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2644_divide__pos__pos,axiom,
% 9.49/5.25      ! [X: rat,Y: rat] :
% 9.49/5.25        ( ( ord_less_rat @ zero_zero_rat @ X )
% 9.49/5.25       => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 9.49/5.25         => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2645_divide__less__0__iff,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 9.49/5.25        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.25            & ( ord_less_real @ B @ zero_zero_real ) )
% 9.49/5.25          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.49/5.25            & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2646_divide__less__0__iff,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 9.49/5.25        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.25            & ( ord_less_rat @ B @ zero_zero_rat ) )
% 9.49/5.25          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.49/5.25            & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2647_divide__less__cancel,axiom,
% 9.49/5.25      ! [A: real,C: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 9.49/5.25        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.25           => ( ord_less_real @ A @ B ) )
% 9.49/5.25          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.25           => ( ord_less_real @ B @ A ) )
% 9.49/5.25          & ( C != zero_zero_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2648_divide__less__cancel,axiom,
% 9.49/5.25      ! [A: rat,C: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 9.49/5.25        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.25           => ( ord_less_rat @ A @ B ) )
% 9.49/5.25          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.25           => ( ord_less_rat @ B @ A ) )
% 9.49/5.25          & ( C != zero_zero_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2649_zero__less__divide__iff,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 9.49/5.25        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.25            & ( ord_less_real @ zero_zero_real @ B ) )
% 9.49/5.25          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.49/5.25            & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2650_zero__less__divide__iff,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 9.49/5.25        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.25            & ( ord_less_rat @ zero_zero_rat @ B ) )
% 9.49/5.25          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.49/5.25            & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2651_divide__strict__right__mono,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real] :
% 9.49/5.25        ( ( ord_less_real @ A @ B )
% 9.49/5.25       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.25         => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2652_divide__strict__right__mono,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_rat @ A @ B )
% 9.49/5.25       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.25         => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2653_divide__strict__right__mono__neg,axiom,
% 9.49/5.25      ! [B: real,A: real,C: real] :
% 9.49/5.25        ( ( ord_less_real @ B @ A )
% 9.49/5.25       => ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.25         => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2654_divide__strict__right__mono__neg,axiom,
% 9.49/5.25      ! [B: rat,A: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_rat @ B @ A )
% 9.49/5.25       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.25         => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2655_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.49/5.25         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2656_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.49/5.25         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2657_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.49/5.25         => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2658_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.49/5.25         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2659_zero__le__mult__iff,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.49/5.25        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.25            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 9.49/5.25          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.49/5.25            & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2660_zero__le__mult__iff,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.49/5.25        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.25            & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 9.49/5.25          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.49/5.25            & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2661_zero__le__mult__iff,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 9.49/5.25        = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.25            & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 9.49/5.25          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.49/5.25            & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2662_mult__nonneg__nonpos2,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.25       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 9.49/5.25         => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2663_mult__nonneg__nonpos2,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.25       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 9.49/5.25         => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2664_mult__nonneg__nonpos2,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.25       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 9.49/5.25         => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2665_mult__nonneg__nonpos2,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.25       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 9.49/5.25         => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2666_mult__nonpos__nonneg,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.49/5.25       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.49/5.25         => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2667_mult__nonpos__nonneg,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.49/5.25       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.49/5.25         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2668_mult__nonpos__nonneg,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.49/5.25       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.49/5.25         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2669_mult__nonpos__nonneg,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.49/5.25       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.49/5.25         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2670_mult__nonneg__nonpos,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.25       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 9.49/5.25         => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2671_mult__nonneg__nonpos,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.25       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 9.49/5.25         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2672_mult__nonneg__nonpos,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.25       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 9.49/5.25         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2673_mult__nonneg__nonpos,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.25       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 9.49/5.25         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2674_mult__nonneg__nonneg,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.25       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.49/5.25         => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2675_mult__nonneg__nonneg,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.25       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.49/5.25         => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2676_mult__nonneg__nonneg,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.25       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.49/5.25         => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2677_mult__nonneg__nonneg,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.25       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.49/5.25         => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2678_split__mult__neg__le,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.25            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 9.49/5.25          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.49/5.25            & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
% 9.49/5.25       => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2679_split__mult__neg__le,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.25            & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 9.49/5.25          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.49/5.25            & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
% 9.49/5.25       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2680_split__mult__neg__le,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.25            & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 9.49/5.25          | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.49/5.25            & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 9.49/5.25       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2681_split__mult__neg__le,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.25            & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 9.49/5.25          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.49/5.25            & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
% 9.49/5.25       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2682_mult__le__0__iff,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 9.49/5.25        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.25            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 9.49/5.25          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.49/5.25            & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2683_mult__le__0__iff,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 9.49/5.25        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.25            & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 9.49/5.25          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.49/5.25            & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2684_mult__le__0__iff,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 9.49/5.25        = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.25            & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 9.49/5.25          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.49/5.25            & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2685_mult__right__mono,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.49/5.25         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2686_mult__right__mono,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.49/5.25         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2687_mult__right__mono,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.49/5.25         => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2688_mult__right__mono,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.49/5.25         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2689_mult__right__mono__neg,axiom,
% 9.49/5.25      ! [B: real,A: real,C: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ B @ A )
% 9.49/5.25       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.49/5.25         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2690_mult__right__mono__neg,axiom,
% 9.49/5.25      ! [B: rat,A: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ B @ A )
% 9.49/5.25       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.49/5.25         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2691_mult__right__mono__neg,axiom,
% 9.49/5.25      ! [B: int,A: int,C: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ B @ A )
% 9.49/5.25       => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.49/5.25         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2692_mult__left__mono,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.49/5.25         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2693_mult__left__mono,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.49/5.25         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2694_mult__left__mono,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.49/5.25         => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2695_mult__left__mono,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.49/5.25         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2696_mult__nonpos__nonpos,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.49/5.25       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 9.49/5.25         => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2697_mult__nonpos__nonpos,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.49/5.25       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 9.49/5.25         => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2698_mult__nonpos__nonpos,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.49/5.25       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 9.49/5.25         => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2699_mult__left__mono__neg,axiom,
% 9.49/5.25      ! [B: real,A: real,C: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ B @ A )
% 9.49/5.25       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.49/5.25         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2700_mult__left__mono__neg,axiom,
% 9.49/5.25      ! [B: rat,A: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ B @ A )
% 9.49/5.25       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.49/5.25         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2701_mult__left__mono__neg,axiom,
% 9.49/5.25      ! [B: int,A: int,C: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ B @ A )
% 9.49/5.25       => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.49/5.25         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2702_split__mult__pos__le,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.25            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 9.49/5.25          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.49/5.25            & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 9.49/5.25       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2703_split__mult__pos__le,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.25            & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 9.49/5.25          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.49/5.25            & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 9.49/5.25       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2704_split__mult__pos__le,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.25            & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 9.49/5.25          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.49/5.25            & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 9.49/5.25       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2705_zero__le__square,axiom,
% 9.49/5.25      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2706_zero__le__square,axiom,
% 9.49/5.25      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2707_zero__le__square,axiom,
% 9.49/5.25      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2708_mult__mono_H,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real,D: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_real @ C @ D )
% 9.49/5.25         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.25           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.49/5.25             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2709_mult__mono_H,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_rat @ C @ D )
% 9.49/5.25         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.25           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.49/5.25             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2710_mult__mono_H,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_nat @ C @ D )
% 9.49/5.25         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.25           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.49/5.25             => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2711_mult__mono_H,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int,D: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_int @ C @ D )
% 9.49/5.25         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.25           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.49/5.25             => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2712_mult__mono,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real,D: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_real @ C @ D )
% 9.49/5.25         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.49/5.25           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.49/5.25             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2713_mult__mono,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_rat @ C @ D )
% 9.49/5.25         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.49/5.25           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.49/5.25             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2714_mult__mono,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_nat @ C @ D )
% 9.49/5.25         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.49/5.25           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.49/5.25             => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2715_mult__mono,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int,D: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_int @ C @ D )
% 9.49/5.25         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.49/5.25           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.49/5.25             => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2716_mult__neg__neg,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ A @ zero_zero_real )
% 9.49/5.25       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.49/5.25         => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2717_mult__neg__neg,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.49/5.25       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.49/5.25         => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2718_mult__neg__neg,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ A @ zero_zero_int )
% 9.49/5.25       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.49/5.25         => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2719_not__square__less__zero,axiom,
% 9.49/5.25      ! [A: real] :
% 9.49/5.25        ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2720_not__square__less__zero,axiom,
% 9.49/5.25      ! [A: rat] :
% 9.49/5.25        ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2721_not__square__less__zero,axiom,
% 9.49/5.25      ! [A: int] :
% 9.49/5.25        ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2722_mult__less__0__iff,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 9.49/5.25        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.25            & ( ord_less_real @ B @ zero_zero_real ) )
% 9.49/5.25          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.49/5.25            & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2723_mult__less__0__iff,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 9.49/5.25        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.25            & ( ord_less_rat @ B @ zero_zero_rat ) )
% 9.49/5.25          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.49/5.25            & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2724_mult__less__0__iff,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 9.49/5.25        = ( ( ( ord_less_int @ zero_zero_int @ A )
% 9.49/5.25            & ( ord_less_int @ B @ zero_zero_int ) )
% 9.49/5.25          | ( ( ord_less_int @ A @ zero_zero_int )
% 9.49/5.25            & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2725_mult__neg__pos,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ A @ zero_zero_real )
% 9.49/5.25       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.49/5.25         => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2726_mult__neg__pos,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.49/5.25       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.49/5.25         => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2727_mult__neg__pos,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( ord_less_nat @ A @ zero_zero_nat )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.49/5.25         => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2728_mult__neg__pos,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ A @ zero_zero_int )
% 9.49/5.25       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.25         => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2729_mult__pos__neg,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.25       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.49/5.25         => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2730_mult__pos__neg,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.25       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.49/5.25         => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2731_mult__pos__neg,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.49/5.25       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 9.49/5.25         => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2732_mult__pos__neg,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ zero_zero_int @ A )
% 9.49/5.25       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.49/5.25         => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2733_mult__pos__pos,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.25       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.49/5.25         => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2734_mult__pos__pos,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.25       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.49/5.25         => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2735_mult__pos__pos,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.49/5.25         => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2736_mult__pos__pos,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ zero_zero_int @ A )
% 9.49/5.25       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.25         => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2737_mult__pos__neg2,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.25       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.49/5.25         => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2738_mult__pos__neg2,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.25       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.49/5.25         => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2739_mult__pos__neg2,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.49/5.25       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 9.49/5.25         => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2740_mult__pos__neg2,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ zero_zero_int @ A )
% 9.49/5.25       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.49/5.25         => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2741_zero__less__mult__iff,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.49/5.25        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.25            & ( ord_less_real @ zero_zero_real @ B ) )
% 9.49/5.25          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.49/5.25            & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2742_zero__less__mult__iff,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.49/5.25        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.25            & ( ord_less_rat @ zero_zero_rat @ B ) )
% 9.49/5.25          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.49/5.25            & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2743_zero__less__mult__iff,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 9.49/5.25        = ( ( ( ord_less_int @ zero_zero_int @ A )
% 9.49/5.25            & ( ord_less_int @ zero_zero_int @ B ) )
% 9.49/5.25          | ( ( ord_less_int @ A @ zero_zero_int )
% 9.49/5.25            & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2744_zero__less__mult__pos,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.49/5.25       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.25         => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2745_zero__less__mult__pos,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.49/5.25       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.25         => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2746_zero__less__mult__pos,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.49/5.25         => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2747_zero__less__mult__pos,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 9.49/5.25       => ( ( ord_less_int @ zero_zero_int @ A )
% 9.49/5.25         => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2748_zero__less__mult__pos2,axiom,
% 9.49/5.25      ! [B: real,A: real] :
% 9.49/5.25        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
% 9.49/5.25       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.25         => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2749_zero__less__mult__pos2,axiom,
% 9.49/5.25      ! [B: rat,A: rat] :
% 9.49/5.25        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
% 9.49/5.25       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.25         => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2750_zero__less__mult__pos2,axiom,
% 9.49/5.25      ! [B: nat,A: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.49/5.25         => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2751_zero__less__mult__pos2,axiom,
% 9.49/5.25      ! [B: int,A: int] :
% 9.49/5.25        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
% 9.49/5.25       => ( ( ord_less_int @ zero_zero_int @ A )
% 9.49/5.25         => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2752_mult__less__cancel__left__neg,axiom,
% 9.49/5.25      ! [C: real,A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.25       => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.49/5.25          = ( ord_less_real @ B @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2753_mult__less__cancel__left__neg,axiom,
% 9.49/5.25      ! [C: rat,A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.25       => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.49/5.25          = ( ord_less_rat @ B @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2754_mult__less__cancel__left__neg,axiom,
% 9.49/5.25      ! [C: int,A: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ C @ zero_zero_int )
% 9.49/5.25       => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.49/5.25          = ( ord_less_int @ B @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2755_mult__less__cancel__left__pos,axiom,
% 9.49/5.25      ! [C: real,A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.25       => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.49/5.25          = ( ord_less_real @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2756_mult__less__cancel__left__pos,axiom,
% 9.49/5.25      ! [C: rat,A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.25       => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.49/5.25          = ( ord_less_rat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2757_mult__less__cancel__left__pos,axiom,
% 9.49/5.25      ! [C: int,A: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ zero_zero_int @ C )
% 9.49/5.25       => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.49/5.25          = ( ord_less_int @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2758_mult__strict__left__mono__neg,axiom,
% 9.49/5.25      ! [B: real,A: real,C: real] :
% 9.49/5.25        ( ( ord_less_real @ B @ A )
% 9.49/5.25       => ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.25         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2759_mult__strict__left__mono__neg,axiom,
% 9.49/5.25      ! [B: rat,A: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_rat @ B @ A )
% 9.49/5.25       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.25         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2760_mult__strict__left__mono__neg,axiom,
% 9.49/5.25      ! [B: int,A: int,C: int] :
% 9.49/5.25        ( ( ord_less_int @ B @ A )
% 9.49/5.25       => ( ( ord_less_int @ C @ zero_zero_int )
% 9.49/5.25         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2761_mult__strict__left__mono,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real] :
% 9.49/5.25        ( ( ord_less_real @ A @ B )
% 9.49/5.25       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.25         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2762_mult__strict__left__mono,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_rat @ A @ B )
% 9.49/5.25       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.25         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2763_mult__strict__left__mono,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat] :
% 9.49/5.25        ( ( ord_less_nat @ A @ B )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.49/5.25         => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2764_mult__strict__left__mono,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int] :
% 9.49/5.25        ( ( ord_less_int @ A @ B )
% 9.49/5.25       => ( ( ord_less_int @ zero_zero_int @ C )
% 9.49/5.25         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2765_mult__less__cancel__left__disj,axiom,
% 9.49/5.25      ! [C: real,A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.49/5.25        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.25            & ( ord_less_real @ A @ B ) )
% 9.49/5.25          | ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.25            & ( ord_less_real @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2766_mult__less__cancel__left__disj,axiom,
% 9.49/5.25      ! [C: rat,A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.49/5.25        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.25            & ( ord_less_rat @ A @ B ) )
% 9.49/5.25          | ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.25            & ( ord_less_rat @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2767_mult__less__cancel__left__disj,axiom,
% 9.49/5.25      ! [C: int,A: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.49/5.25        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.49/5.25            & ( ord_less_int @ A @ B ) )
% 9.49/5.25          | ( ( ord_less_int @ C @ zero_zero_int )
% 9.49/5.25            & ( ord_less_int @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2768_mult__strict__right__mono__neg,axiom,
% 9.49/5.25      ! [B: real,A: real,C: real] :
% 9.49/5.25        ( ( ord_less_real @ B @ A )
% 9.49/5.25       => ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.25         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2769_mult__strict__right__mono__neg,axiom,
% 9.49/5.25      ! [B: rat,A: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_rat @ B @ A )
% 9.49/5.25       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.25         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2770_mult__strict__right__mono__neg,axiom,
% 9.49/5.25      ! [B: int,A: int,C: int] :
% 9.49/5.25        ( ( ord_less_int @ B @ A )
% 9.49/5.25       => ( ( ord_less_int @ C @ zero_zero_int )
% 9.49/5.25         => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2771_mult__strict__right__mono,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real] :
% 9.49/5.25        ( ( ord_less_real @ A @ B )
% 9.49/5.25       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.25         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2772_mult__strict__right__mono,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_rat @ A @ B )
% 9.49/5.25       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.25         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2773_mult__strict__right__mono,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat] :
% 9.49/5.25        ( ( ord_less_nat @ A @ B )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.49/5.25         => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2774_mult__strict__right__mono,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int] :
% 9.49/5.25        ( ( ord_less_int @ A @ B )
% 9.49/5.25       => ( ( ord_less_int @ zero_zero_int @ C )
% 9.49/5.25         => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2775_mult__less__cancel__right__disj,axiom,
% 9.49/5.25      ! [A: real,C: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.49/5.25        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.25            & ( ord_less_real @ A @ B ) )
% 9.49/5.25          | ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.25            & ( ord_less_real @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2776_mult__less__cancel__right__disj,axiom,
% 9.49/5.25      ! [A: rat,C: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.49/5.25        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.25            & ( ord_less_rat @ A @ B ) )
% 9.49/5.25          | ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.25            & ( ord_less_rat @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2777_mult__less__cancel__right__disj,axiom,
% 9.49/5.25      ! [A: int,C: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.49/5.25        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.49/5.25            & ( ord_less_int @ A @ B ) )
% 9.49/5.25          | ( ( ord_less_int @ C @ zero_zero_int )
% 9.49/5.25            & ( ord_less_int @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2778_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real] :
% 9.49/5.25        ( ( ord_less_real @ A @ B )
% 9.49/5.25       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.25         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2779_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_rat @ A @ B )
% 9.49/5.25       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.25         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2780_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat] :
% 9.49/5.25        ( ( ord_less_nat @ A @ B )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.49/5.25         => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2781_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int] :
% 9.49/5.25        ( ( ord_less_int @ A @ B )
% 9.49/5.25       => ( ( ord_less_int @ zero_zero_int @ C )
% 9.49/5.25         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2782_add__scale__eq__noteq,axiom,
% 9.49/5.25      ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
% 9.49/5.25        ( ( R2 != zero_zero_complex )
% 9.49/5.25       => ( ( ( A = B )
% 9.49/5.25            & ( C != D ) )
% 9.49/5.25         => ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C ) )
% 9.49/5.25           != ( plus_plus_complex @ B @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2783_add__scale__eq__noteq,axiom,
% 9.49/5.25      ! [R2: real,A: real,B: real,C: real,D: real] :
% 9.49/5.25        ( ( R2 != zero_zero_real )
% 9.49/5.25       => ( ( ( A = B )
% 9.49/5.25            & ( C != D ) )
% 9.49/5.25         => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
% 9.49/5.25           != ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2784_add__scale__eq__noteq,axiom,
% 9.49/5.25      ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
% 9.49/5.25        ( ( R2 != zero_zero_rat )
% 9.49/5.25       => ( ( ( A = B )
% 9.49/5.25            & ( C != D ) )
% 9.49/5.25         => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
% 9.49/5.25           != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2785_add__scale__eq__noteq,axiom,
% 9.49/5.25      ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
% 9.49/5.25        ( ( R2 != zero_zero_nat )
% 9.49/5.25       => ( ( ( A = B )
% 9.49/5.25            & ( C != D ) )
% 9.49/5.25         => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
% 9.49/5.25           != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2786_add__scale__eq__noteq,axiom,
% 9.49/5.25      ! [R2: int,A: int,B: int,C: int,D: int] :
% 9.49/5.25        ( ( R2 != zero_zero_int )
% 9.49/5.25       => ( ( ( A = B )
% 9.49/5.25            & ( C != D ) )
% 9.49/5.25         => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
% 9.49/5.25           != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2787_nat__mult__le__cancel1,axiom,
% 9.49/5.25      ! [K: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.25       => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.49/5.25          = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2788_Suc__mult__less__cancel1,axiom,
% 9.49/5.25      ! [K: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 9.49/5.25        = ( ord_less_nat @ M @ N ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2789_not__one__le__zero,axiom,
% 9.49/5.25      ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2790_not__one__le__zero,axiom,
% 9.49/5.25      ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2791_not__one__le__zero,axiom,
% 9.49/5.25      ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2792_not__one__le__zero,axiom,
% 9.49/5.25      ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2793_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 9.49/5.25      ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2794_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 9.49/5.25      ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2795_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 9.49/5.25      ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2796_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 9.49/5.25      ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2797_zero__less__one__class_Ozero__le__one,axiom,
% 9.49/5.25      ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2798_zero__less__one__class_Ozero__le__one,axiom,
% 9.49/5.25      ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2799_zero__less__one__class_Ozero__le__one,axiom,
% 9.49/5.25      ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2800_zero__less__one__class_Ozero__le__one,axiom,
% 9.49/5.25      ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2801_less__numeral__extra_I1_J,axiom,
% 9.49/5.25      ord_less_real @ zero_zero_real @ one_one_real ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2802_less__numeral__extra_I1_J,axiom,
% 9.49/5.25      ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2803_less__numeral__extra_I1_J,axiom,
% 9.49/5.25      ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2804_less__numeral__extra_I1_J,axiom,
% 9.49/5.25      ord_less_int @ zero_zero_int @ one_one_int ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2805_zero__less__one,axiom,
% 9.49/5.25      ord_less_real @ zero_zero_real @ one_one_real ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2806_zero__less__one,axiom,
% 9.49/5.25      ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2807_zero__less__one,axiom,
% 9.49/5.25      ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2808_zero__less__one,axiom,
% 9.49/5.25      ord_less_int @ zero_zero_int @ one_one_int ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2809_not__one__less__zero,axiom,
% 9.49/5.25      ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2810_not__one__less__zero,axiom,
% 9.49/5.25      ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2811_not__one__less__zero,axiom,
% 9.49/5.25      ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2812_not__one__less__zero,axiom,
% 9.49/5.25      ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2813_mult__less__mono1,axiom,
% 9.49/5.25      ! [I: nat,J: nat,K: nat] :
% 9.49/5.25        ( ( ord_less_nat @ I @ J )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.25         => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2814_mult__less__mono2,axiom,
% 9.49/5.25      ! [I: nat,J: nat,K: nat] :
% 9.49/5.25        ( ( ord_less_nat @ I @ J )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.25         => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2815_Suc__mult__le__cancel1,axiom,
% 9.49/5.25      ! [K: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 9.49/5.25        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2816_one__le__numeral,axiom,
% 9.49/5.25      ! [N: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2817_one__le__numeral,axiom,
% 9.49/5.25      ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2818_one__le__numeral,axiom,
% 9.49/5.25      ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2819_one__le__numeral,axiom,
% 9.49/5.25      ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2820_one__le__numeral,axiom,
% 9.49/5.25      ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2821_not__numeral__less__one,axiom,
% 9.49/5.25      ! [N: num] :
% 9.49/5.25        ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2822_not__numeral__less__one,axiom,
% 9.49/5.25      ! [N: num] :
% 9.49/5.25        ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2823_not__numeral__less__one,axiom,
% 9.49/5.25      ! [N: num] :
% 9.49/5.25        ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2824_not__numeral__less__one,axiom,
% 9.49/5.25      ! [N: num] :
% 9.49/5.25        ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2825_not__numeral__less__one,axiom,
% 9.49/5.25      ! [N: num] :
% 9.49/5.25        ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2826_mult__Suc,axiom,
% 9.49/5.25      ! [M: nat,N: nat] :
% 9.49/5.25        ( ( times_times_nat @ ( suc @ M ) @ N )
% 9.49/5.25        = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2827_add__mono1,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ A @ B )
% 9.49/5.25       => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2828_add__mono1,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ A @ B )
% 9.49/5.25       => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2829_add__mono1,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( ord_less_nat @ A @ B )
% 9.49/5.25       => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2830_add__mono1,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ A @ B )
% 9.49/5.25       => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2831_less__add__one,axiom,
% 9.49/5.25      ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2832_less__add__one,axiom,
% 9.49/5.25      ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2833_less__add__one,axiom,
% 9.49/5.25      ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2834_less__add__one,axiom,
% 9.49/5.25      ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2835_one__le__power,axiom,
% 9.49/5.25      ! [A: real,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_real @ one_one_real @ A )
% 9.49/5.25       => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2836_one__le__power,axiom,
% 9.49/5.25      ! [A: rat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ one_one_rat @ A )
% 9.49/5.25       => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2837_one__le__power,axiom,
% 9.49/5.25      ! [A: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ one_one_nat @ A )
% 9.49/5.25       => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2838_one__le__power,axiom,
% 9.49/5.25      ! [A: int,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_int @ one_one_int @ A )
% 9.49/5.25       => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2839_power__0,axiom,
% 9.49/5.25      ! [A: rat] :
% 9.49/5.25        ( ( power_power_rat @ A @ zero_zero_nat )
% 9.49/5.25        = one_one_rat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2840_power__0,axiom,
% 9.49/5.25      ! [A: nat] :
% 9.49/5.25        ( ( power_power_nat @ A @ zero_zero_nat )
% 9.49/5.25        = one_one_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2841_power__0,axiom,
% 9.49/5.25      ! [A: real] :
% 9.49/5.25        ( ( power_power_real @ A @ zero_zero_nat )
% 9.49/5.25        = one_one_real ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2842_power__0,axiom,
% 9.49/5.25      ! [A: int] :
% 9.49/5.25        ( ( power_power_int @ A @ zero_zero_nat )
% 9.49/5.25        = one_one_int ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2843_power__0,axiom,
% 9.49/5.25      ! [A: complex] :
% 9.49/5.25        ( ( power_power_complex @ A @ zero_zero_nat )
% 9.49/5.25        = one_one_complex ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2844_One__nat__def,axiom,
% 9.49/5.25      ( one_one_nat
% 9.49/5.25      = ( suc @ zero_zero_nat ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2845_Suc__eq__plus1__left,axiom,
% 9.49/5.25      ( suc
% 9.49/5.25      = ( plus_plus_nat @ one_one_nat ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2846_plus__1__eq__Suc,axiom,
% 9.49/5.25      ( ( plus_plus_nat @ one_one_nat )
% 9.49/5.25      = suc ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2847_Suc__eq__plus1,axiom,
% 9.49/5.25      ( suc
% 9.49/5.25      = ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2848_frac__le,axiom,
% 9.49/5.25      ! [Y: real,X: real,W2: real,Z2: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.49/5.25       => ( ( ord_less_eq_real @ X @ Y )
% 9.49/5.25         => ( ( ord_less_real @ zero_zero_real @ W2 )
% 9.49/5.25           => ( ( ord_less_eq_real @ W2 @ Z2 )
% 9.49/5.25             => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2849_frac__le,axiom,
% 9.49/5.25      ! [Y: rat,X: rat,W2: rat,Z2: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.49/5.25       => ( ( ord_less_eq_rat @ X @ Y )
% 9.49/5.25         => ( ( ord_less_rat @ zero_zero_rat @ W2 )
% 9.49/5.25           => ( ( ord_less_eq_rat @ W2 @ Z2 )
% 9.49/5.25             => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2850_frac__less,axiom,
% 9.49/5.25      ! [X: real,Y: real,W2: real,Z2: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.49/5.25       => ( ( ord_less_real @ X @ Y )
% 9.49/5.25         => ( ( ord_less_real @ zero_zero_real @ W2 )
% 9.49/5.25           => ( ( ord_less_eq_real @ W2 @ Z2 )
% 9.49/5.25             => ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2851_frac__less,axiom,
% 9.49/5.25      ! [X: rat,Y: rat,W2: rat,Z2: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.49/5.25       => ( ( ord_less_rat @ X @ Y )
% 9.49/5.25         => ( ( ord_less_rat @ zero_zero_rat @ W2 )
% 9.49/5.25           => ( ( ord_less_eq_rat @ W2 @ Z2 )
% 9.49/5.25             => ( ord_less_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2852_frac__less2,axiom,
% 9.49/5.25      ! [X: real,Y: real,W2: real,Z2: real] :
% 9.49/5.25        ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.25       => ( ( ord_less_eq_real @ X @ Y )
% 9.49/5.25         => ( ( ord_less_real @ zero_zero_real @ W2 )
% 9.49/5.25           => ( ( ord_less_real @ W2 @ Z2 )
% 9.49/5.25             => ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2853_frac__less2,axiom,
% 9.49/5.25      ! [X: rat,Y: rat,W2: rat,Z2: rat] :
% 9.49/5.25        ( ( ord_less_rat @ zero_zero_rat @ X )
% 9.49/5.25       => ( ( ord_less_eq_rat @ X @ Y )
% 9.49/5.25         => ( ( ord_less_rat @ zero_zero_rat @ W2 )
% 9.49/5.25           => ( ( ord_less_rat @ W2 @ Z2 )
% 9.49/5.25             => ( ord_less_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2854_divide__le__cancel,axiom,
% 9.49/5.25      ! [A: real,C: real,B: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 9.49/5.25        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.25           => ( ord_less_eq_real @ A @ B ) )
% 9.49/5.25          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.25           => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2855_divide__le__cancel,axiom,
% 9.49/5.25      ! [A: rat,C: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 9.49/5.25        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.25           => ( ord_less_eq_rat @ A @ B ) )
% 9.49/5.25          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.25           => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2856_divide__nonneg__neg,axiom,
% 9.49/5.25      ! [X: real,Y: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.49/5.25       => ( ( ord_less_real @ Y @ zero_zero_real )
% 9.49/5.25         => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2857_divide__nonneg__neg,axiom,
% 9.49/5.25      ! [X: rat,Y: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.49/5.25       => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 9.49/5.25         => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2858_divide__nonneg__pos,axiom,
% 9.49/5.25      ! [X: real,Y: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.49/5.25       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.49/5.25         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2859_divide__nonneg__pos,axiom,
% 9.49/5.25      ! [X: rat,Y: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.49/5.25       => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 9.49/5.25         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2860_divide__nonpos__neg,axiom,
% 9.49/5.25      ! [X: real,Y: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.49/5.25       => ( ( ord_less_real @ Y @ zero_zero_real )
% 9.49/5.25         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2861_divide__nonpos__neg,axiom,
% 9.49/5.25      ! [X: rat,Y: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 9.49/5.25       => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 9.49/5.25         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2862_divide__nonpos__pos,axiom,
% 9.49/5.25      ! [X: real,Y: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.49/5.25       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.49/5.25         => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2863_divide__nonpos__pos,axiom,
% 9.49/5.25      ! [X: rat,Y: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 9.49/5.25       => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 9.49/5.25         => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2864_mult__le__cancel__left,axiom,
% 9.49/5.25      ! [C: real,A: real,B: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.49/5.25        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.25           => ( ord_less_eq_real @ A @ B ) )
% 9.49/5.25          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.25           => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2865_mult__le__cancel__left,axiom,
% 9.49/5.25      ! [C: rat,A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.49/5.25        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.25           => ( ord_less_eq_rat @ A @ B ) )
% 9.49/5.25          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.25           => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2866_mult__le__cancel__left,axiom,
% 9.49/5.25      ! [C: int,A: int,B: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.49/5.25        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.49/5.25           => ( ord_less_eq_int @ A @ B ) )
% 9.49/5.25          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.49/5.25           => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2867_mult__le__cancel__right,axiom,
% 9.49/5.25      ! [A: real,C: real,B: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.49/5.25        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.25           => ( ord_less_eq_real @ A @ B ) )
% 9.49/5.25          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.25           => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2868_mult__le__cancel__right,axiom,
% 9.49/5.25      ! [A: rat,C: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.49/5.25        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.25           => ( ord_less_eq_rat @ A @ B ) )
% 9.49/5.25          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.25           => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2869_mult__le__cancel__right,axiom,
% 9.49/5.25      ! [A: int,C: int,B: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.49/5.25        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.49/5.25           => ( ord_less_eq_int @ A @ B ) )
% 9.49/5.25          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.49/5.25           => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2870_mult__left__less__imp__less,axiom,
% 9.49/5.25      ! [C: real,A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.49/5.25       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.49/5.25         => ( ord_less_real @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2871_mult__left__less__imp__less,axiom,
% 9.49/5.25      ! [C: rat,A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.49/5.25       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.49/5.25         => ( ord_less_rat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2872_mult__left__less__imp__less,axiom,
% 9.49/5.25      ! [C: nat,A: nat,B: nat] :
% 9.49/5.25        ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 9.49/5.25       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.49/5.25         => ( ord_less_nat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2873_mult__left__less__imp__less,axiom,
% 9.49/5.25      ! [C: int,A: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.49/5.25       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.49/5.25         => ( ord_less_int @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2874_mult__strict__mono,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real,D: real] :
% 9.49/5.25        ( ( ord_less_real @ A @ B )
% 9.49/5.25       => ( ( ord_less_real @ C @ D )
% 9.49/5.25         => ( ( ord_less_real @ zero_zero_real @ B )
% 9.49/5.25           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.49/5.25             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2875_mult__strict__mono,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.49/5.25        ( ( ord_less_rat @ A @ B )
% 9.49/5.25       => ( ( ord_less_rat @ C @ D )
% 9.49/5.25         => ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.49/5.25           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.49/5.25             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2876_mult__strict__mono,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.49/5.25        ( ( ord_less_nat @ A @ B )
% 9.49/5.25       => ( ( ord_less_nat @ C @ D )
% 9.49/5.25         => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.49/5.25           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.49/5.25             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2877_mult__strict__mono,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int,D: int] :
% 9.49/5.25        ( ( ord_less_int @ A @ B )
% 9.49/5.25       => ( ( ord_less_int @ C @ D )
% 9.49/5.25         => ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.25           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.49/5.25             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2878_mult__less__cancel__left,axiom,
% 9.49/5.25      ! [C: real,A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.49/5.25        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.49/5.25           => ( ord_less_real @ A @ B ) )
% 9.49/5.25          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.49/5.25           => ( ord_less_real @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2879_mult__less__cancel__left,axiom,
% 9.49/5.25      ! [C: rat,A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.49/5.25        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.49/5.25           => ( ord_less_rat @ A @ B ) )
% 9.49/5.25          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.49/5.25           => ( ord_less_rat @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2880_mult__less__cancel__left,axiom,
% 9.49/5.25      ! [C: int,A: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.49/5.25        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.49/5.25           => ( ord_less_int @ A @ B ) )
% 9.49/5.25          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.49/5.25           => ( ord_less_int @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2881_mult__right__less__imp__less,axiom,
% 9.49/5.25      ! [A: real,C: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.49/5.25       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.49/5.25         => ( ord_less_real @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2882_mult__right__less__imp__less,axiom,
% 9.49/5.25      ! [A: rat,C: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.49/5.25       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.49/5.25         => ( ord_less_rat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2883_mult__right__less__imp__less,axiom,
% 9.49/5.25      ! [A: nat,C: nat,B: nat] :
% 9.49/5.25        ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 9.49/5.25       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.49/5.25         => ( ord_less_nat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2884_mult__right__less__imp__less,axiom,
% 9.49/5.25      ! [A: int,C: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.49/5.25       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.49/5.25         => ( ord_less_int @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2885_mult__strict__mono_H,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real,D: real] :
% 9.49/5.25        ( ( ord_less_real @ A @ B )
% 9.49/5.25       => ( ( ord_less_real @ C @ D )
% 9.49/5.25         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.25           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.49/5.25             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2886_mult__strict__mono_H,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.49/5.25        ( ( ord_less_rat @ A @ B )
% 9.49/5.25       => ( ( ord_less_rat @ C @ D )
% 9.49/5.25         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.25           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.49/5.25             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2887_mult__strict__mono_H,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.49/5.25        ( ( ord_less_nat @ A @ B )
% 9.49/5.25       => ( ( ord_less_nat @ C @ D )
% 9.49/5.25         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.25           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.49/5.25             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2888_mult__strict__mono_H,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int,D: int] :
% 9.49/5.25        ( ( ord_less_int @ A @ B )
% 9.49/5.25       => ( ( ord_less_int @ C @ D )
% 9.49/5.25         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.25           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.49/5.25             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2889_mult__less__cancel__right,axiom,
% 9.49/5.25      ! [A: real,C: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.49/5.25        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.49/5.25           => ( ord_less_real @ A @ B ) )
% 9.49/5.25          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.49/5.25           => ( ord_less_real @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2890_mult__less__cancel__right,axiom,
% 9.49/5.25      ! [A: rat,C: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.49/5.25        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.49/5.25           => ( ord_less_rat @ A @ B ) )
% 9.49/5.25          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.49/5.25           => ( ord_less_rat @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2891_mult__less__cancel__right,axiom,
% 9.49/5.25      ! [A: int,C: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.49/5.25        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.49/5.25           => ( ord_less_int @ A @ B ) )
% 9.49/5.25          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.49/5.25           => ( ord_less_int @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2892_mult__le__cancel__left__neg,axiom,
% 9.49/5.25      ! [C: real,A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.25       => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.49/5.25          = ( ord_less_eq_real @ B @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2893_mult__le__cancel__left__neg,axiom,
% 9.49/5.25      ! [C: rat,A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.25       => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.49/5.25          = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2894_mult__le__cancel__left__neg,axiom,
% 9.49/5.25      ! [C: int,A: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ C @ zero_zero_int )
% 9.49/5.25       => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.49/5.25          = ( ord_less_eq_int @ B @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2895_mult__le__cancel__left__pos,axiom,
% 9.49/5.25      ! [C: real,A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.25       => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.49/5.25          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2896_mult__le__cancel__left__pos,axiom,
% 9.49/5.25      ! [C: rat,A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.25       => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.49/5.25          = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2897_mult__le__cancel__left__pos,axiom,
% 9.49/5.25      ! [C: int,A: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ zero_zero_int @ C )
% 9.49/5.25       => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.49/5.25          = ( ord_less_eq_int @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2898_mult__left__le__imp__le,axiom,
% 9.49/5.25      ! [C: real,A: real,B: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.49/5.25       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.25         => ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2899_mult__left__le__imp__le,axiom,
% 9.49/5.25      ! [C: rat,A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.49/5.25       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.25         => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2900_mult__left__le__imp__le,axiom,
% 9.49/5.25      ! [C: nat,A: nat,B: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.49/5.25         => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2901_mult__left__le__imp__le,axiom,
% 9.49/5.25      ! [C: int,A: int,B: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.49/5.25       => ( ( ord_less_int @ zero_zero_int @ C )
% 9.49/5.25         => ( ord_less_eq_int @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2902_mult__right__le__imp__le,axiom,
% 9.49/5.25      ! [A: real,C: real,B: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.49/5.25       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.25         => ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2903_mult__right__le__imp__le,axiom,
% 9.49/5.25      ! [A: rat,C: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.49/5.25       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.25         => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2904_mult__right__le__imp__le,axiom,
% 9.49/5.25      ! [A: nat,C: nat,B: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.49/5.25         => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2905_mult__right__le__imp__le,axiom,
% 9.49/5.25      ! [A: int,C: int,B: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.49/5.25       => ( ( ord_less_int @ zero_zero_int @ C )
% 9.49/5.25         => ( ord_less_eq_int @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2906_mult__le__less__imp__less,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real,D: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.25       => ( ( ord_less_real @ C @ D )
% 9.49/5.25         => ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.25           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.49/5.25             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2907_mult__le__less__imp__less,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.25       => ( ( ord_less_rat @ C @ D )
% 9.49/5.25         => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.25           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.49/5.25             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2908_mult__le__less__imp__less,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.25       => ( ( ord_less_nat @ C @ D )
% 9.49/5.25         => ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.49/5.25           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.49/5.25             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2909_mult__le__less__imp__less,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int,D: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.25       => ( ( ord_less_int @ C @ D )
% 9.49/5.25         => ( ( ord_less_int @ zero_zero_int @ A )
% 9.49/5.25           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.49/5.25             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2910_mult__less__le__imp__less,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real,D: real] :
% 9.49/5.25        ( ( ord_less_real @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_real @ C @ D )
% 9.49/5.25         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.25           => ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.25             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2911_mult__less__le__imp__less,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.49/5.25        ( ( ord_less_rat @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_rat @ C @ D )
% 9.49/5.25         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.25           => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.25             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2912_mult__less__le__imp__less,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.49/5.25        ( ( ord_less_nat @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_nat @ C @ D )
% 9.49/5.25         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.25           => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.49/5.25             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2913_mult__less__le__imp__less,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int,D: int] :
% 9.49/5.25        ( ( ord_less_int @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_int @ C @ D )
% 9.49/5.25         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.25           => ( ( ord_less_int @ zero_zero_int @ C )
% 9.49/5.25             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2914_sum__squares__ge__zero,axiom,
% 9.49/5.25      ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2915_sum__squares__ge__zero,axiom,
% 9.49/5.25      ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2916_sum__squares__ge__zero,axiom,
% 9.49/5.25      ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2917_sum__squares__le__zero__iff,axiom,
% 9.49/5.25      ! [X: real,Y: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
% 9.49/5.25        = ( ( X = zero_zero_real )
% 9.49/5.25          & ( Y = zero_zero_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2918_sum__squares__le__zero__iff,axiom,
% 9.49/5.25      ! [X: rat,Y: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
% 9.49/5.25        = ( ( X = zero_zero_rat )
% 9.49/5.25          & ( Y = zero_zero_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2919_sum__squares__le__zero__iff,axiom,
% 9.49/5.25      ! [X: int,Y: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
% 9.49/5.25        = ( ( X = zero_zero_int )
% 9.49/5.25          & ( Y = zero_zero_int ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2920_not__sum__squares__lt__zero,axiom,
% 9.49/5.25      ! [X: real,Y: real] :
% 9.49/5.25        ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2921_not__sum__squares__lt__zero,axiom,
% 9.49/5.25      ! [X: rat,Y: rat] :
% 9.49/5.25        ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2922_not__sum__squares__lt__zero,axiom,
% 9.49/5.25      ! [X: int,Y: int] :
% 9.49/5.25        ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2923_sum__squares__gt__zero__iff,axiom,
% 9.49/5.25      ! [X: real,Y: real] :
% 9.49/5.25        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
% 9.49/5.25        = ( ( X != zero_zero_real )
% 9.49/5.25          | ( Y != zero_zero_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2924_sum__squares__gt__zero__iff,axiom,
% 9.49/5.25      ! [X: rat,Y: rat] :
% 9.49/5.25        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
% 9.49/5.25        = ( ( X != zero_zero_rat )
% 9.49/5.25          | ( Y != zero_zero_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2925_sum__squares__gt__zero__iff,axiom,
% 9.49/5.25      ! [X: int,Y: int] :
% 9.49/5.25        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
% 9.49/5.25        = ( ( X != zero_zero_int )
% 9.49/5.25          | ( Y != zero_zero_int ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2926_one__less__mult,axiom,
% 9.49/5.25      ! [N: nat,M: nat] :
% 9.49/5.25        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.49/5.25       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.49/5.25         => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2927_n__less__m__mult__n,axiom,
% 9.49/5.25      ! [N: nat,M: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.49/5.25         => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2928_n__less__n__mult__m,axiom,
% 9.49/5.25      ! [N: nat,M: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.49/5.25         => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2929_zero__less__two,axiom,
% 9.49/5.25      ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2930_zero__less__two,axiom,
% 9.49/5.25      ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2931_zero__less__two,axiom,
% 9.49/5.25      ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2932_zero__less__two,axiom,
% 9.49/5.25      ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2933_power__le__one,axiom,
% 9.49/5.25      ! [A: real,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.25       => ( ( ord_less_eq_real @ A @ one_one_real )
% 9.49/5.25         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2934_power__le__one,axiom,
% 9.49/5.25      ! [A: rat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.25       => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 9.49/5.25         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2935_power__le__one,axiom,
% 9.49/5.25      ! [A: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.25       => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 9.49/5.25         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2936_power__le__one,axiom,
% 9.49/5.25      ! [A: int,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.25       => ( ( ord_less_eq_int @ A @ one_one_int )
% 9.49/5.25         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2937_power__gt1,axiom,
% 9.49/5.25      ! [A: real,N: nat] :
% 9.49/5.25        ( ( ord_less_real @ one_one_real @ A )
% 9.49/5.25       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2938_power__gt1,axiom,
% 9.49/5.25      ! [A: rat,N: nat] :
% 9.49/5.25        ( ( ord_less_rat @ one_one_rat @ A )
% 9.49/5.25       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2939_power__gt1,axiom,
% 9.49/5.25      ! [A: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_nat @ one_one_nat @ A )
% 9.49/5.25       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2940_power__gt1,axiom,
% 9.49/5.25      ! [A: int,N: nat] :
% 9.49/5.25        ( ( ord_less_int @ one_one_int @ A )
% 9.49/5.25       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2941_power__0__left,axiom,
% 9.49/5.25      ! [N: nat] :
% 9.49/5.25        ( ( ( N = zero_zero_nat )
% 9.49/5.25         => ( ( power_power_rat @ zero_zero_rat @ N )
% 9.49/5.25            = one_one_rat ) )
% 9.49/5.25        & ( ( N != zero_zero_nat )
% 9.49/5.25         => ( ( power_power_rat @ zero_zero_rat @ N )
% 9.49/5.25            = zero_zero_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2942_power__0__left,axiom,
% 9.49/5.25      ! [N: nat] :
% 9.49/5.25        ( ( ( N = zero_zero_nat )
% 9.49/5.25         => ( ( power_power_nat @ zero_zero_nat @ N )
% 9.49/5.25            = one_one_nat ) )
% 9.49/5.25        & ( ( N != zero_zero_nat )
% 9.49/5.25         => ( ( power_power_nat @ zero_zero_nat @ N )
% 9.49/5.25            = zero_zero_nat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2943_power__0__left,axiom,
% 9.49/5.25      ! [N: nat] :
% 9.49/5.25        ( ( ( N = zero_zero_nat )
% 9.49/5.25         => ( ( power_power_real @ zero_zero_real @ N )
% 9.49/5.25            = one_one_real ) )
% 9.49/5.25        & ( ( N != zero_zero_nat )
% 9.49/5.25         => ( ( power_power_real @ zero_zero_real @ N )
% 9.49/5.25            = zero_zero_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2944_power__0__left,axiom,
% 9.49/5.25      ! [N: nat] :
% 9.49/5.25        ( ( ( N = zero_zero_nat )
% 9.49/5.25         => ( ( power_power_int @ zero_zero_int @ N )
% 9.49/5.25            = one_one_int ) )
% 9.49/5.25        & ( ( N != zero_zero_nat )
% 9.49/5.25         => ( ( power_power_int @ zero_zero_int @ N )
% 9.49/5.25            = zero_zero_int ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2945_power__0__left,axiom,
% 9.49/5.25      ! [N: nat] :
% 9.49/5.25        ( ( ( N = zero_zero_nat )
% 9.49/5.25         => ( ( power_power_complex @ zero_zero_complex @ N )
% 9.49/5.25            = one_one_complex ) )
% 9.49/5.25        & ( ( N != zero_zero_nat )
% 9.49/5.25         => ( ( power_power_complex @ zero_zero_complex @ N )
% 9.49/5.25            = zero_zero_complex ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2946_power__less__imp__less__exp,axiom,
% 9.49/5.25      ! [A: real,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_real @ one_one_real @ A )
% 9.49/5.25       => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 9.49/5.25         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2947_power__less__imp__less__exp,axiom,
% 9.49/5.25      ! [A: rat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_rat @ one_one_rat @ A )
% 9.49/5.25       => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 9.49/5.25         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2948_power__less__imp__less__exp,axiom,
% 9.49/5.25      ! [A: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_nat @ one_one_nat @ A )
% 9.49/5.25       => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 9.49/5.25         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2949_power__less__imp__less__exp,axiom,
% 9.49/5.25      ! [A: int,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_int @ one_one_int @ A )
% 9.49/5.25       => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 9.49/5.25         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2950_power__strict__increasing,axiom,
% 9.49/5.25      ! [N: nat,N6: nat,A: real] :
% 9.49/5.25        ( ( ord_less_nat @ N @ N6 )
% 9.49/5.25       => ( ( ord_less_real @ one_one_real @ A )
% 9.49/5.25         => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N6 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2951_power__strict__increasing,axiom,
% 9.49/5.25      ! [N: nat,N6: nat,A: rat] :
% 9.49/5.25        ( ( ord_less_nat @ N @ N6 )
% 9.49/5.25       => ( ( ord_less_rat @ one_one_rat @ A )
% 9.49/5.25         => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N6 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2952_power__strict__increasing,axiom,
% 9.49/5.25      ! [N: nat,N6: nat,A: nat] :
% 9.49/5.25        ( ( ord_less_nat @ N @ N6 )
% 9.49/5.25       => ( ( ord_less_nat @ one_one_nat @ A )
% 9.49/5.25         => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2953_power__strict__increasing,axiom,
% 9.49/5.25      ! [N: nat,N6: nat,A: int] :
% 9.49/5.25        ( ( ord_less_nat @ N @ N6 )
% 9.49/5.25       => ( ( ord_less_int @ one_one_int @ A )
% 9.49/5.25         => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N6 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2954_power__increasing,axiom,
% 9.49/5.25      ! [N: nat,N6: nat,A: real] :
% 9.49/5.25        ( ( ord_less_eq_nat @ N @ N6 )
% 9.49/5.25       => ( ( ord_less_eq_real @ one_one_real @ A )
% 9.49/5.25         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N6 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2955_power__increasing,axiom,
% 9.49/5.25      ! [N: nat,N6: nat,A: rat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ N @ N6 )
% 9.49/5.25       => ( ( ord_less_eq_rat @ one_one_rat @ A )
% 9.49/5.25         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N6 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2956_power__increasing,axiom,
% 9.49/5.25      ! [N: nat,N6: nat,A: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ N @ N6 )
% 9.49/5.25       => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 9.49/5.25         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2957_power__increasing,axiom,
% 9.49/5.25      ! [N: nat,N6: nat,A: int] :
% 9.49/5.25        ( ( ord_less_eq_nat @ N @ N6 )
% 9.49/5.25       => ( ( ord_less_eq_int @ one_one_int @ A )
% 9.49/5.25         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N6 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2958_nat__induct__non__zero,axiom,
% 9.49/5.25      ! [N: nat,P: nat > $o] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ( P @ one_one_nat )
% 9.49/5.25         => ( ! [N3: nat] :
% 9.49/5.25                ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.49/5.25               => ( ( P @ N3 )
% 9.49/5.25                 => ( P @ ( suc @ N3 ) ) ) )
% 9.49/5.25           => ( P @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2959_power__Suc__le__self,axiom,
% 9.49/5.25      ! [A: real,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.25       => ( ( ord_less_eq_real @ A @ one_one_real )
% 9.49/5.25         => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2960_power__Suc__le__self,axiom,
% 9.49/5.25      ! [A: rat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.25       => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 9.49/5.25         => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2961_power__Suc__le__self,axiom,
% 9.49/5.25      ! [A: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.25       => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 9.49/5.25         => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2962_power__Suc__le__self,axiom,
% 9.49/5.25      ! [A: int,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.25       => ( ( ord_less_eq_int @ A @ one_one_int )
% 9.49/5.25         => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2963_power__Suc__less__one,axiom,
% 9.49/5.25      ! [A: real,N: nat] :
% 9.49/5.25        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.25       => ( ( ord_less_real @ A @ one_one_real )
% 9.49/5.25         => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2964_power__Suc__less__one,axiom,
% 9.49/5.25      ! [A: rat,N: nat] :
% 9.49/5.25        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.25       => ( ( ord_less_rat @ A @ one_one_rat )
% 9.49/5.25         => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2965_power__Suc__less__one,axiom,
% 9.49/5.25      ! [A: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.49/5.25       => ( ( ord_less_nat @ A @ one_one_nat )
% 9.49/5.25         => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2966_power__Suc__less__one,axiom,
% 9.49/5.25      ! [A: int,N: nat] :
% 9.49/5.25        ( ( ord_less_int @ zero_zero_int @ A )
% 9.49/5.25       => ( ( ord_less_int @ A @ one_one_int )
% 9.49/5.25         => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2967_power__strict__decreasing,axiom,
% 9.49/5.25      ! [N: nat,N6: nat,A: real] :
% 9.49/5.25        ( ( ord_less_nat @ N @ N6 )
% 9.49/5.25       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.25         => ( ( ord_less_real @ A @ one_one_real )
% 9.49/5.25           => ( ord_less_real @ ( power_power_real @ A @ N6 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2968_power__strict__decreasing,axiom,
% 9.49/5.25      ! [N: nat,N6: nat,A: rat] :
% 9.49/5.25        ( ( ord_less_nat @ N @ N6 )
% 9.49/5.25       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.25         => ( ( ord_less_rat @ A @ one_one_rat )
% 9.49/5.25           => ( ord_less_rat @ ( power_power_rat @ A @ N6 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2969_power__strict__decreasing,axiom,
% 9.49/5.25      ! [N: nat,N6: nat,A: nat] :
% 9.49/5.25        ( ( ord_less_nat @ N @ N6 )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.49/5.25         => ( ( ord_less_nat @ A @ one_one_nat )
% 9.49/5.25           => ( ord_less_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2970_power__strict__decreasing,axiom,
% 9.49/5.25      ! [N: nat,N6: nat,A: int] :
% 9.49/5.25        ( ( ord_less_nat @ N @ N6 )
% 9.49/5.25       => ( ( ord_less_int @ zero_zero_int @ A )
% 9.49/5.25         => ( ( ord_less_int @ A @ one_one_int )
% 9.49/5.25           => ( ord_less_int @ ( power_power_int @ A @ N6 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2971_power__decreasing,axiom,
% 9.49/5.25      ! [N: nat,N6: nat,A: real] :
% 9.49/5.25        ( ( ord_less_eq_nat @ N @ N6 )
% 9.49/5.25       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.25         => ( ( ord_less_eq_real @ A @ one_one_real )
% 9.49/5.25           => ( ord_less_eq_real @ ( power_power_real @ A @ N6 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2972_power__decreasing,axiom,
% 9.49/5.25      ! [N: nat,N6: nat,A: rat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ N @ N6 )
% 9.49/5.25       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.25         => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 9.49/5.25           => ( ord_less_eq_rat @ ( power_power_rat @ A @ N6 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2973_power__decreasing,axiom,
% 9.49/5.25      ! [N: nat,N6: nat,A: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ N @ N6 )
% 9.49/5.25       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.25         => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 9.49/5.25           => ( ord_less_eq_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2974_power__decreasing,axiom,
% 9.49/5.25      ! [N: nat,N6: nat,A: int] :
% 9.49/5.25        ( ( ord_less_eq_nat @ N @ N6 )
% 9.49/5.25       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.25         => ( ( ord_less_eq_int @ A @ one_one_int )
% 9.49/5.25           => ( ord_less_eq_int @ ( power_power_int @ A @ N6 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2975_power__le__imp__le__exp,axiom,
% 9.49/5.25      ! [A: real,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_real @ one_one_real @ A )
% 9.49/5.25       => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 9.49/5.25         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2976_power__le__imp__le__exp,axiom,
% 9.49/5.25      ! [A: rat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_rat @ one_one_rat @ A )
% 9.49/5.25       => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 9.49/5.25         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2977_power__le__imp__le__exp,axiom,
% 9.49/5.25      ! [A: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_nat @ one_one_nat @ A )
% 9.49/5.25       => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 9.49/5.25         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2978_power__le__imp__le__exp,axiom,
% 9.49/5.25      ! [A: int,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_int @ one_one_int @ A )
% 9.49/5.25       => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 9.49/5.25         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2979_self__le__power,axiom,
% 9.49/5.25      ! [A: real,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_real @ one_one_real @ A )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25         => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2980_self__le__power,axiom,
% 9.49/5.25      ! [A: rat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ one_one_rat @ A )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25         => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2981_self__le__power,axiom,
% 9.49/5.25      ! [A: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ one_one_nat @ A )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25         => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2982_self__le__power,axiom,
% 9.49/5.25      ! [A: int,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_int @ one_one_int @ A )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25         => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2983_one__less__power,axiom,
% 9.49/5.25      ! [A: real,N: nat] :
% 9.49/5.25        ( ( ord_less_real @ one_one_real @ A )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25         => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2984_one__less__power,axiom,
% 9.49/5.25      ! [A: rat,N: nat] :
% 9.49/5.25        ( ( ord_less_rat @ one_one_rat @ A )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25         => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2985_one__less__power,axiom,
% 9.49/5.25      ! [A: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_nat @ one_one_nat @ A )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25         => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2986_one__less__power,axiom,
% 9.49/5.25      ! [A: int,N: nat] :
% 9.49/5.25        ( ( ord_less_int @ one_one_int @ A )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25         => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2987_nat__bit__induct,axiom,
% 9.49/5.25      ! [P: nat > $o,N: nat] :
% 9.49/5.25        ( ( P @ zero_zero_nat )
% 9.49/5.25       => ( ! [N3: nat] :
% 9.49/5.25              ( ( P @ N3 )
% 9.49/5.25             => ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.49/5.25               => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 9.49/5.25         => ( ! [N3: nat] :
% 9.49/5.25                ( ( P @ N3 )
% 9.49/5.25               => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 9.49/5.25           => ( P @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2988_half__gt__zero,axiom,
% 9.49/5.25      ! [A: real] :
% 9.49/5.25        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.25       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2989_half__gt__zero,axiom,
% 9.49/5.25      ! [A: rat] :
% 9.49/5.25        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.25       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2990_half__gt__zero__iff,axiom,
% 9.49/5.25      ! [A: real] :
% 9.49/5.25        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.49/5.25        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2991_half__gt__zero__iff,axiom,
% 9.49/5.25      ! [A: rat] :
% 9.49/5.25        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 9.49/5.25        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2992_field__less__half__sum,axiom,
% 9.49/5.25      ! [X: real,Y: real] :
% 9.49/5.25        ( ( ord_less_real @ X @ Y )
% 9.49/5.25       => ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2993_field__less__half__sum,axiom,
% 9.49/5.25      ! [X: rat,Y: rat] :
% 9.49/5.25        ( ( ord_less_rat @ X @ Y )
% 9.49/5.25       => ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2994_zero__le__even__power_H,axiom,
% 9.49/5.25      ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2995_zero__le__even__power_H,axiom,
% 9.49/5.25      ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2996_zero__le__even__power_H,axiom,
% 9.49/5.25      ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2997_odd__0__le__power__imp__0__le,axiom,
% 9.49/5.25      ! [A: real,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.49/5.25       => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2998_odd__0__le__power__imp__0__le,axiom,
% 9.49/5.25      ! [A: rat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.49/5.25       => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_2999_odd__0__le__power__imp__0__le,axiom,
% 9.49/5.25      ! [A: int,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.49/5.25       => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3000_odd__power__less__zero,axiom,
% 9.49/5.25      ! [A: real,N: nat] :
% 9.49/5.25        ( ( ord_less_real @ A @ zero_zero_real )
% 9.49/5.25       => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3001_odd__power__less__zero,axiom,
% 9.49/5.25      ! [A: rat,N: nat] :
% 9.49/5.25        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.49/5.25       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3002_odd__power__less__zero,axiom,
% 9.49/5.25      ! [A: int,N: nat] :
% 9.49/5.25        ( ( ord_less_int @ A @ zero_zero_int )
% 9.49/5.25       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3003_ex__power__ivl2,axiom,
% 9.49/5.25      ! [B: nat,K: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 9.49/5.25       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.49/5.25         => ? [N3: nat] :
% 9.49/5.25              ( ( ord_less_nat @ ( power_power_nat @ B @ N3 ) @ K )
% 9.49/5.25              & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3004_ex__power__ivl1,axiom,
% 9.49/5.25      ! [B: nat,K: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 9.49/5.25       => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 9.49/5.25         => ? [N3: nat] :
% 9.49/5.25              ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N3 ) @ K )
% 9.49/5.25              & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3005_vebt__insert_Osimps_I2_J,axiom,
% 9.49/5.25      ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
% 9.49/5.25        ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) @ X )
% 9.49/5.25        = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3006_vebt__insert_Osimps_I3_J,axiom,
% 9.49/5.25      ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
% 9.49/5.25        ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X )
% 9.49/5.25        = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3007_aaaa,axiom,
% 9.49/5.25      ( ( ya = mi )
% 9.49/5.25      | ( ( ( xa = ma )
% 9.49/5.25         => ( ya
% 9.49/5.25            = ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ) ) )
% 9.49/5.25        & ( ( xa != ma )
% 9.49/5.25         => ( ya = ma ) ) )
% 9.49/5.25      | ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( vEBT_VEBT_high @ ya @ na ) ) @ ( vEBT_VEBT_low @ ya @ na ) )
% 9.49/5.25        & ( ord_less_nat @ ( vEBT_VEBT_high @ ya @ na ) @ ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3008__092_060open_062vebt__delete_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_061_ANode_A_ISome_A_Imi_M_Aif_Ax_A_061_Ama_Athen_Ahigh_Ax_An_A_K_A2_A_094_A_Ideg_Adiv_A2_J_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Ahigh_Ax_An_J_J_Aelse_Ama_J_J_Adeg_A_ItreeList_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_J_Asummary_092_060close_062,axiom,
% 9.49/5.25      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 9.49/5.25      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ( if_nat @ ( xa = ma ) @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ) @ ma ) ) ) @ deg @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ summary ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3009_one__div__two__eq__zero,axiom,
% 9.49/5.25      ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.25      = zero_zero_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3010_one__div__two__eq__zero,axiom,
% 9.49/5.25      ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.25      = zero_zero_int ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3011_one__div__two__eq__zero,axiom,
% 9.49/5.25      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.49/5.25      = zero_z3403309356797280102nteger ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3012__092_060open_062_Iif_Ax_A_061_Ama_Athen_Ahigh_Ax_An_A_K_A2_A_094_A_Ideg_Adiv_A2_J_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Ahigh_Ax_An_J_J_Aelse_Ama_J_A_061_Ahigh_Ax_An_A_K_A2_A_094_A_Ideg_Adiv_A2_J_A_L_Athe_A_Ivebt__maxt_A_ItreeList_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Ahigh_Ax_An_J_J_092_060close_062,axiom,
% 9.49/5.25      ( ( xa != ma )
% 9.49/5.25     => ( ma
% 9.49/5.25        = ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3013_del__x__mi__lets__in__not__minNull,axiom,
% 9.49/5.25      ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
% 9.49/5.25        ( ( ( X = Mi )
% 9.49/5.25          & ( ord_less_nat @ X @ Ma ) )
% 9.49/5.25       => ( ( Mi != Ma )
% 9.49/5.25         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.25           => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.25                = H2 )
% 9.49/5.25             => ( ( Xn
% 9.49/5.25                  = ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
% 9.49/5.25               => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.25                    = L )
% 9.49/5.25                 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.25                   => ( ( Newnode
% 9.49/5.25                        = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
% 9.49/5.25                     => ( ( Newlist
% 9.49/5.25                          = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
% 9.49/5.25                       => ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 9.49/5.25                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.25                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3014_del__x__not__mi__newnode__not__nil,axiom,
% 9.49/5.25      ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.49/5.25        ( ( ( ord_less_nat @ Mi @ X )
% 9.49/5.25          & ( ord_less_eq_nat @ X @ Ma ) )
% 9.49/5.25       => ( ( Mi != Ma )
% 9.49/5.25         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.25           => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.25                = H2 )
% 9.49/5.25             => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.25                  = L )
% 9.49/5.25               => ( ( Newnode
% 9.49/5.25                    = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
% 9.49/5.25                 => ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 9.49/5.25                   => ( ( Newlist
% 9.49/5.25                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
% 9.49/5.25                     => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.25                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.25                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3015_insert__simp__excp,axiom,
% 9.49/5.25      ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
% 9.49/5.25        ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.25       => ( ( ord_less_nat @ X @ Mi )
% 9.49/5.25         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.25           => ( ( X != Ma )
% 9.49/5.25             => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.25                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3016_insert__simp__norm,axiom,
% 9.49/5.25      ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 9.49/5.25        ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.25       => ( ( ord_less_nat @ Mi @ X )
% 9.49/5.25         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.25           => ( ( X != Ma )
% 9.49/5.25             => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.25                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3017_nested__mint,axiom,
% 9.49/5.25      ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
% 9.49/5.25        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 9.49/5.25       => ( ( N
% 9.49/5.25            = ( suc @ ( suc @ Va ) ) )
% 9.49/5.25         => ( ~ ( ord_less_nat @ Ma @ Mi )
% 9.49/5.25           => ( ( Ma != Mi )
% 9.49/5.25             => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3018_div__mult__self__is__m,axiom,
% 9.49/5.25      ! [N: nat,M: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
% 9.49/5.25          = M ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3019_half__negative__int__iff,axiom,
% 9.49/5.25      ! [K: int] :
% 9.49/5.25        ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 9.49/5.25        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3020_max__bot2,axiom,
% 9.49/5.25      ! [X: set_Pr1261947904930325089at_nat] :
% 9.49/5.25        ( ( ord_ma7524802468073614006at_nat @ X @ bot_bo2099793752762293965at_nat )
% 9.49/5.25        = X ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3021_max__bot2,axiom,
% 9.49/5.25      ! [X: set_real] :
% 9.49/5.25        ( ( ord_max_set_real @ X @ bot_bot_set_real )
% 9.49/5.25        = X ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3022_max__bot2,axiom,
% 9.49/5.25      ! [X: set_nat] :
% 9.49/5.25        ( ( ord_max_set_nat @ X @ bot_bot_set_nat )
% 9.49/5.25        = X ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3023_max__bot2,axiom,
% 9.49/5.25      ! [X: set_int] :
% 9.49/5.25        ( ( ord_max_set_int @ X @ bot_bot_set_int )
% 9.49/5.25        = X ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3024_max__bot2,axiom,
% 9.49/5.25      ! [X: nat] :
% 9.49/5.25        ( ( ord_max_nat @ X @ bot_bot_nat )
% 9.49/5.25        = X ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3025_max__bot,axiom,
% 9.49/5.25      ! [X: set_Pr1261947904930325089at_nat] :
% 9.49/5.25        ( ( ord_ma7524802468073614006at_nat @ bot_bo2099793752762293965at_nat @ X )
% 9.49/5.25        = X ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3026_max__bot,axiom,
% 9.49/5.25      ! [X: set_real] :
% 9.49/5.25        ( ( ord_max_set_real @ bot_bot_set_real @ X )
% 9.49/5.25        = X ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3027_max__bot,axiom,
% 9.49/5.25      ! [X: set_nat] :
% 9.49/5.25        ( ( ord_max_set_nat @ bot_bot_set_nat @ X )
% 9.49/5.25        = X ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3028_max__bot,axiom,
% 9.49/5.25      ! [X: set_int] :
% 9.49/5.25        ( ( ord_max_set_int @ bot_bot_set_int @ X )
% 9.49/5.25        = X ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3029_max__bot,axiom,
% 9.49/5.25      ! [X: nat] :
% 9.49/5.25        ( ( ord_max_nat @ bot_bot_nat @ X )
% 9.49/5.25        = X ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3030_max__Suc__Suc,axiom,
% 9.49/5.25      ! [M: nat,N: nat] :
% 9.49/5.25        ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
% 9.49/5.25        = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3031_max__0R,axiom,
% 9.49/5.25      ! [N: nat] :
% 9.49/5.25        ( ( ord_max_nat @ N @ zero_zero_nat )
% 9.49/5.25        = N ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3032_max__0L,axiom,
% 9.49/5.25      ! [N: nat] :
% 9.49/5.25        ( ( ord_max_nat @ zero_zero_nat @ N )
% 9.49/5.25        = N ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3033_max__nat_Oright__neutral,axiom,
% 9.49/5.25      ! [A: nat] :
% 9.49/5.25        ( ( ord_max_nat @ A @ zero_zero_nat )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3034_max__nat_Oneutr__eq__iff,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( zero_zero_nat
% 9.49/5.25          = ( ord_max_nat @ A @ B ) )
% 9.49/5.25        = ( ( A = zero_zero_nat )
% 9.49/5.25          & ( B = zero_zero_nat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3035_max__nat_Oleft__neutral,axiom,
% 9.49/5.25      ! [A: nat] :
% 9.49/5.25        ( ( ord_max_nat @ zero_zero_nat @ A )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3036_max__nat_Oeq__neutr__iff,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( ( ord_max_nat @ A @ B )
% 9.49/5.25          = zero_zero_nat )
% 9.49/5.25        = ( ( A = zero_zero_nat )
% 9.49/5.25          & ( B = zero_zero_nat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3037_summaxma,axiom,
% 9.49/5.25      ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.49/5.25        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
% 9.49/5.25       => ( ( Mi != Ma )
% 9.49/5.25         => ( ( the_nat2 @ ( vEBT_vebt_maxt @ Summary ) )
% 9.49/5.25            = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3038_div__mult__mult1__if,axiom,
% 9.49/5.25      ! [C: nat,A: nat,B: nat] :
% 9.49/5.25        ( ( ( C = zero_zero_nat )
% 9.49/5.25         => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 9.49/5.25            = zero_zero_nat ) )
% 9.49/5.25        & ( ( C != zero_zero_nat )
% 9.49/5.25         => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 9.49/5.25            = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3039_div__mult__mult1__if,axiom,
% 9.49/5.25      ! [C: int,A: int,B: int] :
% 9.49/5.25        ( ( ( C = zero_zero_int )
% 9.49/5.25         => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.49/5.25            = zero_zero_int ) )
% 9.49/5.25        & ( ( C != zero_zero_int )
% 9.49/5.25         => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.49/5.25            = ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3040_div__mult__mult1__if,axiom,
% 9.49/5.25      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.49/5.25        ( ( ( C = zero_z3403309356797280102nteger )
% 9.49/5.25         => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 9.49/5.25            = zero_z3403309356797280102nteger ) )
% 9.49/5.25        & ( ( C != zero_z3403309356797280102nteger )
% 9.49/5.25         => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 9.49/5.25            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3041_div__mult__mult2,axiom,
% 9.49/5.25      ! [C: nat,A: nat,B: nat] :
% 9.49/5.25        ( ( C != zero_zero_nat )
% 9.49/5.25       => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 9.49/5.25          = ( divide_divide_nat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3042_div__mult__mult2,axiom,
% 9.49/5.25      ! [C: int,A: int,B: int] :
% 9.49/5.25        ( ( C != zero_zero_int )
% 9.49/5.25       => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.49/5.25          = ( divide_divide_int @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3043_div__mult__mult2,axiom,
% 9.49/5.25      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.49/5.25        ( ( C != zero_z3403309356797280102nteger )
% 9.49/5.25       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 9.49/5.25          = ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3044_div__mult__mult1,axiom,
% 9.49/5.25      ! [C: nat,A: nat,B: nat] :
% 9.49/5.25        ( ( C != zero_zero_nat )
% 9.49/5.25       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 9.49/5.25          = ( divide_divide_nat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3045_div__mult__mult1,axiom,
% 9.49/5.25      ! [C: int,A: int,B: int] :
% 9.49/5.25        ( ( C != zero_zero_int )
% 9.49/5.25       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.49/5.25          = ( divide_divide_int @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3046_div__mult__mult1,axiom,
% 9.49/5.25      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.49/5.25        ( ( C != zero_z3403309356797280102nteger )
% 9.49/5.25       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 9.49/5.25          = ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3047_max__number__of_I1_J,axiom,
% 9.49/5.25      ! [U: num,V: num] :
% 9.49/5.25        ( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 9.49/5.25         => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 9.49/5.25            = ( numera1916890842035813515d_enat @ V ) ) )
% 9.49/5.25        & ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 9.49/5.25         => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 9.49/5.25            = ( numera1916890842035813515d_enat @ U ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3048_max__number__of_I1_J,axiom,
% 9.49/5.25      ! [U: num,V: num] :
% 9.49/5.25        ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 9.49/5.25         => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 9.49/5.25            = ( numeral_numeral_real @ V ) ) )
% 9.49/5.25        & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 9.49/5.25         => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 9.49/5.25            = ( numeral_numeral_real @ U ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3049_max__number__of_I1_J,axiom,
% 9.49/5.25      ! [U: num,V: num] :
% 9.49/5.25        ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 9.49/5.25         => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 9.49/5.25            = ( numeral_numeral_rat @ V ) ) )
% 9.49/5.25        & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 9.49/5.25         => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 9.49/5.25            = ( numeral_numeral_rat @ U ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3050_max__number__of_I1_J,axiom,
% 9.49/5.25      ! [U: num,V: num] :
% 9.49/5.25        ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 9.49/5.25         => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 9.49/5.25            = ( numeral_numeral_nat @ V ) ) )
% 9.49/5.25        & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 9.49/5.25         => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 9.49/5.25            = ( numeral_numeral_nat @ U ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3051_max__number__of_I1_J,axiom,
% 9.49/5.25      ! [U: num,V: num] :
% 9.49/5.25        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 9.49/5.25         => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 9.49/5.25            = ( numeral_numeral_int @ V ) ) )
% 9.49/5.25        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 9.49/5.25         => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 9.49/5.25            = ( numeral_numeral_int @ U ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3052_max__0__1_I3_J,axiom,
% 9.49/5.25      ! [X: num] :
% 9.49/5.25        ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X ) )
% 9.49/5.25        = ( numera1916890842035813515d_enat @ X ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3053_max__0__1_I3_J,axiom,
% 9.49/5.25      ! [X: num] :
% 9.49/5.25        ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X ) )
% 9.49/5.25        = ( numeral_numeral_real @ X ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3054_max__0__1_I3_J,axiom,
% 9.49/5.25      ! [X: num] :
% 9.49/5.25        ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
% 9.49/5.25        = ( numeral_numeral_rat @ X ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3055_max__0__1_I3_J,axiom,
% 9.49/5.25      ! [X: num] :
% 9.49/5.25        ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
% 9.49/5.25        = ( numeral_numeral_nat @ X ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3056_max__0__1_I3_J,axiom,
% 9.49/5.25      ! [X: num] :
% 9.49/5.25        ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
% 9.49/5.25        = ( numeral_numeral_int @ X ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3057_max__0__1_I4_J,axiom,
% 9.49/5.25      ! [X: num] :
% 9.49/5.25        ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ zero_z5237406670263579293d_enat )
% 9.49/5.25        = ( numera1916890842035813515d_enat @ X ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3058_max__0__1_I4_J,axiom,
% 9.49/5.25      ! [X: num] :
% 9.49/5.25        ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ zero_zero_real )
% 9.49/5.25        = ( numeral_numeral_real @ X ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3059_max__0__1_I4_J,axiom,
% 9.49/5.25      ! [X: num] :
% 9.49/5.25        ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
% 9.49/5.25        = ( numeral_numeral_rat @ X ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3060_max__0__1_I4_J,axiom,
% 9.49/5.25      ! [X: num] :
% 9.49/5.25        ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
% 9.49/5.25        = ( numeral_numeral_nat @ X ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3061_max__0__1_I4_J,axiom,
% 9.49/5.25      ! [X: num] :
% 9.49/5.25        ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
% 9.49/5.25        = ( numeral_numeral_int @ X ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3062_max__0__1_I2_J,axiom,
% 9.49/5.25      ( ( ord_max_real @ one_one_real @ zero_zero_real )
% 9.49/5.25      = one_one_real ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3063_max__0__1_I2_J,axiom,
% 9.49/5.25      ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
% 9.49/5.25      = one_one_rat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3064_max__0__1_I2_J,axiom,
% 9.49/5.25      ( ( ord_max_int @ one_one_int @ zero_zero_int )
% 9.49/5.25      = one_one_int ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3065_max__0__1_I2_J,axiom,
% 9.49/5.25      ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
% 9.49/5.25      = one_one_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3066_max__0__1_I1_J,axiom,
% 9.49/5.25      ( ( ord_max_real @ zero_zero_real @ one_one_real )
% 9.49/5.25      = one_one_real ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3067_max__0__1_I1_J,axiom,
% 9.49/5.25      ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
% 9.49/5.25      = one_one_rat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3068_max__0__1_I1_J,axiom,
% 9.49/5.25      ( ( ord_max_int @ zero_zero_int @ one_one_int )
% 9.49/5.25      = one_one_int ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3069_max__0__1_I1_J,axiom,
% 9.49/5.25      ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
% 9.49/5.25      = one_one_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3070_div__by__Suc__0,axiom,
% 9.49/5.25      ! [M: nat] :
% 9.49/5.25        ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 9.49/5.25        = M ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3071_div__less,axiom,
% 9.49/5.25      ! [M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_nat @ M @ N )
% 9.49/5.25       => ( ( divide_divide_nat @ M @ N )
% 9.49/5.25          = zero_zero_nat ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3072_option_Ocollapse,axiom,
% 9.49/5.25      ! [Option: option_nat] :
% 9.49/5.25        ( ( Option != none_nat )
% 9.49/5.25       => ( ( some_nat @ ( the_nat2 @ Option ) )
% 9.49/5.25          = Option ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3073_option_Ocollapse,axiom,
% 9.49/5.25      ! [Option: option4927543243414619207at_nat] :
% 9.49/5.25        ( ( Option != none_P5556105721700978146at_nat )
% 9.49/5.25       => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
% 9.49/5.25          = Option ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3074_option_Ocollapse,axiom,
% 9.49/5.25      ! [Option: option_num] :
% 9.49/5.25        ( ( Option != none_num )
% 9.49/5.25       => ( ( some_num @ ( the_num @ Option ) )
% 9.49/5.25          = Option ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3075_div__mult__self4,axiom,
% 9.49/5.25      ! [B: nat,C: nat,A: nat] :
% 9.49/5.25        ( ( B != zero_zero_nat )
% 9.49/5.25       => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 9.49/5.25          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3076_div__mult__self4,axiom,
% 9.49/5.25      ! [B: int,C: int,A: int] :
% 9.49/5.25        ( ( B != zero_zero_int )
% 9.49/5.25       => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 9.49/5.25          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3077_div__mult__self4,axiom,
% 9.49/5.25      ! [B: code_integer,C: code_integer,A: code_integer] :
% 9.49/5.25        ( ( B != zero_z3403309356797280102nteger )
% 9.49/5.25       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
% 9.49/5.25          = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3078_div__mult__self3,axiom,
% 9.49/5.25      ! [B: nat,C: nat,A: nat] :
% 9.49/5.25        ( ( B != zero_zero_nat )
% 9.49/5.25       => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 9.49/5.25          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3079_div__mult__self3,axiom,
% 9.49/5.25      ! [B: int,C: int,A: int] :
% 9.49/5.25        ( ( B != zero_zero_int )
% 9.49/5.25       => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 9.49/5.25          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3080_div__mult__self3,axiom,
% 9.49/5.25      ! [B: code_integer,C: code_integer,A: code_integer] :
% 9.49/5.25        ( ( B != zero_z3403309356797280102nteger )
% 9.49/5.25       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
% 9.49/5.25          = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3081_div__mult__self2,axiom,
% 9.49/5.25      ! [B: nat,A: nat,C: nat] :
% 9.49/5.25        ( ( B != zero_zero_nat )
% 9.49/5.25       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 9.49/5.25          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3082_div__mult__self2,axiom,
% 9.49/5.25      ! [B: int,A: int,C: int] :
% 9.49/5.25        ( ( B != zero_zero_int )
% 9.49/5.25       => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 9.49/5.25          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3083_div__mult__self2,axiom,
% 9.49/5.25      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.49/5.25        ( ( B != zero_z3403309356797280102nteger )
% 9.49/5.25       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
% 9.49/5.25          = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3084_div__mult__self1,axiom,
% 9.49/5.25      ! [B: nat,A: nat,C: nat] :
% 9.49/5.25        ( ( B != zero_zero_nat )
% 9.49/5.25       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 9.49/5.25          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3085_div__mult__self1,axiom,
% 9.49/5.25      ! [B: int,A: int,C: int] :
% 9.49/5.25        ( ( B != zero_zero_int )
% 9.49/5.25       => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 9.49/5.25          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3086_div__mult__self1,axiom,
% 9.49/5.25      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.49/5.25        ( ( B != zero_z3403309356797280102nteger )
% 9.49/5.25       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
% 9.49/5.25          = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3087_div__mult__self1__is__m,axiom,
% 9.49/5.25      ! [N: nat,M: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
% 9.49/5.25          = M ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3088_max__absorb2,axiom,
% 9.49/5.25      ! [X: set_int,Y: set_int] :
% 9.49/5.25        ( ( ord_less_eq_set_int @ X @ Y )
% 9.49/5.25       => ( ( ord_max_set_int @ X @ Y )
% 9.49/5.25          = Y ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3089_max__absorb2,axiom,
% 9.49/5.25      ! [X: rat,Y: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ X @ Y )
% 9.49/5.25       => ( ( ord_max_rat @ X @ Y )
% 9.49/5.25          = Y ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3090_max__absorb2,axiom,
% 9.49/5.25      ! [X: num,Y: num] :
% 9.49/5.25        ( ( ord_less_eq_num @ X @ Y )
% 9.49/5.25       => ( ( ord_max_num @ X @ Y )
% 9.49/5.25          = Y ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3091_max__absorb2,axiom,
% 9.49/5.25      ! [X: nat,Y: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ X @ Y )
% 9.49/5.25       => ( ( ord_max_nat @ X @ Y )
% 9.49/5.25          = Y ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3092_max__absorb2,axiom,
% 9.49/5.25      ! [X: int,Y: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ X @ Y )
% 9.49/5.25       => ( ( ord_max_int @ X @ Y )
% 9.49/5.25          = Y ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3093_max__absorb1,axiom,
% 9.49/5.25      ! [Y: set_int,X: set_int] :
% 9.49/5.25        ( ( ord_less_eq_set_int @ Y @ X )
% 9.49/5.25       => ( ( ord_max_set_int @ X @ Y )
% 9.49/5.25          = X ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3094_max__absorb1,axiom,
% 9.49/5.25      ! [Y: rat,X: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ Y @ X )
% 9.49/5.25       => ( ( ord_max_rat @ X @ Y )
% 9.49/5.25          = X ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3095_max__absorb1,axiom,
% 9.49/5.25      ! [Y: num,X: num] :
% 9.49/5.25        ( ( ord_less_eq_num @ Y @ X )
% 9.49/5.25       => ( ( ord_max_num @ X @ Y )
% 9.49/5.25          = X ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3096_max__absorb1,axiom,
% 9.49/5.25      ! [Y: nat,X: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ Y @ X )
% 9.49/5.25       => ( ( ord_max_nat @ X @ Y )
% 9.49/5.25          = X ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3097_max__absorb1,axiom,
% 9.49/5.25      ! [Y: int,X: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ Y @ X )
% 9.49/5.25       => ( ( ord_max_int @ X @ Y )
% 9.49/5.25          = X ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3098_max__def,axiom,
% 9.49/5.25      ( ord_max_set_int
% 9.49/5.25      = ( ^ [A4: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3099_max__def,axiom,
% 9.49/5.25      ( ord_max_rat
% 9.49/5.25      = ( ^ [A4: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3100_max__def,axiom,
% 9.49/5.25      ( ord_max_num
% 9.49/5.25      = ( ^ [A4: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3101_max__def,axiom,
% 9.49/5.25      ( ord_max_nat
% 9.49/5.25      = ( ^ [A4: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3102_max__def,axiom,
% 9.49/5.25      ( ord_max_int
% 9.49/5.25      = ( ^ [A4: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3103_max__add__distrib__left,axiom,
% 9.49/5.25      ! [X: real,Y: real,Z2: real] :
% 9.49/5.25        ( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z2 )
% 9.49/5.25        = ( ord_max_real @ ( plus_plus_real @ X @ Z2 ) @ ( plus_plus_real @ Y @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3104_max__add__distrib__left,axiom,
% 9.49/5.25      ! [X: rat,Y: rat,Z2: rat] :
% 9.49/5.25        ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z2 )
% 9.49/5.25        = ( ord_max_rat @ ( plus_plus_rat @ X @ Z2 ) @ ( plus_plus_rat @ Y @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3105_max__add__distrib__left,axiom,
% 9.49/5.25      ! [X: int,Y: int,Z2: int] :
% 9.49/5.25        ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z2 )
% 9.49/5.25        = ( ord_max_int @ ( plus_plus_int @ X @ Z2 ) @ ( plus_plus_int @ Y @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3106_max__add__distrib__left,axiom,
% 9.49/5.25      ! [X: nat,Y: nat,Z2: nat] :
% 9.49/5.25        ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z2 )
% 9.49/5.25        = ( ord_max_nat @ ( plus_plus_nat @ X @ Z2 ) @ ( plus_plus_nat @ Y @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3107_max__add__distrib__right,axiom,
% 9.49/5.25      ! [X: real,Y: real,Z2: real] :
% 9.49/5.25        ( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z2 ) )
% 9.49/5.25        = ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3108_max__add__distrib__right,axiom,
% 9.49/5.25      ! [X: rat,Y: rat,Z2: rat] :
% 9.49/5.25        ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z2 ) )
% 9.49/5.25        = ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3109_max__add__distrib__right,axiom,
% 9.49/5.25      ! [X: int,Y: int,Z2: int] :
% 9.49/5.25        ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z2 ) )
% 9.49/5.25        = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3110_max__add__distrib__right,axiom,
% 9.49/5.25      ! [X: nat,Y: nat,Z2: nat] :
% 9.49/5.25        ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z2 ) )
% 9.49/5.25        = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3111_nat__add__max__left,axiom,
% 9.49/5.25      ! [M: nat,N: nat,Q4: nat] :
% 9.49/5.25        ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q4 )
% 9.49/5.25        = ( ord_max_nat @ ( plus_plus_nat @ M @ Q4 ) @ ( plus_plus_nat @ N @ Q4 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3112_nat__add__max__right,axiom,
% 9.49/5.25      ! [M: nat,N: nat,Q4: nat] :
% 9.49/5.25        ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q4 ) )
% 9.49/5.25        = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q4 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3113_nat__mult__max__left,axiom,
% 9.49/5.25      ! [M: nat,N: nat,Q4: nat] :
% 9.49/5.25        ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q4 )
% 9.49/5.25        = ( ord_max_nat @ ( times_times_nat @ M @ Q4 ) @ ( times_times_nat @ N @ Q4 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3114_nat__mult__max__right,axiom,
% 9.49/5.25      ! [M: nat,N: nat,Q4: nat] :
% 9.49/5.25        ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q4 ) )
% 9.49/5.25        = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q4 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3115_option_Osel,axiom,
% 9.49/5.25      ! [X22: nat] :
% 9.49/5.25        ( ( the_nat2 @ ( some_nat @ X22 ) )
% 9.49/5.25        = X22 ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3116_option_Osel,axiom,
% 9.49/5.25      ! [X22: product_prod_nat_nat] :
% 9.49/5.25        ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 9.49/5.25        = X22 ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3117_option_Osel,axiom,
% 9.49/5.25      ! [X22: num] :
% 9.49/5.25        ( ( the_num @ ( some_num @ X22 ) )
% 9.49/5.25        = X22 ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3118_option_Oexpand,axiom,
% 9.49/5.25      ! [Option: option_nat,Option2: option_nat] :
% 9.49/5.25        ( ( ( Option = none_nat )
% 9.49/5.25          = ( Option2 = none_nat ) )
% 9.49/5.25       => ( ( ( Option != none_nat )
% 9.49/5.25           => ( ( Option2 != none_nat )
% 9.49/5.25             => ( ( the_nat2 @ Option )
% 9.49/5.25                = ( the_nat2 @ Option2 ) ) ) )
% 9.49/5.25         => ( Option = Option2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3119_option_Oexpand,axiom,
% 9.49/5.25      ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
% 9.49/5.25        ( ( ( Option = none_P5556105721700978146at_nat )
% 9.49/5.25          = ( Option2 = none_P5556105721700978146at_nat ) )
% 9.49/5.25       => ( ( ( Option != none_P5556105721700978146at_nat )
% 9.49/5.25           => ( ( Option2 != none_P5556105721700978146at_nat )
% 9.49/5.25             => ( ( the_Pr8591224930841456533at_nat @ Option )
% 9.49/5.25                = ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
% 9.49/5.25         => ( Option = Option2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3120_option_Oexpand,axiom,
% 9.49/5.25      ! [Option: option_num,Option2: option_num] :
% 9.49/5.25        ( ( ( Option = none_num )
% 9.49/5.25          = ( Option2 = none_num ) )
% 9.49/5.25       => ( ( ( Option != none_num )
% 9.49/5.25           => ( ( Option2 != none_num )
% 9.49/5.25             => ( ( the_num @ Option )
% 9.49/5.25                = ( the_num @ Option2 ) ) ) )
% 9.49/5.25         => ( Option = Option2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3121_option_Oexhaust__sel,axiom,
% 9.49/5.25      ! [Option: option_nat] :
% 9.49/5.25        ( ( Option != none_nat )
% 9.49/5.25       => ( Option
% 9.49/5.25          = ( some_nat @ ( the_nat2 @ Option ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3122_option_Oexhaust__sel,axiom,
% 9.49/5.25      ! [Option: option4927543243414619207at_nat] :
% 9.49/5.25        ( ( Option != none_P5556105721700978146at_nat )
% 9.49/5.25       => ( Option
% 9.49/5.25          = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3123_option_Oexhaust__sel,axiom,
% 9.49/5.25      ! [Option: option_num] :
% 9.49/5.25        ( ( Option != none_num )
% 9.49/5.25       => ( Option
% 9.49/5.25          = ( some_num @ ( the_num @ Option ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3124_div__le__dividend,axiom,
% 9.49/5.25      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3125_div__le__mono,axiom,
% 9.49/5.25      ! [M: nat,N: nat,K: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.25       => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3126_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 9.49/5.25      ! [M: nat,N: nat] :
% 9.49/5.25        ( ( ( divide_divide_nat @ M @ N )
% 9.49/5.25          = zero_zero_nat )
% 9.49/5.25        = ( ( ord_less_nat @ M @ N )
% 9.49/5.25          | ( N = zero_zero_nat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3127_Suc__div__le__mono,axiom,
% 9.49/5.25      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3128_less__mult__imp__div__less,axiom,
% 9.49/5.25      ! [M: nat,I: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
% 9.49/5.25       => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3129_div__times__less__eq__dividend,axiom,
% 9.49/5.25      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3130_times__div__less__eq__dividend,axiom,
% 9.49/5.25      ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3131_div__add__self2,axiom,
% 9.49/5.25      ! [B: nat,A: nat] :
% 9.49/5.25        ( ( B != zero_zero_nat )
% 9.49/5.25       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 9.49/5.25          = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3132_div__add__self2,axiom,
% 9.49/5.25      ! [B: int,A: int] :
% 9.49/5.25        ( ( B != zero_zero_int )
% 9.49/5.25       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
% 9.49/5.25          = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3133_div__add__self2,axiom,
% 9.49/5.25      ! [B: code_integer,A: code_integer] :
% 9.49/5.25        ( ( B != zero_z3403309356797280102nteger )
% 9.49/5.25       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
% 9.49/5.25          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3134_div__add__self1,axiom,
% 9.49/5.25      ! [B: nat,A: nat] :
% 9.49/5.25        ( ( B != zero_zero_nat )
% 9.49/5.25       => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 9.49/5.25          = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3135_div__add__self1,axiom,
% 9.49/5.25      ! [B: int,A: int] :
% 9.49/5.25        ( ( B != zero_zero_int )
% 9.49/5.25       => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
% 9.49/5.25          = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3136_div__add__self1,axiom,
% 9.49/5.25      ! [B: code_integer,A: code_integer] :
% 9.49/5.25        ( ( B != zero_z3403309356797280102nteger )
% 9.49/5.25       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
% 9.49/5.25          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3137_div__greater__zero__iff,axiom,
% 9.49/5.25      ! [M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
% 9.49/5.25        = ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.25          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3138_div__le__mono2,axiom,
% 9.49/5.25      ! [M: nat,N: nat,K: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.49/5.25       => ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.25         => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3139_div__less__iff__less__mult,axiom,
% 9.49/5.25      ! [Q4: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ Q4 )
% 9.49/5.25       => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q4 ) @ N )
% 9.49/5.25          = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q4 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3140_div__eq__dividend__iff,axiom,
% 9.49/5.25      ! [M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.49/5.25       => ( ( ( divide_divide_nat @ M @ N )
% 9.49/5.25            = M )
% 9.49/5.25          = ( N = one_one_nat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3141_div__less__dividend,axiom,
% 9.49/5.25      ! [N: nat,M: nat] :
% 9.49/5.25        ( ( ord_less_nat @ one_one_nat @ N )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.49/5.25         => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3142_div__nat__eqI,axiom,
% 9.49/5.25      ! [N: nat,Q4: nat,M: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
% 9.49/5.25       => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
% 9.49/5.25         => ( ( divide_divide_nat @ M @ N )
% 9.49/5.25            = Q4 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3143_less__eq__div__iff__mult__less__eq,axiom,
% 9.49/5.25      ! [Q4: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ Q4 )
% 9.49/5.25       => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q4 ) )
% 9.49/5.25          = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q4 ) @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3144_dividend__less__times__div,axiom,
% 9.49/5.25      ! [N: nat,M: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3145_dividend__less__div__times,axiom,
% 9.49/5.25      ! [N: nat,M: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3146_split__div,axiom,
% 9.49/5.25      ! [P: nat > $o,M: nat,N: nat] :
% 9.49/5.25        ( ( P @ ( divide_divide_nat @ M @ N ) )
% 9.49/5.25        = ( ( ( N = zero_zero_nat )
% 9.49/5.25           => ( P @ zero_zero_nat ) )
% 9.49/5.25          & ( ( N != zero_zero_nat )
% 9.49/5.25           => ! [I4: nat,J3: nat] :
% 9.49/5.25                ( ( ord_less_nat @ J3 @ N )
% 9.49/5.25               => ( ( M
% 9.49/5.25                    = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 9.49/5.25                 => ( P @ I4 ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3147_split__div_H,axiom,
% 9.49/5.25      ! [P: nat > $o,M: nat,N: nat] :
% 9.49/5.25        ( ( P @ ( divide_divide_nat @ M @ N ) )
% 9.49/5.25        = ( ( ( N = zero_zero_nat )
% 9.49/5.25            & ( P @ zero_zero_nat ) )
% 9.49/5.25          | ? [Q5: nat] :
% 9.49/5.25              ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q5 ) @ M )
% 9.49/5.25              & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q5 ) ) )
% 9.49/5.25              & ( P @ Q5 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3148_div__2__gt__zero,axiom,
% 9.49/5.25      ! [N: nat] :
% 9.49/5.25        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.49/5.25       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3149_Suc__n__div__2__gt__zero,axiom,
% 9.49/5.25      ! [N: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3150_arith__geo__mean,axiom,
% 9.49/5.25      ! [U: real,X: real,Y: real] :
% 9.49/5.25        ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.25          = ( times_times_real @ X @ Y ) )
% 9.49/5.25       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.49/5.25         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.49/5.25           => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3151_arith__geo__mean,axiom,
% 9.49/5.25      ! [U: rat,X: rat,Y: rat] :
% 9.49/5.25        ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.25          = ( times_times_rat @ X @ Y ) )
% 9.49/5.25       => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.49/5.25         => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.49/5.25           => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3152_mintlistlength,axiom,
% 9.49/5.25      ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 9.49/5.25        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 9.49/5.25       => ( ( Mi != Ma )
% 9.49/5.25         => ( ( ord_less_nat @ Mi @ Ma )
% 9.49/5.25            & ? [M5: nat] :
% 9.49/5.25                ( ( ( some_nat @ M5 )
% 9.49/5.25                  = ( vEBT_vebt_mint @ Summary ) )
% 9.49/5.25                & ( ord_less_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3153_sum__squares__bound,axiom,
% 9.49/5.25      ! [X: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3154_sum__squares__bound,axiom,
% 9.49/5.25      ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3155_mul__def,axiom,
% 9.49/5.25      ( vEBT_VEBT_mul
% 9.49/5.25      = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3156_nat__induct2,axiom,
% 9.49/5.25      ! [P: nat > $o,N: nat] :
% 9.49/5.25        ( ( P @ zero_zero_nat )
% 9.49/5.25       => ( ( P @ one_one_nat )
% 9.49/5.25         => ( ! [N3: nat] :
% 9.49/5.25                ( ( P @ N3 )
% 9.49/5.25               => ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.49/5.25           => ( P @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3157_set__bit__0,axiom,
% 9.49/5.25      ! [A: nat] :
% 9.49/5.25        ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
% 9.49/5.25        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3158_set__bit__0,axiom,
% 9.49/5.25      ! [A: code_integer] :
% 9.49/5.25        ( ( bit_se2793503036327961859nteger @ zero_zero_nat @ A )
% 9.49/5.25        = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3159_set__bit__0,axiom,
% 9.49/5.25      ! [A: int] :
% 9.49/5.25        ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
% 9.49/5.25        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3160_invar__vebt_Osimps,axiom,
% 9.49/5.25      ( vEBT_invar_vebt
% 9.49/5.25      = ( ^ [A1: vEBT_VEBT,A22: nat] :
% 9.49/5.25            ( ( ? [A4: $o,B4: $o] :
% 9.49/5.25                  ( A1
% 9.49/5.25                  = ( vEBT_Leaf @ A4 @ B4 ) )
% 9.49/5.25              & ( A22
% 9.49/5.25                = ( suc @ zero_zero_nat ) ) )
% 9.49/5.25            | ? [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT] :
% 9.49/5.25                ( ( A1
% 9.49/5.25                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList2 @ Summary2 ) )
% 9.49/5.25                & ! [X3: vEBT_VEBT] :
% 9.49/5.25                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.49/5.25                   => ( vEBT_invar_vebt @ X3 @ N2 ) )
% 9.49/5.25                & ( vEBT_invar_vebt @ Summary2 @ N2 )
% 9.49/5.25                & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.49/5.25                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.49/5.25                & ( A22
% 9.49/5.25                  = ( plus_plus_nat @ N2 @ N2 ) )
% 9.49/5.25                & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 9.49/5.25                & ! [X3: vEBT_VEBT] :
% 9.49/5.25                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.49/5.25                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 9.49/5.25            | ? [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT] :
% 9.49/5.25                ( ( A1
% 9.49/5.25                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList2 @ Summary2 ) )
% 9.49/5.25                & ! [X3: vEBT_VEBT] :
% 9.49/5.25                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.49/5.25                   => ( vEBT_invar_vebt @ X3 @ N2 ) )
% 9.49/5.25                & ( vEBT_invar_vebt @ Summary2 @ ( suc @ N2 ) )
% 9.49/5.25                & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.49/5.25                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 9.49/5.25                & ( A22
% 9.49/5.25                  = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
% 9.49/5.25                & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 9.49/5.25                & ! [X3: vEBT_VEBT] :
% 9.49/5.25                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.49/5.25                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 9.49/5.25            | ? [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
% 9.49/5.25                ( ( A1
% 9.49/5.25                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ A22 @ TreeList2 @ Summary2 ) )
% 9.49/5.25                & ! [X3: vEBT_VEBT] :
% 9.49/5.25                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.49/5.25                   => ( vEBT_invar_vebt @ X3 @ N2 ) )
% 9.49/5.25                & ( vEBT_invar_vebt @ Summary2 @ N2 )
% 9.49/5.25                & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.49/5.25                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.49/5.25                & ( A22
% 9.49/5.25                  = ( plus_plus_nat @ N2 @ N2 ) )
% 9.49/5.25                & ! [I4: nat] :
% 9.49/5.25                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.49/5.25                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
% 9.49/5.25                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 9.49/5.25                & ( ( Mi2 = Ma2 )
% 9.49/5.25                 => ! [X3: vEBT_VEBT] :
% 9.49/5.25                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.49/5.25                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 9.49/5.25                & ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 9.49/5.25                & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 9.49/5.25                & ( ( Mi2 != Ma2 )
% 9.49/5.25                 => ! [I4: nat] :
% 9.49/5.25                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.49/5.25                     => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
% 9.49/5.25                            = I4 )
% 9.49/5.25                         => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
% 9.49/5.25                        & ! [X3: nat] :
% 9.49/5.25                            ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
% 9.49/5.25                                = I4 )
% 9.49/5.25                              & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
% 9.49/5.25                           => ( ( ord_less_nat @ Mi2 @ X3 )
% 9.49/5.25                              & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) )
% 9.49/5.25            | ? [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
% 9.49/5.25                ( ( A1
% 9.49/5.25                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ A22 @ TreeList2 @ Summary2 ) )
% 9.49/5.25                & ! [X3: vEBT_VEBT] :
% 9.49/5.25                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.49/5.25                   => ( vEBT_invar_vebt @ X3 @ N2 ) )
% 9.49/5.25                & ( vEBT_invar_vebt @ Summary2 @ ( suc @ N2 ) )
% 9.49/5.25                & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.49/5.25                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 9.49/5.25                & ( A22
% 9.49/5.25                  = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
% 9.49/5.25                & ! [I4: nat] :
% 9.49/5.25                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 9.49/5.25                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
% 9.49/5.25                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 9.49/5.25                & ( ( Mi2 = Ma2 )
% 9.49/5.25                 => ! [X3: vEBT_VEBT] :
% 9.49/5.25                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.49/5.25                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 9.49/5.25                & ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 9.49/5.25                & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 9.49/5.25                & ( ( Mi2 != Ma2 )
% 9.49/5.25                 => ! [I4: nat] :
% 9.49/5.25                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 9.49/5.25                     => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
% 9.49/5.25                            = I4 )
% 9.49/5.25                         => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
% 9.49/5.25                        & ! [X3: nat] :
% 9.49/5.25                            ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
% 9.49/5.25                                = I4 )
% 9.49/5.25                              & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
% 9.49/5.25                           => ( ( ord_less_nat @ Mi2 @ X3 )
% 9.49/5.25                              & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3161_Leaf__0__not,axiom,
% 9.49/5.25      ! [A: $o,B: $o] :
% 9.49/5.25        ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3162_deg__1__Leafy,axiom,
% 9.49/5.25      ! [T: vEBT_VEBT,N: nat] :
% 9.49/5.25        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.25       => ( ( N = one_one_nat )
% 9.49/5.25         => ? [A3: $o,B3: $o] :
% 9.49/5.25              ( T
% 9.49/5.25              = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3163_deg__1__Leaf,axiom,
% 9.49/5.25      ! [T: vEBT_VEBT] :
% 9.49/5.25        ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 9.49/5.25       => ? [A3: $o,B3: $o] :
% 9.49/5.25            ( T
% 9.49/5.25            = ( vEBT_Leaf @ A3 @ B3 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3164_deg1Leaf,axiom,
% 9.49/5.25      ! [T: vEBT_VEBT] :
% 9.49/5.25        ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 9.49/5.25        = ( ? [A4: $o,B4: $o] :
% 9.49/5.25              ( T
% 9.49/5.25              = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3165_mul__shift,axiom,
% 9.49/5.25      ! [X: nat,Y: nat,Z2: nat] :
% 9.49/5.25        ( ( ( times_times_nat @ X @ Y )
% 9.49/5.25          = Z2 )
% 9.49/5.25        = ( ( vEBT_VEBT_mul @ ( some_nat @ X ) @ ( some_nat @ Y ) )
% 9.49/5.25          = ( some_nat @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3166_zle__add1__eq__le,axiom,
% 9.49/5.25      ! [W2: int,Z2: int] :
% 9.49/5.25        ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
% 9.49/5.25        = ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3167_VEBT_Oinject_I2_J,axiom,
% 9.49/5.25      ! [X21: $o,X222: $o,Y21: $o,Y222: $o] :
% 9.49/5.25        ( ( ( vEBT_Leaf @ X21 @ X222 )
% 9.49/5.25          = ( vEBT_Leaf @ Y21 @ Y222 ) )
% 9.49/5.25        = ( ( X21 = Y21 )
% 9.49/5.25          & ( X222 = Y222 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3168_power__minus__is__div,axiom,
% 9.49/5.25      ! [B: nat,A: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ B @ A )
% 9.49/5.25       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B ) )
% 9.49/5.25          = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3169_diff__self,axiom,
% 9.49/5.25      ! [A: complex] :
% 9.49/5.25        ( ( minus_minus_complex @ A @ A )
% 9.49/5.25        = zero_zero_complex ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3170_diff__self,axiom,
% 9.49/5.25      ! [A: real] :
% 9.49/5.25        ( ( minus_minus_real @ A @ A )
% 9.49/5.25        = zero_zero_real ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3171_diff__self,axiom,
% 9.49/5.25      ! [A: rat] :
% 9.49/5.25        ( ( minus_minus_rat @ A @ A )
% 9.49/5.25        = zero_zero_rat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3172_diff__self,axiom,
% 9.49/5.25      ! [A: int] :
% 9.49/5.25        ( ( minus_minus_int @ A @ A )
% 9.49/5.25        = zero_zero_int ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3173_diff__0__right,axiom,
% 9.49/5.25      ! [A: complex] :
% 9.49/5.25        ( ( minus_minus_complex @ A @ zero_zero_complex )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3174_diff__0__right,axiom,
% 9.49/5.25      ! [A: real] :
% 9.49/5.25        ( ( minus_minus_real @ A @ zero_zero_real )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3175_diff__0__right,axiom,
% 9.49/5.25      ! [A: rat] :
% 9.49/5.25        ( ( minus_minus_rat @ A @ zero_zero_rat )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3176_diff__0__right,axiom,
% 9.49/5.25      ! [A: int] :
% 9.49/5.25        ( ( minus_minus_int @ A @ zero_zero_int )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3177_zero__diff,axiom,
% 9.49/5.25      ! [A: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ zero_zero_nat @ A )
% 9.49/5.25        = zero_zero_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3178_diff__zero,axiom,
% 9.49/5.25      ! [A: complex] :
% 9.49/5.25        ( ( minus_minus_complex @ A @ zero_zero_complex )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3179_diff__zero,axiom,
% 9.49/5.25      ! [A: real] :
% 9.49/5.25        ( ( minus_minus_real @ A @ zero_zero_real )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3180_diff__zero,axiom,
% 9.49/5.25      ! [A: rat] :
% 9.49/5.25        ( ( minus_minus_rat @ A @ zero_zero_rat )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3181_diff__zero,axiom,
% 9.49/5.25      ! [A: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ A @ zero_zero_nat )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3182_diff__zero,axiom,
% 9.49/5.25      ! [A: int] :
% 9.49/5.25        ( ( minus_minus_int @ A @ zero_zero_int )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3183_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.49/5.25      ! [A: complex] :
% 9.49/5.25        ( ( minus_minus_complex @ A @ A )
% 9.49/5.25        = zero_zero_complex ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3184_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.49/5.25      ! [A: real] :
% 9.49/5.25        ( ( minus_minus_real @ A @ A )
% 9.49/5.25        = zero_zero_real ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3185_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.49/5.25      ! [A: rat] :
% 9.49/5.25        ( ( minus_minus_rat @ A @ A )
% 9.49/5.25        = zero_zero_rat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3186_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.49/5.25      ! [A: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ A @ A )
% 9.49/5.25        = zero_zero_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3187_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.49/5.25      ! [A: int] :
% 9.49/5.25        ( ( minus_minus_int @ A @ A )
% 9.49/5.25        = zero_zero_int ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3188_add__diff__cancel__right_H,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3189_add__diff__cancel__right_H,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3190_add__diff__cancel__right_H,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3191_add__diff__cancel__right_H,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3192_add__diff__cancel__right,axiom,
% 9.49/5.25      ! [A: real,C: real,B: real] :
% 9.49/5.25        ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 9.49/5.25        = ( minus_minus_real @ A @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3193_add__diff__cancel__right,axiom,
% 9.49/5.25      ! [A: rat,C: rat,B: rat] :
% 9.49/5.25        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 9.49/5.25        = ( minus_minus_rat @ A @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3194_add__diff__cancel__right,axiom,
% 9.49/5.25      ! [A: nat,C: nat,B: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 9.49/5.25        = ( minus_minus_nat @ A @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3195_add__diff__cancel__right,axiom,
% 9.49/5.25      ! [A: int,C: int,B: int] :
% 9.49/5.25        ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 9.49/5.25        = ( minus_minus_int @ A @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3196_add__diff__cancel__left_H,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
% 9.49/5.25        = B ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3197_add__diff__cancel__left_H,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
% 9.49/5.25        = B ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3198_add__diff__cancel__left_H,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 9.49/5.25        = B ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3199_add__diff__cancel__left_H,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 9.49/5.25        = B ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3200_add__diff__cancel__left,axiom,
% 9.49/5.25      ! [C: real,A: real,B: real] :
% 9.49/5.25        ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 9.49/5.25        = ( minus_minus_real @ A @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3201_add__diff__cancel__left,axiom,
% 9.49/5.25      ! [C: rat,A: rat,B: rat] :
% 9.49/5.25        ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 9.49/5.25        = ( minus_minus_rat @ A @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3202_add__diff__cancel__left,axiom,
% 9.49/5.25      ! [C: nat,A: nat,B: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 9.49/5.25        = ( minus_minus_nat @ A @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3203_add__diff__cancel__left,axiom,
% 9.49/5.25      ! [C: int,A: int,B: int] :
% 9.49/5.25        ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 9.49/5.25        = ( minus_minus_int @ A @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3204_diff__add__cancel,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3205_diff__add__cancel,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3206_diff__add__cancel,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3207_add__diff__cancel,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3208_add__diff__cancel,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3209_add__diff__cancel,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 9.49/5.25        = A ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3210_Suc__diff__diff,axiom,
% 9.49/5.25      ! [M: nat,N: nat,K: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 9.49/5.25        = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3211_diff__Suc__Suc,axiom,
% 9.49/5.25      ! [M: nat,N: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 9.49/5.25        = ( minus_minus_nat @ M @ N ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3212_diff__0__eq__0,axiom,
% 9.49/5.25      ! [N: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ zero_zero_nat @ N )
% 9.49/5.25        = zero_zero_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3213_diff__self__eq__0,axiom,
% 9.49/5.25      ! [M: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ M @ M )
% 9.49/5.25        = zero_zero_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3214_diff__diff__cancel,axiom,
% 9.49/5.25      ! [I: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ I @ N )
% 9.49/5.25       => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 9.49/5.25          = I ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3215_diff__diff__left,axiom,
% 9.49/5.25      ! [I: nat,J: nat,K: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 9.49/5.25        = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3216_set__bit__negative__int__iff,axiom,
% 9.49/5.25      ! [N: nat,K: int] :
% 9.49/5.25        ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
% 9.49/5.25        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3217_diff__ge__0__iff__ge,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 9.49/5.25        = ( ord_less_eq_real @ B @ A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3218_diff__ge__0__iff__ge,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 9.49/5.25        = ( ord_less_eq_rat @ B @ A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3219_diff__ge__0__iff__ge,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 9.49/5.25        = ( ord_less_eq_int @ B @ A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3220_diff__gt__0__iff__gt,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 9.49/5.25        = ( ord_less_real @ B @ A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3221_diff__gt__0__iff__gt,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 9.49/5.25        = ( ord_less_rat @ B @ A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3222_diff__gt__0__iff__gt,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 9.49/5.25        = ( ord_less_int @ B @ A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3223_le__add__diff__inverse2,axiom,
% 9.49/5.25      ! [B: real,A: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ B @ A )
% 9.49/5.25       => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3224_le__add__diff__inverse2,axiom,
% 9.49/5.25      ! [B: rat,A: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ B @ A )
% 9.49/5.25       => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3225_le__add__diff__inverse2,axiom,
% 9.49/5.25      ! [B: nat,A: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ B @ A )
% 9.49/5.25       => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3226_le__add__diff__inverse2,axiom,
% 9.49/5.25      ! [B: int,A: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ B @ A )
% 9.49/5.25       => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3227_le__add__diff__inverse,axiom,
% 9.49/5.25      ! [B: real,A: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ B @ A )
% 9.49/5.25       => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3228_le__add__diff__inverse,axiom,
% 9.49/5.25      ! [B: rat,A: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ B @ A )
% 9.49/5.25       => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3229_le__add__diff__inverse,axiom,
% 9.49/5.25      ! [B: nat,A: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ B @ A )
% 9.49/5.25       => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3230_le__add__diff__inverse,axiom,
% 9.49/5.25      ! [B: int,A: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ B @ A )
% 9.49/5.25       => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3231_diff__add__zero,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 9.49/5.25        = zero_zero_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3232_diff__numeral__special_I9_J,axiom,
% 9.49/5.25      ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 9.49/5.25      = zero_zero_complex ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3233_diff__numeral__special_I9_J,axiom,
% 9.49/5.25      ( ( minus_minus_real @ one_one_real @ one_one_real )
% 9.49/5.25      = zero_zero_real ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3234_diff__numeral__special_I9_J,axiom,
% 9.49/5.25      ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 9.49/5.25      = zero_zero_rat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3235_diff__numeral__special_I9_J,axiom,
% 9.49/5.25      ( ( minus_minus_int @ one_one_int @ one_one_int )
% 9.49/5.25      = zero_zero_int ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3236_zero__less__diff,axiom,
% 9.49/5.25      ! [N: nat,M: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 9.49/5.25        = ( ord_less_nat @ M @ N ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3237_diff__is__0__eq_H,axiom,
% 9.49/5.25      ! [M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.25       => ( ( minus_minus_nat @ M @ N )
% 9.49/5.25          = zero_zero_nat ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3238_diff__is__0__eq,axiom,
% 9.49/5.25      ! [M: nat,N: nat] :
% 9.49/5.25        ( ( ( minus_minus_nat @ M @ N )
% 9.49/5.25          = zero_zero_nat )
% 9.49/5.25        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3239_Nat_Oadd__diff__assoc,axiom,
% 9.49/5.25      ! [K: nat,J: nat,I: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ K @ J )
% 9.49/5.25       => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 9.49/5.25          = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3240_Nat_Oadd__diff__assoc2,axiom,
% 9.49/5.25      ! [K: nat,J: nat,I: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ K @ J )
% 9.49/5.25       => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 9.49/5.25          = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3241_Nat_Odiff__diff__right,axiom,
% 9.49/5.25      ! [K: nat,J: nat,I: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ K @ J )
% 9.49/5.25       => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 9.49/5.25          = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3242_diff__Suc__1,axiom,
% 9.49/5.25      ! [N: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
% 9.49/5.25        = N ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3243_div__pos__pos__trivial,axiom,
% 9.49/5.25      ! [K: int,L: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.49/5.25       => ( ( ord_less_int @ K @ L )
% 9.49/5.25         => ( ( divide_divide_int @ K @ L )
% 9.49/5.25            = zero_zero_int ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3244_div__neg__neg__trivial,axiom,
% 9.49/5.25      ! [K: int,L: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ K @ zero_zero_int )
% 9.49/5.25       => ( ( ord_less_int @ L @ K )
% 9.49/5.25         => ( ( divide_divide_int @ K @ L )
% 9.49/5.25            = zero_zero_int ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3245_Suc__pred,axiom,
% 9.49/5.25      ! [N: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 9.49/5.25          = N ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3246_diff__Suc__diff__eq2,axiom,
% 9.49/5.25      ! [K: nat,J: nat,I: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ K @ J )
% 9.49/5.25       => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
% 9.49/5.25          = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3247_diff__Suc__diff__eq1,axiom,
% 9.49/5.25      ! [K: nat,J: nat,I: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ K @ J )
% 9.49/5.25       => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 9.49/5.25          = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3248_Suc__diff__1,axiom,
% 9.49/5.25      ! [N: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
% 9.49/5.25          = N ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3249_pos__zmult__eq__1__iff,axiom,
% 9.49/5.25      ! [M: int,N: int] :
% 9.49/5.25        ( ( ord_less_int @ zero_zero_int @ M )
% 9.49/5.25       => ( ( ( times_times_int @ M @ N )
% 9.49/5.25            = one_one_int )
% 9.49/5.25          = ( ( M = one_one_int )
% 9.49/5.25            & ( N = one_one_int ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3250_zmult__zless__mono2,axiom,
% 9.49/5.25      ! [I: int,J: int,K: int] :
% 9.49/5.25        ( ( ord_less_int @ I @ J )
% 9.49/5.25       => ( ( ord_less_int @ zero_zero_int @ K )
% 9.49/5.25         => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3251_int__one__le__iff__zero__less,axiom,
% 9.49/5.25      ! [Z2: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ one_one_int @ Z2 )
% 9.49/5.25        = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3252_add1__zle__eq,axiom,
% 9.49/5.25      ! [W2: int,Z2: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 )
% 9.49/5.25        = ( ord_less_int @ W2 @ Z2 ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3253_le__imp__0__less,axiom,
% 9.49/5.25      ! [Z2: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 9.49/5.25       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3254_zless__imp__add1__zle,axiom,
% 9.49/5.25      ! [W2: int,Z2: int] :
% 9.49/5.25        ( ( ord_less_int @ W2 @ Z2 )
% 9.49/5.25       => ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3255_incr__mult__lemma,axiom,
% 9.49/5.25      ! [D: int,P: int > $o,K: int] :
% 9.49/5.25        ( ( ord_less_int @ zero_zero_int @ D )
% 9.49/5.25       => ( ! [X4: int] :
% 9.49/5.25              ( ( P @ X4 )
% 9.49/5.25             => ( P @ ( plus_plus_int @ X4 @ D ) ) )
% 9.49/5.25         => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.49/5.25           => ! [X2: int] :
% 9.49/5.25                ( ( P @ X2 )
% 9.49/5.25               => ( P @ ( plus_plus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3256_int__gr__induct,axiom,
% 9.49/5.25      ! [K: int,I: int,P: int > $o] :
% 9.49/5.25        ( ( ord_less_int @ K @ I )
% 9.49/5.25       => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 9.49/5.25         => ( ! [I2: int] :
% 9.49/5.25                ( ( ord_less_int @ K @ I2 )
% 9.49/5.25               => ( ( P @ I2 )
% 9.49/5.25                 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 9.49/5.25           => ( P @ I ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3257_zless__add1__eq,axiom,
% 9.49/5.25      ! [W2: int,Z2: int] :
% 9.49/5.25        ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
% 9.49/5.25        = ( ( ord_less_int @ W2 @ Z2 )
% 9.49/5.25          | ( W2 = Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3258_odd__less__0__iff,axiom,
% 9.49/5.25      ! [Z2: int] :
% 9.49/5.25        ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
% 9.49/5.25        = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3259_less__int__code_I1_J,axiom,
% 9.49/5.25      ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3260_diff__commute,axiom,
% 9.49/5.25      ! [I: nat,J: nat,K: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 9.49/5.25        = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3261_diff__right__commute,axiom,
% 9.49/5.25      ! [A: real,C: real,B: real] :
% 9.49/5.25        ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 9.49/5.25        = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3262_diff__right__commute,axiom,
% 9.49/5.25      ! [A: rat,C: rat,B: rat] :
% 9.49/5.25        ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
% 9.49/5.25        = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3263_diff__right__commute,axiom,
% 9.49/5.25      ! [A: nat,C: nat,B: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 9.49/5.25        = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3264_diff__right__commute,axiom,
% 9.49/5.25      ! [A: int,C: int,B: int] :
% 9.49/5.25        ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 9.49/5.25        = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3265_diff__eq__diff__eq,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real,D: real] :
% 9.49/5.25        ( ( ( minus_minus_real @ A @ B )
% 9.49/5.25          = ( minus_minus_real @ C @ D ) )
% 9.49/5.25       => ( ( A = B )
% 9.49/5.25          = ( C = D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3266_diff__eq__diff__eq,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.49/5.25        ( ( ( minus_minus_rat @ A @ B )
% 9.49/5.25          = ( minus_minus_rat @ C @ D ) )
% 9.49/5.25       => ( ( A = B )
% 9.49/5.25          = ( C = D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3267_diff__eq__diff__eq,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int,D: int] :
% 9.49/5.25        ( ( ( minus_minus_int @ A @ B )
% 9.49/5.25          = ( minus_minus_int @ C @ D ) )
% 9.49/5.25       => ( ( A = B )
% 9.49/5.25          = ( C = D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3268_VEBT_Osize_I4_J,axiom,
% 9.49/5.25      ! [X21: $o,X222: $o] :
% 9.49/5.25        ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 9.49/5.25        = zero_zero_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3269_diff__mono,axiom,
% 9.49/5.25      ! [A: real,B: real,D: real,C: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_real @ D @ C )
% 9.49/5.25         => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3270_diff__mono,axiom,
% 9.49/5.25      ! [A: rat,B: rat,D: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_rat @ D @ C )
% 9.49/5.25         => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3271_diff__mono,axiom,
% 9.49/5.25      ! [A: int,B: int,D: int,C: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_int @ D @ C )
% 9.49/5.25         => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3272_diff__left__mono,axiom,
% 9.49/5.25      ! [B: real,A: real,C: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ B @ A )
% 9.49/5.25       => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3273_diff__left__mono,axiom,
% 9.49/5.25      ! [B: rat,A: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ B @ A )
% 9.49/5.25       => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3274_diff__left__mono,axiom,
% 9.49/5.25      ! [B: int,A: int,C: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ B @ A )
% 9.49/5.25       => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3275_diff__right__mono,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.25       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3276_diff__right__mono,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.25       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3277_diff__right__mono,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.25       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3278_diff__eq__diff__less__eq,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real,D: real] :
% 9.49/5.25        ( ( ( minus_minus_real @ A @ B )
% 9.49/5.25          = ( minus_minus_real @ C @ D ) )
% 9.49/5.25       => ( ( ord_less_eq_real @ A @ B )
% 9.49/5.25          = ( ord_less_eq_real @ C @ D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3279_diff__eq__diff__less__eq,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.49/5.25        ( ( ( minus_minus_rat @ A @ B )
% 9.49/5.25          = ( minus_minus_rat @ C @ D ) )
% 9.49/5.25       => ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.25          = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3280_diff__eq__diff__less__eq,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int,D: int] :
% 9.49/5.25        ( ( ( minus_minus_int @ A @ B )
% 9.49/5.25          = ( minus_minus_int @ C @ D ) )
% 9.49/5.25       => ( ( ord_less_eq_int @ A @ B )
% 9.49/5.25          = ( ord_less_eq_int @ C @ D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3281_eq__iff__diff__eq__0,axiom,
% 9.49/5.25      ( ( ^ [Y5: complex,Z: complex] : ( Y5 = Z ) )
% 9.49/5.25      = ( ^ [A4: complex,B4: complex] :
% 9.49/5.25            ( ( minus_minus_complex @ A4 @ B4 )
% 9.49/5.25            = zero_zero_complex ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3282_eq__iff__diff__eq__0,axiom,
% 9.49/5.25      ( ( ^ [Y5: real,Z: real] : ( Y5 = Z ) )
% 9.49/5.25      = ( ^ [A4: real,B4: real] :
% 9.49/5.25            ( ( minus_minus_real @ A4 @ B4 )
% 9.49/5.25            = zero_zero_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3283_eq__iff__diff__eq__0,axiom,
% 9.49/5.25      ( ( ^ [Y5: rat,Z: rat] : ( Y5 = Z ) )
% 9.49/5.25      = ( ^ [A4: rat,B4: rat] :
% 9.49/5.25            ( ( minus_minus_rat @ A4 @ B4 )
% 9.49/5.25            = zero_zero_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3284_eq__iff__diff__eq__0,axiom,
% 9.49/5.25      ( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
% 9.49/5.25      = ( ^ [A4: int,B4: int] :
% 9.49/5.25            ( ( minus_minus_int @ A4 @ B4 )
% 9.49/5.25            = zero_zero_int ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3285_VEBT_Odistinct_I1_J,axiom,
% 9.49/5.25      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
% 9.49/5.25        ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 9.49/5.25       != ( vEBT_Leaf @ X21 @ X222 ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3286_VEBT_Oexhaust,axiom,
% 9.49/5.25      ! [Y: vEBT_VEBT] :
% 9.49/5.25        ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
% 9.49/5.25            ( Y
% 9.49/5.25           != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
% 9.49/5.25       => ~ ! [X212: $o,X223: $o] :
% 9.49/5.25              ( Y
% 9.49/5.25             != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3287_VEBT__internal_Ovalid_H_Ocases,axiom,
% 9.49/5.25      ! [X: produc9072475918466114483BT_nat] :
% 9.49/5.25        ( ! [Uu2: $o,Uv2: $o,D5: nat] :
% 9.49/5.25            ( X
% 9.49/5.25           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D5 ) )
% 9.49/5.25       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,Deg3: nat] :
% 9.49/5.25              ( X
% 9.49/5.25             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) @ Deg3 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3288_diff__strict__mono,axiom,
% 9.49/5.25      ! [A: real,B: real,D: real,C: real] :
% 9.49/5.25        ( ( ord_less_real @ A @ B )
% 9.49/5.25       => ( ( ord_less_real @ D @ C )
% 9.49/5.25         => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3289_diff__strict__mono,axiom,
% 9.49/5.25      ! [A: rat,B: rat,D: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_rat @ A @ B )
% 9.49/5.25       => ( ( ord_less_rat @ D @ C )
% 9.49/5.25         => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3290_diff__strict__mono,axiom,
% 9.49/5.25      ! [A: int,B: int,D: int,C: int] :
% 9.49/5.25        ( ( ord_less_int @ A @ B )
% 9.49/5.25       => ( ( ord_less_int @ D @ C )
% 9.49/5.25         => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3291_diff__eq__diff__less,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real,D: real] :
% 9.49/5.25        ( ( ( minus_minus_real @ A @ B )
% 9.49/5.25          = ( minus_minus_real @ C @ D ) )
% 9.49/5.25       => ( ( ord_less_real @ A @ B )
% 9.49/5.25          = ( ord_less_real @ C @ D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3292_diff__eq__diff__less,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.49/5.25        ( ( ( minus_minus_rat @ A @ B )
% 9.49/5.25          = ( minus_minus_rat @ C @ D ) )
% 9.49/5.25       => ( ( ord_less_rat @ A @ B )
% 9.49/5.25          = ( ord_less_rat @ C @ D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3293_diff__eq__diff__less,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int,D: int] :
% 9.49/5.25        ( ( ( minus_minus_int @ A @ B )
% 9.49/5.25          = ( minus_minus_int @ C @ D ) )
% 9.49/5.25       => ( ( ord_less_int @ A @ B )
% 9.49/5.25          = ( ord_less_int @ C @ D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3294_diff__strict__left__mono,axiom,
% 9.49/5.25      ! [B: real,A: real,C: real] :
% 9.49/5.25        ( ( ord_less_real @ B @ A )
% 9.49/5.25       => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3295_diff__strict__left__mono,axiom,
% 9.49/5.25      ! [B: rat,A: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_rat @ B @ A )
% 9.49/5.25       => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3296_diff__strict__left__mono,axiom,
% 9.49/5.25      ! [B: int,A: int,C: int] :
% 9.49/5.25        ( ( ord_less_int @ B @ A )
% 9.49/5.25       => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3297_diff__strict__right__mono,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real] :
% 9.49/5.25        ( ( ord_less_real @ A @ B )
% 9.49/5.25       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3298_diff__strict__right__mono,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_rat @ A @ B )
% 9.49/5.25       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3299_diff__strict__right__mono,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int] :
% 9.49/5.25        ( ( ord_less_int @ A @ B )
% 9.49/5.25       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3300_diff__diff__eq,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real] :
% 9.49/5.25        ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
% 9.49/5.25        = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3301_diff__diff__eq,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat] :
% 9.49/5.25        ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 9.49/5.25        = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3302_diff__diff__eq,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 9.49/5.25        = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3303_diff__diff__eq,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int] :
% 9.49/5.25        ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 9.49/5.25        = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3304_add__implies__diff,axiom,
% 9.49/5.25      ! [C: real,B: real,A: real] :
% 9.49/5.25        ( ( ( plus_plus_real @ C @ B )
% 9.49/5.25          = A )
% 9.49/5.25       => ( C
% 9.49/5.25          = ( minus_minus_real @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3305_add__implies__diff,axiom,
% 9.49/5.25      ! [C: rat,B: rat,A: rat] :
% 9.49/5.25        ( ( ( plus_plus_rat @ C @ B )
% 9.49/5.25          = A )
% 9.49/5.25       => ( C
% 9.49/5.25          = ( minus_minus_rat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3306_add__implies__diff,axiom,
% 9.49/5.25      ! [C: nat,B: nat,A: nat] :
% 9.49/5.25        ( ( ( plus_plus_nat @ C @ B )
% 9.49/5.25          = A )
% 9.49/5.25       => ( C
% 9.49/5.25          = ( minus_minus_nat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3307_add__implies__diff,axiom,
% 9.49/5.25      ! [C: int,B: int,A: int] :
% 9.49/5.25        ( ( ( plus_plus_int @ C @ B )
% 9.49/5.25          = A )
% 9.49/5.25       => ( C
% 9.49/5.25          = ( minus_minus_int @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3308_diff__add__eq__diff__diff__swap,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real] :
% 9.49/5.25        ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
% 9.49/5.25        = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3309_diff__add__eq__diff__diff__swap,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat] :
% 9.49/5.25        ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 9.49/5.25        = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3310_diff__add__eq__diff__diff__swap,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int] :
% 9.49/5.25        ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 9.49/5.25        = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3311_diff__add__eq,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real] :
% 9.49/5.25        ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
% 9.49/5.25        = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3312_diff__add__eq,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat] :
% 9.49/5.25        ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 9.49/5.25        = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3313_diff__add__eq,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int] :
% 9.49/5.25        ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 9.49/5.25        = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3314_diff__diff__eq2,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real] :
% 9.49/5.25        ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
% 9.49/5.25        = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3315_diff__diff__eq2,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat] :
% 9.49/5.25        ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 9.49/5.25        = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3316_diff__diff__eq2,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int] :
% 9.49/5.25        ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 9.49/5.25        = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3317_add__diff__eq,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real] :
% 9.49/5.25        ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
% 9.49/5.25        = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3318_add__diff__eq,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat] :
% 9.49/5.25        ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 9.49/5.25        = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3319_add__diff__eq,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int] :
% 9.49/5.25        ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 9.49/5.25        = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3320_eq__diff__eq,axiom,
% 9.49/5.25      ! [A: real,C: real,B: real] :
% 9.49/5.25        ( ( A
% 9.49/5.25          = ( minus_minus_real @ C @ B ) )
% 9.49/5.25        = ( ( plus_plus_real @ A @ B )
% 9.49/5.25          = C ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3321_eq__diff__eq,axiom,
% 9.49/5.25      ! [A: rat,C: rat,B: rat] :
% 9.49/5.25        ( ( A
% 9.49/5.25          = ( minus_minus_rat @ C @ B ) )
% 9.49/5.25        = ( ( plus_plus_rat @ A @ B )
% 9.49/5.25          = C ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3322_eq__diff__eq,axiom,
% 9.49/5.25      ! [A: int,C: int,B: int] :
% 9.49/5.25        ( ( A
% 9.49/5.25          = ( minus_minus_int @ C @ B ) )
% 9.49/5.25        = ( ( plus_plus_int @ A @ B )
% 9.49/5.25          = C ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3323_diff__eq__eq,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real] :
% 9.49/5.25        ( ( ( minus_minus_real @ A @ B )
% 9.49/5.25          = C )
% 9.49/5.25        = ( A
% 9.49/5.25          = ( plus_plus_real @ C @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3324_diff__eq__eq,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat] :
% 9.49/5.25        ( ( ( minus_minus_rat @ A @ B )
% 9.49/5.25          = C )
% 9.49/5.25        = ( A
% 9.49/5.25          = ( plus_plus_rat @ C @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3325_diff__eq__eq,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int] :
% 9.49/5.25        ( ( ( minus_minus_int @ A @ B )
% 9.49/5.25          = C )
% 9.49/5.25        = ( A
% 9.49/5.25          = ( plus_plus_int @ C @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3326_group__cancel_Osub1,axiom,
% 9.49/5.25      ! [A2: real,K: real,A: real,B: real] :
% 9.49/5.25        ( ( A2
% 9.49/5.25          = ( plus_plus_real @ K @ A ) )
% 9.49/5.25       => ( ( minus_minus_real @ A2 @ B )
% 9.49/5.25          = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3327_group__cancel_Osub1,axiom,
% 9.49/5.25      ! [A2: rat,K: rat,A: rat,B: rat] :
% 9.49/5.25        ( ( A2
% 9.49/5.25          = ( plus_plus_rat @ K @ A ) )
% 9.49/5.25       => ( ( minus_minus_rat @ A2 @ B )
% 9.49/5.25          = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3328_group__cancel_Osub1,axiom,
% 9.49/5.25      ! [A2: int,K: int,A: int,B: int] :
% 9.49/5.25        ( ( A2
% 9.49/5.25          = ( plus_plus_int @ K @ A ) )
% 9.49/5.25       => ( ( minus_minus_int @ A2 @ B )
% 9.49/5.25          = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3329_zero__induct__lemma,axiom,
% 9.49/5.25      ! [P: nat > $o,K: nat,I: nat] :
% 9.49/5.25        ( ( P @ K )
% 9.49/5.25       => ( ! [N3: nat] :
% 9.49/5.25              ( ( P @ ( suc @ N3 ) )
% 9.49/5.25             => ( P @ N3 ) )
% 9.49/5.25         => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3330_minus__nat_Odiff__0,axiom,
% 9.49/5.25      ! [M: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ M @ zero_zero_nat )
% 9.49/5.25        = M ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3331_diffs0__imp__equal,axiom,
% 9.49/5.25      ! [M: nat,N: nat] :
% 9.49/5.25        ( ( ( minus_minus_nat @ M @ N )
% 9.49/5.25          = zero_zero_nat )
% 9.49/5.25       => ( ( ( minus_minus_nat @ N @ M )
% 9.49/5.25            = zero_zero_nat )
% 9.49/5.25         => ( M = N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3332_diff__less__mono2,axiom,
% 9.49/5.25      ! [M: nat,N: nat,L: nat] :
% 9.49/5.25        ( ( ord_less_nat @ M @ N )
% 9.49/5.25       => ( ( ord_less_nat @ M @ L )
% 9.49/5.25         => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3333_less__imp__diff__less,axiom,
% 9.49/5.25      ! [J: nat,K: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_nat @ J @ K )
% 9.49/5.25       => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3334_eq__diff__iff,axiom,
% 9.49/5.25      ! [K: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ K @ M )
% 9.49/5.25       => ( ( ord_less_eq_nat @ K @ N )
% 9.49/5.25         => ( ( ( minus_minus_nat @ M @ K )
% 9.49/5.25              = ( minus_minus_nat @ N @ K ) )
% 9.49/5.25            = ( M = N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3335_le__diff__iff,axiom,
% 9.49/5.25      ! [K: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ K @ M )
% 9.49/5.25       => ( ( ord_less_eq_nat @ K @ N )
% 9.49/5.25         => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 9.49/5.25            = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3336_Nat_Odiff__diff__eq,axiom,
% 9.49/5.25      ! [K: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ K @ M )
% 9.49/5.25       => ( ( ord_less_eq_nat @ K @ N )
% 9.49/5.25         => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 9.49/5.25            = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3337_diff__le__mono,axiom,
% 9.49/5.25      ! [M: nat,N: nat,L: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.25       => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3338_diff__le__self,axiom,
% 9.49/5.25      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3339_le__diff__iff_H,axiom,
% 9.49/5.25      ! [A: nat,C: nat,B: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ A @ C )
% 9.49/5.25       => ( ( ord_less_eq_nat @ B @ C )
% 9.49/5.25         => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 9.49/5.25            = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3340_diff__le__mono2,axiom,
% 9.49/5.25      ! [M: nat,N: nat,L: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.25       => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3341_diff__add__inverse2,axiom,
% 9.49/5.25      ! [M: nat,N: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 9.49/5.25        = M ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3342_diff__add__inverse,axiom,
% 9.49/5.25      ! [N: nat,M: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 9.49/5.25        = M ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3343_diff__cancel2,axiom,
% 9.49/5.25      ! [M: nat,K: nat,N: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 9.49/5.25        = ( minus_minus_nat @ M @ N ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3344_Nat_Odiff__cancel,axiom,
% 9.49/5.25      ! [K: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 9.49/5.25        = ( minus_minus_nat @ M @ N ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3345_diff__mult__distrib2,axiom,
% 9.49/5.25      ! [K: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 9.49/5.25        = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3346_diff__mult__distrib,axiom,
% 9.49/5.25      ! [M: nat,N: nat,K: nat] :
% 9.49/5.25        ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 9.49/5.25        = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3347_max__diff__distrib__left,axiom,
% 9.49/5.25      ! [X: real,Y: real,Z2: real] :
% 9.49/5.25        ( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z2 )
% 9.49/5.25        = ( ord_max_real @ ( minus_minus_real @ X @ Z2 ) @ ( minus_minus_real @ Y @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3348_max__diff__distrib__left,axiom,
% 9.49/5.25      ! [X: rat,Y: rat,Z2: rat] :
% 9.49/5.25        ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z2 )
% 9.49/5.25        = ( ord_max_rat @ ( minus_minus_rat @ X @ Z2 ) @ ( minus_minus_rat @ Y @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3349_max__diff__distrib__left,axiom,
% 9.49/5.25      ! [X: int,Y: int,Z2: int] :
% 9.49/5.25        ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z2 )
% 9.49/5.25        = ( ord_max_int @ ( minus_minus_int @ X @ Z2 ) @ ( minus_minus_int @ Y @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3350_VEBT__internal_OminNull_Osimps_I3_J,axiom,
% 9.49/5.25      ! [Uu: $o] :
% 9.49/5.25        ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3351_VEBT__internal_OminNull_Osimps_I2_J,axiom,
% 9.49/5.25      ! [Uv: $o] :
% 9.49/5.25        ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3352_VEBT__internal_OminNull_Osimps_I1_J,axiom,
% 9.49/5.25      vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3353_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
% 9.49/5.25      ! [Uu: $o,Uv: $o,Uw: nat] :
% 9.49/5.25        ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3354_le__iff__diff__le__0,axiom,
% 9.49/5.25      ( ord_less_eq_real
% 9.49/5.25      = ( ^ [A4: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3355_le__iff__diff__le__0,axiom,
% 9.49/5.25      ( ord_less_eq_rat
% 9.49/5.25      = ( ^ [A4: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A4 @ B4 ) @ zero_zero_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3356_le__iff__diff__le__0,axiom,
% 9.49/5.25      ( ord_less_eq_int
% 9.49/5.25      = ( ^ [A4: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3357_less__iff__diff__less__0,axiom,
% 9.49/5.25      ( ord_less_real
% 9.49/5.25      = ( ^ [A4: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3358_less__iff__diff__less__0,axiom,
% 9.49/5.25      ( ord_less_rat
% 9.49/5.25      = ( ^ [A4: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A4 @ B4 ) @ zero_zero_rat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3359_less__iff__diff__less__0,axiom,
% 9.49/5.25      ( ord_less_int
% 9.49/5.25      = ( ^ [A4: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3360_diff__le__eq,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
% 9.49/5.25        = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3361_diff__le__eq,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 9.49/5.25        = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3362_diff__le__eq,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
% 9.49/5.25        = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3363_le__diff__eq,axiom,
% 9.49/5.25      ! [A: real,C: real,B: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
% 9.49/5.25        = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3364_le__diff__eq,axiom,
% 9.49/5.25      ! [A: rat,C: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 9.49/5.25        = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3365_le__diff__eq,axiom,
% 9.49/5.25      ! [A: int,C: int,B: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
% 9.49/5.25        = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3366_diff__add,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.25       => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 9.49/5.25          = B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3367_le__add__diff,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.25       => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3368_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 9.49/5.25          = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3369_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.25       => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 9.49/5.25          = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3370_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.25       => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 9.49/5.25          = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3371_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.25       => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 9.49/5.25          = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3372_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.25       => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 9.49/5.25          = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3373_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.25       => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 9.49/5.25          = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3374_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.25       => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 9.49/5.25          = B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3375_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.25         => ( ( ( minus_minus_nat @ B @ A )
% 9.49/5.25              = C )
% 9.49/5.25            = ( B
% 9.49/5.25              = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3376_add__le__add__imp__diff__le,axiom,
% 9.49/5.25      ! [I: real,K: real,N: real,J: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 9.49/5.25       => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 9.49/5.25         => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 9.49/5.25           => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 9.49/5.25             => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3377_add__le__add__imp__diff__le,axiom,
% 9.49/5.25      ! [I: rat,K: rat,N: rat,J: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 9.49/5.25       => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 9.49/5.25         => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 9.49/5.25           => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 9.49/5.25             => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3378_add__le__add__imp__diff__le,axiom,
% 9.49/5.25      ! [I: nat,K: nat,N: nat,J: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 9.49/5.25       => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 9.49/5.25         => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 9.49/5.25           => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 9.49/5.25             => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3379_add__le__add__imp__diff__le,axiom,
% 9.49/5.25      ! [I: int,K: int,N: int,J: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 9.49/5.25       => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 9.49/5.25         => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 9.49/5.25           => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 9.49/5.25             => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3380_add__le__imp__le__diff,axiom,
% 9.49/5.25      ! [I: real,K: real,N: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 9.49/5.25       => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3381_add__le__imp__le__diff,axiom,
% 9.49/5.25      ! [I: rat,K: rat,N: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 9.49/5.25       => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3382_add__le__imp__le__diff,axiom,
% 9.49/5.25      ! [I: nat,K: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 9.49/5.25       => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3383_add__le__imp__le__diff,axiom,
% 9.49/5.25      ! [I: int,K: int,N: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 9.49/5.25       => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3384_less__diff__eq,axiom,
% 9.49/5.25      ! [A: real,C: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
% 9.49/5.25        = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3385_less__diff__eq,axiom,
% 9.49/5.25      ! [A: rat,C: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 9.49/5.25        = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3386_less__diff__eq,axiom,
% 9.49/5.25      ! [A: int,C: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
% 9.49/5.25        = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3387_diff__less__eq,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real] :
% 9.49/5.25        ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
% 9.49/5.25        = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3388_diff__less__eq,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 9.49/5.25        = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3389_diff__less__eq,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int] :
% 9.49/5.25        ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
% 9.49/5.25        = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3390_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ~ ( ord_less_real @ A @ B )
% 9.49/5.25       => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3391_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ~ ( ord_less_rat @ A @ B )
% 9.49/5.25       => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3392_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ~ ( ord_less_nat @ A @ B )
% 9.49/5.25       => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3393_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ~ ( ord_less_int @ A @ B )
% 9.49/5.25       => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3394_vebt__delete_Osimps_I3_J,axiom,
% 9.49/5.25      ! [A: $o,B: $o,N: nat] :
% 9.49/5.25        ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N ) ) )
% 9.49/5.25        = ( vEBT_Leaf @ A @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3395_vebt__delete_Osimps_I1_J,axiom,
% 9.49/5.25      ! [A: $o,B: $o] :
% 9.49/5.25        ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat )
% 9.49/5.25        = ( vEBT_Leaf @ $false @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3396_diff__less__Suc,axiom,
% 9.49/5.25      ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3397_Suc__diff__Suc,axiom,
% 9.49/5.25      ! [N: nat,M: nat] :
% 9.49/5.25        ( ( ord_less_nat @ N @ M )
% 9.49/5.25       => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
% 9.49/5.25          = ( minus_minus_nat @ M @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3398_diff__less,axiom,
% 9.49/5.25      ! [N: nat,M: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.49/5.25         => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3399_Suc__diff__le,axiom,
% 9.49/5.25      ! [N: nat,M: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.25       => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 9.49/5.25          = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3400_diff__less__mono,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat] :
% 9.49/5.25        ( ( ord_less_nat @ A @ B )
% 9.49/5.25       => ( ( ord_less_eq_nat @ C @ A )
% 9.49/5.25         => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3401_less__diff__iff,axiom,
% 9.49/5.25      ! [K: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ K @ M )
% 9.49/5.25       => ( ( ord_less_eq_nat @ K @ N )
% 9.49/5.25         => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 9.49/5.25            = ( ord_less_nat @ M @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3402_diff__add__0,axiom,
% 9.49/5.25      ! [N: nat,M: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 9.49/5.25        = zero_zero_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3403_add__diff__inverse__nat,axiom,
% 9.49/5.25      ! [M: nat,N: nat] :
% 9.49/5.25        ( ~ ( ord_less_nat @ M @ N )
% 9.49/5.25       => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 9.49/5.25          = M ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3404_less__diff__conv,axiom,
% 9.49/5.25      ! [I: nat,J: nat,K: nat] :
% 9.49/5.25        ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 9.49/5.25        = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3405_le__diff__conv,axiom,
% 9.49/5.25      ! [J: nat,K: nat,I: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 9.49/5.25        = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3406_Nat_Ole__diff__conv2,axiom,
% 9.49/5.25      ! [K: nat,J: nat,I: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ K @ J )
% 9.49/5.25       => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 9.49/5.25          = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3407_Nat_Odiff__add__assoc,axiom,
% 9.49/5.25      ! [K: nat,J: nat,I: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ K @ J )
% 9.49/5.25       => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 9.49/5.25          = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3408_Nat_Odiff__add__assoc2,axiom,
% 9.49/5.25      ! [K: nat,J: nat,I: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ K @ J )
% 9.49/5.25       => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 9.49/5.25          = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3409_Nat_Ole__imp__diff__is__add,axiom,
% 9.49/5.25      ! [I: nat,J: nat,K: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.25       => ( ( ( minus_minus_nat @ J @ I )
% 9.49/5.25            = K )
% 9.49/5.25          = ( J
% 9.49/5.25            = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3410_diff__Suc__eq__diff__pred,axiom,
% 9.49/5.25      ! [M: nat,N: nat] :
% 9.49/5.25        ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 9.49/5.25        = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3411_vebt__buildup_Osimps_I1_J,axiom,
% 9.49/5.25      ( ( vEBT_vebt_buildup @ zero_zero_nat )
% 9.49/5.25      = ( vEBT_Leaf @ $false @ $false ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3412_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
% 9.49/5.25      ! [Uu: $o,Uv: $o,D: nat] :
% 9.49/5.25        ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
% 9.49/5.25        = ( D = one_one_nat ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3413_nat__minus__add__max,axiom,
% 9.49/5.25      ! [N: nat,M: nat] :
% 9.49/5.25        ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
% 9.49/5.25        = ( ord_max_nat @ N @ M ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3414_VEBT__internal_Onaive__member_Ocases,axiom,
% 9.49/5.25      ! [X: produc9072475918466114483BT_nat] :
% 9.49/5.25        ( ! [A3: $o,B3: $o,X4: nat] :
% 9.49/5.25            ( X
% 9.49/5.25           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X4 ) )
% 9.49/5.25       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
% 9.49/5.25              ( X
% 9.49/5.25             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
% 9.49/5.25         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S4: vEBT_VEBT,X4: nat] :
% 9.49/5.25                ( X
% 9.49/5.25               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S4 ) @ X4 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3415_ordered__ring__class_Ole__add__iff2,axiom,
% 9.49/5.25      ! [A: real,E2: real,C: real,B: real,D: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 9.49/5.25        = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3416_ordered__ring__class_Ole__add__iff2,axiom,
% 9.49/5.25      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 9.49/5.25        = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3417_ordered__ring__class_Ole__add__iff2,axiom,
% 9.49/5.25      ! [A: int,E2: int,C: int,B: int,D: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 9.49/5.25        = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3418_ordered__ring__class_Ole__add__iff1,axiom,
% 9.49/5.25      ! [A: real,E2: real,C: real,B: real,D: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 9.49/5.25        = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3419_ordered__ring__class_Ole__add__iff1,axiom,
% 9.49/5.25      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 9.49/5.25        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3420_ordered__ring__class_Ole__add__iff1,axiom,
% 9.49/5.25      ! [A: int,E2: int,C: int,B: int,D: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 9.49/5.25        = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3421_less__add__iff2,axiom,
% 9.49/5.25      ! [A: real,E2: real,C: real,B: real,D: real] :
% 9.49/5.25        ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 9.49/5.25        = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3422_less__add__iff2,axiom,
% 9.49/5.25      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 9.49/5.25        ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 9.49/5.25        = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3423_less__add__iff2,axiom,
% 9.49/5.25      ! [A: int,E2: int,C: int,B: int,D: int] :
% 9.49/5.25        ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 9.49/5.25        = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3424_less__add__iff1,axiom,
% 9.49/5.25      ! [A: real,E2: real,C: real,B: real,D: real] :
% 9.49/5.25        ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 9.49/5.25        = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3425_less__add__iff1,axiom,
% 9.49/5.25      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 9.49/5.25        ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 9.49/5.25        = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3426_less__add__iff1,axiom,
% 9.49/5.25      ! [A: int,E2: int,C: int,B: int,D: int] :
% 9.49/5.25        ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 9.49/5.25        = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3427_invar__vebt_Ointros_I1_J,axiom,
% 9.49/5.25      ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3428_divide__diff__eq__iff,axiom,
% 9.49/5.25      ! [Z2: complex,X: complex,Y: complex] :
% 9.49/5.25        ( ( Z2 != zero_zero_complex )
% 9.49/5.25       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z2 ) @ Y )
% 9.49/5.25          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3429_divide__diff__eq__iff,axiom,
% 9.49/5.25      ! [Z2: real,X: real,Y: real] :
% 9.49/5.25        ( ( Z2 != zero_zero_real )
% 9.49/5.25       => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z2 ) @ Y )
% 9.49/5.25          = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3430_divide__diff__eq__iff,axiom,
% 9.49/5.25      ! [Z2: rat,X: rat,Y: rat] :
% 9.49/5.25        ( ( Z2 != zero_zero_rat )
% 9.49/5.25       => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z2 ) @ Y )
% 9.49/5.25          = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3431_diff__divide__eq__iff,axiom,
% 9.49/5.25      ! [Z2: complex,X: complex,Y: complex] :
% 9.49/5.25        ( ( Z2 != zero_zero_complex )
% 9.49/5.25       => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z2 ) )
% 9.49/5.25          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3432_diff__divide__eq__iff,axiom,
% 9.49/5.25      ! [Z2: real,X: real,Y: real] :
% 9.49/5.25        ( ( Z2 != zero_zero_real )
% 9.49/5.25       => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z2 ) )
% 9.49/5.25          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3433_diff__divide__eq__iff,axiom,
% 9.49/5.25      ! [Z2: rat,X: rat,Y: rat] :
% 9.49/5.25        ( ( Z2 != zero_zero_rat )
% 9.49/5.25       => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z2 ) )
% 9.49/5.25          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3434_diff__frac__eq,axiom,
% 9.49/5.25      ! [Y: complex,Z2: complex,X: complex,W2: complex] :
% 9.49/5.25        ( ( Y != zero_zero_complex )
% 9.49/5.25       => ( ( Z2 != zero_zero_complex )
% 9.49/5.25         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W2 @ Z2 ) )
% 9.49/5.25            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ W2 @ Y ) ) @ ( times_times_complex @ Y @ Z2 ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3435_diff__frac__eq,axiom,
% 9.49/5.25      ! [Y: real,Z2: real,X: real,W2: real] :
% 9.49/5.25        ( ( Y != zero_zero_real )
% 9.49/5.25       => ( ( Z2 != zero_zero_real )
% 9.49/5.25         => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z2 ) )
% 9.49/5.25            = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3436_diff__frac__eq,axiom,
% 9.49/5.25      ! [Y: rat,Z2: rat,X: rat,W2: rat] :
% 9.49/5.25        ( ( Y != zero_zero_rat )
% 9.49/5.25       => ( ( Z2 != zero_zero_rat )
% 9.49/5.25         => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z2 ) )
% 9.49/5.25            = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3437_add__divide__eq__if__simps_I4_J,axiom,
% 9.49/5.25      ! [Z2: complex,A: complex,B: complex] :
% 9.49/5.25        ( ( ( Z2 = zero_zero_complex )
% 9.49/5.25         => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 9.49/5.25            = A ) )
% 9.49/5.25        & ( ( Z2 != zero_zero_complex )
% 9.49/5.25         => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 9.49/5.25            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3438_add__divide__eq__if__simps_I4_J,axiom,
% 9.49/5.25      ! [Z2: real,A: real,B: real] :
% 9.49/5.25        ( ( ( Z2 = zero_zero_real )
% 9.49/5.25         => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 9.49/5.25            = A ) )
% 9.49/5.25        & ( ( Z2 != zero_zero_real )
% 9.49/5.25         => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 9.49/5.25            = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3439_add__divide__eq__if__simps_I4_J,axiom,
% 9.49/5.25      ! [Z2: rat,A: rat,B: rat] :
% 9.49/5.25        ( ( ( Z2 = zero_zero_rat )
% 9.49/5.25         => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 9.49/5.25            = A ) )
% 9.49/5.25        & ( ( Z2 != zero_zero_rat )
% 9.49/5.25         => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 9.49/5.25            = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3440_vebt__delete_Osimps_I2_J,axiom,
% 9.49/5.25      ! [A: $o,B: $o] :
% 9.49/5.25        ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) )
% 9.49/5.25        = ( vEBT_Leaf @ A @ $false ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3441_VEBT__internal_OminNull_Ocases,axiom,
% 9.49/5.25      ! [X: vEBT_VEBT] :
% 9.49/5.25        ( ( X
% 9.49/5.25         != ( vEBT_Leaf @ $false @ $false ) )
% 9.49/5.25       => ( ! [Uv2: $o] :
% 9.49/5.25              ( X
% 9.49/5.25             != ( vEBT_Leaf @ $true @ Uv2 ) )
% 9.49/5.25         => ( ! [Uu2: $o] :
% 9.49/5.25                ( X
% 9.49/5.25               != ( vEBT_Leaf @ Uu2 @ $true ) )
% 9.49/5.25           => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.49/5.25                  ( X
% 9.49/5.25                 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 9.49/5.25             => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.49/5.25                    ( X
% 9.49/5.25                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3442_diff__Suc__less,axiom,
% 9.49/5.25      ! [N: nat,I: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3443_vebt__member_Osimps_I1_J,axiom,
% 9.49/5.25      ! [A: $o,B: $o,X: nat] :
% 9.49/5.25        ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X )
% 9.49/5.25        = ( ( ( X = zero_zero_nat )
% 9.49/5.25           => A )
% 9.49/5.25          & ( ( X != zero_zero_nat )
% 9.49/5.25           => ( ( ( X = one_one_nat )
% 9.49/5.25               => B )
% 9.49/5.25              & ( X = one_one_nat ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3444_vebt__buildup_Osimps_I2_J,axiom,
% 9.49/5.25      ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
% 9.49/5.25      = ( vEBT_Leaf @ $false @ $false ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3445_nat__diff__split,axiom,
% 9.49/5.25      ! [P: nat > $o,A: nat,B: nat] :
% 9.49/5.25        ( ( P @ ( minus_minus_nat @ A @ B ) )
% 9.49/5.25        = ( ( ( ord_less_nat @ A @ B )
% 9.49/5.25           => ( P @ zero_zero_nat ) )
% 9.49/5.25          & ! [D4: nat] :
% 9.49/5.25              ( ( A
% 9.49/5.25                = ( plus_plus_nat @ B @ D4 ) )
% 9.49/5.25             => ( P @ D4 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3446_nat__diff__split__asm,axiom,
% 9.49/5.25      ! [P: nat > $o,A: nat,B: nat] :
% 9.49/5.25        ( ( P @ ( minus_minus_nat @ A @ B ) )
% 9.49/5.25        = ( ~ ( ( ( ord_less_nat @ A @ B )
% 9.49/5.25                & ~ ( P @ zero_zero_nat ) )
% 9.49/5.25              | ? [D4: nat] :
% 9.49/5.25                  ( ( A
% 9.49/5.25                    = ( plus_plus_nat @ B @ D4 ) )
% 9.49/5.25                  & ~ ( P @ D4 ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3447_less__diff__conv2,axiom,
% 9.49/5.25      ! [K: nat,J: nat,I: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ K @ J )
% 9.49/5.25       => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 9.49/5.25          = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3448_VEBT__internal_OminNull_Oelims_I3_J,axiom,
% 9.49/5.25      ! [X: vEBT_VEBT] :
% 9.49/5.25        ( ~ ( vEBT_VEBT_minNull @ X )
% 9.49/5.25       => ( ! [Uv2: $o] :
% 9.49/5.25              ( X
% 9.49/5.25             != ( vEBT_Leaf @ $true @ Uv2 ) )
% 9.49/5.25         => ( ! [Uu2: $o] :
% 9.49/5.25                ( X
% 9.49/5.25               != ( vEBT_Leaf @ Uu2 @ $true ) )
% 9.49/5.25           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.49/5.25                  ( X
% 9.49/5.25                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3449_vebt__insert_Osimps_I1_J,axiom,
% 9.49/5.25      ! [X: nat,A: $o,B: $o] :
% 9.49/5.25        ( ( ( X = zero_zero_nat )
% 9.49/5.25         => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
% 9.49/5.25            = ( vEBT_Leaf @ $true @ B ) ) )
% 9.49/5.25        & ( ( X != zero_zero_nat )
% 9.49/5.25         => ( ( ( X = one_one_nat )
% 9.49/5.25             => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
% 9.49/5.25                = ( vEBT_Leaf @ A @ $true ) ) )
% 9.49/5.25            & ( ( X != one_one_nat )
% 9.49/5.25             => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
% 9.49/5.25                = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3450_nat__diff__add__eq2,axiom,
% 9.49/5.25      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.25       => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.49/5.25          = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3451_nat__diff__add__eq1,axiom,
% 9.49/5.25      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ J @ I )
% 9.49/5.25       => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.49/5.25          = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3452_nat__le__add__iff2,axiom,
% 9.49/5.25      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.25       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.49/5.25          = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3453_nat__le__add__iff1,axiom,
% 9.49/5.25      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ J @ I )
% 9.49/5.25       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.49/5.25          = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3454_nat__eq__add__iff2,axiom,
% 9.49/5.25      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.25       => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 9.49/5.25            = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.49/5.25          = ( M
% 9.49/5.25            = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3455_nat__eq__add__iff1,axiom,
% 9.49/5.25      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ J @ I )
% 9.49/5.25       => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 9.49/5.25            = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.49/5.25          = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
% 9.49/5.25            = N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3456_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
% 9.49/5.25      ! [A: $o,B: $o,X: nat] :
% 9.49/5.25        ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
% 9.49/5.25        = ( ( ( X = zero_zero_nat )
% 9.49/5.25           => A )
% 9.49/5.25          & ( ( X != zero_zero_nat )
% 9.49/5.25           => ( ( ( X = one_one_nat )
% 9.49/5.25               => B )
% 9.49/5.25              & ( X = one_one_nat ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3457_vebt__succ_Osimps_I2_J,axiom,
% 9.49/5.25      ! [Uv: $o,Uw: $o,N: nat] :
% 9.49/5.25        ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
% 9.49/5.25        = none_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3458_vebt__pred_Osimps_I1_J,axiom,
% 9.49/5.25      ! [Uu: $o,Uv: $o] :
% 9.49/5.25        ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
% 9.49/5.25        = none_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3459_VEBT__internal_OminNull_Oelims_I2_J,axiom,
% 9.49/5.25      ! [X: vEBT_VEBT] :
% 9.49/5.25        ( ( vEBT_VEBT_minNull @ X )
% 9.49/5.25       => ( ( X
% 9.49/5.25           != ( vEBT_Leaf @ $false @ $false ) )
% 9.49/5.25         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.49/5.25                ( X
% 9.49/5.25               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3460_realpow__pos__nth2,axiom,
% 9.49/5.25      ! [A: real,N: nat] :
% 9.49/5.25        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.25       => ? [R3: real] :
% 9.49/5.25            ( ( ord_less_real @ zero_zero_real @ R3 )
% 9.49/5.25            & ( ( power_power_real @ R3 @ ( suc @ N ) )
% 9.49/5.25              = A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3461_frac__le__eq,axiom,
% 9.49/5.25      ! [Y: real,Z2: real,X: real,W2: real] :
% 9.49/5.25        ( ( Y != zero_zero_real )
% 9.49/5.25       => ( ( Z2 != zero_zero_real )
% 9.49/5.25         => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z2 ) )
% 9.49/5.25            = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3462_frac__le__eq,axiom,
% 9.49/5.25      ! [Y: rat,Z2: rat,X: rat,W2: rat] :
% 9.49/5.25        ( ( Y != zero_zero_rat )
% 9.49/5.25       => ( ( Z2 != zero_zero_rat )
% 9.49/5.25         => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z2 ) )
% 9.49/5.25            = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3463_frac__less__eq,axiom,
% 9.49/5.25      ! [Y: real,Z2: real,X: real,W2: real] :
% 9.49/5.25        ( ( Y != zero_zero_real )
% 9.49/5.25       => ( ( Z2 != zero_zero_real )
% 9.49/5.25         => ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z2 ) )
% 9.49/5.25            = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3464_frac__less__eq,axiom,
% 9.49/5.25      ! [Y: rat,Z2: rat,X: rat,W2: rat] :
% 9.49/5.25        ( ( Y != zero_zero_rat )
% 9.49/5.25       => ( ( Z2 != zero_zero_rat )
% 9.49/5.25         => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z2 ) )
% 9.49/5.25            = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3465_power__diff,axiom,
% 9.49/5.25      ! [A: complex,N: nat,M: nat] :
% 9.49/5.25        ( ( A != zero_zero_complex )
% 9.49/5.25       => ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.25         => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
% 9.49/5.25            = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3466_power__diff,axiom,
% 9.49/5.25      ! [A: nat,N: nat,M: nat] :
% 9.49/5.25        ( ( A != zero_zero_nat )
% 9.49/5.25       => ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.25         => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
% 9.49/5.25            = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3467_power__diff,axiom,
% 9.49/5.25      ! [A: int,N: nat,M: nat] :
% 9.49/5.25        ( ( A != zero_zero_int )
% 9.49/5.25       => ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.25         => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
% 9.49/5.25            = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3468_power__diff,axiom,
% 9.49/5.25      ! [A: real,N: nat,M: nat] :
% 9.49/5.25        ( ( A != zero_zero_real )
% 9.49/5.25       => ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.25         => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
% 9.49/5.25            = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3469_power__diff,axiom,
% 9.49/5.25      ! [A: rat,N: nat,M: nat] :
% 9.49/5.25        ( ( A != zero_zero_rat )
% 9.49/5.25       => ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.25         => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
% 9.49/5.25            = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3470_power__diff,axiom,
% 9.49/5.25      ! [A: code_integer,N: nat,M: nat] :
% 9.49/5.25        ( ( A != zero_z3403309356797280102nteger )
% 9.49/5.25       => ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.25         => ( ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M @ N ) )
% 9.49/5.25            = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3471_vebt__mint_Ocases,axiom,
% 9.49/5.25      ! [X: vEBT_VEBT] :
% 9.49/5.25        ( ! [A3: $o,B3: $o] :
% 9.49/5.25            ( X
% 9.49/5.25           != ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.25       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.49/5.25              ( X
% 9.49/5.25             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.49/5.25         => ~ ! [Mi3: nat,Ma3: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.49/5.25                ( X
% 9.49/5.25               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3472_div__if,axiom,
% 9.49/5.25      ( divide_divide_nat
% 9.49/5.25      = ( ^ [M2: nat,N2: nat] :
% 9.49/5.25            ( if_nat
% 9.49/5.25            @ ( ( ord_less_nat @ M2 @ N2 )
% 9.49/5.25              | ( N2 = zero_zero_nat ) )
% 9.49/5.25            @ zero_zero_nat
% 9.49/5.25            @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3473_Suc__pred_H,axiom,
% 9.49/5.25      ! [N: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( N
% 9.49/5.25          = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3474_Suc__diff__eq__diff__pred,axiom,
% 9.49/5.25      ! [N: nat,M: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 9.49/5.25          = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3475_add__eq__if,axiom,
% 9.49/5.25      ( plus_plus_nat
% 9.49/5.25      = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3476_nat__less__add__iff2,axiom,
% 9.49/5.25      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.25       => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.49/5.25          = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3477_nat__less__add__iff1,axiom,
% 9.49/5.25      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ J @ I )
% 9.49/5.25       => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.49/5.25          = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3478_mult__eq__if,axiom,
% 9.49/5.25      ( times_times_nat
% 9.49/5.25      = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3479_VEBT__internal_OminNull_Oelims_I1_J,axiom,
% 9.49/5.25      ! [X: vEBT_VEBT,Y: $o] :
% 9.49/5.25        ( ( ( vEBT_VEBT_minNull @ X )
% 9.49/5.25          = Y )
% 9.49/5.25       => ( ( ( X
% 9.49/5.25              = ( vEBT_Leaf @ $false @ $false ) )
% 9.49/5.25           => ~ Y )
% 9.49/5.25         => ( ( ? [Uv2: $o] :
% 9.49/5.25                  ( X
% 9.49/5.25                  = ( vEBT_Leaf @ $true @ Uv2 ) )
% 9.49/5.25             => Y )
% 9.49/5.25           => ( ( ? [Uu2: $o] :
% 9.49/5.25                    ( X
% 9.49/5.25                    = ( vEBT_Leaf @ Uu2 @ $true ) )
% 9.49/5.25               => Y )
% 9.49/5.25             => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.49/5.25                      ( X
% 9.49/5.25                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 9.49/5.25                 => ~ Y )
% 9.49/5.25               => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.49/5.25                        ( X
% 9.49/5.25                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 9.49/5.25                   => Y ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3480_scaling__mono,axiom,
% 9.49/5.25      ! [U: real,V: real,R2: real,S: real] :
% 9.49/5.25        ( ( ord_less_eq_real @ U @ V )
% 9.49/5.25       => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 9.49/5.25         => ( ( ord_less_eq_real @ R2 @ S )
% 9.49/5.25           => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3481_scaling__mono,axiom,
% 9.49/5.25      ! [U: rat,V: rat,R2: rat,S: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ U @ V )
% 9.49/5.25       => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 9.49/5.25         => ( ( ord_less_eq_rat @ R2 @ S )
% 9.49/5.25           => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3482_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 9.49/5.25      ! [N: nat,M: nat] :
% 9.49/5.25        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.25         != zero_zero_nat )
% 9.49/5.25       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 9.49/5.25         != zero_zero_nat ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3483_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 9.49/5.25      ! [N: nat,M: nat] :
% 9.49/5.25        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 9.49/5.25         != zero_zero_int )
% 9.49/5.25       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 9.49/5.25         != zero_zero_int ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3484_power__diff__power__eq,axiom,
% 9.49/5.25      ! [A: nat,N: nat,M: nat] :
% 9.49/5.25        ( ( A != zero_zero_nat )
% 9.49/5.25       => ( ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.25           => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 9.49/5.25              = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 9.49/5.25          & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.49/5.25           => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 9.49/5.25              = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3485_power__diff__power__eq,axiom,
% 9.49/5.25      ! [A: int,N: nat,M: nat] :
% 9.49/5.25        ( ( A != zero_zero_int )
% 9.49/5.25       => ( ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.25           => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 9.49/5.25              = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 9.49/5.25          & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.49/5.25           => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 9.49/5.25              = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3486_power__diff__power__eq,axiom,
% 9.49/5.25      ! [A: code_integer,N: nat,M: nat] :
% 9.49/5.25        ( ( A != zero_z3403309356797280102nteger )
% 9.49/5.25       => ( ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.25           => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) )
% 9.49/5.25              = ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 9.49/5.25          & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.49/5.25           => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) )
% 9.49/5.25              = ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3487_power__eq__if,axiom,
% 9.49/5.25      ( power_power_complex
% 9.49/5.25      = ( ^ [P6: complex,M2: nat] : ( if_complex @ ( M2 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P6 @ ( power_power_complex @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3488_power__eq__if,axiom,
% 9.49/5.25      ( power_power_real
% 9.49/5.25      = ( ^ [P6: real,M2: nat] : ( if_real @ ( M2 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P6 @ ( power_power_real @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3489_power__eq__if,axiom,
% 9.49/5.25      ( power_power_rat
% 9.49/5.25      = ( ^ [P6: rat,M2: nat] : ( if_rat @ ( M2 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P6 @ ( power_power_rat @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3490_power__eq__if,axiom,
% 9.49/5.25      ( power_power_nat
% 9.49/5.25      = ( ^ [P6: nat,M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P6 @ ( power_power_nat @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3491_power__eq__if,axiom,
% 9.49/5.25      ( power_power_int
% 9.49/5.25      = ( ^ [P6: int,M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P6 @ ( power_power_int @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3492_vebt__maxt_Osimps_I1_J,axiom,
% 9.49/5.25      ! [B: $o,A: $o] :
% 9.49/5.25        ( ( B
% 9.49/5.25         => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 9.49/5.25            = ( some_nat @ one_one_nat ) ) )
% 9.49/5.25        & ( ~ B
% 9.49/5.25         => ( ( A
% 9.49/5.25             => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 9.49/5.25                = ( some_nat @ zero_zero_nat ) ) )
% 9.49/5.25            & ( ~ A
% 9.49/5.25             => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 9.49/5.25                = none_nat ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3493_power__minus__mult,axiom,
% 9.49/5.25      ! [N: nat,A: complex] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 9.49/5.25          = ( power_power_complex @ A @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3494_power__minus__mult,axiom,
% 9.49/5.25      ! [N: nat,A: real] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 9.49/5.25          = ( power_power_real @ A @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3495_power__minus__mult,axiom,
% 9.49/5.25      ! [N: nat,A: rat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 9.49/5.25          = ( power_power_rat @ A @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3496_power__minus__mult,axiom,
% 9.49/5.25      ! [N: nat,A: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 9.49/5.25          = ( power_power_nat @ A @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3497_power__minus__mult,axiom,
% 9.49/5.25      ! [N: nat,A: int] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 9.49/5.25          = ( power_power_int @ A @ N ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3498_vebt__mint_Osimps_I1_J,axiom,
% 9.49/5.25      ! [A: $o,B: $o] :
% 9.49/5.25        ( ( A
% 9.49/5.25         => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 9.49/5.25            = ( some_nat @ zero_zero_nat ) ) )
% 9.49/5.25        & ( ~ A
% 9.49/5.25         => ( ( B
% 9.49/5.25             => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 9.49/5.25                = ( some_nat @ one_one_nat ) ) )
% 9.49/5.25            & ( ~ B
% 9.49/5.25             => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 9.49/5.25                = none_nat ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3499_diff__le__diff__pow,axiom,
% 9.49/5.25      ! [K: nat,M: nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.49/5.25       => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3500_le__div__geq,axiom,
% 9.49/5.25      ! [N: nat,M: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.25         => ( ( divide_divide_nat @ M @ N )
% 9.49/5.25            = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3501_vebt__pred_Osimps_I2_J,axiom,
% 9.49/5.25      ! [A: $o,Uw: $o] :
% 9.49/5.25        ( ( A
% 9.49/5.25         => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
% 9.49/5.25            = ( some_nat @ zero_zero_nat ) ) )
% 9.49/5.25        & ( ~ A
% 9.49/5.25         => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
% 9.49/5.25            = none_nat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3502_vebt__succ_Osimps_I1_J,axiom,
% 9.49/5.25      ! [B: $o,Uu: $o] :
% 9.49/5.25        ( ( B
% 9.49/5.25         => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
% 9.49/5.25            = ( some_nat @ one_one_nat ) ) )
% 9.49/5.25        & ( ~ B
% 9.49/5.25         => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
% 9.49/5.25            = none_nat ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3503_vebt__insert_Ocases,axiom,
% 9.49/5.25      ! [X: produc9072475918466114483BT_nat] :
% 9.49/5.25        ( ! [A3: $o,B3: $o,X4: nat] :
% 9.49/5.25            ( X
% 9.49/5.25           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X4 ) )
% 9.49/5.25       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT,X4: nat] :
% 9.49/5.25              ( X
% 9.49/5.25             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S4 ) @ X4 ) )
% 9.49/5.25         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT,X4: nat] :
% 9.49/5.25                ( X
% 9.49/5.25               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S4 ) @ X4 ) )
% 9.49/5.25           => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,X4: nat] :
% 9.49/5.25                  ( X
% 9.49/5.25                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary3 ) @ X4 ) )
% 9.49/5.25             => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,X4: nat] :
% 9.49/5.25                    ( X
% 9.49/5.25                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ X4 ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3504_vebt__pred_Ocases,axiom,
% 9.49/5.25      ! [X: produc9072475918466114483BT_nat] :
% 9.49/5.25        ( ! [Uu2: $o,Uv2: $o] :
% 9.49/5.25            ( X
% 9.49/5.25           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
% 9.49/5.25       => ( ! [A3: $o,Uw2: $o] :
% 9.49/5.25              ( X
% 9.49/5.25             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
% 9.49/5.25         => ( ! [A3: $o,B3: $o,Va2: nat] :
% 9.49/5.25                ( X
% 9.49/5.25               != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va2 ) ) ) )
% 9.49/5.25           => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
% 9.49/5.25                  ( X
% 9.49/5.25                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Vb2 ) )
% 9.49/5.25             => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
% 9.49/5.25                    ( X
% 9.49/5.25                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 ) )
% 9.49/5.25               => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
% 9.49/5.25                      ( X
% 9.49/5.25                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 ) )
% 9.49/5.25                 => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,X4: nat] :
% 9.49/5.25                        ( X
% 9.49/5.25                       != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ X4 ) ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3505_vebt__succ_Ocases,axiom,
% 9.49/5.25      ! [X: produc9072475918466114483BT_nat] :
% 9.49/5.25        ( ! [Uu2: $o,B3: $o] :
% 9.49/5.25            ( X
% 9.49/5.25           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat ) )
% 9.49/5.25       => ( ! [Uv2: $o,Uw2: $o,N3: nat] :
% 9.49/5.25              ( X
% 9.49/5.25             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) ) )
% 9.49/5.25         => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] :
% 9.49/5.25                ( X
% 9.49/5.25               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va3 ) )
% 9.49/5.25           => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
% 9.49/5.25                  ( X
% 9.49/5.25                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 ) )
% 9.49/5.25             => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
% 9.49/5.25                    ( X
% 9.49/5.25                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 ) )
% 9.49/5.25               => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,X4: nat] :
% 9.49/5.25                      ( X
% 9.49/5.25                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ X4 ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3506_vebt__delete_Ocases,axiom,
% 9.49/5.25      ! [X: produc9072475918466114483BT_nat] :
% 9.49/5.25        ( ! [A3: $o,B3: $o] :
% 9.49/5.25            ( X
% 9.49/5.25           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat ) )
% 9.49/5.25       => ( ! [A3: $o,B3: $o] :
% 9.49/5.25              ( X
% 9.49/5.25             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) ) )
% 9.49/5.25         => ( ! [A3: $o,B3: $o,N3: nat] :
% 9.49/5.25                ( X
% 9.49/5.25               != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N3 ) ) ) )
% 9.49/5.25           => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,Uu2: nat] :
% 9.49/5.25                  ( X
% 9.49/5.25                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary3 ) @ Uu2 ) )
% 9.49/5.25             => ( ! [Mi3: nat,Ma3: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,X4: nat] :
% 9.49/5.25                    ( X
% 9.49/5.25                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ X4 ) )
% 9.49/5.25               => ( ! [Mi3: nat,Ma3: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,X4: nat] :
% 9.49/5.25                      ( X
% 9.49/5.25                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ X4 ) )
% 9.49/5.25                 => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,X4: nat] :
% 9.49/5.25                        ( X
% 9.49/5.25                       != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ X4 ) ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3507_vebt__member_Ocases,axiom,
% 9.49/5.25      ! [X: produc9072475918466114483BT_nat] :
% 9.49/5.25        ( ! [A3: $o,B3: $o,X4: nat] :
% 9.49/5.25            ( X
% 9.49/5.25           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X4 ) )
% 9.49/5.25       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] :
% 9.49/5.25              ( X
% 9.49/5.25             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X4 ) )
% 9.49/5.25         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] :
% 9.49/5.25                ( X
% 9.49/5.25               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X4 ) )
% 9.49/5.25           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
% 9.49/5.25                  ( X
% 9.49/5.25                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X4 ) )
% 9.49/5.25             => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,X4: nat] :
% 9.49/5.25                    ( X
% 9.49/5.25                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ X4 ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3508_VEBT__internal_Omembermima_Ocases,axiom,
% 9.49/5.25      ! [X: produc9072475918466114483BT_nat] :
% 9.49/5.25        ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
% 9.49/5.25            ( X
% 9.49/5.25           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
% 9.49/5.25       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
% 9.49/5.25              ( X
% 9.49/5.25             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
% 9.49/5.25         => ( ! [Mi3: nat,Ma3: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] :
% 9.49/5.25                ( X
% 9.49/5.25               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X4 ) )
% 9.49/5.25           => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
% 9.49/5.25                  ( X
% 9.49/5.25                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ X4 ) )
% 9.49/5.25             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT,X4: nat] :
% 9.49/5.25                    ( X
% 9.49/5.25                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ X4 ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3509_vebt__pred_Osimps_I3_J,axiom,
% 9.49/5.25      ! [B: $o,A: $o,Va: nat] :
% 9.49/5.25        ( ( B
% 9.49/5.25         => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
% 9.49/5.25            = ( some_nat @ one_one_nat ) ) )
% 9.49/5.25        & ( ~ B
% 9.49/5.25         => ( ( A
% 9.49/5.25             => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
% 9.49/5.25                = ( some_nat @ zero_zero_nat ) ) )
% 9.49/5.25            & ( ~ A
% 9.49/5.25             => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
% 9.49/5.25                = none_nat ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3510_realpow__pos__nth,axiom,
% 9.49/5.25      ! [N: nat,A: real] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.25         => ? [R3: real] :
% 9.49/5.25              ( ( ord_less_real @ zero_zero_real @ R3 )
% 9.49/5.25              & ( ( power_power_real @ R3 @ N )
% 9.49/5.25                = A ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3511_realpow__pos__nth__unique,axiom,
% 9.49/5.25      ! [N: nat,A: real] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.25         => ? [X4: real] :
% 9.49/5.25              ( ( ord_less_real @ zero_zero_real @ X4 )
% 9.49/5.25              & ( ( power_power_real @ X4 @ N )
% 9.49/5.25                = A )
% 9.49/5.25              & ! [Y4: real] :
% 9.49/5.25                  ( ( ( ord_less_real @ zero_zero_real @ Y4 )
% 9.49/5.25                    & ( ( power_power_real @ Y4 @ N )
% 9.49/5.25                      = A ) )
% 9.49/5.25                 => ( Y4 = X4 ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3512_vebt__maxt_Oelims,axiom,
% 9.49/5.25      ! [X: vEBT_VEBT,Y: option_nat] :
% 9.49/5.25        ( ( ( vEBT_vebt_maxt @ X )
% 9.49/5.25          = Y )
% 9.49/5.25       => ( ! [A3: $o,B3: $o] :
% 9.49/5.25              ( ( X
% 9.49/5.25                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.25             => ~ ( ( B3
% 9.49/5.25                   => ( Y
% 9.49/5.25                      = ( some_nat @ one_one_nat ) ) )
% 9.49/5.25                  & ( ~ B3
% 9.49/5.25                   => ( ( A3
% 9.49/5.25                       => ( Y
% 9.49/5.25                          = ( some_nat @ zero_zero_nat ) ) )
% 9.49/5.25                      & ( ~ A3
% 9.49/5.25                       => ( Y = none_nat ) ) ) ) ) )
% 9.49/5.25         => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.49/5.25                  ( X
% 9.49/5.25                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.49/5.25             => ( Y != none_nat ) )
% 9.49/5.25           => ~ ! [Mi3: nat,Ma3: nat] :
% 9.49/5.25                  ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.49/5.25                      ( X
% 9.49/5.25                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 9.49/5.25                 => ( Y
% 9.49/5.25                   != ( some_nat @ Ma3 ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3513_vebt__mint_Oelims,axiom,
% 9.49/5.25      ! [X: vEBT_VEBT,Y: option_nat] :
% 9.49/5.25        ( ( ( vEBT_vebt_mint @ X )
% 9.49/5.25          = Y )
% 9.49/5.25       => ( ! [A3: $o,B3: $o] :
% 9.49/5.25              ( ( X
% 9.49/5.25                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.25             => ~ ( ( A3
% 9.49/5.25                   => ( Y
% 9.49/5.25                      = ( some_nat @ zero_zero_nat ) ) )
% 9.49/5.25                  & ( ~ A3
% 9.49/5.25                   => ( ( B3
% 9.49/5.25                       => ( Y
% 9.49/5.25                          = ( some_nat @ one_one_nat ) ) )
% 9.49/5.25                      & ( ~ B3
% 9.49/5.25                       => ( Y = none_nat ) ) ) ) ) )
% 9.49/5.25         => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.49/5.25                  ( X
% 9.49/5.25                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.49/5.25             => ( Y != none_nat ) )
% 9.49/5.25           => ~ ! [Mi3: nat] :
% 9.49/5.25                  ( ? [Ma3: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.49/5.25                      ( X
% 9.49/5.25                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 9.49/5.25                 => ( Y
% 9.49/5.25                   != ( some_nat @ Mi3 ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3514_pos2,axiom,
% 9.49/5.25      ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3515_invar__vebt_Ocases,axiom,
% 9.49/5.25      ! [A12: vEBT_VEBT,A23: nat] :
% 9.49/5.25        ( ( vEBT_invar_vebt @ A12 @ A23 )
% 9.49/5.25       => ( ( ? [A3: $o,B3: $o] :
% 9.49/5.25                ( A12
% 9.49/5.25                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.25           => ( A23
% 9.49/5.25             != ( suc @ zero_zero_nat ) ) )
% 9.49/5.25         => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat] :
% 9.49/5.25                ( ( A12
% 9.49/5.25                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary3 ) )
% 9.49/5.25               => ( ( A23 = Deg2 )
% 9.49/5.25                 => ( ! [X2: vEBT_VEBT] :
% 9.49/5.25                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.49/5.25                       => ( vEBT_invar_vebt @ X2 @ N3 ) )
% 9.49/5.25                   => ( ( vEBT_invar_vebt @ Summary3 @ M5 )
% 9.49/5.25                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.49/5.25                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 9.49/5.25                       => ( ( M5 = N3 )
% 9.49/5.25                         => ( ( Deg2
% 9.49/5.25                              = ( plus_plus_nat @ N3 @ M5 ) )
% 9.49/5.25                           => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X_12 )
% 9.49/5.25                             => ~ ! [X2: vEBT_VEBT] :
% 9.49/5.25                                    ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.49/5.25                                   => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ) ) ) ) ) ) )
% 9.49/5.25           => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat] :
% 9.49/5.25                  ( ( A12
% 9.49/5.25                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary3 ) )
% 9.49/5.25                 => ( ( A23 = Deg2 )
% 9.49/5.25                   => ( ! [X2: vEBT_VEBT] :
% 9.49/5.25                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.49/5.25                         => ( vEBT_invar_vebt @ X2 @ N3 ) )
% 9.49/5.25                     => ( ( vEBT_invar_vebt @ Summary3 @ M5 )
% 9.49/5.25                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.49/5.25                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 9.49/5.25                         => ( ( M5
% 9.49/5.25                              = ( suc @ N3 ) )
% 9.49/5.25                           => ( ( Deg2
% 9.49/5.25                                = ( plus_plus_nat @ N3 @ M5 ) )
% 9.49/5.25                             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X_12 )
% 9.49/5.25                               => ~ ! [X2: vEBT_VEBT] :
% 9.49/5.25                                      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.49/5.25                                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ) ) ) ) ) ) )
% 9.49/5.25             => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat,Mi3: nat,Ma3: nat] :
% 9.49/5.25                    ( ( A12
% 9.49/5.25                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg2 @ TreeList3 @ Summary3 ) )
% 9.49/5.25                   => ( ( A23 = Deg2 )
% 9.49/5.25                     => ( ! [X2: vEBT_VEBT] :
% 9.49/5.25                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.49/5.25                           => ( vEBT_invar_vebt @ X2 @ N3 ) )
% 9.49/5.25                       => ( ( vEBT_invar_vebt @ Summary3 @ M5 )
% 9.49/5.25                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.49/5.25                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 9.49/5.25                           => ( ( M5 = N3 )
% 9.49/5.25                             => ( ( Deg2
% 9.49/5.25                                  = ( plus_plus_nat @ N3 @ M5 ) )
% 9.49/5.25                               => ( ! [I3: nat] :
% 9.49/5.25                                      ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 9.49/5.25                                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
% 9.49/5.25                                        = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
% 9.49/5.25                                 => ( ( ( Mi3 = Ma3 )
% 9.49/5.25                                     => ! [X2: vEBT_VEBT] :
% 9.49/5.25                                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.49/5.25                                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) )
% 9.49/5.25                                   => ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.49/5.25                                     => ( ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.49/5.25                                       => ~ ( ( Mi3 != Ma3 )
% 9.49/5.25                                           => ! [I3: nat] :
% 9.49/5.25                                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 9.49/5.25                                               => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
% 9.49/5.25                                                      = I3 )
% 9.49/5.25                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
% 9.49/5.25                                                  & ! [X2: nat] :
% 9.49/5.25                                                      ( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
% 9.49/5.25                                                          = I3 )
% 9.49/5.25                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
% 9.49/5.25                                                     => ( ( ord_less_nat @ Mi3 @ X2 )
% 9.49/5.25                                                        & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 9.49/5.25               => ~ ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat,Mi3: nat,Ma3: nat] :
% 9.49/5.25                      ( ( A12
% 9.49/5.25                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg2 @ TreeList3 @ Summary3 ) )
% 9.49/5.25                     => ( ( A23 = Deg2 )
% 9.49/5.25                       => ( ! [X2: vEBT_VEBT] :
% 9.49/5.25                              ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.49/5.25                             => ( vEBT_invar_vebt @ X2 @ N3 ) )
% 9.49/5.25                         => ( ( vEBT_invar_vebt @ Summary3 @ M5 )
% 9.49/5.25                           => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.49/5.25                                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 9.49/5.25                             => ( ( M5
% 9.49/5.25                                  = ( suc @ N3 ) )
% 9.49/5.25                               => ( ( Deg2
% 9.49/5.25                                    = ( plus_plus_nat @ N3 @ M5 ) )
% 9.49/5.25                                 => ( ! [I3: nat] :
% 9.49/5.25                                        ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 9.49/5.25                                       => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
% 9.49/5.25                                          = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
% 9.49/5.25                                   => ( ( ( Mi3 = Ma3 )
% 9.49/5.25                                       => ! [X2: vEBT_VEBT] :
% 9.49/5.25                                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.49/5.25                                           => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) )
% 9.49/5.25                                     => ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.49/5.25                                       => ( ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.49/5.25                                         => ~ ( ( Mi3 != Ma3 )
% 9.49/5.25                                             => ! [I3: nat] :
% 9.49/5.25                                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
% 9.49/5.25                                                 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
% 9.49/5.25                                                        = I3 )
% 9.49/5.25                                                     => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
% 9.49/5.25                                                    & ! [X2: nat] :
% 9.49/5.25                                                        ( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
% 9.49/5.25                                                            = I3 )
% 9.49/5.25                                                          & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
% 9.49/5.25                                                       => ( ( ord_less_nat @ Mi3 @ X2 )
% 9.49/5.25                                                          & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3516_int__power__div__base,axiom,
% 9.49/5.25      ! [M: nat,K: int] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.49/5.25       => ( ( ord_less_int @ zero_zero_int @ K )
% 9.49/5.25         => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 9.49/5.25            = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3517_inrange,axiom,
% 9.49/5.25      ! [T: vEBT_VEBT,N: nat] :
% 9.49/5.25        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.25       => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3518_max_Oabsorb3,axiom,
% 9.49/5.25      ! [B: real,A: real] :
% 9.49/5.25        ( ( ord_less_real @ B @ A )
% 9.49/5.25       => ( ( ord_max_real @ A @ B )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3519_max_Oabsorb3,axiom,
% 9.49/5.25      ! [B: rat,A: rat] :
% 9.49/5.25        ( ( ord_less_rat @ B @ A )
% 9.49/5.25       => ( ( ord_max_rat @ A @ B )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3520_max_Oabsorb3,axiom,
% 9.49/5.25      ! [B: num,A: num] :
% 9.49/5.25        ( ( ord_less_num @ B @ A )
% 9.49/5.25       => ( ( ord_max_num @ A @ B )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3521_max_Oabsorb3,axiom,
% 9.49/5.25      ! [B: nat,A: nat] :
% 9.49/5.25        ( ( ord_less_nat @ B @ A )
% 9.49/5.25       => ( ( ord_max_nat @ A @ B )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3522_max_Oabsorb3,axiom,
% 9.49/5.25      ! [B: int,A: int] :
% 9.49/5.25        ( ( ord_less_int @ B @ A )
% 9.49/5.25       => ( ( ord_max_int @ A @ B )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3523_max_Oabsorb4,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ A @ B )
% 9.49/5.25       => ( ( ord_max_real @ A @ B )
% 9.49/5.25          = B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3524_max_Oabsorb4,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ A @ B )
% 9.49/5.25       => ( ( ord_max_rat @ A @ B )
% 9.49/5.25          = B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3525_max_Oabsorb4,axiom,
% 9.49/5.25      ! [A: num,B: num] :
% 9.49/5.25        ( ( ord_less_num @ A @ B )
% 9.49/5.25       => ( ( ord_max_num @ A @ B )
% 9.49/5.25          = B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3526_max_Oabsorb4,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( ord_less_nat @ A @ B )
% 9.49/5.25       => ( ( ord_max_nat @ A @ B )
% 9.49/5.25          = B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3527_max_Oabsorb4,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ord_less_int @ A @ B )
% 9.49/5.25       => ( ( ord_max_int @ A @ B )
% 9.49/5.25          = B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3528_max__less__iff__conj,axiom,
% 9.49/5.25      ! [X: real,Y: real,Z2: real] :
% 9.49/5.25        ( ( ord_less_real @ ( ord_max_real @ X @ Y ) @ Z2 )
% 9.49/5.25        = ( ( ord_less_real @ X @ Z2 )
% 9.49/5.25          & ( ord_less_real @ Y @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3529_max__less__iff__conj,axiom,
% 9.49/5.25      ! [X: rat,Y: rat,Z2: rat] :
% 9.49/5.25        ( ( ord_less_rat @ ( ord_max_rat @ X @ Y ) @ Z2 )
% 9.49/5.25        = ( ( ord_less_rat @ X @ Z2 )
% 9.49/5.25          & ( ord_less_rat @ Y @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3530_max__less__iff__conj,axiom,
% 9.49/5.25      ! [X: num,Y: num,Z2: num] :
% 9.49/5.25        ( ( ord_less_num @ ( ord_max_num @ X @ Y ) @ Z2 )
% 9.49/5.25        = ( ( ord_less_num @ X @ Z2 )
% 9.49/5.25          & ( ord_less_num @ Y @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3531_max__less__iff__conj,axiom,
% 9.49/5.25      ! [X: nat,Y: nat,Z2: nat] :
% 9.49/5.25        ( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z2 )
% 9.49/5.25        = ( ( ord_less_nat @ X @ Z2 )
% 9.49/5.25          & ( ord_less_nat @ Y @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3532_max__less__iff__conj,axiom,
% 9.49/5.25      ! [X: int,Y: int,Z2: int] :
% 9.49/5.25        ( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z2 )
% 9.49/5.25        = ( ( ord_less_int @ X @ Z2 )
% 9.49/5.25          & ( ord_less_int @ Y @ Z2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3533_max_Oabsorb1,axiom,
% 9.49/5.25      ! [B: rat,A: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ B @ A )
% 9.49/5.25       => ( ( ord_max_rat @ A @ B )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3534_max_Oabsorb1,axiom,
% 9.49/5.25      ! [B: num,A: num] :
% 9.49/5.25        ( ( ord_less_eq_num @ B @ A )
% 9.49/5.25       => ( ( ord_max_num @ A @ B )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3535_max_Oabsorb1,axiom,
% 9.49/5.25      ! [B: nat,A: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ B @ A )
% 9.49/5.25       => ( ( ord_max_nat @ A @ B )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3536_max_Oabsorb1,axiom,
% 9.49/5.25      ! [B: int,A: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ B @ A )
% 9.49/5.25       => ( ( ord_max_int @ A @ B )
% 9.49/5.25          = A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3537_max_Oabsorb2,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.25       => ( ( ord_max_rat @ A @ B )
% 9.49/5.25          = B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3538_max_Oabsorb2,axiom,
% 9.49/5.25      ! [A: num,B: num] :
% 9.49/5.25        ( ( ord_less_eq_num @ A @ B )
% 9.49/5.25       => ( ( ord_max_num @ A @ B )
% 9.49/5.25          = B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3539_max_Oabsorb2,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ A @ B )
% 9.49/5.25       => ( ( ord_max_nat @ A @ B )
% 9.49/5.25          = B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3540_max_Oabsorb2,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.25       => ( ( ord_max_int @ A @ B )
% 9.49/5.25          = B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3541_max_Obounded__iff,axiom,
% 9.49/5.25      ! [B: rat,C: rat,A: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 9.49/5.25        = ( ( ord_less_eq_rat @ B @ A )
% 9.49/5.25          & ( ord_less_eq_rat @ C @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3542_max_Obounded__iff,axiom,
% 9.49/5.25      ! [B: num,C: num,A: num] :
% 9.49/5.25        ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 9.49/5.25        = ( ( ord_less_eq_num @ B @ A )
% 9.49/5.25          & ( ord_less_eq_num @ C @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3543_max_Obounded__iff,axiom,
% 9.49/5.25      ! [B: nat,C: nat,A: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 9.49/5.25        = ( ( ord_less_eq_nat @ B @ A )
% 9.49/5.25          & ( ord_less_eq_nat @ C @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3544_max_Obounded__iff,axiom,
% 9.49/5.25      ! [B: int,C: int,A: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 9.49/5.25        = ( ( ord_less_eq_int @ B @ A )
% 9.49/5.25          & ( ord_less_eq_int @ C @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3545_div__geq,axiom,
% 9.49/5.25      ! [N: nat,M: nat] :
% 9.49/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.25       => ( ~ ( ord_less_nat @ M @ N )
% 9.49/5.25         => ( ( divide_divide_nat @ M @ N )
% 9.49/5.25            = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3546_Diff__cancel,axiom,
% 9.49/5.25      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.25        ( ( minus_1356011639430497352at_nat @ A2 @ A2 )
% 9.49/5.25        = bot_bo2099793752762293965at_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3547_Diff__cancel,axiom,
% 9.49/5.25      ! [A2: set_real] :
% 9.49/5.25        ( ( minus_minus_set_real @ A2 @ A2 )
% 9.49/5.25        = bot_bot_set_real ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3548_Diff__cancel,axiom,
% 9.49/5.25      ! [A2: set_int] :
% 9.49/5.25        ( ( minus_minus_set_int @ A2 @ A2 )
% 9.49/5.25        = bot_bot_set_int ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3549_Diff__cancel,axiom,
% 9.49/5.25      ! [A2: set_nat] :
% 9.49/5.25        ( ( minus_minus_set_nat @ A2 @ A2 )
% 9.49/5.25        = bot_bot_set_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3550_empty__Diff,axiom,
% 9.49/5.25      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.25        ( ( minus_1356011639430497352at_nat @ bot_bo2099793752762293965at_nat @ A2 )
% 9.49/5.25        = bot_bo2099793752762293965at_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3551_empty__Diff,axiom,
% 9.49/5.25      ! [A2: set_real] :
% 9.49/5.25        ( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
% 9.49/5.25        = bot_bot_set_real ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3552_empty__Diff,axiom,
% 9.49/5.25      ! [A2: set_int] :
% 9.49/5.25        ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
% 9.49/5.25        = bot_bot_set_int ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3553_empty__Diff,axiom,
% 9.49/5.25      ! [A2: set_nat] :
% 9.49/5.25        ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
% 9.49/5.25        = bot_bot_set_nat ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3554_Diff__empty,axiom,
% 9.49/5.25      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.25        ( ( minus_1356011639430497352at_nat @ A2 @ bot_bo2099793752762293965at_nat )
% 9.49/5.25        = A2 ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3555_Diff__empty,axiom,
% 9.49/5.25      ! [A2: set_real] :
% 9.49/5.25        ( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
% 9.49/5.25        = A2 ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3556_Diff__empty,axiom,
% 9.49/5.25      ! [A2: set_int] :
% 9.49/5.25        ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
% 9.49/5.25        = A2 ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3557_Diff__empty,axiom,
% 9.49/5.25      ! [A2: set_nat] :
% 9.49/5.25        ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
% 9.49/5.25        = A2 ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3558_finite__Diff2,axiom,
% 9.49/5.25      ! [B2: set_int,A2: set_int] :
% 9.49/5.25        ( ( finite_finite_int @ B2 )
% 9.49/5.25       => ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.49/5.25          = ( finite_finite_int @ A2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3559_finite__Diff2,axiom,
% 9.49/5.25      ! [B2: set_complex,A2: set_complex] :
% 9.49/5.25        ( ( finite3207457112153483333omplex @ B2 )
% 9.49/5.25       => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 9.49/5.25          = ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3560_finite__Diff2,axiom,
% 9.49/5.25      ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.25        ( ( finite6177210948735845034at_nat @ B2 )
% 9.49/5.25       => ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 9.49/5.25          = ( finite6177210948735845034at_nat @ A2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3561_finite__Diff2,axiom,
% 9.49/5.25      ! [B2: set_nat,A2: set_nat] :
% 9.49/5.25        ( ( finite_finite_nat @ B2 )
% 9.49/5.25       => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.49/5.25          = ( finite_finite_nat @ A2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3562_finite__Diff,axiom,
% 9.49/5.25      ! [A2: set_int,B2: set_int] :
% 9.49/5.25        ( ( finite_finite_int @ A2 )
% 9.49/5.25       => ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3563_finite__Diff,axiom,
% 9.49/5.25      ! [A2: set_complex,B2: set_complex] :
% 9.49/5.25        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.25       => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3564_finite__Diff,axiom,
% 9.49/5.25      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.25        ( ( finite6177210948735845034at_nat @ A2 )
% 9.49/5.25       => ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3565_finite__Diff,axiom,
% 9.49/5.25      ! [A2: set_nat,B2: set_nat] :
% 9.49/5.25        ( ( finite_finite_nat @ A2 )
% 9.49/5.25       => ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3566_Diff__eq__empty__iff,axiom,
% 9.49/5.25      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.25        ( ( ( minus_1356011639430497352at_nat @ A2 @ B2 )
% 9.49/5.25          = bot_bo2099793752762293965at_nat )
% 9.49/5.25        = ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3567_Diff__eq__empty__iff,axiom,
% 9.49/5.25      ! [A2: set_real,B2: set_real] :
% 9.49/5.25        ( ( ( minus_minus_set_real @ A2 @ B2 )
% 9.49/5.25          = bot_bot_set_real )
% 9.49/5.25        = ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3568_Diff__eq__empty__iff,axiom,
% 9.49/5.25      ! [A2: set_nat,B2: set_nat] :
% 9.49/5.25        ( ( ( minus_minus_set_nat @ A2 @ B2 )
% 9.49/5.25          = bot_bot_set_nat )
% 9.49/5.25        = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3569_Diff__eq__empty__iff,axiom,
% 9.49/5.25      ! [A2: set_int,B2: set_int] :
% 9.49/5.25        ( ( ( minus_minus_set_int @ A2 @ B2 )
% 9.49/5.25          = bot_bot_set_int )
% 9.49/5.25        = ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3570_atLeastAtMost__iff,axiom,
% 9.49/5.25      ! [I: set_nat,L: set_nat,U: set_nat] :
% 9.49/5.25        ( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
% 9.49/5.25        = ( ( ord_less_eq_set_nat @ L @ I )
% 9.49/5.25          & ( ord_less_eq_set_nat @ I @ U ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3571_atLeastAtMost__iff,axiom,
% 9.49/5.25      ! [I: set_int,L: set_int,U: set_int] :
% 9.49/5.25        ( ( member_set_int @ I @ ( set_or370866239135849197et_int @ L @ U ) )
% 9.49/5.25        = ( ( ord_less_eq_set_int @ L @ I )
% 9.49/5.25          & ( ord_less_eq_set_int @ I @ U ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3572_atLeastAtMost__iff,axiom,
% 9.49/5.25      ! [I: rat,L: rat,U: rat] :
% 9.49/5.25        ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
% 9.49/5.25        = ( ( ord_less_eq_rat @ L @ I )
% 9.49/5.25          & ( ord_less_eq_rat @ I @ U ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3573_atLeastAtMost__iff,axiom,
% 9.49/5.25      ! [I: num,L: num,U: num] :
% 9.49/5.25        ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L @ U ) )
% 9.49/5.25        = ( ( ord_less_eq_num @ L @ I )
% 9.49/5.25          & ( ord_less_eq_num @ I @ U ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3574_atLeastAtMost__iff,axiom,
% 9.49/5.25      ! [I: nat,L: nat,U: nat] :
% 9.49/5.25        ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 9.49/5.25        = ( ( ord_less_eq_nat @ L @ I )
% 9.49/5.25          & ( ord_less_eq_nat @ I @ U ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3575_atLeastAtMost__iff,axiom,
% 9.49/5.25      ! [I: int,L: int,U: int] :
% 9.49/5.25        ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
% 9.49/5.25        = ( ( ord_less_eq_int @ L @ I )
% 9.49/5.25          & ( ord_less_eq_int @ I @ U ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3576_atLeastAtMost__iff,axiom,
% 9.49/5.25      ! [I: real,L: real,U: real] :
% 9.49/5.25        ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
% 9.49/5.25        = ( ( ord_less_eq_real @ L @ I )
% 9.49/5.25          & ( ord_less_eq_real @ I @ U ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3577_Icc__eq__Icc,axiom,
% 9.49/5.25      ! [L: set_int,H2: set_int,L3: set_int,H3: set_int] :
% 9.49/5.25        ( ( ( set_or370866239135849197et_int @ L @ H2 )
% 9.49/5.25          = ( set_or370866239135849197et_int @ L3 @ H3 ) )
% 9.49/5.25        = ( ( ( L = L3 )
% 9.49/5.25            & ( H2 = H3 ) )
% 9.49/5.25          | ( ~ ( ord_less_eq_set_int @ L @ H2 )
% 9.49/5.25            & ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3578_Icc__eq__Icc,axiom,
% 9.49/5.25      ! [L: rat,H2: rat,L3: rat,H3: rat] :
% 9.49/5.25        ( ( ( set_or633870826150836451st_rat @ L @ H2 )
% 9.49/5.25          = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
% 9.49/5.25        = ( ( ( L = L3 )
% 9.49/5.25            & ( H2 = H3 ) )
% 9.49/5.25          | ( ~ ( ord_less_eq_rat @ L @ H2 )
% 9.49/5.25            & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3579_Icc__eq__Icc,axiom,
% 9.49/5.25      ! [L: num,H2: num,L3: num,H3: num] :
% 9.49/5.25        ( ( ( set_or7049704709247886629st_num @ L @ H2 )
% 9.49/5.25          = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
% 9.49/5.25        = ( ( ( L = L3 )
% 9.49/5.25            & ( H2 = H3 ) )
% 9.49/5.25          | ( ~ ( ord_less_eq_num @ L @ H2 )
% 9.49/5.25            & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3580_Icc__eq__Icc,axiom,
% 9.49/5.25      ! [L: nat,H2: nat,L3: nat,H3: nat] :
% 9.49/5.25        ( ( ( set_or1269000886237332187st_nat @ L @ H2 )
% 9.49/5.25          = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
% 9.49/5.25        = ( ( ( L = L3 )
% 9.49/5.25            & ( H2 = H3 ) )
% 9.49/5.25          | ( ~ ( ord_less_eq_nat @ L @ H2 )
% 9.49/5.25            & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3581_Icc__eq__Icc,axiom,
% 9.49/5.25      ! [L: int,H2: int,L3: int,H3: int] :
% 9.49/5.25        ( ( ( set_or1266510415728281911st_int @ L @ H2 )
% 9.49/5.25          = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
% 9.49/5.25        = ( ( ( L = L3 )
% 9.49/5.25            & ( H2 = H3 ) )
% 9.49/5.25          | ( ~ ( ord_less_eq_int @ L @ H2 )
% 9.49/5.25            & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3582_Icc__eq__Icc,axiom,
% 9.49/5.25      ! [L: real,H2: real,L3: real,H3: real] :
% 9.49/5.25        ( ( ( set_or1222579329274155063t_real @ L @ H2 )
% 9.49/5.25          = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
% 9.49/5.25        = ( ( ( L = L3 )
% 9.49/5.25            & ( H2 = H3 ) )
% 9.49/5.25          | ( ~ ( ord_less_eq_real @ L @ H2 )
% 9.49/5.25            & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3583_not__real__square__gt__zero,axiom,
% 9.49/5.25      ! [X: real] :
% 9.49/5.25        ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
% 9.49/5.25        = ( X = zero_zero_real ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3584_finite__atLeastAtMost,axiom,
% 9.49/5.25      ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3585_atLeastatMost__empty__iff,axiom,
% 9.49/5.25      ! [A: set_int,B: set_int] :
% 9.49/5.25        ( ( ( set_or370866239135849197et_int @ A @ B )
% 9.49/5.25          = bot_bot_set_set_int )
% 9.49/5.25        = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3586_atLeastatMost__empty__iff,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ( set_or633870826150836451st_rat @ A @ B )
% 9.49/5.25          = bot_bot_set_rat )
% 9.49/5.25        = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3587_atLeastatMost__empty__iff,axiom,
% 9.49/5.25      ! [A: num,B: num] :
% 9.49/5.25        ( ( ( set_or7049704709247886629st_num @ A @ B )
% 9.49/5.25          = bot_bot_set_num )
% 9.49/5.25        = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3588_atLeastatMost__empty__iff,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( ( set_or1269000886237332187st_nat @ A @ B )
% 9.49/5.25          = bot_bot_set_nat )
% 9.49/5.25        = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3589_atLeastatMost__empty__iff,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( ( set_or1266510415728281911st_int @ A @ B )
% 9.49/5.25          = bot_bot_set_int )
% 9.49/5.25        = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3590_atLeastatMost__empty__iff,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ( set_or1222579329274155063t_real @ A @ B )
% 9.49/5.25          = bot_bot_set_real )
% 9.49/5.25        = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3591_atLeastatMost__empty__iff2,axiom,
% 9.49/5.25      ! [A: set_int,B: set_int] :
% 9.49/5.25        ( ( bot_bot_set_set_int
% 9.49/5.25          = ( set_or370866239135849197et_int @ A @ B ) )
% 9.49/5.25        = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3592_atLeastatMost__empty__iff2,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( bot_bot_set_rat
% 9.49/5.25          = ( set_or633870826150836451st_rat @ A @ B ) )
% 9.49/5.25        = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3593_atLeastatMost__empty__iff2,axiom,
% 9.49/5.25      ! [A: num,B: num] :
% 9.49/5.25        ( ( bot_bot_set_num
% 9.49/5.25          = ( set_or7049704709247886629st_num @ A @ B ) )
% 9.49/5.25        = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3594_atLeastatMost__empty__iff2,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( bot_bot_set_nat
% 9.49/5.25          = ( set_or1269000886237332187st_nat @ A @ B ) )
% 9.49/5.25        = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3595_atLeastatMost__empty__iff2,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( bot_bot_set_int
% 9.49/5.25          = ( set_or1266510415728281911st_int @ A @ B ) )
% 9.49/5.25        = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3596_atLeastatMost__empty__iff2,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( bot_bot_set_real
% 9.49/5.25          = ( set_or1222579329274155063t_real @ A @ B ) )
% 9.49/5.25        = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3597_atLeastatMost__subset__iff,axiom,
% 9.49/5.25      ! [A: set_int,B: set_int,C: set_int,D: set_int] :
% 9.49/5.25        ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 9.49/5.25        = ( ~ ( ord_less_eq_set_int @ A @ B )
% 9.49/5.25          | ( ( ord_less_eq_set_int @ C @ A )
% 9.49/5.25            & ( ord_less_eq_set_int @ B @ D ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3598_atLeastatMost__subset__iff,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.49/5.25        ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 9.49/5.25        = ( ~ ( ord_less_eq_rat @ A @ B )
% 9.49/5.25          | ( ( ord_less_eq_rat @ C @ A )
% 9.49/5.25            & ( ord_less_eq_rat @ B @ D ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3599_atLeastatMost__subset__iff,axiom,
% 9.49/5.25      ! [A: num,B: num,C: num,D: num] :
% 9.49/5.25        ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 9.49/5.25        = ( ~ ( ord_less_eq_num @ A @ B )
% 9.49/5.25          | ( ( ord_less_eq_num @ C @ A )
% 9.49/5.25            & ( ord_less_eq_num @ B @ D ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3600_atLeastatMost__subset__iff,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.49/5.25        ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 9.49/5.25        = ( ~ ( ord_less_eq_nat @ A @ B )
% 9.49/5.25          | ( ( ord_less_eq_nat @ C @ A )
% 9.49/5.25            & ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3601_atLeastatMost__subset__iff,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int,D: int] :
% 9.49/5.25        ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 9.49/5.25        = ( ~ ( ord_less_eq_int @ A @ B )
% 9.49/5.25          | ( ( ord_less_eq_int @ C @ A )
% 9.49/5.25            & ( ord_less_eq_int @ B @ D ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3602_atLeastatMost__subset__iff,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real,D: real] :
% 9.49/5.25        ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 9.49/5.25        = ( ~ ( ord_less_eq_real @ A @ B )
% 9.49/5.25          | ( ( ord_less_eq_real @ C @ A )
% 9.49/5.25            & ( ord_less_eq_real @ B @ D ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3603_atLeastatMost__empty,axiom,
% 9.49/5.25      ! [B: rat,A: rat] :
% 9.49/5.25        ( ( ord_less_rat @ B @ A )
% 9.49/5.25       => ( ( set_or633870826150836451st_rat @ A @ B )
% 9.49/5.25          = bot_bot_set_rat ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3604_atLeastatMost__empty,axiom,
% 9.49/5.25      ! [B: num,A: num] :
% 9.49/5.25        ( ( ord_less_num @ B @ A )
% 9.49/5.25       => ( ( set_or7049704709247886629st_num @ A @ B )
% 9.49/5.25          = bot_bot_set_num ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3605_atLeastatMost__empty,axiom,
% 9.49/5.25      ! [B: nat,A: nat] :
% 9.49/5.25        ( ( ord_less_nat @ B @ A )
% 9.49/5.25       => ( ( set_or1269000886237332187st_nat @ A @ B )
% 9.49/5.25          = bot_bot_set_nat ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3606_atLeastatMost__empty,axiom,
% 9.49/5.25      ! [B: int,A: int] :
% 9.49/5.25        ( ( ord_less_int @ B @ A )
% 9.49/5.25       => ( ( set_or1266510415728281911st_int @ A @ B )
% 9.49/5.25          = bot_bot_set_int ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3607_atLeastatMost__empty,axiom,
% 9.49/5.25      ! [B: real,A: real] :
% 9.49/5.25        ( ( ord_less_real @ B @ A )
% 9.49/5.25       => ( ( set_or1222579329274155063t_real @ A @ B )
% 9.49/5.25          = bot_bot_set_real ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3608_infinite__Icc__iff,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
% 9.49/5.25        = ( ord_less_rat @ A @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3609_infinite__Icc__iff,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
% 9.49/5.25        = ( ord_less_real @ A @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3610_zle__diff1__eq,axiom,
% 9.49/5.25      ! [W2: int,Z2: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z2 @ one_one_int ) )
% 9.49/5.25        = ( ord_less_int @ W2 @ Z2 ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3611_real__arch__pow,axiom,
% 9.49/5.25      ! [X: real,Y: real] :
% 9.49/5.25        ( ( ord_less_real @ one_one_real @ X )
% 9.49/5.25       => ? [N3: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N3 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3612_real__arch__pow__inv,axiom,
% 9.49/5.25      ! [Y: real,X: real] :
% 9.49/5.25        ( ( ord_less_real @ zero_zero_real @ Y )
% 9.49/5.25       => ( ( ord_less_real @ X @ one_one_real )
% 9.49/5.25         => ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X @ N3 ) @ Y ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3613_less__eq__real__def,axiom,
% 9.49/5.25      ( ord_less_eq_real
% 9.49/5.25      = ( ^ [X3: real,Y2: real] :
% 9.49/5.25            ( ( ord_less_real @ X3 @ Y2 )
% 9.49/5.25            | ( X3 = Y2 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3614_Diff__infinite__finite,axiom,
% 9.49/5.25      ! [T3: set_int,S2: set_int] :
% 9.49/5.25        ( ( finite_finite_int @ T3 )
% 9.49/5.25       => ( ~ ( finite_finite_int @ S2 )
% 9.49/5.25         => ~ ( finite_finite_int @ ( minus_minus_set_int @ S2 @ T3 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3615_Diff__infinite__finite,axiom,
% 9.49/5.25      ! [T3: set_complex,S2: set_complex] :
% 9.49/5.25        ( ( finite3207457112153483333omplex @ T3 )
% 9.49/5.25       => ( ~ ( finite3207457112153483333omplex @ S2 )
% 9.49/5.25         => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S2 @ T3 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3616_Diff__infinite__finite,axiom,
% 9.49/5.25      ! [T3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 9.49/5.25        ( ( finite6177210948735845034at_nat @ T3 )
% 9.49/5.25       => ( ~ ( finite6177210948735845034at_nat @ S2 )
% 9.49/5.25         => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S2 @ T3 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3617_Diff__infinite__finite,axiom,
% 9.49/5.25      ! [T3: set_nat,S2: set_nat] :
% 9.49/5.25        ( ( finite_finite_nat @ T3 )
% 9.49/5.25       => ( ~ ( finite_finite_nat @ S2 )
% 9.49/5.25         => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T3 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3618_double__diff,axiom,
% 9.49/5.25      ! [A2: set_nat,B2: set_nat,C2: set_nat] :
% 9.49/5.25        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.49/5.25       => ( ( ord_less_eq_set_nat @ B2 @ C2 )
% 9.49/5.25         => ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C2 @ A2 ) )
% 9.49/5.25            = A2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3619_double__diff,axiom,
% 9.49/5.25      ! [A2: set_int,B2: set_int,C2: set_int] :
% 9.49/5.25        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.49/5.25       => ( ( ord_less_eq_set_int @ B2 @ C2 )
% 9.49/5.25         => ( ( minus_minus_set_int @ B2 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 9.49/5.25            = A2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3620_Diff__subset,axiom,
% 9.49/5.25      ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3621_Diff__subset,axiom,
% 9.49/5.25      ! [A2: set_int,B2: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ A2 ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3622_Diff__mono,axiom,
% 9.49/5.25      ! [A2: set_nat,C2: set_nat,D6: set_nat,B2: set_nat] :
% 9.49/5.25        ( ( ord_less_eq_set_nat @ A2 @ C2 )
% 9.49/5.25       => ( ( ord_less_eq_set_nat @ D6 @ B2 )
% 9.49/5.25         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C2 @ D6 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3623_Diff__mono,axiom,
% 9.49/5.25      ! [A2: set_int,C2: set_int,D6: set_int,B2: set_int] :
% 9.49/5.25        ( ( ord_less_eq_set_int @ A2 @ C2 )
% 9.49/5.25       => ( ( ord_less_eq_set_int @ D6 @ B2 )
% 9.49/5.25         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ ( minus_minus_set_int @ C2 @ D6 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3624_int__less__induct,axiom,
% 9.49/5.25      ! [I: int,K: int,P: int > $o] :
% 9.49/5.25        ( ( ord_less_int @ I @ K )
% 9.49/5.25       => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 9.49/5.25         => ( ! [I2: int] :
% 9.49/5.25                ( ( ord_less_int @ I2 @ K )
% 9.49/5.25               => ( ( P @ I2 )
% 9.49/5.25                 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 9.49/5.25           => ( P @ I ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3625_psubset__imp__ex__mem,axiom,
% 9.49/5.25      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.25        ( ( ord_le7866589430770878221at_nat @ A2 @ B2 )
% 9.49/5.25       => ? [B3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ B2 @ A2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3626_psubset__imp__ex__mem,axiom,
% 9.49/5.25      ! [A2: set_real,B2: set_real] :
% 9.49/5.25        ( ( ord_less_set_real @ A2 @ B2 )
% 9.49/5.25       => ? [B3: real] : ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3627_psubset__imp__ex__mem,axiom,
% 9.49/5.25      ! [A2: set_set_nat,B2: set_set_nat] :
% 9.49/5.25        ( ( ord_less_set_set_nat @ A2 @ B2 )
% 9.49/5.25       => ? [B3: set_nat] : ( member_set_nat @ B3 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3628_psubset__imp__ex__mem,axiom,
% 9.49/5.25      ! [A2: set_int,B2: set_int] :
% 9.49/5.25        ( ( ord_less_set_int @ A2 @ B2 )
% 9.49/5.25       => ? [B3: int] : ( member_int @ B3 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3629_psubset__imp__ex__mem,axiom,
% 9.49/5.25      ! [A2: set_nat,B2: set_nat] :
% 9.49/5.25        ( ( ord_less_set_nat @ A2 @ B2 )
% 9.49/5.25       => ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3630_infinite__Icc,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_rat @ A @ B )
% 9.49/5.25       => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3631_infinite__Icc,axiom,
% 9.49/5.25      ! [A: real,B: real] :
% 9.49/5.25        ( ( ord_less_real @ A @ B )
% 9.49/5.25       => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3632_ex__nat__less,axiom,
% 9.49/5.25      ! [N: nat,P: nat > $o] :
% 9.49/5.25        ( ( ? [M2: nat] :
% 9.49/5.25              ( ( ord_less_eq_nat @ M2 @ N )
% 9.49/5.25              & ( P @ M2 ) ) )
% 9.49/5.25        = ( ? [X3: nat] :
% 9.49/5.25              ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.49/5.25              & ( P @ X3 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3633_all__nat__less,axiom,
% 9.49/5.25      ! [N: nat,P: nat > $o] :
% 9.49/5.25        ( ( ! [M2: nat] :
% 9.49/5.25              ( ( ord_less_eq_nat @ M2 @ N )
% 9.49/5.25             => ( P @ M2 ) ) )
% 9.49/5.25        = ( ! [X3: nat] :
% 9.49/5.25              ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.49/5.25             => ( P @ X3 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3634_subset__eq__atLeast0__atMost__finite,axiom,
% 9.49/5.25      ! [N6: set_nat,N: nat] :
% 9.49/5.25        ( ( ord_less_eq_set_nat @ N6 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.49/5.25       => ( finite_finite_nat @ N6 ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3635_decr__mult__lemma,axiom,
% 9.49/5.25      ! [D: int,P: int > $o,K: int] :
% 9.49/5.25        ( ( ord_less_int @ zero_zero_int @ D )
% 9.49/5.25       => ( ! [X4: int] :
% 9.49/5.25              ( ( P @ X4 )
% 9.49/5.25             => ( P @ ( minus_minus_int @ X4 @ D ) ) )
% 9.49/5.25         => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.49/5.25           => ! [X2: int] :
% 9.49/5.25                ( ( P @ X2 )
% 9.49/5.25               => ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3636_plusinfinity,axiom,
% 9.49/5.25      ! [D: int,P5: int > $o,P: int > $o] :
% 9.49/5.25        ( ( ord_less_int @ zero_zero_int @ D )
% 9.49/5.25       => ( ! [X4: int,K2: int] :
% 9.49/5.25              ( ( P5 @ X4 )
% 9.49/5.25              = ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 9.49/5.25         => ( ? [Z5: int] :
% 9.49/5.25              ! [X4: int] :
% 9.49/5.25                ( ( ord_less_int @ Z5 @ X4 )
% 9.49/5.25               => ( ( P @ X4 )
% 9.49/5.25                  = ( P5 @ X4 ) ) )
% 9.49/5.25           => ( ? [X_12: int] : ( P5 @ X_12 )
% 9.49/5.25             => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3637_minusinfinity,axiom,
% 9.49/5.25      ! [D: int,P1: int > $o,P: int > $o] :
% 9.49/5.25        ( ( ord_less_int @ zero_zero_int @ D )
% 9.49/5.25       => ( ! [X4: int,K2: int] :
% 9.49/5.25              ( ( P1 @ X4 )
% 9.49/5.25              = ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 9.49/5.25         => ( ? [Z5: int] :
% 9.49/5.25              ! [X4: int] :
% 9.49/5.25                ( ( ord_less_int @ X4 @ Z5 )
% 9.49/5.25               => ( ( P @ X4 )
% 9.49/5.25                  = ( P1 @ X4 ) ) )
% 9.49/5.25           => ( ? [X_12: int] : ( P1 @ X_12 )
% 9.49/5.25             => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3638_div__pos__geq,axiom,
% 9.49/5.25      ! [L: int,K: int] :
% 9.49/5.25        ( ( ord_less_int @ zero_zero_int @ L )
% 9.49/5.25       => ( ( ord_less_eq_int @ L @ K )
% 9.49/5.25         => ( ( divide_divide_int @ K @ L )
% 9.49/5.25            = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3639_atLeastatMost__psubset__iff,axiom,
% 9.49/5.25      ! [A: set_int,B: set_int,C: set_int,D: set_int] :
% 9.49/5.25        ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 9.49/5.25        = ( ( ~ ( ord_less_eq_set_int @ A @ B )
% 9.49/5.25            | ( ( ord_less_eq_set_int @ C @ A )
% 9.49/5.25              & ( ord_less_eq_set_int @ B @ D )
% 9.49/5.25              & ( ( ord_less_set_int @ C @ A )
% 9.49/5.25                | ( ord_less_set_int @ B @ D ) ) ) )
% 9.49/5.25          & ( ord_less_eq_set_int @ C @ D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3640_atLeastatMost__psubset__iff,axiom,
% 9.49/5.25      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.49/5.25        ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 9.49/5.25        = ( ( ~ ( ord_less_eq_rat @ A @ B )
% 9.49/5.25            | ( ( ord_less_eq_rat @ C @ A )
% 9.49/5.25              & ( ord_less_eq_rat @ B @ D )
% 9.49/5.25              & ( ( ord_less_rat @ C @ A )
% 9.49/5.25                | ( ord_less_rat @ B @ D ) ) ) )
% 9.49/5.25          & ( ord_less_eq_rat @ C @ D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3641_atLeastatMost__psubset__iff,axiom,
% 9.49/5.25      ! [A: num,B: num,C: num,D: num] :
% 9.49/5.25        ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 9.49/5.25        = ( ( ~ ( ord_less_eq_num @ A @ B )
% 9.49/5.25            | ( ( ord_less_eq_num @ C @ A )
% 9.49/5.25              & ( ord_less_eq_num @ B @ D )
% 9.49/5.25              & ( ( ord_less_num @ C @ A )
% 9.49/5.25                | ( ord_less_num @ B @ D ) ) ) )
% 9.49/5.25          & ( ord_less_eq_num @ C @ D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3642_atLeastatMost__psubset__iff,axiom,
% 9.49/5.25      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.49/5.25        ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 9.49/5.25        = ( ( ~ ( ord_less_eq_nat @ A @ B )
% 9.49/5.25            | ( ( ord_less_eq_nat @ C @ A )
% 9.49/5.25              & ( ord_less_eq_nat @ B @ D )
% 9.49/5.25              & ( ( ord_less_nat @ C @ A )
% 9.49/5.25                | ( ord_less_nat @ B @ D ) ) ) )
% 9.49/5.25          & ( ord_less_eq_nat @ C @ D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3643_atLeastatMost__psubset__iff,axiom,
% 9.49/5.25      ! [A: int,B: int,C: int,D: int] :
% 9.49/5.25        ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 9.49/5.25        = ( ( ~ ( ord_less_eq_int @ A @ B )
% 9.49/5.25            | ( ( ord_less_eq_int @ C @ A )
% 9.49/5.25              & ( ord_less_eq_int @ B @ D )
% 9.49/5.25              & ( ( ord_less_int @ C @ A )
% 9.49/5.25                | ( ord_less_int @ B @ D ) ) ) )
% 9.49/5.25          & ( ord_less_eq_int @ C @ D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3644_atLeastatMost__psubset__iff,axiom,
% 9.49/5.25      ! [A: real,B: real,C: real,D: real] :
% 9.49/5.25        ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 9.49/5.25        = ( ( ~ ( ord_less_eq_real @ A @ B )
% 9.49/5.25            | ( ( ord_less_eq_real @ C @ A )
% 9.49/5.25              & ( ord_less_eq_real @ B @ D )
% 9.49/5.25              & ( ( ord_less_real @ C @ A )
% 9.49/5.25                | ( ord_less_real @ B @ D ) ) ) )
% 9.49/5.25          & ( ord_less_eq_real @ C @ D ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3645_max_Omono,axiom,
% 9.49/5.25      ! [C: rat,A: rat,D: rat,B: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ C @ A )
% 9.49/5.25       => ( ( ord_less_eq_rat @ D @ B )
% 9.49/5.25         => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3646_max_Omono,axiom,
% 9.49/5.25      ! [C: num,A: num,D: num,B: num] :
% 9.49/5.25        ( ( ord_less_eq_num @ C @ A )
% 9.49/5.25       => ( ( ord_less_eq_num @ D @ B )
% 9.49/5.25         => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3647_max_Omono,axiom,
% 9.49/5.25      ! [C: nat,A: nat,D: nat,B: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ C @ A )
% 9.49/5.25       => ( ( ord_less_eq_nat @ D @ B )
% 9.49/5.25         => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3648_max_Omono,axiom,
% 9.49/5.25      ! [C: int,A: int,D: int,B: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ C @ A )
% 9.49/5.25       => ( ( ord_less_eq_int @ D @ B )
% 9.49/5.25         => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3649_max_OorderE,axiom,
% 9.49/5.25      ! [B: rat,A: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ B @ A )
% 9.49/5.25       => ( A
% 9.49/5.25          = ( ord_max_rat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3650_max_OorderE,axiom,
% 9.49/5.25      ! [B: num,A: num] :
% 9.49/5.25        ( ( ord_less_eq_num @ B @ A )
% 9.49/5.25       => ( A
% 9.49/5.25          = ( ord_max_num @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3651_max_OorderE,axiom,
% 9.49/5.25      ! [B: nat,A: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ B @ A )
% 9.49/5.25       => ( A
% 9.49/5.25          = ( ord_max_nat @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3652_max_OorderE,axiom,
% 9.49/5.25      ! [B: int,A: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ B @ A )
% 9.49/5.25       => ( A
% 9.49/5.25          = ( ord_max_int @ A @ B ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3653_max_OorderI,axiom,
% 9.49/5.25      ! [A: rat,B: rat] :
% 9.49/5.25        ( ( A
% 9.49/5.25          = ( ord_max_rat @ A @ B ) )
% 9.49/5.25       => ( ord_less_eq_rat @ B @ A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3654_max_OorderI,axiom,
% 9.49/5.25      ! [A: num,B: num] :
% 9.49/5.25        ( ( A
% 9.49/5.25          = ( ord_max_num @ A @ B ) )
% 9.49/5.25       => ( ord_less_eq_num @ B @ A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3655_max_OorderI,axiom,
% 9.49/5.25      ! [A: nat,B: nat] :
% 9.49/5.25        ( ( A
% 9.49/5.25          = ( ord_max_nat @ A @ B ) )
% 9.49/5.25       => ( ord_less_eq_nat @ B @ A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3656_max_OorderI,axiom,
% 9.49/5.25      ! [A: int,B: int] :
% 9.49/5.25        ( ( A
% 9.49/5.25          = ( ord_max_int @ A @ B ) )
% 9.49/5.25       => ( ord_less_eq_int @ B @ A ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3657_max_OboundedE,axiom,
% 9.49/5.25      ! [B: rat,C: rat,A: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 9.49/5.25       => ~ ( ( ord_less_eq_rat @ B @ A )
% 9.49/5.25           => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3658_max_OboundedE,axiom,
% 9.49/5.25      ! [B: num,C: num,A: num] :
% 9.49/5.25        ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 9.49/5.25       => ~ ( ( ord_less_eq_num @ B @ A )
% 9.49/5.25           => ~ ( ord_less_eq_num @ C @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3659_max_OboundedE,axiom,
% 9.49/5.25      ! [B: nat,C: nat,A: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 9.49/5.25       => ~ ( ( ord_less_eq_nat @ B @ A )
% 9.49/5.25           => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3660_max_OboundedE,axiom,
% 9.49/5.25      ! [B: int,C: int,A: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 9.49/5.25       => ~ ( ( ord_less_eq_int @ B @ A )
% 9.49/5.25           => ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3661_max_OboundedI,axiom,
% 9.49/5.25      ! [B: rat,A: rat,C: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ B @ A )
% 9.49/5.25       => ( ( ord_less_eq_rat @ C @ A )
% 9.49/5.25         => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3662_max_OboundedI,axiom,
% 9.49/5.25      ! [B: num,A: num,C: num] :
% 9.49/5.25        ( ( ord_less_eq_num @ B @ A )
% 9.49/5.25       => ( ( ord_less_eq_num @ C @ A )
% 9.49/5.25         => ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3663_max_OboundedI,axiom,
% 9.49/5.25      ! [B: nat,A: nat,C: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ B @ A )
% 9.49/5.25       => ( ( ord_less_eq_nat @ C @ A )
% 9.49/5.25         => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3664_max_OboundedI,axiom,
% 9.49/5.25      ! [B: int,A: int,C: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ B @ A )
% 9.49/5.25       => ( ( ord_less_eq_int @ C @ A )
% 9.49/5.25         => ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3665_max_Oorder__iff,axiom,
% 9.49/5.25      ( ord_less_eq_rat
% 9.49/5.25      = ( ^ [B4: rat,A4: rat] :
% 9.49/5.25            ( A4
% 9.49/5.25            = ( ord_max_rat @ A4 @ B4 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3666_max_Oorder__iff,axiom,
% 9.49/5.25      ( ord_less_eq_num
% 9.49/5.25      = ( ^ [B4: num,A4: num] :
% 9.49/5.25            ( A4
% 9.49/5.25            = ( ord_max_num @ A4 @ B4 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3667_max_Oorder__iff,axiom,
% 9.49/5.25      ( ord_less_eq_nat
% 9.49/5.25      = ( ^ [B4: nat,A4: nat] :
% 9.49/5.25            ( A4
% 9.49/5.25            = ( ord_max_nat @ A4 @ B4 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3668_max_Oorder__iff,axiom,
% 9.49/5.25      ( ord_less_eq_int
% 9.49/5.25      = ( ^ [B4: int,A4: int] :
% 9.49/5.25            ( A4
% 9.49/5.25            = ( ord_max_int @ A4 @ B4 ) ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3669_max_Ocobounded1,axiom,
% 9.49/5.25      ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3670_max_Ocobounded1,axiom,
% 9.49/5.25      ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3671_max_Ocobounded1,axiom,
% 9.49/5.25      ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3672_max_Ocobounded1,axiom,
% 9.49/5.25      ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3673_max_Ocobounded2,axiom,
% 9.49/5.25      ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3674_max_Ocobounded2,axiom,
% 9.49/5.25      ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3675_max_Ocobounded2,axiom,
% 9.49/5.25      ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3676_max_Ocobounded2,axiom,
% 9.49/5.25      ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3677_le__max__iff__disj,axiom,
% 9.49/5.25      ! [Z2: rat,X: rat,Y: rat] :
% 9.49/5.25        ( ( ord_less_eq_rat @ Z2 @ ( ord_max_rat @ X @ Y ) )
% 9.49/5.25        = ( ( ord_less_eq_rat @ Z2 @ X )
% 9.49/5.25          | ( ord_less_eq_rat @ Z2 @ Y ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3678_le__max__iff__disj,axiom,
% 9.49/5.25      ! [Z2: num,X: num,Y: num] :
% 9.49/5.25        ( ( ord_less_eq_num @ Z2 @ ( ord_max_num @ X @ Y ) )
% 9.49/5.25        = ( ( ord_less_eq_num @ Z2 @ X )
% 9.49/5.25          | ( ord_less_eq_num @ Z2 @ Y ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3679_le__max__iff__disj,axiom,
% 9.49/5.25      ! [Z2: nat,X: nat,Y: nat] :
% 9.49/5.25        ( ( ord_less_eq_nat @ Z2 @ ( ord_max_nat @ X @ Y ) )
% 9.49/5.25        = ( ( ord_less_eq_nat @ Z2 @ X )
% 9.49/5.25          | ( ord_less_eq_nat @ Z2 @ Y ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3680_le__max__iff__disj,axiom,
% 9.49/5.25      ! [Z2: int,X: int,Y: int] :
% 9.49/5.25        ( ( ord_less_eq_int @ Z2 @ ( ord_max_int @ X @ Y ) )
% 9.49/5.25        = ( ( ord_less_eq_int @ Z2 @ X )
% 9.49/5.25          | ( ord_less_eq_int @ Z2 @ Y ) ) ) ).
% 9.49/5.25  
% 9.49/5.25  thf(fact_3681_max_Oabsorb__iff1,axiom,
% 9.49/5.25      ( ord_less_eq_rat
% 9.49/5.25      = ( ^ [B4: rat,A4: rat] :
% 9.49/5.25            ( ( ord_max_rat @ A4 @ B4 )
% 9.49/5.26            = A4 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3682_max_Oabsorb__iff1,axiom,
% 9.49/5.26      ( ord_less_eq_num
% 9.49/5.26      = ( ^ [B4: num,A4: num] :
% 9.49/5.26            ( ( ord_max_num @ A4 @ B4 )
% 9.49/5.26            = A4 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3683_max_Oabsorb__iff1,axiom,
% 9.49/5.26      ( ord_less_eq_nat
% 9.49/5.26      = ( ^ [B4: nat,A4: nat] :
% 9.49/5.26            ( ( ord_max_nat @ A4 @ B4 )
% 9.49/5.26            = A4 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3684_max_Oabsorb__iff1,axiom,
% 9.49/5.26      ( ord_less_eq_int
% 9.49/5.26      = ( ^ [B4: int,A4: int] :
% 9.49/5.26            ( ( ord_max_int @ A4 @ B4 )
% 9.49/5.26            = A4 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3685_max_Oabsorb__iff2,axiom,
% 9.49/5.26      ( ord_less_eq_rat
% 9.49/5.26      = ( ^ [A4: rat,B4: rat] :
% 9.49/5.26            ( ( ord_max_rat @ A4 @ B4 )
% 9.49/5.26            = B4 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3686_max_Oabsorb__iff2,axiom,
% 9.49/5.26      ( ord_less_eq_num
% 9.49/5.26      = ( ^ [A4: num,B4: num] :
% 9.49/5.26            ( ( ord_max_num @ A4 @ B4 )
% 9.49/5.26            = B4 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3687_max_Oabsorb__iff2,axiom,
% 9.49/5.26      ( ord_less_eq_nat
% 9.49/5.26      = ( ^ [A4: nat,B4: nat] :
% 9.49/5.26            ( ( ord_max_nat @ A4 @ B4 )
% 9.49/5.26            = B4 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3688_max_Oabsorb__iff2,axiom,
% 9.49/5.26      ( ord_less_eq_int
% 9.49/5.26      = ( ^ [A4: int,B4: int] :
% 9.49/5.26            ( ( ord_max_int @ A4 @ B4 )
% 9.49/5.26            = B4 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3689_max_OcoboundedI1,axiom,
% 9.49/5.26      ! [C: rat,A: rat,B: rat] :
% 9.49/5.26        ( ( ord_less_eq_rat @ C @ A )
% 9.49/5.26       => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3690_max_OcoboundedI1,axiom,
% 9.49/5.26      ! [C: num,A: num,B: num] :
% 9.49/5.26        ( ( ord_less_eq_num @ C @ A )
% 9.49/5.26       => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3691_max_OcoboundedI1,axiom,
% 9.49/5.26      ! [C: nat,A: nat,B: nat] :
% 9.49/5.26        ( ( ord_less_eq_nat @ C @ A )
% 9.49/5.26       => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3692_max_OcoboundedI1,axiom,
% 9.49/5.26      ! [C: int,A: int,B: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ C @ A )
% 9.49/5.26       => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3693_max_OcoboundedI2,axiom,
% 9.49/5.26      ! [C: rat,B: rat,A: rat] :
% 9.49/5.26        ( ( ord_less_eq_rat @ C @ B )
% 9.49/5.26       => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3694_max_OcoboundedI2,axiom,
% 9.49/5.26      ! [C: num,B: num,A: num] :
% 9.49/5.26        ( ( ord_less_eq_num @ C @ B )
% 9.49/5.26       => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3695_max_OcoboundedI2,axiom,
% 9.49/5.26      ! [C: nat,B: nat,A: nat] :
% 9.49/5.26        ( ( ord_less_eq_nat @ C @ B )
% 9.49/5.26       => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3696_max_OcoboundedI2,axiom,
% 9.49/5.26      ! [C: int,B: int,A: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ C @ B )
% 9.49/5.26       => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3697_less__max__iff__disj,axiom,
% 9.49/5.26      ! [Z2: real,X: real,Y: real] :
% 9.49/5.26        ( ( ord_less_real @ Z2 @ ( ord_max_real @ X @ Y ) )
% 9.49/5.26        = ( ( ord_less_real @ Z2 @ X )
% 9.49/5.26          | ( ord_less_real @ Z2 @ Y ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3698_less__max__iff__disj,axiom,
% 9.49/5.26      ! [Z2: rat,X: rat,Y: rat] :
% 9.49/5.26        ( ( ord_less_rat @ Z2 @ ( ord_max_rat @ X @ Y ) )
% 9.49/5.26        = ( ( ord_less_rat @ Z2 @ X )
% 9.49/5.26          | ( ord_less_rat @ Z2 @ Y ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3699_less__max__iff__disj,axiom,
% 9.49/5.26      ! [Z2: num,X: num,Y: num] :
% 9.49/5.26        ( ( ord_less_num @ Z2 @ ( ord_max_num @ X @ Y ) )
% 9.49/5.26        = ( ( ord_less_num @ Z2 @ X )
% 9.49/5.26          | ( ord_less_num @ Z2 @ Y ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3700_less__max__iff__disj,axiom,
% 9.49/5.26      ! [Z2: nat,X: nat,Y: nat] :
% 9.49/5.26        ( ( ord_less_nat @ Z2 @ ( ord_max_nat @ X @ Y ) )
% 9.49/5.26        = ( ( ord_less_nat @ Z2 @ X )
% 9.49/5.26          | ( ord_less_nat @ Z2 @ Y ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3701_less__max__iff__disj,axiom,
% 9.49/5.26      ! [Z2: int,X: int,Y: int] :
% 9.49/5.26        ( ( ord_less_int @ Z2 @ ( ord_max_int @ X @ Y ) )
% 9.49/5.26        = ( ( ord_less_int @ Z2 @ X )
% 9.49/5.26          | ( ord_less_int @ Z2 @ Y ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3702_max_Ostrict__boundedE,axiom,
% 9.49/5.26      ! [B: real,C: real,A: real] :
% 9.49/5.26        ( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
% 9.49/5.26       => ~ ( ( ord_less_real @ B @ A )
% 9.49/5.26           => ~ ( ord_less_real @ C @ A ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3703_max_Ostrict__boundedE,axiom,
% 9.49/5.26      ! [B: rat,C: rat,A: rat] :
% 9.49/5.26        ( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
% 9.49/5.26       => ~ ( ( ord_less_rat @ B @ A )
% 9.49/5.26           => ~ ( ord_less_rat @ C @ A ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3704_max_Ostrict__boundedE,axiom,
% 9.49/5.26      ! [B: num,C: num,A: num] :
% 9.49/5.26        ( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
% 9.49/5.26       => ~ ( ( ord_less_num @ B @ A )
% 9.49/5.26           => ~ ( ord_less_num @ C @ A ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3705_max_Ostrict__boundedE,axiom,
% 9.49/5.26      ! [B: nat,C: nat,A: nat] :
% 9.49/5.26        ( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
% 9.49/5.26       => ~ ( ( ord_less_nat @ B @ A )
% 9.49/5.26           => ~ ( ord_less_nat @ C @ A ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3706_max_Ostrict__boundedE,axiom,
% 9.49/5.26      ! [B: int,C: int,A: int] :
% 9.49/5.26        ( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
% 9.49/5.26       => ~ ( ( ord_less_int @ B @ A )
% 9.49/5.26           => ~ ( ord_less_int @ C @ A ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3707_max_Ostrict__order__iff,axiom,
% 9.49/5.26      ( ord_less_real
% 9.49/5.26      = ( ^ [B4: real,A4: real] :
% 9.49/5.26            ( ( A4
% 9.49/5.26              = ( ord_max_real @ A4 @ B4 ) )
% 9.49/5.26            & ( A4 != B4 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3708_max_Ostrict__order__iff,axiom,
% 9.49/5.26      ( ord_less_rat
% 9.49/5.26      = ( ^ [B4: rat,A4: rat] :
% 9.49/5.26            ( ( A4
% 9.49/5.26              = ( ord_max_rat @ A4 @ B4 ) )
% 9.49/5.26            & ( A4 != B4 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3709_max_Ostrict__order__iff,axiom,
% 9.49/5.26      ( ord_less_num
% 9.49/5.26      = ( ^ [B4: num,A4: num] :
% 9.49/5.26            ( ( A4
% 9.49/5.26              = ( ord_max_num @ A4 @ B4 ) )
% 9.49/5.26            & ( A4 != B4 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3710_max_Ostrict__order__iff,axiom,
% 9.49/5.26      ( ord_less_nat
% 9.49/5.26      = ( ^ [B4: nat,A4: nat] :
% 9.49/5.26            ( ( A4
% 9.49/5.26              = ( ord_max_nat @ A4 @ B4 ) )
% 9.49/5.26            & ( A4 != B4 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3711_max_Ostrict__order__iff,axiom,
% 9.49/5.26      ( ord_less_int
% 9.49/5.26      = ( ^ [B4: int,A4: int] :
% 9.49/5.26            ( ( A4
% 9.49/5.26              = ( ord_max_int @ A4 @ B4 ) )
% 9.49/5.26            & ( A4 != B4 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3712_max_Ostrict__coboundedI1,axiom,
% 9.49/5.26      ! [C: real,A: real,B: real] :
% 9.49/5.26        ( ( ord_less_real @ C @ A )
% 9.49/5.26       => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3713_max_Ostrict__coboundedI1,axiom,
% 9.49/5.26      ! [C: rat,A: rat,B: rat] :
% 9.49/5.26        ( ( ord_less_rat @ C @ A )
% 9.49/5.26       => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3714_max_Ostrict__coboundedI1,axiom,
% 9.49/5.26      ! [C: num,A: num,B: num] :
% 9.49/5.26        ( ( ord_less_num @ C @ A )
% 9.49/5.26       => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3715_max_Ostrict__coboundedI1,axiom,
% 9.49/5.26      ! [C: nat,A: nat,B: nat] :
% 9.49/5.26        ( ( ord_less_nat @ C @ A )
% 9.49/5.26       => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3716_max_Ostrict__coboundedI1,axiom,
% 9.49/5.26      ! [C: int,A: int,B: int] :
% 9.49/5.26        ( ( ord_less_int @ C @ A )
% 9.49/5.26       => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3717_max_Ostrict__coboundedI2,axiom,
% 9.49/5.26      ! [C: real,B: real,A: real] :
% 9.49/5.26        ( ( ord_less_real @ C @ B )
% 9.49/5.26       => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3718_max_Ostrict__coboundedI2,axiom,
% 9.49/5.26      ! [C: rat,B: rat,A: rat] :
% 9.49/5.26        ( ( ord_less_rat @ C @ B )
% 9.49/5.26       => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3719_max_Ostrict__coboundedI2,axiom,
% 9.49/5.26      ! [C: num,B: num,A: num] :
% 9.49/5.26        ( ( ord_less_num @ C @ B )
% 9.49/5.26       => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3720_max_Ostrict__coboundedI2,axiom,
% 9.49/5.26      ! [C: nat,B: nat,A: nat] :
% 9.49/5.26        ( ( ord_less_nat @ C @ B )
% 9.49/5.26       => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3721_max_Ostrict__coboundedI2,axiom,
% 9.49/5.26      ! [C: int,B: int,A: int] :
% 9.49/5.26        ( ( ord_less_int @ C @ B )
% 9.49/5.26       => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3722_unique__quotient__lemma__neg,axiom,
% 9.49/5.26      ! [B: int,Q6: int,R4: int,Q4: int,R2: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 9.49/5.26       => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 9.49/5.26         => ( ( ord_less_int @ B @ R2 )
% 9.49/5.26           => ( ( ord_less_int @ B @ R4 )
% 9.49/5.26             => ( ord_less_eq_int @ Q4 @ Q6 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3723_unique__quotient__lemma,axiom,
% 9.49/5.26      ! [B: int,Q6: int,R4: int,Q4: int,R2: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 9.49/5.26       => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 9.49/5.26         => ( ( ord_less_int @ R4 @ B )
% 9.49/5.26           => ( ( ord_less_int @ R2 @ B )
% 9.49/5.26             => ( ord_less_eq_int @ Q6 @ Q4 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3724_zdiv__mono2__neg__lemma,axiom,
% 9.49/5.26      ! [B: int,Q4: int,R2: int,B7: int,Q6: int,R4: int] :
% 9.49/5.26        ( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
% 9.49/5.26          = ( plus_plus_int @ ( times_times_int @ B7 @ Q6 ) @ R4 ) )
% 9.49/5.26       => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q6 ) @ R4 ) @ zero_zero_int )
% 9.49/5.26         => ( ( ord_less_int @ R2 @ B )
% 9.49/5.26           => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 9.49/5.26             => ( ( ord_less_int @ zero_zero_int @ B7 )
% 9.49/5.26               => ( ( ord_less_eq_int @ B7 @ B )
% 9.49/5.26                 => ( ord_less_eq_int @ Q6 @ Q4 ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3725_zdiv__mono2__lemma,axiom,
% 9.49/5.26      ! [B: int,Q4: int,R2: int,B7: int,Q6: int,R4: int] :
% 9.49/5.26        ( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
% 9.49/5.26          = ( plus_plus_int @ ( times_times_int @ B7 @ Q6 ) @ R4 ) )
% 9.49/5.26       => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q6 ) @ R4 ) )
% 9.49/5.26         => ( ( ord_less_int @ R4 @ B7 )
% 9.49/5.26           => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 9.49/5.26             => ( ( ord_less_int @ zero_zero_int @ B7 )
% 9.49/5.26               => ( ( ord_less_eq_int @ B7 @ B )
% 9.49/5.26                 => ( ord_less_eq_int @ Q4 @ Q6 ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3726_q__pos__lemma,axiom,
% 9.49/5.26      ! [B7: int,Q6: int,R4: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q6 ) @ R4 ) )
% 9.49/5.26       => ( ( ord_less_int @ R4 @ B7 )
% 9.49/5.26         => ( ( ord_less_int @ zero_zero_int @ B7 )
% 9.49/5.26           => ( ord_less_eq_int @ zero_zero_int @ Q6 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3727_int__div__pos__eq,axiom,
% 9.49/5.26      ! [A: int,B: int,Q4: int,R2: int] :
% 9.49/5.26        ( ( A
% 9.49/5.26          = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 9.49/5.26       => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 9.49/5.26         => ( ( ord_less_int @ R2 @ B )
% 9.49/5.26           => ( ( divide_divide_int @ A @ B )
% 9.49/5.26              = Q4 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3728_int__div__neg__eq,axiom,
% 9.49/5.26      ! [A: int,B: int,Q4: int,R2: int] :
% 9.49/5.26        ( ( A
% 9.49/5.26          = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 9.49/5.26       => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 9.49/5.26         => ( ( ord_less_int @ B @ R2 )
% 9.49/5.26           => ( ( divide_divide_int @ A @ B )
% 9.49/5.26              = Q4 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3729_split__zdiv,axiom,
% 9.49/5.26      ! [P: int > $o,N: int,K: int] :
% 9.49/5.26        ( ( P @ ( divide_divide_int @ N @ K ) )
% 9.49/5.26        = ( ( ( K = zero_zero_int )
% 9.49/5.26           => ( P @ zero_zero_int ) )
% 9.49/5.26          & ( ( ord_less_int @ zero_zero_int @ K )
% 9.49/5.26           => ! [I4: int,J3: int] :
% 9.49/5.26                ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 9.49/5.26                  & ( ord_less_int @ J3 @ K )
% 9.49/5.26                  & ( N
% 9.49/5.26                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 9.49/5.26               => ( P @ I4 ) ) )
% 9.49/5.26          & ( ( ord_less_int @ K @ zero_zero_int )
% 9.49/5.26           => ! [I4: int,J3: int] :
% 9.49/5.26                ( ( ( ord_less_int @ K @ J3 )
% 9.49/5.26                  & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 9.49/5.26                  & ( N
% 9.49/5.26                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 9.49/5.26               => ( P @ I4 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3730_zdiv__mono1,axiom,
% 9.49/5.26      ! [A: int,A7: int,B: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ A @ A7 )
% 9.49/5.26       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.26         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A7 @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3731_zdiv__mono2,axiom,
% 9.49/5.26      ! [A: int,B7: int,B: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.26       => ( ( ord_less_int @ zero_zero_int @ B7 )
% 9.49/5.26         => ( ( ord_less_eq_int @ B7 @ B )
% 9.49/5.26           => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B7 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3732_zdiv__eq__0__iff,axiom,
% 9.49/5.26      ! [I: int,K: int] :
% 9.49/5.26        ( ( ( divide_divide_int @ I @ K )
% 9.49/5.26          = zero_zero_int )
% 9.49/5.26        = ( ( K = zero_zero_int )
% 9.49/5.26          | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 9.49/5.26            & ( ord_less_int @ I @ K ) )
% 9.49/5.26          | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 9.49/5.26            & ( ord_less_int @ K @ I ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3733_zdiv__mono1__neg,axiom,
% 9.49/5.26      ! [A: int,A7: int,B: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ A @ A7 )
% 9.49/5.26       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.49/5.26         => ( ord_less_eq_int @ ( divide_divide_int @ A7 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3734_zdiv__mono2__neg,axiom,
% 9.49/5.26      ! [A: int,B7: int,B: int] :
% 9.49/5.26        ( ( ord_less_int @ A @ zero_zero_int )
% 9.49/5.26       => ( ( ord_less_int @ zero_zero_int @ B7 )
% 9.49/5.26         => ( ( ord_less_eq_int @ B7 @ B )
% 9.49/5.26           => ( ord_less_eq_int @ ( divide_divide_int @ A @ B7 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3735_div__int__pos__iff,axiom,
% 9.49/5.26      ! [K: int,L: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
% 9.49/5.26        = ( ( K = zero_zero_int )
% 9.49/5.26          | ( L = zero_zero_int )
% 9.49/5.26          | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.49/5.26            & ( ord_less_eq_int @ zero_zero_int @ L ) )
% 9.49/5.26          | ( ( ord_less_int @ K @ zero_zero_int )
% 9.49/5.26            & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3736_div__positive__int,axiom,
% 9.49/5.26      ! [L: int,K: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ L @ K )
% 9.49/5.26       => ( ( ord_less_int @ zero_zero_int @ L )
% 9.49/5.26         => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3737_div__nonneg__neg__le0,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.26       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.49/5.26         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3738_div__nonpos__pos__le0,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.49/5.26       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.26         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3739_pos__imp__zdiv__pos__iff,axiom,
% 9.49/5.26      ! [K: int,I: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ K )
% 9.49/5.26       => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
% 9.49/5.26          = ( ord_less_eq_int @ K @ I ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3740_neg__imp__zdiv__nonneg__iff,axiom,
% 9.49/5.26      ! [B: int,A: int] :
% 9.49/5.26        ( ( ord_less_int @ B @ zero_zero_int )
% 9.49/5.26       => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 9.49/5.26          = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3741_pos__imp__zdiv__nonneg__iff,axiom,
% 9.49/5.26      ! [B: int,A: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.26       => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 9.49/5.26          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3742_nonneg1__imp__zdiv__pos__iff,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.26       => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 9.49/5.26          = ( ( ord_less_eq_int @ B @ A )
% 9.49/5.26            & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3743_pos__imp__zdiv__neg__iff,axiom,
% 9.49/5.26      ! [B: int,A: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.26       => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 9.49/5.26          = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3744_neg__imp__zdiv__neg__iff,axiom,
% 9.49/5.26      ! [B: int,A: int] :
% 9.49/5.26        ( ( ord_less_int @ B @ zero_zero_int )
% 9.49/5.26       => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 9.49/5.26          = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3745_int__div__less__self,axiom,
% 9.49/5.26      ! [X: int,K: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ X )
% 9.49/5.26       => ( ( ord_less_int @ one_one_int @ K )
% 9.49/5.26         => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3746_div__neg__pos__less0,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( ord_less_int @ A @ zero_zero_int )
% 9.49/5.26       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.26         => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3747_div__positive,axiom,
% 9.49/5.26      ! [B: code_integer,A: code_integer] :
% 9.49/5.26        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 9.49/5.26       => ( ( ord_le3102999989581377725nteger @ B @ A )
% 9.49/5.26         => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3748_div__positive,axiom,
% 9.49/5.26      ! [B: nat,A: nat] :
% 9.49/5.26        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.49/5.26       => ( ( ord_less_eq_nat @ B @ A )
% 9.49/5.26         => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3749_div__positive,axiom,
% 9.49/5.26      ! [B: int,A: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.26       => ( ( ord_less_eq_int @ B @ A )
% 9.49/5.26         => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3750_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 9.49/5.26      ! [A: code_integer,B: code_integer] :
% 9.49/5.26        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.49/5.26       => ( ( ord_le6747313008572928689nteger @ A @ B )
% 9.49/5.26         => ( ( divide6298287555418463151nteger @ A @ B )
% 9.49/5.26            = zero_z3403309356797280102nteger ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3751_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 9.49/5.26      ! [A: nat,B: nat] :
% 9.49/5.26        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.26       => ( ( ord_less_nat @ A @ B )
% 9.49/5.26         => ( ( divide_divide_nat @ A @ B )
% 9.49/5.26            = zero_zero_nat ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3752_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.26       => ( ( ord_less_int @ A @ B )
% 9.49/5.26         => ( ( divide_divide_int @ A @ B )
% 9.49/5.26            = zero_zero_int ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3753_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 9.49/5.26      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.49/5.26        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 9.49/5.26       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 9.49/5.26          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3754_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 9.49/5.26      ! [C: nat,A: nat,B: nat] :
% 9.49/5.26        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.49/5.26       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 9.49/5.26          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3755_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 9.49/5.26      ! [C: int,A: int,B: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.49/5.26       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 9.49/5.26          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3756_discrete,axiom,
% 9.49/5.26      ( ord_less_nat
% 9.49/5.26      = ( ^ [A4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ one_one_nat ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3757_discrete,axiom,
% 9.49/5.26      ( ord_less_int
% 9.49/5.26      = ( ^ [A4: int] : ( ord_less_eq_int @ ( plus_plus_int @ A4 @ one_one_int ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3758_divmod__step__eq,axiom,
% 9.49/5.26      ! [L: num,R2: nat,Q4: nat] :
% 9.49/5.26        ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
% 9.49/5.26         => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
% 9.49/5.26            = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
% 9.49/5.26        & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
% 9.49/5.26         => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
% 9.49/5.26            = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3759_divmod__step__eq,axiom,
% 9.49/5.26      ! [L: num,R2: int,Q4: int] :
% 9.49/5.26        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
% 9.49/5.26         => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.49/5.26            = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
% 9.49/5.26        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
% 9.49/5.26         => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.49/5.26            = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3760_divmod__step__eq,axiom,
% 9.49/5.26      ! [L: num,R2: code_integer,Q4: code_integer] :
% 9.49/5.26        ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
% 9.49/5.26         => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
% 9.49/5.26            = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
% 9.49/5.26        & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
% 9.49/5.26         => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
% 9.49/5.26            = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3761_enat__ord__number_I1_J,axiom,
% 9.49/5.26      ! [M: num,N: num] :
% 9.49/5.26        ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 9.49/5.26        = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3762_enat__ord__number_I2_J,axiom,
% 9.49/5.26      ! [M: num,N: num] :
% 9.49/5.26        ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 9.49/5.26        = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3763_unset__bit__0,axiom,
% 9.49/5.26      ! [A: nat] :
% 9.49/5.26        ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
% 9.49/5.26        = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3764_unset__bit__0,axiom,
% 9.49/5.26      ! [A: code_integer] :
% 9.49/5.26        ( ( bit_se8260200283734997820nteger @ zero_zero_nat @ A )
% 9.49/5.26        = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3765_unset__bit__0,axiom,
% 9.49/5.26      ! [A: int] :
% 9.49/5.26        ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
% 9.49/5.26        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3766_succ__less__length__list,axiom,
% 9.49/5.26      ! [Deg: nat,Mi: nat,X: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 9.49/5.26        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.26       => ( ( ord_less_eq_nat @ Mi @ X )
% 9.49/5.26         => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.26           => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.26              = ( if_option_nat
% 9.49/5.26                @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                   != none_nat )
% 9.49/5.26                  & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.49/5.26                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                @ ( if_option_nat
% 9.49/5.26                  @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.49/5.26                    = none_nat )
% 9.49/5.26                  @ none_nat
% 9.49/5.26                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3767_succ__greatereq__min,axiom,
% 9.49/5.26      ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.49/5.26        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.26       => ( ( ord_less_eq_nat @ Mi @ X )
% 9.49/5.26         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.26            = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.26              @ ( if_option_nat
% 9.49/5.26                @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                   != none_nat )
% 9.49/5.26                  & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.49/5.26                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                @ ( if_option_nat
% 9.49/5.26                  @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.49/5.26                    = none_nat )
% 9.49/5.26                  @ none_nat
% 9.49/5.26                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.49/5.26              @ none_nat ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3768_pred__lesseq__max,axiom,
% 9.49/5.26      ! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.49/5.26        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.26       => ( ( ord_less_eq_nat @ X @ Ma )
% 9.49/5.26         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.26            = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.26              @ ( if_option_nat
% 9.49/5.26                @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                   != none_nat )
% 9.49/5.26                  & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.49/5.26                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                @ ( if_option_nat
% 9.49/5.26                  @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.49/5.26                    = none_nat )
% 9.49/5.26                  @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
% 9.49/5.26                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.49/5.26              @ none_nat ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3769_pred__less__length__list,axiom,
% 9.49/5.26      ! [Deg: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
% 9.49/5.26        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.26       => ( ( ord_less_eq_nat @ X @ Ma )
% 9.49/5.26         => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.26           => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.26              = ( if_option_nat
% 9.49/5.26                @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                   != none_nat )
% 9.49/5.26                  & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.49/5.26                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                @ ( if_option_nat
% 9.49/5.26                  @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.49/5.26                    = none_nat )
% 9.49/5.26                  @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
% 9.49/5.26                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3770_set__vebt_H__def,axiom,
% 9.49/5.26      ( vEBT_VEBT_set_vebt
% 9.49/5.26      = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3771_Diff__idemp,axiom,
% 9.49/5.26      ! [A2: set_nat,B2: set_nat] :
% 9.49/5.26        ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ B2 )
% 9.49/5.26        = ( minus_minus_set_nat @ A2 @ B2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3772_Diff__iff,axiom,
% 9.49/5.26      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.26        ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 9.49/5.26        = ( ( member8440522571783428010at_nat @ C @ A2 )
% 9.49/5.26          & ~ ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3773_Diff__iff,axiom,
% 9.49/5.26      ! [C: real,A2: set_real,B2: set_real] :
% 9.49/5.26        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 9.49/5.26        = ( ( member_real @ C @ A2 )
% 9.49/5.26          & ~ ( member_real @ C @ B2 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3774_Diff__iff,axiom,
% 9.49/5.26      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.49/5.26        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 9.49/5.26        = ( ( member_set_nat @ C @ A2 )
% 9.49/5.26          & ~ ( member_set_nat @ C @ B2 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3775_Diff__iff,axiom,
% 9.49/5.26      ! [C: int,A2: set_int,B2: set_int] :
% 9.49/5.26        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.49/5.26        = ( ( member_int @ C @ A2 )
% 9.49/5.26          & ~ ( member_int @ C @ B2 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3776_Diff__iff,axiom,
% 9.49/5.26      ! [C: nat,A2: set_nat,B2: set_nat] :
% 9.49/5.26        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.49/5.26        = ( ( member_nat @ C @ A2 )
% 9.49/5.26          & ~ ( member_nat @ C @ B2 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3777_DiffI,axiom,
% 9.49/5.26      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.26        ( ( member8440522571783428010at_nat @ C @ A2 )
% 9.49/5.26       => ( ~ ( member8440522571783428010at_nat @ C @ B2 )
% 9.49/5.26         => ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3778_DiffI,axiom,
% 9.49/5.26      ! [C: real,A2: set_real,B2: set_real] :
% 9.49/5.26        ( ( member_real @ C @ A2 )
% 9.49/5.26       => ( ~ ( member_real @ C @ B2 )
% 9.49/5.26         => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3779_DiffI,axiom,
% 9.49/5.26      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.49/5.26        ( ( member_set_nat @ C @ A2 )
% 9.49/5.26       => ( ~ ( member_set_nat @ C @ B2 )
% 9.49/5.26         => ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3780_DiffI,axiom,
% 9.49/5.26      ! [C: int,A2: set_int,B2: set_int] :
% 9.49/5.26        ( ( member_int @ C @ A2 )
% 9.49/5.26       => ( ~ ( member_int @ C @ B2 )
% 9.49/5.26         => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3781_DiffI,axiom,
% 9.49/5.26      ! [C: nat,A2: set_nat,B2: set_nat] :
% 9.49/5.26        ( ( member_nat @ C @ A2 )
% 9.49/5.26       => ( ~ ( member_nat @ C @ B2 )
% 9.49/5.26         => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3782_i0__less,axiom,
% 9.49/5.26      ! [N: extended_enat] :
% 9.49/5.26        ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 9.49/5.26        = ( N != zero_z5237406670263579293d_enat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3783_finite__Collect__conjI,axiom,
% 9.49/5.26      ! [P: real > $o,Q: real > $o] :
% 9.49/5.26        ( ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.49/5.26          | ( finite_finite_real @ ( collect_real @ Q ) ) )
% 9.49/5.26       => ( finite_finite_real
% 9.49/5.26          @ ( collect_real
% 9.49/5.26            @ ^ [X3: real] :
% 9.49/5.26                ( ( P @ X3 )
% 9.49/5.26                & ( Q @ X3 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3784_finite__Collect__conjI,axiom,
% 9.49/5.26      ! [P: list_nat > $o,Q: list_nat > $o] :
% 9.49/5.26        ( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
% 9.49/5.26          | ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) )
% 9.49/5.26       => ( finite8100373058378681591st_nat
% 9.49/5.26          @ ( collect_list_nat
% 9.49/5.26            @ ^ [X3: list_nat] :
% 9.49/5.26                ( ( P @ X3 )
% 9.49/5.26                & ( Q @ X3 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3785_finite__Collect__conjI,axiom,
% 9.49/5.26      ! [P: set_nat > $o,Q: set_nat > $o] :
% 9.49/5.26        ( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 9.49/5.26          | ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
% 9.49/5.26       => ( finite1152437895449049373et_nat
% 9.49/5.26          @ ( collect_set_nat
% 9.49/5.26            @ ^ [X3: set_nat] :
% 9.49/5.26                ( ( P @ X3 )
% 9.49/5.26                & ( Q @ X3 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3786_finite__Collect__conjI,axiom,
% 9.49/5.26      ! [P: nat > $o,Q: nat > $o] :
% 9.49/5.26        ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.49/5.26          | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
% 9.49/5.26       => ( finite_finite_nat
% 9.49/5.26          @ ( collect_nat
% 9.49/5.26            @ ^ [X3: nat] :
% 9.49/5.26                ( ( P @ X3 )
% 9.49/5.26                & ( Q @ X3 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3787_finite__Collect__conjI,axiom,
% 9.49/5.26      ! [P: int > $o,Q: int > $o] :
% 9.49/5.26        ( ( ( finite_finite_int @ ( collect_int @ P ) )
% 9.49/5.26          | ( finite_finite_int @ ( collect_int @ Q ) ) )
% 9.49/5.26       => ( finite_finite_int
% 9.49/5.26          @ ( collect_int
% 9.49/5.26            @ ^ [X3: int] :
% 9.49/5.26                ( ( P @ X3 )
% 9.49/5.26                & ( Q @ X3 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3788_finite__Collect__conjI,axiom,
% 9.49/5.26      ! [P: complex > $o,Q: complex > $o] :
% 9.49/5.26        ( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 9.49/5.26          | ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
% 9.49/5.26       => ( finite3207457112153483333omplex
% 9.49/5.26          @ ( collect_complex
% 9.49/5.26            @ ^ [X3: complex] :
% 9.49/5.26                ( ( P @ X3 )
% 9.49/5.26                & ( Q @ X3 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3789_finite__Collect__conjI,axiom,
% 9.49/5.26      ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 9.49/5.26        ( ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
% 9.49/5.26          | ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) )
% 9.49/5.26       => ( finite6177210948735845034at_nat
% 9.49/5.26          @ ( collec3392354462482085612at_nat
% 9.49/5.26            @ ^ [X3: product_prod_nat_nat] :
% 9.49/5.26                ( ( P @ X3 )
% 9.49/5.26                & ( Q @ X3 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3790_finite__Collect__disjI,axiom,
% 9.49/5.26      ! [P: real > $o,Q: real > $o] :
% 9.49/5.26        ( ( finite_finite_real
% 9.49/5.26          @ ( collect_real
% 9.49/5.26            @ ^ [X3: real] :
% 9.49/5.26                ( ( P @ X3 )
% 9.49/5.26                | ( Q @ X3 ) ) ) )
% 9.49/5.26        = ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.49/5.26          & ( finite_finite_real @ ( collect_real @ Q ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3791_finite__Collect__disjI,axiom,
% 9.49/5.26      ! [P: list_nat > $o,Q: list_nat > $o] :
% 9.49/5.26        ( ( finite8100373058378681591st_nat
% 9.49/5.26          @ ( collect_list_nat
% 9.49/5.26            @ ^ [X3: list_nat] :
% 9.49/5.26                ( ( P @ X3 )
% 9.49/5.26                | ( Q @ X3 ) ) ) )
% 9.49/5.26        = ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
% 9.49/5.26          & ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3792_finite__Collect__disjI,axiom,
% 9.49/5.26      ! [P: set_nat > $o,Q: set_nat > $o] :
% 9.49/5.26        ( ( finite1152437895449049373et_nat
% 9.49/5.26          @ ( collect_set_nat
% 9.49/5.26            @ ^ [X3: set_nat] :
% 9.49/5.26                ( ( P @ X3 )
% 9.49/5.26                | ( Q @ X3 ) ) ) )
% 9.49/5.26        = ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 9.49/5.26          & ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3793_finite__Collect__disjI,axiom,
% 9.49/5.26      ! [P: nat > $o,Q: nat > $o] :
% 9.49/5.26        ( ( finite_finite_nat
% 9.49/5.26          @ ( collect_nat
% 9.49/5.26            @ ^ [X3: nat] :
% 9.49/5.26                ( ( P @ X3 )
% 9.49/5.26                | ( Q @ X3 ) ) ) )
% 9.49/5.26        = ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.49/5.26          & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3794_finite__Collect__disjI,axiom,
% 9.49/5.26      ! [P: int > $o,Q: int > $o] :
% 9.49/5.26        ( ( finite_finite_int
% 9.49/5.26          @ ( collect_int
% 9.49/5.26            @ ^ [X3: int] :
% 9.49/5.26                ( ( P @ X3 )
% 9.49/5.26                | ( Q @ X3 ) ) ) )
% 9.49/5.26        = ( ( finite_finite_int @ ( collect_int @ P ) )
% 9.49/5.26          & ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3795_finite__Collect__disjI,axiom,
% 9.49/5.26      ! [P: complex > $o,Q: complex > $o] :
% 9.49/5.26        ( ( finite3207457112153483333omplex
% 9.49/5.26          @ ( collect_complex
% 9.49/5.26            @ ^ [X3: complex] :
% 9.49/5.26                ( ( P @ X3 )
% 9.49/5.26                | ( Q @ X3 ) ) ) )
% 9.49/5.26        = ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 9.49/5.26          & ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3796_finite__Collect__disjI,axiom,
% 9.49/5.26      ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 9.49/5.26        ( ( finite6177210948735845034at_nat
% 9.49/5.26          @ ( collec3392354462482085612at_nat
% 9.49/5.26            @ ^ [X3: product_prod_nat_nat] :
% 9.49/5.26                ( ( P @ X3 )
% 9.49/5.26                | ( Q @ X3 ) ) ) )
% 9.49/5.26        = ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
% 9.49/5.26          & ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3797_finite__interval__int1,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( finite_finite_int
% 9.49/5.26        @ ( collect_int
% 9.49/5.26          @ ^ [I4: int] :
% 9.49/5.26              ( ( ord_less_eq_int @ A @ I4 )
% 9.49/5.26              & ( ord_less_eq_int @ I4 @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3798_finite__interval__int4,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( finite_finite_int
% 9.49/5.26        @ ( collect_int
% 9.49/5.26          @ ^ [I4: int] :
% 9.49/5.26              ( ( ord_less_int @ A @ I4 )
% 9.49/5.26              & ( ord_less_int @ I4 @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3799_pred__empty,axiom,
% 9.49/5.26      ! [T: vEBT_VEBT,N: nat,X: nat] :
% 9.49/5.26        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.26       => ( ( ( vEBT_vebt_pred @ T @ X )
% 9.49/5.26            = none_nat )
% 9.49/5.26          = ( ( collect_nat
% 9.49/5.26              @ ^ [Y2: nat] :
% 9.49/5.26                  ( ( vEBT_vebt_member @ T @ Y2 )
% 9.49/5.26                  & ( ord_less_nat @ Y2 @ X ) ) )
% 9.49/5.26            = bot_bot_set_nat ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3800_succ__empty,axiom,
% 9.49/5.26      ! [T: vEBT_VEBT,N: nat,X: nat] :
% 9.49/5.26        ( ( vEBT_invar_vebt @ T @ N )
% 9.49/5.26       => ( ( ( vEBT_vebt_succ @ T @ X )
% 9.49/5.26            = none_nat )
% 9.49/5.26          = ( ( collect_nat
% 9.49/5.26              @ ^ [Y2: nat] :
% 9.49/5.26                  ( ( vEBT_vebt_member @ T @ Y2 )
% 9.49/5.26                  & ( ord_less_nat @ X @ Y2 ) ) )
% 9.49/5.26            = bot_bot_set_nat ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3801_unset__bit__negative__int__iff,axiom,
% 9.49/5.26      ! [N: nat,K: int] :
% 9.49/5.26        ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
% 9.49/5.26        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3802_finite__Collect__subsets,axiom,
% 9.49/5.26      ! [A2: set_nat] :
% 9.49/5.26        ( ( finite_finite_nat @ A2 )
% 9.49/5.26       => ( finite1152437895449049373et_nat
% 9.49/5.26          @ ( collect_set_nat
% 9.49/5.26            @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A2 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3803_finite__Collect__subsets,axiom,
% 9.49/5.26      ! [A2: set_complex] :
% 9.49/5.26        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.26       => ( finite6551019134538273531omplex
% 9.49/5.26          @ ( collect_set_complex
% 9.49/5.26            @ ^ [B6: set_complex] : ( ord_le211207098394363844omplex @ B6 @ A2 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3804_finite__Collect__subsets,axiom,
% 9.49/5.26      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.26        ( ( finite6177210948735845034at_nat @ A2 )
% 9.49/5.26       => ( finite9047747110432174090at_nat
% 9.49/5.26          @ ( collec5514110066124741708at_nat
% 9.49/5.26            @ ^ [B6: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ B6 @ A2 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3805_finite__Collect__subsets,axiom,
% 9.49/5.26      ! [A2: set_int] :
% 9.49/5.26        ( ( finite_finite_int @ A2 )
% 9.49/5.26       => ( finite6197958912794628473et_int
% 9.49/5.26          @ ( collect_set_int
% 9.49/5.26            @ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A2 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3806_finite__interval__int2,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( finite_finite_int
% 9.49/5.26        @ ( collect_int
% 9.49/5.26          @ ^ [I4: int] :
% 9.49/5.26              ( ( ord_less_eq_int @ A @ I4 )
% 9.49/5.26              & ( ord_less_int @ I4 @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3807_finite__interval__int3,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( finite_finite_int
% 9.49/5.26        @ ( collect_int
% 9.49/5.26          @ ^ [I4: int] :
% 9.49/5.26              ( ( ord_less_int @ A @ I4 )
% 9.49/5.26              & ( ord_less_eq_int @ I4 @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3808_finite__Collect__less__nat,axiom,
% 9.49/5.26      ! [K: nat] :
% 9.49/5.26        ( finite_finite_nat
% 9.49/5.26        @ ( collect_nat
% 9.49/5.26          @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3809_finite__Collect__le__nat,axiom,
% 9.49/5.26      ! [K: nat] :
% 9.49/5.26        ( finite_finite_nat
% 9.49/5.26        @ ( collect_nat
% 9.49/5.26          @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3810_del__x__not__mi,axiom,
% 9.49/5.26      ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.49/5.26        ( ( ( ord_less_nat @ Mi @ X )
% 9.49/5.26          & ( ord_less_eq_nat @ X @ Ma ) )
% 9.49/5.26       => ( ( Mi != Ma )
% 9.49/5.26         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.26           => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.26                = H2 )
% 9.49/5.26             => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.26                  = L )
% 9.49/5.26               => ( ( Newnode
% 9.49/5.26                    = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
% 9.49/5.26                 => ( ( Newlist
% 9.49/5.26                      = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
% 9.49/5.26                   => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.26                     => ( ( ( vEBT_VEBT_minNull @ Newnode )
% 9.49/5.26                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.26                            = ( vEBT_Node
% 9.49/5.26                              @ ( some_P7363390416028606310at_nat
% 9.49/5.26                                @ ( product_Pair_nat_nat @ Mi
% 9.49/5.26                                  @ ( if_nat @ ( X = Ma )
% 9.49/5.26                                    @ ( if_nat
% 9.49/5.26                                      @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 9.49/5.26                                        = none_nat )
% 9.49/5.26                                      @ Mi
% 9.49/5.26                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
% 9.49/5.26                                    @ Ma ) ) )
% 9.49/5.26                              @ Deg
% 9.49/5.26                              @ Newlist
% 9.49/5.26                              @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
% 9.49/5.26                        & ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 9.49/5.26                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.26                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3811_del__x__not__mi__new__node__nil,axiom,
% 9.49/5.26      ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
% 9.49/5.26        ( ( ( ord_less_nat @ Mi @ X )
% 9.49/5.26          & ( ord_less_eq_nat @ X @ Ma ) )
% 9.49/5.26       => ( ( Mi != Ma )
% 9.49/5.26         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.26           => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.26                = H2 )
% 9.49/5.26             => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.26                  = L )
% 9.49/5.26               => ( ( Newnode
% 9.49/5.26                    = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
% 9.49/5.26                 => ( ( vEBT_VEBT_minNull @ Newnode )
% 9.49/5.26                   => ( ( Sn
% 9.49/5.26                        = ( vEBT_vebt_delete @ Summary @ H2 ) )
% 9.49/5.26                     => ( ( Newlist
% 9.49/5.26                          = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
% 9.49/5.26                       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.26                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.26                            = ( vEBT_Node
% 9.49/5.26                              @ ( some_P7363390416028606310at_nat
% 9.49/5.26                                @ ( product_Pair_nat_nat @ Mi
% 9.49/5.26                                  @ ( if_nat @ ( X = Ma )
% 9.49/5.26                                    @ ( if_nat
% 9.49/5.26                                      @ ( ( vEBT_vebt_maxt @ Sn )
% 9.49/5.26                                        = none_nat )
% 9.49/5.26                                      @ Mi
% 9.49/5.26                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat2 @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
% 9.49/5.26                                    @ Ma ) ) )
% 9.49/5.26                              @ Deg
% 9.49/5.26                              @ Newlist
% 9.49/5.26                              @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3812_del__x__not__mia,axiom,
% 9.49/5.26      ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.49/5.26        ( ( ( ord_less_nat @ Mi @ X )
% 9.49/5.26          & ( ord_less_eq_nat @ X @ Ma ) )
% 9.49/5.26       => ( ( Mi != Ma )
% 9.49/5.26         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.26           => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.26                = H2 )
% 9.49/5.26             => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.26                  = L )
% 9.49/5.26               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.26                 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.26                    = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
% 9.49/5.26                      @ ( vEBT_Node
% 9.49/5.26                        @ ( some_P7363390416028606310at_nat
% 9.49/5.26                          @ ( product_Pair_nat_nat @ Mi
% 9.49/5.26                            @ ( if_nat @ ( X = Ma )
% 9.49/5.26                              @ ( if_nat
% 9.49/5.26                                @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 9.49/5.26                                  = none_nat )
% 9.49/5.26                                @ Mi
% 9.49/5.26                                @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
% 9.49/5.26                              @ Ma ) ) )
% 9.49/5.26                        @ Deg
% 9.49/5.26                        @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
% 9.49/5.26                        @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 9.49/5.26                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3813_del__in__range,axiom,
% 9.49/5.26      ! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.49/5.26        ( ( ( ord_less_eq_nat @ Mi @ X )
% 9.49/5.26          & ( ord_less_eq_nat @ X @ Ma ) )
% 9.49/5.26       => ( ( Mi != Ma )
% 9.49/5.26         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.26           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.26              = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.26                @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                  @ ( vEBT_Node
% 9.49/5.26                    @ ( some_P7363390416028606310at_nat
% 9.49/5.26                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
% 9.49/5.26                        @ ( if_nat
% 9.49/5.26                          @ ( ( ( X = Mi )
% 9.49/5.26                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 9.49/5.26                                = Ma ) )
% 9.49/5.26                            & ( ( X != Mi )
% 9.49/5.26                             => ( X = Ma ) ) )
% 9.49/5.26                          @ ( if_nat
% 9.49/5.26                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                              = none_nat )
% 9.49/5.26                            @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
% 9.49/5.26                            @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 9.49/5.26                          @ Ma ) ) )
% 9.49/5.26                    @ Deg
% 9.49/5.26                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                    @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                  @ ( vEBT_Node
% 9.49/5.26                    @ ( some_P7363390416028606310at_nat
% 9.49/5.26                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
% 9.49/5.26                        @ ( if_nat
% 9.49/5.26                          @ ( ( ( X = Mi )
% 9.49/5.26                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 9.49/5.26                                = Ma ) )
% 9.49/5.26                            & ( ( X != Mi )
% 9.49/5.26                             => ( X = Ma ) ) )
% 9.49/5.26                          @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.49/5.26                          @ Ma ) ) )
% 9.49/5.26                    @ Deg
% 9.49/5.26                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                    @ Summary ) )
% 9.49/5.26                @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3814_del__x__mi,axiom,
% 9.49/5.26      ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat] :
% 9.49/5.26        ( ( ( X = Mi )
% 9.49/5.26          & ( ord_less_nat @ X @ Ma ) )
% 9.49/5.26       => ( ( Mi != Ma )
% 9.49/5.26         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.26           => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.26                = H2 )
% 9.49/5.26             => ( ( Xn
% 9.49/5.26                  = ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
% 9.49/5.26               => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.26                    = L )
% 9.49/5.26                 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.26                   => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.26                      = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
% 9.49/5.26                        @ ( vEBT_Node
% 9.49/5.26                          @ ( some_P7363390416028606310at_nat
% 9.49/5.26                            @ ( product_Pair_nat_nat @ Xn
% 9.49/5.26                              @ ( if_nat @ ( Xn = Ma )
% 9.49/5.26                                @ ( if_nat
% 9.49/5.26                                  @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 9.49/5.26                                    = none_nat )
% 9.49/5.26                                  @ Xn
% 9.49/5.26                                  @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
% 9.49/5.26                                @ Ma ) ) )
% 9.49/5.26                          @ Deg
% 9.49/5.26                          @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
% 9.49/5.26                          @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 9.49/5.26                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3815_del__x__mi__lets__in,axiom,
% 9.49/5.26      ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
% 9.49/5.26        ( ( ( X = Mi )
% 9.49/5.26          & ( ord_less_nat @ X @ Ma ) )
% 9.49/5.26       => ( ( Mi != Ma )
% 9.49/5.26         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.26           => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.26                = H2 )
% 9.49/5.26             => ( ( Xn
% 9.49/5.26                  = ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
% 9.49/5.26               => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.26                    = L )
% 9.49/5.26                 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.26                   => ( ( Newnode
% 9.49/5.26                        = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
% 9.49/5.26                     => ( ( Newlist
% 9.49/5.26                          = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
% 9.49/5.26                       => ( ( ( vEBT_VEBT_minNull @ Newnode )
% 9.49/5.26                           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.26                              = ( vEBT_Node
% 9.49/5.26                                @ ( some_P7363390416028606310at_nat
% 9.49/5.26                                  @ ( product_Pair_nat_nat @ Xn
% 9.49/5.26                                    @ ( if_nat @ ( Xn = Ma )
% 9.49/5.26                                      @ ( if_nat
% 9.49/5.26                                        @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 9.49/5.26                                          = none_nat )
% 9.49/5.26                                        @ Xn
% 9.49/5.26                                        @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
% 9.49/5.26                                      @ Ma ) ) )
% 9.49/5.26                                @ Deg
% 9.49/5.26                                @ Newlist
% 9.49/5.26                                @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
% 9.49/5.26                          & ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 9.49/5.26                           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.26                              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3816_del__x__mi__lets__in__minNull,axiom,
% 9.49/5.26      ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
% 9.49/5.26        ( ( ( X = Mi )
% 9.49/5.26          & ( ord_less_nat @ X @ Ma ) )
% 9.49/5.26       => ( ( Mi != Ma )
% 9.49/5.26         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.26           => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.26                = H2 )
% 9.49/5.26             => ( ( Xn
% 9.49/5.26                  = ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
% 9.49/5.26               => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.26                    = L )
% 9.49/5.26                 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.26                   => ( ( Newnode
% 9.49/5.26                        = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
% 9.49/5.26                     => ( ( Newlist
% 9.49/5.26                          = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
% 9.49/5.26                       => ( ( vEBT_VEBT_minNull @ Newnode )
% 9.49/5.26                         => ( ( Sn
% 9.49/5.26                              = ( vEBT_vebt_delete @ Summary @ H2 ) )
% 9.49/5.26                           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.26                              = ( vEBT_Node
% 9.49/5.26                                @ ( some_P7363390416028606310at_nat
% 9.49/5.26                                  @ ( product_Pair_nat_nat @ Xn
% 9.49/5.26                                    @ ( if_nat @ ( Xn = Ma )
% 9.49/5.26                                      @ ( if_nat
% 9.49/5.26                                        @ ( ( vEBT_vebt_maxt @ Sn )
% 9.49/5.26                                          = none_nat )
% 9.49/5.26                                        @ Xn
% 9.49/5.26                                        @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat2 @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
% 9.49/5.26                                      @ Ma ) ) )
% 9.49/5.26                                @ Deg
% 9.49/5.26                                @ Newlist
% 9.49/5.26                                @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3817_del__x__mia,axiom,
% 9.49/5.26      ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.49/5.26        ( ( ( X = Mi )
% 9.49/5.26          & ( ord_less_nat @ X @ Ma ) )
% 9.49/5.26       => ( ( Mi != Ma )
% 9.49/5.26         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.49/5.26           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
% 9.49/5.26              = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.26                @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                  @ ( vEBT_Node
% 9.49/5.26                    @ ( some_P7363390416028606310at_nat
% 9.49/5.26                      @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 9.49/5.26                        @ ( if_nat
% 9.49/5.26                          @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 9.49/5.26                            = Ma )
% 9.49/5.26                          @ ( if_nat
% 9.49/5.26                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                              = none_nat )
% 9.49/5.26                            @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 9.49/5.26                            @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 9.49/5.26                          @ Ma ) ) )
% 9.49/5.26                    @ Deg
% 9.49/5.26                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                    @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                  @ ( vEBT_Node
% 9.49/5.26                    @ ( some_P7363390416028606310at_nat
% 9.49/5.26                      @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 9.49/5.26                        @ ( if_nat
% 9.49/5.26                          @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 9.49/5.26                            = Ma )
% 9.49/5.26                          @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.49/5.26                          @ Ma ) ) )
% 9.49/5.26                    @ Deg
% 9.49/5.26                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                    @ Summary ) )
% 9.49/5.26                @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3818_enat__0__less__mult__iff,axiom,
% 9.49/5.26      ! [M: extended_enat,N: extended_enat] :
% 9.49/5.26        ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
% 9.49/5.26        = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 9.49/5.26          & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3819_not__iless0,axiom,
% 9.49/5.26      ! [N: extended_enat] :
% 9.49/5.26        ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3820_enat__less__induct,axiom,
% 9.49/5.26      ! [P: extended_enat > $o,N: extended_enat] :
% 9.49/5.26        ( ! [N3: extended_enat] :
% 9.49/5.26            ( ! [M3: extended_enat] :
% 9.49/5.26                ( ( ord_le72135733267957522d_enat @ M3 @ N3 )
% 9.49/5.26               => ( P @ M3 ) )
% 9.49/5.26           => ( P @ N3 ) )
% 9.49/5.26       => ( P @ N ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3821_less__set__def,axiom,
% 9.49/5.26      ( ord_le7866589430770878221at_nat
% 9.49/5.26      = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 9.49/5.26            ( ord_le549003669493604880_nat_o
% 9.49/5.26            @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A6 )
% 9.49/5.26            @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ B6 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3822_less__set__def,axiom,
% 9.49/5.26      ( ord_less_set_real
% 9.49/5.26      = ( ^ [A6: set_real,B6: set_real] :
% 9.49/5.26            ( ord_less_real_o
% 9.49/5.26            @ ^ [X3: real] : ( member_real @ X3 @ A6 )
% 9.49/5.26            @ ^ [X3: real] : ( member_real @ X3 @ B6 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3823_less__set__def,axiom,
% 9.49/5.26      ( ord_less_set_set_nat
% 9.49/5.26      = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 9.49/5.26            ( ord_less_set_nat_o
% 9.49/5.26            @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 )
% 9.49/5.26            @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B6 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3824_less__set__def,axiom,
% 9.49/5.26      ( ord_less_set_nat
% 9.49/5.26      = ( ^ [A6: set_nat,B6: set_nat] :
% 9.49/5.26            ( ord_less_nat_o
% 9.49/5.26            @ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
% 9.49/5.26            @ ^ [X3: nat] : ( member_nat @ X3 @ B6 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3825_less__set__def,axiom,
% 9.49/5.26      ( ord_less_set_int
% 9.49/5.26      = ( ^ [A6: set_int,B6: set_int] :
% 9.49/5.26            ( ord_less_int_o
% 9.49/5.26            @ ^ [X3: int] : ( member_int @ X3 @ A6 )
% 9.49/5.26            @ ^ [X3: int] : ( member_int @ X3 @ B6 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3826_minus__set__def,axiom,
% 9.49/5.26      ( minus_1356011639430497352at_nat
% 9.49/5.26      = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 9.49/5.26            ( collec3392354462482085612at_nat
% 9.49/5.26            @ ( minus_2270307095948843157_nat_o
% 9.49/5.26              @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A6 )
% 9.49/5.26              @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ B6 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3827_minus__set__def,axiom,
% 9.49/5.26      ( minus_minus_set_real
% 9.49/5.26      = ( ^ [A6: set_real,B6: set_real] :
% 9.49/5.26            ( collect_real
% 9.49/5.26            @ ( minus_minus_real_o
% 9.49/5.26              @ ^ [X3: real] : ( member_real @ X3 @ A6 )
% 9.49/5.26              @ ^ [X3: real] : ( member_real @ X3 @ B6 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3828_minus__set__def,axiom,
% 9.49/5.26      ( minus_7954133019191499631st_nat
% 9.49/5.26      = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 9.49/5.26            ( collect_list_nat
% 9.49/5.26            @ ( minus_1139252259498527702_nat_o
% 9.49/5.26              @ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A6 )
% 9.49/5.26              @ ^ [X3: list_nat] : ( member_list_nat @ X3 @ B6 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3829_minus__set__def,axiom,
% 9.49/5.26      ( minus_2163939370556025621et_nat
% 9.49/5.26      = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 9.49/5.26            ( collect_set_nat
% 9.49/5.26            @ ( minus_6910147592129066416_nat_o
% 9.49/5.26              @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 )
% 9.49/5.26              @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B6 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3830_minus__set__def,axiom,
% 9.49/5.26      ( minus_minus_set_int
% 9.49/5.26      = ( ^ [A6: set_int,B6: set_int] :
% 9.49/5.26            ( collect_int
% 9.49/5.26            @ ( minus_minus_int_o
% 9.49/5.26              @ ^ [X3: int] : ( member_int @ X3 @ A6 )
% 9.49/5.26              @ ^ [X3: int] : ( member_int @ X3 @ B6 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3831_minus__set__def,axiom,
% 9.49/5.26      ( minus_minus_set_nat
% 9.49/5.26      = ( ^ [A6: set_nat,B6: set_nat] :
% 9.49/5.26            ( collect_nat
% 9.49/5.26            @ ( minus_minus_nat_o
% 9.49/5.26              @ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
% 9.49/5.26              @ ^ [X3: nat] : ( member_nat @ X3 @ B6 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3832_set__diff__eq,axiom,
% 9.49/5.26      ( minus_1356011639430497352at_nat
% 9.49/5.26      = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 9.49/5.26            ( collec3392354462482085612at_nat
% 9.49/5.26            @ ^ [X3: product_prod_nat_nat] :
% 9.49/5.26                ( ( member8440522571783428010at_nat @ X3 @ A6 )
% 9.49/5.26                & ~ ( member8440522571783428010at_nat @ X3 @ B6 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3833_set__diff__eq,axiom,
% 9.49/5.26      ( minus_minus_set_real
% 9.49/5.26      = ( ^ [A6: set_real,B6: set_real] :
% 9.49/5.26            ( collect_real
% 9.49/5.26            @ ^ [X3: real] :
% 9.49/5.26                ( ( member_real @ X3 @ A6 )
% 9.49/5.26                & ~ ( member_real @ X3 @ B6 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3834_set__diff__eq,axiom,
% 9.49/5.26      ( minus_7954133019191499631st_nat
% 9.49/5.26      = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 9.49/5.26            ( collect_list_nat
% 9.49/5.26            @ ^ [X3: list_nat] :
% 9.49/5.26                ( ( member_list_nat @ X3 @ A6 )
% 9.49/5.26                & ~ ( member_list_nat @ X3 @ B6 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3835_set__diff__eq,axiom,
% 9.49/5.26      ( minus_2163939370556025621et_nat
% 9.49/5.26      = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 9.49/5.26            ( collect_set_nat
% 9.49/5.26            @ ^ [X3: set_nat] :
% 9.49/5.26                ( ( member_set_nat @ X3 @ A6 )
% 9.49/5.26                & ~ ( member_set_nat @ X3 @ B6 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3836_set__diff__eq,axiom,
% 9.49/5.26      ( minus_minus_set_int
% 9.49/5.26      = ( ^ [A6: set_int,B6: set_int] :
% 9.49/5.26            ( collect_int
% 9.49/5.26            @ ^ [X3: int] :
% 9.49/5.26                ( ( member_int @ X3 @ A6 )
% 9.49/5.26                & ~ ( member_int @ X3 @ B6 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3837_set__diff__eq,axiom,
% 9.49/5.26      ( minus_minus_set_nat
% 9.49/5.26      = ( ^ [A6: set_nat,B6: set_nat] :
% 9.49/5.26            ( collect_nat
% 9.49/5.26            @ ^ [X3: nat] :
% 9.49/5.26                ( ( member_nat @ X3 @ A6 )
% 9.49/5.26                & ~ ( member_nat @ X3 @ B6 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3838_DiffD2,axiom,
% 9.49/5.26      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.26        ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 9.49/5.26       => ~ ( member8440522571783428010at_nat @ C @ B2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3839_DiffD2,axiom,
% 9.49/5.26      ! [C: real,A2: set_real,B2: set_real] :
% 9.49/5.26        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 9.49/5.26       => ~ ( member_real @ C @ B2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3840_DiffD2,axiom,
% 9.49/5.26      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.49/5.26        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 9.49/5.26       => ~ ( member_set_nat @ C @ B2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3841_DiffD2,axiom,
% 9.49/5.26      ! [C: int,A2: set_int,B2: set_int] :
% 9.49/5.26        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.49/5.26       => ~ ( member_int @ C @ B2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3842_DiffD2,axiom,
% 9.49/5.26      ! [C: nat,A2: set_nat,B2: set_nat] :
% 9.49/5.26        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.49/5.26       => ~ ( member_nat @ C @ B2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3843_DiffD1,axiom,
% 9.49/5.26      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.26        ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 9.49/5.26       => ( member8440522571783428010at_nat @ C @ A2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3844_DiffD1,axiom,
% 9.49/5.26      ! [C: real,A2: set_real,B2: set_real] :
% 9.49/5.26        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 9.49/5.26       => ( member_real @ C @ A2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3845_DiffD1,axiom,
% 9.49/5.26      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.49/5.26        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 9.49/5.26       => ( member_set_nat @ C @ A2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3846_DiffD1,axiom,
% 9.49/5.26      ! [C: int,A2: set_int,B2: set_int] :
% 9.49/5.26        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.49/5.26       => ( member_int @ C @ A2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3847_DiffD1,axiom,
% 9.49/5.26      ! [C: nat,A2: set_nat,B2: set_nat] :
% 9.49/5.26        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.49/5.26       => ( member_nat @ C @ A2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3848_DiffE,axiom,
% 9.49/5.26      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.26        ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 9.49/5.26       => ~ ( ( member8440522571783428010at_nat @ C @ A2 )
% 9.49/5.26           => ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3849_DiffE,axiom,
% 9.49/5.26      ! [C: real,A2: set_real,B2: set_real] :
% 9.49/5.26        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 9.49/5.26       => ~ ( ( member_real @ C @ A2 )
% 9.49/5.26           => ( member_real @ C @ B2 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3850_DiffE,axiom,
% 9.49/5.26      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.49/5.26        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 9.49/5.26       => ~ ( ( member_set_nat @ C @ A2 )
% 9.49/5.26           => ( member_set_nat @ C @ B2 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3851_DiffE,axiom,
% 9.49/5.26      ! [C: int,A2: set_int,B2: set_int] :
% 9.49/5.26        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.49/5.26       => ~ ( ( member_int @ C @ A2 )
% 9.49/5.26           => ( member_int @ C @ B2 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3852_DiffE,axiom,
% 9.49/5.26      ! [C: nat,A2: set_nat,B2: set_nat] :
% 9.49/5.26        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.49/5.26       => ~ ( ( member_nat @ C @ A2 )
% 9.49/5.26           => ( member_nat @ C @ B2 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3853_pred__equals__eq2,axiom,
% 9.49/5.26      ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 9.49/5.26        ( ( ( ^ [X3: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R ) )
% 9.49/5.26          = ( ^ [X3: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ S2 ) ) )
% 9.49/5.26        = ( R = S2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3854_pred__equals__eq2,axiom,
% 9.49/5.26      ! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 9.49/5.26        ( ( ( ^ [X3: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R ) )
% 9.49/5.26          = ( ^ [X3: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ S2 ) ) )
% 9.49/5.26        = ( R = S2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3855_pred__equals__eq2,axiom,
% 9.49/5.26      ! [R: set_Pr8056137968301705908nteger,S2: set_Pr8056137968301705908nteger] :
% 9.49/5.26        ( ( ( ^ [X3: code_integer > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y2 ) @ R ) )
% 9.49/5.26          = ( ^ [X3: code_integer > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y2 ) @ S2 ) ) )
% 9.49/5.26        = ( R = S2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3856_pred__equals__eq2,axiom,
% 9.49/5.26      ! [R: set_Pr1281608226676607948nteger,S2: set_Pr1281608226676607948nteger] :
% 9.49/5.26        ( ( ( ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y2 ) @ R ) )
% 9.49/5.26          = ( ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y2 ) @ S2 ) ) )
% 9.49/5.26        = ( R = S2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3857_pred__equals__eq2,axiom,
% 9.49/5.26      ! [R: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
% 9.49/5.26        ( ( ( ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y2 ) @ R ) )
% 9.49/5.26          = ( ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y2 ) @ S2 ) ) )
% 9.49/5.26        = ( R = S2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3858_pred__equals__eq2,axiom,
% 9.49/5.26      ! [R: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
% 9.49/5.26        ( ( ( ^ [X3: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y2 ) @ R ) )
% 9.49/5.26          = ( ^ [X3: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y2 ) @ S2 ) ) )
% 9.49/5.26        = ( R = S2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3859_Collect__restrict,axiom,
% 9.49/5.26      ! [X8: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
% 9.49/5.26        ( ord_le3146513528884898305at_nat
% 9.49/5.26        @ ( collec3392354462482085612at_nat
% 9.49/5.26          @ ^ [X3: product_prod_nat_nat] :
% 9.49/5.26              ( ( member8440522571783428010at_nat @ X3 @ X8 )
% 9.49/5.26              & ( P @ X3 ) ) )
% 9.49/5.26        @ X8 ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3860_Collect__restrict,axiom,
% 9.49/5.26      ! [X8: set_real,P: real > $o] :
% 9.49/5.26        ( ord_less_eq_set_real
% 9.49/5.26        @ ( collect_real
% 9.49/5.26          @ ^ [X3: real] :
% 9.49/5.26              ( ( member_real @ X3 @ X8 )
% 9.49/5.26              & ( P @ X3 ) ) )
% 9.49/5.26        @ X8 ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3861_Collect__restrict,axiom,
% 9.49/5.26      ! [X8: set_list_nat,P: list_nat > $o] :
% 9.49/5.26        ( ord_le6045566169113846134st_nat
% 9.49/5.26        @ ( collect_list_nat
% 9.49/5.26          @ ^ [X3: list_nat] :
% 9.49/5.26              ( ( member_list_nat @ X3 @ X8 )
% 9.49/5.26              & ( P @ X3 ) ) )
% 9.49/5.26        @ X8 ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3862_Collect__restrict,axiom,
% 9.49/5.26      ! [X8: set_set_nat,P: set_nat > $o] :
% 9.49/5.26        ( ord_le6893508408891458716et_nat
% 9.49/5.26        @ ( collect_set_nat
% 9.49/5.26          @ ^ [X3: set_nat] :
% 9.49/5.26              ( ( member_set_nat @ X3 @ X8 )
% 9.49/5.26              & ( P @ X3 ) ) )
% 9.49/5.26        @ X8 ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3863_Collect__restrict,axiom,
% 9.49/5.26      ! [X8: set_nat,P: nat > $o] :
% 9.49/5.26        ( ord_less_eq_set_nat
% 9.49/5.26        @ ( collect_nat
% 9.49/5.26          @ ^ [X3: nat] :
% 9.49/5.26              ( ( member_nat @ X3 @ X8 )
% 9.49/5.26              & ( P @ X3 ) ) )
% 9.49/5.26        @ X8 ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3864_Collect__restrict,axiom,
% 9.49/5.26      ! [X8: set_int,P: int > $o] :
% 9.49/5.26        ( ord_less_eq_set_int
% 9.49/5.26        @ ( collect_int
% 9.49/5.26          @ ^ [X3: int] :
% 9.49/5.26              ( ( member_int @ X3 @ X8 )
% 9.49/5.26              & ( P @ X3 ) ) )
% 9.49/5.26        @ X8 ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3865_prop__restrict,axiom,
% 9.49/5.26      ! [X: product_prod_nat_nat,Z6: set_Pr1261947904930325089at_nat,X8: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
% 9.49/5.26        ( ( member8440522571783428010at_nat @ X @ Z6 )
% 9.49/5.26       => ( ( ord_le3146513528884898305at_nat @ Z6
% 9.49/5.26            @ ( collec3392354462482085612at_nat
% 9.49/5.26              @ ^ [X3: product_prod_nat_nat] :
% 9.49/5.26                  ( ( member8440522571783428010at_nat @ X3 @ X8 )
% 9.49/5.26                  & ( P @ X3 ) ) ) )
% 9.49/5.26         => ( P @ X ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3866_prop__restrict,axiom,
% 9.49/5.26      ! [X: real,Z6: set_real,X8: set_real,P: real > $o] :
% 9.49/5.26        ( ( member_real @ X @ Z6 )
% 9.49/5.26       => ( ( ord_less_eq_set_real @ Z6
% 9.49/5.26            @ ( collect_real
% 9.49/5.26              @ ^ [X3: real] :
% 9.49/5.26                  ( ( member_real @ X3 @ X8 )
% 9.49/5.26                  & ( P @ X3 ) ) ) )
% 9.49/5.26         => ( P @ X ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3867_prop__restrict,axiom,
% 9.49/5.26      ! [X: list_nat,Z6: set_list_nat,X8: set_list_nat,P: list_nat > $o] :
% 9.49/5.26        ( ( member_list_nat @ X @ Z6 )
% 9.49/5.26       => ( ( ord_le6045566169113846134st_nat @ Z6
% 9.49/5.26            @ ( collect_list_nat
% 9.49/5.26              @ ^ [X3: list_nat] :
% 9.49/5.26                  ( ( member_list_nat @ X3 @ X8 )
% 9.49/5.26                  & ( P @ X3 ) ) ) )
% 9.49/5.26         => ( P @ X ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3868_prop__restrict,axiom,
% 9.49/5.26      ! [X: set_nat,Z6: set_set_nat,X8: set_set_nat,P: set_nat > $o] :
% 9.49/5.26        ( ( member_set_nat @ X @ Z6 )
% 9.49/5.26       => ( ( ord_le6893508408891458716et_nat @ Z6
% 9.49/5.26            @ ( collect_set_nat
% 9.49/5.26              @ ^ [X3: set_nat] :
% 9.49/5.26                  ( ( member_set_nat @ X3 @ X8 )
% 9.49/5.26                  & ( P @ X3 ) ) ) )
% 9.49/5.26         => ( P @ X ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3869_prop__restrict,axiom,
% 9.49/5.26      ! [X: nat,Z6: set_nat,X8: set_nat,P: nat > $o] :
% 9.49/5.26        ( ( member_nat @ X @ Z6 )
% 9.49/5.26       => ( ( ord_less_eq_set_nat @ Z6
% 9.49/5.26            @ ( collect_nat
% 9.49/5.26              @ ^ [X3: nat] :
% 9.49/5.26                  ( ( member_nat @ X3 @ X8 )
% 9.49/5.26                  & ( P @ X3 ) ) ) )
% 9.49/5.26         => ( P @ X ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3870_prop__restrict,axiom,
% 9.49/5.26      ! [X: int,Z6: set_int,X8: set_int,P: int > $o] :
% 9.49/5.26        ( ( member_int @ X @ Z6 )
% 9.49/5.26       => ( ( ord_less_eq_set_int @ Z6
% 9.49/5.26            @ ( collect_int
% 9.49/5.26              @ ^ [X3: int] :
% 9.49/5.26                  ( ( member_int @ X3 @ X8 )
% 9.49/5.26                  & ( P @ X3 ) ) ) )
% 9.49/5.26         => ( P @ X ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3871_pred__subset__eq,axiom,
% 9.49/5.26      ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 9.49/5.26        ( ( ord_le704812498762024988_nat_o
% 9.49/5.26          @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ R )
% 9.49/5.26          @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ S2 ) )
% 9.49/5.26        = ( ord_le3146513528884898305at_nat @ R @ S2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3872_pred__subset__eq,axiom,
% 9.49/5.26      ! [R: set_real,S2: set_real] :
% 9.49/5.26        ( ( ord_less_eq_real_o
% 9.49/5.26          @ ^ [X3: real] : ( member_real @ X3 @ R )
% 9.49/5.26          @ ^ [X3: real] : ( member_real @ X3 @ S2 ) )
% 9.49/5.26        = ( ord_less_eq_set_real @ R @ S2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3873_pred__subset__eq,axiom,
% 9.49/5.26      ! [R: set_set_nat,S2: set_set_nat] :
% 9.49/5.26        ( ( ord_le3964352015994296041_nat_o
% 9.49/5.26          @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ R )
% 9.49/5.26          @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ S2 ) )
% 9.49/5.26        = ( ord_le6893508408891458716et_nat @ R @ S2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3874_pred__subset__eq,axiom,
% 9.49/5.26      ! [R: set_nat,S2: set_nat] :
% 9.49/5.26        ( ( ord_less_eq_nat_o
% 9.49/5.26          @ ^ [X3: nat] : ( member_nat @ X3 @ R )
% 9.49/5.26          @ ^ [X3: nat] : ( member_nat @ X3 @ S2 ) )
% 9.49/5.26        = ( ord_less_eq_set_nat @ R @ S2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3875_pred__subset__eq,axiom,
% 9.49/5.26      ! [R: set_int,S2: set_int] :
% 9.49/5.26        ( ( ord_less_eq_int_o
% 9.49/5.26          @ ^ [X3: int] : ( member_int @ X3 @ R )
% 9.49/5.26          @ ^ [X3: int] : ( member_int @ X3 @ S2 ) )
% 9.49/5.26        = ( ord_less_eq_set_int @ R @ S2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3876_less__eq__set__def,axiom,
% 9.49/5.26      ( ord_le3146513528884898305at_nat
% 9.49/5.26      = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 9.49/5.26            ( ord_le704812498762024988_nat_o
% 9.49/5.26            @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A6 )
% 9.49/5.26            @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ B6 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3877_less__eq__set__def,axiom,
% 9.49/5.26      ( ord_less_eq_set_real
% 9.49/5.26      = ( ^ [A6: set_real,B6: set_real] :
% 9.49/5.26            ( ord_less_eq_real_o
% 9.49/5.26            @ ^ [X3: real] : ( member_real @ X3 @ A6 )
% 9.49/5.26            @ ^ [X3: real] : ( member_real @ X3 @ B6 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3878_less__eq__set__def,axiom,
% 9.49/5.26      ( ord_le6893508408891458716et_nat
% 9.49/5.26      = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 9.49/5.26            ( ord_le3964352015994296041_nat_o
% 9.49/5.26            @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 )
% 9.49/5.26            @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B6 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3879_less__eq__set__def,axiom,
% 9.49/5.26      ( ord_less_eq_set_nat
% 9.49/5.26      = ( ^ [A6: set_nat,B6: set_nat] :
% 9.49/5.26            ( ord_less_eq_nat_o
% 9.49/5.26            @ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
% 9.49/5.26            @ ^ [X3: nat] : ( member_nat @ X3 @ B6 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3880_less__eq__set__def,axiom,
% 9.49/5.26      ( ord_less_eq_set_int
% 9.49/5.26      = ( ^ [A6: set_int,B6: set_int] :
% 9.49/5.26            ( ord_less_eq_int_o
% 9.49/5.26            @ ^ [X3: int] : ( member_int @ X3 @ A6 )
% 9.49/5.26            @ ^ [X3: int] : ( member_int @ X3 @ B6 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3881_Collect__subset,axiom,
% 9.49/5.26      ! [A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
% 9.49/5.26        ( ord_le3146513528884898305at_nat
% 9.49/5.26        @ ( collec3392354462482085612at_nat
% 9.49/5.26          @ ^ [X3: product_prod_nat_nat] :
% 9.49/5.26              ( ( member8440522571783428010at_nat @ X3 @ A2 )
% 9.49/5.26              & ( P @ X3 ) ) )
% 9.49/5.26        @ A2 ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3882_Collect__subset,axiom,
% 9.49/5.26      ! [A2: set_real,P: real > $o] :
% 9.49/5.26        ( ord_less_eq_set_real
% 9.49/5.26        @ ( collect_real
% 9.49/5.26          @ ^ [X3: real] :
% 9.49/5.26              ( ( member_real @ X3 @ A2 )
% 9.49/5.26              & ( P @ X3 ) ) )
% 9.49/5.26        @ A2 ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3883_Collect__subset,axiom,
% 9.49/5.26      ! [A2: set_list_nat,P: list_nat > $o] :
% 9.49/5.26        ( ord_le6045566169113846134st_nat
% 9.49/5.26        @ ( collect_list_nat
% 9.49/5.26          @ ^ [X3: list_nat] :
% 9.49/5.26              ( ( member_list_nat @ X3 @ A2 )
% 9.49/5.26              & ( P @ X3 ) ) )
% 9.49/5.26        @ A2 ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3884_Collect__subset,axiom,
% 9.49/5.26      ! [A2: set_set_nat,P: set_nat > $o] :
% 9.49/5.26        ( ord_le6893508408891458716et_nat
% 9.49/5.26        @ ( collect_set_nat
% 9.49/5.26          @ ^ [X3: set_nat] :
% 9.49/5.26              ( ( member_set_nat @ X3 @ A2 )
% 9.49/5.26              & ( P @ X3 ) ) )
% 9.49/5.26        @ A2 ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3885_Collect__subset,axiom,
% 9.49/5.26      ! [A2: set_nat,P: nat > $o] :
% 9.49/5.26        ( ord_less_eq_set_nat
% 9.49/5.26        @ ( collect_nat
% 9.49/5.26          @ ^ [X3: nat] :
% 9.49/5.26              ( ( member_nat @ X3 @ A2 )
% 9.49/5.26              & ( P @ X3 ) ) )
% 9.49/5.26        @ A2 ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3886_Collect__subset,axiom,
% 9.49/5.26      ! [A2: set_int,P: int > $o] :
% 9.49/5.26        ( ord_less_eq_set_int
% 9.49/5.26        @ ( collect_int
% 9.49/5.26          @ ^ [X3: int] :
% 9.49/5.26              ( ( member_int @ X3 @ A2 )
% 9.49/5.26              & ( P @ X3 ) ) )
% 9.49/5.26        @ A2 ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3887_pigeonhole__infinite__rel,axiom,
% 9.49/5.26      ! [A2: set_real,B2: set_nat,R: real > nat > $o] :
% 9.49/5.26        ( ~ ( finite_finite_real @ A2 )
% 9.49/5.26       => ( ( finite_finite_nat @ B2 )
% 9.49/5.26         => ( ! [X4: real] :
% 9.49/5.26                ( ( member_real @ X4 @ A2 )
% 9.49/5.26               => ? [Xa: nat] :
% 9.49/5.26                    ( ( member_nat @ Xa @ B2 )
% 9.49/5.26                    & ( R @ X4 @ Xa ) ) )
% 9.49/5.26           => ? [X4: nat] :
% 9.49/5.26                ( ( member_nat @ X4 @ B2 )
% 9.49/5.26                & ~ ( finite_finite_real
% 9.49/5.26                    @ ( collect_real
% 9.49/5.26                      @ ^ [A4: real] :
% 9.49/5.26                          ( ( member_real @ A4 @ A2 )
% 9.49/5.26                          & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3888_pigeonhole__infinite__rel,axiom,
% 9.49/5.26      ! [A2: set_real,B2: set_int,R: real > int > $o] :
% 9.49/5.26        ( ~ ( finite_finite_real @ A2 )
% 9.49/5.26       => ( ( finite_finite_int @ B2 )
% 9.49/5.26         => ( ! [X4: real] :
% 9.49/5.26                ( ( member_real @ X4 @ A2 )
% 9.49/5.26               => ? [Xa: int] :
% 9.49/5.26                    ( ( member_int @ Xa @ B2 )
% 9.49/5.26                    & ( R @ X4 @ Xa ) ) )
% 9.49/5.26           => ? [X4: int] :
% 9.49/5.26                ( ( member_int @ X4 @ B2 )
% 9.49/5.26                & ~ ( finite_finite_real
% 9.49/5.26                    @ ( collect_real
% 9.49/5.26                      @ ^ [A4: real] :
% 9.49/5.26                          ( ( member_real @ A4 @ A2 )
% 9.49/5.26                          & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3889_pigeonhole__infinite__rel,axiom,
% 9.49/5.26      ! [A2: set_real,B2: set_complex,R: real > complex > $o] :
% 9.49/5.26        ( ~ ( finite_finite_real @ A2 )
% 9.49/5.26       => ( ( finite3207457112153483333omplex @ B2 )
% 9.49/5.26         => ( ! [X4: real] :
% 9.49/5.26                ( ( member_real @ X4 @ A2 )
% 9.49/5.26               => ? [Xa: complex] :
% 9.49/5.26                    ( ( member_complex @ Xa @ B2 )
% 9.49/5.26                    & ( R @ X4 @ Xa ) ) )
% 9.49/5.26           => ? [X4: complex] :
% 9.49/5.26                ( ( member_complex @ X4 @ B2 )
% 9.49/5.26                & ~ ( finite_finite_real
% 9.49/5.26                    @ ( collect_real
% 9.49/5.26                      @ ^ [A4: real] :
% 9.49/5.26                          ( ( member_real @ A4 @ A2 )
% 9.49/5.26                          & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3890_pigeonhole__infinite__rel,axiom,
% 9.49/5.26      ! [A2: set_nat,B2: set_nat,R: nat > nat > $o] :
% 9.49/5.26        ( ~ ( finite_finite_nat @ A2 )
% 9.49/5.26       => ( ( finite_finite_nat @ B2 )
% 9.49/5.26         => ( ! [X4: nat] :
% 9.49/5.26                ( ( member_nat @ X4 @ A2 )
% 9.49/5.26               => ? [Xa: nat] :
% 9.49/5.26                    ( ( member_nat @ Xa @ B2 )
% 9.49/5.26                    & ( R @ X4 @ Xa ) ) )
% 9.49/5.26           => ? [X4: nat] :
% 9.49/5.26                ( ( member_nat @ X4 @ B2 )
% 9.49/5.26                & ~ ( finite_finite_nat
% 9.49/5.26                    @ ( collect_nat
% 9.49/5.26                      @ ^ [A4: nat] :
% 9.49/5.26                          ( ( member_nat @ A4 @ A2 )
% 9.49/5.26                          & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3891_pigeonhole__infinite__rel,axiom,
% 9.49/5.26      ! [A2: set_nat,B2: set_int,R: nat > int > $o] :
% 9.49/5.26        ( ~ ( finite_finite_nat @ A2 )
% 9.49/5.26       => ( ( finite_finite_int @ B2 )
% 9.49/5.26         => ( ! [X4: nat] :
% 9.49/5.26                ( ( member_nat @ X4 @ A2 )
% 9.49/5.26               => ? [Xa: int] :
% 9.49/5.26                    ( ( member_int @ Xa @ B2 )
% 9.49/5.26                    & ( R @ X4 @ Xa ) ) )
% 9.49/5.26           => ? [X4: int] :
% 9.49/5.26                ( ( member_int @ X4 @ B2 )
% 9.49/5.26                & ~ ( finite_finite_nat
% 9.49/5.26                    @ ( collect_nat
% 9.49/5.26                      @ ^ [A4: nat] :
% 9.49/5.26                          ( ( member_nat @ A4 @ A2 )
% 9.49/5.26                          & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3892_pigeonhole__infinite__rel,axiom,
% 9.49/5.26      ! [A2: set_nat,B2: set_complex,R: nat > complex > $o] :
% 9.49/5.26        ( ~ ( finite_finite_nat @ A2 )
% 9.49/5.26       => ( ( finite3207457112153483333omplex @ B2 )
% 9.49/5.26         => ( ! [X4: nat] :
% 9.49/5.26                ( ( member_nat @ X4 @ A2 )
% 9.49/5.26               => ? [Xa: complex] :
% 9.49/5.26                    ( ( member_complex @ Xa @ B2 )
% 9.49/5.26                    & ( R @ X4 @ Xa ) ) )
% 9.49/5.26           => ? [X4: complex] :
% 9.49/5.26                ( ( member_complex @ X4 @ B2 )
% 9.49/5.26                & ~ ( finite_finite_nat
% 9.49/5.26                    @ ( collect_nat
% 9.49/5.26                      @ ^ [A4: nat] :
% 9.49/5.26                          ( ( member_nat @ A4 @ A2 )
% 9.49/5.26                          & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3893_pigeonhole__infinite__rel,axiom,
% 9.49/5.26      ! [A2: set_int,B2: set_nat,R: int > nat > $o] :
% 9.49/5.26        ( ~ ( finite_finite_int @ A2 )
% 9.49/5.26       => ( ( finite_finite_nat @ B2 )
% 9.49/5.26         => ( ! [X4: int] :
% 9.49/5.26                ( ( member_int @ X4 @ A2 )
% 9.49/5.26               => ? [Xa: nat] :
% 9.49/5.26                    ( ( member_nat @ Xa @ B2 )
% 9.49/5.26                    & ( R @ X4 @ Xa ) ) )
% 9.49/5.26           => ? [X4: nat] :
% 9.49/5.26                ( ( member_nat @ X4 @ B2 )
% 9.49/5.26                & ~ ( finite_finite_int
% 9.49/5.26                    @ ( collect_int
% 9.49/5.26                      @ ^ [A4: int] :
% 9.49/5.26                          ( ( member_int @ A4 @ A2 )
% 9.49/5.26                          & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3894_pigeonhole__infinite__rel,axiom,
% 9.49/5.26      ! [A2: set_int,B2: set_int,R: int > int > $o] :
% 9.49/5.26        ( ~ ( finite_finite_int @ A2 )
% 9.49/5.26       => ( ( finite_finite_int @ B2 )
% 9.49/5.26         => ( ! [X4: int] :
% 9.49/5.26                ( ( member_int @ X4 @ A2 )
% 9.49/5.26               => ? [Xa: int] :
% 9.49/5.26                    ( ( member_int @ Xa @ B2 )
% 9.49/5.26                    & ( R @ X4 @ Xa ) ) )
% 9.49/5.26           => ? [X4: int] :
% 9.49/5.26                ( ( member_int @ X4 @ B2 )
% 9.49/5.26                & ~ ( finite_finite_int
% 9.49/5.26                    @ ( collect_int
% 9.49/5.26                      @ ^ [A4: int] :
% 9.49/5.26                          ( ( member_int @ A4 @ A2 )
% 9.49/5.26                          & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3895_pigeonhole__infinite__rel,axiom,
% 9.49/5.26      ! [A2: set_int,B2: set_complex,R: int > complex > $o] :
% 9.49/5.26        ( ~ ( finite_finite_int @ A2 )
% 9.49/5.26       => ( ( finite3207457112153483333omplex @ B2 )
% 9.49/5.26         => ( ! [X4: int] :
% 9.49/5.26                ( ( member_int @ X4 @ A2 )
% 9.49/5.26               => ? [Xa: complex] :
% 9.49/5.26                    ( ( member_complex @ Xa @ B2 )
% 9.49/5.26                    & ( R @ X4 @ Xa ) ) )
% 9.49/5.26           => ? [X4: complex] :
% 9.49/5.26                ( ( member_complex @ X4 @ B2 )
% 9.49/5.26                & ~ ( finite_finite_int
% 9.49/5.26                    @ ( collect_int
% 9.49/5.26                      @ ^ [A4: int] :
% 9.49/5.26                          ( ( member_int @ A4 @ A2 )
% 9.49/5.26                          & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3896_pigeonhole__infinite__rel,axiom,
% 9.49/5.26      ! [A2: set_complex,B2: set_nat,R: complex > nat > $o] :
% 9.49/5.26        ( ~ ( finite3207457112153483333omplex @ A2 )
% 9.49/5.26       => ( ( finite_finite_nat @ B2 )
% 9.49/5.26         => ( ! [X4: complex] :
% 9.49/5.26                ( ( member_complex @ X4 @ A2 )
% 9.49/5.26               => ? [Xa: nat] :
% 9.49/5.26                    ( ( member_nat @ Xa @ B2 )
% 9.49/5.26                    & ( R @ X4 @ Xa ) ) )
% 9.49/5.26           => ? [X4: nat] :
% 9.49/5.26                ( ( member_nat @ X4 @ B2 )
% 9.49/5.26                & ~ ( finite3207457112153483333omplex
% 9.49/5.26                    @ ( collect_complex
% 9.49/5.26                      @ ^ [A4: complex] :
% 9.49/5.26                          ( ( member_complex @ A4 @ A2 )
% 9.49/5.26                          & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3897_not__finite__existsD,axiom,
% 9.49/5.26      ! [P: real > $o] :
% 9.49/5.26        ( ~ ( finite_finite_real @ ( collect_real @ P ) )
% 9.49/5.26       => ? [X_1: real] : ( P @ X_1 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3898_not__finite__existsD,axiom,
% 9.49/5.26      ! [P: list_nat > $o] :
% 9.49/5.26        ( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
% 9.49/5.26       => ? [X_1: list_nat] : ( P @ X_1 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3899_not__finite__existsD,axiom,
% 9.49/5.26      ! [P: set_nat > $o] :
% 9.49/5.26        ( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 9.49/5.26       => ? [X_1: set_nat] : ( P @ X_1 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3900_not__finite__existsD,axiom,
% 9.49/5.26      ! [P: nat > $o] :
% 9.49/5.26        ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.49/5.26       => ? [X_1: nat] : ( P @ X_1 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3901_not__finite__existsD,axiom,
% 9.49/5.26      ! [P: int > $o] :
% 9.49/5.26        ( ~ ( finite_finite_int @ ( collect_int @ P ) )
% 9.49/5.26       => ? [X_1: int] : ( P @ X_1 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3902_not__finite__existsD,axiom,
% 9.49/5.26      ! [P: complex > $o] :
% 9.49/5.26        ( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 9.49/5.26       => ? [X_1: complex] : ( P @ X_1 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3903_not__finite__existsD,axiom,
% 9.49/5.26      ! [P: product_prod_nat_nat > $o] :
% 9.49/5.26        ( ~ ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
% 9.49/5.26       => ? [X_1: product_prod_nat_nat] : ( P @ X_1 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3904_empty__def,axiom,
% 9.49/5.26      ( bot_bot_set_list_nat
% 9.49/5.26      = ( collect_list_nat
% 9.49/5.26        @ ^ [X3: list_nat] : $false ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3905_empty__def,axiom,
% 9.49/5.26      ( bot_bot_set_set_nat
% 9.49/5.26      = ( collect_set_nat
% 9.49/5.26        @ ^ [X3: set_nat] : $false ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3906_empty__def,axiom,
% 9.49/5.26      ( bot_bo2099793752762293965at_nat
% 9.49/5.26      = ( collec3392354462482085612at_nat
% 9.49/5.26        @ ^ [X3: product_prod_nat_nat] : $false ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3907_empty__def,axiom,
% 9.49/5.26      ( bot_bot_set_real
% 9.49/5.26      = ( collect_real
% 9.49/5.26        @ ^ [X3: real] : $false ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3908_empty__def,axiom,
% 9.49/5.26      ( bot_bot_set_nat
% 9.49/5.26      = ( collect_nat
% 9.49/5.26        @ ^ [X3: nat] : $false ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3909_empty__def,axiom,
% 9.49/5.26      ( bot_bot_set_int
% 9.49/5.26      = ( collect_int
% 9.49/5.26        @ ^ [X3: int] : $false ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3910_lambda__zero,axiom,
% 9.49/5.26      ( ( ^ [H: complex] : zero_zero_complex )
% 9.49/5.26      = ( times_times_complex @ zero_zero_complex ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3911_lambda__zero,axiom,
% 9.49/5.26      ( ( ^ [H: real] : zero_zero_real )
% 9.49/5.26      = ( times_times_real @ zero_zero_real ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3912_lambda__zero,axiom,
% 9.49/5.26      ( ( ^ [H: rat] : zero_zero_rat )
% 9.49/5.26      = ( times_times_rat @ zero_zero_rat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3913_lambda__zero,axiom,
% 9.49/5.26      ( ( ^ [H: nat] : zero_zero_nat )
% 9.49/5.26      = ( times_times_nat @ zero_zero_nat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3914_lambda__zero,axiom,
% 9.49/5.26      ( ( ^ [H: int] : zero_zero_int )
% 9.49/5.26      = ( times_times_int @ zero_zero_int ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3915_max__def__raw,axiom,
% 9.49/5.26      ( ord_max_set_int
% 9.49/5.26      = ( ^ [A4: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3916_max__def__raw,axiom,
% 9.49/5.26      ( ord_max_rat
% 9.49/5.26      = ( ^ [A4: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3917_max__def__raw,axiom,
% 9.49/5.26      ( ord_max_num
% 9.49/5.26      = ( ^ [A4: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3918_max__def__raw,axiom,
% 9.49/5.26      ( ord_max_nat
% 9.49/5.26      = ( ^ [A4: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3919_max__def__raw,axiom,
% 9.49/5.26      ( ord_max_int
% 9.49/5.26      = ( ^ [A4: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3920_pred__subset__eq2,axiom,
% 9.49/5.26      ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 9.49/5.26        ( ( ord_le2646555220125990790_nat_o
% 9.49/5.26          @ ^ [X3: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R )
% 9.49/5.26          @ ^ [X3: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ S2 ) )
% 9.49/5.26        = ( ord_le3146513528884898305at_nat @ R @ S2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3921_pred__subset__eq2,axiom,
% 9.49/5.26      ! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 9.49/5.26        ( ( ord_le6741204236512500942_int_o
% 9.49/5.26          @ ^ [X3: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R )
% 9.49/5.26          @ ^ [X3: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ S2 ) )
% 9.49/5.26        = ( ord_le2843351958646193337nt_int @ R @ S2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3922_pred__subset__eq2,axiom,
% 9.49/5.26      ! [R: set_Pr8056137968301705908nteger,S2: set_Pr8056137968301705908nteger] :
% 9.49/5.26        ( ( ord_le3636971675376928563eger_o
% 9.49/5.26          @ ^ [X3: code_integer > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y2 ) @ R )
% 9.49/5.26          @ ^ [X3: code_integer > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y2 ) @ S2 ) )
% 9.49/5.26        = ( ord_le3216752416896350996nteger @ R @ S2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3923_pred__subset__eq2,axiom,
% 9.49/5.26      ! [R: set_Pr1281608226676607948nteger,S2: set_Pr1281608226676607948nteger] :
% 9.49/5.26        ( ( ord_le4340812435750786203eger_o
% 9.49/5.26          @ ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y2 ) @ R )
% 9.49/5.26          @ ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y2 ) @ S2 ) )
% 9.49/5.26        = ( ord_le653643898420964396nteger @ R @ S2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3924_pred__subset__eq2,axiom,
% 9.49/5.26      ! [R: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
% 9.49/5.26        ( ( ord_le5643404153117327598_int_o
% 9.49/5.26          @ ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y2 ) @ R )
% 9.49/5.26          @ ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y2 ) @ S2 ) )
% 9.49/5.26        = ( ord_le8725513860283290265nt_int @ R @ S2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3925_pred__subset__eq2,axiom,
% 9.49/5.26      ! [R: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
% 9.49/5.26        ( ( ord_le2124322318746777828_int_o
% 9.49/5.26          @ ^ [X3: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y2 ) @ R )
% 9.49/5.26          @ ^ [X3: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y2 ) @ S2 ) )
% 9.49/5.26        = ( ord_le135402666524580259nt_int @ R @ S2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3926_bot__empty__eq2,axiom,
% 9.49/5.26      ( bot_bot_int_int_o
% 9.49/5.26      = ( ^ [X3: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ bot_bo1796632182523588997nt_int ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3927_bot__empty__eq2,axiom,
% 9.49/5.26      ( bot_bo5358457235160185703eger_o
% 9.49/5.26      = ( ^ [X3: code_integer > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y2 ) @ bot_bo3145834390647256904nteger ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3928_bot__empty__eq2,axiom,
% 9.49/5.26      ( bot_bo3000040243691356879eger_o
% 9.49/5.26      = ( ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y2 ) @ bot_bo5443222936135328352nteger ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3929_bot__empty__eq2,axiom,
% 9.49/5.26      ( bot_bo8662317086119403298_int_o
% 9.49/5.26      = ( ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y2 ) @ bot_bo572930865798478029nt_int ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3930_bot__empty__eq2,axiom,
% 9.49/5.26      ( bot_bo1403522918969695512_int_o
% 9.49/5.26      = ( ^ [X3: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y2 ) @ bot_bo4508923176915781079nt_int ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3931_bot__empty__eq2,axiom,
% 9.49/5.26      ( bot_bot_nat_nat_o
% 9.49/5.26      = ( ^ [X3: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ bot_bo2099793752762293965at_nat ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3932_finite__M__bounded__by__nat,axiom,
% 9.49/5.26      ! [P: nat > $o,I: nat] :
% 9.49/5.26        ( finite_finite_nat
% 9.49/5.26        @ ( collect_nat
% 9.49/5.26          @ ^ [K3: nat] :
% 9.49/5.26              ( ( P @ K3 )
% 9.49/5.26              & ( ord_less_nat @ K3 @ I ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3933_finite__less__ub,axiom,
% 9.49/5.26      ! [F: nat > nat,U: nat] :
% 9.49/5.26        ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
% 9.49/5.26       => ( finite_finite_nat
% 9.49/5.26          @ ( collect_nat
% 9.49/5.26            @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3934_set__vebt__def,axiom,
% 9.49/5.26      ( vEBT_set_vebt
% 9.49/5.26      = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3935_finite__lists__length__eq,axiom,
% 9.49/5.26      ! [A2: set_complex,N: nat] :
% 9.49/5.26        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.26       => ( finite8712137658972009173omplex
% 9.49/5.26          @ ( collect_list_complex
% 9.49/5.26            @ ^ [Xs2: list_complex] :
% 9.49/5.26                ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
% 9.49/5.26                & ( ( size_s3451745648224563538omplex @ Xs2 )
% 9.49/5.26                  = N ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3936_finite__lists__length__eq,axiom,
% 9.49/5.26      ! [A2: set_Pr1261947904930325089at_nat,N: nat] :
% 9.49/5.26        ( ( finite6177210948735845034at_nat @ A2 )
% 9.49/5.26       => ( finite500796754983035824at_nat
% 9.49/5.26          @ ( collec3343600615725829874at_nat
% 9.49/5.26            @ ^ [Xs2: list_P6011104703257516679at_nat] :
% 9.49/5.26                ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 )
% 9.49/5.26                & ( ( size_s5460976970255530739at_nat @ Xs2 )
% 9.49/5.26                  = N ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3937_finite__lists__length__eq,axiom,
% 9.49/5.26      ! [A2: set_VEBT_VEBT,N: nat] :
% 9.49/5.26        ( ( finite5795047828879050333T_VEBT @ A2 )
% 9.49/5.26       => ( finite3004134309566078307T_VEBT
% 9.49/5.26          @ ( collec5608196760682091941T_VEBT
% 9.49/5.26            @ ^ [Xs2: list_VEBT_VEBT] :
% 9.49/5.26                ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
% 9.49/5.26                & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 9.49/5.26                  = N ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3938_finite__lists__length__eq,axiom,
% 9.49/5.26      ! [A2: set_o,N: nat] :
% 9.49/5.26        ( ( finite_finite_o @ A2 )
% 9.49/5.26       => ( finite_finite_list_o
% 9.49/5.26          @ ( collect_list_o
% 9.49/5.26            @ ^ [Xs2: list_o] :
% 9.49/5.26                ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A2 )
% 9.49/5.26                & ( ( size_size_list_o @ Xs2 )
% 9.49/5.26                  = N ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3939_finite__lists__length__eq,axiom,
% 9.49/5.26      ! [A2: set_nat,N: nat] :
% 9.49/5.26        ( ( finite_finite_nat @ A2 )
% 9.49/5.26       => ( finite8100373058378681591st_nat
% 9.49/5.26          @ ( collect_list_nat
% 9.49/5.26            @ ^ [Xs2: list_nat] :
% 9.49/5.26                ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
% 9.49/5.26                & ( ( size_size_list_nat @ Xs2 )
% 9.49/5.26                  = N ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3940_finite__lists__length__eq,axiom,
% 9.49/5.26      ! [A2: set_int,N: nat] :
% 9.49/5.26        ( ( finite_finite_int @ A2 )
% 9.49/5.26       => ( finite3922522038869484883st_int
% 9.49/5.26          @ ( collect_list_int
% 9.49/5.26            @ ^ [Xs2: list_int] :
% 9.49/5.26                ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
% 9.49/5.26                & ( ( size_size_list_int @ Xs2 )
% 9.49/5.26                  = N ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3941_finite__lists__length__le,axiom,
% 9.49/5.26      ! [A2: set_complex,N: nat] :
% 9.49/5.26        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.26       => ( finite8712137658972009173omplex
% 9.49/5.26          @ ( collect_list_complex
% 9.49/5.26            @ ^ [Xs2: list_complex] :
% 9.49/5.26                ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
% 9.49/5.26                & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs2 ) @ N ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3942_finite__lists__length__le,axiom,
% 9.49/5.26      ! [A2: set_Pr1261947904930325089at_nat,N: nat] :
% 9.49/5.26        ( ( finite6177210948735845034at_nat @ A2 )
% 9.49/5.26       => ( finite500796754983035824at_nat
% 9.49/5.26          @ ( collec3343600615725829874at_nat
% 9.49/5.26            @ ^ [Xs2: list_P6011104703257516679at_nat] :
% 9.49/5.26                ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 )
% 9.49/5.26                & ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) @ N ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3943_finite__lists__length__le,axiom,
% 9.49/5.26      ! [A2: set_VEBT_VEBT,N: nat] :
% 9.49/5.26        ( ( finite5795047828879050333T_VEBT @ A2 )
% 9.49/5.26       => ( finite3004134309566078307T_VEBT
% 9.49/5.26          @ ( collec5608196760682091941T_VEBT
% 9.49/5.26            @ ^ [Xs2: list_VEBT_VEBT] :
% 9.49/5.26                ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
% 9.49/5.26                & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ N ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3944_finite__lists__length__le,axiom,
% 9.49/5.26      ! [A2: set_o,N: nat] :
% 9.49/5.26        ( ( finite_finite_o @ A2 )
% 9.49/5.26       => ( finite_finite_list_o
% 9.49/5.26          @ ( collect_list_o
% 9.49/5.26            @ ^ [Xs2: list_o] :
% 9.49/5.26                ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A2 )
% 9.49/5.26                & ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ N ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3945_finite__lists__length__le,axiom,
% 9.49/5.26      ! [A2: set_nat,N: nat] :
% 9.49/5.26        ( ( finite_finite_nat @ A2 )
% 9.49/5.26       => ( finite8100373058378681591st_nat
% 9.49/5.26          @ ( collect_list_nat
% 9.49/5.26            @ ^ [Xs2: list_nat] :
% 9.49/5.26                ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
% 9.49/5.26                & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3946_finite__lists__length__le,axiom,
% 9.49/5.26      ! [A2: set_int,N: nat] :
% 9.49/5.26        ( ( finite_finite_int @ A2 )
% 9.49/5.26       => ( finite3922522038869484883st_int
% 9.49/5.26          @ ( collect_list_int
% 9.49/5.26            @ ^ [Xs2: list_int] :
% 9.49/5.26                ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
% 9.49/5.26                & ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3947_periodic__finite__ex,axiom,
% 9.49/5.26      ! [D: int,P: int > $o] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ D )
% 9.49/5.26       => ( ! [X4: int,K2: int] :
% 9.49/5.26              ( ( P @ X4 )
% 9.49/5.26              = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 9.49/5.26         => ( ( ? [X6: int] : ( P @ X6 ) )
% 9.49/5.26            = ( ? [X3: int] :
% 9.49/5.26                  ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 9.49/5.26                  & ( P @ X3 ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3948_aset_I7_J,axiom,
% 9.49/5.26      ! [D6: int,A2: set_int,T: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.49/5.26       => ! [X2: int] :
% 9.49/5.26            ( ! [Xa3: int] :
% 9.49/5.26                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.49/5.26               => ! [Xb3: int] :
% 9.49/5.26                    ( ( member_int @ Xb3 @ A2 )
% 9.49/5.26                   => ( X2
% 9.49/5.26                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.49/5.26           => ( ( ord_less_int @ T @ X2 )
% 9.49/5.26             => ( ord_less_int @ T @ ( plus_plus_int @ X2 @ D6 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3949_aset_I5_J,axiom,
% 9.49/5.26      ! [D6: int,T: int,A2: set_int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.49/5.26       => ( ( member_int @ T @ A2 )
% 9.49/5.26         => ! [X2: int] :
% 9.49/5.26              ( ! [Xa3: int] :
% 9.49/5.26                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.49/5.26                 => ! [Xb3: int] :
% 9.49/5.26                      ( ( member_int @ Xb3 @ A2 )
% 9.49/5.26                     => ( X2
% 9.49/5.26                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.49/5.26             => ( ( ord_less_int @ X2 @ T )
% 9.49/5.26               => ( ord_less_int @ ( plus_plus_int @ X2 @ D6 ) @ T ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3950_aset_I4_J,axiom,
% 9.49/5.26      ! [D6: int,T: int,A2: set_int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.49/5.26       => ( ( member_int @ T @ A2 )
% 9.49/5.26         => ! [X2: int] :
% 9.49/5.26              ( ! [Xa3: int] :
% 9.49/5.26                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.49/5.26                 => ! [Xb3: int] :
% 9.49/5.26                      ( ( member_int @ Xb3 @ A2 )
% 9.49/5.26                     => ( X2
% 9.49/5.26                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.49/5.26             => ( ( X2 != T )
% 9.49/5.26               => ( ( plus_plus_int @ X2 @ D6 )
% 9.49/5.26                 != T ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3951_aset_I3_J,axiom,
% 9.49/5.26      ! [D6: int,T: int,A2: set_int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.49/5.26       => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 9.49/5.26         => ! [X2: int] :
% 9.49/5.26              ( ! [Xa3: int] :
% 9.49/5.26                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.49/5.26                 => ! [Xb3: int] :
% 9.49/5.26                      ( ( member_int @ Xb3 @ A2 )
% 9.49/5.26                     => ( X2
% 9.49/5.26                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.49/5.26             => ( ( X2 = T )
% 9.49/5.26               => ( ( plus_plus_int @ X2 @ D6 )
% 9.49/5.26                  = T ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3952_bset_I7_J,axiom,
% 9.49/5.26      ! [D6: int,T: int,B2: set_int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.49/5.26       => ( ( member_int @ T @ B2 )
% 9.49/5.26         => ! [X2: int] :
% 9.49/5.26              ( ! [Xa3: int] :
% 9.49/5.26                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.49/5.26                 => ! [Xb3: int] :
% 9.49/5.26                      ( ( member_int @ Xb3 @ B2 )
% 9.49/5.26                     => ( X2
% 9.49/5.26                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.49/5.26             => ( ( ord_less_int @ T @ X2 )
% 9.49/5.26               => ( ord_less_int @ T @ ( minus_minus_int @ X2 @ D6 ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3953_bset_I5_J,axiom,
% 9.49/5.26      ! [D6: int,B2: set_int,T: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.49/5.26       => ! [X2: int] :
% 9.49/5.26            ( ! [Xa3: int] :
% 9.49/5.26                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.49/5.26               => ! [Xb3: int] :
% 9.49/5.26                    ( ( member_int @ Xb3 @ B2 )
% 9.49/5.26                   => ( X2
% 9.49/5.26                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.49/5.26           => ( ( ord_less_int @ X2 @ T )
% 9.49/5.26             => ( ord_less_int @ ( minus_minus_int @ X2 @ D6 ) @ T ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3954_bset_I4_J,axiom,
% 9.49/5.26      ! [D6: int,T: int,B2: set_int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.49/5.26       => ( ( member_int @ T @ B2 )
% 9.49/5.26         => ! [X2: int] :
% 9.49/5.26              ( ! [Xa3: int] :
% 9.49/5.26                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.49/5.26                 => ! [Xb3: int] :
% 9.49/5.26                      ( ( member_int @ Xb3 @ B2 )
% 9.49/5.26                     => ( X2
% 9.49/5.26                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.49/5.26             => ( ( X2 != T )
% 9.49/5.26               => ( ( minus_minus_int @ X2 @ D6 )
% 9.49/5.26                 != T ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3955_bset_I3_J,axiom,
% 9.49/5.26      ! [D6: int,T: int,B2: set_int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.49/5.26       => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
% 9.49/5.26         => ! [X2: int] :
% 9.49/5.26              ( ! [Xa3: int] :
% 9.49/5.26                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.49/5.26                 => ! [Xb3: int] :
% 9.49/5.26                      ( ( member_int @ Xb3 @ B2 )
% 9.49/5.26                     => ( X2
% 9.49/5.26                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.49/5.26             => ( ( X2 = T )
% 9.49/5.26               => ( ( minus_minus_int @ X2 @ D6 )
% 9.49/5.26                  = T ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3956_aset_I8_J,axiom,
% 9.49/5.26      ! [D6: int,A2: set_int,T: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.49/5.26       => ! [X2: int] :
% 9.49/5.26            ( ! [Xa3: int] :
% 9.49/5.26                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.49/5.26               => ! [Xb3: int] :
% 9.49/5.26                    ( ( member_int @ Xb3 @ A2 )
% 9.49/5.26                   => ( X2
% 9.49/5.26                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.49/5.26           => ( ( ord_less_eq_int @ T @ X2 )
% 9.49/5.26             => ( ord_less_eq_int @ T @ ( plus_plus_int @ X2 @ D6 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3957_aset_I6_J,axiom,
% 9.49/5.26      ! [D6: int,T: int,A2: set_int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.49/5.26       => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 9.49/5.26         => ! [X2: int] :
% 9.49/5.26              ( ! [Xa3: int] :
% 9.49/5.26                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.49/5.26                 => ! [Xb3: int] :
% 9.49/5.26                      ( ( member_int @ Xb3 @ A2 )
% 9.49/5.26                     => ( X2
% 9.49/5.26                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.49/5.26             => ( ( ord_less_eq_int @ X2 @ T )
% 9.49/5.26               => ( ord_less_eq_int @ ( plus_plus_int @ X2 @ D6 ) @ T ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3958_bset_I8_J,axiom,
% 9.49/5.26      ! [D6: int,T: int,B2: set_int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.49/5.26       => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
% 9.49/5.26         => ! [X2: int] :
% 9.49/5.26              ( ! [Xa3: int] :
% 9.49/5.26                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.49/5.26                 => ! [Xb3: int] :
% 9.49/5.26                      ( ( member_int @ Xb3 @ B2 )
% 9.49/5.26                     => ( X2
% 9.49/5.26                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.49/5.26             => ( ( ord_less_eq_int @ T @ X2 )
% 9.49/5.26               => ( ord_less_eq_int @ T @ ( minus_minus_int @ X2 @ D6 ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3959_bset_I6_J,axiom,
% 9.49/5.26      ! [D6: int,B2: set_int,T: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.49/5.26       => ! [X2: int] :
% 9.49/5.26            ( ! [Xa3: int] :
% 9.49/5.26                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.49/5.26               => ! [Xb3: int] :
% 9.49/5.26                    ( ( member_int @ Xb3 @ B2 )
% 9.49/5.26                   => ( X2
% 9.49/5.26                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.49/5.26           => ( ( ord_less_eq_int @ X2 @ T )
% 9.49/5.26             => ( ord_less_eq_int @ ( minus_minus_int @ X2 @ D6 ) @ T ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3960_cpmi,axiom,
% 9.49/5.26      ! [D6: int,P: int > $o,P5: int > $o,B2: set_int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.49/5.26       => ( ? [Z5: int] :
% 9.49/5.26            ! [X4: int] :
% 9.49/5.26              ( ( ord_less_int @ X4 @ Z5 )
% 9.49/5.26             => ( ( P @ X4 )
% 9.49/5.26                = ( P5 @ X4 ) ) )
% 9.49/5.26         => ( ! [X4: int] :
% 9.49/5.26                ( ! [Xa: int] :
% 9.49/5.26                    ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.49/5.26                   => ! [Xb: int] :
% 9.49/5.26                        ( ( member_int @ Xb @ B2 )
% 9.49/5.26                       => ( X4
% 9.49/5.26                         != ( plus_plus_int @ Xb @ Xa ) ) ) )
% 9.49/5.26               => ( ( P @ X4 )
% 9.49/5.26                 => ( P @ ( minus_minus_int @ X4 @ D6 ) ) ) )
% 9.49/5.26           => ( ! [X4: int,K2: int] :
% 9.49/5.26                  ( ( P5 @ X4 )
% 9.49/5.26                  = ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D6 ) ) ) )
% 9.49/5.26             => ( ( ? [X6: int] : ( P @ X6 ) )
% 9.49/5.26                = ( ? [X3: int] :
% 9.49/5.26                      ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.49/5.26                      & ( P5 @ X3 ) )
% 9.49/5.26                  | ? [X3: int] :
% 9.49/5.26                      ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.49/5.26                      & ? [Y2: int] :
% 9.49/5.26                          ( ( member_int @ Y2 @ B2 )
% 9.49/5.26                          & ( P @ ( plus_plus_int @ Y2 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3961_cppi,axiom,
% 9.49/5.26      ! [D6: int,P: int > $o,P5: int > $o,A2: set_int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.49/5.26       => ( ? [Z5: int] :
% 9.49/5.26            ! [X4: int] :
% 9.49/5.26              ( ( ord_less_int @ Z5 @ X4 )
% 9.49/5.26             => ( ( P @ X4 )
% 9.49/5.26                = ( P5 @ X4 ) ) )
% 9.49/5.26         => ( ! [X4: int] :
% 9.49/5.26                ( ! [Xa: int] :
% 9.49/5.26                    ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.49/5.26                   => ! [Xb: int] :
% 9.49/5.26                        ( ( member_int @ Xb @ A2 )
% 9.49/5.26                       => ( X4
% 9.49/5.26                         != ( minus_minus_int @ Xb @ Xa ) ) ) )
% 9.49/5.26               => ( ( P @ X4 )
% 9.49/5.26                 => ( P @ ( plus_plus_int @ X4 @ D6 ) ) ) )
% 9.49/5.26           => ( ! [X4: int,K2: int] :
% 9.49/5.26                  ( ( P5 @ X4 )
% 9.49/5.26                  = ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D6 ) ) ) )
% 9.49/5.26             => ( ( ? [X6: int] : ( P @ X6 ) )
% 9.49/5.26                = ( ? [X3: int] :
% 9.49/5.26                      ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.49/5.26                      & ( P5 @ X3 ) )
% 9.49/5.26                  | ? [X3: int] :
% 9.49/5.26                      ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.49/5.26                      & ? [Y2: int] :
% 9.49/5.26                          ( ( member_int @ Y2 @ A2 )
% 9.49/5.26                          & ( P @ ( minus_minus_int @ Y2 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3962_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 9.49/5.26      ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
% 9.49/5.26        ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) @ X )
% 9.49/5.26        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.26           => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26          & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3963_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 9.49/5.26      ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
% 9.49/5.26        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) @ X )
% 9.49/5.26        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.26           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26          & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3964_vebt__member_Osimps_I5_J,axiom,
% 9.49/5.26      ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 9.49/5.26        ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
% 9.49/5.26        = ( ( X != Mi )
% 9.49/5.26         => ( ( X != Ma )
% 9.49/5.26           => ( ~ ( ord_less_nat @ X @ Mi )
% 9.49/5.26              & ( ~ ( ord_less_nat @ X @ Mi )
% 9.49/5.26               => ( ~ ( ord_less_nat @ Ma @ X )
% 9.49/5.26                  & ( ~ ( ord_less_nat @ Ma @ X )
% 9.49/5.26                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.26                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                      & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3965_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 9.49/5.26      ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
% 9.49/5.26        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc ) @ X )
% 9.49/5.26        = ( ( X = Mi )
% 9.49/5.26          | ( X = Ma )
% 9.49/5.26          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.26             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26            & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3966_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat] :
% 9.49/5.26        ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 9.49/5.26       => ( ! [A3: $o,B3: $o] :
% 9.49/5.26              ( ( X
% 9.49/5.26                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26             => ( ( ( Xa2 = zero_zero_nat )
% 9.49/5.26                 => A3 )
% 9.49/5.26                & ( ( Xa2 != zero_zero_nat )
% 9.49/5.26                 => ( ( ( Xa2 = one_one_nat )
% 9.49/5.26                     => B3 )
% 9.49/5.26                    & ( Xa2 = one_one_nat ) ) ) ) )
% 9.49/5.26         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.49/5.26                ( X
% 9.49/5.26               != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 9.49/5.26           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.49/5.26                  ( ? [S4: vEBT_VEBT] :
% 9.49/5.26                      ( X
% 9.49/5.26                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S4 ) )
% 9.49/5.26                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                     => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3967_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat] :
% 9.49/5.26        ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 9.49/5.26       => ( ! [A3: $o,B3: $o] :
% 9.49/5.26              ( ( X
% 9.49/5.26                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26             => ~ ( ( ( Xa2 = zero_zero_nat )
% 9.49/5.26                   => A3 )
% 9.49/5.26                  & ( ( Xa2 != zero_zero_nat )
% 9.49/5.26                   => ( ( ( Xa2 = one_one_nat )
% 9.49/5.26                       => B3 )
% 9.49/5.26                      & ( Xa2 = one_one_nat ) ) ) ) )
% 9.49/5.26         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.49/5.26                ( ? [S4: vEBT_VEBT] :
% 9.49/5.26                    ( X
% 9.49/5.26                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S4 ) )
% 9.49/5.26               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                     => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3968_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 9.49/5.26        ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 9.49/5.26          = Y )
% 9.49/5.26       => ( ! [A3: $o,B3: $o] :
% 9.49/5.26              ( ( X
% 9.49/5.26                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26             => ( Y
% 9.49/5.26                = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 9.49/5.26                       => A3 )
% 9.49/5.26                      & ( ( Xa2 != zero_zero_nat )
% 9.49/5.26                       => ( ( ( Xa2 = one_one_nat )
% 9.49/5.26                           => B3 )
% 9.49/5.26                          & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 9.49/5.26         => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.49/5.26                  ( X
% 9.49/5.26                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 9.49/5.26             => Y )
% 9.49/5.26           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.49/5.26                  ( ? [S4: vEBT_VEBT] :
% 9.49/5.26                      ( X
% 9.49/5.26                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S4 ) )
% 9.49/5.26                 => ( Y
% 9.49/5.26                    = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                           => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3969_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat] :
% 9.49/5.26        ( ( vEBT_VEBT_membermima @ X @ Xa2 )
% 9.49/5.26       => ( ! [Mi3: nat,Ma3: nat] :
% 9.49/5.26              ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.49/5.26                  ( X
% 9.49/5.26                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.49/5.26             => ~ ( ( Xa2 = Mi3 )
% 9.49/5.26                  | ( Xa2 = Ma3 ) ) )
% 9.49/5.26         => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.49/5.26                ( ? [Vc2: vEBT_VEBT] :
% 9.49/5.26                    ( X
% 9.49/5.26                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 9.49/5.26               => ~ ( ( Xa2 = Mi3 )
% 9.49/5.26                    | ( Xa2 = Ma3 )
% 9.49/5.26                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 9.49/5.26           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.49/5.26                  ( ? [Vd2: vEBT_VEBT] :
% 9.49/5.26                      ( X
% 9.49/5.26                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 9.49/5.26                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3970_vebt__member_Oelims_I2_J,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat] :
% 9.49/5.26        ( ( vEBT_vebt_member @ X @ Xa2 )
% 9.49/5.26       => ( ! [A3: $o,B3: $o] :
% 9.49/5.26              ( ( X
% 9.49/5.26                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26             => ~ ( ( ( Xa2 = zero_zero_nat )
% 9.49/5.26                   => A3 )
% 9.49/5.26                  & ( ( Xa2 != zero_zero_nat )
% 9.49/5.26                   => ( ( ( Xa2 = one_one_nat )
% 9.49/5.26                       => B3 )
% 9.49/5.26                      & ( Xa2 = one_one_nat ) ) ) ) )
% 9.49/5.26         => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 9.49/5.26                ( ? [Summary3: vEBT_VEBT] :
% 9.49/5.26                    ( X
% 9.49/5.26                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 9.49/5.26               => ~ ( ( Xa2 != Mi3 )
% 9.49/5.26                   => ( ( Xa2 != Ma3 )
% 9.49/5.26                     => ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 9.49/5.26                        & ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 9.49/5.26                         => ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 9.49/5.26                            & ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 9.49/5.26                             => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                                 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3971_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat] :
% 9.49/5.26        ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
% 9.49/5.26       => ( ! [Uu2: $o,Uv2: $o] :
% 9.49/5.26              ( X
% 9.49/5.26             != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.49/5.26         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.49/5.26                ( X
% 9.49/5.26               != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 9.49/5.26           => ( ! [Mi3: nat,Ma3: nat] :
% 9.49/5.26                  ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.49/5.26                      ( X
% 9.49/5.26                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.49/5.26                 => ( ( Xa2 = Mi3 )
% 9.49/5.26                    | ( Xa2 = Ma3 ) ) )
% 9.49/5.26             => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.49/5.26                    ( ? [Vc2: vEBT_VEBT] :
% 9.49/5.26                        ( X
% 9.49/5.26                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 9.49/5.26                   => ( ( Xa2 = Mi3 )
% 9.49/5.26                      | ( Xa2 = Ma3 )
% 9.49/5.26                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 9.49/5.26               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.49/5.26                      ( ? [Vd2: vEBT_VEBT] :
% 9.49/5.26                          ( X
% 9.49/5.26                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 9.49/5.26                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3972_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 9.49/5.26        ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
% 9.49/5.26          = Y )
% 9.49/5.26       => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.49/5.26                ( X
% 9.49/5.26                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.49/5.26           => Y )
% 9.49/5.26         => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.49/5.26                  ( X
% 9.49/5.26                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 9.49/5.26             => Y )
% 9.49/5.26           => ( ! [Mi3: nat,Ma3: nat] :
% 9.49/5.26                  ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.49/5.26                      ( X
% 9.49/5.26                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.49/5.26                 => ( Y
% 9.49/5.26                    = ( ~ ( ( Xa2 = Mi3 )
% 9.49/5.26                          | ( Xa2 = Ma3 ) ) ) ) )
% 9.49/5.26             => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.49/5.26                    ( ? [Vc2: vEBT_VEBT] :
% 9.49/5.26                        ( X
% 9.49/5.26                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 9.49/5.26                   => ( Y
% 9.49/5.26                      = ( ~ ( ( Xa2 = Mi3 )
% 9.49/5.26                            | ( Xa2 = Ma3 )
% 9.49/5.26                            | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                               => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
% 9.49/5.26               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.49/5.26                      ( ? [Vd2: vEBT_VEBT] :
% 9.49/5.26                          ( X
% 9.49/5.26                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 9.49/5.26                     => ( Y
% 9.49/5.26                        = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                               => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3973_vebt__insert_Osimps_I5_J,axiom,
% 9.49/5.26      ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 9.49/5.26        ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
% 9.49/5.26        = ( if_VEBT_VEBT
% 9.49/5.26          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.26            & ~ ( ( X = Mi )
% 9.49/5.26                | ( X = Ma ) ) )
% 9.49/5.26          @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
% 9.49/5.26          @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3974_vebt__member_Oelims_I3_J,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat] :
% 9.49/5.26        ( ~ ( vEBT_vebt_member @ X @ Xa2 )
% 9.49/5.26       => ( ! [A3: $o,B3: $o] :
% 9.49/5.26              ( ( X
% 9.49/5.26                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26             => ( ( ( Xa2 = zero_zero_nat )
% 9.49/5.26                 => A3 )
% 9.49/5.26                & ( ( Xa2 != zero_zero_nat )
% 9.49/5.26                 => ( ( ( Xa2 = one_one_nat )
% 9.49/5.26                     => B3 )
% 9.49/5.26                    & ( Xa2 = one_one_nat ) ) ) ) )
% 9.49/5.26         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.49/5.26                ( X
% 9.49/5.26               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.49/5.26           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.49/5.26                  ( X
% 9.49/5.26                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 9.49/5.26             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.49/5.26                    ( X
% 9.49/5.26                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 9.49/5.26               => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 9.49/5.26                      ( ? [Summary3: vEBT_VEBT] :
% 9.49/5.26                          ( X
% 9.49/5.26                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 9.49/5.26                     => ( ( Xa2 != Mi3 )
% 9.49/5.26                       => ( ( Xa2 != Ma3 )
% 9.49/5.26                         => ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 9.49/5.26                            & ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 9.49/5.26                             => ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 9.49/5.26                                & ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 9.49/5.26                                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3975_vebt__member_Oelims_I1_J,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 9.49/5.26        ( ( ( vEBT_vebt_member @ X @ Xa2 )
% 9.49/5.26          = Y )
% 9.49/5.26       => ( ! [A3: $o,B3: $o] :
% 9.49/5.26              ( ( X
% 9.49/5.26                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26             => ( Y
% 9.49/5.26                = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 9.49/5.26                       => A3 )
% 9.49/5.26                      & ( ( Xa2 != zero_zero_nat )
% 9.49/5.26                       => ( ( ( Xa2 = one_one_nat )
% 9.49/5.26                           => B3 )
% 9.49/5.26                          & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 9.49/5.26         => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.49/5.26                  ( X
% 9.49/5.26                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.49/5.26             => Y )
% 9.49/5.26           => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.49/5.26                    ( X
% 9.49/5.26                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 9.49/5.26               => Y )
% 9.49/5.26             => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.49/5.26                      ( X
% 9.49/5.26                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 9.49/5.26                 => Y )
% 9.49/5.26               => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 9.49/5.26                      ( ? [Summary3: vEBT_VEBT] :
% 9.49/5.26                          ( X
% 9.49/5.26                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 9.49/5.26                     => ( Y
% 9.49/5.26                        = ( ~ ( ( Xa2 != Mi3 )
% 9.49/5.26                             => ( ( Xa2 != Ma3 )
% 9.49/5.26                               => ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 9.49/5.26                                  & ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 9.49/5.26                                   => ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 9.49/5.26                                      & ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 9.49/5.26                                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                                           => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3976_vebt__insert_Oelims,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
% 9.49/5.26        ( ( ( vEBT_vebt_insert @ X @ Xa2 )
% 9.49/5.26          = Y )
% 9.49/5.26       => ( ! [A3: $o,B3: $o] :
% 9.49/5.26              ( ( X
% 9.49/5.26                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26             => ~ ( ( ( Xa2 = zero_zero_nat )
% 9.49/5.26                   => ( Y
% 9.49/5.26                      = ( vEBT_Leaf @ $true @ B3 ) ) )
% 9.49/5.26                  & ( ( Xa2 != zero_zero_nat )
% 9.49/5.26                   => ( ( ( Xa2 = one_one_nat )
% 9.49/5.26                       => ( Y
% 9.49/5.26                          = ( vEBT_Leaf @ A3 @ $true ) ) )
% 9.49/5.26                      & ( ( Xa2 != one_one_nat )
% 9.49/5.26                       => ( Y
% 9.49/5.26                          = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) ) )
% 9.49/5.26         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT] :
% 9.49/5.26                ( ( X
% 9.49/5.26                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S4 ) )
% 9.49/5.26               => ( Y
% 9.49/5.26                 != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S4 ) ) )
% 9.49/5.26           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT] :
% 9.49/5.26                  ( ( X
% 9.49/5.26                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S4 ) )
% 9.49/5.26                 => ( Y
% 9.49/5.26                   != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S4 ) ) )
% 9.49/5.26             => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.49/5.26                    ( ( X
% 9.49/5.26                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary3 ) )
% 9.49/5.26                   => ( Y
% 9.49/5.26                     != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary3 ) ) )
% 9.49/5.26               => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.49/5.26                      ( ( X
% 9.49/5.26                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 9.49/5.26                     => ( Y
% 9.49/5.26                       != ( if_VEBT_VEBT
% 9.49/5.26                          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                            & ~ ( ( Xa2 = Mi3 )
% 9.49/5.26                                | ( Xa2 = Ma3 ) ) )
% 9.49/5.26                          @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Xa2 @ Mi3 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ Ma3 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary3 ) )
% 9.49/5.26                          @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3977_vebt__succ_Osimps_I6_J,axiom,
% 9.49/5.26      ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.49/5.26        ( ( ( ord_less_nat @ X @ Mi )
% 9.49/5.26         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
% 9.49/5.26            = ( some_nat @ Mi ) ) )
% 9.49/5.26        & ( ~ ( ord_less_nat @ X @ Mi )
% 9.49/5.26         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
% 9.49/5.26            = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.26              @ ( if_option_nat
% 9.49/5.26                @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                   != none_nat )
% 9.49/5.26                  & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.49/5.26                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                @ ( if_option_nat
% 9.49/5.26                  @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.49/5.26                    = none_nat )
% 9.49/5.26                  @ none_nat
% 9.49/5.26                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.49/5.26              @ none_nat ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3978_vebt__pred_Osimps_I7_J,axiom,
% 9.49/5.26      ! [Ma: nat,X: nat,Mi: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.49/5.26        ( ( ( ord_less_nat @ Ma @ X )
% 9.49/5.26         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
% 9.49/5.26            = ( some_nat @ Ma ) ) )
% 9.49/5.26        & ( ~ ( ord_less_nat @ Ma @ X )
% 9.49/5.26         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
% 9.49/5.26            = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.26              @ ( if_option_nat
% 9.49/5.26                @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                   != none_nat )
% 9.49/5.26                  & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.49/5.26                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                @ ( if_option_nat
% 9.49/5.26                  @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.49/5.26                    = none_nat )
% 9.49/5.26                  @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
% 9.49/5.26                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.49/5.26              @ none_nat ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3979_vebt__delete_Osimps_I7_J,axiom,
% 9.49/5.26      ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.49/5.26        ( ( ( ( ord_less_nat @ X @ Mi )
% 9.49/5.26            | ( ord_less_nat @ Ma @ X ) )
% 9.49/5.26         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
% 9.49/5.26            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
% 9.49/5.26        & ( ~ ( ( ord_less_nat @ X @ Mi )
% 9.49/5.26              | ( ord_less_nat @ Ma @ X ) )
% 9.49/5.26         => ( ( ( ( X = Mi )
% 9.49/5.26                & ( X = Ma ) )
% 9.49/5.26             => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
% 9.49/5.26                = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
% 9.49/5.26            & ( ~ ( ( X = Mi )
% 9.49/5.26                  & ( X = Ma ) )
% 9.49/5.26             => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
% 9.49/5.26                = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.49/5.26                  @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                    @ ( vEBT_Node
% 9.49/5.26                      @ ( some_P7363390416028606310at_nat
% 9.49/5.26                        @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
% 9.49/5.26                          @ ( if_nat
% 9.49/5.26                            @ ( ( ( X = Mi )
% 9.49/5.26                               => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 9.49/5.26                                  = Ma ) )
% 9.49/5.26                              & ( ( X != Mi )
% 9.49/5.26                               => ( X = Ma ) ) )
% 9.49/5.26                            @ ( if_nat
% 9.49/5.26                              @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                = none_nat )
% 9.49/5.26                              @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
% 9.49/5.26                              @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 9.49/5.26                            @ Ma ) ) )
% 9.49/5.26                      @ ( suc @ ( suc @ Va ) )
% 9.49/5.26                      @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                      @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                    @ ( vEBT_Node
% 9.49/5.26                      @ ( some_P7363390416028606310at_nat
% 9.49/5.26                        @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
% 9.49/5.26                          @ ( if_nat
% 9.49/5.26                            @ ( ( ( X = Mi )
% 9.49/5.26                               => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 9.49/5.26                                  = Ma ) )
% 9.49/5.26                              & ( ( X != Mi )
% 9.49/5.26                               => ( X = Ma ) ) )
% 9.49/5.26                            @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.49/5.26                            @ Ma ) ) )
% 9.49/5.26                      @ ( suc @ ( suc @ Va ) )
% 9.49/5.26                      @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                      @ Summary ) )
% 9.49/5.26                  @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3980_vebt__delete_Oelims,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
% 9.49/5.26        ( ( ( vEBT_vebt_delete @ X @ Xa2 )
% 9.49/5.26          = Y )
% 9.49/5.26       => ( ! [A3: $o,B3: $o] :
% 9.49/5.26              ( ( X
% 9.49/5.26                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26             => ( ( Xa2 = zero_zero_nat )
% 9.49/5.26               => ( Y
% 9.49/5.26                 != ( vEBT_Leaf @ $false @ B3 ) ) ) )
% 9.49/5.26         => ( ! [A3: $o] :
% 9.49/5.26                ( ? [B3: $o] :
% 9.49/5.26                    ( X
% 9.49/5.26                    = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26               => ( ( Xa2
% 9.49/5.26                    = ( suc @ zero_zero_nat ) )
% 9.49/5.26                 => ( Y
% 9.49/5.26                   != ( vEBT_Leaf @ A3 @ $false ) ) ) )
% 9.49/5.26           => ( ! [A3: $o,B3: $o] :
% 9.49/5.26                  ( ( X
% 9.49/5.26                    = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26                 => ( ? [N3: nat] :
% 9.49/5.26                        ( Xa2
% 9.49/5.26                        = ( suc @ ( suc @ N3 ) ) )
% 9.49/5.26                   => ( Y
% 9.49/5.26                     != ( vEBT_Leaf @ A3 @ B3 ) ) ) )
% 9.49/5.26             => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.49/5.26                    ( ( X
% 9.49/5.26                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary3 ) )
% 9.49/5.26                   => ( Y
% 9.49/5.26                     != ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary3 ) ) )
% 9.49/5.26               => ( ! [Mi3: nat,Ma3: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
% 9.49/5.26                      ( ( X
% 9.49/5.26                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
% 9.49/5.26                     => ( Y
% 9.49/5.26                       != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) ) )
% 9.49/5.26                 => ( ! [Mi3: nat,Ma3: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
% 9.49/5.26                        ( ( X
% 9.49/5.26                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
% 9.49/5.26                       => ( Y
% 9.49/5.26                         != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) ) )
% 9.49/5.26                   => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.49/5.26                          ( ( X
% 9.49/5.26                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 9.49/5.26                         => ~ ( ( ( ( ord_less_nat @ Xa2 @ Mi3 )
% 9.49/5.26                                  | ( ord_less_nat @ Ma3 @ Xa2 ) )
% 9.49/5.26                               => ( Y
% 9.49/5.26                                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) ) )
% 9.49/5.26                              & ( ~ ( ( ord_less_nat @ Xa2 @ Mi3 )
% 9.49/5.26                                    | ( ord_less_nat @ Ma3 @ Xa2 ) )
% 9.49/5.26                               => ( ( ( ( Xa2 = Mi3 )
% 9.49/5.26                                      & ( Xa2 = Ma3 ) )
% 9.49/5.26                                   => ( Y
% 9.49/5.26                                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) ) )
% 9.49/5.26                                  & ( ~ ( ( Xa2 = Mi3 )
% 9.49/5.26                                        & ( Xa2 = Ma3 ) )
% 9.49/5.26                                   => ( Y
% 9.49/5.26                                      = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                                        @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                          @ ( vEBT_Node
% 9.49/5.26                                            @ ( some_P7363390416028606310at_nat
% 9.49/5.26                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi3 ) @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi3 )
% 9.49/5.26                                                @ ( if_nat
% 9.49/5.26                                                  @ ( ( ( Xa2 = Mi3 )
% 9.49/5.26                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) )
% 9.49/5.26                                                        = Ma3 ) )
% 9.49/5.26                                                    & ( ( Xa2 != Mi3 )
% 9.49/5.26                                                     => ( Xa2 = Ma3 ) ) )
% 9.49/5.26                                                  @ ( if_nat
% 9.49/5.26                                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                                      = none_nat )
% 9.49/5.26                                                    @ ( if_nat @ ( Xa2 = Mi3 ) @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi3 )
% 9.49/5.26                                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 9.49/5.26                                                  @ Ma3 ) ) )
% 9.49/5.26                                            @ ( suc @ ( suc @ Va2 ) )
% 9.49/5.26                                            @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                            @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                          @ ( vEBT_Node
% 9.49/5.26                                            @ ( some_P7363390416028606310at_nat
% 9.49/5.26                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi3 ) @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi3 )
% 9.49/5.26                                                @ ( if_nat
% 9.49/5.26                                                  @ ( ( ( Xa2 = Mi3 )
% 9.49/5.26                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) )
% 9.49/5.26                                                        = Ma3 ) )
% 9.49/5.26                                                    & ( ( Xa2 != Mi3 )
% 9.49/5.26                                                     => ( Xa2 = Ma3 ) ) )
% 9.49/5.26                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.49/5.26                                                  @ Ma3 ) ) )
% 9.49/5.26                                            @ ( suc @ ( suc @ Va2 ) )
% 9.49/5.26                                            @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                            @ Summary3 ) )
% 9.49/5.26                                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3981_vebt__succ_Oelims,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
% 9.49/5.26        ( ( ( vEBT_vebt_succ @ X @ Xa2 )
% 9.49/5.26          = Y )
% 9.49/5.26       => ( ! [Uu2: $o,B3: $o] :
% 9.49/5.26              ( ( X
% 9.49/5.26                = ( vEBT_Leaf @ Uu2 @ B3 ) )
% 9.49/5.26             => ( ( Xa2 = zero_zero_nat )
% 9.49/5.26               => ~ ( ( B3
% 9.49/5.26                     => ( Y
% 9.49/5.26                        = ( some_nat @ one_one_nat ) ) )
% 9.49/5.26                    & ( ~ B3
% 9.49/5.26                     => ( Y = none_nat ) ) ) ) )
% 9.49/5.26         => ( ( ? [Uv2: $o,Uw2: $o] :
% 9.49/5.26                  ( X
% 9.49/5.26                  = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
% 9.49/5.26             => ( ? [N3: nat] :
% 9.49/5.26                    ( Xa2
% 9.49/5.26                    = ( suc @ N3 ) )
% 9.49/5.26               => ( Y != none_nat ) ) )
% 9.49/5.26           => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.49/5.26                    ( X
% 9.49/5.26                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
% 9.49/5.26               => ( Y != none_nat ) )
% 9.49/5.26             => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 9.49/5.26                      ( X
% 9.49/5.26                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 9.49/5.26                 => ( Y != none_nat ) )
% 9.49/5.26               => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
% 9.49/5.26                        ( X
% 9.49/5.26                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
% 9.49/5.26                   => ( Y != none_nat ) )
% 9.49/5.26                 => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.49/5.26                        ( ( X
% 9.49/5.26                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 9.49/5.26                       => ~ ( ( ( ord_less_nat @ Xa2 @ Mi3 )
% 9.49/5.26                             => ( Y
% 9.49/5.26                                = ( some_nat @ Mi3 ) ) )
% 9.49/5.26                            & ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 9.49/5.26                             => ( Y
% 9.49/5.26                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                                  @ ( if_option_nat
% 9.49/5.26                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                       != none_nat )
% 9.49/5.26                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.49/5.26                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                    @ ( if_option_nat
% 9.49/5.26                                      @ ( ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.49/5.26                                        = none_nat )
% 9.49/5.26                                      @ none_nat
% 9.49/5.26                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.49/5.26                                  @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3982_vebt__pred_Oelims,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
% 9.49/5.26        ( ( ( vEBT_vebt_pred @ X @ Xa2 )
% 9.49/5.26          = Y )
% 9.49/5.26       => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.49/5.26                ( X
% 9.49/5.26                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.49/5.26           => ( ( Xa2 = zero_zero_nat )
% 9.49/5.26             => ( Y != none_nat ) ) )
% 9.49/5.26         => ( ! [A3: $o] :
% 9.49/5.26                ( ? [Uw2: $o] :
% 9.49/5.26                    ( X
% 9.49/5.26                    = ( vEBT_Leaf @ A3 @ Uw2 ) )
% 9.49/5.26               => ( ( Xa2
% 9.49/5.26                    = ( suc @ zero_zero_nat ) )
% 9.49/5.26                 => ~ ( ( A3
% 9.49/5.26                       => ( Y
% 9.49/5.26                          = ( some_nat @ zero_zero_nat ) ) )
% 9.49/5.26                      & ( ~ A3
% 9.49/5.26                       => ( Y = none_nat ) ) ) ) )
% 9.49/5.26           => ( ! [A3: $o,B3: $o] :
% 9.49/5.26                  ( ( X
% 9.49/5.26                    = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26                 => ( ? [Va2: nat] :
% 9.49/5.26                        ( Xa2
% 9.49/5.26                        = ( suc @ ( suc @ Va2 ) ) )
% 9.49/5.26                   => ~ ( ( B3
% 9.49/5.26                         => ( Y
% 9.49/5.26                            = ( some_nat @ one_one_nat ) ) )
% 9.49/5.26                        & ( ~ B3
% 9.49/5.26                         => ( ( A3
% 9.49/5.26                             => ( Y
% 9.49/5.26                                = ( some_nat @ zero_zero_nat ) ) )
% 9.49/5.26                            & ( ~ A3
% 9.49/5.26                             => ( Y = none_nat ) ) ) ) ) ) )
% 9.49/5.26             => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
% 9.49/5.26                      ( X
% 9.49/5.26                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
% 9.49/5.26                 => ( Y != none_nat ) )
% 9.49/5.26               => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
% 9.49/5.26                        ( X
% 9.49/5.26                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
% 9.49/5.26                   => ( Y != none_nat ) )
% 9.49/5.26                 => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
% 9.49/5.26                          ( X
% 9.49/5.26                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
% 9.49/5.26                     => ( Y != none_nat ) )
% 9.49/5.26                   => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.49/5.26                          ( ( X
% 9.49/5.26                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 9.49/5.26                         => ~ ( ( ( ord_less_nat @ Ma3 @ Xa2 )
% 9.49/5.26                               => ( Y
% 9.49/5.26                                  = ( some_nat @ Ma3 ) ) )
% 9.49/5.26                              & ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 9.49/5.26                               => ( Y
% 9.49/5.26                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                                    @ ( if_option_nat
% 9.49/5.26                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                         != none_nat )
% 9.49/5.26                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.49/5.26                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                      @ ( if_option_nat
% 9.49/5.26                                        @ ( ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.49/5.26                                          = none_nat )
% 9.49/5.26                                        @ ( if_option_nat @ ( ord_less_nat @ Mi3 @ Xa2 ) @ ( some_nat @ Mi3 ) @ none_nat )
% 9.49/5.26                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.49/5.26                                    @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3983_finite__nth__roots,axiom,
% 9.49/5.26      ! [N: nat,C: complex] :
% 9.49/5.26        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.26       => ( finite3207457112153483333omplex
% 9.49/5.26          @ ( collect_complex
% 9.49/5.26            @ ^ [Z3: complex] :
% 9.49/5.26                ( ( power_power_complex @ Z3 @ N )
% 9.49/5.26                = C ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3984_finite__roots__unity,axiom,
% 9.49/5.26      ! [N: nat] :
% 9.49/5.26        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.49/5.26       => ( finite_finite_real
% 9.49/5.26          @ ( collect_real
% 9.49/5.26            @ ^ [Z3: real] :
% 9.49/5.26                ( ( power_power_real @ Z3 @ N )
% 9.49/5.26                = one_one_real ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3985_finite__roots__unity,axiom,
% 9.49/5.26      ! [N: nat] :
% 9.49/5.26        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.49/5.26       => ( finite3207457112153483333omplex
% 9.49/5.26          @ ( collect_complex
% 9.49/5.26            @ ^ [Z3: complex] :
% 9.49/5.26                ( ( power_power_complex @ Z3 @ N )
% 9.49/5.26                = one_one_complex ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3986_vebt__succ_Opelims,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
% 9.49/5.26        ( ( ( vEBT_vebt_succ @ X @ Xa2 )
% 9.49/5.26          = Y )
% 9.49/5.26       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 9.49/5.26         => ( ! [Uu2: $o,B3: $o] :
% 9.49/5.26                ( ( X
% 9.49/5.26                  = ( vEBT_Leaf @ Uu2 @ B3 ) )
% 9.49/5.26               => ( ( Xa2 = zero_zero_nat )
% 9.49/5.26                 => ( ( ( B3
% 9.49/5.26                       => ( Y
% 9.49/5.26                          = ( some_nat @ one_one_nat ) ) )
% 9.49/5.26                      & ( ~ B3
% 9.49/5.26                       => ( Y = none_nat ) ) )
% 9.49/5.26                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat ) ) ) ) )
% 9.49/5.26           => ( ! [Uv2: $o,Uw2: $o] :
% 9.49/5.26                  ( ( X
% 9.49/5.26                    = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
% 9.49/5.26                 => ! [N3: nat] :
% 9.49/5.26                      ( ( Xa2
% 9.49/5.26                        = ( suc @ N3 ) )
% 9.49/5.26                     => ( ( Y = none_nat )
% 9.49/5.26                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) ) ) ) ) )
% 9.49/5.26             => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.49/5.26                    ( ( X
% 9.49/5.26                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
% 9.49/5.26                   => ( ( Y = none_nat )
% 9.49/5.26                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 9.49/5.26               => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 9.49/5.26                      ( ( X
% 9.49/5.26                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 9.49/5.26                     => ( ( Y = none_nat )
% 9.49/5.26                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa2 ) ) ) )
% 9.49/5.26                 => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
% 9.49/5.26                        ( ( X
% 9.49/5.26                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
% 9.49/5.26                       => ( ( Y = none_nat )
% 9.49/5.26                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa2 ) ) ) )
% 9.49/5.26                   => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.49/5.26                          ( ( X
% 9.49/5.26                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 9.49/5.26                         => ( ( ( ( ord_less_nat @ Xa2 @ Mi3 )
% 9.49/5.26                               => ( Y
% 9.49/5.26                                  = ( some_nat @ Mi3 ) ) )
% 9.49/5.26                              & ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 9.49/5.26                               => ( Y
% 9.49/5.26                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                                    @ ( if_option_nat
% 9.49/5.26                                      @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                         != none_nat )
% 9.49/5.26                                        & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.49/5.26                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                      @ ( if_option_nat
% 9.49/5.26                                        @ ( ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.49/5.26                                          = none_nat )
% 9.49/5.26                                        @ none_nat
% 9.49/5.26                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.49/5.26                                    @ none_nat ) ) ) )
% 9.49/5.26                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3987_vebt__pred_Opelims,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
% 9.49/5.26        ( ( ( vEBT_vebt_pred @ X @ Xa2 )
% 9.49/5.26          = Y )
% 9.49/5.26       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 9.49/5.26         => ( ! [Uu2: $o,Uv2: $o] :
% 9.49/5.26                ( ( X
% 9.49/5.26                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.49/5.26               => ( ( Xa2 = zero_zero_nat )
% 9.49/5.26                 => ( ( Y = none_nat )
% 9.49/5.26                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
% 9.49/5.26           => ( ! [A3: $o,Uw2: $o] :
% 9.49/5.26                  ( ( X
% 9.49/5.26                    = ( vEBT_Leaf @ A3 @ Uw2 ) )
% 9.49/5.26                 => ( ( Xa2
% 9.49/5.26                      = ( suc @ zero_zero_nat ) )
% 9.49/5.26                   => ( ( ( A3
% 9.49/5.26                         => ( Y
% 9.49/5.26                            = ( some_nat @ zero_zero_nat ) ) )
% 9.49/5.26                        & ( ~ A3
% 9.49/5.26                         => ( Y = none_nat ) ) )
% 9.49/5.26                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
% 9.49/5.26             => ( ! [A3: $o,B3: $o] :
% 9.49/5.26                    ( ( X
% 9.49/5.26                      = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26                   => ! [Va2: nat] :
% 9.49/5.26                        ( ( Xa2
% 9.49/5.26                          = ( suc @ ( suc @ Va2 ) ) )
% 9.49/5.26                       => ( ( ( B3
% 9.49/5.26                             => ( Y
% 9.49/5.26                                = ( some_nat @ one_one_nat ) ) )
% 9.49/5.26                            & ( ~ B3
% 9.49/5.26                             => ( ( A3
% 9.49/5.26                                 => ( Y
% 9.49/5.26                                    = ( some_nat @ zero_zero_nat ) ) )
% 9.49/5.26                                & ( ~ A3
% 9.49/5.26                                 => ( Y = none_nat ) ) ) ) )
% 9.49/5.26                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) )
% 9.49/5.26               => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
% 9.49/5.26                      ( ( X
% 9.49/5.26                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
% 9.49/5.26                     => ( ( Y = none_nat )
% 9.49/5.26                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
% 9.49/5.26                 => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
% 9.49/5.26                        ( ( X
% 9.49/5.26                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
% 9.49/5.26                       => ( ( Y = none_nat )
% 9.49/5.26                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa2 ) ) ) )
% 9.49/5.26                   => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
% 9.49/5.26                          ( ( X
% 9.49/5.26                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
% 9.49/5.26                         => ( ( Y = none_nat )
% 9.49/5.26                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa2 ) ) ) )
% 9.49/5.26                     => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.49/5.26                            ( ( X
% 9.49/5.26                              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 9.49/5.26                           => ( ( ( ( ord_less_nat @ Ma3 @ Xa2 )
% 9.49/5.26                                 => ( Y
% 9.49/5.26                                    = ( some_nat @ Ma3 ) ) )
% 9.49/5.26                                & ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 9.49/5.26                                 => ( Y
% 9.49/5.26                                    = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                                      @ ( if_option_nat
% 9.49/5.26                                        @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                           != none_nat )
% 9.49/5.26                                          & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.49/5.26                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                        @ ( if_option_nat
% 9.49/5.26                                          @ ( ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.49/5.26                                            = none_nat )
% 9.49/5.26                                          @ ( if_option_nat @ ( ord_less_nat @ Mi3 @ Xa2 ) @ ( some_nat @ Mi3 ) @ none_nat )
% 9.49/5.26                                          @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.49/5.26                                      @ none_nat ) ) ) )
% 9.49/5.26                             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3988_vebt__delete_Opelims,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
% 9.49/5.26        ( ( ( vEBT_vebt_delete @ X @ Xa2 )
% 9.49/5.26          = Y )
% 9.49/5.26       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 9.49/5.26         => ( ! [A3: $o,B3: $o] :
% 9.49/5.26                ( ( X
% 9.49/5.26                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26               => ( ( Xa2 = zero_zero_nat )
% 9.49/5.26                 => ( ( Y
% 9.49/5.26                      = ( vEBT_Leaf @ $false @ B3 ) )
% 9.49/5.26                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat ) ) ) ) )
% 9.49/5.26           => ( ! [A3: $o,B3: $o] :
% 9.49/5.26                  ( ( X
% 9.49/5.26                    = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26                 => ( ( Xa2
% 9.49/5.26                      = ( suc @ zero_zero_nat ) )
% 9.49/5.26                   => ( ( Y
% 9.49/5.26                        = ( vEBT_Leaf @ A3 @ $false ) )
% 9.49/5.26                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
% 9.49/5.26             => ( ! [A3: $o,B3: $o] :
% 9.49/5.26                    ( ( X
% 9.49/5.26                      = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26                   => ! [N3: nat] :
% 9.49/5.26                        ( ( Xa2
% 9.49/5.26                          = ( suc @ ( suc @ N3 ) ) )
% 9.49/5.26                       => ( ( Y
% 9.49/5.26                            = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N3 ) ) ) ) ) ) )
% 9.49/5.26               => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.49/5.26                      ( ( X
% 9.49/5.26                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary3 ) )
% 9.49/5.26                     => ( ( Y
% 9.49/5.26                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary3 ) )
% 9.49/5.26                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary3 ) @ Xa2 ) ) ) )
% 9.49/5.26                 => ( ! [Mi3: nat,Ma3: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
% 9.49/5.26                        ( ( X
% 9.49/5.26                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
% 9.49/5.26                       => ( ( Y
% 9.49/5.26                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
% 9.49/5.26                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ Xa2 ) ) ) )
% 9.49/5.26                   => ( ! [Mi3: nat,Ma3: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
% 9.49/5.26                          ( ( X
% 9.49/5.26                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
% 9.49/5.26                         => ( ( Y
% 9.49/5.26                              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
% 9.49/5.26                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ Xa2 ) ) ) )
% 9.49/5.26                     => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.49/5.26                            ( ( X
% 9.49/5.26                              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 9.49/5.26                           => ( ( ( ( ( ord_less_nat @ Xa2 @ Mi3 )
% 9.49/5.26                                    | ( ord_less_nat @ Ma3 @ Xa2 ) )
% 9.49/5.26                                 => ( Y
% 9.49/5.26                                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) ) )
% 9.49/5.26                                & ( ~ ( ( ord_less_nat @ Xa2 @ Mi3 )
% 9.49/5.26                                      | ( ord_less_nat @ Ma3 @ Xa2 ) )
% 9.49/5.26                                 => ( ( ( ( Xa2 = Mi3 )
% 9.49/5.26                                        & ( Xa2 = Ma3 ) )
% 9.49/5.26                                     => ( Y
% 9.49/5.26                                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) ) )
% 9.49/5.26                                    & ( ~ ( ( Xa2 = Mi3 )
% 9.49/5.26                                          & ( Xa2 = Ma3 ) )
% 9.49/5.26                                     => ( Y
% 9.49/5.26                                        = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                                          @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                            @ ( vEBT_Node
% 9.49/5.26                                              @ ( some_P7363390416028606310at_nat
% 9.49/5.26                                                @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi3 ) @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi3 )
% 9.49/5.26                                                  @ ( if_nat
% 9.49/5.26                                                    @ ( ( ( Xa2 = Mi3 )
% 9.49/5.26                                                       => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) )
% 9.49/5.26                                                          = Ma3 ) )
% 9.49/5.26                                                      & ( ( Xa2 != Mi3 )
% 9.49/5.26                                                       => ( Xa2 = Ma3 ) ) )
% 9.49/5.26                                                    @ ( if_nat
% 9.49/5.26                                                      @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                                        = none_nat )
% 9.49/5.26                                                      @ ( if_nat @ ( Xa2 = Mi3 ) @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi3 )
% 9.49/5.26                                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 9.49/5.26                                                    @ Ma3 ) ) )
% 9.49/5.26                                              @ ( suc @ ( suc @ Va2 ) )
% 9.49/5.26                                              @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                              @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                            @ ( vEBT_Node
% 9.49/5.26                                              @ ( some_P7363390416028606310at_nat
% 9.49/5.26                                                @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi3 ) @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi3 )
% 9.49/5.26                                                  @ ( if_nat
% 9.49/5.26                                                    @ ( ( ( Xa2 = Mi3 )
% 9.49/5.26                                                       => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) )
% 9.49/5.26                                                          = Ma3 ) )
% 9.49/5.26                                                      & ( ( Xa2 != Mi3 )
% 9.49/5.26                                                       => ( Xa2 = Ma3 ) ) )
% 9.49/5.26                                                    @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.49/5.26                                                    @ Ma3 ) ) )
% 9.49/5.26                                              @ ( suc @ ( suc @ Va2 ) )
% 9.49/5.26                                              @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat2 @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat2 @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                              @ Summary3 ) )
% 9.49/5.26                                          @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) ) ) ) ) ) )
% 9.49/5.26                             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3989_vebt__insert_Opelims,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
% 9.49/5.26        ( ( ( vEBT_vebt_insert @ X @ Xa2 )
% 9.49/5.26          = Y )
% 9.49/5.26       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 9.49/5.26         => ( ! [A3: $o,B3: $o] :
% 9.49/5.26                ( ( X
% 9.49/5.26                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26               => ( ( ( ( Xa2 = zero_zero_nat )
% 9.49/5.26                     => ( Y
% 9.49/5.26                        = ( vEBT_Leaf @ $true @ B3 ) ) )
% 9.49/5.26                    & ( ( Xa2 != zero_zero_nat )
% 9.49/5.26                     => ( ( ( Xa2 = one_one_nat )
% 9.49/5.26                         => ( Y
% 9.49/5.26                            = ( vEBT_Leaf @ A3 @ $true ) ) )
% 9.49/5.26                        & ( ( Xa2 != one_one_nat )
% 9.49/5.26                         => ( Y
% 9.49/5.26                            = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) )
% 9.49/5.26                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
% 9.49/5.26           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT] :
% 9.49/5.26                  ( ( X
% 9.49/5.26                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S4 ) )
% 9.49/5.26                 => ( ( Y
% 9.49/5.26                      = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S4 ) )
% 9.49/5.26                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S4 ) @ Xa2 ) ) ) )
% 9.49/5.26             => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S4: vEBT_VEBT] :
% 9.49/5.26                    ( ( X
% 9.49/5.26                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S4 ) )
% 9.49/5.26                   => ( ( Y
% 9.49/5.26                        = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S4 ) )
% 9.49/5.26                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S4 ) @ Xa2 ) ) ) )
% 9.49/5.26               => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.49/5.26                      ( ( X
% 9.49/5.26                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary3 ) )
% 9.49/5.26                     => ( ( Y
% 9.49/5.26                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary3 ) )
% 9.49/5.26                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary3 ) @ Xa2 ) ) ) )
% 9.49/5.26                 => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.49/5.26                        ( ( X
% 9.49/5.26                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 9.49/5.26                       => ( ( Y
% 9.49/5.26                            = ( if_VEBT_VEBT
% 9.49/5.26                              @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                                & ~ ( ( Xa2 = Mi3 )
% 9.49/5.26                                    | ( Xa2 = Ma3 ) ) )
% 9.49/5.26                              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Xa2 @ Mi3 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ Ma3 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi3 ) @ Mi3 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary3 ) )
% 9.49/5.26                              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) ) )
% 9.49/5.26                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3990_finite__atLeastAtMost__int,axiom,
% 9.49/5.26      ! [L: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L @ U ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3991_bot__enat__def,axiom,
% 9.49/5.26      bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3992_vebt__member_Opelims_I1_J,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 9.49/5.26        ( ( ( vEBT_vebt_member @ X @ Xa2 )
% 9.49/5.26          = Y )
% 9.49/5.26       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 9.49/5.26         => ( ! [A3: $o,B3: $o] :
% 9.49/5.26                ( ( X
% 9.49/5.26                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26               => ( ( Y
% 9.49/5.26                    = ( ( ( Xa2 = zero_zero_nat )
% 9.49/5.26                       => A3 )
% 9.49/5.26                      & ( ( Xa2 != zero_zero_nat )
% 9.49/5.26                       => ( ( ( Xa2 = one_one_nat )
% 9.49/5.26                           => B3 )
% 9.49/5.26                          & ( Xa2 = one_one_nat ) ) ) ) )
% 9.49/5.26                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
% 9.49/5.26           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.49/5.26                  ( ( X
% 9.49/5.26                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.49/5.26                 => ( ~ Y
% 9.49/5.26                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 9.49/5.26             => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.49/5.26                    ( ( X
% 9.49/5.26                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 9.49/5.26                   => ( ~ Y
% 9.49/5.26                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 9.49/5.26               => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.49/5.26                      ( ( X
% 9.49/5.26                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 9.49/5.26                     => ( ~ Y
% 9.49/5.26                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
% 9.49/5.26                 => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.49/5.26                        ( ( X
% 9.49/5.26                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 9.49/5.26                       => ( ( Y
% 9.49/5.26                            = ( ( Xa2 != Mi3 )
% 9.49/5.26                             => ( ( Xa2 != Ma3 )
% 9.49/5.26                               => ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 9.49/5.26                                  & ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 9.49/5.26                                   => ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 9.49/5.26                                      & ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 9.49/5.26                                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                                           => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) )
% 9.49/5.26                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3993_vebt__member_Opelims_I3_J,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat] :
% 9.49/5.26        ( ~ ( vEBT_vebt_member @ X @ Xa2 )
% 9.49/5.26       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 9.49/5.26         => ( ! [A3: $o,B3: $o] :
% 9.49/5.26                ( ( X
% 9.49/5.26                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
% 9.49/5.26                 => ( ( ( Xa2 = zero_zero_nat )
% 9.49/5.26                     => A3 )
% 9.49/5.26                    & ( ( Xa2 != zero_zero_nat )
% 9.49/5.26                     => ( ( ( Xa2 = one_one_nat )
% 9.49/5.26                         => B3 )
% 9.49/5.26                        & ( Xa2 = one_one_nat ) ) ) ) ) )
% 9.49/5.26           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.49/5.26                  ( ( X
% 9.49/5.26                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.49/5.26                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 9.49/5.26             => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.49/5.26                    ( ( X
% 9.49/5.26                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 9.49/5.26                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
% 9.49/5.26               => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.49/5.26                      ( ( X
% 9.49/5.26                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 9.49/5.26                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
% 9.49/5.26                 => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.49/5.26                        ( ( X
% 9.49/5.26                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 9.49/5.26                       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ Xa2 ) )
% 9.49/5.26                         => ( ( Xa2 != Mi3 )
% 9.49/5.26                           => ( ( Xa2 != Ma3 )
% 9.49/5.26                             => ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 9.49/5.26                                & ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 9.49/5.26                                 => ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 9.49/5.26                                    & ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 9.49/5.26                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3994_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat] :
% 9.49/5.26        ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 9.49/5.26       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 9.49/5.26         => ( ! [A3: $o,B3: $o] :
% 9.49/5.26                ( ( X
% 9.49/5.26                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
% 9.49/5.26                 => ( ( ( Xa2 = zero_zero_nat )
% 9.49/5.26                     => A3 )
% 9.49/5.26                    & ( ( Xa2 != zero_zero_nat )
% 9.49/5.26                     => ( ( ( Xa2 = one_one_nat )
% 9.49/5.26                         => B3 )
% 9.49/5.26                        & ( Xa2 = one_one_nat ) ) ) ) ) )
% 9.49/5.26           => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.49/5.26                  ( ( X
% 9.49/5.26                    = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 9.49/5.26                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 9.49/5.26             => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S4: vEBT_VEBT] :
% 9.49/5.26                    ( ( X
% 9.49/5.26                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S4 ) )
% 9.49/5.26                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S4 ) @ Xa2 ) )
% 9.49/5.26                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3995_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat] :
% 9.49/5.26        ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 9.49/5.26       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 9.49/5.26         => ( ! [A3: $o,B3: $o] :
% 9.49/5.26                ( ( X
% 9.49/5.26                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
% 9.49/5.26                 => ~ ( ( ( Xa2 = zero_zero_nat )
% 9.49/5.26                       => A3 )
% 9.49/5.26                      & ( ( Xa2 != zero_zero_nat )
% 9.49/5.26                       => ( ( ( Xa2 = one_one_nat )
% 9.49/5.26                           => B3 )
% 9.49/5.26                          & ( Xa2 = one_one_nat ) ) ) ) ) )
% 9.49/5.26           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S4: vEBT_VEBT] :
% 9.49/5.26                  ( ( X
% 9.49/5.26                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S4 ) )
% 9.49/5.26                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S4 ) @ Xa2 ) )
% 9.49/5.26                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3996_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 9.49/5.26        ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 9.49/5.26          = Y )
% 9.49/5.26       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 9.49/5.26         => ( ! [A3: $o,B3: $o] :
% 9.49/5.26                ( ( X
% 9.49/5.26                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26               => ( ( Y
% 9.49/5.26                    = ( ( ( Xa2 = zero_zero_nat )
% 9.49/5.26                       => A3 )
% 9.49/5.26                      & ( ( Xa2 != zero_zero_nat )
% 9.49/5.26                       => ( ( ( Xa2 = one_one_nat )
% 9.49/5.26                           => B3 )
% 9.49/5.26                          & ( Xa2 = one_one_nat ) ) ) ) )
% 9.49/5.26                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
% 9.49/5.26           => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.49/5.26                  ( ( X
% 9.49/5.26                    = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 9.49/5.26                 => ( ~ Y
% 9.49/5.26                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 9.49/5.26             => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S4: vEBT_VEBT] :
% 9.49/5.26                    ( ( X
% 9.49/5.26                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S4 ) )
% 9.49/5.26                   => ( ( Y
% 9.49/5.26                        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                           => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 9.49/5.26                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S4 ) @ Xa2 ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3997_vebt__member_Opelims_I2_J,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat] :
% 9.49/5.26        ( ( vEBT_vebt_member @ X @ Xa2 )
% 9.49/5.26       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 9.49/5.26         => ( ! [A3: $o,B3: $o] :
% 9.49/5.26                ( ( X
% 9.49/5.26                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.49/5.26               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
% 9.49/5.26                 => ~ ( ( ( Xa2 = zero_zero_nat )
% 9.49/5.26                       => A3 )
% 9.49/5.26                      & ( ( Xa2 != zero_zero_nat )
% 9.49/5.26                       => ( ( ( Xa2 = one_one_nat )
% 9.49/5.26                           => B3 )
% 9.49/5.26                          & ( Xa2 = one_one_nat ) ) ) ) ) )
% 9.49/5.26           => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.49/5.26                  ( ( X
% 9.49/5.26                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) )
% 9.49/5.26                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary3 ) @ Xa2 ) )
% 9.49/5.26                   => ~ ( ( Xa2 != Mi3 )
% 9.49/5.26                       => ( ( Xa2 != Ma3 )
% 9.49/5.26                         => ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 9.49/5.26                            & ( ~ ( ord_less_nat @ Xa2 @ Mi3 )
% 9.49/5.26                             => ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 9.49/5.26                                & ( ~ ( ord_less_nat @ Ma3 @ Xa2 )
% 9.49/5.26                                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3998_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 9.49/5.26        ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
% 9.49/5.26          = Y )
% 9.49/5.26       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 9.49/5.26         => ( ! [Uu2: $o,Uv2: $o] :
% 9.49/5.26                ( ( X
% 9.49/5.26                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.49/5.26               => ( ~ Y
% 9.49/5.26                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 9.49/5.26           => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.49/5.26                  ( ( X
% 9.49/5.26                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 9.49/5.26                 => ( ~ Y
% 9.49/5.26                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
% 9.49/5.26             => ( ! [Mi3: nat,Ma3: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.49/5.26                    ( ( X
% 9.49/5.26                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.49/5.26                   => ( ( Y
% 9.49/5.26                        = ( ( Xa2 = Mi3 )
% 9.49/5.26                          | ( Xa2 = Ma3 ) ) )
% 9.49/5.26                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
% 9.49/5.26               => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.49/5.26                      ( ( X
% 9.49/5.26                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 9.49/5.26                     => ( ( Y
% 9.49/5.26                          = ( ( Xa2 = Mi3 )
% 9.49/5.26                            | ( Xa2 = Ma3 )
% 9.49/5.26                            | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                               => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 9.49/5.26                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) ) )
% 9.49/5.26                 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 9.49/5.26                        ( ( X
% 9.49/5.26                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 9.49/5.26                       => ( ( Y
% 9.49/5.26                            = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                               => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 9.49/5.26                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_3999_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat] :
% 9.49/5.26        ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
% 9.49/5.26       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 9.49/5.26         => ( ! [Uu2: $o,Uv2: $o] :
% 9.49/5.26                ( ( X
% 9.49/5.26                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.49/5.26               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
% 9.49/5.26           => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.49/5.26                  ( ( X
% 9.49/5.26                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 9.49/5.26                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
% 9.49/5.26             => ( ! [Mi3: nat,Ma3: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.49/5.26                    ( ( X
% 9.49/5.26                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.49/5.26                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
% 9.49/5.26                     => ( ( Xa2 = Mi3 )
% 9.49/5.26                        | ( Xa2 = Ma3 ) ) ) )
% 9.49/5.26               => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.49/5.26                      ( ( X
% 9.49/5.26                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 9.49/5.26                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
% 9.49/5.26                       => ( ( Xa2 = Mi3 )
% 9.49/5.26                          | ( Xa2 = Ma3 )
% 9.49/5.26                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 9.49/5.26                 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 9.49/5.26                        ( ( X
% 9.49/5.26                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 9.49/5.26                       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
% 9.49/5.26                         => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4000_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 9.49/5.26      ! [X: vEBT_VEBT,Xa2: nat] :
% 9.49/5.26        ( ( vEBT_VEBT_membermima @ X @ Xa2 )
% 9.49/5.26       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 9.49/5.26         => ( ! [Mi3: nat,Ma3: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.49/5.26                ( ( X
% 9.49/5.26                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.49/5.26               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
% 9.49/5.26                 => ~ ( ( Xa2 = Mi3 )
% 9.49/5.26                      | ( Xa2 = Ma3 ) ) ) )
% 9.49/5.26           => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.49/5.26                  ( ( X
% 9.49/5.26                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 9.49/5.26                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
% 9.49/5.26                   => ~ ( ( Xa2 = Mi3 )
% 9.49/5.26                        | ( Xa2 = Ma3 )
% 9.49/5.26                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 9.49/5.26             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 9.49/5.26                    ( ( X
% 9.49/5.26                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 9.49/5.26                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
% 9.49/5.26                     => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.49/5.26                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.49/5.26                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4001_arcosh__1,axiom,
% 9.49/5.26      ( ( arcosh_real @ one_one_real )
% 9.49/5.26      = zero_zero_real ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4002_mult__le__cancel__iff1,axiom,
% 9.49/5.26      ! [Z2: real,X: real,Y: real] :
% 9.49/5.26        ( ( ord_less_real @ zero_zero_real @ Z2 )
% 9.49/5.26       => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
% 9.49/5.26          = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4003_mult__le__cancel__iff1,axiom,
% 9.49/5.26      ! [Z2: rat,X: rat,Y: rat] :
% 9.49/5.26        ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 9.49/5.26       => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y @ Z2 ) )
% 9.49/5.26          = ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4004_mult__le__cancel__iff1,axiom,
% 9.49/5.26      ! [Z2: int,X: int,Y: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 9.49/5.26       => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
% 9.49/5.26          = ( ord_less_eq_int @ X @ Y ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4005_mult__le__cancel__iff2,axiom,
% 9.49/5.26      ! [Z2: real,X: real,Y: real] :
% 9.49/5.26        ( ( ord_less_real @ zero_zero_real @ Z2 )
% 9.49/5.26       => ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ ( times_times_real @ Z2 @ Y ) )
% 9.49/5.26          = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4006_mult__le__cancel__iff2,axiom,
% 9.49/5.26      ! [Z2: rat,X: rat,Y: rat] :
% 9.49/5.26        ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 9.49/5.26       => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X ) @ ( times_times_rat @ Z2 @ Y ) )
% 9.49/5.26          = ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4007_mult__le__cancel__iff2,axiom,
% 9.49/5.26      ! [Z2: int,X: int,Y: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 9.49/5.26       => ( ( ord_less_eq_int @ ( times_times_int @ Z2 @ X ) @ ( times_times_int @ Z2 @ Y ) )
% 9.49/5.26          = ( ord_less_eq_int @ X @ Y ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4008_divides__aux__eq,axiom,
% 9.49/5.26      ! [Q4: nat,R2: nat] :
% 9.49/5.26        ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
% 9.49/5.26        = ( R2 = zero_zero_nat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4009_divides__aux__eq,axiom,
% 9.49/5.26      ! [Q4: int,R2: int] :
% 9.49/5.26        ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.49/5.26        = ( R2 = zero_zero_int ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4010_arsinh__0,axiom,
% 9.49/5.26      ( ( arsinh_real @ zero_zero_real )
% 9.49/5.26      = zero_zero_real ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4011_artanh__0,axiom,
% 9.49/5.26      ( ( artanh_real @ zero_zero_real )
% 9.49/5.26      = zero_zero_real ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4012_neg__eucl__rel__int__mult__2,axiom,
% 9.49/5.26      ! [B: int,A: int,Q4: int,R2: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ B @ zero_zero_int )
% 9.49/5.26       => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.49/5.26         => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4013_low__def,axiom,
% 9.49/5.26      ( vEBT_VEBT_low
% 9.49/5.26      = ( ^ [X3: nat,N2: nat] : ( modulo_modulo_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4014_mod__self,axiom,
% 9.49/5.26      ! [A: nat] :
% 9.49/5.26        ( ( modulo_modulo_nat @ A @ A )
% 9.49/5.26        = zero_zero_nat ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4015_mod__self,axiom,
% 9.49/5.26      ! [A: int] :
% 9.49/5.26        ( ( modulo_modulo_int @ A @ A )
% 9.49/5.26        = zero_zero_int ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4016_mod__self,axiom,
% 9.49/5.26      ! [A: code_integer] :
% 9.49/5.26        ( ( modulo364778990260209775nteger @ A @ A )
% 9.49/5.26        = zero_z3403309356797280102nteger ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4017_mod__by__0,axiom,
% 9.49/5.26      ! [A: nat] :
% 9.49/5.26        ( ( modulo_modulo_nat @ A @ zero_zero_nat )
% 9.49/5.26        = A ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4018_mod__by__0,axiom,
% 9.49/5.26      ! [A: int] :
% 9.49/5.26        ( ( modulo_modulo_int @ A @ zero_zero_int )
% 9.49/5.26        = A ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4019_mod__by__0,axiom,
% 9.49/5.26      ! [A: code_integer] :
% 9.49/5.26        ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
% 9.49/5.26        = A ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4020_mod__0,axiom,
% 9.49/5.26      ! [A: nat] :
% 9.49/5.26        ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 9.49/5.26        = zero_zero_nat ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4021_mod__0,axiom,
% 9.49/5.26      ! [A: int] :
% 9.49/5.26        ( ( modulo_modulo_int @ zero_zero_int @ A )
% 9.49/5.26        = zero_zero_int ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4022_mod__0,axiom,
% 9.49/5.26      ! [A: code_integer] :
% 9.49/5.26        ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 9.49/5.26        = zero_z3403309356797280102nteger ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4023_bits__mod__0,axiom,
% 9.49/5.26      ! [A: nat] :
% 9.49/5.26        ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 9.49/5.26        = zero_zero_nat ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4024_bits__mod__0,axiom,
% 9.49/5.26      ! [A: int] :
% 9.49/5.26        ( ( modulo_modulo_int @ zero_zero_int @ A )
% 9.49/5.26        = zero_zero_int ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4025_bits__mod__0,axiom,
% 9.49/5.26      ! [A: code_integer] :
% 9.49/5.26        ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 9.49/5.26        = zero_z3403309356797280102nteger ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4026_mod__less,axiom,
% 9.49/5.26      ! [M: nat,N: nat] :
% 9.49/5.26        ( ( ord_less_nat @ M @ N )
% 9.49/5.26       => ( ( modulo_modulo_nat @ M @ N )
% 9.49/5.26          = M ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4027_mod__mult__self2__is__0,axiom,
% 9.49/5.26      ! [A: nat,B: nat] :
% 9.49/5.26        ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
% 9.49/5.26        = zero_zero_nat ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4028_mod__mult__self2__is__0,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
% 9.49/5.26        = zero_zero_int ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4029_mod__mult__self2__is__0,axiom,
% 9.49/5.26      ! [A: code_integer,B: code_integer] :
% 9.49/5.26        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
% 9.49/5.26        = zero_z3403309356797280102nteger ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4030_mod__mult__self1__is__0,axiom,
% 9.49/5.26      ! [B: nat,A: nat] :
% 9.49/5.26        ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
% 9.49/5.26        = zero_zero_nat ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4031_mod__mult__self1__is__0,axiom,
% 9.49/5.26      ! [B: int,A: int] :
% 9.49/5.26        ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
% 9.49/5.26        = zero_zero_int ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4032_mod__mult__self1__is__0,axiom,
% 9.49/5.26      ! [B: code_integer,A: code_integer] :
% 9.49/5.26        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
% 9.49/5.26        = zero_z3403309356797280102nteger ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4033_mod__by__1,axiom,
% 9.49/5.26      ! [A: nat] :
% 9.49/5.26        ( ( modulo_modulo_nat @ A @ one_one_nat )
% 9.49/5.26        = zero_zero_nat ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4034_mod__by__1,axiom,
% 9.49/5.26      ! [A: int] :
% 9.49/5.26        ( ( modulo_modulo_int @ A @ one_one_int )
% 9.49/5.26        = zero_zero_int ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4035_mod__by__1,axiom,
% 9.49/5.26      ! [A: code_integer] :
% 9.49/5.26        ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 9.49/5.26        = zero_z3403309356797280102nteger ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4036_bits__mod__by__1,axiom,
% 9.49/5.26      ! [A: nat] :
% 9.49/5.26        ( ( modulo_modulo_nat @ A @ one_one_nat )
% 9.49/5.26        = zero_zero_nat ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4037_bits__mod__by__1,axiom,
% 9.49/5.26      ! [A: int] :
% 9.49/5.26        ( ( modulo_modulo_int @ A @ one_one_int )
% 9.49/5.26        = zero_zero_int ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4038_bits__mod__by__1,axiom,
% 9.49/5.26      ! [A: code_integer] :
% 9.49/5.26        ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 9.49/5.26        = zero_z3403309356797280102nteger ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4039_mod__div__trivial,axiom,
% 9.49/5.26      ! [A: nat,B: nat] :
% 9.49/5.26        ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 9.49/5.26        = zero_zero_nat ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4040_mod__div__trivial,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 9.49/5.26        = zero_zero_int ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4041_mod__div__trivial,axiom,
% 9.49/5.26      ! [A: code_integer,B: code_integer] :
% 9.49/5.26        ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 9.49/5.26        = zero_z3403309356797280102nteger ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4042_bits__mod__div__trivial,axiom,
% 9.49/5.26      ! [A: nat,B: nat] :
% 9.49/5.26        ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 9.49/5.26        = zero_zero_nat ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4043_bits__mod__div__trivial,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 9.49/5.26        = zero_zero_int ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4044_bits__mod__div__trivial,axiom,
% 9.49/5.26      ! [A: code_integer,B: code_integer] :
% 9.49/5.26        ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 9.49/5.26        = zero_z3403309356797280102nteger ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4045_mod__by__Suc__0,axiom,
% 9.49/5.26      ! [M: nat] :
% 9.49/5.26        ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 9.49/5.26        = zero_zero_nat ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4046_not__mod__2__eq__1__eq__0,axiom,
% 9.49/5.26      ! [A: nat] :
% 9.49/5.26        ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.26         != one_one_nat )
% 9.49/5.26        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.26          = zero_zero_nat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4047_not__mod__2__eq__1__eq__0,axiom,
% 9.49/5.26      ! [A: int] :
% 9.49/5.26        ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.26         != one_one_int )
% 9.49/5.26        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.26          = zero_zero_int ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4048_not__mod__2__eq__1__eq__0,axiom,
% 9.49/5.26      ! [A: code_integer] :
% 9.49/5.26        ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.49/5.26         != one_one_Code_integer )
% 9.49/5.26        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.49/5.26          = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4049_not__mod__2__eq__0__eq__1,axiom,
% 9.49/5.26      ! [A: nat] :
% 9.49/5.26        ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.26         != zero_zero_nat )
% 9.49/5.26        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.26          = one_one_nat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4050_not__mod__2__eq__0__eq__1,axiom,
% 9.49/5.26      ! [A: int] :
% 9.49/5.26        ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.26         != zero_zero_int )
% 9.49/5.26        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.26          = one_one_int ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4051_not__mod__2__eq__0__eq__1,axiom,
% 9.49/5.26      ! [A: code_integer] :
% 9.49/5.26        ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.49/5.26         != zero_z3403309356797280102nteger )
% 9.49/5.26        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.49/5.26          = one_one_Code_integer ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4052_not__mod2__eq__Suc__0__eq__0,axiom,
% 9.49/5.26      ! [N: nat] :
% 9.49/5.26        ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.26         != ( suc @ zero_zero_nat ) )
% 9.49/5.26        = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.26          = zero_zero_nat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4053_add__self__mod__2,axiom,
% 9.49/5.26      ! [M: nat] :
% 9.49/5.26        ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.26        = zero_zero_nat ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4054_mod2__gr__0,axiom,
% 9.49/5.26      ! [M: nat] :
% 9.49/5.26        ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.26        = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.26          = one_one_nat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4055_unique__quotient,axiom,
% 9.49/5.26      ! [A: int,B: int,Q4: int,R2: int,Q6: int,R4: int] :
% 9.49/5.26        ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.49/5.26       => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R4 ) )
% 9.49/5.26         => ( Q4 = Q6 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4056_unique__remainder,axiom,
% 9.49/5.26      ! [A: int,B: int,Q4: int,R2: int,Q6: int,R4: int] :
% 9.49/5.26        ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.49/5.26       => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R4 ) )
% 9.49/5.26         => ( R2 = R4 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4057_mod__less__eq__dividend,axiom,
% 9.49/5.26      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4058_eucl__rel__int__by0,axiom,
% 9.49/5.26      ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4059_div__int__unique,axiom,
% 9.49/5.26      ! [K: int,L: int,Q4: int,R2: int] :
% 9.49/5.26        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.49/5.26       => ( ( divide_divide_int @ K @ L )
% 9.49/5.26          = Q4 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4060_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 9.49/5.26      ! [A: code_integer,B: code_integer] :
% 9.49/5.26        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.49/5.26       => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4061_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 9.49/5.26      ! [A: nat,B: nat] :
% 9.49/5.26        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.26       => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4062_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.26       => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4063_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 9.49/5.26      ! [B: nat,A: nat] :
% 9.49/5.26        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.49/5.26       => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4064_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 9.49/5.26      ! [B: int,A: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.26       => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4065_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 9.49/5.26      ! [B: code_integer,A: code_integer] :
% 9.49/5.26        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 9.49/5.26       => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4066_mod__eq__self__iff__div__eq__0,axiom,
% 9.49/5.26      ! [A: nat,B: nat] :
% 9.49/5.26        ( ( ( modulo_modulo_nat @ A @ B )
% 9.49/5.26          = A )
% 9.49/5.26        = ( ( divide_divide_nat @ A @ B )
% 9.49/5.26          = zero_zero_nat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4067_mod__eq__self__iff__div__eq__0,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( ( modulo_modulo_int @ A @ B )
% 9.49/5.26          = A )
% 9.49/5.26        = ( ( divide_divide_int @ A @ B )
% 9.49/5.26          = zero_zero_int ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4068_mod__eq__self__iff__div__eq__0,axiom,
% 9.49/5.26      ! [A: code_integer,B: code_integer] :
% 9.49/5.26        ( ( ( modulo364778990260209775nteger @ A @ B )
% 9.49/5.26          = A )
% 9.49/5.26        = ( ( divide6298287555418463151nteger @ A @ B )
% 9.49/5.26          = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4069_mod__Suc,axiom,
% 9.49/5.26      ! [M: nat,N: nat] :
% 9.49/5.26        ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 9.49/5.26            = N )
% 9.49/5.26         => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 9.49/5.26            = zero_zero_nat ) )
% 9.49/5.26        & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 9.49/5.26           != N )
% 9.49/5.26         => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 9.49/5.26            = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4070_mod__induct,axiom,
% 9.49/5.26      ! [P: nat > $o,N: nat,P4: nat,M: nat] :
% 9.49/5.26        ( ( P @ N )
% 9.49/5.26       => ( ( ord_less_nat @ N @ P4 )
% 9.49/5.26         => ( ( ord_less_nat @ M @ P4 )
% 9.49/5.26           => ( ! [N3: nat] :
% 9.49/5.26                  ( ( ord_less_nat @ N3 @ P4 )
% 9.49/5.26                 => ( ( P @ N3 )
% 9.49/5.26                   => ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P4 ) ) ) )
% 9.49/5.26             => ( P @ M ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4071_mod__less__divisor,axiom,
% 9.49/5.26      ! [N: nat,M: nat] :
% 9.49/5.26        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.26       => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4072_gcd__nat__induct,axiom,
% 9.49/5.26      ! [P: nat > nat > $o,M: nat,N: nat] :
% 9.49/5.26        ( ! [M5: nat] : ( P @ M5 @ zero_zero_nat )
% 9.49/5.26       => ( ! [M5: nat,N3: nat] :
% 9.49/5.26              ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.49/5.26             => ( ( P @ N3 @ ( modulo_modulo_nat @ M5 @ N3 ) )
% 9.49/5.26               => ( P @ M5 @ N3 ) ) )
% 9.49/5.26         => ( P @ M @ N ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4073_mod__Suc__le__divisor,axiom,
% 9.49/5.26      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4074_mod__eq__0D,axiom,
% 9.49/5.26      ! [M: nat,D: nat] :
% 9.49/5.26        ( ( ( modulo_modulo_nat @ M @ D )
% 9.49/5.26          = zero_zero_nat )
% 9.49/5.26       => ? [Q3: nat] :
% 9.49/5.26            ( M
% 9.49/5.26            = ( times_times_nat @ D @ Q3 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4075_mod__geq,axiom,
% 9.49/5.26      ! [M: nat,N: nat] :
% 9.49/5.26        ( ~ ( ord_less_nat @ M @ N )
% 9.49/5.26       => ( ( modulo_modulo_nat @ M @ N )
% 9.49/5.26          = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4076_mod__if,axiom,
% 9.49/5.26      ( modulo_modulo_nat
% 9.49/5.26      = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M2 @ N2 ) @ M2 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4077_le__mod__geq,axiom,
% 9.49/5.26      ! [N: nat,M: nat] :
% 9.49/5.26        ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.26       => ( ( modulo_modulo_nat @ M @ N )
% 9.49/5.26          = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4078_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 9.49/5.26      ! [B: code_integer,A: code_integer] :
% 9.49/5.26        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 9.49/5.26       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4079_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 9.49/5.26      ! [B: nat,A: nat] :
% 9.49/5.26        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.49/5.26       => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4080_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 9.49/5.26      ! [B: int,A: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.26       => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4081_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 9.49/5.26      ! [A: code_integer,B: code_integer] :
% 9.49/5.26        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.49/5.26       => ( ( ord_le6747313008572928689nteger @ A @ B )
% 9.49/5.26         => ( ( modulo364778990260209775nteger @ A @ B )
% 9.49/5.26            = A ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4082_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 9.49/5.26      ! [A: nat,B: nat] :
% 9.49/5.26        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.26       => ( ( ord_less_nat @ A @ B )
% 9.49/5.26         => ( ( modulo_modulo_nat @ A @ B )
% 9.49/5.26            = A ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4083_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.26       => ( ( ord_less_int @ A @ B )
% 9.49/5.26         => ( ( modulo_modulo_int @ A @ B )
% 9.49/5.26            = A ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4084_cong__exp__iff__simps_I2_J,axiom,
% 9.49/5.26      ! [N: num,Q4: num] :
% 9.49/5.26        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
% 9.49/5.26          = zero_zero_nat )
% 9.49/5.26        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q4 ) )
% 9.49/5.26          = zero_zero_nat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4085_cong__exp__iff__simps_I2_J,axiom,
% 9.49/5.26      ! [N: num,Q4: num] :
% 9.49/5.26        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
% 9.49/5.26          = zero_zero_int )
% 9.49/5.26        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q4 ) )
% 9.49/5.26          = zero_zero_int ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4086_cong__exp__iff__simps_I2_J,axiom,
% 9.49/5.26      ! [N: num,Q4: num] :
% 9.49/5.26        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
% 9.49/5.26          = zero_z3403309356797280102nteger )
% 9.49/5.26        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q4 ) )
% 9.49/5.26          = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4087_cong__exp__iff__simps_I1_J,axiom,
% 9.49/5.26      ! [N: num] :
% 9.49/5.26        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
% 9.49/5.26        = zero_zero_nat ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4088_cong__exp__iff__simps_I1_J,axiom,
% 9.49/5.26      ! [N: num] :
% 9.49/5.26        ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
% 9.49/5.26        = zero_zero_int ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4089_cong__exp__iff__simps_I1_J,axiom,
% 9.49/5.26      ! [N: num] :
% 9.49/5.26        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
% 9.49/5.26        = zero_z3403309356797280102nteger ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4090_mod__le__divisor,axiom,
% 9.49/5.26      ! [N: nat,M: nat] :
% 9.49/5.26        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.26       => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4091_div__less__mono,axiom,
% 9.49/5.26      ! [A2: nat,B2: nat,N: nat] :
% 9.49/5.26        ( ( ord_less_nat @ A2 @ B2 )
% 9.49/5.26       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.26         => ( ( ( modulo_modulo_nat @ A2 @ N )
% 9.49/5.26              = zero_zero_nat )
% 9.49/5.26           => ( ( ( modulo_modulo_nat @ B2 @ N )
% 9.49/5.26                = zero_zero_nat )
% 9.49/5.26             => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N ) @ ( divide_divide_nat @ B2 @ N ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4092_mod__eq__nat1E,axiom,
% 9.49/5.26      ! [M: nat,Q4: nat,N: nat] :
% 9.49/5.26        ( ( ( modulo_modulo_nat @ M @ Q4 )
% 9.49/5.26          = ( modulo_modulo_nat @ N @ Q4 ) )
% 9.49/5.26       => ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.26         => ~ ! [S4: nat] :
% 9.49/5.26                ( M
% 9.49/5.26               != ( plus_plus_nat @ N @ ( times_times_nat @ Q4 @ S4 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4093_mod__eq__nat2E,axiom,
% 9.49/5.26      ! [M: nat,Q4: nat,N: nat] :
% 9.49/5.26        ( ( ( modulo_modulo_nat @ M @ Q4 )
% 9.49/5.26          = ( modulo_modulo_nat @ N @ Q4 ) )
% 9.49/5.26       => ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.26         => ~ ! [S4: nat] :
% 9.49/5.26                ( N
% 9.49/5.26               != ( plus_plus_nat @ M @ ( times_times_nat @ Q4 @ S4 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4094_nat__mod__eq__lemma,axiom,
% 9.49/5.26      ! [X: nat,N: nat,Y: nat] :
% 9.49/5.26        ( ( ( modulo_modulo_nat @ X @ N )
% 9.49/5.26          = ( modulo_modulo_nat @ Y @ N ) )
% 9.49/5.26       => ( ( ord_less_eq_nat @ Y @ X )
% 9.49/5.26         => ? [Q3: nat] :
% 9.49/5.26              ( X
% 9.49/5.26              = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q3 ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4095_eucl__rel__int__dividesI,axiom,
% 9.49/5.26      ! [L: int,K: int,Q4: int] :
% 9.49/5.26        ( ( L != zero_zero_int )
% 9.49/5.26       => ( ( K
% 9.49/5.26            = ( times_times_int @ Q4 @ L ) )
% 9.49/5.26         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ zero_zero_int ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4096_split__mod,axiom,
% 9.49/5.26      ! [P: nat > $o,M: nat,N: nat] :
% 9.49/5.26        ( ( P @ ( modulo_modulo_nat @ M @ N ) )
% 9.49/5.26        = ( ( ( N = zero_zero_nat )
% 9.49/5.26           => ( P @ M ) )
% 9.49/5.26          & ( ( N != zero_zero_nat )
% 9.49/5.26           => ! [I4: nat,J3: nat] :
% 9.49/5.26                ( ( ord_less_nat @ J3 @ N )
% 9.49/5.26               => ( ( M
% 9.49/5.26                    = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 9.49/5.26                 => ( P @ J3 ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4097_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 9.49/5.26      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.49/5.26        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 9.49/5.26       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 9.49/5.26          = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4098_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 9.49/5.26      ! [C: nat,A: nat,B: nat] :
% 9.49/5.26        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.49/5.26       => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
% 9.49/5.26          = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4099_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 9.49/5.26      ! [C: int,A: int,B: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.49/5.26       => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 9.49/5.26          = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4100_Suc__times__mod__eq,axiom,
% 9.49/5.26      ! [M: nat,N: nat] :
% 9.49/5.26        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.49/5.26       => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
% 9.49/5.26          = one_one_nat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4101_divmod__digit__0_I2_J,axiom,
% 9.49/5.26      ! [B: nat,A: nat] :
% 9.49/5.26        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.49/5.26       => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.49/5.26         => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
% 9.49/5.26            = ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4102_divmod__digit__0_I2_J,axiom,
% 9.49/5.26      ! [B: int,A: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.26       => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.49/5.26         => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
% 9.49/5.26            = ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4103_divmod__digit__0_I2_J,axiom,
% 9.49/5.26      ! [B: code_integer,A: code_integer] :
% 9.49/5.26        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 9.49/5.26       => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.49/5.26         => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
% 9.49/5.26            = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4104_bits__stable__imp__add__self,axiom,
% 9.49/5.26      ! [A: nat] :
% 9.49/5.26        ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.26          = A )
% 9.49/5.26       => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.26          = zero_zero_nat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4105_bits__stable__imp__add__self,axiom,
% 9.49/5.26      ! [A: int] :
% 9.49/5.26        ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.26          = A )
% 9.49/5.26       => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 9.49/5.26          = zero_zero_int ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4106_bits__stable__imp__add__self,axiom,
% 9.49/5.26      ! [A: code_integer] :
% 9.49/5.26        ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.49/5.26          = A )
% 9.49/5.26       => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 9.49/5.26          = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4107_verit__le__mono__div,axiom,
% 9.49/5.26      ! [A2: nat,B2: nat,N: nat] :
% 9.49/5.26        ( ( ord_less_nat @ A2 @ B2 )
% 9.49/5.26       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.26         => ( ord_less_eq_nat
% 9.49/5.26            @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N )
% 9.49/5.26              @ ( if_nat
% 9.49/5.26                @ ( ( modulo_modulo_nat @ B2 @ N )
% 9.49/5.26                  = zero_zero_nat )
% 9.49/5.26                @ one_one_nat
% 9.49/5.26                @ zero_zero_nat ) )
% 9.49/5.26            @ ( divide_divide_nat @ B2 @ N ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4108_divmod__digit__0_I1_J,axiom,
% 9.49/5.26      ! [B: nat,A: nat] :
% 9.49/5.26        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.49/5.26       => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.49/5.26         => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 9.49/5.26            = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4109_divmod__digit__0_I1_J,axiom,
% 9.49/5.26      ! [B: int,A: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.26       => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.49/5.26         => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 9.49/5.26            = ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4110_divmod__digit__0_I1_J,axiom,
% 9.49/5.26      ! [B: code_integer,A: code_integer] :
% 9.49/5.26        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 9.49/5.26       => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.49/5.26         => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 9.49/5.26            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4111_mult__exp__mod__exp__eq,axiom,
% 9.49/5.26      ! [M: nat,N: nat,A: nat] :
% 9.49/5.26        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.26       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.26          = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4112_mult__exp__mod__exp__eq,axiom,
% 9.49/5.26      ! [M: nat,N: nat,A: int] :
% 9.49/5.26        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.26       => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.26          = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4113_mult__exp__mod__exp__eq,axiom,
% 9.49/5.26      ! [M: nat,N: nat,A: code_integer] :
% 9.49/5.26        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.26       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.26          = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4114_eucl__rel__int__iff,axiom,
% 9.49/5.26      ! [K: int,L: int,Q4: int,R2: int] :
% 9.49/5.26        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.49/5.26        = ( ( K
% 9.49/5.26            = ( plus_plus_int @ ( times_times_int @ L @ Q4 ) @ R2 ) )
% 9.49/5.26          & ( ( ord_less_int @ zero_zero_int @ L )
% 9.49/5.26           => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 9.49/5.26              & ( ord_less_int @ R2 @ L ) ) )
% 9.49/5.26          & ( ~ ( ord_less_int @ zero_zero_int @ L )
% 9.49/5.26           => ( ( ( ord_less_int @ L @ zero_zero_int )
% 9.49/5.26               => ( ( ord_less_int @ L @ R2 )
% 9.49/5.26                  & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
% 9.49/5.26              & ( ~ ( ord_less_int @ L @ zero_zero_int )
% 9.49/5.26               => ( Q4 = zero_zero_int ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4115_mod__double__modulus,axiom,
% 9.49/5.26      ! [M: code_integer,X: code_integer] :
% 9.49/5.26        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 9.49/5.26       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 9.49/5.26         => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 9.49/5.26              = ( modulo364778990260209775nteger @ X @ M ) )
% 9.49/5.26            | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 9.49/5.26              = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4116_mod__double__modulus,axiom,
% 9.49/5.26      ! [M: nat,X: nat] :
% 9.49/5.26        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.49/5.26       => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 9.49/5.26         => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.49/5.26              = ( modulo_modulo_nat @ X @ M ) )
% 9.49/5.26            | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.49/5.26              = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4117_mod__double__modulus,axiom,
% 9.49/5.26      ! [M: int,X: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ M )
% 9.49/5.26       => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.49/5.26         => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 9.49/5.26              = ( modulo_modulo_int @ X @ M ) )
% 9.49/5.26            | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 9.49/5.26              = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4118_divmod__digit__1_I2_J,axiom,
% 9.49/5.26      ! [A: code_integer,B: code_integer] :
% 9.49/5.26        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.49/5.26       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 9.49/5.26         => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 9.49/5.26           => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.49/5.26              = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4119_divmod__digit__1_I2_J,axiom,
% 9.49/5.26      ! [A: nat,B: nat] :
% 9.49/5.26        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.26       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.49/5.26         => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 9.49/5.26           => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.49/5.26              = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4120_divmod__digit__1_I2_J,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.26       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.26         => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 9.49/5.26           => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.49/5.26              = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4121_divmod__digit__1_I1_J,axiom,
% 9.49/5.26      ! [A: code_integer,B: code_integer] :
% 9.49/5.26        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.49/5.26       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 9.49/5.26         => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 9.49/5.26           => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
% 9.49/5.26              = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4122_divmod__digit__1_I1_J,axiom,
% 9.49/5.26      ! [A: nat,B: nat] :
% 9.49/5.26        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.49/5.26       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.49/5.26         => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 9.49/5.26           => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
% 9.49/5.26              = ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4123_divmod__digit__1_I1_J,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.26       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.26         => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 9.49/5.26           => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
% 9.49/5.26              = ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4124_mult__less__iff1,axiom,
% 9.49/5.26      ! [Z2: real,X: real,Y: real] :
% 9.49/5.26        ( ( ord_less_real @ zero_zero_real @ Z2 )
% 9.49/5.26       => ( ( ord_less_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
% 9.49/5.26          = ( ord_less_real @ X @ Y ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4125_mult__less__iff1,axiom,
% 9.49/5.26      ! [Z2: rat,X: rat,Y: rat] :
% 9.49/5.26        ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 9.49/5.26       => ( ( ord_less_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y @ Z2 ) )
% 9.49/5.26          = ( ord_less_rat @ X @ Y ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4126_mult__less__iff1,axiom,
% 9.49/5.26      ! [Z2: int,X: int,Y: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 9.49/5.26       => ( ( ord_less_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
% 9.49/5.26          = ( ord_less_int @ X @ Y ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4127_pos__eucl__rel__int__mult__2,axiom,
% 9.49/5.26      ! [B: int,A: int,Q4: int,R2: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.49/5.26       => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.49/5.26         => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4128_product__nth,axiom,
% 9.49/5.26      ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 9.49/5.26        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 9.49/5.26       => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) @ N )
% 9.49/5.26          = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4129_product__nth,axiom,
% 9.49/5.26      ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
% 9.49/5.26        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) )
% 9.49/5.26       => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) @ N )
% 9.49/5.26          = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4130_product__nth,axiom,
% 9.49/5.26      ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
% 9.49/5.26        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
% 9.49/5.26       => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys ) @ N )
% 9.49/5.26          = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4131_product__nth,axiom,
% 9.49/5.26      ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
% 9.49/5.26        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) )
% 9.49/5.26       => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) @ N )
% 9.49/5.26          = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4132_product__nth,axiom,
% 9.49/5.26      ! [N: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
% 9.49/5.26        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 9.49/5.26       => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys ) @ N )
% 9.49/5.26          = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4133_product__nth,axiom,
% 9.49/5.26      ! [N: nat,Xs: list_o,Ys: list_o] :
% 9.49/5.26        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) )
% 9.49/5.26       => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs @ Ys ) @ N )
% 9.49/5.26          = ( product_Pair_o_o @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4134_product__nth,axiom,
% 9.49/5.26      ! [N: nat,Xs: list_o,Ys: list_nat] :
% 9.49/5.26        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
% 9.49/5.26       => ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs @ Ys ) @ N )
% 9.49/5.26          = ( product_Pair_o_nat @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4135_product__nth,axiom,
% 9.49/5.26      ! [N: nat,Xs: list_o,Ys: list_int] :
% 9.49/5.26        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) )
% 9.49/5.26       => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs @ Ys ) @ N )
% 9.49/5.26          = ( product_Pair_o_int @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4136_product__nth,axiom,
% 9.49/5.26      ! [N: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
% 9.49/5.26        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 9.49/5.26       => ( ( nth_Pr744662078594809490T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys ) @ N )
% 9.49/5.26          = ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4137_product__nth,axiom,
% 9.49/5.26      ! [N: nat,Xs: list_nat,Ys: list_o] :
% 9.49/5.26        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys ) ) )
% 9.49/5.26       => ( ( nth_Pr112076138515278198_nat_o @ ( product_nat_o @ Xs @ Ys ) @ N )
% 9.49/5.26          = ( product_Pair_nat_o @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4138_prod_Ofinite__Collect__op,axiom,
% 9.49/5.26      ! [I6: set_real,X: real > complex,Y: real > complex] :
% 9.49/5.26        ( ( finite_finite_real
% 9.49/5.26          @ ( collect_real
% 9.49/5.26            @ ^ [I4: real] :
% 9.49/5.26                ( ( member_real @ I4 @ I6 )
% 9.49/5.26                & ( ( X @ I4 )
% 9.49/5.26                 != one_one_complex ) ) ) )
% 9.49/5.26       => ( ( finite_finite_real
% 9.49/5.26            @ ( collect_real
% 9.49/5.26              @ ^ [I4: real] :
% 9.49/5.26                  ( ( member_real @ I4 @ I6 )
% 9.49/5.26                  & ( ( Y @ I4 )
% 9.49/5.26                   != one_one_complex ) ) ) )
% 9.49/5.26         => ( finite_finite_real
% 9.49/5.26            @ ( collect_real
% 9.49/5.26              @ ^ [I4: real] :
% 9.49/5.26                  ( ( member_real @ I4 @ I6 )
% 9.49/5.26                  & ( ( times_times_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
% 9.49/5.26                   != one_one_complex ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4139_prod_Ofinite__Collect__op,axiom,
% 9.49/5.26      ! [I6: set_nat,X: nat > complex,Y: nat > complex] :
% 9.49/5.26        ( ( finite_finite_nat
% 9.49/5.26          @ ( collect_nat
% 9.49/5.26            @ ^ [I4: nat] :
% 9.49/5.26                ( ( member_nat @ I4 @ I6 )
% 9.49/5.26                & ( ( X @ I4 )
% 9.49/5.26                 != one_one_complex ) ) ) )
% 9.49/5.26       => ( ( finite_finite_nat
% 9.49/5.26            @ ( collect_nat
% 9.49/5.26              @ ^ [I4: nat] :
% 9.49/5.26                  ( ( member_nat @ I4 @ I6 )
% 9.49/5.26                  & ( ( Y @ I4 )
% 9.49/5.26                   != one_one_complex ) ) ) )
% 9.49/5.26         => ( finite_finite_nat
% 9.49/5.26            @ ( collect_nat
% 9.49/5.26              @ ^ [I4: nat] :
% 9.49/5.26                  ( ( member_nat @ I4 @ I6 )
% 9.49/5.26                  & ( ( times_times_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
% 9.49/5.26                   != one_one_complex ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4140_prod_Ofinite__Collect__op,axiom,
% 9.49/5.26      ! [I6: set_int,X: int > complex,Y: int > complex] :
% 9.49/5.26        ( ( finite_finite_int
% 9.49/5.26          @ ( collect_int
% 9.49/5.26            @ ^ [I4: int] :
% 9.49/5.26                ( ( member_int @ I4 @ I6 )
% 9.49/5.26                & ( ( X @ I4 )
% 9.49/5.26                 != one_one_complex ) ) ) )
% 9.49/5.26       => ( ( finite_finite_int
% 9.49/5.26            @ ( collect_int
% 9.49/5.26              @ ^ [I4: int] :
% 9.49/5.26                  ( ( member_int @ I4 @ I6 )
% 9.49/5.26                  & ( ( Y @ I4 )
% 9.49/5.26                   != one_one_complex ) ) ) )
% 9.49/5.26         => ( finite_finite_int
% 9.49/5.26            @ ( collect_int
% 9.49/5.26              @ ^ [I4: int] :
% 9.49/5.26                  ( ( member_int @ I4 @ I6 )
% 9.49/5.26                  & ( ( times_times_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
% 9.49/5.26                   != one_one_complex ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4141_prod_Ofinite__Collect__op,axiom,
% 9.49/5.26      ! [I6: set_complex,X: complex > complex,Y: complex > complex] :
% 9.49/5.26        ( ( finite3207457112153483333omplex
% 9.49/5.26          @ ( collect_complex
% 9.49/5.26            @ ^ [I4: complex] :
% 9.49/5.26                ( ( member_complex @ I4 @ I6 )
% 9.49/5.26                & ( ( X @ I4 )
% 9.49/5.26                 != one_one_complex ) ) ) )
% 9.49/5.26       => ( ( finite3207457112153483333omplex
% 9.49/5.26            @ ( collect_complex
% 9.49/5.26              @ ^ [I4: complex] :
% 9.49/5.26                  ( ( member_complex @ I4 @ I6 )
% 9.49/5.26                  & ( ( Y @ I4 )
% 9.49/5.26                   != one_one_complex ) ) ) )
% 9.49/5.26         => ( finite3207457112153483333omplex
% 9.49/5.26            @ ( collect_complex
% 9.49/5.26              @ ^ [I4: complex] :
% 9.49/5.26                  ( ( member_complex @ I4 @ I6 )
% 9.49/5.26                  & ( ( times_times_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
% 9.49/5.26                   != one_one_complex ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4142_prod_Ofinite__Collect__op,axiom,
% 9.49/5.26      ! [I6: set_real,X: real > real,Y: real > real] :
% 9.49/5.26        ( ( finite_finite_real
% 9.49/5.26          @ ( collect_real
% 9.49/5.26            @ ^ [I4: real] :
% 9.49/5.26                ( ( member_real @ I4 @ I6 )
% 9.49/5.26                & ( ( X @ I4 )
% 9.49/5.26                 != one_one_real ) ) ) )
% 9.49/5.26       => ( ( finite_finite_real
% 9.49/5.26            @ ( collect_real
% 9.49/5.26              @ ^ [I4: real] :
% 9.49/5.26                  ( ( member_real @ I4 @ I6 )
% 9.49/5.26                  & ( ( Y @ I4 )
% 9.49/5.26                   != one_one_real ) ) ) )
% 9.49/5.26         => ( finite_finite_real
% 9.49/5.26            @ ( collect_real
% 9.49/5.26              @ ^ [I4: real] :
% 9.49/5.26                  ( ( member_real @ I4 @ I6 )
% 9.49/5.26                  & ( ( times_times_real @ ( X @ I4 ) @ ( Y @ I4 ) )
% 9.49/5.26                   != one_one_real ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4143_prod_Ofinite__Collect__op,axiom,
% 9.49/5.26      ! [I6: set_nat,X: nat > real,Y: nat > real] :
% 9.49/5.26        ( ( finite_finite_nat
% 9.49/5.26          @ ( collect_nat
% 9.49/5.26            @ ^ [I4: nat] :
% 9.49/5.26                ( ( member_nat @ I4 @ I6 )
% 9.49/5.26                & ( ( X @ I4 )
% 9.49/5.26                 != one_one_real ) ) ) )
% 9.49/5.26       => ( ( finite_finite_nat
% 9.49/5.26            @ ( collect_nat
% 9.49/5.26              @ ^ [I4: nat] :
% 9.49/5.26                  ( ( member_nat @ I4 @ I6 )
% 9.49/5.26                  & ( ( Y @ I4 )
% 9.49/5.26                   != one_one_real ) ) ) )
% 9.49/5.26         => ( finite_finite_nat
% 9.49/5.26            @ ( collect_nat
% 9.49/5.26              @ ^ [I4: nat] :
% 9.49/5.26                  ( ( member_nat @ I4 @ I6 )
% 9.49/5.26                  & ( ( times_times_real @ ( X @ I4 ) @ ( Y @ I4 ) )
% 9.49/5.26                   != one_one_real ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4144_prod_Ofinite__Collect__op,axiom,
% 9.49/5.26      ! [I6: set_int,X: int > real,Y: int > real] :
% 9.49/5.26        ( ( finite_finite_int
% 9.49/5.26          @ ( collect_int
% 9.49/5.26            @ ^ [I4: int] :
% 9.49/5.26                ( ( member_int @ I4 @ I6 )
% 9.49/5.26                & ( ( X @ I4 )
% 9.49/5.26                 != one_one_real ) ) ) )
% 9.49/5.26       => ( ( finite_finite_int
% 9.49/5.26            @ ( collect_int
% 9.49/5.26              @ ^ [I4: int] :
% 9.49/5.26                  ( ( member_int @ I4 @ I6 )
% 9.49/5.26                  & ( ( Y @ I4 )
% 9.49/5.26                   != one_one_real ) ) ) )
% 9.49/5.26         => ( finite_finite_int
% 9.49/5.26            @ ( collect_int
% 9.49/5.26              @ ^ [I4: int] :
% 9.49/5.26                  ( ( member_int @ I4 @ I6 )
% 9.49/5.26                  & ( ( times_times_real @ ( X @ I4 ) @ ( Y @ I4 ) )
% 9.49/5.26                   != one_one_real ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4145_prod_Ofinite__Collect__op,axiom,
% 9.49/5.26      ! [I6: set_complex,X: complex > real,Y: complex > real] :
% 9.49/5.26        ( ( finite3207457112153483333omplex
% 9.49/5.26          @ ( collect_complex
% 9.49/5.26            @ ^ [I4: complex] :
% 9.49/5.26                ( ( member_complex @ I4 @ I6 )
% 9.49/5.26                & ( ( X @ I4 )
% 9.49/5.26                 != one_one_real ) ) ) )
% 9.49/5.26       => ( ( finite3207457112153483333omplex
% 9.49/5.26            @ ( collect_complex
% 9.49/5.26              @ ^ [I4: complex] :
% 9.49/5.26                  ( ( member_complex @ I4 @ I6 )
% 9.49/5.26                  & ( ( Y @ I4 )
% 9.49/5.26                   != one_one_real ) ) ) )
% 9.49/5.26         => ( finite3207457112153483333omplex
% 9.49/5.26            @ ( collect_complex
% 9.49/5.26              @ ^ [I4: complex] :
% 9.49/5.26                  ( ( member_complex @ I4 @ I6 )
% 9.49/5.26                  & ( ( times_times_real @ ( X @ I4 ) @ ( Y @ I4 ) )
% 9.49/5.26                   != one_one_real ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4146_prod_Ofinite__Collect__op,axiom,
% 9.49/5.26      ! [I6: set_real,X: real > rat,Y: real > rat] :
% 9.49/5.26        ( ( finite_finite_real
% 9.49/5.26          @ ( collect_real
% 9.49/5.26            @ ^ [I4: real] :
% 9.49/5.26                ( ( member_real @ I4 @ I6 )
% 9.49/5.26                & ( ( X @ I4 )
% 9.49/5.26                 != one_one_rat ) ) ) )
% 9.49/5.26       => ( ( finite_finite_real
% 9.49/5.26            @ ( collect_real
% 9.49/5.26              @ ^ [I4: real] :
% 9.49/5.26                  ( ( member_real @ I4 @ I6 )
% 9.49/5.26                  & ( ( Y @ I4 )
% 9.49/5.26                   != one_one_rat ) ) ) )
% 9.49/5.26         => ( finite_finite_real
% 9.49/5.26            @ ( collect_real
% 9.49/5.26              @ ^ [I4: real] :
% 9.49/5.26                  ( ( member_real @ I4 @ I6 )
% 9.49/5.26                  & ( ( times_times_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
% 9.49/5.26                   != one_one_rat ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4147_prod_Ofinite__Collect__op,axiom,
% 9.49/5.26      ! [I6: set_nat,X: nat > rat,Y: nat > rat] :
% 9.49/5.26        ( ( finite_finite_nat
% 9.49/5.26          @ ( collect_nat
% 9.49/5.26            @ ^ [I4: nat] :
% 9.49/5.26                ( ( member_nat @ I4 @ I6 )
% 9.49/5.26                & ( ( X @ I4 )
% 9.49/5.26                 != one_one_rat ) ) ) )
% 9.49/5.26       => ( ( finite_finite_nat
% 9.49/5.26            @ ( collect_nat
% 9.49/5.26              @ ^ [I4: nat] :
% 9.49/5.26                  ( ( member_nat @ I4 @ I6 )
% 9.49/5.26                  & ( ( Y @ I4 )
% 9.49/5.26                   != one_one_rat ) ) ) )
% 9.49/5.26         => ( finite_finite_nat
% 9.49/5.26            @ ( collect_nat
% 9.49/5.26              @ ^ [I4: nat] :
% 9.49/5.26                  ( ( member_nat @ I4 @ I6 )
% 9.49/5.26                  & ( ( times_times_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
% 9.49/5.26                   != one_one_rat ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4148_sum_Ofinite__Collect__op,axiom,
% 9.49/5.26      ! [I6: set_real,X: real > complex,Y: real > complex] :
% 9.49/5.26        ( ( finite_finite_real
% 9.49/5.26          @ ( collect_real
% 9.49/5.26            @ ^ [I4: real] :
% 9.49/5.26                ( ( member_real @ I4 @ I6 )
% 9.49/5.26                & ( ( X @ I4 )
% 9.49/5.26                 != zero_zero_complex ) ) ) )
% 9.49/5.26       => ( ( finite_finite_real
% 9.49/5.26            @ ( collect_real
% 9.49/5.26              @ ^ [I4: real] :
% 9.49/5.26                  ( ( member_real @ I4 @ I6 )
% 9.49/5.26                  & ( ( Y @ I4 )
% 9.49/5.26                   != zero_zero_complex ) ) ) )
% 9.49/5.26         => ( finite_finite_real
% 9.49/5.26            @ ( collect_real
% 9.49/5.26              @ ^ [I4: real] :
% 9.49/5.26                  ( ( member_real @ I4 @ I6 )
% 9.49/5.26                  & ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
% 9.49/5.26                   != zero_zero_complex ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4149_sum_Ofinite__Collect__op,axiom,
% 9.49/5.26      ! [I6: set_nat,X: nat > complex,Y: nat > complex] :
% 9.49/5.26        ( ( finite_finite_nat
% 9.49/5.26          @ ( collect_nat
% 9.49/5.26            @ ^ [I4: nat] :
% 9.49/5.26                ( ( member_nat @ I4 @ I6 )
% 9.49/5.26                & ( ( X @ I4 )
% 9.49/5.26                 != zero_zero_complex ) ) ) )
% 9.49/5.26       => ( ( finite_finite_nat
% 9.49/5.26            @ ( collect_nat
% 9.49/5.26              @ ^ [I4: nat] :
% 9.49/5.26                  ( ( member_nat @ I4 @ I6 )
% 9.49/5.26                  & ( ( Y @ I4 )
% 9.49/5.26                   != zero_zero_complex ) ) ) )
% 9.49/5.26         => ( finite_finite_nat
% 9.49/5.26            @ ( collect_nat
% 9.49/5.26              @ ^ [I4: nat] :
% 9.49/5.26                  ( ( member_nat @ I4 @ I6 )
% 9.49/5.26                  & ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
% 9.49/5.26                   != zero_zero_complex ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4150_sum_Ofinite__Collect__op,axiom,
% 9.49/5.26      ! [I6: set_int,X: int > complex,Y: int > complex] :
% 9.49/5.26        ( ( finite_finite_int
% 9.49/5.26          @ ( collect_int
% 9.49/5.26            @ ^ [I4: int] :
% 9.49/5.26                ( ( member_int @ I4 @ I6 )
% 9.49/5.26                & ( ( X @ I4 )
% 9.49/5.26                 != zero_zero_complex ) ) ) )
% 9.49/5.26       => ( ( finite_finite_int
% 9.49/5.26            @ ( collect_int
% 9.49/5.26              @ ^ [I4: int] :
% 9.49/5.26                  ( ( member_int @ I4 @ I6 )
% 9.49/5.26                  & ( ( Y @ I4 )
% 9.49/5.26                   != zero_zero_complex ) ) ) )
% 9.49/5.26         => ( finite_finite_int
% 9.49/5.26            @ ( collect_int
% 9.49/5.26              @ ^ [I4: int] :
% 9.49/5.26                  ( ( member_int @ I4 @ I6 )
% 9.49/5.26                  & ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
% 9.49/5.26                   != zero_zero_complex ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4151_sum_Ofinite__Collect__op,axiom,
% 9.49/5.26      ! [I6: set_complex,X: complex > complex,Y: complex > complex] :
% 9.49/5.26        ( ( finite3207457112153483333omplex
% 9.49/5.26          @ ( collect_complex
% 9.49/5.26            @ ^ [I4: complex] :
% 9.49/5.26                ( ( member_complex @ I4 @ I6 )
% 9.49/5.26                & ( ( X @ I4 )
% 9.49/5.26                 != zero_zero_complex ) ) ) )
% 9.49/5.26       => ( ( finite3207457112153483333omplex
% 9.49/5.26            @ ( collect_complex
% 9.49/5.26              @ ^ [I4: complex] :
% 9.49/5.26                  ( ( member_complex @ I4 @ I6 )
% 9.49/5.26                  & ( ( Y @ I4 )
% 9.49/5.26                   != zero_zero_complex ) ) ) )
% 9.49/5.26         => ( finite3207457112153483333omplex
% 9.49/5.26            @ ( collect_complex
% 9.49/5.26              @ ^ [I4: complex] :
% 9.49/5.26                  ( ( member_complex @ I4 @ I6 )
% 9.49/5.26                  & ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y @ I4 ) )
% 9.49/5.26                   != zero_zero_complex ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4152_sum_Ofinite__Collect__op,axiom,
% 9.49/5.26      ! [I6: set_real,X: real > real,Y: real > real] :
% 9.49/5.26        ( ( finite_finite_real
% 9.49/5.26          @ ( collect_real
% 9.49/5.26            @ ^ [I4: real] :
% 9.49/5.26                ( ( member_real @ I4 @ I6 )
% 9.49/5.26                & ( ( X @ I4 )
% 9.49/5.26                 != zero_zero_real ) ) ) )
% 9.49/5.26       => ( ( finite_finite_real
% 9.49/5.26            @ ( collect_real
% 9.49/5.26              @ ^ [I4: real] :
% 9.49/5.26                  ( ( member_real @ I4 @ I6 )
% 9.49/5.26                  & ( ( Y @ I4 )
% 9.49/5.26                   != zero_zero_real ) ) ) )
% 9.49/5.26         => ( finite_finite_real
% 9.49/5.26            @ ( collect_real
% 9.49/5.26              @ ^ [I4: real] :
% 9.49/5.26                  ( ( member_real @ I4 @ I6 )
% 9.49/5.26                  & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y @ I4 ) )
% 9.49/5.26                   != zero_zero_real ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4153_sum_Ofinite__Collect__op,axiom,
% 9.49/5.26      ! [I6: set_nat,X: nat > real,Y: nat > real] :
% 9.49/5.26        ( ( finite_finite_nat
% 9.49/5.26          @ ( collect_nat
% 9.49/5.26            @ ^ [I4: nat] :
% 9.49/5.26                ( ( member_nat @ I4 @ I6 )
% 9.49/5.26                & ( ( X @ I4 )
% 9.49/5.26                 != zero_zero_real ) ) ) )
% 9.49/5.26       => ( ( finite_finite_nat
% 9.49/5.26            @ ( collect_nat
% 9.49/5.26              @ ^ [I4: nat] :
% 9.49/5.26                  ( ( member_nat @ I4 @ I6 )
% 9.49/5.26                  & ( ( Y @ I4 )
% 9.49/5.26                   != zero_zero_real ) ) ) )
% 9.49/5.26         => ( finite_finite_nat
% 9.49/5.26            @ ( collect_nat
% 9.49/5.26              @ ^ [I4: nat] :
% 9.49/5.26                  ( ( member_nat @ I4 @ I6 )
% 9.49/5.26                  & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y @ I4 ) )
% 9.49/5.26                   != zero_zero_real ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4154_sum_Ofinite__Collect__op,axiom,
% 9.49/5.26      ! [I6: set_int,X: int > real,Y: int > real] :
% 9.49/5.26        ( ( finite_finite_int
% 9.49/5.26          @ ( collect_int
% 9.49/5.26            @ ^ [I4: int] :
% 9.49/5.26                ( ( member_int @ I4 @ I6 )
% 9.49/5.26                & ( ( X @ I4 )
% 9.49/5.26                 != zero_zero_real ) ) ) )
% 9.49/5.26       => ( ( finite_finite_int
% 9.49/5.26            @ ( collect_int
% 9.49/5.26              @ ^ [I4: int] :
% 9.49/5.26                  ( ( member_int @ I4 @ I6 )
% 9.49/5.26                  & ( ( Y @ I4 )
% 9.49/5.26                   != zero_zero_real ) ) ) )
% 9.49/5.26         => ( finite_finite_int
% 9.49/5.26            @ ( collect_int
% 9.49/5.26              @ ^ [I4: int] :
% 9.49/5.26                  ( ( member_int @ I4 @ I6 )
% 9.49/5.26                  & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y @ I4 ) )
% 9.49/5.26                   != zero_zero_real ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4155_sum_Ofinite__Collect__op,axiom,
% 9.49/5.26      ! [I6: set_complex,X: complex > real,Y: complex > real] :
% 9.49/5.26        ( ( finite3207457112153483333omplex
% 9.49/5.26          @ ( collect_complex
% 9.49/5.26            @ ^ [I4: complex] :
% 9.49/5.26                ( ( member_complex @ I4 @ I6 )
% 9.49/5.26                & ( ( X @ I4 )
% 9.49/5.26                 != zero_zero_real ) ) ) )
% 9.49/5.26       => ( ( finite3207457112153483333omplex
% 9.49/5.26            @ ( collect_complex
% 9.49/5.26              @ ^ [I4: complex] :
% 9.49/5.26                  ( ( member_complex @ I4 @ I6 )
% 9.49/5.26                  & ( ( Y @ I4 )
% 9.49/5.26                   != zero_zero_real ) ) ) )
% 9.49/5.26         => ( finite3207457112153483333omplex
% 9.49/5.26            @ ( collect_complex
% 9.49/5.26              @ ^ [I4: complex] :
% 9.49/5.26                  ( ( member_complex @ I4 @ I6 )
% 9.49/5.26                  & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y @ I4 ) )
% 9.49/5.26                   != zero_zero_real ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4156_sum_Ofinite__Collect__op,axiom,
% 9.49/5.26      ! [I6: set_real,X: real > rat,Y: real > rat] :
% 9.49/5.26        ( ( finite_finite_real
% 9.49/5.26          @ ( collect_real
% 9.49/5.26            @ ^ [I4: real] :
% 9.49/5.26                ( ( member_real @ I4 @ I6 )
% 9.49/5.26                & ( ( X @ I4 )
% 9.49/5.26                 != zero_zero_rat ) ) ) )
% 9.49/5.26       => ( ( finite_finite_real
% 9.49/5.26            @ ( collect_real
% 9.49/5.26              @ ^ [I4: real] :
% 9.49/5.26                  ( ( member_real @ I4 @ I6 )
% 9.49/5.26                  & ( ( Y @ I4 )
% 9.49/5.26                   != zero_zero_rat ) ) ) )
% 9.49/5.26         => ( finite_finite_real
% 9.49/5.26            @ ( collect_real
% 9.49/5.26              @ ^ [I4: real] :
% 9.49/5.26                  ( ( member_real @ I4 @ I6 )
% 9.49/5.26                  & ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
% 9.49/5.26                   != zero_zero_rat ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4157_sum_Ofinite__Collect__op,axiom,
% 9.49/5.26      ! [I6: set_nat,X: nat > rat,Y: nat > rat] :
% 9.49/5.26        ( ( finite_finite_nat
% 9.49/5.26          @ ( collect_nat
% 9.49/5.26            @ ^ [I4: nat] :
% 9.49/5.26                ( ( member_nat @ I4 @ I6 )
% 9.49/5.26                & ( ( X @ I4 )
% 9.49/5.26                 != zero_zero_rat ) ) ) )
% 9.49/5.26       => ( ( finite_finite_nat
% 9.49/5.26            @ ( collect_nat
% 9.49/5.26              @ ^ [I4: nat] :
% 9.49/5.26                  ( ( member_nat @ I4 @ I6 )
% 9.49/5.26                  & ( ( Y @ I4 )
% 9.49/5.26                   != zero_zero_rat ) ) ) )
% 9.49/5.26         => ( finite_finite_nat
% 9.49/5.26            @ ( collect_nat
% 9.49/5.26              @ ^ [I4: nat] :
% 9.49/5.26                  ( ( member_nat @ I4 @ I6 )
% 9.49/5.26                  & ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
% 9.49/5.26                   != zero_zero_rat ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4158_even__succ__mod__exp,axiom,
% 9.49/5.26      ! [A: nat,N: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.49/5.26       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.26         => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.26            = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4159_even__succ__mod__exp,axiom,
% 9.49/5.26      ! [A: int,N: nat] :
% 9.49/5.26        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.49/5.26       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.26         => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.26            = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4160_even__succ__mod__exp,axiom,
% 9.49/5.26      ! [A: code_integer,N: nat] :
% 9.49/5.26        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.49/5.26       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.26         => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.26            = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4161_Bolzano,axiom,
% 9.49/5.26      ! [A: real,B: real,P: real > real > $o] :
% 9.49/5.26        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.26       => ( ! [A3: real,B3: real,C3: real] :
% 9.49/5.26              ( ( P @ A3 @ B3 )
% 9.49/5.26             => ( ( P @ B3 @ C3 )
% 9.49/5.26               => ( ( ord_less_eq_real @ A3 @ B3 )
% 9.49/5.26                 => ( ( ord_less_eq_real @ B3 @ C3 )
% 9.49/5.26                   => ( P @ A3 @ C3 ) ) ) ) )
% 9.49/5.26         => ( ! [X4: real] :
% 9.49/5.26                ( ( ord_less_eq_real @ A @ X4 )
% 9.49/5.26               => ( ( ord_less_eq_real @ X4 @ B )
% 9.49/5.26                 => ? [D3: real] :
% 9.49/5.26                      ( ( ord_less_real @ zero_zero_real @ D3 )
% 9.49/5.26                      & ! [A3: real,B3: real] :
% 9.49/5.26                          ( ( ( ord_less_eq_real @ A3 @ X4 )
% 9.49/5.26                            & ( ord_less_eq_real @ X4 @ B3 )
% 9.49/5.26                            & ( ord_less_real @ ( minus_minus_real @ B3 @ A3 ) @ D3 ) )
% 9.49/5.26                         => ( P @ A3 @ B3 ) ) ) ) )
% 9.49/5.26           => ( P @ A @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4162_nat__dvd__1__iff__1,axiom,
% 9.49/5.26      ! [M: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ M @ one_one_nat )
% 9.49/5.26        = ( M = one_one_nat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4163_dvd__0__right,axiom,
% 9.49/5.26      ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4164_dvd__0__right,axiom,
% 9.49/5.26      ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4165_dvd__0__right,axiom,
% 9.49/5.26      ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4166_dvd__0__right,axiom,
% 9.49/5.26      ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4167_dvd__0__right,axiom,
% 9.49/5.26      ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4168_dvd__0__right,axiom,
% 9.49/5.26      ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4169_dvd__0__left__iff,axiom,
% 9.49/5.26      ! [A: code_integer] :
% 9.49/5.26        ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 9.49/5.26        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4170_dvd__0__left__iff,axiom,
% 9.49/5.26      ! [A: complex] :
% 9.49/5.26        ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 9.49/5.26        = ( A = zero_zero_complex ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4171_dvd__0__left__iff,axiom,
% 9.49/5.26      ! [A: real] :
% 9.49/5.26        ( ( dvd_dvd_real @ zero_zero_real @ A )
% 9.49/5.26        = ( A = zero_zero_real ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4172_dvd__0__left__iff,axiom,
% 9.49/5.26      ! [A: rat] :
% 9.49/5.26        ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 9.49/5.26        = ( A = zero_zero_rat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4173_dvd__0__left__iff,axiom,
% 9.49/5.26      ! [A: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 9.49/5.26        = ( A = zero_zero_nat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4174_dvd__0__left__iff,axiom,
% 9.49/5.26      ! [A: int] :
% 9.49/5.26        ( ( dvd_dvd_int @ zero_zero_int @ A )
% 9.49/5.26        = ( A = zero_zero_int ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4175_dvd__1__iff__1,axiom,
% 9.49/5.26      ! [M: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 9.49/5.26        = ( M
% 9.49/5.26          = ( suc @ zero_zero_nat ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4176_dvd__1__left,axiom,
% 9.49/5.26      ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4177_nat__mult__dvd__cancel__disj,axiom,
% 9.49/5.26      ! [K: nat,M: nat,N: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.49/5.26        = ( ( K = zero_zero_nat )
% 9.49/5.26          | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4178_dvd__mult__cancel__left,axiom,
% 9.49/5.26      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.49/5.26        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 9.49/5.26        = ( ( C = zero_z3403309356797280102nteger )
% 9.49/5.26          | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4179_dvd__mult__cancel__left,axiom,
% 9.49/5.26      ! [C: complex,A: complex,B: complex] :
% 9.49/5.26        ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 9.49/5.26        = ( ( C = zero_zero_complex )
% 9.49/5.26          | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4180_dvd__mult__cancel__left,axiom,
% 9.49/5.26      ! [C: real,A: real,B: real] :
% 9.49/5.26        ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.49/5.26        = ( ( C = zero_zero_real )
% 9.49/5.26          | ( dvd_dvd_real @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4181_dvd__mult__cancel__left,axiom,
% 9.49/5.26      ! [C: rat,A: rat,B: rat] :
% 9.49/5.26        ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.49/5.26        = ( ( C = zero_zero_rat )
% 9.49/5.26          | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4182_dvd__mult__cancel__left,axiom,
% 9.49/5.26      ! [C: int,A: int,B: int] :
% 9.49/5.26        ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.49/5.26        = ( ( C = zero_zero_int )
% 9.49/5.26          | ( dvd_dvd_int @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4183_dvd__mult__cancel__right,axiom,
% 9.49/5.26      ! [A: code_integer,C: code_integer,B: code_integer] :
% 9.49/5.26        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 9.49/5.26        = ( ( C = zero_z3403309356797280102nteger )
% 9.49/5.26          | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4184_dvd__mult__cancel__right,axiom,
% 9.49/5.26      ! [A: complex,C: complex,B: complex] :
% 9.49/5.26        ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 9.49/5.26        = ( ( C = zero_zero_complex )
% 9.49/5.26          | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4185_dvd__mult__cancel__right,axiom,
% 9.49/5.26      ! [A: real,C: real,B: real] :
% 9.49/5.26        ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.49/5.26        = ( ( C = zero_zero_real )
% 9.49/5.26          | ( dvd_dvd_real @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4186_dvd__mult__cancel__right,axiom,
% 9.49/5.26      ! [A: rat,C: rat,B: rat] :
% 9.49/5.26        ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.49/5.26        = ( ( C = zero_zero_rat )
% 9.49/5.26          | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4187_dvd__mult__cancel__right,axiom,
% 9.49/5.26      ! [A: int,C: int,B: int] :
% 9.49/5.26        ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.49/5.26        = ( ( C = zero_zero_int )
% 9.49/5.26          | ( dvd_dvd_int @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4188_dvd__times__left__cancel__iff,axiom,
% 9.49/5.26      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.49/5.26        ( ( A != zero_z3403309356797280102nteger )
% 9.49/5.26       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
% 9.49/5.26          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4189_dvd__times__left__cancel__iff,axiom,
% 9.49/5.26      ! [A: nat,B: nat,C: nat] :
% 9.49/5.26        ( ( A != zero_zero_nat )
% 9.49/5.26       => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
% 9.49/5.26          = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4190_dvd__times__left__cancel__iff,axiom,
% 9.49/5.26      ! [A: int,B: int,C: int] :
% 9.49/5.26        ( ( A != zero_zero_int )
% 9.49/5.26       => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
% 9.49/5.26          = ( dvd_dvd_int @ B @ C ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4191_dvd__times__right__cancel__iff,axiom,
% 9.49/5.26      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.49/5.26        ( ( A != zero_z3403309356797280102nteger )
% 9.49/5.26       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
% 9.49/5.26          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4192_dvd__times__right__cancel__iff,axiom,
% 9.49/5.26      ! [A: nat,B: nat,C: nat] :
% 9.49/5.26        ( ( A != zero_zero_nat )
% 9.49/5.26       => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
% 9.49/5.26          = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4193_dvd__times__right__cancel__iff,axiom,
% 9.49/5.26      ! [A: int,B: int,C: int] :
% 9.49/5.26        ( ( A != zero_zero_int )
% 9.49/5.26       => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
% 9.49/5.26          = ( dvd_dvd_int @ B @ C ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4194_dvd__imp__mod__0,axiom,
% 9.49/5.26      ! [A: nat,B: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ A @ B )
% 9.49/5.26       => ( ( modulo_modulo_nat @ B @ A )
% 9.49/5.26          = zero_zero_nat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4195_dvd__imp__mod__0,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( dvd_dvd_int @ A @ B )
% 9.49/5.26       => ( ( modulo_modulo_int @ B @ A )
% 9.49/5.26          = zero_zero_int ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4196_dvd__imp__mod__0,axiom,
% 9.49/5.26      ! [A: code_integer,B: code_integer] :
% 9.49/5.26        ( ( dvd_dvd_Code_integer @ A @ B )
% 9.49/5.26       => ( ( modulo364778990260209775nteger @ B @ A )
% 9.49/5.26          = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4197_mod__neg__neg__trivial,axiom,
% 9.49/5.26      ! [K: int,L: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ K @ zero_zero_int )
% 9.49/5.26       => ( ( ord_less_int @ L @ K )
% 9.49/5.26         => ( ( modulo_modulo_int @ K @ L )
% 9.49/5.26            = K ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4198_mod__pos__pos__trivial,axiom,
% 9.49/5.26      ! [K: int,L: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.49/5.26       => ( ( ord_less_int @ K @ L )
% 9.49/5.26         => ( ( modulo_modulo_int @ K @ L )
% 9.49/5.26            = K ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4199_pow__divides__pow__iff,axiom,
% 9.49/5.26      ! [N: nat,A: nat,B: nat] :
% 9.49/5.26        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.26       => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 9.49/5.26          = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4200_pow__divides__pow__iff,axiom,
% 9.49/5.26      ! [N: nat,A: int,B: int] :
% 9.49/5.26        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.26       => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 9.49/5.26          = ( dvd_dvd_int @ A @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4201_length__product,axiom,
% 9.49/5.26      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 9.49/5.26        ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) )
% 9.49/5.26        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4202_length__product,axiom,
% 9.49/5.26      ! [Xs: list_VEBT_VEBT,Ys: list_o] :
% 9.49/5.26        ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) )
% 9.49/5.26        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4203_length__product,axiom,
% 9.49/5.26      ! [Xs: list_VEBT_VEBT,Ys: list_nat] :
% 9.49/5.26        ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys ) )
% 9.49/5.26        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4204_length__product,axiom,
% 9.49/5.26      ! [Xs: list_VEBT_VEBT,Ys: list_int] :
% 9.49/5.26        ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) )
% 9.49/5.26        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4205_length__product,axiom,
% 9.49/5.26      ! [Xs: list_o,Ys: list_VEBT_VEBT] :
% 9.49/5.26        ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys ) )
% 9.49/5.26        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4206_length__product,axiom,
% 9.49/5.26      ! [Xs: list_o,Ys: list_o] :
% 9.49/5.26        ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs @ Ys ) )
% 9.49/5.26        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4207_length__product,axiom,
% 9.49/5.26      ! [Xs: list_o,Ys: list_nat] :
% 9.49/5.26        ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs @ Ys ) )
% 9.49/5.26        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4208_length__product,axiom,
% 9.49/5.26      ! [Xs: list_o,Ys: list_int] :
% 9.49/5.26        ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs @ Ys ) )
% 9.49/5.26        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4209_length__product,axiom,
% 9.49/5.26      ! [Xs: list_nat,Ys: list_VEBT_VEBT] :
% 9.49/5.26        ( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys ) )
% 9.49/5.26        = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4210_length__product,axiom,
% 9.49/5.26      ! [Xs: list_nat,Ys: list_o] :
% 9.49/5.26        ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs @ Ys ) )
% 9.49/5.26        = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4211_zero__le__power__eq__numeral,axiom,
% 9.49/5.26      ! [A: real,W2: num] :
% 9.49/5.26        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 9.49/5.26        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26            & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4212_zero__le__power__eq__numeral,axiom,
% 9.49/5.26      ! [A: rat,W2: num] :
% 9.49/5.26        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 9.49/5.26        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26            & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4213_zero__le__power__eq__numeral,axiom,
% 9.49/5.26      ! [A: int,W2: num] :
% 9.49/5.26        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 9.49/5.26        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26            & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4214_power__less__zero__eq__numeral,axiom,
% 9.49/5.26      ! [A: real,W2: num] :
% 9.49/5.26        ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
% 9.49/5.26        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26          & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4215_power__less__zero__eq__numeral,axiom,
% 9.49/5.26      ! [A: rat,W2: num] :
% 9.49/5.26        ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
% 9.49/5.26        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26          & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4216_power__less__zero__eq__numeral,axiom,
% 9.49/5.26      ! [A: int,W2: num] :
% 9.49/5.26        ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
% 9.49/5.26        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26          & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4217_power__less__zero__eq,axiom,
% 9.49/5.26      ! [A: real,N: nat] :
% 9.49/5.26        ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 9.49/5.26        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.26          & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4218_power__less__zero__eq,axiom,
% 9.49/5.26      ! [A: rat,N: nat] :
% 9.49/5.26        ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 9.49/5.26        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.26          & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4219_power__less__zero__eq,axiom,
% 9.49/5.26      ! [A: int,N: nat] :
% 9.49/5.26        ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 9.49/5.26        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.26          & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4220_odd__Suc__minus__one,axiom,
% 9.49/5.26      ! [N: nat] :
% 9.49/5.26        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.26       => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 9.49/5.26          = N ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4221_even__diff__nat,axiom,
% 9.49/5.26      ! [M: nat,N: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
% 9.49/5.26        = ( ( ord_less_nat @ M @ N )
% 9.49/5.26          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4222_zero__less__power__eq__numeral,axiom,
% 9.49/5.26      ! [A: real,W2: num] :
% 9.49/5.26        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 9.49/5.26        = ( ( ( numeral_numeral_nat @ W2 )
% 9.49/5.26            = zero_zero_nat )
% 9.49/5.26          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26            & ( A != zero_zero_real ) )
% 9.49/5.26          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26            & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4223_zero__less__power__eq__numeral,axiom,
% 9.49/5.26      ! [A: rat,W2: num] :
% 9.49/5.26        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 9.49/5.26        = ( ( ( numeral_numeral_nat @ W2 )
% 9.49/5.26            = zero_zero_nat )
% 9.49/5.26          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26            & ( A != zero_zero_rat ) )
% 9.49/5.26          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26            & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4224_zero__less__power__eq__numeral,axiom,
% 9.49/5.26      ! [A: int,W2: num] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 9.49/5.26        = ( ( ( numeral_numeral_nat @ W2 )
% 9.49/5.26            = zero_zero_nat )
% 9.49/5.26          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26            & ( A != zero_zero_int ) )
% 9.49/5.26          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26            & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4225_even__power,axiom,
% 9.49/5.26      ! [A: code_integer,N: nat] :
% 9.49/5.26        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
% 9.49/5.26        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.49/5.26          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4226_even__power,axiom,
% 9.49/5.26      ! [A: nat,N: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
% 9.49/5.26        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.49/5.26          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4227_even__power,axiom,
% 9.49/5.26      ! [A: int,N: nat] :
% 9.49/5.26        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
% 9.49/5.26        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.49/5.26          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4228_power__le__zero__eq__numeral,axiom,
% 9.49/5.26      ! [A: real,W2: num] :
% 9.49/5.26        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
% 9.49/5.26        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26              & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 9.49/5.26            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26              & ( A = zero_zero_real ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4229_power__le__zero__eq__numeral,axiom,
% 9.49/5.26      ! [A: rat,W2: num] :
% 9.49/5.26        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
% 9.49/5.26        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26              & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 9.49/5.26            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26              & ( A = zero_zero_rat ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4230_power__le__zero__eq__numeral,axiom,
% 9.49/5.26      ! [A: int,W2: num] :
% 9.49/5.26        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
% 9.49/5.26        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26              & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 9.49/5.26            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.49/5.26              & ( A = zero_zero_int ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4231_semiring__parity__class_Oeven__mask__iff,axiom,
% 9.49/5.26      ! [N: nat] :
% 9.49/5.26        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
% 9.49/5.26        = ( N = zero_zero_nat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4232_semiring__parity__class_Oeven__mask__iff,axiom,
% 9.49/5.26      ! [N: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
% 9.49/5.26        = ( N = zero_zero_nat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4233_semiring__parity__class_Oeven__mask__iff,axiom,
% 9.49/5.26      ! [N: nat] :
% 9.49/5.26        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 9.49/5.26        = ( N = zero_zero_nat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4234_even__succ__div__exp,axiom,
% 9.49/5.26      ! [A: nat,N: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.49/5.26       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.26         => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.26            = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4235_even__succ__div__exp,axiom,
% 9.49/5.26      ! [A: int,N: nat] :
% 9.49/5.26        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.49/5.26       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.26         => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.26            = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4236_even__succ__div__exp,axiom,
% 9.49/5.26      ! [A: code_integer,N: nat] :
% 9.49/5.26        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.49/5.26       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.26         => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.26            = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4237_dvd__field__iff,axiom,
% 9.49/5.26      ( dvd_dvd_complex
% 9.49/5.26      = ( ^ [A4: complex,B4: complex] :
% 9.49/5.26            ( ( A4 = zero_zero_complex )
% 9.49/5.26           => ( B4 = zero_zero_complex ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4238_dvd__field__iff,axiom,
% 9.49/5.26      ( dvd_dvd_real
% 9.49/5.26      = ( ^ [A4: real,B4: real] :
% 9.49/5.26            ( ( A4 = zero_zero_real )
% 9.49/5.26           => ( B4 = zero_zero_real ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4239_dvd__field__iff,axiom,
% 9.49/5.26      ( dvd_dvd_rat
% 9.49/5.26      = ( ^ [A4: rat,B4: rat] :
% 9.49/5.26            ( ( A4 = zero_zero_rat )
% 9.49/5.26           => ( B4 = zero_zero_rat ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4240_dvd__0__left,axiom,
% 9.49/5.26      ! [A: code_integer] :
% 9.49/5.26        ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 9.49/5.26       => ( A = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4241_dvd__0__left,axiom,
% 9.49/5.26      ! [A: complex] :
% 9.49/5.26        ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 9.49/5.26       => ( A = zero_zero_complex ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4242_dvd__0__left,axiom,
% 9.49/5.26      ! [A: real] :
% 9.49/5.26        ( ( dvd_dvd_real @ zero_zero_real @ A )
% 9.49/5.26       => ( A = zero_zero_real ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4243_dvd__0__left,axiom,
% 9.49/5.26      ! [A: rat] :
% 9.49/5.26        ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 9.49/5.26       => ( A = zero_zero_rat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4244_dvd__0__left,axiom,
% 9.49/5.26      ! [A: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 9.49/5.26       => ( A = zero_zero_nat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4245_dvd__0__left,axiom,
% 9.49/5.26      ! [A: int] :
% 9.49/5.26        ( ( dvd_dvd_int @ zero_zero_int @ A )
% 9.49/5.26       => ( A = zero_zero_int ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4246_gcd__nat_Oextremum,axiom,
% 9.49/5.26      ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4247_gcd__nat_Oextremum__strict,axiom,
% 9.49/5.26      ! [A: nat] :
% 9.49/5.26        ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 9.49/5.26          & ( zero_zero_nat != A ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4248_gcd__nat_Oextremum__unique,axiom,
% 9.49/5.26      ! [A: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 9.49/5.26        = ( A = zero_zero_nat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4249_gcd__nat_Onot__eq__extremum,axiom,
% 9.49/5.26      ! [A: nat] :
% 9.49/5.26        ( ( A != zero_zero_nat )
% 9.49/5.26        = ( ( dvd_dvd_nat @ A @ zero_zero_nat )
% 9.49/5.26          & ( A != zero_zero_nat ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4250_gcd__nat_Oextremum__uniqueI,axiom,
% 9.49/5.26      ! [A: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 9.49/5.26       => ( A = zero_zero_nat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4251_dvd__diff__nat,axiom,
% 9.49/5.26      ! [K: nat,M: nat,N: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ K @ M )
% 9.49/5.26       => ( ( dvd_dvd_nat @ K @ N )
% 9.49/5.26         => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4252_mod__int__pos__iff,axiom,
% 9.49/5.26      ! [K: int,L: int] :
% 9.49/5.26        ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
% 9.49/5.26        = ( ( dvd_dvd_int @ L @ K )
% 9.49/5.26          | ( ( L = zero_zero_int )
% 9.49/5.26            & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 9.49/5.26          | ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4253_subset__divisors__dvd,axiom,
% 9.49/5.26      ! [A: real,B: real] :
% 9.49/5.26        ( ( ord_less_eq_set_real
% 9.49/5.26          @ ( collect_real
% 9.49/5.26            @ ^ [C4: real] : ( dvd_dvd_real @ C4 @ A ) )
% 9.49/5.26          @ ( collect_real
% 9.49/5.26            @ ^ [C4: real] : ( dvd_dvd_real @ C4 @ B ) ) )
% 9.49/5.26        = ( dvd_dvd_real @ A @ B ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4254_subset__divisors__dvd,axiom,
% 9.49/5.26      ! [A: nat,B: nat] :
% 9.49/5.26        ( ( ord_less_eq_set_nat
% 9.49/5.26          @ ( collect_nat
% 9.49/5.26            @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ A ) )
% 9.49/5.26          @ ( collect_nat
% 9.49/5.26            @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ B ) ) )
% 9.49/5.26        = ( dvd_dvd_nat @ A @ B ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4255_subset__divisors__dvd,axiom,
% 9.49/5.26      ! [A: code_integer,B: code_integer] :
% 9.49/5.26        ( ( ord_le7084787975880047091nteger
% 9.49/5.26          @ ( collect_Code_integer
% 9.49/5.26            @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ A ) )
% 9.49/5.26          @ ( collect_Code_integer
% 9.49/5.26            @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ B ) ) )
% 9.49/5.26        = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4256_subset__divisors__dvd,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( ord_less_eq_set_int
% 9.49/5.26          @ ( collect_int
% 9.49/5.26            @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ A ) )
% 9.49/5.26          @ ( collect_int
% 9.49/5.26            @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ B ) ) )
% 9.49/5.26        = ( dvd_dvd_int @ A @ B ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4257_strict__subset__divisors__dvd,axiom,
% 9.49/5.26      ! [A: real,B: real] :
% 9.49/5.26        ( ( ord_less_set_real
% 9.49/5.26          @ ( collect_real
% 9.49/5.26            @ ^ [C4: real] : ( dvd_dvd_real @ C4 @ A ) )
% 9.49/5.26          @ ( collect_real
% 9.49/5.26            @ ^ [C4: real] : ( dvd_dvd_real @ C4 @ B ) ) )
% 9.49/5.26        = ( ( dvd_dvd_real @ A @ B )
% 9.49/5.26          & ~ ( dvd_dvd_real @ B @ A ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4258_strict__subset__divisors__dvd,axiom,
% 9.49/5.26      ! [A: nat,B: nat] :
% 9.49/5.26        ( ( ord_less_set_nat
% 9.49/5.26          @ ( collect_nat
% 9.49/5.26            @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ A ) )
% 9.49/5.26          @ ( collect_nat
% 9.49/5.26            @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ B ) ) )
% 9.49/5.26        = ( ( dvd_dvd_nat @ A @ B )
% 9.49/5.26          & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4259_strict__subset__divisors__dvd,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( ord_less_set_int
% 9.49/5.26          @ ( collect_int
% 9.49/5.26            @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ A ) )
% 9.49/5.26          @ ( collect_int
% 9.49/5.26            @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ B ) ) )
% 9.49/5.26        = ( ( dvd_dvd_int @ A @ B )
% 9.49/5.26          & ~ ( dvd_dvd_int @ B @ A ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4260_strict__subset__divisors__dvd,axiom,
% 9.49/5.26      ! [A: code_integer,B: code_integer] :
% 9.49/5.26        ( ( ord_le1307284697595431911nteger
% 9.49/5.26          @ ( collect_Code_integer
% 9.49/5.26            @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ A ) )
% 9.49/5.26          @ ( collect_Code_integer
% 9.49/5.26            @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ B ) ) )
% 9.49/5.26        = ( ( dvd_dvd_Code_integer @ A @ B )
% 9.49/5.26          & ~ ( dvd_dvd_Code_integer @ B @ A ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4261_not__is__unit__0,axiom,
% 9.49/5.26      ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4262_not__is__unit__0,axiom,
% 9.49/5.26      ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4263_not__is__unit__0,axiom,
% 9.49/5.26      ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4264_pinf_I9_J,axiom,
% 9.49/5.26      ! [D: code_integer,S: code_integer] :
% 9.49/5.26      ? [Z4: code_integer] :
% 9.49/5.26      ! [X2: code_integer] :
% 9.49/5.26        ( ( ord_le6747313008572928689nteger @ Z4 @ X2 )
% 9.49/5.26       => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) )
% 9.49/5.26          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4265_pinf_I9_J,axiom,
% 9.49/5.26      ! [D: real,S: real] :
% 9.49/5.26      ? [Z4: real] :
% 9.49/5.26      ! [X2: real] :
% 9.49/5.26        ( ( ord_less_real @ Z4 @ X2 )
% 9.49/5.26       => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) )
% 9.49/5.26          = ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4266_pinf_I9_J,axiom,
% 9.49/5.26      ! [D: rat,S: rat] :
% 9.49/5.26      ? [Z4: rat] :
% 9.49/5.26      ! [X2: rat] :
% 9.49/5.26        ( ( ord_less_rat @ Z4 @ X2 )
% 9.49/5.26       => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) )
% 9.49/5.26          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4267_pinf_I9_J,axiom,
% 9.49/5.26      ! [D: nat,S: nat] :
% 9.49/5.26      ? [Z4: nat] :
% 9.49/5.26      ! [X2: nat] :
% 9.49/5.26        ( ( ord_less_nat @ Z4 @ X2 )
% 9.49/5.26       => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) )
% 9.49/5.26          = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4268_pinf_I9_J,axiom,
% 9.49/5.26      ! [D: int,S: int] :
% 9.49/5.26      ? [Z4: int] :
% 9.49/5.26      ! [X2: int] :
% 9.49/5.26        ( ( ord_less_int @ Z4 @ X2 )
% 9.49/5.26       => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) )
% 9.49/5.26          = ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4269_pinf_I10_J,axiom,
% 9.49/5.26      ! [D: code_integer,S: code_integer] :
% 9.49/5.26      ? [Z4: code_integer] :
% 9.49/5.26      ! [X2: code_integer] :
% 9.49/5.26        ( ( ord_le6747313008572928689nteger @ Z4 @ X2 )
% 9.49/5.26       => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) )
% 9.49/5.26          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4270_pinf_I10_J,axiom,
% 9.49/5.26      ! [D: real,S: real] :
% 9.49/5.26      ? [Z4: real] :
% 9.49/5.26      ! [X2: real] :
% 9.49/5.26        ( ( ord_less_real @ Z4 @ X2 )
% 9.49/5.26       => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) )
% 9.49/5.26          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4271_pinf_I10_J,axiom,
% 9.49/5.26      ! [D: rat,S: rat] :
% 9.49/5.26      ? [Z4: rat] :
% 9.49/5.26      ! [X2: rat] :
% 9.49/5.26        ( ( ord_less_rat @ Z4 @ X2 )
% 9.49/5.26       => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) )
% 9.49/5.26          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4272_pinf_I10_J,axiom,
% 9.49/5.26      ! [D: nat,S: nat] :
% 9.49/5.26      ? [Z4: nat] :
% 9.49/5.26      ! [X2: nat] :
% 9.49/5.26        ( ( ord_less_nat @ Z4 @ X2 )
% 9.49/5.26       => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) )
% 9.49/5.26          = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4273_pinf_I10_J,axiom,
% 9.49/5.26      ! [D: int,S: int] :
% 9.49/5.26      ? [Z4: int] :
% 9.49/5.26      ! [X2: int] :
% 9.49/5.26        ( ( ord_less_int @ Z4 @ X2 )
% 9.49/5.26       => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) )
% 9.49/5.26          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4274_minf_I9_J,axiom,
% 9.49/5.26      ! [D: code_integer,S: code_integer] :
% 9.49/5.26      ? [Z4: code_integer] :
% 9.49/5.26      ! [X2: code_integer] :
% 9.49/5.26        ( ( ord_le6747313008572928689nteger @ X2 @ Z4 )
% 9.49/5.26       => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) )
% 9.49/5.26          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4275_minf_I9_J,axiom,
% 9.49/5.26      ! [D: real,S: real] :
% 9.49/5.26      ? [Z4: real] :
% 9.49/5.26      ! [X2: real] :
% 9.49/5.26        ( ( ord_less_real @ X2 @ Z4 )
% 9.49/5.26       => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) )
% 9.49/5.26          = ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4276_minf_I9_J,axiom,
% 9.49/5.26      ! [D: rat,S: rat] :
% 9.49/5.26      ? [Z4: rat] :
% 9.49/5.26      ! [X2: rat] :
% 9.49/5.26        ( ( ord_less_rat @ X2 @ Z4 )
% 9.49/5.26       => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) )
% 9.49/5.26          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4277_minf_I9_J,axiom,
% 9.49/5.26      ! [D: nat,S: nat] :
% 9.49/5.26      ? [Z4: nat] :
% 9.49/5.26      ! [X2: nat] :
% 9.49/5.26        ( ( ord_less_nat @ X2 @ Z4 )
% 9.49/5.26       => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) )
% 9.49/5.26          = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4278_minf_I9_J,axiom,
% 9.49/5.26      ! [D: int,S: int] :
% 9.49/5.26      ? [Z4: int] :
% 9.49/5.26      ! [X2: int] :
% 9.49/5.26        ( ( ord_less_int @ X2 @ Z4 )
% 9.49/5.26       => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) )
% 9.49/5.26          = ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4279_minf_I10_J,axiom,
% 9.49/5.26      ! [D: code_integer,S: code_integer] :
% 9.49/5.26      ? [Z4: code_integer] :
% 9.49/5.26      ! [X2: code_integer] :
% 9.49/5.26        ( ( ord_le6747313008572928689nteger @ X2 @ Z4 )
% 9.49/5.26       => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) )
% 9.49/5.26          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4280_minf_I10_J,axiom,
% 9.49/5.26      ! [D: real,S: real] :
% 9.49/5.26      ? [Z4: real] :
% 9.49/5.26      ! [X2: real] :
% 9.49/5.26        ( ( ord_less_real @ X2 @ Z4 )
% 9.49/5.26       => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) )
% 9.49/5.26          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4281_minf_I10_J,axiom,
% 9.49/5.26      ! [D: rat,S: rat] :
% 9.49/5.26      ? [Z4: rat] :
% 9.49/5.26      ! [X2: rat] :
% 9.49/5.26        ( ( ord_less_rat @ X2 @ Z4 )
% 9.49/5.26       => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) )
% 9.49/5.26          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4282_minf_I10_J,axiom,
% 9.49/5.26      ! [D: nat,S: nat] :
% 9.49/5.26      ? [Z4: nat] :
% 9.49/5.26      ! [X2: nat] :
% 9.49/5.26        ( ( ord_less_nat @ X2 @ Z4 )
% 9.49/5.26       => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) )
% 9.49/5.26          = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4283_minf_I10_J,axiom,
% 9.49/5.26      ! [D: int,S: int] :
% 9.49/5.26      ? [Z4: int] :
% 9.49/5.26      ! [X2: int] :
% 9.49/5.26        ( ( ord_less_int @ X2 @ Z4 )
% 9.49/5.26       => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) )
% 9.49/5.26          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4284_dvd__div__eq__0__iff,axiom,
% 9.49/5.26      ! [B: complex,A: complex] :
% 9.49/5.26        ( ( dvd_dvd_complex @ B @ A )
% 9.49/5.26       => ( ( ( divide1717551699836669952omplex @ A @ B )
% 9.49/5.26            = zero_zero_complex )
% 9.49/5.26          = ( A = zero_zero_complex ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4285_dvd__div__eq__0__iff,axiom,
% 9.49/5.26      ! [B: nat,A: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ B @ A )
% 9.49/5.26       => ( ( ( divide_divide_nat @ A @ B )
% 9.49/5.26            = zero_zero_nat )
% 9.49/5.26          = ( A = zero_zero_nat ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4286_dvd__div__eq__0__iff,axiom,
% 9.49/5.26      ! [B: int,A: int] :
% 9.49/5.26        ( ( dvd_dvd_int @ B @ A )
% 9.49/5.26       => ( ( ( divide_divide_int @ A @ B )
% 9.49/5.26            = zero_zero_int )
% 9.49/5.26          = ( A = zero_zero_int ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4287_dvd__div__eq__0__iff,axiom,
% 9.49/5.26      ! [B: real,A: real] :
% 9.49/5.26        ( ( dvd_dvd_real @ B @ A )
% 9.49/5.26       => ( ( ( divide_divide_real @ A @ B )
% 9.49/5.26            = zero_zero_real )
% 9.49/5.26          = ( A = zero_zero_real ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4288_dvd__div__eq__0__iff,axiom,
% 9.49/5.26      ! [B: rat,A: rat] :
% 9.49/5.26        ( ( dvd_dvd_rat @ B @ A )
% 9.49/5.26       => ( ( ( divide_divide_rat @ A @ B )
% 9.49/5.26            = zero_zero_rat )
% 9.49/5.26          = ( A = zero_zero_rat ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4289_dvd__div__eq__0__iff,axiom,
% 9.49/5.26      ! [B: code_integer,A: code_integer] :
% 9.49/5.26        ( ( dvd_dvd_Code_integer @ B @ A )
% 9.49/5.26       => ( ( ( divide6298287555418463151nteger @ A @ B )
% 9.49/5.26            = zero_z3403309356797280102nteger )
% 9.49/5.26          = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4290_mod__0__imp__dvd,axiom,
% 9.49/5.26      ! [A: nat,B: nat] :
% 9.49/5.26        ( ( ( modulo_modulo_nat @ A @ B )
% 9.49/5.26          = zero_zero_nat )
% 9.49/5.26       => ( dvd_dvd_nat @ B @ A ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4291_mod__0__imp__dvd,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( ( modulo_modulo_int @ A @ B )
% 9.49/5.26          = zero_zero_int )
% 9.49/5.26       => ( dvd_dvd_int @ B @ A ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4292_mod__0__imp__dvd,axiom,
% 9.49/5.26      ! [A: code_integer,B: code_integer] :
% 9.49/5.26        ( ( ( modulo364778990260209775nteger @ A @ B )
% 9.49/5.26          = zero_z3403309356797280102nteger )
% 9.49/5.26       => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4293_dvd__eq__mod__eq__0,axiom,
% 9.49/5.26      ( dvd_dvd_nat
% 9.49/5.26      = ( ^ [A4: nat,B4: nat] :
% 9.49/5.26            ( ( modulo_modulo_nat @ B4 @ A4 )
% 9.49/5.26            = zero_zero_nat ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4294_dvd__eq__mod__eq__0,axiom,
% 9.49/5.26      ( dvd_dvd_int
% 9.49/5.26      = ( ^ [A4: int,B4: int] :
% 9.49/5.26            ( ( modulo_modulo_int @ B4 @ A4 )
% 9.49/5.26            = zero_zero_int ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4295_dvd__eq__mod__eq__0,axiom,
% 9.49/5.26      ( dvd_dvd_Code_integer
% 9.49/5.26      = ( ^ [A4: code_integer,B4: code_integer] :
% 9.49/5.26            ( ( modulo364778990260209775nteger @ B4 @ A4 )
% 9.49/5.26            = zero_z3403309356797280102nteger ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4296_mod__eq__0__iff__dvd,axiom,
% 9.49/5.26      ! [A: nat,B: nat] :
% 9.49/5.26        ( ( ( modulo_modulo_nat @ A @ B )
% 9.49/5.26          = zero_zero_nat )
% 9.49/5.26        = ( dvd_dvd_nat @ B @ A ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4297_mod__eq__0__iff__dvd,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( ( modulo_modulo_int @ A @ B )
% 9.49/5.26          = zero_zero_int )
% 9.49/5.26        = ( dvd_dvd_int @ B @ A ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4298_mod__eq__0__iff__dvd,axiom,
% 9.49/5.26      ! [A: code_integer,B: code_integer] :
% 9.49/5.26        ( ( ( modulo364778990260209775nteger @ A @ B )
% 9.49/5.26          = zero_z3403309356797280102nteger )
% 9.49/5.26        = ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4299_dvd__power__le,axiom,
% 9.49/5.26      ! [X: code_integer,Y: code_integer,N: nat,M: nat] :
% 9.49/5.26        ( ( dvd_dvd_Code_integer @ X @ Y )
% 9.49/5.26       => ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.26         => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ M ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4300_dvd__power__le,axiom,
% 9.49/5.26      ! [X: nat,Y: nat,N: nat,M: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ X @ Y )
% 9.49/5.26       => ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.26         => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ M ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4301_dvd__power__le,axiom,
% 9.49/5.26      ! [X: real,Y: real,N: nat,M: nat] :
% 9.49/5.26        ( ( dvd_dvd_real @ X @ Y )
% 9.49/5.26       => ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.26         => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ M ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4302_dvd__power__le,axiom,
% 9.49/5.26      ! [X: int,Y: int,N: nat,M: nat] :
% 9.49/5.26        ( ( dvd_dvd_int @ X @ Y )
% 9.49/5.26       => ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.26         => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ M ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4303_dvd__power__le,axiom,
% 9.49/5.26      ! [X: complex,Y: complex,N: nat,M: nat] :
% 9.49/5.26        ( ( dvd_dvd_complex @ X @ Y )
% 9.49/5.26       => ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.26         => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ M ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4304_power__le__dvd,axiom,
% 9.49/5.26      ! [A: code_integer,N: nat,B: code_integer,M: nat] :
% 9.49/5.26        ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
% 9.49/5.26       => ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.26         => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4305_power__le__dvd,axiom,
% 9.49/5.26      ! [A: nat,N: nat,B: nat,M: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
% 9.49/5.26       => ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.26         => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4306_power__le__dvd,axiom,
% 9.49/5.26      ! [A: real,N: nat,B: real,M: nat] :
% 9.49/5.26        ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
% 9.49/5.26       => ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.26         => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4307_power__le__dvd,axiom,
% 9.49/5.26      ! [A: int,N: nat,B: int,M: nat] :
% 9.49/5.26        ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
% 9.49/5.26       => ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.26         => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4308_power__le__dvd,axiom,
% 9.49/5.26      ! [A: complex,N: nat,B: complex,M: nat] :
% 9.49/5.26        ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
% 9.49/5.26       => ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.26         => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4309_le__imp__power__dvd,axiom,
% 9.49/5.26      ! [M: nat,N: nat,A: code_integer] :
% 9.49/5.26        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.26       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4310_le__imp__power__dvd,axiom,
% 9.49/5.26      ! [M: nat,N: nat,A: nat] :
% 9.49/5.26        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.26       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4311_le__imp__power__dvd,axiom,
% 9.49/5.26      ! [M: nat,N: nat,A: real] :
% 9.49/5.26        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.26       => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4312_le__imp__power__dvd,axiom,
% 9.49/5.26      ! [M: nat,N: nat,A: int] :
% 9.49/5.26        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.26       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4313_le__imp__power__dvd,axiom,
% 9.49/5.26      ! [M: nat,N: nat,A: complex] :
% 9.49/5.26        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.26       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4314_Euclidean__Division_Opos__mod__bound,axiom,
% 9.49/5.26      ! [L: int,K: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ L )
% 9.49/5.26       => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4315_neg__mod__bound,axiom,
% 9.49/5.26      ! [L: int,K: int] :
% 9.49/5.26        ( ( ord_less_int @ L @ zero_zero_int )
% 9.49/5.26       => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4316_nat__dvd__not__less,axiom,
% 9.49/5.26      ! [M: nat,N: nat] :
% 9.49/5.26        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.49/5.26       => ( ( ord_less_nat @ M @ N )
% 9.49/5.26         => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4317_dvd__pos__nat,axiom,
% 9.49/5.26      ! [N: nat,M: nat] :
% 9.49/5.26        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.26       => ( ( dvd_dvd_nat @ M @ N )
% 9.49/5.26         => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4318_dvd__minus__self,axiom,
% 9.49/5.26      ! [M: nat,N: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
% 9.49/5.26        = ( ( ord_less_nat @ N @ M )
% 9.49/5.26          | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4319_less__eq__dvd__minus,axiom,
% 9.49/5.26      ! [M: nat,N: nat] :
% 9.49/5.26        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.26       => ( ( dvd_dvd_nat @ M @ N )
% 9.49/5.26          = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4320_dvd__diffD1,axiom,
% 9.49/5.26      ! [K: nat,M: nat,N: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 9.49/5.26       => ( ( dvd_dvd_nat @ K @ M )
% 9.49/5.26         => ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.26           => ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4321_dvd__diffD,axiom,
% 9.49/5.26      ! [K: nat,M: nat,N: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 9.49/5.26       => ( ( dvd_dvd_nat @ K @ N )
% 9.49/5.26         => ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.26           => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4322_zdvd__not__zless,axiom,
% 9.49/5.26      ! [M: int,N: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ M )
% 9.49/5.26       => ( ( ord_less_int @ M @ N )
% 9.49/5.26         => ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4323_mod__int__unique,axiom,
% 9.49/5.26      ! [K: int,L: int,Q4: int,R2: int] :
% 9.49/5.26        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.49/5.26       => ( ( modulo_modulo_int @ K @ L )
% 9.49/5.26          = R2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4324_finite__divisors__int,axiom,
% 9.49/5.26      ! [I: int] :
% 9.49/5.26        ( ( I != zero_zero_int )
% 9.49/5.26       => ( finite_finite_int
% 9.49/5.26          @ ( collect_int
% 9.49/5.26            @ ^ [D4: int] : ( dvd_dvd_int @ D4 @ I ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4325_unity__coeff__ex,axiom,
% 9.49/5.26      ! [P: code_integer > $o,L: code_integer] :
% 9.49/5.26        ( ( ? [X3: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X3 ) ) )
% 9.49/5.26        = ( ? [X3: code_integer] :
% 9.49/5.26              ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X3 @ zero_z3403309356797280102nteger ) )
% 9.49/5.26              & ( P @ X3 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4326_unity__coeff__ex,axiom,
% 9.49/5.26      ! [P: complex > $o,L: complex] :
% 9.49/5.26        ( ( ? [X3: complex] : ( P @ ( times_times_complex @ L @ X3 ) ) )
% 9.49/5.26        = ( ? [X3: complex] :
% 9.49/5.26              ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X3 @ zero_zero_complex ) )
% 9.49/5.26              & ( P @ X3 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4327_unity__coeff__ex,axiom,
% 9.49/5.26      ! [P: real > $o,L: real] :
% 9.49/5.26        ( ( ? [X3: real] : ( P @ ( times_times_real @ L @ X3 ) ) )
% 9.49/5.26        = ( ? [X3: real] :
% 9.49/5.26              ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X3 @ zero_zero_real ) )
% 9.49/5.26              & ( P @ X3 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4328_unity__coeff__ex,axiom,
% 9.49/5.26      ! [P: rat > $o,L: rat] :
% 9.49/5.26        ( ( ? [X3: rat] : ( P @ ( times_times_rat @ L @ X3 ) ) )
% 9.49/5.26        = ( ? [X3: rat] :
% 9.49/5.26              ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X3 @ zero_zero_rat ) )
% 9.49/5.26              & ( P @ X3 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4329_unity__coeff__ex,axiom,
% 9.49/5.26      ! [P: nat > $o,L: nat] :
% 9.49/5.26        ( ( ? [X3: nat] : ( P @ ( times_times_nat @ L @ X3 ) ) )
% 9.49/5.26        = ( ? [X3: nat] :
% 9.49/5.26              ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X3 @ zero_zero_nat ) )
% 9.49/5.26              & ( P @ X3 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4330_unity__coeff__ex,axiom,
% 9.49/5.26      ! [P: int > $o,L: int] :
% 9.49/5.26        ( ( ? [X3: int] : ( P @ ( times_times_int @ L @ X3 ) ) )
% 9.49/5.26        = ( ? [X3: int] :
% 9.49/5.26              ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X3 @ zero_zero_int ) )
% 9.49/5.26              & ( P @ X3 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4331_unit__dvdE,axiom,
% 9.49/5.26      ! [A: code_integer,B: code_integer] :
% 9.49/5.26        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.49/5.26       => ~ ( ( A != zero_z3403309356797280102nteger )
% 9.49/5.26           => ! [C3: code_integer] :
% 9.49/5.26                ( B
% 9.49/5.26               != ( times_3573771949741848930nteger @ A @ C3 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4332_unit__dvdE,axiom,
% 9.49/5.26      ! [A: nat,B: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.49/5.26       => ~ ( ( A != zero_zero_nat )
% 9.49/5.26           => ! [C3: nat] :
% 9.49/5.26                ( B
% 9.49/5.26               != ( times_times_nat @ A @ C3 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4333_unit__dvdE,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.49/5.26       => ~ ( ( A != zero_zero_int )
% 9.49/5.26           => ! [C3: int] :
% 9.49/5.26                ( B
% 9.49/5.26               != ( times_times_int @ A @ C3 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4334_dvd__div__div__eq__mult,axiom,
% 9.49/5.26      ! [A: nat,C: nat,B: nat,D: nat] :
% 9.49/5.26        ( ( A != zero_zero_nat )
% 9.49/5.26       => ( ( C != zero_zero_nat )
% 9.49/5.26         => ( ( dvd_dvd_nat @ A @ B )
% 9.49/5.26           => ( ( dvd_dvd_nat @ C @ D )
% 9.49/5.26             => ( ( ( divide_divide_nat @ B @ A )
% 9.49/5.26                  = ( divide_divide_nat @ D @ C ) )
% 9.49/5.26                = ( ( times_times_nat @ B @ C )
% 9.49/5.26                  = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4335_dvd__div__div__eq__mult,axiom,
% 9.49/5.26      ! [A: int,C: int,B: int,D: int] :
% 9.49/5.26        ( ( A != zero_zero_int )
% 9.49/5.26       => ( ( C != zero_zero_int )
% 9.49/5.26         => ( ( dvd_dvd_int @ A @ B )
% 9.49/5.26           => ( ( dvd_dvd_int @ C @ D )
% 9.49/5.26             => ( ( ( divide_divide_int @ B @ A )
% 9.49/5.26                  = ( divide_divide_int @ D @ C ) )
% 9.49/5.26                = ( ( times_times_int @ B @ C )
% 9.49/5.26                  = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4336_dvd__div__div__eq__mult,axiom,
% 9.49/5.26      ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 9.49/5.26        ( ( A != zero_z3403309356797280102nteger )
% 9.49/5.26       => ( ( C != zero_z3403309356797280102nteger )
% 9.49/5.26         => ( ( dvd_dvd_Code_integer @ A @ B )
% 9.49/5.26           => ( ( dvd_dvd_Code_integer @ C @ D )
% 9.49/5.26             => ( ( ( divide6298287555418463151nteger @ B @ A )
% 9.49/5.26                  = ( divide6298287555418463151nteger @ D @ C ) )
% 9.49/5.26                = ( ( times_3573771949741848930nteger @ B @ C )
% 9.49/5.26                  = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4337_dvd__div__iff__mult,axiom,
% 9.49/5.26      ! [C: nat,B: nat,A: nat] :
% 9.49/5.26        ( ( C != zero_zero_nat )
% 9.49/5.26       => ( ( dvd_dvd_nat @ C @ B )
% 9.49/5.26         => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 9.49/5.26            = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4338_dvd__div__iff__mult,axiom,
% 9.49/5.26      ! [C: int,B: int,A: int] :
% 9.49/5.26        ( ( C != zero_zero_int )
% 9.49/5.26       => ( ( dvd_dvd_int @ C @ B )
% 9.49/5.26         => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
% 9.49/5.26            = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4339_dvd__div__iff__mult,axiom,
% 9.49/5.26      ! [C: code_integer,B: code_integer,A: code_integer] :
% 9.49/5.26        ( ( C != zero_z3403309356797280102nteger )
% 9.49/5.26       => ( ( dvd_dvd_Code_integer @ C @ B )
% 9.49/5.26         => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 9.49/5.26            = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4340_div__dvd__iff__mult,axiom,
% 9.49/5.26      ! [B: nat,A: nat,C: nat] :
% 9.49/5.26        ( ( B != zero_zero_nat )
% 9.49/5.26       => ( ( dvd_dvd_nat @ B @ A )
% 9.49/5.26         => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 9.49/5.26            = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4341_div__dvd__iff__mult,axiom,
% 9.49/5.26      ! [B: int,A: int,C: int] :
% 9.49/5.26        ( ( B != zero_zero_int )
% 9.49/5.26       => ( ( dvd_dvd_int @ B @ A )
% 9.49/5.26         => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 9.49/5.26            = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4342_div__dvd__iff__mult,axiom,
% 9.49/5.26      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.49/5.26        ( ( B != zero_z3403309356797280102nteger )
% 9.49/5.26       => ( ( dvd_dvd_Code_integer @ B @ A )
% 9.49/5.26         => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 9.49/5.26            = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4343_dvd__div__eq__mult,axiom,
% 9.49/5.26      ! [A: nat,B: nat,C: nat] :
% 9.49/5.26        ( ( A != zero_zero_nat )
% 9.49/5.26       => ( ( dvd_dvd_nat @ A @ B )
% 9.49/5.26         => ( ( ( divide_divide_nat @ B @ A )
% 9.49/5.26              = C )
% 9.49/5.26            = ( B
% 9.49/5.26              = ( times_times_nat @ C @ A ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4344_dvd__div__eq__mult,axiom,
% 9.49/5.26      ! [A: int,B: int,C: int] :
% 9.49/5.26        ( ( A != zero_zero_int )
% 9.49/5.26       => ( ( dvd_dvd_int @ A @ B )
% 9.49/5.26         => ( ( ( divide_divide_int @ B @ A )
% 9.49/5.26              = C )
% 9.49/5.26            = ( B
% 9.49/5.26              = ( times_times_int @ C @ A ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4345_dvd__div__eq__mult,axiom,
% 9.49/5.26      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.49/5.26        ( ( A != zero_z3403309356797280102nteger )
% 9.49/5.26       => ( ( dvd_dvd_Code_integer @ A @ B )
% 9.49/5.26         => ( ( ( divide6298287555418463151nteger @ B @ A )
% 9.49/5.26              = C )
% 9.49/5.26            = ( B
% 9.49/5.26              = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4346_unit__div__eq__0__iff,axiom,
% 9.49/5.26      ! [B: nat,A: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.49/5.26       => ( ( ( divide_divide_nat @ A @ B )
% 9.49/5.26            = zero_zero_nat )
% 9.49/5.26          = ( A = zero_zero_nat ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4347_unit__div__eq__0__iff,axiom,
% 9.49/5.26      ! [B: int,A: int] :
% 9.49/5.26        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.49/5.26       => ( ( ( divide_divide_int @ A @ B )
% 9.49/5.26            = zero_zero_int )
% 9.49/5.26          = ( A = zero_zero_int ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4348_unit__div__eq__0__iff,axiom,
% 9.49/5.26      ! [B: code_integer,A: code_integer] :
% 9.49/5.26        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.49/5.26       => ( ( ( divide6298287555418463151nteger @ A @ B )
% 9.49/5.26            = zero_z3403309356797280102nteger )
% 9.49/5.26          = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4349_is__unit__power__iff,axiom,
% 9.49/5.26      ! [A: code_integer,N: nat] :
% 9.49/5.26        ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
% 9.49/5.26        = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.49/5.26          | ( N = zero_zero_nat ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4350_is__unit__power__iff,axiom,
% 9.49/5.26      ! [A: nat,N: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
% 9.49/5.26        = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.49/5.26          | ( N = zero_zero_nat ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4351_is__unit__power__iff,axiom,
% 9.49/5.26      ! [A: int,N: nat] :
% 9.49/5.26        ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
% 9.49/5.26        = ( ( dvd_dvd_int @ A @ one_one_int )
% 9.49/5.26          | ( N = zero_zero_nat ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4352_unit__imp__mod__eq__0,axiom,
% 9.49/5.26      ! [B: nat,A: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.49/5.26       => ( ( modulo_modulo_nat @ A @ B )
% 9.49/5.26          = zero_zero_nat ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4353_unit__imp__mod__eq__0,axiom,
% 9.49/5.26      ! [B: int,A: int] :
% 9.49/5.26        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.49/5.26       => ( ( modulo_modulo_int @ A @ B )
% 9.49/5.26          = zero_zero_int ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4354_unit__imp__mod__eq__0,axiom,
% 9.49/5.26      ! [B: code_integer,A: code_integer] :
% 9.49/5.26        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.49/5.26       => ( ( modulo364778990260209775nteger @ A @ B )
% 9.49/5.26          = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4355_neg__mod__sign,axiom,
% 9.49/5.26      ! [L: int,K: int] :
% 9.49/5.26        ( ( ord_less_int @ L @ zero_zero_int )
% 9.49/5.26       => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4356_Euclidean__Division_Opos__mod__sign,axiom,
% 9.49/5.26      ! [L: int,K: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ L )
% 9.49/5.26       => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4357_zmod__trivial__iff,axiom,
% 9.49/5.26      ! [I: int,K: int] :
% 9.49/5.26        ( ( ( modulo_modulo_int @ I @ K )
% 9.49/5.26          = I )
% 9.49/5.26        = ( ( K = zero_zero_int )
% 9.49/5.26          | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 9.49/5.26            & ( ord_less_int @ I @ K ) )
% 9.49/5.26          | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 9.49/5.26            & ( ord_less_int @ K @ I ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4358_pos__mod__conj,axiom,
% 9.49/5.26      ! [B: int,A: int] :
% 9.49/5.26        ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.26       => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
% 9.49/5.26          & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4359_neg__mod__conj,axiom,
% 9.49/5.26      ! [B: int,A: int] :
% 9.49/5.26        ( ( ord_less_int @ B @ zero_zero_int )
% 9.49/5.26       => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
% 9.49/5.26          & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4360_dvd__imp__le,axiom,
% 9.49/5.26      ! [K: nat,N: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ K @ N )
% 9.49/5.26       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.26         => ( ord_less_eq_nat @ K @ N ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4361_nat__mult__dvd__cancel1,axiom,
% 9.49/5.26      ! [K: nat,M: nat,N: nat] :
% 9.49/5.26        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.26       => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.49/5.26          = ( dvd_dvd_nat @ M @ N ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4362_dvd__mult__cancel,axiom,
% 9.49/5.26      ! [K: nat,M: nat,N: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.49/5.26       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.26         => ( dvd_dvd_nat @ M @ N ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4363_zdiv__mono__strict,axiom,
% 9.49/5.26      ! [A2: int,B2: int,N: int] :
% 9.49/5.26        ( ( ord_less_int @ A2 @ B2 )
% 9.49/5.26       => ( ( ord_less_int @ zero_zero_int @ N )
% 9.49/5.26         => ( ( ( modulo_modulo_int @ A2 @ N )
% 9.49/5.26              = zero_zero_int )
% 9.49/5.26           => ( ( ( modulo_modulo_int @ B2 @ N )
% 9.49/5.26                = zero_zero_int )
% 9.49/5.26             => ( ord_less_int @ ( divide_divide_int @ A2 @ N ) @ ( divide_divide_int @ B2 @ N ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4364_bezout__add__strong__nat,axiom,
% 9.49/5.26      ! [A: nat,B: nat] :
% 9.49/5.26        ( ( A != zero_zero_nat )
% 9.49/5.26       => ? [D5: nat,X4: nat,Y3: nat] :
% 9.49/5.26            ( ( dvd_dvd_nat @ D5 @ A )
% 9.49/5.26            & ( dvd_dvd_nat @ D5 @ B )
% 9.49/5.26            & ( ( times_times_nat @ A @ X4 )
% 9.49/5.26              = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D5 ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4365_zdvd__imp__le,axiom,
% 9.49/5.26      ! [Z2: int,N: int] :
% 9.49/5.26        ( ( dvd_dvd_int @ Z2 @ N )
% 9.49/5.26       => ( ( ord_less_int @ zero_zero_int @ N )
% 9.49/5.26         => ( ord_less_eq_int @ Z2 @ N ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4366_mod__greater__zero__iff__not__dvd,axiom,
% 9.49/5.26      ! [M: nat,N: nat] :
% 9.49/5.26        ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
% 9.49/5.26        = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4367_mod__eq__dvd__iff__nat,axiom,
% 9.49/5.26      ! [N: nat,M: nat,Q4: nat] :
% 9.49/5.26        ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.26       => ( ( ( modulo_modulo_nat @ M @ Q4 )
% 9.49/5.26            = ( modulo_modulo_nat @ N @ Q4 ) )
% 9.49/5.26          = ( dvd_dvd_nat @ Q4 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4368_eucl__rel__int,axiom,
% 9.49/5.26      ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4369_prod__decode__aux_Ocases,axiom,
% 9.49/5.26      ! [X: product_prod_nat_nat] :
% 9.49/5.26        ~ ! [K2: nat,M5: nat] :
% 9.49/5.26            ( X
% 9.49/5.26           != ( product_Pair_nat_nat @ K2 @ M5 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4370_finite__divisors__nat,axiom,
% 9.49/5.26      ! [M: nat] :
% 9.49/5.26        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.49/5.26       => ( finite_finite_nat
% 9.49/5.26          @ ( collect_nat
% 9.49/5.26            @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4371_full__exhaustive__int_H_Ocases,axiom,
% 9.49/5.26      ! [X: produc2285326912895808259nt_int] :
% 9.49/5.26        ~ ! [F2: produc8551481072490612790e_term > option6357759511663192854e_term,D5: int,I2: int] :
% 9.49/5.26            ( X
% 9.49/5.26           != ( produc5700946648718959541nt_int @ F2 @ ( product_Pair_int_int @ D5 @ I2 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4372_exhaustive__int_H_Ocases,axiom,
% 9.49/5.26      ! [X: produc7773217078559923341nt_int] :
% 9.49/5.26        ~ ! [F2: int > option6357759511663192854e_term,D5: int,I2: int] :
% 9.49/5.26            ( X
% 9.49/5.26           != ( produc4305682042979456191nt_int @ F2 @ ( product_Pair_int_int @ D5 @ I2 ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4373_small__lazy_H_Ocases,axiom,
% 9.49/5.26      ! [X: product_prod_int_int] :
% 9.49/5.26        ~ ! [D5: int,I2: int] :
% 9.49/5.26            ( X
% 9.49/5.26           != ( product_Pair_int_int @ D5 @ I2 ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4374_even__zero,axiom,
% 9.49/5.26      dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4375_even__zero,axiom,
% 9.49/5.26      dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4376_even__zero,axiom,
% 9.49/5.26      dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4377_is__unit__div__mult__cancel__right,axiom,
% 9.49/5.26      ! [A: nat,B: nat] :
% 9.49/5.26        ( ( A != zero_zero_nat )
% 9.49/5.26       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.49/5.26         => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
% 9.49/5.26            = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4378_is__unit__div__mult__cancel__right,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( A != zero_zero_int )
% 9.49/5.26       => ( ( dvd_dvd_int @ B @ one_one_int )
% 9.49/5.26         => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
% 9.49/5.26            = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4379_is__unit__div__mult__cancel__right,axiom,
% 9.49/5.26      ! [A: code_integer,B: code_integer] :
% 9.49/5.26        ( ( A != zero_z3403309356797280102nteger )
% 9.49/5.26       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.49/5.26         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
% 9.49/5.26            = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4380_is__unit__div__mult__cancel__left,axiom,
% 9.49/5.26      ! [A: nat,B: nat] :
% 9.49/5.26        ( ( A != zero_zero_nat )
% 9.49/5.26       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.49/5.26         => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
% 9.49/5.26            = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4381_is__unit__div__mult__cancel__left,axiom,
% 9.49/5.26      ! [A: int,B: int] :
% 9.49/5.26        ( ( A != zero_zero_int )
% 9.49/5.26       => ( ( dvd_dvd_int @ B @ one_one_int )
% 9.49/5.26         => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
% 9.49/5.26            = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4382_is__unit__div__mult__cancel__left,axiom,
% 9.49/5.26      ! [A: code_integer,B: code_integer] :
% 9.49/5.26        ( ( A != zero_z3403309356797280102nteger )
% 9.49/5.26       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.49/5.26         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
% 9.49/5.26            = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4383_is__unitE,axiom,
% 9.49/5.26      ! [A: nat,C: nat] :
% 9.49/5.26        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.49/5.26       => ~ ( ( A != zero_zero_nat )
% 9.49/5.26           => ! [B3: nat] :
% 9.49/5.26                ( ( B3 != zero_zero_nat )
% 9.49/5.26               => ( ( dvd_dvd_nat @ B3 @ one_one_nat )
% 9.49/5.26                 => ( ( ( divide_divide_nat @ one_one_nat @ A )
% 9.49/5.26                      = B3 )
% 9.49/5.26                   => ( ( ( divide_divide_nat @ one_one_nat @ B3 )
% 9.49/5.26                        = A )
% 9.49/5.26                     => ( ( ( times_times_nat @ A @ B3 )
% 9.49/5.26                          = one_one_nat )
% 9.49/5.26                       => ( ( divide_divide_nat @ C @ A )
% 9.49/5.26                         != ( times_times_nat @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 9.49/5.26  
% 9.49/5.26  thf(fact_4384_is__unitE,axiom,
% 9.49/5.26      ! [A: int,C: int] :
% 9.49/5.26        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.49/5.26       => ~ ( ( A != zero_zero_int )
% 9.49/5.26           => ! [B3: int] :
% 9.49/5.26                ( ( B3 != zero_zero_int )
% 9.49/5.26               => ( ( dvd_dvd_int @ B3 @ one_one_int )
% 9.49/5.26                 => ( ( ( divide_divide_int @ one_one_int @ A )
% 9.49/5.26                      = B3 )
% 9.49/5.27                   => ( ( ( divide_divide_int @ one_one_int @ B3 )
% 9.49/5.27                        = A )
% 9.49/5.27                     => ( ( ( times_times_int @ A @ B3 )
% 9.49/5.27                          = one_one_int )
% 9.49/5.27                       => ( ( divide_divide_int @ C @ A )
% 9.49/5.27                         != ( times_times_int @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4385_is__unitE,axiom,
% 9.49/5.27      ! [A: code_integer,C: code_integer] :
% 9.49/5.27        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.49/5.27       => ~ ( ( A != zero_z3403309356797280102nteger )
% 9.49/5.27           => ! [B3: code_integer] :
% 9.49/5.27                ( ( B3 != zero_z3403309356797280102nteger )
% 9.49/5.27               => ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
% 9.49/5.27                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
% 9.49/5.27                      = B3 )
% 9.49/5.27                   => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B3 )
% 9.49/5.27                        = A )
% 9.49/5.27                     => ( ( ( times_3573771949741848930nteger @ A @ B3 )
% 9.49/5.27                          = one_one_Code_integer )
% 9.49/5.27                       => ( ( divide6298287555418463151nteger @ C @ A )
% 9.49/5.27                         != ( times_3573771949741848930nteger @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4386_dvd__power__iff,axiom,
% 9.49/5.27      ! [X: code_integer,M: nat,N: nat] :
% 9.49/5.27        ( ( X != zero_z3403309356797280102nteger )
% 9.49/5.27       => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N ) )
% 9.49/5.27          = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
% 9.49/5.27            | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4387_dvd__power__iff,axiom,
% 9.49/5.27      ! [X: nat,M: nat,N: nat] :
% 9.49/5.27        ( ( X != zero_zero_nat )
% 9.49/5.27       => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
% 9.49/5.27          = ( ( dvd_dvd_nat @ X @ one_one_nat )
% 9.49/5.27            | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4388_dvd__power__iff,axiom,
% 9.49/5.27      ! [X: int,M: nat,N: nat] :
% 9.49/5.27        ( ( X != zero_zero_int )
% 9.49/5.27       => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
% 9.49/5.27          = ( ( dvd_dvd_int @ X @ one_one_int )
% 9.49/5.27            | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4389_dvd__power,axiom,
% 9.49/5.27      ! [N: nat,X: code_integer] :
% 9.49/5.27        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.27          | ( X = one_one_Code_integer ) )
% 9.49/5.27       => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4390_dvd__power,axiom,
% 9.49/5.27      ! [N: nat,X: rat] :
% 9.49/5.27        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.27          | ( X = one_one_rat ) )
% 9.49/5.27       => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4391_dvd__power,axiom,
% 9.49/5.27      ! [N: nat,X: nat] :
% 9.49/5.27        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.27          | ( X = one_one_nat ) )
% 9.49/5.27       => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4392_dvd__power,axiom,
% 9.49/5.27      ! [N: nat,X: real] :
% 9.49/5.27        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.27          | ( X = one_one_real ) )
% 9.49/5.27       => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4393_dvd__power,axiom,
% 9.49/5.27      ! [N: nat,X: int] :
% 9.49/5.27        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.27          | ( X = one_one_int ) )
% 9.49/5.27       => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4394_dvd__power,axiom,
% 9.49/5.27      ! [N: nat,X: complex] :
% 9.49/5.27        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.27          | ( X = one_one_complex ) )
% 9.49/5.27       => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4395_mod__pos__neg__trivial,axiom,
% 9.49/5.27      ! [K: int,L: int] :
% 9.49/5.27        ( ( ord_less_int @ zero_zero_int @ K )
% 9.49/5.27       => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 9.49/5.27         => ( ( modulo_modulo_int @ K @ L )
% 9.49/5.27            = ( plus_plus_int @ K @ L ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4396_mod__pos__geq,axiom,
% 9.49/5.27      ! [L: int,K: int] :
% 9.49/5.27        ( ( ord_less_int @ zero_zero_int @ L )
% 9.49/5.27       => ( ( ord_less_eq_int @ L @ K )
% 9.49/5.27         => ( ( modulo_modulo_int @ K @ L )
% 9.49/5.27            = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4397_dvd__mult__cancel1,axiom,
% 9.49/5.27      ! [M: nat,N: nat] :
% 9.49/5.27        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.49/5.27       => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
% 9.49/5.27          = ( N = one_one_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4398_dvd__mult__cancel2,axiom,
% 9.49/5.27      ! [M: nat,N: nat] :
% 9.49/5.27        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.49/5.27       => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
% 9.49/5.27          = ( N = one_one_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4399_dvd__minus__add,axiom,
% 9.49/5.27      ! [Q4: nat,N: nat,R2: nat,M: nat] :
% 9.49/5.27        ( ( ord_less_eq_nat @ Q4 @ N )
% 9.49/5.27       => ( ( ord_less_eq_nat @ Q4 @ ( times_times_nat @ R2 @ M ) )
% 9.49/5.27         => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q4 ) )
% 9.49/5.27            = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q4 ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4400_power__dvd__imp__le,axiom,
% 9.49/5.27      ! [I: nat,M: nat,N: nat] :
% 9.49/5.27        ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 9.49/5.27       => ( ( ord_less_nat @ one_one_nat @ I )
% 9.49/5.27         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4401_mod__nat__eqI,axiom,
% 9.49/5.27      ! [R2: nat,N: nat,M: nat] :
% 9.49/5.27        ( ( ord_less_nat @ R2 @ N )
% 9.49/5.27       => ( ( ord_less_eq_nat @ R2 @ M )
% 9.49/5.27         => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R2 ) )
% 9.49/5.27           => ( ( modulo_modulo_nat @ M @ N )
% 9.49/5.27              = R2 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4402_even__iff__mod__2__eq__zero,axiom,
% 9.49/5.27      ! [A: nat] :
% 9.49/5.27        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.49/5.27        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.27          = zero_zero_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4403_even__iff__mod__2__eq__zero,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.49/5.27        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.27          = zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4404_even__iff__mod__2__eq__zero,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.49/5.27        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.49/5.27          = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4405_power__mono__odd,axiom,
% 9.49/5.27      ! [N: nat,A: real,B: real] :
% 9.49/5.27        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27       => ( ( ord_less_eq_real @ A @ B )
% 9.49/5.27         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4406_power__mono__odd,axiom,
% 9.49/5.27      ! [N: nat,A: rat,B: rat] :
% 9.49/5.27        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27       => ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.27         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4407_power__mono__odd,axiom,
% 9.49/5.27      ! [N: nat,A: int,B: int] :
% 9.49/5.27        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27       => ( ( ord_less_eq_int @ A @ B )
% 9.49/5.27         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4408_odd__pos,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4409_dvd__power__iff__le,axiom,
% 9.49/5.27      ! [K: nat,M: nat,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.49/5.27       => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
% 9.49/5.27          = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4410_split__zmod,axiom,
% 9.49/5.27      ! [P: int > $o,N: int,K: int] :
% 9.49/5.27        ( ( P @ ( modulo_modulo_int @ N @ K ) )
% 9.49/5.27        = ( ( ( K = zero_zero_int )
% 9.49/5.27           => ( P @ N ) )
% 9.49/5.27          & ( ( ord_less_int @ zero_zero_int @ K )
% 9.49/5.27           => ! [I4: int,J3: int] :
% 9.49/5.27                ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 9.49/5.27                  & ( ord_less_int @ J3 @ K )
% 9.49/5.27                  & ( N
% 9.49/5.27                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 9.49/5.27               => ( P @ J3 ) ) )
% 9.49/5.27          & ( ( ord_less_int @ K @ zero_zero_int )
% 9.49/5.27           => ! [I4: int,J3: int] :
% 9.49/5.27                ( ( ( ord_less_int @ K @ J3 )
% 9.49/5.27                  & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 9.49/5.27                  & ( N
% 9.49/5.27                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 9.49/5.27               => ( P @ J3 ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4411_int__mod__neg__eq,axiom,
% 9.49/5.27      ! [A: int,B: int,Q4: int,R2: int] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 9.49/5.27       => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 9.49/5.27         => ( ( ord_less_int @ B @ R2 )
% 9.49/5.27           => ( ( modulo_modulo_int @ A @ B )
% 9.49/5.27              = R2 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4412_int__mod__pos__eq,axiom,
% 9.49/5.27      ! [A: int,B: int,Q4: int,R2: int] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 9.49/5.27       => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 9.49/5.27         => ( ( ord_less_int @ R2 @ B )
% 9.49/5.27           => ( ( modulo_modulo_int @ A @ B )
% 9.49/5.27              = R2 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4413_even__unset__bit__iff,axiom,
% 9.49/5.27      ! [M: nat,A: code_integer] :
% 9.49/5.27        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 9.49/5.27        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.49/5.27          | ( M = zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4414_even__unset__bit__iff,axiom,
% 9.49/5.27      ! [M: nat,A: nat] :
% 9.49/5.27        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 9.49/5.27        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.49/5.27          | ( M = zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4415_even__unset__bit__iff,axiom,
% 9.49/5.27      ! [M: nat,A: int] :
% 9.49/5.27        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 9.49/5.27        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.49/5.27          | ( M = zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4416_even__set__bit__iff,axiom,
% 9.49/5.27      ! [M: nat,A: code_integer] :
% 9.49/5.27        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 9.49/5.27        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.49/5.27          & ( M != zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4417_even__set__bit__iff,axiom,
% 9.49/5.27      ! [M: nat,A: nat] :
% 9.49/5.27        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 9.49/5.27        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.49/5.27          & ( M != zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4418_even__set__bit__iff,axiom,
% 9.49/5.27      ! [M: nat,A: int] :
% 9.49/5.27        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 9.49/5.27        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.49/5.27          & ( M != zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4419_mod2__eq__if,axiom,
% 9.49/5.27      ! [A: nat] :
% 9.49/5.27        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.49/5.27         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.27            = zero_zero_nat ) )
% 9.49/5.27        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.49/5.27         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.27            = one_one_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4420_mod2__eq__if,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.49/5.27         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.27            = zero_zero_int ) )
% 9.49/5.27        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.49/5.27         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.27            = one_one_int ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4421_mod2__eq__if,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.49/5.27         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.49/5.27            = zero_z3403309356797280102nteger ) )
% 9.49/5.27        & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.49/5.27         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.49/5.27            = one_one_Code_integer ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4422_parity__cases,axiom,
% 9.49/5.27      ! [A: nat] :
% 9.49/5.27        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.49/5.27         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.27           != zero_zero_nat ) )
% 9.49/5.27       => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.49/5.27           => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.27             != one_one_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4423_parity__cases,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.49/5.27         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.27           != zero_zero_int ) )
% 9.49/5.27       => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.49/5.27           => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.27             != one_one_int ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4424_parity__cases,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.49/5.27         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.49/5.27           != zero_z3403309356797280102nteger ) )
% 9.49/5.27       => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.49/5.27           => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.49/5.27             != one_one_Code_integer ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4425_zero__le__even__power,axiom,
% 9.49/5.27      ! [N: nat,A: real] :
% 9.49/5.27        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27       => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4426_zero__le__even__power,axiom,
% 9.49/5.27      ! [N: nat,A: rat] :
% 9.49/5.27        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27       => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4427_zero__le__even__power,axiom,
% 9.49/5.27      ! [N: nat,A: int] :
% 9.49/5.27        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27       => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4428_zero__le__odd__power,axiom,
% 9.49/5.27      ! [N: nat,A: real] :
% 9.49/5.27        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27       => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 9.49/5.27          = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4429_zero__le__odd__power,axiom,
% 9.49/5.27      ! [N: nat,A: rat] :
% 9.49/5.27        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27       => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 9.49/5.27          = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4430_zero__le__odd__power,axiom,
% 9.49/5.27      ! [N: nat,A: int] :
% 9.49/5.27        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27       => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 9.49/5.27          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4431_zero__le__power__eq,axiom,
% 9.49/5.27      ! [A: real,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 9.49/5.27        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27            & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4432_zero__le__power__eq,axiom,
% 9.49/5.27      ! [A: rat,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 9.49/5.27        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27            & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4433_zero__le__power__eq,axiom,
% 9.49/5.27      ! [A: int,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 9.49/5.27        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27            & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4434_verit__le__mono__div__int,axiom,
% 9.49/5.27      ! [A2: int,B2: int,N: int] :
% 9.49/5.27        ( ( ord_less_int @ A2 @ B2 )
% 9.49/5.27       => ( ( ord_less_int @ zero_zero_int @ N )
% 9.49/5.27         => ( ord_less_eq_int
% 9.49/5.27            @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N )
% 9.49/5.27              @ ( if_int
% 9.49/5.27                @ ( ( modulo_modulo_int @ B2 @ N )
% 9.49/5.27                  = zero_zero_int )
% 9.49/5.27                @ one_one_int
% 9.49/5.27                @ zero_zero_int ) )
% 9.49/5.27            @ ( divide_divide_int @ B2 @ N ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4435_split__neg__lemma,axiom,
% 9.49/5.27      ! [K: int,P: int > int > $o,N: int] :
% 9.49/5.27        ( ( ord_less_int @ K @ zero_zero_int )
% 9.49/5.27       => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 9.49/5.27          = ( ! [I4: int,J3: int] :
% 9.49/5.27                ( ( ( ord_less_int @ K @ J3 )
% 9.49/5.27                  & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 9.49/5.27                  & ( N
% 9.49/5.27                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 9.49/5.27               => ( P @ I4 @ J3 ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4436_split__pos__lemma,axiom,
% 9.49/5.27      ! [K: int,P: int > int > $o,N: int] :
% 9.49/5.27        ( ( ord_less_int @ zero_zero_int @ K )
% 9.49/5.27       => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 9.49/5.27          = ( ! [I4: int,J3: int] :
% 9.49/5.27                ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 9.49/5.27                  & ( ord_less_int @ J3 @ K )
% 9.49/5.27                  & ( N
% 9.49/5.27                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 9.49/5.27               => ( P @ I4 @ J3 ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4437_list__decode_Ocases,axiom,
% 9.49/5.27      ! [X: nat] :
% 9.49/5.27        ( ( X != zero_zero_nat )
% 9.49/5.27       => ~ ! [N3: nat] :
% 9.49/5.27              ( X
% 9.49/5.27             != ( suc @ N3 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4438_zero__less__power__eq,axiom,
% 9.49/5.27      ! [A: real,N: nat] :
% 9.49/5.27        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 9.49/5.27        = ( ( N = zero_zero_nat )
% 9.49/5.27          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27            & ( A != zero_zero_real ) )
% 9.49/5.27          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27            & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4439_zero__less__power__eq,axiom,
% 9.49/5.27      ! [A: rat,N: nat] :
% 9.49/5.27        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 9.49/5.27        = ( ( N = zero_zero_nat )
% 9.49/5.27          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27            & ( A != zero_zero_rat ) )
% 9.49/5.27          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27            & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4440_zero__less__power__eq,axiom,
% 9.49/5.27      ! [A: int,N: nat] :
% 9.49/5.27        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 9.49/5.27        = ( ( N = zero_zero_nat )
% 9.49/5.27          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27            & ( A != zero_zero_int ) )
% 9.49/5.27          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27            & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4441_even__mask__div__iff_H,axiom,
% 9.49/5.27      ! [M: nat,N: nat] :
% 9.49/5.27        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.49/5.27        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4442_even__mask__div__iff_H,axiom,
% 9.49/5.27      ! [M: nat,N: nat] :
% 9.49/5.27        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 9.49/5.27        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4443_even__mask__div__iff_H,axiom,
% 9.49/5.27      ! [M: nat,N: nat] :
% 9.49/5.27        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 9.49/5.27        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4444_power__le__zero__eq,axiom,
% 9.49/5.27      ! [A: real,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 9.49/5.27        = ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.27          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27              & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 9.49/5.27            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27              & ( A = zero_zero_real ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4445_power__le__zero__eq,axiom,
% 9.49/5.27      ! [A: rat,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 9.49/5.27        = ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.27          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27              & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 9.49/5.27            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27              & ( A = zero_zero_rat ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4446_power__le__zero__eq,axiom,
% 9.49/5.27      ! [A: int,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 9.49/5.27        = ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.27          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27              & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 9.49/5.27            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27              & ( A = zero_zero_int ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4447_even__mod__4__div__2,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.49/5.27          = ( suc @ zero_zero_nat ) )
% 9.49/5.27       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4448_even__mask__div__iff,axiom,
% 9.49/5.27      ! [M: nat,N: nat] :
% 9.49/5.27        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.49/5.27        = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27            = zero_zero_nat )
% 9.49/5.27          | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4449_even__mask__div__iff,axiom,
% 9.49/5.27      ! [M: nat,N: nat] :
% 9.49/5.27        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 9.49/5.27        = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 9.49/5.27            = zero_zero_int )
% 9.49/5.27          | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4450_even__mask__div__iff,axiom,
% 9.49/5.27      ! [M: nat,N: nat] :
% 9.49/5.27        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 9.49/5.27        = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 9.49/5.27            = zero_z3403309356797280102nteger )
% 9.49/5.27          | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4451_even__mult__exp__div__exp__iff,axiom,
% 9.49/5.27      ! [A: nat,M: nat,N: nat] :
% 9.49/5.27        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.49/5.27        = ( ( ord_less_nat @ N @ M )
% 9.49/5.27          | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.27            = zero_zero_nat )
% 9.49/5.27          | ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.27            & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4452_even__mult__exp__div__exp__iff,axiom,
% 9.49/5.27      ! [A: int,M: nat,N: nat] :
% 9.49/5.27        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 9.49/5.27        = ( ( ord_less_nat @ N @ M )
% 9.49/5.27          | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 9.49/5.27            = zero_zero_int )
% 9.49/5.27          | ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.27            & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4453_even__mult__exp__div__exp__iff,axiom,
% 9.49/5.27      ! [A: code_integer,M: nat,N: nat] :
% 9.49/5.27        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 9.49/5.27        = ( ( ord_less_nat @ N @ M )
% 9.49/5.27          | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 9.49/5.27            = zero_z3403309356797280102nteger )
% 9.49/5.27          | ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.27            & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4454_vebt__buildup_Oelims,axiom,
% 9.49/5.27      ! [X: nat,Y: vEBT_VEBT] :
% 9.49/5.27        ( ( ( vEBT_vebt_buildup @ X )
% 9.49/5.27          = Y )
% 9.49/5.27       => ( ( ( X = zero_zero_nat )
% 9.49/5.27           => ( Y
% 9.49/5.27             != ( vEBT_Leaf @ $false @ $false ) ) )
% 9.49/5.27         => ( ( ( X
% 9.49/5.27                = ( suc @ zero_zero_nat ) )
% 9.49/5.27             => ( Y
% 9.49/5.27               != ( vEBT_Leaf @ $false @ $false ) ) )
% 9.49/5.27           => ~ ! [Va2: nat] :
% 9.49/5.27                  ( ( X
% 9.49/5.27                    = ( suc @ ( suc @ Va2 ) ) )
% 9.49/5.27                 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 9.49/5.27                       => ( Y
% 9.49/5.27                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 9.49/5.27                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 9.49/5.27                       => ( Y
% 9.49/5.27                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4455_option_Osize__gen_I2_J,axiom,
% 9.49/5.27      ! [X: nat > nat,X22: nat] :
% 9.49/5.27        ( ( size_option_nat @ X @ ( some_nat @ X22 ) )
% 9.49/5.27        = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4456_option_Osize__gen_I2_J,axiom,
% 9.49/5.27      ! [X: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
% 9.49/5.27        ( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X22 ) )
% 9.49/5.27        = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4457_option_Osize__gen_I2_J,axiom,
% 9.49/5.27      ! [X: num > nat,X22: num] :
% 9.49/5.27        ( ( size_option_num @ X @ ( some_num @ X22 ) )
% 9.49/5.27        = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4458_diff__shunt__var,axiom,
% 9.49/5.27      ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
% 9.49/5.27        ( ( ( minus_1356011639430497352at_nat @ X @ Y )
% 9.49/5.27          = bot_bo2099793752762293965at_nat )
% 9.49/5.27        = ( ord_le3146513528884898305at_nat @ X @ Y ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4459_diff__shunt__var,axiom,
% 9.49/5.27      ! [X: set_real,Y: set_real] :
% 9.49/5.27        ( ( ( minus_minus_set_real @ X @ Y )
% 9.49/5.27          = bot_bot_set_real )
% 9.49/5.27        = ( ord_less_eq_set_real @ X @ Y ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4460_diff__shunt__var,axiom,
% 9.49/5.27      ! [X: set_nat,Y: set_nat] :
% 9.49/5.27        ( ( ( minus_minus_set_nat @ X @ Y )
% 9.49/5.27          = bot_bot_set_nat )
% 9.49/5.27        = ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4461_diff__shunt__var,axiom,
% 9.49/5.27      ! [X: set_int,Y: set_int] :
% 9.49/5.27        ( ( ( minus_minus_set_int @ X @ Y )
% 9.49/5.27          = bot_bot_set_int )
% 9.49/5.27        = ( ord_less_eq_set_int @ X @ Y ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4462_divmod__step__def,axiom,
% 9.49/5.27      ( unique5026877609467782581ep_nat
% 9.49/5.27      = ( ^ [L2: num] :
% 9.49/5.27            ( produc2626176000494625587at_nat
% 9.49/5.27            @ ^ [Q5: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4463_divmod__step__def,axiom,
% 9.49/5.27      ( unique5024387138958732305ep_int
% 9.49/5.27      = ( ^ [L2: num] :
% 9.49/5.27            ( produc4245557441103728435nt_int
% 9.49/5.27            @ ^ [Q5: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4464_divmod__step__def,axiom,
% 9.49/5.27      ( unique4921790084139445826nteger
% 9.49/5.27      = ( ^ [L2: num] :
% 9.49/5.27            ( produc6916734918728496179nteger
% 9.49/5.27            @ ^ [Q5: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4465_take__bit__rec,axiom,
% 9.49/5.27      ( bit_se1745604003318907178nteger
% 9.49/5.27      = ( ^ [N2: nat,A4: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4466_take__bit__rec,axiom,
% 9.49/5.27      ( bit_se2925701944663578781it_nat
% 9.49/5.27      = ( ^ [N2: nat,A4: nat] : ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4467_take__bit__rec,axiom,
% 9.49/5.27      ( bit_se2923211474154528505it_int
% 9.49/5.27      = ( ^ [N2: nat,A4: int] : ( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4468_num_Osize__gen_I2_J,axiom,
% 9.49/5.27      ! [X22: num] :
% 9.49/5.27        ( ( size_num @ ( bit0 @ X22 ) )
% 9.49/5.27        = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4469_vebt__buildup_Osimps_I3_J,axiom,
% 9.49/5.27      ! [Va: nat] :
% 9.49/5.27        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 9.49/5.27         => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 9.49/5.27            = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 9.49/5.27        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 9.49/5.27         => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 9.49/5.27            = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4470_even__flip__bit__iff,axiom,
% 9.49/5.27      ! [M: nat,A: code_integer] :
% 9.49/5.27        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 9.49/5.27        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.49/5.27         != ( M = zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4471_even__flip__bit__iff,axiom,
% 9.49/5.27      ! [M: nat,A: nat] :
% 9.49/5.27        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 9.49/5.27        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.49/5.27         != ( M = zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4472_even__flip__bit__iff,axiom,
% 9.49/5.27      ! [M: nat,A: int] :
% 9.49/5.27        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 9.49/5.27        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.49/5.27         != ( M = zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4473_intind,axiom,
% 9.49/5.27      ! [I: nat,N: nat,P: nat > $o,X: nat] :
% 9.49/5.27        ( ( ord_less_nat @ I @ N )
% 9.49/5.27       => ( ( P @ X )
% 9.49/5.27         => ( P @ ( nth_nat @ ( replicate_nat @ N @ X ) @ I ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4474_intind,axiom,
% 9.49/5.27      ! [I: nat,N: nat,P: int > $o,X: int] :
% 9.49/5.27        ( ( ord_less_nat @ I @ N )
% 9.49/5.27       => ( ( P @ X )
% 9.49/5.27         => ( P @ ( nth_int @ ( replicate_int @ N @ X ) @ I ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4475_intind,axiom,
% 9.49/5.27      ! [I: nat,N: nat,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
% 9.49/5.27        ( ( ord_less_nat @ I @ N )
% 9.49/5.27       => ( ( P @ X )
% 9.49/5.27         => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4476_take__bit__of__0,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
% 9.49/5.27        = zero_zero_nat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4477_take__bit__of__0,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
% 9.49/5.27        = zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4478_flip__bit__negative__int__iff,axiom,
% 9.49/5.27      ! [N: nat,K: int] :
% 9.49/5.27        ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
% 9.49/5.27        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4479_replicate__eq__replicate,axiom,
% 9.49/5.27      ! [M: nat,X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
% 9.49/5.27        ( ( ( replicate_VEBT_VEBT @ M @ X )
% 9.49/5.27          = ( replicate_VEBT_VEBT @ N @ Y ) )
% 9.49/5.27        = ( ( M = N )
% 9.49/5.27          & ( ( M != zero_zero_nat )
% 9.49/5.27           => ( X = Y ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4480_length__replicate,axiom,
% 9.49/5.27      ! [N: nat,X: vEBT_VEBT] :
% 9.49/5.27        ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) )
% 9.49/5.27        = N ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4481_length__replicate,axiom,
% 9.49/5.27      ! [N: nat,X: $o] :
% 9.49/5.27        ( ( size_size_list_o @ ( replicate_o @ N @ X ) )
% 9.49/5.27        = N ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4482_length__replicate,axiom,
% 9.49/5.27      ! [N: nat,X: nat] :
% 9.49/5.27        ( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
% 9.49/5.27        = N ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4483_length__replicate,axiom,
% 9.49/5.27      ! [N: nat,X: int] :
% 9.49/5.27        ( ( size_size_list_int @ ( replicate_int @ N @ X ) )
% 9.49/5.27        = N ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4484_case__prod__conv,axiom,
% 9.49/5.27      ! [F: nat > nat > $o,A: nat,B: nat] :
% 9.49/5.27        ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 9.49/5.27        = ( F @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4485_case__prod__conv,axiom,
% 9.49/5.27      ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
% 9.49/5.27        ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 9.49/5.27        = ( F @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4486_case__prod__conv,axiom,
% 9.49/5.27      ! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
% 9.49/5.27        ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 9.49/5.27        = ( F @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4487_case__prod__conv,axiom,
% 9.49/5.27      ! [F: int > int > product_prod_int_int,A: int,B: int] :
% 9.49/5.27        ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
% 9.49/5.27        = ( F @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4488_case__prod__conv,axiom,
% 9.49/5.27      ! [F: int > int > $o,A: int,B: int] :
% 9.49/5.27        ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
% 9.49/5.27        = ( F @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4489_take__bit__0,axiom,
% 9.49/5.27      ! [A: nat] :
% 9.49/5.27        ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
% 9.49/5.27        = zero_zero_nat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4490_take__bit__0,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
% 9.49/5.27        = zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4491_in__set__replicate,axiom,
% 9.49/5.27      ! [X: product_prod_nat_nat,N: nat,Y: product_prod_nat_nat] :
% 9.49/5.27        ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ Y ) ) )
% 9.49/5.27        = ( ( X = Y )
% 9.49/5.27          & ( N != zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4492_in__set__replicate,axiom,
% 9.49/5.27      ! [X: real,N: nat,Y: real] :
% 9.49/5.27        ( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N @ Y ) ) )
% 9.49/5.27        = ( ( X = Y )
% 9.49/5.27          & ( N != zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4493_in__set__replicate,axiom,
% 9.49/5.27      ! [X: set_nat,N: nat,Y: set_nat] :
% 9.49/5.27        ( ( member_set_nat @ X @ ( set_set_nat2 @ ( replicate_set_nat @ N @ Y ) ) )
% 9.49/5.27        = ( ( X = Y )
% 9.49/5.27          & ( N != zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4494_in__set__replicate,axiom,
% 9.49/5.27      ! [X: int,N: nat,Y: int] :
% 9.49/5.27        ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ Y ) ) )
% 9.49/5.27        = ( ( X = Y )
% 9.49/5.27          & ( N != zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4495_in__set__replicate,axiom,
% 9.49/5.27      ! [X: nat,N: nat,Y: nat] :
% 9.49/5.27        ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
% 9.49/5.27        = ( ( X = Y )
% 9.49/5.27          & ( N != zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4496_in__set__replicate,axiom,
% 9.49/5.27      ! [X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
% 9.49/5.27        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y ) ) )
% 9.49/5.27        = ( ( X = Y )
% 9.49/5.27          & ( N != zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4497_Bex__set__replicate,axiom,
% 9.49/5.27      ! [N: nat,A: int,P: int > $o] :
% 9.49/5.27        ( ( ? [X3: int] :
% 9.49/5.27              ( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
% 9.49/5.27              & ( P @ X3 ) ) )
% 9.49/5.27        = ( ( P @ A )
% 9.49/5.27          & ( N != zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4498_Bex__set__replicate,axiom,
% 9.49/5.27      ! [N: nat,A: nat,P: nat > $o] :
% 9.49/5.27        ( ( ? [X3: nat] :
% 9.49/5.27              ( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
% 9.49/5.27              & ( P @ X3 ) ) )
% 9.49/5.27        = ( ( P @ A )
% 9.49/5.27          & ( N != zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4499_Bex__set__replicate,axiom,
% 9.49/5.27      ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 9.49/5.27        ( ( ? [X3: vEBT_VEBT] :
% 9.49/5.27              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
% 9.49/5.27              & ( P @ X3 ) ) )
% 9.49/5.27        = ( ( P @ A )
% 9.49/5.27          & ( N != zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4500_Ball__set__replicate,axiom,
% 9.49/5.27      ! [N: nat,A: int,P: int > $o] :
% 9.49/5.27        ( ( ! [X3: int] :
% 9.49/5.27              ( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
% 9.49/5.27             => ( P @ X3 ) ) )
% 9.49/5.27        = ( ( P @ A )
% 9.49/5.27          | ( N = zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4501_Ball__set__replicate,axiom,
% 9.49/5.27      ! [N: nat,A: nat,P: nat > $o] :
% 9.49/5.27        ( ( ! [X3: nat] :
% 9.49/5.27              ( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
% 9.49/5.27             => ( P @ X3 ) ) )
% 9.49/5.27        = ( ( P @ A )
% 9.49/5.27          | ( N = zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4502_Ball__set__replicate,axiom,
% 9.49/5.27      ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 9.49/5.27        ( ( ! [X3: vEBT_VEBT] :
% 9.49/5.27              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
% 9.49/5.27             => ( P @ X3 ) ) )
% 9.49/5.27        = ( ( P @ A )
% 9.49/5.27          | ( N = zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4503_nth__replicate,axiom,
% 9.49/5.27      ! [I: nat,N: nat,X: nat] :
% 9.49/5.27        ( ( ord_less_nat @ I @ N )
% 9.49/5.27       => ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I )
% 9.49/5.27          = X ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4504_nth__replicate,axiom,
% 9.49/5.27      ! [I: nat,N: nat,X: int] :
% 9.49/5.27        ( ( ord_less_nat @ I @ N )
% 9.49/5.27       => ( ( nth_int @ ( replicate_int @ N @ X ) @ I )
% 9.49/5.27          = X ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4505_nth__replicate,axiom,
% 9.49/5.27      ! [I: nat,N: nat,X: vEBT_VEBT] :
% 9.49/5.27        ( ( ord_less_nat @ I @ N )
% 9.49/5.27       => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I )
% 9.49/5.27          = X ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4506_take__bit__of__1__eq__0__iff,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 9.49/5.27          = zero_zero_nat )
% 9.49/5.27        = ( N = zero_zero_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4507_take__bit__of__1__eq__0__iff,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 9.49/5.27          = zero_zero_int )
% 9.49/5.27        = ( N = zero_zero_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4508_even__take__bit__eq,axiom,
% 9.49/5.27      ! [N: nat,A: code_integer] :
% 9.49/5.27        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N @ A ) )
% 9.49/5.27        = ( ( N = zero_zero_nat )
% 9.49/5.27          | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4509_even__take__bit__eq,axiom,
% 9.49/5.27      ! [N: nat,A: nat] :
% 9.49/5.27        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 9.49/5.27        = ( ( N = zero_zero_nat )
% 9.49/5.27          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4510_even__take__bit__eq,axiom,
% 9.49/5.27      ! [N: nat,A: int] :
% 9.49/5.27        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 9.49/5.27        = ( ( N = zero_zero_nat )
% 9.49/5.27          | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4511_take__bit__Suc__0,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A )
% 9.49/5.27        = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4512_take__bit__Suc__0,axiom,
% 9.49/5.27      ! [A: nat] :
% 9.49/5.27        ( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
% 9.49/5.27        = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4513_take__bit__Suc__0,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
% 9.49/5.27        = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4514_dvd__antisym,axiom,
% 9.49/5.27      ! [M: nat,N: nat] :
% 9.49/5.27        ( ( dvd_dvd_nat @ M @ N )
% 9.49/5.27       => ( ( dvd_dvd_nat @ N @ M )
% 9.49/5.27         => ( M = N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4515_take__bit__flip__bit__eq,axiom,
% 9.49/5.27      ! [N: nat,M: nat,A: nat] :
% 9.49/5.27        ( ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.27         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 9.49/5.27            = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 9.49/5.27        & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.49/5.27         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 9.49/5.27            = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4516_take__bit__flip__bit__eq,axiom,
% 9.49/5.27      ! [N: nat,M: nat,A: int] :
% 9.49/5.27        ( ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.27         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 9.49/5.27            = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 9.49/5.27        & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.49/5.27         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 9.49/5.27            = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4517_take__bit__tightened,axiom,
% 9.49/5.27      ! [N: nat,A: nat,B: nat,M: nat] :
% 9.49/5.27        ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 9.49/5.27          = ( bit_se2925701944663578781it_nat @ N @ B ) )
% 9.49/5.27       => ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.27         => ( ( bit_se2925701944663578781it_nat @ M @ A )
% 9.49/5.27            = ( bit_se2925701944663578781it_nat @ M @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4518_take__bit__tightened,axiom,
% 9.49/5.27      ! [N: nat,A: int,B: int,M: nat] :
% 9.49/5.27        ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 9.49/5.27          = ( bit_se2923211474154528505it_int @ N @ B ) )
% 9.49/5.27       => ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.27         => ( ( bit_se2923211474154528505it_int @ M @ A )
% 9.49/5.27            = ( bit_se2923211474154528505it_int @ M @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4519_take__bit__tightened__less__eq__nat,axiom,
% 9.49/5.27      ! [M: nat,N: nat,Q4: nat] :
% 9.49/5.27        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.27       => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q4 ) @ ( bit_se2925701944663578781it_nat @ N @ Q4 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4520_take__bit__nat__less__eq__self,axiom,
% 9.49/5.27      ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4521_old_Oprod_Ocase,axiom,
% 9.49/5.27      ! [F: nat > nat > $o,X1: nat,X22: nat] :
% 9.49/5.27        ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 9.49/5.27        = ( F @ X1 @ X22 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4522_old_Oprod_Ocase,axiom,
% 9.49/5.27      ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X22: nat] :
% 9.49/5.27        ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 9.49/5.27        = ( F @ X1 @ X22 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4523_old_Oprod_Ocase,axiom,
% 9.49/5.27      ! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X22: nat] :
% 9.49/5.27        ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 9.49/5.27        = ( F @ X1 @ X22 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4524_old_Oprod_Ocase,axiom,
% 9.49/5.27      ! [F: int > int > product_prod_int_int,X1: int,X22: int] :
% 9.49/5.27        ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
% 9.49/5.27        = ( F @ X1 @ X22 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4525_old_Oprod_Ocase,axiom,
% 9.49/5.27      ! [F: int > int > $o,X1: int,X22: int] :
% 9.49/5.27        ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
% 9.49/5.27        = ( F @ X1 @ X22 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4526_split__cong,axiom,
% 9.49/5.27      ! [Q4: product_prod_nat_nat,F: nat > nat > $o,G: nat > nat > $o,P4: product_prod_nat_nat] :
% 9.49/5.27        ( ! [X4: nat,Y3: nat] :
% 9.49/5.27            ( ( ( product_Pair_nat_nat @ X4 @ Y3 )
% 9.49/5.27              = Q4 )
% 9.49/5.27           => ( ( F @ X4 @ Y3 )
% 9.49/5.27              = ( G @ X4 @ Y3 ) ) )
% 9.49/5.27       => ( ( P4 = Q4 )
% 9.49/5.27         => ( ( produc6081775807080527818_nat_o @ F @ P4 )
% 9.49/5.27            = ( produc6081775807080527818_nat_o @ G @ Q4 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4527_split__cong,axiom,
% 9.49/5.27      ! [Q4: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: nat > nat > product_prod_nat_nat > product_prod_nat_nat,P4: product_prod_nat_nat] :
% 9.49/5.27        ( ! [X4: nat,Y3: nat] :
% 9.49/5.27            ( ( ( product_Pair_nat_nat @ X4 @ Y3 )
% 9.49/5.27              = Q4 )
% 9.49/5.27           => ( ( F @ X4 @ Y3 )
% 9.49/5.27              = ( G @ X4 @ Y3 ) ) )
% 9.49/5.27       => ( ( P4 = Q4 )
% 9.49/5.27         => ( ( produc27273713700761075at_nat @ F @ P4 )
% 9.49/5.27            = ( produc27273713700761075at_nat @ G @ Q4 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4528_split__cong,axiom,
% 9.49/5.27      ! [Q4: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > $o,G: nat > nat > product_prod_nat_nat > $o,P4: product_prod_nat_nat] :
% 9.49/5.27        ( ! [X4: nat,Y3: nat] :
% 9.49/5.27            ( ( ( product_Pair_nat_nat @ X4 @ Y3 )
% 9.49/5.27              = Q4 )
% 9.49/5.27           => ( ( F @ X4 @ Y3 )
% 9.49/5.27              = ( G @ X4 @ Y3 ) ) )
% 9.49/5.27       => ( ( P4 = Q4 )
% 9.49/5.27         => ( ( produc8739625826339149834_nat_o @ F @ P4 )
% 9.49/5.27            = ( produc8739625826339149834_nat_o @ G @ Q4 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4529_split__cong,axiom,
% 9.49/5.27      ! [Q4: product_prod_int_int,F: int > int > product_prod_int_int,G: int > int > product_prod_int_int,P4: product_prod_int_int] :
% 9.49/5.27        ( ! [X4: int,Y3: int] :
% 9.49/5.27            ( ( ( product_Pair_int_int @ X4 @ Y3 )
% 9.49/5.27              = Q4 )
% 9.49/5.27           => ( ( F @ X4 @ Y3 )
% 9.49/5.27              = ( G @ X4 @ Y3 ) ) )
% 9.49/5.27       => ( ( P4 = Q4 )
% 9.49/5.27         => ( ( produc4245557441103728435nt_int @ F @ P4 )
% 9.49/5.27            = ( produc4245557441103728435nt_int @ G @ Q4 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4530_split__cong,axiom,
% 9.49/5.27      ! [Q4: product_prod_int_int,F: int > int > $o,G: int > int > $o,P4: product_prod_int_int] :
% 9.49/5.27        ( ! [X4: int,Y3: int] :
% 9.49/5.27            ( ( ( product_Pair_int_int @ X4 @ Y3 )
% 9.49/5.27              = Q4 )
% 9.49/5.27           => ( ( F @ X4 @ Y3 )
% 9.49/5.27              = ( G @ X4 @ Y3 ) ) )
% 9.49/5.27       => ( ( P4 = Q4 )
% 9.49/5.27         => ( ( produc4947309494688390418_int_o @ F @ P4 )
% 9.49/5.27            = ( produc4947309494688390418_int_o @ G @ Q4 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4531_case__prodE2,axiom,
% 9.49/5.27      ! [Q: $o > $o,P: nat > nat > $o,Z2: product_prod_nat_nat] :
% 9.49/5.27        ( ( Q @ ( produc6081775807080527818_nat_o @ P @ Z2 ) )
% 9.49/5.27       => ~ ! [X4: nat,Y3: nat] :
% 9.49/5.27              ( ( Z2
% 9.49/5.27                = ( product_Pair_nat_nat @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( Q @ ( P @ X4 @ Y3 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4532_case__prodE2,axiom,
% 9.49/5.27      ! [Q: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z2: product_prod_nat_nat] :
% 9.49/5.27        ( ( Q @ ( produc27273713700761075at_nat @ P @ Z2 ) )
% 9.49/5.27       => ~ ! [X4: nat,Y3: nat] :
% 9.49/5.27              ( ( Z2
% 9.49/5.27                = ( product_Pair_nat_nat @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( Q @ ( P @ X4 @ Y3 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4533_case__prodE2,axiom,
% 9.49/5.27      ! [Q: ( product_prod_nat_nat > $o ) > $o,P: nat > nat > product_prod_nat_nat > $o,Z2: product_prod_nat_nat] :
% 9.49/5.27        ( ( Q @ ( produc8739625826339149834_nat_o @ P @ Z2 ) )
% 9.49/5.27       => ~ ! [X4: nat,Y3: nat] :
% 9.49/5.27              ( ( Z2
% 9.49/5.27                = ( product_Pair_nat_nat @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( Q @ ( P @ X4 @ Y3 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4534_case__prodE2,axiom,
% 9.49/5.27      ! [Q: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z2: product_prod_int_int] :
% 9.49/5.27        ( ( Q @ ( produc4245557441103728435nt_int @ P @ Z2 ) )
% 9.49/5.27       => ~ ! [X4: int,Y3: int] :
% 9.49/5.27              ( ( Z2
% 9.49/5.27                = ( product_Pair_int_int @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( Q @ ( P @ X4 @ Y3 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4535_case__prodE2,axiom,
% 9.49/5.27      ! [Q: $o > $o,P: int > int > $o,Z2: product_prod_int_int] :
% 9.49/5.27        ( ( Q @ ( produc4947309494688390418_int_o @ P @ Z2 ) )
% 9.49/5.27       => ~ ! [X4: int,Y3: int] :
% 9.49/5.27              ( ( Z2
% 9.49/5.27                = ( product_Pair_int_int @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( Q @ ( P @ X4 @ Y3 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4536_case__prod__eta,axiom,
% 9.49/5.27      ! [F: product_prod_nat_nat > $o] :
% 9.49/5.27        ( ( produc6081775807080527818_nat_o
% 9.49/5.27          @ ^ [X3: nat,Y2: nat] : ( F @ ( product_Pair_nat_nat @ X3 @ Y2 ) ) )
% 9.49/5.27        = F ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4537_case__prod__eta,axiom,
% 9.49/5.27      ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 9.49/5.27        ( ( produc27273713700761075at_nat
% 9.49/5.27          @ ^ [X3: nat,Y2: nat] : ( F @ ( product_Pair_nat_nat @ X3 @ Y2 ) ) )
% 9.49/5.27        = F ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4538_case__prod__eta,axiom,
% 9.49/5.27      ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 9.49/5.27        ( ( produc8739625826339149834_nat_o
% 9.49/5.27          @ ^ [X3: nat,Y2: nat] : ( F @ ( product_Pair_nat_nat @ X3 @ Y2 ) ) )
% 9.49/5.27        = F ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4539_case__prod__eta,axiom,
% 9.49/5.27      ! [F: product_prod_int_int > product_prod_int_int] :
% 9.49/5.27        ( ( produc4245557441103728435nt_int
% 9.49/5.27          @ ^ [X3: int,Y2: int] : ( F @ ( product_Pair_int_int @ X3 @ Y2 ) ) )
% 9.49/5.27        = F ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4540_case__prod__eta,axiom,
% 9.49/5.27      ! [F: product_prod_int_int > $o] :
% 9.49/5.27        ( ( produc4947309494688390418_int_o
% 9.49/5.27          @ ^ [X3: int,Y2: int] : ( F @ ( product_Pair_int_int @ X3 @ Y2 ) ) )
% 9.49/5.27        = F ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4541_cond__case__prod__eta,axiom,
% 9.49/5.27      ! [F: nat > nat > $o,G: product_prod_nat_nat > $o] :
% 9.49/5.27        ( ! [X4: nat,Y3: nat] :
% 9.49/5.27            ( ( F @ X4 @ Y3 )
% 9.49/5.27            = ( G @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
% 9.49/5.27       => ( ( produc6081775807080527818_nat_o @ F )
% 9.49/5.27          = G ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4542_cond__case__prod__eta,axiom,
% 9.49/5.27      ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 9.49/5.27        ( ! [X4: nat,Y3: nat] :
% 9.49/5.27            ( ( F @ X4 @ Y3 )
% 9.49/5.27            = ( G @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
% 9.49/5.27       => ( ( produc27273713700761075at_nat @ F )
% 9.49/5.27          = G ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4543_cond__case__prod__eta,axiom,
% 9.49/5.27      ! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 9.49/5.27        ( ! [X4: nat,Y3: nat] :
% 9.49/5.27            ( ( F @ X4 @ Y3 )
% 9.49/5.27            = ( G @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
% 9.49/5.27       => ( ( produc8739625826339149834_nat_o @ F )
% 9.49/5.27          = G ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4544_cond__case__prod__eta,axiom,
% 9.49/5.27      ! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
% 9.49/5.27        ( ! [X4: int,Y3: int] :
% 9.49/5.27            ( ( F @ X4 @ Y3 )
% 9.49/5.27            = ( G @ ( product_Pair_int_int @ X4 @ Y3 ) ) )
% 9.49/5.27       => ( ( produc4245557441103728435nt_int @ F )
% 9.49/5.27          = G ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4545_cond__case__prod__eta,axiom,
% 9.49/5.27      ! [F: int > int > $o,G: product_prod_int_int > $o] :
% 9.49/5.27        ( ! [X4: int,Y3: int] :
% 9.49/5.27            ( ( F @ X4 @ Y3 )
% 9.49/5.27            = ( G @ ( product_Pair_int_int @ X4 @ Y3 ) ) )
% 9.49/5.27       => ( ( produc4947309494688390418_int_o @ F )
% 9.49/5.27          = G ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4546_take__bit__tightened__less__eq__int,axiom,
% 9.49/5.27      ! [M: nat,N: nat,K: int] :
% 9.49/5.27        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.27       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4547_not__take__bit__negative,axiom,
% 9.49/5.27      ! [N: nat,K: int] :
% 9.49/5.27        ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4548_take__bit__int__greater__self__iff,axiom,
% 9.49/5.27      ! [K: int,N: nat] :
% 9.49/5.27        ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 9.49/5.27        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4549_take__bit__unset__bit__eq,axiom,
% 9.49/5.27      ! [N: nat,M: nat,A: nat] :
% 9.49/5.27        ( ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.27         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 9.49/5.27            = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 9.49/5.27        & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.49/5.27         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 9.49/5.27            = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4550_take__bit__unset__bit__eq,axiom,
% 9.49/5.27      ! [N: nat,M: nat,A: int] :
% 9.49/5.27        ( ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.27         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 9.49/5.27            = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 9.49/5.27        & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.49/5.27         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 9.49/5.27            = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4551_take__bit__set__bit__eq,axiom,
% 9.49/5.27      ! [N: nat,M: nat,A: nat] :
% 9.49/5.27        ( ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.27         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 9.49/5.27            = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 9.49/5.27        & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.49/5.27         => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 9.49/5.27            = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4552_take__bit__set__bit__eq,axiom,
% 9.49/5.27      ! [N: nat,M: nat,A: int] :
% 9.49/5.27        ( ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.27         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 9.49/5.27            = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 9.49/5.27        & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.49/5.27         => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 9.49/5.27            = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4553_replicate__eqI,axiom,
% 9.49/5.27      ! [Xs: list_P6011104703257516679at_nat,N: nat,X: product_prod_nat_nat] :
% 9.49/5.27        ( ( ( size_s5460976970255530739at_nat @ Xs )
% 9.49/5.27          = N )
% 9.49/5.27       => ( ! [Y3: product_prod_nat_nat] :
% 9.49/5.27              ( ( member8440522571783428010at_nat @ Y3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.49/5.27             => ( Y3 = X ) )
% 9.49/5.27         => ( Xs
% 9.49/5.27            = ( replic4235873036481779905at_nat @ N @ X ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4554_replicate__eqI,axiom,
% 9.49/5.27      ! [Xs: list_real,N: nat,X: real] :
% 9.49/5.27        ( ( ( size_size_list_real @ Xs )
% 9.49/5.27          = N )
% 9.49/5.27       => ( ! [Y3: real] :
% 9.49/5.27              ( ( member_real @ Y3 @ ( set_real2 @ Xs ) )
% 9.49/5.27             => ( Y3 = X ) )
% 9.49/5.27         => ( Xs
% 9.49/5.27            = ( replicate_real @ N @ X ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4555_replicate__eqI,axiom,
% 9.49/5.27      ! [Xs: list_set_nat,N: nat,X: set_nat] :
% 9.49/5.27        ( ( ( size_s3254054031482475050et_nat @ Xs )
% 9.49/5.27          = N )
% 9.49/5.27       => ( ! [Y3: set_nat] :
% 9.49/5.27              ( ( member_set_nat @ Y3 @ ( set_set_nat2 @ Xs ) )
% 9.49/5.27             => ( Y3 = X ) )
% 9.49/5.27         => ( Xs
% 9.49/5.27            = ( replicate_set_nat @ N @ X ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4556_replicate__eqI,axiom,
% 9.49/5.27      ! [Xs: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
% 9.49/5.27        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.49/5.27          = N )
% 9.49/5.27       => ( ! [Y3: vEBT_VEBT] :
% 9.49/5.27              ( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.27             => ( Y3 = X ) )
% 9.49/5.27         => ( Xs
% 9.49/5.27            = ( replicate_VEBT_VEBT @ N @ X ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4557_replicate__eqI,axiom,
% 9.49/5.27      ! [Xs: list_o,N: nat,X: $o] :
% 9.49/5.27        ( ( ( size_size_list_o @ Xs )
% 9.49/5.27          = N )
% 9.49/5.27       => ( ! [Y3: $o] :
% 9.49/5.27              ( ( member_o @ Y3 @ ( set_o2 @ Xs ) )
% 9.49/5.27             => ( Y3 = X ) )
% 9.49/5.27         => ( Xs
% 9.49/5.27            = ( replicate_o @ N @ X ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4558_replicate__eqI,axiom,
% 9.49/5.27      ! [Xs: list_nat,N: nat,X: nat] :
% 9.49/5.27        ( ( ( size_size_list_nat @ Xs )
% 9.49/5.27          = N )
% 9.49/5.27       => ( ! [Y3: nat] :
% 9.49/5.27              ( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
% 9.49/5.27             => ( Y3 = X ) )
% 9.49/5.27         => ( Xs
% 9.49/5.27            = ( replicate_nat @ N @ X ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4559_replicate__eqI,axiom,
% 9.49/5.27      ! [Xs: list_int,N: nat,X: int] :
% 9.49/5.27        ( ( ( size_size_list_int @ Xs )
% 9.49/5.27          = N )
% 9.49/5.27       => ( ! [Y3: int] :
% 9.49/5.27              ( ( member_int @ Y3 @ ( set_int2 @ Xs ) )
% 9.49/5.27             => ( Y3 = X ) )
% 9.49/5.27         => ( Xs
% 9.49/5.27            = ( replicate_int @ N @ X ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4560_replicate__length__same,axiom,
% 9.49/5.27      ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 9.49/5.27        ( ! [X4: vEBT_VEBT] :
% 9.49/5.27            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.49/5.27           => ( X4 = X ) )
% 9.49/5.27       => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs ) @ X )
% 9.49/5.27          = Xs ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4561_replicate__length__same,axiom,
% 9.49/5.27      ! [Xs: list_o,X: $o] :
% 9.49/5.27        ( ! [X4: $o] :
% 9.49/5.27            ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 9.49/5.27           => ( X4 = X ) )
% 9.49/5.27       => ( ( replicate_o @ ( size_size_list_o @ Xs ) @ X )
% 9.49/5.27          = Xs ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4562_replicate__length__same,axiom,
% 9.49/5.27      ! [Xs: list_nat,X: nat] :
% 9.49/5.27        ( ! [X4: nat] :
% 9.49/5.27            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 9.49/5.27           => ( X4 = X ) )
% 9.49/5.27       => ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X )
% 9.49/5.27          = Xs ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4563_replicate__length__same,axiom,
% 9.49/5.27      ! [Xs: list_int,X: int] :
% 9.49/5.27        ( ! [X4: int] :
% 9.49/5.27            ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 9.49/5.27           => ( X4 = X ) )
% 9.49/5.27       => ( ( replicate_int @ ( size_size_list_int @ Xs ) @ X )
% 9.49/5.27          = Xs ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4564_take__bit__nat__eq__self,axiom,
% 9.49/5.27      ! [M: nat,N: nat] :
% 9.49/5.27        ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27       => ( ( bit_se2925701944663578781it_nat @ N @ M )
% 9.49/5.27          = M ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4565_take__bit__nat__less__exp,axiom,
% 9.49/5.27      ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4566_take__bit__nat__eq__self__iff,axiom,
% 9.49/5.27      ! [N: nat,M: nat] :
% 9.49/5.27        ( ( ( bit_se2925701944663578781it_nat @ N @ M )
% 9.49/5.27          = M )
% 9.49/5.27        = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4567_take__bit__int__less__exp,axiom,
% 9.49/5.27      ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4568_num_Osize__gen_I1_J,axiom,
% 9.49/5.27      ( ( size_num @ one )
% 9.49/5.27      = zero_zero_nat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4569_take__bit__eq__0__iff,axiom,
% 9.49/5.27      ! [N: nat,A: code_integer] :
% 9.49/5.27        ( ( ( bit_se1745604003318907178nteger @ N @ A )
% 9.49/5.27          = zero_z3403309356797280102nteger )
% 9.49/5.27        = ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4570_take__bit__eq__0__iff,axiom,
% 9.49/5.27      ! [N: nat,A: nat] :
% 9.49/5.27        ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 9.49/5.27          = zero_zero_nat )
% 9.49/5.27        = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4571_take__bit__eq__0__iff,axiom,
% 9.49/5.27      ! [N: nat,A: int] :
% 9.49/5.27        ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 9.49/5.27          = zero_zero_int )
% 9.49/5.27        = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4572_take__bit__nat__less__self__iff,axiom,
% 9.49/5.27      ! [N: nat,M: nat] :
% 9.49/5.27        ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
% 9.49/5.27        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4573_take__bit__int__less__self__iff,axiom,
% 9.49/5.27      ! [N: nat,K: int] :
% 9.49/5.27        ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 9.49/5.27        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4574_take__bit__int__greater__eq__self__iff,axiom,
% 9.49/5.27      ! [K: int,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 9.49/5.27        = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4575_take__bit__int__eq__self,axiom,
% 9.49/5.27      ! [K: int,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.49/5.27       => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27         => ( ( bit_se2923211474154528505it_int @ N @ K )
% 9.49/5.27            = K ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4576_take__bit__int__eq__self__iff,axiom,
% 9.49/5.27      ! [N: nat,K: int] :
% 9.49/5.27        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 9.49/5.27          = K )
% 9.49/5.27        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.49/5.27          & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4577_take__bit__int__less__eq,axiom,
% 9.49/5.27      ! [N: nat,K: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 9.49/5.27       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.27         => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4578_take__bit__int__greater__eq,axiom,
% 9.49/5.27      ! [K: int,N: nat] :
% 9.49/5.27        ( ( ord_less_int @ K @ zero_zero_int )
% 9.49/5.27       => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4579_stable__imp__take__bit__eq,axiom,
% 9.49/5.27      ! [A: code_integer,N: nat] :
% 9.49/5.27        ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.49/5.27          = A )
% 9.49/5.27       => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.49/5.27           => ( ( bit_se1745604003318907178nteger @ N @ A )
% 9.49/5.27              = zero_z3403309356797280102nteger ) )
% 9.49/5.27          & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.49/5.27           => ( ( bit_se1745604003318907178nteger @ N @ A )
% 9.49/5.27              = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4580_stable__imp__take__bit__eq,axiom,
% 9.49/5.27      ! [A: nat,N: nat] :
% 9.49/5.27        ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.27          = A )
% 9.49/5.27       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.49/5.27           => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 9.49/5.27              = zero_zero_nat ) )
% 9.49/5.27          & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.49/5.27           => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 9.49/5.27              = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4581_stable__imp__take__bit__eq,axiom,
% 9.49/5.27      ! [A: int,N: nat] :
% 9.49/5.27        ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.27          = A )
% 9.49/5.27       => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.49/5.27           => ( ( bit_se2923211474154528505it_int @ N @ A )
% 9.49/5.27              = zero_zero_int ) )
% 9.49/5.27          & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.49/5.27           => ( ( bit_se2923211474154528505it_int @ N @ A )
% 9.49/5.27              = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4582_option_Osize__gen_I1_J,axiom,
% 9.49/5.27      ! [X: nat > nat] :
% 9.49/5.27        ( ( size_option_nat @ X @ none_nat )
% 9.49/5.27        = ( suc @ zero_zero_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4583_option_Osize__gen_I1_J,axiom,
% 9.49/5.27      ! [X: product_prod_nat_nat > nat] :
% 9.49/5.27        ( ( size_o8335143837870341156at_nat @ X @ none_P5556105721700978146at_nat )
% 9.49/5.27        = ( suc @ zero_zero_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4584_option_Osize__gen_I1_J,axiom,
% 9.49/5.27      ! [X: num > nat] :
% 9.49/5.27        ( ( size_option_num @ X @ none_num )
% 9.49/5.27        = ( suc @ zero_zero_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4585_divmod__step__nat__def,axiom,
% 9.49/5.27      ( unique5026877609467782581ep_nat
% 9.49/5.27      = ( ^ [L2: num] :
% 9.49/5.27            ( produc2626176000494625587at_nat
% 9.49/5.27            @ ^ [Q5: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4586_divmod__step__int__def,axiom,
% 9.49/5.27      ( unique5024387138958732305ep_int
% 9.49/5.27      = ( ^ [L2: num] :
% 9.49/5.27            ( produc4245557441103728435nt_int
% 9.49/5.27            @ ^ [Q5: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4587_divmod__algorithm__code_I5_J,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.49/5.27        = ( produc4245557441103728435nt_int
% 9.49/5.27          @ ^ [Q5: int,R5: int] : ( product_Pair_int_int @ Q5 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
% 9.49/5.27          @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4588_divmod__algorithm__code_I5_J,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.49/5.27        = ( produc2626176000494625587at_nat
% 9.49/5.27          @ ^ [Q5: nat,R5: nat] : ( product_Pair_nat_nat @ Q5 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
% 9.49/5.27          @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4589_divmod__algorithm__code_I5_J,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.49/5.27        = ( produc6916734918728496179nteger
% 9.49/5.27          @ ^ [Q5: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
% 9.49/5.27          @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4590_divmod__nat__if,axiom,
% 9.49/5.27      ( divmod_nat
% 9.49/5.27      = ( ^ [M2: nat,N2: nat] :
% 9.49/5.27            ( if_Pro6206227464963214023at_nat
% 9.49/5.27            @ ( ( N2 = zero_zero_nat )
% 9.49/5.27              | ( ord_less_nat @ M2 @ N2 ) )
% 9.49/5.27            @ ( product_Pair_nat_nat @ zero_zero_nat @ M2 )
% 9.49/5.27            @ ( produc2626176000494625587at_nat
% 9.49/5.27              @ ^ [Q5: nat] : ( product_Pair_nat_nat @ ( suc @ Q5 ) )
% 9.49/5.27              @ ( divmod_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4591_vebt__buildup_Opelims,axiom,
% 9.49/5.27      ! [X: nat,Y: vEBT_VEBT] :
% 9.49/5.27        ( ( ( vEBT_vebt_buildup @ X )
% 9.49/5.27          = Y )
% 9.49/5.27       => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
% 9.49/5.27         => ( ( ( X = zero_zero_nat )
% 9.49/5.27             => ( ( Y
% 9.49/5.27                  = ( vEBT_Leaf @ $false @ $false ) )
% 9.49/5.27               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 9.49/5.27           => ( ( ( X
% 9.49/5.27                  = ( suc @ zero_zero_nat ) )
% 9.49/5.27               => ( ( Y
% 9.49/5.27                    = ( vEBT_Leaf @ $false @ $false ) )
% 9.49/5.27                 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 9.49/5.27             => ~ ! [Va2: nat] :
% 9.49/5.27                    ( ( X
% 9.49/5.27                      = ( suc @ ( suc @ Va2 ) ) )
% 9.49/5.27                   => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 9.49/5.27                         => ( Y
% 9.49/5.27                            = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 9.49/5.27                        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 9.49/5.27                         => ( Y
% 9.49/5.27                            = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 9.49/5.27                     => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4592_flip__bit__0,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
% 9.49/5.27        = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4593_flip__bit__0,axiom,
% 9.49/5.27      ! [A: nat] :
% 9.49/5.27        ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
% 9.49/5.27        = ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4594_flip__bit__0,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
% 9.49/5.27        = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4595_set__decode__0,axiom,
% 9.49/5.27      ! [X: nat] :
% 9.49/5.27        ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
% 9.49/5.27        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4596_even__set__encode__iff,axiom,
% 9.49/5.27      ! [A2: set_nat] :
% 9.49/5.27        ( ( finite_finite_nat @ A2 )
% 9.49/5.27       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
% 9.49/5.27          = ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4597_one__mod__2__pow__eq,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27        = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4598_one__mod__2__pow__eq,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4599_one__mod__2__pow__eq,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27        = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4600_case__prodI2,axiom,
% 9.49/5.27      ! [P4: produc8763457246119570046nteger,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
% 9.49/5.27        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 9.49/5.27            ( ( P4
% 9.49/5.27              = ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 9.49/5.27           => ( C @ A3 @ B3 ) )
% 9.49/5.27       => ( produc127349428274296955eger_o @ C @ P4 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4601_case__prodI2,axiom,
% 9.49/5.27      ! [P4: produc1908205239877642774nteger,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
% 9.49/5.27        ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 9.49/5.27            ( ( P4
% 9.49/5.27              = ( produc8603105652947943368nteger @ A3 @ B3 ) )
% 9.49/5.27           => ( C @ A3 @ B3 ) )
% 9.49/5.27       => ( produc6253627499356882019eger_o @ C @ P4 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4602_case__prodI2,axiom,
% 9.49/5.27      ! [P4: produc2285326912895808259nt_int,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
% 9.49/5.27        ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
% 9.49/5.27            ( ( P4
% 9.49/5.27              = ( produc5700946648718959541nt_int @ A3 @ B3 ) )
% 9.49/5.27           => ( C @ A3 @ B3 ) )
% 9.49/5.27       => ( produc1573362020775583542_int_o @ C @ P4 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4603_case__prodI2,axiom,
% 9.49/5.27      ! [P4: produc7773217078559923341nt_int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
% 9.49/5.27        ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
% 9.49/5.27            ( ( P4
% 9.49/5.27              = ( produc4305682042979456191nt_int @ A3 @ B3 ) )
% 9.49/5.27           => ( C @ A3 @ B3 ) )
% 9.49/5.27       => ( produc2558449545302689196_int_o @ C @ P4 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4604_case__prodI2,axiom,
% 9.49/5.27      ! [P4: product_prod_nat_nat,C: nat > nat > $o] :
% 9.49/5.27        ( ! [A3: nat,B3: nat] :
% 9.49/5.27            ( ( P4
% 9.49/5.27              = ( product_Pair_nat_nat @ A3 @ B3 ) )
% 9.49/5.27           => ( C @ A3 @ B3 ) )
% 9.49/5.27       => ( produc6081775807080527818_nat_o @ C @ P4 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4605_case__prodI2,axiom,
% 9.49/5.27      ! [P4: product_prod_int_int,C: int > int > $o] :
% 9.49/5.27        ( ! [A3: int,B3: int] :
% 9.49/5.27            ( ( P4
% 9.49/5.27              = ( product_Pair_int_int @ A3 @ B3 ) )
% 9.49/5.27           => ( C @ A3 @ B3 ) )
% 9.49/5.27       => ( produc4947309494688390418_int_o @ C @ P4 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4606_case__prodI,axiom,
% 9.49/5.27      ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 9.49/5.27        ( ( F @ A @ B )
% 9.49/5.27       => ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4607_case__prodI,axiom,
% 9.49/5.27      ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 9.49/5.27        ( ( F @ A @ B )
% 9.49/5.27       => ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4608_case__prodI,axiom,
% 9.49/5.27      ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
% 9.49/5.27        ( ( F @ A @ B )
% 9.49/5.27       => ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4609_case__prodI,axiom,
% 9.49/5.27      ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
% 9.49/5.27        ( ( F @ A @ B )
% 9.49/5.27       => ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4610_case__prodI,axiom,
% 9.49/5.27      ! [F: nat > nat > $o,A: nat,B: nat] :
% 9.49/5.27        ( ( F @ A @ B )
% 9.49/5.27       => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4611_case__prodI,axiom,
% 9.49/5.27      ! [F: int > int > $o,A: int,B: int] :
% 9.49/5.27        ( ( F @ A @ B )
% 9.49/5.27       => ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4612_mem__case__prodI2,axiom,
% 9.49/5.27      ! [P4: product_prod_int_int,Z2: real,C: int > int > set_real] :
% 9.49/5.27        ( ! [A3: int,B3: int] :
% 9.49/5.27            ( ( P4
% 9.49/5.27              = ( product_Pair_int_int @ A3 @ B3 ) )
% 9.49/5.27           => ( member_real @ Z2 @ ( C @ A3 @ B3 ) ) )
% 9.49/5.27       => ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C @ P4 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4613_mem__case__prodI2,axiom,
% 9.49/5.27      ! [P4: product_prod_int_int,Z2: nat,C: int > int > set_nat] :
% 9.49/5.27        ( ! [A3: int,B3: int] :
% 9.49/5.27            ( ( P4
% 9.49/5.27              = ( product_Pair_int_int @ A3 @ B3 ) )
% 9.49/5.27           => ( member_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
% 9.49/5.27       => ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ P4 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4614_mem__case__prodI2,axiom,
% 9.49/5.27      ! [P4: product_prod_int_int,Z2: int,C: int > int > set_int] :
% 9.49/5.27        ( ! [A3: int,B3: int] :
% 9.49/5.27            ( ( P4
% 9.49/5.27              = ( product_Pair_int_int @ A3 @ B3 ) )
% 9.49/5.27           => ( member_int @ Z2 @ ( C @ A3 @ B3 ) ) )
% 9.49/5.27       => ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ P4 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4615_mem__case__prodI2,axiom,
% 9.49/5.27      ! [P4: product_prod_int_int,Z2: set_nat,C: int > int > set_set_nat] :
% 9.49/5.27        ( ! [A3: int,B3: int] :
% 9.49/5.27            ( ( P4
% 9.49/5.27              = ( product_Pair_int_int @ A3 @ B3 ) )
% 9.49/5.27           => ( member_set_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
% 9.49/5.27       => ( member_set_nat @ Z2 @ ( produc5233655623923918146et_nat @ C @ P4 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4616_mem__case__prodI2,axiom,
% 9.49/5.27      ! [P4: product_prod_int_int,Z2: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat] :
% 9.49/5.27        ( ! [A3: int,B3: int] :
% 9.49/5.27            ( ( P4
% 9.49/5.27              = ( product_Pair_int_int @ A3 @ B3 ) )
% 9.49/5.27           => ( member8440522571783428010at_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
% 9.49/5.27       => ( member8440522571783428010at_nat @ Z2 @ ( produc1656060378719767003at_nat @ C @ P4 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4617_mem__case__prodI2,axiom,
% 9.49/5.27      ! [P4: produc8763457246119570046nteger,Z2: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real] :
% 9.49/5.27        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 9.49/5.27            ( ( P4
% 9.49/5.27              = ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 9.49/5.27           => ( member_real @ Z2 @ ( C @ A3 @ B3 ) ) )
% 9.49/5.27       => ( member_real @ Z2 @ ( produc815715089573277247t_real @ C @ P4 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4618_mem__case__prodI2,axiom,
% 9.49/5.27      ! [P4: produc8763457246119570046nteger,Z2: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat] :
% 9.49/5.27        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 9.49/5.27            ( ( P4
% 9.49/5.27              = ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 9.49/5.27           => ( member_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
% 9.49/5.27       => ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C @ P4 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4619_mem__case__prodI2,axiom,
% 9.49/5.27      ! [P4: produc8763457246119570046nteger,Z2: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int] :
% 9.49/5.27        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 9.49/5.27            ( ( P4
% 9.49/5.27              = ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 9.49/5.27           => ( member_int @ Z2 @ ( C @ A3 @ B3 ) ) )
% 9.49/5.27       => ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C @ P4 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4620_mem__case__prodI2,axiom,
% 9.49/5.27      ! [P4: produc7773217078559923341nt_int,Z2: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real] :
% 9.49/5.27        ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
% 9.49/5.27            ( ( P4
% 9.49/5.27              = ( produc4305682042979456191nt_int @ A3 @ B3 ) )
% 9.49/5.27           => ( member_real @ Z2 @ ( C @ A3 @ B3 ) ) )
% 9.49/5.27       => ( member_real @ Z2 @ ( produc8709739885379107790t_real @ C @ P4 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4621_mem__case__prodI2,axiom,
% 9.49/5.27      ! [P4: produc7773217078559923341nt_int,Z2: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat] :
% 9.49/5.27        ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
% 9.49/5.27            ( ( P4
% 9.49/5.27              = ( produc4305682042979456191nt_int @ A3 @ B3 ) )
% 9.49/5.27           => ( member_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
% 9.49/5.27       => ( member_nat @ Z2 @ ( produc8289552606927098482et_nat @ C @ P4 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4622_mem__case__prodI,axiom,
% 9.49/5.27      ! [Z2: real,C: int > int > set_real,A: int,B: int] :
% 9.49/5.27        ( ( member_real @ Z2 @ ( C @ A @ B ) )
% 9.49/5.27       => ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4623_mem__case__prodI,axiom,
% 9.49/5.27      ! [Z2: nat,C: int > int > set_nat,A: int,B: int] :
% 9.49/5.27        ( ( member_nat @ Z2 @ ( C @ A @ B ) )
% 9.49/5.27       => ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4624_mem__case__prodI,axiom,
% 9.49/5.27      ! [Z2: int,C: int > int > set_int,A: int,B: int] :
% 9.49/5.27        ( ( member_int @ Z2 @ ( C @ A @ B ) )
% 9.49/5.27       => ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4625_mem__case__prodI,axiom,
% 9.49/5.27      ! [Z2: set_nat,C: int > int > set_set_nat,A: int,B: int] :
% 9.49/5.27        ( ( member_set_nat @ Z2 @ ( C @ A @ B ) )
% 9.49/5.27       => ( member_set_nat @ Z2 @ ( produc5233655623923918146et_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4626_mem__case__prodI,axiom,
% 9.49/5.27      ! [Z2: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat,A: int,B: int] :
% 9.49/5.27        ( ( member8440522571783428010at_nat @ Z2 @ ( C @ A @ B ) )
% 9.49/5.27       => ( member8440522571783428010at_nat @ Z2 @ ( produc1656060378719767003at_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4627_mem__case__prodI,axiom,
% 9.49/5.27      ! [Z2: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 9.49/5.27        ( ( member_real @ Z2 @ ( C @ A @ B ) )
% 9.49/5.27       => ( member_real @ Z2 @ ( produc815715089573277247t_real @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4628_mem__case__prodI,axiom,
% 9.49/5.27      ! [Z2: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 9.49/5.27        ( ( member_nat @ Z2 @ ( C @ A @ B ) )
% 9.49/5.27       => ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4629_mem__case__prodI,axiom,
% 9.49/5.27      ! [Z2: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 9.49/5.27        ( ( member_int @ Z2 @ ( C @ A @ B ) )
% 9.49/5.27       => ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4630_mem__case__prodI,axiom,
% 9.49/5.27      ! [Z2: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
% 9.49/5.27        ( ( member_real @ Z2 @ ( C @ A @ B ) )
% 9.49/5.27       => ( member_real @ Z2 @ ( produc8709739885379107790t_real @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4631_mem__case__prodI,axiom,
% 9.49/5.27      ! [Z2: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
% 9.49/5.27        ( ( member_nat @ Z2 @ ( C @ A @ B ) )
% 9.49/5.27       => ( member_nat @ Z2 @ ( produc8289552606927098482et_nat @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4632_case__prodI2_H,axiom,
% 9.49/5.27      ! [P4: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
% 9.49/5.27        ( ! [A3: nat,B3: nat] :
% 9.49/5.27            ( ( ( product_Pair_nat_nat @ A3 @ B3 )
% 9.49/5.27              = P4 )
% 9.49/5.27           => ( C @ A3 @ B3 @ X ) )
% 9.49/5.27       => ( produc8739625826339149834_nat_o @ C @ P4 @ X ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4633_of__bool__less__eq__iff,axiom,
% 9.49/5.27      ! [P: $o,Q: $o] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 9.49/5.27        = ( P
% 9.49/5.27         => Q ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4634_of__bool__less__eq__iff,axiom,
% 9.49/5.27      ! [P: $o,Q: $o] :
% 9.49/5.27        ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 9.49/5.27        = ( P
% 9.49/5.27         => Q ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4635_of__bool__less__eq__iff,axiom,
% 9.49/5.27      ! [P: $o,Q: $o] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 9.49/5.27        = ( P
% 9.49/5.27         => Q ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4636_of__bool__eq_I1_J,axiom,
% 9.49/5.27      ( ( zero_n1201886186963655149omplex @ $false )
% 9.49/5.27      = zero_zero_complex ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4637_of__bool__eq_I1_J,axiom,
% 9.49/5.27      ( ( zero_n3304061248610475627l_real @ $false )
% 9.49/5.27      = zero_zero_real ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4638_of__bool__eq_I1_J,axiom,
% 9.49/5.27      ( ( zero_n2052037380579107095ol_rat @ $false )
% 9.49/5.27      = zero_zero_rat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4639_of__bool__eq_I1_J,axiom,
% 9.49/5.27      ( ( zero_n2687167440665602831ol_nat @ $false )
% 9.49/5.27      = zero_zero_nat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4640_of__bool__eq_I1_J,axiom,
% 9.49/5.27      ( ( zero_n2684676970156552555ol_int @ $false )
% 9.49/5.27      = zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4641_of__bool__eq__0__iff,axiom,
% 9.49/5.27      ! [P: $o] :
% 9.49/5.27        ( ( ( zero_n1201886186963655149omplex @ P )
% 9.49/5.27          = zero_zero_complex )
% 9.49/5.27        = ~ P ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4642_of__bool__eq__0__iff,axiom,
% 9.49/5.27      ! [P: $o] :
% 9.49/5.27        ( ( ( zero_n3304061248610475627l_real @ P )
% 9.49/5.27          = zero_zero_real )
% 9.49/5.27        = ~ P ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4643_of__bool__eq__0__iff,axiom,
% 9.49/5.27      ! [P: $o] :
% 9.49/5.27        ( ( ( zero_n2052037380579107095ol_rat @ P )
% 9.49/5.27          = zero_zero_rat )
% 9.49/5.27        = ~ P ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4644_of__bool__eq__0__iff,axiom,
% 9.49/5.27      ! [P: $o] :
% 9.49/5.27        ( ( ( zero_n2687167440665602831ol_nat @ P )
% 9.49/5.27          = zero_zero_nat )
% 9.49/5.27        = ~ P ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4645_of__bool__eq__0__iff,axiom,
% 9.49/5.27      ! [P: $o] :
% 9.49/5.27        ( ( ( zero_n2684676970156552555ol_int @ P )
% 9.49/5.27          = zero_zero_int )
% 9.49/5.27        = ~ P ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4646_of__bool__less__iff,axiom,
% 9.49/5.27      ! [P: $o,Q: $o] :
% 9.49/5.27        ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 9.49/5.27        = ( ~ P
% 9.49/5.27          & Q ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4647_of__bool__less__iff,axiom,
% 9.49/5.27      ! [P: $o,Q: $o] :
% 9.49/5.27        ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 9.49/5.27        = ( ~ P
% 9.49/5.27          & Q ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4648_of__bool__less__iff,axiom,
% 9.49/5.27      ! [P: $o,Q: $o] :
% 9.49/5.27        ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 9.49/5.27        = ( ~ P
% 9.49/5.27          & Q ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4649_of__bool__less__iff,axiom,
% 9.49/5.27      ! [P: $o,Q: $o] :
% 9.49/5.27        ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 9.49/5.27        = ( ~ P
% 9.49/5.27          & Q ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4650_zero__less__of__bool__iff,axiom,
% 9.49/5.27      ! [P: $o] :
% 9.49/5.27        ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
% 9.49/5.27        = P ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4651_zero__less__of__bool__iff,axiom,
% 9.49/5.27      ! [P: $o] :
% 9.49/5.27        ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 9.49/5.27        = P ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4652_zero__less__of__bool__iff,axiom,
% 9.49/5.27      ! [P: $o] :
% 9.49/5.27        ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 9.49/5.27        = P ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4653_zero__less__of__bool__iff,axiom,
% 9.49/5.27      ! [P: $o] :
% 9.49/5.27        ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 9.49/5.27        = P ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4654_of__bool__less__one__iff,axiom,
% 9.49/5.27      ! [P: $o] :
% 9.49/5.27        ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
% 9.49/5.27        = ~ P ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4655_of__bool__less__one__iff,axiom,
% 9.49/5.27      ! [P: $o] :
% 9.49/5.27        ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
% 9.49/5.27        = ~ P ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4656_of__bool__less__one__iff,axiom,
% 9.49/5.27      ! [P: $o] :
% 9.49/5.27        ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
% 9.49/5.27        = ~ P ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4657_of__bool__less__one__iff,axiom,
% 9.49/5.27      ! [P: $o] :
% 9.49/5.27        ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
% 9.49/5.27        = ~ P ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4658_Suc__0__mod__eq,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.49/5.27        = ( zero_n2687167440665602831ol_nat
% 9.49/5.27          @ ( N
% 9.49/5.27           != ( suc @ zero_zero_nat ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4659_set__decode__zero,axiom,
% 9.49/5.27      ( ( nat_set_decode @ zero_zero_nat )
% 9.49/5.27      = bot_bot_set_nat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4660_set__encode__empty,axiom,
% 9.49/5.27      ( ( nat_set_encode @ bot_bot_set_nat )
% 9.49/5.27      = zero_zero_nat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4661_set__encode__inverse,axiom,
% 9.49/5.27      ! [A2: set_nat] :
% 9.49/5.27        ( ( finite_finite_nat @ A2 )
% 9.49/5.27       => ( ( nat_set_decode @ ( nat_set_encode @ A2 ) )
% 9.49/5.27          = A2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4662_take__bit__of__Suc__0,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.49/5.27        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4663_divmod__algorithm__code_I2_J,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ( ( unique5052692396658037445od_int @ M @ one )
% 9.49/5.27        = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4664_divmod__algorithm__code_I2_J,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ( ( unique5055182867167087721od_nat @ M @ one )
% 9.49/5.27        = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4665_divmod__algorithm__code_I2_J,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ( ( unique3479559517661332726nteger @ M @ one )
% 9.49/5.27        = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4666_take__bit__of__1,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 9.49/5.27        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4667_take__bit__of__1,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 9.49/5.27        = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4668_of__bool__half__eq__0,axiom,
% 9.49/5.27      ! [B: $o] :
% 9.49/5.27        ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.49/5.27        = zero_z3403309356797280102nteger ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4669_of__bool__half__eq__0,axiom,
% 9.49/5.27      ! [B: $o] :
% 9.49/5.27        ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.27        = zero_zero_nat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4670_of__bool__half__eq__0,axiom,
% 9.49/5.27      ! [B: $o] :
% 9.49/5.27        ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.27        = zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4671_divmod__algorithm__code_I3_J,axiom,
% 9.49/5.27      ! [N: num] :
% 9.49/5.27        ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
% 9.49/5.27        = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4672_divmod__algorithm__code_I3_J,axiom,
% 9.49/5.27      ! [N: num] :
% 9.49/5.27        ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
% 9.49/5.27        = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4673_divmod__algorithm__code_I3_J,axiom,
% 9.49/5.27      ! [N: num] :
% 9.49/5.27        ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
% 9.49/5.27        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4674_bits__1__div__exp,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27        = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4675_bits__1__div__exp,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27        = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4676_bits__1__div__exp,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27        = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4677_one__div__2__pow__eq,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27        = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4678_one__div__2__pow__eq,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27        = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4679_one__div__2__pow__eq,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27        = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4680_take__bit__of__exp,axiom,
% 9.49/5.27      ! [M: nat,N: nat] :
% 9.49/5.27        ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4681_take__bit__of__exp,axiom,
% 9.49/5.27      ! [M: nat,N: nat] :
% 9.49/5.27        ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4682_take__bit__of__2,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.27        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4683_take__bit__of__2,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.27        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4684_mem__case__prodE,axiom,
% 9.49/5.27      ! [Z2: real,C: int > int > set_real,P4: product_prod_int_int] :
% 9.49/5.27        ( ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C @ P4 ) )
% 9.49/5.27       => ~ ! [X4: int,Y3: int] :
% 9.49/5.27              ( ( P4
% 9.49/5.27                = ( product_Pair_int_int @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( member_real @ Z2 @ ( C @ X4 @ Y3 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4685_mem__case__prodE,axiom,
% 9.49/5.27      ! [Z2: nat,C: int > int > set_nat,P4: product_prod_int_int] :
% 9.49/5.27        ( ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ P4 ) )
% 9.49/5.27       => ~ ! [X4: int,Y3: int] :
% 9.49/5.27              ( ( P4
% 9.49/5.27                = ( product_Pair_int_int @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( member_nat @ Z2 @ ( C @ X4 @ Y3 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4686_mem__case__prodE,axiom,
% 9.49/5.27      ! [Z2: int,C: int > int > set_int,P4: product_prod_int_int] :
% 9.49/5.27        ( ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ P4 ) )
% 9.49/5.27       => ~ ! [X4: int,Y3: int] :
% 9.49/5.27              ( ( P4
% 9.49/5.27                = ( product_Pair_int_int @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( member_int @ Z2 @ ( C @ X4 @ Y3 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4687_mem__case__prodE,axiom,
% 9.49/5.27      ! [Z2: set_nat,C: int > int > set_set_nat,P4: product_prod_int_int] :
% 9.49/5.27        ( ( member_set_nat @ Z2 @ ( produc5233655623923918146et_nat @ C @ P4 ) )
% 9.49/5.27       => ~ ! [X4: int,Y3: int] :
% 9.49/5.27              ( ( P4
% 9.49/5.27                = ( product_Pair_int_int @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( member_set_nat @ Z2 @ ( C @ X4 @ Y3 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4688_mem__case__prodE,axiom,
% 9.49/5.27      ! [Z2: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat,P4: product_prod_int_int] :
% 9.49/5.27        ( ( member8440522571783428010at_nat @ Z2 @ ( produc1656060378719767003at_nat @ C @ P4 ) )
% 9.49/5.27       => ~ ! [X4: int,Y3: int] :
% 9.49/5.27              ( ( P4
% 9.49/5.27                = ( product_Pair_int_int @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( member8440522571783428010at_nat @ Z2 @ ( C @ X4 @ Y3 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4689_mem__case__prodE,axiom,
% 9.49/5.27      ! [Z2: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,P4: produc8763457246119570046nteger] :
% 9.49/5.27        ( ( member_real @ Z2 @ ( produc815715089573277247t_real @ C @ P4 ) )
% 9.49/5.27       => ~ ! [X4: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
% 9.49/5.27              ( ( P4
% 9.49/5.27                = ( produc6137756002093451184nteger @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( member_real @ Z2 @ ( C @ X4 @ Y3 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4690_mem__case__prodE,axiom,
% 9.49/5.27      ! [Z2: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,P4: produc8763457246119570046nteger] :
% 9.49/5.27        ( ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C @ P4 ) )
% 9.49/5.27       => ~ ! [X4: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
% 9.49/5.27              ( ( P4
% 9.49/5.27                = ( produc6137756002093451184nteger @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( member_nat @ Z2 @ ( C @ X4 @ Y3 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4691_mem__case__prodE,axiom,
% 9.49/5.27      ! [Z2: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,P4: produc8763457246119570046nteger] :
% 9.49/5.27        ( ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C @ P4 ) )
% 9.49/5.27       => ~ ! [X4: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
% 9.49/5.27              ( ( P4
% 9.49/5.27                = ( produc6137756002093451184nteger @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( member_int @ Z2 @ ( C @ X4 @ Y3 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4692_mem__case__prodE,axiom,
% 9.49/5.27      ! [Z2: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real,P4: produc7773217078559923341nt_int] :
% 9.49/5.27        ( ( member_real @ Z2 @ ( produc8709739885379107790t_real @ C @ P4 ) )
% 9.49/5.27       => ~ ! [X4: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
% 9.49/5.27              ( ( P4
% 9.49/5.27                = ( produc4305682042979456191nt_int @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( member_real @ Z2 @ ( C @ X4 @ Y3 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4693_mem__case__prodE,axiom,
% 9.49/5.27      ! [Z2: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,P4: produc7773217078559923341nt_int] :
% 9.49/5.27        ( ( member_nat @ Z2 @ ( produc8289552606927098482et_nat @ C @ P4 ) )
% 9.49/5.27       => ~ ! [X4: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
% 9.49/5.27              ( ( P4
% 9.49/5.27                = ( produc4305682042979456191nt_int @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( member_nat @ Z2 @ ( C @ X4 @ Y3 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4694_case__prodE,axiom,
% 9.49/5.27      ! [C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc8763457246119570046nteger] :
% 9.49/5.27        ( ( produc127349428274296955eger_o @ C @ P4 )
% 9.49/5.27       => ~ ! [X4: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
% 9.49/5.27              ( ( P4
% 9.49/5.27                = ( produc6137756002093451184nteger @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( C @ X4 @ Y3 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4695_case__prodE,axiom,
% 9.49/5.27      ! [C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc1908205239877642774nteger] :
% 9.49/5.27        ( ( produc6253627499356882019eger_o @ C @ P4 )
% 9.49/5.27       => ~ ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
% 9.49/5.27              ( ( P4
% 9.49/5.27                = ( produc8603105652947943368nteger @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( C @ X4 @ Y3 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4696_case__prodE,axiom,
% 9.49/5.27      ! [C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc2285326912895808259nt_int] :
% 9.49/5.27        ( ( produc1573362020775583542_int_o @ C @ P4 )
% 9.49/5.27       => ~ ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
% 9.49/5.27              ( ( P4
% 9.49/5.27                = ( produc5700946648718959541nt_int @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( C @ X4 @ Y3 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4697_case__prodE,axiom,
% 9.49/5.27      ! [C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc7773217078559923341nt_int] :
% 9.49/5.27        ( ( produc2558449545302689196_int_o @ C @ P4 )
% 9.49/5.27       => ~ ! [X4: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
% 9.49/5.27              ( ( P4
% 9.49/5.27                = ( produc4305682042979456191nt_int @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( C @ X4 @ Y3 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4698_case__prodE,axiom,
% 9.49/5.27      ! [C: nat > nat > $o,P4: product_prod_nat_nat] :
% 9.49/5.27        ( ( produc6081775807080527818_nat_o @ C @ P4 )
% 9.49/5.27       => ~ ! [X4: nat,Y3: nat] :
% 9.49/5.27              ( ( P4
% 9.49/5.27                = ( product_Pair_nat_nat @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( C @ X4 @ Y3 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4699_case__prodE,axiom,
% 9.49/5.27      ! [C: int > int > $o,P4: product_prod_int_int] :
% 9.49/5.27        ( ( produc4947309494688390418_int_o @ C @ P4 )
% 9.49/5.27       => ~ ! [X4: int,Y3: int] :
% 9.49/5.27              ( ( P4
% 9.49/5.27                = ( product_Pair_int_int @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( C @ X4 @ Y3 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4700_case__prodD,axiom,
% 9.49/5.27      ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 9.49/5.27        ( ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B ) )
% 9.49/5.27       => ( F @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4701_case__prodD,axiom,
% 9.49/5.27      ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 9.49/5.27        ( ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) )
% 9.49/5.27       => ( F @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4702_case__prodD,axiom,
% 9.49/5.27      ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
% 9.49/5.27        ( ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) )
% 9.49/5.27       => ( F @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4703_case__prodD,axiom,
% 9.49/5.27      ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
% 9.49/5.27        ( ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) )
% 9.49/5.27       => ( F @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4704_case__prodD,axiom,
% 9.49/5.27      ! [F: nat > nat > $o,A: nat,B: nat] :
% 9.49/5.27        ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 9.49/5.27       => ( F @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4705_case__prodD,axiom,
% 9.49/5.27      ! [F: int > int > $o,A: int,B: int] :
% 9.49/5.27        ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
% 9.49/5.27       => ( F @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4706_case__prodE_H,axiom,
% 9.49/5.27      ! [C: nat > nat > product_prod_nat_nat > $o,P4: product_prod_nat_nat,Z2: product_prod_nat_nat] :
% 9.49/5.27        ( ( produc8739625826339149834_nat_o @ C @ P4 @ Z2 )
% 9.49/5.27       => ~ ! [X4: nat,Y3: nat] :
% 9.49/5.27              ( ( P4
% 9.49/5.27                = ( product_Pair_nat_nat @ X4 @ Y3 ) )
% 9.49/5.27             => ~ ( C @ X4 @ Y3 @ Z2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4707_case__prodD_H,axiom,
% 9.49/5.27      ! [R: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
% 9.49/5.27        ( ( produc8739625826339149834_nat_o @ R @ ( product_Pair_nat_nat @ A @ B ) @ C )
% 9.49/5.27       => ( R @ A @ B @ C ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4708_zero__less__eq__of__bool,axiom,
% 9.49/5.27      ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4709_zero__less__eq__of__bool,axiom,
% 9.49/5.27      ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4710_zero__less__eq__of__bool,axiom,
% 9.49/5.27      ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4711_zero__less__eq__of__bool,axiom,
% 9.49/5.27      ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4712_of__bool__less__eq__one,axiom,
% 9.49/5.27      ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4713_of__bool__less__eq__one,axiom,
% 9.49/5.27      ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4714_of__bool__less__eq__one,axiom,
% 9.49/5.27      ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4715_of__bool__less__eq__one,axiom,
% 9.49/5.27      ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4716_split__of__bool__asm,axiom,
% 9.49/5.27      ! [P: complex > $o,P4: $o] :
% 9.49/5.27        ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
% 9.49/5.27        = ( ~ ( ( P4
% 9.49/5.27                & ~ ( P @ one_one_complex ) )
% 9.49/5.27              | ( ~ P4
% 9.49/5.27                & ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4717_split__of__bool__asm,axiom,
% 9.49/5.27      ! [P: real > $o,P4: $o] :
% 9.49/5.27        ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
% 9.49/5.27        = ( ~ ( ( P4
% 9.49/5.27                & ~ ( P @ one_one_real ) )
% 9.49/5.27              | ( ~ P4
% 9.49/5.27                & ~ ( P @ zero_zero_real ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4718_split__of__bool__asm,axiom,
% 9.49/5.27      ! [P: rat > $o,P4: $o] :
% 9.49/5.27        ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
% 9.49/5.27        = ( ~ ( ( P4
% 9.49/5.27                & ~ ( P @ one_one_rat ) )
% 9.49/5.27              | ( ~ P4
% 9.49/5.27                & ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4719_split__of__bool__asm,axiom,
% 9.49/5.27      ! [P: nat > $o,P4: $o] :
% 9.49/5.27        ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
% 9.49/5.27        = ( ~ ( ( P4
% 9.49/5.27                & ~ ( P @ one_one_nat ) )
% 9.49/5.27              | ( ~ P4
% 9.49/5.27                & ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4720_split__of__bool__asm,axiom,
% 9.49/5.27      ! [P: int > $o,P4: $o] :
% 9.49/5.27        ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
% 9.49/5.27        = ( ~ ( ( P4
% 9.49/5.27                & ~ ( P @ one_one_int ) )
% 9.49/5.27              | ( ~ P4
% 9.49/5.27                & ~ ( P @ zero_zero_int ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4721_split__of__bool,axiom,
% 9.49/5.27      ! [P: complex > $o,P4: $o] :
% 9.49/5.27        ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
% 9.49/5.27        = ( ( P4
% 9.49/5.27           => ( P @ one_one_complex ) )
% 9.49/5.27          & ( ~ P4
% 9.49/5.27           => ( P @ zero_zero_complex ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4722_split__of__bool,axiom,
% 9.49/5.27      ! [P: real > $o,P4: $o] :
% 9.49/5.27        ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
% 9.49/5.27        = ( ( P4
% 9.49/5.27           => ( P @ one_one_real ) )
% 9.49/5.27          & ( ~ P4
% 9.49/5.27           => ( P @ zero_zero_real ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4723_split__of__bool,axiom,
% 9.49/5.27      ! [P: rat > $o,P4: $o] :
% 9.49/5.27        ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
% 9.49/5.27        = ( ( P4
% 9.49/5.27           => ( P @ one_one_rat ) )
% 9.49/5.27          & ( ~ P4
% 9.49/5.27           => ( P @ zero_zero_rat ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4724_split__of__bool,axiom,
% 9.49/5.27      ! [P: nat > $o,P4: $o] :
% 9.49/5.27        ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
% 9.49/5.27        = ( ( P4
% 9.49/5.27           => ( P @ one_one_nat ) )
% 9.49/5.27          & ( ~ P4
% 9.49/5.27           => ( P @ zero_zero_nat ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4725_split__of__bool,axiom,
% 9.49/5.27      ! [P: int > $o,P4: $o] :
% 9.49/5.27        ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
% 9.49/5.27        = ( ( P4
% 9.49/5.27           => ( P @ one_one_int ) )
% 9.49/5.27          & ( ~ P4
% 9.49/5.27           => ( P @ zero_zero_int ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4726_of__bool__def,axiom,
% 9.49/5.27      ( zero_n1201886186963655149omplex
% 9.49/5.27      = ( ^ [P6: $o] : ( if_complex @ P6 @ one_one_complex @ zero_zero_complex ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4727_of__bool__def,axiom,
% 9.49/5.27      ( zero_n3304061248610475627l_real
% 9.49/5.27      = ( ^ [P6: $o] : ( if_real @ P6 @ one_one_real @ zero_zero_real ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4728_of__bool__def,axiom,
% 9.49/5.27      ( zero_n2052037380579107095ol_rat
% 9.49/5.27      = ( ^ [P6: $o] : ( if_rat @ P6 @ one_one_rat @ zero_zero_rat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4729_of__bool__def,axiom,
% 9.49/5.27      ( zero_n2687167440665602831ol_nat
% 9.49/5.27      = ( ^ [P6: $o] : ( if_nat @ P6 @ one_one_nat @ zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4730_of__bool__def,axiom,
% 9.49/5.27      ( zero_n2684676970156552555ol_int
% 9.49/5.27      = ( ^ [P6: $o] : ( if_int @ P6 @ one_one_int @ zero_zero_int ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4731_finite__set__decode,axiom,
% 9.49/5.27      ! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4732_set__encode__eq,axiom,
% 9.49/5.27      ! [A2: set_nat,B2: set_nat] :
% 9.49/5.27        ( ( finite_finite_nat @ A2 )
% 9.49/5.27       => ( ( finite_finite_nat @ B2 )
% 9.49/5.27         => ( ( ( nat_set_encode @ A2 )
% 9.49/5.27              = ( nat_set_encode @ B2 ) )
% 9.49/5.27            = ( A2 = B2 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4733_set__encode__inf,axiom,
% 9.49/5.27      ! [A2: set_nat] :
% 9.49/5.27        ( ~ ( finite_finite_nat @ A2 )
% 9.49/5.27       => ( ( nat_set_encode @ A2 )
% 9.49/5.27          = zero_zero_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4734_subset__decode__imp__le,axiom,
% 9.49/5.27      ! [M: nat,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
% 9.49/5.27       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4735_divmod__int__def,axiom,
% 9.49/5.27      ( unique5052692396658037445od_int
% 9.49/5.27      = ( ^ [M2: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4736_divmod__def,axiom,
% 9.49/5.27      ( unique5052692396658037445od_int
% 9.49/5.27      = ( ^ [M2: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4737_divmod__def,axiom,
% 9.49/5.27      ( unique5055182867167087721od_nat
% 9.49/5.27      = ( ^ [M2: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4738_divmod__def,axiom,
% 9.49/5.27      ( unique3479559517661332726nteger
% 9.49/5.27      = ( ^ [M2: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4739_divmod_H__nat__def,axiom,
% 9.49/5.27      ( unique5055182867167087721od_nat
% 9.49/5.27      = ( ^ [M2: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4740_exp__mod__exp,axiom,
% 9.49/5.27      ! [M: nat,N: nat] :
% 9.49/5.27        ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27        = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4741_exp__mod__exp,axiom,
% 9.49/5.27      ! [M: nat,N: nat] :
% 9.49/5.27        ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4742_exp__mod__exp,axiom,
% 9.49/5.27      ! [M: nat,N: nat] :
% 9.49/5.27        ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4743_divmod__nat__def,axiom,
% 9.49/5.27      ( divmod_nat
% 9.49/5.27      = ( ^ [M2: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M2 @ N2 ) @ ( modulo_modulo_nat @ M2 @ N2 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4744_exp__div__exp__eq,axiom,
% 9.49/5.27      ! [M: nat,N: nat] :
% 9.49/5.27        ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27        = ( times_3573771949741848930nteger
% 9.49/5.27          @ ( zero_n356916108424825756nteger
% 9.49/5.27            @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 9.49/5.27               != zero_z3403309356797280102nteger )
% 9.49/5.27              & ( ord_less_eq_nat @ N @ M ) ) )
% 9.49/5.27          @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4745_exp__div__exp__eq,axiom,
% 9.49/5.27      ! [M: nat,N: nat] :
% 9.49/5.27        ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27        = ( times_times_nat
% 9.49/5.27          @ ( zero_n2687167440665602831ol_nat
% 9.49/5.27            @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 9.49/5.27               != zero_zero_nat )
% 9.49/5.27              & ( ord_less_eq_nat @ N @ M ) ) )
% 9.49/5.27          @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4746_exp__div__exp__eq,axiom,
% 9.49/5.27      ! [M: nat,N: nat] :
% 9.49/5.27        ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27        = ( times_times_int
% 9.49/5.27          @ ( zero_n2684676970156552555ol_int
% 9.49/5.27            @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 9.49/5.27               != zero_zero_int )
% 9.49/5.27              & ( ord_less_eq_nat @ N @ M ) ) )
% 9.49/5.27          @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4747_divmod__divmod__step,axiom,
% 9.49/5.27      ( unique5055182867167087721od_nat
% 9.49/5.27      = ( ^ [M2: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M2 @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M2 ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4748_divmod__divmod__step,axiom,
% 9.49/5.27      ( unique5052692396658037445od_int
% 9.49/5.27      = ( ^ [M2: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M2 @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M2 ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4749_divmod__divmod__step,axiom,
% 9.49/5.27      ( unique3479559517661332726nteger
% 9.49/5.27      = ( ^ [M2: num,N2: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M2 @ N2 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( unique4921790084139445826nteger @ N2 @ ( unique3479559517661332726nteger @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4750_listrel1p__def,axiom,
% 9.49/5.27      ( listrel1p_nat
% 9.49/5.27      = ( ^ [R5: nat > nat > $o,Xs2: list_nat,Ys3: list_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys3 ) @ ( listrel1_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R5 ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4751_listrel1p__def,axiom,
% 9.49/5.27      ( listrel1p_int
% 9.49/5.27      = ( ^ [R5: int > int > $o,Xs2: list_int,Ys3: list_int] : ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys3 ) @ ( listrel1_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R5 ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4752_divmod__algorithm__code_I6_J,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.49/5.27        = ( produc4245557441103728435nt_int
% 9.49/5.27          @ ^ [Q5: int,R5: int] : ( product_Pair_int_int @ Q5 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
% 9.49/5.27          @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4753_divmod__algorithm__code_I6_J,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.49/5.27        = ( produc2626176000494625587at_nat
% 9.49/5.27          @ ^ [Q5: nat,R5: nat] : ( product_Pair_nat_nat @ Q5 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
% 9.49/5.27          @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4754_divmod__algorithm__code_I6_J,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.49/5.27        = ( produc6916734918728496179nteger
% 9.49/5.27          @ ^ [Q5: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
% 9.49/5.27          @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4755_Divides_Oadjust__div__eq,axiom,
% 9.49/5.27      ! [Q4: int,R2: int] :
% 9.49/5.27        ( ( adjust_div @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.49/5.27        = ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4756_int__ge__less__than2__def,axiom,
% 9.49/5.27      ( int_ge_less_than2
% 9.49/5.27      = ( ^ [D4: int] :
% 9.49/5.27            ( collec213857154873943460nt_int
% 9.49/5.27            @ ( produc4947309494688390418_int_o
% 9.49/5.27              @ ^ [Z7: int,Z3: int] :
% 9.49/5.27                  ( ( ord_less_eq_int @ D4 @ Z3 )
% 9.49/5.27                  & ( ord_less_int @ Z7 @ Z3 ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4757_int__ge__less__than__def,axiom,
% 9.49/5.27      ( int_ge_less_than
% 9.49/5.27      = ( ^ [D4: int] :
% 9.49/5.27            ( collec213857154873943460nt_int
% 9.49/5.27            @ ( produc4947309494688390418_int_o
% 9.49/5.27              @ ^ [Z7: int,Z3: int] :
% 9.49/5.27                  ( ( ord_less_eq_int @ D4 @ Z7 )
% 9.49/5.27                  & ( ord_less_int @ Z7 @ Z3 ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4758_take__bit__minus__small__eq,axiom,
% 9.49/5.27      ! [K: int,N: nat] :
% 9.49/5.27        ( ( ord_less_int @ zero_zero_int @ K )
% 9.49/5.27       => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27         => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
% 9.49/5.27            = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4759_divmod__algorithm__code_I7_J,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( ( ord_less_eq_num @ M @ N )
% 9.49/5.27         => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.49/5.27            = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
% 9.49/5.27        & ( ~ ( ord_less_eq_num @ M @ N )
% 9.49/5.27         => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.49/5.27            = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4760_divmod__algorithm__code_I7_J,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( ( ord_less_eq_num @ M @ N )
% 9.49/5.27         => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.49/5.27            = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
% 9.49/5.27        & ( ~ ( ord_less_eq_num @ M @ N )
% 9.49/5.27         => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.49/5.27            = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4761_divmod__algorithm__code_I7_J,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( ( ord_less_eq_num @ M @ N )
% 9.49/5.27         => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.49/5.27            = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
% 9.49/5.27        & ( ~ ( ord_less_eq_num @ M @ N )
% 9.49/5.27         => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.49/5.27            = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4762_add_Oinverse__inverse,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 9.49/5.27        = A ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4763_add_Oinverse__inverse,axiom,
% 9.49/5.27      ! [A: real] :
% 9.49/5.27        ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 9.49/5.27        = A ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4764_add_Oinverse__inverse,axiom,
% 9.49/5.27      ! [A: rat] :
% 9.49/5.27        ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
% 9.49/5.27        = A ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4765_add_Oinverse__inverse,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
% 9.49/5.27        = A ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4766_neg__equal__iff__equal,axiom,
% 9.49/5.27      ! [A: int,B: int] :
% 9.49/5.27        ( ( ( uminus_uminus_int @ A )
% 9.49/5.27          = ( uminus_uminus_int @ B ) )
% 9.49/5.27        = ( A = B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4767_neg__equal__iff__equal,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( ( uminus_uminus_real @ A )
% 9.49/5.27          = ( uminus_uminus_real @ B ) )
% 9.49/5.27        = ( A = B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4768_neg__equal__iff__equal,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( ( uminus_uminus_rat @ A )
% 9.49/5.27          = ( uminus_uminus_rat @ B ) )
% 9.49/5.27        = ( A = B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4769_neg__equal__iff__equal,axiom,
% 9.49/5.27      ! [A: code_integer,B: code_integer] :
% 9.49/5.27        ( ( ( uminus1351360451143612070nteger @ A )
% 9.49/5.27          = ( uminus1351360451143612070nteger @ B ) )
% 9.49/5.27        = ( A = B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4770_compl__le__compl__iff,axiom,
% 9.49/5.27      ! [X: set_int,Y: set_int] :
% 9.49/5.27        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( uminus1532241313380277803et_int @ Y ) )
% 9.49/5.27        = ( ord_less_eq_set_int @ Y @ X ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4771_neg__le__iff__le,axiom,
% 9.49/5.27      ! [B: real,A: real] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 9.49/5.27        = ( ord_less_eq_real @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4772_neg__le__iff__le,axiom,
% 9.49/5.27      ! [B: code_integer,A: code_integer] :
% 9.49/5.27        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 9.49/5.27        = ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4773_neg__le__iff__le,axiom,
% 9.49/5.27      ! [B: rat,A: rat] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 9.49/5.27        = ( ord_less_eq_rat @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4774_neg__le__iff__le,axiom,
% 9.49/5.27      ! [B: int,A: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 9.49/5.27        = ( ord_less_eq_int @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4775_neg__equal__zero,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( ( uminus_uminus_int @ A )
% 9.49/5.27          = A )
% 9.49/5.27        = ( A = zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4776_neg__equal__zero,axiom,
% 9.49/5.27      ! [A: real] :
% 9.49/5.27        ( ( ( uminus_uminus_real @ A )
% 9.49/5.27          = A )
% 9.49/5.27        = ( A = zero_zero_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4777_neg__equal__zero,axiom,
% 9.49/5.27      ! [A: rat] :
% 9.49/5.27        ( ( ( uminus_uminus_rat @ A )
% 9.49/5.27          = A )
% 9.49/5.27        = ( A = zero_zero_rat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4778_neg__equal__zero,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( ( uminus1351360451143612070nteger @ A )
% 9.49/5.27          = A )
% 9.49/5.27        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4779_equal__neg__zero,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( uminus_uminus_int @ A ) )
% 9.49/5.27        = ( A = zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4780_equal__neg__zero,axiom,
% 9.49/5.27      ! [A: real] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( uminus_uminus_real @ A ) )
% 9.49/5.27        = ( A = zero_zero_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4781_equal__neg__zero,axiom,
% 9.49/5.27      ! [A: rat] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( uminus_uminus_rat @ A ) )
% 9.49/5.27        = ( A = zero_zero_rat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4782_equal__neg__zero,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( uminus1351360451143612070nteger @ A ) )
% 9.49/5.27        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4783_neg__equal__0__iff__equal,axiom,
% 9.49/5.27      ! [A: complex] :
% 9.49/5.27        ( ( ( uminus1482373934393186551omplex @ A )
% 9.49/5.27          = zero_zero_complex )
% 9.49/5.27        = ( A = zero_zero_complex ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4784_neg__equal__0__iff__equal,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( ( uminus_uminus_int @ A )
% 9.49/5.27          = zero_zero_int )
% 9.49/5.27        = ( A = zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4785_neg__equal__0__iff__equal,axiom,
% 9.49/5.27      ! [A: real] :
% 9.49/5.27        ( ( ( uminus_uminus_real @ A )
% 9.49/5.27          = zero_zero_real )
% 9.49/5.27        = ( A = zero_zero_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4786_neg__equal__0__iff__equal,axiom,
% 9.49/5.27      ! [A: rat] :
% 9.49/5.27        ( ( ( uminus_uminus_rat @ A )
% 9.49/5.27          = zero_zero_rat )
% 9.49/5.27        = ( A = zero_zero_rat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4787_neg__equal__0__iff__equal,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( ( uminus1351360451143612070nteger @ A )
% 9.49/5.27          = zero_z3403309356797280102nteger )
% 9.49/5.27        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4788_neg__0__equal__iff__equal,axiom,
% 9.49/5.27      ! [A: complex] :
% 9.49/5.27        ( ( zero_zero_complex
% 9.49/5.27          = ( uminus1482373934393186551omplex @ A ) )
% 9.49/5.27        = ( zero_zero_complex = A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4789_neg__0__equal__iff__equal,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( zero_zero_int
% 9.49/5.27          = ( uminus_uminus_int @ A ) )
% 9.49/5.27        = ( zero_zero_int = A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4790_neg__0__equal__iff__equal,axiom,
% 9.49/5.27      ! [A: real] :
% 9.49/5.27        ( ( zero_zero_real
% 9.49/5.27          = ( uminus_uminus_real @ A ) )
% 9.49/5.27        = ( zero_zero_real = A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4791_neg__0__equal__iff__equal,axiom,
% 9.49/5.27      ! [A: rat] :
% 9.49/5.27        ( ( zero_zero_rat
% 9.49/5.27          = ( uminus_uminus_rat @ A ) )
% 9.49/5.27        = ( zero_zero_rat = A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4792_neg__0__equal__iff__equal,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( zero_z3403309356797280102nteger
% 9.49/5.27          = ( uminus1351360451143612070nteger @ A ) )
% 9.49/5.27        = ( zero_z3403309356797280102nteger = A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4793_add_Oinverse__neutral,axiom,
% 9.49/5.27      ( ( uminus1482373934393186551omplex @ zero_zero_complex )
% 9.49/5.27      = zero_zero_complex ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4794_add_Oinverse__neutral,axiom,
% 9.49/5.27      ( ( uminus_uminus_int @ zero_zero_int )
% 9.49/5.27      = zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4795_add_Oinverse__neutral,axiom,
% 9.49/5.27      ( ( uminus_uminus_real @ zero_zero_real )
% 9.49/5.27      = zero_zero_real ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4796_add_Oinverse__neutral,axiom,
% 9.49/5.27      ( ( uminus_uminus_rat @ zero_zero_rat )
% 9.49/5.27      = zero_zero_rat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4797_add_Oinverse__neutral,axiom,
% 9.49/5.27      ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 9.49/5.27      = zero_z3403309356797280102nteger ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4798_neg__less__iff__less,axiom,
% 9.49/5.27      ! [B: int,A: int] :
% 9.49/5.27        ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 9.49/5.27        = ( ord_less_int @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4799_neg__less__iff__less,axiom,
% 9.49/5.27      ! [B: real,A: real] :
% 9.49/5.27        ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 9.49/5.27        = ( ord_less_real @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4800_neg__less__iff__less,axiom,
% 9.49/5.27      ! [B: rat,A: rat] :
% 9.49/5.27        ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 9.49/5.27        = ( ord_less_rat @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4801_neg__less__iff__less,axiom,
% 9.49/5.27      ! [B: code_integer,A: code_integer] :
% 9.49/5.27        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 9.49/5.27        = ( ord_le6747313008572928689nteger @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4802_add__minus__cancel,axiom,
% 9.49/5.27      ! [A: int,B: int] :
% 9.49/5.27        ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 9.49/5.27        = B ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4803_add__minus__cancel,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
% 9.49/5.27        = B ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4804_add__minus__cancel,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
% 9.49/5.27        = B ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4805_add__minus__cancel,axiom,
% 9.49/5.27      ! [A: code_integer,B: code_integer] :
% 9.49/5.27        ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
% 9.49/5.27        = B ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4806_minus__add__cancel,axiom,
% 9.49/5.27      ! [A: int,B: int] :
% 9.49/5.27        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 9.49/5.27        = B ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4807_minus__add__cancel,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
% 9.49/5.27        = B ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4808_minus__add__cancel,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
% 9.49/5.27        = B ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4809_minus__add__cancel,axiom,
% 9.49/5.27      ! [A: code_integer,B: code_integer] :
% 9.49/5.27        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 9.49/5.27        = B ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4810_minus__add__distrib,axiom,
% 9.49/5.27      ! [A: int,B: int] :
% 9.49/5.27        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 9.49/5.27        = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4811_minus__add__distrib,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 9.49/5.27        = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4812_minus__add__distrib,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 9.49/5.27        = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4813_minus__add__distrib,axiom,
% 9.49/5.27      ! [A: code_integer,B: code_integer] :
% 9.49/5.27        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 9.49/5.27        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4814_minus__diff__eq,axiom,
% 9.49/5.27      ! [A: int,B: int] :
% 9.49/5.27        ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 9.49/5.27        = ( minus_minus_int @ B @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4815_minus__diff__eq,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
% 9.49/5.27        = ( minus_minus_real @ B @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4816_minus__diff__eq,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
% 9.49/5.27        = ( minus_minus_rat @ B @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4817_minus__diff__eq,axiom,
% 9.49/5.27      ! [A: code_integer,B: code_integer] :
% 9.49/5.27        ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
% 9.49/5.27        = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4818_semiring__norm_I80_J,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.49/5.27        = ( ord_less_num @ M @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4819_neg__0__le__iff__le,axiom,
% 9.49/5.27      ! [A: real] :
% 9.49/5.27        ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 9.49/5.27        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4820_neg__0__le__iff__le,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 9.49/5.27        = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4821_neg__0__le__iff__le,axiom,
% 9.49/5.27      ! [A: rat] :
% 9.49/5.27        ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 9.49/5.27        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4822_neg__0__le__iff__le,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 9.49/5.27        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4823_neg__le__0__iff__le,axiom,
% 9.49/5.27      ! [A: real] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 9.49/5.27        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4824_neg__le__0__iff__le,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 9.49/5.27        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4825_neg__le__0__iff__le,axiom,
% 9.49/5.27      ! [A: rat] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 9.49/5.27        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4826_neg__le__0__iff__le,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 9.49/5.27        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4827_less__eq__neg__nonpos,axiom,
% 9.49/5.27      ! [A: real] :
% 9.49/5.27        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 9.49/5.27        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4828_less__eq__neg__nonpos,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 9.49/5.27        = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4829_less__eq__neg__nonpos,axiom,
% 9.49/5.27      ! [A: rat] :
% 9.49/5.27        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
% 9.49/5.27        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4830_less__eq__neg__nonpos,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 9.49/5.27        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4831_neg__less__eq__nonneg,axiom,
% 9.49/5.27      ! [A: real] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 9.49/5.27        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4832_neg__less__eq__nonneg,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 9.49/5.27        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4833_neg__less__eq__nonneg,axiom,
% 9.49/5.27      ! [A: rat] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
% 9.49/5.27        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4834_neg__less__eq__nonneg,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 9.49/5.27        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4835_neg__less__0__iff__less,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 9.49/5.27        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4836_neg__less__0__iff__less,axiom,
% 9.49/5.27      ! [A: real] :
% 9.49/5.27        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 9.49/5.27        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4837_neg__less__0__iff__less,axiom,
% 9.49/5.27      ! [A: rat] :
% 9.49/5.27        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 9.49/5.27        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4838_neg__less__0__iff__less,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 9.49/5.27        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4839_neg__0__less__iff__less,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 9.49/5.27        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4840_neg__0__less__iff__less,axiom,
% 9.49/5.27      ! [A: real] :
% 9.49/5.27        ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 9.49/5.27        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4841_neg__0__less__iff__less,axiom,
% 9.49/5.27      ! [A: rat] :
% 9.49/5.27        ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 9.49/5.27        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4842_neg__0__less__iff__less,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 9.49/5.27        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4843_neg__less__pos,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 9.49/5.27        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4844_neg__less__pos,axiom,
% 9.49/5.27      ! [A: real] :
% 9.49/5.27        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 9.49/5.27        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4845_neg__less__pos,axiom,
% 9.49/5.27      ! [A: rat] :
% 9.49/5.27        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
% 9.49/5.27        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4846_neg__less__pos,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 9.49/5.27        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4847_less__neg__neg,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 9.49/5.27        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4848_less__neg__neg,axiom,
% 9.49/5.27      ! [A: real] :
% 9.49/5.27        ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 9.49/5.27        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4849_less__neg__neg,axiom,
% 9.49/5.27      ! [A: rat] :
% 9.49/5.27        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
% 9.49/5.27        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4850_less__neg__neg,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 9.49/5.27        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4851_add_Oright__inverse,axiom,
% 9.49/5.27      ! [A: complex] :
% 9.49/5.27        ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
% 9.49/5.27        = zero_zero_complex ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4852_add_Oright__inverse,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
% 9.49/5.27        = zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4853_add_Oright__inverse,axiom,
% 9.49/5.27      ! [A: real] :
% 9.49/5.27        ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 9.49/5.27        = zero_zero_real ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4854_add_Oright__inverse,axiom,
% 9.49/5.27      ! [A: rat] :
% 9.49/5.27        ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
% 9.49/5.27        = zero_zero_rat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4855_add_Oright__inverse,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 9.49/5.27        = zero_z3403309356797280102nteger ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4856_ab__left__minus,axiom,
% 9.49/5.27      ! [A: complex] :
% 9.49/5.27        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 9.49/5.27        = zero_zero_complex ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4857_ab__left__minus,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 9.49/5.27        = zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4858_ab__left__minus,axiom,
% 9.49/5.27      ! [A: real] :
% 9.49/5.27        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 9.49/5.27        = zero_zero_real ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4859_ab__left__minus,axiom,
% 9.49/5.27      ! [A: rat] :
% 9.49/5.27        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 9.49/5.27        = zero_zero_rat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4860_ab__left__minus,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 9.49/5.27        = zero_z3403309356797280102nteger ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4861_verit__minus__simplify_I3_J,axiom,
% 9.49/5.27      ! [B: complex] :
% 9.49/5.27        ( ( minus_minus_complex @ zero_zero_complex @ B )
% 9.49/5.27        = ( uminus1482373934393186551omplex @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4862_verit__minus__simplify_I3_J,axiom,
% 9.49/5.27      ! [B: int] :
% 9.49/5.27        ( ( minus_minus_int @ zero_zero_int @ B )
% 9.49/5.27        = ( uminus_uminus_int @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4863_verit__minus__simplify_I3_J,axiom,
% 9.49/5.27      ! [B: real] :
% 9.49/5.27        ( ( minus_minus_real @ zero_zero_real @ B )
% 9.49/5.27        = ( uminus_uminus_real @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4864_verit__minus__simplify_I3_J,axiom,
% 9.49/5.27      ! [B: rat] :
% 9.49/5.27        ( ( minus_minus_rat @ zero_zero_rat @ B )
% 9.49/5.27        = ( uminus_uminus_rat @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4865_verit__minus__simplify_I3_J,axiom,
% 9.49/5.27      ! [B: code_integer] :
% 9.49/5.27        ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
% 9.49/5.27        = ( uminus1351360451143612070nteger @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4866_diff__0,axiom,
% 9.49/5.27      ! [A: complex] :
% 9.49/5.27        ( ( minus_minus_complex @ zero_zero_complex @ A )
% 9.49/5.27        = ( uminus1482373934393186551omplex @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4867_diff__0,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( minus_minus_int @ zero_zero_int @ A )
% 9.49/5.27        = ( uminus_uminus_int @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4868_diff__0,axiom,
% 9.49/5.27      ! [A: real] :
% 9.49/5.27        ( ( minus_minus_real @ zero_zero_real @ A )
% 9.49/5.27        = ( uminus_uminus_real @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4869_diff__0,axiom,
% 9.49/5.27      ! [A: rat] :
% 9.49/5.27        ( ( minus_minus_rat @ zero_zero_rat @ A )
% 9.49/5.27        = ( uminus_uminus_rat @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4870_diff__0,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
% 9.49/5.27        = ( uminus1351360451143612070nteger @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4871_uminus__add__conv__diff,axiom,
% 9.49/5.27      ! [A: int,B: int] :
% 9.49/5.27        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 9.49/5.27        = ( minus_minus_int @ B @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4872_uminus__add__conv__diff,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
% 9.49/5.27        = ( minus_minus_real @ B @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4873_uminus__add__conv__diff,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
% 9.49/5.27        = ( minus_minus_rat @ B @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4874_uminus__add__conv__diff,axiom,
% 9.49/5.27      ! [A: code_integer,B: code_integer] :
% 9.49/5.27        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 9.49/5.27        = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4875_diff__minus__eq__add,axiom,
% 9.49/5.27      ! [A: int,B: int] :
% 9.49/5.27        ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 9.49/5.27        = ( plus_plus_int @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4876_diff__minus__eq__add,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
% 9.49/5.27        = ( plus_plus_real @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4877_diff__minus__eq__add,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
% 9.49/5.27        = ( plus_plus_rat @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4878_diff__minus__eq__add,axiom,
% 9.49/5.27      ! [A: code_integer,B: code_integer] :
% 9.49/5.27        ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 9.49/5.27        = ( plus_p5714425477246183910nteger @ A @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4879_semiring__norm_I81_J,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.49/5.27        = ( ord_less_num @ M @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4880_semiring__norm_I77_J,axiom,
% 9.49/5.27      ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4881_add__neg__numeral__special_I7_J,axiom,
% 9.49/5.27      ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.49/5.27      = zero_zero_complex ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4882_add__neg__numeral__special_I7_J,axiom,
% 9.49/5.27      ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 9.49/5.27      = zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4883_add__neg__numeral__special_I7_J,axiom,
% 9.49/5.27      ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 9.49/5.27      = zero_zero_real ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4884_add__neg__numeral__special_I7_J,axiom,
% 9.49/5.27      ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.49/5.27      = zero_zero_rat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4885_add__neg__numeral__special_I7_J,axiom,
% 9.49/5.27      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.49/5.27      = zero_z3403309356797280102nteger ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4886_add__neg__numeral__special_I8_J,axiom,
% 9.49/5.27      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 9.49/5.27      = zero_zero_complex ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4887_add__neg__numeral__special_I8_J,axiom,
% 9.49/5.27      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 9.49/5.27      = zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4888_add__neg__numeral__special_I8_J,axiom,
% 9.49/5.27      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 9.49/5.27      = zero_zero_real ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4889_add__neg__numeral__special_I8_J,axiom,
% 9.49/5.27      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 9.49/5.27      = zero_zero_rat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4890_add__neg__numeral__special_I8_J,axiom,
% 9.49/5.27      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 9.49/5.27      = zero_z3403309356797280102nteger ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4891_diff__numeral__special_I12_J,axiom,
% 9.49/5.27      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.49/5.27      = zero_zero_complex ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4892_diff__numeral__special_I12_J,axiom,
% 9.49/5.27      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 9.49/5.27      = zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4893_diff__numeral__special_I12_J,axiom,
% 9.49/5.27      ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 9.49/5.27      = zero_zero_real ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4894_diff__numeral__special_I12_J,axiom,
% 9.49/5.27      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.49/5.27      = zero_zero_rat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4895_diff__numeral__special_I12_J,axiom,
% 9.49/5.27      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.49/5.27      = zero_z3403309356797280102nteger ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4896_mod__minus1__right,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 9.49/5.27        = zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4897_mod__minus1__right,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.49/5.27        = zero_z3403309356797280102nteger ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4898_max__number__of_I4_J,axiom,
% 9.49/5.27      ! [U: num,V: num] :
% 9.49/5.27        ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.49/5.27         => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.49/5.27            = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 9.49/5.27        & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.49/5.27         => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.49/5.27            = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4899_max__number__of_I4_J,axiom,
% 9.49/5.27      ! [U: num,V: num] :
% 9.49/5.27        ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.49/5.27         => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.49/5.27            = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 9.49/5.27        & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.49/5.27         => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.49/5.27            = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4900_max__number__of_I4_J,axiom,
% 9.49/5.27      ! [U: num,V: num] :
% 9.49/5.27        ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.49/5.27         => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.49/5.27            = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 9.49/5.27        & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.49/5.27         => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.49/5.27            = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4901_max__number__of_I4_J,axiom,
% 9.49/5.27      ! [U: num,V: num] :
% 9.49/5.27        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.49/5.27         => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.49/5.27            = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 9.49/5.27        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.49/5.27         => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.49/5.27            = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4902_max__number__of_I3_J,axiom,
% 9.49/5.27      ! [U: num,V: num] :
% 9.49/5.27        ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 9.49/5.27         => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 9.49/5.27            = ( numeral_numeral_real @ V ) ) )
% 9.49/5.27        & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 9.49/5.27         => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 9.49/5.27            = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4903_max__number__of_I3_J,axiom,
% 9.49/5.27      ! [U: num,V: num] :
% 9.49/5.27        ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 9.49/5.27         => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 9.49/5.27            = ( numera6620942414471956472nteger @ V ) ) )
% 9.49/5.27        & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 9.49/5.27         => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 9.49/5.27            = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4904_max__number__of_I3_J,axiom,
% 9.49/5.27      ! [U: num,V: num] :
% 9.49/5.27        ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 9.49/5.27         => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 9.49/5.27            = ( numeral_numeral_rat @ V ) ) )
% 9.49/5.27        & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 9.49/5.27         => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 9.49/5.27            = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4905_max__number__of_I3_J,axiom,
% 9.49/5.27      ! [U: num,V: num] :
% 9.49/5.27        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 9.49/5.27         => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 9.49/5.27            = ( numeral_numeral_int @ V ) ) )
% 9.49/5.27        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 9.49/5.27         => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 9.49/5.27            = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4906_max__number__of_I2_J,axiom,
% 9.49/5.27      ! [U: num,V: num] :
% 9.49/5.27        ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.49/5.27         => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.49/5.27            = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 9.49/5.27        & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.49/5.27         => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.49/5.27            = ( numeral_numeral_real @ U ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4907_max__number__of_I2_J,axiom,
% 9.49/5.27      ! [U: num,V: num] :
% 9.49/5.27        ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.49/5.27         => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.49/5.27            = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 9.49/5.27        & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.49/5.27         => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.49/5.27            = ( numera6620942414471956472nteger @ U ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4908_max__number__of_I2_J,axiom,
% 9.49/5.27      ! [U: num,V: num] :
% 9.49/5.27        ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.49/5.27         => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.49/5.27            = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 9.49/5.27        & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.49/5.27         => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.49/5.27            = ( numeral_numeral_rat @ U ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4909_max__number__of_I2_J,axiom,
% 9.49/5.27      ! [U: num,V: num] :
% 9.49/5.27        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.49/5.27         => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.49/5.27            = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 9.49/5.27        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.49/5.27         => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.49/5.27            = ( numeral_numeral_int @ U ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4910_neg__numeral__le__iff,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.49/5.27        = ( ord_less_eq_num @ N @ M ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4911_neg__numeral__le__iff,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.49/5.27        = ( ord_less_eq_num @ N @ M ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4912_neg__numeral__le__iff,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.49/5.27        = ( ord_less_eq_num @ N @ M ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4913_neg__numeral__le__iff,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.49/5.27        = ( ord_less_eq_num @ N @ M ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4914_neg__numeral__less__iff,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.49/5.27        = ( ord_less_num @ N @ M ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4915_neg__numeral__less__iff,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.49/5.27        = ( ord_less_num @ N @ M ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4916_neg__numeral__less__iff,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.49/5.27        = ( ord_less_num @ N @ M ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4917_neg__numeral__less__iff,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.49/5.27        = ( ord_less_num @ N @ M ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4918_semiring__norm_I74_J,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.49/5.27        = ( ord_less_num @ M @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4919_semiring__norm_I79_J,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.49/5.27        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4920_not__neg__one__le__neg__numeral__iff,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 9.49/5.27        = ( M != one ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4921_not__neg__one__le__neg__numeral__iff,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 9.49/5.27        = ( M != one ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4922_not__neg__one__le__neg__numeral__iff,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 9.49/5.27        = ( M != one ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4923_not__neg__one__le__neg__numeral__iff,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 9.49/5.27        = ( M != one ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4924_divide__le__eq__numeral1_I2_J,axiom,
% 9.49/5.27      ! [B: real,W2: num,A: real] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
% 9.49/5.27        = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4925_divide__le__eq__numeral1_I2_J,axiom,
% 9.49/5.27      ! [B: rat,W2: num,A: rat] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
% 9.49/5.27        = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4926_le__divide__eq__numeral1_I2_J,axiom,
% 9.49/5.27      ! [A: real,B: real,W2: num] :
% 9.49/5.27        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
% 9.49/5.27        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4927_le__divide__eq__numeral1_I2_J,axiom,
% 9.49/5.27      ! [A: rat,B: rat,W2: num] :
% 9.49/5.27        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
% 9.49/5.27        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4928_divide__eq__eq__numeral1_I2_J,axiom,
% 9.49/5.27      ! [B: complex,W2: num,A: complex] :
% 9.49/5.27        ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
% 9.49/5.27          = A )
% 9.49/5.27        = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.49/5.27             != zero_zero_complex )
% 9.49/5.27           => ( B
% 9.49/5.27              = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) )
% 9.49/5.27          & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.49/5.27              = zero_zero_complex )
% 9.49/5.27           => ( A = zero_zero_complex ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4929_divide__eq__eq__numeral1_I2_J,axiom,
% 9.49/5.27      ! [B: real,W2: num,A: real] :
% 9.49/5.27        ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 9.49/5.27          = A )
% 9.49/5.27        = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.49/5.27             != zero_zero_real )
% 9.49/5.27           => ( B
% 9.49/5.27              = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) )
% 9.49/5.27          & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.49/5.27              = zero_zero_real )
% 9.49/5.27           => ( A = zero_zero_real ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4930_divide__eq__eq__numeral1_I2_J,axiom,
% 9.49/5.27      ! [B: rat,W2: num,A: rat] :
% 9.49/5.27        ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 9.49/5.27          = A )
% 9.49/5.27        = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.49/5.27             != zero_zero_rat )
% 9.49/5.27           => ( B
% 9.49/5.27              = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) )
% 9.49/5.27          & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.49/5.27              = zero_zero_rat )
% 9.49/5.27           => ( A = zero_zero_rat ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4931_eq__divide__eq__numeral1_I2_J,axiom,
% 9.49/5.27      ! [A: complex,B: complex,W2: num] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) )
% 9.49/5.27        = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.49/5.27             != zero_zero_complex )
% 9.49/5.27           => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
% 9.49/5.27              = B ) )
% 9.49/5.27          & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.49/5.27              = zero_zero_complex )
% 9.49/5.27           => ( A = zero_zero_complex ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4932_eq__divide__eq__numeral1_I2_J,axiom,
% 9.49/5.27      ! [A: real,B: real,W2: num] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
% 9.49/5.27        = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.49/5.27             != zero_zero_real )
% 9.49/5.27           => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 9.49/5.27              = B ) )
% 9.49/5.27          & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.49/5.27              = zero_zero_real )
% 9.49/5.27           => ( A = zero_zero_real ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4933_eq__divide__eq__numeral1_I2_J,axiom,
% 9.49/5.27      ! [A: rat,B: rat,W2: num] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
% 9.49/5.27        = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.49/5.27             != zero_zero_rat )
% 9.49/5.27           => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 9.49/5.27              = B ) )
% 9.49/5.27          & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.49/5.27              = zero_zero_rat )
% 9.49/5.27           => ( A = zero_zero_rat ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4934_neg__numeral__less__neg__one__iff,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 9.49/5.27        = ( M != one ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4935_neg__numeral__less__neg__one__iff,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 9.49/5.27        = ( M != one ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4936_neg__numeral__less__neg__one__iff,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.49/5.27        = ( M != one ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4937_neg__numeral__less__neg__one__iff,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.49/5.27        = ( M != one ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4938_less__divide__eq__numeral1_I2_J,axiom,
% 9.49/5.27      ! [A: real,B: real,W2: num] :
% 9.49/5.27        ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
% 9.49/5.27        = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4939_less__divide__eq__numeral1_I2_J,axiom,
% 9.49/5.27      ! [A: rat,B: rat,W2: num] :
% 9.49/5.27        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
% 9.49/5.27        = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4940_divide__less__eq__numeral1_I2_J,axiom,
% 9.49/5.27      ! [B: real,W2: num,A: real] :
% 9.49/5.27        ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
% 9.49/5.27        = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4941_divide__less__eq__numeral1_I2_J,axiom,
% 9.49/5.27      ! [B: rat,W2: num,A: rat] :
% 9.49/5.27        ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
% 9.49/5.27        = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4942_divmod__algorithm__code_I4_J,axiom,
% 9.49/5.27      ! [N: num] :
% 9.49/5.27        ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
% 9.49/5.27        = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4943_divmod__algorithm__code_I4_J,axiom,
% 9.49/5.27      ! [N: num] :
% 9.49/5.27        ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
% 9.49/5.27        = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4944_divmod__algorithm__code_I4_J,axiom,
% 9.49/5.27      ! [N: num] :
% 9.49/5.27        ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
% 9.49/5.27        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4945_divmod__algorithm__code_I8_J,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( ( ord_less_num @ M @ N )
% 9.49/5.27         => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.49/5.27            = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
% 9.49/5.27        & ( ~ ( ord_less_num @ M @ N )
% 9.49/5.27         => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.49/5.27            = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4946_divmod__algorithm__code_I8_J,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( ( ord_less_num @ M @ N )
% 9.49/5.27         => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.49/5.27            = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
% 9.49/5.27        & ( ~ ( ord_less_num @ M @ N )
% 9.49/5.27         => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.49/5.27            = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4947_divmod__algorithm__code_I8_J,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ( ( ( ord_less_num @ M @ N )
% 9.49/5.27         => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.49/5.27            = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
% 9.49/5.27        & ( ~ ( ord_less_num @ M @ N )
% 9.49/5.27         => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.49/5.27            = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4948_equation__minus__iff,axiom,
% 9.49/5.27      ! [A: int,B: int] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( uminus_uminus_int @ B ) )
% 9.49/5.27        = ( B
% 9.49/5.27          = ( uminus_uminus_int @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4949_equation__minus__iff,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( uminus_uminus_real @ B ) )
% 9.49/5.27        = ( B
% 9.49/5.27          = ( uminus_uminus_real @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4950_equation__minus__iff,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( uminus_uminus_rat @ B ) )
% 9.49/5.27        = ( B
% 9.49/5.27          = ( uminus_uminus_rat @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4951_equation__minus__iff,axiom,
% 9.49/5.27      ! [A: code_integer,B: code_integer] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( uminus1351360451143612070nteger @ B ) )
% 9.49/5.27        = ( B
% 9.49/5.27          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4952_minus__equation__iff,axiom,
% 9.49/5.27      ! [A: int,B: int] :
% 9.49/5.27        ( ( ( uminus_uminus_int @ A )
% 9.49/5.27          = B )
% 9.49/5.27        = ( ( uminus_uminus_int @ B )
% 9.49/5.27          = A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4953_minus__equation__iff,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( ( uminus_uminus_real @ A )
% 9.49/5.27          = B )
% 9.49/5.27        = ( ( uminus_uminus_real @ B )
% 9.49/5.27          = A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4954_minus__equation__iff,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( ( uminus_uminus_rat @ A )
% 9.49/5.27          = B )
% 9.49/5.27        = ( ( uminus_uminus_rat @ B )
% 9.49/5.27          = A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4955_minus__equation__iff,axiom,
% 9.49/5.27      ! [A: code_integer,B: code_integer] :
% 9.49/5.27        ( ( ( uminus1351360451143612070nteger @ A )
% 9.49/5.27          = B )
% 9.49/5.27        = ( ( uminus1351360451143612070nteger @ B )
% 9.49/5.27          = A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4956_compl__le__swap2,axiom,
% 9.49/5.27      ! [Y: set_int,X: set_int] :
% 9.49/5.27        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ X )
% 9.49/5.27       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4957_compl__le__swap1,axiom,
% 9.49/5.27      ! [Y: set_int,X: set_int] :
% 9.49/5.27        ( ( ord_less_eq_set_int @ Y @ ( uminus1532241313380277803et_int @ X ) )
% 9.49/5.27       => ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4958_compl__mono,axiom,
% 9.49/5.27      ! [X: set_int,Y: set_int] :
% 9.49/5.27        ( ( ord_less_eq_set_int @ X @ Y )
% 9.49/5.27       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ ( uminus1532241313380277803et_int @ X ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4959_le__imp__neg__le,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.27       => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4960_le__imp__neg__le,axiom,
% 9.49/5.27      ! [A: code_integer,B: code_integer] :
% 9.49/5.27        ( ( ord_le3102999989581377725nteger @ A @ B )
% 9.49/5.27       => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4961_le__imp__neg__le,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.27       => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4962_le__imp__neg__le,axiom,
% 9.49/5.27      ! [A: int,B: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.27       => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4963_minus__le__iff,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 9.49/5.27        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4964_minus__le__iff,axiom,
% 9.49/5.27      ! [A: code_integer,B: code_integer] :
% 9.49/5.27        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 9.49/5.27        = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4965_minus__le__iff,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 9.49/5.27        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4966_minus__le__iff,axiom,
% 9.49/5.27      ! [A: int,B: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 9.49/5.27        = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4967_le__minus__iff,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 9.49/5.27        = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4968_le__minus__iff,axiom,
% 9.49/5.27      ! [A: code_integer,B: code_integer] :
% 9.49/5.27        ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 9.49/5.27        = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4969_le__minus__iff,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
% 9.49/5.27        = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4970_le__minus__iff,axiom,
% 9.49/5.27      ! [A: int,B: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 9.49/5.27        = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4971_minus__less__iff,axiom,
% 9.49/5.27      ! [A: int,B: int] :
% 9.49/5.27        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
% 9.49/5.27        = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4972_minus__less__iff,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
% 9.49/5.27        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4973_minus__less__iff,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
% 9.49/5.27        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4974_minus__less__iff,axiom,
% 9.49/5.27      ! [A: code_integer,B: code_integer] :
% 9.49/5.27        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 9.49/5.27        = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4975_less__minus__iff,axiom,
% 9.49/5.27      ! [A: int,B: int] :
% 9.49/5.27        ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
% 9.49/5.27        = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4976_less__minus__iff,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
% 9.49/5.27        = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4977_less__minus__iff,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
% 9.49/5.27        = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4978_less__minus__iff,axiom,
% 9.49/5.27      ! [A: code_integer,B: code_integer] :
% 9.49/5.27        ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 9.49/5.27        = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4979_verit__negate__coefficient_I2_J,axiom,
% 9.49/5.27      ! [A: int,B: int] :
% 9.49/5.27        ( ( ord_less_int @ A @ B )
% 9.49/5.27       => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4980_verit__negate__coefficient_I2_J,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( ord_less_real @ A @ B )
% 9.49/5.27       => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4981_verit__negate__coefficient_I2_J,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( ord_less_rat @ A @ B )
% 9.49/5.27       => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4982_verit__negate__coefficient_I2_J,axiom,
% 9.49/5.27      ! [A: code_integer,B: code_integer] :
% 9.49/5.27        ( ( ord_le6747313008572928689nteger @ A @ B )
% 9.49/5.27       => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4983_group__cancel_Oneg1,axiom,
% 9.49/5.27      ! [A2: int,K: int,A: int] :
% 9.49/5.27        ( ( A2
% 9.49/5.27          = ( plus_plus_int @ K @ A ) )
% 9.49/5.27       => ( ( uminus_uminus_int @ A2 )
% 9.49/5.27          = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4984_group__cancel_Oneg1,axiom,
% 9.49/5.27      ! [A2: real,K: real,A: real] :
% 9.49/5.27        ( ( A2
% 9.49/5.27          = ( plus_plus_real @ K @ A ) )
% 9.49/5.27       => ( ( uminus_uminus_real @ A2 )
% 9.49/5.27          = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4985_group__cancel_Oneg1,axiom,
% 9.49/5.27      ! [A2: rat,K: rat,A: rat] :
% 9.49/5.27        ( ( A2
% 9.49/5.27          = ( plus_plus_rat @ K @ A ) )
% 9.49/5.27       => ( ( uminus_uminus_rat @ A2 )
% 9.49/5.27          = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4986_group__cancel_Oneg1,axiom,
% 9.49/5.27      ! [A2: code_integer,K: code_integer,A: code_integer] :
% 9.49/5.27        ( ( A2
% 9.49/5.27          = ( plus_p5714425477246183910nteger @ K @ A ) )
% 9.49/5.27       => ( ( uminus1351360451143612070nteger @ A2 )
% 9.49/5.27          = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4987_add_Oinverse__distrib__swap,axiom,
% 9.49/5.27      ! [A: int,B: int] :
% 9.49/5.27        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 9.49/5.27        = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4988_add_Oinverse__distrib__swap,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 9.49/5.27        = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4989_add_Oinverse__distrib__swap,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 9.49/5.27        = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4990_add_Oinverse__distrib__swap,axiom,
% 9.49/5.27      ! [A: code_integer,B: code_integer] :
% 9.49/5.27        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 9.49/5.27        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4991_minus__diff__commute,axiom,
% 9.49/5.27      ! [B: int,A: int] :
% 9.49/5.27        ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 9.49/5.27        = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4992_minus__diff__commute,axiom,
% 9.49/5.27      ! [B: real,A: real] :
% 9.49/5.27        ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
% 9.49/5.27        = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4993_minus__diff__commute,axiom,
% 9.49/5.27      ! [B: rat,A: rat] :
% 9.49/5.27        ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
% 9.49/5.27        = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4994_minus__diff__commute,axiom,
% 9.49/5.27      ! [B: code_integer,A: code_integer] :
% 9.49/5.27        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
% 9.49/5.27        = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4995_not__numeral__le__neg__numeral,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4996_not__numeral__le__neg__numeral,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4997_not__numeral__le__neg__numeral,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4998_not__numeral__le__neg__numeral,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_4999_neg__numeral__le__numeral,axiom,
% 9.49/5.27      ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5000_neg__numeral__le__numeral,axiom,
% 9.49/5.27      ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5001_neg__numeral__le__numeral,axiom,
% 9.49/5.27      ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5002_neg__numeral__le__numeral,axiom,
% 9.49/5.27      ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5003_zero__neq__neg__numeral,axiom,
% 9.49/5.27      ! [N: num] :
% 9.49/5.27        ( zero_zero_complex
% 9.49/5.27       != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5004_zero__neq__neg__numeral,axiom,
% 9.49/5.27      ! [N: num] :
% 9.49/5.27        ( zero_zero_int
% 9.49/5.27       != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5005_zero__neq__neg__numeral,axiom,
% 9.49/5.27      ! [N: num] :
% 9.49/5.27        ( zero_zero_real
% 9.49/5.27       != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5006_zero__neq__neg__numeral,axiom,
% 9.49/5.27      ! [N: num] :
% 9.49/5.27        ( zero_zero_rat
% 9.49/5.27       != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5007_zero__neq__neg__numeral,axiom,
% 9.49/5.27      ! [N: num] :
% 9.49/5.27        ( zero_z3403309356797280102nteger
% 9.49/5.27       != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5008_not__numeral__less__neg__numeral,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5009_not__numeral__less__neg__numeral,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5010_not__numeral__less__neg__numeral,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5011_not__numeral__less__neg__numeral,axiom,
% 9.49/5.27      ! [M: num,N: num] :
% 9.49/5.27        ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5012_neg__numeral__less__numeral,axiom,
% 9.49/5.27      ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5013_neg__numeral__less__numeral,axiom,
% 9.49/5.27      ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5014_neg__numeral__less__numeral,axiom,
% 9.49/5.27      ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5015_neg__numeral__less__numeral,axiom,
% 9.49/5.27      ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5016_xor__num_Ocases,axiom,
% 9.49/5.27      ! [X: product_prod_num_num] :
% 9.49/5.27        ( ( X
% 9.49/5.27         != ( product_Pair_num_num @ one @ one ) )
% 9.49/5.27       => ( ! [N3: num] :
% 9.49/5.27              ( X
% 9.49/5.27             != ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) )
% 9.49/5.27         => ( ! [N3: num] :
% 9.49/5.27                ( X
% 9.49/5.27               != ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) )
% 9.49/5.27           => ( ! [M5: num] :
% 9.49/5.27                  ( X
% 9.49/5.27                 != ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) )
% 9.49/5.27             => ( ! [M5: num,N3: num] :
% 9.49/5.27                    ( X
% 9.49/5.27                   != ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) )
% 9.49/5.27               => ( ! [M5: num,N3: num] :
% 9.49/5.27                      ( X
% 9.49/5.27                     != ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) )
% 9.49/5.27                 => ( ! [M5: num] :
% 9.49/5.27                        ( X
% 9.49/5.27                       != ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) )
% 9.49/5.27                   => ( ! [M5: num,N3: num] :
% 9.49/5.27                          ( X
% 9.49/5.27                         != ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) )
% 9.49/5.27                     => ~ ! [M5: num,N3: num] :
% 9.49/5.27                            ( X
% 9.49/5.27                           != ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5017_le__minus__one__simps_I2_J,axiom,
% 9.49/5.27      ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5018_le__minus__one__simps_I2_J,axiom,
% 9.49/5.27      ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5019_le__minus__one__simps_I2_J,axiom,
% 9.49/5.27      ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5020_le__minus__one__simps_I2_J,axiom,
% 9.49/5.27      ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5021_le__minus__one__simps_I4_J,axiom,
% 9.49/5.27      ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5022_le__minus__one__simps_I4_J,axiom,
% 9.49/5.27      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5023_le__minus__one__simps_I4_J,axiom,
% 9.49/5.27      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5024_le__minus__one__simps_I4_J,axiom,
% 9.49/5.27      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5025_add__eq__0__iff,axiom,
% 9.49/5.27      ! [A: complex,B: complex] :
% 9.49/5.27        ( ( ( plus_plus_complex @ A @ B )
% 9.49/5.27          = zero_zero_complex )
% 9.49/5.27        = ( B
% 9.49/5.27          = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5026_add__eq__0__iff,axiom,
% 9.49/5.27      ! [A: int,B: int] :
% 9.49/5.27        ( ( ( plus_plus_int @ A @ B )
% 9.49/5.27          = zero_zero_int )
% 9.49/5.27        = ( B
% 9.49/5.27          = ( uminus_uminus_int @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5027_add__eq__0__iff,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( ( plus_plus_real @ A @ B )
% 9.49/5.27          = zero_zero_real )
% 9.49/5.27        = ( B
% 9.49/5.27          = ( uminus_uminus_real @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5028_add__eq__0__iff,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( ( plus_plus_rat @ A @ B )
% 9.49/5.27          = zero_zero_rat )
% 9.49/5.27        = ( B
% 9.49/5.27          = ( uminus_uminus_rat @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5029_add__eq__0__iff,axiom,
% 9.49/5.27      ! [A: code_integer,B: code_integer] :
% 9.49/5.27        ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 9.49/5.27          = zero_z3403309356797280102nteger )
% 9.49/5.27        = ( B
% 9.49/5.27          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5030_ab__group__add__class_Oab__left__minus,axiom,
% 9.49/5.27      ! [A: complex] :
% 9.49/5.27        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 9.49/5.27        = zero_zero_complex ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5031_ab__group__add__class_Oab__left__minus,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 9.49/5.27        = zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5032_ab__group__add__class_Oab__left__minus,axiom,
% 9.49/5.27      ! [A: real] :
% 9.49/5.27        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 9.49/5.27        = zero_zero_real ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5033_ab__group__add__class_Oab__left__minus,axiom,
% 9.49/5.27      ! [A: rat] :
% 9.49/5.27        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 9.49/5.27        = zero_zero_rat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5034_ab__group__add__class_Oab__left__minus,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 9.49/5.27        = zero_z3403309356797280102nteger ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5035_add_Oinverse__unique,axiom,
% 9.49/5.27      ! [A: complex,B: complex] :
% 9.49/5.27        ( ( ( plus_plus_complex @ A @ B )
% 9.49/5.27          = zero_zero_complex )
% 9.49/5.27       => ( ( uminus1482373934393186551omplex @ A )
% 9.49/5.27          = B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5036_add_Oinverse__unique,axiom,
% 9.49/5.27      ! [A: int,B: int] :
% 9.49/5.27        ( ( ( plus_plus_int @ A @ B )
% 9.49/5.27          = zero_zero_int )
% 9.49/5.27       => ( ( uminus_uminus_int @ A )
% 9.49/5.27          = B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5037_add_Oinverse__unique,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( ( plus_plus_real @ A @ B )
% 9.49/5.27          = zero_zero_real )
% 9.49/5.27       => ( ( uminus_uminus_real @ A )
% 9.49/5.27          = B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5038_add_Oinverse__unique,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( ( plus_plus_rat @ A @ B )
% 9.49/5.27          = zero_zero_rat )
% 9.49/5.27       => ( ( uminus_uminus_rat @ A )
% 9.49/5.27          = B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5039_add_Oinverse__unique,axiom,
% 9.49/5.27      ! [A: code_integer,B: code_integer] :
% 9.49/5.27        ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 9.49/5.27          = zero_z3403309356797280102nteger )
% 9.49/5.27       => ( ( uminus1351360451143612070nteger @ A )
% 9.49/5.27          = B ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5040_eq__neg__iff__add__eq__0,axiom,
% 9.49/5.27      ! [A: complex,B: complex] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( uminus1482373934393186551omplex @ B ) )
% 9.49/5.27        = ( ( plus_plus_complex @ A @ B )
% 9.49/5.27          = zero_zero_complex ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5041_eq__neg__iff__add__eq__0,axiom,
% 9.49/5.27      ! [A: int,B: int] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( uminus_uminus_int @ B ) )
% 9.49/5.27        = ( ( plus_plus_int @ A @ B )
% 9.49/5.27          = zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5042_eq__neg__iff__add__eq__0,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( uminus_uminus_real @ B ) )
% 9.49/5.27        = ( ( plus_plus_real @ A @ B )
% 9.49/5.27          = zero_zero_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5043_eq__neg__iff__add__eq__0,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( uminus_uminus_rat @ B ) )
% 9.49/5.27        = ( ( plus_plus_rat @ A @ B )
% 9.49/5.27          = zero_zero_rat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5044_eq__neg__iff__add__eq__0,axiom,
% 9.49/5.27      ! [A: code_integer,B: code_integer] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( uminus1351360451143612070nteger @ B ) )
% 9.49/5.27        = ( ( plus_p5714425477246183910nteger @ A @ B )
% 9.49/5.27          = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5045_neg__eq__iff__add__eq__0,axiom,
% 9.49/5.27      ! [A: complex,B: complex] :
% 9.49/5.27        ( ( ( uminus1482373934393186551omplex @ A )
% 9.49/5.27          = B )
% 9.49/5.27        = ( ( plus_plus_complex @ A @ B )
% 9.49/5.27          = zero_zero_complex ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5046_neg__eq__iff__add__eq__0,axiom,
% 9.49/5.27      ! [A: int,B: int] :
% 9.49/5.27        ( ( ( uminus_uminus_int @ A )
% 9.49/5.27          = B )
% 9.49/5.27        = ( ( plus_plus_int @ A @ B )
% 9.49/5.27          = zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5047_neg__eq__iff__add__eq__0,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( ( uminus_uminus_real @ A )
% 9.49/5.27          = B )
% 9.49/5.27        = ( ( plus_plus_real @ A @ B )
% 9.49/5.27          = zero_zero_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5048_neg__eq__iff__add__eq__0,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( ( uminus_uminus_rat @ A )
% 9.49/5.27          = B )
% 9.49/5.27        = ( ( plus_plus_rat @ A @ B )
% 9.49/5.27          = zero_zero_rat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5049_neg__eq__iff__add__eq__0,axiom,
% 9.49/5.27      ! [A: code_integer,B: code_integer] :
% 9.49/5.27        ( ( ( uminus1351360451143612070nteger @ A )
% 9.49/5.27          = B )
% 9.49/5.27        = ( ( plus_p5714425477246183910nteger @ A @ B )
% 9.49/5.27          = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5050_zero__neq__neg__one,axiom,
% 9.49/5.27      ( zero_zero_complex
% 9.49/5.27     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5051_zero__neq__neg__one,axiom,
% 9.49/5.27      ( zero_zero_int
% 9.49/5.27     != ( uminus_uminus_int @ one_one_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5052_zero__neq__neg__one,axiom,
% 9.49/5.27      ( zero_zero_real
% 9.49/5.27     != ( uminus_uminus_real @ one_one_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5053_zero__neq__neg__one,axiom,
% 9.49/5.27      ( zero_zero_rat
% 9.49/5.27     != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5054_zero__neq__neg__one,axiom,
% 9.49/5.27      ( zero_z3403309356797280102nteger
% 9.49/5.27     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5055_less__minus__one__simps_I2_J,axiom,
% 9.49/5.27      ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5056_less__minus__one__simps_I2_J,axiom,
% 9.49/5.27      ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5057_less__minus__one__simps_I2_J,axiom,
% 9.49/5.27      ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5058_less__minus__one__simps_I2_J,axiom,
% 9.49/5.27      ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5059_less__minus__one__simps_I4_J,axiom,
% 9.49/5.27      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5060_less__minus__one__simps_I4_J,axiom,
% 9.49/5.27      ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5061_less__minus__one__simps_I4_J,axiom,
% 9.49/5.27      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5062_less__minus__one__simps_I4_J,axiom,
% 9.49/5.27      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5063_nonzero__minus__divide__right,axiom,
% 9.49/5.27      ! [B: complex,A: complex] :
% 9.49/5.27        ( ( B != zero_zero_complex )
% 9.49/5.27       => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 9.49/5.27          = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5064_nonzero__minus__divide__right,axiom,
% 9.49/5.27      ! [B: real,A: real] :
% 9.49/5.27        ( ( B != zero_zero_real )
% 9.49/5.27       => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 9.49/5.27          = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5065_nonzero__minus__divide__right,axiom,
% 9.49/5.27      ! [B: rat,A: rat] :
% 9.49/5.27        ( ( B != zero_zero_rat )
% 9.49/5.27       => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 9.49/5.27          = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5066_nonzero__minus__divide__divide,axiom,
% 9.49/5.27      ! [B: complex,A: complex] :
% 9.49/5.27        ( ( B != zero_zero_complex )
% 9.49/5.27       => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 9.49/5.27          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5067_nonzero__minus__divide__divide,axiom,
% 9.49/5.27      ! [B: real,A: real] :
% 9.49/5.27        ( ( B != zero_zero_real )
% 9.49/5.27       => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 9.49/5.27          = ( divide_divide_real @ A @ B ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5068_nonzero__minus__divide__divide,axiom,
% 9.49/5.27      ! [B: rat,A: rat] :
% 9.49/5.27        ( ( B != zero_zero_rat )
% 9.49/5.27       => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 9.49/5.27          = ( divide_divide_rat @ A @ B ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5069_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.49/5.27      ( minus_minus_int
% 9.49/5.27      = ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5070_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.49/5.27      ( minus_minus_real
% 9.49/5.27      = ( ^ [A4: real,B4: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5071_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.49/5.27      ( minus_minus_rat
% 9.49/5.27      = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5072_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.49/5.27      ( minus_8373710615458151222nteger
% 9.49/5.27      = ( ^ [A4: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5073_diff__conv__add__uminus,axiom,
% 9.49/5.27      ( minus_minus_int
% 9.49/5.27      = ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5074_diff__conv__add__uminus,axiom,
% 9.49/5.27      ( minus_minus_real
% 9.49/5.27      = ( ^ [A4: real,B4: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5075_diff__conv__add__uminus,axiom,
% 9.49/5.27      ( minus_minus_rat
% 9.49/5.27      = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5076_diff__conv__add__uminus,axiom,
% 9.49/5.27      ( minus_8373710615458151222nteger
% 9.49/5.27      = ( ^ [A4: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5077_group__cancel_Osub2,axiom,
% 9.49/5.27      ! [B2: int,K: int,B: int,A: int] :
% 9.49/5.27        ( ( B2
% 9.49/5.27          = ( plus_plus_int @ K @ B ) )
% 9.49/5.27       => ( ( minus_minus_int @ A @ B2 )
% 9.49/5.27          = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5078_group__cancel_Osub2,axiom,
% 9.49/5.27      ! [B2: real,K: real,B: real,A: real] :
% 9.49/5.27        ( ( B2
% 9.49/5.27          = ( plus_plus_real @ K @ B ) )
% 9.49/5.27       => ( ( minus_minus_real @ A @ B2 )
% 9.49/5.27          = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5079_group__cancel_Osub2,axiom,
% 9.49/5.27      ! [B2: rat,K: rat,B: rat,A: rat] :
% 9.49/5.27        ( ( B2
% 9.49/5.27          = ( plus_plus_rat @ K @ B ) )
% 9.49/5.27       => ( ( minus_minus_rat @ A @ B2 )
% 9.49/5.27          = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5080_group__cancel_Osub2,axiom,
% 9.49/5.27      ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
% 9.49/5.27        ( ( B2
% 9.49/5.27          = ( plus_p5714425477246183910nteger @ K @ B ) )
% 9.49/5.27       => ( ( minus_8373710615458151222nteger @ A @ B2 )
% 9.49/5.27          = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5081_neg__numeral__le__zero,axiom,
% 9.49/5.27      ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5082_neg__numeral__le__zero,axiom,
% 9.49/5.27      ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5083_neg__numeral__le__zero,axiom,
% 9.49/5.27      ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5084_neg__numeral__le__zero,axiom,
% 9.49/5.27      ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5085_not__zero__le__neg__numeral,axiom,
% 9.49/5.27      ! [N: num] :
% 9.49/5.27        ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5086_not__zero__le__neg__numeral,axiom,
% 9.49/5.27      ! [N: num] :
% 9.49/5.27        ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5087_not__zero__le__neg__numeral,axiom,
% 9.49/5.27      ! [N: num] :
% 9.49/5.27        ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5088_not__zero__le__neg__numeral,axiom,
% 9.49/5.27      ! [N: num] :
% 9.49/5.27        ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5089_not__zero__less__neg__numeral,axiom,
% 9.49/5.27      ! [N: num] :
% 9.49/5.27        ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5090_not__zero__less__neg__numeral,axiom,
% 9.49/5.27      ! [N: num] :
% 9.49/5.27        ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5091_not__zero__less__neg__numeral,axiom,
% 9.49/5.27      ! [N: num] :
% 9.49/5.27        ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5092_not__zero__less__neg__numeral,axiom,
% 9.49/5.27      ! [N: num] :
% 9.49/5.27        ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5093_neg__numeral__less__zero,axiom,
% 9.49/5.27      ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5094_neg__numeral__less__zero,axiom,
% 9.49/5.27      ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5095_neg__numeral__less__zero,axiom,
% 9.49/5.27      ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5096_neg__numeral__less__zero,axiom,
% 9.49/5.27      ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5097_le__minus__one__simps_I3_J,axiom,
% 9.49/5.27      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5098_le__minus__one__simps_I3_J,axiom,
% 9.49/5.27      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5099_le__minus__one__simps_I3_J,axiom,
% 9.49/5.27      ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5100_le__minus__one__simps_I3_J,axiom,
% 9.49/5.27      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5101_le__minus__one__simps_I1_J,axiom,
% 9.49/5.27      ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5102_le__minus__one__simps_I1_J,axiom,
% 9.49/5.27      ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5103_le__minus__one__simps_I1_J,axiom,
% 9.49/5.27      ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5104_le__minus__one__simps_I1_J,axiom,
% 9.49/5.27      ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5105_less__minus__one__simps_I1_J,axiom,
% 9.49/5.27      ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5106_less__minus__one__simps_I1_J,axiom,
% 9.49/5.27      ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5107_less__minus__one__simps_I1_J,axiom,
% 9.49/5.27      ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5108_less__minus__one__simps_I1_J,axiom,
% 9.49/5.27      ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5109_less__minus__one__simps_I3_J,axiom,
% 9.49/5.27      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5110_less__minus__one__simps_I3_J,axiom,
% 9.49/5.27      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5111_less__minus__one__simps_I3_J,axiom,
% 9.49/5.27      ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5112_less__minus__one__simps_I3_J,axiom,
% 9.49/5.27      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5113_neg__numeral__le__one,axiom,
% 9.49/5.27      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5114_neg__numeral__le__one,axiom,
% 9.49/5.27      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5115_neg__numeral__le__one,axiom,
% 9.49/5.27      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5116_neg__numeral__le__one,axiom,
% 9.49/5.27      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5117_neg__one__le__numeral,axiom,
% 9.49/5.27      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5118_neg__one__le__numeral,axiom,
% 9.49/5.27      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5119_neg__one__le__numeral,axiom,
% 9.49/5.27      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5120_neg__one__le__numeral,axiom,
% 9.49/5.27      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5121_neg__numeral__le__neg__one,axiom,
% 9.49/5.27      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5122_neg__numeral__le__neg__one,axiom,
% 9.49/5.27      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5123_neg__numeral__le__neg__one,axiom,
% 9.49/5.27      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5124_neg__numeral__le__neg__one,axiom,
% 9.49/5.27      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5125_not__numeral__le__neg__one,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5126_not__numeral__le__neg__one,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5127_not__numeral__le__neg__one,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5128_not__numeral__le__neg__one,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5129_not__one__le__neg__numeral,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5130_not__one__le__neg__numeral,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5131_not__one__le__neg__numeral,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5132_not__one__le__neg__numeral,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5133_not__neg__one__less__neg__numeral,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5134_not__neg__one__less__neg__numeral,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5135_not__neg__one__less__neg__numeral,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5136_not__neg__one__less__neg__numeral,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5137_not__one__less__neg__numeral,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5138_not__one__less__neg__numeral,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5139_not__one__less__neg__numeral,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5140_not__one__less__neg__numeral,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5141_not__numeral__less__neg__one,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5142_not__numeral__less__neg__one,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5143_not__numeral__less__neg__one,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5144_not__numeral__less__neg__one,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5145_neg__one__less__numeral,axiom,
% 9.49/5.27      ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5146_neg__one__less__numeral,axiom,
% 9.49/5.27      ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5147_neg__one__less__numeral,axiom,
% 9.49/5.27      ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5148_neg__one__less__numeral,axiom,
% 9.49/5.27      ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5149_neg__numeral__less__one,axiom,
% 9.49/5.27      ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5150_neg__numeral__less__one,axiom,
% 9.49/5.27      ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5151_neg__numeral__less__one,axiom,
% 9.49/5.27      ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5152_neg__numeral__less__one,axiom,
% 9.49/5.27      ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5153_eq__minus__divide__eq,axiom,
% 9.49/5.27      ! [A: complex,B: complex,C: complex] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
% 9.49/5.27        = ( ( ( C != zero_zero_complex )
% 9.49/5.27           => ( ( times_times_complex @ A @ C )
% 9.49/5.27              = ( uminus1482373934393186551omplex @ B ) ) )
% 9.49/5.27          & ( ( C = zero_zero_complex )
% 9.49/5.27           => ( A = zero_zero_complex ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5154_eq__minus__divide__eq,axiom,
% 9.49/5.27      ! [A: real,B: real,C: real] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.49/5.27        = ( ( ( C != zero_zero_real )
% 9.49/5.27           => ( ( times_times_real @ A @ C )
% 9.49/5.27              = ( uminus_uminus_real @ B ) ) )
% 9.49/5.27          & ( ( C = zero_zero_real )
% 9.49/5.27           => ( A = zero_zero_real ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5155_eq__minus__divide__eq,axiom,
% 9.49/5.27      ! [A: rat,B: rat,C: rat] :
% 9.49/5.27        ( ( A
% 9.49/5.27          = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.49/5.27        = ( ( ( C != zero_zero_rat )
% 9.49/5.27           => ( ( times_times_rat @ A @ C )
% 9.49/5.27              = ( uminus_uminus_rat @ B ) ) )
% 9.49/5.27          & ( ( C = zero_zero_rat )
% 9.49/5.27           => ( A = zero_zero_rat ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5156_minus__divide__eq__eq,axiom,
% 9.49/5.27      ! [B: complex,C: complex,A: complex] :
% 9.49/5.27        ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
% 9.49/5.27          = A )
% 9.49/5.27        = ( ( ( C != zero_zero_complex )
% 9.49/5.27           => ( ( uminus1482373934393186551omplex @ B )
% 9.49/5.27              = ( times_times_complex @ A @ C ) ) )
% 9.49/5.27          & ( ( C = zero_zero_complex )
% 9.49/5.27           => ( A = zero_zero_complex ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5157_minus__divide__eq__eq,axiom,
% 9.49/5.27      ! [B: real,C: real,A: real] :
% 9.49/5.27        ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
% 9.49/5.27          = A )
% 9.49/5.27        = ( ( ( C != zero_zero_real )
% 9.49/5.27           => ( ( uminus_uminus_real @ B )
% 9.49/5.27              = ( times_times_real @ A @ C ) ) )
% 9.49/5.27          & ( ( C = zero_zero_real )
% 9.49/5.27           => ( A = zero_zero_real ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5158_minus__divide__eq__eq,axiom,
% 9.49/5.27      ! [B: rat,C: rat,A: rat] :
% 9.49/5.27        ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
% 9.49/5.27          = A )
% 9.49/5.27        = ( ( ( C != zero_zero_rat )
% 9.49/5.27           => ( ( uminus_uminus_rat @ B )
% 9.49/5.27              = ( times_times_rat @ A @ C ) ) )
% 9.49/5.27          & ( ( C = zero_zero_rat )
% 9.49/5.27           => ( A = zero_zero_rat ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5159_nonzero__neg__divide__eq__eq,axiom,
% 9.49/5.27      ! [B: complex,A: complex,C: complex] :
% 9.49/5.27        ( ( B != zero_zero_complex )
% 9.49/5.27       => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 9.49/5.27            = C )
% 9.49/5.27          = ( ( uminus1482373934393186551omplex @ A )
% 9.49/5.27            = ( times_times_complex @ C @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5160_nonzero__neg__divide__eq__eq,axiom,
% 9.49/5.27      ! [B: real,A: real,C: real] :
% 9.49/5.27        ( ( B != zero_zero_real )
% 9.49/5.27       => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 9.49/5.27            = C )
% 9.49/5.27          = ( ( uminus_uminus_real @ A )
% 9.49/5.27            = ( times_times_real @ C @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5161_nonzero__neg__divide__eq__eq,axiom,
% 9.49/5.27      ! [B: rat,A: rat,C: rat] :
% 9.49/5.27        ( ( B != zero_zero_rat )
% 9.49/5.27       => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 9.49/5.27            = C )
% 9.49/5.27          = ( ( uminus_uminus_rat @ A )
% 9.49/5.27            = ( times_times_rat @ C @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5162_nonzero__neg__divide__eq__eq2,axiom,
% 9.49/5.27      ! [B: complex,C: complex,A: complex] :
% 9.49/5.27        ( ( B != zero_zero_complex )
% 9.49/5.27       => ( ( C
% 9.49/5.27            = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 9.49/5.27          = ( ( times_times_complex @ C @ B )
% 9.49/5.27            = ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5163_nonzero__neg__divide__eq__eq2,axiom,
% 9.49/5.27      ! [B: real,C: real,A: real] :
% 9.49/5.27        ( ( B != zero_zero_real )
% 9.49/5.27       => ( ( C
% 9.49/5.27            = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
% 9.49/5.27          = ( ( times_times_real @ C @ B )
% 9.49/5.27            = ( uminus_uminus_real @ A ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5164_nonzero__neg__divide__eq__eq2,axiom,
% 9.49/5.27      ! [B: rat,C: rat,A: rat] :
% 9.49/5.27        ( ( B != zero_zero_rat )
% 9.49/5.27       => ( ( C
% 9.49/5.27            = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
% 9.49/5.27          = ( ( times_times_rat @ C @ B )
% 9.49/5.27            = ( uminus_uminus_rat @ A ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5165_divide__eq__minus__1__iff,axiom,
% 9.49/5.27      ! [A: complex,B: complex] :
% 9.49/5.27        ( ( ( divide1717551699836669952omplex @ A @ B )
% 9.49/5.27          = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.49/5.27        = ( ( B != zero_zero_complex )
% 9.49/5.27          & ( A
% 9.49/5.27            = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5166_divide__eq__minus__1__iff,axiom,
% 9.49/5.27      ! [A: real,B: real] :
% 9.49/5.27        ( ( ( divide_divide_real @ A @ B )
% 9.49/5.27          = ( uminus_uminus_real @ one_one_real ) )
% 9.49/5.27        = ( ( B != zero_zero_real )
% 9.49/5.27          & ( A
% 9.49/5.27            = ( uminus_uminus_real @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5167_divide__eq__minus__1__iff,axiom,
% 9.49/5.27      ! [A: rat,B: rat] :
% 9.49/5.27        ( ( ( divide_divide_rat @ A @ B )
% 9.49/5.27          = ( uminus_uminus_rat @ one_one_rat ) )
% 9.49/5.27        = ( ( B != zero_zero_rat )
% 9.49/5.27          & ( A
% 9.49/5.27            = ( uminus_uminus_rat @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5168_pos__minus__divide__less__eq,axiom,
% 9.49/5.27      ! [C: real,B: real,A: real] :
% 9.49/5.27        ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.27       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 9.49/5.27          = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5169_pos__minus__divide__less__eq,axiom,
% 9.49/5.27      ! [C: rat,B: rat,A: rat] :
% 9.49/5.27        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.27       => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 9.49/5.27          = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5170_pos__less__minus__divide__eq,axiom,
% 9.49/5.27      ! [C: real,A: real,B: real] :
% 9.49/5.27        ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.27       => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.49/5.27          = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5171_pos__less__minus__divide__eq,axiom,
% 9.49/5.27      ! [C: rat,A: rat,B: rat] :
% 9.49/5.27        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.27       => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.49/5.27          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5172_neg__minus__divide__less__eq,axiom,
% 9.49/5.27      ! [C: real,B: real,A: real] :
% 9.49/5.27        ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.27       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 9.49/5.27          = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5173_neg__minus__divide__less__eq,axiom,
% 9.49/5.27      ! [C: rat,B: rat,A: rat] :
% 9.49/5.27        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.27       => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 9.49/5.27          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5174_neg__less__minus__divide__eq,axiom,
% 9.49/5.27      ! [C: real,A: real,B: real] :
% 9.49/5.27        ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.27       => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.49/5.27          = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5175_neg__less__minus__divide__eq,axiom,
% 9.49/5.27      ! [C: rat,A: rat,B: rat] :
% 9.49/5.27        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.27       => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.49/5.27          = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5176_minus__divide__less__eq,axiom,
% 9.49/5.27      ! [B: real,C: real,A: real] :
% 9.49/5.27        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 9.49/5.27        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.27           => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 9.49/5.27          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.27           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.27               => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 9.49/5.27              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.27               => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5177_minus__divide__less__eq,axiom,
% 9.49/5.27      ! [B: rat,C: rat,A: rat] :
% 9.49/5.27        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 9.49/5.27        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.27           => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 9.49/5.27          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.27           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.27               => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 9.49/5.27              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.27               => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5178_less__minus__divide__eq,axiom,
% 9.49/5.27      ! [A: real,B: real,C: real] :
% 9.49/5.27        ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.49/5.27        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.27           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 9.49/5.27          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.27           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.27               => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 9.49/5.27              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.27               => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5179_less__minus__divide__eq,axiom,
% 9.49/5.27      ! [A: rat,B: rat,C: rat] :
% 9.49/5.27        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.49/5.27        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.27           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 9.49/5.27          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.27           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.27               => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 9.49/5.27              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.27               => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5180_divide__eq__eq__numeral_I2_J,axiom,
% 9.49/5.27      ! [B: complex,C: complex,W2: num] :
% 9.49/5.27        ( ( ( divide1717551699836669952omplex @ B @ C )
% 9.49/5.27          = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
% 9.49/5.27        = ( ( ( C != zero_zero_complex )
% 9.49/5.27           => ( B
% 9.49/5.27              = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C ) ) )
% 9.49/5.27          & ( ( C = zero_zero_complex )
% 9.49/5.27           => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.49/5.27              = zero_zero_complex ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5181_divide__eq__eq__numeral_I2_J,axiom,
% 9.49/5.27      ! [B: real,C: real,W2: num] :
% 9.49/5.27        ( ( ( divide_divide_real @ B @ C )
% 9.49/5.27          = ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 9.49/5.27        = ( ( ( C != zero_zero_real )
% 9.49/5.27           => ( B
% 9.49/5.27              = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 9.49/5.27          & ( ( C = zero_zero_real )
% 9.49/5.27           => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.49/5.27              = zero_zero_real ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5182_divide__eq__eq__numeral_I2_J,axiom,
% 9.49/5.27      ! [B: rat,C: rat,W2: num] :
% 9.49/5.27        ( ( ( divide_divide_rat @ B @ C )
% 9.49/5.27          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 9.49/5.27        = ( ( ( C != zero_zero_rat )
% 9.49/5.27           => ( B
% 9.49/5.27              = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 9.49/5.27          & ( ( C = zero_zero_rat )
% 9.49/5.27           => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.49/5.27              = zero_zero_rat ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5183_eq__divide__eq__numeral_I2_J,axiom,
% 9.49/5.27      ! [W2: num,B: complex,C: complex] :
% 9.49/5.27        ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.49/5.27          = ( divide1717551699836669952omplex @ B @ C ) )
% 9.49/5.27        = ( ( ( C != zero_zero_complex )
% 9.49/5.27           => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C )
% 9.49/5.27              = B ) )
% 9.49/5.27          & ( ( C = zero_zero_complex )
% 9.49/5.27           => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.49/5.27              = zero_zero_complex ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5184_eq__divide__eq__numeral_I2_J,axiom,
% 9.49/5.27      ! [W2: num,B: real,C: real] :
% 9.49/5.27        ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.49/5.27          = ( divide_divide_real @ B @ C ) )
% 9.49/5.27        = ( ( ( C != zero_zero_real )
% 9.49/5.27           => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C )
% 9.49/5.27              = B ) )
% 9.49/5.27          & ( ( C = zero_zero_real )
% 9.49/5.27           => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.49/5.27              = zero_zero_real ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5185_eq__divide__eq__numeral_I2_J,axiom,
% 9.49/5.27      ! [W2: num,B: rat,C: rat] :
% 9.49/5.27        ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.49/5.27          = ( divide_divide_rat @ B @ C ) )
% 9.49/5.27        = ( ( ( C != zero_zero_rat )
% 9.49/5.27           => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C )
% 9.49/5.27              = B ) )
% 9.49/5.27          & ( ( C = zero_zero_rat )
% 9.49/5.27           => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.49/5.27              = zero_zero_rat ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5186_cong__exp__iff__simps_I3_J,axiom,
% 9.49/5.27      ! [N: num,Q4: num] :
% 9.49/5.27        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
% 9.49/5.27       != zero_zero_nat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5187_cong__exp__iff__simps_I3_J,axiom,
% 9.49/5.27      ! [N: num,Q4: num] :
% 9.49/5.27        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
% 9.49/5.27       != zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5188_cong__exp__iff__simps_I3_J,axiom,
% 9.49/5.27      ! [N: num,Q4: num] :
% 9.49/5.27        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
% 9.49/5.27       != zero_z3403309356797280102nteger ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5189_add__divide__eq__if__simps_I3_J,axiom,
% 9.49/5.27      ! [Z2: complex,A: complex,B: complex] :
% 9.49/5.27        ( ( ( Z2 = zero_zero_complex )
% 9.49/5.27         => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 9.49/5.27            = B ) )
% 9.49/5.27        & ( ( Z2 != zero_zero_complex )
% 9.49/5.27         => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 9.49/5.27            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5190_add__divide__eq__if__simps_I3_J,axiom,
% 9.49/5.27      ! [Z2: real,A: real,B: real] :
% 9.49/5.27        ( ( ( Z2 = zero_zero_real )
% 9.49/5.27         => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 9.49/5.27            = B ) )
% 9.49/5.27        & ( ( Z2 != zero_zero_real )
% 9.49/5.27         => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 9.49/5.27            = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5191_add__divide__eq__if__simps_I3_J,axiom,
% 9.49/5.27      ! [Z2: rat,A: rat,B: rat] :
% 9.49/5.27        ( ( ( Z2 = zero_zero_rat )
% 9.49/5.27         => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 9.49/5.27            = B ) )
% 9.49/5.27        & ( ( Z2 != zero_zero_rat )
% 9.49/5.27         => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 9.49/5.27            = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5192_minus__divide__add__eq__iff,axiom,
% 9.49/5.27      ! [Z2: complex,X: complex,Y: complex] :
% 9.49/5.27        ( ( Z2 != zero_zero_complex )
% 9.49/5.27       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z2 ) ) @ Y )
% 9.49/5.27          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5193_minus__divide__add__eq__iff,axiom,
% 9.49/5.27      ! [Z2: real,X: real,Y: real] :
% 9.49/5.27        ( ( Z2 != zero_zero_real )
% 9.49/5.27       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z2 ) ) @ Y )
% 9.49/5.27          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5194_minus__divide__add__eq__iff,axiom,
% 9.49/5.27      ! [Z2: rat,X: rat,Y: rat] :
% 9.49/5.27        ( ( Z2 != zero_zero_rat )
% 9.49/5.27       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z2 ) ) @ Y )
% 9.49/5.27          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5195_add__divide__eq__if__simps_I6_J,axiom,
% 9.49/5.27      ! [Z2: complex,A: complex,B: complex] :
% 9.49/5.27        ( ( ( Z2 = zero_zero_complex )
% 9.49/5.27         => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 9.49/5.27            = ( uminus1482373934393186551omplex @ B ) ) )
% 9.49/5.27        & ( ( Z2 != zero_zero_complex )
% 9.49/5.27         => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 9.49/5.27            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5196_add__divide__eq__if__simps_I6_J,axiom,
% 9.49/5.27      ! [Z2: real,A: real,B: real] :
% 9.49/5.27        ( ( ( Z2 = zero_zero_real )
% 9.49/5.27         => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 9.49/5.27            = ( uminus_uminus_real @ B ) ) )
% 9.49/5.27        & ( ( Z2 != zero_zero_real )
% 9.49/5.27         => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 9.49/5.27            = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5197_add__divide__eq__if__simps_I6_J,axiom,
% 9.49/5.27      ! [Z2: rat,A: rat,B: rat] :
% 9.49/5.27        ( ( ( Z2 = zero_zero_rat )
% 9.49/5.27         => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 9.49/5.27            = ( uminus_uminus_rat @ B ) ) )
% 9.49/5.27        & ( ( Z2 != zero_zero_rat )
% 9.49/5.27         => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 9.49/5.27            = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5198_add__divide__eq__if__simps_I5_J,axiom,
% 9.49/5.27      ! [Z2: complex,A: complex,B: complex] :
% 9.49/5.27        ( ( ( Z2 = zero_zero_complex )
% 9.49/5.27         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 9.49/5.27            = ( uminus1482373934393186551omplex @ B ) ) )
% 9.49/5.27        & ( ( Z2 != zero_zero_complex )
% 9.49/5.27         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 9.49/5.27            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5199_add__divide__eq__if__simps_I5_J,axiom,
% 9.49/5.27      ! [Z2: real,A: real,B: real] :
% 9.49/5.27        ( ( ( Z2 = zero_zero_real )
% 9.49/5.27         => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 9.49/5.27            = ( uminus_uminus_real @ B ) ) )
% 9.49/5.27        & ( ( Z2 != zero_zero_real )
% 9.49/5.27         => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 9.49/5.27            = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5200_add__divide__eq__if__simps_I5_J,axiom,
% 9.49/5.27      ! [Z2: rat,A: rat,B: rat] :
% 9.49/5.27        ( ( ( Z2 = zero_zero_rat )
% 9.49/5.27         => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 9.49/5.27            = ( uminus_uminus_rat @ B ) ) )
% 9.49/5.27        & ( ( Z2 != zero_zero_rat )
% 9.49/5.27         => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 9.49/5.27            = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5201_minus__divide__diff__eq__iff,axiom,
% 9.49/5.27      ! [Z2: complex,X: complex,Y: complex] :
% 9.49/5.27        ( ( Z2 != zero_zero_complex )
% 9.49/5.27       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z2 ) ) @ Y )
% 9.49/5.27          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5202_minus__divide__diff__eq__iff,axiom,
% 9.49/5.27      ! [Z2: real,X: real,Y: real] :
% 9.49/5.27        ( ( Z2 != zero_zero_real )
% 9.49/5.27       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z2 ) ) @ Y )
% 9.49/5.27          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5203_minus__divide__diff__eq__iff,axiom,
% 9.49/5.27      ! [Z2: rat,X: rat,Y: rat] :
% 9.49/5.27        ( ( Z2 != zero_zero_rat )
% 9.49/5.27       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z2 ) ) @ Y )
% 9.49/5.27          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5204_numeral__3__eq__3,axiom,
% 9.49/5.27      ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 9.49/5.27      = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5205_verit__less__mono__div__int2,axiom,
% 9.49/5.27      ! [A2: int,B2: int,N: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ A2 @ B2 )
% 9.49/5.27       => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
% 9.49/5.27         => ( ord_less_eq_int @ ( divide_divide_int @ B2 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5206_div__eq__minus1,axiom,
% 9.49/5.27      ! [B: int] :
% 9.49/5.27        ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.27       => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 9.49/5.27          = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5207_num_Osize__gen_I3_J,axiom,
% 9.49/5.27      ! [X32: num] :
% 9.49/5.27        ( ( size_num @ ( bit1 @ X32 ) )
% 9.49/5.27        = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5208_num_Osize_I6_J,axiom,
% 9.49/5.27      ! [X32: num] :
% 9.49/5.27        ( ( size_size_num @ ( bit1 @ X32 ) )
% 9.49/5.27        = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5209_le__minus__divide__eq,axiom,
% 9.49/5.27      ! [A: real,B: real,C: real] :
% 9.49/5.27        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.49/5.27        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.27           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 9.49/5.27          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.27           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.27               => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 9.49/5.27              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.27               => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5210_le__minus__divide__eq,axiom,
% 9.49/5.27      ! [A: rat,B: rat,C: rat] :
% 9.49/5.27        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.49/5.27        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.27           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 9.49/5.27          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.27           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.27               => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 9.49/5.27              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.27               => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5211_minus__divide__le__eq,axiom,
% 9.49/5.27      ! [B: real,C: real,A: real] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 9.49/5.27        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.27           => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 9.49/5.27          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.27           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.27               => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 9.49/5.27              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.27               => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5212_minus__divide__le__eq,axiom,
% 9.49/5.27      ! [B: rat,C: rat,A: rat] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 9.49/5.27        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.27           => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 9.49/5.27          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.27           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.27               => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 9.49/5.27              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.27               => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5213_neg__le__minus__divide__eq,axiom,
% 9.49/5.27      ! [C: real,A: real,B: real] :
% 9.49/5.27        ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.27       => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.49/5.27          = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5214_neg__le__minus__divide__eq,axiom,
% 9.49/5.27      ! [C: rat,A: rat,B: rat] :
% 9.49/5.27        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.27       => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.49/5.27          = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5215_neg__minus__divide__le__eq,axiom,
% 9.49/5.27      ! [C: real,B: real,A: real] :
% 9.49/5.27        ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.27       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 9.49/5.27          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5216_neg__minus__divide__le__eq,axiom,
% 9.49/5.27      ! [C: rat,B: rat,A: rat] :
% 9.49/5.27        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.27       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 9.49/5.27          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5217_pos__le__minus__divide__eq,axiom,
% 9.49/5.27      ! [C: real,A: real,B: real] :
% 9.49/5.27        ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.27       => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.49/5.27          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5218_pos__le__minus__divide__eq,axiom,
% 9.49/5.27      ! [C: rat,A: rat,B: rat] :
% 9.49/5.27        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.27       => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.49/5.27          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5219_pos__minus__divide__le__eq,axiom,
% 9.49/5.27      ! [C: real,B: real,A: real] :
% 9.49/5.27        ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.27       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 9.49/5.27          = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5220_pos__minus__divide__le__eq,axiom,
% 9.49/5.27      ! [C: rat,B: rat,A: rat] :
% 9.49/5.27        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.27       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 9.49/5.27          = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5221_less__divide__eq__numeral_I2_J,axiom,
% 9.49/5.27      ! [W2: num,B: real,C: real] :
% 9.49/5.27        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C ) )
% 9.49/5.27        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.27           => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
% 9.49/5.27          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.27           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.27               => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 9.49/5.27              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.27               => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5222_less__divide__eq__numeral_I2_J,axiom,
% 9.49/5.27      ! [W2: num,B: rat,C: rat] :
% 9.49/5.27        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C ) )
% 9.49/5.27        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.27           => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
% 9.49/5.27          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.27           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.27               => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 9.49/5.27              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.27               => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5223_divide__less__eq__numeral_I2_J,axiom,
% 9.49/5.27      ! [B: real,C: real,W2: num] :
% 9.49/5.27        ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 9.49/5.27        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.27           => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 9.49/5.27          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.27           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.27               => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
% 9.49/5.27              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.27               => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5224_divide__less__eq__numeral_I2_J,axiom,
% 9.49/5.27      ! [B: rat,C: rat,W2: num] :
% 9.49/5.27        ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 9.49/5.27        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.27           => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 9.49/5.27          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.27           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.27               => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
% 9.49/5.27              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.27               => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5225_cong__exp__iff__simps_I7_J,axiom,
% 9.49/5.27      ! [Q4: num,N: num] :
% 9.49/5.27        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
% 9.49/5.27          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) )
% 9.49/5.27        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q4 ) )
% 9.49/5.27          = zero_zero_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5226_cong__exp__iff__simps_I7_J,axiom,
% 9.49/5.27      ! [Q4: num,N: num] :
% 9.49/5.27        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
% 9.49/5.27          = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) )
% 9.49/5.27        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q4 ) )
% 9.49/5.27          = zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5227_cong__exp__iff__simps_I7_J,axiom,
% 9.49/5.27      ! [Q4: num,N: num] :
% 9.49/5.27        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
% 9.49/5.27          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) )
% 9.49/5.27        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q4 ) )
% 9.49/5.27          = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5228_cong__exp__iff__simps_I11_J,axiom,
% 9.49/5.27      ! [M: num,Q4: num] :
% 9.49/5.27        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
% 9.49/5.27          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) )
% 9.49/5.27        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q4 ) )
% 9.49/5.27          = zero_zero_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5229_cong__exp__iff__simps_I11_J,axiom,
% 9.49/5.27      ! [M: num,Q4: num] :
% 9.49/5.27        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
% 9.49/5.27          = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) )
% 9.49/5.27        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q4 ) )
% 9.49/5.27          = zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5230_cong__exp__iff__simps_I11_J,axiom,
% 9.49/5.27      ! [M: num,Q4: num] :
% 9.49/5.27        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
% 9.49/5.27          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) )
% 9.49/5.27        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q4 ) )
% 9.49/5.27          = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5231_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.49/5.27      ! [K: nat,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_nat @ K @ N )
% 9.49/5.27       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
% 9.49/5.27          = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5232_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.49/5.27      ! [K: nat,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_nat @ K @ N )
% 9.49/5.27       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
% 9.49/5.27          = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5233_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.49/5.27      ! [K: nat,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_nat @ K @ N )
% 9.49/5.27       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
% 9.49/5.27          = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5234_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.49/5.27      ! [K: nat,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_nat @ K @ N )
% 9.49/5.27       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
% 9.49/5.27          = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5235_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.49/5.27      ! [K: nat,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_nat @ K @ N )
% 9.49/5.27       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
% 9.49/5.27          = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5236_zmod__minus1,axiom,
% 9.49/5.27      ! [B: int] :
% 9.49/5.27        ( ( ord_less_int @ zero_zero_int @ B )
% 9.49/5.27       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 9.49/5.27          = ( minus_minus_int @ B @ one_one_int ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5237_zminus1__lemma,axiom,
% 9.49/5.27      ! [A: int,B: int,Q4: int,R2: int] :
% 9.49/5.27        ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.49/5.27       => ( ( B != zero_zero_int )
% 9.49/5.27         => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q4 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q4 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R2 ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5238_divide__le__eq__numeral_I2_J,axiom,
% 9.49/5.27      ! [B: real,C: real,W2: num] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 9.49/5.27        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.27           => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 9.49/5.27          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.27           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.27               => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
% 9.49/5.27              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.27               => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5239_divide__le__eq__numeral_I2_J,axiom,
% 9.49/5.27      ! [B: rat,C: rat,W2: num] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 9.49/5.27        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.27           => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 9.49/5.27          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.27           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.27               => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
% 9.49/5.27              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.27               => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5240_le__divide__eq__numeral_I2_J,axiom,
% 9.49/5.27      ! [W2: num,B: real,C: real] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C ) )
% 9.49/5.27        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.27           => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
% 9.49/5.27          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.49/5.27           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.27               => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 9.49/5.27              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.49/5.27               => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5241_le__divide__eq__numeral_I2_J,axiom,
% 9.49/5.27      ! [W2: num,B: rat,C: rat] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C ) )
% 9.49/5.27        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.27           => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
% 9.49/5.27          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.49/5.27           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.27               => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 9.49/5.27              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.49/5.27               => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5242_square__le__1,axiom,
% 9.49/5.27      ! [X: real] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.49/5.27       => ( ( ord_less_eq_real @ X @ one_one_real )
% 9.49/5.27         => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5243_square__le__1,axiom,
% 9.49/5.27      ! [X: code_integer] :
% 9.49/5.27        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 9.49/5.27       => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
% 9.49/5.27         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5244_square__le__1,axiom,
% 9.49/5.27      ! [X: rat] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
% 9.49/5.27       => ( ( ord_less_eq_rat @ X @ one_one_rat )
% 9.49/5.27         => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5245_square__le__1,axiom,
% 9.49/5.27      ! [X: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 9.49/5.27       => ( ( ord_less_eq_int @ X @ one_one_int )
% 9.49/5.27         => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5246_div__pos__neg__trivial,axiom,
% 9.49/5.27      ! [K: int,L: int] :
% 9.49/5.27        ( ( ord_less_int @ zero_zero_int @ K )
% 9.49/5.27       => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 9.49/5.27         => ( ( divide_divide_int @ K @ L )
% 9.49/5.27            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5247_odd__mod__4__div__2,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.49/5.27          = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.49/5.27       => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5248_mod__exhaust__less__4,axiom,
% 9.49/5.27      ! [M: nat] :
% 9.49/5.27        ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.49/5.27          = zero_zero_nat )
% 9.49/5.27        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.49/5.27          = one_one_nat )
% 9.49/5.27        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.49/5.27          = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.27        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.49/5.27          = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5249_signed__take__bit__rec,axiom,
% 9.49/5.27      ( bit_ri6519982836138164636nteger
% 9.49/5.27      = ( ^ [N2: nat,A4: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5250_signed__take__bit__rec,axiom,
% 9.49/5.27      ( bit_ri631733984087533419it_int
% 9.49/5.27      = ( ^ [N2: nat,A4: int] : ( if_int @ ( N2 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5251_divmod__BitM__2__eq,axiom,
% 9.49/5.27      ! [M: num] :
% 9.49/5.27        ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
% 9.49/5.27        = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5252_of__int__code__if,axiom,
% 9.49/5.27      ( ring_17405671764205052669omplex
% 9.49/5.27      = ( ^ [K3: int] :
% 9.49/5.27            ( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
% 9.49/5.27            @ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
% 9.49/5.27              @ ( if_complex
% 9.49/5.27                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.27                  = zero_zero_int )
% 9.49/5.27                @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.49/5.27                @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5253_of__int__code__if,axiom,
% 9.49/5.27      ( ring_1_of_int_int
% 9.49/5.27      = ( ^ [K3: int] :
% 9.49/5.27            ( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
% 9.49/5.27            @ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
% 9.49/5.27              @ ( if_int
% 9.49/5.27                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.27                  = zero_zero_int )
% 9.49/5.27                @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.49/5.27                @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5254_of__int__code__if,axiom,
% 9.49/5.27      ( ring_1_of_int_real
% 9.49/5.27      = ( ^ [K3: int] :
% 9.49/5.27            ( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
% 9.49/5.27            @ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
% 9.49/5.27              @ ( if_real
% 9.49/5.27                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.27                  = zero_zero_int )
% 9.49/5.27                @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.49/5.27                @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5255_of__int__code__if,axiom,
% 9.49/5.27      ( ring_1_of_int_rat
% 9.49/5.27      = ( ^ [K3: int] :
% 9.49/5.27            ( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
% 9.49/5.27            @ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
% 9.49/5.27              @ ( if_rat
% 9.49/5.27                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.27                  = zero_zero_int )
% 9.49/5.27                @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.49/5.27                @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5256_of__int__code__if,axiom,
% 9.49/5.27      ( ring_18347121197199848620nteger
% 9.49/5.27      = ( ^ [K3: int] :
% 9.49/5.27            ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 9.49/5.27            @ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
% 9.49/5.27              @ ( if_Code_integer
% 9.49/5.27                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.27                  = zero_zero_int )
% 9.49/5.27                @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.49/5.27                @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5257_signed__take__bit__0,axiom,
% 9.49/5.27      ! [A: code_integer] :
% 9.49/5.27        ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
% 9.49/5.27        = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5258_signed__take__bit__0,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
% 9.49/5.27        = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5259_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 9.49/5.27      ! [A: int,X: num,N: nat] :
% 9.49/5.27        ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 9.49/5.27        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5260_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 9.49/5.27      ! [A: int,X: num,N: nat] :
% 9.49/5.27        ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 9.49/5.27        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5261_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 9.49/5.27      ! [A: int,X: num,N: nat] :
% 9.49/5.27        ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 9.49/5.27        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5262_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 9.49/5.27      ! [A: int,X: num,N: nat] :
% 9.49/5.27        ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 9.49/5.27        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5263_Compl__anti__mono,axiom,
% 9.49/5.27      ! [A2: set_int,B2: set_int] :
% 9.49/5.27        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.49/5.27       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B2 ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5264_Compl__subset__Compl__iff,axiom,
% 9.49/5.27      ! [A2: set_int,B2: set_int] :
% 9.49/5.27        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B2 ) )
% 9.49/5.27        = ( ord_less_eq_set_int @ B2 @ A2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5265_signed__take__bit__of__0,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
% 9.49/5.27        = zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5266_of__int__0,axiom,
% 9.49/5.27      ( ( ring_17405671764205052669omplex @ zero_zero_int )
% 9.49/5.27      = zero_zero_complex ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5267_of__int__0,axiom,
% 9.49/5.27      ( ( ring_1_of_int_int @ zero_zero_int )
% 9.49/5.27      = zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5268_of__int__0,axiom,
% 9.49/5.27      ( ( ring_1_of_int_real @ zero_zero_int )
% 9.49/5.27      = zero_zero_real ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5269_of__int__0,axiom,
% 9.49/5.27      ( ( ring_1_of_int_rat @ zero_zero_int )
% 9.49/5.27      = zero_zero_rat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5270_of__int__0__eq__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( zero_zero_complex
% 9.49/5.27          = ( ring_17405671764205052669omplex @ Z2 ) )
% 9.49/5.27        = ( Z2 = zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5271_of__int__0__eq__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( zero_zero_int
% 9.49/5.27          = ( ring_1_of_int_int @ Z2 ) )
% 9.49/5.27        = ( Z2 = zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5272_of__int__0__eq__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( zero_zero_real
% 9.49/5.27          = ( ring_1_of_int_real @ Z2 ) )
% 9.49/5.27        = ( Z2 = zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5273_of__int__0__eq__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( zero_zero_rat
% 9.49/5.27          = ( ring_1_of_int_rat @ Z2 ) )
% 9.49/5.27        = ( Z2 = zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5274_of__int__eq__0__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ( ring_17405671764205052669omplex @ Z2 )
% 9.49/5.27          = zero_zero_complex )
% 9.49/5.27        = ( Z2 = zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5275_of__int__eq__0__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ( ring_1_of_int_int @ Z2 )
% 9.49/5.27          = zero_zero_int )
% 9.49/5.27        = ( Z2 = zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5276_of__int__eq__0__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ( ring_1_of_int_real @ Z2 )
% 9.49/5.27          = zero_zero_real )
% 9.49/5.27        = ( Z2 = zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5277_of__int__eq__0__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ( ring_1_of_int_rat @ Z2 )
% 9.49/5.27          = zero_zero_rat )
% 9.49/5.27        = ( Z2 = zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5278_of__int__le__iff,axiom,
% 9.49/5.27      ! [W2: int,Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) )
% 9.49/5.27        = ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5279_of__int__le__iff,axiom,
% 9.49/5.27      ! [W2: int,Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) )
% 9.49/5.27        = ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5280_of__int__le__iff,axiom,
% 9.49/5.27      ! [W2: int,Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
% 9.49/5.27        = ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5281_of__int__less__iff,axiom,
% 9.49/5.27      ! [W2: int,Z2: int] :
% 9.49/5.27        ( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) )
% 9.49/5.27        = ( ord_less_int @ W2 @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5282_of__int__less__iff,axiom,
% 9.49/5.27      ! [W2: int,Z2: int] :
% 9.49/5.27        ( ( ord_less_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) )
% 9.49/5.27        = ( ord_less_int @ W2 @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5283_of__int__less__iff,axiom,
% 9.49/5.27      ! [W2: int,Z2: int] :
% 9.49/5.27        ( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
% 9.49/5.27        = ( ord_less_int @ W2 @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5284_of__int__le__0__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
% 9.49/5.27        = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5285_of__int__le__0__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
% 9.49/5.27        = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5286_of__int__le__0__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
% 9.49/5.27        = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5287_of__int__0__le__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
% 9.49/5.27        = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5288_of__int__0__le__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 9.49/5.27        = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5289_of__int__0__le__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
% 9.49/5.27        = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5290_of__int__0__less__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
% 9.49/5.27        = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5291_of__int__0__less__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 9.49/5.27        = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5292_of__int__0__less__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
% 9.49/5.27        = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5293_of__int__less__0__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
% 9.49/5.27        = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5294_of__int__less__0__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
% 9.49/5.27        = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5295_of__int__less__0__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
% 9.49/5.27        = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5296_of__int__le__numeral__iff,axiom,
% 9.49/5.27      ! [Z2: int,N: num] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N ) )
% 9.49/5.27        = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5297_of__int__le__numeral__iff,axiom,
% 9.49/5.27      ! [Z2: int,N: num] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N ) )
% 9.49/5.27        = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5298_of__int__le__numeral__iff,axiom,
% 9.49/5.27      ! [Z2: int,N: num] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N ) )
% 9.49/5.27        = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5299_of__int__numeral__le__iff,axiom,
% 9.49/5.27      ! [N: num,Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z2 ) )
% 9.49/5.27        = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5300_of__int__numeral__le__iff,axiom,
% 9.49/5.27      ! [N: num,Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z2 ) )
% 9.49/5.27        = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5301_of__int__numeral__le__iff,axiom,
% 9.49/5.27      ! [N: num,Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z2 ) )
% 9.49/5.27        = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5302_of__int__less__numeral__iff,axiom,
% 9.49/5.27      ! [Z2: int,N: num] :
% 9.49/5.27        ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N ) )
% 9.49/5.27        = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5303_of__int__less__numeral__iff,axiom,
% 9.49/5.27      ! [Z2: int,N: num] :
% 9.49/5.27        ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N ) )
% 9.49/5.27        = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5304_of__int__less__numeral__iff,axiom,
% 9.49/5.27      ! [Z2: int,N: num] :
% 9.49/5.27        ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N ) )
% 9.49/5.27        = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5305_of__int__numeral__less__iff,axiom,
% 9.49/5.27      ! [N: num,Z2: int] :
% 9.49/5.27        ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z2 ) )
% 9.49/5.27        = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5306_of__int__numeral__less__iff,axiom,
% 9.49/5.27      ! [N: num,Z2: int] :
% 9.49/5.27        ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z2 ) )
% 9.49/5.27        = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5307_of__int__numeral__less__iff,axiom,
% 9.49/5.27      ! [N: num,Z2: int] :
% 9.49/5.27        ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z2 ) )
% 9.49/5.27        = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5308_of__int__le__1__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
% 9.49/5.27        = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5309_of__int__le__1__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
% 9.49/5.27        = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5310_of__int__le__1__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
% 9.49/5.27        = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5311_of__int__1__le__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
% 9.49/5.27        = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5312_of__int__1__le__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 9.49/5.27        = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5313_of__int__1__le__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
% 9.49/5.27        = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5314_of__int__less__1__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
% 9.49/5.27        = ( ord_less_int @ Z2 @ one_one_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5315_of__int__less__1__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
% 9.49/5.27        = ( ord_less_int @ Z2 @ one_one_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5316_of__int__less__1__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
% 9.49/5.27        = ( ord_less_int @ Z2 @ one_one_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5317_of__int__1__less__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
% 9.49/5.27        = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5318_of__int__1__less__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 9.49/5.27        = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5319_of__int__1__less__iff,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
% 9.49/5.27        = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5320_of__int__le__of__int__power__cancel__iff,axiom,
% 9.49/5.27      ! [B: int,W2: nat,X: int] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X ) )
% 9.49/5.27        = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5321_of__int__le__of__int__power__cancel__iff,axiom,
% 9.49/5.27      ! [B: int,W2: nat,X: int] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X ) )
% 9.49/5.27        = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5322_of__int__le__of__int__power__cancel__iff,axiom,
% 9.49/5.27      ! [B: int,W2: nat,X: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X ) )
% 9.49/5.27        = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5323_of__int__power__le__of__int__cancel__iff,axiom,
% 9.49/5.27      ! [X: int,B: int,W2: nat] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
% 9.49/5.27        = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5324_of__int__power__le__of__int__cancel__iff,axiom,
% 9.49/5.27      ! [X: int,B: int,W2: nat] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
% 9.49/5.27        = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5325_of__int__power__le__of__int__cancel__iff,axiom,
% 9.49/5.27      ! [X: int,B: int,W2: nat] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
% 9.49/5.27        = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5326_of__int__less__of__int__power__cancel__iff,axiom,
% 9.49/5.27      ! [B: int,W2: nat,X: int] :
% 9.49/5.27        ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X ) )
% 9.49/5.27        = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5327_of__int__less__of__int__power__cancel__iff,axiom,
% 9.49/5.27      ! [B: int,W2: nat,X: int] :
% 9.49/5.27        ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X ) )
% 9.49/5.27        = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5328_of__int__less__of__int__power__cancel__iff,axiom,
% 9.49/5.27      ! [B: int,W2: nat,X: int] :
% 9.49/5.27        ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X ) )
% 9.49/5.27        = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5329_of__int__power__less__of__int__cancel__iff,axiom,
% 9.49/5.27      ! [X: int,B: int,W2: nat] :
% 9.49/5.27        ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
% 9.49/5.27        = ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5330_of__int__power__less__of__int__cancel__iff,axiom,
% 9.49/5.27      ! [X: int,B: int,W2: nat] :
% 9.49/5.27        ( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
% 9.49/5.27        = ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5331_of__int__power__less__of__int__cancel__iff,axiom,
% 9.49/5.27      ! [X: int,B: int,W2: nat] :
% 9.49/5.27        ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
% 9.49/5.27        = ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5332_numeral__power__le__of__int__cancel__iff,axiom,
% 9.49/5.27      ! [X: num,N: nat,A: int] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 9.49/5.27        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5333_numeral__power__le__of__int__cancel__iff,axiom,
% 9.49/5.27      ! [X: num,N: nat,A: int] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 9.49/5.27        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5334_numeral__power__le__of__int__cancel__iff,axiom,
% 9.49/5.27      ! [X: num,N: nat,A: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 9.49/5.27        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5335_of__int__le__numeral__power__cancel__iff,axiom,
% 9.49/5.27      ! [A: int,X: num,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 9.49/5.27        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5336_of__int__le__numeral__power__cancel__iff,axiom,
% 9.49/5.27      ! [A: int,X: num,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 9.49/5.27        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5337_of__int__le__numeral__power__cancel__iff,axiom,
% 9.49/5.27      ! [A: int,X: num,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 9.49/5.27        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5338_numeral__power__less__of__int__cancel__iff,axiom,
% 9.49/5.27      ! [X: num,N: nat,A: int] :
% 9.49/5.27        ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 9.49/5.27        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5339_numeral__power__less__of__int__cancel__iff,axiom,
% 9.49/5.27      ! [X: num,N: nat,A: int] :
% 9.49/5.27        ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 9.49/5.27        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5340_numeral__power__less__of__int__cancel__iff,axiom,
% 9.49/5.27      ! [X: num,N: nat,A: int] :
% 9.49/5.27        ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 9.49/5.27        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5341_of__int__less__numeral__power__cancel__iff,axiom,
% 9.49/5.27      ! [A: int,X: num,N: nat] :
% 9.49/5.27        ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 9.49/5.27        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5342_of__int__less__numeral__power__cancel__iff,axiom,
% 9.49/5.27      ! [A: int,X: num,N: nat] :
% 9.49/5.27        ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 9.49/5.27        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5343_of__int__less__numeral__power__cancel__iff,axiom,
% 9.49/5.27      ! [A: int,X: num,N: nat] :
% 9.49/5.27        ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 9.49/5.27        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5344_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 9.49/5.27      ! [X: num,N: nat,A: int] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 9.49/5.27        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5345_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 9.49/5.27      ! [X: num,N: nat,A: int] :
% 9.49/5.27        ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 9.49/5.27        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5346_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 9.49/5.27      ! [X: num,N: nat,A: int] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 9.49/5.27        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5347_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 9.49/5.27      ! [X: num,N: nat,A: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 9.49/5.27        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5348_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 9.49/5.27      ! [A: int,X: num,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 9.49/5.27        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5349_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 9.49/5.27      ! [A: int,X: num,N: nat] :
% 9.49/5.27        ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 9.49/5.27        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5350_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 9.49/5.27      ! [A: int,X: num,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 9.49/5.27        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5351_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 9.49/5.27      ! [A: int,X: num,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 9.49/5.27        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5352_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 9.49/5.27      ! [X: num,N: nat,A: int] :
% 9.49/5.27        ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 9.49/5.27        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5353_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 9.49/5.27      ! [X: num,N: nat,A: int] :
% 9.49/5.27        ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 9.49/5.27        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5354_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 9.49/5.27      ! [X: num,N: nat,A: int] :
% 9.49/5.27        ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 9.49/5.27        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5355_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 9.49/5.27      ! [X: num,N: nat,A: int] :
% 9.49/5.27        ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 9.49/5.27        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5356_subset__Compl__self__eq,axiom,
% 9.49/5.27      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.27        ( ( ord_le3146513528884898305at_nat @ A2 @ ( uminus6524753893492686040at_nat @ A2 ) )
% 9.49/5.27        = ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5357_subset__Compl__self__eq,axiom,
% 9.49/5.27      ! [A2: set_real] :
% 9.49/5.27        ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ A2 ) )
% 9.49/5.27        = ( A2 = bot_bot_set_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5358_subset__Compl__self__eq,axiom,
% 9.49/5.27      ! [A2: set_nat] :
% 9.49/5.27        ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
% 9.49/5.27        = ( A2 = bot_bot_set_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5359_subset__Compl__self__eq,axiom,
% 9.49/5.27      ! [A2: set_int] :
% 9.49/5.27        ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
% 9.49/5.27        = ( A2 = bot_bot_set_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5360_real__add__less__0__iff,axiom,
% 9.49/5.27      ! [X: real,Y: real] :
% 9.49/5.27        ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 9.49/5.27        = ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5361_real__0__less__add__iff,axiom,
% 9.49/5.27      ! [X: real,Y: real] :
% 9.49/5.27        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
% 9.49/5.27        = ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5362_signed__take__bit__take__bit,axiom,
% 9.49/5.27      ! [M: nat,N: nat,A: int] :
% 9.49/5.27        ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 9.49/5.27        = ( if_int_int @ ( ord_less_eq_nat @ N @ M ) @ ( bit_se2923211474154528505it_int @ N ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5363_int__le__real__less,axiom,
% 9.49/5.27      ( ord_less_eq_int
% 9.49/5.27      = ( ^ [N2: int,M2: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M2 ) @ one_one_real ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5364_int__less__real__le,axiom,
% 9.49/5.27      ( ord_less_int
% 9.49/5.27      = ( ^ [N2: int,M2: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M2 ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5365_of__int__nonneg,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 9.49/5.27       => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5366_of__int__nonneg,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 9.49/5.27       => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5367_of__int__nonneg,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 9.49/5.27       => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5368_of__int__pos,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 9.49/5.27       => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5369_of__int__pos,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 9.49/5.27       => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5370_of__int__pos,axiom,
% 9.49/5.27      ! [Z2: int] :
% 9.49/5.27        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 9.49/5.27       => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5371_take__bit__signed__take__bit,axiom,
% 9.49/5.27      ! [M: nat,N: nat,A: int] :
% 9.49/5.27        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.49/5.27       => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N @ A ) )
% 9.49/5.27          = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5372_signed__take__bit__int__less__exp,axiom,
% 9.49/5.27      ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5373_signed__take__bit__int__greater__eq__self__iff,axiom,
% 9.49/5.27      ! [K: int,N: nat] :
% 9.49/5.27        ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 9.49/5.27        = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5374_signed__take__bit__int__less__self__iff,axiom,
% 9.49/5.27      ! [N: nat,K: int] :
% 9.49/5.27        ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 9.49/5.27        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5375_signed__take__bit__int__greater__self__iff,axiom,
% 9.49/5.27      ! [K: int,N: nat] :
% 9.49/5.27        ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 9.49/5.27        = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5376_signed__take__bit__int__eq__self__iff,axiom,
% 9.49/5.27      ! [N: nat,K: int] :
% 9.49/5.27        ( ( ( bit_ri631733984087533419it_int @ N @ K )
% 9.49/5.27          = K )
% 9.49/5.27        = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 9.49/5.27          & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5377_signed__take__bit__int__eq__self,axiom,
% 9.49/5.27      ! [N: nat,K: int] :
% 9.49/5.27        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 9.49/5.27       => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.27         => ( ( bit_ri631733984087533419it_int @ N @ K )
% 9.49/5.27            = K ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5378_signed__take__bit__int__greater__eq,axiom,
% 9.49/5.27      ! [K: int,N: nat] :
% 9.49/5.27        ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 9.49/5.27       => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5379_floor__exists1,axiom,
% 9.49/5.27      ! [X: real] :
% 9.49/5.27      ? [X4: int] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ X )
% 9.49/5.27        & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X4 @ one_one_int ) ) )
% 9.49/5.27        & ! [Y4: int] :
% 9.49/5.27            ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y4 ) @ X )
% 9.49/5.27              & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
% 9.49/5.27           => ( Y4 = X4 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5380_floor__exists1,axiom,
% 9.49/5.27      ! [X: rat] :
% 9.49/5.27      ? [X4: int] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ X )
% 9.49/5.27        & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X4 @ one_one_int ) ) )
% 9.49/5.27        & ! [Y4: int] :
% 9.49/5.27            ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y4 ) @ X )
% 9.49/5.27              & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
% 9.49/5.27           => ( Y4 = X4 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5381_floor__exists,axiom,
% 9.49/5.27      ! [X: real] :
% 9.49/5.27      ? [Z4: int] :
% 9.49/5.27        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z4 ) @ X )
% 9.49/5.27        & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5382_floor__exists,axiom,
% 9.49/5.27      ! [X: rat] :
% 9.49/5.27      ? [Z4: int] :
% 9.49/5.27        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z4 ) @ X )
% 9.49/5.27        & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5383_round__unique,axiom,
% 9.49/5.27      ! [X: real,Y: int] :
% 9.49/5.27        ( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y ) )
% 9.49/5.27       => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.49/5.27         => ( ( archim8280529875227126926d_real @ X )
% 9.49/5.27            = Y ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5384_round__unique,axiom,
% 9.49/5.27      ! [X: rat,Y: int] :
% 9.49/5.27        ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
% 9.49/5.27       => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
% 9.49/5.27         => ( ( archim7778729529865785530nd_rat @ X )
% 9.49/5.27            = Y ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5385_set__encode__insert,axiom,
% 9.49/5.27      ! [A2: set_nat,N: nat] :
% 9.49/5.27        ( ( finite_finite_nat @ A2 )
% 9.49/5.27       => ( ~ ( member_nat @ N @ A2 )
% 9.49/5.27         => ( ( nat_set_encode @ ( insert_nat @ N @ A2 ) )
% 9.49/5.27            = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5386_arith__series__nat,axiom,
% 9.49/5.27      ! [A: nat,D: nat,N: nat] :
% 9.49/5.27        ( ( groups3542108847815614940at_nat
% 9.49/5.27          @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I4 @ D ) )
% 9.49/5.27          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.49/5.27        = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5387_Suc__0__xor__eq,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.49/5.27        = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.49/5.27          @ ( zero_n2687167440665602831ol_nat
% 9.49/5.27            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5388_xor__Suc__0__eq,axiom,
% 9.49/5.27      ! [N: nat] :
% 9.49/5.27        ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.49/5.27        = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.49/5.27          @ ( zero_n2687167440665602831ol_nat
% 9.49/5.27            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5389_insertCI,axiom,
% 9.49/5.27      ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
% 9.49/5.27        ( ( ~ ( member8757157785044589968at_nat @ A @ B2 )
% 9.49/5.27         => ( A = B ) )
% 9.49/5.27       => ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5390_insertCI,axiom,
% 9.49/5.27      ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
% 9.49/5.27        ( ( ~ ( member8440522571783428010at_nat @ A @ B2 )
% 9.49/5.27         => ( A = B ) )
% 9.49/5.27       => ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5391_insertCI,axiom,
% 9.49/5.27      ! [A: real,B2: set_real,B: real] :
% 9.49/5.27        ( ( ~ ( member_real @ A @ B2 )
% 9.49/5.27         => ( A = B ) )
% 9.49/5.27       => ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5392_insertCI,axiom,
% 9.49/5.27      ! [A: set_nat,B2: set_set_nat,B: set_nat] :
% 9.49/5.27        ( ( ~ ( member_set_nat @ A @ B2 )
% 9.49/5.27         => ( A = B ) )
% 9.49/5.27       => ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5393_insertCI,axiom,
% 9.49/5.27      ! [A: nat,B2: set_nat,B: nat] :
% 9.49/5.27        ( ( ~ ( member_nat @ A @ B2 )
% 9.49/5.27         => ( A = B ) )
% 9.49/5.27       => ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5394_insertCI,axiom,
% 9.49/5.27      ! [A: int,B2: set_int,B: int] :
% 9.49/5.27        ( ( ~ ( member_int @ A @ B2 )
% 9.49/5.27         => ( A = B ) )
% 9.49/5.27       => ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5395_insert__iff,axiom,
% 9.49/5.27      ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.49/5.27        ( ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ A2 ) )
% 9.49/5.27        = ( ( A = B )
% 9.49/5.27          | ( member8757157785044589968at_nat @ A @ A2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5396_insert__iff,axiom,
% 9.49/5.27      ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.27        ( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A2 ) )
% 9.49/5.27        = ( ( A = B )
% 9.49/5.27          | ( member8440522571783428010at_nat @ A @ A2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5397_insert__iff,axiom,
% 9.49/5.27      ! [A: real,B: real,A2: set_real] :
% 9.49/5.27        ( ( member_real @ A @ ( insert_real @ B @ A2 ) )
% 9.49/5.27        = ( ( A = B )
% 9.49/5.27          | ( member_real @ A @ A2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5398_insert__iff,axiom,
% 9.49/5.27      ! [A: set_nat,B: set_nat,A2: set_set_nat] :
% 9.49/5.27        ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
% 9.49/5.27        = ( ( A = B )
% 9.49/5.27          | ( member_set_nat @ A @ A2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5399_insert__iff,axiom,
% 9.49/5.27      ! [A: nat,B: nat,A2: set_nat] :
% 9.49/5.27        ( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
% 9.49/5.27        = ( ( A = B )
% 9.49/5.27          | ( member_nat @ A @ A2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5400_insert__iff,axiom,
% 9.49/5.27      ! [A: int,B: int,A2: set_int] :
% 9.49/5.27        ( ( member_int @ A @ ( insert_int @ B @ A2 ) )
% 9.49/5.27        = ( ( A = B )
% 9.49/5.27          | ( member_int @ A @ A2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5401_insert__absorb2,axiom,
% 9.49/5.27      ! [X: nat,A2: set_nat] :
% 9.49/5.27        ( ( insert_nat @ X @ ( insert_nat @ X @ A2 ) )
% 9.49/5.27        = ( insert_nat @ X @ A2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5402_insert__absorb2,axiom,
% 9.49/5.27      ! [X: int,A2: set_int] :
% 9.49/5.27        ( ( insert_int @ X @ ( insert_int @ X @ A2 ) )
% 9.49/5.27        = ( insert_int @ X @ A2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5403_insert__absorb2,axiom,
% 9.49/5.27      ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.27        ( ( insert8211810215607154385at_nat @ X @ ( insert8211810215607154385at_nat @ X @ A2 ) )
% 9.49/5.27        = ( insert8211810215607154385at_nat @ X @ A2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5404_insert__absorb2,axiom,
% 9.49/5.27      ! [X: real,A2: set_real] :
% 9.49/5.27        ( ( insert_real @ X @ ( insert_real @ X @ A2 ) )
% 9.49/5.27        = ( insert_real @ X @ A2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5405_insert__absorb2,axiom,
% 9.49/5.27      ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.49/5.27        ( ( insert9069300056098147895at_nat @ X @ ( insert9069300056098147895at_nat @ X @ A2 ) )
% 9.49/5.27        = ( insert9069300056098147895at_nat @ X @ A2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5406_ComplI,axiom,
% 9.49/5.27      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.27        ( ~ ( member8440522571783428010at_nat @ C @ A2 )
% 9.49/5.27       => ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5407_ComplI,axiom,
% 9.49/5.27      ! [C: real,A2: set_real] :
% 9.49/5.27        ( ~ ( member_real @ C @ A2 )
% 9.49/5.27       => ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5408_ComplI,axiom,
% 9.49/5.27      ! [C: set_nat,A2: set_set_nat] :
% 9.49/5.27        ( ~ ( member_set_nat @ C @ A2 )
% 9.49/5.27       => ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5409_ComplI,axiom,
% 9.49/5.27      ! [C: nat,A2: set_nat] :
% 9.49/5.27        ( ~ ( member_nat @ C @ A2 )
% 9.49/5.27       => ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5410_ComplI,axiom,
% 9.49/5.27      ! [C: int,A2: set_int] :
% 9.49/5.27        ( ~ ( member_int @ C @ A2 )
% 9.49/5.27       => ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5411_Compl__iff,axiom,
% 9.49/5.27      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.27        ( ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A2 ) )
% 9.49/5.27        = ( ~ ( member8440522571783428010at_nat @ C @ A2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5412_Compl__iff,axiom,
% 9.49/5.27      ! [C: real,A2: set_real] :
% 9.49/5.27        ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
% 9.49/5.27        = ( ~ ( member_real @ C @ A2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5413_Compl__iff,axiom,
% 9.49/5.27      ! [C: set_nat,A2: set_set_nat] :
% 9.49/5.27        ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
% 9.49/5.27        = ( ~ ( member_set_nat @ C @ A2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5414_Compl__iff,axiom,
% 9.49/5.27      ! [C: nat,A2: set_nat] :
% 9.49/5.27        ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
% 9.49/5.27        = ( ~ ( member_nat @ C @ A2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5415_Compl__iff,axiom,
% 9.49/5.27      ! [C: int,A2: set_int] :
% 9.49/5.27        ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
% 9.49/5.27        = ( ~ ( member_int @ C @ A2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5416_singletonI,axiom,
% 9.49/5.27      ! [A: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5417_singletonI,axiom,
% 9.49/5.27      ! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5418_singletonI,axiom,
% 9.49/5.27      ! [A: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5419_singletonI,axiom,
% 9.49/5.27      ! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5420_singletonI,axiom,
% 9.49/5.27      ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5421_singletonI,axiom,
% 9.49/5.27      ! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5422_finite__insert,axiom,
% 9.49/5.27      ! [A: real,A2: set_real] :
% 9.49/5.27        ( ( finite_finite_real @ ( insert_real @ A @ A2 ) )
% 9.49/5.27        = ( finite_finite_real @ A2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5423_finite__insert,axiom,
% 9.49/5.27      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.49/5.27        ( ( finite4343798906461161616at_nat @ ( insert9069300056098147895at_nat @ A @ A2 ) )
% 9.49/5.27        = ( finite4343798906461161616at_nat @ A2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5424_finite__insert,axiom,
% 9.49/5.27      ! [A: nat,A2: set_nat] :
% 9.49/5.27        ( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
% 9.49/5.27        = ( finite_finite_nat @ A2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5425_finite__insert,axiom,
% 9.49/5.27      ! [A: int,A2: set_int] :
% 9.49/5.27        ( ( finite_finite_int @ ( insert_int @ A @ A2 ) )
% 9.49/5.27        = ( finite_finite_int @ A2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5426_finite__insert,axiom,
% 9.49/5.27      ! [A: complex,A2: set_complex] :
% 9.49/5.27        ( ( finite3207457112153483333omplex @ ( insert_complex @ A @ A2 ) )
% 9.49/5.27        = ( finite3207457112153483333omplex @ A2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5427_finite__insert,axiom,
% 9.49/5.27      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.27        ( ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) )
% 9.49/5.27        = ( finite6177210948735845034at_nat @ A2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5428_insert__subset,axiom,
% 9.49/5.27      ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.49/5.27        ( ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X @ A2 ) @ B2 )
% 9.49/5.27        = ( ( member8757157785044589968at_nat @ X @ B2 )
% 9.49/5.27          & ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5429_insert__subset,axiom,
% 9.49/5.27      ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.27        ( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) @ B2 )
% 9.49/5.27        = ( ( member8440522571783428010at_nat @ X @ B2 )
% 9.49/5.27          & ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5430_insert__subset,axiom,
% 9.49/5.27      ! [X: real,A2: set_real,B2: set_real] :
% 9.49/5.27        ( ( ord_less_eq_set_real @ ( insert_real @ X @ A2 ) @ B2 )
% 9.49/5.27        = ( ( member_real @ X @ B2 )
% 9.49/5.27          & ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5431_insert__subset,axiom,
% 9.49/5.27      ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.49/5.27        ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
% 9.49/5.27        = ( ( member_set_nat @ X @ B2 )
% 9.49/5.27          & ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5432_insert__subset,axiom,
% 9.49/5.27      ! [X: nat,A2: set_nat,B2: set_nat] :
% 9.49/5.27        ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
% 9.49/5.27        = ( ( member_nat @ X @ B2 )
% 9.49/5.27          & ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5433_insert__subset,axiom,
% 9.49/5.27      ! [X: int,A2: set_int,B2: set_int] :
% 9.49/5.27        ( ( ord_less_eq_set_int @ ( insert_int @ X @ A2 ) @ B2 )
% 9.49/5.27        = ( ( member_int @ X @ B2 )
% 9.49/5.27          & ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5434_Diff__insert0,axiom,
% 9.49/5.27      ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.49/5.27        ( ~ ( member8757157785044589968at_nat @ X @ A2 )
% 9.49/5.27       => ( ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ B2 ) )
% 9.49/5.27          = ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5435_Diff__insert0,axiom,
% 9.49/5.27      ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.27        ( ~ ( member8440522571783428010at_nat @ X @ A2 )
% 9.49/5.27       => ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B2 ) )
% 9.49/5.27          = ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5436_Diff__insert0,axiom,
% 9.49/5.27      ! [X: real,A2: set_real,B2: set_real] :
% 9.49/5.27        ( ~ ( member_real @ X @ A2 )
% 9.49/5.27       => ( ( minus_minus_set_real @ A2 @ ( insert_real @ X @ B2 ) )
% 9.49/5.27          = ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5437_Diff__insert0,axiom,
% 9.49/5.27      ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.49/5.27        ( ~ ( member_set_nat @ X @ A2 )
% 9.49/5.27       => ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
% 9.49/5.27          = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5438_Diff__insert0,axiom,
% 9.49/5.27      ! [X: int,A2: set_int,B2: set_int] :
% 9.49/5.27        ( ~ ( member_int @ X @ A2 )
% 9.49/5.27       => ( ( minus_minus_set_int @ A2 @ ( insert_int @ X @ B2 ) )
% 9.49/5.27          = ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5439_Diff__insert0,axiom,
% 9.49/5.27      ! [X: nat,A2: set_nat,B2: set_nat] :
% 9.49/5.27        ( ~ ( member_nat @ X @ A2 )
% 9.49/5.27       => ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
% 9.49/5.27          = ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5440_insert__Diff1,axiom,
% 9.49/5.27      ! [X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.49/5.27        ( ( member8757157785044589968at_nat @ X @ B2 )
% 9.49/5.27       => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A2 ) @ B2 )
% 9.49/5.27          = ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5441_insert__Diff1,axiom,
% 9.49/5.27      ! [X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.27        ( ( member8440522571783428010at_nat @ X @ B2 )
% 9.49/5.27       => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) @ B2 )
% 9.49/5.27          = ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5442_insert__Diff1,axiom,
% 9.49/5.27      ! [X: real,B2: set_real,A2: set_real] :
% 9.49/5.27        ( ( member_real @ X @ B2 )
% 9.49/5.27       => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
% 9.49/5.27          = ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5443_insert__Diff1,axiom,
% 9.49/5.27      ! [X: set_nat,B2: set_set_nat,A2: set_set_nat] :
% 9.49/5.27        ( ( member_set_nat @ X @ B2 )
% 9.49/5.27       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
% 9.49/5.27          = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5444_insert__Diff1,axiom,
% 9.49/5.27      ! [X: int,B2: set_int,A2: set_int] :
% 9.49/5.27        ( ( member_int @ X @ B2 )
% 9.49/5.27       => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B2 )
% 9.49/5.27          = ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5445_insert__Diff1,axiom,
% 9.49/5.27      ! [X: nat,B2: set_nat,A2: set_nat] :
% 9.49/5.27        ( ( member_nat @ X @ B2 )
% 9.49/5.27       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
% 9.49/5.27          = ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5446_xor_Oright__neutral,axiom,
% 9.49/5.27      ! [A: nat] :
% 9.49/5.27        ( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
% 9.49/5.27        = A ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5447_xor_Oright__neutral,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( bit_se6526347334894502574or_int @ A @ zero_zero_int )
% 9.49/5.27        = A ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5448_xor_Oleft__neutral,axiom,
% 9.49/5.27      ! [A: nat] :
% 9.49/5.27        ( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
% 9.49/5.27        = A ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5449_xor_Oleft__neutral,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( bit_se6526347334894502574or_int @ zero_zero_int @ A )
% 9.49/5.27        = A ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5450_xor__self__eq,axiom,
% 9.49/5.27      ! [A: nat] :
% 9.49/5.27        ( ( bit_se6528837805403552850or_nat @ A @ A )
% 9.49/5.27        = zero_zero_nat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5451_xor__self__eq,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( bit_se6526347334894502574or_int @ A @ A )
% 9.49/5.27        = zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5452_bit_Oxor__self,axiom,
% 9.49/5.27      ! [X: int] :
% 9.49/5.27        ( ( bit_se6526347334894502574or_int @ X @ X )
% 9.49/5.27        = zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5453_sum_Oneutral__const,axiom,
% 9.49/5.27      ! [A2: set_nat] :
% 9.49/5.27        ( ( groups3542108847815614940at_nat
% 9.49/5.27          @ ^ [Uu3: nat] : zero_zero_nat
% 9.49/5.27          @ A2 )
% 9.49/5.27        = zero_zero_nat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5454_sum_Oneutral__const,axiom,
% 9.49/5.27      ! [A2: set_complex] :
% 9.49/5.27        ( ( groups7754918857620584856omplex
% 9.49/5.27          @ ^ [Uu3: complex] : zero_zero_complex
% 9.49/5.27          @ A2 )
% 9.49/5.27        = zero_zero_complex ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5455_sum_Oneutral__const,axiom,
% 9.49/5.27      ! [A2: set_nat] :
% 9.49/5.27        ( ( groups6591440286371151544t_real
% 9.49/5.27          @ ^ [Uu3: nat] : zero_zero_real
% 9.49/5.27          @ A2 )
% 9.49/5.27        = zero_zero_real ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5456_singleton__conv,axiom,
% 9.49/5.27      ! [A: produc3843707927480180839at_nat] :
% 9.49/5.27        ( ( collec6321179662152712658at_nat
% 9.49/5.27          @ ^ [X3: produc3843707927480180839at_nat] : ( X3 = A ) )
% 9.49/5.27        = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5457_singleton__conv,axiom,
% 9.49/5.27      ! [A: list_nat] :
% 9.49/5.27        ( ( collect_list_nat
% 9.49/5.27          @ ^ [X3: list_nat] : ( X3 = A ) )
% 9.49/5.27        = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5458_singleton__conv,axiom,
% 9.49/5.27      ! [A: set_nat] :
% 9.49/5.27        ( ( collect_set_nat
% 9.49/5.27          @ ^ [X3: set_nat] : ( X3 = A ) )
% 9.49/5.27        = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5459_singleton__conv,axiom,
% 9.49/5.27      ! [A: product_prod_nat_nat] :
% 9.49/5.27        ( ( collec3392354462482085612at_nat
% 9.49/5.27          @ ^ [X3: product_prod_nat_nat] : ( X3 = A ) )
% 9.49/5.27        = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5460_singleton__conv,axiom,
% 9.49/5.27      ! [A: real] :
% 9.49/5.27        ( ( collect_real
% 9.49/5.27          @ ^ [X3: real] : ( X3 = A ) )
% 9.49/5.27        = ( insert_real @ A @ bot_bot_set_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5461_singleton__conv,axiom,
% 9.49/5.27      ! [A: nat] :
% 9.49/5.27        ( ( collect_nat
% 9.49/5.27          @ ^ [X3: nat] : ( X3 = A ) )
% 9.49/5.27        = ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5462_singleton__conv,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( collect_int
% 9.49/5.27          @ ^ [X3: int] : ( X3 = A ) )
% 9.49/5.27        = ( insert_int @ A @ bot_bot_set_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5463_singleton__conv2,axiom,
% 9.49/5.27      ! [A: produc3843707927480180839at_nat] :
% 9.49/5.27        ( ( collec6321179662152712658at_nat
% 9.49/5.27          @ ( ^ [Y5: produc3843707927480180839at_nat,Z: produc3843707927480180839at_nat] : ( Y5 = Z )
% 9.49/5.27            @ A ) )
% 9.49/5.27        = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5464_singleton__conv2,axiom,
% 9.49/5.27      ! [A: list_nat] :
% 9.49/5.27        ( ( collect_list_nat
% 9.49/5.27          @ ( ^ [Y5: list_nat,Z: list_nat] : ( Y5 = Z )
% 9.49/5.27            @ A ) )
% 9.49/5.27        = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5465_singleton__conv2,axiom,
% 9.49/5.27      ! [A: set_nat] :
% 9.49/5.27        ( ( collect_set_nat
% 9.49/5.27          @ ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z )
% 9.49/5.27            @ A ) )
% 9.49/5.27        = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5466_singleton__conv2,axiom,
% 9.49/5.27      ! [A: product_prod_nat_nat] :
% 9.49/5.27        ( ( collec3392354462482085612at_nat
% 9.49/5.27          @ ( ^ [Y5: product_prod_nat_nat,Z: product_prod_nat_nat] : ( Y5 = Z )
% 9.49/5.27            @ A ) )
% 9.49/5.27        = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5467_singleton__conv2,axiom,
% 9.49/5.27      ! [A: real] :
% 9.49/5.27        ( ( collect_real
% 9.49/5.27          @ ( ^ [Y5: real,Z: real] : ( Y5 = Z )
% 9.49/5.27            @ A ) )
% 9.49/5.27        = ( insert_real @ A @ bot_bot_set_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5468_singleton__conv2,axiom,
% 9.49/5.27      ! [A: nat] :
% 9.49/5.27        ( ( collect_nat
% 9.49/5.27          @ ( ^ [Y5: nat,Z: nat] : ( Y5 = Z )
% 9.49/5.27            @ A ) )
% 9.49/5.27        = ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5469_singleton__conv2,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( collect_int
% 9.49/5.27          @ ( ^ [Y5: int,Z: int] : ( Y5 = Z )
% 9.49/5.27            @ A ) )
% 9.49/5.27        = ( insert_int @ A @ bot_bot_set_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5470_sum_Oempty,axiom,
% 9.49/5.27      ! [G: real > complex] :
% 9.49/5.27        ( ( groups5754745047067104278omplex @ G @ bot_bot_set_real )
% 9.49/5.27        = zero_zero_complex ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5471_sum_Oempty,axiom,
% 9.49/5.27      ! [G: real > real] :
% 9.49/5.27        ( ( groups8097168146408367636l_real @ G @ bot_bot_set_real )
% 9.49/5.27        = zero_zero_real ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5472_sum_Oempty,axiom,
% 9.49/5.27      ! [G: real > rat] :
% 9.49/5.27        ( ( groups1300246762558778688al_rat @ G @ bot_bot_set_real )
% 9.49/5.27        = zero_zero_rat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5473_sum_Oempty,axiom,
% 9.49/5.27      ! [G: real > nat] :
% 9.49/5.27        ( ( groups1935376822645274424al_nat @ G @ bot_bot_set_real )
% 9.49/5.27        = zero_zero_nat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5474_sum_Oempty,axiom,
% 9.49/5.27      ! [G: real > int] :
% 9.49/5.27        ( ( groups1932886352136224148al_int @ G @ bot_bot_set_real )
% 9.49/5.27        = zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5475_sum_Oempty,axiom,
% 9.49/5.27      ! [G: nat > complex] :
% 9.49/5.27        ( ( groups2073611262835488442omplex @ G @ bot_bot_set_nat )
% 9.49/5.27        = zero_zero_complex ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5476_sum_Oempty,axiom,
% 9.49/5.27      ! [G: nat > rat] :
% 9.49/5.27        ( ( groups2906978787729119204at_rat @ G @ bot_bot_set_nat )
% 9.49/5.27        = zero_zero_rat ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5477_sum_Oempty,axiom,
% 9.49/5.27      ! [G: nat > int] :
% 9.49/5.27        ( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
% 9.49/5.27        = zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5478_sum_Oempty,axiom,
% 9.49/5.27      ! [G: int > complex] :
% 9.49/5.27        ( ( groups3049146728041665814omplex @ G @ bot_bot_set_int )
% 9.49/5.27        = zero_zero_complex ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5479_sum_Oempty,axiom,
% 9.49/5.27      ! [G: int > real] :
% 9.49/5.27        ( ( groups8778361861064173332t_real @ G @ bot_bot_set_int )
% 9.49/5.27        = zero_zero_real ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5480_sum__eq__0__iff,axiom,
% 9.49/5.27      ! [F3: set_int,F: int > nat] :
% 9.49/5.27        ( ( finite_finite_int @ F3 )
% 9.49/5.27       => ( ( ( groups4541462559716669496nt_nat @ F @ F3 )
% 9.49/5.27            = zero_zero_nat )
% 9.49/5.27          = ( ! [X3: int] :
% 9.49/5.27                ( ( member_int @ X3 @ F3 )
% 9.49/5.27               => ( ( F @ X3 )
% 9.49/5.27                  = zero_zero_nat ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5481_sum__eq__0__iff,axiom,
% 9.49/5.27      ! [F3: set_complex,F: complex > nat] :
% 9.49/5.27        ( ( finite3207457112153483333omplex @ F3 )
% 9.49/5.27       => ( ( ( groups5693394587270226106ex_nat @ F @ F3 )
% 9.49/5.27            = zero_zero_nat )
% 9.49/5.27          = ( ! [X3: complex] :
% 9.49/5.27                ( ( member_complex @ X3 @ F3 )
% 9.49/5.27               => ( ( F @ X3 )
% 9.49/5.27                  = zero_zero_nat ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5482_sum__eq__0__iff,axiom,
% 9.49/5.27      ! [F3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 9.49/5.27        ( ( finite6177210948735845034at_nat @ F3 )
% 9.49/5.27       => ( ( ( groups977919841031483927at_nat @ F @ F3 )
% 9.49/5.27            = zero_zero_nat )
% 9.49/5.27          = ( ! [X3: product_prod_nat_nat] :
% 9.49/5.27                ( ( member8440522571783428010at_nat @ X3 @ F3 )
% 9.49/5.27               => ( ( F @ X3 )
% 9.49/5.27                  = zero_zero_nat ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5483_sum__eq__0__iff,axiom,
% 9.49/5.27      ! [F3: set_nat,F: nat > nat] :
% 9.49/5.27        ( ( finite_finite_nat @ F3 )
% 9.49/5.27       => ( ( ( groups3542108847815614940at_nat @ F @ F3 )
% 9.49/5.27            = zero_zero_nat )
% 9.49/5.27          = ( ! [X3: nat] :
% 9.49/5.27                ( ( member_nat @ X3 @ F3 )
% 9.49/5.27               => ( ( F @ X3 )
% 9.49/5.27                  = zero_zero_nat ) ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5484_sum_Oinfinite,axiom,
% 9.49/5.27      ! [A2: set_nat,G: nat > complex] :
% 9.49/5.27        ( ~ ( finite_finite_nat @ A2 )
% 9.49/5.27       => ( ( groups2073611262835488442omplex @ G @ A2 )
% 9.49/5.27          = zero_zero_complex ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5485_sum_Oinfinite,axiom,
% 9.49/5.27      ! [A2: set_int,G: int > complex] :
% 9.49/5.27        ( ~ ( finite_finite_int @ A2 )
% 9.49/5.27       => ( ( groups3049146728041665814omplex @ G @ A2 )
% 9.49/5.27          = zero_zero_complex ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5486_sum_Oinfinite,axiom,
% 9.49/5.27      ! [A2: set_int,G: int > real] :
% 9.49/5.27        ( ~ ( finite_finite_int @ A2 )
% 9.49/5.27       => ( ( groups8778361861064173332t_real @ G @ A2 )
% 9.49/5.27          = zero_zero_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5487_sum_Oinfinite,axiom,
% 9.49/5.27      ! [A2: set_complex,G: complex > real] :
% 9.49/5.27        ( ~ ( finite3207457112153483333omplex @ A2 )
% 9.49/5.27       => ( ( groups5808333547571424918x_real @ G @ A2 )
% 9.49/5.27          = zero_zero_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5488_sum_Oinfinite,axiom,
% 9.49/5.27      ! [A2: set_nat,G: nat > rat] :
% 9.49/5.27        ( ~ ( finite_finite_nat @ A2 )
% 9.49/5.27       => ( ( groups2906978787729119204at_rat @ G @ A2 )
% 9.49/5.27          = zero_zero_rat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5489_sum_Oinfinite,axiom,
% 9.49/5.27      ! [A2: set_int,G: int > rat] :
% 9.49/5.27        ( ~ ( finite_finite_int @ A2 )
% 9.49/5.27       => ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 9.49/5.27          = zero_zero_rat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5490_sum_Oinfinite,axiom,
% 9.49/5.27      ! [A2: set_complex,G: complex > rat] :
% 9.49/5.27        ( ~ ( finite3207457112153483333omplex @ A2 )
% 9.49/5.27       => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 9.49/5.27          = zero_zero_rat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5491_sum_Oinfinite,axiom,
% 9.49/5.27      ! [A2: set_int,G: int > nat] :
% 9.49/5.27        ( ~ ( finite_finite_int @ A2 )
% 9.49/5.27       => ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 9.49/5.27          = zero_zero_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5492_sum_Oinfinite,axiom,
% 9.49/5.27      ! [A2: set_complex,G: complex > nat] :
% 9.49/5.27        ( ~ ( finite3207457112153483333omplex @ A2 )
% 9.49/5.27       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 9.49/5.27          = zero_zero_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5493_sum_Oinfinite,axiom,
% 9.49/5.27      ! [A2: set_nat,G: nat > int] :
% 9.49/5.27        ( ~ ( finite_finite_nat @ A2 )
% 9.49/5.27       => ( ( groups3539618377306564664at_int @ G @ A2 )
% 9.49/5.27          = zero_zero_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5494_singleton__insert__inj__eq,axiom,
% 9.49/5.27      ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.49/5.27        ( ( ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat )
% 9.49/5.27          = ( insert9069300056098147895at_nat @ A @ A2 ) )
% 9.49/5.27        = ( ( A = B )
% 9.49/5.27          & ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5495_singleton__insert__inj__eq,axiom,
% 9.49/5.27      ! [B: product_prod_nat_nat,A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.27        ( ( ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat )
% 9.49/5.27          = ( insert8211810215607154385at_nat @ A @ A2 ) )
% 9.49/5.27        = ( ( A = B )
% 9.49/5.27          & ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5496_singleton__insert__inj__eq,axiom,
% 9.49/5.27      ! [B: real,A: real,A2: set_real] :
% 9.49/5.27        ( ( ( insert_real @ B @ bot_bot_set_real )
% 9.49/5.27          = ( insert_real @ A @ A2 ) )
% 9.49/5.27        = ( ( A = B )
% 9.49/5.27          & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5497_singleton__insert__inj__eq,axiom,
% 9.49/5.27      ! [B: nat,A: nat,A2: set_nat] :
% 9.49/5.27        ( ( ( insert_nat @ B @ bot_bot_set_nat )
% 9.49/5.27          = ( insert_nat @ A @ A2 ) )
% 9.49/5.27        = ( ( A = B )
% 9.49/5.27          & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5498_singleton__insert__inj__eq,axiom,
% 9.49/5.27      ! [B: int,A: int,A2: set_int] :
% 9.49/5.27        ( ( ( insert_int @ B @ bot_bot_set_int )
% 9.49/5.27          = ( insert_int @ A @ A2 ) )
% 9.49/5.27        = ( ( A = B )
% 9.49/5.27          & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5499_singleton__insert__inj__eq_H,axiom,
% 9.49/5.27      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
% 9.49/5.27        ( ( ( insert9069300056098147895at_nat @ A @ A2 )
% 9.49/5.27          = ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
% 9.49/5.27        = ( ( A = B )
% 9.49/5.27          & ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5500_singleton__insert__inj__eq_H,axiom,
% 9.49/5.27      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
% 9.49/5.27        ( ( ( insert8211810215607154385at_nat @ A @ A2 )
% 9.49/5.27          = ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
% 9.49/5.27        = ( ( A = B )
% 9.49/5.27          & ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5501_singleton__insert__inj__eq_H,axiom,
% 9.49/5.27      ! [A: real,A2: set_real,B: real] :
% 9.49/5.27        ( ( ( insert_real @ A @ A2 )
% 9.49/5.27          = ( insert_real @ B @ bot_bot_set_real ) )
% 9.49/5.27        = ( ( A = B )
% 9.49/5.27          & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5502_singleton__insert__inj__eq_H,axiom,
% 9.49/5.27      ! [A: nat,A2: set_nat,B: nat] :
% 9.49/5.27        ( ( ( insert_nat @ A @ A2 )
% 9.49/5.27          = ( insert_nat @ B @ bot_bot_set_nat ) )
% 9.49/5.27        = ( ( A = B )
% 9.49/5.27          & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5503_singleton__insert__inj__eq_H,axiom,
% 9.49/5.27      ! [A: int,A2: set_int,B: int] :
% 9.49/5.27        ( ( ( insert_int @ A @ A2 )
% 9.49/5.27          = ( insert_int @ B @ bot_bot_set_int ) )
% 9.49/5.27        = ( ( A = B )
% 9.49/5.27          & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5504_atLeastAtMost__singleton__iff,axiom,
% 9.49/5.27      ! [A: nat,B: nat,C: nat] :
% 9.49/5.27        ( ( ( set_or1269000886237332187st_nat @ A @ B )
% 9.49/5.27          = ( insert_nat @ C @ bot_bot_set_nat ) )
% 9.49/5.27        = ( ( A = B )
% 9.49/5.27          & ( B = C ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5505_atLeastAtMost__singleton__iff,axiom,
% 9.49/5.27      ! [A: int,B: int,C: int] :
% 9.49/5.27        ( ( ( set_or1266510415728281911st_int @ A @ B )
% 9.49/5.27          = ( insert_int @ C @ bot_bot_set_int ) )
% 9.49/5.27        = ( ( A = B )
% 9.49/5.27          & ( B = C ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5506_atLeastAtMost__singleton__iff,axiom,
% 9.49/5.27      ! [A: real,B: real,C: real] :
% 9.49/5.27        ( ( ( set_or1222579329274155063t_real @ A @ B )
% 9.49/5.27          = ( insert_real @ C @ bot_bot_set_real ) )
% 9.49/5.27        = ( ( A = B )
% 9.49/5.27          & ( B = C ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5507_atLeastAtMost__singleton,axiom,
% 9.49/5.27      ! [A: nat] :
% 9.49/5.27        ( ( set_or1269000886237332187st_nat @ A @ A )
% 9.49/5.27        = ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5508_atLeastAtMost__singleton,axiom,
% 9.49/5.27      ! [A: int] :
% 9.49/5.27        ( ( set_or1266510415728281911st_int @ A @ A )
% 9.49/5.27        = ( insert_int @ A @ bot_bot_set_int ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5509_atLeastAtMost__singleton,axiom,
% 9.49/5.27      ! [A: real] :
% 9.49/5.27        ( ( set_or1222579329274155063t_real @ A @ A )
% 9.49/5.27        = ( insert_real @ A @ bot_bot_set_real ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5510_insert__Diff__single,axiom,
% 9.49/5.27      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.49/5.27        ( ( insert9069300056098147895at_nat @ A @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 9.49/5.27        = ( insert9069300056098147895at_nat @ A @ A2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5511_insert__Diff__single,axiom,
% 9.49/5.27      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.27        ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 9.49/5.27        = ( insert8211810215607154385at_nat @ A @ A2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5512_insert__Diff__single,axiom,
% 9.49/5.27      ! [A: real,A2: set_real] :
% 9.49/5.27        ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.49/5.27        = ( insert_real @ A @ A2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5513_insert__Diff__single,axiom,
% 9.49/5.27      ! [A: int,A2: set_int] :
% 9.49/5.27        ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.49/5.27        = ( insert_int @ A @ A2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5514_insert__Diff__single,axiom,
% 9.49/5.27      ! [A: nat,A2: set_nat] :
% 9.49/5.27        ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 9.49/5.27        = ( insert_nat @ A @ A2 ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5515_finite__Diff__insert,axiom,
% 9.49/5.27      ! [A2: set_real,A: real,B2: set_real] :
% 9.49/5.27        ( ( finite_finite_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) ) )
% 9.49/5.27        = ( finite_finite_real @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5516_finite__Diff__insert,axiom,
% 9.49/5.27      ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.49/5.27        ( ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) ) )
% 9.49/5.27        = ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5517_finite__Diff__insert,axiom,
% 9.49/5.27      ! [A2: set_int,A: int,B2: set_int] :
% 9.49/5.27        ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) ) )
% 9.49/5.27        = ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5518_finite__Diff__insert,axiom,
% 9.49/5.27      ! [A2: set_complex,A: complex,B2: set_complex] :
% 9.49/5.27        ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ B2 ) ) )
% 9.49/5.27        = ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5519_finite__Diff__insert,axiom,
% 9.49/5.27      ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.27        ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) ) )
% 9.49/5.27        = ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5520_finite__Diff__insert,axiom,
% 9.49/5.27      ! [A2: set_nat,A: nat,B2: set_nat] :
% 9.49/5.27        ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
% 9.49/5.27        = ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5521_round__0,axiom,
% 9.49/5.27      ( ( archim8280529875227126926d_real @ zero_zero_real )
% 9.49/5.27      = zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5522_round__0,axiom,
% 9.49/5.27      ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
% 9.49/5.27      = zero_zero_int ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5523_sum_Odelta,axiom,
% 9.49/5.27      ! [S2: set_real,A: real,B: real > complex] :
% 9.49/5.27        ( ( finite_finite_real @ S2 )
% 9.49/5.27       => ( ( ( member_real @ A @ S2 )
% 9.49/5.27           => ( ( groups5754745047067104278omplex
% 9.49/5.27                @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.49/5.27                @ S2 )
% 9.49/5.27              = ( B @ A ) ) )
% 9.49/5.27          & ( ~ ( member_real @ A @ S2 )
% 9.49/5.27           => ( ( groups5754745047067104278omplex
% 9.49/5.27                @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.49/5.27                @ S2 )
% 9.49/5.27              = zero_zero_complex ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5524_sum_Odelta,axiom,
% 9.49/5.27      ! [S2: set_nat,A: nat,B: nat > complex] :
% 9.49/5.27        ( ( finite_finite_nat @ S2 )
% 9.49/5.27       => ( ( ( member_nat @ A @ S2 )
% 9.49/5.27           => ( ( groups2073611262835488442omplex
% 9.49/5.27                @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.49/5.27                @ S2 )
% 9.49/5.27              = ( B @ A ) ) )
% 9.49/5.27          & ( ~ ( member_nat @ A @ S2 )
% 9.49/5.27           => ( ( groups2073611262835488442omplex
% 9.49/5.27                @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.49/5.27                @ S2 )
% 9.49/5.27              = zero_zero_complex ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5525_sum_Odelta,axiom,
% 9.49/5.27      ! [S2: set_int,A: int,B: int > complex] :
% 9.49/5.27        ( ( finite_finite_int @ S2 )
% 9.49/5.27       => ( ( ( member_int @ A @ S2 )
% 9.49/5.27           => ( ( groups3049146728041665814omplex
% 9.49/5.27                @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.49/5.27                @ S2 )
% 9.49/5.27              = ( B @ A ) ) )
% 9.49/5.27          & ( ~ ( member_int @ A @ S2 )
% 9.49/5.27           => ( ( groups3049146728041665814omplex
% 9.49/5.27                @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.49/5.27                @ S2 )
% 9.49/5.27              = zero_zero_complex ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5526_sum_Odelta,axiom,
% 9.49/5.27      ! [S2: set_real,A: real,B: real > real] :
% 9.49/5.27        ( ( finite_finite_real @ S2 )
% 9.49/5.27       => ( ( ( member_real @ A @ S2 )
% 9.49/5.27           => ( ( groups8097168146408367636l_real
% 9.49/5.27                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 9.49/5.27                @ S2 )
% 9.49/5.27              = ( B @ A ) ) )
% 9.49/5.27          & ( ~ ( member_real @ A @ S2 )
% 9.49/5.27           => ( ( groups8097168146408367636l_real
% 9.49/5.27                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 9.49/5.27                @ S2 )
% 9.49/5.27              = zero_zero_real ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5527_sum_Odelta,axiom,
% 9.49/5.27      ! [S2: set_int,A: int,B: int > real] :
% 9.49/5.27        ( ( finite_finite_int @ S2 )
% 9.49/5.27       => ( ( ( member_int @ A @ S2 )
% 9.49/5.27           => ( ( groups8778361861064173332t_real
% 9.49/5.27                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 9.49/5.27                @ S2 )
% 9.49/5.27              = ( B @ A ) ) )
% 9.49/5.27          & ( ~ ( member_int @ A @ S2 )
% 9.49/5.27           => ( ( groups8778361861064173332t_real
% 9.49/5.27                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 9.49/5.27                @ S2 )
% 9.49/5.27              = zero_zero_real ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5528_sum_Odelta,axiom,
% 9.49/5.27      ! [S2: set_complex,A: complex,B: complex > real] :
% 9.49/5.27        ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.27       => ( ( ( member_complex @ A @ S2 )
% 9.49/5.27           => ( ( groups5808333547571424918x_real
% 9.49/5.27                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 9.49/5.27                @ S2 )
% 9.49/5.27              = ( B @ A ) ) )
% 9.49/5.27          & ( ~ ( member_complex @ A @ S2 )
% 9.49/5.27           => ( ( groups5808333547571424918x_real
% 9.49/5.27                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 9.49/5.27                @ S2 )
% 9.49/5.27              = zero_zero_real ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5529_sum_Odelta,axiom,
% 9.49/5.27      ! [S2: set_real,A: real,B: real > rat] :
% 9.49/5.27        ( ( finite_finite_real @ S2 )
% 9.49/5.27       => ( ( ( member_real @ A @ S2 )
% 9.49/5.27           => ( ( groups1300246762558778688al_rat
% 9.49/5.27                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.49/5.27                @ S2 )
% 9.49/5.27              = ( B @ A ) ) )
% 9.49/5.27          & ( ~ ( member_real @ A @ S2 )
% 9.49/5.27           => ( ( groups1300246762558778688al_rat
% 9.49/5.27                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.49/5.27                @ S2 )
% 9.49/5.27              = zero_zero_rat ) ) ) ) ).
% 9.49/5.27  
% 9.49/5.27  thf(fact_5530_sum_Odelta,axiom,
% 9.49/5.27      ! [S2: set_nat,A: nat,B: nat > rat] :
% 9.49/5.27        ( ( finite_finite_nat @ S2 )
% 9.49/5.27       => ( ( ( member_nat @ A @ S2 )
% 9.49/5.27           => ( ( groups2906978787729119204at_rat
% 9.49/5.27                @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.49/5.27                @ S2 )
% 9.49/5.27              = ( B @ A ) ) )
% 9.49/5.28          & ( ~ ( member_nat @ A @ S2 )
% 9.49/5.28           => ( ( groups2906978787729119204at_rat
% 9.49/5.28                @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = zero_zero_rat ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5531_sum_Odelta,axiom,
% 9.49/5.28      ! [S2: set_int,A: int,B: int > rat] :
% 9.49/5.28        ( ( finite_finite_int @ S2 )
% 9.49/5.28       => ( ( ( member_int @ A @ S2 )
% 9.49/5.28           => ( ( groups3906332499630173760nt_rat
% 9.49/5.28                @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( B @ A ) ) )
% 9.49/5.28          & ( ~ ( member_int @ A @ S2 )
% 9.49/5.28           => ( ( groups3906332499630173760nt_rat
% 9.49/5.28                @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = zero_zero_rat ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5532_sum_Odelta,axiom,
% 9.49/5.28      ! [S2: set_complex,A: complex,B: complex > rat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.28       => ( ( ( member_complex @ A @ S2 )
% 9.49/5.28           => ( ( groups5058264527183730370ex_rat
% 9.49/5.28                @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( B @ A ) ) )
% 9.49/5.28          & ( ~ ( member_complex @ A @ S2 )
% 9.49/5.28           => ( ( groups5058264527183730370ex_rat
% 9.49/5.28                @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = zero_zero_rat ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5533_sum_Odelta_H,axiom,
% 9.49/5.28      ! [S2: set_real,A: real,B: real > complex] :
% 9.49/5.28        ( ( finite_finite_real @ S2 )
% 9.49/5.28       => ( ( ( member_real @ A @ S2 )
% 9.49/5.28           => ( ( groups5754745047067104278omplex
% 9.49/5.28                @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( B @ A ) ) )
% 9.49/5.28          & ( ~ ( member_real @ A @ S2 )
% 9.49/5.28           => ( ( groups5754745047067104278omplex
% 9.49/5.28                @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = zero_zero_complex ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5534_sum_Odelta_H,axiom,
% 9.49/5.28      ! [S2: set_nat,A: nat,B: nat > complex] :
% 9.49/5.28        ( ( finite_finite_nat @ S2 )
% 9.49/5.28       => ( ( ( member_nat @ A @ S2 )
% 9.49/5.28           => ( ( groups2073611262835488442omplex
% 9.49/5.28                @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( B @ A ) ) )
% 9.49/5.28          & ( ~ ( member_nat @ A @ S2 )
% 9.49/5.28           => ( ( groups2073611262835488442omplex
% 9.49/5.28                @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = zero_zero_complex ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5535_sum_Odelta_H,axiom,
% 9.49/5.28      ! [S2: set_int,A: int,B: int > complex] :
% 9.49/5.28        ( ( finite_finite_int @ S2 )
% 9.49/5.28       => ( ( ( member_int @ A @ S2 )
% 9.49/5.28           => ( ( groups3049146728041665814omplex
% 9.49/5.28                @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( B @ A ) ) )
% 9.49/5.28          & ( ~ ( member_int @ A @ S2 )
% 9.49/5.28           => ( ( groups3049146728041665814omplex
% 9.49/5.28                @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = zero_zero_complex ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5536_sum_Odelta_H,axiom,
% 9.49/5.28      ! [S2: set_real,A: real,B: real > real] :
% 9.49/5.28        ( ( finite_finite_real @ S2 )
% 9.49/5.28       => ( ( ( member_real @ A @ S2 )
% 9.49/5.28           => ( ( groups8097168146408367636l_real
% 9.49/5.28                @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( B @ A ) ) )
% 9.49/5.28          & ( ~ ( member_real @ A @ S2 )
% 9.49/5.28           => ( ( groups8097168146408367636l_real
% 9.49/5.28                @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = zero_zero_real ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5537_sum_Odelta_H,axiom,
% 9.49/5.28      ! [S2: set_int,A: int,B: int > real] :
% 9.49/5.28        ( ( finite_finite_int @ S2 )
% 9.49/5.28       => ( ( ( member_int @ A @ S2 )
% 9.49/5.28           => ( ( groups8778361861064173332t_real
% 9.49/5.28                @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( B @ A ) ) )
% 9.49/5.28          & ( ~ ( member_int @ A @ S2 )
% 9.49/5.28           => ( ( groups8778361861064173332t_real
% 9.49/5.28                @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = zero_zero_real ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5538_sum_Odelta_H,axiom,
% 9.49/5.28      ! [S2: set_complex,A: complex,B: complex > real] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.28       => ( ( ( member_complex @ A @ S2 )
% 9.49/5.28           => ( ( groups5808333547571424918x_real
% 9.49/5.28                @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( B @ A ) ) )
% 9.49/5.28          & ( ~ ( member_complex @ A @ S2 )
% 9.49/5.28           => ( ( groups5808333547571424918x_real
% 9.49/5.28                @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = zero_zero_real ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5539_sum_Odelta_H,axiom,
% 9.49/5.28      ! [S2: set_real,A: real,B: real > rat] :
% 9.49/5.28        ( ( finite_finite_real @ S2 )
% 9.49/5.28       => ( ( ( member_real @ A @ S2 )
% 9.49/5.28           => ( ( groups1300246762558778688al_rat
% 9.49/5.28                @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( B @ A ) ) )
% 9.49/5.28          & ( ~ ( member_real @ A @ S2 )
% 9.49/5.28           => ( ( groups1300246762558778688al_rat
% 9.49/5.28                @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = zero_zero_rat ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5540_sum_Odelta_H,axiom,
% 9.49/5.28      ! [S2: set_nat,A: nat,B: nat > rat] :
% 9.49/5.28        ( ( finite_finite_nat @ S2 )
% 9.49/5.28       => ( ( ( member_nat @ A @ S2 )
% 9.49/5.28           => ( ( groups2906978787729119204at_rat
% 9.49/5.28                @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( B @ A ) ) )
% 9.49/5.28          & ( ~ ( member_nat @ A @ S2 )
% 9.49/5.28           => ( ( groups2906978787729119204at_rat
% 9.49/5.28                @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = zero_zero_rat ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5541_sum_Odelta_H,axiom,
% 9.49/5.28      ! [S2: set_int,A: int,B: int > rat] :
% 9.49/5.28        ( ( finite_finite_int @ S2 )
% 9.49/5.28       => ( ( ( member_int @ A @ S2 )
% 9.49/5.28           => ( ( groups3906332499630173760nt_rat
% 9.49/5.28                @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( B @ A ) ) )
% 9.49/5.28          & ( ~ ( member_int @ A @ S2 )
% 9.49/5.28           => ( ( groups3906332499630173760nt_rat
% 9.49/5.28                @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = zero_zero_rat ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5542_sum_Odelta_H,axiom,
% 9.49/5.28      ! [S2: set_complex,A: complex,B: complex > rat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.28       => ( ( ( member_complex @ A @ S2 )
% 9.49/5.28           => ( ( groups5058264527183730370ex_rat
% 9.49/5.28                @ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( B @ A ) ) )
% 9.49/5.28          & ( ~ ( member_complex @ A @ S2 )
% 9.49/5.28           => ( ( groups5058264527183730370ex_rat
% 9.49/5.28                @ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = zero_zero_rat ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5543_sum_Oinsert,axiom,
% 9.49/5.28      ! [A2: set_real,X: real,G: real > real] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ~ ( member_real @ X @ A2 )
% 9.49/5.28         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
% 9.49/5.28            = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5544_sum_Oinsert,axiom,
% 9.49/5.28      ! [A2: set_int,X: int,G: int > real] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ~ ( member_int @ X @ A2 )
% 9.49/5.28         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
% 9.49/5.28            = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5545_sum_Oinsert,axiom,
% 9.49/5.28      ! [A2: set_complex,X: complex,G: complex > real] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ~ ( member_complex @ X @ A2 )
% 9.49/5.28         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
% 9.49/5.28            = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5546_sum_Oinsert,axiom,
% 9.49/5.28      ! [A2: set_real,X: real,G: real > rat] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ~ ( member_real @ X @ A2 )
% 9.49/5.28         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
% 9.49/5.28            = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5547_sum_Oinsert,axiom,
% 9.49/5.28      ! [A2: set_nat,X: nat,G: nat > rat] :
% 9.49/5.28        ( ( finite_finite_nat @ A2 )
% 9.49/5.28       => ( ~ ( member_nat @ X @ A2 )
% 9.49/5.28         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
% 9.49/5.28            = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5548_sum_Oinsert,axiom,
% 9.49/5.28      ! [A2: set_int,X: int,G: int > rat] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ~ ( member_int @ X @ A2 )
% 9.49/5.28         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
% 9.49/5.28            = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5549_sum_Oinsert,axiom,
% 9.49/5.28      ! [A2: set_complex,X: complex,G: complex > rat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ~ ( member_complex @ X @ A2 )
% 9.49/5.28         => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
% 9.49/5.28            = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5550_sum_Oinsert,axiom,
% 9.49/5.28      ! [A2: set_real,X: real,G: real > nat] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ~ ( member_real @ X @ A2 )
% 9.49/5.28         => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A2 ) )
% 9.49/5.28            = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5551_sum_Oinsert,axiom,
% 9.49/5.28      ! [A2: set_int,X: int,G: int > nat] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ~ ( member_int @ X @ A2 )
% 9.49/5.28         => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A2 ) )
% 9.49/5.28            = ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5552_sum_Oinsert,axiom,
% 9.49/5.28      ! [A2: set_complex,X: complex,G: complex > nat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ~ ( member_complex @ X @ A2 )
% 9.49/5.28         => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
% 9.49/5.28            = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5553_subset__Compl__singleton,axiom,
% 9.49/5.28      ! [A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
% 9.49/5.28        ( ( ord_le1268244103169919719at_nat @ A2 @ ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) )
% 9.49/5.28        = ( ~ ( member8757157785044589968at_nat @ B @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5554_subset__Compl__singleton,axiom,
% 9.49/5.28      ! [A2: set_set_nat,B: set_nat] :
% 9.49/5.28        ( ( ord_le6893508408891458716et_nat @ A2 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
% 9.49/5.28        = ( ~ ( member_set_nat @ B @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5555_subset__Compl__singleton,axiom,
% 9.49/5.28      ! [A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
% 9.49/5.28        ( ( ord_le3146513528884898305at_nat @ A2 @ ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) )
% 9.49/5.28        = ( ~ ( member8440522571783428010at_nat @ B @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5556_subset__Compl__singleton,axiom,
% 9.49/5.28      ! [A2: set_real,B: real] :
% 9.49/5.28        ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ ( insert_real @ B @ bot_bot_set_real ) ) )
% 9.49/5.28        = ( ~ ( member_real @ B @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5557_subset__Compl__singleton,axiom,
% 9.49/5.28      ! [A2: set_nat,B: nat] :
% 9.49/5.28        ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 9.49/5.28        = ( ~ ( member_nat @ B @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5558_subset__Compl__singleton,axiom,
% 9.49/5.28      ! [A2: set_int,B: int] :
% 9.49/5.28        ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
% 9.49/5.28        = ( ~ ( member_int @ B @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5559_set__replicate,axiom,
% 9.49/5.28      ! [N: nat,X: produc3843707927480180839at_nat] :
% 9.49/5.28        ( ( N != zero_zero_nat )
% 9.49/5.28       => ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N @ X ) )
% 9.49/5.28          = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5560_set__replicate,axiom,
% 9.49/5.28      ! [N: nat,X: vEBT_VEBT] :
% 9.49/5.28        ( ( N != zero_zero_nat )
% 9.49/5.28       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
% 9.49/5.28          = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5561_set__replicate,axiom,
% 9.49/5.28      ! [N: nat,X: product_prod_nat_nat] :
% 9.49/5.28        ( ( N != zero_zero_nat )
% 9.49/5.28       => ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ X ) )
% 9.49/5.28          = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5562_set__replicate,axiom,
% 9.49/5.28      ! [N: nat,X: real] :
% 9.49/5.28        ( ( N != zero_zero_nat )
% 9.49/5.28       => ( ( set_real2 @ ( replicate_real @ N @ X ) )
% 9.49/5.28          = ( insert_real @ X @ bot_bot_set_real ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5563_set__replicate,axiom,
% 9.49/5.28      ! [N: nat,X: nat] :
% 9.49/5.28        ( ( N != zero_zero_nat )
% 9.49/5.28       => ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
% 9.49/5.28          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5564_set__replicate,axiom,
% 9.49/5.28      ! [N: nat,X: int] :
% 9.49/5.28        ( ( N != zero_zero_nat )
% 9.49/5.28       => ( ( set_int2 @ ( replicate_int @ N @ X ) )
% 9.49/5.28          = ( insert_int @ X @ bot_bot_set_int ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5565_sum_Ocl__ivl__Suc,axiom,
% 9.49/5.28      ! [N: nat,M: nat,G: nat > complex] :
% 9.49/5.28        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.49/5.28         => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.49/5.28            = zero_zero_complex ) )
% 9.49/5.28        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.49/5.28         => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.49/5.28            = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5566_sum_Ocl__ivl__Suc,axiom,
% 9.49/5.28      ! [N: nat,M: nat,G: nat > rat] :
% 9.49/5.28        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.49/5.28         => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.49/5.28            = zero_zero_rat ) )
% 9.49/5.28        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.49/5.28         => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.49/5.28            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5567_sum_Ocl__ivl__Suc,axiom,
% 9.49/5.28      ! [N: nat,M: nat,G: nat > int] :
% 9.49/5.28        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.49/5.28         => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.49/5.28            = zero_zero_int ) )
% 9.49/5.28        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.49/5.28         => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.49/5.28            = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5568_sum_Ocl__ivl__Suc,axiom,
% 9.49/5.28      ! [N: nat,M: nat,G: nat > nat] :
% 9.49/5.28        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.49/5.28         => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.49/5.28            = zero_zero_nat ) )
% 9.49/5.28        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.49/5.28         => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.49/5.28            = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5569_sum_Ocl__ivl__Suc,axiom,
% 9.49/5.28      ! [N: nat,M: nat,G: nat > real] :
% 9.49/5.28        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.49/5.28         => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.49/5.28            = zero_zero_real ) )
% 9.49/5.28        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.49/5.28         => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.49/5.28            = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5570_xor__nat__numerals_I1_J,axiom,
% 9.49/5.28      ! [Y: num] :
% 9.49/5.28        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 9.49/5.28        = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5571_xor__nat__numerals_I2_J,axiom,
% 9.49/5.28      ! [Y: num] :
% 9.49/5.28        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 9.49/5.28        = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5572_xor__nat__numerals_I3_J,axiom,
% 9.49/5.28      ! [X: num] :
% 9.49/5.28        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 9.49/5.28        = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5573_xor__nat__numerals_I4_J,axiom,
% 9.49/5.28      ! [X: num] :
% 9.49/5.28        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 9.49/5.28        = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5574_sum__zero__power,axiom,
% 9.49/5.28      ! [A2: set_nat,C: nat > complex] :
% 9.49/5.28        ( ( ( ( finite_finite_nat @ A2 )
% 9.49/5.28            & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.49/5.28         => ( ( groups2073611262835488442omplex
% 9.49/5.28              @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = ( C @ zero_zero_nat ) ) )
% 9.49/5.28        & ( ~ ( ( finite_finite_nat @ A2 )
% 9.49/5.28              & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.49/5.28         => ( ( groups2073611262835488442omplex
% 9.49/5.28              @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = zero_zero_complex ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5575_sum__zero__power,axiom,
% 9.49/5.28      ! [A2: set_nat,C: nat > rat] :
% 9.49/5.28        ( ( ( ( finite_finite_nat @ A2 )
% 9.49/5.28            & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.49/5.28         => ( ( groups2906978787729119204at_rat
% 9.49/5.28              @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = ( C @ zero_zero_nat ) ) )
% 9.49/5.28        & ( ~ ( ( finite_finite_nat @ A2 )
% 9.49/5.28              & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.49/5.28         => ( ( groups2906978787729119204at_rat
% 9.49/5.28              @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = zero_zero_rat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5576_sum__zero__power,axiom,
% 9.49/5.28      ! [A2: set_nat,C: nat > real] :
% 9.49/5.28        ( ( ( ( finite_finite_nat @ A2 )
% 9.49/5.28            & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.49/5.28         => ( ( groups6591440286371151544t_real
% 9.49/5.28              @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = ( C @ zero_zero_nat ) ) )
% 9.49/5.28        & ( ~ ( ( finite_finite_nat @ A2 )
% 9.49/5.28              & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.49/5.28         => ( ( groups6591440286371151544t_real
% 9.49/5.28              @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = zero_zero_real ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5577_sum__zero__power_H,axiom,
% 9.49/5.28      ! [A2: set_nat,C: nat > complex,D: nat > complex] :
% 9.49/5.28        ( ( ( ( finite_finite_nat @ A2 )
% 9.49/5.28            & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.49/5.28         => ( ( groups2073611262835488442omplex
% 9.49/5.28              @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 9.49/5.28        & ( ~ ( ( finite_finite_nat @ A2 )
% 9.49/5.28              & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.49/5.28         => ( ( groups2073611262835488442omplex
% 9.49/5.28              @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = zero_zero_complex ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5578_sum__zero__power_H,axiom,
% 9.49/5.28      ! [A2: set_nat,C: nat > rat,D: nat > rat] :
% 9.49/5.28        ( ( ( ( finite_finite_nat @ A2 )
% 9.49/5.28            & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.49/5.28         => ( ( groups2906978787729119204at_rat
% 9.49/5.28              @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 9.49/5.28        & ( ~ ( ( finite_finite_nat @ A2 )
% 9.49/5.28              & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.49/5.28         => ( ( groups2906978787729119204at_rat
% 9.49/5.28              @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = zero_zero_rat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5579_sum__zero__power_H,axiom,
% 9.49/5.28      ! [A2: set_nat,C: nat > real,D: nat > real] :
% 9.49/5.28        ( ( ( ( finite_finite_nat @ A2 )
% 9.49/5.28            & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.49/5.28         => ( ( groups6591440286371151544t_real
% 9.49/5.28              @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 9.49/5.28        & ( ~ ( ( finite_finite_nat @ A2 )
% 9.49/5.28              & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.49/5.28         => ( ( groups6591440286371151544t_real
% 9.49/5.28              @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = zero_zero_real ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5580_uminus__set__def,axiom,
% 9.49/5.28      ( uminus6524753893492686040at_nat
% 9.49/5.28      = ( ^ [A6: set_Pr1261947904930325089at_nat] :
% 9.49/5.28            ( collec3392354462482085612at_nat
% 9.49/5.28            @ ( uminus8676089048583255045_nat_o
% 9.49/5.28              @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5581_uminus__set__def,axiom,
% 9.49/5.28      ( uminus612125837232591019t_real
% 9.49/5.28      = ( ^ [A6: set_real] :
% 9.49/5.28            ( collect_real
% 9.49/5.28            @ ( uminus_uminus_real_o
% 9.49/5.28              @ ^ [X3: real] : ( member_real @ X3 @ A6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5582_uminus__set__def,axiom,
% 9.49/5.28      ( uminus3195874150345416415st_nat
% 9.49/5.28      = ( ^ [A6: set_list_nat] :
% 9.49/5.28            ( collect_list_nat
% 9.49/5.28            @ ( uminus5770388063884162150_nat_o
% 9.49/5.28              @ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5583_uminus__set__def,axiom,
% 9.49/5.28      ( uminus613421341184616069et_nat
% 9.49/5.28      = ( ^ [A6: set_set_nat] :
% 9.49/5.28            ( collect_set_nat
% 9.49/5.28            @ ( uminus6401447641752708672_nat_o
% 9.49/5.28              @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5584_uminus__set__def,axiom,
% 9.49/5.28      ( uminus5710092332889474511et_nat
% 9.49/5.28      = ( ^ [A6: set_nat] :
% 9.49/5.28            ( collect_nat
% 9.49/5.28            @ ( uminus_uminus_nat_o
% 9.49/5.28              @ ^ [X3: nat] : ( member_nat @ X3 @ A6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5585_uminus__set__def,axiom,
% 9.49/5.28      ( uminus1532241313380277803et_int
% 9.49/5.28      = ( ^ [A6: set_int] :
% 9.49/5.28            ( collect_int
% 9.49/5.28            @ ( uminus_uminus_int_o
% 9.49/5.28              @ ^ [X3: int] : ( member_int @ X3 @ A6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5586_Collect__neg__eq,axiom,
% 9.49/5.28      ! [P: real > $o] :
% 9.49/5.28        ( ( collect_real
% 9.49/5.28          @ ^ [X3: real] :
% 9.49/5.28              ~ ( P @ X3 ) )
% 9.49/5.28        = ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5587_Collect__neg__eq,axiom,
% 9.49/5.28      ! [P: list_nat > $o] :
% 9.49/5.28        ( ( collect_list_nat
% 9.49/5.28          @ ^ [X3: list_nat] :
% 9.49/5.28              ~ ( P @ X3 ) )
% 9.49/5.28        = ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5588_Collect__neg__eq,axiom,
% 9.49/5.28      ! [P: set_nat > $o] :
% 9.49/5.28        ( ( collect_set_nat
% 9.49/5.28          @ ^ [X3: set_nat] :
% 9.49/5.28              ~ ( P @ X3 ) )
% 9.49/5.28        = ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5589_Collect__neg__eq,axiom,
% 9.49/5.28      ! [P: nat > $o] :
% 9.49/5.28        ( ( collect_nat
% 9.49/5.28          @ ^ [X3: nat] :
% 9.49/5.28              ~ ( P @ X3 ) )
% 9.49/5.28        = ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5590_Collect__neg__eq,axiom,
% 9.49/5.28      ! [P: int > $o] :
% 9.49/5.28        ( ( collect_int
% 9.49/5.28          @ ^ [X3: int] :
% 9.49/5.28              ~ ( P @ X3 ) )
% 9.49/5.28        = ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5591_Compl__eq,axiom,
% 9.49/5.28      ( uminus6524753893492686040at_nat
% 9.49/5.28      = ( ^ [A6: set_Pr1261947904930325089at_nat] :
% 9.49/5.28            ( collec3392354462482085612at_nat
% 9.49/5.28            @ ^ [X3: product_prod_nat_nat] :
% 9.49/5.28                ~ ( member8440522571783428010at_nat @ X3 @ A6 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5592_Compl__eq,axiom,
% 9.49/5.28      ( uminus612125837232591019t_real
% 9.49/5.28      = ( ^ [A6: set_real] :
% 9.49/5.28            ( collect_real
% 9.49/5.28            @ ^ [X3: real] :
% 9.49/5.28                ~ ( member_real @ X3 @ A6 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5593_Compl__eq,axiom,
% 9.49/5.28      ( uminus3195874150345416415st_nat
% 9.49/5.28      = ( ^ [A6: set_list_nat] :
% 9.49/5.28            ( collect_list_nat
% 9.49/5.28            @ ^ [X3: list_nat] :
% 9.49/5.28                ~ ( member_list_nat @ X3 @ A6 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5594_Compl__eq,axiom,
% 9.49/5.28      ( uminus613421341184616069et_nat
% 9.49/5.28      = ( ^ [A6: set_set_nat] :
% 9.49/5.28            ( collect_set_nat
% 9.49/5.28            @ ^ [X3: set_nat] :
% 9.49/5.28                ~ ( member_set_nat @ X3 @ A6 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5595_Compl__eq,axiom,
% 9.49/5.28      ( uminus5710092332889474511et_nat
% 9.49/5.28      = ( ^ [A6: set_nat] :
% 9.49/5.28            ( collect_nat
% 9.49/5.28            @ ^ [X3: nat] :
% 9.49/5.28                ~ ( member_nat @ X3 @ A6 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5596_Compl__eq,axiom,
% 9.49/5.28      ( uminus1532241313380277803et_int
% 9.49/5.28      = ( ^ [A6: set_int] :
% 9.49/5.28            ( collect_int
% 9.49/5.28            @ ^ [X3: int] :
% 9.49/5.28                ~ ( member_int @ X3 @ A6 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5597_ComplD,axiom,
% 9.49/5.28      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A2 ) )
% 9.49/5.28       => ~ ( member8440522571783428010at_nat @ C @ A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5598_ComplD,axiom,
% 9.49/5.28      ! [C: real,A2: set_real] :
% 9.49/5.28        ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
% 9.49/5.28       => ~ ( member_real @ C @ A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5599_ComplD,axiom,
% 9.49/5.28      ! [C: set_nat,A2: set_set_nat] :
% 9.49/5.28        ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
% 9.49/5.28       => ~ ( member_set_nat @ C @ A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5600_ComplD,axiom,
% 9.49/5.28      ! [C: nat,A2: set_nat] :
% 9.49/5.28        ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
% 9.49/5.28       => ~ ( member_nat @ C @ A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5601_ComplD,axiom,
% 9.49/5.28      ! [C: int,A2: set_int] :
% 9.49/5.28        ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
% 9.49/5.28       => ~ ( member_int @ C @ A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5602_insert__Collect,axiom,
% 9.49/5.28      ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
% 9.49/5.28        ( ( insert8211810215607154385at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
% 9.49/5.28        = ( collec3392354462482085612at_nat
% 9.49/5.28          @ ^ [U2: product_prod_nat_nat] :
% 9.49/5.28              ( ( U2 != A )
% 9.49/5.28             => ( P @ U2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5603_insert__Collect,axiom,
% 9.49/5.28      ! [A: produc3843707927480180839at_nat,P: produc3843707927480180839at_nat > $o] :
% 9.49/5.28        ( ( insert9069300056098147895at_nat @ A @ ( collec6321179662152712658at_nat @ P ) )
% 9.49/5.28        = ( collec6321179662152712658at_nat
% 9.49/5.28          @ ^ [U2: produc3843707927480180839at_nat] :
% 9.49/5.28              ( ( U2 != A )
% 9.49/5.28             => ( P @ U2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5604_insert__Collect,axiom,
% 9.49/5.28      ! [A: real,P: real > $o] :
% 9.49/5.28        ( ( insert_real @ A @ ( collect_real @ P ) )
% 9.49/5.28        = ( collect_real
% 9.49/5.28          @ ^ [U2: real] :
% 9.49/5.28              ( ( U2 != A )
% 9.49/5.28             => ( P @ U2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5605_insert__Collect,axiom,
% 9.49/5.28      ! [A: list_nat,P: list_nat > $o] :
% 9.49/5.28        ( ( insert_list_nat @ A @ ( collect_list_nat @ P ) )
% 9.49/5.28        = ( collect_list_nat
% 9.49/5.28          @ ^ [U2: list_nat] :
% 9.49/5.28              ( ( U2 != A )
% 9.49/5.28             => ( P @ U2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5606_insert__Collect,axiom,
% 9.49/5.28      ! [A: set_nat,P: set_nat > $o] :
% 9.49/5.28        ( ( insert_set_nat @ A @ ( collect_set_nat @ P ) )
% 9.49/5.28        = ( collect_set_nat
% 9.49/5.28          @ ^ [U2: set_nat] :
% 9.49/5.28              ( ( U2 != A )
% 9.49/5.28             => ( P @ U2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5607_insert__Collect,axiom,
% 9.49/5.28      ! [A: nat,P: nat > $o] :
% 9.49/5.28        ( ( insert_nat @ A @ ( collect_nat @ P ) )
% 9.49/5.28        = ( collect_nat
% 9.49/5.28          @ ^ [U2: nat] :
% 9.49/5.28              ( ( U2 != A )
% 9.49/5.28             => ( P @ U2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5608_insert__Collect,axiom,
% 9.49/5.28      ! [A: int,P: int > $o] :
% 9.49/5.28        ( ( insert_int @ A @ ( collect_int @ P ) )
% 9.49/5.28        = ( collect_int
% 9.49/5.28          @ ^ [U2: int] :
% 9.49/5.28              ( ( U2 != A )
% 9.49/5.28             => ( P @ U2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5609_insert__compr,axiom,
% 9.49/5.28      ( insert9069300056098147895at_nat
% 9.49/5.28      = ( ^ [A4: produc3843707927480180839at_nat,B6: set_Pr4329608150637261639at_nat] :
% 9.49/5.28            ( collec6321179662152712658at_nat
% 9.49/5.28            @ ^ [X3: produc3843707927480180839at_nat] :
% 9.49/5.28                ( ( X3 = A4 )
% 9.49/5.28                | ( member8757157785044589968at_nat @ X3 @ B6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5610_insert__compr,axiom,
% 9.49/5.28      ( insert8211810215607154385at_nat
% 9.49/5.28      = ( ^ [A4: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
% 9.49/5.28            ( collec3392354462482085612at_nat
% 9.49/5.28            @ ^ [X3: product_prod_nat_nat] :
% 9.49/5.28                ( ( X3 = A4 )
% 9.49/5.28                | ( member8440522571783428010at_nat @ X3 @ B6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5611_insert__compr,axiom,
% 9.49/5.28      ( insert_real
% 9.49/5.28      = ( ^ [A4: real,B6: set_real] :
% 9.49/5.28            ( collect_real
% 9.49/5.28            @ ^ [X3: real] :
% 9.49/5.28                ( ( X3 = A4 )
% 9.49/5.28                | ( member_real @ X3 @ B6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5612_insert__compr,axiom,
% 9.49/5.28      ( insert_list_nat
% 9.49/5.28      = ( ^ [A4: list_nat,B6: set_list_nat] :
% 9.49/5.28            ( collect_list_nat
% 9.49/5.28            @ ^ [X3: list_nat] :
% 9.49/5.28                ( ( X3 = A4 )
% 9.49/5.28                | ( member_list_nat @ X3 @ B6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5613_insert__compr,axiom,
% 9.49/5.28      ( insert_set_nat
% 9.49/5.28      = ( ^ [A4: set_nat,B6: set_set_nat] :
% 9.49/5.28            ( collect_set_nat
% 9.49/5.28            @ ^ [X3: set_nat] :
% 9.49/5.28                ( ( X3 = A4 )
% 9.49/5.28                | ( member_set_nat @ X3 @ B6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5614_insert__compr,axiom,
% 9.49/5.28      ( insert_nat
% 9.49/5.28      = ( ^ [A4: nat,B6: set_nat] :
% 9.49/5.28            ( collect_nat
% 9.49/5.28            @ ^ [X3: nat] :
% 9.49/5.28                ( ( X3 = A4 )
% 9.49/5.28                | ( member_nat @ X3 @ B6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5615_insert__compr,axiom,
% 9.49/5.28      ( insert_int
% 9.49/5.28      = ( ^ [A4: int,B6: set_int] :
% 9.49/5.28            ( collect_int
% 9.49/5.28            @ ^ [X3: int] :
% 9.49/5.28                ( ( X3 = A4 )
% 9.49/5.28                | ( member_int @ X3 @ B6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5616_insertE,axiom,
% 9.49/5.28      ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ A2 ) )
% 9.49/5.28       => ( ( A != B )
% 9.49/5.28         => ( member8757157785044589968at_nat @ A @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5617_insertE,axiom,
% 9.49/5.28      ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A2 ) )
% 9.49/5.28       => ( ( A != B )
% 9.49/5.28         => ( member8440522571783428010at_nat @ A @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5618_insertE,axiom,
% 9.49/5.28      ! [A: real,B: real,A2: set_real] :
% 9.49/5.28        ( ( member_real @ A @ ( insert_real @ B @ A2 ) )
% 9.49/5.28       => ( ( A != B )
% 9.49/5.28         => ( member_real @ A @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5619_insertE,axiom,
% 9.49/5.28      ! [A: set_nat,B: set_nat,A2: set_set_nat] :
% 9.49/5.28        ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
% 9.49/5.28       => ( ( A != B )
% 9.49/5.28         => ( member_set_nat @ A @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5620_insertE,axiom,
% 9.49/5.28      ! [A: nat,B: nat,A2: set_nat] :
% 9.49/5.28        ( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
% 9.49/5.28       => ( ( A != B )
% 9.49/5.28         => ( member_nat @ A @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5621_insertE,axiom,
% 9.49/5.28      ! [A: int,B: int,A2: set_int] :
% 9.49/5.28        ( ( member_int @ A @ ( insert_int @ B @ A2 ) )
% 9.49/5.28       => ( ( A != B )
% 9.49/5.28         => ( member_int @ A @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5622_insertI1,axiom,
% 9.49/5.28      ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ A @ B2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5623_insertI1,axiom,
% 9.49/5.28      ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ B2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5624_insertI1,axiom,
% 9.49/5.28      ! [A: real,B2: set_real] : ( member_real @ A @ ( insert_real @ A @ B2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5625_insertI1,axiom,
% 9.49/5.28      ! [A: set_nat,B2: set_set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ B2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5626_insertI1,axiom,
% 9.49/5.28      ! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5627_insertI1,axiom,
% 9.49/5.28      ! [A: int,B2: set_int] : ( member_int @ A @ ( insert_int @ A @ B2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5628_insertI2,axiom,
% 9.49/5.28      ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
% 9.49/5.28        ( ( member8757157785044589968at_nat @ A @ B2 )
% 9.49/5.28       => ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5629_insertI2,axiom,
% 9.49/5.28      ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
% 9.49/5.28        ( ( member8440522571783428010at_nat @ A @ B2 )
% 9.49/5.28       => ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5630_insertI2,axiom,
% 9.49/5.28      ! [A: real,B2: set_real,B: real] :
% 9.49/5.28        ( ( member_real @ A @ B2 )
% 9.49/5.28       => ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5631_insertI2,axiom,
% 9.49/5.28      ! [A: set_nat,B2: set_set_nat,B: set_nat] :
% 9.49/5.28        ( ( member_set_nat @ A @ B2 )
% 9.49/5.28       => ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5632_insertI2,axiom,
% 9.49/5.28      ! [A: nat,B2: set_nat,B: nat] :
% 9.49/5.28        ( ( member_nat @ A @ B2 )
% 9.49/5.28       => ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5633_insertI2,axiom,
% 9.49/5.28      ! [A: int,B2: set_int,B: int] :
% 9.49/5.28        ( ( member_int @ A @ B2 )
% 9.49/5.28       => ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5634_Set_Oset__insert,axiom,
% 9.49/5.28      ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ( member8757157785044589968at_nat @ X @ A2 )
% 9.49/5.28       => ~ ! [B8: set_Pr4329608150637261639at_nat] :
% 9.49/5.28              ( ( A2
% 9.49/5.28                = ( insert9069300056098147895at_nat @ X @ B8 ) )
% 9.49/5.28             => ( member8757157785044589968at_nat @ X @ B8 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5635_Set_Oset__insert,axiom,
% 9.49/5.28      ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ( member8440522571783428010at_nat @ X @ A2 )
% 9.49/5.28       => ~ ! [B8: set_Pr1261947904930325089at_nat] :
% 9.49/5.28              ( ( A2
% 9.49/5.28                = ( insert8211810215607154385at_nat @ X @ B8 ) )
% 9.49/5.28             => ( member8440522571783428010at_nat @ X @ B8 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5636_Set_Oset__insert,axiom,
% 9.49/5.28      ! [X: real,A2: set_real] :
% 9.49/5.28        ( ( member_real @ X @ A2 )
% 9.49/5.28       => ~ ! [B8: set_real] :
% 9.49/5.28              ( ( A2
% 9.49/5.28                = ( insert_real @ X @ B8 ) )
% 9.49/5.28             => ( member_real @ X @ B8 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5637_Set_Oset__insert,axiom,
% 9.49/5.28      ! [X: set_nat,A2: set_set_nat] :
% 9.49/5.28        ( ( member_set_nat @ X @ A2 )
% 9.49/5.28       => ~ ! [B8: set_set_nat] :
% 9.49/5.28              ( ( A2
% 9.49/5.28                = ( insert_set_nat @ X @ B8 ) )
% 9.49/5.28             => ( member_set_nat @ X @ B8 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5638_Set_Oset__insert,axiom,
% 9.49/5.28      ! [X: nat,A2: set_nat] :
% 9.49/5.28        ( ( member_nat @ X @ A2 )
% 9.49/5.28       => ~ ! [B8: set_nat] :
% 9.49/5.28              ( ( A2
% 9.49/5.28                = ( insert_nat @ X @ B8 ) )
% 9.49/5.28             => ( member_nat @ X @ B8 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5639_Set_Oset__insert,axiom,
% 9.49/5.28      ! [X: int,A2: set_int] :
% 9.49/5.28        ( ( member_int @ X @ A2 )
% 9.49/5.28       => ~ ! [B8: set_int] :
% 9.49/5.28              ( ( A2
% 9.49/5.28                = ( insert_int @ X @ B8 ) )
% 9.49/5.28             => ( member_int @ X @ B8 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5640_insert__ident,axiom,
% 9.49/5.28      ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ~ ( member8757157785044589968at_nat @ X @ A2 )
% 9.49/5.28       => ( ~ ( member8757157785044589968at_nat @ X @ B2 )
% 9.49/5.28         => ( ( ( insert9069300056098147895at_nat @ X @ A2 )
% 9.49/5.28              = ( insert9069300056098147895at_nat @ X @ B2 ) )
% 9.49/5.28            = ( A2 = B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5641_insert__ident,axiom,
% 9.49/5.28      ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ~ ( member8440522571783428010at_nat @ X @ A2 )
% 9.49/5.28       => ( ~ ( member8440522571783428010at_nat @ X @ B2 )
% 9.49/5.28         => ( ( ( insert8211810215607154385at_nat @ X @ A2 )
% 9.49/5.28              = ( insert8211810215607154385at_nat @ X @ B2 ) )
% 9.49/5.28            = ( A2 = B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5642_insert__ident,axiom,
% 9.49/5.28      ! [X: real,A2: set_real,B2: set_real] :
% 9.49/5.28        ( ~ ( member_real @ X @ A2 )
% 9.49/5.28       => ( ~ ( member_real @ X @ B2 )
% 9.49/5.28         => ( ( ( insert_real @ X @ A2 )
% 9.49/5.28              = ( insert_real @ X @ B2 ) )
% 9.49/5.28            = ( A2 = B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5643_insert__ident,axiom,
% 9.49/5.28      ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.49/5.28        ( ~ ( member_set_nat @ X @ A2 )
% 9.49/5.28       => ( ~ ( member_set_nat @ X @ B2 )
% 9.49/5.28         => ( ( ( insert_set_nat @ X @ A2 )
% 9.49/5.28              = ( insert_set_nat @ X @ B2 ) )
% 9.49/5.28            = ( A2 = B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5644_insert__ident,axiom,
% 9.49/5.28      ! [X: nat,A2: set_nat,B2: set_nat] :
% 9.49/5.28        ( ~ ( member_nat @ X @ A2 )
% 9.49/5.28       => ( ~ ( member_nat @ X @ B2 )
% 9.49/5.28         => ( ( ( insert_nat @ X @ A2 )
% 9.49/5.28              = ( insert_nat @ X @ B2 ) )
% 9.49/5.28            = ( A2 = B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5645_insert__ident,axiom,
% 9.49/5.28      ! [X: int,A2: set_int,B2: set_int] :
% 9.49/5.28        ( ~ ( member_int @ X @ A2 )
% 9.49/5.28       => ( ~ ( member_int @ X @ B2 )
% 9.49/5.28         => ( ( ( insert_int @ X @ A2 )
% 9.49/5.28              = ( insert_int @ X @ B2 ) )
% 9.49/5.28            = ( A2 = B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5646_insert__absorb,axiom,
% 9.49/5.28      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ( member8757157785044589968at_nat @ A @ A2 )
% 9.49/5.28       => ( ( insert9069300056098147895at_nat @ A @ A2 )
% 9.49/5.28          = A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5647_insert__absorb,axiom,
% 9.49/5.28      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ( member8440522571783428010at_nat @ A @ A2 )
% 9.49/5.28       => ( ( insert8211810215607154385at_nat @ A @ A2 )
% 9.49/5.28          = A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5648_insert__absorb,axiom,
% 9.49/5.28      ! [A: real,A2: set_real] :
% 9.49/5.28        ( ( member_real @ A @ A2 )
% 9.49/5.28       => ( ( insert_real @ A @ A2 )
% 9.49/5.28          = A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5649_insert__absorb,axiom,
% 9.49/5.28      ! [A: set_nat,A2: set_set_nat] :
% 9.49/5.28        ( ( member_set_nat @ A @ A2 )
% 9.49/5.28       => ( ( insert_set_nat @ A @ A2 )
% 9.49/5.28          = A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5650_insert__absorb,axiom,
% 9.49/5.28      ! [A: nat,A2: set_nat] :
% 9.49/5.28        ( ( member_nat @ A @ A2 )
% 9.49/5.28       => ( ( insert_nat @ A @ A2 )
% 9.49/5.28          = A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5651_insert__absorb,axiom,
% 9.49/5.28      ! [A: int,A2: set_int] :
% 9.49/5.28        ( ( member_int @ A @ A2 )
% 9.49/5.28       => ( ( insert_int @ A @ A2 )
% 9.49/5.28          = A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5652_insert__eq__iff,axiom,
% 9.49/5.28      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ~ ( member8757157785044589968at_nat @ A @ A2 )
% 9.49/5.28       => ( ~ ( member8757157785044589968at_nat @ B @ B2 )
% 9.49/5.28         => ( ( ( insert9069300056098147895at_nat @ A @ A2 )
% 9.49/5.28              = ( insert9069300056098147895at_nat @ B @ B2 ) )
% 9.49/5.28            = ( ( ( A = B )
% 9.49/5.28               => ( A2 = B2 ) )
% 9.49/5.28              & ( ( A != B )
% 9.49/5.28               => ? [C5: set_Pr4329608150637261639at_nat] :
% 9.49/5.28                    ( ( A2
% 9.49/5.28                      = ( insert9069300056098147895at_nat @ B @ C5 ) )
% 9.49/5.28                    & ~ ( member8757157785044589968at_nat @ B @ C5 )
% 9.49/5.28                    & ( B2
% 9.49/5.28                      = ( insert9069300056098147895at_nat @ A @ C5 ) )
% 9.49/5.28                    & ~ ( member8757157785044589968at_nat @ A @ C5 ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5653_insert__eq__iff,axiom,
% 9.49/5.28      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ~ ( member8440522571783428010at_nat @ A @ A2 )
% 9.49/5.28       => ( ~ ( member8440522571783428010at_nat @ B @ B2 )
% 9.49/5.28         => ( ( ( insert8211810215607154385at_nat @ A @ A2 )
% 9.49/5.28              = ( insert8211810215607154385at_nat @ B @ B2 ) )
% 9.49/5.28            = ( ( ( A = B )
% 9.49/5.28               => ( A2 = B2 ) )
% 9.49/5.28              & ( ( A != B )
% 9.49/5.28               => ? [C5: set_Pr1261947904930325089at_nat] :
% 9.49/5.28                    ( ( A2
% 9.49/5.28                      = ( insert8211810215607154385at_nat @ B @ C5 ) )
% 9.49/5.28                    & ~ ( member8440522571783428010at_nat @ B @ C5 )
% 9.49/5.28                    & ( B2
% 9.49/5.28                      = ( insert8211810215607154385at_nat @ A @ C5 ) )
% 9.49/5.28                    & ~ ( member8440522571783428010at_nat @ A @ C5 ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5654_insert__eq__iff,axiom,
% 9.49/5.28      ! [A: real,A2: set_real,B: real,B2: set_real] :
% 9.49/5.28        ( ~ ( member_real @ A @ A2 )
% 9.49/5.28       => ( ~ ( member_real @ B @ B2 )
% 9.49/5.28         => ( ( ( insert_real @ A @ A2 )
% 9.49/5.28              = ( insert_real @ B @ B2 ) )
% 9.49/5.28            = ( ( ( A = B )
% 9.49/5.28               => ( A2 = B2 ) )
% 9.49/5.28              & ( ( A != B )
% 9.49/5.28               => ? [C5: set_real] :
% 9.49/5.28                    ( ( A2
% 9.49/5.28                      = ( insert_real @ B @ C5 ) )
% 9.49/5.28                    & ~ ( member_real @ B @ C5 )
% 9.49/5.28                    & ( B2
% 9.49/5.28                      = ( insert_real @ A @ C5 ) )
% 9.49/5.28                    & ~ ( member_real @ A @ C5 ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5655_insert__eq__iff,axiom,
% 9.49/5.28      ! [A: set_nat,A2: set_set_nat,B: set_nat,B2: set_set_nat] :
% 9.49/5.28        ( ~ ( member_set_nat @ A @ A2 )
% 9.49/5.28       => ( ~ ( member_set_nat @ B @ B2 )
% 9.49/5.28         => ( ( ( insert_set_nat @ A @ A2 )
% 9.49/5.28              = ( insert_set_nat @ B @ B2 ) )
% 9.49/5.28            = ( ( ( A = B )
% 9.49/5.28               => ( A2 = B2 ) )
% 9.49/5.28              & ( ( A != B )
% 9.49/5.28               => ? [C5: set_set_nat] :
% 9.49/5.28                    ( ( A2
% 9.49/5.28                      = ( insert_set_nat @ B @ C5 ) )
% 9.49/5.28                    & ~ ( member_set_nat @ B @ C5 )
% 9.49/5.28                    & ( B2
% 9.49/5.28                      = ( insert_set_nat @ A @ C5 ) )
% 9.49/5.28                    & ~ ( member_set_nat @ A @ C5 ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5656_insert__eq__iff,axiom,
% 9.49/5.28      ! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
% 9.49/5.28        ( ~ ( member_nat @ A @ A2 )
% 9.49/5.28       => ( ~ ( member_nat @ B @ B2 )
% 9.49/5.28         => ( ( ( insert_nat @ A @ A2 )
% 9.49/5.28              = ( insert_nat @ B @ B2 ) )
% 9.49/5.28            = ( ( ( A = B )
% 9.49/5.28               => ( A2 = B2 ) )
% 9.49/5.28              & ( ( A != B )
% 9.49/5.28               => ? [C5: set_nat] :
% 9.49/5.28                    ( ( A2
% 9.49/5.28                      = ( insert_nat @ B @ C5 ) )
% 9.49/5.28                    & ~ ( member_nat @ B @ C5 )
% 9.49/5.28                    & ( B2
% 9.49/5.28                      = ( insert_nat @ A @ C5 ) )
% 9.49/5.28                    & ~ ( member_nat @ A @ C5 ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5657_insert__eq__iff,axiom,
% 9.49/5.28      ! [A: int,A2: set_int,B: int,B2: set_int] :
% 9.49/5.28        ( ~ ( member_int @ A @ A2 )
% 9.49/5.28       => ( ~ ( member_int @ B @ B2 )
% 9.49/5.28         => ( ( ( insert_int @ A @ A2 )
% 9.49/5.28              = ( insert_int @ B @ B2 ) )
% 9.49/5.28            = ( ( ( A = B )
% 9.49/5.28               => ( A2 = B2 ) )
% 9.49/5.28              & ( ( A != B )
% 9.49/5.28               => ? [C5: set_int] :
% 9.49/5.28                    ( ( A2
% 9.49/5.28                      = ( insert_int @ B @ C5 ) )
% 9.49/5.28                    & ~ ( member_int @ B @ C5 )
% 9.49/5.28                    & ( B2
% 9.49/5.28                      = ( insert_int @ A @ C5 ) )
% 9.49/5.28                    & ~ ( member_int @ A @ C5 ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5658_insert__commute,axiom,
% 9.49/5.28      ! [X: nat,Y: nat,A2: set_nat] :
% 9.49/5.28        ( ( insert_nat @ X @ ( insert_nat @ Y @ A2 ) )
% 9.49/5.28        = ( insert_nat @ Y @ ( insert_nat @ X @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5659_insert__commute,axiom,
% 9.49/5.28      ! [X: int,Y: int,A2: set_int] :
% 9.49/5.28        ( ( insert_int @ X @ ( insert_int @ Y @ A2 ) )
% 9.49/5.28        = ( insert_int @ Y @ ( insert_int @ X @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5660_insert__commute,axiom,
% 9.49/5.28      ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ( insert8211810215607154385at_nat @ X @ ( insert8211810215607154385at_nat @ Y @ A2 ) )
% 9.49/5.28        = ( insert8211810215607154385at_nat @ Y @ ( insert8211810215607154385at_nat @ X @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5661_insert__commute,axiom,
% 9.49/5.28      ! [X: real,Y: real,A2: set_real] :
% 9.49/5.28        ( ( insert_real @ X @ ( insert_real @ Y @ A2 ) )
% 9.49/5.28        = ( insert_real @ Y @ ( insert_real @ X @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5662_insert__commute,axiom,
% 9.49/5.28      ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ( insert9069300056098147895at_nat @ X @ ( insert9069300056098147895at_nat @ Y @ A2 ) )
% 9.49/5.28        = ( insert9069300056098147895at_nat @ Y @ ( insert9069300056098147895at_nat @ X @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5663_mk__disjoint__insert,axiom,
% 9.49/5.28      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ( member8757157785044589968at_nat @ A @ A2 )
% 9.49/5.28       => ? [B8: set_Pr4329608150637261639at_nat] :
% 9.49/5.28            ( ( A2
% 9.49/5.28              = ( insert9069300056098147895at_nat @ A @ B8 ) )
% 9.49/5.28            & ~ ( member8757157785044589968at_nat @ A @ B8 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5664_mk__disjoint__insert,axiom,
% 9.49/5.28      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ( member8440522571783428010at_nat @ A @ A2 )
% 9.49/5.28       => ? [B8: set_Pr1261947904930325089at_nat] :
% 9.49/5.28            ( ( A2
% 9.49/5.28              = ( insert8211810215607154385at_nat @ A @ B8 ) )
% 9.49/5.28            & ~ ( member8440522571783428010at_nat @ A @ B8 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5665_mk__disjoint__insert,axiom,
% 9.49/5.28      ! [A: real,A2: set_real] :
% 9.49/5.28        ( ( member_real @ A @ A2 )
% 9.49/5.28       => ? [B8: set_real] :
% 9.49/5.28            ( ( A2
% 9.49/5.28              = ( insert_real @ A @ B8 ) )
% 9.49/5.28            & ~ ( member_real @ A @ B8 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5666_mk__disjoint__insert,axiom,
% 9.49/5.28      ! [A: set_nat,A2: set_set_nat] :
% 9.49/5.28        ( ( member_set_nat @ A @ A2 )
% 9.49/5.28       => ? [B8: set_set_nat] :
% 9.49/5.28            ( ( A2
% 9.49/5.28              = ( insert_set_nat @ A @ B8 ) )
% 9.49/5.28            & ~ ( member_set_nat @ A @ B8 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5667_mk__disjoint__insert,axiom,
% 9.49/5.28      ! [A: nat,A2: set_nat] :
% 9.49/5.28        ( ( member_nat @ A @ A2 )
% 9.49/5.28       => ? [B8: set_nat] :
% 9.49/5.28            ( ( A2
% 9.49/5.28              = ( insert_nat @ A @ B8 ) )
% 9.49/5.28            & ~ ( member_nat @ A @ B8 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5668_mk__disjoint__insert,axiom,
% 9.49/5.28      ! [A: int,A2: set_int] :
% 9.49/5.28        ( ( member_int @ A @ A2 )
% 9.49/5.28       => ? [B8: set_int] :
% 9.49/5.28            ( ( A2
% 9.49/5.28              = ( insert_int @ A @ B8 ) )
% 9.49/5.28            & ~ ( member_int @ A @ B8 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5669_sum_Oinsert__if,axiom,
% 9.49/5.28      ! [A2: set_real,X: real,G: real > real] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( ( member_real @ X @ A2 )
% 9.49/5.28           => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
% 9.49/5.28              = ( groups8097168146408367636l_real @ G @ A2 ) ) )
% 9.49/5.28          & ( ~ ( member_real @ X @ A2 )
% 9.49/5.28           => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
% 9.49/5.28              = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5670_sum_Oinsert__if,axiom,
% 9.49/5.28      ! [A2: set_int,X: int,G: int > real] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( ( member_int @ X @ A2 )
% 9.49/5.28           => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
% 9.49/5.28              = ( groups8778361861064173332t_real @ G @ A2 ) ) )
% 9.49/5.28          & ( ~ ( member_int @ X @ A2 )
% 9.49/5.28           => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
% 9.49/5.28              = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5671_sum_Oinsert__if,axiom,
% 9.49/5.28      ! [A2: set_complex,X: complex,G: complex > real] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( ( member_complex @ X @ A2 )
% 9.49/5.28           => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
% 9.49/5.28              = ( groups5808333547571424918x_real @ G @ A2 ) ) )
% 9.49/5.28          & ( ~ ( member_complex @ X @ A2 )
% 9.49/5.28           => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
% 9.49/5.28              = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5672_sum_Oinsert__if,axiom,
% 9.49/5.28      ! [A2: set_real,X: real,G: real > rat] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( ( member_real @ X @ A2 )
% 9.49/5.28           => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
% 9.49/5.28              = ( groups1300246762558778688al_rat @ G @ A2 ) ) )
% 9.49/5.28          & ( ~ ( member_real @ X @ A2 )
% 9.49/5.28           => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
% 9.49/5.28              = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5673_sum_Oinsert__if,axiom,
% 9.49/5.28      ! [A2: set_nat,X: nat,G: nat > rat] :
% 9.49/5.28        ( ( finite_finite_nat @ A2 )
% 9.49/5.28       => ( ( ( member_nat @ X @ A2 )
% 9.49/5.28           => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
% 9.49/5.28              = ( groups2906978787729119204at_rat @ G @ A2 ) ) )
% 9.49/5.28          & ( ~ ( member_nat @ X @ A2 )
% 9.49/5.28           => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
% 9.49/5.28              = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5674_sum_Oinsert__if,axiom,
% 9.49/5.28      ! [A2: set_int,X: int,G: int > rat] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( ( member_int @ X @ A2 )
% 9.49/5.28           => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
% 9.49/5.28              = ( groups3906332499630173760nt_rat @ G @ A2 ) ) )
% 9.49/5.28          & ( ~ ( member_int @ X @ A2 )
% 9.49/5.28           => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
% 9.49/5.28              = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5675_sum_Oinsert__if,axiom,
% 9.49/5.28      ! [A2: set_complex,X: complex,G: complex > rat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( ( member_complex @ X @ A2 )
% 9.49/5.28           => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
% 9.49/5.28              = ( groups5058264527183730370ex_rat @ G @ A2 ) ) )
% 9.49/5.28          & ( ~ ( member_complex @ X @ A2 )
% 9.49/5.28           => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
% 9.49/5.28              = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5676_sum_Oinsert__if,axiom,
% 9.49/5.28      ! [A2: set_real,X: real,G: real > nat] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( ( member_real @ X @ A2 )
% 9.49/5.28           => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A2 ) )
% 9.49/5.28              = ( groups1935376822645274424al_nat @ G @ A2 ) ) )
% 9.49/5.28          & ( ~ ( member_real @ X @ A2 )
% 9.49/5.28           => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A2 ) )
% 9.49/5.28              = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5677_sum_Oinsert__if,axiom,
% 9.49/5.28      ! [A2: set_int,X: int,G: int > nat] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( ( member_int @ X @ A2 )
% 9.49/5.28           => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A2 ) )
% 9.49/5.28              = ( groups4541462559716669496nt_nat @ G @ A2 ) ) )
% 9.49/5.28          & ( ~ ( member_int @ X @ A2 )
% 9.49/5.28           => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A2 ) )
% 9.49/5.28              = ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5678_sum_Oinsert__if,axiom,
% 9.49/5.28      ! [A2: set_complex,X: complex,G: complex > nat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( ( member_complex @ X @ A2 )
% 9.49/5.28           => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
% 9.49/5.28              = ( groups5693394587270226106ex_nat @ G @ A2 ) ) )
% 9.49/5.28          & ( ~ ( member_complex @ X @ A2 )
% 9.49/5.28           => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
% 9.49/5.28              = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5679_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.49/5.28      ! [G: real > complex,A2: set_real] :
% 9.49/5.28        ( ( ( groups5754745047067104278omplex @ G @ A2 )
% 9.49/5.28         != zero_zero_complex )
% 9.49/5.28       => ~ ! [A3: real] :
% 9.49/5.28              ( ( member_real @ A3 @ A2 )
% 9.49/5.28             => ( ( G @ A3 )
% 9.49/5.28                = zero_zero_complex ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5680_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.49/5.28      ! [G: nat > complex,A2: set_nat] :
% 9.49/5.28        ( ( ( groups2073611262835488442omplex @ G @ A2 )
% 9.49/5.28         != zero_zero_complex )
% 9.49/5.28       => ~ ! [A3: nat] :
% 9.49/5.28              ( ( member_nat @ A3 @ A2 )
% 9.49/5.28             => ( ( G @ A3 )
% 9.49/5.28                = zero_zero_complex ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5681_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.49/5.28      ! [G: int > complex,A2: set_int] :
% 9.49/5.28        ( ( ( groups3049146728041665814omplex @ G @ A2 )
% 9.49/5.28         != zero_zero_complex )
% 9.49/5.28       => ~ ! [A3: int] :
% 9.49/5.28              ( ( member_int @ A3 @ A2 )
% 9.49/5.28             => ( ( G @ A3 )
% 9.49/5.28                = zero_zero_complex ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5682_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.49/5.28      ! [G: real > real,A2: set_real] :
% 9.49/5.28        ( ( ( groups8097168146408367636l_real @ G @ A2 )
% 9.49/5.28         != zero_zero_real )
% 9.49/5.28       => ~ ! [A3: real] :
% 9.49/5.28              ( ( member_real @ A3 @ A2 )
% 9.49/5.28             => ( ( G @ A3 )
% 9.49/5.28                = zero_zero_real ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5683_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.49/5.28      ! [G: int > real,A2: set_int] :
% 9.49/5.28        ( ( ( groups8778361861064173332t_real @ G @ A2 )
% 9.49/5.28         != zero_zero_real )
% 9.49/5.28       => ~ ! [A3: int] :
% 9.49/5.28              ( ( member_int @ A3 @ A2 )
% 9.49/5.28             => ( ( G @ A3 )
% 9.49/5.28                = zero_zero_real ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5684_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.49/5.28      ! [G: real > rat,A2: set_real] :
% 9.49/5.28        ( ( ( groups1300246762558778688al_rat @ G @ A2 )
% 9.49/5.28         != zero_zero_rat )
% 9.49/5.28       => ~ ! [A3: real] :
% 9.49/5.28              ( ( member_real @ A3 @ A2 )
% 9.49/5.28             => ( ( G @ A3 )
% 9.49/5.28                = zero_zero_rat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5685_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.49/5.28      ! [G: nat > rat,A2: set_nat] :
% 9.49/5.28        ( ( ( groups2906978787729119204at_rat @ G @ A2 )
% 9.49/5.28         != zero_zero_rat )
% 9.49/5.28       => ~ ! [A3: nat] :
% 9.49/5.28              ( ( member_nat @ A3 @ A2 )
% 9.49/5.28             => ( ( G @ A3 )
% 9.49/5.28                = zero_zero_rat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5686_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.49/5.28      ! [G: int > rat,A2: set_int] :
% 9.49/5.28        ( ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 9.49/5.28         != zero_zero_rat )
% 9.49/5.28       => ~ ! [A3: int] :
% 9.49/5.28              ( ( member_int @ A3 @ A2 )
% 9.49/5.28             => ( ( G @ A3 )
% 9.49/5.28                = zero_zero_rat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5687_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.49/5.28      ! [G: real > nat,A2: set_real] :
% 9.49/5.28        ( ( ( groups1935376822645274424al_nat @ G @ A2 )
% 9.49/5.28         != zero_zero_nat )
% 9.49/5.28       => ~ ! [A3: real] :
% 9.49/5.28              ( ( member_real @ A3 @ A2 )
% 9.49/5.28             => ( ( G @ A3 )
% 9.49/5.28                = zero_zero_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5688_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.49/5.28      ! [G: int > nat,A2: set_int] :
% 9.49/5.28        ( ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 9.49/5.28         != zero_zero_nat )
% 9.49/5.28       => ~ ! [A3: int] :
% 9.49/5.28              ( ( member_int @ A3 @ A2 )
% 9.49/5.28             => ( ( G @ A3 )
% 9.49/5.28                = zero_zero_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5689_sum_Oneutral,axiom,
% 9.49/5.28      ! [A2: set_nat,G: nat > nat] :
% 9.49/5.28        ( ! [X4: nat] :
% 9.49/5.28            ( ( member_nat @ X4 @ A2 )
% 9.49/5.28           => ( ( G @ X4 )
% 9.49/5.28              = zero_zero_nat ) )
% 9.49/5.28       => ( ( groups3542108847815614940at_nat @ G @ A2 )
% 9.49/5.28          = zero_zero_nat ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5690_sum_Oneutral,axiom,
% 9.49/5.28      ! [A2: set_complex,G: complex > complex] :
% 9.49/5.28        ( ! [X4: complex] :
% 9.49/5.28            ( ( member_complex @ X4 @ A2 )
% 9.49/5.28           => ( ( G @ X4 )
% 9.49/5.28              = zero_zero_complex ) )
% 9.49/5.28       => ( ( groups7754918857620584856omplex @ G @ A2 )
% 9.49/5.28          = zero_zero_complex ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5691_sum_Oneutral,axiom,
% 9.49/5.28      ! [A2: set_nat,G: nat > real] :
% 9.49/5.28        ( ! [X4: nat] :
% 9.49/5.28            ( ( member_nat @ X4 @ A2 )
% 9.49/5.28           => ( ( G @ X4 )
% 9.49/5.28              = zero_zero_real ) )
% 9.49/5.28       => ( ( groups6591440286371151544t_real @ G @ A2 )
% 9.49/5.28          = zero_zero_real ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5692_singleton__inject,axiom,
% 9.49/5.28      ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
% 9.49/5.28        ( ( ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat )
% 9.49/5.28          = ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
% 9.49/5.28       => ( A = B ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5693_singleton__inject,axiom,
% 9.49/5.28      ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
% 9.49/5.28        ( ( ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat )
% 9.49/5.28          = ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
% 9.49/5.28       => ( A = B ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5694_singleton__inject,axiom,
% 9.49/5.28      ! [A: real,B: real] :
% 9.49/5.28        ( ( ( insert_real @ A @ bot_bot_set_real )
% 9.49/5.28          = ( insert_real @ B @ bot_bot_set_real ) )
% 9.49/5.28       => ( A = B ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5695_singleton__inject,axiom,
% 9.49/5.28      ! [A: nat,B: nat] :
% 9.49/5.28        ( ( ( insert_nat @ A @ bot_bot_set_nat )
% 9.49/5.28          = ( insert_nat @ B @ bot_bot_set_nat ) )
% 9.49/5.28       => ( A = B ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5696_singleton__inject,axiom,
% 9.49/5.28      ! [A: int,B: int] :
% 9.49/5.28        ( ( ( insert_int @ A @ bot_bot_set_int )
% 9.49/5.28          = ( insert_int @ B @ bot_bot_set_int ) )
% 9.49/5.28       => ( A = B ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5697_insert__not__empty,axiom,
% 9.49/5.28      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ( insert9069300056098147895at_nat @ A @ A2 )
% 9.49/5.28       != bot_bo228742789529271731at_nat ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5698_insert__not__empty,axiom,
% 9.49/5.28      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ( insert8211810215607154385at_nat @ A @ A2 )
% 9.49/5.28       != bot_bo2099793752762293965at_nat ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5699_insert__not__empty,axiom,
% 9.49/5.28      ! [A: real,A2: set_real] :
% 9.49/5.28        ( ( insert_real @ A @ A2 )
% 9.49/5.28       != bot_bot_set_real ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5700_insert__not__empty,axiom,
% 9.49/5.28      ! [A: nat,A2: set_nat] :
% 9.49/5.28        ( ( insert_nat @ A @ A2 )
% 9.49/5.28       != bot_bot_set_nat ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5701_insert__not__empty,axiom,
% 9.49/5.28      ! [A: int,A2: set_int] :
% 9.49/5.28        ( ( insert_int @ A @ A2 )
% 9.49/5.28       != bot_bot_set_int ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5702_doubleton__eq__iff,axiom,
% 9.49/5.28      ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,C: produc3843707927480180839at_nat,D: produc3843707927480180839at_nat] :
% 9.49/5.28        ( ( ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
% 9.49/5.28          = ( insert9069300056098147895at_nat @ C @ ( insert9069300056098147895at_nat @ D @ bot_bo228742789529271731at_nat ) ) )
% 9.49/5.28        = ( ( ( A = C )
% 9.49/5.28            & ( B = D ) )
% 9.49/5.28          | ( ( A = D )
% 9.49/5.28            & ( B = C ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5703_doubleton__eq__iff,axiom,
% 9.49/5.28      ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat,D: product_prod_nat_nat] :
% 9.49/5.28        ( ( ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
% 9.49/5.28          = ( insert8211810215607154385at_nat @ C @ ( insert8211810215607154385at_nat @ D @ bot_bo2099793752762293965at_nat ) ) )
% 9.49/5.28        = ( ( ( A = C )
% 9.49/5.28            & ( B = D ) )
% 9.49/5.28          | ( ( A = D )
% 9.49/5.28            & ( B = C ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5704_doubleton__eq__iff,axiom,
% 9.49/5.28      ! [A: real,B: real,C: real,D: real] :
% 9.49/5.28        ( ( ( insert_real @ A @ ( insert_real @ B @ bot_bot_set_real ) )
% 9.49/5.28          = ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
% 9.49/5.28        = ( ( ( A = C )
% 9.49/5.28            & ( B = D ) )
% 9.49/5.28          | ( ( A = D )
% 9.49/5.28            & ( B = C ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5705_doubleton__eq__iff,axiom,
% 9.49/5.28      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.49/5.28        ( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
% 9.49/5.28          = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
% 9.49/5.28        = ( ( ( A = C )
% 9.49/5.28            & ( B = D ) )
% 9.49/5.28          | ( ( A = D )
% 9.49/5.28            & ( B = C ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5706_doubleton__eq__iff,axiom,
% 9.49/5.28      ! [A: int,B: int,C: int,D: int] :
% 9.49/5.28        ( ( ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) )
% 9.49/5.28          = ( insert_int @ C @ ( insert_int @ D @ bot_bot_set_int ) ) )
% 9.49/5.28        = ( ( ( A = C )
% 9.49/5.28            & ( B = D ) )
% 9.49/5.28          | ( ( A = D )
% 9.49/5.28            & ( B = C ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5707_singleton__iff,axiom,
% 9.49/5.28      ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
% 9.49/5.28        ( ( member8757157785044589968at_nat @ B @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
% 9.49/5.28        = ( B = A ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5708_singleton__iff,axiom,
% 9.49/5.28      ! [B: set_nat,A: set_nat] :
% 9.49/5.28        ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
% 9.49/5.28        = ( B = A ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5709_singleton__iff,axiom,
% 9.49/5.28      ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
% 9.49/5.28        ( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
% 9.49/5.28        = ( B = A ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5710_singleton__iff,axiom,
% 9.49/5.28      ! [B: real,A: real] :
% 9.49/5.28        ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
% 9.49/5.28        = ( B = A ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5711_singleton__iff,axiom,
% 9.49/5.28      ! [B: nat,A: nat] :
% 9.49/5.28        ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
% 9.49/5.28        = ( B = A ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5712_singleton__iff,axiom,
% 9.49/5.28      ! [B: int,A: int] :
% 9.49/5.28        ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
% 9.49/5.28        = ( B = A ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5713_singletonD,axiom,
% 9.49/5.28      ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
% 9.49/5.28        ( ( member8757157785044589968at_nat @ B @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
% 9.49/5.28       => ( B = A ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5714_singletonD,axiom,
% 9.49/5.28      ! [B: set_nat,A: set_nat] :
% 9.49/5.28        ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
% 9.49/5.28       => ( B = A ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5715_singletonD,axiom,
% 9.49/5.28      ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
% 9.49/5.28        ( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
% 9.49/5.28       => ( B = A ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5716_singletonD,axiom,
% 9.49/5.28      ! [B: real,A: real] :
% 9.49/5.28        ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
% 9.49/5.28       => ( B = A ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5717_singletonD,axiom,
% 9.49/5.28      ! [B: nat,A: nat] :
% 9.49/5.28        ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
% 9.49/5.28       => ( B = A ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5718_singletonD,axiom,
% 9.49/5.28      ! [B: int,A: int] :
% 9.49/5.28        ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
% 9.49/5.28       => ( B = A ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5719_finite_OinsertI,axiom,
% 9.49/5.28      ! [A2: set_real,A: real] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( finite_finite_real @ ( insert_real @ A @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5720_finite_OinsertI,axiom,
% 9.49/5.28      ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
% 9.49/5.28        ( ( finite4343798906461161616at_nat @ A2 )
% 9.49/5.28       => ( finite4343798906461161616at_nat @ ( insert9069300056098147895at_nat @ A @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5721_finite_OinsertI,axiom,
% 9.49/5.28      ! [A2: set_nat,A: nat] :
% 9.49/5.28        ( ( finite_finite_nat @ A2 )
% 9.49/5.28       => ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5722_finite_OinsertI,axiom,
% 9.49/5.28      ! [A2: set_int,A: int] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( finite_finite_int @ ( insert_int @ A @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5723_finite_OinsertI,axiom,
% 9.49/5.28      ! [A2: set_complex,A: complex] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( finite3207457112153483333omplex @ ( insert_complex @ A @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5724_finite_OinsertI,axiom,
% 9.49/5.28      ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
% 9.49/5.28        ( ( finite6177210948735845034at_nat @ A2 )
% 9.49/5.28       => ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5725_subset__insertI2,axiom,
% 9.49/5.28      ! [A2: set_nat,B2: set_nat,B: nat] :
% 9.49/5.28        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.49/5.28       => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5726_subset__insertI2,axiom,
% 9.49/5.28      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
% 9.49/5.28        ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 9.49/5.28       => ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5727_subset__insertI2,axiom,
% 9.49/5.28      ! [A2: set_real,B2: set_real,B: real] :
% 9.49/5.28        ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.49/5.28       => ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5728_subset__insertI2,axiom,
% 9.49/5.28      ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
% 9.49/5.28        ( ( ord_le1268244103169919719at_nat @ A2 @ B2 )
% 9.49/5.28       => ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ B @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5729_subset__insertI2,axiom,
% 9.49/5.28      ! [A2: set_int,B2: set_int,B: int] :
% 9.49/5.28        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.49/5.28       => ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5730_subset__insertI,axiom,
% 9.49/5.28      ! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A @ B2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5731_subset__insertI,axiom,
% 9.49/5.28      ! [B2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ B2 @ ( insert8211810215607154385at_nat @ A @ B2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5732_subset__insertI,axiom,
% 9.49/5.28      ! [B2: set_real,A: real] : ( ord_less_eq_set_real @ B2 @ ( insert_real @ A @ B2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5733_subset__insertI,axiom,
% 9.49/5.28      ! [B2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] : ( ord_le1268244103169919719at_nat @ B2 @ ( insert9069300056098147895at_nat @ A @ B2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5734_subset__insertI,axiom,
% 9.49/5.28      ! [B2: set_int,A: int] : ( ord_less_eq_set_int @ B2 @ ( insert_int @ A @ B2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5735_subset__insert,axiom,
% 9.49/5.28      ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ~ ( member8757157785044589968at_nat @ X @ A2 )
% 9.49/5.28       => ( ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ B2 ) )
% 9.49/5.28          = ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5736_subset__insert,axiom,
% 9.49/5.28      ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ~ ( member8440522571783428010at_nat @ X @ A2 )
% 9.49/5.28       => ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B2 ) )
% 9.49/5.28          = ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5737_subset__insert,axiom,
% 9.49/5.28      ! [X: real,A2: set_real,B2: set_real] :
% 9.49/5.28        ( ~ ( member_real @ X @ A2 )
% 9.49/5.28       => ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) )
% 9.49/5.28          = ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5738_subset__insert,axiom,
% 9.49/5.28      ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.49/5.28        ( ~ ( member_set_nat @ X @ A2 )
% 9.49/5.28       => ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
% 9.49/5.28          = ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5739_subset__insert,axiom,
% 9.49/5.28      ! [X: nat,A2: set_nat,B2: set_nat] :
% 9.49/5.28        ( ~ ( member_nat @ X @ A2 )
% 9.49/5.28       => ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
% 9.49/5.28          = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5740_subset__insert,axiom,
% 9.49/5.28      ! [X: int,A2: set_int,B2: set_int] :
% 9.49/5.28        ( ~ ( member_int @ X @ A2 )
% 9.49/5.28       => ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) )
% 9.49/5.28          = ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5741_insert__mono,axiom,
% 9.49/5.28      ! [C2: set_nat,D6: set_nat,A: nat] :
% 9.49/5.28        ( ( ord_less_eq_set_nat @ C2 @ D6 )
% 9.49/5.28       => ( ord_less_eq_set_nat @ ( insert_nat @ A @ C2 ) @ ( insert_nat @ A @ D6 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5742_insert__mono,axiom,
% 9.49/5.28      ! [C2: set_Pr1261947904930325089at_nat,D6: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
% 9.49/5.28        ( ( ord_le3146513528884898305at_nat @ C2 @ D6 )
% 9.49/5.28       => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ A @ C2 ) @ ( insert8211810215607154385at_nat @ A @ D6 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5743_insert__mono,axiom,
% 9.49/5.28      ! [C2: set_real,D6: set_real,A: real] :
% 9.49/5.28        ( ( ord_less_eq_set_real @ C2 @ D6 )
% 9.49/5.28       => ( ord_less_eq_set_real @ ( insert_real @ A @ C2 ) @ ( insert_real @ A @ D6 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5744_insert__mono,axiom,
% 9.49/5.28      ! [C2: set_Pr4329608150637261639at_nat,D6: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
% 9.49/5.28        ( ( ord_le1268244103169919719at_nat @ C2 @ D6 )
% 9.49/5.28       => ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ A @ C2 ) @ ( insert9069300056098147895at_nat @ A @ D6 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5745_insert__mono,axiom,
% 9.49/5.28      ! [C2: set_int,D6: set_int,A: int] :
% 9.49/5.28        ( ( ord_less_eq_set_int @ C2 @ D6 )
% 9.49/5.28       => ( ord_less_eq_set_int @ ( insert_int @ A @ C2 ) @ ( insert_int @ A @ D6 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5746_insert__subsetI,axiom,
% 9.49/5.28      ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,X8: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ( member8757157785044589968at_nat @ X @ A2 )
% 9.49/5.28       => ( ( ord_le1268244103169919719at_nat @ X8 @ A2 )
% 9.49/5.28         => ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X @ X8 ) @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5747_insert__subsetI,axiom,
% 9.49/5.28      ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,X8: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ( member8440522571783428010at_nat @ X @ A2 )
% 9.49/5.28       => ( ( ord_le3146513528884898305at_nat @ X8 @ A2 )
% 9.49/5.28         => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X @ X8 ) @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5748_insert__subsetI,axiom,
% 9.49/5.28      ! [X: real,A2: set_real,X8: set_real] :
% 9.49/5.28        ( ( member_real @ X @ A2 )
% 9.49/5.28       => ( ( ord_less_eq_set_real @ X8 @ A2 )
% 9.49/5.28         => ( ord_less_eq_set_real @ ( insert_real @ X @ X8 ) @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5749_insert__subsetI,axiom,
% 9.49/5.28      ! [X: set_nat,A2: set_set_nat,X8: set_set_nat] :
% 9.49/5.28        ( ( member_set_nat @ X @ A2 )
% 9.49/5.28       => ( ( ord_le6893508408891458716et_nat @ X8 @ A2 )
% 9.49/5.28         => ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ X8 ) @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5750_insert__subsetI,axiom,
% 9.49/5.28      ! [X: nat,A2: set_nat,X8: set_nat] :
% 9.49/5.28        ( ( member_nat @ X @ A2 )
% 9.49/5.28       => ( ( ord_less_eq_set_nat @ X8 @ A2 )
% 9.49/5.28         => ( ord_less_eq_set_nat @ ( insert_nat @ X @ X8 ) @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5751_insert__subsetI,axiom,
% 9.49/5.28      ! [X: int,A2: set_int,X8: set_int] :
% 9.49/5.28        ( ( member_int @ X @ A2 )
% 9.49/5.28       => ( ( ord_less_eq_set_int @ X8 @ A2 )
% 9.49/5.28         => ( ord_less_eq_set_int @ ( insert_int @ X @ X8 ) @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5752_insert__Diff__if,axiom,
% 9.49/5.28      ! [X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ( ( member8757157785044589968at_nat @ X @ B2 )
% 9.49/5.28         => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A2 ) @ B2 )
% 9.49/5.28            = ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) )
% 9.49/5.28        & ( ~ ( member8757157785044589968at_nat @ X @ B2 )
% 9.49/5.28         => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A2 ) @ B2 )
% 9.49/5.28            = ( insert9069300056098147895at_nat @ X @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5753_insert__Diff__if,axiom,
% 9.49/5.28      ! [X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ( ( member8440522571783428010at_nat @ X @ B2 )
% 9.49/5.28         => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) @ B2 )
% 9.49/5.28            = ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) )
% 9.49/5.28        & ( ~ ( member8440522571783428010at_nat @ X @ B2 )
% 9.49/5.28         => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) @ B2 )
% 9.49/5.28            = ( insert8211810215607154385at_nat @ X @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5754_insert__Diff__if,axiom,
% 9.49/5.28      ! [X: real,B2: set_real,A2: set_real] :
% 9.49/5.28        ( ( ( member_real @ X @ B2 )
% 9.49/5.28         => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
% 9.49/5.28            = ( minus_minus_set_real @ A2 @ B2 ) ) )
% 9.49/5.28        & ( ~ ( member_real @ X @ B2 )
% 9.49/5.28         => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
% 9.49/5.28            = ( insert_real @ X @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5755_insert__Diff__if,axiom,
% 9.49/5.28      ! [X: set_nat,B2: set_set_nat,A2: set_set_nat] :
% 9.49/5.28        ( ( ( member_set_nat @ X @ B2 )
% 9.49/5.28         => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
% 9.49/5.28            = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) )
% 9.49/5.28        & ( ~ ( member_set_nat @ X @ B2 )
% 9.49/5.28         => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
% 9.49/5.28            = ( insert_set_nat @ X @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5756_insert__Diff__if,axiom,
% 9.49/5.28      ! [X: int,B2: set_int,A2: set_int] :
% 9.49/5.28        ( ( ( member_int @ X @ B2 )
% 9.49/5.28         => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B2 )
% 9.49/5.28            = ( minus_minus_set_int @ A2 @ B2 ) ) )
% 9.49/5.28        & ( ~ ( member_int @ X @ B2 )
% 9.49/5.28         => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B2 )
% 9.49/5.28            = ( insert_int @ X @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5757_insert__Diff__if,axiom,
% 9.49/5.28      ! [X: nat,B2: set_nat,A2: set_nat] :
% 9.49/5.28        ( ( ( member_nat @ X @ B2 )
% 9.49/5.28         => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
% 9.49/5.28            = ( minus_minus_set_nat @ A2 @ B2 ) ) )
% 9.49/5.28        & ( ~ ( member_nat @ X @ B2 )
% 9.49/5.28         => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
% 9.49/5.28            = ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5758_sum__mono,axiom,
% 9.49/5.28      ! [K4: set_real,F: real > rat,G: real > rat] :
% 9.49/5.28        ( ! [I2: real] :
% 9.49/5.28            ( ( member_real @ I2 @ K4 )
% 9.49/5.28           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 9.49/5.28       => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K4 ) @ ( groups1300246762558778688al_rat @ G @ K4 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5759_sum__mono,axiom,
% 9.49/5.28      ! [K4: set_nat,F: nat > rat,G: nat > rat] :
% 9.49/5.28        ( ! [I2: nat] :
% 9.49/5.28            ( ( member_nat @ I2 @ K4 )
% 9.49/5.28           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 9.49/5.28       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K4 ) @ ( groups2906978787729119204at_rat @ G @ K4 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5760_sum__mono,axiom,
% 9.49/5.28      ! [K4: set_int,F: int > rat,G: int > rat] :
% 9.49/5.28        ( ! [I2: int] :
% 9.49/5.28            ( ( member_int @ I2 @ K4 )
% 9.49/5.28           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 9.49/5.28       => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K4 ) @ ( groups3906332499630173760nt_rat @ G @ K4 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5761_sum__mono,axiom,
% 9.49/5.28      ! [K4: set_real,F: real > nat,G: real > nat] :
% 9.49/5.28        ( ! [I2: real] :
% 9.49/5.28            ( ( member_real @ I2 @ K4 )
% 9.49/5.28           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 9.49/5.28       => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K4 ) @ ( groups1935376822645274424al_nat @ G @ K4 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5762_sum__mono,axiom,
% 9.49/5.28      ! [K4: set_int,F: int > nat,G: int > nat] :
% 9.49/5.28        ( ! [I2: int] :
% 9.49/5.28            ( ( member_int @ I2 @ K4 )
% 9.49/5.28           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 9.49/5.28       => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K4 ) @ ( groups4541462559716669496nt_nat @ G @ K4 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5763_sum__mono,axiom,
% 9.49/5.28      ! [K4: set_real,F: real > int,G: real > int] :
% 9.49/5.28        ( ! [I2: real] :
% 9.49/5.28            ( ( member_real @ I2 @ K4 )
% 9.49/5.28           => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 9.49/5.28       => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K4 ) @ ( groups1932886352136224148al_int @ G @ K4 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5764_sum__mono,axiom,
% 9.49/5.28      ! [K4: set_nat,F: nat > int,G: nat > int] :
% 9.49/5.28        ( ! [I2: nat] :
% 9.49/5.28            ( ( member_nat @ I2 @ K4 )
% 9.49/5.28           => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 9.49/5.28       => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K4 ) @ ( groups3539618377306564664at_int @ G @ K4 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5765_sum__mono,axiom,
% 9.49/5.28      ! [K4: set_int,F: int > int,G: int > int] :
% 9.49/5.28        ( ! [I2: int] :
% 9.49/5.28            ( ( member_int @ I2 @ K4 )
% 9.49/5.28           => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 9.49/5.28       => ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ K4 ) @ ( groups4538972089207619220nt_int @ G @ K4 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5766_sum__mono,axiom,
% 9.49/5.28      ! [K4: set_nat,F: nat > nat,G: nat > nat] :
% 9.49/5.28        ( ! [I2: nat] :
% 9.49/5.28            ( ( member_nat @ I2 @ K4 )
% 9.49/5.28           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 9.49/5.28       => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K4 ) @ ( groups3542108847815614940at_nat @ G @ K4 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5767_sum__mono,axiom,
% 9.49/5.28      ! [K4: set_nat,F: nat > real,G: nat > real] :
% 9.49/5.28        ( ! [I2: nat] :
% 9.49/5.28            ( ( member_nat @ I2 @ K4 )
% 9.49/5.28           => ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 9.49/5.28       => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K4 ) @ ( groups6591440286371151544t_real @ G @ K4 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5768_sum_Oswap__restrict,axiom,
% 9.49/5.28      ! [A2: set_real,B2: set_nat,G: real > nat > nat,R: real > nat > $o] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( finite_finite_nat @ B2 )
% 9.49/5.28         => ( ( groups1935376822645274424al_nat
% 9.49/5.28              @ ^ [X3: real] :
% 9.49/5.28                  ( groups3542108847815614940at_nat @ ( G @ X3 )
% 9.49/5.28                  @ ( collect_nat
% 9.49/5.28                    @ ^ [Y2: nat] :
% 9.49/5.28                        ( ( member_nat @ Y2 @ B2 )
% 9.49/5.28                        & ( R @ X3 @ Y2 ) ) ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = ( groups3542108847815614940at_nat
% 9.49/5.28              @ ^ [Y2: nat] :
% 9.49/5.28                  ( groups1935376822645274424al_nat
% 9.49/5.28                  @ ^ [X3: real] : ( G @ X3 @ Y2 )
% 9.49/5.28                  @ ( collect_real
% 9.49/5.28                    @ ^ [X3: real] :
% 9.49/5.28                        ( ( member_real @ X3 @ A2 )
% 9.49/5.28                        & ( R @ X3 @ Y2 ) ) ) )
% 9.49/5.28              @ B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5769_sum_Oswap__restrict,axiom,
% 9.49/5.28      ! [A2: set_int,B2: set_nat,G: int > nat > nat,R: int > nat > $o] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( finite_finite_nat @ B2 )
% 9.49/5.28         => ( ( groups4541462559716669496nt_nat
% 9.49/5.28              @ ^ [X3: int] :
% 9.49/5.28                  ( groups3542108847815614940at_nat @ ( G @ X3 )
% 9.49/5.28                  @ ( collect_nat
% 9.49/5.28                    @ ^ [Y2: nat] :
% 9.49/5.28                        ( ( member_nat @ Y2 @ B2 )
% 9.49/5.28                        & ( R @ X3 @ Y2 ) ) ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = ( groups3542108847815614940at_nat
% 9.49/5.28              @ ^ [Y2: nat] :
% 9.49/5.28                  ( groups4541462559716669496nt_nat
% 9.49/5.28                  @ ^ [X3: int] : ( G @ X3 @ Y2 )
% 9.49/5.28                  @ ( collect_int
% 9.49/5.28                    @ ^ [X3: int] :
% 9.49/5.28                        ( ( member_int @ X3 @ A2 )
% 9.49/5.28                        & ( R @ X3 @ Y2 ) ) ) )
% 9.49/5.28              @ B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5770_sum_Oswap__restrict,axiom,
% 9.49/5.28      ! [A2: set_complex,B2: set_nat,G: complex > nat > nat,R: complex > nat > $o] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( finite_finite_nat @ B2 )
% 9.49/5.28         => ( ( groups5693394587270226106ex_nat
% 9.49/5.28              @ ^ [X3: complex] :
% 9.49/5.28                  ( groups3542108847815614940at_nat @ ( G @ X3 )
% 9.49/5.28                  @ ( collect_nat
% 9.49/5.28                    @ ^ [Y2: nat] :
% 9.49/5.28                        ( ( member_nat @ Y2 @ B2 )
% 9.49/5.28                        & ( R @ X3 @ Y2 ) ) ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = ( groups3542108847815614940at_nat
% 9.49/5.28              @ ^ [Y2: nat] :
% 9.49/5.28                  ( groups5693394587270226106ex_nat
% 9.49/5.28                  @ ^ [X3: complex] : ( G @ X3 @ Y2 )
% 9.49/5.28                  @ ( collect_complex
% 9.49/5.28                    @ ^ [X3: complex] :
% 9.49/5.28                        ( ( member_complex @ X3 @ A2 )
% 9.49/5.28                        & ( R @ X3 @ Y2 ) ) ) )
% 9.49/5.28              @ B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5771_sum_Oswap__restrict,axiom,
% 9.49/5.28      ! [A2: set_real,B2: set_complex,G: real > complex > complex,R: real > complex > $o] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( finite3207457112153483333omplex @ B2 )
% 9.49/5.28         => ( ( groups5754745047067104278omplex
% 9.49/5.28              @ ^ [X3: real] :
% 9.49/5.28                  ( groups7754918857620584856omplex @ ( G @ X3 )
% 9.49/5.28                  @ ( collect_complex
% 9.49/5.28                    @ ^ [Y2: complex] :
% 9.49/5.28                        ( ( member_complex @ Y2 @ B2 )
% 9.49/5.28                        & ( R @ X3 @ Y2 ) ) ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = ( groups7754918857620584856omplex
% 9.49/5.28              @ ^ [Y2: complex] :
% 9.49/5.28                  ( groups5754745047067104278omplex
% 9.49/5.28                  @ ^ [X3: real] : ( G @ X3 @ Y2 )
% 9.49/5.28                  @ ( collect_real
% 9.49/5.28                    @ ^ [X3: real] :
% 9.49/5.28                        ( ( member_real @ X3 @ A2 )
% 9.49/5.28                        & ( R @ X3 @ Y2 ) ) ) )
% 9.49/5.28              @ B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5772_sum_Oswap__restrict,axiom,
% 9.49/5.28      ! [A2: set_nat,B2: set_complex,G: nat > complex > complex,R: nat > complex > $o] :
% 9.49/5.28        ( ( finite_finite_nat @ A2 )
% 9.49/5.28       => ( ( finite3207457112153483333omplex @ B2 )
% 9.49/5.28         => ( ( groups2073611262835488442omplex
% 9.49/5.28              @ ^ [X3: nat] :
% 9.49/5.28                  ( groups7754918857620584856omplex @ ( G @ X3 )
% 9.49/5.28                  @ ( collect_complex
% 9.49/5.28                    @ ^ [Y2: complex] :
% 9.49/5.28                        ( ( member_complex @ Y2 @ B2 )
% 9.49/5.28                        & ( R @ X3 @ Y2 ) ) ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = ( groups7754918857620584856omplex
% 9.49/5.28              @ ^ [Y2: complex] :
% 9.49/5.28                  ( groups2073611262835488442omplex
% 9.49/5.28                  @ ^ [X3: nat] : ( G @ X3 @ Y2 )
% 9.49/5.28                  @ ( collect_nat
% 9.49/5.28                    @ ^ [X3: nat] :
% 9.49/5.28                        ( ( member_nat @ X3 @ A2 )
% 9.49/5.28                        & ( R @ X3 @ Y2 ) ) ) )
% 9.49/5.28              @ B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5773_sum_Oswap__restrict,axiom,
% 9.49/5.28      ! [A2: set_int,B2: set_complex,G: int > complex > complex,R: int > complex > $o] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( finite3207457112153483333omplex @ B2 )
% 9.49/5.28         => ( ( groups3049146728041665814omplex
% 9.49/5.28              @ ^ [X3: int] :
% 9.49/5.28                  ( groups7754918857620584856omplex @ ( G @ X3 )
% 9.49/5.28                  @ ( collect_complex
% 9.49/5.28                    @ ^ [Y2: complex] :
% 9.49/5.28                        ( ( member_complex @ Y2 @ B2 )
% 9.49/5.28                        & ( R @ X3 @ Y2 ) ) ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = ( groups7754918857620584856omplex
% 9.49/5.28              @ ^ [Y2: complex] :
% 9.49/5.28                  ( groups3049146728041665814omplex
% 9.49/5.28                  @ ^ [X3: int] : ( G @ X3 @ Y2 )
% 9.49/5.28                  @ ( collect_int
% 9.49/5.28                    @ ^ [X3: int] :
% 9.49/5.28                        ( ( member_int @ X3 @ A2 )
% 9.49/5.28                        & ( R @ X3 @ Y2 ) ) ) )
% 9.49/5.28              @ B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5774_sum_Oswap__restrict,axiom,
% 9.49/5.28      ! [A2: set_real,B2: set_nat,G: real > nat > real,R: real > nat > $o] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( finite_finite_nat @ B2 )
% 9.49/5.28         => ( ( groups8097168146408367636l_real
% 9.49/5.28              @ ^ [X3: real] :
% 9.49/5.28                  ( groups6591440286371151544t_real @ ( G @ X3 )
% 9.49/5.28                  @ ( collect_nat
% 9.49/5.28                    @ ^ [Y2: nat] :
% 9.49/5.28                        ( ( member_nat @ Y2 @ B2 )
% 9.49/5.28                        & ( R @ X3 @ Y2 ) ) ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = ( groups6591440286371151544t_real
% 9.49/5.28              @ ^ [Y2: nat] :
% 9.49/5.28                  ( groups8097168146408367636l_real
% 9.49/5.28                  @ ^ [X3: real] : ( G @ X3 @ Y2 )
% 9.49/5.28                  @ ( collect_real
% 9.49/5.28                    @ ^ [X3: real] :
% 9.49/5.28                        ( ( member_real @ X3 @ A2 )
% 9.49/5.28                        & ( R @ X3 @ Y2 ) ) ) )
% 9.49/5.28              @ B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5775_sum_Oswap__restrict,axiom,
% 9.49/5.28      ! [A2: set_int,B2: set_nat,G: int > nat > real,R: int > nat > $o] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( finite_finite_nat @ B2 )
% 9.49/5.28         => ( ( groups8778361861064173332t_real
% 9.49/5.28              @ ^ [X3: int] :
% 9.49/5.28                  ( groups6591440286371151544t_real @ ( G @ X3 )
% 9.49/5.28                  @ ( collect_nat
% 9.49/5.28                    @ ^ [Y2: nat] :
% 9.49/5.28                        ( ( member_nat @ Y2 @ B2 )
% 9.49/5.28                        & ( R @ X3 @ Y2 ) ) ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = ( groups6591440286371151544t_real
% 9.49/5.28              @ ^ [Y2: nat] :
% 9.49/5.28                  ( groups8778361861064173332t_real
% 9.49/5.28                  @ ^ [X3: int] : ( G @ X3 @ Y2 )
% 9.49/5.28                  @ ( collect_int
% 9.49/5.28                    @ ^ [X3: int] :
% 9.49/5.28                        ( ( member_int @ X3 @ A2 )
% 9.49/5.28                        & ( R @ X3 @ Y2 ) ) ) )
% 9.49/5.28              @ B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5776_sum_Oswap__restrict,axiom,
% 9.49/5.28      ! [A2: set_complex,B2: set_nat,G: complex > nat > real,R: complex > nat > $o] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( finite_finite_nat @ B2 )
% 9.49/5.28         => ( ( groups5808333547571424918x_real
% 9.49/5.28              @ ^ [X3: complex] :
% 9.49/5.28                  ( groups6591440286371151544t_real @ ( G @ X3 )
% 9.49/5.28                  @ ( collect_nat
% 9.49/5.28                    @ ^ [Y2: nat] :
% 9.49/5.28                        ( ( member_nat @ Y2 @ B2 )
% 9.49/5.28                        & ( R @ X3 @ Y2 ) ) ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = ( groups6591440286371151544t_real
% 9.49/5.28              @ ^ [Y2: nat] :
% 9.49/5.28                  ( groups5808333547571424918x_real
% 9.49/5.28                  @ ^ [X3: complex] : ( G @ X3 @ Y2 )
% 9.49/5.28                  @ ( collect_complex
% 9.49/5.28                    @ ^ [X3: complex] :
% 9.49/5.28                        ( ( member_complex @ X3 @ A2 )
% 9.49/5.28                        & ( R @ X3 @ Y2 ) ) ) )
% 9.49/5.28              @ B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5777_sum_Oswap__restrict,axiom,
% 9.49/5.28      ! [A2: set_nat,B2: set_real,G: nat > real > nat,R: nat > real > $o] :
% 9.49/5.28        ( ( finite_finite_nat @ A2 )
% 9.49/5.28       => ( ( finite_finite_real @ B2 )
% 9.49/5.28         => ( ( groups3542108847815614940at_nat
% 9.49/5.28              @ ^ [X3: nat] :
% 9.49/5.28                  ( groups1935376822645274424al_nat @ ( G @ X3 )
% 9.49/5.28                  @ ( collect_real
% 9.49/5.28                    @ ^ [Y2: real] :
% 9.49/5.28                        ( ( member_real @ Y2 @ B2 )
% 9.49/5.28                        & ( R @ X3 @ Y2 ) ) ) )
% 9.49/5.28              @ A2 )
% 9.49/5.28            = ( groups1935376822645274424al_nat
% 9.49/5.28              @ ^ [Y2: real] :
% 9.49/5.28                  ( groups3542108847815614940at_nat
% 9.49/5.28                  @ ^ [X3: nat] : ( G @ X3 @ Y2 )
% 9.49/5.28                  @ ( collect_nat
% 9.49/5.28                    @ ^ [X3: nat] :
% 9.49/5.28                        ( ( member_nat @ X3 @ A2 )
% 9.49/5.28                        & ( R @ X3 @ Y2 ) ) ) )
% 9.49/5.28              @ B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5778_sum__diff1__nat,axiom,
% 9.49/5.28      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > nat] :
% 9.49/5.28        ( ( ( member8757157785044589968at_nat @ A @ A2 )
% 9.49/5.28         => ( ( groups3860910324918113789at_nat @ F @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 9.49/5.28            = ( minus_minus_nat @ ( groups3860910324918113789at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.49/5.28        & ( ~ ( member8757157785044589968at_nat @ A @ A2 )
% 9.49/5.28         => ( ( groups3860910324918113789at_nat @ F @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 9.49/5.28            = ( groups3860910324918113789at_nat @ F @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5779_sum__diff1__nat,axiom,
% 9.49/5.28      ! [A: set_nat,A2: set_set_nat,F: set_nat > nat] :
% 9.49/5.28        ( ( ( member_set_nat @ A @ A2 )
% 9.49/5.28         => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 9.49/5.28            = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.49/5.28        & ( ~ ( member_set_nat @ A @ A2 )
% 9.49/5.28         => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 9.49/5.28            = ( groups8294997508430121362at_nat @ F @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5780_sum__diff1__nat,axiom,
% 9.49/5.28      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 9.49/5.28        ( ( ( member8440522571783428010at_nat @ A @ A2 )
% 9.49/5.28         => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 9.49/5.28            = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.49/5.28        & ( ~ ( member8440522571783428010at_nat @ A @ A2 )
% 9.49/5.28         => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 9.49/5.28            = ( groups977919841031483927at_nat @ F @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5781_sum__diff1__nat,axiom,
% 9.49/5.28      ! [A: real,A2: set_real,F: real > nat] :
% 9.49/5.28        ( ( ( member_real @ A @ A2 )
% 9.49/5.28         => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.49/5.28            = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.49/5.28        & ( ~ ( member_real @ A @ A2 )
% 9.49/5.28         => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.49/5.28            = ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5782_sum__diff1__nat,axiom,
% 9.49/5.28      ! [A: int,A2: set_int,F: int > nat] :
% 9.49/5.28        ( ( ( member_int @ A @ A2 )
% 9.49/5.28         => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.49/5.28            = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.49/5.28        & ( ~ ( member_int @ A @ A2 )
% 9.49/5.28         => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.49/5.28            = ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5783_sum__diff1__nat,axiom,
% 9.49/5.28      ! [A: nat,A2: set_nat,F: nat > nat] :
% 9.49/5.28        ( ( ( member_nat @ A @ A2 )
% 9.49/5.28         => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 9.49/5.28            = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.49/5.28        & ( ~ ( member_nat @ A @ A2 )
% 9.49/5.28         => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 9.49/5.28            = ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5784_Collect__conv__if,axiom,
% 9.49/5.28      ! [P: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
% 9.49/5.28        ( ( ( P @ A )
% 9.49/5.28         => ( ( collec6321179662152712658at_nat
% 9.49/5.28              @ ^ [X3: produc3843707927480180839at_nat] :
% 9.49/5.28                  ( ( X3 = A )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 9.49/5.28        & ( ~ ( P @ A )
% 9.49/5.28         => ( ( collec6321179662152712658at_nat
% 9.49/5.28              @ ^ [X3: produc3843707927480180839at_nat] :
% 9.49/5.28                  ( ( X3 = A )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = bot_bo228742789529271731at_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5785_Collect__conv__if,axiom,
% 9.49/5.28      ! [P: list_nat > $o,A: list_nat] :
% 9.49/5.28        ( ( ( P @ A )
% 9.49/5.28         => ( ( collect_list_nat
% 9.49/5.28              @ ^ [X3: list_nat] :
% 9.49/5.28                  ( ( X3 = A )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
% 9.49/5.28        & ( ~ ( P @ A )
% 9.49/5.28         => ( ( collect_list_nat
% 9.49/5.28              @ ^ [X3: list_nat] :
% 9.49/5.28                  ( ( X3 = A )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = bot_bot_set_list_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5786_Collect__conv__if,axiom,
% 9.49/5.28      ! [P: set_nat > $o,A: set_nat] :
% 9.49/5.28        ( ( ( P @ A )
% 9.49/5.28         => ( ( collect_set_nat
% 9.49/5.28              @ ^ [X3: set_nat] :
% 9.49/5.28                  ( ( X3 = A )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 9.49/5.28        & ( ~ ( P @ A )
% 9.49/5.28         => ( ( collect_set_nat
% 9.49/5.28              @ ^ [X3: set_nat] :
% 9.49/5.28                  ( ( X3 = A )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = bot_bot_set_set_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5787_Collect__conv__if,axiom,
% 9.49/5.28      ! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
% 9.49/5.28        ( ( ( P @ A )
% 9.49/5.28         => ( ( collec3392354462482085612at_nat
% 9.49/5.28              @ ^ [X3: product_prod_nat_nat] :
% 9.49/5.28                  ( ( X3 = A )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 9.49/5.28        & ( ~ ( P @ A )
% 9.49/5.28         => ( ( collec3392354462482085612at_nat
% 9.49/5.28              @ ^ [X3: product_prod_nat_nat] :
% 9.49/5.28                  ( ( X3 = A )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = bot_bo2099793752762293965at_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5788_Collect__conv__if,axiom,
% 9.49/5.28      ! [P: real > $o,A: real] :
% 9.49/5.28        ( ( ( P @ A )
% 9.49/5.28         => ( ( collect_real
% 9.49/5.28              @ ^ [X3: real] :
% 9.49/5.28                  ( ( X3 = A )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.49/5.28        & ( ~ ( P @ A )
% 9.49/5.28         => ( ( collect_real
% 9.49/5.28              @ ^ [X3: real] :
% 9.49/5.28                  ( ( X3 = A )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = bot_bot_set_real ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5789_Collect__conv__if,axiom,
% 9.49/5.28      ! [P: nat > $o,A: nat] :
% 9.49/5.28        ( ( ( P @ A )
% 9.49/5.28         => ( ( collect_nat
% 9.49/5.28              @ ^ [X3: nat] :
% 9.49/5.28                  ( ( X3 = A )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 9.49/5.28        & ( ~ ( P @ A )
% 9.49/5.28         => ( ( collect_nat
% 9.49/5.28              @ ^ [X3: nat] :
% 9.49/5.28                  ( ( X3 = A )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = bot_bot_set_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5790_Collect__conv__if,axiom,
% 9.49/5.28      ! [P: int > $o,A: int] :
% 9.49/5.28        ( ( ( P @ A )
% 9.49/5.28         => ( ( collect_int
% 9.49/5.28              @ ^ [X3: int] :
% 9.49/5.28                  ( ( X3 = A )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.49/5.28        & ( ~ ( P @ A )
% 9.49/5.28         => ( ( collect_int
% 9.49/5.28              @ ^ [X3: int] :
% 9.49/5.28                  ( ( X3 = A )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = bot_bot_set_int ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5791_Collect__conv__if2,axiom,
% 9.49/5.28      ! [P: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
% 9.49/5.28        ( ( ( P @ A )
% 9.49/5.28         => ( ( collec6321179662152712658at_nat
% 9.49/5.28              @ ^ [X3: produc3843707927480180839at_nat] :
% 9.49/5.28                  ( ( A = X3 )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 9.49/5.28        & ( ~ ( P @ A )
% 9.49/5.28         => ( ( collec6321179662152712658at_nat
% 9.49/5.28              @ ^ [X3: produc3843707927480180839at_nat] :
% 9.49/5.28                  ( ( A = X3 )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = bot_bo228742789529271731at_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5792_Collect__conv__if2,axiom,
% 9.49/5.28      ! [P: list_nat > $o,A: list_nat] :
% 9.49/5.28        ( ( ( P @ A )
% 9.49/5.28         => ( ( collect_list_nat
% 9.49/5.28              @ ^ [X3: list_nat] :
% 9.49/5.28                  ( ( A = X3 )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
% 9.49/5.28        & ( ~ ( P @ A )
% 9.49/5.28         => ( ( collect_list_nat
% 9.49/5.28              @ ^ [X3: list_nat] :
% 9.49/5.28                  ( ( A = X3 )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = bot_bot_set_list_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5793_Collect__conv__if2,axiom,
% 9.49/5.28      ! [P: set_nat > $o,A: set_nat] :
% 9.49/5.28        ( ( ( P @ A )
% 9.49/5.28         => ( ( collect_set_nat
% 9.49/5.28              @ ^ [X3: set_nat] :
% 9.49/5.28                  ( ( A = X3 )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 9.49/5.28        & ( ~ ( P @ A )
% 9.49/5.28         => ( ( collect_set_nat
% 9.49/5.28              @ ^ [X3: set_nat] :
% 9.49/5.28                  ( ( A = X3 )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = bot_bot_set_set_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5794_Collect__conv__if2,axiom,
% 9.49/5.28      ! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
% 9.49/5.28        ( ( ( P @ A )
% 9.49/5.28         => ( ( collec3392354462482085612at_nat
% 9.49/5.28              @ ^ [X3: product_prod_nat_nat] :
% 9.49/5.28                  ( ( A = X3 )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 9.49/5.28        & ( ~ ( P @ A )
% 9.49/5.28         => ( ( collec3392354462482085612at_nat
% 9.49/5.28              @ ^ [X3: product_prod_nat_nat] :
% 9.49/5.28                  ( ( A = X3 )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = bot_bo2099793752762293965at_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5795_Collect__conv__if2,axiom,
% 9.49/5.28      ! [P: real > $o,A: real] :
% 9.49/5.28        ( ( ( P @ A )
% 9.49/5.28         => ( ( collect_real
% 9.49/5.28              @ ^ [X3: real] :
% 9.49/5.28                  ( ( A = X3 )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.49/5.28        & ( ~ ( P @ A )
% 9.49/5.28         => ( ( collect_real
% 9.49/5.28              @ ^ [X3: real] :
% 9.49/5.28                  ( ( A = X3 )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = bot_bot_set_real ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5796_Collect__conv__if2,axiom,
% 9.49/5.28      ! [P: nat > $o,A: nat] :
% 9.49/5.28        ( ( ( P @ A )
% 9.49/5.28         => ( ( collect_nat
% 9.49/5.28              @ ^ [X3: nat] :
% 9.49/5.28                  ( ( A = X3 )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 9.49/5.28        & ( ~ ( P @ A )
% 9.49/5.28         => ( ( collect_nat
% 9.49/5.28              @ ^ [X3: nat] :
% 9.49/5.28                  ( ( A = X3 )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = bot_bot_set_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5797_Collect__conv__if2,axiom,
% 9.49/5.28      ! [P: int > $o,A: int] :
% 9.49/5.28        ( ( ( P @ A )
% 9.49/5.28         => ( ( collect_int
% 9.49/5.28              @ ^ [X3: int] :
% 9.49/5.28                  ( ( A = X3 )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.49/5.28        & ( ~ ( P @ A )
% 9.49/5.28         => ( ( collect_int
% 9.49/5.28              @ ^ [X3: int] :
% 9.49/5.28                  ( ( A = X3 )
% 9.49/5.28                  & ( P @ X3 ) ) )
% 9.49/5.28            = bot_bot_set_int ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5798_round__mono,axiom,
% 9.49/5.28      ! [X: rat,Y: rat] :
% 9.49/5.28        ( ( ord_less_eq_rat @ X @ Y )
% 9.49/5.28       => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5799_sum_Oinsert__remove,axiom,
% 9.49/5.28      ! [A2: set_complex,G: complex > real,X: complex] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
% 9.49/5.28          = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5800_sum_Oinsert__remove,axiom,
% 9.49/5.28      ! [A2: set_complex,G: complex > rat,X: complex] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
% 9.49/5.28          = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5801_sum_Oinsert__remove,axiom,
% 9.49/5.28      ! [A2: set_complex,G: complex > nat,X: complex] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
% 9.49/5.28          = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5802_sum_Oinsert__remove,axiom,
% 9.49/5.28      ! [A2: set_complex,G: complex > int,X: complex] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X @ A2 ) )
% 9.49/5.28          = ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5803_sum_Oinsert__remove,axiom,
% 9.49/5.28      ! [A2: set_real,G: real > real,X: real] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
% 9.49/5.28          = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5804_sum_Oinsert__remove,axiom,
% 9.49/5.28      ! [A2: set_real,G: real > rat,X: real] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
% 9.49/5.28          = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5805_sum_Oinsert__remove,axiom,
% 9.49/5.28      ! [A2: set_real,G: real > nat,X: real] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A2 ) )
% 9.49/5.28          = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5806_sum_Oinsert__remove,axiom,
% 9.49/5.28      ! [A2: set_real,G: real > int,X: real] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( groups1932886352136224148al_int @ G @ ( insert_real @ X @ A2 ) )
% 9.49/5.28          = ( plus_plus_int @ ( G @ X ) @ ( groups1932886352136224148al_int @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5807_sum_Oinsert__remove,axiom,
% 9.49/5.28      ! [A2: set_int,G: int > real,X: int] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
% 9.49/5.28          = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5808_sum_Oinsert__remove,axiom,
% 9.49/5.28      ! [A2: set_int,G: int > rat,X: int] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
% 9.49/5.28          = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5809_sum_Oremove,axiom,
% 9.49/5.28      ! [A2: set_complex,X: complex,G: complex > real] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( member_complex @ X @ A2 )
% 9.49/5.28         => ( ( groups5808333547571424918x_real @ G @ A2 )
% 9.49/5.28            = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5810_sum_Oremove,axiom,
% 9.49/5.28      ! [A2: set_complex,X: complex,G: complex > rat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( member_complex @ X @ A2 )
% 9.49/5.28         => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 9.49/5.28            = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5811_sum_Oremove,axiom,
% 9.49/5.28      ! [A2: set_complex,X: complex,G: complex > nat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( member_complex @ X @ A2 )
% 9.49/5.28         => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 9.49/5.28            = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5812_sum_Oremove,axiom,
% 9.49/5.28      ! [A2: set_complex,X: complex,G: complex > int] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( member_complex @ X @ A2 )
% 9.49/5.28         => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 9.49/5.28            = ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5813_sum_Oremove,axiom,
% 9.49/5.28      ! [A2: set_real,X: real,G: real > real] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( member_real @ X @ A2 )
% 9.49/5.28         => ( ( groups8097168146408367636l_real @ G @ A2 )
% 9.49/5.28            = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5814_sum_Oremove,axiom,
% 9.49/5.28      ! [A2: set_real,X: real,G: real > rat] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( member_real @ X @ A2 )
% 9.49/5.28         => ( ( groups1300246762558778688al_rat @ G @ A2 )
% 9.49/5.28            = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5815_sum_Oremove,axiom,
% 9.49/5.28      ! [A2: set_real,X: real,G: real > nat] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( member_real @ X @ A2 )
% 9.49/5.28         => ( ( groups1935376822645274424al_nat @ G @ A2 )
% 9.49/5.28            = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5816_sum_Oremove,axiom,
% 9.49/5.28      ! [A2: set_real,X: real,G: real > int] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( member_real @ X @ A2 )
% 9.49/5.28         => ( ( groups1932886352136224148al_int @ G @ A2 )
% 9.49/5.28            = ( plus_plus_int @ ( G @ X ) @ ( groups1932886352136224148al_int @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5817_sum_Oremove,axiom,
% 9.49/5.28      ! [A2: set_int,X: int,G: int > real] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( member_int @ X @ A2 )
% 9.49/5.28         => ( ( groups8778361861064173332t_real @ G @ A2 )
% 9.49/5.28            = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5818_sum_Oremove,axiom,
% 9.49/5.28      ! [A2: set_int,X: int,G: int > rat] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( member_int @ X @ A2 )
% 9.49/5.28         => ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 9.49/5.28            = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5819_sum__diff1,axiom,
% 9.49/5.28      ! [A2: set_complex,A: complex,F: complex > real] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( ( member_complex @ A @ A2 )
% 9.49/5.28           => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 9.49/5.28              = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.49/5.28          & ( ~ ( member_complex @ A @ A2 )
% 9.49/5.28           => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 9.49/5.28              = ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5820_sum__diff1,axiom,
% 9.49/5.28      ! [A2: set_real,A: real,F: real > real] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( ( member_real @ A @ A2 )
% 9.49/5.28           => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.49/5.28              = ( minus_minus_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.49/5.28          & ( ~ ( member_real @ A @ A2 )
% 9.49/5.28           => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.49/5.28              = ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5821_sum__diff1,axiom,
% 9.49/5.28      ! [A2: set_int,A: int,F: int > real] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( ( member_int @ A @ A2 )
% 9.49/5.28           => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.49/5.28              = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.49/5.28          & ( ~ ( member_int @ A @ A2 )
% 9.49/5.28           => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.49/5.28              = ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5822_sum__diff1,axiom,
% 9.49/5.28      ! [A2: set_complex,A: complex,F: complex > rat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( ( member_complex @ A @ A2 )
% 9.49/5.28           => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 9.49/5.28              = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.49/5.28          & ( ~ ( member_complex @ A @ A2 )
% 9.49/5.28           => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 9.49/5.28              = ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5823_sum__diff1,axiom,
% 9.49/5.28      ! [A2: set_real,A: real,F: real > rat] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( ( member_real @ A @ A2 )
% 9.49/5.28           => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.49/5.28              = ( minus_minus_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.49/5.28          & ( ~ ( member_real @ A @ A2 )
% 9.49/5.28           => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.49/5.28              = ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5824_sum__diff1,axiom,
% 9.49/5.28      ! [A2: set_int,A: int,F: int > rat] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( ( member_int @ A @ A2 )
% 9.49/5.28           => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.49/5.28              = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.49/5.28          & ( ~ ( member_int @ A @ A2 )
% 9.49/5.28           => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.49/5.28              = ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5825_sum__diff1,axiom,
% 9.49/5.28      ! [A2: set_complex,A: complex,F: complex > int] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( ( member_complex @ A @ A2 )
% 9.49/5.28           => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 9.49/5.28              = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.49/5.28          & ( ~ ( member_complex @ A @ A2 )
% 9.49/5.28           => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 9.49/5.28              = ( groups5690904116761175830ex_int @ F @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5826_sum__diff1,axiom,
% 9.49/5.28      ! [A2: set_real,A: real,F: real > int] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( ( member_real @ A @ A2 )
% 9.49/5.28           => ( ( groups1932886352136224148al_int @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.49/5.28              = ( minus_minus_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.49/5.28          & ( ~ ( member_real @ A @ A2 )
% 9.49/5.28           => ( ( groups1932886352136224148al_int @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.49/5.28              = ( groups1932886352136224148al_int @ F @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5827_sum__diff1,axiom,
% 9.49/5.28      ! [A2: set_int,A: int,F: int > int] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( ( member_int @ A @ A2 )
% 9.49/5.28           => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.49/5.28              = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.49/5.28          & ( ~ ( member_int @ A @ A2 )
% 9.49/5.28           => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.49/5.28              = ( groups4538972089207619220nt_int @ F @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5828_sum__diff1,axiom,
% 9.49/5.28      ! [A2: set_nat,A: nat,F: nat > rat] :
% 9.49/5.28        ( ( finite_finite_nat @ A2 )
% 9.49/5.28       => ( ( ( member_nat @ A @ A2 )
% 9.49/5.28           => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 9.49/5.28              = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.49/5.28          & ( ~ ( member_nat @ A @ A2 )
% 9.49/5.28           => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 9.49/5.28              = ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5829_sum__nonneg,axiom,
% 9.49/5.28      ! [A2: set_real,F: real > real] :
% 9.49/5.28        ( ! [X4: real] :
% 9.49/5.28            ( ( member_real @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.49/5.28       => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5830_sum__nonneg,axiom,
% 9.49/5.28      ! [A2: set_int,F: int > real] :
% 9.49/5.28        ( ! [X4: int] :
% 9.49/5.28            ( ( member_int @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.49/5.28       => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5831_sum__nonneg,axiom,
% 9.49/5.28      ! [A2: set_real,F: real > rat] :
% 9.49/5.28        ( ! [X4: real] :
% 9.49/5.28            ( ( member_real @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.49/5.28       => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5832_sum__nonneg,axiom,
% 9.49/5.28      ! [A2: set_nat,F: nat > rat] :
% 9.49/5.28        ( ! [X4: nat] :
% 9.49/5.28            ( ( member_nat @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.49/5.28       => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5833_sum__nonneg,axiom,
% 9.49/5.28      ! [A2: set_int,F: int > rat] :
% 9.49/5.28        ( ! [X4: int] :
% 9.49/5.28            ( ( member_int @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.49/5.28       => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5834_sum__nonneg,axiom,
% 9.49/5.28      ! [A2: set_real,F: real > nat] :
% 9.49/5.28        ( ! [X4: real] :
% 9.49/5.28            ( ( member_real @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.49/5.28       => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5835_sum__nonneg,axiom,
% 9.49/5.28      ! [A2: set_int,F: int > nat] :
% 9.49/5.28        ( ! [X4: int] :
% 9.49/5.28            ( ( member_int @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.49/5.28       => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5836_sum__nonneg,axiom,
% 9.49/5.28      ! [A2: set_real,F: real > int] :
% 9.49/5.28        ( ! [X4: real] :
% 9.49/5.28            ( ( member_real @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 9.49/5.28       => ( ord_less_eq_int @ zero_zero_int @ ( groups1932886352136224148al_int @ F @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5837_sum__nonneg,axiom,
% 9.49/5.28      ! [A2: set_nat,F: nat > int] :
% 9.49/5.28        ( ! [X4: nat] :
% 9.49/5.28            ( ( member_nat @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 9.49/5.28       => ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5838_sum__nonneg,axiom,
% 9.49/5.28      ! [A2: set_int,F: int > int] :
% 9.49/5.28        ( ! [X4: int] :
% 9.49/5.28            ( ( member_int @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 9.49/5.28       => ( ord_less_eq_int @ zero_zero_int @ ( groups4538972089207619220nt_int @ F @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5839_sum__nonpos,axiom,
% 9.49/5.28      ! [A2: set_real,F: real > real] :
% 9.49/5.28        ( ! [X4: real] :
% 9.49/5.28            ( ( member_real @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
% 9.49/5.28       => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5840_sum__nonpos,axiom,
% 9.49/5.28      ! [A2: set_int,F: int > real] :
% 9.49/5.28        ( ! [X4: int] :
% 9.49/5.28            ( ( member_int @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
% 9.49/5.28       => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5841_sum__nonpos,axiom,
% 9.49/5.28      ! [A2: set_real,F: real > rat] :
% 9.49/5.28        ( ! [X4: real] :
% 9.49/5.28            ( ( member_real @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
% 9.49/5.28       => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5842_sum__nonpos,axiom,
% 9.49/5.28      ! [A2: set_nat,F: nat > rat] :
% 9.49/5.28        ( ! [X4: nat] :
% 9.49/5.28            ( ( member_nat @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
% 9.49/5.28       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5843_sum__nonpos,axiom,
% 9.49/5.28      ! [A2: set_int,F: int > rat] :
% 9.49/5.28        ( ! [X4: int] :
% 9.49/5.28            ( ( member_int @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
% 9.49/5.28       => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5844_sum__nonpos,axiom,
% 9.49/5.28      ! [A2: set_real,F: real > nat] :
% 9.49/5.28        ( ! [X4: real] :
% 9.49/5.28            ( ( member_real @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
% 9.49/5.28       => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5845_sum__nonpos,axiom,
% 9.49/5.28      ! [A2: set_int,F: int > nat] :
% 9.49/5.28        ( ! [X4: int] :
% 9.49/5.28            ( ( member_int @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
% 9.49/5.28       => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5846_sum__nonpos,axiom,
% 9.49/5.28      ! [A2: set_real,F: real > int] :
% 9.49/5.28        ( ! [X4: real] :
% 9.49/5.28            ( ( member_real @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
% 9.49/5.28       => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ zero_zero_int ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5847_sum__nonpos,axiom,
% 9.49/5.28      ! [A2: set_nat,F: nat > int] :
% 9.49/5.28        ( ! [X4: nat] :
% 9.49/5.28            ( ( member_nat @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
% 9.49/5.28       => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ zero_zero_int ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5848_sum__nonpos,axiom,
% 9.49/5.28      ! [A2: set_int,F: int > int] :
% 9.49/5.28        ( ! [X4: int] :
% 9.49/5.28            ( ( member_int @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
% 9.49/5.28       => ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ zero_zero_int ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5849_sum__mono__inv,axiom,
% 9.49/5.28      ! [F: real > rat,I6: set_real,G: real > rat,I: real] :
% 9.49/5.28        ( ( ( groups1300246762558778688al_rat @ F @ I6 )
% 9.49/5.28          = ( groups1300246762558778688al_rat @ G @ I6 ) )
% 9.49/5.28       => ( ! [I2: real] :
% 9.49/5.28              ( ( member_real @ I2 @ I6 )
% 9.49/5.28             => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 9.49/5.28         => ( ( member_real @ I @ I6 )
% 9.49/5.28           => ( ( finite_finite_real @ I6 )
% 9.49/5.28             => ( ( F @ I )
% 9.49/5.28                = ( G @ I ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5850_sum__mono__inv,axiom,
% 9.49/5.28      ! [F: nat > rat,I6: set_nat,G: nat > rat,I: nat] :
% 9.49/5.28        ( ( ( groups2906978787729119204at_rat @ F @ I6 )
% 9.49/5.28          = ( groups2906978787729119204at_rat @ G @ I6 ) )
% 9.49/5.28       => ( ! [I2: nat] :
% 9.49/5.28              ( ( member_nat @ I2 @ I6 )
% 9.49/5.28             => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 9.49/5.28         => ( ( member_nat @ I @ I6 )
% 9.49/5.28           => ( ( finite_finite_nat @ I6 )
% 9.49/5.28             => ( ( F @ I )
% 9.49/5.28                = ( G @ I ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5851_sum__mono__inv,axiom,
% 9.49/5.28      ! [F: int > rat,I6: set_int,G: int > rat,I: int] :
% 9.49/5.28        ( ( ( groups3906332499630173760nt_rat @ F @ I6 )
% 9.49/5.28          = ( groups3906332499630173760nt_rat @ G @ I6 ) )
% 9.49/5.28       => ( ! [I2: int] :
% 9.49/5.28              ( ( member_int @ I2 @ I6 )
% 9.49/5.28             => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 9.49/5.28         => ( ( member_int @ I @ I6 )
% 9.49/5.28           => ( ( finite_finite_int @ I6 )
% 9.49/5.28             => ( ( F @ I )
% 9.49/5.28                = ( G @ I ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5852_sum__mono__inv,axiom,
% 9.49/5.28      ! [F: complex > rat,I6: set_complex,G: complex > rat,I: complex] :
% 9.49/5.28        ( ( ( groups5058264527183730370ex_rat @ F @ I6 )
% 9.49/5.28          = ( groups5058264527183730370ex_rat @ G @ I6 ) )
% 9.49/5.28       => ( ! [I2: complex] :
% 9.49/5.28              ( ( member_complex @ I2 @ I6 )
% 9.49/5.28             => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 9.49/5.28         => ( ( member_complex @ I @ I6 )
% 9.49/5.28           => ( ( finite3207457112153483333omplex @ I6 )
% 9.49/5.28             => ( ( F @ I )
% 9.49/5.28                = ( G @ I ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5853_sum__mono__inv,axiom,
% 9.49/5.28      ! [F: real > nat,I6: set_real,G: real > nat,I: real] :
% 9.49/5.28        ( ( ( groups1935376822645274424al_nat @ F @ I6 )
% 9.49/5.28          = ( groups1935376822645274424al_nat @ G @ I6 ) )
% 9.49/5.28       => ( ! [I2: real] :
% 9.49/5.28              ( ( member_real @ I2 @ I6 )
% 9.49/5.28             => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 9.49/5.28         => ( ( member_real @ I @ I6 )
% 9.49/5.28           => ( ( finite_finite_real @ I6 )
% 9.49/5.28             => ( ( F @ I )
% 9.49/5.28                = ( G @ I ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5854_sum__mono__inv,axiom,
% 9.49/5.28      ! [F: int > nat,I6: set_int,G: int > nat,I: int] :
% 9.49/5.28        ( ( ( groups4541462559716669496nt_nat @ F @ I6 )
% 9.49/5.28          = ( groups4541462559716669496nt_nat @ G @ I6 ) )
% 9.49/5.28       => ( ! [I2: int] :
% 9.49/5.28              ( ( member_int @ I2 @ I6 )
% 9.49/5.28             => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 9.49/5.28         => ( ( member_int @ I @ I6 )
% 9.49/5.28           => ( ( finite_finite_int @ I6 )
% 9.49/5.28             => ( ( F @ I )
% 9.49/5.28                = ( G @ I ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5855_sum__mono__inv,axiom,
% 9.49/5.28      ! [F: complex > nat,I6: set_complex,G: complex > nat,I: complex] :
% 9.49/5.28        ( ( ( groups5693394587270226106ex_nat @ F @ I6 )
% 9.49/5.28          = ( groups5693394587270226106ex_nat @ G @ I6 ) )
% 9.49/5.28       => ( ! [I2: complex] :
% 9.49/5.28              ( ( member_complex @ I2 @ I6 )
% 9.49/5.28             => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 9.49/5.28         => ( ( member_complex @ I @ I6 )
% 9.49/5.28           => ( ( finite3207457112153483333omplex @ I6 )
% 9.49/5.28             => ( ( F @ I )
% 9.49/5.28                = ( G @ I ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5856_sum__mono__inv,axiom,
% 9.49/5.28      ! [F: real > int,I6: set_real,G: real > int,I: real] :
% 9.49/5.28        ( ( ( groups1932886352136224148al_int @ F @ I6 )
% 9.49/5.28          = ( groups1932886352136224148al_int @ G @ I6 ) )
% 9.49/5.28       => ( ! [I2: real] :
% 9.49/5.28              ( ( member_real @ I2 @ I6 )
% 9.49/5.28             => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 9.49/5.28         => ( ( member_real @ I @ I6 )
% 9.49/5.28           => ( ( finite_finite_real @ I6 )
% 9.49/5.28             => ( ( F @ I )
% 9.49/5.28                = ( G @ I ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5857_sum__mono__inv,axiom,
% 9.49/5.28      ! [F: nat > int,I6: set_nat,G: nat > int,I: nat] :
% 9.49/5.28        ( ( ( groups3539618377306564664at_int @ F @ I6 )
% 9.49/5.28          = ( groups3539618377306564664at_int @ G @ I6 ) )
% 9.49/5.28       => ( ! [I2: nat] :
% 9.49/5.28              ( ( member_nat @ I2 @ I6 )
% 9.49/5.28             => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 9.49/5.28         => ( ( member_nat @ I @ I6 )
% 9.49/5.28           => ( ( finite_finite_nat @ I6 )
% 9.49/5.28             => ( ( F @ I )
% 9.49/5.28                = ( G @ I ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5858_sum__mono__inv,axiom,
% 9.49/5.28      ! [F: int > int,I6: set_int,G: int > int,I: int] :
% 9.49/5.28        ( ( ( groups4538972089207619220nt_int @ F @ I6 )
% 9.49/5.28          = ( groups4538972089207619220nt_int @ G @ I6 ) )
% 9.49/5.28       => ( ! [I2: int] :
% 9.49/5.28              ( ( member_int @ I2 @ I6 )
% 9.49/5.28             => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 9.49/5.28         => ( ( member_int @ I @ I6 )
% 9.49/5.28           => ( ( finite_finite_int @ I6 )
% 9.49/5.28             => ( ( F @ I )
% 9.49/5.28                = ( G @ I ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5859_sum__cong__Suc,axiom,
% 9.49/5.28      ! [A2: set_nat,F: nat > nat,G: nat > nat] :
% 9.49/5.28        ( ~ ( member_nat @ zero_zero_nat @ A2 )
% 9.49/5.28       => ( ! [X4: nat] :
% 9.49/5.28              ( ( member_nat @ ( suc @ X4 ) @ A2 )
% 9.49/5.28             => ( ( F @ ( suc @ X4 ) )
% 9.49/5.28                = ( G @ ( suc @ X4 ) ) ) )
% 9.49/5.28         => ( ( groups3542108847815614940at_nat @ F @ A2 )
% 9.49/5.28            = ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5860_sum__cong__Suc,axiom,
% 9.49/5.28      ! [A2: set_nat,F: nat > real,G: nat > real] :
% 9.49/5.28        ( ~ ( member_nat @ zero_zero_nat @ A2 )
% 9.49/5.28       => ( ! [X4: nat] :
% 9.49/5.28              ( ( member_nat @ ( suc @ X4 ) @ A2 )
% 9.49/5.28             => ( ( F @ ( suc @ X4 ) )
% 9.49/5.28                = ( G @ ( suc @ X4 ) ) ) )
% 9.49/5.28         => ( ( groups6591440286371151544t_real @ F @ A2 )
% 9.49/5.28            = ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5861_sum_Odelta__remove,axiom,
% 9.49/5.28      ! [S2: set_complex,A: complex,B: complex > real,C: complex > real] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.28       => ( ( ( member_complex @ A @ S2 )
% 9.49/5.28           => ( ( groups5808333547571424918x_real
% 9.49/5.28                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( plus_plus_real @ ( B @ A ) @ ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
% 9.49/5.28          & ( ~ ( member_complex @ A @ S2 )
% 9.49/5.28           => ( ( groups5808333547571424918x_real
% 9.49/5.28                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5862_sum_Odelta__remove,axiom,
% 9.49/5.28      ! [S2: set_complex,A: complex,B: complex > rat,C: complex > rat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.28       => ( ( ( member_complex @ A @ S2 )
% 9.49/5.28           => ( ( groups5058264527183730370ex_rat
% 9.49/5.28                @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( plus_plus_rat @ ( B @ A ) @ ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
% 9.49/5.28          & ( ~ ( member_complex @ A @ S2 )
% 9.49/5.28           => ( ( groups5058264527183730370ex_rat
% 9.49/5.28                @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5863_sum_Odelta__remove,axiom,
% 9.49/5.28      ! [S2: set_complex,A: complex,B: complex > nat,C: complex > nat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.28       => ( ( ( member_complex @ A @ S2 )
% 9.49/5.28           => ( ( groups5693394587270226106ex_nat
% 9.49/5.28                @ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( plus_plus_nat @ ( B @ A ) @ ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
% 9.49/5.28          & ( ~ ( member_complex @ A @ S2 )
% 9.49/5.28           => ( ( groups5693394587270226106ex_nat
% 9.49/5.28                @ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5864_sum_Odelta__remove,axiom,
% 9.49/5.28      ! [S2: set_complex,A: complex,B: complex > int,C: complex > int] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.28       => ( ( ( member_complex @ A @ S2 )
% 9.49/5.28           => ( ( groups5690904116761175830ex_int
% 9.49/5.28                @ ^ [K3: complex] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( plus_plus_int @ ( B @ A ) @ ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
% 9.49/5.28          & ( ~ ( member_complex @ A @ S2 )
% 9.49/5.28           => ( ( groups5690904116761175830ex_int
% 9.49/5.28                @ ^ [K3: complex] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5865_sum_Odelta__remove,axiom,
% 9.49/5.28      ! [S2: set_real,A: real,B: real > real,C: real > real] :
% 9.49/5.28        ( ( finite_finite_real @ S2 )
% 9.49/5.28       => ( ( ( member_real @ A @ S2 )
% 9.49/5.28           => ( ( groups8097168146408367636l_real
% 9.49/5.28                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( plus_plus_real @ ( B @ A ) @ ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
% 9.49/5.28          & ( ~ ( member_real @ A @ S2 )
% 9.49/5.28           => ( ( groups8097168146408367636l_real
% 9.49/5.28                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5866_sum_Odelta__remove,axiom,
% 9.49/5.28      ! [S2: set_real,A: real,B: real > rat,C: real > rat] :
% 9.49/5.28        ( ( finite_finite_real @ S2 )
% 9.49/5.28       => ( ( ( member_real @ A @ S2 )
% 9.49/5.28           => ( ( groups1300246762558778688al_rat
% 9.49/5.28                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( plus_plus_rat @ ( B @ A ) @ ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
% 9.49/5.28          & ( ~ ( member_real @ A @ S2 )
% 9.49/5.28           => ( ( groups1300246762558778688al_rat
% 9.49/5.28                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5867_sum_Odelta__remove,axiom,
% 9.49/5.28      ! [S2: set_real,A: real,B: real > nat,C: real > nat] :
% 9.49/5.28        ( ( finite_finite_real @ S2 )
% 9.49/5.28       => ( ( ( member_real @ A @ S2 )
% 9.49/5.28           => ( ( groups1935376822645274424al_nat
% 9.49/5.28                @ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( plus_plus_nat @ ( B @ A ) @ ( groups1935376822645274424al_nat @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
% 9.49/5.28          & ( ~ ( member_real @ A @ S2 )
% 9.49/5.28           => ( ( groups1935376822645274424al_nat
% 9.49/5.28                @ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( groups1935376822645274424al_nat @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5868_sum_Odelta__remove,axiom,
% 9.49/5.28      ! [S2: set_real,A: real,B: real > int,C: real > int] :
% 9.49/5.28        ( ( finite_finite_real @ S2 )
% 9.49/5.28       => ( ( ( member_real @ A @ S2 )
% 9.49/5.28           => ( ( groups1932886352136224148al_int
% 9.49/5.28                @ ^ [K3: real] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( plus_plus_int @ ( B @ A ) @ ( groups1932886352136224148al_int @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
% 9.49/5.28          & ( ~ ( member_real @ A @ S2 )
% 9.49/5.28           => ( ( groups1932886352136224148al_int
% 9.49/5.28                @ ^ [K3: real] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( groups1932886352136224148al_int @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5869_sum_Odelta__remove,axiom,
% 9.49/5.28      ! [S2: set_int,A: int,B: int > real,C: int > real] :
% 9.49/5.28        ( ( finite_finite_int @ S2 )
% 9.49/5.28       => ( ( ( member_int @ A @ S2 )
% 9.49/5.28           => ( ( groups8778361861064173332t_real
% 9.49/5.28                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( plus_plus_real @ ( B @ A ) @ ( groups8778361861064173332t_real @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
% 9.49/5.28          & ( ~ ( member_int @ A @ S2 )
% 9.49/5.28           => ( ( groups8778361861064173332t_real
% 9.49/5.28                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( groups8778361861064173332t_real @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5870_sum_Odelta__remove,axiom,
% 9.49/5.28      ! [S2: set_int,A: int,B: int > rat,C: int > rat] :
% 9.49/5.28        ( ( finite_finite_int @ S2 )
% 9.49/5.28       => ( ( ( member_int @ A @ S2 )
% 9.49/5.28           => ( ( groups3906332499630173760nt_rat
% 9.49/5.28                @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( plus_plus_rat @ ( B @ A ) @ ( groups3906332499630173760nt_rat @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
% 9.49/5.28          & ( ~ ( member_int @ A @ S2 )
% 9.49/5.28           => ( ( groups3906332499630173760nt_rat
% 9.49/5.28                @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.49/5.28                @ S2 )
% 9.49/5.28              = ( groups3906332499630173760nt_rat @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5871_infinite__finite__induct,axiom,
% 9.49/5.28      ! [P: set_Pr4329608150637261639at_nat > $o,A2: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ! [A5: set_Pr4329608150637261639at_nat] :
% 9.49/5.28            ( ~ ( finite4343798906461161616at_nat @ A5 )
% 9.49/5.28           => ( P @ A5 ) )
% 9.49/5.28       => ( ( P @ bot_bo228742789529271731at_nat )
% 9.49/5.28         => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
% 9.49/5.28                ( ( finite4343798906461161616at_nat @ F4 )
% 9.49/5.28               => ( ~ ( member8757157785044589968at_nat @ X4 @ F4 )
% 9.49/5.28                 => ( ( P @ F4 )
% 9.49/5.28                   => ( P @ ( insert9069300056098147895at_nat @ X4 @ F4 ) ) ) ) )
% 9.49/5.28           => ( P @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5872_infinite__finite__induct,axiom,
% 9.49/5.28      ! [P: set_set_nat > $o,A2: set_set_nat] :
% 9.49/5.28        ( ! [A5: set_set_nat] :
% 9.49/5.28            ( ~ ( finite1152437895449049373et_nat @ A5 )
% 9.49/5.28           => ( P @ A5 ) )
% 9.49/5.28       => ( ( P @ bot_bot_set_set_nat )
% 9.49/5.28         => ( ! [X4: set_nat,F4: set_set_nat] :
% 9.49/5.28                ( ( finite1152437895449049373et_nat @ F4 )
% 9.49/5.28               => ( ~ ( member_set_nat @ X4 @ F4 )
% 9.49/5.28                 => ( ( P @ F4 )
% 9.49/5.28                   => ( P @ ( insert_set_nat @ X4 @ F4 ) ) ) ) )
% 9.49/5.28           => ( P @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5873_infinite__finite__induct,axiom,
% 9.49/5.28      ! [P: set_complex > $o,A2: set_complex] :
% 9.49/5.28        ( ! [A5: set_complex] :
% 9.49/5.28            ( ~ ( finite3207457112153483333omplex @ A5 )
% 9.49/5.28           => ( P @ A5 ) )
% 9.49/5.28       => ( ( P @ bot_bot_set_complex )
% 9.49/5.28         => ( ! [X4: complex,F4: set_complex] :
% 9.49/5.28                ( ( finite3207457112153483333omplex @ F4 )
% 9.49/5.28               => ( ~ ( member_complex @ X4 @ F4 )
% 9.49/5.28                 => ( ( P @ F4 )
% 9.49/5.28                   => ( P @ ( insert_complex @ X4 @ F4 ) ) ) ) )
% 9.49/5.28           => ( P @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5874_infinite__finite__induct,axiom,
% 9.49/5.28      ! [P: set_Pr1261947904930325089at_nat > $o,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ! [A5: set_Pr1261947904930325089at_nat] :
% 9.49/5.28            ( ~ ( finite6177210948735845034at_nat @ A5 )
% 9.49/5.28           => ( P @ A5 ) )
% 9.49/5.28       => ( ( P @ bot_bo2099793752762293965at_nat )
% 9.49/5.28         => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 9.49/5.28                ( ( finite6177210948735845034at_nat @ F4 )
% 9.49/5.28               => ( ~ ( member8440522571783428010at_nat @ X4 @ F4 )
% 9.49/5.28                 => ( ( P @ F4 )
% 9.49/5.28                   => ( P @ ( insert8211810215607154385at_nat @ X4 @ F4 ) ) ) ) )
% 9.49/5.28           => ( P @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5875_infinite__finite__induct,axiom,
% 9.49/5.28      ! [P: set_real > $o,A2: set_real] :
% 9.49/5.28        ( ! [A5: set_real] :
% 9.49/5.28            ( ~ ( finite_finite_real @ A5 )
% 9.49/5.28           => ( P @ A5 ) )
% 9.49/5.28       => ( ( P @ bot_bot_set_real )
% 9.49/5.28         => ( ! [X4: real,F4: set_real] :
% 9.49/5.28                ( ( finite_finite_real @ F4 )
% 9.49/5.28               => ( ~ ( member_real @ X4 @ F4 )
% 9.49/5.28                 => ( ( P @ F4 )
% 9.49/5.28                   => ( P @ ( insert_real @ X4 @ F4 ) ) ) ) )
% 9.49/5.28           => ( P @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5876_infinite__finite__induct,axiom,
% 9.49/5.28      ! [P: set_nat > $o,A2: set_nat] :
% 9.49/5.28        ( ! [A5: set_nat] :
% 9.49/5.28            ( ~ ( finite_finite_nat @ A5 )
% 9.49/5.28           => ( P @ A5 ) )
% 9.49/5.28       => ( ( P @ bot_bot_set_nat )
% 9.49/5.28         => ( ! [X4: nat,F4: set_nat] :
% 9.49/5.28                ( ( finite_finite_nat @ F4 )
% 9.49/5.28               => ( ~ ( member_nat @ X4 @ F4 )
% 9.49/5.28                 => ( ( P @ F4 )
% 9.49/5.28                   => ( P @ ( insert_nat @ X4 @ F4 ) ) ) ) )
% 9.49/5.28           => ( P @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5877_infinite__finite__induct,axiom,
% 9.49/5.28      ! [P: set_int > $o,A2: set_int] :
% 9.49/5.28        ( ! [A5: set_int] :
% 9.49/5.28            ( ~ ( finite_finite_int @ A5 )
% 9.49/5.28           => ( P @ A5 ) )
% 9.49/5.28       => ( ( P @ bot_bot_set_int )
% 9.49/5.28         => ( ! [X4: int,F4: set_int] :
% 9.49/5.28                ( ( finite_finite_int @ F4 )
% 9.49/5.28               => ( ~ ( member_int @ X4 @ F4 )
% 9.49/5.28                 => ( ( P @ F4 )
% 9.49/5.28                   => ( P @ ( insert_int @ X4 @ F4 ) ) ) ) )
% 9.49/5.28           => ( P @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5878_finite__ne__induct,axiom,
% 9.49/5.28      ! [F3: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 9.49/5.28        ( ( finite4343798906461161616at_nat @ F3 )
% 9.49/5.28       => ( ( F3 != bot_bo228742789529271731at_nat )
% 9.49/5.28         => ( ! [X4: produc3843707927480180839at_nat] : ( P @ ( insert9069300056098147895at_nat @ X4 @ bot_bo228742789529271731at_nat ) )
% 9.49/5.28           => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
% 9.49/5.28                  ( ( finite4343798906461161616at_nat @ F4 )
% 9.49/5.28                 => ( ( F4 != bot_bo228742789529271731at_nat )
% 9.49/5.28                   => ( ~ ( member8757157785044589968at_nat @ X4 @ F4 )
% 9.49/5.28                     => ( ( P @ F4 )
% 9.49/5.28                       => ( P @ ( insert9069300056098147895at_nat @ X4 @ F4 ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5879_finite__ne__induct,axiom,
% 9.49/5.28      ! [F3: set_set_nat,P: set_set_nat > $o] :
% 9.49/5.28        ( ( finite1152437895449049373et_nat @ F3 )
% 9.49/5.28       => ( ( F3 != bot_bot_set_set_nat )
% 9.49/5.28         => ( ! [X4: set_nat] : ( P @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
% 9.49/5.28           => ( ! [X4: set_nat,F4: set_set_nat] :
% 9.49/5.28                  ( ( finite1152437895449049373et_nat @ F4 )
% 9.49/5.28                 => ( ( F4 != bot_bot_set_set_nat )
% 9.49/5.28                   => ( ~ ( member_set_nat @ X4 @ F4 )
% 9.49/5.28                     => ( ( P @ F4 )
% 9.49/5.28                       => ( P @ ( insert_set_nat @ X4 @ F4 ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5880_finite__ne__induct,axiom,
% 9.49/5.28      ! [F3: set_complex,P: set_complex > $o] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ F3 )
% 9.49/5.28       => ( ( F3 != bot_bot_set_complex )
% 9.49/5.28         => ( ! [X4: complex] : ( P @ ( insert_complex @ X4 @ bot_bot_set_complex ) )
% 9.49/5.28           => ( ! [X4: complex,F4: set_complex] :
% 9.49/5.28                  ( ( finite3207457112153483333omplex @ F4 )
% 9.49/5.28                 => ( ( F4 != bot_bot_set_complex )
% 9.49/5.28                   => ( ~ ( member_complex @ X4 @ F4 )
% 9.49/5.28                     => ( ( P @ F4 )
% 9.49/5.28                       => ( P @ ( insert_complex @ X4 @ F4 ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5881_finite__ne__induct,axiom,
% 9.49/5.28      ! [F3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.49/5.28        ( ( finite6177210948735845034at_nat @ F3 )
% 9.49/5.28       => ( ( F3 != bot_bo2099793752762293965at_nat )
% 9.49/5.28         => ( ! [X4: product_prod_nat_nat] : ( P @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) )
% 9.49/5.28           => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 9.49/5.28                  ( ( finite6177210948735845034at_nat @ F4 )
% 9.49/5.28                 => ( ( F4 != bot_bo2099793752762293965at_nat )
% 9.49/5.28                   => ( ~ ( member8440522571783428010at_nat @ X4 @ F4 )
% 9.49/5.28                     => ( ( P @ F4 )
% 9.49/5.28                       => ( P @ ( insert8211810215607154385at_nat @ X4 @ F4 ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5882_finite__ne__induct,axiom,
% 9.49/5.28      ! [F3: set_real,P: set_real > $o] :
% 9.49/5.28        ( ( finite_finite_real @ F3 )
% 9.49/5.28       => ( ( F3 != bot_bot_set_real )
% 9.49/5.28         => ( ! [X4: real] : ( P @ ( insert_real @ X4 @ bot_bot_set_real ) )
% 9.49/5.28           => ( ! [X4: real,F4: set_real] :
% 9.49/5.28                  ( ( finite_finite_real @ F4 )
% 9.49/5.28                 => ( ( F4 != bot_bot_set_real )
% 9.49/5.28                   => ( ~ ( member_real @ X4 @ F4 )
% 9.49/5.28                     => ( ( P @ F4 )
% 9.49/5.28                       => ( P @ ( insert_real @ X4 @ F4 ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5883_finite__ne__induct,axiom,
% 9.49/5.28      ! [F3: set_nat,P: set_nat > $o] :
% 9.49/5.28        ( ( finite_finite_nat @ F3 )
% 9.49/5.28       => ( ( F3 != bot_bot_set_nat )
% 9.49/5.28         => ( ! [X4: nat] : ( P @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
% 9.49/5.28           => ( ! [X4: nat,F4: set_nat] :
% 9.49/5.28                  ( ( finite_finite_nat @ F4 )
% 9.49/5.28                 => ( ( F4 != bot_bot_set_nat )
% 9.49/5.28                   => ( ~ ( member_nat @ X4 @ F4 )
% 9.49/5.28                     => ( ( P @ F4 )
% 9.49/5.28                       => ( P @ ( insert_nat @ X4 @ F4 ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5884_finite__ne__induct,axiom,
% 9.49/5.28      ! [F3: set_int,P: set_int > $o] :
% 9.49/5.28        ( ( finite_finite_int @ F3 )
% 9.49/5.28       => ( ( F3 != bot_bot_set_int )
% 9.49/5.28         => ( ! [X4: int] : ( P @ ( insert_int @ X4 @ bot_bot_set_int ) )
% 9.49/5.28           => ( ! [X4: int,F4: set_int] :
% 9.49/5.28                  ( ( finite_finite_int @ F4 )
% 9.49/5.28                 => ( ( F4 != bot_bot_set_int )
% 9.49/5.28                   => ( ~ ( member_int @ X4 @ F4 )
% 9.49/5.28                     => ( ( P @ F4 )
% 9.49/5.28                       => ( P @ ( insert_int @ X4 @ F4 ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5885_finite__induct,axiom,
% 9.49/5.28      ! [F3: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 9.49/5.28        ( ( finite4343798906461161616at_nat @ F3 )
% 9.49/5.28       => ( ( P @ bot_bo228742789529271731at_nat )
% 9.49/5.28         => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
% 9.49/5.28                ( ( finite4343798906461161616at_nat @ F4 )
% 9.49/5.28               => ( ~ ( member8757157785044589968at_nat @ X4 @ F4 )
% 9.49/5.28                 => ( ( P @ F4 )
% 9.49/5.28                   => ( P @ ( insert9069300056098147895at_nat @ X4 @ F4 ) ) ) ) )
% 9.49/5.28           => ( P @ F3 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5886_finite__induct,axiom,
% 9.49/5.28      ! [F3: set_set_nat,P: set_set_nat > $o] :
% 9.49/5.28        ( ( finite1152437895449049373et_nat @ F3 )
% 9.49/5.28       => ( ( P @ bot_bot_set_set_nat )
% 9.49/5.28         => ( ! [X4: set_nat,F4: set_set_nat] :
% 9.49/5.28                ( ( finite1152437895449049373et_nat @ F4 )
% 9.49/5.28               => ( ~ ( member_set_nat @ X4 @ F4 )
% 9.49/5.28                 => ( ( P @ F4 )
% 9.49/5.28                   => ( P @ ( insert_set_nat @ X4 @ F4 ) ) ) ) )
% 9.49/5.28           => ( P @ F3 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5887_finite__induct,axiom,
% 9.49/5.28      ! [F3: set_complex,P: set_complex > $o] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ F3 )
% 9.49/5.28       => ( ( P @ bot_bot_set_complex )
% 9.49/5.28         => ( ! [X4: complex,F4: set_complex] :
% 9.49/5.28                ( ( finite3207457112153483333omplex @ F4 )
% 9.49/5.28               => ( ~ ( member_complex @ X4 @ F4 )
% 9.49/5.28                 => ( ( P @ F4 )
% 9.49/5.28                   => ( P @ ( insert_complex @ X4 @ F4 ) ) ) ) )
% 9.49/5.28           => ( P @ F3 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5888_finite__induct,axiom,
% 9.49/5.28      ! [F3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.49/5.28        ( ( finite6177210948735845034at_nat @ F3 )
% 9.49/5.28       => ( ( P @ bot_bo2099793752762293965at_nat )
% 9.49/5.28         => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 9.49/5.28                ( ( finite6177210948735845034at_nat @ F4 )
% 9.49/5.28               => ( ~ ( member8440522571783428010at_nat @ X4 @ F4 )
% 9.49/5.28                 => ( ( P @ F4 )
% 9.49/5.28                   => ( P @ ( insert8211810215607154385at_nat @ X4 @ F4 ) ) ) ) )
% 9.49/5.28           => ( P @ F3 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5889_finite__induct,axiom,
% 9.49/5.28      ! [F3: set_real,P: set_real > $o] :
% 9.49/5.28        ( ( finite_finite_real @ F3 )
% 9.49/5.28       => ( ( P @ bot_bot_set_real )
% 9.49/5.28         => ( ! [X4: real,F4: set_real] :
% 9.49/5.28                ( ( finite_finite_real @ F4 )
% 9.49/5.28               => ( ~ ( member_real @ X4 @ F4 )
% 9.49/5.28                 => ( ( P @ F4 )
% 9.49/5.28                   => ( P @ ( insert_real @ X4 @ F4 ) ) ) ) )
% 9.49/5.28           => ( P @ F3 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5890_finite__induct,axiom,
% 9.49/5.28      ! [F3: set_nat,P: set_nat > $o] :
% 9.49/5.28        ( ( finite_finite_nat @ F3 )
% 9.49/5.28       => ( ( P @ bot_bot_set_nat )
% 9.49/5.28         => ( ! [X4: nat,F4: set_nat] :
% 9.49/5.28                ( ( finite_finite_nat @ F4 )
% 9.49/5.28               => ( ~ ( member_nat @ X4 @ F4 )
% 9.49/5.28                 => ( ( P @ F4 )
% 9.49/5.28                   => ( P @ ( insert_nat @ X4 @ F4 ) ) ) ) )
% 9.49/5.28           => ( P @ F3 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5891_finite__induct,axiom,
% 9.49/5.28      ! [F3: set_int,P: set_int > $o] :
% 9.49/5.28        ( ( finite_finite_int @ F3 )
% 9.49/5.28       => ( ( P @ bot_bot_set_int )
% 9.49/5.28         => ( ! [X4: int,F4: set_int] :
% 9.49/5.28                ( ( finite_finite_int @ F4 )
% 9.49/5.28               => ( ~ ( member_int @ X4 @ F4 )
% 9.49/5.28                 => ( ( P @ F4 )
% 9.49/5.28                   => ( P @ ( insert_int @ X4 @ F4 ) ) ) ) )
% 9.49/5.28           => ( P @ F3 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5892_finite_Osimps,axiom,
% 9.49/5.28      ( finite4343798906461161616at_nat
% 9.49/5.28      = ( ^ [A4: set_Pr4329608150637261639at_nat] :
% 9.49/5.28            ( ( A4 = bot_bo228742789529271731at_nat )
% 9.49/5.28            | ? [A6: set_Pr4329608150637261639at_nat,B4: produc3843707927480180839at_nat] :
% 9.49/5.28                ( ( A4
% 9.49/5.28                  = ( insert9069300056098147895at_nat @ B4 @ A6 ) )
% 9.49/5.28                & ( finite4343798906461161616at_nat @ A6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5893_finite_Osimps,axiom,
% 9.49/5.28      ( finite3207457112153483333omplex
% 9.49/5.28      = ( ^ [A4: set_complex] :
% 9.49/5.28            ( ( A4 = bot_bot_set_complex )
% 9.49/5.28            | ? [A6: set_complex,B4: complex] :
% 9.49/5.28                ( ( A4
% 9.49/5.28                  = ( insert_complex @ B4 @ A6 ) )
% 9.49/5.28                & ( finite3207457112153483333omplex @ A6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5894_finite_Osimps,axiom,
% 9.49/5.28      ( finite6177210948735845034at_nat
% 9.49/5.28      = ( ^ [A4: set_Pr1261947904930325089at_nat] :
% 9.49/5.28            ( ( A4 = bot_bo2099793752762293965at_nat )
% 9.49/5.28            | ? [A6: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat] :
% 9.49/5.28                ( ( A4
% 9.49/5.28                  = ( insert8211810215607154385at_nat @ B4 @ A6 ) )
% 9.49/5.28                & ( finite6177210948735845034at_nat @ A6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5895_finite_Osimps,axiom,
% 9.49/5.28      ( finite_finite_real
% 9.49/5.28      = ( ^ [A4: set_real] :
% 9.49/5.28            ( ( A4 = bot_bot_set_real )
% 9.49/5.28            | ? [A6: set_real,B4: real] :
% 9.49/5.28                ( ( A4
% 9.49/5.28                  = ( insert_real @ B4 @ A6 ) )
% 9.49/5.28                & ( finite_finite_real @ A6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5896_finite_Osimps,axiom,
% 9.49/5.28      ( finite_finite_nat
% 9.49/5.28      = ( ^ [A4: set_nat] :
% 9.49/5.28            ( ( A4 = bot_bot_set_nat )
% 9.49/5.28            | ? [A6: set_nat,B4: nat] :
% 9.49/5.28                ( ( A4
% 9.49/5.28                  = ( insert_nat @ B4 @ A6 ) )
% 9.49/5.28                & ( finite_finite_nat @ A6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5897_finite_Osimps,axiom,
% 9.49/5.28      ( finite_finite_int
% 9.49/5.28      = ( ^ [A4: set_int] :
% 9.49/5.28            ( ( A4 = bot_bot_set_int )
% 9.49/5.28            | ? [A6: set_int,B4: int] :
% 9.49/5.28                ( ( A4
% 9.49/5.28                  = ( insert_int @ B4 @ A6 ) )
% 9.49/5.28                & ( finite_finite_int @ A6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5898_finite_Ocases,axiom,
% 9.49/5.28      ! [A: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ( finite4343798906461161616at_nat @ A )
% 9.49/5.28       => ( ( A != bot_bo228742789529271731at_nat )
% 9.49/5.28         => ~ ! [A5: set_Pr4329608150637261639at_nat] :
% 9.49/5.28                ( ? [A3: produc3843707927480180839at_nat] :
% 9.49/5.28                    ( A
% 9.49/5.28                    = ( insert9069300056098147895at_nat @ A3 @ A5 ) )
% 9.49/5.28               => ~ ( finite4343798906461161616at_nat @ A5 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5899_finite_Ocases,axiom,
% 9.49/5.28      ! [A: set_complex] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A )
% 9.49/5.28       => ( ( A != bot_bot_set_complex )
% 9.49/5.28         => ~ ! [A5: set_complex] :
% 9.49/5.28                ( ? [A3: complex] :
% 9.49/5.28                    ( A
% 9.49/5.28                    = ( insert_complex @ A3 @ A5 ) )
% 9.49/5.28               => ~ ( finite3207457112153483333omplex @ A5 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5900_finite_Ocases,axiom,
% 9.49/5.28      ! [A: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ( finite6177210948735845034at_nat @ A )
% 9.49/5.28       => ( ( A != bot_bo2099793752762293965at_nat )
% 9.49/5.28         => ~ ! [A5: set_Pr1261947904930325089at_nat] :
% 9.49/5.28                ( ? [A3: product_prod_nat_nat] :
% 9.49/5.28                    ( A
% 9.49/5.28                    = ( insert8211810215607154385at_nat @ A3 @ A5 ) )
% 9.49/5.28               => ~ ( finite6177210948735845034at_nat @ A5 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5901_finite_Ocases,axiom,
% 9.49/5.28      ! [A: set_real] :
% 9.49/5.28        ( ( finite_finite_real @ A )
% 9.49/5.28       => ( ( A != bot_bot_set_real )
% 9.49/5.28         => ~ ! [A5: set_real] :
% 9.49/5.28                ( ? [A3: real] :
% 9.49/5.28                    ( A
% 9.49/5.28                    = ( insert_real @ A3 @ A5 ) )
% 9.49/5.28               => ~ ( finite_finite_real @ A5 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5902_finite_Ocases,axiom,
% 9.49/5.28      ! [A: set_nat] :
% 9.49/5.28        ( ( finite_finite_nat @ A )
% 9.49/5.28       => ( ( A != bot_bot_set_nat )
% 9.49/5.28         => ~ ! [A5: set_nat] :
% 9.49/5.28                ( ? [A3: nat] :
% 9.49/5.28                    ( A
% 9.49/5.28                    = ( insert_nat @ A3 @ A5 ) )
% 9.49/5.28               => ~ ( finite_finite_nat @ A5 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5903_finite_Ocases,axiom,
% 9.49/5.28      ! [A: set_int] :
% 9.49/5.28        ( ( finite_finite_int @ A )
% 9.49/5.28       => ( ( A != bot_bot_set_int )
% 9.49/5.28         => ~ ! [A5: set_int] :
% 9.49/5.28                ( ? [A3: int] :
% 9.49/5.28                    ( A
% 9.49/5.28                    = ( insert_int @ A3 @ A5 ) )
% 9.49/5.28               => ~ ( finite_finite_int @ A5 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5904_subset__singletonD,axiom,
% 9.49/5.28      ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
% 9.49/5.28        ( ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
% 9.49/5.28       => ( ( A2 = bot_bo228742789529271731at_nat )
% 9.49/5.28          | ( A2
% 9.49/5.28            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5905_subset__singletonD,axiom,
% 9.49/5.28      ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
% 9.49/5.28        ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
% 9.49/5.28       => ( ( A2 = bot_bo2099793752762293965at_nat )
% 9.49/5.28          | ( A2
% 9.49/5.28            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5906_subset__singletonD,axiom,
% 9.49/5.28      ! [A2: set_real,X: real] :
% 9.49/5.28        ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) )
% 9.49/5.28       => ( ( A2 = bot_bot_set_real )
% 9.49/5.28          | ( A2
% 9.49/5.28            = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5907_subset__singletonD,axiom,
% 9.49/5.28      ! [A2: set_nat,X: nat] :
% 9.49/5.28        ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 9.49/5.28       => ( ( A2 = bot_bot_set_nat )
% 9.49/5.28          | ( A2
% 9.49/5.28            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5908_subset__singletonD,axiom,
% 9.49/5.28      ! [A2: set_int,X: int] :
% 9.49/5.28        ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) )
% 9.49/5.28       => ( ( A2 = bot_bot_set_int )
% 9.49/5.28          | ( A2
% 9.49/5.28            = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5909_subset__singleton__iff,axiom,
% 9.49/5.28      ! [X8: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
% 9.49/5.28        ( ( ord_le1268244103169919719at_nat @ X8 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
% 9.49/5.28        = ( ( X8 = bot_bo228742789529271731at_nat )
% 9.49/5.28          | ( X8
% 9.49/5.28            = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5910_subset__singleton__iff,axiom,
% 9.49/5.28      ! [X8: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
% 9.49/5.28        ( ( ord_le3146513528884898305at_nat @ X8 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
% 9.49/5.28        = ( ( X8 = bot_bo2099793752762293965at_nat )
% 9.49/5.28          | ( X8
% 9.49/5.28            = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5911_subset__singleton__iff,axiom,
% 9.49/5.28      ! [X8: set_real,A: real] :
% 9.49/5.28        ( ( ord_less_eq_set_real @ X8 @ ( insert_real @ A @ bot_bot_set_real ) )
% 9.49/5.28        = ( ( X8 = bot_bot_set_real )
% 9.49/5.28          | ( X8
% 9.49/5.28            = ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5912_subset__singleton__iff,axiom,
% 9.49/5.28      ! [X8: set_nat,A: nat] :
% 9.49/5.28        ( ( ord_less_eq_set_nat @ X8 @ ( insert_nat @ A @ bot_bot_set_nat ) )
% 9.49/5.28        = ( ( X8 = bot_bot_set_nat )
% 9.49/5.28          | ( X8
% 9.49/5.28            = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5913_subset__singleton__iff,axiom,
% 9.49/5.28      ! [X8: set_int,A: int] :
% 9.49/5.28        ( ( ord_less_eq_set_int @ X8 @ ( insert_int @ A @ bot_bot_set_int ) )
% 9.49/5.28        = ( ( X8 = bot_bot_set_int )
% 9.49/5.28          | ( X8
% 9.49/5.28            = ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5914_atLeastAtMost__singleton_H,axiom,
% 9.49/5.28      ! [A: nat,B: nat] :
% 9.49/5.28        ( ( A = B )
% 9.49/5.28       => ( ( set_or1269000886237332187st_nat @ A @ B )
% 9.49/5.28          = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5915_atLeastAtMost__singleton_H,axiom,
% 9.49/5.28      ! [A: int,B: int] :
% 9.49/5.28        ( ( A = B )
% 9.49/5.28       => ( ( set_or1266510415728281911st_int @ A @ B )
% 9.49/5.28          = ( insert_int @ A @ bot_bot_set_int ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5916_atLeastAtMost__singleton_H,axiom,
% 9.49/5.28      ! [A: real,B: real] :
% 9.49/5.28        ( ( A = B )
% 9.49/5.28       => ( ( set_or1222579329274155063t_real @ A @ B )
% 9.49/5.28          = ( insert_real @ A @ bot_bot_set_real ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5917_Diff__insert__absorb,axiom,
% 9.49/5.28      ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ~ ( member8757157785044589968at_nat @ X @ A2 )
% 9.49/5.28       => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A2 ) @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
% 9.49/5.28          = A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5918_Diff__insert__absorb,axiom,
% 9.49/5.28      ! [X: set_nat,A2: set_set_nat] :
% 9.49/5.28        ( ~ ( member_set_nat @ X @ A2 )
% 9.49/5.28       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 9.49/5.28          = A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5919_Diff__insert__absorb,axiom,
% 9.49/5.28      ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ~ ( member8440522571783428010at_nat @ X @ A2 )
% 9.49/5.28       => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
% 9.49/5.28          = A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5920_Diff__insert__absorb,axiom,
% 9.49/5.28      ! [X: real,A2: set_real] :
% 9.49/5.28        ( ~ ( member_real @ X @ A2 )
% 9.49/5.28       => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ ( insert_real @ X @ bot_bot_set_real ) )
% 9.49/5.28          = A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5921_Diff__insert__absorb,axiom,
% 9.49/5.28      ! [X: int,A2: set_int] :
% 9.49/5.28        ( ~ ( member_int @ X @ A2 )
% 9.49/5.28       => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) )
% 9.49/5.28          = A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5922_Diff__insert__absorb,axiom,
% 9.49/5.28      ! [X: nat,A2: set_nat] :
% 9.49/5.28        ( ~ ( member_nat @ X @ A2 )
% 9.49/5.28       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 9.49/5.28          = A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5923_Diff__insert2,axiom,
% 9.49/5.28      ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) )
% 9.49/5.28        = ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ B2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5924_Diff__insert2,axiom,
% 9.49/5.28      ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) )
% 9.49/5.28        = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ B2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5925_Diff__insert2,axiom,
% 9.49/5.28      ! [A2: set_real,A: real,B2: set_real] :
% 9.49/5.28        ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
% 9.49/5.28        = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) @ B2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5926_Diff__insert2,axiom,
% 9.49/5.28      ! [A2: set_int,A: int,B2: set_int] :
% 9.49/5.28        ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
% 9.49/5.28        = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5927_Diff__insert2,axiom,
% 9.49/5.28      ! [A2: set_nat,A: nat,B2: set_nat] :
% 9.49/5.28        ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
% 9.49/5.28        = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5928_insert__Diff,axiom,
% 9.49/5.28      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ( member8757157785044589968at_nat @ A @ A2 )
% 9.49/5.28       => ( ( insert9069300056098147895at_nat @ A @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 9.49/5.28          = A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5929_insert__Diff,axiom,
% 9.49/5.28      ! [A: set_nat,A2: set_set_nat] :
% 9.49/5.28        ( ( member_set_nat @ A @ A2 )
% 9.49/5.28       => ( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 9.49/5.28          = A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5930_insert__Diff,axiom,
% 9.49/5.28      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ( member8440522571783428010at_nat @ A @ A2 )
% 9.49/5.28       => ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 9.49/5.28          = A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5931_insert__Diff,axiom,
% 9.49/5.28      ! [A: real,A2: set_real] :
% 9.49/5.28        ( ( member_real @ A @ A2 )
% 9.49/5.28       => ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.49/5.28          = A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5932_insert__Diff,axiom,
% 9.49/5.28      ! [A: int,A2: set_int] :
% 9.49/5.28        ( ( member_int @ A @ A2 )
% 9.49/5.28       => ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.49/5.28          = A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5933_insert__Diff,axiom,
% 9.49/5.28      ! [A: nat,A2: set_nat] :
% 9.49/5.28        ( ( member_nat @ A @ A2 )
% 9.49/5.28       => ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 9.49/5.28          = A2 ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5934_Diff__insert,axiom,
% 9.49/5.28      ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) )
% 9.49/5.28        = ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5935_Diff__insert,axiom,
% 9.49/5.28      ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) )
% 9.49/5.28        = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5936_Diff__insert,axiom,
% 9.49/5.28      ! [A2: set_real,A: real,B2: set_real] :
% 9.49/5.28        ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
% 9.49/5.28        = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ B2 ) @ ( insert_real @ A @ bot_bot_set_real ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5937_Diff__insert,axiom,
% 9.49/5.28      ! [A2: set_int,A: int,B2: set_int] :
% 9.49/5.28        ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
% 9.49/5.28        = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5938_Diff__insert,axiom,
% 9.49/5.28      ! [A2: set_nat,A: nat,B2: set_nat] :
% 9.49/5.28        ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
% 9.49/5.28        = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5939_Compl__insert,axiom,
% 9.49/5.28      ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ X @ A2 ) )
% 9.49/5.28        = ( minus_3314409938677909166at_nat @ ( uminus935396558254630718at_nat @ A2 ) @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5940_Compl__insert,axiom,
% 9.49/5.28      ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) )
% 9.49/5.28        = ( minus_1356011639430497352at_nat @ ( uminus6524753893492686040at_nat @ A2 ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5941_Compl__insert,axiom,
% 9.49/5.28      ! [X: real,A2: set_real] :
% 9.49/5.28        ( ( uminus612125837232591019t_real @ ( insert_real @ X @ A2 ) )
% 9.49/5.28        = ( minus_minus_set_real @ ( uminus612125837232591019t_real @ A2 ) @ ( insert_real @ X @ bot_bot_set_real ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5942_Compl__insert,axiom,
% 9.49/5.28      ! [X: int,A2: set_int] :
% 9.49/5.28        ( ( uminus1532241313380277803et_int @ ( insert_int @ X @ A2 ) )
% 9.49/5.28        = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5943_Compl__insert,axiom,
% 9.49/5.28      ! [X: nat,A2: set_nat] :
% 9.49/5.28        ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A2 ) )
% 9.49/5.28        = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5944_subset__Diff__insert,axiom,
% 9.49/5.28      ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ( ord_le1268244103169919719at_nat @ A2 @ ( minus_3314409938677909166at_nat @ B2 @ ( insert9069300056098147895at_nat @ X @ C2 ) ) )
% 9.49/5.28        = ( ( ord_le1268244103169919719at_nat @ A2 @ ( minus_3314409938677909166at_nat @ B2 @ C2 ) )
% 9.49/5.28          & ~ ( member8757157785044589968at_nat @ X @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5945_subset__Diff__insert,axiom,
% 9.49/5.28      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ( ord_le3146513528884898305at_nat @ A2 @ ( minus_1356011639430497352at_nat @ B2 @ ( insert8211810215607154385at_nat @ X @ C2 ) ) )
% 9.49/5.28        = ( ( ord_le3146513528884898305at_nat @ A2 @ ( minus_1356011639430497352at_nat @ B2 @ C2 ) )
% 9.49/5.28          & ~ ( member8440522571783428010at_nat @ X @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5946_subset__Diff__insert,axiom,
% 9.49/5.28      ! [A2: set_real,B2: set_real,X: real,C2: set_real] :
% 9.49/5.28        ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ ( insert_real @ X @ C2 ) ) )
% 9.49/5.28        = ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ C2 ) )
% 9.49/5.28          & ~ ( member_real @ X @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5947_subset__Diff__insert,axiom,
% 9.49/5.28      ! [A2: set_set_nat,B2: set_set_nat,X: set_nat,C2: set_set_nat] :
% 9.49/5.28        ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ ( insert_set_nat @ X @ C2 ) ) )
% 9.49/5.28        = ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ C2 ) )
% 9.49/5.28          & ~ ( member_set_nat @ X @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5948_subset__Diff__insert,axiom,
% 9.49/5.28      ! [A2: set_nat,B2: set_nat,X: nat,C2: set_nat] :
% 9.49/5.28        ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X @ C2 ) ) )
% 9.49/5.28        = ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C2 ) )
% 9.49/5.28          & ~ ( member_nat @ X @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5949_subset__Diff__insert,axiom,
% 9.49/5.28      ! [A2: set_int,B2: set_int,X: int,C2: set_int] :
% 9.49/5.28        ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ ( insert_int @ X @ C2 ) ) )
% 9.49/5.28        = ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ C2 ) )
% 9.49/5.28          & ~ ( member_int @ X @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5950_sum_Ointer__filter,axiom,
% 9.49/5.28      ! [A2: set_real,G: real > complex,P: real > $o] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( groups5754745047067104278omplex @ G
% 9.49/5.28            @ ( collect_real
% 9.49/5.28              @ ^ [X3: real] :
% 9.49/5.28                  ( ( member_real @ X3 @ A2 )
% 9.49/5.28                  & ( P @ X3 ) ) ) )
% 9.49/5.28          = ( groups5754745047067104278omplex
% 9.49/5.28            @ ^ [X3: real] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_complex )
% 9.49/5.28            @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5951_sum_Ointer__filter,axiom,
% 9.49/5.28      ! [A2: set_nat,G: nat > complex,P: nat > $o] :
% 9.49/5.28        ( ( finite_finite_nat @ A2 )
% 9.49/5.28       => ( ( groups2073611262835488442omplex @ G
% 9.49/5.28            @ ( collect_nat
% 9.49/5.28              @ ^ [X3: nat] :
% 9.49/5.28                  ( ( member_nat @ X3 @ A2 )
% 9.49/5.28                  & ( P @ X3 ) ) ) )
% 9.49/5.28          = ( groups2073611262835488442omplex
% 9.49/5.28            @ ^ [X3: nat] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_complex )
% 9.49/5.28            @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5952_sum_Ointer__filter,axiom,
% 9.49/5.28      ! [A2: set_int,G: int > complex,P: int > $o] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( groups3049146728041665814omplex @ G
% 9.49/5.28            @ ( collect_int
% 9.49/5.28              @ ^ [X3: int] :
% 9.49/5.28                  ( ( member_int @ X3 @ A2 )
% 9.49/5.28                  & ( P @ X3 ) ) ) )
% 9.49/5.28          = ( groups3049146728041665814omplex
% 9.49/5.28            @ ^ [X3: int] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_complex )
% 9.49/5.28            @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5953_sum_Ointer__filter,axiom,
% 9.49/5.28      ! [A2: set_real,G: real > real,P: real > $o] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( groups8097168146408367636l_real @ G
% 9.49/5.28            @ ( collect_real
% 9.49/5.28              @ ^ [X3: real] :
% 9.49/5.28                  ( ( member_real @ X3 @ A2 )
% 9.49/5.28                  & ( P @ X3 ) ) ) )
% 9.49/5.28          = ( groups8097168146408367636l_real
% 9.49/5.28            @ ^ [X3: real] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
% 9.49/5.28            @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5954_sum_Ointer__filter,axiom,
% 9.49/5.28      ! [A2: set_int,G: int > real,P: int > $o] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( groups8778361861064173332t_real @ G
% 9.49/5.28            @ ( collect_int
% 9.49/5.28              @ ^ [X3: int] :
% 9.49/5.28                  ( ( member_int @ X3 @ A2 )
% 9.49/5.28                  & ( P @ X3 ) ) ) )
% 9.49/5.28          = ( groups8778361861064173332t_real
% 9.49/5.28            @ ^ [X3: int] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
% 9.49/5.28            @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5955_sum_Ointer__filter,axiom,
% 9.49/5.28      ! [A2: set_complex,G: complex > real,P: complex > $o] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( groups5808333547571424918x_real @ G
% 9.49/5.28            @ ( collect_complex
% 9.49/5.28              @ ^ [X3: complex] :
% 9.49/5.28                  ( ( member_complex @ X3 @ A2 )
% 9.49/5.28                  & ( P @ X3 ) ) ) )
% 9.49/5.28          = ( groups5808333547571424918x_real
% 9.49/5.28            @ ^ [X3: complex] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
% 9.49/5.28            @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5956_sum_Ointer__filter,axiom,
% 9.49/5.28      ! [A2: set_real,G: real > rat,P: real > $o] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( groups1300246762558778688al_rat @ G
% 9.49/5.28            @ ( collect_real
% 9.49/5.28              @ ^ [X3: real] :
% 9.49/5.28                  ( ( member_real @ X3 @ A2 )
% 9.49/5.28                  & ( P @ X3 ) ) ) )
% 9.49/5.28          = ( groups1300246762558778688al_rat
% 9.49/5.28            @ ^ [X3: real] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
% 9.49/5.28            @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5957_sum_Ointer__filter,axiom,
% 9.49/5.28      ! [A2: set_nat,G: nat > rat,P: nat > $o] :
% 9.49/5.28        ( ( finite_finite_nat @ A2 )
% 9.49/5.28       => ( ( groups2906978787729119204at_rat @ G
% 9.49/5.28            @ ( collect_nat
% 9.49/5.28              @ ^ [X3: nat] :
% 9.49/5.28                  ( ( member_nat @ X3 @ A2 )
% 9.49/5.28                  & ( P @ X3 ) ) ) )
% 9.49/5.28          = ( groups2906978787729119204at_rat
% 9.49/5.28            @ ^ [X3: nat] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
% 9.49/5.28            @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5958_sum_Ointer__filter,axiom,
% 9.49/5.28      ! [A2: set_int,G: int > rat,P: int > $o] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( groups3906332499630173760nt_rat @ G
% 9.49/5.28            @ ( collect_int
% 9.49/5.28              @ ^ [X3: int] :
% 9.49/5.28                  ( ( member_int @ X3 @ A2 )
% 9.49/5.28                  & ( P @ X3 ) ) ) )
% 9.49/5.28          = ( groups3906332499630173760nt_rat
% 9.49/5.28            @ ^ [X3: int] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
% 9.49/5.28            @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5959_sum_Ointer__filter,axiom,
% 9.49/5.28      ! [A2: set_complex,G: complex > rat,P: complex > $o] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( groups5058264527183730370ex_rat @ G
% 9.49/5.28            @ ( collect_complex
% 9.49/5.28              @ ^ [X3: complex] :
% 9.49/5.28                  ( ( member_complex @ X3 @ A2 )
% 9.49/5.28                  & ( P @ X3 ) ) ) )
% 9.49/5.28          = ( groups5058264527183730370ex_rat
% 9.49/5.28            @ ^ [X3: complex] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
% 9.49/5.28            @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5960_member__le__sum,axiom,
% 9.49/5.28      ! [I: complex,A2: set_complex,F: complex > real] :
% 9.49/5.28        ( ( member_complex @ I @ A2 )
% 9.49/5.28       => ( ! [X4: complex] :
% 9.49/5.28              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.49/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.49/5.28         => ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28           => ( ord_less_eq_real @ ( F @ I ) @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5961_member__le__sum,axiom,
% 9.49/5.28      ! [I: real,A2: set_real,F: real > real] :
% 9.49/5.28        ( ( member_real @ I @ A2 )
% 9.49/5.28       => ( ! [X4: real] :
% 9.49/5.28              ( ( member_real @ X4 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.49/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.49/5.28         => ( ( finite_finite_real @ A2 )
% 9.49/5.28           => ( ord_less_eq_real @ ( F @ I ) @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5962_member__le__sum,axiom,
% 9.49/5.28      ! [I: int,A2: set_int,F: int > real] :
% 9.49/5.28        ( ( member_int @ I @ A2 )
% 9.49/5.28       => ( ! [X4: int] :
% 9.49/5.28              ( ( member_int @ X4 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 9.49/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.49/5.28         => ( ( finite_finite_int @ A2 )
% 9.49/5.28           => ( ord_less_eq_real @ ( F @ I ) @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5963_member__le__sum,axiom,
% 9.49/5.28      ! [I: complex,A2: set_complex,F: complex > rat] :
% 9.49/5.28        ( ( member_complex @ I @ A2 )
% 9.49/5.28       => ( ! [X4: complex] :
% 9.49/5.28              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.49/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.49/5.28         => ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28           => ( ord_less_eq_rat @ ( F @ I ) @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5964_member__le__sum,axiom,
% 9.49/5.28      ! [I: real,A2: set_real,F: real > rat] :
% 9.49/5.28        ( ( member_real @ I @ A2 )
% 9.49/5.28       => ( ! [X4: real] :
% 9.49/5.28              ( ( member_real @ X4 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.49/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.49/5.28         => ( ( finite_finite_real @ A2 )
% 9.49/5.28           => ( ord_less_eq_rat @ ( F @ I ) @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5965_member__le__sum,axiom,
% 9.49/5.28      ! [I: int,A2: set_int,F: int > rat] :
% 9.49/5.28        ( ( member_int @ I @ A2 )
% 9.49/5.28       => ( ! [X4: int] :
% 9.49/5.28              ( ( member_int @ X4 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 9.49/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.49/5.28         => ( ( finite_finite_int @ A2 )
% 9.49/5.28           => ( ord_less_eq_rat @ ( F @ I ) @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5966_member__le__sum,axiom,
% 9.49/5.28      ! [I: nat,A2: set_nat,F: nat > rat] :
% 9.49/5.28        ( ( member_nat @ I @ A2 )
% 9.49/5.28       => ( ! [X4: nat] :
% 9.49/5.28              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
% 9.49/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.49/5.28         => ( ( finite_finite_nat @ A2 )
% 9.49/5.28           => ( ord_less_eq_rat @ ( F @ I ) @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5967_member__le__sum,axiom,
% 9.49/5.28      ! [I: complex,A2: set_complex,F: complex > nat] :
% 9.49/5.28        ( ( member_complex @ I @ A2 )
% 9.49/5.28       => ( ! [X4: complex] :
% 9.49/5.28              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.49/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.49/5.28         => ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28           => ( ord_less_eq_nat @ ( F @ I ) @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5968_member__le__sum,axiom,
% 9.49/5.28      ! [I: real,A2: set_real,F: real > nat] :
% 9.49/5.28        ( ( member_real @ I @ A2 )
% 9.49/5.28       => ( ! [X4: real] :
% 9.49/5.28              ( ( member_real @ X4 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.49/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.49/5.28         => ( ( finite_finite_real @ A2 )
% 9.49/5.28           => ( ord_less_eq_nat @ ( F @ I ) @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5969_member__le__sum,axiom,
% 9.49/5.28      ! [I: int,A2: set_int,F: int > nat] :
% 9.49/5.28        ( ( member_int @ I @ A2 )
% 9.49/5.28       => ( ! [X4: int] :
% 9.49/5.28              ( ( member_int @ X4 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 9.49/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.49/5.28         => ( ( finite_finite_int @ A2 )
% 9.49/5.28           => ( ord_less_eq_nat @ ( F @ I ) @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5970_sum__subtractf__nat,axiom,
% 9.49/5.28      ! [A2: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,F: product_prod_nat_nat > nat] :
% 9.49/5.28        ( ! [X4: product_prod_nat_nat] :
% 9.49/5.28            ( ( member8440522571783428010at_nat @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 9.49/5.28       => ( ( groups977919841031483927at_nat
% 9.49/5.28            @ ^ [X3: product_prod_nat_nat] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
% 9.49/5.28            @ A2 )
% 9.49/5.28          = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A2 ) @ ( groups977919841031483927at_nat @ G @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5971_sum__subtractf__nat,axiom,
% 9.49/5.28      ! [A2: set_real,G: real > nat,F: real > nat] :
% 9.49/5.28        ( ! [X4: real] :
% 9.49/5.28            ( ( member_real @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 9.49/5.28       => ( ( groups1935376822645274424al_nat
% 9.49/5.28            @ ^ [X3: real] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
% 9.49/5.28            @ A2 )
% 9.49/5.28          = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5972_sum__subtractf__nat,axiom,
% 9.49/5.28      ! [A2: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
% 9.49/5.28        ( ! [X4: set_nat] :
% 9.49/5.28            ( ( member_set_nat @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 9.49/5.28       => ( ( groups8294997508430121362at_nat
% 9.49/5.28            @ ^ [X3: set_nat] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
% 9.49/5.28            @ A2 )
% 9.49/5.28          = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( groups8294997508430121362at_nat @ G @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5973_sum__subtractf__nat,axiom,
% 9.49/5.28      ! [A2: set_int,G: int > nat,F: int > nat] :
% 9.49/5.28        ( ! [X4: int] :
% 9.49/5.28            ( ( member_int @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 9.49/5.28       => ( ( groups4541462559716669496nt_nat
% 9.49/5.28            @ ^ [X3: int] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
% 9.49/5.28            @ A2 )
% 9.49/5.28          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5974_sum__subtractf__nat,axiom,
% 9.49/5.28      ! [A2: set_nat,G: nat > nat,F: nat > nat] :
% 9.49/5.28        ( ! [X4: nat] :
% 9.49/5.28            ( ( member_nat @ X4 @ A2 )
% 9.49/5.28           => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 9.49/5.28       => ( ( groups3542108847815614940at_nat
% 9.49/5.28            @ ^ [X3: nat] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
% 9.49/5.28            @ A2 )
% 9.49/5.28          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5975_sum__le__included,axiom,
% 9.49/5.28      ! [S: set_int,T: set_int,G: int > real,I: int > int,F: int > real] :
% 9.49/5.28        ( ( finite_finite_int @ S )
% 9.49/5.28       => ( ( finite_finite_int @ T )
% 9.49/5.28         => ( ! [X4: int] :
% 9.49/5.28                ( ( member_int @ X4 @ T )
% 9.49/5.28               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 9.49/5.28           => ( ! [X4: int] :
% 9.49/5.28                  ( ( member_int @ X4 @ S )
% 9.49/5.28                 => ? [Xa: int] :
% 9.49/5.28                      ( ( member_int @ Xa @ T )
% 9.49/5.28                      & ( ( I @ Xa )
% 9.49/5.28                        = X4 )
% 9.49/5.28                      & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 9.49/5.28             => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5976_sum__le__included,axiom,
% 9.49/5.28      ! [S: set_int,T: set_complex,G: complex > real,I: complex > int,F: int > real] :
% 9.49/5.28        ( ( finite_finite_int @ S )
% 9.49/5.28       => ( ( finite3207457112153483333omplex @ T )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ T )
% 9.49/5.28               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 9.49/5.28           => ( ! [X4: int] :
% 9.49/5.28                  ( ( member_int @ X4 @ S )
% 9.49/5.28                 => ? [Xa: complex] :
% 9.49/5.28                      ( ( member_complex @ Xa @ T )
% 9.49/5.28                      & ( ( I @ Xa )
% 9.49/5.28                        = X4 )
% 9.49/5.28                      & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 9.49/5.28             => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5977_sum__le__included,axiom,
% 9.49/5.28      ! [S: set_complex,T: set_int,G: int > real,I: int > complex,F: complex > real] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ S )
% 9.49/5.28       => ( ( finite_finite_int @ T )
% 9.49/5.28         => ( ! [X4: int] :
% 9.49/5.28                ( ( member_int @ X4 @ T )
% 9.49/5.28               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 9.49/5.28           => ( ! [X4: complex] :
% 9.49/5.28                  ( ( member_complex @ X4 @ S )
% 9.49/5.28                 => ? [Xa: int] :
% 9.49/5.28                      ( ( member_int @ Xa @ T )
% 9.49/5.28                      & ( ( I @ Xa )
% 9.49/5.28                        = X4 )
% 9.49/5.28                      & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 9.49/5.28             => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5978_sum__le__included,axiom,
% 9.49/5.28      ! [S: set_complex,T: set_complex,G: complex > real,I: complex > complex,F: complex > real] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ S )
% 9.49/5.28       => ( ( finite3207457112153483333omplex @ T )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ T )
% 9.49/5.28               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 9.49/5.28           => ( ! [X4: complex] :
% 9.49/5.28                  ( ( member_complex @ X4 @ S )
% 9.49/5.28                 => ? [Xa: complex] :
% 9.49/5.28                      ( ( member_complex @ Xa @ T )
% 9.49/5.28                      & ( ( I @ Xa )
% 9.49/5.28                        = X4 )
% 9.49/5.28                      & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 9.49/5.28             => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5979_sum__le__included,axiom,
% 9.49/5.28      ! [S: set_nat,T: set_nat,G: nat > rat,I: nat > nat,F: nat > rat] :
% 9.49/5.28        ( ( finite_finite_nat @ S )
% 9.49/5.28       => ( ( finite_finite_nat @ T )
% 9.49/5.28         => ( ! [X4: nat] :
% 9.49/5.28                ( ( member_nat @ X4 @ T )
% 9.49/5.28               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 9.49/5.28           => ( ! [X4: nat] :
% 9.49/5.28                  ( ( member_nat @ X4 @ S )
% 9.49/5.28                 => ? [Xa: nat] :
% 9.49/5.28                      ( ( member_nat @ Xa @ T )
% 9.49/5.28                      & ( ( I @ Xa )
% 9.49/5.28                        = X4 )
% 9.49/5.28                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 9.49/5.28             => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5980_sum__le__included,axiom,
% 9.49/5.28      ! [S: set_nat,T: set_int,G: int > rat,I: int > nat,F: nat > rat] :
% 9.49/5.28        ( ( finite_finite_nat @ S )
% 9.49/5.28       => ( ( finite_finite_int @ T )
% 9.49/5.28         => ( ! [X4: int] :
% 9.49/5.28                ( ( member_int @ X4 @ T )
% 9.49/5.28               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 9.49/5.28           => ( ! [X4: nat] :
% 9.49/5.28                  ( ( member_nat @ X4 @ S )
% 9.49/5.28                 => ? [Xa: int] :
% 9.49/5.28                      ( ( member_int @ Xa @ T )
% 9.49/5.28                      & ( ( I @ Xa )
% 9.49/5.28                        = X4 )
% 9.49/5.28                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 9.49/5.28             => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5981_sum__le__included,axiom,
% 9.49/5.28      ! [S: set_nat,T: set_complex,G: complex > rat,I: complex > nat,F: nat > rat] :
% 9.49/5.28        ( ( finite_finite_nat @ S )
% 9.49/5.28       => ( ( finite3207457112153483333omplex @ T )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ T )
% 9.49/5.28               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 9.49/5.28           => ( ! [X4: nat] :
% 9.49/5.28                  ( ( member_nat @ X4 @ S )
% 9.49/5.28                 => ? [Xa: complex] :
% 9.49/5.28                      ( ( member_complex @ Xa @ T )
% 9.49/5.28                      & ( ( I @ Xa )
% 9.49/5.28                        = X4 )
% 9.49/5.28                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 9.49/5.28             => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5982_sum__le__included,axiom,
% 9.49/5.28      ! [S: set_int,T: set_nat,G: nat > rat,I: nat > int,F: int > rat] :
% 9.49/5.28        ( ( finite_finite_int @ S )
% 9.49/5.28       => ( ( finite_finite_nat @ T )
% 9.49/5.28         => ( ! [X4: nat] :
% 9.49/5.28                ( ( member_nat @ X4 @ T )
% 9.49/5.28               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 9.49/5.28           => ( ! [X4: int] :
% 9.49/5.28                  ( ( member_int @ X4 @ S )
% 9.49/5.28                 => ? [Xa: nat] :
% 9.49/5.28                      ( ( member_nat @ Xa @ T )
% 9.49/5.28                      & ( ( I @ Xa )
% 9.49/5.28                        = X4 )
% 9.49/5.28                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 9.49/5.28             => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5983_sum__le__included,axiom,
% 9.49/5.28      ! [S: set_int,T: set_int,G: int > rat,I: int > int,F: int > rat] :
% 9.49/5.28        ( ( finite_finite_int @ S )
% 9.49/5.28       => ( ( finite_finite_int @ T )
% 9.49/5.28         => ( ! [X4: int] :
% 9.49/5.28                ( ( member_int @ X4 @ T )
% 9.49/5.28               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 9.49/5.28           => ( ! [X4: int] :
% 9.49/5.28                  ( ( member_int @ X4 @ S )
% 9.49/5.28                 => ? [Xa: int] :
% 9.49/5.28                      ( ( member_int @ Xa @ T )
% 9.49/5.28                      & ( ( I @ Xa )
% 9.49/5.28                        = X4 )
% 9.49/5.28                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 9.49/5.28             => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5984_sum__le__included,axiom,
% 9.49/5.28      ! [S: set_int,T: set_complex,G: complex > rat,I: complex > int,F: int > rat] :
% 9.49/5.28        ( ( finite_finite_int @ S )
% 9.49/5.28       => ( ( finite3207457112153483333omplex @ T )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ T )
% 9.49/5.28               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 9.49/5.28           => ( ! [X4: int] :
% 9.49/5.28                  ( ( member_int @ X4 @ S )
% 9.49/5.28                 => ? [Xa: complex] :
% 9.49/5.28                      ( ( member_complex @ Xa @ T )
% 9.49/5.28                      & ( ( I @ Xa )
% 9.49/5.28                        = X4 )
% 9.49/5.28                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 9.49/5.28             => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5985_sum__nonneg__eq__0__iff,axiom,
% 9.49/5.28      ! [A2: set_real,F: real > real] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ! [X4: real] :
% 9.49/5.28              ( ( member_real @ X4 @ A2 )
% 9.49/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.49/5.28         => ( ( ( groups8097168146408367636l_real @ F @ A2 )
% 9.49/5.28              = zero_zero_real )
% 9.49/5.28            = ( ! [X3: real] :
% 9.49/5.28                  ( ( member_real @ X3 @ A2 )
% 9.49/5.28                 => ( ( F @ X3 )
% 9.49/5.28                    = zero_zero_real ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5986_sum__nonneg__eq__0__iff,axiom,
% 9.49/5.28      ! [A2: set_int,F: int > real] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ! [X4: int] :
% 9.49/5.28              ( ( member_int @ X4 @ A2 )
% 9.49/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.49/5.28         => ( ( ( groups8778361861064173332t_real @ F @ A2 )
% 9.49/5.28              = zero_zero_real )
% 9.49/5.28            = ( ! [X3: int] :
% 9.49/5.28                  ( ( member_int @ X3 @ A2 )
% 9.49/5.28                 => ( ( F @ X3 )
% 9.49/5.28                    = zero_zero_real ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5987_sum__nonneg__eq__0__iff,axiom,
% 9.49/5.28      ! [A2: set_complex,F: complex > real] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ! [X4: complex] :
% 9.49/5.28              ( ( member_complex @ X4 @ A2 )
% 9.49/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.49/5.28         => ( ( ( groups5808333547571424918x_real @ F @ A2 )
% 9.49/5.28              = zero_zero_real )
% 9.49/5.28            = ( ! [X3: complex] :
% 9.49/5.28                  ( ( member_complex @ X3 @ A2 )
% 9.49/5.28                 => ( ( F @ X3 )
% 9.49/5.28                    = zero_zero_real ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5988_sum__nonneg__eq__0__iff,axiom,
% 9.49/5.28      ! [A2: set_real,F: real > rat] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ! [X4: real] :
% 9.49/5.28              ( ( member_real @ X4 @ A2 )
% 9.49/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.49/5.28         => ( ( ( groups1300246762558778688al_rat @ F @ A2 )
% 9.49/5.28              = zero_zero_rat )
% 9.49/5.28            = ( ! [X3: real] :
% 9.49/5.28                  ( ( member_real @ X3 @ A2 )
% 9.49/5.28                 => ( ( F @ X3 )
% 9.49/5.28                    = zero_zero_rat ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5989_sum__nonneg__eq__0__iff,axiom,
% 9.49/5.28      ! [A2: set_nat,F: nat > rat] :
% 9.49/5.28        ( ( finite_finite_nat @ A2 )
% 9.49/5.28       => ( ! [X4: nat] :
% 9.49/5.28              ( ( member_nat @ X4 @ A2 )
% 9.49/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.49/5.28         => ( ( ( groups2906978787729119204at_rat @ F @ A2 )
% 9.49/5.28              = zero_zero_rat )
% 9.49/5.28            = ( ! [X3: nat] :
% 9.49/5.28                  ( ( member_nat @ X3 @ A2 )
% 9.49/5.28                 => ( ( F @ X3 )
% 9.49/5.28                    = zero_zero_rat ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5990_sum__nonneg__eq__0__iff,axiom,
% 9.49/5.28      ! [A2: set_int,F: int > rat] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ! [X4: int] :
% 9.49/5.28              ( ( member_int @ X4 @ A2 )
% 9.49/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.49/5.28         => ( ( ( groups3906332499630173760nt_rat @ F @ A2 )
% 9.49/5.28              = zero_zero_rat )
% 9.49/5.28            = ( ! [X3: int] :
% 9.49/5.28                  ( ( member_int @ X3 @ A2 )
% 9.49/5.28                 => ( ( F @ X3 )
% 9.49/5.28                    = zero_zero_rat ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5991_sum__nonneg__eq__0__iff,axiom,
% 9.49/5.28      ! [A2: set_complex,F: complex > rat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ! [X4: complex] :
% 9.49/5.28              ( ( member_complex @ X4 @ A2 )
% 9.49/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.49/5.28         => ( ( ( groups5058264527183730370ex_rat @ F @ A2 )
% 9.49/5.28              = zero_zero_rat )
% 9.49/5.28            = ( ! [X3: complex] :
% 9.49/5.28                  ( ( member_complex @ X3 @ A2 )
% 9.49/5.28                 => ( ( F @ X3 )
% 9.49/5.28                    = zero_zero_rat ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5992_sum__nonneg__eq__0__iff,axiom,
% 9.49/5.28      ! [A2: set_real,F: real > nat] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ! [X4: real] :
% 9.49/5.28              ( ( member_real @ X4 @ A2 )
% 9.49/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.49/5.28         => ( ( ( groups1935376822645274424al_nat @ F @ A2 )
% 9.49/5.28              = zero_zero_nat )
% 9.49/5.28            = ( ! [X3: real] :
% 9.49/5.28                  ( ( member_real @ X3 @ A2 )
% 9.49/5.28                 => ( ( F @ X3 )
% 9.49/5.28                    = zero_zero_nat ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5993_sum__nonneg__eq__0__iff,axiom,
% 9.49/5.28      ! [A2: set_int,F: int > nat] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ! [X4: int] :
% 9.49/5.28              ( ( member_int @ X4 @ A2 )
% 9.49/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.49/5.28         => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 9.49/5.28              = zero_zero_nat )
% 9.49/5.28            = ( ! [X3: int] :
% 9.49/5.28                  ( ( member_int @ X3 @ A2 )
% 9.49/5.28                 => ( ( F @ X3 )
% 9.49/5.28                    = zero_zero_nat ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5994_sum__nonneg__eq__0__iff,axiom,
% 9.49/5.28      ! [A2: set_complex,F: complex > nat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ! [X4: complex] :
% 9.49/5.28              ( ( member_complex @ X4 @ A2 )
% 9.49/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.49/5.28         => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 9.49/5.28              = zero_zero_nat )
% 9.49/5.28            = ( ! [X3: complex] :
% 9.49/5.28                  ( ( member_complex @ X3 @ A2 )
% 9.49/5.28                 => ( ( F @ X3 )
% 9.49/5.28                    = zero_zero_nat ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5995_sum__strict__mono__ex1,axiom,
% 9.49/5.28      ! [A2: set_int,F: int > real,G: int > real] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ! [X4: int] :
% 9.49/5.28              ( ( member_int @ X4 @ A2 )
% 9.49/5.28             => ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28         => ( ? [X2: int] :
% 9.49/5.28                ( ( member_int @ X2 @ A2 )
% 9.49/5.28                & ( ord_less_real @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 9.49/5.28           => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5996_sum__strict__mono__ex1,axiom,
% 9.49/5.28      ! [A2: set_complex,F: complex > real,G: complex > real] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ! [X4: complex] :
% 9.49/5.28              ( ( member_complex @ X4 @ A2 )
% 9.49/5.28             => ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28         => ( ? [X2: complex] :
% 9.49/5.28                ( ( member_complex @ X2 @ A2 )
% 9.49/5.28                & ( ord_less_real @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 9.49/5.28           => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5997_sum__strict__mono__ex1,axiom,
% 9.49/5.28      ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 9.49/5.28        ( ( finite_finite_nat @ A2 )
% 9.49/5.28       => ( ! [X4: nat] :
% 9.49/5.28              ( ( member_nat @ X4 @ A2 )
% 9.49/5.28             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28         => ( ? [X2: nat] :
% 9.49/5.28                ( ( member_nat @ X2 @ A2 )
% 9.49/5.28                & ( ord_less_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 9.49/5.28           => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5998_sum__strict__mono__ex1,axiom,
% 9.49/5.28      ! [A2: set_int,F: int > rat,G: int > rat] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ! [X4: int] :
% 9.49/5.28              ( ( member_int @ X4 @ A2 )
% 9.49/5.28             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28         => ( ? [X2: int] :
% 9.49/5.28                ( ( member_int @ X2 @ A2 )
% 9.49/5.28                & ( ord_less_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 9.49/5.28           => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_5999_sum__strict__mono__ex1,axiom,
% 9.49/5.28      ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ! [X4: complex] :
% 9.49/5.28              ( ( member_complex @ X4 @ A2 )
% 9.49/5.28             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28         => ( ? [X2: complex] :
% 9.49/5.28                ( ( member_complex @ X2 @ A2 )
% 9.49/5.28                & ( ord_less_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 9.49/5.28           => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6000_sum__strict__mono__ex1,axiom,
% 9.49/5.28      ! [A2: set_int,F: int > nat,G: int > nat] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ! [X4: int] :
% 9.49/5.28              ( ( member_int @ X4 @ A2 )
% 9.49/5.28             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28         => ( ? [X2: int] :
% 9.49/5.28                ( ( member_int @ X2 @ A2 )
% 9.49/5.28                & ( ord_less_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 9.49/5.28           => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6001_sum__strict__mono__ex1,axiom,
% 9.49/5.28      ! [A2: set_complex,F: complex > nat,G: complex > nat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ! [X4: complex] :
% 9.49/5.28              ( ( member_complex @ X4 @ A2 )
% 9.49/5.28             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28         => ( ? [X2: complex] :
% 9.49/5.28                ( ( member_complex @ X2 @ A2 )
% 9.49/5.28                & ( ord_less_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 9.49/5.28           => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6002_sum__strict__mono__ex1,axiom,
% 9.49/5.28      ! [A2: set_nat,F: nat > int,G: nat > int] :
% 9.49/5.28        ( ( finite_finite_nat @ A2 )
% 9.49/5.28       => ( ! [X4: nat] :
% 9.49/5.28              ( ( member_nat @ X4 @ A2 )
% 9.49/5.28             => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28         => ( ? [X2: nat] :
% 9.49/5.28                ( ( member_nat @ X2 @ A2 )
% 9.49/5.28                & ( ord_less_int @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 9.49/5.28           => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6003_sum__strict__mono__ex1,axiom,
% 9.49/5.28      ! [A2: set_int,F: int > int,G: int > int] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ! [X4: int] :
% 9.49/5.28              ( ( member_int @ X4 @ A2 )
% 9.49/5.28             => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28         => ( ? [X2: int] :
% 9.49/5.28                ( ( member_int @ X2 @ A2 )
% 9.49/5.28                & ( ord_less_int @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 9.49/5.28           => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6004_sum__strict__mono__ex1,axiom,
% 9.49/5.28      ! [A2: set_complex,F: complex > int,G: complex > int] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ! [X4: complex] :
% 9.49/5.28              ( ( member_complex @ X4 @ A2 )
% 9.49/5.28             => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28         => ( ? [X2: complex] :
% 9.49/5.28                ( ( member_complex @ X2 @ A2 )
% 9.49/5.28                & ( ord_less_int @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 9.49/5.28           => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6005_sum_Orelated,axiom,
% 9.49/5.28      ! [R: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
% 9.49/5.28        ( ( R @ zero_zero_complex @ zero_zero_complex )
% 9.49/5.28       => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
% 9.49/5.28              ( ( ( R @ X15 @ X23 )
% 9.49/5.28                & ( R @ Y15 @ Y23 ) )
% 9.49/5.28             => ( R @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
% 9.49/5.28         => ( ( finite_finite_nat @ S2 )
% 9.49/5.28           => ( ! [X4: nat] :
% 9.49/5.28                  ( ( member_nat @ X4 @ S2 )
% 9.49/5.28                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28             => ( R @ ( groups2073611262835488442omplex @ H2 @ S2 ) @ ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6006_sum_Orelated,axiom,
% 9.49/5.28      ! [R: complex > complex > $o,S2: set_int,H2: int > complex,G: int > complex] :
% 9.49/5.28        ( ( R @ zero_zero_complex @ zero_zero_complex )
% 9.49/5.28       => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
% 9.49/5.28              ( ( ( R @ X15 @ X23 )
% 9.49/5.28                & ( R @ Y15 @ Y23 ) )
% 9.49/5.28             => ( R @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
% 9.49/5.28         => ( ( finite_finite_int @ S2 )
% 9.49/5.28           => ( ! [X4: int] :
% 9.49/5.28                  ( ( member_int @ X4 @ S2 )
% 9.49/5.28                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28             => ( R @ ( groups3049146728041665814omplex @ H2 @ S2 ) @ ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6007_sum_Orelated,axiom,
% 9.49/5.28      ! [R: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
% 9.49/5.28        ( ( R @ zero_zero_real @ zero_zero_real )
% 9.49/5.28       => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
% 9.49/5.28              ( ( ( R @ X15 @ X23 )
% 9.49/5.28                & ( R @ Y15 @ Y23 ) )
% 9.49/5.28             => ( R @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
% 9.49/5.28         => ( ( finite_finite_int @ S2 )
% 9.49/5.28           => ( ! [X4: int] :
% 9.49/5.28                  ( ( member_int @ X4 @ S2 )
% 9.49/5.28                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28             => ( R @ ( groups8778361861064173332t_real @ H2 @ S2 ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6008_sum_Orelated,axiom,
% 9.49/5.28      ! [R: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
% 9.49/5.28        ( ( R @ zero_zero_real @ zero_zero_real )
% 9.49/5.28       => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
% 9.49/5.28              ( ( ( R @ X15 @ X23 )
% 9.49/5.28                & ( R @ Y15 @ Y23 ) )
% 9.49/5.28             => ( R @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
% 9.49/5.28         => ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.28           => ( ! [X4: complex] :
% 9.49/5.28                  ( ( member_complex @ X4 @ S2 )
% 9.49/5.28                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28             => ( R @ ( groups5808333547571424918x_real @ H2 @ S2 ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6009_sum_Orelated,axiom,
% 9.49/5.28      ! [R: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
% 9.49/5.28        ( ( R @ zero_zero_rat @ zero_zero_rat )
% 9.49/5.28       => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
% 9.49/5.28              ( ( ( R @ X15 @ X23 )
% 9.49/5.28                & ( R @ Y15 @ Y23 ) )
% 9.49/5.28             => ( R @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 9.49/5.28         => ( ( finite_finite_nat @ S2 )
% 9.49/5.28           => ( ! [X4: nat] :
% 9.49/5.28                  ( ( member_nat @ X4 @ S2 )
% 9.49/5.28                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28             => ( R @ ( groups2906978787729119204at_rat @ H2 @ S2 ) @ ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6010_sum_Orelated,axiom,
% 9.49/5.28      ! [R: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
% 9.49/5.28        ( ( R @ zero_zero_rat @ zero_zero_rat )
% 9.49/5.28       => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
% 9.49/5.28              ( ( ( R @ X15 @ X23 )
% 9.49/5.28                & ( R @ Y15 @ Y23 ) )
% 9.49/5.28             => ( R @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 9.49/5.28         => ( ( finite_finite_int @ S2 )
% 9.49/5.28           => ( ! [X4: int] :
% 9.49/5.28                  ( ( member_int @ X4 @ S2 )
% 9.49/5.28                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28             => ( R @ ( groups3906332499630173760nt_rat @ H2 @ S2 ) @ ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6011_sum_Orelated,axiom,
% 9.49/5.28      ! [R: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
% 9.49/5.28        ( ( R @ zero_zero_rat @ zero_zero_rat )
% 9.49/5.28       => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
% 9.49/5.28              ( ( ( R @ X15 @ X23 )
% 9.49/5.28                & ( R @ Y15 @ Y23 ) )
% 9.49/5.28             => ( R @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 9.49/5.28         => ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.28           => ( ! [X4: complex] :
% 9.49/5.28                  ( ( member_complex @ X4 @ S2 )
% 9.49/5.28                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28             => ( R @ ( groups5058264527183730370ex_rat @ H2 @ S2 ) @ ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6012_sum_Orelated,axiom,
% 9.49/5.28      ! [R: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
% 9.49/5.28        ( ( R @ zero_zero_nat @ zero_zero_nat )
% 9.49/5.28       => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
% 9.49/5.28              ( ( ( R @ X15 @ X23 )
% 9.49/5.28                & ( R @ Y15 @ Y23 ) )
% 9.49/5.28             => ( R @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
% 9.49/5.28         => ( ( finite_finite_int @ S2 )
% 9.49/5.28           => ( ! [X4: int] :
% 9.49/5.28                  ( ( member_int @ X4 @ S2 )
% 9.49/5.28                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28             => ( R @ ( groups4541462559716669496nt_nat @ H2 @ S2 ) @ ( groups4541462559716669496nt_nat @ G @ S2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6013_sum_Orelated,axiom,
% 9.49/5.28      ! [R: nat > nat > $o,S2: set_complex,H2: complex > nat,G: complex > nat] :
% 9.49/5.28        ( ( R @ zero_zero_nat @ zero_zero_nat )
% 9.49/5.28       => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
% 9.49/5.28              ( ( ( R @ X15 @ X23 )
% 9.49/5.28                & ( R @ Y15 @ Y23 ) )
% 9.49/5.28             => ( R @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
% 9.49/5.28         => ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.28           => ( ! [X4: complex] :
% 9.49/5.28                  ( ( member_complex @ X4 @ S2 )
% 9.49/5.28                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28             => ( R @ ( groups5693394587270226106ex_nat @ H2 @ S2 ) @ ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6014_sum_Orelated,axiom,
% 9.49/5.28      ! [R: int > int > $o,S2: set_nat,H2: nat > int,G: nat > int] :
% 9.49/5.28        ( ( R @ zero_zero_int @ zero_zero_int )
% 9.49/5.28       => ( ! [X15: int,Y15: int,X23: int,Y23: int] :
% 9.49/5.28              ( ( ( R @ X15 @ X23 )
% 9.49/5.28                & ( R @ Y15 @ Y23 ) )
% 9.49/5.28             => ( R @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
% 9.49/5.28         => ( ( finite_finite_nat @ S2 )
% 9.49/5.28           => ( ! [X4: nat] :
% 9.49/5.28                  ( ( member_nat @ X4 @ S2 )
% 9.49/5.28                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28             => ( R @ ( groups3539618377306564664at_int @ H2 @ S2 ) @ ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6015_sum__strict__mono,axiom,
% 9.49/5.28      ! [A2: set_complex,F: complex > real,G: complex > real] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( A2 != bot_bot_set_complex )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ A2 )
% 9.49/5.28               => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28           => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6016_sum__strict__mono,axiom,
% 9.49/5.28      ! [A2: set_real,F: real > real,G: real > real] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( A2 != bot_bot_set_real )
% 9.49/5.28         => ( ! [X4: real] :
% 9.49/5.28                ( ( member_real @ X4 @ A2 )
% 9.49/5.28               => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28           => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6017_sum__strict__mono,axiom,
% 9.49/5.28      ! [A2: set_int,F: int > real,G: int > real] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( A2 != bot_bot_set_int )
% 9.49/5.28         => ( ! [X4: int] :
% 9.49/5.28                ( ( member_int @ X4 @ A2 )
% 9.49/5.28               => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28           => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6018_sum__strict__mono,axiom,
% 9.49/5.28      ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( A2 != bot_bot_set_complex )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ A2 )
% 9.49/5.28               => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28           => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6019_sum__strict__mono,axiom,
% 9.49/5.28      ! [A2: set_real,F: real > rat,G: real > rat] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( A2 != bot_bot_set_real )
% 9.49/5.28         => ( ! [X4: real] :
% 9.49/5.28                ( ( member_real @ X4 @ A2 )
% 9.49/5.28               => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28           => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6020_sum__strict__mono,axiom,
% 9.49/5.28      ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 9.49/5.28        ( ( finite_finite_nat @ A2 )
% 9.49/5.28       => ( ( A2 != bot_bot_set_nat )
% 9.49/5.28         => ( ! [X4: nat] :
% 9.49/5.28                ( ( member_nat @ X4 @ A2 )
% 9.49/5.28               => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28           => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6021_sum__strict__mono,axiom,
% 9.49/5.28      ! [A2: set_int,F: int > rat,G: int > rat] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( A2 != bot_bot_set_int )
% 9.49/5.28         => ( ! [X4: int] :
% 9.49/5.28                ( ( member_int @ X4 @ A2 )
% 9.49/5.28               => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28           => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6022_sum__strict__mono,axiom,
% 9.49/5.28      ! [A2: set_complex,F: complex > nat,G: complex > nat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( A2 != bot_bot_set_complex )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ A2 )
% 9.49/5.28               => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28           => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6023_sum__strict__mono,axiom,
% 9.49/5.28      ! [A2: set_real,F: real > nat,G: real > nat] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( A2 != bot_bot_set_real )
% 9.49/5.28         => ( ! [X4: real] :
% 9.49/5.28                ( ( member_real @ X4 @ A2 )
% 9.49/5.28               => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28           => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6024_sum__strict__mono,axiom,
% 9.49/5.28      ! [A2: set_int,F: int > nat,G: int > nat] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( A2 != bot_bot_set_int )
% 9.49/5.28         => ( ! [X4: int] :
% 9.49/5.28                ( ( member_int @ X4 @ A2 )
% 9.49/5.28               => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.49/5.28           => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6025_sum__eq__Suc0__iff,axiom,
% 9.49/5.28      ! [A2: set_int,F: int > nat] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 9.49/5.28            = ( suc @ zero_zero_nat ) )
% 9.49/5.28          = ( ? [X3: int] :
% 9.49/5.28                ( ( member_int @ X3 @ A2 )
% 9.49/5.28                & ( ( F @ X3 )
% 9.49/5.28                  = ( suc @ zero_zero_nat ) )
% 9.49/5.28                & ! [Y2: int] :
% 9.49/5.28                    ( ( member_int @ Y2 @ A2 )
% 9.49/5.28                   => ( ( X3 != Y2 )
% 9.49/5.28                     => ( ( F @ Y2 )
% 9.49/5.28                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6026_sum__eq__Suc0__iff,axiom,
% 9.49/5.28      ! [A2: set_complex,F: complex > nat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 9.49/5.28            = ( suc @ zero_zero_nat ) )
% 9.49/5.28          = ( ? [X3: complex] :
% 9.49/5.28                ( ( member_complex @ X3 @ A2 )
% 9.49/5.28                & ( ( F @ X3 )
% 9.49/5.28                  = ( suc @ zero_zero_nat ) )
% 9.49/5.28                & ! [Y2: complex] :
% 9.49/5.28                    ( ( member_complex @ Y2 @ A2 )
% 9.49/5.28                   => ( ( X3 != Y2 )
% 9.49/5.28                     => ( ( F @ Y2 )
% 9.49/5.28                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6027_sum__eq__Suc0__iff,axiom,
% 9.49/5.28      ! [A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 9.49/5.28        ( ( finite6177210948735845034at_nat @ A2 )
% 9.49/5.28       => ( ( ( groups977919841031483927at_nat @ F @ A2 )
% 9.49/5.28            = ( suc @ zero_zero_nat ) )
% 9.49/5.28          = ( ? [X3: product_prod_nat_nat] :
% 9.49/5.28                ( ( member8440522571783428010at_nat @ X3 @ A2 )
% 9.49/5.28                & ( ( F @ X3 )
% 9.49/5.28                  = ( suc @ zero_zero_nat ) )
% 9.49/5.28                & ! [Y2: product_prod_nat_nat] :
% 9.49/5.28                    ( ( member8440522571783428010at_nat @ Y2 @ A2 )
% 9.49/5.28                   => ( ( X3 != Y2 )
% 9.49/5.28                     => ( ( F @ Y2 )
% 9.49/5.28                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6028_sum__eq__Suc0__iff,axiom,
% 9.49/5.28      ! [A2: set_nat,F: nat > nat] :
% 9.49/5.28        ( ( finite_finite_nat @ A2 )
% 9.49/5.28       => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 9.49/5.28            = ( suc @ zero_zero_nat ) )
% 9.49/5.28          = ( ? [X3: nat] :
% 9.49/5.28                ( ( member_nat @ X3 @ A2 )
% 9.49/5.28                & ( ( F @ X3 )
% 9.49/5.28                  = ( suc @ zero_zero_nat ) )
% 9.49/5.28                & ! [Y2: nat] :
% 9.49/5.28                    ( ( member_nat @ Y2 @ A2 )
% 9.49/5.28                   => ( ( X3 != Y2 )
% 9.49/5.28                     => ( ( F @ Y2 )
% 9.49/5.28                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6029_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.49/5.28      ! [S5: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
% 9.49/5.28        ( ( finite_finite_real @ S5 )
% 9.49/5.28       => ( ( finite_finite_real @ T4 )
% 9.49/5.28         => ( ! [A3: real] :
% 9.49/5.28                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 9.49/5.28               => ( ( I @ ( J @ A3 ) )
% 9.49/5.28                  = A3 ) )
% 9.49/5.28           => ( ! [A3: real] :
% 9.49/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 9.49/5.28                 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 9.49/5.28             => ( ! [B3: real] :
% 9.49/5.28                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 9.49/5.28                   => ( ( J @ ( I @ B3 ) )
% 9.49/5.28                      = B3 ) )
% 9.49/5.28               => ( ! [B3: real] :
% 9.49/5.28                      ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 9.49/5.28                     => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
% 9.49/5.28                 => ( ! [A3: real] :
% 9.49/5.28                        ( ( member_real @ A3 @ S5 )
% 9.49/5.28                       => ( ( G @ A3 )
% 9.49/5.28                          = zero_zero_complex ) )
% 9.49/5.28                   => ( ! [B3: real] :
% 9.49/5.28                          ( ( member_real @ B3 @ T4 )
% 9.49/5.28                         => ( ( H2 @ B3 )
% 9.49/5.28                            = zero_zero_complex ) )
% 9.49/5.28                     => ( ! [A3: real] :
% 9.49/5.28                            ( ( member_real @ A3 @ S2 )
% 9.49/5.28                           => ( ( H2 @ ( J @ A3 ) )
% 9.49/5.28                              = ( G @ A3 ) ) )
% 9.49/5.28                       => ( ( groups5754745047067104278omplex @ G @ S2 )
% 9.49/5.28                          = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6030_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.49/5.28      ! [S5: set_real,T4: set_int,S2: set_real,I: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
% 9.49/5.28        ( ( finite_finite_real @ S5 )
% 9.49/5.28       => ( ( finite_finite_int @ T4 )
% 9.49/5.28         => ( ! [A3: real] :
% 9.49/5.28                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 9.49/5.28               => ( ( I @ ( J @ A3 ) )
% 9.49/5.28                  = A3 ) )
% 9.49/5.28           => ( ! [A3: real] :
% 9.49/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 9.49/5.28                 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 9.49/5.28             => ( ! [B3: int] :
% 9.49/5.28                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 9.49/5.28                   => ( ( J @ ( I @ B3 ) )
% 9.49/5.28                      = B3 ) )
% 9.49/5.28               => ( ! [B3: int] :
% 9.49/5.28                      ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 9.49/5.28                     => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
% 9.49/5.28                 => ( ! [A3: real] :
% 9.49/5.28                        ( ( member_real @ A3 @ S5 )
% 9.49/5.28                       => ( ( G @ A3 )
% 9.49/5.28                          = zero_zero_complex ) )
% 9.49/5.28                   => ( ! [B3: int] :
% 9.49/5.28                          ( ( member_int @ B3 @ T4 )
% 9.49/5.28                         => ( ( H2 @ B3 )
% 9.49/5.28                            = zero_zero_complex ) )
% 9.49/5.28                     => ( ! [A3: real] :
% 9.49/5.28                            ( ( member_real @ A3 @ S2 )
% 9.49/5.28                           => ( ( H2 @ ( J @ A3 ) )
% 9.49/5.28                              = ( G @ A3 ) ) )
% 9.49/5.28                       => ( ( groups5754745047067104278omplex @ G @ S2 )
% 9.49/5.28                          = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6031_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.49/5.28      ! [S5: set_int,T4: set_real,S2: set_int,I: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
% 9.49/5.28        ( ( finite_finite_int @ S5 )
% 9.49/5.28       => ( ( finite_finite_real @ T4 )
% 9.49/5.28         => ( ! [A3: int] :
% 9.49/5.28                ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 9.49/5.28               => ( ( I @ ( J @ A3 ) )
% 9.49/5.28                  = A3 ) )
% 9.49/5.28           => ( ! [A3: int] :
% 9.49/5.28                  ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 9.49/5.28                 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 9.49/5.28             => ( ! [B3: real] :
% 9.49/5.28                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 9.49/5.28                   => ( ( J @ ( I @ B3 ) )
% 9.49/5.28                      = B3 ) )
% 9.49/5.28               => ( ! [B3: real] :
% 9.49/5.28                      ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 9.49/5.28                     => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
% 9.49/5.28                 => ( ! [A3: int] :
% 9.49/5.28                        ( ( member_int @ A3 @ S5 )
% 9.49/5.28                       => ( ( G @ A3 )
% 9.49/5.28                          = zero_zero_complex ) )
% 9.49/5.28                   => ( ! [B3: real] :
% 9.49/5.28                          ( ( member_real @ B3 @ T4 )
% 9.49/5.28                         => ( ( H2 @ B3 )
% 9.49/5.28                            = zero_zero_complex ) )
% 9.49/5.28                     => ( ! [A3: int] :
% 9.49/5.28                            ( ( member_int @ A3 @ S2 )
% 9.49/5.28                           => ( ( H2 @ ( J @ A3 ) )
% 9.49/5.28                              = ( G @ A3 ) ) )
% 9.49/5.28                       => ( ( groups3049146728041665814omplex @ G @ S2 )
% 9.49/5.28                          = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6032_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.49/5.28      ! [S5: set_int,T4: set_int,S2: set_int,I: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
% 9.49/5.28        ( ( finite_finite_int @ S5 )
% 9.49/5.28       => ( ( finite_finite_int @ T4 )
% 9.49/5.28         => ( ! [A3: int] :
% 9.49/5.28                ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 9.49/5.28               => ( ( I @ ( J @ A3 ) )
% 9.49/5.28                  = A3 ) )
% 9.49/5.28           => ( ! [A3: int] :
% 9.49/5.28                  ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 9.49/5.28                 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 9.49/5.28             => ( ! [B3: int] :
% 9.49/5.28                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 9.49/5.28                   => ( ( J @ ( I @ B3 ) )
% 9.49/5.28                      = B3 ) )
% 9.49/5.28               => ( ! [B3: int] :
% 9.49/5.28                      ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 9.49/5.28                     => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
% 9.49/5.28                 => ( ! [A3: int] :
% 9.49/5.28                        ( ( member_int @ A3 @ S5 )
% 9.49/5.28                       => ( ( G @ A3 )
% 9.49/5.28                          = zero_zero_complex ) )
% 9.49/5.28                   => ( ! [B3: int] :
% 9.49/5.28                          ( ( member_int @ B3 @ T4 )
% 9.49/5.28                         => ( ( H2 @ B3 )
% 9.49/5.28                            = zero_zero_complex ) )
% 9.49/5.28                     => ( ! [A3: int] :
% 9.49/5.28                            ( ( member_int @ A3 @ S2 )
% 9.49/5.28                           => ( ( H2 @ ( J @ A3 ) )
% 9.49/5.28                              = ( G @ A3 ) ) )
% 9.49/5.28                       => ( ( groups3049146728041665814omplex @ G @ S2 )
% 9.49/5.28                          = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6033_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.49/5.28      ! [S5: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T3: set_real,G: real > real,H2: real > real] :
% 9.49/5.28        ( ( finite_finite_real @ S5 )
% 9.49/5.28       => ( ( finite_finite_real @ T4 )
% 9.49/5.28         => ( ! [A3: real] :
% 9.49/5.28                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 9.49/5.28               => ( ( I @ ( J @ A3 ) )
% 9.49/5.28                  = A3 ) )
% 9.49/5.28           => ( ! [A3: real] :
% 9.49/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 9.49/5.28                 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 9.49/5.28             => ( ! [B3: real] :
% 9.49/5.28                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 9.49/5.28                   => ( ( J @ ( I @ B3 ) )
% 9.49/5.28                      = B3 ) )
% 9.49/5.28               => ( ! [B3: real] :
% 9.49/5.28                      ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 9.49/5.28                     => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
% 9.49/5.28                 => ( ! [A3: real] :
% 9.49/5.28                        ( ( member_real @ A3 @ S5 )
% 9.49/5.28                       => ( ( G @ A3 )
% 9.49/5.28                          = zero_zero_real ) )
% 9.49/5.28                   => ( ! [B3: real] :
% 9.49/5.28                          ( ( member_real @ B3 @ T4 )
% 9.49/5.28                         => ( ( H2 @ B3 )
% 9.49/5.28                            = zero_zero_real ) )
% 9.49/5.28                     => ( ! [A3: real] :
% 9.49/5.28                            ( ( member_real @ A3 @ S2 )
% 9.49/5.28                           => ( ( H2 @ ( J @ A3 ) )
% 9.49/5.28                              = ( G @ A3 ) ) )
% 9.49/5.28                       => ( ( groups8097168146408367636l_real @ G @ S2 )
% 9.49/5.28                          = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6034_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.49/5.28      ! [S5: set_real,T4: set_int,S2: set_real,I: int > real,J: real > int,T3: set_int,G: real > real,H2: int > real] :
% 9.49/5.28        ( ( finite_finite_real @ S5 )
% 9.49/5.28       => ( ( finite_finite_int @ T4 )
% 9.49/5.28         => ( ! [A3: real] :
% 9.49/5.28                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 9.49/5.28               => ( ( I @ ( J @ A3 ) )
% 9.49/5.28                  = A3 ) )
% 9.49/5.28           => ( ! [A3: real] :
% 9.49/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 9.49/5.28                 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 9.49/5.28             => ( ! [B3: int] :
% 9.49/5.28                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 9.49/5.28                   => ( ( J @ ( I @ B3 ) )
% 9.49/5.28                      = B3 ) )
% 9.49/5.28               => ( ! [B3: int] :
% 9.49/5.28                      ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 9.49/5.28                     => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
% 9.49/5.28                 => ( ! [A3: real] :
% 9.49/5.28                        ( ( member_real @ A3 @ S5 )
% 9.49/5.28                       => ( ( G @ A3 )
% 9.49/5.28                          = zero_zero_real ) )
% 9.49/5.28                   => ( ! [B3: int] :
% 9.49/5.28                          ( ( member_int @ B3 @ T4 )
% 9.49/5.28                         => ( ( H2 @ B3 )
% 9.49/5.28                            = zero_zero_real ) )
% 9.49/5.28                     => ( ! [A3: real] :
% 9.49/5.28                            ( ( member_real @ A3 @ S2 )
% 9.49/5.28                           => ( ( H2 @ ( J @ A3 ) )
% 9.49/5.28                              = ( G @ A3 ) ) )
% 9.49/5.28                       => ( ( groups8097168146408367636l_real @ G @ S2 )
% 9.49/5.28                          = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6035_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.49/5.28      ! [S5: set_real,T4: set_complex,S2: set_real,I: complex > real,J: real > complex,T3: set_complex,G: real > real,H2: complex > real] :
% 9.49/5.28        ( ( finite_finite_real @ S5 )
% 9.49/5.28       => ( ( finite3207457112153483333omplex @ T4 )
% 9.49/5.28         => ( ! [A3: real] :
% 9.49/5.28                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 9.49/5.28               => ( ( I @ ( J @ A3 ) )
% 9.49/5.28                  = A3 ) )
% 9.49/5.28           => ( ! [A3: real] :
% 9.49/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 9.49/5.28                 => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 9.49/5.28             => ( ! [B3: complex] :
% 9.49/5.28                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 9.49/5.28                   => ( ( J @ ( I @ B3 ) )
% 9.49/5.28                      = B3 ) )
% 9.49/5.28               => ( ! [B3: complex] :
% 9.49/5.28                      ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 9.49/5.28                     => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
% 9.49/5.28                 => ( ! [A3: real] :
% 9.49/5.28                        ( ( member_real @ A3 @ S5 )
% 9.49/5.28                       => ( ( G @ A3 )
% 9.49/5.28                          = zero_zero_real ) )
% 9.49/5.28                   => ( ! [B3: complex] :
% 9.49/5.28                          ( ( member_complex @ B3 @ T4 )
% 9.49/5.28                         => ( ( H2 @ B3 )
% 9.49/5.28                            = zero_zero_real ) )
% 9.49/5.28                     => ( ! [A3: real] :
% 9.49/5.28                            ( ( member_real @ A3 @ S2 )
% 9.49/5.28                           => ( ( H2 @ ( J @ A3 ) )
% 9.49/5.28                              = ( G @ A3 ) ) )
% 9.49/5.28                       => ( ( groups8097168146408367636l_real @ G @ S2 )
% 9.49/5.28                          = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6036_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.49/5.28      ! [S5: set_int,T4: set_real,S2: set_int,I: real > int,J: int > real,T3: set_real,G: int > real,H2: real > real] :
% 9.49/5.28        ( ( finite_finite_int @ S5 )
% 9.49/5.28       => ( ( finite_finite_real @ T4 )
% 9.49/5.28         => ( ! [A3: int] :
% 9.49/5.28                ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 9.49/5.28               => ( ( I @ ( J @ A3 ) )
% 9.49/5.28                  = A3 ) )
% 9.49/5.28           => ( ! [A3: int] :
% 9.49/5.28                  ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 9.49/5.28                 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 9.49/5.28             => ( ! [B3: real] :
% 9.49/5.28                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 9.49/5.28                   => ( ( J @ ( I @ B3 ) )
% 9.49/5.28                      = B3 ) )
% 9.49/5.28               => ( ! [B3: real] :
% 9.49/5.28                      ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 9.49/5.28                     => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
% 9.49/5.28                 => ( ! [A3: int] :
% 9.49/5.28                        ( ( member_int @ A3 @ S5 )
% 9.49/5.28                       => ( ( G @ A3 )
% 9.49/5.28                          = zero_zero_real ) )
% 9.49/5.28                   => ( ! [B3: real] :
% 9.49/5.28                          ( ( member_real @ B3 @ T4 )
% 9.49/5.28                         => ( ( H2 @ B3 )
% 9.49/5.28                            = zero_zero_real ) )
% 9.49/5.28                     => ( ! [A3: int] :
% 9.49/5.28                            ( ( member_int @ A3 @ S2 )
% 9.49/5.28                           => ( ( H2 @ ( J @ A3 ) )
% 9.49/5.28                              = ( G @ A3 ) ) )
% 9.49/5.28                       => ( ( groups8778361861064173332t_real @ G @ S2 )
% 9.49/5.28                          = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6037_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.49/5.28      ! [S5: set_int,T4: set_int,S2: set_int,I: int > int,J: int > int,T3: set_int,G: int > real,H2: int > real] :
% 9.49/5.28        ( ( finite_finite_int @ S5 )
% 9.49/5.28       => ( ( finite_finite_int @ T4 )
% 9.49/5.28         => ( ! [A3: int] :
% 9.49/5.28                ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 9.49/5.28               => ( ( I @ ( J @ A3 ) )
% 9.49/5.28                  = A3 ) )
% 9.49/5.28           => ( ! [A3: int] :
% 9.49/5.28                  ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 9.49/5.28                 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 9.49/5.28             => ( ! [B3: int] :
% 9.49/5.28                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 9.49/5.28                   => ( ( J @ ( I @ B3 ) )
% 9.49/5.28                      = B3 ) )
% 9.49/5.28               => ( ! [B3: int] :
% 9.49/5.28                      ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 9.49/5.28                     => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
% 9.49/5.28                 => ( ! [A3: int] :
% 9.49/5.28                        ( ( member_int @ A3 @ S5 )
% 9.49/5.28                       => ( ( G @ A3 )
% 9.49/5.28                          = zero_zero_real ) )
% 9.49/5.28                   => ( ! [B3: int] :
% 9.49/5.28                          ( ( member_int @ B3 @ T4 )
% 9.49/5.28                         => ( ( H2 @ B3 )
% 9.49/5.28                            = zero_zero_real ) )
% 9.49/5.28                     => ( ! [A3: int] :
% 9.49/5.28                            ( ( member_int @ A3 @ S2 )
% 9.49/5.28                           => ( ( H2 @ ( J @ A3 ) )
% 9.49/5.28                              = ( G @ A3 ) ) )
% 9.49/5.28                       => ( ( groups8778361861064173332t_real @ G @ S2 )
% 9.49/5.28                          = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6038_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.49/5.28      ! [S5: set_int,T4: set_complex,S2: set_int,I: complex > int,J: int > complex,T3: set_complex,G: int > real,H2: complex > real] :
% 9.49/5.28        ( ( finite_finite_int @ S5 )
% 9.49/5.28       => ( ( finite3207457112153483333omplex @ T4 )
% 9.49/5.28         => ( ! [A3: int] :
% 9.49/5.28                ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 9.49/5.28               => ( ( I @ ( J @ A3 ) )
% 9.49/5.28                  = A3 ) )
% 9.49/5.28           => ( ! [A3: int] :
% 9.49/5.28                  ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 9.49/5.28                 => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 9.49/5.28             => ( ! [B3: complex] :
% 9.49/5.28                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 9.49/5.28                   => ( ( J @ ( I @ B3 ) )
% 9.49/5.28                      = B3 ) )
% 9.49/5.28               => ( ! [B3: complex] :
% 9.49/5.28                      ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 9.49/5.28                     => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
% 9.49/5.28                 => ( ! [A3: int] :
% 9.49/5.28                        ( ( member_int @ A3 @ S5 )
% 9.49/5.28                       => ( ( G @ A3 )
% 9.49/5.28                          = zero_zero_real ) )
% 9.49/5.28                   => ( ! [B3: complex] :
% 9.49/5.28                          ( ( member_complex @ B3 @ T4 )
% 9.49/5.28                         => ( ( H2 @ B3 )
% 9.49/5.28                            = zero_zero_real ) )
% 9.49/5.28                     => ( ! [A3: int] :
% 9.49/5.28                            ( ( member_int @ A3 @ S2 )
% 9.49/5.28                           => ( ( H2 @ ( J @ A3 ) )
% 9.49/5.28                              = ( G @ A3 ) ) )
% 9.49/5.28                       => ( ( groups8778361861064173332t_real @ G @ S2 )
% 9.49/5.28                          = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6039_sum__SucD,axiom,
% 9.49/5.28      ! [F: nat > nat,A2: set_nat,N: nat] :
% 9.49/5.28        ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 9.49/5.28          = ( suc @ N ) )
% 9.49/5.28       => ? [X4: nat] :
% 9.49/5.28            ( ( member_nat @ X4 @ A2 )
% 9.49/5.28            & ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6040_sum__eq__1__iff,axiom,
% 9.49/5.28      ! [A2: set_int,F: int > nat] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 9.49/5.28            = one_one_nat )
% 9.49/5.28          = ( ? [X3: int] :
% 9.49/5.28                ( ( member_int @ X3 @ A2 )
% 9.49/5.28                & ( ( F @ X3 )
% 9.49/5.28                  = one_one_nat )
% 9.49/5.28                & ! [Y2: int] :
% 9.49/5.28                    ( ( member_int @ Y2 @ A2 )
% 9.49/5.28                   => ( ( X3 != Y2 )
% 9.49/5.28                     => ( ( F @ Y2 )
% 9.49/5.28                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6041_sum__eq__1__iff,axiom,
% 9.49/5.28      ! [A2: set_complex,F: complex > nat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 9.49/5.28            = one_one_nat )
% 9.49/5.28          = ( ? [X3: complex] :
% 9.49/5.28                ( ( member_complex @ X3 @ A2 )
% 9.49/5.28                & ( ( F @ X3 )
% 9.49/5.28                  = one_one_nat )
% 9.49/5.28                & ! [Y2: complex] :
% 9.49/5.28                    ( ( member_complex @ Y2 @ A2 )
% 9.49/5.28                   => ( ( X3 != Y2 )
% 9.49/5.28                     => ( ( F @ Y2 )
% 9.49/5.28                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6042_sum__eq__1__iff,axiom,
% 9.49/5.28      ! [A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 9.49/5.28        ( ( finite6177210948735845034at_nat @ A2 )
% 9.49/5.28       => ( ( ( groups977919841031483927at_nat @ F @ A2 )
% 9.49/5.28            = one_one_nat )
% 9.49/5.28          = ( ? [X3: product_prod_nat_nat] :
% 9.49/5.28                ( ( member8440522571783428010at_nat @ X3 @ A2 )
% 9.49/5.28                & ( ( F @ X3 )
% 9.49/5.28                  = one_one_nat )
% 9.49/5.28                & ! [Y2: product_prod_nat_nat] :
% 9.49/5.28                    ( ( member8440522571783428010at_nat @ Y2 @ A2 )
% 9.49/5.28                   => ( ( X3 != Y2 )
% 9.49/5.28                     => ( ( F @ Y2 )
% 9.49/5.28                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6043_sum__eq__1__iff,axiom,
% 9.49/5.28      ! [A2: set_nat,F: nat > nat] :
% 9.49/5.28        ( ( finite_finite_nat @ A2 )
% 9.49/5.28       => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 9.49/5.28            = one_one_nat )
% 9.49/5.28          = ( ? [X3: nat] :
% 9.49/5.28                ( ( member_nat @ X3 @ A2 )
% 9.49/5.28                & ( ( F @ X3 )
% 9.49/5.28                  = one_one_nat )
% 9.49/5.28                & ! [Y2: nat] :
% 9.49/5.28                    ( ( member_nat @ Y2 @ A2 )
% 9.49/5.28                   => ( ( X3 != Y2 )
% 9.49/5.28                     => ( ( F @ Y2 )
% 9.49/5.28                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6044_finite__ranking__induct,axiom,
% 9.49/5.28      ! [S2: set_complex,P: set_complex > $o,F: complex > rat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.28       => ( ( P @ bot_bot_set_complex )
% 9.49/5.28         => ( ! [X4: complex,S6: set_complex] :
% 9.49/5.28                ( ( finite3207457112153483333omplex @ S6 )
% 9.49/5.28               => ( ! [Y4: complex] :
% 9.49/5.28                      ( ( member_complex @ Y4 @ S6 )
% 9.49/5.28                     => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 9.49/5.28                 => ( ( P @ S6 )
% 9.49/5.28                   => ( P @ ( insert_complex @ X4 @ S6 ) ) ) ) )
% 9.49/5.28           => ( P @ S2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6045_finite__ranking__induct,axiom,
% 9.49/5.28      ! [S2: set_real,P: set_real > $o,F: real > rat] :
% 9.49/5.28        ( ( finite_finite_real @ S2 )
% 9.49/5.28       => ( ( P @ bot_bot_set_real )
% 9.49/5.28         => ( ! [X4: real,S6: set_real] :
% 9.49/5.28                ( ( finite_finite_real @ S6 )
% 9.49/5.28               => ( ! [Y4: real] :
% 9.49/5.28                      ( ( member_real @ Y4 @ S6 )
% 9.49/5.28                     => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 9.49/5.28                 => ( ( P @ S6 )
% 9.49/5.28                   => ( P @ ( insert_real @ X4 @ S6 ) ) ) ) )
% 9.49/5.28           => ( P @ S2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6046_finite__ranking__induct,axiom,
% 9.49/5.28      ! [S2: set_nat,P: set_nat > $o,F: nat > rat] :
% 9.49/5.28        ( ( finite_finite_nat @ S2 )
% 9.49/5.28       => ( ( P @ bot_bot_set_nat )
% 9.49/5.28         => ( ! [X4: nat,S6: set_nat] :
% 9.49/5.28                ( ( finite_finite_nat @ S6 )
% 9.49/5.28               => ( ! [Y4: nat] :
% 9.49/5.28                      ( ( member_nat @ Y4 @ S6 )
% 9.49/5.28                     => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 9.49/5.28                 => ( ( P @ S6 )
% 9.49/5.28                   => ( P @ ( insert_nat @ X4 @ S6 ) ) ) ) )
% 9.49/5.28           => ( P @ S2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6047_finite__ranking__induct,axiom,
% 9.49/5.28      ! [S2: set_int,P: set_int > $o,F: int > rat] :
% 9.49/5.28        ( ( finite_finite_int @ S2 )
% 9.49/5.28       => ( ( P @ bot_bot_set_int )
% 9.49/5.28         => ( ! [X4: int,S6: set_int] :
% 9.49/5.28                ( ( finite_finite_int @ S6 )
% 9.49/5.28               => ( ! [Y4: int] :
% 9.49/5.28                      ( ( member_int @ Y4 @ S6 )
% 9.49/5.28                     => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 9.49/5.28                 => ( ( P @ S6 )
% 9.49/5.28                   => ( P @ ( insert_int @ X4 @ S6 ) ) ) ) )
% 9.49/5.28           => ( P @ S2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6048_finite__ranking__induct,axiom,
% 9.49/5.28      ! [S2: set_complex,P: set_complex > $o,F: complex > num] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.28       => ( ( P @ bot_bot_set_complex )
% 9.49/5.28         => ( ! [X4: complex,S6: set_complex] :
% 9.49/5.28                ( ( finite3207457112153483333omplex @ S6 )
% 9.49/5.28               => ( ! [Y4: complex] :
% 9.49/5.28                      ( ( member_complex @ Y4 @ S6 )
% 9.49/5.28                     => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 9.49/5.28                 => ( ( P @ S6 )
% 9.49/5.28                   => ( P @ ( insert_complex @ X4 @ S6 ) ) ) ) )
% 9.49/5.28           => ( P @ S2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6049_finite__ranking__induct,axiom,
% 9.49/5.28      ! [S2: set_real,P: set_real > $o,F: real > num] :
% 9.49/5.28        ( ( finite_finite_real @ S2 )
% 9.49/5.28       => ( ( P @ bot_bot_set_real )
% 9.49/5.28         => ( ! [X4: real,S6: set_real] :
% 9.49/5.28                ( ( finite_finite_real @ S6 )
% 9.49/5.28               => ( ! [Y4: real] :
% 9.49/5.28                      ( ( member_real @ Y4 @ S6 )
% 9.49/5.28                     => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 9.49/5.28                 => ( ( P @ S6 )
% 9.49/5.28                   => ( P @ ( insert_real @ X4 @ S6 ) ) ) ) )
% 9.49/5.28           => ( P @ S2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6050_finite__ranking__induct,axiom,
% 9.49/5.28      ! [S2: set_nat,P: set_nat > $o,F: nat > num] :
% 9.49/5.28        ( ( finite_finite_nat @ S2 )
% 9.49/5.28       => ( ( P @ bot_bot_set_nat )
% 9.49/5.28         => ( ! [X4: nat,S6: set_nat] :
% 9.49/5.28                ( ( finite_finite_nat @ S6 )
% 9.49/5.28               => ( ! [Y4: nat] :
% 9.49/5.28                      ( ( member_nat @ Y4 @ S6 )
% 9.49/5.28                     => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 9.49/5.28                 => ( ( P @ S6 )
% 9.49/5.28                   => ( P @ ( insert_nat @ X4 @ S6 ) ) ) ) )
% 9.49/5.28           => ( P @ S2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6051_finite__ranking__induct,axiom,
% 9.49/5.28      ! [S2: set_int,P: set_int > $o,F: int > num] :
% 9.49/5.28        ( ( finite_finite_int @ S2 )
% 9.49/5.28       => ( ( P @ bot_bot_set_int )
% 9.49/5.28         => ( ! [X4: int,S6: set_int] :
% 9.49/5.28                ( ( finite_finite_int @ S6 )
% 9.49/5.28               => ( ! [Y4: int] :
% 9.49/5.28                      ( ( member_int @ Y4 @ S6 )
% 9.49/5.28                     => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 9.49/5.28                 => ( ( P @ S6 )
% 9.49/5.28                   => ( P @ ( insert_int @ X4 @ S6 ) ) ) ) )
% 9.49/5.28           => ( P @ S2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6052_finite__ranking__induct,axiom,
% 9.49/5.28      ! [S2: set_complex,P: set_complex > $o,F: complex > nat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.28       => ( ( P @ bot_bot_set_complex )
% 9.49/5.28         => ( ! [X4: complex,S6: set_complex] :
% 9.49/5.28                ( ( finite3207457112153483333omplex @ S6 )
% 9.49/5.28               => ( ! [Y4: complex] :
% 9.49/5.28                      ( ( member_complex @ Y4 @ S6 )
% 9.49/5.28                     => ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 9.49/5.28                 => ( ( P @ S6 )
% 9.49/5.28                   => ( P @ ( insert_complex @ X4 @ S6 ) ) ) ) )
% 9.49/5.28           => ( P @ S2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6053_finite__ranking__induct,axiom,
% 9.49/5.28      ! [S2: set_real,P: set_real > $o,F: real > nat] :
% 9.49/5.28        ( ( finite_finite_real @ S2 )
% 9.49/5.28       => ( ( P @ bot_bot_set_real )
% 9.49/5.28         => ( ! [X4: real,S6: set_real] :
% 9.49/5.28                ( ( finite_finite_real @ S6 )
% 9.49/5.28               => ( ! [Y4: real] :
% 9.49/5.28                      ( ( member_real @ Y4 @ S6 )
% 9.49/5.28                     => ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X4 ) ) )
% 9.49/5.28                 => ( ( P @ S6 )
% 9.49/5.28                   => ( P @ ( insert_real @ X4 @ S6 ) ) ) ) )
% 9.49/5.28           => ( P @ S2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6054_finite__linorder__min__induct,axiom,
% 9.49/5.28      ! [A2: set_real,P: set_real > $o] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( P @ bot_bot_set_real )
% 9.49/5.28         => ( ! [B3: real,A5: set_real] :
% 9.49/5.28                ( ( finite_finite_real @ A5 )
% 9.49/5.28               => ( ! [X2: real] :
% 9.49/5.28                      ( ( member_real @ X2 @ A5 )
% 9.49/5.28                     => ( ord_less_real @ B3 @ X2 ) )
% 9.49/5.28                 => ( ( P @ A5 )
% 9.49/5.28                   => ( P @ ( insert_real @ B3 @ A5 ) ) ) ) )
% 9.49/5.28           => ( P @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6055_finite__linorder__min__induct,axiom,
% 9.49/5.28      ! [A2: set_rat,P: set_rat > $o] :
% 9.49/5.28        ( ( finite_finite_rat @ A2 )
% 9.49/5.28       => ( ( P @ bot_bot_set_rat )
% 9.49/5.28         => ( ! [B3: rat,A5: set_rat] :
% 9.49/5.28                ( ( finite_finite_rat @ A5 )
% 9.49/5.28               => ( ! [X2: rat] :
% 9.49/5.28                      ( ( member_rat @ X2 @ A5 )
% 9.49/5.28                     => ( ord_less_rat @ B3 @ X2 ) )
% 9.49/5.28                 => ( ( P @ A5 )
% 9.49/5.28                   => ( P @ ( insert_rat @ B3 @ A5 ) ) ) ) )
% 9.49/5.28           => ( P @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6056_finite__linorder__min__induct,axiom,
% 9.49/5.28      ! [A2: set_num,P: set_num > $o] :
% 9.49/5.28        ( ( finite_finite_num @ A2 )
% 9.49/5.28       => ( ( P @ bot_bot_set_num )
% 9.49/5.28         => ( ! [B3: num,A5: set_num] :
% 9.49/5.28                ( ( finite_finite_num @ A5 )
% 9.49/5.28               => ( ! [X2: num] :
% 9.49/5.28                      ( ( member_num @ X2 @ A5 )
% 9.49/5.28                     => ( ord_less_num @ B3 @ X2 ) )
% 9.49/5.28                 => ( ( P @ A5 )
% 9.49/5.28                   => ( P @ ( insert_num @ B3 @ A5 ) ) ) ) )
% 9.49/5.28           => ( P @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6057_finite__linorder__min__induct,axiom,
% 9.49/5.28      ! [A2: set_nat,P: set_nat > $o] :
% 9.49/5.28        ( ( finite_finite_nat @ A2 )
% 9.49/5.28       => ( ( P @ bot_bot_set_nat )
% 9.49/5.28         => ( ! [B3: nat,A5: set_nat] :
% 9.49/5.28                ( ( finite_finite_nat @ A5 )
% 9.49/5.28               => ( ! [X2: nat] :
% 9.49/5.28                      ( ( member_nat @ X2 @ A5 )
% 9.49/5.28                     => ( ord_less_nat @ B3 @ X2 ) )
% 9.49/5.28                 => ( ( P @ A5 )
% 9.49/5.28                   => ( P @ ( insert_nat @ B3 @ A5 ) ) ) ) )
% 9.49/5.28           => ( P @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6058_finite__linorder__min__induct,axiom,
% 9.49/5.28      ! [A2: set_int,P: set_int > $o] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( P @ bot_bot_set_int )
% 9.49/5.28         => ( ! [B3: int,A5: set_int] :
% 9.49/5.28                ( ( finite_finite_int @ A5 )
% 9.49/5.28               => ( ! [X2: int] :
% 9.49/5.28                      ( ( member_int @ X2 @ A5 )
% 9.49/5.28                     => ( ord_less_int @ B3 @ X2 ) )
% 9.49/5.28                 => ( ( P @ A5 )
% 9.49/5.28                   => ( P @ ( insert_int @ B3 @ A5 ) ) ) ) )
% 9.49/5.28           => ( P @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6059_finite__linorder__max__induct,axiom,
% 9.49/5.28      ! [A2: set_real,P: set_real > $o] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( P @ bot_bot_set_real )
% 9.49/5.28         => ( ! [B3: real,A5: set_real] :
% 9.49/5.28                ( ( finite_finite_real @ A5 )
% 9.49/5.28               => ( ! [X2: real] :
% 9.49/5.28                      ( ( member_real @ X2 @ A5 )
% 9.49/5.28                     => ( ord_less_real @ X2 @ B3 ) )
% 9.49/5.28                 => ( ( P @ A5 )
% 9.49/5.28                   => ( P @ ( insert_real @ B3 @ A5 ) ) ) ) )
% 9.49/5.28           => ( P @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6060_finite__linorder__max__induct,axiom,
% 9.49/5.28      ! [A2: set_rat,P: set_rat > $o] :
% 9.49/5.28        ( ( finite_finite_rat @ A2 )
% 9.49/5.28       => ( ( P @ bot_bot_set_rat )
% 9.49/5.28         => ( ! [B3: rat,A5: set_rat] :
% 9.49/5.28                ( ( finite_finite_rat @ A5 )
% 9.49/5.28               => ( ! [X2: rat] :
% 9.49/5.28                      ( ( member_rat @ X2 @ A5 )
% 9.49/5.28                     => ( ord_less_rat @ X2 @ B3 ) )
% 9.49/5.28                 => ( ( P @ A5 )
% 9.49/5.28                   => ( P @ ( insert_rat @ B3 @ A5 ) ) ) ) )
% 9.49/5.28           => ( P @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6061_finite__linorder__max__induct,axiom,
% 9.49/5.28      ! [A2: set_num,P: set_num > $o] :
% 9.49/5.28        ( ( finite_finite_num @ A2 )
% 9.49/5.28       => ( ( P @ bot_bot_set_num )
% 9.49/5.28         => ( ! [B3: num,A5: set_num] :
% 9.49/5.28                ( ( finite_finite_num @ A5 )
% 9.49/5.28               => ( ! [X2: num] :
% 9.49/5.28                      ( ( member_num @ X2 @ A5 )
% 9.49/5.28                     => ( ord_less_num @ X2 @ B3 ) )
% 9.49/5.28                 => ( ( P @ A5 )
% 9.49/5.28                   => ( P @ ( insert_num @ B3 @ A5 ) ) ) ) )
% 9.49/5.28           => ( P @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6062_finite__linorder__max__induct,axiom,
% 9.49/5.28      ! [A2: set_nat,P: set_nat > $o] :
% 9.49/5.28        ( ( finite_finite_nat @ A2 )
% 9.49/5.28       => ( ( P @ bot_bot_set_nat )
% 9.49/5.28         => ( ! [B3: nat,A5: set_nat] :
% 9.49/5.28                ( ( finite_finite_nat @ A5 )
% 9.49/5.28               => ( ! [X2: nat] :
% 9.49/5.28                      ( ( member_nat @ X2 @ A5 )
% 9.49/5.28                     => ( ord_less_nat @ X2 @ B3 ) )
% 9.49/5.28                 => ( ( P @ A5 )
% 9.49/5.28                   => ( P @ ( insert_nat @ B3 @ A5 ) ) ) ) )
% 9.49/5.28           => ( P @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6063_finite__linorder__max__induct,axiom,
% 9.49/5.28      ! [A2: set_int,P: set_int > $o] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( P @ bot_bot_set_int )
% 9.49/5.28         => ( ! [B3: int,A5: set_int] :
% 9.49/5.28                ( ( finite_finite_int @ A5 )
% 9.49/5.28               => ( ! [X2: int] :
% 9.49/5.28                      ( ( member_int @ X2 @ A5 )
% 9.49/5.28                     => ( ord_less_int @ X2 @ B3 ) )
% 9.49/5.28                 => ( ( P @ A5 )
% 9.49/5.28                   => ( P @ ( insert_int @ B3 @ A5 ) ) ) ) )
% 9.49/5.28           => ( P @ A2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6064_finite__subset__induct,axiom,
% 9.49/5.28      ! [F3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 9.49/5.28        ( ( finite4343798906461161616at_nat @ F3 )
% 9.49/5.28       => ( ( ord_le1268244103169919719at_nat @ F3 @ A2 )
% 9.49/5.28         => ( ( P @ bot_bo228742789529271731at_nat )
% 9.49/5.28           => ( ! [A3: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
% 9.49/5.28                  ( ( finite4343798906461161616at_nat @ F4 )
% 9.49/5.28                 => ( ( member8757157785044589968at_nat @ A3 @ A2 )
% 9.49/5.28                   => ( ~ ( member8757157785044589968at_nat @ A3 @ F4 )
% 9.49/5.28                     => ( ( P @ F4 )
% 9.49/5.28                       => ( P @ ( insert9069300056098147895at_nat @ A3 @ F4 ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6065_finite__subset__induct,axiom,
% 9.49/5.28      ! [F3: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
% 9.49/5.28        ( ( finite1152437895449049373et_nat @ F3 )
% 9.49/5.28       => ( ( ord_le6893508408891458716et_nat @ F3 @ A2 )
% 9.49/5.28         => ( ( P @ bot_bot_set_set_nat )
% 9.49/5.28           => ( ! [A3: set_nat,F4: set_set_nat] :
% 9.49/5.28                  ( ( finite1152437895449049373et_nat @ F4 )
% 9.49/5.28                 => ( ( member_set_nat @ A3 @ A2 )
% 9.49/5.28                   => ( ~ ( member_set_nat @ A3 @ F4 )
% 9.49/5.28                     => ( ( P @ F4 )
% 9.49/5.28                       => ( P @ ( insert_set_nat @ A3 @ F4 ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6066_finite__subset__induct,axiom,
% 9.49/5.28      ! [F3: set_complex,A2: set_complex,P: set_complex > $o] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ F3 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ F3 @ A2 )
% 9.49/5.28         => ( ( P @ bot_bot_set_complex )
% 9.49/5.28           => ( ! [A3: complex,F4: set_complex] :
% 9.49/5.28                  ( ( finite3207457112153483333omplex @ F4 )
% 9.49/5.28                 => ( ( member_complex @ A3 @ A2 )
% 9.49/5.28                   => ( ~ ( member_complex @ A3 @ F4 )
% 9.49/5.28                     => ( ( P @ F4 )
% 9.49/5.28                       => ( P @ ( insert_complex @ A3 @ F4 ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6067_finite__subset__induct,axiom,
% 9.49/5.28      ! [F3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.49/5.28        ( ( finite6177210948735845034at_nat @ F3 )
% 9.49/5.28       => ( ( ord_le3146513528884898305at_nat @ F3 @ A2 )
% 9.49/5.28         => ( ( P @ bot_bo2099793752762293965at_nat )
% 9.49/5.28           => ( ! [A3: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 9.49/5.28                  ( ( finite6177210948735845034at_nat @ F4 )
% 9.49/5.28                 => ( ( member8440522571783428010at_nat @ A3 @ A2 )
% 9.49/5.28                   => ( ~ ( member8440522571783428010at_nat @ A3 @ F4 )
% 9.49/5.28                     => ( ( P @ F4 )
% 9.49/5.28                       => ( P @ ( insert8211810215607154385at_nat @ A3 @ F4 ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6068_finite__subset__induct,axiom,
% 9.49/5.28      ! [F3: set_real,A2: set_real,P: set_real > $o] :
% 9.49/5.28        ( ( finite_finite_real @ F3 )
% 9.49/5.28       => ( ( ord_less_eq_set_real @ F3 @ A2 )
% 9.49/5.28         => ( ( P @ bot_bot_set_real )
% 9.49/5.28           => ( ! [A3: real,F4: set_real] :
% 9.49/5.28                  ( ( finite_finite_real @ F4 )
% 9.49/5.28                 => ( ( member_real @ A3 @ A2 )
% 9.49/5.28                   => ( ~ ( member_real @ A3 @ F4 )
% 9.49/5.28                     => ( ( P @ F4 )
% 9.49/5.28                       => ( P @ ( insert_real @ A3 @ F4 ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6069_finite__subset__induct,axiom,
% 9.49/5.28      ! [F3: set_nat,A2: set_nat,P: set_nat > $o] :
% 9.49/5.28        ( ( finite_finite_nat @ F3 )
% 9.49/5.28       => ( ( ord_less_eq_set_nat @ F3 @ A2 )
% 9.49/5.28         => ( ( P @ bot_bot_set_nat )
% 9.49/5.28           => ( ! [A3: nat,F4: set_nat] :
% 9.49/5.28                  ( ( finite_finite_nat @ F4 )
% 9.49/5.28                 => ( ( member_nat @ A3 @ A2 )
% 9.49/5.28                   => ( ~ ( member_nat @ A3 @ F4 )
% 9.49/5.28                     => ( ( P @ F4 )
% 9.49/5.28                       => ( P @ ( insert_nat @ A3 @ F4 ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6070_finite__subset__induct,axiom,
% 9.49/5.28      ! [F3: set_int,A2: set_int,P: set_int > $o] :
% 9.49/5.28        ( ( finite_finite_int @ F3 )
% 9.49/5.28       => ( ( ord_less_eq_set_int @ F3 @ A2 )
% 9.49/5.28         => ( ( P @ bot_bot_set_int )
% 9.49/5.28           => ( ! [A3: int,F4: set_int] :
% 9.49/5.28                  ( ( finite_finite_int @ F4 )
% 9.49/5.28                 => ( ( member_int @ A3 @ A2 )
% 9.49/5.28                   => ( ~ ( member_int @ A3 @ F4 )
% 9.49/5.28                     => ( ( P @ F4 )
% 9.49/5.28                       => ( P @ ( insert_int @ A3 @ F4 ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6071_finite__subset__induct_H,axiom,
% 9.49/5.28      ! [F3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 9.49/5.28        ( ( finite4343798906461161616at_nat @ F3 )
% 9.49/5.28       => ( ( ord_le1268244103169919719at_nat @ F3 @ A2 )
% 9.49/5.28         => ( ( P @ bot_bo228742789529271731at_nat )
% 9.49/5.28           => ( ! [A3: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
% 9.49/5.28                  ( ( finite4343798906461161616at_nat @ F4 )
% 9.49/5.28                 => ( ( member8757157785044589968at_nat @ A3 @ A2 )
% 9.49/5.28                   => ( ( ord_le1268244103169919719at_nat @ F4 @ A2 )
% 9.49/5.28                     => ( ~ ( member8757157785044589968at_nat @ A3 @ F4 )
% 9.49/5.28                       => ( ( P @ F4 )
% 9.49/5.28                         => ( P @ ( insert9069300056098147895at_nat @ A3 @ F4 ) ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6072_finite__subset__induct_H,axiom,
% 9.49/5.28      ! [F3: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
% 9.49/5.28        ( ( finite1152437895449049373et_nat @ F3 )
% 9.49/5.28       => ( ( ord_le6893508408891458716et_nat @ F3 @ A2 )
% 9.49/5.28         => ( ( P @ bot_bot_set_set_nat )
% 9.49/5.28           => ( ! [A3: set_nat,F4: set_set_nat] :
% 9.49/5.28                  ( ( finite1152437895449049373et_nat @ F4 )
% 9.49/5.28                 => ( ( member_set_nat @ A3 @ A2 )
% 9.49/5.28                   => ( ( ord_le6893508408891458716et_nat @ F4 @ A2 )
% 9.49/5.28                     => ( ~ ( member_set_nat @ A3 @ F4 )
% 9.49/5.28                       => ( ( P @ F4 )
% 9.49/5.28                         => ( P @ ( insert_set_nat @ A3 @ F4 ) ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6073_finite__subset__induct_H,axiom,
% 9.49/5.28      ! [F3: set_complex,A2: set_complex,P: set_complex > $o] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ F3 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ F3 @ A2 )
% 9.49/5.28         => ( ( P @ bot_bot_set_complex )
% 9.49/5.28           => ( ! [A3: complex,F4: set_complex] :
% 9.49/5.28                  ( ( finite3207457112153483333omplex @ F4 )
% 9.49/5.28                 => ( ( member_complex @ A3 @ A2 )
% 9.49/5.28                   => ( ( ord_le211207098394363844omplex @ F4 @ A2 )
% 9.49/5.28                     => ( ~ ( member_complex @ A3 @ F4 )
% 9.49/5.28                       => ( ( P @ F4 )
% 9.49/5.28                         => ( P @ ( insert_complex @ A3 @ F4 ) ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6074_finite__subset__induct_H,axiom,
% 9.49/5.28      ! [F3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.49/5.28        ( ( finite6177210948735845034at_nat @ F3 )
% 9.49/5.28       => ( ( ord_le3146513528884898305at_nat @ F3 @ A2 )
% 9.49/5.28         => ( ( P @ bot_bo2099793752762293965at_nat )
% 9.49/5.28           => ( ! [A3: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 9.49/5.28                  ( ( finite6177210948735845034at_nat @ F4 )
% 9.49/5.28                 => ( ( member8440522571783428010at_nat @ A3 @ A2 )
% 9.49/5.28                   => ( ( ord_le3146513528884898305at_nat @ F4 @ A2 )
% 9.49/5.28                     => ( ~ ( member8440522571783428010at_nat @ A3 @ F4 )
% 9.49/5.28                       => ( ( P @ F4 )
% 9.49/5.28                         => ( P @ ( insert8211810215607154385at_nat @ A3 @ F4 ) ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6075_finite__subset__induct_H,axiom,
% 9.49/5.28      ! [F3: set_real,A2: set_real,P: set_real > $o] :
% 9.49/5.28        ( ( finite_finite_real @ F3 )
% 9.49/5.28       => ( ( ord_less_eq_set_real @ F3 @ A2 )
% 9.49/5.28         => ( ( P @ bot_bot_set_real )
% 9.49/5.28           => ( ! [A3: real,F4: set_real] :
% 9.49/5.28                  ( ( finite_finite_real @ F4 )
% 9.49/5.28                 => ( ( member_real @ A3 @ A2 )
% 9.49/5.28                   => ( ( ord_less_eq_set_real @ F4 @ A2 )
% 9.49/5.28                     => ( ~ ( member_real @ A3 @ F4 )
% 9.49/5.28                       => ( ( P @ F4 )
% 9.49/5.28                         => ( P @ ( insert_real @ A3 @ F4 ) ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6076_finite__subset__induct_H,axiom,
% 9.49/5.28      ! [F3: set_nat,A2: set_nat,P: set_nat > $o] :
% 9.49/5.28        ( ( finite_finite_nat @ F3 )
% 9.49/5.28       => ( ( ord_less_eq_set_nat @ F3 @ A2 )
% 9.49/5.28         => ( ( P @ bot_bot_set_nat )
% 9.49/5.28           => ( ! [A3: nat,F4: set_nat] :
% 9.49/5.28                  ( ( finite_finite_nat @ F4 )
% 9.49/5.28                 => ( ( member_nat @ A3 @ A2 )
% 9.49/5.28                   => ( ( ord_less_eq_set_nat @ F4 @ A2 )
% 9.49/5.28                     => ( ~ ( member_nat @ A3 @ F4 )
% 9.49/5.28                       => ( ( P @ F4 )
% 9.49/5.28                         => ( P @ ( insert_nat @ A3 @ F4 ) ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6077_finite__subset__induct_H,axiom,
% 9.49/5.28      ! [F3: set_int,A2: set_int,P: set_int > $o] :
% 9.49/5.28        ( ( finite_finite_int @ F3 )
% 9.49/5.28       => ( ( ord_less_eq_set_int @ F3 @ A2 )
% 9.49/5.28         => ( ( P @ bot_bot_set_int )
% 9.49/5.28           => ( ! [A3: int,F4: set_int] :
% 9.49/5.28                  ( ( finite_finite_int @ F4 )
% 9.49/5.28                 => ( ( member_int @ A3 @ A2 )
% 9.49/5.28                   => ( ( ord_less_eq_set_int @ F4 @ A2 )
% 9.49/5.28                     => ( ~ ( member_int @ A3 @ F4 )
% 9.49/5.28                       => ( ( P @ F4 )
% 9.49/5.28                         => ( P @ ( insert_int @ A3 @ F4 ) ) ) ) ) ) )
% 9.49/5.28             => ( P @ F3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6078_sum__nonneg__leq__bound,axiom,
% 9.49/5.28      ! [S: set_real,F: real > real,B2: real,I: real] :
% 9.49/5.28        ( ( finite_finite_real @ S )
% 9.49/5.28       => ( ! [I2: real] :
% 9.49/5.28              ( ( member_real @ I2 @ S )
% 9.49/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.49/5.28         => ( ( ( groups8097168146408367636l_real @ F @ S )
% 9.49/5.28              = B2 )
% 9.49/5.28           => ( ( member_real @ I @ S )
% 9.49/5.28             => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6079_sum__nonneg__leq__bound,axiom,
% 9.49/5.28      ! [S: set_int,F: int > real,B2: real,I: int] :
% 9.49/5.28        ( ( finite_finite_int @ S )
% 9.49/5.28       => ( ! [I2: int] :
% 9.49/5.28              ( ( member_int @ I2 @ S )
% 9.49/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.49/5.28         => ( ( ( groups8778361861064173332t_real @ F @ S )
% 9.49/5.28              = B2 )
% 9.49/5.28           => ( ( member_int @ I @ S )
% 9.49/5.28             => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6080_sum__nonneg__leq__bound,axiom,
% 9.49/5.28      ! [S: set_complex,F: complex > real,B2: real,I: complex] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ S )
% 9.49/5.28       => ( ! [I2: complex] :
% 9.49/5.28              ( ( member_complex @ I2 @ S )
% 9.49/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.49/5.28         => ( ( ( groups5808333547571424918x_real @ F @ S )
% 9.49/5.28              = B2 )
% 9.49/5.28           => ( ( member_complex @ I @ S )
% 9.49/5.28             => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6081_sum__nonneg__leq__bound,axiom,
% 9.49/5.28      ! [S: set_real,F: real > rat,B2: rat,I: real] :
% 9.49/5.28        ( ( finite_finite_real @ S )
% 9.49/5.28       => ( ! [I2: real] :
% 9.49/5.28              ( ( member_real @ I2 @ S )
% 9.49/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.49/5.28         => ( ( ( groups1300246762558778688al_rat @ F @ S )
% 9.49/5.28              = B2 )
% 9.49/5.28           => ( ( member_real @ I @ S )
% 9.49/5.28             => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6082_sum__nonneg__leq__bound,axiom,
% 9.49/5.28      ! [S: set_nat,F: nat > rat,B2: rat,I: nat] :
% 9.49/5.28        ( ( finite_finite_nat @ S )
% 9.49/5.28       => ( ! [I2: nat] :
% 9.49/5.28              ( ( member_nat @ I2 @ S )
% 9.49/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.49/5.28         => ( ( ( groups2906978787729119204at_rat @ F @ S )
% 9.49/5.28              = B2 )
% 9.49/5.28           => ( ( member_nat @ I @ S )
% 9.49/5.28             => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6083_sum__nonneg__leq__bound,axiom,
% 9.49/5.28      ! [S: set_int,F: int > rat,B2: rat,I: int] :
% 9.49/5.28        ( ( finite_finite_int @ S )
% 9.49/5.28       => ( ! [I2: int] :
% 9.49/5.28              ( ( member_int @ I2 @ S )
% 9.49/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.49/5.28         => ( ( ( groups3906332499630173760nt_rat @ F @ S )
% 9.49/5.28              = B2 )
% 9.49/5.28           => ( ( member_int @ I @ S )
% 9.49/5.28             => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6084_sum__nonneg__leq__bound,axiom,
% 9.49/5.28      ! [S: set_complex,F: complex > rat,B2: rat,I: complex] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ S )
% 9.49/5.28       => ( ! [I2: complex] :
% 9.49/5.28              ( ( member_complex @ I2 @ S )
% 9.49/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.49/5.28         => ( ( ( groups5058264527183730370ex_rat @ F @ S )
% 9.49/5.28              = B2 )
% 9.49/5.28           => ( ( member_complex @ I @ S )
% 9.49/5.28             => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6085_sum__nonneg__leq__bound,axiom,
% 9.49/5.28      ! [S: set_real,F: real > nat,B2: nat,I: real] :
% 9.49/5.28        ( ( finite_finite_real @ S )
% 9.49/5.28       => ( ! [I2: real] :
% 9.49/5.28              ( ( member_real @ I2 @ S )
% 9.49/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 9.49/5.28         => ( ( ( groups1935376822645274424al_nat @ F @ S )
% 9.49/5.28              = B2 )
% 9.49/5.28           => ( ( member_real @ I @ S )
% 9.49/5.28             => ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6086_sum__nonneg__leq__bound,axiom,
% 9.49/5.28      ! [S: set_int,F: int > nat,B2: nat,I: int] :
% 9.49/5.28        ( ( finite_finite_int @ S )
% 9.49/5.28       => ( ! [I2: int] :
% 9.49/5.28              ( ( member_int @ I2 @ S )
% 9.49/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 9.49/5.28         => ( ( ( groups4541462559716669496nt_nat @ F @ S )
% 9.49/5.28              = B2 )
% 9.49/5.28           => ( ( member_int @ I @ S )
% 9.49/5.28             => ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6087_sum__nonneg__leq__bound,axiom,
% 9.49/5.28      ! [S: set_complex,F: complex > nat,B2: nat,I: complex] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ S )
% 9.49/5.28       => ( ! [I2: complex] :
% 9.49/5.28              ( ( member_complex @ I2 @ S )
% 9.49/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 9.49/5.28         => ( ( ( groups5693394587270226106ex_nat @ F @ S )
% 9.49/5.28              = B2 )
% 9.49/5.28           => ( ( member_complex @ I @ S )
% 9.49/5.28             => ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6088_sum__nonneg__0,axiom,
% 9.49/5.28      ! [S: set_real,F: real > real,I: real] :
% 9.49/5.28        ( ( finite_finite_real @ S )
% 9.49/5.28       => ( ! [I2: real] :
% 9.49/5.28              ( ( member_real @ I2 @ S )
% 9.49/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.49/5.28         => ( ( ( groups8097168146408367636l_real @ F @ S )
% 9.49/5.28              = zero_zero_real )
% 9.49/5.28           => ( ( member_real @ I @ S )
% 9.49/5.28             => ( ( F @ I )
% 9.49/5.28                = zero_zero_real ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6089_sum__nonneg__0,axiom,
% 9.49/5.28      ! [S: set_int,F: int > real,I: int] :
% 9.49/5.28        ( ( finite_finite_int @ S )
% 9.49/5.28       => ( ! [I2: int] :
% 9.49/5.28              ( ( member_int @ I2 @ S )
% 9.49/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.49/5.28         => ( ( ( groups8778361861064173332t_real @ F @ S )
% 9.49/5.28              = zero_zero_real )
% 9.49/5.28           => ( ( member_int @ I @ S )
% 9.49/5.28             => ( ( F @ I )
% 9.49/5.28                = zero_zero_real ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6090_sum__nonneg__0,axiom,
% 9.49/5.28      ! [S: set_complex,F: complex > real,I: complex] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ S )
% 9.49/5.28       => ( ! [I2: complex] :
% 9.49/5.28              ( ( member_complex @ I2 @ S )
% 9.49/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.49/5.28         => ( ( ( groups5808333547571424918x_real @ F @ S )
% 9.49/5.28              = zero_zero_real )
% 9.49/5.28           => ( ( member_complex @ I @ S )
% 9.49/5.28             => ( ( F @ I )
% 9.49/5.28                = zero_zero_real ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6091_sum__nonneg__0,axiom,
% 9.49/5.28      ! [S: set_real,F: real > rat,I: real] :
% 9.49/5.28        ( ( finite_finite_real @ S )
% 9.49/5.28       => ( ! [I2: real] :
% 9.49/5.28              ( ( member_real @ I2 @ S )
% 9.49/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.49/5.28         => ( ( ( groups1300246762558778688al_rat @ F @ S )
% 9.49/5.28              = zero_zero_rat )
% 9.49/5.28           => ( ( member_real @ I @ S )
% 9.49/5.28             => ( ( F @ I )
% 9.49/5.28                = zero_zero_rat ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6092_sum__nonneg__0,axiom,
% 9.49/5.28      ! [S: set_nat,F: nat > rat,I: nat] :
% 9.49/5.28        ( ( finite_finite_nat @ S )
% 9.49/5.28       => ( ! [I2: nat] :
% 9.49/5.28              ( ( member_nat @ I2 @ S )
% 9.49/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.49/5.28         => ( ( ( groups2906978787729119204at_rat @ F @ S )
% 9.49/5.28              = zero_zero_rat )
% 9.49/5.28           => ( ( member_nat @ I @ S )
% 9.49/5.28             => ( ( F @ I )
% 9.49/5.28                = zero_zero_rat ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6093_sum__nonneg__0,axiom,
% 9.49/5.28      ! [S: set_int,F: int > rat,I: int] :
% 9.49/5.28        ( ( finite_finite_int @ S )
% 9.49/5.28       => ( ! [I2: int] :
% 9.49/5.28              ( ( member_int @ I2 @ S )
% 9.49/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.49/5.28         => ( ( ( groups3906332499630173760nt_rat @ F @ S )
% 9.49/5.28              = zero_zero_rat )
% 9.49/5.28           => ( ( member_int @ I @ S )
% 9.49/5.28             => ( ( F @ I )
% 9.49/5.28                = zero_zero_rat ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6094_sum__nonneg__0,axiom,
% 9.49/5.28      ! [S: set_complex,F: complex > rat,I: complex] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ S )
% 9.49/5.28       => ( ! [I2: complex] :
% 9.49/5.28              ( ( member_complex @ I2 @ S )
% 9.49/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.49/5.28         => ( ( ( groups5058264527183730370ex_rat @ F @ S )
% 9.49/5.28              = zero_zero_rat )
% 9.49/5.28           => ( ( member_complex @ I @ S )
% 9.49/5.28             => ( ( F @ I )
% 9.49/5.28                = zero_zero_rat ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6095_sum__nonneg__0,axiom,
% 9.49/5.28      ! [S: set_real,F: real > nat,I: real] :
% 9.49/5.28        ( ( finite_finite_real @ S )
% 9.49/5.28       => ( ! [I2: real] :
% 9.49/5.28              ( ( member_real @ I2 @ S )
% 9.49/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 9.49/5.28         => ( ( ( groups1935376822645274424al_nat @ F @ S )
% 9.49/5.28              = zero_zero_nat )
% 9.49/5.28           => ( ( member_real @ I @ S )
% 9.49/5.28             => ( ( F @ I )
% 9.49/5.28                = zero_zero_nat ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6096_sum__nonneg__0,axiom,
% 9.49/5.28      ! [S: set_int,F: int > nat,I: int] :
% 9.49/5.28        ( ( finite_finite_int @ S )
% 9.49/5.28       => ( ! [I2: int] :
% 9.49/5.28              ( ( member_int @ I2 @ S )
% 9.49/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 9.49/5.28         => ( ( ( groups4541462559716669496nt_nat @ F @ S )
% 9.49/5.28              = zero_zero_nat )
% 9.49/5.28           => ( ( member_int @ I @ S )
% 9.49/5.28             => ( ( F @ I )
% 9.49/5.28                = zero_zero_nat ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6097_sum__nonneg__0,axiom,
% 9.49/5.28      ! [S: set_complex,F: complex > nat,I: complex] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ S )
% 9.49/5.28       => ( ! [I2: complex] :
% 9.49/5.28              ( ( member_complex @ I2 @ S )
% 9.49/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 9.49/5.28         => ( ( ( groups5693394587270226106ex_nat @ F @ S )
% 9.49/5.28              = zero_zero_nat )
% 9.49/5.28           => ( ( member_complex @ I @ S )
% 9.49/5.28             => ( ( F @ I )
% 9.49/5.28                = zero_zero_nat ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6098_finite__empty__induct,axiom,
% 9.49/5.28      ! [A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 9.49/5.28        ( ( finite4343798906461161616at_nat @ A2 )
% 9.49/5.28       => ( ( P @ A2 )
% 9.49/5.28         => ( ! [A3: produc3843707927480180839at_nat,A5: set_Pr4329608150637261639at_nat] :
% 9.49/5.28                ( ( finite4343798906461161616at_nat @ A5 )
% 9.49/5.28               => ( ( member8757157785044589968at_nat @ A3 @ A5 )
% 9.49/5.28                 => ( ( P @ A5 )
% 9.49/5.28                   => ( P @ ( minus_3314409938677909166at_nat @ A5 @ ( insert9069300056098147895at_nat @ A3 @ bot_bo228742789529271731at_nat ) ) ) ) ) )
% 9.49/5.28           => ( P @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6099_finite__empty__induct,axiom,
% 9.49/5.28      ! [A2: set_set_nat,P: set_set_nat > $o] :
% 9.49/5.28        ( ( finite1152437895449049373et_nat @ A2 )
% 9.49/5.28       => ( ( P @ A2 )
% 9.49/5.28         => ( ! [A3: set_nat,A5: set_set_nat] :
% 9.49/5.28                ( ( finite1152437895449049373et_nat @ A5 )
% 9.49/5.28               => ( ( member_set_nat @ A3 @ A5 )
% 9.49/5.28                 => ( ( P @ A5 )
% 9.49/5.28                   => ( P @ ( minus_2163939370556025621et_nat @ A5 @ ( insert_set_nat @ A3 @ bot_bot_set_set_nat ) ) ) ) ) )
% 9.49/5.28           => ( P @ bot_bot_set_set_nat ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6100_finite__empty__induct,axiom,
% 9.49/5.28      ! [A2: set_complex,P: set_complex > $o] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( P @ A2 )
% 9.49/5.28         => ( ! [A3: complex,A5: set_complex] :
% 9.49/5.28                ( ( finite3207457112153483333omplex @ A5 )
% 9.49/5.28               => ( ( member_complex @ A3 @ A5 )
% 9.49/5.28                 => ( ( P @ A5 )
% 9.49/5.28                   => ( P @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ A3 @ bot_bot_set_complex ) ) ) ) ) )
% 9.49/5.28           => ( P @ bot_bot_set_complex ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6101_finite__empty__induct,axiom,
% 9.49/5.28      ! [A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.49/5.28        ( ( finite6177210948735845034at_nat @ A2 )
% 9.49/5.28       => ( ( P @ A2 )
% 9.49/5.28         => ( ! [A3: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
% 9.49/5.28                ( ( finite6177210948735845034at_nat @ A5 )
% 9.49/5.28               => ( ( member8440522571783428010at_nat @ A3 @ A5 )
% 9.49/5.28                 => ( ( P @ A5 )
% 9.49/5.28                   => ( P @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ A3 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
% 9.49/5.28           => ( P @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6102_finite__empty__induct,axiom,
% 9.49/5.28      ! [A2: set_real,P: set_real > $o] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( P @ A2 )
% 9.49/5.28         => ( ! [A3: real,A5: set_real] :
% 9.49/5.28                ( ( finite_finite_real @ A5 )
% 9.49/5.28               => ( ( member_real @ A3 @ A5 )
% 9.49/5.28                 => ( ( P @ A5 )
% 9.49/5.28                   => ( P @ ( minus_minus_set_real @ A5 @ ( insert_real @ A3 @ bot_bot_set_real ) ) ) ) ) )
% 9.49/5.28           => ( P @ bot_bot_set_real ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6103_finite__empty__induct,axiom,
% 9.49/5.28      ! [A2: set_int,P: set_int > $o] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( P @ A2 )
% 9.49/5.28         => ( ! [A3: int,A5: set_int] :
% 9.49/5.28                ( ( finite_finite_int @ A5 )
% 9.49/5.28               => ( ( member_int @ A3 @ A5 )
% 9.49/5.28                 => ( ( P @ A5 )
% 9.49/5.28                   => ( P @ ( minus_minus_set_int @ A5 @ ( insert_int @ A3 @ bot_bot_set_int ) ) ) ) ) )
% 9.49/5.28           => ( P @ bot_bot_set_int ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6104_finite__empty__induct,axiom,
% 9.49/5.28      ! [A2: set_nat,P: set_nat > $o] :
% 9.49/5.28        ( ( finite_finite_nat @ A2 )
% 9.49/5.28       => ( ( P @ A2 )
% 9.49/5.28         => ( ! [A3: nat,A5: set_nat] :
% 9.49/5.28                ( ( finite_finite_nat @ A5 )
% 9.49/5.28               => ( ( member_nat @ A3 @ A5 )
% 9.49/5.28                 => ( ( P @ A5 )
% 9.49/5.28                   => ( P @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ) )
% 9.49/5.28           => ( P @ bot_bot_set_nat ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6105_infinite__coinduct,axiom,
% 9.49/5.28      ! [X8: set_Pr4329608150637261639at_nat > $o,A2: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ( X8 @ A2 )
% 9.49/5.28       => ( ! [A5: set_Pr4329608150637261639at_nat] :
% 9.49/5.28              ( ( X8 @ A5 )
% 9.49/5.28             => ? [X2: produc3843707927480180839at_nat] :
% 9.49/5.28                  ( ( member8757157785044589968at_nat @ X2 @ A5 )
% 9.49/5.28                  & ( ( X8 @ ( minus_3314409938677909166at_nat @ A5 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) )
% 9.49/5.28                    | ~ ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A5 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) ) ) )
% 9.49/5.28         => ~ ( finite4343798906461161616at_nat @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6106_infinite__coinduct,axiom,
% 9.49/5.28      ! [X8: set_complex > $o,A2: set_complex] :
% 9.49/5.28        ( ( X8 @ A2 )
% 9.49/5.28       => ( ! [A5: set_complex] :
% 9.49/5.28              ( ( X8 @ A5 )
% 9.49/5.28             => ? [X2: complex] :
% 9.49/5.28                  ( ( member_complex @ X2 @ A5 )
% 9.49/5.28                  & ( ( X8 @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) )
% 9.49/5.28                    | ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) )
% 9.49/5.28         => ~ ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6107_infinite__coinduct,axiom,
% 9.49/5.28      ! [X8: set_Pr1261947904930325089at_nat > $o,A2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ( X8 @ A2 )
% 9.49/5.28       => ( ! [A5: set_Pr1261947904930325089at_nat] :
% 9.49/5.28              ( ( X8 @ A5 )
% 9.49/5.28             => ? [X2: product_prod_nat_nat] :
% 9.49/5.28                  ( ( member8440522571783428010at_nat @ X2 @ A5 )
% 9.49/5.28                  & ( ( X8 @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
% 9.49/5.28                    | ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
% 9.49/5.28         => ~ ( finite6177210948735845034at_nat @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6108_infinite__coinduct,axiom,
% 9.49/5.28      ! [X8: set_real > $o,A2: set_real] :
% 9.49/5.28        ( ( X8 @ A2 )
% 9.49/5.28       => ( ! [A5: set_real] :
% 9.49/5.28              ( ( X8 @ A5 )
% 9.49/5.28             => ? [X2: real] :
% 9.49/5.28                  ( ( member_real @ X2 @ A5 )
% 9.49/5.28                  & ( ( X8 @ ( minus_minus_set_real @ A5 @ ( insert_real @ X2 @ bot_bot_set_real ) ) )
% 9.49/5.28                    | ~ ( finite_finite_real @ ( minus_minus_set_real @ A5 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) )
% 9.49/5.28         => ~ ( finite_finite_real @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6109_infinite__coinduct,axiom,
% 9.49/5.28      ! [X8: set_int > $o,A2: set_int] :
% 9.49/5.28        ( ( X8 @ A2 )
% 9.49/5.28       => ( ! [A5: set_int] :
% 9.49/5.28              ( ( X8 @ A5 )
% 9.49/5.28             => ? [X2: int] :
% 9.49/5.28                  ( ( member_int @ X2 @ A5 )
% 9.49/5.28                  & ( ( X8 @ ( minus_minus_set_int @ A5 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
% 9.49/5.28                    | ~ ( finite_finite_int @ ( minus_minus_set_int @ A5 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) )
% 9.49/5.28         => ~ ( finite_finite_int @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6110_infinite__coinduct,axiom,
% 9.49/5.28      ! [X8: set_nat > $o,A2: set_nat] :
% 9.49/5.28        ( ( X8 @ A2 )
% 9.49/5.28       => ( ! [A5: set_nat] :
% 9.49/5.28              ( ( X8 @ A5 )
% 9.49/5.28             => ? [X2: nat] :
% 9.49/5.28                  ( ( member_nat @ X2 @ A5 )
% 9.49/5.28                  & ( ( X8 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
% 9.49/5.28                    | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) )
% 9.49/5.28         => ~ ( finite_finite_nat @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6111_infinite__remove,axiom,
% 9.49/5.28      ! [S2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
% 9.49/5.28        ( ~ ( finite4343798906461161616at_nat @ S2 )
% 9.49/5.28       => ~ ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ S2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6112_infinite__remove,axiom,
% 9.49/5.28      ! [S2: set_complex,A: complex] :
% 9.49/5.28        ( ~ ( finite3207457112153483333omplex @ S2 )
% 9.49/5.28       => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6113_infinite__remove,axiom,
% 9.49/5.28      ! [S2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
% 9.49/5.28        ( ~ ( finite6177210948735845034at_nat @ S2 )
% 9.49/5.28       => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6114_infinite__remove,axiom,
% 9.49/5.28      ! [S2: set_real,A: real] :
% 9.49/5.28        ( ~ ( finite_finite_real @ S2 )
% 9.49/5.28       => ~ ( finite_finite_real @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6115_infinite__remove,axiom,
% 9.49/5.28      ! [S2: set_int,A: int] :
% 9.49/5.28        ( ~ ( finite_finite_int @ S2 )
% 9.49/5.28       => ~ ( finite_finite_int @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6116_infinite__remove,axiom,
% 9.49/5.28      ! [S2: set_nat,A: nat] :
% 9.49/5.28        ( ~ ( finite_finite_nat @ S2 )
% 9.49/5.28       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6117_Diff__single__insert,axiom,
% 9.49/5.28      ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B2 )
% 9.49/5.28       => ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6118_Diff__single__insert,axiom,
% 9.49/5.28      ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B2 )
% 9.49/5.28       => ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6119_Diff__single__insert,axiom,
% 9.49/5.28      ! [A2: set_real,X: real,B2: set_real] :
% 9.49/5.28        ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 )
% 9.49/5.28       => ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6120_Diff__single__insert,axiom,
% 9.49/5.28      ! [A2: set_nat,X: nat,B2: set_nat] :
% 9.49/5.28        ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 )
% 9.49/5.28       => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6121_Diff__single__insert,axiom,
% 9.49/5.28      ! [A2: set_int,X: int,B2: set_int] :
% 9.49/5.28        ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 )
% 9.49/5.28       => ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6122_subset__insert__iff,axiom,
% 9.49/5.28      ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.49/5.28        ( ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ B2 ) )
% 9.49/5.28        = ( ( ( member8757157785044589968at_nat @ X @ A2 )
% 9.49/5.28           => ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B2 ) )
% 9.49/5.28          & ( ~ ( member8757157785044589968at_nat @ X @ A2 )
% 9.49/5.28           => ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6123_subset__insert__iff,axiom,
% 9.49/5.28      ! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
% 9.49/5.28        ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
% 9.49/5.28        = ( ( ( member_set_nat @ X @ A2 )
% 9.49/5.28           => ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B2 ) )
% 9.49/5.28          & ( ~ ( member_set_nat @ X @ A2 )
% 9.49/5.28           => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6124_subset__insert__iff,axiom,
% 9.49/5.28      ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.28        ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B2 ) )
% 9.49/5.28        = ( ( ( member8440522571783428010at_nat @ X @ A2 )
% 9.49/5.28           => ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B2 ) )
% 9.49/5.28          & ( ~ ( member8440522571783428010at_nat @ X @ A2 )
% 9.49/5.28           => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6125_subset__insert__iff,axiom,
% 9.49/5.28      ! [A2: set_real,X: real,B2: set_real] :
% 9.49/5.28        ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) )
% 9.49/5.28        = ( ( ( member_real @ X @ A2 )
% 9.49/5.28           => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 ) )
% 9.49/5.28          & ( ~ ( member_real @ X @ A2 )
% 9.49/5.28           => ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6126_subset__insert__iff,axiom,
% 9.49/5.28      ! [A2: set_nat,X: nat,B2: set_nat] :
% 9.49/5.28        ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
% 9.49/5.28        = ( ( ( member_nat @ X @ A2 )
% 9.49/5.28           => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
% 9.49/5.28          & ( ~ ( member_nat @ X @ A2 )
% 9.49/5.28           => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6127_subset__insert__iff,axiom,
% 9.49/5.28      ! [A2: set_int,X: int,B2: set_int] :
% 9.49/5.28        ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) )
% 9.49/5.28        = ( ( ( member_int @ X @ A2 )
% 9.49/5.28           => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 ) )
% 9.49/5.28          & ( ~ ( member_int @ X @ A2 )
% 9.49/5.28           => ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6128_atLeast0__atMost__Suc,axiom,
% 9.49/5.28      ! [N: nat] :
% 9.49/5.28        ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 9.49/5.28        = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6129_Icc__eq__insert__lb__nat,axiom,
% 9.49/5.28      ! [M: nat,N: nat] :
% 9.49/5.28        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.28       => ( ( set_or1269000886237332187st_nat @ M @ N )
% 9.49/5.28          = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6130_atLeastAtMostSuc__conv,axiom,
% 9.49/5.28      ! [M: nat,N: nat] :
% 9.49/5.28        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.49/5.28       => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
% 9.49/5.28          = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6131_atLeastAtMost__insertL,axiom,
% 9.49/5.28      ! [M: nat,N: nat] :
% 9.49/5.28        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.28       => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 9.49/5.28          = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6132_sum_Osetdiff__irrelevant,axiom,
% 9.49/5.28      ! [A2: set_real,G: real > complex] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( groups5754745047067104278omplex @ G
% 9.49/5.28            @ ( minus_minus_set_real @ A2
% 9.49/5.28              @ ( collect_real
% 9.49/5.28                @ ^ [X3: real] :
% 9.49/5.28                    ( ( G @ X3 )
% 9.49/5.28                    = zero_zero_complex ) ) ) )
% 9.49/5.28          = ( groups5754745047067104278omplex @ G @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6133_sum_Osetdiff__irrelevant,axiom,
% 9.49/5.28      ! [A2: set_int,G: int > complex] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( groups3049146728041665814omplex @ G
% 9.49/5.28            @ ( minus_minus_set_int @ A2
% 9.49/5.28              @ ( collect_int
% 9.49/5.28                @ ^ [X3: int] :
% 9.49/5.28                    ( ( G @ X3 )
% 9.49/5.28                    = zero_zero_complex ) ) ) )
% 9.49/5.28          = ( groups3049146728041665814omplex @ G @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6134_sum_Osetdiff__irrelevant,axiom,
% 9.49/5.28      ! [A2: set_real,G: real > real] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( groups8097168146408367636l_real @ G
% 9.49/5.28            @ ( minus_minus_set_real @ A2
% 9.49/5.28              @ ( collect_real
% 9.49/5.28                @ ^ [X3: real] :
% 9.49/5.28                    ( ( G @ X3 )
% 9.49/5.28                    = zero_zero_real ) ) ) )
% 9.49/5.28          = ( groups8097168146408367636l_real @ G @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6135_sum_Osetdiff__irrelevant,axiom,
% 9.49/5.28      ! [A2: set_int,G: int > real] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( groups8778361861064173332t_real @ G
% 9.49/5.28            @ ( minus_minus_set_int @ A2
% 9.49/5.28              @ ( collect_int
% 9.49/5.28                @ ^ [X3: int] :
% 9.49/5.28                    ( ( G @ X3 )
% 9.49/5.28                    = zero_zero_real ) ) ) )
% 9.49/5.28          = ( groups8778361861064173332t_real @ G @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6136_sum_Osetdiff__irrelevant,axiom,
% 9.49/5.28      ! [A2: set_complex,G: complex > real] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( groups5808333547571424918x_real @ G
% 9.49/5.28            @ ( minus_811609699411566653omplex @ A2
% 9.49/5.28              @ ( collect_complex
% 9.49/5.28                @ ^ [X3: complex] :
% 9.49/5.28                    ( ( G @ X3 )
% 9.49/5.28                    = zero_zero_real ) ) ) )
% 9.49/5.28          = ( groups5808333547571424918x_real @ G @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6137_sum_Osetdiff__irrelevant,axiom,
% 9.49/5.28      ! [A2: set_real,G: real > rat] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( groups1300246762558778688al_rat @ G
% 9.49/5.28            @ ( minus_minus_set_real @ A2
% 9.49/5.28              @ ( collect_real
% 9.49/5.28                @ ^ [X3: real] :
% 9.49/5.28                    ( ( G @ X3 )
% 9.49/5.28                    = zero_zero_rat ) ) ) )
% 9.49/5.28          = ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6138_sum_Osetdiff__irrelevant,axiom,
% 9.49/5.28      ! [A2: set_int,G: int > rat] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( groups3906332499630173760nt_rat @ G
% 9.49/5.28            @ ( minus_minus_set_int @ A2
% 9.49/5.28              @ ( collect_int
% 9.49/5.28                @ ^ [X3: int] :
% 9.49/5.28                    ( ( G @ X3 )
% 9.49/5.28                    = zero_zero_rat ) ) ) )
% 9.49/5.28          = ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6139_sum_Osetdiff__irrelevant,axiom,
% 9.49/5.28      ! [A2: set_complex,G: complex > rat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.28       => ( ( groups5058264527183730370ex_rat @ G
% 9.49/5.28            @ ( minus_811609699411566653omplex @ A2
% 9.49/5.28              @ ( collect_complex
% 9.49/5.28                @ ^ [X3: complex] :
% 9.49/5.28                    ( ( G @ X3 )
% 9.49/5.28                    = zero_zero_rat ) ) ) )
% 9.49/5.28          = ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6140_sum_Osetdiff__irrelevant,axiom,
% 9.49/5.28      ! [A2: set_real,G: real > nat] :
% 9.49/5.28        ( ( finite_finite_real @ A2 )
% 9.49/5.28       => ( ( groups1935376822645274424al_nat @ G
% 9.49/5.28            @ ( minus_minus_set_real @ A2
% 9.49/5.28              @ ( collect_real
% 9.49/5.28                @ ^ [X3: real] :
% 9.49/5.28                    ( ( G @ X3 )
% 9.49/5.28                    = zero_zero_nat ) ) ) )
% 9.49/5.28          = ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6141_sum_Osetdiff__irrelevant,axiom,
% 9.49/5.28      ! [A2: set_int,G: int > nat] :
% 9.49/5.28        ( ( finite_finite_int @ A2 )
% 9.49/5.28       => ( ( groups4541462559716669496nt_nat @ G
% 9.49/5.28            @ ( minus_minus_set_int @ A2
% 9.49/5.28              @ ( collect_int
% 9.49/5.28                @ ^ [X3: int] :
% 9.49/5.28                    ( ( G @ X3 )
% 9.49/5.28                    = zero_zero_nat ) ) ) )
% 9.49/5.28          = ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6142_set__update__subset__insert,axiom,
% 9.49/5.28      ! [Xs: list_P6011104703257516679at_nat,I: nat,X: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ X ) ) @ ( insert8211810215607154385at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6143_set__update__subset__insert,axiom,
% 9.49/5.28      ! [Xs: list_real,I: nat,X: real] : ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X ) ) @ ( insert_real @ X @ ( set_real2 @ Xs ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6144_set__update__subset__insert,axiom,
% 9.49/5.28      ! [Xs: list_P5464809261938338413at_nat,I: nat,X: produc3843707927480180839at_nat] : ( ord_le1268244103169919719at_nat @ ( set_Pr3765526544606949372at_nat @ ( list_u4696772448584712917at_nat @ Xs @ I @ X ) ) @ ( insert9069300056098147895at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6145_set__update__subset__insert,axiom,
% 9.49/5.28      ! [Xs: list_nat,I: nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ ( insert_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6146_set__update__subset__insert,axiom,
% 9.49/5.28      ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) ) @ ( insert_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6147_set__update__subset__insert,axiom,
% 9.49/5.28      ! [Xs: list_int,I: nat,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X ) ) @ ( insert_int @ X @ ( set_int2 @ Xs ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6148_sum__pos2,axiom,
% 9.49/5.28      ! [I6: set_real,I: real,F: real > real] :
% 9.49/5.28        ( ( finite_finite_real @ I6 )
% 9.49/5.28       => ( ( member_real @ I @ I6 )
% 9.49/5.28         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.49/5.28           => ( ! [I2: real] :
% 9.49/5.28                  ( ( member_real @ I2 @ I6 )
% 9.49/5.28                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.49/5.28             => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I6 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6149_sum__pos2,axiom,
% 9.49/5.28      ! [I6: set_int,I: int,F: int > real] :
% 9.49/5.28        ( ( finite_finite_int @ I6 )
% 9.49/5.28       => ( ( member_int @ I @ I6 )
% 9.49/5.28         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.49/5.28           => ( ! [I2: int] :
% 9.49/5.28                  ( ( member_int @ I2 @ I6 )
% 9.49/5.28                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.49/5.28             => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I6 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6150_sum__pos2,axiom,
% 9.49/5.28      ! [I6: set_complex,I: complex,F: complex > real] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ I6 )
% 9.49/5.28       => ( ( member_complex @ I @ I6 )
% 9.49/5.28         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.49/5.28           => ( ! [I2: complex] :
% 9.49/5.28                  ( ( member_complex @ I2 @ I6 )
% 9.49/5.28                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.49/5.28             => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6151_sum__pos2,axiom,
% 9.49/5.28      ! [I6: set_real,I: real,F: real > rat] :
% 9.49/5.28        ( ( finite_finite_real @ I6 )
% 9.49/5.28       => ( ( member_real @ I @ I6 )
% 9.49/5.28         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 9.49/5.28           => ( ! [I2: real] :
% 9.49/5.28                  ( ( member_real @ I2 @ I6 )
% 9.49/5.28                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.49/5.28             => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I6 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6152_sum__pos2,axiom,
% 9.49/5.28      ! [I6: set_nat,I: nat,F: nat > rat] :
% 9.49/5.28        ( ( finite_finite_nat @ I6 )
% 9.49/5.28       => ( ( member_nat @ I @ I6 )
% 9.49/5.28         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 9.49/5.28           => ( ! [I2: nat] :
% 9.49/5.28                  ( ( member_nat @ I2 @ I6 )
% 9.49/5.28                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.49/5.28             => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I6 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6153_sum__pos2,axiom,
% 9.49/5.28      ! [I6: set_int,I: int,F: int > rat] :
% 9.49/5.28        ( ( finite_finite_int @ I6 )
% 9.49/5.28       => ( ( member_int @ I @ I6 )
% 9.49/5.28         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 9.49/5.28           => ( ! [I2: int] :
% 9.49/5.28                  ( ( member_int @ I2 @ I6 )
% 9.49/5.28                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.49/5.28             => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I6 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6154_sum__pos2,axiom,
% 9.49/5.28      ! [I6: set_complex,I: complex,F: complex > rat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ I6 )
% 9.49/5.28       => ( ( member_complex @ I @ I6 )
% 9.49/5.28         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 9.49/5.28           => ( ! [I2: complex] :
% 9.49/5.28                  ( ( member_complex @ I2 @ I6 )
% 9.49/5.28                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.49/5.28             => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I6 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6155_sum__pos2,axiom,
% 9.49/5.28      ! [I6: set_real,I: real,F: real > nat] :
% 9.49/5.28        ( ( finite_finite_real @ I6 )
% 9.49/5.28       => ( ( member_real @ I @ I6 )
% 9.49/5.28         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 9.49/5.28           => ( ! [I2: real] :
% 9.49/5.28                  ( ( member_real @ I2 @ I6 )
% 9.49/5.28                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 9.49/5.28             => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I6 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6156_sum__pos2,axiom,
% 9.49/5.28      ! [I6: set_int,I: int,F: int > nat] :
% 9.49/5.28        ( ( finite_finite_int @ I6 )
% 9.49/5.28       => ( ( member_int @ I @ I6 )
% 9.49/5.28         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 9.49/5.28           => ( ! [I2: int] :
% 9.49/5.28                  ( ( member_int @ I2 @ I6 )
% 9.49/5.28                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 9.49/5.28             => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I6 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6157_sum__pos2,axiom,
% 9.49/5.28      ! [I6: set_complex,I: complex,F: complex > nat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ I6 )
% 9.49/5.28       => ( ( member_complex @ I @ I6 )
% 9.49/5.28         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 9.49/5.28           => ( ! [I2: complex] :
% 9.49/5.28                  ( ( member_complex @ I2 @ I6 )
% 9.49/5.28                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 9.49/5.28             => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I6 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6158_sum__pos,axiom,
% 9.49/5.28      ! [I6: set_complex,F: complex > real] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ I6 )
% 9.49/5.28       => ( ( I6 != bot_bot_set_complex )
% 9.49/5.28         => ( ! [I2: complex] :
% 9.49/5.28                ( ( member_complex @ I2 @ I6 )
% 9.49/5.28               => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.49/5.28           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6159_sum__pos,axiom,
% 9.49/5.28      ! [I6: set_real,F: real > real] :
% 9.49/5.28        ( ( finite_finite_real @ I6 )
% 9.49/5.28       => ( ( I6 != bot_bot_set_real )
% 9.49/5.28         => ( ! [I2: real] :
% 9.49/5.28                ( ( member_real @ I2 @ I6 )
% 9.49/5.28               => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.49/5.28           => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6160_sum__pos,axiom,
% 9.49/5.28      ! [I6: set_int,F: int > real] :
% 9.49/5.28        ( ( finite_finite_int @ I6 )
% 9.49/5.28       => ( ( I6 != bot_bot_set_int )
% 9.49/5.28         => ( ! [I2: int] :
% 9.49/5.28                ( ( member_int @ I2 @ I6 )
% 9.49/5.28               => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.49/5.28           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6161_sum__pos,axiom,
% 9.49/5.28      ! [I6: set_complex,F: complex > rat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ I6 )
% 9.49/5.28       => ( ( I6 != bot_bot_set_complex )
% 9.49/5.28         => ( ! [I2: complex] :
% 9.49/5.28                ( ( member_complex @ I2 @ I6 )
% 9.49/5.28               => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.49/5.28           => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6162_sum__pos,axiom,
% 9.49/5.28      ! [I6: set_real,F: real > rat] :
% 9.49/5.28        ( ( finite_finite_real @ I6 )
% 9.49/5.28       => ( ( I6 != bot_bot_set_real )
% 9.49/5.28         => ( ! [I2: real] :
% 9.49/5.28                ( ( member_real @ I2 @ I6 )
% 9.49/5.28               => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.49/5.28           => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6163_sum__pos,axiom,
% 9.49/5.28      ! [I6: set_nat,F: nat > rat] :
% 9.49/5.28        ( ( finite_finite_nat @ I6 )
% 9.49/5.28       => ( ( I6 != bot_bot_set_nat )
% 9.49/5.28         => ( ! [I2: nat] :
% 9.49/5.28                ( ( member_nat @ I2 @ I6 )
% 9.49/5.28               => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.49/5.28           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6164_sum__pos,axiom,
% 9.49/5.28      ! [I6: set_int,F: int > rat] :
% 9.49/5.28        ( ( finite_finite_int @ I6 )
% 9.49/5.28       => ( ( I6 != bot_bot_set_int )
% 9.49/5.28         => ( ! [I2: int] :
% 9.49/5.28                ( ( member_int @ I2 @ I6 )
% 9.49/5.28               => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.49/5.28           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6165_sum__pos,axiom,
% 9.49/5.28      ! [I6: set_complex,F: complex > nat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ I6 )
% 9.49/5.28       => ( ( I6 != bot_bot_set_complex )
% 9.49/5.28         => ( ! [I2: complex] :
% 9.49/5.28                ( ( member_complex @ I2 @ I6 )
% 9.49/5.28               => ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 9.49/5.28           => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6166_sum__pos,axiom,
% 9.49/5.28      ! [I6: set_real,F: real > nat] :
% 9.49/5.28        ( ( finite_finite_real @ I6 )
% 9.49/5.28       => ( ( I6 != bot_bot_set_real )
% 9.49/5.28         => ( ! [I2: real] :
% 9.49/5.28                ( ( member_real @ I2 @ I6 )
% 9.49/5.28               => ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 9.49/5.28           => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6167_sum__pos,axiom,
% 9.49/5.28      ! [I6: set_int,F: int > nat] :
% 9.49/5.28        ( ( finite_finite_int @ I6 )
% 9.49/5.28       => ( ( I6 != bot_bot_set_int )
% 9.49/5.28         => ( ! [I2: int] :
% 9.49/5.28                ( ( member_int @ I2 @ I6 )
% 9.49/5.28               => ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 9.49/5.28           => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I6 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6168_sum_Omono__neutral__cong__right,axiom,
% 9.49/5.28      ! [T3: set_real,S2: set_real,G: real > complex,H2: real > complex] :
% 9.49/5.28        ( ( finite_finite_real @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: real] :
% 9.49/5.28                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_complex ) )
% 9.49/5.28           => ( ! [X4: real] :
% 9.49/5.28                  ( ( member_real @ X4 @ S2 )
% 9.49/5.28                 => ( ( G @ X4 )
% 9.49/5.28                    = ( H2 @ X4 ) ) )
% 9.49/5.28             => ( ( groups5754745047067104278omplex @ G @ T3 )
% 9.49/5.28                = ( groups5754745047067104278omplex @ H2 @ S2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6169_sum_Omono__neutral__cong__right,axiom,
% 9.49/5.28      ! [T3: set_real,S2: set_real,G: real > real,H2: real > real] :
% 9.49/5.28        ( ( finite_finite_real @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: real] :
% 9.49/5.28                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_real ) )
% 9.49/5.28           => ( ! [X4: real] :
% 9.49/5.28                  ( ( member_real @ X4 @ S2 )
% 9.49/5.28                 => ( ( G @ X4 )
% 9.49/5.28                    = ( H2 @ X4 ) ) )
% 9.49/5.28             => ( ( groups8097168146408367636l_real @ G @ T3 )
% 9.49/5.28                = ( groups8097168146408367636l_real @ H2 @ S2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6170_sum_Omono__neutral__cong__right,axiom,
% 9.49/5.28      ! [T3: set_complex,S2: set_complex,G: complex > real,H2: complex > real] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_real ) )
% 9.49/5.28           => ( ! [X4: complex] :
% 9.49/5.28                  ( ( member_complex @ X4 @ S2 )
% 9.49/5.28                 => ( ( G @ X4 )
% 9.49/5.28                    = ( H2 @ X4 ) ) )
% 9.49/5.28             => ( ( groups5808333547571424918x_real @ G @ T3 )
% 9.49/5.28                = ( groups5808333547571424918x_real @ H2 @ S2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6171_sum_Omono__neutral__cong__right,axiom,
% 9.49/5.28      ! [T3: set_real,S2: set_real,G: real > rat,H2: real > rat] :
% 9.49/5.28        ( ( finite_finite_real @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: real] :
% 9.49/5.28                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_rat ) )
% 9.49/5.28           => ( ! [X4: real] :
% 9.49/5.28                  ( ( member_real @ X4 @ S2 )
% 9.49/5.28                 => ( ( G @ X4 )
% 9.49/5.28                    = ( H2 @ X4 ) ) )
% 9.49/5.28             => ( ( groups1300246762558778688al_rat @ G @ T3 )
% 9.49/5.28                = ( groups1300246762558778688al_rat @ H2 @ S2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6172_sum_Omono__neutral__cong__right,axiom,
% 9.49/5.28      ! [T3: set_complex,S2: set_complex,G: complex > rat,H2: complex > rat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_rat ) )
% 9.49/5.28           => ( ! [X4: complex] :
% 9.49/5.28                  ( ( member_complex @ X4 @ S2 )
% 9.49/5.28                 => ( ( G @ X4 )
% 9.49/5.28                    = ( H2 @ X4 ) ) )
% 9.49/5.28             => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 9.49/5.28                = ( groups5058264527183730370ex_rat @ H2 @ S2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6173_sum_Omono__neutral__cong__right,axiom,
% 9.49/5.28      ! [T3: set_real,S2: set_real,G: real > nat,H2: real > nat] :
% 9.49/5.28        ( ( finite_finite_real @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: real] :
% 9.49/5.28                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_nat ) )
% 9.49/5.28           => ( ! [X4: real] :
% 9.49/5.28                  ( ( member_real @ X4 @ S2 )
% 9.49/5.28                 => ( ( G @ X4 )
% 9.49/5.28                    = ( H2 @ X4 ) ) )
% 9.49/5.28             => ( ( groups1935376822645274424al_nat @ G @ T3 )
% 9.49/5.28                = ( groups1935376822645274424al_nat @ H2 @ S2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6174_sum_Omono__neutral__cong__right,axiom,
% 9.49/5.28      ! [T3: set_complex,S2: set_complex,G: complex > nat,H2: complex > nat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_nat ) )
% 9.49/5.28           => ( ! [X4: complex] :
% 9.49/5.28                  ( ( member_complex @ X4 @ S2 )
% 9.49/5.28                 => ( ( G @ X4 )
% 9.49/5.28                    = ( H2 @ X4 ) ) )
% 9.49/5.28             => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 9.49/5.28                = ( groups5693394587270226106ex_nat @ H2 @ S2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6175_sum_Omono__neutral__cong__right,axiom,
% 9.49/5.28      ! [T3: set_real,S2: set_real,G: real > int,H2: real > int] :
% 9.49/5.28        ( ( finite_finite_real @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: real] :
% 9.49/5.28                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_int ) )
% 9.49/5.28           => ( ! [X4: real] :
% 9.49/5.28                  ( ( member_real @ X4 @ S2 )
% 9.49/5.28                 => ( ( G @ X4 )
% 9.49/5.28                    = ( H2 @ X4 ) ) )
% 9.49/5.28             => ( ( groups1932886352136224148al_int @ G @ T3 )
% 9.49/5.28                = ( groups1932886352136224148al_int @ H2 @ S2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6176_sum_Omono__neutral__cong__right,axiom,
% 9.49/5.28      ! [T3: set_complex,S2: set_complex,G: complex > int,H2: complex > int] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_int ) )
% 9.49/5.28           => ( ! [X4: complex] :
% 9.49/5.28                  ( ( member_complex @ X4 @ S2 )
% 9.49/5.28                 => ( ( G @ X4 )
% 9.49/5.28                    = ( H2 @ X4 ) ) )
% 9.49/5.28             => ( ( groups5690904116761175830ex_int @ G @ T3 )
% 9.49/5.28                = ( groups5690904116761175830ex_int @ H2 @ S2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6177_sum_Omono__neutral__cong__right,axiom,
% 9.49/5.28      ! [T3: set_nat,S2: set_nat,G: nat > complex,H2: nat > complex] :
% 9.49/5.28        ( ( finite_finite_nat @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: nat] :
% 9.49/5.28                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_complex ) )
% 9.49/5.28           => ( ! [X4: nat] :
% 9.49/5.28                  ( ( member_nat @ X4 @ S2 )
% 9.49/5.28                 => ( ( G @ X4 )
% 9.49/5.28                    = ( H2 @ X4 ) ) )
% 9.49/5.28             => ( ( groups2073611262835488442omplex @ G @ T3 )
% 9.49/5.28                = ( groups2073611262835488442omplex @ H2 @ S2 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6178_sum_Omono__neutral__cong__left,axiom,
% 9.49/5.28      ! [T3: set_real,S2: set_real,H2: real > complex,G: real > complex] :
% 9.49/5.28        ( ( finite_finite_real @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: real] :
% 9.49/5.28                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.49/5.28               => ( ( H2 @ X4 )
% 9.49/5.28                  = zero_zero_complex ) )
% 9.49/5.28           => ( ! [X4: real] :
% 9.49/5.28                  ( ( member_real @ X4 @ S2 )
% 9.49/5.28                 => ( ( G @ X4 )
% 9.49/5.28                    = ( H2 @ X4 ) ) )
% 9.49/5.28             => ( ( groups5754745047067104278omplex @ G @ S2 )
% 9.49/5.28                = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6179_sum_Omono__neutral__cong__left,axiom,
% 9.49/5.28      ! [T3: set_real,S2: set_real,H2: real > real,G: real > real] :
% 9.49/5.28        ( ( finite_finite_real @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: real] :
% 9.49/5.28                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.49/5.28               => ( ( H2 @ X4 )
% 9.49/5.28                  = zero_zero_real ) )
% 9.49/5.28           => ( ! [X4: real] :
% 9.49/5.28                  ( ( member_real @ X4 @ S2 )
% 9.49/5.28                 => ( ( G @ X4 )
% 9.49/5.28                    = ( H2 @ X4 ) ) )
% 9.49/5.28             => ( ( groups8097168146408367636l_real @ G @ S2 )
% 9.49/5.28                = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6180_sum_Omono__neutral__cong__left,axiom,
% 9.49/5.28      ! [T3: set_complex,S2: set_complex,H2: complex > real,G: complex > real] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.49/5.28               => ( ( H2 @ X4 )
% 9.49/5.28                  = zero_zero_real ) )
% 9.49/5.28           => ( ! [X4: complex] :
% 9.49/5.28                  ( ( member_complex @ X4 @ S2 )
% 9.49/5.28                 => ( ( G @ X4 )
% 9.49/5.28                    = ( H2 @ X4 ) ) )
% 9.49/5.28             => ( ( groups5808333547571424918x_real @ G @ S2 )
% 9.49/5.28                = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6181_sum_Omono__neutral__cong__left,axiom,
% 9.49/5.28      ! [T3: set_real,S2: set_real,H2: real > rat,G: real > rat] :
% 9.49/5.28        ( ( finite_finite_real @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: real] :
% 9.49/5.28                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.49/5.28               => ( ( H2 @ X4 )
% 9.49/5.28                  = zero_zero_rat ) )
% 9.49/5.28           => ( ! [X4: real] :
% 9.49/5.28                  ( ( member_real @ X4 @ S2 )
% 9.49/5.28                 => ( ( G @ X4 )
% 9.49/5.28                    = ( H2 @ X4 ) ) )
% 9.49/5.28             => ( ( groups1300246762558778688al_rat @ G @ S2 )
% 9.49/5.28                = ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6182_sum_Omono__neutral__cong__left,axiom,
% 9.49/5.28      ! [T3: set_complex,S2: set_complex,H2: complex > rat,G: complex > rat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.49/5.28               => ( ( H2 @ X4 )
% 9.49/5.28                  = zero_zero_rat ) )
% 9.49/5.28           => ( ! [X4: complex] :
% 9.49/5.28                  ( ( member_complex @ X4 @ S2 )
% 9.49/5.28                 => ( ( G @ X4 )
% 9.49/5.28                    = ( H2 @ X4 ) ) )
% 9.49/5.28             => ( ( groups5058264527183730370ex_rat @ G @ S2 )
% 9.49/5.28                = ( groups5058264527183730370ex_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6183_sum_Omono__neutral__cong__left,axiom,
% 9.49/5.28      ! [T3: set_real,S2: set_real,H2: real > nat,G: real > nat] :
% 9.49/5.28        ( ( finite_finite_real @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: real] :
% 9.49/5.28                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.49/5.28               => ( ( H2 @ X4 )
% 9.49/5.28                  = zero_zero_nat ) )
% 9.49/5.28           => ( ! [X4: real] :
% 9.49/5.28                  ( ( member_real @ X4 @ S2 )
% 9.49/5.28                 => ( ( G @ X4 )
% 9.49/5.28                    = ( H2 @ X4 ) ) )
% 9.49/5.28             => ( ( groups1935376822645274424al_nat @ G @ S2 )
% 9.49/5.28                = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6184_sum_Omono__neutral__cong__left,axiom,
% 9.49/5.28      ! [T3: set_complex,S2: set_complex,H2: complex > nat,G: complex > nat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.49/5.28               => ( ( H2 @ X4 )
% 9.49/5.28                  = zero_zero_nat ) )
% 9.49/5.28           => ( ! [X4: complex] :
% 9.49/5.28                  ( ( member_complex @ X4 @ S2 )
% 9.49/5.28                 => ( ( G @ X4 )
% 9.49/5.28                    = ( H2 @ X4 ) ) )
% 9.49/5.28             => ( ( groups5693394587270226106ex_nat @ G @ S2 )
% 9.49/5.28                = ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6185_sum_Omono__neutral__cong__left,axiom,
% 9.49/5.28      ! [T3: set_real,S2: set_real,H2: real > int,G: real > int] :
% 9.49/5.28        ( ( finite_finite_real @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: real] :
% 9.49/5.28                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.49/5.28               => ( ( H2 @ X4 )
% 9.49/5.28                  = zero_zero_int ) )
% 9.49/5.28           => ( ! [X4: real] :
% 9.49/5.28                  ( ( member_real @ X4 @ S2 )
% 9.49/5.28                 => ( ( G @ X4 )
% 9.49/5.28                    = ( H2 @ X4 ) ) )
% 9.49/5.28             => ( ( groups1932886352136224148al_int @ G @ S2 )
% 9.49/5.28                = ( groups1932886352136224148al_int @ H2 @ T3 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6186_sum_Omono__neutral__cong__left,axiom,
% 9.49/5.28      ! [T3: set_complex,S2: set_complex,H2: complex > int,G: complex > int] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.49/5.28               => ( ( H2 @ X4 )
% 9.49/5.28                  = zero_zero_int ) )
% 9.49/5.28           => ( ! [X4: complex] :
% 9.49/5.28                  ( ( member_complex @ X4 @ S2 )
% 9.49/5.28                 => ( ( G @ X4 )
% 9.49/5.28                    = ( H2 @ X4 ) ) )
% 9.49/5.28             => ( ( groups5690904116761175830ex_int @ G @ S2 )
% 9.49/5.28                = ( groups5690904116761175830ex_int @ H2 @ T3 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6187_sum_Omono__neutral__cong__left,axiom,
% 9.49/5.28      ! [T3: set_nat,S2: set_nat,H2: nat > complex,G: nat > complex] :
% 9.49/5.28        ( ( finite_finite_nat @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: nat] :
% 9.49/5.28                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.49/5.28               => ( ( H2 @ X4 )
% 9.49/5.28                  = zero_zero_complex ) )
% 9.49/5.28           => ( ! [X4: nat] :
% 9.49/5.28                  ( ( member_nat @ X4 @ S2 )
% 9.49/5.28                 => ( ( G @ X4 )
% 9.49/5.28                    = ( H2 @ X4 ) ) )
% 9.49/5.28             => ( ( groups2073611262835488442omplex @ G @ S2 )
% 9.49/5.28                = ( groups2073611262835488442omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6188_sum_Omono__neutral__right,axiom,
% 9.49/5.28      ! [T3: set_complex,S2: set_complex,G: complex > real] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_real ) )
% 9.49/5.28           => ( ( groups5808333547571424918x_real @ G @ T3 )
% 9.49/5.28              = ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6189_sum_Omono__neutral__right,axiom,
% 9.49/5.28      ! [T3: set_complex,S2: set_complex,G: complex > rat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_rat ) )
% 9.49/5.28           => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 9.49/5.28              = ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6190_sum_Omono__neutral__right,axiom,
% 9.49/5.28      ! [T3: set_complex,S2: set_complex,G: complex > nat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_nat ) )
% 9.49/5.28           => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 9.49/5.28              = ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6191_sum_Omono__neutral__right,axiom,
% 9.49/5.28      ! [T3: set_complex,S2: set_complex,G: complex > int] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_int ) )
% 9.49/5.28           => ( ( groups5690904116761175830ex_int @ G @ T3 )
% 9.49/5.28              = ( groups5690904116761175830ex_int @ G @ S2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6192_sum_Omono__neutral__right,axiom,
% 9.49/5.28      ! [T3: set_nat,S2: set_nat,G: nat > complex] :
% 9.49/5.28        ( ( finite_finite_nat @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: nat] :
% 9.49/5.28                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_complex ) )
% 9.49/5.28           => ( ( groups2073611262835488442omplex @ G @ T3 )
% 9.49/5.28              = ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6193_sum_Omono__neutral__right,axiom,
% 9.49/5.28      ! [T3: set_nat,S2: set_nat,G: nat > rat] :
% 9.49/5.28        ( ( finite_finite_nat @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: nat] :
% 9.49/5.28                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_rat ) )
% 9.49/5.28           => ( ( groups2906978787729119204at_rat @ G @ T3 )
% 9.49/5.28              = ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6194_sum_Omono__neutral__right,axiom,
% 9.49/5.28      ! [T3: set_nat,S2: set_nat,G: nat > int] :
% 9.49/5.28        ( ( finite_finite_nat @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: nat] :
% 9.49/5.28                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_int ) )
% 9.49/5.28           => ( ( groups3539618377306564664at_int @ G @ T3 )
% 9.49/5.28              = ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6195_sum_Omono__neutral__right,axiom,
% 9.49/5.28      ! [T3: set_int,S2: set_int,G: int > complex] :
% 9.49/5.28        ( ( finite_finite_int @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: int] :
% 9.49/5.28                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_complex ) )
% 9.49/5.28           => ( ( groups3049146728041665814omplex @ G @ T3 )
% 9.49/5.28              = ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6196_sum_Omono__neutral__right,axiom,
% 9.49/5.28      ! [T3: set_int,S2: set_int,G: int > real] :
% 9.49/5.28        ( ( finite_finite_int @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: int] :
% 9.49/5.28                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_real ) )
% 9.49/5.28           => ( ( groups8778361861064173332t_real @ G @ T3 )
% 9.49/5.28              = ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6197_sum_Omono__neutral__right,axiom,
% 9.49/5.28      ! [T3: set_int,S2: set_int,G: int > rat] :
% 9.49/5.28        ( ( finite_finite_int @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: int] :
% 9.49/5.28                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_rat ) )
% 9.49/5.28           => ( ( groups3906332499630173760nt_rat @ G @ T3 )
% 9.49/5.28              = ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6198_sum_Omono__neutral__left,axiom,
% 9.49/5.28      ! [T3: set_complex,S2: set_complex,G: complex > real] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_real ) )
% 9.49/5.28           => ( ( groups5808333547571424918x_real @ G @ S2 )
% 9.49/5.28              = ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6199_sum_Omono__neutral__left,axiom,
% 9.49/5.28      ! [T3: set_complex,S2: set_complex,G: complex > rat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_rat ) )
% 9.49/5.28           => ( ( groups5058264527183730370ex_rat @ G @ S2 )
% 9.49/5.28              = ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6200_sum_Omono__neutral__left,axiom,
% 9.49/5.28      ! [T3: set_complex,S2: set_complex,G: complex > nat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_nat ) )
% 9.49/5.28           => ( ( groups5693394587270226106ex_nat @ G @ S2 )
% 9.49/5.28              = ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6201_sum_Omono__neutral__left,axiom,
% 9.49/5.28      ! [T3: set_complex,S2: set_complex,G: complex > int] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: complex] :
% 9.49/5.28                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_int ) )
% 9.49/5.28           => ( ( groups5690904116761175830ex_int @ G @ S2 )
% 9.49/5.28              = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6202_sum_Omono__neutral__left,axiom,
% 9.49/5.28      ! [T3: set_nat,S2: set_nat,G: nat > complex] :
% 9.49/5.28        ( ( finite_finite_nat @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: nat] :
% 9.49/5.28                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_complex ) )
% 9.49/5.28           => ( ( groups2073611262835488442omplex @ G @ S2 )
% 9.49/5.28              = ( groups2073611262835488442omplex @ G @ T3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6203_sum_Omono__neutral__left,axiom,
% 9.49/5.28      ! [T3: set_nat,S2: set_nat,G: nat > rat] :
% 9.49/5.28        ( ( finite_finite_nat @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: nat] :
% 9.49/5.28                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_rat ) )
% 9.49/5.28           => ( ( groups2906978787729119204at_rat @ G @ S2 )
% 9.49/5.28              = ( groups2906978787729119204at_rat @ G @ T3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6204_sum_Omono__neutral__left,axiom,
% 9.49/5.28      ! [T3: set_nat,S2: set_nat,G: nat > int] :
% 9.49/5.28        ( ( finite_finite_nat @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: nat] :
% 9.49/5.28                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_int ) )
% 9.49/5.28           => ( ( groups3539618377306564664at_int @ G @ S2 )
% 9.49/5.28              = ( groups3539618377306564664at_int @ G @ T3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6205_sum_Omono__neutral__left,axiom,
% 9.49/5.28      ! [T3: set_int,S2: set_int,G: int > complex] :
% 9.49/5.28        ( ( finite_finite_int @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: int] :
% 9.49/5.28                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_complex ) )
% 9.49/5.28           => ( ( groups3049146728041665814omplex @ G @ S2 )
% 9.49/5.28              = ( groups3049146728041665814omplex @ G @ T3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6206_sum_Omono__neutral__left,axiom,
% 9.49/5.28      ! [T3: set_int,S2: set_int,G: int > real] :
% 9.49/5.28        ( ( finite_finite_int @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: int] :
% 9.49/5.28                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_real ) )
% 9.49/5.28           => ( ( groups8778361861064173332t_real @ G @ S2 )
% 9.49/5.28              = ( groups8778361861064173332t_real @ G @ T3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6207_sum_Omono__neutral__left,axiom,
% 9.49/5.28      ! [T3: set_int,S2: set_int,G: int > rat] :
% 9.49/5.28        ( ( finite_finite_int @ T3 )
% 9.49/5.28       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.49/5.28         => ( ! [X4: int] :
% 9.49/5.28                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.49/5.28               => ( ( G @ X4 )
% 9.49/5.28                  = zero_zero_rat ) )
% 9.49/5.28           => ( ( groups3906332499630173760nt_rat @ G @ S2 )
% 9.49/5.28              = ( groups3906332499630173760nt_rat @ G @ T3 ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6208_sum_Osame__carrierI,axiom,
% 9.49/5.28      ! [C2: set_real,A2: set_real,B2: set_real,G: real > complex,H2: real > complex] :
% 9.49/5.28        ( ( finite_finite_real @ C2 )
% 9.49/5.28       => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 9.49/5.28         => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 9.49/5.28           => ( ! [A3: real] :
% 9.49/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 9.49/5.28                 => ( ( G @ A3 )
% 9.49/5.28                    = zero_zero_complex ) )
% 9.49/5.28             => ( ! [B3: real] :
% 9.49/5.28                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
% 9.49/5.28                   => ( ( H2 @ B3 )
% 9.49/5.28                      = zero_zero_complex ) )
% 9.49/5.28               => ( ( ( groups5754745047067104278omplex @ G @ C2 )
% 9.49/5.28                    = ( groups5754745047067104278omplex @ H2 @ C2 ) )
% 9.49/5.28                 => ( ( groups5754745047067104278omplex @ G @ A2 )
% 9.49/5.28                    = ( groups5754745047067104278omplex @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6209_sum_Osame__carrierI,axiom,
% 9.49/5.28      ! [C2: set_real,A2: set_real,B2: set_real,G: real > real,H2: real > real] :
% 9.49/5.28        ( ( finite_finite_real @ C2 )
% 9.49/5.28       => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 9.49/5.28         => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 9.49/5.28           => ( ! [A3: real] :
% 9.49/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 9.49/5.28                 => ( ( G @ A3 )
% 9.49/5.28                    = zero_zero_real ) )
% 9.49/5.28             => ( ! [B3: real] :
% 9.49/5.28                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
% 9.49/5.28                   => ( ( H2 @ B3 )
% 9.49/5.28                      = zero_zero_real ) )
% 9.49/5.28               => ( ( ( groups8097168146408367636l_real @ G @ C2 )
% 9.49/5.28                    = ( groups8097168146408367636l_real @ H2 @ C2 ) )
% 9.49/5.28                 => ( ( groups8097168146408367636l_real @ G @ A2 )
% 9.49/5.28                    = ( groups8097168146408367636l_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6210_sum_Osame__carrierI,axiom,
% 9.49/5.28      ! [C2: set_complex,A2: set_complex,B2: set_complex,G: complex > real,H2: complex > real] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ C2 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 9.49/5.28         => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 9.49/5.28           => ( ! [A3: complex] :
% 9.49/5.28                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 9.49/5.28                 => ( ( G @ A3 )
% 9.49/5.28                    = zero_zero_real ) )
% 9.49/5.28             => ( ! [B3: complex] :
% 9.49/5.28                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 9.49/5.28                   => ( ( H2 @ B3 )
% 9.49/5.28                      = zero_zero_real ) )
% 9.49/5.28               => ( ( ( groups5808333547571424918x_real @ G @ C2 )
% 9.49/5.28                    = ( groups5808333547571424918x_real @ H2 @ C2 ) )
% 9.49/5.28                 => ( ( groups5808333547571424918x_real @ G @ A2 )
% 9.49/5.28                    = ( groups5808333547571424918x_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6211_sum_Osame__carrierI,axiom,
% 9.49/5.28      ! [C2: set_real,A2: set_real,B2: set_real,G: real > rat,H2: real > rat] :
% 9.49/5.28        ( ( finite_finite_real @ C2 )
% 9.49/5.28       => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 9.49/5.28         => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 9.49/5.28           => ( ! [A3: real] :
% 9.49/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 9.49/5.28                 => ( ( G @ A3 )
% 9.49/5.28                    = zero_zero_rat ) )
% 9.49/5.28             => ( ! [B3: real] :
% 9.49/5.28                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
% 9.49/5.28                   => ( ( H2 @ B3 )
% 9.49/5.28                      = zero_zero_rat ) )
% 9.49/5.28               => ( ( ( groups1300246762558778688al_rat @ G @ C2 )
% 9.49/5.28                    = ( groups1300246762558778688al_rat @ H2 @ C2 ) )
% 9.49/5.28                 => ( ( groups1300246762558778688al_rat @ G @ A2 )
% 9.49/5.28                    = ( groups1300246762558778688al_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6212_sum_Osame__carrierI,axiom,
% 9.49/5.28      ! [C2: set_complex,A2: set_complex,B2: set_complex,G: complex > rat,H2: complex > rat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ C2 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 9.49/5.28         => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 9.49/5.28           => ( ! [A3: complex] :
% 9.49/5.28                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 9.49/5.28                 => ( ( G @ A3 )
% 9.49/5.28                    = zero_zero_rat ) )
% 9.49/5.28             => ( ! [B3: complex] :
% 9.49/5.28                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 9.49/5.28                   => ( ( H2 @ B3 )
% 9.49/5.28                      = zero_zero_rat ) )
% 9.49/5.28               => ( ( ( groups5058264527183730370ex_rat @ G @ C2 )
% 9.49/5.28                    = ( groups5058264527183730370ex_rat @ H2 @ C2 ) )
% 9.49/5.28                 => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 9.49/5.28                    = ( groups5058264527183730370ex_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6213_sum_Osame__carrierI,axiom,
% 9.49/5.28      ! [C2: set_real,A2: set_real,B2: set_real,G: real > nat,H2: real > nat] :
% 9.49/5.28        ( ( finite_finite_real @ C2 )
% 9.49/5.28       => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 9.49/5.28         => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 9.49/5.28           => ( ! [A3: real] :
% 9.49/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 9.49/5.28                 => ( ( G @ A3 )
% 9.49/5.28                    = zero_zero_nat ) )
% 9.49/5.28             => ( ! [B3: real] :
% 9.49/5.28                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
% 9.49/5.28                   => ( ( H2 @ B3 )
% 9.49/5.28                      = zero_zero_nat ) )
% 9.49/5.28               => ( ( ( groups1935376822645274424al_nat @ G @ C2 )
% 9.49/5.28                    = ( groups1935376822645274424al_nat @ H2 @ C2 ) )
% 9.49/5.28                 => ( ( groups1935376822645274424al_nat @ G @ A2 )
% 9.49/5.28                    = ( groups1935376822645274424al_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6214_sum_Osame__carrierI,axiom,
% 9.49/5.28      ! [C2: set_complex,A2: set_complex,B2: set_complex,G: complex > nat,H2: complex > nat] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ C2 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 9.49/5.28         => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 9.49/5.28           => ( ! [A3: complex] :
% 9.49/5.28                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 9.49/5.28                 => ( ( G @ A3 )
% 9.49/5.28                    = zero_zero_nat ) )
% 9.49/5.28             => ( ! [B3: complex] :
% 9.49/5.28                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 9.49/5.28                   => ( ( H2 @ B3 )
% 9.49/5.28                      = zero_zero_nat ) )
% 9.49/5.28               => ( ( ( groups5693394587270226106ex_nat @ G @ C2 )
% 9.49/5.28                    = ( groups5693394587270226106ex_nat @ H2 @ C2 ) )
% 9.49/5.28                 => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 9.49/5.28                    = ( groups5693394587270226106ex_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6215_sum_Osame__carrierI,axiom,
% 9.49/5.28      ! [C2: set_real,A2: set_real,B2: set_real,G: real > int,H2: real > int] :
% 9.49/5.28        ( ( finite_finite_real @ C2 )
% 9.49/5.28       => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 9.49/5.28         => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 9.49/5.28           => ( ! [A3: real] :
% 9.49/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 9.49/5.28                 => ( ( G @ A3 )
% 9.49/5.28                    = zero_zero_int ) )
% 9.49/5.28             => ( ! [B3: real] :
% 9.49/5.28                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
% 9.49/5.28                   => ( ( H2 @ B3 )
% 9.49/5.28                      = zero_zero_int ) )
% 9.49/5.28               => ( ( ( groups1932886352136224148al_int @ G @ C2 )
% 9.49/5.28                    = ( groups1932886352136224148al_int @ H2 @ C2 ) )
% 9.49/5.28                 => ( ( groups1932886352136224148al_int @ G @ A2 )
% 9.49/5.28                    = ( groups1932886352136224148al_int @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6216_sum_Osame__carrierI,axiom,
% 9.49/5.28      ! [C2: set_complex,A2: set_complex,B2: set_complex,G: complex > int,H2: complex > int] :
% 9.49/5.28        ( ( finite3207457112153483333omplex @ C2 )
% 9.49/5.28       => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 9.49/5.28         => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 9.49/5.28           => ( ! [A3: complex] :
% 9.49/5.28                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 9.49/5.28                 => ( ( G @ A3 )
% 9.49/5.28                    = zero_zero_int ) )
% 9.49/5.28             => ( ! [B3: complex] :
% 9.49/5.28                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 9.49/5.28                   => ( ( H2 @ B3 )
% 9.49/5.28                      = zero_zero_int ) )
% 9.49/5.28               => ( ( ( groups5690904116761175830ex_int @ G @ C2 )
% 9.49/5.28                    = ( groups5690904116761175830ex_int @ H2 @ C2 ) )
% 9.49/5.28                 => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 9.49/5.28                    = ( groups5690904116761175830ex_int @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6217_sum_Osame__carrierI,axiom,
% 9.49/5.28      ! [C2: set_nat,A2: set_nat,B2: set_nat,G: nat > complex,H2: nat > complex] :
% 9.49/5.28        ( ( finite_finite_nat @ C2 )
% 9.49/5.28       => ( ( ord_less_eq_set_nat @ A2 @ C2 )
% 9.49/5.28         => ( ( ord_less_eq_set_nat @ B2 @ C2 )
% 9.49/5.28           => ( ! [A3: nat] :
% 9.49/5.28                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C2 @ A2 ) )
% 9.49/5.28                 => ( ( G @ A3 )
% 9.49/5.28                    = zero_zero_complex ) )
% 9.49/5.28             => ( ! [B3: nat] :
% 9.49/5.28                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C2 @ B2 ) )
% 9.49/5.28                   => ( ( H2 @ B3 )
% 9.49/5.28                      = zero_zero_complex ) )
% 9.49/5.28               => ( ( ( groups2073611262835488442omplex @ G @ C2 )
% 9.49/5.28                    = ( groups2073611262835488442omplex @ H2 @ C2 ) )
% 9.49/5.28                 => ( ( groups2073611262835488442omplex @ G @ A2 )
% 9.49/5.28                    = ( groups2073611262835488442omplex @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6218_sum_Osame__carrier,axiom,
% 9.49/5.28      ! [C2: set_real,A2: set_real,B2: set_real,G: real > complex,H2: real > complex] :
% 9.49/5.28        ( ( finite_finite_real @ C2 )
% 9.49/5.28       => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 9.49/5.28         => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 9.49/5.28           => ( ! [A3: real] :
% 9.49/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 9.49/5.28                 => ( ( G @ A3 )
% 9.49/5.28                    = zero_zero_complex ) )
% 9.49/5.28             => ( ! [B3: real] :
% 9.49/5.28                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
% 9.49/5.28                   => ( ( H2 @ B3 )
% 9.49/5.28                      = zero_zero_complex ) )
% 9.49/5.28               => ( ( ( groups5754745047067104278omplex @ G @ A2 )
% 9.49/5.28                    = ( groups5754745047067104278omplex @ H2 @ B2 ) )
% 9.49/5.28                  = ( ( groups5754745047067104278omplex @ G @ C2 )
% 9.49/5.28                    = ( groups5754745047067104278omplex @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 9.49/5.28  
% 9.49/5.28  thf(fact_6219_sum_Osame__carrier,axiom,
% 9.49/5.29      ! [C2: set_real,A2: set_real,B2: set_real,G: real > real,H2: real > real] :
% 9.49/5.29        ( ( finite_finite_real @ C2 )
% 9.49/5.29       => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 9.49/5.29         => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 9.49/5.29           => ( ! [A3: real] :
% 9.49/5.29                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 9.49/5.29                 => ( ( G @ A3 )
% 9.49/5.29                    = zero_zero_real ) )
% 9.49/5.29             => ( ! [B3: real] :
% 9.49/5.29                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
% 9.49/5.29                   => ( ( H2 @ B3 )
% 9.49/5.29                      = zero_zero_real ) )
% 9.49/5.29               => ( ( ( groups8097168146408367636l_real @ G @ A2 )
% 9.49/5.29                    = ( groups8097168146408367636l_real @ H2 @ B2 ) )
% 9.49/5.29                  = ( ( groups8097168146408367636l_real @ G @ C2 )
% 9.49/5.29                    = ( groups8097168146408367636l_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6220_sum_Osame__carrier,axiom,
% 9.49/5.29      ! [C2: set_complex,A2: set_complex,B2: set_complex,G: complex > real,H2: complex > real] :
% 9.49/5.29        ( ( finite3207457112153483333omplex @ C2 )
% 9.49/5.29       => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 9.49/5.29         => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 9.49/5.29           => ( ! [A3: complex] :
% 9.49/5.29                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 9.49/5.29                 => ( ( G @ A3 )
% 9.49/5.29                    = zero_zero_real ) )
% 9.49/5.29             => ( ! [B3: complex] :
% 9.49/5.29                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 9.49/5.29                   => ( ( H2 @ B3 )
% 9.49/5.29                      = zero_zero_real ) )
% 9.49/5.29               => ( ( ( groups5808333547571424918x_real @ G @ A2 )
% 9.49/5.29                    = ( groups5808333547571424918x_real @ H2 @ B2 ) )
% 9.49/5.29                  = ( ( groups5808333547571424918x_real @ G @ C2 )
% 9.49/5.29                    = ( groups5808333547571424918x_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6221_sum_Osame__carrier,axiom,
% 9.49/5.29      ! [C2: set_real,A2: set_real,B2: set_real,G: real > rat,H2: real > rat] :
% 9.49/5.29        ( ( finite_finite_real @ C2 )
% 9.49/5.29       => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 9.49/5.29         => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 9.49/5.29           => ( ! [A3: real] :
% 9.49/5.29                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 9.49/5.29                 => ( ( G @ A3 )
% 9.49/5.29                    = zero_zero_rat ) )
% 9.49/5.29             => ( ! [B3: real] :
% 9.49/5.29                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
% 9.49/5.29                   => ( ( H2 @ B3 )
% 9.49/5.29                      = zero_zero_rat ) )
% 9.49/5.29               => ( ( ( groups1300246762558778688al_rat @ G @ A2 )
% 9.49/5.29                    = ( groups1300246762558778688al_rat @ H2 @ B2 ) )
% 9.49/5.29                  = ( ( groups1300246762558778688al_rat @ G @ C2 )
% 9.49/5.29                    = ( groups1300246762558778688al_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6222_sum_Osame__carrier,axiom,
% 9.49/5.29      ! [C2: set_complex,A2: set_complex,B2: set_complex,G: complex > rat,H2: complex > rat] :
% 9.49/5.29        ( ( finite3207457112153483333omplex @ C2 )
% 9.49/5.29       => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 9.49/5.29         => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 9.49/5.29           => ( ! [A3: complex] :
% 9.49/5.29                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 9.49/5.29                 => ( ( G @ A3 )
% 9.49/5.29                    = zero_zero_rat ) )
% 9.49/5.29             => ( ! [B3: complex] :
% 9.49/5.29                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 9.49/5.29                   => ( ( H2 @ B3 )
% 9.49/5.29                      = zero_zero_rat ) )
% 9.49/5.29               => ( ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 9.49/5.29                    = ( groups5058264527183730370ex_rat @ H2 @ B2 ) )
% 9.49/5.29                  = ( ( groups5058264527183730370ex_rat @ G @ C2 )
% 9.49/5.29                    = ( groups5058264527183730370ex_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6223_sum_Osame__carrier,axiom,
% 9.49/5.29      ! [C2: set_real,A2: set_real,B2: set_real,G: real > nat,H2: real > nat] :
% 9.49/5.29        ( ( finite_finite_real @ C2 )
% 9.49/5.29       => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 9.49/5.29         => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 9.49/5.29           => ( ! [A3: real] :
% 9.49/5.29                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 9.49/5.29                 => ( ( G @ A3 )
% 9.49/5.29                    = zero_zero_nat ) )
% 9.49/5.29             => ( ! [B3: real] :
% 9.49/5.29                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
% 9.49/5.29                   => ( ( H2 @ B3 )
% 9.49/5.29                      = zero_zero_nat ) )
% 9.49/5.29               => ( ( ( groups1935376822645274424al_nat @ G @ A2 )
% 9.49/5.29                    = ( groups1935376822645274424al_nat @ H2 @ B2 ) )
% 9.49/5.29                  = ( ( groups1935376822645274424al_nat @ G @ C2 )
% 9.49/5.29                    = ( groups1935376822645274424al_nat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6224_sum_Osame__carrier,axiom,
% 9.49/5.29      ! [C2: set_complex,A2: set_complex,B2: set_complex,G: complex > nat,H2: complex > nat] :
% 9.49/5.29        ( ( finite3207457112153483333omplex @ C2 )
% 9.49/5.29       => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 9.49/5.29         => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 9.49/5.29           => ( ! [A3: complex] :
% 9.49/5.29                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 9.49/5.29                 => ( ( G @ A3 )
% 9.49/5.29                    = zero_zero_nat ) )
% 9.49/5.29             => ( ! [B3: complex] :
% 9.49/5.29                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 9.49/5.29                   => ( ( H2 @ B3 )
% 9.49/5.29                      = zero_zero_nat ) )
% 9.49/5.29               => ( ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 9.49/5.29                    = ( groups5693394587270226106ex_nat @ H2 @ B2 ) )
% 9.49/5.29                  = ( ( groups5693394587270226106ex_nat @ G @ C2 )
% 9.49/5.29                    = ( groups5693394587270226106ex_nat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6225_sum_Osame__carrier,axiom,
% 9.49/5.29      ! [C2: set_real,A2: set_real,B2: set_real,G: real > int,H2: real > int] :
% 9.49/5.29        ( ( finite_finite_real @ C2 )
% 9.49/5.29       => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 9.49/5.29         => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 9.49/5.29           => ( ! [A3: real] :
% 9.49/5.29                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 9.49/5.29                 => ( ( G @ A3 )
% 9.49/5.29                    = zero_zero_int ) )
% 9.49/5.29             => ( ! [B3: real] :
% 9.49/5.29                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
% 9.49/5.29                   => ( ( H2 @ B3 )
% 9.49/5.29                      = zero_zero_int ) )
% 9.49/5.29               => ( ( ( groups1932886352136224148al_int @ G @ A2 )
% 9.49/5.29                    = ( groups1932886352136224148al_int @ H2 @ B2 ) )
% 9.49/5.29                  = ( ( groups1932886352136224148al_int @ G @ C2 )
% 9.49/5.29                    = ( groups1932886352136224148al_int @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6226_sum_Osame__carrier,axiom,
% 9.49/5.29      ! [C2: set_complex,A2: set_complex,B2: set_complex,G: complex > int,H2: complex > int] :
% 9.49/5.29        ( ( finite3207457112153483333omplex @ C2 )
% 9.49/5.29       => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 9.49/5.29         => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 9.49/5.29           => ( ! [A3: complex] :
% 9.49/5.29                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 9.49/5.29                 => ( ( G @ A3 )
% 9.49/5.29                    = zero_zero_int ) )
% 9.49/5.29             => ( ! [B3: complex] :
% 9.49/5.29                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 9.49/5.29                   => ( ( H2 @ B3 )
% 9.49/5.29                      = zero_zero_int ) )
% 9.49/5.29               => ( ( ( groups5690904116761175830ex_int @ G @ A2 )
% 9.49/5.29                    = ( groups5690904116761175830ex_int @ H2 @ B2 ) )
% 9.49/5.29                  = ( ( groups5690904116761175830ex_int @ G @ C2 )
% 9.49/5.29                    = ( groups5690904116761175830ex_int @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6227_sum_Osame__carrier,axiom,
% 9.49/5.29      ! [C2: set_nat,A2: set_nat,B2: set_nat,G: nat > complex,H2: nat > complex] :
% 9.49/5.29        ( ( finite_finite_nat @ C2 )
% 9.49/5.29       => ( ( ord_less_eq_set_nat @ A2 @ C2 )
% 9.49/5.29         => ( ( ord_less_eq_set_nat @ B2 @ C2 )
% 9.49/5.29           => ( ! [A3: nat] :
% 9.49/5.29                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C2 @ A2 ) )
% 9.49/5.29                 => ( ( G @ A3 )
% 9.49/5.29                    = zero_zero_complex ) )
% 9.49/5.29             => ( ! [B3: nat] :
% 9.49/5.29                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C2 @ B2 ) )
% 9.49/5.29                   => ( ( H2 @ B3 )
% 9.49/5.29                      = zero_zero_complex ) )
% 9.49/5.29               => ( ( ( groups2073611262835488442omplex @ G @ A2 )
% 9.49/5.29                    = ( groups2073611262835488442omplex @ H2 @ B2 ) )
% 9.49/5.29                  = ( ( groups2073611262835488442omplex @ G @ C2 )
% 9.49/5.29                    = ( groups2073611262835488442omplex @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6228_sum_Osubset__diff,axiom,
% 9.49/5.29      ! [B2: set_complex,A2: set_complex,G: complex > real] :
% 9.49/5.29        ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 9.49/5.29       => ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.29         => ( ( groups5808333547571424918x_real @ G @ A2 )
% 9.49/5.29            = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5808333547571424918x_real @ G @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6229_sum_Osubset__diff,axiom,
% 9.49/5.29      ! [B2: set_complex,A2: set_complex,G: complex > rat] :
% 9.49/5.29        ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 9.49/5.29       => ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.29         => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 9.49/5.29            = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6230_sum_Osubset__diff,axiom,
% 9.49/5.29      ! [B2: set_complex,A2: set_complex,G: complex > nat] :
% 9.49/5.29        ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 9.49/5.29       => ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.29         => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 9.49/5.29            = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6231_sum_Osubset__diff,axiom,
% 9.49/5.29      ! [B2: set_complex,A2: set_complex,G: complex > int] :
% 9.49/5.29        ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 9.49/5.29       => ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.29         => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 9.49/5.29            = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6232_sum_Osubset__diff,axiom,
% 9.49/5.29      ! [B2: set_nat,A2: set_nat,G: nat > rat] :
% 9.49/5.29        ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 9.49/5.29       => ( ( finite_finite_nat @ A2 )
% 9.49/5.29         => ( ( groups2906978787729119204at_rat @ G @ A2 )
% 9.49/5.29            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups2906978787729119204at_rat @ G @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6233_sum_Osubset__diff,axiom,
% 9.49/5.29      ! [B2: set_nat,A2: set_nat,G: nat > int] :
% 9.49/5.29        ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 9.49/5.29       => ( ( finite_finite_nat @ A2 )
% 9.49/5.29         => ( ( groups3539618377306564664at_int @ G @ A2 )
% 9.49/5.29            = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups3539618377306564664at_int @ G @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6234_sum_Osubset__diff,axiom,
% 9.49/5.29      ! [B2: set_int,A2: set_int,G: int > real] :
% 9.49/5.29        ( ( ord_less_eq_set_int @ B2 @ A2 )
% 9.49/5.29       => ( ( finite_finite_int @ A2 )
% 9.49/5.29         => ( ( groups8778361861064173332t_real @ G @ A2 )
% 9.49/5.29            = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups8778361861064173332t_real @ G @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6235_sum_Osubset__diff,axiom,
% 9.49/5.29      ! [B2: set_int,A2: set_int,G: int > rat] :
% 9.49/5.29        ( ( ord_less_eq_set_int @ B2 @ A2 )
% 9.49/5.29       => ( ( finite_finite_int @ A2 )
% 9.49/5.29         => ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 9.49/5.29            = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6236_sum_Osubset__diff,axiom,
% 9.49/5.29      ! [B2: set_int,A2: set_int,G: int > nat] :
% 9.49/5.29        ( ( ord_less_eq_set_int @ B2 @ A2 )
% 9.49/5.29       => ( ( finite_finite_int @ A2 )
% 9.49/5.29         => ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 9.49/5.29            = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6237_sum_Osubset__diff,axiom,
% 9.49/5.29      ! [B2: set_int,A2: set_int,G: int > int] :
% 9.49/5.29        ( ( ord_less_eq_set_int @ B2 @ A2 )
% 9.49/5.29       => ( ( finite_finite_int @ A2 )
% 9.49/5.29         => ( ( groups4538972089207619220nt_int @ G @ A2 )
% 9.49/5.29            = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4538972089207619220nt_int @ G @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6238_sum__diff,axiom,
% 9.49/5.29      ! [A2: set_complex,B2: set_complex,F: complex > real] :
% 9.49/5.29        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.29       => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 9.49/5.29         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 9.49/5.29            = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6239_sum__diff,axiom,
% 9.49/5.29      ! [A2: set_complex,B2: set_complex,F: complex > rat] :
% 9.49/5.29        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.29       => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 9.49/5.29         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 9.49/5.29            = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6240_sum__diff,axiom,
% 9.49/5.29      ! [A2: set_complex,B2: set_complex,F: complex > int] :
% 9.49/5.29        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.29       => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 9.49/5.29         => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 9.49/5.29            = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6241_sum__diff,axiom,
% 9.49/5.29      ! [A2: set_nat,B2: set_nat,F: nat > rat] :
% 9.49/5.29        ( ( finite_finite_nat @ A2 )
% 9.49/5.29       => ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 9.49/5.29         => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.49/5.29            = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6242_sum__diff,axiom,
% 9.49/5.29      ! [A2: set_nat,B2: set_nat,F: nat > int] :
% 9.49/5.29        ( ( finite_finite_nat @ A2 )
% 9.49/5.29       => ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 9.49/5.29         => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.49/5.29            = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6243_sum__diff,axiom,
% 9.49/5.29      ! [A2: set_int,B2: set_int,F: int > real] :
% 9.49/5.29        ( ( finite_finite_int @ A2 )
% 9.49/5.29       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 9.49/5.29         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.49/5.29            = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6244_sum__diff,axiom,
% 9.49/5.29      ! [A2: set_int,B2: set_int,F: int > rat] :
% 9.49/5.29        ( ( finite_finite_int @ A2 )
% 9.49/5.29       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 9.49/5.29         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.49/5.29            = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6245_sum__diff,axiom,
% 9.49/5.29      ! [A2: set_int,B2: set_int,F: int > int] :
% 9.49/5.29        ( ( finite_finite_int @ A2 )
% 9.49/5.29       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 9.49/5.29         => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.49/5.29            = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6246_sum__diff,axiom,
% 9.49/5.29      ! [A2: set_complex,B2: set_complex,F: complex > complex] :
% 9.49/5.29        ( ( finite3207457112153483333omplex @ A2 )
% 9.49/5.29       => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 9.49/5.29         => ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 9.49/5.29            = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6247_sum__diff,axiom,
% 9.49/5.29      ! [A2: set_nat,B2: set_nat,F: nat > real] :
% 9.49/5.29        ( ( finite_finite_nat @ A2 )
% 9.49/5.29       => ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 9.49/5.29         => ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.49/5.29            = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6248_sum__diff__nat,axiom,
% 9.49/5.29      ! [B2: set_complex,A2: set_complex,F: complex > nat] :
% 9.49/5.29        ( ( finite3207457112153483333omplex @ B2 )
% 9.49/5.29       => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 9.49/5.29         => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 9.49/5.29            = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6249_sum__diff__nat,axiom,
% 9.49/5.29      ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 9.49/5.29        ( ( finite6177210948735845034at_nat @ B2 )
% 9.49/5.29       => ( ( ord_le3146513528884898305at_nat @ B2 @ A2 )
% 9.49/5.29         => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 9.49/5.29            = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A2 ) @ ( groups977919841031483927at_nat @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6250_sum__diff__nat,axiom,
% 9.49/5.29      ! [B2: set_int,A2: set_int,F: int > nat] :
% 9.49/5.29        ( ( finite_finite_int @ B2 )
% 9.49/5.29       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 9.49/5.29         => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.49/5.29            = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6251_sum__diff__nat,axiom,
% 9.49/5.29      ! [B2: set_nat,A2: set_nat,F: nat > nat] :
% 9.49/5.29        ( ( finite_finite_nat @ B2 )
% 9.49/5.29       => ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 9.49/5.29         => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.49/5.29            = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6252_sum__shift__lb__Suc0__0,axiom,
% 9.49/5.29      ! [F: nat > complex,K: nat] :
% 9.49/5.29        ( ( ( F @ zero_zero_nat )
% 9.49/5.29          = zero_zero_complex )
% 9.49/5.29       => ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 9.49/5.29          = ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6253_sum__shift__lb__Suc0__0,axiom,
% 9.49/5.29      ! [F: nat > rat,K: nat] :
% 9.49/5.29        ( ( ( F @ zero_zero_nat )
% 9.49/5.29          = zero_zero_rat )
% 9.49/5.29       => ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 9.49/5.29          = ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6254_sum__shift__lb__Suc0__0,axiom,
% 9.49/5.29      ! [F: nat > int,K: nat] :
% 9.49/5.29        ( ( ( F @ zero_zero_nat )
% 9.49/5.29          = zero_zero_int )
% 9.49/5.29       => ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 9.49/5.29          = ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6255_sum__shift__lb__Suc0__0,axiom,
% 9.49/5.29      ! [F: nat > nat,K: nat] :
% 9.49/5.29        ( ( ( F @ zero_zero_nat )
% 9.49/5.29          = zero_zero_nat )
% 9.49/5.29       => ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 9.49/5.29          = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6256_sum__shift__lb__Suc0__0,axiom,
% 9.49/5.29      ! [F: nat > real,K: nat] :
% 9.49/5.29        ( ( ( F @ zero_zero_nat )
% 9.49/5.29          = zero_zero_real )
% 9.49/5.29       => ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 9.49/5.29          = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6257_sum_OatLeast0__atMost__Suc,axiom,
% 9.49/5.29      ! [G: nat > rat,N: nat] :
% 9.49/5.29        ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.49/5.29        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6258_sum_OatLeast0__atMost__Suc,axiom,
% 9.49/5.29      ! [G: nat > int,N: nat] :
% 9.49/5.29        ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.49/5.29        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6259_sum_OatLeast0__atMost__Suc,axiom,
% 9.49/5.29      ! [G: nat > nat,N: nat] :
% 9.49/5.29        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.49/5.29        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6260_sum_OatLeast0__atMost__Suc,axiom,
% 9.49/5.29      ! [G: nat > real,N: nat] :
% 9.49/5.29        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.49/5.29        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6261_sum_OatLeast__Suc__atMost,axiom,
% 9.49/5.29      ! [M: nat,N: nat,G: nat > rat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29          = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6262_sum_OatLeast__Suc__atMost,axiom,
% 9.49/5.29      ! [M: nat,N: nat,G: nat > int] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29          = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6263_sum_OatLeast__Suc__atMost,axiom,
% 9.49/5.29      ! [M: nat,N: nat,G: nat > nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29          = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6264_sum_OatLeast__Suc__atMost,axiom,
% 9.49/5.29      ! [M: nat,N: nat,G: nat > real] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29          = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6265_sum_Onat__ivl__Suc_H,axiom,
% 9.49/5.29      ! [M: nat,N: nat,G: nat > rat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.49/5.29       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.49/5.29          = ( plus_plus_rat @ ( G @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6266_sum_Onat__ivl__Suc_H,axiom,
% 9.49/5.29      ! [M: nat,N: nat,G: nat > int] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.49/5.29       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.49/5.29          = ( plus_plus_int @ ( G @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6267_sum_Onat__ivl__Suc_H,axiom,
% 9.49/5.29      ! [M: nat,N: nat,G: nat > nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.49/5.29       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.49/5.29          = ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6268_sum_Onat__ivl__Suc_H,axiom,
% 9.49/5.29      ! [M: nat,N: nat,G: nat > real] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.49/5.29       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.49/5.29          = ( plus_plus_real @ ( G @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6269_remove__induct,axiom,
% 9.49/5.29      ! [P: set_Pr4329608150637261639at_nat > $o,B2: set_Pr4329608150637261639at_nat] :
% 9.49/5.29        ( ( P @ bot_bo228742789529271731at_nat )
% 9.49/5.29       => ( ( ~ ( finite4343798906461161616at_nat @ B2 )
% 9.49/5.29           => ( P @ B2 ) )
% 9.49/5.29         => ( ! [A5: set_Pr4329608150637261639at_nat] :
% 9.49/5.29                ( ( finite4343798906461161616at_nat @ A5 )
% 9.49/5.29               => ( ( A5 != bot_bo228742789529271731at_nat )
% 9.49/5.29                 => ( ( ord_le1268244103169919719at_nat @ A5 @ B2 )
% 9.49/5.29                   => ( ! [X2: produc3843707927480180839at_nat] :
% 9.49/5.29                          ( ( member8757157785044589968at_nat @ X2 @ A5 )
% 9.49/5.29                         => ( P @ ( minus_3314409938677909166at_nat @ A5 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) )
% 9.49/5.29                     => ( P @ A5 ) ) ) ) )
% 9.49/5.29           => ( P @ B2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6270_remove__induct,axiom,
% 9.49/5.29      ! [P: set_set_nat > $o,B2: set_set_nat] :
% 9.49/5.29        ( ( P @ bot_bot_set_set_nat )
% 9.49/5.29       => ( ( ~ ( finite1152437895449049373et_nat @ B2 )
% 9.49/5.29           => ( P @ B2 ) )
% 9.49/5.29         => ( ! [A5: set_set_nat] :
% 9.49/5.29                ( ( finite1152437895449049373et_nat @ A5 )
% 9.49/5.29               => ( ( A5 != bot_bot_set_set_nat )
% 9.49/5.29                 => ( ( ord_le6893508408891458716et_nat @ A5 @ B2 )
% 9.49/5.29                   => ( ! [X2: set_nat] :
% 9.49/5.29                          ( ( member_set_nat @ X2 @ A5 )
% 9.49/5.29                         => ( P @ ( minus_2163939370556025621et_nat @ A5 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) )
% 9.49/5.29                     => ( P @ A5 ) ) ) ) )
% 9.49/5.29           => ( P @ B2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6271_remove__induct,axiom,
% 9.49/5.29      ! [P: set_complex > $o,B2: set_complex] :
% 9.49/5.29        ( ( P @ bot_bot_set_complex )
% 9.49/5.29       => ( ( ~ ( finite3207457112153483333omplex @ B2 )
% 9.49/5.29           => ( P @ B2 ) )
% 9.49/5.29         => ( ! [A5: set_complex] :
% 9.49/5.29                ( ( finite3207457112153483333omplex @ A5 )
% 9.49/5.29               => ( ( A5 != bot_bot_set_complex )
% 9.49/5.29                 => ( ( ord_le211207098394363844omplex @ A5 @ B2 )
% 9.49/5.29                   => ( ! [X2: complex] :
% 9.49/5.29                          ( ( member_complex @ X2 @ A5 )
% 9.49/5.29                         => ( P @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) )
% 9.49/5.29                     => ( P @ A5 ) ) ) ) )
% 9.49/5.29           => ( P @ B2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6272_remove__induct,axiom,
% 9.49/5.29      ! [P: set_Pr1261947904930325089at_nat > $o,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.29        ( ( P @ bot_bo2099793752762293965at_nat )
% 9.49/5.29       => ( ( ~ ( finite6177210948735845034at_nat @ B2 )
% 9.49/5.29           => ( P @ B2 ) )
% 9.49/5.29         => ( ! [A5: set_Pr1261947904930325089at_nat] :
% 9.49/5.29                ( ( finite6177210948735845034at_nat @ A5 )
% 9.49/5.29               => ( ( A5 != bot_bo2099793752762293965at_nat )
% 9.49/5.29                 => ( ( ord_le3146513528884898305at_nat @ A5 @ B2 )
% 9.49/5.29                   => ( ! [X2: product_prod_nat_nat] :
% 9.49/5.29                          ( ( member8440522571783428010at_nat @ X2 @ A5 )
% 9.49/5.29                         => ( P @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) )
% 9.49/5.29                     => ( P @ A5 ) ) ) ) )
% 9.49/5.29           => ( P @ B2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6273_remove__induct,axiom,
% 9.49/5.29      ! [P: set_real > $o,B2: set_real] :
% 9.49/5.29        ( ( P @ bot_bot_set_real )
% 9.49/5.29       => ( ( ~ ( finite_finite_real @ B2 )
% 9.49/5.29           => ( P @ B2 ) )
% 9.49/5.29         => ( ! [A5: set_real] :
% 9.49/5.29                ( ( finite_finite_real @ A5 )
% 9.49/5.29               => ( ( A5 != bot_bot_set_real )
% 9.49/5.29                 => ( ( ord_less_eq_set_real @ A5 @ B2 )
% 9.49/5.29                   => ( ! [X2: real] :
% 9.49/5.29                          ( ( member_real @ X2 @ A5 )
% 9.49/5.29                         => ( P @ ( minus_minus_set_real @ A5 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) )
% 9.49/5.29                     => ( P @ A5 ) ) ) ) )
% 9.49/5.29           => ( P @ B2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6274_remove__induct,axiom,
% 9.49/5.29      ! [P: set_nat > $o,B2: set_nat] :
% 9.49/5.29        ( ( P @ bot_bot_set_nat )
% 9.49/5.29       => ( ( ~ ( finite_finite_nat @ B2 )
% 9.49/5.29           => ( P @ B2 ) )
% 9.49/5.29         => ( ! [A5: set_nat] :
% 9.49/5.29                ( ( finite_finite_nat @ A5 )
% 9.49/5.29               => ( ( A5 != bot_bot_set_nat )
% 9.49/5.29                 => ( ( ord_less_eq_set_nat @ A5 @ B2 )
% 9.49/5.29                   => ( ! [X2: nat] :
% 9.49/5.29                          ( ( member_nat @ X2 @ A5 )
% 9.49/5.29                         => ( P @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
% 9.49/5.29                     => ( P @ A5 ) ) ) ) )
% 9.49/5.29           => ( P @ B2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6275_remove__induct,axiom,
% 9.49/5.29      ! [P: set_int > $o,B2: set_int] :
% 9.49/5.29        ( ( P @ bot_bot_set_int )
% 9.49/5.29       => ( ( ~ ( finite_finite_int @ B2 )
% 9.49/5.29           => ( P @ B2 ) )
% 9.49/5.29         => ( ! [A5: set_int] :
% 9.49/5.29                ( ( finite_finite_int @ A5 )
% 9.49/5.29               => ( ( A5 != bot_bot_set_int )
% 9.49/5.29                 => ( ( ord_less_eq_set_int @ A5 @ B2 )
% 9.49/5.29                   => ( ! [X2: int] :
% 9.49/5.29                          ( ( member_int @ X2 @ A5 )
% 9.49/5.29                         => ( P @ ( minus_minus_set_int @ A5 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) )
% 9.49/5.29                     => ( P @ A5 ) ) ) ) )
% 9.49/5.29           => ( P @ B2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6276_finite__remove__induct,axiom,
% 9.49/5.29      ! [B2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 9.49/5.29        ( ( finite4343798906461161616at_nat @ B2 )
% 9.49/5.29       => ( ( P @ bot_bo228742789529271731at_nat )
% 9.49/5.29         => ( ! [A5: set_Pr4329608150637261639at_nat] :
% 9.49/5.29                ( ( finite4343798906461161616at_nat @ A5 )
% 9.49/5.29               => ( ( A5 != bot_bo228742789529271731at_nat )
% 9.49/5.29                 => ( ( ord_le1268244103169919719at_nat @ A5 @ B2 )
% 9.49/5.29                   => ( ! [X2: produc3843707927480180839at_nat] :
% 9.49/5.29                          ( ( member8757157785044589968at_nat @ X2 @ A5 )
% 9.49/5.29                         => ( P @ ( minus_3314409938677909166at_nat @ A5 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) )
% 9.49/5.29                     => ( P @ A5 ) ) ) ) )
% 9.49/5.29           => ( P @ B2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6277_finite__remove__induct,axiom,
% 9.49/5.29      ! [B2: set_set_nat,P: set_set_nat > $o] :
% 9.49/5.29        ( ( finite1152437895449049373et_nat @ B2 )
% 9.49/5.29       => ( ( P @ bot_bot_set_set_nat )
% 9.49/5.29         => ( ! [A5: set_set_nat] :
% 9.49/5.29                ( ( finite1152437895449049373et_nat @ A5 )
% 9.49/5.29               => ( ( A5 != bot_bot_set_set_nat )
% 9.49/5.29                 => ( ( ord_le6893508408891458716et_nat @ A5 @ B2 )
% 9.49/5.29                   => ( ! [X2: set_nat] :
% 9.49/5.29                          ( ( member_set_nat @ X2 @ A5 )
% 9.49/5.29                         => ( P @ ( minus_2163939370556025621et_nat @ A5 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) )
% 9.49/5.29                     => ( P @ A5 ) ) ) ) )
% 9.49/5.29           => ( P @ B2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6278_finite__remove__induct,axiom,
% 9.49/5.29      ! [B2: set_complex,P: set_complex > $o] :
% 9.49/5.29        ( ( finite3207457112153483333omplex @ B2 )
% 9.49/5.29       => ( ( P @ bot_bot_set_complex )
% 9.49/5.29         => ( ! [A5: set_complex] :
% 9.49/5.29                ( ( finite3207457112153483333omplex @ A5 )
% 9.49/5.29               => ( ( A5 != bot_bot_set_complex )
% 9.49/5.29                 => ( ( ord_le211207098394363844omplex @ A5 @ B2 )
% 9.49/5.29                   => ( ! [X2: complex] :
% 9.49/5.29                          ( ( member_complex @ X2 @ A5 )
% 9.49/5.29                         => ( P @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) )
% 9.49/5.29                     => ( P @ A5 ) ) ) ) )
% 9.49/5.29           => ( P @ B2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6279_finite__remove__induct,axiom,
% 9.49/5.29      ! [B2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.49/5.29        ( ( finite6177210948735845034at_nat @ B2 )
% 9.49/5.29       => ( ( P @ bot_bo2099793752762293965at_nat )
% 9.49/5.29         => ( ! [A5: set_Pr1261947904930325089at_nat] :
% 9.49/5.29                ( ( finite6177210948735845034at_nat @ A5 )
% 9.49/5.29               => ( ( A5 != bot_bo2099793752762293965at_nat )
% 9.49/5.29                 => ( ( ord_le3146513528884898305at_nat @ A5 @ B2 )
% 9.49/5.29                   => ( ! [X2: product_prod_nat_nat] :
% 9.49/5.29                          ( ( member8440522571783428010at_nat @ X2 @ A5 )
% 9.49/5.29                         => ( P @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) )
% 9.49/5.29                     => ( P @ A5 ) ) ) ) )
% 9.49/5.29           => ( P @ B2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6280_finite__remove__induct,axiom,
% 9.49/5.29      ! [B2: set_real,P: set_real > $o] :
% 9.49/5.29        ( ( finite_finite_real @ B2 )
% 9.49/5.29       => ( ( P @ bot_bot_set_real )
% 9.49/5.29         => ( ! [A5: set_real] :
% 9.49/5.29                ( ( finite_finite_real @ A5 )
% 9.49/5.29               => ( ( A5 != bot_bot_set_real )
% 9.49/5.29                 => ( ( ord_less_eq_set_real @ A5 @ B2 )
% 9.49/5.29                   => ( ! [X2: real] :
% 9.49/5.29                          ( ( member_real @ X2 @ A5 )
% 9.49/5.29                         => ( P @ ( minus_minus_set_real @ A5 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) )
% 9.49/5.29                     => ( P @ A5 ) ) ) ) )
% 9.49/5.29           => ( P @ B2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6281_finite__remove__induct,axiom,
% 9.49/5.29      ! [B2: set_nat,P: set_nat > $o] :
% 9.49/5.29        ( ( finite_finite_nat @ B2 )
% 9.49/5.29       => ( ( P @ bot_bot_set_nat )
% 9.49/5.29         => ( ! [A5: set_nat] :
% 9.49/5.29                ( ( finite_finite_nat @ A5 )
% 9.49/5.29               => ( ( A5 != bot_bot_set_nat )
% 9.49/5.29                 => ( ( ord_less_eq_set_nat @ A5 @ B2 )
% 9.49/5.29                   => ( ! [X2: nat] :
% 9.49/5.29                          ( ( member_nat @ X2 @ A5 )
% 9.49/5.29                         => ( P @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
% 9.49/5.29                     => ( P @ A5 ) ) ) ) )
% 9.49/5.29           => ( P @ B2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6282_finite__remove__induct,axiom,
% 9.49/5.29      ! [B2: set_int,P: set_int > $o] :
% 9.49/5.29        ( ( finite_finite_int @ B2 )
% 9.49/5.29       => ( ( P @ bot_bot_set_int )
% 9.49/5.29         => ( ! [A5: set_int] :
% 9.49/5.29                ( ( finite_finite_int @ A5 )
% 9.49/5.29               => ( ( A5 != bot_bot_set_int )
% 9.49/5.29                 => ( ( ord_less_eq_set_int @ A5 @ B2 )
% 9.49/5.29                   => ( ! [X2: int] :
% 9.49/5.29                          ( ( member_int @ X2 @ A5 )
% 9.49/5.29                         => ( P @ ( minus_minus_set_int @ A5 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) )
% 9.49/5.29                     => ( P @ A5 ) ) ) ) )
% 9.49/5.29           => ( P @ B2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6283_finite__induct__select,axiom,
% 9.49/5.29      ! [S2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 9.49/5.29        ( ( finite4343798906461161616at_nat @ S2 )
% 9.49/5.29       => ( ( P @ bot_bo228742789529271731at_nat )
% 9.49/5.29         => ( ! [T5: set_Pr4329608150637261639at_nat] :
% 9.49/5.29                ( ( ord_le2604355607129572851at_nat @ T5 @ S2 )
% 9.49/5.29               => ( ( P @ T5 )
% 9.49/5.29                 => ? [X2: produc3843707927480180839at_nat] :
% 9.49/5.29                      ( ( member8757157785044589968at_nat @ X2 @ ( minus_3314409938677909166at_nat @ S2 @ T5 ) )
% 9.49/5.29                      & ( P @ ( insert9069300056098147895at_nat @ X2 @ T5 ) ) ) ) )
% 9.49/5.29           => ( P @ S2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6284_finite__induct__select,axiom,
% 9.49/5.29      ! [S2: set_complex,P: set_complex > $o] :
% 9.49/5.29        ( ( finite3207457112153483333omplex @ S2 )
% 9.49/5.29       => ( ( P @ bot_bot_set_complex )
% 9.49/5.29         => ( ! [T5: set_complex] :
% 9.49/5.29                ( ( ord_less_set_complex @ T5 @ S2 )
% 9.49/5.29               => ( ( P @ T5 )
% 9.49/5.29                 => ? [X2: complex] :
% 9.49/5.29                      ( ( member_complex @ X2 @ ( minus_811609699411566653omplex @ S2 @ T5 ) )
% 9.49/5.29                      & ( P @ ( insert_complex @ X2 @ T5 ) ) ) ) )
% 9.49/5.29           => ( P @ S2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6285_finite__induct__select,axiom,
% 9.49/5.29      ! [S2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.49/5.29        ( ( finite6177210948735845034at_nat @ S2 )
% 9.49/5.29       => ( ( P @ bot_bo2099793752762293965at_nat )
% 9.49/5.29         => ( ! [T5: set_Pr1261947904930325089at_nat] :
% 9.49/5.29                ( ( ord_le7866589430770878221at_nat @ T5 @ S2 )
% 9.49/5.29               => ( ( P @ T5 )
% 9.49/5.29                 => ? [X2: product_prod_nat_nat] :
% 9.49/5.29                      ( ( member8440522571783428010at_nat @ X2 @ ( minus_1356011639430497352at_nat @ S2 @ T5 ) )
% 9.49/5.29                      & ( P @ ( insert8211810215607154385at_nat @ X2 @ T5 ) ) ) ) )
% 9.49/5.29           => ( P @ S2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6286_finite__induct__select,axiom,
% 9.49/5.29      ! [S2: set_real,P: set_real > $o] :
% 9.49/5.29        ( ( finite_finite_real @ S2 )
% 9.49/5.29       => ( ( P @ bot_bot_set_real )
% 9.49/5.29         => ( ! [T5: set_real] :
% 9.49/5.29                ( ( ord_less_set_real @ T5 @ S2 )
% 9.49/5.29               => ( ( P @ T5 )
% 9.49/5.29                 => ? [X2: real] :
% 9.49/5.29                      ( ( member_real @ X2 @ ( minus_minus_set_real @ S2 @ T5 ) )
% 9.49/5.29                      & ( P @ ( insert_real @ X2 @ T5 ) ) ) ) )
% 9.49/5.29           => ( P @ S2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6287_finite__induct__select,axiom,
% 9.49/5.29      ! [S2: set_int,P: set_int > $o] :
% 9.49/5.29        ( ( finite_finite_int @ S2 )
% 9.49/5.29       => ( ( P @ bot_bot_set_int )
% 9.49/5.29         => ( ! [T5: set_int] :
% 9.49/5.29                ( ( ord_less_set_int @ T5 @ S2 )
% 9.49/5.29               => ( ( P @ T5 )
% 9.49/5.29                 => ? [X2: int] :
% 9.49/5.29                      ( ( member_int @ X2 @ ( minus_minus_set_int @ S2 @ T5 ) )
% 9.49/5.29                      & ( P @ ( insert_int @ X2 @ T5 ) ) ) ) )
% 9.49/5.29           => ( P @ S2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6288_finite__induct__select,axiom,
% 9.49/5.29      ! [S2: set_nat,P: set_nat > $o] :
% 9.49/5.29        ( ( finite_finite_nat @ S2 )
% 9.49/5.29       => ( ( P @ bot_bot_set_nat )
% 9.49/5.29         => ( ! [T5: set_nat] :
% 9.49/5.29                ( ( ord_less_set_nat @ T5 @ S2 )
% 9.49/5.29               => ( ( P @ T5 )
% 9.49/5.29                 => ? [X2: nat] :
% 9.49/5.29                      ( ( member_nat @ X2 @ ( minus_minus_set_nat @ S2 @ T5 ) )
% 9.49/5.29                      & ( P @ ( insert_nat @ X2 @ T5 ) ) ) ) )
% 9.49/5.29           => ( P @ S2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6289_psubset__insert__iff,axiom,
% 9.49/5.29      ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.49/5.29        ( ( ord_le2604355607129572851at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ B2 ) )
% 9.49/5.29        = ( ( ( member8757157785044589968at_nat @ X @ B2 )
% 9.49/5.29           => ( ord_le2604355607129572851at_nat @ A2 @ B2 ) )
% 9.49/5.29          & ( ~ ( member8757157785044589968at_nat @ X @ B2 )
% 9.49/5.29           => ( ( ( member8757157785044589968at_nat @ X @ A2 )
% 9.49/5.29               => ( ord_le2604355607129572851at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B2 ) )
% 9.49/5.29              & ( ~ ( member8757157785044589968at_nat @ X @ A2 )
% 9.49/5.29               => ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6290_psubset__insert__iff,axiom,
% 9.49/5.29      ! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
% 9.49/5.29        ( ( ord_less_set_set_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
% 9.49/5.29        = ( ( ( member_set_nat @ X @ B2 )
% 9.49/5.29           => ( ord_less_set_set_nat @ A2 @ B2 ) )
% 9.49/5.29          & ( ~ ( member_set_nat @ X @ B2 )
% 9.49/5.29           => ( ( ( member_set_nat @ X @ A2 )
% 9.49/5.29               => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B2 ) )
% 9.49/5.29              & ( ~ ( member_set_nat @ X @ A2 )
% 9.49/5.29               => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6291_psubset__insert__iff,axiom,
% 9.49/5.29      ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.49/5.29        ( ( ord_le7866589430770878221at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B2 ) )
% 9.49/5.29        = ( ( ( member8440522571783428010at_nat @ X @ B2 )
% 9.49/5.29           => ( ord_le7866589430770878221at_nat @ A2 @ B2 ) )
% 9.49/5.29          & ( ~ ( member8440522571783428010at_nat @ X @ B2 )
% 9.49/5.29           => ( ( ( member8440522571783428010at_nat @ X @ A2 )
% 9.49/5.29               => ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B2 ) )
% 9.49/5.29              & ( ~ ( member8440522571783428010at_nat @ X @ A2 )
% 9.49/5.29               => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6292_psubset__insert__iff,axiom,
% 9.49/5.29      ! [A2: set_real,X: real,B2: set_real] :
% 9.49/5.29        ( ( ord_less_set_real @ A2 @ ( insert_real @ X @ B2 ) )
% 9.49/5.29        = ( ( ( member_real @ X @ B2 )
% 9.49/5.29           => ( ord_less_set_real @ A2 @ B2 ) )
% 9.49/5.29          & ( ~ ( member_real @ X @ B2 )
% 9.49/5.29           => ( ( ( member_real @ X @ A2 )
% 9.49/5.29               => ( ord_less_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 ) )
% 9.49/5.29              & ( ~ ( member_real @ X @ A2 )
% 9.49/5.29               => ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6293_psubset__insert__iff,axiom,
% 9.49/5.29      ! [A2: set_nat,X: nat,B2: set_nat] :
% 9.49/5.29        ( ( ord_less_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
% 9.49/5.29        = ( ( ( member_nat @ X @ B2 )
% 9.49/5.29           => ( ord_less_set_nat @ A2 @ B2 ) )
% 9.49/5.29          & ( ~ ( member_nat @ X @ B2 )
% 9.49/5.29           => ( ( ( member_nat @ X @ A2 )
% 9.49/5.29               => ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
% 9.49/5.29              & ( ~ ( member_nat @ X @ A2 )
% 9.49/5.29               => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6294_psubset__insert__iff,axiom,
% 9.49/5.29      ! [A2: set_int,X: int,B2: set_int] :
% 9.49/5.29        ( ( ord_less_set_int @ A2 @ ( insert_int @ X @ B2 ) )
% 9.49/5.29        = ( ( ( member_int @ X @ B2 )
% 9.49/5.29           => ( ord_less_set_int @ A2 @ B2 ) )
% 9.49/5.29          & ( ~ ( member_int @ X @ B2 )
% 9.49/5.29           => ( ( ( member_int @ X @ A2 )
% 9.49/5.29               => ( ord_less_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 ) )
% 9.49/5.29              & ( ~ ( member_int @ X @ A2 )
% 9.49/5.29               => ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6295_set__replicate__Suc,axiom,
% 9.49/5.29      ! [N: nat,X: produc3843707927480180839at_nat] :
% 9.49/5.29        ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ ( suc @ N ) @ X ) )
% 9.49/5.29        = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6296_set__replicate__Suc,axiom,
% 9.49/5.29      ! [N: nat,X: vEBT_VEBT] :
% 9.49/5.29        ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ ( suc @ N ) @ X ) )
% 9.49/5.29        = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6297_set__replicate__Suc,axiom,
% 9.49/5.29      ! [N: nat,X: product_prod_nat_nat] :
% 9.49/5.29        ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ ( suc @ N ) @ X ) )
% 9.49/5.29        = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6298_set__replicate__Suc,axiom,
% 9.49/5.29      ! [N: nat,X: real] :
% 9.49/5.29        ( ( set_real2 @ ( replicate_real @ ( suc @ N ) @ X ) )
% 9.49/5.29        = ( insert_real @ X @ bot_bot_set_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6299_set__replicate__Suc,axiom,
% 9.49/5.29      ! [N: nat,X: nat] :
% 9.49/5.29        ( ( set_nat2 @ ( replicate_nat @ ( suc @ N ) @ X ) )
% 9.49/5.29        = ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6300_set__replicate__Suc,axiom,
% 9.49/5.29      ! [N: nat,X: int] :
% 9.49/5.29        ( ( set_int2 @ ( replicate_int @ ( suc @ N ) @ X ) )
% 9.49/5.29        = ( insert_int @ X @ bot_bot_set_int ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6301_set__replicate__conv__if,axiom,
% 9.49/5.29      ! [N: nat,X: produc3843707927480180839at_nat] :
% 9.49/5.29        ( ( ( N = zero_zero_nat )
% 9.49/5.29         => ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N @ X ) )
% 9.49/5.29            = bot_bo228742789529271731at_nat ) )
% 9.49/5.29        & ( ( N != zero_zero_nat )
% 9.49/5.29         => ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N @ X ) )
% 9.49/5.29            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6302_set__replicate__conv__if,axiom,
% 9.49/5.29      ! [N: nat,X: vEBT_VEBT] :
% 9.49/5.29        ( ( ( N = zero_zero_nat )
% 9.49/5.29         => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
% 9.49/5.29            = bot_bo8194388402131092736T_VEBT ) )
% 9.49/5.29        & ( ( N != zero_zero_nat )
% 9.49/5.29         => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
% 9.49/5.29            = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6303_set__replicate__conv__if,axiom,
% 9.49/5.29      ! [N: nat,X: product_prod_nat_nat] :
% 9.49/5.29        ( ( ( N = zero_zero_nat )
% 9.49/5.29         => ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ X ) )
% 9.49/5.29            = bot_bo2099793752762293965at_nat ) )
% 9.49/5.29        & ( ( N != zero_zero_nat )
% 9.49/5.29         => ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ X ) )
% 9.49/5.29            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6304_set__replicate__conv__if,axiom,
% 9.49/5.29      ! [N: nat,X: real] :
% 9.49/5.29        ( ( ( N = zero_zero_nat )
% 9.49/5.29         => ( ( set_real2 @ ( replicate_real @ N @ X ) )
% 9.49/5.29            = bot_bot_set_real ) )
% 9.49/5.29        & ( ( N != zero_zero_nat )
% 9.49/5.29         => ( ( set_real2 @ ( replicate_real @ N @ X ) )
% 9.49/5.29            = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6305_set__replicate__conv__if,axiom,
% 9.49/5.29      ! [N: nat,X: nat] :
% 9.49/5.29        ( ( ( N = zero_zero_nat )
% 9.49/5.29         => ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
% 9.49/5.29            = bot_bot_set_nat ) )
% 9.49/5.29        & ( ( N != zero_zero_nat )
% 9.49/5.29         => ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
% 9.49/5.29            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6306_set__replicate__conv__if,axiom,
% 9.49/5.29      ! [N: nat,X: int] :
% 9.49/5.29        ( ( ( N = zero_zero_nat )
% 9.49/5.29         => ( ( set_int2 @ ( replicate_int @ N @ X ) )
% 9.49/5.29            = bot_bot_set_int ) )
% 9.49/5.29        & ( ( N != zero_zero_nat )
% 9.49/5.29         => ( ( set_int2 @ ( replicate_int @ N @ X ) )
% 9.49/5.29            = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6307_sum_OSuc__reindex__ivl,axiom,
% 9.49/5.29      ! [M: nat,N: nat,G: nat > rat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29       => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 9.49/5.29          = ( plus_plus_rat @ ( G @ M )
% 9.49/5.29            @ ( groups2906978787729119204at_rat
% 9.49/5.29              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.49/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6308_sum_OSuc__reindex__ivl,axiom,
% 9.49/5.29      ! [M: nat,N: nat,G: nat > int] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29       => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 9.49/5.29          = ( plus_plus_int @ ( G @ M )
% 9.49/5.29            @ ( groups3539618377306564664at_int
% 9.49/5.29              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.49/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6309_sum_OSuc__reindex__ivl,axiom,
% 9.49/5.29      ! [M: nat,N: nat,G: nat > nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29       => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 9.49/5.29          = ( plus_plus_nat @ ( G @ M )
% 9.49/5.29            @ ( groups3542108847815614940at_nat
% 9.49/5.29              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.49/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6310_sum_OSuc__reindex__ivl,axiom,
% 9.49/5.29      ! [M: nat,N: nat,G: nat > real] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29       => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 9.49/5.29          = ( plus_plus_real @ ( G @ M )
% 9.49/5.29            @ ( groups6591440286371151544t_real
% 9.49/5.29              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.49/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6311_sum__Suc__diff,axiom,
% 9.49/5.29      ! [M: nat,N: nat,F: nat > rat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.49/5.29       => ( ( groups2906978787729119204at_rat
% 9.49/5.29            @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 9.49/5.29            @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29          = ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6312_sum__Suc__diff,axiom,
% 9.49/5.29      ! [M: nat,N: nat,F: nat > int] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.49/5.29       => ( ( groups3539618377306564664at_int
% 9.49/5.29            @ ^ [I4: nat] : ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 9.49/5.29            @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29          = ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6313_sum__Suc__diff,axiom,
% 9.49/5.29      ! [M: nat,N: nat,F: nat > real] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.49/5.29       => ( ( groups6591440286371151544t_real
% 9.49/5.29            @ ^ [I4: nat] : ( minus_minus_real @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 9.49/5.29            @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29          = ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6314_sum__mono2,axiom,
% 9.49/5.29      ! [B2: set_real,A2: set_real,F: real > real] :
% 9.49/5.29        ( ( finite_finite_real @ B2 )
% 9.49/5.29       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.49/5.29         => ( ! [B3: real] :
% 9.49/5.29                ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) )
% 9.49/5.29               => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
% 9.49/5.29           => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6315_sum__mono2,axiom,
% 9.49/5.29      ! [B2: set_complex,A2: set_complex,F: complex > real] :
% 9.49/5.29        ( ( finite3207457112153483333omplex @ B2 )
% 9.49/5.29       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.49/5.29         => ( ! [B3: complex] :
% 9.49/5.29                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 9.49/5.29               => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
% 9.49/5.29           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6316_sum__mono2,axiom,
% 9.49/5.29      ! [B2: set_real,A2: set_real,F: real > rat] :
% 9.49/5.29        ( ( finite_finite_real @ B2 )
% 9.49/5.29       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.49/5.29         => ( ! [B3: real] :
% 9.49/5.29                ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) )
% 9.49/5.29               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
% 9.49/5.29           => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6317_sum__mono2,axiom,
% 9.49/5.29      ! [B2: set_complex,A2: set_complex,F: complex > rat] :
% 9.49/5.29        ( ( finite3207457112153483333omplex @ B2 )
% 9.49/5.29       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.49/5.29         => ( ! [B3: complex] :
% 9.49/5.29                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 9.49/5.29               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
% 9.49/5.29           => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6318_sum__mono2,axiom,
% 9.49/5.29      ! [B2: set_nat,A2: set_nat,F: nat > rat] :
% 9.49/5.29        ( ( finite_finite_nat @ B2 )
% 9.49/5.29       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.49/5.29         => ( ! [B3: nat] :
% 9.49/5.29                ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B2 @ A2 ) )
% 9.49/5.29               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
% 9.49/5.29           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6319_sum__mono2,axiom,
% 9.49/5.29      ! [B2: set_real,A2: set_real,F: real > nat] :
% 9.49/5.29        ( ( finite_finite_real @ B2 )
% 9.49/5.29       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.49/5.29         => ( ! [B3: real] :
% 9.49/5.29                ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) )
% 9.49/5.29               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
% 9.49/5.29           => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6320_sum__mono2,axiom,
% 9.49/5.29      ! [B2: set_complex,A2: set_complex,F: complex > nat] :
% 9.49/5.29        ( ( finite3207457112153483333omplex @ B2 )
% 9.49/5.29       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.49/5.29         => ( ! [B3: complex] :
% 9.49/5.29                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 9.49/5.29               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
% 9.49/5.29           => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6321_sum__mono2,axiom,
% 9.49/5.29      ! [B2: set_real,A2: set_real,F: real > int] :
% 9.49/5.29        ( ( finite_finite_real @ B2 )
% 9.49/5.29       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.49/5.29         => ( ! [B3: real] :
% 9.49/5.29                ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) )
% 9.49/5.29               => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
% 9.49/5.29           => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6322_sum__mono2,axiom,
% 9.49/5.29      ! [B2: set_complex,A2: set_complex,F: complex > int] :
% 9.49/5.29        ( ( finite3207457112153483333omplex @ B2 )
% 9.49/5.29       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.49/5.29         => ( ! [B3: complex] :
% 9.49/5.29                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 9.49/5.29               => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
% 9.49/5.29           => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6323_sum__mono2,axiom,
% 9.49/5.29      ! [B2: set_nat,A2: set_nat,F: nat > int] :
% 9.49/5.29        ( ( finite_finite_nat @ B2 )
% 9.49/5.29       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.49/5.29         => ( ! [B3: nat] :
% 9.49/5.29                ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B2 @ A2 ) )
% 9.49/5.29               => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
% 9.49/5.29           => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6324_sum_Oub__add__nat,axiom,
% 9.49/5.29      ! [M: nat,N: nat,G: nat > rat,P4: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 9.49/5.29       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 9.49/5.29          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6325_sum_Oub__add__nat,axiom,
% 9.49/5.29      ! [M: nat,N: nat,G: nat > int,P4: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 9.49/5.29       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 9.49/5.29          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6326_sum_Oub__add__nat,axiom,
% 9.49/5.29      ! [M: nat,N: nat,G: nat > nat,P4: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 9.49/5.29       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 9.49/5.29          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6327_sum_Oub__add__nat,axiom,
% 9.49/5.29      ! [M: nat,N: nat,G: nat > real,P4: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 9.49/5.29       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 9.49/5.29          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6328_sum__count__set,axiom,
% 9.49/5.29      ! [Xs: list_complex,X8: set_complex] :
% 9.49/5.29        ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ X8 )
% 9.49/5.29       => ( ( finite3207457112153483333omplex @ X8 )
% 9.49/5.29         => ( ( groups5693394587270226106ex_nat @ ( count_list_complex @ Xs ) @ X8 )
% 9.49/5.29            = ( size_s3451745648224563538omplex @ Xs ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6329_sum__count__set,axiom,
% 9.49/5.29      ! [Xs: list_P6011104703257516679at_nat,X8: set_Pr1261947904930325089at_nat] :
% 9.49/5.29        ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ X8 )
% 9.49/5.29       => ( ( finite6177210948735845034at_nat @ X8 )
% 9.49/5.29         => ( ( groups977919841031483927at_nat @ ( count_4203492906077236349at_nat @ Xs ) @ X8 )
% 9.49/5.29            = ( size_s5460976970255530739at_nat @ Xs ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6330_sum__count__set,axiom,
% 9.49/5.29      ! [Xs: list_VEBT_VEBT,X8: set_VEBT_VEBT] :
% 9.49/5.29        ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ X8 )
% 9.49/5.29       => ( ( finite5795047828879050333T_VEBT @ X8 )
% 9.49/5.29         => ( ( groups771621172384141258BT_nat @ ( count_list_VEBT_VEBT @ Xs ) @ X8 )
% 9.49/5.29            = ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6331_sum__count__set,axiom,
% 9.49/5.29      ! [Xs: list_o,X8: set_o] :
% 9.49/5.29        ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ X8 )
% 9.49/5.29       => ( ( finite_finite_o @ X8 )
% 9.49/5.29         => ( ( groups8507830703676809646_o_nat @ ( count_list_o @ Xs ) @ X8 )
% 9.49/5.29            = ( size_size_list_o @ Xs ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6332_sum__count__set,axiom,
% 9.49/5.29      ! [Xs: list_int,X8: set_int] :
% 9.49/5.29        ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ X8 )
% 9.49/5.29       => ( ( finite_finite_int @ X8 )
% 9.49/5.29         => ( ( groups4541462559716669496nt_nat @ ( count_list_int @ Xs ) @ X8 )
% 9.49/5.29            = ( size_size_list_int @ Xs ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6333_sum__count__set,axiom,
% 9.49/5.29      ! [Xs: list_nat,X8: set_nat] :
% 9.49/5.29        ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ X8 )
% 9.49/5.29       => ( ( finite_finite_nat @ X8 )
% 9.49/5.29         => ( ( groups3542108847815614940at_nat @ ( count_list_nat @ Xs ) @ X8 )
% 9.49/5.29            = ( size_size_list_nat @ Xs ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6334_sum__strict__mono2,axiom,
% 9.49/5.29      ! [B2: set_real,A2: set_real,B: real,F: real > real] :
% 9.49/5.29        ( ( finite_finite_real @ B2 )
% 9.49/5.29       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.49/5.29         => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
% 9.49/5.29           => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 9.49/5.29             => ( ! [X4: real] :
% 9.49/5.29                    ( ( member_real @ X4 @ B2 )
% 9.49/5.29                   => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.49/5.29               => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6335_sum__strict__mono2,axiom,
% 9.49/5.29      ! [B2: set_complex,A2: set_complex,B: complex,F: complex > real] :
% 9.49/5.29        ( ( finite3207457112153483333omplex @ B2 )
% 9.49/5.29       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.49/5.29         => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 9.49/5.29           => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 9.49/5.29             => ( ! [X4: complex] :
% 9.49/5.29                    ( ( member_complex @ X4 @ B2 )
% 9.49/5.29                   => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.49/5.29               => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6336_sum__strict__mono2,axiom,
% 9.49/5.29      ! [B2: set_real,A2: set_real,B: real,F: real > rat] :
% 9.49/5.29        ( ( finite_finite_real @ B2 )
% 9.49/5.29       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.49/5.29         => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
% 9.49/5.29           => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 9.49/5.29             => ( ! [X4: real] :
% 9.49/5.29                    ( ( member_real @ X4 @ B2 )
% 9.49/5.29                   => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.49/5.29               => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6337_sum__strict__mono2,axiom,
% 9.49/5.29      ! [B2: set_complex,A2: set_complex,B: complex,F: complex > rat] :
% 9.49/5.29        ( ( finite3207457112153483333omplex @ B2 )
% 9.49/5.29       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.49/5.29         => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 9.49/5.29           => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 9.49/5.29             => ( ! [X4: complex] :
% 9.49/5.29                    ( ( member_complex @ X4 @ B2 )
% 9.49/5.29                   => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.49/5.29               => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6338_sum__strict__mono2,axiom,
% 9.49/5.29      ! [B2: set_nat,A2: set_nat,B: nat,F: nat > rat] :
% 9.49/5.29        ( ( finite_finite_nat @ B2 )
% 9.49/5.29       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.49/5.29         => ( ( member_nat @ B @ ( minus_minus_set_nat @ B2 @ A2 ) )
% 9.49/5.29           => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 9.49/5.29             => ( ! [X4: nat] :
% 9.49/5.29                    ( ( member_nat @ X4 @ B2 )
% 9.49/5.29                   => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.49/5.29               => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6339_sum__strict__mono2,axiom,
% 9.49/5.29      ! [B2: set_real,A2: set_real,B: real,F: real > nat] :
% 9.49/5.29        ( ( finite_finite_real @ B2 )
% 9.49/5.29       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.49/5.29         => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
% 9.49/5.29           => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 9.49/5.29             => ( ! [X4: real] :
% 9.49/5.29                    ( ( member_real @ X4 @ B2 )
% 9.49/5.29                   => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.49/5.29               => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6340_sum__strict__mono2,axiom,
% 9.49/5.29      ! [B2: set_complex,A2: set_complex,B: complex,F: complex > nat] :
% 9.49/5.29        ( ( finite3207457112153483333omplex @ B2 )
% 9.49/5.29       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.49/5.29         => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 9.49/5.29           => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 9.49/5.29             => ( ! [X4: complex] :
% 9.49/5.29                    ( ( member_complex @ X4 @ B2 )
% 9.49/5.29                   => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.49/5.29               => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6341_sum__strict__mono2,axiom,
% 9.49/5.29      ! [B2: set_real,A2: set_real,B: real,F: real > int] :
% 9.49/5.29        ( ( finite_finite_real @ B2 )
% 9.49/5.29       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.49/5.29         => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
% 9.49/5.29           => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 9.49/5.29             => ( ! [X4: real] :
% 9.49/5.29                    ( ( member_real @ X4 @ B2 )
% 9.49/5.29                   => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 9.49/5.29               => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B2 ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6342_sum__strict__mono2,axiom,
% 9.49/5.29      ! [B2: set_complex,A2: set_complex,B: complex,F: complex > int] :
% 9.49/5.29        ( ( finite3207457112153483333omplex @ B2 )
% 9.49/5.29       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.49/5.29         => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 9.49/5.29           => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 9.49/5.29             => ( ! [X4: complex] :
% 9.49/5.29                    ( ( member_complex @ X4 @ B2 )
% 9.49/5.29                   => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 9.49/5.29               => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6343_sum__strict__mono2,axiom,
% 9.49/5.29      ! [B2: set_nat,A2: set_nat,B: nat,F: nat > int] :
% 9.49/5.29        ( ( finite_finite_nat @ B2 )
% 9.49/5.29       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.49/5.29         => ( ( member_nat @ B @ ( minus_minus_set_nat @ B2 @ A2 ) )
% 9.49/5.29           => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 9.49/5.29             => ( ! [X4: nat] :
% 9.49/5.29                    ( ( member_nat @ X4 @ B2 )
% 9.49/5.29                   => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 9.49/5.29               => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6344_sum__natinterval__diff,axiom,
% 9.49/5.29      ! [M: nat,N: nat,F: nat > complex] :
% 9.49/5.29        ( ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29         => ( ( groups2073611262835488442omplex
% 9.49/5.29              @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.49/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29            = ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 9.49/5.29        & ( ~ ( ord_less_eq_nat @ M @ N )
% 9.49/5.29         => ( ( groups2073611262835488442omplex
% 9.49/5.29              @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.49/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29            = zero_zero_complex ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6345_sum__natinterval__diff,axiom,
% 9.49/5.29      ! [M: nat,N: nat,F: nat > rat] :
% 9.49/5.29        ( ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29         => ( ( groups2906978787729119204at_rat
% 9.49/5.29              @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.49/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29            = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 9.49/5.29        & ( ~ ( ord_less_eq_nat @ M @ N )
% 9.49/5.29         => ( ( groups2906978787729119204at_rat
% 9.49/5.29              @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.49/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29            = zero_zero_rat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6346_sum__natinterval__diff,axiom,
% 9.49/5.29      ! [M: nat,N: nat,F: nat > int] :
% 9.49/5.29        ( ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29         => ( ( groups3539618377306564664at_int
% 9.49/5.29              @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.49/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29            = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 9.49/5.29        & ( ~ ( ord_less_eq_nat @ M @ N )
% 9.49/5.29         => ( ( groups3539618377306564664at_int
% 9.49/5.29              @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.49/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29            = zero_zero_int ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6347_sum__natinterval__diff,axiom,
% 9.49/5.29      ! [M: nat,N: nat,F: nat > real] :
% 9.49/5.29        ( ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29         => ( ( groups6591440286371151544t_real
% 9.49/5.29              @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.49/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29            = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 9.49/5.29        & ( ~ ( ord_less_eq_nat @ M @ N )
% 9.49/5.29         => ( ( groups6591440286371151544t_real
% 9.49/5.29              @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.49/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29            = zero_zero_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6348_sum__telescope_H_H,axiom,
% 9.49/5.29      ! [M: nat,N: nat,F: nat > rat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29       => ( ( groups2906978787729119204at_rat
% 9.49/5.29            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 9.49/5.29            @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 9.49/5.29          = ( minus_minus_rat @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6349_sum__telescope_H_H,axiom,
% 9.49/5.29      ! [M: nat,N: nat,F: nat > int] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29       => ( ( groups3539618377306564664at_int
% 9.49/5.29            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 9.49/5.29            @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 9.49/5.29          = ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6350_sum__telescope_H_H,axiom,
% 9.49/5.29      ! [M: nat,N: nat,F: nat > real] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29       => ( ( groups6591440286371151544t_real
% 9.49/5.29            @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 9.49/5.29            @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 9.49/5.29          = ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6351_exists__least__lemma,axiom,
% 9.49/5.29      ! [P: nat > $o] :
% 9.49/5.29        ( ~ ( P @ zero_zero_nat )
% 9.49/5.29       => ( ? [X_12: nat] : ( P @ X_12 )
% 9.49/5.29         => ? [N3: nat] :
% 9.49/5.29              ( ~ ( P @ N3 )
% 9.49/5.29              & ( P @ ( suc @ N3 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6352_mask__eq__sum__exp,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int )
% 9.49/5.29        = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.49/5.29          @ ( collect_nat
% 9.49/5.29            @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6353_mask__eq__sum__exp,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat )
% 9.49/5.29        = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.29          @ ( collect_nat
% 9.49/5.29            @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6354_ex__le__of__int,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29      ? [Z4: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z4 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6355_ex__le__of__int,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29      ? [Z4: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z4 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6356_ex__less__of__int,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29      ? [Z4: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z4 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6357_ex__less__of__int,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29      ? [Z4: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z4 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6358_ex__of__int__less,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29      ? [Z4: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z4 ) @ X ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6359_ex__of__int__less,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29      ? [Z4: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z4 ) @ X ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6360_sum__gp__multiplied,axiom,
% 9.49/5.29      ! [M: nat,N: nat,X: complex] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29       => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 9.49/5.29          = ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6361_sum__gp__multiplied,axiom,
% 9.49/5.29      ! [M: nat,N: nat,X: rat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29       => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 9.49/5.29          = ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6362_sum__gp__multiplied,axiom,
% 9.49/5.29      ! [M: nat,N: nat,X: int] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29       => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 9.49/5.29          = ( minus_minus_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6363_sum__gp__multiplied,axiom,
% 9.49/5.29      ! [M: nat,N: nat,X: real] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29       => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 9.49/5.29          = ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6364_of__int__round__le,axiom,
% 9.49/5.29      ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6365_of__int__round__le,axiom,
% 9.49/5.29      ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6366_of__int__round__ge,axiom,
% 9.49/5.29      ! [X: real] : ( ord_less_eq_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6367_of__int__round__ge,axiom,
% 9.49/5.29      ! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6368_of__int__round__gt,axiom,
% 9.49/5.29      ! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6369_of__int__round__gt,axiom,
% 9.49/5.29      ! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6370_xor__nat__unfold,axiom,
% 9.49/5.29      ( bit_se6528837805403552850or_nat
% 9.49/5.29      = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M2 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6371_mask__eq__sum__exp__nat,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
% 9.49/5.29        = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.49/5.29          @ ( collect_nat
% 9.49/5.29            @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6372_gauss__sum__nat,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( groups3542108847815614940at_nat
% 9.49/5.29          @ ^ [X3: nat] : X3
% 9.49/5.29          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.49/5.29        = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6373_sum__gp,axiom,
% 9.49/5.29      ! [N: nat,M: nat,X: complex] :
% 9.49/5.29        ( ( ( ord_less_nat @ N @ M )
% 9.49/5.29         => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29            = zero_zero_complex ) )
% 9.49/5.29        & ( ~ ( ord_less_nat @ N @ M )
% 9.49/5.29         => ( ( ( X = one_one_complex )
% 9.49/5.29             => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29                = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 9.49/5.29            & ( ( X != one_one_complex )
% 9.49/5.29             => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29                = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6374_sum__gp,axiom,
% 9.49/5.29      ! [N: nat,M: nat,X: rat] :
% 9.49/5.29        ( ( ( ord_less_nat @ N @ M )
% 9.49/5.29         => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29            = zero_zero_rat ) )
% 9.49/5.29        & ( ~ ( ord_less_nat @ N @ M )
% 9.49/5.29         => ( ( ( X = one_one_rat )
% 9.49/5.29             => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29                = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 9.49/5.29            & ( ( X != one_one_rat )
% 9.49/5.29             => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29                = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6375_sum__gp,axiom,
% 9.49/5.29      ! [N: nat,M: nat,X: real] :
% 9.49/5.29        ( ( ( ord_less_nat @ N @ M )
% 9.49/5.29         => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29            = zero_zero_real ) )
% 9.49/5.29        & ( ~ ( ord_less_nat @ N @ M )
% 9.49/5.29         => ( ( ( X = one_one_real )
% 9.49/5.29             => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29                = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 9.49/5.29            & ( ( X != one_one_real )
% 9.49/5.29             => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29                = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6376_gauss__sum__from__Suc__0,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 9.49/5.29        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6377_gauss__sum__from__Suc__0,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 9.49/5.29        = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6378_gauss__sum__from__Suc__0,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 9.49/5.29        = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6379_double__gauss__sum__from__Suc__0,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 9.49/5.29        = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6380_double__gauss__sum__from__Suc__0,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 9.49/5.29        = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6381_double__gauss__sum__from__Suc__0,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 9.49/5.29        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6382_double__gauss__sum__from__Suc__0,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 9.49/5.29        = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6383_double__gauss__sum__from__Suc__0,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 9.49/5.29        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6384_double__gauss__sum__from__Suc__0,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 9.49/5.29        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6385_gauss__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.49/5.29        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6386_gauss__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.49/5.29        = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6387_gauss__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.49/5.29        = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6388_arith__series,axiom,
% 9.49/5.29      ! [A: code_integer,D: code_integer,N: nat] :
% 9.49/5.29        ( ( groups7501900531339628137nteger
% 9.49/5.29          @ ^ [I4: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I4 ) @ D ) )
% 9.49/5.29          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.49/5.29        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ D ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6389_arith__series,axiom,
% 9.49/5.29      ! [A: int,D: int,N: nat] :
% 9.49/5.29        ( ( groups3539618377306564664at_int
% 9.49/5.29          @ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
% 9.49/5.29          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.49/5.29        = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6390_arith__series,axiom,
% 9.49/5.29      ! [A: nat,D: nat,N: nat] :
% 9.49/5.29        ( ( groups3542108847815614940at_nat
% 9.49/5.29          @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
% 9.49/5.29          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.49/5.29        = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6391_double__gauss__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.49/5.29        = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6392_double__gauss__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.49/5.29        = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6393_double__gauss__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.49/5.29        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6394_double__gauss__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.49/5.29        = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6395_double__gauss__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.49/5.29        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6396_double__gauss__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.49/5.29        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6397_double__arith__series,axiom,
% 9.49/5.29      ! [A: complex,D: complex,N: nat] :
% 9.49/5.29        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 9.49/5.29          @ ( groups2073611262835488442omplex
% 9.49/5.29            @ ^ [I4: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I4 ) @ D ) )
% 9.49/5.29            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.49/5.29        = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ D ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6398_double__arith__series,axiom,
% 9.49/5.29      ! [A: extended_enat,D: extended_enat,N: nat] :
% 9.49/5.29        ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) )
% 9.49/5.29          @ ( groups7108830773950497114d_enat
% 9.49/5.29            @ ^ [I4: nat] : ( plus_p3455044024723400733d_enat @ A @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ I4 ) @ D ) )
% 9.49/5.29            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.49/5.29        = ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A ) @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ D ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6399_double__arith__series,axiom,
% 9.49/5.29      ! [A: rat,D: rat,N: nat] :
% 9.49/5.29        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 9.49/5.29          @ ( groups2906978787729119204at_rat
% 9.49/5.29            @ ^ [I4: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I4 ) @ D ) )
% 9.49/5.29            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.49/5.29        = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ D ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6400_double__arith__series,axiom,
% 9.49/5.29      ! [A: int,D: int,N: nat] :
% 9.49/5.29        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 9.49/5.29          @ ( groups3539618377306564664at_int
% 9.49/5.29            @ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
% 9.49/5.29            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.49/5.29        = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6401_double__arith__series,axiom,
% 9.49/5.29      ! [A: nat,D: nat,N: nat] :
% 9.49/5.29        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
% 9.49/5.29          @ ( groups3542108847815614940at_nat
% 9.49/5.29            @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
% 9.49/5.29            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.49/5.29        = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6402_double__arith__series,axiom,
% 9.49/5.29      ! [A: real,D: real,N: nat] :
% 9.49/5.29        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 9.49/5.29          @ ( groups6591440286371151544t_real
% 9.49/5.29            @ ^ [I4: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I4 ) @ D ) )
% 9.49/5.29            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.49/5.29        = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ D ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6403_of__nat__eq__iff,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ( semiri1314217659103216013at_int @ M )
% 9.49/5.29          = ( semiri1314217659103216013at_int @ N ) )
% 9.49/5.29        = ( M = N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6404_of__nat__eq__iff,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ( semiri5074537144036343181t_real @ M )
% 9.49/5.29          = ( semiri5074537144036343181t_real @ N ) )
% 9.49/5.29        = ( M = N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6405_negative__eq__positive,axiom,
% 9.49/5.29      ! [N: nat,M: nat] :
% 9.49/5.29        ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
% 9.49/5.29          = ( semiri1314217659103216013at_int @ M ) )
% 9.49/5.29        = ( ( N = zero_zero_nat )
% 9.49/5.29          & ( M = zero_zero_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6406_xor__negative__int__iff,axiom,
% 9.49/5.29      ! [K: int,L: int] :
% 9.49/5.29        ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
% 9.49/5.29        = ( ( ord_less_int @ K @ zero_zero_int )
% 9.49/5.29         != ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6407_of__nat__0,axiom,
% 9.49/5.29      ( ( semiri8010041392384452111omplex @ zero_zero_nat )
% 9.49/5.29      = zero_zero_complex ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6408_of__nat__0,axiom,
% 9.49/5.29      ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 9.49/5.29      = zero_zero_rat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6409_of__nat__0,axiom,
% 9.49/5.29      ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 9.49/5.29      = zero_zero_nat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6410_of__nat__0,axiom,
% 9.49/5.29      ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 9.49/5.29      = zero_zero_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6411_of__nat__0,axiom,
% 9.49/5.29      ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 9.49/5.29      = zero_zero_real ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6412_of__nat__0__eq__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( zero_zero_complex
% 9.49/5.29          = ( semiri8010041392384452111omplex @ N ) )
% 9.49/5.29        = ( zero_zero_nat = N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6413_of__nat__0__eq__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( zero_zero_rat
% 9.49/5.29          = ( semiri681578069525770553at_rat @ N ) )
% 9.49/5.29        = ( zero_zero_nat = N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6414_of__nat__0__eq__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( zero_zero_nat
% 9.49/5.29          = ( semiri1316708129612266289at_nat @ N ) )
% 9.49/5.29        = ( zero_zero_nat = N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6415_of__nat__0__eq__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( zero_zero_int
% 9.49/5.29          = ( semiri1314217659103216013at_int @ N ) )
% 9.49/5.29        = ( zero_zero_nat = N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6416_of__nat__0__eq__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( zero_zero_real
% 9.49/5.29          = ( semiri5074537144036343181t_real @ N ) )
% 9.49/5.29        = ( zero_zero_nat = N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6417_of__nat__eq__0__iff,axiom,
% 9.49/5.29      ! [M: nat] :
% 9.49/5.29        ( ( ( semiri8010041392384452111omplex @ M )
% 9.49/5.29          = zero_zero_complex )
% 9.49/5.29        = ( M = zero_zero_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6418_of__nat__eq__0__iff,axiom,
% 9.49/5.29      ! [M: nat] :
% 9.49/5.29        ( ( ( semiri681578069525770553at_rat @ M )
% 9.49/5.29          = zero_zero_rat )
% 9.49/5.29        = ( M = zero_zero_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6419_of__nat__eq__0__iff,axiom,
% 9.49/5.29      ! [M: nat] :
% 9.49/5.29        ( ( ( semiri1316708129612266289at_nat @ M )
% 9.49/5.29          = zero_zero_nat )
% 9.49/5.29        = ( M = zero_zero_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6420_of__nat__eq__0__iff,axiom,
% 9.49/5.29      ! [M: nat] :
% 9.49/5.29        ( ( ( semiri1314217659103216013at_int @ M )
% 9.49/5.29          = zero_zero_int )
% 9.49/5.29        = ( M = zero_zero_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6421_of__nat__eq__0__iff,axiom,
% 9.49/5.29      ! [M: nat] :
% 9.49/5.29        ( ( ( semiri5074537144036343181t_real @ M )
% 9.49/5.29          = zero_zero_real )
% 9.49/5.29        = ( M = zero_zero_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6422_of__nat__less__iff,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 9.49/5.29        = ( ord_less_nat @ M @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6423_of__nat__less__iff,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 9.49/5.29        = ( ord_less_nat @ M @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6424_of__nat__less__iff,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.49/5.29        = ( ord_less_nat @ M @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6425_of__nat__less__iff,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 9.49/5.29        = ( ord_less_nat @ M @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6426_of__nat__le__iff,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 9.49/5.29        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6427_of__nat__le__iff,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 9.49/5.29        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6428_of__nat__le__iff,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 9.49/5.29        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6429_of__nat__le__iff,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.49/5.29        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6430_numeral__less__real__of__nat__iff,axiom,
% 9.49/5.29      ! [W2: num,N: nat] :
% 9.49/5.29        ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( semiri5074537144036343181t_real @ N ) )
% 9.49/5.29        = ( ord_less_nat @ ( numeral_numeral_nat @ W2 ) @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6431_real__of__nat__less__numeral__iff,axiom,
% 9.49/5.29      ! [N: nat,W2: num] :
% 9.49/5.29        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W2 ) )
% 9.49/5.29        = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6432_of__nat__add,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
% 9.49/5.29        = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6433_of__nat__add,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
% 9.49/5.29        = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6434_of__nat__add,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
% 9.49/5.29        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6435_of__nat__add,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
% 9.49/5.29        = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6436_numeral__le__real__of__nat__iff,axiom,
% 9.49/5.29      ! [N: num,M: nat] :
% 9.49/5.29        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
% 9.49/5.29        = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6437_of__nat__mult,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
% 9.49/5.29        = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6438_of__nat__mult,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
% 9.49/5.29        = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6439_of__nat__mult,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
% 9.49/5.29        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6440_of__nat__mult,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
% 9.49/5.29        = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6441_of__nat__mult,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
% 9.49/5.29        = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6442_of__nat__eq__1__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ( semiri8010041392384452111omplex @ N )
% 9.49/5.29          = one_one_complex )
% 9.49/5.29        = ( N = one_one_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6443_of__nat__eq__1__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ( semiri681578069525770553at_rat @ N )
% 9.49/5.29          = one_one_rat )
% 9.49/5.29        = ( N = one_one_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6444_of__nat__eq__1__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ( semiri1316708129612266289at_nat @ N )
% 9.49/5.29          = one_one_nat )
% 9.49/5.29        = ( N = one_one_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6445_of__nat__eq__1__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ( semiri1314217659103216013at_int @ N )
% 9.49/5.29          = one_one_int )
% 9.49/5.29        = ( N = one_one_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6446_of__nat__eq__1__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ( semiri5074537144036343181t_real @ N )
% 9.49/5.29          = one_one_real )
% 9.49/5.29        = ( N = one_one_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6447_of__nat__1__eq__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( one_one_complex
% 9.49/5.29          = ( semiri8010041392384452111omplex @ N ) )
% 9.49/5.29        = ( N = one_one_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6448_of__nat__1__eq__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( one_one_rat
% 9.49/5.29          = ( semiri681578069525770553at_rat @ N ) )
% 9.49/5.29        = ( N = one_one_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6449_of__nat__1__eq__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( one_one_nat
% 9.49/5.29          = ( semiri1316708129612266289at_nat @ N ) )
% 9.49/5.29        = ( N = one_one_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6450_of__nat__1__eq__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( one_one_int
% 9.49/5.29          = ( semiri1314217659103216013at_int @ N ) )
% 9.49/5.29        = ( N = one_one_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6451_of__nat__1__eq__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( one_one_real
% 9.49/5.29          = ( semiri5074537144036343181t_real @ N ) )
% 9.49/5.29        = ( N = one_one_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6452_of__nat__1,axiom,
% 9.49/5.29      ( ( semiri8010041392384452111omplex @ one_one_nat )
% 9.49/5.29      = one_one_complex ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6453_of__nat__1,axiom,
% 9.49/5.29      ( ( semiri681578069525770553at_rat @ one_one_nat )
% 9.49/5.29      = one_one_rat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6454_of__nat__1,axiom,
% 9.49/5.29      ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 9.49/5.29      = one_one_nat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6455_of__nat__1,axiom,
% 9.49/5.29      ( ( semiri1314217659103216013at_int @ one_one_nat )
% 9.49/5.29      = one_one_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6456_of__nat__1,axiom,
% 9.49/5.29      ( ( semiri5074537144036343181t_real @ one_one_nat )
% 9.49/5.29      = one_one_real ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6457_negative__zless,axiom,
% 9.49/5.29      ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6458_of__nat__of__bool,axiom,
% 9.49/5.29      ! [P: $o] :
% 9.49/5.29        ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
% 9.49/5.29        = ( zero_n3304061248610475627l_real @ P ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6459_of__nat__of__bool,axiom,
% 9.49/5.29      ! [P: $o] :
% 9.49/5.29        ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 9.49/5.29        = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6460_of__nat__of__bool,axiom,
% 9.49/5.29      ! [P: $o] :
% 9.49/5.29        ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
% 9.49/5.29        = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6461_of__nat__le__0__iff,axiom,
% 9.49/5.29      ! [M: nat] :
% 9.49/5.29        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 9.49/5.29        = ( M = zero_zero_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6462_of__nat__le__0__iff,axiom,
% 9.49/5.29      ! [M: nat] :
% 9.49/5.29        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 9.49/5.29        = ( M = zero_zero_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6463_of__nat__le__0__iff,axiom,
% 9.49/5.29      ! [M: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 9.49/5.29        = ( M = zero_zero_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6464_of__nat__le__0__iff,axiom,
% 9.49/5.29      ! [M: nat] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 9.49/5.29        = ( M = zero_zero_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6465_of__nat__Suc,axiom,
% 9.49/5.29      ! [M: nat] :
% 9.49/5.29        ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 9.49/5.29        = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6466_of__nat__Suc,axiom,
% 9.49/5.29      ! [M: nat] :
% 9.49/5.29        ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 9.49/5.29        = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6467_of__nat__Suc,axiom,
% 9.49/5.29      ! [M: nat] :
% 9.49/5.29        ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 9.49/5.29        = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6468_of__nat__Suc,axiom,
% 9.49/5.29      ! [M: nat] :
% 9.49/5.29        ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 9.49/5.29        = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6469_of__nat__Suc,axiom,
% 9.49/5.29      ! [M: nat] :
% 9.49/5.29        ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 9.49/5.29        = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6470_of__nat__0__less__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
% 9.49/5.29        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6471_of__nat__0__less__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
% 9.49/5.29        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6472_of__nat__0__less__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
% 9.49/5.29        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6473_of__nat__0__less__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
% 9.49/5.29        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6474_of__nat__less__of__nat__power__cancel__iff,axiom,
% 9.49/5.29      ! [B: nat,W2: nat,X: nat] :
% 9.49/5.29        ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X ) )
% 9.49/5.29        = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6475_of__nat__less__of__nat__power__cancel__iff,axiom,
% 9.49/5.29      ! [B: nat,W2: nat,X: nat] :
% 9.49/5.29        ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
% 9.49/5.29        = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6476_of__nat__less__of__nat__power__cancel__iff,axiom,
% 9.49/5.29      ! [B: nat,W2: nat,X: nat] :
% 9.49/5.29        ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
% 9.49/5.29        = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6477_of__nat__less__of__nat__power__cancel__iff,axiom,
% 9.49/5.29      ! [B: nat,W2: nat,X: nat] :
% 9.49/5.29        ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
% 9.49/5.29        = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6478_of__nat__power__less__of__nat__cancel__iff,axiom,
% 9.49/5.29      ! [X: nat,B: nat,W2: nat] :
% 9.49/5.29        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
% 9.49/5.29        = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6479_of__nat__power__less__of__nat__cancel__iff,axiom,
% 9.49/5.29      ! [X: nat,B: nat,W2: nat] :
% 9.49/5.29        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
% 9.49/5.29        = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6480_of__nat__power__less__of__nat__cancel__iff,axiom,
% 9.49/5.29      ! [X: nat,B: nat,W2: nat] :
% 9.49/5.29        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
% 9.49/5.29        = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6481_of__nat__power__less__of__nat__cancel__iff,axiom,
% 9.49/5.29      ! [X: nat,B: nat,W2: nat] :
% 9.49/5.29        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
% 9.49/5.29        = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6482_of__nat__le__of__nat__power__cancel__iff,axiom,
% 9.49/5.29      ! [B: nat,W2: nat,X: nat] :
% 9.49/5.29        ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
% 9.49/5.29        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6483_of__nat__le__of__nat__power__cancel__iff,axiom,
% 9.49/5.29      ! [B: nat,W2: nat,X: nat] :
% 9.49/5.29        ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X ) )
% 9.49/5.29        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6484_of__nat__le__of__nat__power__cancel__iff,axiom,
% 9.49/5.29      ! [B: nat,W2: nat,X: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
% 9.49/5.29        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6485_of__nat__le__of__nat__power__cancel__iff,axiom,
% 9.49/5.29      ! [B: nat,W2: nat,X: nat] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
% 9.49/5.29        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6486_of__nat__power__le__of__nat__cancel__iff,axiom,
% 9.49/5.29      ! [X: nat,B: nat,W2: nat] :
% 9.49/5.29        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
% 9.49/5.29        = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6487_of__nat__power__le__of__nat__cancel__iff,axiom,
% 9.49/5.29      ! [X: nat,B: nat,W2: nat] :
% 9.49/5.29        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
% 9.49/5.29        = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6488_of__nat__power__le__of__nat__cancel__iff,axiom,
% 9.49/5.29      ! [X: nat,B: nat,W2: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
% 9.49/5.29        = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6489_of__nat__power__le__of__nat__cancel__iff,axiom,
% 9.49/5.29      ! [X: nat,B: nat,W2: nat] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
% 9.49/5.29        = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6490_of__nat__zero__less__power__iff,axiom,
% 9.49/5.29      ! [X: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
% 9.49/5.29        = ( ( ord_less_nat @ zero_zero_nat @ X )
% 9.49/5.29          | ( N = zero_zero_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6491_of__nat__zero__less__power__iff,axiom,
% 9.49/5.29      ! [X: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
% 9.49/5.29        = ( ( ord_less_nat @ zero_zero_nat @ X )
% 9.49/5.29          | ( N = zero_zero_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6492_of__nat__zero__less__power__iff,axiom,
% 9.49/5.29      ! [X: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
% 9.49/5.29        = ( ( ord_less_nat @ zero_zero_nat @ X )
% 9.49/5.29          | ( N = zero_zero_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6493_of__nat__zero__less__power__iff,axiom,
% 9.49/5.29      ! [X: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
% 9.49/5.29        = ( ( ord_less_nat @ zero_zero_nat @ X )
% 9.49/5.29          | ( N = zero_zero_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6494_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.49/5.29      ! [X: nat,I: num,N: nat] :
% 9.49/5.29        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 9.49/5.29        = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6495_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.49/5.29      ! [X: nat,I: num,N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 9.49/5.29        = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6496_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.49/5.29      ! [X: nat,I: num,N: nat] :
% 9.49/5.29        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 9.49/5.29        = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6497_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.49/5.29      ! [X: nat,I: num,N: nat] :
% 9.49/5.29        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 9.49/5.29        = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6498_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.49/5.29      ! [I: num,N: nat,X: nat] :
% 9.49/5.29        ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
% 9.49/5.29        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6499_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.49/5.29      ! [I: num,N: nat,X: nat] :
% 9.49/5.29        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
% 9.49/5.29        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6500_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.49/5.29      ! [I: num,N: nat,X: nat] :
% 9.49/5.29        ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
% 9.49/5.29        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6501_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.49/5.29      ! [I: num,N: nat,X: nat] :
% 9.49/5.29        ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
% 9.49/5.29        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6502_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.49/5.29      ! [I: num,N: nat,X: nat] :
% 9.49/5.29        ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
% 9.49/5.29        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6503_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.49/5.29      ! [I: num,N: nat,X: nat] :
% 9.49/5.29        ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
% 9.49/5.29        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6504_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.49/5.29      ! [I: num,N: nat,X: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
% 9.49/5.29        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6505_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.49/5.29      ! [I: num,N: nat,X: nat] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
% 9.49/5.29        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6506_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.49/5.29      ! [X: nat,I: num,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 9.49/5.29        = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6507_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.49/5.29      ! [X: nat,I: num,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 9.49/5.29        = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6508_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.49/5.29      ! [X: nat,I: num,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 9.49/5.29        = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6509_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.49/5.29      ! [X: nat,I: num,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 9.49/5.29        = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6510_real__arch__simple,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29      ? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6511_real__arch__simple,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29      ? [N3: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6512_reals__Archimedean2,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29      ? [N3: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6513_reals__Archimedean2,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29      ? [N3: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6514_mult__of__nat__commute,axiom,
% 9.49/5.29      ! [X: nat,Y: complex] :
% 9.49/5.29        ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y )
% 9.49/5.29        = ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6515_mult__of__nat__commute,axiom,
% 9.49/5.29      ! [X: nat,Y: rat] :
% 9.49/5.29        ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
% 9.49/5.29        = ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6516_mult__of__nat__commute,axiom,
% 9.49/5.29      ! [X: nat,Y: nat] :
% 9.49/5.29        ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
% 9.49/5.29        = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6517_mult__of__nat__commute,axiom,
% 9.49/5.29      ! [X: nat,Y: int] :
% 9.49/5.29        ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
% 9.49/5.29        = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6518_mult__of__nat__commute,axiom,
% 9.49/5.29      ! [X: nat,Y: real] :
% 9.49/5.29        ( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
% 9.49/5.29        = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6519_of__nat__less__of__int__iff,axiom,
% 9.49/5.29      ! [N: nat,X: int] :
% 9.49/5.29        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X ) )
% 9.49/5.29        = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6520_of__nat__less__of__int__iff,axiom,
% 9.49/5.29      ! [N: nat,X: int] :
% 9.49/5.29        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
% 9.49/5.29        = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6521_of__nat__less__of__int__iff,axiom,
% 9.49/5.29      ! [N: nat,X: int] :
% 9.49/5.29        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
% 9.49/5.29        = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6522_of__nat__0__le__iff,axiom,
% 9.49/5.29      ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6523_of__nat__0__le__iff,axiom,
% 9.49/5.29      ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6524_of__nat__0__le__iff,axiom,
% 9.49/5.29      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6525_of__nat__0__le__iff,axiom,
% 9.49/5.29      ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6526_of__nat__less__0__iff,axiom,
% 9.49/5.29      ! [M: nat] :
% 9.49/5.29        ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6527_of__nat__less__0__iff,axiom,
% 9.49/5.29      ! [M: nat] :
% 9.49/5.29        ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6528_of__nat__less__0__iff,axiom,
% 9.49/5.29      ! [M: nat] :
% 9.49/5.29        ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6529_of__nat__less__0__iff,axiom,
% 9.49/5.29      ! [M: nat] :
% 9.49/5.29        ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6530_of__nat__neq__0,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
% 9.49/5.29       != zero_zero_complex ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6531_of__nat__neq__0,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
% 9.49/5.29       != zero_zero_rat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6532_of__nat__neq__0,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
% 9.49/5.29       != zero_zero_nat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6533_of__nat__neq__0,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 9.49/5.29       != zero_zero_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6534_of__nat__neq__0,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
% 9.49/5.29       != zero_zero_real ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6535_less__imp__of__nat__less,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ M @ N )
% 9.49/5.29       => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6536_less__imp__of__nat__less,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ M @ N )
% 9.49/5.29       => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6537_less__imp__of__nat__less,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ M @ N )
% 9.49/5.29       => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6538_less__imp__of__nat__less,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ M @ N )
% 9.49/5.29       => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6539_of__nat__less__imp__less,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 9.49/5.29       => ( ord_less_nat @ M @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6540_of__nat__less__imp__less,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 9.49/5.29       => ( ord_less_nat @ M @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6541_of__nat__less__imp__less,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.49/5.29       => ( ord_less_nat @ M @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6542_of__nat__less__imp__less,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 9.49/5.29       => ( ord_less_nat @ M @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6543_of__nat__mono,axiom,
% 9.49/5.29      ! [I: nat,J: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.29       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6544_of__nat__mono,axiom,
% 9.49/5.29      ! [I: nat,J: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.29       => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6545_of__nat__mono,axiom,
% 9.49/5.29      ! [I: nat,J: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.29       => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6546_of__nat__mono,axiom,
% 9.49/5.29      ! [I: nat,J: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ I @ J )
% 9.49/5.29       => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6547_int__ops_I1_J,axiom,
% 9.49/5.29      ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 9.49/5.29      = zero_zero_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6548_nat__int__comparison_I2_J,axiom,
% 9.49/5.29      ( ord_less_nat
% 9.49/5.29      = ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6549_nat__int__comparison_I3_J,axiom,
% 9.49/5.29      ( ord_less_eq_nat
% 9.49/5.29      = ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6550_zle__int,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.49/5.29        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6551_not__int__zless__negative,axiom,
% 9.49/5.29      ! [N: nat,M: nat] :
% 9.49/5.29        ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6552_of__nat__max,axiom,
% 9.49/5.29      ! [X: nat,Y: nat] :
% 9.49/5.29        ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y ) )
% 9.49/5.29        = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6553_of__nat__max,axiom,
% 9.49/5.29      ! [X: nat,Y: nat] :
% 9.49/5.29        ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y ) )
% 9.49/5.29        = ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6554_of__nat__max,axiom,
% 9.49/5.29      ! [X: nat,Y: nat] :
% 9.49/5.29        ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y ) )
% 9.49/5.29        = ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6555_nat__less__as__int,axiom,
% 9.49/5.29      ( ord_less_nat
% 9.49/5.29      = ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6556_nat__leq__as__int,axiom,
% 9.49/5.29      ( ord_less_eq_nat
% 9.49/5.29      = ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6557_ex__less__of__nat__mult,axiom,
% 9.49/5.29      ! [X: rat,Y: rat] :
% 9.49/5.29        ( ( ord_less_rat @ zero_zero_rat @ X )
% 9.49/5.29       => ? [N3: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6558_ex__less__of__nat__mult,axiom,
% 9.49/5.29      ! [X: real,Y: real] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.29       => ? [N3: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6559_of__nat__diff,axiom,
% 9.49/5.29      ! [N: nat,M: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.29       => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
% 9.49/5.29          = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6560_of__nat__diff,axiom,
% 9.49/5.29      ! [N: nat,M: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.29       => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
% 9.49/5.29          = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6561_of__nat__diff,axiom,
% 9.49/5.29      ! [N: nat,M: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.29       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
% 9.49/5.29          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6562_of__nat__diff,axiom,
% 9.49/5.29      ! [N: nat,M: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.29       => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
% 9.49/5.29          = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6563_reals__Archimedean3,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.29       => ! [Y4: real] :
% 9.49/5.29          ? [N3: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6564_int__cases4,axiom,
% 9.49/5.29      ! [M: int] :
% 9.49/5.29        ( ! [N3: nat] :
% 9.49/5.29            ( M
% 9.49/5.29           != ( semiri1314217659103216013at_int @ N3 ) )
% 9.49/5.29       => ~ ! [N3: nat] :
% 9.49/5.29              ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.49/5.29             => ( M
% 9.49/5.29               != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6565_zless__iff__Suc__zadd,axiom,
% 9.49/5.29      ( ord_less_int
% 9.49/5.29      = ( ^ [W3: int,Z3: int] :
% 9.49/5.29          ? [N2: nat] :
% 9.49/5.29            ( Z3
% 9.49/5.29            = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6566_int__zle__neg,axiom,
% 9.49/5.29      ! [N: nat,M: nat] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
% 9.49/5.29        = ( ( N = zero_zero_nat )
% 9.49/5.29          & ( M = zero_zero_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6567_sum__roots__unity,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ one_one_nat @ N )
% 9.49/5.29       => ( ( groups7754918857620584856omplex
% 9.49/5.29            @ ^ [X3: complex] : X3
% 9.49/5.29            @ ( collect_complex
% 9.49/5.29              @ ^ [Z3: complex] :
% 9.49/5.29                  ( ( power_power_complex @ Z3 @ N )
% 9.49/5.29                  = one_one_complex ) ) )
% 9.49/5.29          = zero_zero_complex ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6568_sum__nth__roots,axiom,
% 9.49/5.29      ! [N: nat,C: complex] :
% 9.49/5.29        ( ( ord_less_nat @ one_one_nat @ N )
% 9.49/5.29       => ( ( groups7754918857620584856omplex
% 9.49/5.29            @ ^ [X3: complex] : X3
% 9.49/5.29            @ ( collect_complex
% 9.49/5.29              @ ^ [Z3: complex] :
% 9.49/5.29                  ( ( power_power_complex @ Z3 @ N )
% 9.49/5.29                  = C ) ) )
% 9.49/5.29          = zero_zero_complex ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6569_pos__int__cases,axiom,
% 9.49/5.29      ! [K: int] :
% 9.49/5.29        ( ( ord_less_int @ zero_zero_int @ K )
% 9.49/5.29       => ~ ! [N3: nat] :
% 9.49/5.29              ( ( K
% 9.49/5.29                = ( semiri1314217659103216013at_int @ N3 ) )
% 9.49/5.29             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6570_zero__less__imp__eq__int,axiom,
% 9.49/5.29      ! [K: int] :
% 9.49/5.29        ( ( ord_less_int @ zero_zero_int @ K )
% 9.49/5.29       => ? [N3: nat] :
% 9.49/5.29            ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.49/5.29            & ( K
% 9.49/5.29              = ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6571_int__cases3,axiom,
% 9.49/5.29      ! [K: int] :
% 9.49/5.29        ( ( K != zero_zero_int )
% 9.49/5.29       => ( ! [N3: nat] :
% 9.49/5.29              ( ( K
% 9.49/5.29                = ( semiri1314217659103216013at_int @ N3 ) )
% 9.49/5.29             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
% 9.49/5.29         => ~ ! [N3: nat] :
% 9.49/5.29                ( ( K
% 9.49/5.29                  = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
% 9.49/5.29               => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6572_nat__less__real__le,axiom,
% 9.49/5.29      ( ord_less_nat
% 9.49/5.29      = ( ^ [N2: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6573_nat__le__real__less,axiom,
% 9.49/5.29      ( ord_less_eq_nat
% 9.49/5.29      = ( ^ [N2: nat,M2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6574_zmult__zless__mono2__lemma,axiom,
% 9.49/5.29      ! [I: int,J: int,K: nat] :
% 9.49/5.29        ( ( ord_less_int @ I @ J )
% 9.49/5.29       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.29         => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6575_negD,axiom,
% 9.49/5.29      ! [X: int] :
% 9.49/5.29        ( ( ord_less_int @ X @ zero_zero_int )
% 9.49/5.29       => ? [N3: nat] :
% 9.49/5.29            ( X
% 9.49/5.29            = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6576_negative__zless__0,axiom,
% 9.49/5.29      ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6577_int__ops_I6_J,axiom,
% 9.49/5.29      ! [A: nat,B: nat] :
% 9.49/5.29        ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 9.49/5.29         => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 9.49/5.29            = zero_zero_int ) )
% 9.49/5.29        & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 9.49/5.29         => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 9.49/5.29            = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6578_simp__from__to,axiom,
% 9.49/5.29      ( set_or1266510415728281911st_int
% 9.49/5.29      = ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6579_nat__approx__posE,axiom,
% 9.49/5.29      ! [E2: rat] :
% 9.49/5.29        ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 9.49/5.29       => ~ ! [N3: nat] :
% 9.49/5.29              ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ E2 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6580_nat__approx__posE,axiom,
% 9.49/5.29      ! [E2: real] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.49/5.29       => ~ ! [N3: nat] :
% 9.49/5.29              ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ E2 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6581_of__nat__less__two__power,axiom,
% 9.49/5.29      ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6582_of__nat__less__two__power,axiom,
% 9.49/5.29      ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6583_of__nat__less__two__power,axiom,
% 9.49/5.29      ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6584_inverse__of__nat__le,axiom,
% 9.49/5.29      ! [N: nat,M: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.29       => ( ( N != zero_zero_nat )
% 9.49/5.29         => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6585_inverse__of__nat__le,axiom,
% 9.49/5.29      ! [N: nat,M: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ N @ M )
% 9.49/5.29       => ( ( N != zero_zero_nat )
% 9.49/5.29         => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6586_real__archimedian__rdiv__eq__0,axiom,
% 9.49/5.29      ! [X: real,C: real] :
% 9.49/5.29        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.49/5.29       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.49/5.29         => ( ! [M5: nat] :
% 9.49/5.29                ( ( ord_less_nat @ zero_zero_nat @ M5 )
% 9.49/5.29               => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ X ) @ C ) )
% 9.49/5.29           => ( X = zero_zero_real ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6587_neg__int__cases,axiom,
% 9.49/5.29      ! [K: int] :
% 9.49/5.29        ( ( ord_less_int @ K @ zero_zero_int )
% 9.49/5.29       => ~ ! [N3: nat] :
% 9.49/5.29              ( ( K
% 9.49/5.29                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
% 9.49/5.29             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6588_zdiff__int__split,axiom,
% 9.49/5.29      ! [P: int > $o,X: nat,Y: nat] :
% 9.49/5.29        ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
% 9.49/5.29        = ( ( ( ord_less_eq_nat @ Y @ X )
% 9.49/5.29           => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
% 9.49/5.29          & ( ( ord_less_nat @ X @ Y )
% 9.49/5.29           => ( P @ zero_zero_int ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6589_XOR__upper,axiom,
% 9.49/5.29      ! [X: int,N: nat,Y: int] :
% 9.49/5.29        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.49/5.29       => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.29         => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.29           => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6590_of__nat__code__if,axiom,
% 9.49/5.29      ( semiri8010041392384452111omplex
% 9.49/5.29      = ( ^ [N2: nat] :
% 9.49/5.29            ( if_complex @ ( N2 = zero_zero_nat ) @ zero_zero_complex
% 9.49/5.29            @ ( produc1917071388513777916omplex
% 9.49/5.29              @ ^ [M2: nat,Q5: nat] : ( if_complex @ ( Q5 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M2 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M2 ) ) @ one_one_complex ) )
% 9.49/5.29              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6591_of__nat__code__if,axiom,
% 9.49/5.29      ( semiri4216267220026989637d_enat
% 9.49/5.29      = ( ^ [N2: nat] :
% 9.49/5.29            ( if_Extended_enat @ ( N2 = zero_zero_nat ) @ zero_z5237406670263579293d_enat
% 9.49/5.29            @ ( produc2676513652042109336d_enat
% 9.49/5.29              @ ^ [M2: nat,Q5: nat] : ( if_Extended_enat @ ( Q5 = zero_zero_nat ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( semiri4216267220026989637d_enat @ M2 ) ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( semiri4216267220026989637d_enat @ M2 ) ) @ one_on7984719198319812577d_enat ) )
% 9.49/5.29              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6592_of__nat__code__if,axiom,
% 9.49/5.29      ( semiri681578069525770553at_rat
% 9.49/5.29      = ( ^ [N2: nat] :
% 9.49/5.29            ( if_rat @ ( N2 = zero_zero_nat ) @ zero_zero_rat
% 9.49/5.29            @ ( produc6207742614233964070at_rat
% 9.49/5.29              @ ^ [M2: nat,Q5: nat] : ( if_rat @ ( Q5 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M2 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M2 ) ) @ one_one_rat ) )
% 9.49/5.29              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6593_of__nat__code__if,axiom,
% 9.49/5.29      ( semiri1316708129612266289at_nat
% 9.49/5.29      = ( ^ [N2: nat] :
% 9.49/5.29            ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
% 9.49/5.29            @ ( produc6842872674320459806at_nat
% 9.49/5.29              @ ^ [M2: nat,Q5: nat] : ( if_nat @ ( Q5 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M2 ) ) @ one_one_nat ) )
% 9.49/5.29              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6594_of__nat__code__if,axiom,
% 9.49/5.29      ( semiri1314217659103216013at_int
% 9.49/5.29      = ( ^ [N2: nat] :
% 9.49/5.29            ( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int
% 9.49/5.29            @ ( produc6840382203811409530at_int
% 9.49/5.29              @ ^ [M2: nat,Q5: nat] : ( if_int @ ( Q5 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ one_one_int ) )
% 9.49/5.29              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6595_of__nat__code__if,axiom,
% 9.49/5.29      ( semiri5074537144036343181t_real
% 9.49/5.29      = ( ^ [N2: nat] :
% 9.49/5.29            ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
% 9.49/5.29            @ ( produc1703576794950452218t_real
% 9.49/5.29              @ ^ [M2: nat,Q5: nat] : ( if_real @ ( Q5 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ one_one_real ) )
% 9.49/5.29              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6596_lemma__termdiff3,axiom,
% 9.49/5.29      ! [H2: real,Z2: real,K4: real,N: nat] :
% 9.49/5.29        ( ( H2 != zero_zero_real )
% 9.49/5.29       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ K4 )
% 9.49/5.29         => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z2 @ H2 ) ) @ K4 )
% 9.49/5.29           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ N ) @ ( power_power_real @ Z2 @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K4 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6597_lemma__termdiff3,axiom,
% 9.49/5.29      ! [H2: complex,Z2: complex,K4: real,N: nat] :
% 9.49/5.29        ( ( H2 != zero_zero_complex )
% 9.49/5.29       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ K4 )
% 9.49/5.29         => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z2 @ H2 ) ) @ K4 )
% 9.49/5.29           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ N ) @ ( power_power_complex @ Z2 @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K4 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6598_lemma__termdiff2,axiom,
% 9.49/5.29      ! [H2: complex,Z2: complex,N: nat] :
% 9.49/5.29        ( ( H2 != zero_zero_complex )
% 9.49/5.29       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ N ) @ ( power_power_complex @ Z2 @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.49/5.29          = ( times_times_complex @ H2
% 9.49/5.29            @ ( groups2073611262835488442omplex
% 9.49/5.29              @ ^ [P6: nat] :
% 9.49/5.29                  ( groups2073611262835488442omplex
% 9.49/5.29                  @ ^ [Q5: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ Q5 ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
% 9.49/5.29                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
% 9.49/5.29              @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6599_lemma__termdiff2,axiom,
% 9.49/5.29      ! [H2: rat,Z2: rat,N: nat] :
% 9.49/5.29        ( ( H2 != zero_zero_rat )
% 9.49/5.29       => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ N ) @ ( power_power_rat @ Z2 @ N ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.49/5.29          = ( times_times_rat @ H2
% 9.49/5.29            @ ( groups2906978787729119204at_rat
% 9.49/5.29              @ ^ [P6: nat] :
% 9.49/5.29                  ( groups2906978787729119204at_rat
% 9.49/5.29                  @ ^ [Q5: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ Q5 ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
% 9.49/5.29                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
% 9.49/5.29              @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6600_lemma__termdiff2,axiom,
% 9.49/5.29      ! [H2: real,Z2: real,N: nat] :
% 9.49/5.29        ( ( H2 != zero_zero_real )
% 9.49/5.29       => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ N ) @ ( power_power_real @ Z2 @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.49/5.29          = ( times_times_real @ H2
% 9.49/5.29            @ ( groups6591440286371151544t_real
% 9.49/5.29              @ ^ [P6: nat] :
% 9.49/5.29                  ( groups6591440286371151544t_real
% 9.49/5.29                  @ ^ [Q5: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ Q5 ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
% 9.49/5.29                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
% 9.49/5.29              @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6601_of__nat__code,axiom,
% 9.49/5.29      ( semiri8010041392384452111omplex
% 9.49/5.29      = ( ^ [N2: nat] :
% 9.49/5.29            ( semiri2816024913162550771omplex
% 9.49/5.29            @ ^ [I4: complex] : ( plus_plus_complex @ I4 @ one_one_complex )
% 9.49/5.29            @ N2
% 9.49/5.29            @ zero_zero_complex ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6602_of__nat__code,axiom,
% 9.49/5.29      ( semiri681578069525770553at_rat
% 9.49/5.29      = ( ^ [N2: nat] :
% 9.49/5.29            ( semiri7787848453975740701ux_rat
% 9.49/5.29            @ ^ [I4: rat] : ( plus_plus_rat @ I4 @ one_one_rat )
% 9.49/5.29            @ N2
% 9.49/5.29            @ zero_zero_rat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6603_of__nat__code,axiom,
% 9.49/5.29      ( semiri1316708129612266289at_nat
% 9.49/5.29      = ( ^ [N2: nat] :
% 9.49/5.29            ( semiri8422978514062236437ux_nat
% 9.49/5.29            @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ one_one_nat )
% 9.49/5.29            @ N2
% 9.49/5.29            @ zero_zero_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6604_of__nat__code,axiom,
% 9.49/5.29      ( semiri1314217659103216013at_int
% 9.49/5.29      = ( ^ [N2: nat] :
% 9.49/5.29            ( semiri8420488043553186161ux_int
% 9.49/5.29            @ ^ [I4: int] : ( plus_plus_int @ I4 @ one_one_int )
% 9.49/5.29            @ N2
% 9.49/5.29            @ zero_zero_int ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6605_of__nat__code,axiom,
% 9.49/5.29      ( semiri5074537144036343181t_real
% 9.49/5.29      = ( ^ [N2: nat] :
% 9.49/5.29            ( semiri7260567687927622513x_real
% 9.49/5.29            @ ^ [I4: real] : ( plus_plus_real @ I4 @ one_one_real )
% 9.49/5.29            @ N2
% 9.49/5.29            @ zero_zero_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6606_and__int_Oelims,axiom,
% 9.49/5.29      ! [X: int,Xa2: int,Y: int] :
% 9.49/5.29        ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
% 9.49/5.29          = Y )
% 9.49/5.29       => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.49/5.29              & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.49/5.29           => ( Y
% 9.49/5.29              = ( uminus_uminus_int
% 9.49/5.29                @ ( zero_n2684676970156552555ol_int
% 9.49/5.29                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 9.49/5.29                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 9.49/5.29          & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.49/5.29                & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.49/5.29           => ( Y
% 9.49/5.29              = ( plus_plus_int
% 9.49/5.29                @ ( zero_n2684676970156552555ol_int
% 9.49/5.29                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 9.49/5.29                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 9.49/5.29                @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6607_and__int_Osimps,axiom,
% 9.49/5.29      ( bit_se725231765392027082nd_int
% 9.49/5.29      = ( ^ [K3: int,L2: int] :
% 9.49/5.29            ( if_int
% 9.49/5.29            @ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.49/5.29              & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.49/5.29            @ ( uminus_uminus_int
% 9.49/5.29              @ ( zero_n2684676970156552555ol_int
% 9.49/5.29                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 9.49/5.29                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 9.49/5.29            @ ( plus_plus_int
% 9.49/5.29              @ ( zero_n2684676970156552555ol_int
% 9.49/5.29                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 9.49/5.29                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 9.49/5.29              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6608_lessThan__iff,axiom,
% 9.49/5.29      ! [I: set_nat,K: set_nat] :
% 9.49/5.29        ( ( member_set_nat @ I @ ( set_or890127255671739683et_nat @ K ) )
% 9.49/5.29        = ( ord_less_set_nat @ I @ K ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6609_lessThan__iff,axiom,
% 9.49/5.29      ! [I: rat,K: rat] :
% 9.49/5.29        ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
% 9.49/5.29        = ( ord_less_rat @ I @ K ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6610_lessThan__iff,axiom,
% 9.49/5.29      ! [I: num,K: num] :
% 9.49/5.29        ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
% 9.49/5.29        = ( ord_less_num @ I @ K ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6611_lessThan__iff,axiom,
% 9.49/5.29      ! [I: nat,K: nat] :
% 9.49/5.29        ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
% 9.49/5.29        = ( ord_less_nat @ I @ K ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6612_lessThan__iff,axiom,
% 9.49/5.29      ! [I: int,K: int] :
% 9.49/5.29        ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
% 9.49/5.29        = ( ord_less_int @ I @ K ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6613_lessThan__iff,axiom,
% 9.49/5.29      ! [I: real,K: real] :
% 9.49/5.29        ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
% 9.49/5.29        = ( ord_less_real @ I @ K ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6614_bit_Oconj__zero__right,axiom,
% 9.49/5.29      ! [X: int] :
% 9.49/5.29        ( ( bit_se725231765392027082nd_int @ X @ zero_zero_int )
% 9.49/5.29        = zero_zero_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6615_bit_Oconj__zero__left,axiom,
% 9.49/5.29      ! [X: int] :
% 9.49/5.29        ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X )
% 9.49/5.29        = zero_zero_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6616_zero__and__eq,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
% 9.49/5.29        = zero_zero_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6617_zero__and__eq,axiom,
% 9.49/5.29      ! [A: nat] :
% 9.49/5.29        ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
% 9.49/5.29        = zero_zero_nat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6618_and__zero__eq,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
% 9.49/5.29        = zero_zero_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6619_and__zero__eq,axiom,
% 9.49/5.29      ! [A: nat] :
% 9.49/5.29        ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
% 9.49/5.29        = zero_zero_nat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6620_finite__lessThan,axiom,
% 9.49/5.29      ! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6621_lessThan__subset__iff,axiom,
% 9.49/5.29      ! [X: rat,Y: rat] :
% 9.49/5.29        ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X ) @ ( set_ord_lessThan_rat @ Y ) )
% 9.49/5.29        = ( ord_less_eq_rat @ X @ Y ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6622_lessThan__subset__iff,axiom,
% 9.49/5.29      ! [X: num,Y: num] :
% 9.49/5.29        ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y ) )
% 9.49/5.29        = ( ord_less_eq_num @ X @ Y ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6623_lessThan__subset__iff,axiom,
% 9.49/5.29      ! [X: nat,Y: nat] :
% 9.49/5.29        ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
% 9.49/5.29        = ( ord_less_eq_nat @ X @ Y ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6624_lessThan__subset__iff,axiom,
% 9.49/5.29      ! [X: int,Y: int] :
% 9.49/5.29        ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
% 9.49/5.29        = ( ord_less_eq_int @ X @ Y ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6625_lessThan__subset__iff,axiom,
% 9.49/5.29      ! [X: real,Y: real] :
% 9.49/5.29        ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y ) )
% 9.49/5.29        = ( ord_less_eq_real @ X @ Y ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6626_and__negative__int__iff,axiom,
% 9.49/5.29      ! [K: int,L: int] :
% 9.49/5.29        ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
% 9.49/5.29        = ( ( ord_less_int @ K @ zero_zero_int )
% 9.49/5.29          & ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6627_lessThan__0,axiom,
% 9.49/5.29      ( ( set_ord_lessThan_nat @ zero_zero_nat )
% 9.49/5.29      = bot_bot_set_nat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6628_single__Diff__lessThan,axiom,
% 9.49/5.29      ! [K: nat] :
% 9.49/5.29        ( ( minus_minus_set_nat @ ( insert_nat @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
% 9.49/5.29        = ( insert_nat @ K @ bot_bot_set_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6629_single__Diff__lessThan,axiom,
% 9.49/5.29      ! [K: int] :
% 9.49/5.29        ( ( minus_minus_set_int @ ( insert_int @ K @ bot_bot_set_int ) @ ( set_ord_lessThan_int @ K ) )
% 9.49/5.29        = ( insert_int @ K @ bot_bot_set_int ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6630_single__Diff__lessThan,axiom,
% 9.49/5.29      ! [K: real] :
% 9.49/5.29        ( ( minus_minus_set_real @ ( insert_real @ K @ bot_bot_set_real ) @ ( set_or5984915006950818249n_real @ K ) )
% 9.49/5.29        = ( insert_real @ K @ bot_bot_set_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6631_and__numerals_I1_J,axiom,
% 9.49/5.29      ! [Y: num] :
% 9.49/5.29        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
% 9.49/5.29        = zero_zero_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6632_and__numerals_I1_J,axiom,
% 9.49/5.29      ! [Y: num] :
% 9.49/5.29        ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 9.49/5.29        = zero_zero_nat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6633_and__numerals_I5_J,axiom,
% 9.49/5.29      ! [X: num] :
% 9.49/5.29        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
% 9.49/5.29        = zero_zero_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6634_and__numerals_I5_J,axiom,
% 9.49/5.29      ! [X: num] :
% 9.49/5.29        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
% 9.49/5.29        = zero_zero_nat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6635_lessThan__non__empty,axiom,
% 9.49/5.29      ! [X: int] :
% 9.49/5.29        ( ( set_ord_lessThan_int @ X )
% 9.49/5.29       != bot_bot_set_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6636_lessThan__non__empty,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( set_or5984915006950818249n_real @ X )
% 9.49/5.29       != bot_bot_set_real ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6637_infinite__Iio,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ~ ( finite_finite_int @ ( set_ord_lessThan_int @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6638_infinite__Iio,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ~ ( finite_finite_real @ ( set_or5984915006950818249n_real @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6639_lessThan__def,axiom,
% 9.49/5.29      ( set_or890127255671739683et_nat
% 9.49/5.29      = ( ^ [U2: set_nat] :
% 9.49/5.29            ( collect_set_nat
% 9.49/5.29            @ ^ [X3: set_nat] : ( ord_less_set_nat @ X3 @ U2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6640_lessThan__def,axiom,
% 9.49/5.29      ( set_ord_lessThan_rat
% 9.49/5.29      = ( ^ [U2: rat] :
% 9.49/5.29            ( collect_rat
% 9.49/5.29            @ ^ [X3: rat] : ( ord_less_rat @ X3 @ U2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6641_lessThan__def,axiom,
% 9.49/5.29      ( set_ord_lessThan_num
% 9.49/5.29      = ( ^ [U2: num] :
% 9.49/5.29            ( collect_num
% 9.49/5.29            @ ^ [X3: num] : ( ord_less_num @ X3 @ U2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6642_lessThan__def,axiom,
% 9.49/5.29      ( set_ord_lessThan_nat
% 9.49/5.29      = ( ^ [U2: nat] :
% 9.49/5.29            ( collect_nat
% 9.49/5.29            @ ^ [X3: nat] : ( ord_less_nat @ X3 @ U2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6643_lessThan__def,axiom,
% 9.49/5.29      ( set_ord_lessThan_int
% 9.49/5.29      = ( ^ [U2: int] :
% 9.49/5.29            ( collect_int
% 9.49/5.29            @ ^ [X3: int] : ( ord_less_int @ X3 @ U2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6644_lessThan__def,axiom,
% 9.49/5.29      ( set_or5984915006950818249n_real
% 9.49/5.29      = ( ^ [U2: real] :
% 9.49/5.29            ( collect_real
% 9.49/5.29            @ ^ [X3: real] : ( ord_less_real @ X3 @ U2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6645_Iio__eq__empty__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ( set_ord_lessThan_nat @ N )
% 9.49/5.29          = bot_bot_set_nat )
% 9.49/5.29        = ( N = bot_bot_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6646_lessThan__strict__subset__iff,axiom,
% 9.49/5.29      ! [M: rat,N: rat] :
% 9.49/5.29        ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N ) )
% 9.49/5.29        = ( ord_less_rat @ M @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6647_lessThan__strict__subset__iff,axiom,
% 9.49/5.29      ! [M: num,N: num] :
% 9.49/5.29        ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
% 9.49/5.29        = ( ord_less_num @ M @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6648_lessThan__strict__subset__iff,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
% 9.49/5.29        = ( ord_less_nat @ M @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6649_lessThan__strict__subset__iff,axiom,
% 9.49/5.29      ! [M: int,N: int] :
% 9.49/5.29        ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
% 9.49/5.29        = ( ord_less_int @ M @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6650_lessThan__strict__subset__iff,axiom,
% 9.49/5.29      ! [M: real,N: real] :
% 9.49/5.29        ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N ) )
% 9.49/5.29        = ( ord_less_real @ M @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6651_lessThan__empty__iff,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ( set_ord_lessThan_nat @ N )
% 9.49/5.29          = bot_bot_set_nat )
% 9.49/5.29        = ( N = zero_zero_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6652_finite__nat__bounded,axiom,
% 9.49/5.29      ! [S2: set_nat] :
% 9.49/5.29        ( ( finite_finite_nat @ S2 )
% 9.49/5.29       => ? [K2: nat] : ( ord_less_eq_set_nat @ S2 @ ( set_ord_lessThan_nat @ K2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6653_finite__nat__iff__bounded,axiom,
% 9.49/5.29      ( finite_finite_nat
% 9.49/5.29      = ( ^ [S7: set_nat] :
% 9.49/5.29          ? [K3: nat] : ( ord_less_eq_set_nat @ S7 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6654_AND__upper2_H_H,axiom,
% 9.49/5.29      ! [Y: int,Z2: int,X: int] :
% 9.49/5.29        ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.49/5.29       => ( ( ord_less_int @ Y @ Z2 )
% 9.49/5.29         => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6655_AND__upper1_H_H,axiom,
% 9.49/5.29      ! [Y: int,Z2: int,Ya: int] :
% 9.49/5.29        ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.49/5.29       => ( ( ord_less_int @ Y @ Z2 )
% 9.49/5.29         => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6656_and__less__eq,axiom,
% 9.49/5.29      ! [L: int,K: int] :
% 9.49/5.29        ( ( ord_less_int @ L @ zero_zero_int )
% 9.49/5.29       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6657_sum__diff__distrib,axiom,
% 9.49/5.29      ! [Q: int > nat,P: int > nat,N: int] :
% 9.49/5.29        ( ! [X4: int] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
% 9.49/5.29       => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N ) ) )
% 9.49/5.29          = ( groups4541462559716669496nt_nat
% 9.49/5.29            @ ^ [X3: int] : ( minus_minus_nat @ ( P @ X3 ) @ ( Q @ X3 ) )
% 9.49/5.29            @ ( set_ord_lessThan_int @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6658_sum__diff__distrib,axiom,
% 9.49/5.29      ! [Q: real > nat,P: real > nat,N: real] :
% 9.49/5.29        ( ! [X4: real] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
% 9.49/5.29       => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N ) ) )
% 9.49/5.29          = ( groups1935376822645274424al_nat
% 9.49/5.29            @ ^ [X3: real] : ( minus_minus_nat @ ( P @ X3 ) @ ( Q @ X3 ) )
% 9.49/5.29            @ ( set_or5984915006950818249n_real @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6659_sum__diff__distrib,axiom,
% 9.49/5.29      ! [Q: nat > nat,P: nat > nat,N: nat] :
% 9.49/5.29        ( ! [X4: nat] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
% 9.49/5.29       => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
% 9.49/5.29          = ( groups3542108847815614940at_nat
% 9.49/5.29            @ ^ [X3: nat] : ( minus_minus_nat @ ( P @ X3 ) @ ( Q @ X3 ) )
% 9.49/5.29            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6660_sum_OlessThan__Suc__shift,axiom,
% 9.49/5.29      ! [G: nat > rat,N: nat] :
% 9.49/5.29        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.49/5.29        = ( plus_plus_rat @ ( G @ zero_zero_nat )
% 9.49/5.29          @ ( groups2906978787729119204at_rat
% 9.49/5.29            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.49/5.29            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6661_sum_OlessThan__Suc__shift,axiom,
% 9.49/5.29      ! [G: nat > int,N: nat] :
% 9.49/5.29        ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.49/5.29        = ( plus_plus_int @ ( G @ zero_zero_nat )
% 9.49/5.29          @ ( groups3539618377306564664at_int
% 9.49/5.29            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.49/5.29            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6662_sum_OlessThan__Suc__shift,axiom,
% 9.49/5.29      ! [G: nat > nat,N: nat] :
% 9.49/5.29        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.49/5.29        = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 9.49/5.29          @ ( groups3542108847815614940at_nat
% 9.49/5.29            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.49/5.29            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6663_sum_OlessThan__Suc__shift,axiom,
% 9.49/5.29      ! [G: nat > real,N: nat] :
% 9.49/5.29        ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.49/5.29        = ( plus_plus_real @ ( G @ zero_zero_nat )
% 9.49/5.29          @ ( groups6591440286371151544t_real
% 9.49/5.29            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.49/5.29            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6664_sum__lessThan__telescope_H,axiom,
% 9.49/5.29      ! [F: nat > rat,M: nat] :
% 9.49/5.29        ( ( groups2906978787729119204at_rat
% 9.49/5.29          @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.49/5.29          @ ( set_ord_lessThan_nat @ M ) )
% 9.49/5.29        = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6665_sum__lessThan__telescope_H,axiom,
% 9.49/5.29      ! [F: nat > int,M: nat] :
% 9.49/5.29        ( ( groups3539618377306564664at_int
% 9.49/5.29          @ ^ [N2: nat] : ( minus_minus_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.49/5.29          @ ( set_ord_lessThan_nat @ M ) )
% 9.49/5.29        = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6666_sum__lessThan__telescope_H,axiom,
% 9.49/5.29      ! [F: nat > real,M: nat] :
% 9.49/5.29        ( ( groups6591440286371151544t_real
% 9.49/5.29          @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.49/5.29          @ ( set_ord_lessThan_nat @ M ) )
% 9.49/5.29        = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6667_sum__lessThan__telescope,axiom,
% 9.49/5.29      ! [F: nat > rat,M: nat] :
% 9.49/5.29        ( ( groups2906978787729119204at_rat
% 9.49/5.29          @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 9.49/5.29          @ ( set_ord_lessThan_nat @ M ) )
% 9.49/5.29        = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6668_sum__lessThan__telescope,axiom,
% 9.49/5.29      ! [F: nat > int,M: nat] :
% 9.49/5.29        ( ( groups3539618377306564664at_int
% 9.49/5.29          @ ^ [N2: nat] : ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 9.49/5.29          @ ( set_ord_lessThan_nat @ M ) )
% 9.49/5.29        = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6669_sum__lessThan__telescope,axiom,
% 9.49/5.29      ! [F: nat > real,M: nat] :
% 9.49/5.29        ( ( groups6591440286371151544t_real
% 9.49/5.29          @ ^ [N2: nat] : ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 9.49/5.29          @ ( set_ord_lessThan_nat @ M ) )
% 9.49/5.29        = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6670_sum_OatLeast1__atMost__eq,axiom,
% 9.49/5.29      ! [G: nat > nat,N: nat] :
% 9.49/5.29        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 9.49/5.29        = ( groups3542108847815614940at_nat
% 9.49/5.29          @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 9.49/5.29          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6671_sum_OatLeast1__atMost__eq,axiom,
% 9.49/5.29      ! [G: nat > real,N: nat] :
% 9.49/5.29        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 9.49/5.29        = ( groups6591440286371151544t_real
% 9.49/5.29          @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 9.49/5.29          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6672_real__sum__nat__ivl__bounded2,axiom,
% 9.49/5.29      ! [N: nat,F: nat > rat,K4: rat,K: nat] :
% 9.49/5.29        ( ! [P7: nat] :
% 9.49/5.29            ( ( ord_less_nat @ P7 @ N )
% 9.49/5.29           => ( ord_less_eq_rat @ ( F @ P7 ) @ K4 ) )
% 9.49/5.29       => ( ( ord_less_eq_rat @ zero_zero_rat @ K4 )
% 9.49/5.29         => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K4 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6673_real__sum__nat__ivl__bounded2,axiom,
% 9.49/5.29      ! [N: nat,F: nat > int,K4: int,K: nat] :
% 9.49/5.29        ( ! [P7: nat] :
% 9.49/5.29            ( ( ord_less_nat @ P7 @ N )
% 9.49/5.29           => ( ord_less_eq_int @ ( F @ P7 ) @ K4 ) )
% 9.49/5.29       => ( ( ord_less_eq_int @ zero_zero_int @ K4 )
% 9.49/5.29         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K4 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6674_real__sum__nat__ivl__bounded2,axiom,
% 9.49/5.29      ! [N: nat,F: nat > nat,K4: nat,K: nat] :
% 9.49/5.29        ( ! [P7: nat] :
% 9.49/5.29            ( ( ord_less_nat @ P7 @ N )
% 9.49/5.29           => ( ord_less_eq_nat @ ( F @ P7 ) @ K4 ) )
% 9.49/5.29       => ( ( ord_less_eq_nat @ zero_zero_nat @ K4 )
% 9.49/5.29         => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K4 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6675_real__sum__nat__ivl__bounded2,axiom,
% 9.49/5.29      ! [N: nat,F: nat > real,K4: real,K: nat] :
% 9.49/5.29        ( ! [P7: nat] :
% 9.49/5.29            ( ( ord_less_nat @ P7 @ N )
% 9.49/5.29           => ( ord_less_eq_real @ ( F @ P7 ) @ K4 ) )
% 9.49/5.29       => ( ( ord_less_eq_real @ zero_zero_real @ K4 )
% 9.49/5.29         => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K4 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6676_norm__le__zero__iff,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
% 9.49/5.29        = ( X = zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6677_norm__le__zero__iff,axiom,
% 9.49/5.29      ! [X: complex] :
% 9.49/5.29        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
% 9.49/5.29        = ( X = zero_zero_complex ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6678_zero__less__norm__iff,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
% 9.49/5.29        = ( X != zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6679_zero__less__norm__iff,axiom,
% 9.49/5.29      ! [X: complex] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
% 9.49/5.29        = ( X != zero_zero_complex ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6680_norm__eq__zero,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ( real_V7735802525324610683m_real @ X )
% 9.49/5.29          = zero_zero_real )
% 9.49/5.29        = ( X = zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6681_norm__eq__zero,axiom,
% 9.49/5.29      ! [X: complex] :
% 9.49/5.29        ( ( ( real_V1022390504157884413omplex @ X )
% 9.49/5.29          = zero_zero_real )
% 9.49/5.29        = ( X = zero_zero_complex ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6682_norm__zero,axiom,
% 9.49/5.29      ( ( real_V7735802525324610683m_real @ zero_zero_real )
% 9.49/5.29      = zero_zero_real ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6683_norm__zero,axiom,
% 9.49/5.29      ( ( real_V1022390504157884413omplex @ zero_zero_complex )
% 9.49/5.29      = zero_zero_real ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6684_power__eq__1__iff,axiom,
% 9.49/5.29      ! [W2: real,N: nat] :
% 9.49/5.29        ( ( ( power_power_real @ W2 @ N )
% 9.49/5.29          = one_one_real )
% 9.49/5.29       => ( ( ( real_V7735802525324610683m_real @ W2 )
% 9.49/5.29            = one_one_real )
% 9.49/5.29          | ( N = zero_zero_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6685_power__eq__1__iff,axiom,
% 9.49/5.29      ! [W2: complex,N: nat] :
% 9.49/5.29        ( ( ( power_power_complex @ W2 @ N )
% 9.49/5.29          = one_one_complex )
% 9.49/5.29       => ( ( ( real_V1022390504157884413omplex @ W2 )
% 9.49/5.29            = one_one_real )
% 9.49/5.29          | ( N = zero_zero_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6686_norm__diff__triangle__less,axiom,
% 9.49/5.29      ! [X: real,Y: real,E1: real,Z2: real,E22: real] :
% 9.49/5.29        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
% 9.49/5.29       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z2 ) ) @ E22 )
% 9.49/5.29         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6687_norm__diff__triangle__less,axiom,
% 9.49/5.29      ! [X: complex,Y: complex,E1: real,Z2: complex,E22: real] :
% 9.49/5.29        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
% 9.49/5.29       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z2 ) ) @ E22 )
% 9.49/5.29         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6688_and__nat__numerals_I3_J,axiom,
% 9.49/5.29      ! [X: num] :
% 9.49/5.29        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 9.49/5.29        = zero_zero_nat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6689_and__nat__numerals_I1_J,axiom,
% 9.49/5.29      ! [Y: num] :
% 9.49/5.29        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 9.49/5.29        = zero_zero_nat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6690_and__nat__numerals_I4_J,axiom,
% 9.49/5.29      ! [X: num] :
% 9.49/5.29        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 9.49/5.29        = one_one_nat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6691_and__nat__numerals_I2_J,axiom,
% 9.49/5.29      ! [Y: num] :
% 9.49/5.29        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 9.49/5.29        = one_one_nat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6692_Suc__0__and__eq,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.49/5.29        = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6693_and__Suc__0__eq,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.49/5.29        = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6694_and__nat__unfold,axiom,
% 9.49/5.29      ( bit_se727722235901077358nd_nat
% 9.49/5.29      = ( ^ [M2: nat,N2: nat] :
% 9.49/5.29            ( if_nat
% 9.49/5.29            @ ( ( M2 = zero_zero_nat )
% 9.49/5.29              | ( N2 = zero_zero_nat ) )
% 9.49/5.29            @ zero_zero_nat
% 9.49/5.29            @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6695_nonzero__norm__divide,axiom,
% 9.49/5.29      ! [B: real,A: real] :
% 9.49/5.29        ( ( B != zero_zero_real )
% 9.49/5.29       => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 9.49/5.29          = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6696_nonzero__norm__divide,axiom,
% 9.49/5.29      ! [B: complex,A: complex] :
% 9.49/5.29        ( ( B != zero_zero_complex )
% 9.49/5.29       => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 9.49/5.29          = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6697_norm__not__less__zero,axiom,
% 9.49/5.29      ! [X: complex] :
% 9.49/5.29        ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6698_power__eq__imp__eq__norm,axiom,
% 9.49/5.29      ! [W2: real,N: nat,Z2: real] :
% 9.49/5.29        ( ( ( power_power_real @ W2 @ N )
% 9.49/5.29          = ( power_power_real @ Z2 @ N ) )
% 9.49/5.29       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.29         => ( ( real_V7735802525324610683m_real @ W2 )
% 9.49/5.29            = ( real_V7735802525324610683m_real @ Z2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6699_power__eq__imp__eq__norm,axiom,
% 9.49/5.29      ! [W2: complex,N: nat,Z2: complex] :
% 9.49/5.29        ( ( ( power_power_complex @ W2 @ N )
% 9.49/5.29          = ( power_power_complex @ Z2 @ N ) )
% 9.49/5.29       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.29         => ( ( real_V1022390504157884413omplex @ W2 )
% 9.49/5.29            = ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6700_norm__mult__less,axiom,
% 9.49/5.29      ! [X: real,R2: real,Y: real,S: real] :
% 9.49/5.29        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
% 9.49/5.29       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
% 9.49/5.29         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6701_norm__mult__less,axiom,
% 9.49/5.29      ! [X: complex,R2: real,Y: complex,S: real] :
% 9.49/5.29        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
% 9.49/5.29       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
% 9.49/5.29         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6702_norm__triangle__lt,axiom,
% 9.49/5.29      ! [X: real,Y: real,E2: real] :
% 9.49/5.29        ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
% 9.49/5.29       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E2 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6703_norm__triangle__lt,axiom,
% 9.49/5.29      ! [X: complex,Y: complex,E2: real] :
% 9.49/5.29        ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
% 9.49/5.29       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E2 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6704_norm__add__less,axiom,
% 9.49/5.29      ! [X: real,R2: real,Y: real,S: real] :
% 9.49/5.29        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
% 9.49/5.29       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
% 9.49/5.29         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6705_norm__add__less,axiom,
% 9.49/5.29      ! [X: complex,R2: real,Y: complex,S: real] :
% 9.49/5.29        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
% 9.49/5.29       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
% 9.49/5.29         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6706_and__int_Opelims,axiom,
% 9.49/5.29      ! [X: int,Xa2: int,Y: int] :
% 9.49/5.29        ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
% 9.49/5.29          = Y )
% 9.49/5.29       => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
% 9.49/5.29         => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.49/5.29                    & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.49/5.29                 => ( Y
% 9.49/5.29                    = ( uminus_uminus_int
% 9.49/5.29                      @ ( zero_n2684676970156552555ol_int
% 9.49/5.29                        @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 9.49/5.29                          & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 9.49/5.29                & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.49/5.29                      & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.49/5.29                 => ( Y
% 9.49/5.29                    = ( plus_plus_int
% 9.49/5.29                      @ ( zero_n2684676970156552555ol_int
% 9.49/5.29                        @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 9.49/5.29                          & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 9.49/5.29                      @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 9.49/5.29             => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6707_and__int_Opsimps,axiom,
% 9.49/5.29      ! [K: int,L: int] :
% 9.49/5.29        ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
% 9.49/5.29       => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.49/5.29              & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.49/5.29           => ( ( bit_se725231765392027082nd_int @ K @ L )
% 9.49/5.29              = ( uminus_uminus_int
% 9.49/5.29                @ ( zero_n2684676970156552555ol_int
% 9.49/5.29                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 9.49/5.29                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
% 9.49/5.29          & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.49/5.29                & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.49/5.29           => ( ( bit_se725231765392027082nd_int @ K @ L )
% 9.49/5.29              = ( plus_plus_int
% 9.49/5.29                @ ( zero_n2684676970156552555ol_int
% 9.49/5.29                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 9.49/5.29                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 9.49/5.29                @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6708_geometric__deriv__sums,axiom,
% 9.49/5.29      ! [Z2: real] :
% 9.49/5.29        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ one_one_real )
% 9.49/5.29       => ( sums_real
% 9.49/5.29          @ ^ [N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( power_power_real @ Z2 @ N2 ) )
% 9.49/5.29          @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6709_geometric__deriv__sums,axiom,
% 9.49/5.29      ! [Z2: complex] :
% 9.49/5.29        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ one_one_real )
% 9.49/5.29       => ( sums_complex
% 9.49/5.29          @ ^ [N2: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( power_power_complex @ Z2 @ N2 ) )
% 9.49/5.29          @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6710_central__binomial__lower__bound,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.29       => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6711_polyfun__diff,axiom,
% 9.49/5.29      ! [N: nat,A: nat > complex,X: complex,Y: complex] :
% 9.49/5.29        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.49/5.29       => ( ( minus_minus_complex
% 9.49/5.29            @ ( groups2073611262835488442omplex
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29            @ ( groups2073611262835488442omplex
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ Y @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29          = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
% 9.49/5.29            @ ( groups2073611262835488442omplex
% 9.49/5.29              @ ^ [J3: nat] :
% 9.49/5.29                  ( times_times_complex
% 9.49/5.29                  @ ( groups2073611262835488442omplex
% 9.49/5.29                    @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
% 9.49/5.29                    @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
% 9.49/5.29                  @ ( power_power_complex @ X @ J3 ) )
% 9.49/5.29              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6712_polyfun__diff,axiom,
% 9.49/5.29      ! [N: nat,A: nat > rat,X: rat,Y: rat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.49/5.29       => ( ( minus_minus_rat
% 9.49/5.29            @ ( groups2906978787729119204at_rat
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ X @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29            @ ( groups2906978787729119204at_rat
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ Y @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29          = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
% 9.49/5.29            @ ( groups2906978787729119204at_rat
% 9.49/5.29              @ ^ [J3: nat] :
% 9.49/5.29                  ( times_times_rat
% 9.49/5.29                  @ ( groups2906978787729119204at_rat
% 9.49/5.29                    @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
% 9.49/5.29                    @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
% 9.49/5.29                  @ ( power_power_rat @ X @ J3 ) )
% 9.49/5.29              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6713_polyfun__diff,axiom,
% 9.49/5.29      ! [N: nat,A: nat > int,X: int,Y: int] :
% 9.49/5.29        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.49/5.29       => ( ( minus_minus_int
% 9.49/5.29            @ ( groups3539618377306564664at_int
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ X @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29            @ ( groups3539618377306564664at_int
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ Y @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29          = ( times_times_int @ ( minus_minus_int @ X @ Y )
% 9.49/5.29            @ ( groups3539618377306564664at_int
% 9.49/5.29              @ ^ [J3: nat] :
% 9.49/5.29                  ( times_times_int
% 9.49/5.29                  @ ( groups3539618377306564664at_int
% 9.49/5.29                    @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
% 9.49/5.29                    @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
% 9.49/5.29                  @ ( power_power_int @ X @ J3 ) )
% 9.49/5.29              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6714_polyfun__diff,axiom,
% 9.49/5.29      ! [N: nat,A: nat > real,X: real,Y: real] :
% 9.49/5.29        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.49/5.29       => ( ( minus_minus_real
% 9.49/5.29            @ ( groups6591440286371151544t_real
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ X @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29            @ ( groups6591440286371151544t_real
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ Y @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29          = ( times_times_real @ ( minus_minus_real @ X @ Y )
% 9.49/5.29            @ ( groups6591440286371151544t_real
% 9.49/5.29              @ ^ [J3: nat] :
% 9.49/5.29                  ( times_times_real
% 9.49/5.29                  @ ( groups6591440286371151544t_real
% 9.49/5.29                    @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
% 9.49/5.29                    @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
% 9.49/5.29                  @ ( power_power_real @ X @ J3 ) )
% 9.49/5.29              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6715_modulo__int__unfold,axiom,
% 9.49/5.29      ! [L: int,K: int,N: nat,M: nat] :
% 9.49/5.29        ( ( ( ( ( sgn_sgn_int @ L )
% 9.49/5.29              = zero_zero_int )
% 9.49/5.29            | ( ( sgn_sgn_int @ K )
% 9.49/5.29              = zero_zero_int )
% 9.49/5.29            | ( N = zero_zero_nat ) )
% 9.49/5.29         => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.49/5.29            = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 9.49/5.29        & ( ~ ( ( ( sgn_sgn_int @ L )
% 9.49/5.29                = zero_zero_int )
% 9.49/5.29              | ( ( sgn_sgn_int @ K )
% 9.49/5.29                = zero_zero_int )
% 9.49/5.29              | ( N = zero_zero_nat ) )
% 9.49/5.29         => ( ( ( ( sgn_sgn_int @ K )
% 9.49/5.29                = ( sgn_sgn_int @ L ) )
% 9.49/5.29             => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.49/5.29                = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
% 9.49/5.29            & ( ( ( sgn_sgn_int @ K )
% 9.49/5.29               != ( sgn_sgn_int @ L ) )
% 9.49/5.29             => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.49/5.29                = ( times_times_int @ ( sgn_sgn_int @ L )
% 9.49/5.29                  @ ( minus_minus_int
% 9.49/5.29                    @ ( semiri1314217659103216013at_int
% 9.49/5.29                      @ ( times_times_nat @ N
% 9.49/5.29                        @ ( zero_n2687167440665602831ol_nat
% 9.49/5.29                          @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
% 9.49/5.29                    @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6716_atMost__iff,axiom,
% 9.49/5.29      ! [I: real,K: real] :
% 9.49/5.29        ( ( member_real @ I @ ( set_ord_atMost_real @ K ) )
% 9.49/5.29        = ( ord_less_eq_real @ I @ K ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6717_atMost__iff,axiom,
% 9.49/5.29      ! [I: set_nat,K: set_nat] :
% 9.49/5.29        ( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K ) )
% 9.49/5.29        = ( ord_less_eq_set_nat @ I @ K ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6718_atMost__iff,axiom,
% 9.49/5.29      ! [I: set_int,K: set_int] :
% 9.49/5.29        ( ( member_set_int @ I @ ( set_or58775011639299419et_int @ K ) )
% 9.49/5.29        = ( ord_less_eq_set_int @ I @ K ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6719_atMost__iff,axiom,
% 9.49/5.29      ! [I: rat,K: rat] :
% 9.49/5.29        ( ( member_rat @ I @ ( set_ord_atMost_rat @ K ) )
% 9.49/5.29        = ( ord_less_eq_rat @ I @ K ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6720_atMost__iff,axiom,
% 9.49/5.29      ! [I: num,K: num] :
% 9.49/5.29        ( ( member_num @ I @ ( set_ord_atMost_num @ K ) )
% 9.49/5.29        = ( ord_less_eq_num @ I @ K ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6721_atMost__iff,axiom,
% 9.49/5.29      ! [I: nat,K: nat] :
% 9.49/5.29        ( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
% 9.49/5.29        = ( ord_less_eq_nat @ I @ K ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6722_atMost__iff,axiom,
% 9.49/5.29      ! [I: int,K: int] :
% 9.49/5.29        ( ( member_int @ I @ ( set_ord_atMost_int @ K ) )
% 9.49/5.29        = ( ord_less_eq_int @ I @ K ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6723_sgn__0,axiom,
% 9.49/5.29      ( ( sgn_sgn_Code_integer @ zero_z3403309356797280102nteger )
% 9.49/5.29      = zero_z3403309356797280102nteger ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6724_sgn__0,axiom,
% 9.49/5.29      ( ( sgn_sgn_complex @ zero_zero_complex )
% 9.49/5.29      = zero_zero_complex ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6725_sgn__0,axiom,
% 9.49/5.29      ( ( sgn_sgn_real @ zero_zero_real )
% 9.49/5.29      = zero_zero_real ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6726_sgn__0,axiom,
% 9.49/5.29      ( ( sgn_sgn_rat @ zero_zero_rat )
% 9.49/5.29      = zero_zero_rat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6727_sgn__0,axiom,
% 9.49/5.29      ( ( sgn_sgn_int @ zero_zero_int )
% 9.49/5.29      = zero_zero_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6728_sgn__zero,axiom,
% 9.49/5.29      ( ( sgn_sgn_complex @ zero_zero_complex )
% 9.49/5.29      = zero_zero_complex ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6729_sgn__zero,axiom,
% 9.49/5.29      ( ( sgn_sgn_real @ zero_zero_real )
% 9.49/5.29      = zero_zero_real ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6730_finite__atMost,axiom,
% 9.49/5.29      ! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6731_sgn__greater,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A ) )
% 9.49/5.29        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6732_sgn__greater,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A ) )
% 9.49/5.29        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6733_sgn__greater,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( ord_less_rat @ zero_zero_rat @ ( sgn_sgn_rat @ A ) )
% 9.49/5.29        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6734_sgn__greater,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A ) )
% 9.49/5.29        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6735_sgn__less,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 9.49/5.29        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6736_sgn__less,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( ord_less_real @ ( sgn_sgn_real @ A ) @ zero_zero_real )
% 9.49/5.29        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6737_sgn__less,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( ord_less_rat @ ( sgn_sgn_rat @ A ) @ zero_zero_rat )
% 9.49/5.29        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6738_sgn__less,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( ord_less_int @ ( sgn_sgn_int @ A ) @ zero_zero_int )
% 9.49/5.29        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6739_atMost__subset__iff,axiom,
% 9.49/5.29      ! [X: set_int,Y: set_int] :
% 9.49/5.29        ( ( ord_le4403425263959731960et_int @ ( set_or58775011639299419et_int @ X ) @ ( set_or58775011639299419et_int @ Y ) )
% 9.49/5.29        = ( ord_less_eq_set_int @ X @ Y ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6740_atMost__subset__iff,axiom,
% 9.49/5.29      ! [X: rat,Y: rat] :
% 9.49/5.29        ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X ) @ ( set_ord_atMost_rat @ Y ) )
% 9.49/5.29        = ( ord_less_eq_rat @ X @ Y ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6741_atMost__subset__iff,axiom,
% 9.49/5.29      ! [X: num,Y: num] :
% 9.49/5.29        ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X ) @ ( set_ord_atMost_num @ Y ) )
% 9.49/5.29        = ( ord_less_eq_num @ X @ Y ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6742_atMost__subset__iff,axiom,
% 9.49/5.29      ! [X: nat,Y: nat] :
% 9.49/5.29        ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
% 9.49/5.29        = ( ord_less_eq_nat @ X @ Y ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6743_atMost__subset__iff,axiom,
% 9.49/5.29      ! [X: int,Y: int] :
% 9.49/5.29        ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y ) )
% 9.49/5.29        = ( ord_less_eq_int @ X @ Y ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6744_sgn__pos,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 9.49/5.29       => ( ( sgn_sgn_Code_integer @ A )
% 9.49/5.29          = one_one_Code_integer ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6745_sgn__pos,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.29       => ( ( sgn_sgn_real @ A )
% 9.49/5.29          = one_one_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6746_sgn__pos,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.29       => ( ( sgn_sgn_rat @ A )
% 9.49/5.29          = one_one_rat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6747_sgn__pos,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( ord_less_int @ zero_zero_int @ A )
% 9.49/5.29       => ( ( sgn_sgn_int @ A )
% 9.49/5.29          = one_one_int ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6748_Icc__subset__Iic__iff,axiom,
% 9.49/5.29      ! [L: set_int,H2: set_int,H3: set_int] :
% 9.49/5.29        ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ L @ H2 ) @ ( set_or58775011639299419et_int @ H3 ) )
% 9.49/5.29        = ( ~ ( ord_less_eq_set_int @ L @ H2 )
% 9.49/5.29          | ( ord_less_eq_set_int @ H2 @ H3 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6749_Icc__subset__Iic__iff,axiom,
% 9.49/5.29      ! [L: rat,H2: rat,H3: rat] :
% 9.49/5.29        ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H2 ) @ ( set_ord_atMost_rat @ H3 ) )
% 9.49/5.29        = ( ~ ( ord_less_eq_rat @ L @ H2 )
% 9.49/5.29          | ( ord_less_eq_rat @ H2 @ H3 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6750_Icc__subset__Iic__iff,axiom,
% 9.49/5.29      ! [L: num,H2: num,H3: num] :
% 9.49/5.29        ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L @ H2 ) @ ( set_ord_atMost_num @ H3 ) )
% 9.49/5.29        = ( ~ ( ord_less_eq_num @ L @ H2 )
% 9.49/5.29          | ( ord_less_eq_num @ H2 @ H3 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6751_Icc__subset__Iic__iff,axiom,
% 9.49/5.29      ! [L: nat,H2: nat,H3: nat] :
% 9.49/5.29        ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H2 ) @ ( set_ord_atMost_nat @ H3 ) )
% 9.49/5.29        = ( ~ ( ord_less_eq_nat @ L @ H2 )
% 9.49/5.29          | ( ord_less_eq_nat @ H2 @ H3 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6752_Icc__subset__Iic__iff,axiom,
% 9.49/5.29      ! [L: int,H2: int,H3: int] :
% 9.49/5.29        ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H2 ) @ ( set_ord_atMost_int @ H3 ) )
% 9.49/5.29        = ( ~ ( ord_less_eq_int @ L @ H2 )
% 9.49/5.29          | ( ord_less_eq_int @ H2 @ H3 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6753_Icc__subset__Iic__iff,axiom,
% 9.49/5.29      ! [L: real,H2: real,H3: real] :
% 9.49/5.29        ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H2 ) @ ( set_ord_atMost_real @ H3 ) )
% 9.49/5.29        = ( ~ ( ord_less_eq_real @ L @ H2 )
% 9.49/5.29          | ( ord_less_eq_real @ H2 @ H3 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6754_sgn__mult__self__eq,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
% 9.49/5.29        = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6755_sgn__mult__self__eq,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ A ) )
% 9.49/5.29        = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6756_sgn__mult__self__eq,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ A ) )
% 9.49/5.29        = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6757_sgn__mult__self__eq,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ A ) )
% 9.49/5.29        = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6758_atMost__0,axiom,
% 9.49/5.29      ( ( set_ord_atMost_nat @ zero_zero_nat )
% 9.49/5.29      = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6759_sgn__neg,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( ord_less_int @ A @ zero_zero_int )
% 9.49/5.29       => ( ( sgn_sgn_int @ A )
% 9.49/5.29          = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6760_sgn__neg,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( ord_less_real @ A @ zero_zero_real )
% 9.49/5.29       => ( ( sgn_sgn_real @ A )
% 9.49/5.29          = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6761_sgn__neg,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.49/5.29       => ( ( sgn_sgn_rat @ A )
% 9.49/5.29          = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6762_sgn__neg,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 9.49/5.29       => ( ( sgn_sgn_Code_integer @ A )
% 9.49/5.29          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6763_sgn__of__nat,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( sgn_sgn_rat @ ( semiri681578069525770553at_rat @ N ) )
% 9.49/5.29        = ( zero_n2052037380579107095ol_rat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6764_sgn__of__nat,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( sgn_sgn_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 9.49/5.29        = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6765_sgn__of__nat,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( sgn_sgn_real @ ( semiri5074537144036343181t_real @ N ) )
% 9.49/5.29        = ( zero_n3304061248610475627l_real @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6766_sgn__of__nat,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( sgn_sgn_int @ ( semiri1314217659103216013at_int @ N ) )
% 9.49/5.29        = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6767_powser__sums__zero__iff,axiom,
% 9.49/5.29      ! [A: nat > complex,X: complex] :
% 9.49/5.29        ( ( sums_complex
% 9.49/5.29          @ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
% 9.49/5.29          @ X )
% 9.49/5.29        = ( ( A @ zero_zero_nat )
% 9.49/5.29          = X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6768_powser__sums__zero__iff,axiom,
% 9.49/5.29      ! [A: nat > real,X: real] :
% 9.49/5.29        ( ( sums_real
% 9.49/5.29          @ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
% 9.49/5.29          @ X )
% 9.49/5.29        = ( ( A @ zero_zero_nat )
% 9.49/5.29          = X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6769_sgn__eq__0__iff,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( ( sgn_sgn_Code_integer @ A )
% 9.49/5.29          = zero_z3403309356797280102nteger )
% 9.49/5.29        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6770_sgn__eq__0__iff,axiom,
% 9.49/5.29      ! [A: complex] :
% 9.49/5.29        ( ( ( sgn_sgn_complex @ A )
% 9.49/5.29          = zero_zero_complex )
% 9.49/5.29        = ( A = zero_zero_complex ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6771_sgn__eq__0__iff,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( ( sgn_sgn_real @ A )
% 9.49/5.29          = zero_zero_real )
% 9.49/5.29        = ( A = zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6772_sgn__eq__0__iff,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( ( sgn_sgn_rat @ A )
% 9.49/5.29          = zero_zero_rat )
% 9.49/5.29        = ( A = zero_zero_rat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6773_sgn__eq__0__iff,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( ( sgn_sgn_int @ A )
% 9.49/5.29          = zero_zero_int )
% 9.49/5.29        = ( A = zero_zero_int ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6774_sgn__0__0,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( ( sgn_sgn_Code_integer @ A )
% 9.49/5.29          = zero_z3403309356797280102nteger )
% 9.49/5.29        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6775_sgn__0__0,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( ( sgn_sgn_real @ A )
% 9.49/5.29          = zero_zero_real )
% 9.49/5.29        = ( A = zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6776_sgn__0__0,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( ( sgn_sgn_rat @ A )
% 9.49/5.29          = zero_zero_rat )
% 9.49/5.29        = ( A = zero_zero_rat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6777_sgn__0__0,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( ( sgn_sgn_int @ A )
% 9.49/5.29          = zero_zero_int )
% 9.49/5.29        = ( A = zero_zero_int ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6778_sgn__zero__iff,axiom,
% 9.49/5.29      ! [X: complex] :
% 9.49/5.29        ( ( ( sgn_sgn_complex @ X )
% 9.49/5.29          = zero_zero_complex )
% 9.49/5.29        = ( X = zero_zero_complex ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6779_sgn__zero__iff,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ( sgn_sgn_real @ X )
% 9.49/5.29          = zero_zero_real )
% 9.49/5.29        = ( X = zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6780_not__empty__eq__Iic__eq__empty,axiom,
% 9.49/5.29      ! [H2: real] :
% 9.49/5.29        ( bot_bot_set_real
% 9.49/5.29       != ( set_ord_atMost_real @ H2 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6781_not__empty__eq__Iic__eq__empty,axiom,
% 9.49/5.29      ! [H2: nat] :
% 9.49/5.29        ( bot_bot_set_nat
% 9.49/5.29       != ( set_ord_atMost_nat @ H2 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6782_not__empty__eq__Iic__eq__empty,axiom,
% 9.49/5.29      ! [H2: int] :
% 9.49/5.29        ( bot_bot_set_int
% 9.49/5.29       != ( set_ord_atMost_int @ H2 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6783_infinite__Iic,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ~ ( finite_finite_int @ ( set_ord_atMost_int @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6784_atMost__def,axiom,
% 9.49/5.29      ( set_ord_atMost_real
% 9.49/5.29      = ( ^ [U2: real] :
% 9.49/5.29            ( collect_real
% 9.49/5.29            @ ^ [X3: real] : ( ord_less_eq_real @ X3 @ U2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6785_atMost__def,axiom,
% 9.49/5.29      ( set_or4236626031148496127et_nat
% 9.49/5.29      = ( ^ [U2: set_nat] :
% 9.49/5.29            ( collect_set_nat
% 9.49/5.29            @ ^ [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ U2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6786_atMost__def,axiom,
% 9.49/5.29      ( set_or58775011639299419et_int
% 9.49/5.29      = ( ^ [U2: set_int] :
% 9.49/5.29            ( collect_set_int
% 9.49/5.29            @ ^ [X3: set_int] : ( ord_less_eq_set_int @ X3 @ U2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6787_atMost__def,axiom,
% 9.49/5.29      ( set_ord_atMost_rat
% 9.49/5.29      = ( ^ [U2: rat] :
% 9.49/5.29            ( collect_rat
% 9.49/5.29            @ ^ [X3: rat] : ( ord_less_eq_rat @ X3 @ U2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6788_atMost__def,axiom,
% 9.49/5.29      ( set_ord_atMost_num
% 9.49/5.29      = ( ^ [U2: num] :
% 9.49/5.29            ( collect_num
% 9.49/5.29            @ ^ [X3: num] : ( ord_less_eq_num @ X3 @ U2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6789_atMost__def,axiom,
% 9.49/5.29      ( set_ord_atMost_nat
% 9.49/5.29      = ( ^ [U2: nat] :
% 9.49/5.29            ( collect_nat
% 9.49/5.29            @ ^ [X3: nat] : ( ord_less_eq_nat @ X3 @ U2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6790_atMost__def,axiom,
% 9.49/5.29      ( set_ord_atMost_int
% 9.49/5.29      = ( ^ [U2: int] :
% 9.49/5.29            ( collect_int
% 9.49/5.29            @ ^ [X3: int] : ( ord_less_eq_int @ X3 @ U2 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6791_sgn__not__eq__imp,axiom,
% 9.49/5.29      ! [B: int,A: int] :
% 9.49/5.29        ( ( ( sgn_sgn_int @ B )
% 9.49/5.29         != ( sgn_sgn_int @ A ) )
% 9.49/5.29       => ( ( ( sgn_sgn_int @ A )
% 9.49/5.29           != zero_zero_int )
% 9.49/5.29         => ( ( ( sgn_sgn_int @ B )
% 9.49/5.29             != zero_zero_int )
% 9.49/5.29           => ( ( sgn_sgn_int @ A )
% 9.49/5.29              = ( uminus_uminus_int @ ( sgn_sgn_int @ B ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6792_sgn__not__eq__imp,axiom,
% 9.49/5.29      ! [B: real,A: real] :
% 9.49/5.29        ( ( ( sgn_sgn_real @ B )
% 9.49/5.29         != ( sgn_sgn_real @ A ) )
% 9.49/5.29       => ( ( ( sgn_sgn_real @ A )
% 9.49/5.29           != zero_zero_real )
% 9.49/5.29         => ( ( ( sgn_sgn_real @ B )
% 9.49/5.29             != zero_zero_real )
% 9.49/5.29           => ( ( sgn_sgn_real @ A )
% 9.49/5.29              = ( uminus_uminus_real @ ( sgn_sgn_real @ B ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6793_sgn__not__eq__imp,axiom,
% 9.49/5.29      ! [B: rat,A: rat] :
% 9.49/5.29        ( ( ( sgn_sgn_rat @ B )
% 9.49/5.29         != ( sgn_sgn_rat @ A ) )
% 9.49/5.29       => ( ( ( sgn_sgn_rat @ A )
% 9.49/5.29           != zero_zero_rat )
% 9.49/5.29         => ( ( ( sgn_sgn_rat @ B )
% 9.49/5.29             != zero_zero_rat )
% 9.49/5.29           => ( ( sgn_sgn_rat @ A )
% 9.49/5.29              = ( uminus_uminus_rat @ ( sgn_sgn_rat @ B ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6794_sgn__not__eq__imp,axiom,
% 9.49/5.29      ! [B: code_integer,A: code_integer] :
% 9.49/5.29        ( ( ( sgn_sgn_Code_integer @ B )
% 9.49/5.29         != ( sgn_sgn_Code_integer @ A ) )
% 9.49/5.29       => ( ( ( sgn_sgn_Code_integer @ A )
% 9.49/5.29           != zero_z3403309356797280102nteger )
% 9.49/5.29         => ( ( ( sgn_sgn_Code_integer @ B )
% 9.49/5.29             != zero_z3403309356797280102nteger )
% 9.49/5.29           => ( ( sgn_sgn_Code_integer @ A )
% 9.49/5.29              = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ B ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6795_atMost__atLeast0,axiom,
% 9.49/5.29      ( set_ord_atMost_nat
% 9.49/5.29      = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6796_not__Iic__le__Icc,axiom,
% 9.49/5.29      ! [H2: int,L3: int,H3: int] :
% 9.49/5.29        ~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H2 ) @ ( set_or1266510415728281911st_int @ L3 @ H3 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6797_not__Iic__le__Icc,axiom,
% 9.49/5.29      ! [H2: real,L3: real,H3: real] :
% 9.49/5.29        ~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H2 ) @ ( set_or1222579329274155063t_real @ L3 @ H3 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6798_finite__nat__iff__bounded__le,axiom,
% 9.49/5.29      ( finite_finite_nat
% 9.49/5.29      = ( ^ [S7: set_nat] :
% 9.49/5.29          ? [K3: nat] : ( ord_less_eq_set_nat @ S7 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6799_sgn__1__pos,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( ( sgn_sgn_Code_integer @ A )
% 9.49/5.29          = one_one_Code_integer )
% 9.49/5.29        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6800_sgn__1__pos,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( ( sgn_sgn_real @ A )
% 9.49/5.29          = one_one_real )
% 9.49/5.29        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6801_sgn__1__pos,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( ( sgn_sgn_rat @ A )
% 9.49/5.29          = one_one_rat )
% 9.49/5.29        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6802_sgn__1__pos,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( ( sgn_sgn_int @ A )
% 9.49/5.29          = one_one_int )
% 9.49/5.29        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6803_Iic__subset__Iio__iff,axiom,
% 9.49/5.29      ! [A: rat,B: rat] :
% 9.49/5.29        ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ A ) @ ( set_ord_lessThan_rat @ B ) )
% 9.49/5.29        = ( ord_less_rat @ A @ B ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6804_Iic__subset__Iio__iff,axiom,
% 9.49/5.29      ! [A: num,B: num] :
% 9.49/5.29        ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ A ) @ ( set_ord_lessThan_num @ B ) )
% 9.49/5.29        = ( ord_less_num @ A @ B ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6805_Iic__subset__Iio__iff,axiom,
% 9.49/5.29      ! [A: nat,B: nat] :
% 9.49/5.29        ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
% 9.49/5.29        = ( ord_less_nat @ A @ B ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6806_Iic__subset__Iio__iff,axiom,
% 9.49/5.29      ! [A: int,B: int] :
% 9.49/5.29        ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B ) )
% 9.49/5.29        = ( ord_less_int @ A @ B ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6807_Iic__subset__Iio__iff,axiom,
% 9.49/5.29      ! [A: real,B: real] :
% 9.49/5.29        ( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ A ) @ ( set_or5984915006950818249n_real @ B ) )
% 9.49/5.29        = ( ord_less_real @ A @ B ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6808_powser__sums__zero,axiom,
% 9.49/5.29      ! [A: nat > complex] :
% 9.49/5.29        ( sums_complex
% 9.49/5.29        @ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
% 9.49/5.29        @ ( A @ zero_zero_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6809_powser__sums__zero,axiom,
% 9.49/5.29      ! [A: nat > real] :
% 9.49/5.29        ( sums_real
% 9.49/5.29        @ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
% 9.49/5.29        @ ( A @ zero_zero_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6810_sgn__1__neg,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( ( sgn_sgn_int @ A )
% 9.49/5.29          = ( uminus_uminus_int @ one_one_int ) )
% 9.49/5.29        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6811_sgn__1__neg,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( ( sgn_sgn_real @ A )
% 9.49/5.29          = ( uminus_uminus_real @ one_one_real ) )
% 9.49/5.29        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6812_sgn__1__neg,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( ( sgn_sgn_rat @ A )
% 9.49/5.29          = ( uminus_uminus_rat @ one_one_rat ) )
% 9.49/5.29        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6813_sgn__1__neg,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( ( sgn_sgn_Code_integer @ A )
% 9.49/5.29          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.49/5.29        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6814_sgn__if,axiom,
% 9.49/5.29      ( sgn_sgn_int
% 9.49/5.29      = ( ^ [X3: int] : ( if_int @ ( X3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6815_sgn__if,axiom,
% 9.49/5.29      ( sgn_sgn_real
% 9.49/5.29      = ( ^ [X3: real] : ( if_real @ ( X3 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X3 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6816_sgn__if,axiom,
% 9.49/5.29      ( sgn_sgn_rat
% 9.49/5.29      = ( ^ [X3: rat] : ( if_rat @ ( X3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6817_sgn__if,axiom,
% 9.49/5.29      ( sgn_sgn_Code_integer
% 9.49/5.29      = ( ^ [X3: code_integer] : ( if_Code_integer @ ( X3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X3 ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6818_zsgn__def,axiom,
% 9.49/5.29      ( sgn_sgn_int
% 9.49/5.29      = ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6819_norm__sgn,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ( X = zero_zero_real )
% 9.49/5.29         => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
% 9.49/5.29            = zero_zero_real ) )
% 9.49/5.29        & ( ( X != zero_zero_real )
% 9.49/5.29         => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
% 9.49/5.29            = one_one_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6820_norm__sgn,axiom,
% 9.49/5.29      ! [X: complex] :
% 9.49/5.29        ( ( ( X = zero_zero_complex )
% 9.49/5.29         => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
% 9.49/5.29            = zero_zero_real ) )
% 9.49/5.29        & ( ( X != zero_zero_complex )
% 9.49/5.29         => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
% 9.49/5.29            = one_one_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6821_sum_OatMost__Suc__shift,axiom,
% 9.49/5.29      ! [G: nat > rat,N: nat] :
% 9.49/5.29        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 9.49/5.29        = ( plus_plus_rat @ ( G @ zero_zero_nat )
% 9.49/5.29          @ ( groups2906978787729119204at_rat
% 9.49/5.29            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.49/5.29            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6822_sum_OatMost__Suc__shift,axiom,
% 9.49/5.29      ! [G: nat > int,N: nat] :
% 9.49/5.29        ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 9.49/5.29        = ( plus_plus_int @ ( G @ zero_zero_nat )
% 9.49/5.29          @ ( groups3539618377306564664at_int
% 9.49/5.29            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.49/5.29            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6823_sum_OatMost__Suc__shift,axiom,
% 9.49/5.29      ! [G: nat > nat,N: nat] :
% 9.49/5.29        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 9.49/5.29        = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 9.49/5.29          @ ( groups3542108847815614940at_nat
% 9.49/5.29            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.49/5.29            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6824_sum_OatMost__Suc__shift,axiom,
% 9.49/5.29      ! [G: nat > real,N: nat] :
% 9.49/5.29        ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 9.49/5.29        = ( plus_plus_real @ ( G @ zero_zero_nat )
% 9.49/5.29          @ ( groups6591440286371151544t_real
% 9.49/5.29            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.49/5.29            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6825_sum__telescope,axiom,
% 9.49/5.29      ! [F: nat > rat,I: nat] :
% 9.49/5.29        ( ( groups2906978787729119204at_rat
% 9.49/5.29          @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
% 9.49/5.29          @ ( set_ord_atMost_nat @ I ) )
% 9.49/5.29        = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6826_sum__telescope,axiom,
% 9.49/5.29      ! [F: nat > int,I: nat] :
% 9.49/5.29        ( ( groups3539618377306564664at_int
% 9.49/5.29          @ ^ [I4: nat] : ( minus_minus_int @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
% 9.49/5.29          @ ( set_ord_atMost_nat @ I ) )
% 9.49/5.29        = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6827_sum__telescope,axiom,
% 9.49/5.29      ! [F: nat > real,I: nat] :
% 9.49/5.29        ( ( groups6591440286371151544t_real
% 9.49/5.29          @ ^ [I4: nat] : ( minus_minus_real @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
% 9.49/5.29          @ ( set_ord_atMost_nat @ I ) )
% 9.49/5.29        = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6828_polyfun__eq__coeffs,axiom,
% 9.49/5.29      ! [C: nat > complex,N: nat,D: nat > complex] :
% 9.49/5.29        ( ( ! [X3: complex] :
% 9.49/5.29              ( ( groups2073611262835488442omplex
% 9.49/5.29                @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X3 @ I4 ) )
% 9.49/5.29                @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29              = ( groups2073611262835488442omplex
% 9.49/5.29                @ ^ [I4: nat] : ( times_times_complex @ ( D @ I4 ) @ ( power_power_complex @ X3 @ I4 ) )
% 9.49/5.29                @ ( set_ord_atMost_nat @ N ) ) ) )
% 9.49/5.29        = ( ! [I4: nat] :
% 9.49/5.29              ( ( ord_less_eq_nat @ I4 @ N )
% 9.49/5.29             => ( ( C @ I4 )
% 9.49/5.29                = ( D @ I4 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6829_polyfun__eq__coeffs,axiom,
% 9.49/5.29      ! [C: nat > real,N: nat,D: nat > real] :
% 9.49/5.29        ( ( ! [X3: real] :
% 9.49/5.29              ( ( groups6591440286371151544t_real
% 9.49/5.29                @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X3 @ I4 ) )
% 9.49/5.29                @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29              = ( groups6591440286371151544t_real
% 9.49/5.29                @ ^ [I4: nat] : ( times_times_real @ ( D @ I4 ) @ ( power_power_real @ X3 @ I4 ) )
% 9.49/5.29                @ ( set_ord_atMost_nat @ N ) ) ) )
% 9.49/5.29        = ( ! [I4: nat] :
% 9.49/5.29              ( ( ord_less_eq_nat @ I4 @ N )
% 9.49/5.29             => ( ( C @ I4 )
% 9.49/5.29                = ( D @ I4 ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6830_binomial__maximum_H,axiom,
% 9.49/5.29      ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6831_binomial__mono,axiom,
% 9.49/5.29      ! [K: nat,K5: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ K @ K5 )
% 9.49/5.29       => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) @ N )
% 9.49/5.29         => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6832_binomial__antimono,axiom,
% 9.49/5.29      ! [K: nat,K5: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ K @ K5 )
% 9.49/5.29       => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 9.49/5.29         => ( ( ord_less_eq_nat @ K5 @ N )
% 9.49/5.29           => ( ord_less_eq_nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6833_binomial__maximum,axiom,
% 9.49/5.29      ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6834_zero__polynom__imp__zero__coeffs,axiom,
% 9.49/5.29      ! [C: nat > complex,N: nat,K: nat] :
% 9.49/5.29        ( ! [W: complex] :
% 9.49/5.29            ( ( groups2073611262835488442omplex
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ W @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29            = zero_zero_complex )
% 9.49/5.29       => ( ( ord_less_eq_nat @ K @ N )
% 9.49/5.29         => ( ( C @ K )
% 9.49/5.29            = zero_zero_complex ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6835_zero__polynom__imp__zero__coeffs,axiom,
% 9.49/5.29      ! [C: nat > real,N: nat,K: nat] :
% 9.49/5.29        ( ! [W: real] :
% 9.49/5.29            ( ( groups6591440286371151544t_real
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ W @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29            = zero_zero_real )
% 9.49/5.29       => ( ( ord_less_eq_nat @ K @ N )
% 9.49/5.29         => ( ( C @ K )
% 9.49/5.29            = zero_zero_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6836_polyfun__eq__0,axiom,
% 9.49/5.29      ! [C: nat > complex,N: nat] :
% 9.49/5.29        ( ( ! [X3: complex] :
% 9.49/5.29              ( ( groups2073611262835488442omplex
% 9.49/5.29                @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X3 @ I4 ) )
% 9.49/5.29                @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29              = zero_zero_complex ) )
% 9.49/5.29        = ( ! [I4: nat] :
% 9.49/5.29              ( ( ord_less_eq_nat @ I4 @ N )
% 9.49/5.29             => ( ( C @ I4 )
% 9.49/5.29                = zero_zero_complex ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6837_polyfun__eq__0,axiom,
% 9.49/5.29      ! [C: nat > real,N: nat] :
% 9.49/5.29        ( ( ! [X3: real] :
% 9.49/5.29              ( ( groups6591440286371151544t_real
% 9.49/5.29                @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X3 @ I4 ) )
% 9.49/5.29                @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29              = zero_zero_real ) )
% 9.49/5.29        = ( ! [I4: nat] :
% 9.49/5.29              ( ( ord_less_eq_nat @ I4 @ N )
% 9.49/5.29             => ( ( C @ I4 )
% 9.49/5.29                = zero_zero_real ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6838_sum_OatMost__shift,axiom,
% 9.49/5.29      ! [G: nat > rat,N: nat] :
% 9.49/5.29        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29        = ( plus_plus_rat @ ( G @ zero_zero_nat )
% 9.49/5.29          @ ( groups2906978787729119204at_rat
% 9.49/5.29            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.49/5.29            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6839_sum_OatMost__shift,axiom,
% 9.49/5.29      ! [G: nat > int,N: nat] :
% 9.49/5.29        ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29        = ( plus_plus_int @ ( G @ zero_zero_nat )
% 9.49/5.29          @ ( groups3539618377306564664at_int
% 9.49/5.29            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.49/5.29            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6840_sum_OatMost__shift,axiom,
% 9.49/5.29      ! [G: nat > nat,N: nat] :
% 9.49/5.29        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29        = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 9.49/5.29          @ ( groups3542108847815614940at_nat
% 9.49/5.29            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.49/5.29            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6841_sum_OatMost__shift,axiom,
% 9.49/5.29      ! [G: nat > real,N: nat] :
% 9.49/5.29        ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29        = ( plus_plus_real @ ( G @ zero_zero_nat )
% 9.49/5.29          @ ( groups6591440286371151544t_real
% 9.49/5.29            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.49/5.29            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6842_atLeast1__atMost__eq__remove0,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.49/5.29        = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6843_sum_Otriangle__reindex__eq,axiom,
% 9.49/5.29      ! [G: nat > nat > nat,N: nat] :
% 9.49/5.29        ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
% 9.49/5.29          @ ( collec3392354462482085612at_nat
% 9.49/5.29            @ ( produc6081775807080527818_nat_o
% 9.49/5.29              @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 9.49/5.29        = ( groups3542108847815614940at_nat
% 9.49/5.29          @ ^ [K3: nat] :
% 9.49/5.29              ( groups3542108847815614940at_nat
% 9.49/5.29              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ K3 ) )
% 9.49/5.29          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6844_sum_Otriangle__reindex__eq,axiom,
% 9.49/5.29      ! [G: nat > nat > real,N: nat] :
% 9.49/5.29        ( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
% 9.49/5.29          @ ( collec3392354462482085612at_nat
% 9.49/5.29            @ ( produc6081775807080527818_nat_o
% 9.49/5.29              @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 9.49/5.29        = ( groups6591440286371151544t_real
% 9.49/5.29          @ ^ [K3: nat] :
% 9.49/5.29              ( groups6591440286371151544t_real
% 9.49/5.29              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ K3 ) )
% 9.49/5.29          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6845_binomial__less__binomial__Suc,axiom,
% 9.49/5.29      ! [K: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.29       => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6846_binomial__strict__antimono,axiom,
% 9.49/5.29      ! [K: nat,K5: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ K @ K5 )
% 9.49/5.29       => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 9.49/5.29         => ( ( ord_less_eq_nat @ K5 @ N )
% 9.49/5.29           => ( ord_less_nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6847_binomial__strict__mono,axiom,
% 9.49/5.29      ! [K: nat,K5: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ K @ K5 )
% 9.49/5.29       => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) @ N )
% 9.49/5.29         => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6848_polyfun__finite__roots,axiom,
% 9.49/5.29      ! [C: nat > complex,N: nat] :
% 9.49/5.29        ( ( finite3207457112153483333omplex
% 9.49/5.29          @ ( collect_complex
% 9.49/5.29            @ ^ [X3: complex] :
% 9.49/5.29                ( ( groups2073611262835488442omplex
% 9.49/5.29                  @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X3 @ I4 ) )
% 9.49/5.29                  @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29                = zero_zero_complex ) ) )
% 9.49/5.29        = ( ? [I4: nat] :
% 9.49/5.29              ( ( ord_less_eq_nat @ I4 @ N )
% 9.49/5.29              & ( ( C @ I4 )
% 9.49/5.29               != zero_zero_complex ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6849_polyfun__finite__roots,axiom,
% 9.49/5.29      ! [C: nat > real,N: nat] :
% 9.49/5.29        ( ( finite_finite_real
% 9.49/5.29          @ ( collect_real
% 9.49/5.29            @ ^ [X3: real] :
% 9.49/5.29                ( ( groups6591440286371151544t_real
% 9.49/5.29                  @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X3 @ I4 ) )
% 9.49/5.29                  @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29                = zero_zero_real ) ) )
% 9.49/5.29        = ( ? [I4: nat] :
% 9.49/5.29              ( ( ord_less_eq_nat @ I4 @ N )
% 9.49/5.29              & ( ( C @ I4 )
% 9.49/5.29               != zero_zero_real ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6850_polyfun__roots__finite,axiom,
% 9.49/5.29      ! [C: nat > complex,K: nat,N: nat] :
% 9.49/5.29        ( ( ( C @ K )
% 9.49/5.29         != zero_zero_complex )
% 9.49/5.29       => ( ( ord_less_eq_nat @ K @ N )
% 9.49/5.29         => ( finite3207457112153483333omplex
% 9.49/5.29            @ ( collect_complex
% 9.49/5.29              @ ^ [Z3: complex] :
% 9.49/5.29                  ( ( groups2073611262835488442omplex
% 9.49/5.29                    @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z3 @ I4 ) )
% 9.49/5.29                    @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29                  = zero_zero_complex ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6851_polyfun__roots__finite,axiom,
% 9.49/5.29      ! [C: nat > real,K: nat,N: nat] :
% 9.49/5.29        ( ( ( C @ K )
% 9.49/5.29         != zero_zero_real )
% 9.49/5.29       => ( ( ord_less_eq_nat @ K @ N )
% 9.49/5.29         => ( finite_finite_real
% 9.49/5.29            @ ( collect_real
% 9.49/5.29              @ ^ [Z3: real] :
% 9.49/5.29                  ( ( groups6591440286371151544t_real
% 9.49/5.29                    @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z3 @ I4 ) )
% 9.49/5.29                    @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29                  = zero_zero_real ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6852_polyfun__linear__factor__root,axiom,
% 9.49/5.29      ! [C: nat > complex,A: complex,N: nat] :
% 9.49/5.29        ( ( ( groups2073611262835488442omplex
% 9.49/5.29            @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ A @ I4 ) )
% 9.49/5.29            @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29          = zero_zero_complex )
% 9.49/5.29       => ~ ! [B3: nat > complex] :
% 9.49/5.29              ~ ! [Z5: complex] :
% 9.49/5.29                  ( ( groups2073611262835488442omplex
% 9.49/5.29                    @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z5 @ I4 ) )
% 9.49/5.29                    @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29                  = ( times_times_complex @ ( minus_minus_complex @ Z5 @ A )
% 9.49/5.29                    @ ( groups2073611262835488442omplex
% 9.49/5.29                      @ ^ [I4: nat] : ( times_times_complex @ ( B3 @ I4 ) @ ( power_power_complex @ Z5 @ I4 ) )
% 9.49/5.29                      @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6853_polyfun__linear__factor__root,axiom,
% 9.49/5.29      ! [C: nat > rat,A: rat,N: nat] :
% 9.49/5.29        ( ( ( groups2906978787729119204at_rat
% 9.49/5.29            @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ A @ I4 ) )
% 9.49/5.29            @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29          = zero_zero_rat )
% 9.49/5.29       => ~ ! [B3: nat > rat] :
% 9.49/5.29              ~ ! [Z5: rat] :
% 9.49/5.29                  ( ( groups2906978787729119204at_rat
% 9.49/5.29                    @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ Z5 @ I4 ) )
% 9.49/5.29                    @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29                  = ( times_times_rat @ ( minus_minus_rat @ Z5 @ A )
% 9.49/5.29                    @ ( groups2906978787729119204at_rat
% 9.49/5.29                      @ ^ [I4: nat] : ( times_times_rat @ ( B3 @ I4 ) @ ( power_power_rat @ Z5 @ I4 ) )
% 9.49/5.29                      @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6854_polyfun__linear__factor__root,axiom,
% 9.49/5.29      ! [C: nat > int,A: int,N: nat] :
% 9.49/5.29        ( ( ( groups3539618377306564664at_int
% 9.49/5.29            @ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ A @ I4 ) )
% 9.49/5.29            @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29          = zero_zero_int )
% 9.49/5.29       => ~ ! [B3: nat > int] :
% 9.49/5.29              ~ ! [Z5: int] :
% 9.49/5.29                  ( ( groups3539618377306564664at_int
% 9.49/5.29                    @ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ Z5 @ I4 ) )
% 9.49/5.29                    @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29                  = ( times_times_int @ ( minus_minus_int @ Z5 @ A )
% 9.49/5.29                    @ ( groups3539618377306564664at_int
% 9.49/5.29                      @ ^ [I4: nat] : ( times_times_int @ ( B3 @ I4 ) @ ( power_power_int @ Z5 @ I4 ) )
% 9.49/5.29                      @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6855_polyfun__linear__factor__root,axiom,
% 9.49/5.29      ! [C: nat > real,A: real,N: nat] :
% 9.49/5.29        ( ( ( groups6591440286371151544t_real
% 9.49/5.29            @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ A @ I4 ) )
% 9.49/5.29            @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29          = zero_zero_real )
% 9.49/5.29       => ~ ! [B3: nat > real] :
% 9.49/5.29              ~ ! [Z5: real] :
% 9.49/5.29                  ( ( groups6591440286371151544t_real
% 9.49/5.29                    @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z5 @ I4 ) )
% 9.49/5.29                    @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29                  = ( times_times_real @ ( minus_minus_real @ Z5 @ A )
% 9.49/5.29                    @ ( groups6591440286371151544t_real
% 9.49/5.29                      @ ^ [I4: nat] : ( times_times_real @ ( B3 @ I4 ) @ ( power_power_real @ Z5 @ I4 ) )
% 9.49/5.29                      @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6856_sum__power__shift,axiom,
% 9.49/5.29      ! [M: nat,N: nat,X: complex] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29          = ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6857_sum__power__shift,axiom,
% 9.49/5.29      ! [M: nat,N: nat,X: rat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29          = ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6858_sum__power__shift,axiom,
% 9.49/5.29      ! [M: nat,N: nat,X: int] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29       => ( ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29          = ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6859_sum__power__shift,axiom,
% 9.49/5.29      ! [M: nat,N: nat,X: real] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.49/5.29          = ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6860_sum_Otriangle__reindex,axiom,
% 9.49/5.29      ! [G: nat > nat > nat,N: nat] :
% 9.49/5.29        ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
% 9.49/5.29          @ ( collec3392354462482085612at_nat
% 9.49/5.29            @ ( produc6081775807080527818_nat_o
% 9.49/5.29              @ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 9.49/5.29        = ( groups3542108847815614940at_nat
% 9.49/5.29          @ ^ [K3: nat] :
% 9.49/5.29              ( groups3542108847815614940at_nat
% 9.49/5.29              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ K3 ) )
% 9.49/5.29          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6861_sum_Otriangle__reindex,axiom,
% 9.49/5.29      ! [G: nat > nat > real,N: nat] :
% 9.49/5.29        ( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
% 9.49/5.29          @ ( collec3392354462482085612at_nat
% 9.49/5.29            @ ( produc6081775807080527818_nat_o
% 9.49/5.29              @ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 9.49/5.29        = ( groups6591440286371151544t_real
% 9.49/5.29          @ ^ [K3: nat] :
% 9.49/5.29              ( groups6591440286371151544t_real
% 9.49/5.29              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ K3 ) )
% 9.49/5.29          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6862_polynomial__product,axiom,
% 9.49/5.29      ! [M: nat,A: nat > complex,N: nat,B: nat > complex,X: complex] :
% 9.49/5.29        ( ! [I2: nat] :
% 9.49/5.29            ( ( ord_less_nat @ M @ I2 )
% 9.49/5.29           => ( ( A @ I2 )
% 9.49/5.29              = zero_zero_complex ) )
% 9.49/5.29       => ( ! [J2: nat] :
% 9.49/5.29              ( ( ord_less_nat @ N @ J2 )
% 9.49/5.29             => ( ( B @ J2 )
% 9.49/5.29                = zero_zero_complex ) )
% 9.49/5.29         => ( ( times_times_complex
% 9.49/5.29              @ ( groups2073611262835488442omplex
% 9.49/5.29                @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X @ I4 ) )
% 9.49/5.29                @ ( set_ord_atMost_nat @ M ) )
% 9.49/5.29              @ ( groups2073611262835488442omplex
% 9.49/5.29                @ ^ [J3: nat] : ( times_times_complex @ ( B @ J3 ) @ ( power_power_complex @ X @ J3 ) )
% 9.49/5.29                @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29            = ( groups2073611262835488442omplex
% 9.49/5.29              @ ^ [R5: nat] :
% 9.49/5.29                  ( times_times_complex
% 9.49/5.29                  @ ( groups2073611262835488442omplex
% 9.49/5.29                    @ ^ [K3: nat] : ( times_times_complex @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
% 9.49/5.29                    @ ( set_ord_atMost_nat @ R5 ) )
% 9.49/5.29                  @ ( power_power_complex @ X @ R5 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6863_polynomial__product,axiom,
% 9.49/5.29      ! [M: nat,A: nat > rat,N: nat,B: nat > rat,X: rat] :
% 9.49/5.29        ( ! [I2: nat] :
% 9.49/5.29            ( ( ord_less_nat @ M @ I2 )
% 9.49/5.29           => ( ( A @ I2 )
% 9.49/5.29              = zero_zero_rat ) )
% 9.49/5.29       => ( ! [J2: nat] :
% 9.49/5.29              ( ( ord_less_nat @ N @ J2 )
% 9.49/5.29             => ( ( B @ J2 )
% 9.49/5.29                = zero_zero_rat ) )
% 9.49/5.29         => ( ( times_times_rat
% 9.49/5.29              @ ( groups2906978787729119204at_rat
% 9.49/5.29                @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ X @ I4 ) )
% 9.49/5.29                @ ( set_ord_atMost_nat @ M ) )
% 9.49/5.29              @ ( groups2906978787729119204at_rat
% 9.49/5.29                @ ^ [J3: nat] : ( times_times_rat @ ( B @ J3 ) @ ( power_power_rat @ X @ J3 ) )
% 9.49/5.29                @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29            = ( groups2906978787729119204at_rat
% 9.49/5.29              @ ^ [R5: nat] :
% 9.49/5.29                  ( times_times_rat
% 9.49/5.29                  @ ( groups2906978787729119204at_rat
% 9.49/5.29                    @ ^ [K3: nat] : ( times_times_rat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
% 9.49/5.29                    @ ( set_ord_atMost_nat @ R5 ) )
% 9.49/5.29                  @ ( power_power_rat @ X @ R5 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6864_polynomial__product,axiom,
% 9.49/5.29      ! [M: nat,A: nat > int,N: nat,B: nat > int,X: int] :
% 9.49/5.29        ( ! [I2: nat] :
% 9.49/5.29            ( ( ord_less_nat @ M @ I2 )
% 9.49/5.29           => ( ( A @ I2 )
% 9.49/5.29              = zero_zero_int ) )
% 9.49/5.29       => ( ! [J2: nat] :
% 9.49/5.29              ( ( ord_less_nat @ N @ J2 )
% 9.49/5.29             => ( ( B @ J2 )
% 9.49/5.29                = zero_zero_int ) )
% 9.49/5.29         => ( ( times_times_int
% 9.49/5.29              @ ( groups3539618377306564664at_int
% 9.49/5.29                @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ X @ I4 ) )
% 9.49/5.29                @ ( set_ord_atMost_nat @ M ) )
% 9.49/5.29              @ ( groups3539618377306564664at_int
% 9.49/5.29                @ ^ [J3: nat] : ( times_times_int @ ( B @ J3 ) @ ( power_power_int @ X @ J3 ) )
% 9.49/5.29                @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29            = ( groups3539618377306564664at_int
% 9.49/5.29              @ ^ [R5: nat] :
% 9.49/5.29                  ( times_times_int
% 9.49/5.29                  @ ( groups3539618377306564664at_int
% 9.49/5.29                    @ ^ [K3: nat] : ( times_times_int @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
% 9.49/5.29                    @ ( set_ord_atMost_nat @ R5 ) )
% 9.49/5.29                  @ ( power_power_int @ X @ R5 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6865_polynomial__product,axiom,
% 9.49/5.29      ! [M: nat,A: nat > real,N: nat,B: nat > real,X: real] :
% 9.49/5.29        ( ! [I2: nat] :
% 9.49/5.29            ( ( ord_less_nat @ M @ I2 )
% 9.49/5.29           => ( ( A @ I2 )
% 9.49/5.29              = zero_zero_real ) )
% 9.49/5.29       => ( ! [J2: nat] :
% 9.49/5.29              ( ( ord_less_nat @ N @ J2 )
% 9.49/5.29             => ( ( B @ J2 )
% 9.49/5.29                = zero_zero_real ) )
% 9.49/5.29         => ( ( times_times_real
% 9.49/5.29              @ ( groups6591440286371151544t_real
% 9.49/5.29                @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ X @ I4 ) )
% 9.49/5.29                @ ( set_ord_atMost_nat @ M ) )
% 9.49/5.29              @ ( groups6591440286371151544t_real
% 9.49/5.29                @ ^ [J3: nat] : ( times_times_real @ ( B @ J3 ) @ ( power_power_real @ X @ J3 ) )
% 9.49/5.29                @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29            = ( groups6591440286371151544t_real
% 9.49/5.29              @ ^ [R5: nat] :
% 9.49/5.29                  ( times_times_real
% 9.49/5.29                  @ ( groups6591440286371151544t_real
% 9.49/5.29                    @ ^ [K3: nat] : ( times_times_real @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
% 9.49/5.29                    @ ( set_ord_atMost_nat @ R5 ) )
% 9.49/5.29                  @ ( power_power_real @ X @ R5 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6866_polyfun__eq__const,axiom,
% 9.49/5.29      ! [C: nat > complex,N: nat,K: complex] :
% 9.49/5.29        ( ( ! [X3: complex] :
% 9.49/5.29              ( ( groups2073611262835488442omplex
% 9.49/5.29                @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X3 @ I4 ) )
% 9.49/5.29                @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29              = K ) )
% 9.49/5.29        = ( ( ( C @ zero_zero_nat )
% 9.49/5.29            = K )
% 9.49/5.29          & ! [X3: nat] :
% 9.49/5.29              ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
% 9.49/5.29             => ( ( C @ X3 )
% 9.49/5.29                = zero_zero_complex ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6867_polyfun__eq__const,axiom,
% 9.49/5.29      ! [C: nat > real,N: nat,K: real] :
% 9.49/5.29        ( ( ! [X3: real] :
% 9.49/5.29              ( ( groups6591440286371151544t_real
% 9.49/5.29                @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X3 @ I4 ) )
% 9.49/5.29                @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29              = K ) )
% 9.49/5.29        = ( ( ( C @ zero_zero_nat )
% 9.49/5.29            = K )
% 9.49/5.29          & ! [X3: nat] :
% 9.49/5.29              ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
% 9.49/5.29             => ( ( C @ X3 )
% 9.49/5.29                = zero_zero_real ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6868_polynomial__product__nat,axiom,
% 9.49/5.29      ! [M: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
% 9.49/5.29        ( ! [I2: nat] :
% 9.49/5.29            ( ( ord_less_nat @ M @ I2 )
% 9.49/5.29           => ( ( A @ I2 )
% 9.49/5.29              = zero_zero_nat ) )
% 9.49/5.29       => ( ! [J2: nat] :
% 9.49/5.29              ( ( ord_less_nat @ N @ J2 )
% 9.49/5.29             => ( ( B @ J2 )
% 9.49/5.29                = zero_zero_nat ) )
% 9.49/5.29         => ( ( times_times_nat
% 9.49/5.29              @ ( groups3542108847815614940at_nat
% 9.49/5.29                @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( power_power_nat @ X @ I4 ) )
% 9.49/5.29                @ ( set_ord_atMost_nat @ M ) )
% 9.49/5.29              @ ( groups3542108847815614940at_nat
% 9.49/5.29                @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
% 9.49/5.29                @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29            = ( groups3542108847815614940at_nat
% 9.49/5.29              @ ^ [R5: nat] :
% 9.49/5.29                  ( times_times_nat
% 9.49/5.29                  @ ( groups3542108847815614940at_nat
% 9.49/5.29                    @ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
% 9.49/5.29                    @ ( set_ord_atMost_nat @ R5 ) )
% 9.49/5.29                  @ ( power_power_nat @ X @ R5 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6869_and__int_Opinduct,axiom,
% 9.49/5.29      ! [A0: int,A12: int,P: int > int > $o] :
% 9.49/5.29        ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 9.49/5.29       => ( ! [K2: int,L4: int] :
% 9.49/5.29              ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L4 ) )
% 9.49/5.29             => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.49/5.29                      & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.49/5.29                 => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.49/5.29               => ( P @ K2 @ L4 ) ) )
% 9.49/5.29         => ( P @ A0 @ A12 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6870_sum_Ozero__middle,axiom,
% 9.49/5.29      ! [P4: nat,K: nat,G: nat > complex,H2: nat > complex] :
% 9.49/5.29        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 9.49/5.29       => ( ( ord_less_eq_nat @ K @ P4 )
% 9.49/5.29         => ( ( groups2073611262835488442omplex
% 9.49/5.29              @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K ) @ zero_zero_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ P4 ) )
% 9.49/5.29            = ( groups2073611262835488442omplex
% 9.49/5.29              @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6871_sum_Ozero__middle,axiom,
% 9.49/5.29      ! [P4: nat,K: nat,G: nat > rat,H2: nat > rat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 9.49/5.29       => ( ( ord_less_eq_nat @ K @ P4 )
% 9.49/5.29         => ( ( groups2906978787729119204at_rat
% 9.49/5.29              @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ zero_zero_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ P4 ) )
% 9.49/5.29            = ( groups2906978787729119204at_rat
% 9.49/5.29              @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6872_sum_Ozero__middle,axiom,
% 9.49/5.29      ! [P4: nat,K: nat,G: nat > int,H2: nat > int] :
% 9.49/5.29        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 9.49/5.29       => ( ( ord_less_eq_nat @ K @ P4 )
% 9.49/5.29         => ( ( groups3539618377306564664at_int
% 9.49/5.29              @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ zero_zero_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ P4 ) )
% 9.49/5.29            = ( groups3539618377306564664at_int
% 9.49/5.29              @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6873_sum_Ozero__middle,axiom,
% 9.49/5.29      ! [P4: nat,K: nat,G: nat > nat,H2: nat > nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 9.49/5.29       => ( ( ord_less_eq_nat @ K @ P4 )
% 9.49/5.29         => ( ( groups3542108847815614940at_nat
% 9.49/5.29              @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ zero_zero_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ P4 ) )
% 9.49/5.29            = ( groups3542108847815614940at_nat
% 9.49/5.29              @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6874_sum_Ozero__middle,axiom,
% 9.49/5.29      ! [P4: nat,K: nat,G: nat > real,H2: nat > real] :
% 9.49/5.29        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 9.49/5.29       => ( ( ord_less_eq_nat @ K @ P4 )
% 9.49/5.29         => ( ( groups6591440286371151544t_real
% 9.49/5.29              @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ zero_zero_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ P4 ) )
% 9.49/5.29            = ( groups6591440286371151544t_real
% 9.49/5.29              @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6875_root__polyfun,axiom,
% 9.49/5.29      ! [N: nat,Z2: complex,A: complex] :
% 9.49/5.29        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.49/5.29       => ( ( ( power_power_complex @ Z2 @ N )
% 9.49/5.29            = A )
% 9.49/5.29          = ( ( groups2073611262835488442omplex
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_complex @ ( if_complex @ ( I4 = zero_zero_nat ) @ ( uminus1482373934393186551omplex @ A ) @ ( if_complex @ ( I4 = N ) @ one_one_complex @ zero_zero_complex ) ) @ ( power_power_complex @ Z2 @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29            = zero_zero_complex ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6876_root__polyfun,axiom,
% 9.49/5.29      ! [N: nat,Z2: int,A: int] :
% 9.49/5.29        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.49/5.29       => ( ( ( power_power_int @ Z2 @ N )
% 9.49/5.29            = A )
% 9.49/5.29          = ( ( groups3539618377306564664at_int
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_int @ ( if_int @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_int @ A ) @ ( if_int @ ( I4 = N ) @ one_one_int @ zero_zero_int ) ) @ ( power_power_int @ Z2 @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29            = zero_zero_int ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6877_root__polyfun,axiom,
% 9.49/5.29      ! [N: nat,Z2: rat,A: rat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.49/5.29       => ( ( ( power_power_rat @ Z2 @ N )
% 9.49/5.29            = A )
% 9.49/5.29          = ( ( groups2906978787729119204at_rat
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_rat @ ( if_rat @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_rat @ A ) @ ( if_rat @ ( I4 = N ) @ one_one_rat @ zero_zero_rat ) ) @ ( power_power_rat @ Z2 @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29            = zero_zero_rat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6878_root__polyfun,axiom,
% 9.49/5.29      ! [N: nat,Z2: code_integer,A: code_integer] :
% 9.49/5.29        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.49/5.29       => ( ( ( power_8256067586552552935nteger @ Z2 @ N )
% 9.49/5.29            = A )
% 9.49/5.29          = ( ( groups7501900531339628137nteger
% 9.49/5.29              @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( if_Code_integer @ ( I4 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ A ) @ ( if_Code_integer @ ( I4 = N ) @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) @ ( power_8256067586552552935nteger @ Z2 @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29            = zero_z3403309356797280102nteger ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6879_root__polyfun,axiom,
% 9.49/5.29      ! [N: nat,Z2: real,A: real] :
% 9.49/5.29        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.49/5.29       => ( ( ( power_power_real @ Z2 @ N )
% 9.49/5.29            = A )
% 9.49/5.29          = ( ( groups6591440286371151544t_real
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_real @ ( if_real @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_real @ A ) @ ( if_real @ ( I4 = N ) @ one_one_real @ zero_zero_real ) ) @ ( power_power_real @ Z2 @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29            = zero_zero_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6880_polyfun__diff__alt,axiom,
% 9.49/5.29      ! [N: nat,A: nat > complex,X: complex,Y: complex] :
% 9.49/5.29        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.49/5.29       => ( ( minus_minus_complex
% 9.49/5.29            @ ( groups2073611262835488442omplex
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29            @ ( groups2073611262835488442omplex
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ Y @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29          = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
% 9.49/5.29            @ ( groups2073611262835488442omplex
% 9.49/5.29              @ ^ [J3: nat] :
% 9.49/5.29                  ( groups2073611262835488442omplex
% 9.49/5.29                  @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_complex @ Y @ K3 ) ) @ ( power_power_complex @ X @ J3 ) )
% 9.49/5.29                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
% 9.49/5.29              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6881_polyfun__diff__alt,axiom,
% 9.49/5.29      ! [N: nat,A: nat > rat,X: rat,Y: rat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.49/5.29       => ( ( minus_minus_rat
% 9.49/5.29            @ ( groups2906978787729119204at_rat
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ X @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29            @ ( groups2906978787729119204at_rat
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ Y @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29          = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
% 9.49/5.29            @ ( groups2906978787729119204at_rat
% 9.49/5.29              @ ^ [J3: nat] :
% 9.49/5.29                  ( groups2906978787729119204at_rat
% 9.49/5.29                  @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_rat @ Y @ K3 ) ) @ ( power_power_rat @ X @ J3 ) )
% 9.49/5.29                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
% 9.49/5.29              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6882_polyfun__diff__alt,axiom,
% 9.49/5.29      ! [N: nat,A: nat > int,X: int,Y: int] :
% 9.49/5.29        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.49/5.29       => ( ( minus_minus_int
% 9.49/5.29            @ ( groups3539618377306564664at_int
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ X @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29            @ ( groups3539618377306564664at_int
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ Y @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29          = ( times_times_int @ ( minus_minus_int @ X @ Y )
% 9.49/5.29            @ ( groups3539618377306564664at_int
% 9.49/5.29              @ ^ [J3: nat] :
% 9.49/5.29                  ( groups3539618377306564664at_int
% 9.49/5.29                  @ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_int @ Y @ K3 ) ) @ ( power_power_int @ X @ J3 ) )
% 9.49/5.29                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
% 9.49/5.29              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6883_polyfun__diff__alt,axiom,
% 9.49/5.29      ! [N: nat,A: nat > real,X: real,Y: real] :
% 9.49/5.29        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.49/5.29       => ( ( minus_minus_real
% 9.49/5.29            @ ( groups6591440286371151544t_real
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ X @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29            @ ( groups6591440286371151544t_real
% 9.49/5.29              @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ Y @ I4 ) )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29          = ( times_times_real @ ( minus_minus_real @ X @ Y )
% 9.49/5.29            @ ( groups6591440286371151544t_real
% 9.49/5.29              @ ^ [J3: nat] :
% 9.49/5.29                  ( groups6591440286371151544t_real
% 9.49/5.29                  @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_real @ Y @ K3 ) ) @ ( power_power_real @ X @ J3 ) )
% 9.49/5.29                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
% 9.49/5.29              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6884_divide__int__unfold,axiom,
% 9.49/5.29      ! [L: int,K: int,N: nat,M: nat] :
% 9.49/5.29        ( ( ( ( ( sgn_sgn_int @ L )
% 9.49/5.29              = zero_zero_int )
% 9.49/5.29            | ( ( sgn_sgn_int @ K )
% 9.49/5.29              = zero_zero_int )
% 9.49/5.29            | ( N = zero_zero_nat ) )
% 9.49/5.29         => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.49/5.29            = zero_zero_int ) )
% 9.49/5.29        & ( ~ ( ( ( sgn_sgn_int @ L )
% 9.49/5.29                = zero_zero_int )
% 9.49/5.29              | ( ( sgn_sgn_int @ K )
% 9.49/5.29                = zero_zero_int )
% 9.49/5.29              | ( N = zero_zero_nat ) )
% 9.49/5.29         => ( ( ( ( sgn_sgn_int @ K )
% 9.49/5.29                = ( sgn_sgn_int @ L ) )
% 9.49/5.29             => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.49/5.29                = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
% 9.49/5.29            & ( ( ( sgn_sgn_int @ K )
% 9.49/5.29               != ( sgn_sgn_int @ L ) )
% 9.49/5.29             => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.49/5.29                = ( uminus_uminus_int
% 9.49/5.29                  @ ( semiri1314217659103216013at_int
% 9.49/5.29                    @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
% 9.49/5.29                      @ ( zero_n2687167440665602831ol_nat
% 9.49/5.29                        @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6885_polyfun__extremal__lemma,axiom,
% 9.49/5.29      ! [E2: real,C: nat > complex,N: nat] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.49/5.29       => ? [M8: real] :
% 9.49/5.29          ! [Z5: complex] :
% 9.49/5.29            ( ( ord_less_eq_real @ M8 @ ( real_V1022390504157884413omplex @ Z5 ) )
% 9.49/5.29           => ( ord_less_eq_real
% 9.49/5.29              @ ( real_V1022390504157884413omplex
% 9.49/5.29                @ ( groups2073611262835488442omplex
% 9.49/5.29                  @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z5 @ I4 ) )
% 9.49/5.29                  @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29              @ ( times_times_real @ E2 @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( suc @ N ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6886_polyfun__extremal__lemma,axiom,
% 9.49/5.29      ! [E2: real,C: nat > real,N: nat] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.49/5.29       => ? [M8: real] :
% 9.49/5.29          ! [Z5: real] :
% 9.49/5.29            ( ( ord_less_eq_real @ M8 @ ( real_V7735802525324610683m_real @ Z5 ) )
% 9.49/5.29           => ( ord_less_eq_real
% 9.49/5.29              @ ( real_V7735802525324610683m_real
% 9.49/5.29                @ ( groups6591440286371151544t_real
% 9.49/5.29                  @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z5 @ I4 ) )
% 9.49/5.29                  @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29              @ ( times_times_real @ E2 @ ( power_power_real @ ( real_V7735802525324610683m_real @ Z5 ) @ ( suc @ N ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6887_choose__odd__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.29       => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 9.49/5.29            @ ( groups2073611262835488442omplex
% 9.49/5.29              @ ^ [I4: nat] :
% 9.49/5.29                  ( if_complex
% 9.49/5.29                  @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 9.49/5.29                  @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) )
% 9.49/5.29                  @ zero_zero_complex )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29          = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6888_choose__odd__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.29       => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 9.49/5.29            @ ( groups2906978787729119204at_rat
% 9.49/5.29              @ ^ [I4: nat] :
% 9.49/5.29                  ( if_rat
% 9.49/5.29                  @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 9.49/5.29                  @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) )
% 9.49/5.29                  @ zero_zero_rat )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29          = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6889_choose__odd__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.29       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 9.49/5.29            @ ( groups3539618377306564664at_int
% 9.49/5.29              @ ^ [I4: nat] :
% 9.49/5.29                  ( if_int
% 9.49/5.29                  @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 9.49/5.29                  @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) )
% 9.49/5.29                  @ zero_zero_int )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29          = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6890_choose__odd__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.29       => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 9.49/5.29            @ ( groups6591440286371151544t_real
% 9.49/5.29              @ ^ [I4: nat] :
% 9.49/5.29                  ( if_real
% 9.49/5.29                  @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 9.49/5.29                  @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) )
% 9.49/5.29                  @ zero_zero_real )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6891_choose__even__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.29       => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 9.49/5.29            @ ( groups2073611262835488442omplex
% 9.49/5.29              @ ^ [I4: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) @ zero_zero_complex )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29          = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6892_choose__even__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.29       => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 9.49/5.29            @ ( groups2906978787729119204at_rat
% 9.49/5.29              @ ^ [I4: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) @ zero_zero_rat )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29          = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6893_choose__even__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.29       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 9.49/5.29            @ ( groups3539618377306564664at_int
% 9.49/5.29              @ ^ [I4: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) @ zero_zero_int )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29          = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6894_choose__even__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.29       => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 9.49/5.29            @ ( groups6591440286371151544t_real
% 9.49/5.29              @ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) @ zero_zero_real )
% 9.49/5.29              @ ( set_ord_atMost_nat @ N ) ) )
% 9.49/5.29          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6895_choose__alternating__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.29       => ( ( groups2073611262835488442omplex
% 9.49/5.29            @ ^ [I4: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I4 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) )
% 9.49/5.29            @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29          = zero_zero_complex ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6896_choose__alternating__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.29       => ( ( groups2906978787729119204at_rat
% 9.49/5.29            @ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I4 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) )
% 9.49/5.29            @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29          = zero_zero_rat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6897_choose__alternating__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.29       => ( ( groups7501900531339628137nteger
% 9.49/5.29            @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I4 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I4 ) ) )
% 9.49/5.29            @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29          = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6898_choose__alternating__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.29       => ( ( groups3539618377306564664at_int
% 9.49/5.29            @ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I4 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) )
% 9.49/5.29            @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29          = zero_zero_int ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6899_choose__alternating__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.29       => ( ( groups6591440286371151544t_real
% 9.49/5.29            @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) )
% 9.49/5.29            @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29          = zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6900_choose__alternating__linear__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( N != one_one_nat )
% 9.49/5.29       => ( ( groups2073611262835488442omplex
% 9.49/5.29            @ ^ [I4: nat] : ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I4 ) @ ( semiri8010041392384452111omplex @ I4 ) ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) )
% 9.49/5.29            @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29          = zero_zero_complex ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6901_choose__alternating__linear__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( N != one_one_nat )
% 9.49/5.29       => ( ( groups2906978787729119204at_rat
% 9.49/5.29            @ ^ [I4: nat] : ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I4 ) @ ( semiri681578069525770553at_rat @ I4 ) ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) )
% 9.49/5.29            @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29          = zero_zero_rat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6902_choose__alternating__linear__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( N != one_one_nat )
% 9.49/5.29       => ( ( groups7501900531339628137nteger
% 9.49/5.29            @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I4 ) @ ( semiri4939895301339042750nteger @ I4 ) ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I4 ) ) )
% 9.49/5.29            @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29          = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6903_choose__alternating__linear__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( N != one_one_nat )
% 9.49/5.29       => ( ( groups3539618377306564664at_int
% 9.49/5.29            @ ^ [I4: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I4 ) @ ( semiri1314217659103216013at_int @ I4 ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) )
% 9.49/5.29            @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29          = zero_zero_int ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6904_choose__alternating__linear__sum,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( N != one_one_nat )
% 9.49/5.29       => ( ( groups6591440286371151544t_real
% 9.49/5.29            @ ^ [I4: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( semiri5074537144036343181t_real @ I4 ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) )
% 9.49/5.29            @ ( set_ord_atMost_nat @ N ) )
% 9.49/5.29          = zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6905_zero__less__binomial__iff,axiom,
% 9.49/5.29      ! [N: nat,K: nat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
% 9.49/5.29        = ( ord_less_eq_nat @ K @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6906_geometric__sums,axiom,
% 9.49/5.29      ! [C: real] :
% 9.49/5.29        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 9.49/5.29       => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6907_geometric__sums,axiom,
% 9.49/5.29      ! [C: complex] :
% 9.49/5.29        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 9.49/5.29       => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6908_sums__zero,axiom,
% 9.49/5.29      ( sums_complex
% 9.49/5.29      @ ^ [N2: nat] : zero_zero_complex
% 9.49/5.29      @ zero_zero_complex ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6909_sums__zero,axiom,
% 9.49/5.29      ( sums_real
% 9.49/5.29      @ ^ [N2: nat] : zero_zero_real
% 9.49/5.29      @ zero_zero_real ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6910_sums__zero,axiom,
% 9.49/5.29      ( sums_nat
% 9.49/5.29      @ ^ [N2: nat] : zero_zero_nat
% 9.49/5.29      @ zero_zero_nat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6911_sums__zero,axiom,
% 9.49/5.29      ( sums_int
% 9.49/5.29      @ ^ [N2: nat] : zero_zero_int
% 9.49/5.29      @ zero_zero_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6912_binomial__1,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
% 9.49/5.29        = N ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6913_binomial__0__Suc,axiom,
% 9.49/5.29      ! [K: nat] :
% 9.49/5.29        ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
% 9.49/5.29        = zero_zero_nat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6914_binomial__eq__0__iff,axiom,
% 9.49/5.29      ! [N: nat,K: nat] :
% 9.49/5.29        ( ( ( binomial @ N @ K )
% 9.49/5.29          = zero_zero_nat )
% 9.49/5.29        = ( ord_less_nat @ N @ K ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6915_binomial__n__0,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( binomial @ N @ zero_zero_nat )
% 9.49/5.29        = one_one_nat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6916_sgn__real__def,axiom,
% 9.49/5.29      ( sgn_sgn_real
% 9.49/5.29      = ( ^ [A4: real] : ( if_real @ ( A4 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A4 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6917_sums__le,axiom,
% 9.49/5.29      ! [F: nat > nat,G: nat > nat,S: nat,T: nat] :
% 9.49/5.29        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
% 9.49/5.29       => ( ( sums_nat @ F @ S )
% 9.49/5.29         => ( ( sums_nat @ G @ T )
% 9.49/5.29           => ( ord_less_eq_nat @ S @ T ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6918_sums__le,axiom,
% 9.49/5.29      ! [F: nat > int,G: nat > int,S: int,T: int] :
% 9.49/5.29        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
% 9.49/5.29       => ( ( sums_int @ F @ S )
% 9.49/5.29         => ( ( sums_int @ G @ T )
% 9.49/5.29           => ( ord_less_eq_int @ S @ T ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6919_sums__0,axiom,
% 9.49/5.29      ! [F: nat > complex] :
% 9.49/5.29        ( ! [N3: nat] :
% 9.49/5.29            ( ( F @ N3 )
% 9.49/5.29            = zero_zero_complex )
% 9.49/5.29       => ( sums_complex @ F @ zero_zero_complex ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6920_sums__0,axiom,
% 9.49/5.29      ! [F: nat > real] :
% 9.49/5.29        ( ! [N3: nat] :
% 9.49/5.29            ( ( F @ N3 )
% 9.49/5.29            = zero_zero_real )
% 9.49/5.29       => ( sums_real @ F @ zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6921_sums__0,axiom,
% 9.49/5.29      ! [F: nat > nat] :
% 9.49/5.29        ( ! [N3: nat] :
% 9.49/5.29            ( ( F @ N3 )
% 9.49/5.29            = zero_zero_nat )
% 9.49/5.29       => ( sums_nat @ F @ zero_zero_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6922_sums__0,axiom,
% 9.49/5.29      ! [F: nat > int] :
% 9.49/5.29        ( ! [N3: nat] :
% 9.49/5.29            ( ( F @ N3 )
% 9.49/5.29            = zero_zero_int )
% 9.49/5.29       => ( sums_int @ F @ zero_zero_int ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6923_sums__single,axiom,
% 9.49/5.29      ! [I: nat,F: nat > complex] :
% 9.49/5.29        ( sums_complex
% 9.49/5.29        @ ^ [R5: nat] : ( if_complex @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_complex )
% 9.49/5.29        @ ( F @ I ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6924_sums__single,axiom,
% 9.49/5.29      ! [I: nat,F: nat > real] :
% 9.49/5.29        ( sums_real
% 9.49/5.29        @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real )
% 9.49/5.29        @ ( F @ I ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6925_sums__single,axiom,
% 9.49/5.29      ! [I: nat,F: nat > nat] :
% 9.49/5.29        ( sums_nat
% 9.49/5.29        @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat )
% 9.49/5.29        @ ( F @ I ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6926_sums__single,axiom,
% 9.49/5.29      ! [I: nat,F: nat > int] :
% 9.49/5.29        ( sums_int
% 9.49/5.29        @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int )
% 9.49/5.29        @ ( F @ I ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6927_binomial__eq__0,axiom,
% 9.49/5.29      ! [N: nat,K: nat] :
% 9.49/5.29        ( ( ord_less_nat @ N @ K )
% 9.49/5.29       => ( ( binomial @ N @ K )
% 9.49/5.29          = zero_zero_nat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6928_binomial__symmetric,axiom,
% 9.49/5.29      ! [K: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ K @ N )
% 9.49/5.29       => ( ( binomial @ N @ K )
% 9.49/5.29          = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6929_binomial__le__pow,axiom,
% 9.49/5.29      ! [R2: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ R2 @ N )
% 9.49/5.29       => ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6930_sums__mult__iff,axiom,
% 9.49/5.29      ! [C: complex,F: nat > complex,D: complex] :
% 9.49/5.29        ( ( C != zero_zero_complex )
% 9.49/5.29       => ( ( sums_complex
% 9.49/5.29            @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
% 9.49/5.29            @ ( times_times_complex @ C @ D ) )
% 9.49/5.29          = ( sums_complex @ F @ D ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6931_sums__mult__iff,axiom,
% 9.49/5.29      ! [C: real,F: nat > real,D: real] :
% 9.49/5.29        ( ( C != zero_zero_real )
% 9.49/5.29       => ( ( sums_real
% 9.49/5.29            @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
% 9.49/5.29            @ ( times_times_real @ C @ D ) )
% 9.49/5.29          = ( sums_real @ F @ D ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6932_sums__mult2__iff,axiom,
% 9.49/5.29      ! [C: complex,F: nat > complex,D: complex] :
% 9.49/5.29        ( ( C != zero_zero_complex )
% 9.49/5.29       => ( ( sums_complex
% 9.49/5.29            @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ C )
% 9.49/5.29            @ ( times_times_complex @ D @ C ) )
% 9.49/5.29          = ( sums_complex @ F @ D ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6933_sums__mult2__iff,axiom,
% 9.49/5.29      ! [C: real,F: nat > real,D: real] :
% 9.49/5.29        ( ( C != zero_zero_real )
% 9.49/5.29       => ( ( sums_real
% 9.49/5.29            @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C )
% 9.49/5.29            @ ( times_times_real @ D @ C ) )
% 9.49/5.29          = ( sums_real @ F @ D ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6934_zero__less__binomial,axiom,
% 9.49/5.29      ! [K: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ K @ N )
% 9.49/5.29       => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6935_choose__mult,axiom,
% 9.49/5.29      ! [K: nat,M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ K @ M )
% 9.49/5.29       => ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29         => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
% 9.49/5.29            = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6936_sums__mult__D,axiom,
% 9.49/5.29      ! [C: complex,F: nat > complex,A: complex] :
% 9.49/5.29        ( ( sums_complex
% 9.49/5.29          @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
% 9.49/5.29          @ A )
% 9.49/5.29       => ( ( C != zero_zero_complex )
% 9.49/5.29         => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6937_sums__mult__D,axiom,
% 9.49/5.29      ! [C: real,F: nat > real,A: real] :
% 9.49/5.29        ( ( sums_real
% 9.49/5.29          @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
% 9.49/5.29          @ A )
% 9.49/5.29       => ( ( C != zero_zero_real )
% 9.49/5.29         => ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6938_sums__Suc__imp,axiom,
% 9.49/5.29      ! [F: nat > complex,S: complex] :
% 9.49/5.29        ( ( ( F @ zero_zero_nat )
% 9.49/5.29          = zero_zero_complex )
% 9.49/5.29       => ( ( sums_complex
% 9.49/5.29            @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 9.49/5.29            @ S )
% 9.49/5.29         => ( sums_complex @ F @ S ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6939_sums__Suc__imp,axiom,
% 9.49/5.29      ! [F: nat > real,S: real] :
% 9.49/5.29        ( ( ( F @ zero_zero_nat )
% 9.49/5.29          = zero_zero_real )
% 9.49/5.29       => ( ( sums_real
% 9.49/5.29            @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 9.49/5.29            @ S )
% 9.49/5.29         => ( sums_real @ F @ S ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6940_sums__Suc,axiom,
% 9.49/5.29      ! [F: nat > real,L: real] :
% 9.49/5.29        ( ( sums_real
% 9.49/5.29          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 9.49/5.29          @ L )
% 9.49/5.29       => ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6941_sums__Suc,axiom,
% 9.49/5.29      ! [F: nat > nat,L: nat] :
% 9.49/5.29        ( ( sums_nat
% 9.49/5.29          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 9.49/5.29          @ L )
% 9.49/5.29       => ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6942_sums__Suc,axiom,
% 9.49/5.29      ! [F: nat > int,L: int] :
% 9.49/5.29        ( ( sums_int
% 9.49/5.29          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 9.49/5.29          @ L )
% 9.49/5.29       => ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6943_sums__Suc__iff,axiom,
% 9.49/5.29      ! [F: nat > real,S: real] :
% 9.49/5.29        ( ( sums_real
% 9.49/5.29          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 9.49/5.29          @ S )
% 9.49/5.29        = ( sums_real @ F @ ( plus_plus_real @ S @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6944_sums__zero__iff__shift,axiom,
% 9.49/5.29      ! [N: nat,F: nat > complex,S: complex] :
% 9.49/5.29        ( ! [I2: nat] :
% 9.49/5.29            ( ( ord_less_nat @ I2 @ N )
% 9.49/5.29           => ( ( F @ I2 )
% 9.49/5.29              = zero_zero_complex ) )
% 9.49/5.29       => ( ( sums_complex
% 9.49/5.29            @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 9.49/5.29            @ S )
% 9.49/5.29          = ( sums_complex @ F @ S ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6945_sums__zero__iff__shift,axiom,
% 9.49/5.29      ! [N: nat,F: nat > real,S: real] :
% 9.49/5.29        ( ! [I2: nat] :
% 9.49/5.29            ( ( ord_less_nat @ I2 @ N )
% 9.49/5.29           => ( ( F @ I2 )
% 9.49/5.29              = zero_zero_real ) )
% 9.49/5.29       => ( ( sums_real
% 9.49/5.29            @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 9.49/5.29            @ S )
% 9.49/5.29          = ( sums_real @ F @ S ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6946_sums__finite,axiom,
% 9.49/5.29      ! [N6: set_nat,F: nat > complex] :
% 9.49/5.29        ( ( finite_finite_nat @ N6 )
% 9.49/5.29       => ( ! [N3: nat] :
% 9.49/5.29              ( ~ ( member_nat @ N3 @ N6 )
% 9.49/5.29             => ( ( F @ N3 )
% 9.49/5.29                = zero_zero_complex ) )
% 9.49/5.29         => ( sums_complex @ F @ ( groups2073611262835488442omplex @ F @ N6 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6947_sums__finite,axiom,
% 9.49/5.29      ! [N6: set_nat,F: nat > int] :
% 9.49/5.29        ( ( finite_finite_nat @ N6 )
% 9.49/5.29       => ( ! [N3: nat] :
% 9.49/5.29              ( ~ ( member_nat @ N3 @ N6 )
% 9.49/5.29             => ( ( F @ N3 )
% 9.49/5.29                = zero_zero_int ) )
% 9.49/5.29         => ( sums_int @ F @ ( groups3539618377306564664at_int @ F @ N6 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6948_sums__finite,axiom,
% 9.49/5.29      ! [N6: set_nat,F: nat > nat] :
% 9.49/5.29        ( ( finite_finite_nat @ N6 )
% 9.49/5.29       => ( ! [N3: nat] :
% 9.49/5.29              ( ~ ( member_nat @ N3 @ N6 )
% 9.49/5.29             => ( ( F @ N3 )
% 9.49/5.29                = zero_zero_nat ) )
% 9.49/5.29         => ( sums_nat @ F @ ( groups3542108847815614940at_nat @ F @ N6 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6949_sums__finite,axiom,
% 9.49/5.29      ! [N6: set_nat,F: nat > real] :
% 9.49/5.29        ( ( finite_finite_nat @ N6 )
% 9.49/5.29       => ( ! [N3: nat] :
% 9.49/5.29              ( ~ ( member_nat @ N3 @ N6 )
% 9.49/5.29             => ( ( F @ N3 )
% 9.49/5.29                = zero_zero_real ) )
% 9.49/5.29         => ( sums_real @ F @ ( groups6591440286371151544t_real @ F @ N6 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6950_sums__If__finite,axiom,
% 9.49/5.29      ! [P: nat > $o,F: nat > complex] :
% 9.49/5.29        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.49/5.29       => ( sums_complex
% 9.49/5.29          @ ^ [R5: nat] : ( if_complex @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_complex )
% 9.49/5.29          @ ( groups2073611262835488442omplex @ F @ ( collect_nat @ P ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6951_sums__If__finite,axiom,
% 9.49/5.29      ! [P: nat > $o,F: nat > int] :
% 9.49/5.29        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.49/5.29       => ( sums_int
% 9.49/5.29          @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int )
% 9.49/5.29          @ ( groups3539618377306564664at_int @ F @ ( collect_nat @ P ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6952_sums__If__finite,axiom,
% 9.49/5.29      ! [P: nat > $o,F: nat > nat] :
% 9.49/5.29        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.49/5.29       => ( sums_nat
% 9.49/5.29          @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat )
% 9.49/5.29          @ ( groups3542108847815614940at_nat @ F @ ( collect_nat @ P ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6953_sums__If__finite,axiom,
% 9.49/5.29      ! [P: nat > $o,F: nat > real] :
% 9.49/5.29        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.49/5.29       => ( sums_real
% 9.49/5.29          @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real )
% 9.49/5.29          @ ( groups6591440286371151544t_real @ F @ ( collect_nat @ P ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6954_sums__If__finite__set,axiom,
% 9.49/5.29      ! [A2: set_nat,F: nat > complex] :
% 9.49/5.29        ( ( finite_finite_nat @ A2 )
% 9.49/5.29       => ( sums_complex
% 9.49/5.29          @ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_complex )
% 9.49/5.29          @ ( groups2073611262835488442omplex @ F @ A2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6955_sums__If__finite__set,axiom,
% 9.49/5.29      ! [A2: set_nat,F: nat > int] :
% 9.49/5.29        ( ( finite_finite_nat @ A2 )
% 9.49/5.29       => ( sums_int
% 9.49/5.29          @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_int )
% 9.49/5.29          @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6956_sums__If__finite__set,axiom,
% 9.49/5.29      ! [A2: set_nat,F: nat > nat] :
% 9.49/5.29        ( ( finite_finite_nat @ A2 )
% 9.49/5.29       => ( sums_nat
% 9.49/5.29          @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_nat )
% 9.49/5.29          @ ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6957_sums__If__finite__set,axiom,
% 9.49/5.29      ! [A2: set_nat,F: nat > real] :
% 9.49/5.29        ( ( finite_finite_nat @ A2 )
% 9.49/5.29       => ( sums_real
% 9.49/5.29          @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_real )
% 9.49/5.29          @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6958_powser__sums__if,axiom,
% 9.49/5.29      ! [M: nat,Z2: complex] :
% 9.49/5.29        ( sums_complex
% 9.49/5.29        @ ^ [N2: nat] : ( times_times_complex @ ( if_complex @ ( N2 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z2 @ N2 ) )
% 9.49/5.29        @ ( power_power_complex @ Z2 @ M ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6959_powser__sums__if,axiom,
% 9.49/5.29      ! [M: nat,Z2: real] :
% 9.49/5.29        ( sums_real
% 9.49/5.29        @ ^ [N2: nat] : ( times_times_real @ ( if_real @ ( N2 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z2 @ N2 ) )
% 9.49/5.29        @ ( power_power_real @ Z2 @ M ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6960_powser__sums__if,axiom,
% 9.49/5.29      ! [M: nat,Z2: int] :
% 9.49/5.29        ( sums_int
% 9.49/5.29        @ ^ [N2: nat] : ( times_times_int @ ( if_int @ ( N2 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z2 @ N2 ) )
% 9.49/5.29        @ ( power_power_int @ Z2 @ M ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6961_sums__If__finite__set_H,axiom,
% 9.49/5.29      ! [G: nat > real,S2: real,A2: set_nat,S5: real,F: nat > real] :
% 9.49/5.29        ( ( sums_real @ G @ S2 )
% 9.49/5.29       => ( ( finite_finite_nat @ A2 )
% 9.49/5.29         => ( ( S5
% 9.49/5.29              = ( plus_plus_real @ S2
% 9.49/5.29                @ ( groups6591440286371151544t_real
% 9.49/5.29                  @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 9.49/5.29                  @ A2 ) ) )
% 9.49/5.29           => ( sums_real
% 9.49/5.29              @ ^ [N2: nat] : ( if_real @ ( member_nat @ N2 @ A2 ) @ ( F @ N2 ) @ ( G @ N2 ) )
% 9.49/5.29              @ S5 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6962_binomial__ge__n__over__k__pow__k,axiom,
% 9.49/5.29      ! [K: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ K @ N )
% 9.49/5.29       => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6963_binomial__ge__n__over__k__pow__k,axiom,
% 9.49/5.29      ! [K: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ K @ N )
% 9.49/5.29       => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6964_binomial__le__pow2,axiom,
% 9.49/5.29      ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6965_choose__reduce__nat,axiom,
% 9.49/5.29      ! [N: nat,K: nat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.29       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.29         => ( ( binomial @ N @ K )
% 9.49/5.29            = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6966_times__binomial__minus1__eq,axiom,
% 9.49/5.29      ! [K: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.29       => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
% 9.49/5.29          = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6967_sum__choose__diagonal,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.49/5.29       => ( ( groups3542108847815614940at_nat
% 9.49/5.29            @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
% 9.49/5.29            @ ( set_ord_atMost_nat @ M ) )
% 9.49/5.29          = ( binomial @ ( suc @ N ) @ M ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6968_binomial__addition__formula,axiom,
% 9.49/5.29      ! [N: nat,K: nat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.49/5.29       => ( ( binomial @ N @ ( suc @ K ) )
% 9.49/5.29          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6969_upto_Opinduct,axiom,
% 9.49/5.29      ! [A0: int,A12: int,P: int > int > $o] :
% 9.49/5.29        ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 9.49/5.29       => ( ! [I2: int,J2: int] :
% 9.49/5.29              ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
% 9.49/5.29             => ( ( ( ord_less_eq_int @ I2 @ J2 )
% 9.49/5.29                 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
% 9.49/5.29               => ( P @ I2 @ J2 ) ) )
% 9.49/5.29         => ( P @ A0 @ A12 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6970_gchoose__row__sum__weighted,axiom,
% 9.49/5.29      ! [R2: complex,M: nat] :
% 9.49/5.29        ( ( groups2073611262835488442omplex
% 9.49/5.29          @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ R2 @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
% 9.49/5.29          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 9.49/5.29        = ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R2 @ ( suc @ M ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6971_gchoose__row__sum__weighted,axiom,
% 9.49/5.29      ! [R2: rat,M: nat] :
% 9.49/5.29        ( ( groups2906978787729119204at_rat
% 9.49/5.29          @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ R2 @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
% 9.49/5.29          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 9.49/5.29        = ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R2 @ ( suc @ M ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6972_gchoose__row__sum__weighted,axiom,
% 9.49/5.29      ! [R2: real,M: nat] :
% 9.49/5.29        ( ( groups6591440286371151544t_real
% 9.49/5.29          @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ R2 @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ R2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
% 9.49/5.29          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 9.49/5.29        = ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R2 @ ( suc @ M ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6973_sqrt__sum__squares__half__less,axiom,
% 9.49/5.29      ! [X: real,U: real,Y: real] :
% 9.49/5.29        ( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.49/5.29       => ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.49/5.29         => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.49/5.29           => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.49/5.29             => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6974_log2__of__power__le,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.29       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.49/5.29         => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6975_neg__numeral__le__ceiling,axiom,
% 9.49/5.29      ! [V: num,X: real] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
% 9.49/5.29        = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6976_neg__numeral__le__ceiling,axiom,
% 9.49/5.29      ! [V: num,X: rat] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
% 9.49/5.29        = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6977_ceiling__less__neg__numeral,axiom,
% 9.49/5.29      ! [X: real,V: num] :
% 9.49/5.29        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.49/5.29        = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6978_ceiling__less__neg__numeral,axiom,
% 9.49/5.29      ! [X: rat,V: num] :
% 9.49/5.29        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.49/5.29        = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6979_real__sqrt__less__iff,axiom,
% 9.49/5.29      ! [X: real,Y: real] :
% 9.49/5.29        ( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
% 9.49/5.29        = ( ord_less_real @ X @ Y ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6980_gbinomial__0_I2_J,axiom,
% 9.49/5.29      ! [K: nat] :
% 9.49/5.29        ( ( gbinomial_complex @ zero_zero_complex @ ( suc @ K ) )
% 9.49/5.29        = zero_zero_complex ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6981_gbinomial__0_I2_J,axiom,
% 9.49/5.29      ! [K: nat] :
% 9.49/5.29        ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
% 9.49/5.29        = zero_zero_real ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6982_gbinomial__0_I2_J,axiom,
% 9.49/5.29      ! [K: nat] :
% 9.49/5.29        ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
% 9.49/5.29        = zero_zero_rat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6983_gbinomial__0_I2_J,axiom,
% 9.49/5.29      ! [K: nat] :
% 9.49/5.29        ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
% 9.49/5.29        = zero_zero_nat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6984_gbinomial__0_I2_J,axiom,
% 9.49/5.29      ! [K: nat] :
% 9.49/5.29        ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
% 9.49/5.29        = zero_zero_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6985_real__sqrt__gt__0__iff,axiom,
% 9.49/5.29      ! [Y: real] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y ) )
% 9.49/5.29        = ( ord_less_real @ zero_zero_real @ Y ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6986_real__sqrt__lt__0__iff,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
% 9.49/5.29        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6987_ceiling__zero,axiom,
% 9.49/5.29      ( ( archim2889992004027027881ng_rat @ zero_zero_rat )
% 9.49/5.29      = zero_zero_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6988_ceiling__zero,axiom,
% 9.49/5.29      ( ( archim7802044766580827645g_real @ zero_zero_real )
% 9.49/5.29      = zero_zero_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6989_gbinomial__0_I1_J,axiom,
% 9.49/5.29      ! [A: complex] :
% 9.49/5.29        ( ( gbinomial_complex @ A @ zero_zero_nat )
% 9.49/5.29        = one_one_complex ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6990_gbinomial__0_I1_J,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( gbinomial_real @ A @ zero_zero_nat )
% 9.49/5.29        = one_one_real ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6991_gbinomial__0_I1_J,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( gbinomial_rat @ A @ zero_zero_nat )
% 9.49/5.29        = one_one_rat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6992_gbinomial__0_I1_J,axiom,
% 9.49/5.29      ! [A: nat] :
% 9.49/5.29        ( ( gbinomial_nat @ A @ zero_zero_nat )
% 9.49/5.29        = one_one_nat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6993_gbinomial__0_I1_J,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( gbinomial_int @ A @ zero_zero_nat )
% 9.49/5.29        = one_one_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6994_real__sqrt__gt__1__iff,axiom,
% 9.49/5.29      ! [Y: real] :
% 9.49/5.29        ( ( ord_less_real @ one_one_real @ ( sqrt @ Y ) )
% 9.49/5.29        = ( ord_less_real @ one_one_real @ Y ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6995_real__sqrt__lt__1__iff,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
% 9.49/5.29        = ( ord_less_real @ X @ one_one_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6996_log__eq__one,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.29       => ( ( A != one_one_real )
% 9.49/5.29         => ( ( log @ A @ A )
% 9.49/5.29            = one_one_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6997_log__less__cancel__iff,axiom,
% 9.49/5.29      ! [A: real,X: real,Y: real] :
% 9.49/5.29        ( ( ord_less_real @ one_one_real @ A )
% 9.49/5.29       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.29         => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.49/5.29           => ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
% 9.49/5.29              = ( ord_less_real @ X @ Y ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6998_log__less__one__cancel__iff,axiom,
% 9.49/5.29      ! [A: real,X: real] :
% 9.49/5.29        ( ( ord_less_real @ one_one_real @ A )
% 9.49/5.29       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.29         => ( ( ord_less_real @ ( log @ A @ X ) @ one_one_real )
% 9.49/5.29            = ( ord_less_real @ X @ A ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_6999_one__less__log__cancel__iff,axiom,
% 9.49/5.29      ! [A: real,X: real] :
% 9.49/5.29        ( ( ord_less_real @ one_one_real @ A )
% 9.49/5.29       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.29         => ( ( ord_less_real @ one_one_real @ ( log @ A @ X ) )
% 9.49/5.29            = ( ord_less_real @ A @ X ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7000_log__less__zero__cancel__iff,axiom,
% 9.49/5.29      ! [A: real,X: real] :
% 9.49/5.29        ( ( ord_less_real @ one_one_real @ A )
% 9.49/5.29       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.29         => ( ( ord_less_real @ ( log @ A @ X ) @ zero_zero_real )
% 9.49/5.29            = ( ord_less_real @ X @ one_one_real ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7001_zero__less__log__cancel__iff,axiom,
% 9.49/5.29      ! [A: real,X: real] :
% 9.49/5.29        ( ( ord_less_real @ one_one_real @ A )
% 9.49/5.29       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.29         => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X ) )
% 9.49/5.29            = ( ord_less_real @ one_one_real @ X ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7002_ceiling__le__zero,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
% 9.49/5.29        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7003_ceiling__le__zero,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
% 9.49/5.29        = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7004_zero__less__ceiling,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29        ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
% 9.49/5.29        = ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7005_zero__less__ceiling,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
% 9.49/5.29        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7006_ceiling__le__numeral,axiom,
% 9.49/5.29      ! [X: real,V: num] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
% 9.49/5.29        = ( ord_less_eq_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7007_ceiling__le__numeral,axiom,
% 9.49/5.29      ! [X: rat,V: num] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 9.49/5.29        = ( ord_less_eq_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7008_ceiling__less__one,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
% 9.49/5.29        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7009_ceiling__less__one,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
% 9.49/5.29        = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7010_numeral__less__ceiling,axiom,
% 9.49/5.29      ! [V: num,X: real] :
% 9.49/5.29        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
% 9.49/5.29        = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7011_numeral__less__ceiling,axiom,
% 9.49/5.29      ! [V: num,X: rat] :
% 9.49/5.29        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
% 9.49/5.29        = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7012_one__le__ceiling,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29        ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
% 9.49/5.29        = ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7013_one__le__ceiling,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
% 9.49/5.29        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7014_ceiling__le__one,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
% 9.49/5.29        = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7015_ceiling__le__one,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
% 9.49/5.29        = ( ord_less_eq_rat @ X @ one_one_rat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7016_one__less__ceiling,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29        ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
% 9.49/5.29        = ( ord_less_rat @ one_one_rat @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7017_one__less__ceiling,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
% 9.49/5.29        = ( ord_less_real @ one_one_real @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7018_zero__le__log__cancel__iff,axiom,
% 9.49/5.29      ! [A: real,X: real] :
% 9.49/5.29        ( ( ord_less_real @ one_one_real @ A )
% 9.49/5.29       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.29         => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
% 9.49/5.29            = ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7019_log__le__zero__cancel__iff,axiom,
% 9.49/5.29      ! [A: real,X: real] :
% 9.49/5.29        ( ( ord_less_real @ one_one_real @ A )
% 9.49/5.29       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.29         => ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
% 9.49/5.29            = ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7020_one__le__log__cancel__iff,axiom,
% 9.49/5.29      ! [A: real,X: real] :
% 9.49/5.29        ( ( ord_less_real @ one_one_real @ A )
% 9.49/5.29       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.29         => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
% 9.49/5.29            = ( ord_less_eq_real @ A @ X ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7021_log__le__one__cancel__iff,axiom,
% 9.49/5.29      ! [A: real,X: real] :
% 9.49/5.29        ( ( ord_less_real @ one_one_real @ A )
% 9.49/5.29       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.29         => ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
% 9.49/5.29            = ( ord_less_eq_real @ X @ A ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7022_log__le__cancel__iff,axiom,
% 9.49/5.29      ! [A: real,X: real,Y: real] :
% 9.49/5.29        ( ( ord_less_real @ one_one_real @ A )
% 9.49/5.29       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.29         => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.49/5.29           => ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
% 9.49/5.29              = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7023_ceiling__less__zero,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
% 9.49/5.29        = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7024_ceiling__less__zero,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
% 9.49/5.29        = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7025_zero__le__ceiling,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
% 9.49/5.29        = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7026_zero__le__ceiling,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29        ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
% 9.49/5.29        = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7027_log__pow__cancel,axiom,
% 9.49/5.29      ! [A: real,B: nat] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.29       => ( ( A != one_one_real )
% 9.49/5.29         => ( ( log @ A @ ( power_power_real @ A @ B ) )
% 9.49/5.29            = ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7028_ceiling__less__numeral,axiom,
% 9.49/5.29      ! [X: real,V: num] :
% 9.49/5.29        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
% 9.49/5.29        = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7029_ceiling__less__numeral,axiom,
% 9.49/5.29      ! [X: rat,V: num] :
% 9.49/5.29        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 9.49/5.29        = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7030_numeral__le__ceiling,axiom,
% 9.49/5.29      ! [V: num,X: real] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
% 9.49/5.29        = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7031_numeral__le__ceiling,axiom,
% 9.49/5.29      ! [V: num,X: rat] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
% 9.49/5.29        = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7032_ceiling__le__neg__numeral,axiom,
% 9.49/5.29      ! [X: real,V: num] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.49/5.29        = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7033_ceiling__le__neg__numeral,axiom,
% 9.49/5.29      ! [X: rat,V: num] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.49/5.29        = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7034_neg__numeral__less__ceiling,axiom,
% 9.49/5.29      ! [V: num,X: real] :
% 9.49/5.29        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
% 9.49/5.29        = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7035_neg__numeral__less__ceiling,axiom,
% 9.49/5.29      ! [V: num,X: rat] :
% 9.49/5.29        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
% 9.49/5.29        = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7036_real__sqrt__less__mono,axiom,
% 9.49/5.29      ! [X: real,Y: real] :
% 9.49/5.29        ( ( ord_less_real @ X @ Y )
% 9.49/5.29       => ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7037_real__sqrt__gt__zero,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.29       => ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7038_ceiling__mono,axiom,
% 9.49/5.29      ! [Y: real,X: real] :
% 9.49/5.29        ( ( ord_less_eq_real @ Y @ X )
% 9.49/5.29       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y ) @ ( archim7802044766580827645g_real @ X ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7039_ceiling__mono,axiom,
% 9.49/5.29      ! [Y: rat,X: rat] :
% 9.49/5.29        ( ( ord_less_eq_rat @ Y @ X )
% 9.49/5.29       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y ) @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7040_le__of__int__ceiling,axiom,
% 9.49/5.29      ! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7041_le__of__int__ceiling,axiom,
% 9.49/5.29      ! [X: rat] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7042_ceiling__less__cancel,axiom,
% 9.49/5.29      ! [X: rat,Y: rat] :
% 9.49/5.29        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) )
% 9.49/5.29       => ( ord_less_rat @ X @ Y ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7043_ceiling__less__cancel,axiom,
% 9.49/5.29      ! [X: real,Y: real] :
% 9.49/5.29        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) )
% 9.49/5.29       => ( ord_less_real @ X @ Y ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7044_ceiling__le__iff,axiom,
% 9.49/5.29      ! [X: real,Z2: int] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z2 )
% 9.49/5.29        = ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7045_ceiling__le__iff,axiom,
% 9.49/5.29      ! [X: rat,Z2: int] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 )
% 9.49/5.29        = ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7046_ceiling__le,axiom,
% 9.49/5.29      ! [X: real,A: int] :
% 9.49/5.29        ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) )
% 9.49/5.29       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7047_ceiling__le,axiom,
% 9.49/5.29      ! [X: rat,A: int] :
% 9.49/5.29        ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) )
% 9.49/5.29       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7048_less__ceiling__iff,axiom,
% 9.49/5.29      ! [Z2: int,X: rat] :
% 9.49/5.29        ( ( ord_less_int @ Z2 @ ( archim2889992004027027881ng_rat @ X ) )
% 9.49/5.29        = ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7049_less__ceiling__iff,axiom,
% 9.49/5.29      ! [Z2: int,X: real] :
% 9.49/5.29        ( ( ord_less_int @ Z2 @ ( archim7802044766580827645g_real @ X ) )
% 9.49/5.29        = ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7050_gbinomial__of__nat__symmetric,axiom,
% 9.49/5.29      ! [K: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ K @ N )
% 9.49/5.29       => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
% 9.49/5.29          = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7051_sqrt2__less__2,axiom,
% 9.49/5.29      ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7052_of__int__ceiling__le__add__one,axiom,
% 9.49/5.29      ! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7053_of__int__ceiling__le__add__one,axiom,
% 9.49/5.29      ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7054_log__base__change,axiom,
% 9.49/5.29      ! [A: real,B: real,X: real] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.29       => ( ( A != one_one_real )
% 9.49/5.29         => ( ( log @ B @ X )
% 9.49/5.29            = ( divide_divide_real @ ( log @ A @ X ) @ ( log @ A @ B ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7055_of__int__ceiling__diff__one__le,axiom,
% 9.49/5.29      ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7056_of__int__ceiling__diff__one__le,axiom,
% 9.49/5.29      ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7057_log__of__power__eq,axiom,
% 9.49/5.29      ! [M: nat,B: real,N: nat] :
% 9.49/5.29        ( ( ( semiri5074537144036343181t_real @ M )
% 9.49/5.29          = ( power_power_real @ B @ N ) )
% 9.49/5.29       => ( ( ord_less_real @ one_one_real @ B )
% 9.49/5.29         => ( ( semiri5074537144036343181t_real @ N )
% 9.49/5.29            = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7058_less__log__of__power,axiom,
% 9.49/5.29      ! [B: real,N: nat,M: real] :
% 9.49/5.29        ( ( ord_less_real @ ( power_power_real @ B @ N ) @ M )
% 9.49/5.29       => ( ( ord_less_real @ one_one_real @ B )
% 9.49/5.29         => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7059_gbinomial__ge__n__over__k__pow__k,axiom,
% 9.49/5.29      ! [K: nat,A: real] :
% 9.49/5.29        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
% 9.49/5.29       => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7060_gbinomial__ge__n__over__k__pow__k,axiom,
% 9.49/5.29      ! [K: nat,A: rat] :
% 9.49/5.29        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
% 9.49/5.29       => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7061_real__less__rsqrt,axiom,
% 9.49/5.29      ! [X: real,Y: real] :
% 9.49/5.29        ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
% 9.49/5.29       => ( ord_less_real @ X @ ( sqrt @ Y ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7062_ceiling__split,axiom,
% 9.49/5.29      ! [P: int > $o,T: real] :
% 9.49/5.29        ( ( P @ ( archim7802044766580827645g_real @ T ) )
% 9.49/5.29        = ( ! [I4: int] :
% 9.49/5.29              ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) @ T )
% 9.49/5.29                & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I4 ) ) )
% 9.49/5.29             => ( P @ I4 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7063_ceiling__split,axiom,
% 9.49/5.29      ! [P: int > $o,T: rat] :
% 9.49/5.29        ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
% 9.49/5.29        = ( ! [I4: int] :
% 9.49/5.29              ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) @ T )
% 9.49/5.29                & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I4 ) ) )
% 9.49/5.29             => ( P @ I4 ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7064_ceiling__eq__iff,axiom,
% 9.49/5.29      ! [X: real,A: int] :
% 9.49/5.29        ( ( ( archim7802044766580827645g_real @ X )
% 9.49/5.29          = A )
% 9.49/5.29        = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X )
% 9.49/5.29          & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7065_ceiling__eq__iff,axiom,
% 9.49/5.29      ! [X: rat,A: int] :
% 9.49/5.29        ( ( ( archim2889992004027027881ng_rat @ X )
% 9.49/5.29          = A )
% 9.49/5.29        = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X )
% 9.49/5.29          & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7066_ceiling__unique,axiom,
% 9.49/5.29      ! [Z2: int,X: real] :
% 9.49/5.29        ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X )
% 9.49/5.29       => ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) )
% 9.49/5.29         => ( ( archim7802044766580827645g_real @ X )
% 9.49/5.29            = Z2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7067_ceiling__unique,axiom,
% 9.49/5.29      ! [Z2: int,X: rat] :
% 9.49/5.29        ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X )
% 9.49/5.29       => ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z2 ) )
% 9.49/5.29         => ( ( archim2889992004027027881ng_rat @ X )
% 9.49/5.29            = Z2 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7068_ceiling__correct,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) @ one_one_real ) @ X )
% 9.49/5.29        & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7069_ceiling__correct,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29        ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) @ one_one_rat ) @ X )
% 9.49/5.29        & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7070_mult__ceiling__le,axiom,
% 9.49/5.29      ! [A: real,B: real] :
% 9.49/5.29        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.29       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.49/5.29         => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7071_mult__ceiling__le,axiom,
% 9.49/5.29      ! [A: rat,B: rat] :
% 9.49/5.29        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.29       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.49/5.29         => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7072_log__mult,axiom,
% 9.49/5.29      ! [A: real,X: real,Y: real] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.29       => ( ( A != one_one_real )
% 9.49/5.29         => ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.29           => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.49/5.29             => ( ( log @ A @ ( times_times_real @ X @ Y ) )
% 9.49/5.29                = ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7073_ceiling__less__iff,axiom,
% 9.49/5.29      ! [X: real,Z2: int] :
% 9.49/5.29        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ Z2 )
% 9.49/5.29        = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7074_ceiling__less__iff,axiom,
% 9.49/5.29      ! [X: rat,Z2: int] :
% 9.49/5.29        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 )
% 9.49/5.29        = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7075_le__ceiling__iff,axiom,
% 9.49/5.29      ! [Z2: int,X: rat] :
% 9.49/5.29        ( ( ord_less_eq_int @ Z2 @ ( archim2889992004027027881ng_rat @ X ) )
% 9.49/5.29        = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7076_le__ceiling__iff,axiom,
% 9.49/5.29      ! [Z2: int,X: real] :
% 9.49/5.29        ( ( ord_less_eq_int @ Z2 @ ( archim7802044766580827645g_real @ X ) )
% 9.49/5.29        = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7077_log__divide,axiom,
% 9.49/5.29      ! [A: real,X: real,Y: real] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.29       => ( ( A != one_one_real )
% 9.49/5.29         => ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.29           => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.49/5.29             => ( ( log @ A @ ( divide_divide_real @ X @ Y ) )
% 9.49/5.29                = ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7078_le__log__of__power,axiom,
% 9.49/5.29      ! [B: real,N: nat,M: real] :
% 9.49/5.29        ( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M )
% 9.49/5.29       => ( ( ord_less_real @ one_one_real @ B )
% 9.49/5.29         => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7079_log__base__pow,axiom,
% 9.49/5.29      ! [A: real,N: nat,X: real] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.29       => ( ( log @ ( power_power_real @ A @ N ) @ X )
% 9.49/5.29          = ( divide_divide_real @ ( log @ A @ X ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7080_log__nat__power,axiom,
% 9.49/5.29      ! [X: real,B: real,N: nat] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.29       => ( ( log @ B @ ( power_power_real @ X @ N ) )
% 9.49/5.29          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7081_gbinomial__trinomial__revision,axiom,
% 9.49/5.29      ! [K: nat,M: nat,A: complex] :
% 9.49/5.29        ( ( ord_less_eq_nat @ K @ M )
% 9.49/5.29       => ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
% 9.49/5.29          = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7082_gbinomial__trinomial__revision,axiom,
% 9.49/5.29      ! [K: nat,M: nat,A: rat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ K @ M )
% 9.49/5.29       => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
% 9.49/5.29          = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7083_gbinomial__trinomial__revision,axiom,
% 9.49/5.29      ! [K: nat,M: nat,A: real] :
% 9.49/5.29        ( ( ord_less_eq_nat @ K @ M )
% 9.49/5.29       => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
% 9.49/5.29          = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7084_ceiling__log2__div2,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.29       => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.49/5.29          = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7085_ceiling__log__nat__eq__if,axiom,
% 9.49/5.29      ! [B: nat,N: nat,K: nat] :
% 9.49/5.29        ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 9.49/5.29       => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 9.49/5.29         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 9.49/5.29           => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 9.49/5.29              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7086_ceiling__divide__upper,axiom,
% 9.49/5.29      ! [Q4: real,P4: real] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ Q4 )
% 9.49/5.29       => ( ord_less_eq_real @ P4 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P4 @ Q4 ) ) ) @ Q4 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7087_ceiling__divide__upper,axiom,
% 9.49/5.29      ! [Q4: rat,P4: rat] :
% 9.49/5.29        ( ( ord_less_rat @ zero_zero_rat @ Q4 )
% 9.49/5.29       => ( ord_less_eq_rat @ P4 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P4 @ Q4 ) ) ) @ Q4 ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7088_lemma__real__divide__sqrt__less,axiom,
% 9.49/5.29      ! [U: real] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ U )
% 9.49/5.29       => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7089_log__of__power__less,axiom,
% 9.49/5.29      ! [M: nat,B: real,N: nat] :
% 9.49/5.29        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 9.49/5.29       => ( ( ord_less_real @ one_one_real @ B )
% 9.49/5.29         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.49/5.29           => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7090_ceiling__log__nat__eq__powr__iff,axiom,
% 9.49/5.29      ! [B: nat,K: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 9.49/5.29       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.29         => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 9.49/5.29              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
% 9.49/5.29            = ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 9.49/5.29              & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7091_real__less__lsqrt,axiom,
% 9.49/5.29      ! [X: real,Y: real] :
% 9.49/5.29        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.49/5.29       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.49/5.29         => ( ( ord_less_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.49/5.29           => ( ord_less_real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7092_ceiling__divide__lower,axiom,
% 9.49/5.29      ! [Q4: real,P4: real] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ Q4 )
% 9.49/5.29       => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P4 @ Q4 ) ) ) @ one_one_real ) @ Q4 ) @ P4 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7093_ceiling__divide__lower,axiom,
% 9.49/5.29      ! [Q4: rat,P4: rat] :
% 9.49/5.29        ( ( ord_less_rat @ zero_zero_rat @ Q4 )
% 9.49/5.29       => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P4 @ Q4 ) ) ) @ one_one_rat ) @ Q4 ) @ P4 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7094_log__of__power__le,axiom,
% 9.49/5.29      ! [M: nat,B: real,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 9.49/5.29       => ( ( ord_less_real @ one_one_real @ B )
% 9.49/5.29         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.49/5.29           => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7095_ceiling__eq,axiom,
% 9.49/5.29      ! [N: int,X: real] :
% 9.49/5.29        ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
% 9.49/5.29       => ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 9.49/5.29         => ( ( archim7802044766580827645g_real @ X )
% 9.49/5.29            = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7096_ceiling__eq,axiom,
% 9.49/5.29      ! [N: int,X: rat] :
% 9.49/5.29        ( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X )
% 9.49/5.29       => ( ( ord_less_eq_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
% 9.49/5.29         => ( ( archim2889992004027027881ng_rat @ X )
% 9.49/5.29            = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7097_gbinomial__reduce__nat,axiom,
% 9.49/5.29      ! [K: nat,A: complex] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.29       => ( ( gbinomial_complex @ A @ K )
% 9.49/5.29          = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7098_gbinomial__reduce__nat,axiom,
% 9.49/5.29      ! [K: nat,A: real] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.29       => ( ( gbinomial_real @ A @ K )
% 9.49/5.29          = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7099_gbinomial__reduce__nat,axiom,
% 9.49/5.29      ! [K: nat,A: rat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.29       => ( ( gbinomial_rat @ A @ K )
% 9.49/5.29          = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7100_arsinh__real__aux,axiom,
% 9.49/5.29      ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7101_less__log2__of__power,axiom,
% 9.49/5.29      ! [N: nat,M: nat] :
% 9.49/5.29        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 9.49/5.29       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7102_le__log2__of__power,axiom,
% 9.49/5.29      ! [N: nat,M: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 9.49/5.29       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7103_gbinomial__sum__up__index,axiom,
% 9.49/5.29      ! [K: nat,N: nat] :
% 9.49/5.29        ( ( groups2073611262835488442omplex
% 9.49/5.29          @ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ K )
% 9.49/5.29          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.49/5.29        = ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7104_gbinomial__sum__up__index,axiom,
% 9.49/5.29      ! [K: nat,N: nat] :
% 9.49/5.29        ( ( groups2906978787729119204at_rat
% 9.49/5.29          @ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K )
% 9.49/5.29          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.49/5.29        = ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7105_gbinomial__sum__up__index,axiom,
% 9.49/5.29      ! [K: nat,N: nat] :
% 9.49/5.29        ( ( groups6591440286371151544t_real
% 9.49/5.29          @ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K )
% 9.49/5.29          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.49/5.29        = ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7106_log2__of__power__less,axiom,
% 9.49/5.29      ! [M: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.49/5.29       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.49/5.29         => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7107_gbinomial__absorption_H,axiom,
% 9.49/5.29      ! [K: nat,A: complex] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.29       => ( ( gbinomial_complex @ A @ K )
% 9.49/5.29          = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7108_gbinomial__absorption_H,axiom,
% 9.49/5.29      ! [K: nat,A: rat] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.29       => ( ( gbinomial_rat @ A @ K )
% 9.49/5.29          = ( times_times_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7109_gbinomial__absorption_H,axiom,
% 9.49/5.29      ! [K: nat,A: real] :
% 9.49/5.29        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.29       => ( ( gbinomial_real @ A @ K )
% 9.49/5.29          = ( times_times_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7110_floor__log__nat__eq__powr__iff,axiom,
% 9.49/5.29      ! [B: nat,K: nat,N: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 9.49/5.29       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.49/5.29         => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 9.49/5.29              = ( semiri1314217659103216013at_int @ N ) )
% 9.49/5.29            = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 9.49/5.29              & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7111_ceiling__log__eq__powr__iff,axiom,
% 9.49/5.29      ! [X: real,B: real,K: nat] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.29       => ( ( ord_less_real @ one_one_real @ B )
% 9.49/5.29         => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
% 9.49/5.29              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 9.49/5.29            = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
% 9.49/5.29              & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7112_floor__log__nat__eq__if,axiom,
% 9.49/5.29      ! [B: nat,N: nat,K: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 9.49/5.29       => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 9.49/5.29         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 9.49/5.29           => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 9.49/5.29              = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7113_floor__log2__div2,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.49/5.29       => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.49/5.29          = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7114_real__sqrt__sum__squares__less,axiom,
% 9.49/5.29      ! [X: real,U: real,Y: real] :
% 9.49/5.29        ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.49/5.29       => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.49/5.29         => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7115_dbl__dec__simps_I2_J,axiom,
% 9.49/5.29      ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 9.49/5.29      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7116_dbl__dec__simps_I2_J,axiom,
% 9.49/5.29      ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 9.49/5.29      = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7117_dbl__dec__simps_I2_J,axiom,
% 9.49/5.29      ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 9.49/5.29      = ( uminus_uminus_real @ one_one_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7118_dbl__dec__simps_I2_J,axiom,
% 9.49/5.29      ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 9.49/5.29      = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7119_dbl__dec__simps_I2_J,axiom,
% 9.49/5.29      ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 9.49/5.29      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7120_abs__idempotent,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 9.49/5.29        = ( abs_abs_real @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7121_abs__idempotent,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 9.49/5.29        = ( abs_abs_int @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7122_abs__idempotent,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 9.49/5.29        = ( abs_abs_rat @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7123_abs__idempotent,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 9.49/5.29        = ( abs_abs_Code_integer @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7124_abs__0__eq,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( zero_z3403309356797280102nteger
% 9.49/5.29          = ( abs_abs_Code_integer @ A ) )
% 9.49/5.29        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7125_abs__0__eq,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( zero_zero_real
% 9.49/5.29          = ( abs_abs_real @ A ) )
% 9.49/5.29        = ( A = zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7126_abs__0__eq,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( zero_zero_rat
% 9.49/5.29          = ( abs_abs_rat @ A ) )
% 9.49/5.29        = ( A = zero_zero_rat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7127_abs__0__eq,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( zero_zero_int
% 9.49/5.29          = ( abs_abs_int @ A ) )
% 9.49/5.29        = ( A = zero_zero_int ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7128_abs__eq__0,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( ( abs_abs_Code_integer @ A )
% 9.49/5.29          = zero_z3403309356797280102nteger )
% 9.49/5.29        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7129_abs__eq__0,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( ( abs_abs_real @ A )
% 9.49/5.29          = zero_zero_real )
% 9.49/5.29        = ( A = zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7130_abs__eq__0,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( ( abs_abs_rat @ A )
% 9.49/5.29          = zero_zero_rat )
% 9.49/5.29        = ( A = zero_zero_rat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7131_abs__eq__0,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( ( abs_abs_int @ A )
% 9.49/5.29          = zero_zero_int )
% 9.49/5.29        = ( A = zero_zero_int ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7132_abs__zero,axiom,
% 9.49/5.29      ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 9.49/5.29      = zero_z3403309356797280102nteger ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7133_abs__zero,axiom,
% 9.49/5.29      ( ( abs_abs_real @ zero_zero_real )
% 9.49/5.29      = zero_zero_real ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7134_abs__zero,axiom,
% 9.49/5.29      ( ( abs_abs_rat @ zero_zero_rat )
% 9.49/5.29      = zero_zero_rat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7135_abs__zero,axiom,
% 9.49/5.29      ( ( abs_abs_int @ zero_zero_int )
% 9.49/5.29      = zero_zero_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7136_abs__0,axiom,
% 9.49/5.29      ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 9.49/5.29      = zero_z3403309356797280102nteger ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7137_abs__0,axiom,
% 9.49/5.29      ( ( abs_abs_complex @ zero_zero_complex )
% 9.49/5.29      = zero_zero_complex ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7138_abs__0,axiom,
% 9.49/5.29      ( ( abs_abs_real @ zero_zero_real )
% 9.49/5.29      = zero_zero_real ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7139_abs__0,axiom,
% 9.49/5.29      ( ( abs_abs_rat @ zero_zero_rat )
% 9.49/5.29      = zero_zero_rat ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7140_abs__0,axiom,
% 9.49/5.29      ( ( abs_abs_int @ zero_zero_int )
% 9.49/5.29      = zero_zero_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7141_abs__add__abs,axiom,
% 9.49/5.29      ! [A: code_integer,B: code_integer] :
% 9.49/5.29        ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
% 9.49/5.29        = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7142_abs__add__abs,axiom,
% 9.49/5.29      ! [A: real,B: real] :
% 9.49/5.29        ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 9.49/5.29        = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7143_abs__add__abs,axiom,
% 9.49/5.29      ! [A: rat,B: rat] :
% 9.49/5.29        ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
% 9.49/5.29        = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7144_abs__add__abs,axiom,
% 9.49/5.29      ! [A: int,B: int] :
% 9.49/5.29        ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
% 9.49/5.29        = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7145_abs__minus__cancel,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 9.49/5.29        = ( abs_abs_int @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7146_abs__minus__cancel,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 9.49/5.29        = ( abs_abs_real @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7147_abs__minus__cancel,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 9.49/5.29        = ( abs_abs_rat @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7148_abs__minus__cancel,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 9.49/5.29        = ( abs_abs_Code_integer @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7149_abs__of__nat,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
% 9.49/5.29        = ( semiri681578069525770553at_rat @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7150_abs__of__nat,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 9.49/5.29        = ( semiri4939895301339042750nteger @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7151_abs__of__nat,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
% 9.49/5.29        = ( semiri1314217659103216013at_int @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7152_abs__of__nat,axiom,
% 9.49/5.29      ! [N: nat] :
% 9.49/5.29        ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
% 9.49/5.29        = ( semiri5074537144036343181t_real @ N ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7153_powr__0,axiom,
% 9.49/5.29      ! [Z2: real] :
% 9.49/5.29        ( ( powr_real @ zero_zero_real @ Z2 )
% 9.49/5.29        = zero_zero_real ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7154_powr__eq__0__iff,axiom,
% 9.49/5.29      ! [W2: real,Z2: real] :
% 9.49/5.29        ( ( ( powr_real @ W2 @ Z2 )
% 9.49/5.29          = zero_zero_real )
% 9.49/5.29        = ( W2 = zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7155_abs__le__zero__iff,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 9.49/5.29        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7156_abs__le__zero__iff,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 9.49/5.29        = ( A = zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7157_abs__le__zero__iff,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
% 9.49/5.29        = ( A = zero_zero_rat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7158_abs__le__zero__iff,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
% 9.49/5.29        = ( A = zero_zero_int ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7159_abs__le__self__iff,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
% 9.49/5.29        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7160_abs__le__self__iff,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 9.49/5.29        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7161_abs__le__self__iff,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
% 9.49/5.29        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7162_abs__le__self__iff,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
% 9.49/5.29        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7163_abs__of__nonneg,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.49/5.29       => ( ( abs_abs_Code_integer @ A )
% 9.49/5.29          = A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7164_abs__of__nonneg,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.29       => ( ( abs_abs_real @ A )
% 9.49/5.29          = A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7165_abs__of__nonneg,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.29       => ( ( abs_abs_rat @ A )
% 9.49/5.29          = A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7166_abs__of__nonneg,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.29       => ( ( abs_abs_int @ A )
% 9.49/5.29          = A ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7167_zero__less__abs__iff,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
% 9.49/5.29        = ( A != zero_z3403309356797280102nteger ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7168_zero__less__abs__iff,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 9.49/5.29        = ( A != zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7169_zero__less__abs__iff,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
% 9.49/5.29        = ( A != zero_zero_rat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7170_zero__less__abs__iff,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
% 9.49/5.29        = ( A != zero_zero_int ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7171_powr__zero__eq__one,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ( X = zero_zero_real )
% 9.49/5.29         => ( ( powr_real @ X @ zero_zero_real )
% 9.49/5.29            = zero_zero_real ) )
% 9.49/5.29        & ( ( X != zero_zero_real )
% 9.49/5.29         => ( ( powr_real @ X @ zero_zero_real )
% 9.49/5.29            = one_one_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7172_floor__zero,axiom,
% 9.49/5.29      ( ( archim6058952711729229775r_real @ zero_zero_real )
% 9.49/5.29      = zero_zero_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7173_floor__zero,axiom,
% 9.49/5.29      ( ( archim3151403230148437115or_rat @ zero_zero_rat )
% 9.49/5.29      = zero_zero_int ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7174_powr__gt__zero,axiom,
% 9.49/5.29      ! [X: real,A: real] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A ) )
% 9.49/5.29        = ( X != zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7175_powr__less__cancel__iff,axiom,
% 9.49/5.29      ! [X: real,A: real,B: real] :
% 9.49/5.29        ( ( ord_less_real @ one_one_real @ X )
% 9.49/5.29       => ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 9.49/5.29          = ( ord_less_real @ A @ B ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7176_sum__abs,axiom,
% 9.49/5.29      ! [F: nat > real,A2: set_nat] :
% 9.49/5.29        ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 9.49/5.29        @ ( groups6591440286371151544t_real
% 9.49/5.29          @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
% 9.49/5.29          @ A2 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7177_zero__le__divide__abs__iff,axiom,
% 9.49/5.29      ! [A: real,B: real] :
% 9.49/5.29        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
% 9.49/5.29        = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.29          | ( B = zero_zero_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7178_zero__le__divide__abs__iff,axiom,
% 9.49/5.29      ! [A: rat,B: rat] :
% 9.49/5.29        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
% 9.49/5.29        = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.29          | ( B = zero_zero_rat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7179_divide__le__0__abs__iff,axiom,
% 9.49/5.29      ! [A: real,B: real] :
% 9.49/5.29        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
% 9.49/5.29        = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.49/5.29          | ( B = zero_zero_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7180_divide__le__0__abs__iff,axiom,
% 9.49/5.29      ! [A: rat,B: rat] :
% 9.49/5.29        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
% 9.49/5.29        = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.49/5.29          | ( B = zero_zero_rat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7181_abs__of__nonpos,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.49/5.29       => ( ( abs_abs_real @ A )
% 9.49/5.29          = ( uminus_uminus_real @ A ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7182_abs__of__nonpos,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
% 9.49/5.29       => ( ( abs_abs_Code_integer @ A )
% 9.49/5.29          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7183_abs__of__nonpos,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.49/5.29       => ( ( abs_abs_rat @ A )
% 9.49/5.29          = ( uminus_uminus_rat @ A ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7184_abs__of__nonpos,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.49/5.29       => ( ( abs_abs_int @ A )
% 9.49/5.29          = ( uminus_uminus_int @ A ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7185_abs__sgn__eq__1,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( A != zero_z3403309356797280102nteger )
% 9.49/5.29       => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 9.49/5.29          = one_one_Code_integer ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7186_abs__sgn__eq__1,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( A != zero_zero_real )
% 9.49/5.29       => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 9.49/5.29          = one_one_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7187_abs__sgn__eq__1,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( A != zero_zero_rat )
% 9.49/5.29       => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 9.49/5.29          = one_one_rat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7188_abs__sgn__eq__1,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( A != zero_zero_int )
% 9.49/5.29       => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 9.49/5.29          = one_one_int ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7189_powr__eq__one__iff,axiom,
% 9.49/5.29      ! [A: real,X: real] :
% 9.49/5.29        ( ( ord_less_real @ one_one_real @ A )
% 9.49/5.29       => ( ( ( powr_real @ A @ X )
% 9.49/5.29            = one_one_real )
% 9.49/5.29          = ( X = zero_zero_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7190_powr__le__cancel__iff,axiom,
% 9.49/5.29      ! [X: real,A: real,B: real] :
% 9.49/5.29        ( ( ord_less_real @ one_one_real @ X )
% 9.49/5.29       => ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 9.49/5.29          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7191_artanh__minus__real,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.49/5.29       => ( ( artanh_real @ ( uminus_uminus_real @ X ) )
% 9.49/5.29          = ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7192_idom__abs__sgn__class_Oabs__sgn,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( sgn_sgn_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 9.49/5.29        = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7193_idom__abs__sgn__class_Oabs__sgn,axiom,
% 9.49/5.29      ! [A: complex] :
% 9.49/5.29        ( ( sgn_sgn_complex @ ( abs_abs_complex @ A ) )
% 9.49/5.29        = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7194_idom__abs__sgn__class_Oabs__sgn,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( sgn_sgn_real @ ( abs_abs_real @ A ) )
% 9.49/5.29        = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7195_idom__abs__sgn__class_Oabs__sgn,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( sgn_sgn_rat @ ( abs_abs_rat @ A ) )
% 9.49/5.29        = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7196_idom__abs__sgn__class_Oabs__sgn,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( sgn_sgn_int @ ( abs_abs_int @ A ) )
% 9.49/5.29        = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7197_sgn__abs,axiom,
% 9.49/5.29      ! [A: code_integer] :
% 9.49/5.29        ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 9.49/5.29        = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7198_sgn__abs,axiom,
% 9.49/5.29      ! [A: complex] :
% 9.49/5.29        ( ( abs_abs_complex @ ( sgn_sgn_complex @ A ) )
% 9.49/5.29        = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7199_sgn__abs,axiom,
% 9.49/5.29      ! [A: real] :
% 9.49/5.29        ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 9.49/5.29        = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7200_sgn__abs,axiom,
% 9.49/5.29      ! [A: rat] :
% 9.49/5.29        ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 9.49/5.29        = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7201_sgn__abs,axiom,
% 9.49/5.29      ! [A: int] :
% 9.49/5.29        ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 9.49/5.29        = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7202_sum__abs__ge__zero,axiom,
% 9.49/5.29      ! [F: nat > real,A2: set_nat] :
% 9.49/5.29        ( ord_less_eq_real @ zero_zero_real
% 9.49/5.29        @ ( groups6591440286371151544t_real
% 9.49/5.29          @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
% 9.49/5.29          @ A2 ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7203_zero__less__power__abs__iff,axiom,
% 9.49/5.29      ! [A: code_integer,N: nat] :
% 9.49/5.29        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
% 9.49/5.29        = ( ( A != zero_z3403309356797280102nteger )
% 9.49/5.29          | ( N = zero_zero_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7204_zero__less__power__abs__iff,axiom,
% 9.49/5.29      ! [A: real,N: nat] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 9.49/5.29        = ( ( A != zero_zero_real )
% 9.49/5.29          | ( N = zero_zero_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7205_zero__less__power__abs__iff,axiom,
% 9.49/5.29      ! [A: rat,N: nat] :
% 9.49/5.29        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
% 9.49/5.29        = ( ( A != zero_zero_rat )
% 9.49/5.29          | ( N = zero_zero_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7206_zero__less__power__abs__iff,axiom,
% 9.49/5.29      ! [A: int,N: nat] :
% 9.49/5.29        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
% 9.49/5.29        = ( ( A != zero_zero_int )
% 9.49/5.29          | ( N = zero_zero_nat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7207_zero__le__floor,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
% 9.49/5.29        = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7208_zero__le__floor,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29        ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
% 9.49/5.29        = ( ord_less_eq_rat @ zero_zero_rat @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7209_floor__less__zero,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
% 9.49/5.29        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7210_floor__less__zero,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
% 9.49/5.29        = ( ord_less_rat @ X @ zero_zero_rat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7211_numeral__le__floor,axiom,
% 9.49/5.29      ! [V: num,X: real] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
% 9.49/5.29        = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7212_numeral__le__floor,axiom,
% 9.49/5.29      ! [V: num,X: rat] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
% 9.49/5.29        = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7213_zero__less__floor,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
% 9.49/5.29        = ( ord_less_eq_real @ one_one_real @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7214_zero__less__floor,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29        ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
% 9.49/5.29        = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7215_floor__le__zero,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
% 9.49/5.29        = ( ord_less_real @ X @ one_one_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7216_floor__le__zero,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
% 9.49/5.29        = ( ord_less_rat @ X @ one_one_rat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7217_floor__less__numeral,axiom,
% 9.49/5.29      ! [X: real,V: num] :
% 9.49/5.29        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
% 9.49/5.29        = ( ord_less_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7218_floor__less__numeral,axiom,
% 9.49/5.29      ! [X: rat,V: num] :
% 9.49/5.29        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 9.49/5.29        = ( ord_less_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7219_one__le__floor,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
% 9.49/5.29        = ( ord_less_eq_real @ one_one_real @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7220_one__le__floor,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29        ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
% 9.49/5.29        = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7221_floor__less__one,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
% 9.49/5.29        = ( ord_less_real @ X @ one_one_real ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7222_floor__less__one,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
% 9.49/5.29        = ( ord_less_rat @ X @ one_one_rat ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7223_log__powr__cancel,axiom,
% 9.49/5.29      ! [A: real,Y: real] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.29       => ( ( A != one_one_real )
% 9.49/5.29         => ( ( log @ A @ ( powr_real @ A @ Y ) )
% 9.49/5.29            = Y ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7224_powr__log__cancel,axiom,
% 9.49/5.29      ! [A: real,X: real] :
% 9.49/5.29        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.29       => ( ( A != one_one_real )
% 9.49/5.29         => ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.29           => ( ( powr_real @ A @ ( log @ A @ X ) )
% 9.49/5.29              = X ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7225_numeral__less__floor,axiom,
% 9.49/5.29      ! [V: num,X: real] :
% 9.49/5.29        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
% 9.49/5.29        = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7226_numeral__less__floor,axiom,
% 9.49/5.29      ! [V: num,X: rat] :
% 9.49/5.29        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
% 9.49/5.29        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7227_floor__le__numeral,axiom,
% 9.49/5.29      ! [X: real,V: num] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
% 9.49/5.29        = ( ord_less_real @ X @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7228_floor__le__numeral,axiom,
% 9.49/5.29      ! [X: rat,V: num] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
% 9.49/5.29        = ( ord_less_rat @ X @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7229_one__less__floor,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
% 9.49/5.29        = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7230_one__less__floor,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29        ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
% 9.49/5.29        = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7231_floor__le__one,axiom,
% 9.49/5.29      ! [X: real] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
% 9.49/5.29        = ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7232_floor__le__one,axiom,
% 9.49/5.29      ! [X: rat] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
% 9.49/5.29        = ( ord_less_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7233_neg__numeral__le__floor,axiom,
% 9.49/5.29      ! [V: num,X: real] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
% 9.49/5.29        = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7234_neg__numeral__le__floor,axiom,
% 9.49/5.29      ! [V: num,X: rat] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
% 9.49/5.29        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7235_floor__less__neg__numeral,axiom,
% 9.49/5.29      ! [X: real,V: num] :
% 9.49/5.29        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.49/5.29        = ( ord_less_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7236_floor__less__neg__numeral,axiom,
% 9.49/5.29      ! [X: rat,V: num] :
% 9.49/5.29        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.49/5.29        = ( ord_less_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7237_neg__numeral__less__floor,axiom,
% 9.49/5.29      ! [V: num,X: real] :
% 9.49/5.29        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
% 9.49/5.29        = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7238_neg__numeral__less__floor,axiom,
% 9.49/5.29      ! [V: num,X: rat] :
% 9.49/5.29        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
% 9.49/5.29        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7239_floor__le__neg__numeral,axiom,
% 9.49/5.29      ! [X: real,V: num] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.49/5.29        = ( ord_less_real @ X @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7240_floor__le__neg__numeral,axiom,
% 9.49/5.29      ! [X: rat,V: num] :
% 9.49/5.29        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.49/5.29        = ( ord_less_rat @ X @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7241_abs__le__D1,axiom,
% 9.49/5.29      ! [A: real,B: real] :
% 9.49/5.29        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 9.49/5.29       => ( ord_less_eq_real @ A @ B ) ) ).
% 9.49/5.29  
% 9.49/5.29  thf(fact_7242_abs__le__D1,axiom,
% 9.49/5.30      ! [A: code_integer,B: code_integer] :
% 9.49/5.30        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 9.49/5.30       => ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7243_abs__le__D1,axiom,
% 9.49/5.30      ! [A: rat,B: rat] :
% 9.49/5.30        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 9.49/5.30       => ( ord_less_eq_rat @ A @ B ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7244_abs__le__D1,axiom,
% 9.49/5.30      ! [A: int,B: int] :
% 9.49/5.30        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 9.49/5.30       => ( ord_less_eq_int @ A @ B ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7245_abs__ge__self,axiom,
% 9.49/5.30      ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7246_abs__ge__self,axiom,
% 9.49/5.30      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7247_abs__ge__self,axiom,
% 9.49/5.30      ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7248_abs__ge__self,axiom,
% 9.49/5.30      ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7249_abs__eq__0__iff,axiom,
% 9.49/5.30      ! [A: code_integer] :
% 9.49/5.30        ( ( ( abs_abs_Code_integer @ A )
% 9.49/5.30          = zero_z3403309356797280102nteger )
% 9.49/5.30        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7250_abs__eq__0__iff,axiom,
% 9.49/5.30      ! [A: complex] :
% 9.49/5.30        ( ( ( abs_abs_complex @ A )
% 9.49/5.30          = zero_zero_complex )
% 9.49/5.30        = ( A = zero_zero_complex ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7251_abs__eq__0__iff,axiom,
% 9.49/5.30      ! [A: real] :
% 9.49/5.30        ( ( ( abs_abs_real @ A )
% 9.49/5.30          = zero_zero_real )
% 9.49/5.30        = ( A = zero_zero_real ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7252_abs__eq__0__iff,axiom,
% 9.49/5.30      ! [A: rat] :
% 9.49/5.30        ( ( ( abs_abs_rat @ A )
% 9.49/5.30          = zero_zero_rat )
% 9.49/5.30        = ( A = zero_zero_rat ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7253_abs__eq__0__iff,axiom,
% 9.49/5.30      ! [A: int] :
% 9.49/5.30        ( ( ( abs_abs_int @ A )
% 9.49/5.30          = zero_zero_int )
% 9.49/5.30        = ( A = zero_zero_int ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7254_abs__minus__commute,axiom,
% 9.49/5.30      ! [A: code_integer,B: code_integer] :
% 9.49/5.30        ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
% 9.49/5.30        = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7255_abs__minus__commute,axiom,
% 9.49/5.30      ! [A: real,B: real] :
% 9.49/5.30        ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 9.49/5.30        = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7256_abs__minus__commute,axiom,
% 9.49/5.30      ! [A: rat,B: rat] :
% 9.49/5.30        ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
% 9.49/5.30        = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7257_abs__minus__commute,axiom,
% 9.49/5.30      ! [A: int,B: int] :
% 9.49/5.30        ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
% 9.49/5.30        = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7258_abs__ge__zero,axiom,
% 9.49/5.30      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7259_abs__ge__zero,axiom,
% 9.49/5.30      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7260_abs__ge__zero,axiom,
% 9.49/5.30      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7261_abs__ge__zero,axiom,
% 9.49/5.30      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7262_floor__mono,axiom,
% 9.49/5.30      ! [X: real,Y: real] :
% 9.49/5.30        ( ( ord_less_eq_real @ X @ Y )
% 9.49/5.30       => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7263_floor__mono,axiom,
% 9.49/5.30      ! [X: rat,Y: rat] :
% 9.49/5.30        ( ( ord_less_eq_rat @ X @ Y )
% 9.49/5.30       => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7264_abs__of__pos,axiom,
% 9.49/5.30      ! [A: code_integer] :
% 9.49/5.30        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 9.49/5.30       => ( ( abs_abs_Code_integer @ A )
% 9.49/5.30          = A ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7265_abs__of__pos,axiom,
% 9.49/5.30      ! [A: real] :
% 9.49/5.30        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.30       => ( ( abs_abs_real @ A )
% 9.49/5.30          = A ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7266_abs__of__pos,axiom,
% 9.49/5.30      ! [A: rat] :
% 9.49/5.30        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.49/5.30       => ( ( abs_abs_rat @ A )
% 9.49/5.30          = A ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7267_abs__of__pos,axiom,
% 9.49/5.30      ! [A: int] :
% 9.49/5.30        ( ( ord_less_int @ zero_zero_int @ A )
% 9.49/5.30       => ( ( abs_abs_int @ A )
% 9.49/5.30          = A ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7268_abs__not__less__zero,axiom,
% 9.49/5.30      ! [A: code_integer] :
% 9.49/5.30        ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7269_abs__not__less__zero,axiom,
% 9.49/5.30      ! [A: real] :
% 9.49/5.30        ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7270_abs__not__less__zero,axiom,
% 9.49/5.30      ! [A: rat] :
% 9.49/5.30        ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7271_abs__not__less__zero,axiom,
% 9.49/5.30      ! [A: int] :
% 9.49/5.30        ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7272_of__int__floor__le,axiom,
% 9.49/5.30      ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7273_of__int__floor__le,axiom,
% 9.49/5.30      ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7274_floor__less__cancel,axiom,
% 9.49/5.30      ! [X: real,Y: real] :
% 9.49/5.30        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) )
% 9.49/5.30       => ( ord_less_real @ X @ Y ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7275_floor__less__cancel,axiom,
% 9.49/5.30      ! [X: rat,Y: rat] :
% 9.49/5.30        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) )
% 9.49/5.30       => ( ord_less_rat @ X @ Y ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7276_abs__triangle__ineq,axiom,
% 9.49/5.30      ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7277_abs__triangle__ineq,axiom,
% 9.49/5.30      ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7278_abs__triangle__ineq,axiom,
% 9.49/5.30      ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7279_abs__triangle__ineq,axiom,
% 9.49/5.30      ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7280_abs__mult__less,axiom,
% 9.49/5.30      ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 9.49/5.30        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
% 9.49/5.30       => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
% 9.49/5.30         => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7281_abs__mult__less,axiom,
% 9.49/5.30      ! [A: real,C: real,B: real,D: real] :
% 9.49/5.30        ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
% 9.49/5.30       => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
% 9.49/5.30         => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7282_abs__mult__less,axiom,
% 9.49/5.30      ! [A: rat,C: rat,B: rat,D: rat] :
% 9.49/5.30        ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
% 9.49/5.30       => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
% 9.49/5.30         => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7283_abs__mult__less,axiom,
% 9.49/5.30      ! [A: int,C: int,B: int,D: int] :
% 9.49/5.30        ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
% 9.49/5.30       => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
% 9.49/5.30         => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7284_abs__triangle__ineq2__sym,axiom,
% 9.49/5.30      ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7285_abs__triangle__ineq2__sym,axiom,
% 9.49/5.30      ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7286_abs__triangle__ineq2__sym,axiom,
% 9.49/5.30      ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7287_abs__triangle__ineq2__sym,axiom,
% 9.49/5.30      ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7288_abs__triangle__ineq3,axiom,
% 9.49/5.30      ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7289_abs__triangle__ineq3,axiom,
% 9.49/5.30      ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7290_abs__triangle__ineq3,axiom,
% 9.49/5.30      ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7291_abs__triangle__ineq3,axiom,
% 9.49/5.30      ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7292_abs__triangle__ineq2,axiom,
% 9.49/5.30      ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7293_abs__triangle__ineq2,axiom,
% 9.49/5.30      ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7294_abs__triangle__ineq2,axiom,
% 9.49/5.30      ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7295_abs__triangle__ineq2,axiom,
% 9.49/5.30      ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7296_nonzero__abs__divide,axiom,
% 9.49/5.30      ! [B: real,A: real] :
% 9.49/5.30        ( ( B != zero_zero_real )
% 9.49/5.30       => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 9.49/5.30          = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7297_nonzero__abs__divide,axiom,
% 9.49/5.30      ! [B: rat,A: rat] :
% 9.49/5.30        ( ( B != zero_zero_rat )
% 9.49/5.30       => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 9.49/5.30          = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7298_abs__leI,axiom,
% 9.49/5.30      ! [A: real,B: real] :
% 9.49/5.30        ( ( ord_less_eq_real @ A @ B )
% 9.49/5.30       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 9.49/5.30         => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7299_abs__leI,axiom,
% 9.49/5.30      ! [A: code_integer,B: code_integer] :
% 9.49/5.30        ( ( ord_le3102999989581377725nteger @ A @ B )
% 9.49/5.30       => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 9.49/5.30         => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7300_abs__leI,axiom,
% 9.49/5.30      ! [A: rat,B: rat] :
% 9.49/5.30        ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.30       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 9.49/5.30         => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7301_abs__leI,axiom,
% 9.49/5.30      ! [A: int,B: int] :
% 9.49/5.30        ( ( ord_less_eq_int @ A @ B )
% 9.49/5.30       => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 9.49/5.30         => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7302_abs__le__D2,axiom,
% 9.49/5.30      ! [A: real,B: real] :
% 9.49/5.30        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 9.49/5.30       => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7303_abs__le__D2,axiom,
% 9.49/5.30      ! [A: code_integer,B: code_integer] :
% 9.49/5.30        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 9.49/5.30       => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7304_abs__le__D2,axiom,
% 9.49/5.30      ! [A: rat,B: rat] :
% 9.49/5.30        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 9.49/5.30       => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7305_abs__le__D2,axiom,
% 9.49/5.30      ! [A: int,B: int] :
% 9.49/5.30        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 9.49/5.30       => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7306_abs__le__iff,axiom,
% 9.49/5.30      ! [A: real,B: real] :
% 9.49/5.30        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 9.49/5.30        = ( ( ord_less_eq_real @ A @ B )
% 9.49/5.30          & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7307_abs__le__iff,axiom,
% 9.49/5.30      ! [A: code_integer,B: code_integer] :
% 9.49/5.30        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 9.49/5.30        = ( ( ord_le3102999989581377725nteger @ A @ B )
% 9.49/5.30          & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7308_abs__le__iff,axiom,
% 9.49/5.30      ! [A: rat,B: rat] :
% 9.49/5.30        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 9.49/5.30        = ( ( ord_less_eq_rat @ A @ B )
% 9.49/5.30          & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7309_abs__le__iff,axiom,
% 9.49/5.30      ! [A: int,B: int] :
% 9.49/5.30        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 9.49/5.30        = ( ( ord_less_eq_int @ A @ B )
% 9.49/5.30          & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7310_abs__ge__minus__self,axiom,
% 9.49/5.30      ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7311_abs__ge__minus__self,axiom,
% 9.49/5.30      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7312_abs__ge__minus__self,axiom,
% 9.49/5.30      ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7313_abs__ge__minus__self,axiom,
% 9.49/5.30      ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7314_abs__less__iff,axiom,
% 9.49/5.30      ! [A: int,B: int] :
% 9.49/5.30        ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
% 9.49/5.30        = ( ( ord_less_int @ A @ B )
% 9.49/5.30          & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7315_abs__less__iff,axiom,
% 9.49/5.30      ! [A: real,B: real] :
% 9.49/5.30        ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
% 9.49/5.30        = ( ( ord_less_real @ A @ B )
% 9.49/5.30          & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7316_abs__less__iff,axiom,
% 9.49/5.30      ! [A: rat,B: rat] :
% 9.49/5.30        ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
% 9.49/5.30        = ( ( ord_less_rat @ A @ B )
% 9.49/5.30          & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7317_abs__less__iff,axiom,
% 9.49/5.30      ! [A: code_integer,B: code_integer] :
% 9.49/5.30        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 9.49/5.30        = ( ( ord_le6747313008572928689nteger @ A @ B )
% 9.49/5.30          & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7318_powr__less__mono2__neg,axiom,
% 9.49/5.30      ! [A: real,X: real,Y: real] :
% 9.49/5.30        ( ( ord_less_real @ A @ zero_zero_real )
% 9.49/5.30       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.30         => ( ( ord_less_real @ X @ Y )
% 9.49/5.30           => ( ord_less_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7319_powr__non__neg,axiom,
% 9.49/5.30      ! [A: real,X: real] :
% 9.49/5.30        ~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7320_powr__less__mono,axiom,
% 9.49/5.30      ! [A: real,B: real,X: real] :
% 9.49/5.30        ( ( ord_less_real @ A @ B )
% 9.49/5.30       => ( ( ord_less_real @ one_one_real @ X )
% 9.49/5.30         => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7321_powr__less__cancel,axiom,
% 9.49/5.30      ! [X: real,A: real,B: real] :
% 9.49/5.30        ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 9.49/5.30       => ( ( ord_less_real @ one_one_real @ X )
% 9.49/5.30         => ( ord_less_real @ A @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7322_dense__eq0__I,axiom,
% 9.49/5.30      ! [X: real] :
% 9.49/5.30        ( ! [E: real] :
% 9.49/5.30            ( ( ord_less_real @ zero_zero_real @ E )
% 9.49/5.30           => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E ) )
% 9.49/5.30       => ( X = zero_zero_real ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7323_dense__eq0__I,axiom,
% 9.49/5.30      ! [X: rat] :
% 9.49/5.30        ( ! [E: rat] :
% 9.49/5.30            ( ( ord_less_rat @ zero_zero_rat @ E )
% 9.49/5.30           => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E ) )
% 9.49/5.30       => ( X = zero_zero_rat ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7324_abs__eq__mult,axiom,
% 9.49/5.30      ! [A: code_integer,B: code_integer] :
% 9.49/5.30        ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.49/5.30            | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
% 9.49/5.30          & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 9.49/5.30            | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
% 9.49/5.30       => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 9.49/5.30          = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7325_abs__eq__mult,axiom,
% 9.49/5.30      ! [A: real,B: real] :
% 9.49/5.30        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.30            | ( ord_less_eq_real @ A @ zero_zero_real ) )
% 9.49/5.30          & ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.49/5.30            | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 9.49/5.30       => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 9.49/5.30          = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7326_abs__eq__mult,axiom,
% 9.49/5.30      ! [A: rat,B: rat] :
% 9.49/5.30        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.30            | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 9.49/5.30          & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.49/5.30            | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 9.49/5.30       => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 9.49/5.30          = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7327_abs__eq__mult,axiom,
% 9.49/5.30      ! [A: int,B: int] :
% 9.49/5.30        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.30            | ( ord_less_eq_int @ A @ zero_zero_int ) )
% 9.49/5.30          & ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.49/5.30            | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 9.49/5.30       => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 9.49/5.30          = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7328_abs__mult__pos,axiom,
% 9.49/5.30      ! [X: code_integer,Y: code_integer] :
% 9.49/5.30        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 9.49/5.30       => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X )
% 9.49/5.30          = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7329_abs__mult__pos,axiom,
% 9.49/5.30      ! [X: real,Y: real] :
% 9.49/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.49/5.30       => ( ( times_times_real @ ( abs_abs_real @ Y ) @ X )
% 9.49/5.30          = ( abs_abs_real @ ( times_times_real @ Y @ X ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7330_abs__mult__pos,axiom,
% 9.49/5.30      ! [X: rat,Y: rat] :
% 9.49/5.30        ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.49/5.30       => ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X )
% 9.49/5.30          = ( abs_abs_rat @ ( times_times_rat @ Y @ X ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7331_abs__mult__pos,axiom,
% 9.49/5.30      ! [X: int,Y: int] :
% 9.49/5.30        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.49/5.30       => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
% 9.49/5.30          = ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7332_abs__minus__le__zero,axiom,
% 9.49/5.30      ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7333_abs__minus__le__zero,axiom,
% 9.49/5.30      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7334_abs__minus__le__zero,axiom,
% 9.49/5.30      ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7335_abs__minus__le__zero,axiom,
% 9.49/5.30      ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7336_eq__abs__iff_H,axiom,
% 9.49/5.30      ! [A: real,B: real] :
% 9.49/5.30        ( ( A
% 9.49/5.30          = ( abs_abs_real @ B ) )
% 9.49/5.30        = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.49/5.30          & ( ( B = A )
% 9.49/5.30            | ( B
% 9.49/5.30              = ( uminus_uminus_real @ A ) ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7337_eq__abs__iff_H,axiom,
% 9.49/5.30      ! [A: code_integer,B: code_integer] :
% 9.49/5.30        ( ( A
% 9.49/5.30          = ( abs_abs_Code_integer @ B ) )
% 9.49/5.30        = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.49/5.30          & ( ( B = A )
% 9.49/5.30            | ( B
% 9.49/5.30              = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7338_eq__abs__iff_H,axiom,
% 9.49/5.30      ! [A: rat,B: rat] :
% 9.49/5.30        ( ( A
% 9.49/5.30          = ( abs_abs_rat @ B ) )
% 9.49/5.30        = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.49/5.30          & ( ( B = A )
% 9.49/5.30            | ( B
% 9.49/5.30              = ( uminus_uminus_rat @ A ) ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7339_eq__abs__iff_H,axiom,
% 9.49/5.30      ! [A: int,B: int] :
% 9.49/5.30        ( ( A
% 9.49/5.30          = ( abs_abs_int @ B ) )
% 9.49/5.30        = ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.49/5.30          & ( ( B = A )
% 9.49/5.30            | ( B
% 9.49/5.30              = ( uminus_uminus_int @ A ) ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7340_abs__eq__iff_H,axiom,
% 9.49/5.30      ! [A: real,B: real] :
% 9.49/5.30        ( ( ( abs_abs_real @ A )
% 9.49/5.30          = B )
% 9.49/5.30        = ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.49/5.30          & ( ( A = B )
% 9.49/5.30            | ( A
% 9.49/5.30              = ( uminus_uminus_real @ B ) ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7341_abs__eq__iff_H,axiom,
% 9.49/5.30      ! [A: code_integer,B: code_integer] :
% 9.49/5.30        ( ( ( abs_abs_Code_integer @ A )
% 9.49/5.30          = B )
% 9.49/5.30        = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 9.49/5.30          & ( ( A = B )
% 9.49/5.30            | ( A
% 9.49/5.30              = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7342_abs__eq__iff_H,axiom,
% 9.49/5.30      ! [A: rat,B: rat] :
% 9.49/5.30        ( ( ( abs_abs_rat @ A )
% 9.49/5.30          = B )
% 9.49/5.30        = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.49/5.30          & ( ( A = B )
% 9.49/5.30            | ( A
% 9.49/5.30              = ( uminus_uminus_rat @ B ) ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7343_abs__eq__iff_H,axiom,
% 9.49/5.30      ! [A: int,B: int] :
% 9.49/5.30        ( ( ( abs_abs_int @ A )
% 9.49/5.30          = B )
% 9.49/5.30        = ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.49/5.30          & ( ( A = B )
% 9.49/5.30            | ( A
% 9.49/5.30              = ( uminus_uminus_int @ B ) ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7344_abs__div__pos,axiom,
% 9.49/5.30      ! [Y: real,X: real] :
% 9.49/5.30        ( ( ord_less_real @ zero_zero_real @ Y )
% 9.49/5.30       => ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y )
% 9.49/5.30          = ( abs_abs_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7345_abs__div__pos,axiom,
% 9.49/5.30      ! [Y: rat,X: rat] :
% 9.49/5.30        ( ( ord_less_rat @ zero_zero_rat @ Y )
% 9.49/5.30       => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y )
% 9.49/5.30          = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7346_zero__le__power__abs,axiom,
% 9.49/5.30      ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7347_zero__le__power__abs,axiom,
% 9.49/5.30      ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7348_zero__le__power__abs,axiom,
% 9.49/5.30      ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7349_zero__le__power__abs,axiom,
% 9.49/5.30      ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7350_abs__if__raw,axiom,
% 9.49/5.30      ( abs_abs_int
% 9.49/5.30      = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7351_abs__if__raw,axiom,
% 9.49/5.30      ( abs_abs_real
% 9.49/5.30      = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7352_abs__if__raw,axiom,
% 9.49/5.30      ( abs_abs_rat
% 9.49/5.30      = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7353_abs__if__raw,axiom,
% 9.49/5.30      ( abs_abs_Code_integer
% 9.49/5.30      = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7354_abs__of__neg,axiom,
% 9.49/5.30      ! [A: int] :
% 9.49/5.30        ( ( ord_less_int @ A @ zero_zero_int )
% 9.49/5.30       => ( ( abs_abs_int @ A )
% 9.49/5.30          = ( uminus_uminus_int @ A ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7355_abs__of__neg,axiom,
% 9.49/5.30      ! [A: real] :
% 9.49/5.30        ( ( ord_less_real @ A @ zero_zero_real )
% 9.49/5.30       => ( ( abs_abs_real @ A )
% 9.49/5.30          = ( uminus_uminus_real @ A ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7356_abs__of__neg,axiom,
% 9.49/5.30      ! [A: rat] :
% 9.49/5.30        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.49/5.30       => ( ( abs_abs_rat @ A )
% 9.49/5.30          = ( uminus_uminus_rat @ A ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7357_abs__of__neg,axiom,
% 9.49/5.30      ! [A: code_integer] :
% 9.49/5.30        ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 9.49/5.30       => ( ( abs_abs_Code_integer @ A )
% 9.49/5.30          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7358_abs__if,axiom,
% 9.49/5.30      ( abs_abs_int
% 9.49/5.30      = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7359_abs__if,axiom,
% 9.49/5.30      ( abs_abs_real
% 9.49/5.30      = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7360_abs__if,axiom,
% 9.49/5.30      ( abs_abs_rat
% 9.49/5.30      = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7361_abs__if,axiom,
% 9.49/5.30      ( abs_abs_Code_integer
% 9.49/5.30      = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7362_abs__diff__triangle__ineq,axiom,
% 9.49/5.30      ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7363_abs__diff__triangle__ineq,axiom,
% 9.49/5.30      ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7364_abs__diff__triangle__ineq,axiom,
% 9.49/5.30      ! [A: rat,B: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7365_abs__diff__triangle__ineq,axiom,
% 9.49/5.30      ! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7366_abs__triangle__ineq4,axiom,
% 9.49/5.30      ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7367_abs__triangle__ineq4,axiom,
% 9.49/5.30      ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7368_abs__triangle__ineq4,axiom,
% 9.49/5.30      ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7369_abs__triangle__ineq4,axiom,
% 9.49/5.30      ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7370_abs__diff__le__iff,axiom,
% 9.49/5.30      ! [X: code_integer,A: code_integer,R2: code_integer] :
% 9.49/5.30        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
% 9.49/5.30        = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
% 9.49/5.30          & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7371_abs__diff__le__iff,axiom,
% 9.49/5.30      ! [X: real,A: real,R2: real] :
% 9.49/5.30        ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
% 9.49/5.30        = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X )
% 9.49/5.30          & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7372_abs__diff__le__iff,axiom,
% 9.49/5.30      ! [X: rat,A: rat,R2: rat] :
% 9.49/5.30        ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
% 9.49/5.30        = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
% 9.49/5.30          & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7373_abs__diff__le__iff,axiom,
% 9.49/5.30      ! [X: int,A: int,R2: int] :
% 9.49/5.30        ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
% 9.49/5.30        = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X )
% 9.49/5.30          & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7374_abs__diff__less__iff,axiom,
% 9.49/5.30      ! [X: code_integer,A: code_integer,R2: code_integer] :
% 9.49/5.30        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
% 9.49/5.30        = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
% 9.49/5.30          & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7375_abs__diff__less__iff,axiom,
% 9.49/5.30      ! [X: real,A: real,R2: real] :
% 9.49/5.30        ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
% 9.49/5.30        = ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X )
% 9.49/5.30          & ( ord_less_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7376_abs__diff__less__iff,axiom,
% 9.49/5.30      ! [X: rat,A: rat,R2: rat] :
% 9.49/5.30        ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
% 9.49/5.30        = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
% 9.49/5.30          & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7377_abs__diff__less__iff,axiom,
% 9.49/5.30      ! [X: int,A: int,R2: int] :
% 9.49/5.30        ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
% 9.49/5.30        = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X )
% 9.49/5.30          & ( ord_less_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7378_le__floor__iff,axiom,
% 9.49/5.30      ! [Z2: int,X: real] :
% 9.49/5.30        ( ( ord_less_eq_int @ Z2 @ ( archim6058952711729229775r_real @ X ) )
% 9.49/5.30        = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7379_le__floor__iff,axiom,
% 9.49/5.30      ! [Z2: int,X: rat] :
% 9.49/5.30        ( ( ord_less_eq_int @ Z2 @ ( archim3151403230148437115or_rat @ X ) )
% 9.49/5.30        = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7380_floor__less__iff,axiom,
% 9.49/5.30      ! [X: real,Z2: int] :
% 9.49/5.30        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ Z2 )
% 9.49/5.30        = ( ord_less_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7381_floor__less__iff,axiom,
% 9.49/5.30      ! [X: rat,Z2: int] :
% 9.49/5.30        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ Z2 )
% 9.49/5.30        = ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7382_powr__mono2_H,axiom,
% 9.49/5.30      ! [A: real,X: real,Y: real] :
% 9.49/5.30        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.49/5.30       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.30         => ( ( ord_less_eq_real @ X @ Y )
% 9.49/5.30           => ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7383_powr__less__mono2,axiom,
% 9.49/5.30      ! [A: real,X: real,Y: real] :
% 9.49/5.30        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.30       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.49/5.30         => ( ( ord_less_real @ X @ Y )
% 9.49/5.30           => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7384_gr__one__powr,axiom,
% 9.49/5.30      ! [X: real,Y: real] :
% 9.49/5.30        ( ( ord_less_real @ one_one_real @ X )
% 9.49/5.30       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.49/5.30         => ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7385_powr__inj,axiom,
% 9.49/5.30      ! [A: real,X: real,Y: real] :
% 9.49/5.30        ( ( ord_less_real @ zero_zero_real @ A )
% 9.49/5.30       => ( ( A != one_one_real )
% 9.49/5.30         => ( ( ( powr_real @ A @ X )
% 9.49/5.30              = ( powr_real @ A @ Y ) )
% 9.49/5.30            = ( X = Y ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7386_abs__sgn__eq,axiom,
% 9.49/5.30      ! [A: code_integer] :
% 9.49/5.30        ( ( ( A = zero_z3403309356797280102nteger )
% 9.49/5.30         => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 9.49/5.30            = zero_z3403309356797280102nteger ) )
% 9.49/5.30        & ( ( A != zero_z3403309356797280102nteger )
% 9.49/5.30         => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
% 9.49/5.30            = one_one_Code_integer ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7387_abs__sgn__eq,axiom,
% 9.49/5.30      ! [A: real] :
% 9.49/5.30        ( ( ( A = zero_zero_real )
% 9.49/5.30         => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 9.49/5.30            = zero_zero_real ) )
% 9.49/5.30        & ( ( A != zero_zero_real )
% 9.49/5.30         => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
% 9.49/5.30            = one_one_real ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7388_abs__sgn__eq,axiom,
% 9.49/5.30      ! [A: rat] :
% 9.49/5.30        ( ( ( A = zero_zero_rat )
% 9.49/5.30         => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 9.49/5.30            = zero_zero_rat ) )
% 9.49/5.30        & ( ( A != zero_zero_rat )
% 9.49/5.30         => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
% 9.49/5.30            = one_one_rat ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7389_abs__sgn__eq,axiom,
% 9.49/5.30      ! [A: int] :
% 9.49/5.30        ( ( ( A = zero_zero_int )
% 9.49/5.30         => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 9.49/5.30            = zero_zero_int ) )
% 9.49/5.30        & ( ( A != zero_zero_int )
% 9.49/5.30         => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
% 9.49/5.30            = one_one_int ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7390_abs__real__def,axiom,
% 9.49/5.30      ( abs_abs_real
% 9.49/5.30      = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7391_abs__add__one__gt__zero,axiom,
% 9.49/5.30      ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7392_abs__add__one__gt__zero,axiom,
% 9.49/5.30      ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7393_abs__add__one__gt__zero,axiom,
% 9.49/5.30      ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7394_abs__add__one__gt__zero,axiom,
% 9.49/5.30      ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7395_floor__log__eq__powr__iff,axiom,
% 9.49/5.30      ! [X: real,B: real,K: int] :
% 9.49/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.49/5.30       => ( ( ord_less_real @ one_one_real @ B )
% 9.49/5.30         => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
% 9.49/5.30              = K )
% 9.49/5.30            = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
% 9.49/5.30              & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7396_of__int__leD,axiom,
% 9.49/5.30      ! [N: int,X: code_integer] :
% 9.49/5.30        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
% 9.49/5.30       => ( ( N = zero_zero_int )
% 9.49/5.30          | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7397_of__int__leD,axiom,
% 9.49/5.30      ! [N: int,X: real] :
% 9.49/5.30        ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 9.49/5.30       => ( ( N = zero_zero_int )
% 9.49/5.30          | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7398_of__int__leD,axiom,
% 9.49/5.30      ! [N: int,X: rat] :
% 9.49/5.30        ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
% 9.49/5.30       => ( ( N = zero_zero_int )
% 9.49/5.30          | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7399_of__int__leD,axiom,
% 9.49/5.30      ! [N: int,X: int] :
% 9.49/5.30        ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 9.49/5.30       => ( ( N = zero_zero_int )
% 9.49/5.30          | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7400_of__int__lessD,axiom,
% 9.49/5.30      ! [N: int,X: code_integer] :
% 9.49/5.30        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
% 9.49/5.30       => ( ( N = zero_zero_int )
% 9.49/5.30          | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7401_of__int__lessD,axiom,
% 9.49/5.30      ! [N: int,X: real] :
% 9.49/5.30        ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 9.49/5.30       => ( ( N = zero_zero_int )
% 9.49/5.30          | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 9.49/5.30  
% 9.49/5.30  thf(fact_7402_of__int__lessD,axiom,
% 9.49/5.30      ! [N: int,X: rat] :
% 9.49/5.30        ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
% 9.49/5.30       => ( ( N = zero_zero_int )
% 9.51/5.30          | ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7403_of__int__lessD,axiom,
% 9.51/5.30      ! [N: int,X: int] :
% 9.51/5.30        ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 9.51/5.30       => ( ( N = zero_zero_int )
% 9.51/5.30          | ( ord_less_int @ one_one_int @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7404_powr__realpow,axiom,
% 9.51/5.30      ! [X: real,N: nat] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
% 9.51/5.30          = ( power_power_real @ X @ N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7405_less__log__iff,axiom,
% 9.51/5.30      ! [B: real,X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ one_one_real @ B )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30         => ( ( ord_less_real @ Y @ ( log @ B @ X ) )
% 9.51/5.30            = ( ord_less_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7406_log__less__iff,axiom,
% 9.51/5.30      ! [B: real,X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ one_one_real @ B )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30         => ( ( ord_less_real @ ( log @ B @ X ) @ Y )
% 9.51/5.30            = ( ord_less_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7407_less__powr__iff,axiom,
% 9.51/5.30      ! [B: real,X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ one_one_real @ B )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30         => ( ( ord_less_real @ X @ ( powr_real @ B @ Y ) )
% 9.51/5.30            = ( ord_less_real @ ( log @ B @ X ) @ Y ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7408_powr__less__iff,axiom,
% 9.51/5.30      ! [B: real,X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ one_one_real @ B )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30         => ( ( ord_less_real @ ( powr_real @ B @ Y ) @ X )
% 9.51/5.30            = ( ord_less_real @ Y @ ( log @ B @ X ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7409_floor__eq,axiom,
% 9.51/5.30      ! [N: int,X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 9.51/5.30         => ( ( archim6058952711729229775r_real @ X )
% 9.51/5.30            = N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7410_real__of__int__floor__add__one__gt,axiom,
% 9.51/5.30      ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7411_real__of__int__floor__gt__diff__one,axiom,
% 9.51/5.30      ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7412_round__diff__minimal,axiom,
% 9.51/5.30      ! [Z2: real,M: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z2 ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ ( ring_1_of_int_real @ M ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7413_round__diff__minimal,axiom,
% 9.51/5.30      ! [Z2: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z2 @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z2 ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z2 @ ( ring_1_of_int_rat @ M ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7414_floor__unique,axiom,
% 9.51/5.30      ! [Z2: int,X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) )
% 9.51/5.30         => ( ( archim6058952711729229775r_real @ X )
% 9.51/5.30            = Z2 ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7415_floor__unique,axiom,
% 9.51/5.30      ! [Z2: int,X: rat] :
% 9.51/5.30        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
% 9.51/5.30       => ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) )
% 9.51/5.30         => ( ( archim3151403230148437115or_rat @ X )
% 9.51/5.30            = Z2 ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7416_floor__eq__iff,axiom,
% 9.51/5.30      ! [X: real,A: int] :
% 9.51/5.30        ( ( ( archim6058952711729229775r_real @ X )
% 9.51/5.30          = A )
% 9.51/5.30        = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X )
% 9.51/5.30          & ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7417_floor__eq__iff,axiom,
% 9.51/5.30      ! [X: rat,A: int] :
% 9.51/5.30        ( ( ( archim3151403230148437115or_rat @ X )
% 9.51/5.30          = A )
% 9.51/5.30        = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X )
% 9.51/5.30          & ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7418_floor__split,axiom,
% 9.51/5.30      ! [P: int > $o,T: real] :
% 9.51/5.30        ( ( P @ ( archim6058952711729229775r_real @ T ) )
% 9.51/5.30        = ( ! [I4: int] :
% 9.51/5.30              ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I4 ) @ T )
% 9.51/5.30                & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) ) )
% 9.51/5.30             => ( P @ I4 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7419_floor__split,axiom,
% 9.51/5.30      ! [P: int > $o,T: rat] :
% 9.51/5.30        ( ( P @ ( archim3151403230148437115or_rat @ T ) )
% 9.51/5.30        = ( ! [I4: int] :
% 9.51/5.30              ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I4 ) @ T )
% 9.51/5.30                & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) ) )
% 9.51/5.30             => ( P @ I4 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7420_le__mult__floor,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.51/5.30       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.51/5.30         => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7421_le__mult__floor,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.51/5.30       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.51/5.30         => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7422_less__floor__iff,axiom,
% 9.51/5.30      ! [Z2: int,X: real] :
% 9.51/5.30        ( ( ord_less_int @ Z2 @ ( archim6058952711729229775r_real @ X ) )
% 9.51/5.30        = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7423_less__floor__iff,axiom,
% 9.51/5.30      ! [Z2: int,X: rat] :
% 9.51/5.30        ( ( ord_less_int @ Z2 @ ( archim3151403230148437115or_rat @ X ) )
% 9.51/5.30        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7424_floor__le__iff,axiom,
% 9.51/5.30      ! [X: real,Z2: int] :
% 9.51/5.30        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ Z2 )
% 9.51/5.30        = ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7425_floor__le__iff,axiom,
% 9.51/5.30      ! [X: rat,Z2: int] :
% 9.51/5.30        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z2 )
% 9.51/5.30        = ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7426_floor__correct,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X )
% 9.51/5.30        & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7427_floor__correct,axiom,
% 9.51/5.30      ! [X: rat] :
% 9.51/5.30        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X )
% 9.51/5.30        & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7428_abs__le__square__iff,axiom,
% 9.51/5.30      ! [X: code_integer,Y: code_integer] :
% 9.51/5.30        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y ) )
% 9.51/5.30        = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7429_abs__le__square__iff,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) )
% 9.51/5.30        = ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7430_abs__le__square__iff,axiom,
% 9.51/5.30      ! [X: rat,Y: rat] :
% 9.51/5.30        ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y ) )
% 9.51/5.30        = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7431_abs__le__square__iff,axiom,
% 9.51/5.30      ! [X: int,Y: int] :
% 9.51/5.30        ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) )
% 9.51/5.30        = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7432_powr__neg__one,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 9.51/5.30          = ( divide_divide_real @ one_one_real @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7433_powr__le__iff,axiom,
% 9.51/5.30      ! [B: real,X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ one_one_real @ B )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30         => ( ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X )
% 9.51/5.30            = ( ord_less_eq_real @ Y @ ( log @ B @ X ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7434_le__powr__iff,axiom,
% 9.51/5.30      ! [B: real,X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ one_one_real @ B )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30         => ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) )
% 9.51/5.30            = ( ord_less_eq_real @ ( log @ B @ X ) @ Y ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7435_log__le__iff,axiom,
% 9.51/5.30      ! [B: real,X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ one_one_real @ B )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30         => ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y )
% 9.51/5.30            = ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7436_le__log__iff,axiom,
% 9.51/5.30      ! [B: real,X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ one_one_real @ B )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30         => ( ( ord_less_eq_real @ Y @ ( log @ B @ X ) )
% 9.51/5.30            = ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7437_floor__eq2,axiom,
% 9.51/5.30      ! [N: int,X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 9.51/5.30         => ( ( archim6058952711729229775r_real @ X )
% 9.51/5.30            = N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7438_sgn__power__injE,axiom,
% 9.51/5.30      ! [A: real,N: nat,X: real,B: real] :
% 9.51/5.30        ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 9.51/5.30          = X )
% 9.51/5.30       => ( ( X
% 9.51/5.30            = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
% 9.51/5.30         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.30           => ( A = B ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7439_floor__divide__lower,axiom,
% 9.51/5.30      ! [Q4: real,P4: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ Q4 )
% 9.51/5.30       => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P4 @ Q4 ) ) ) @ Q4 ) @ P4 ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7440_floor__divide__lower,axiom,
% 9.51/5.30      ! [Q4: rat,P4: rat] :
% 9.51/5.30        ( ( ord_less_rat @ zero_zero_rat @ Q4 )
% 9.51/5.30       => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P4 @ Q4 ) ) ) @ Q4 ) @ P4 ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7441_power2__le__iff__abs__le,axiom,
% 9.51/5.30      ! [Y: code_integer,X: code_integer] :
% 9.51/5.30        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
% 9.51/5.30       => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.51/5.30          = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7442_power2__le__iff__abs__le,axiom,
% 9.51/5.30      ! [Y: real,X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.51/5.30       => ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.51/5.30          = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7443_power2__le__iff__abs__le,axiom,
% 9.51/5.30      ! [Y: rat,X: rat] :
% 9.51/5.30        ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 9.51/5.30       => ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.51/5.30          = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7444_power2__le__iff__abs__le,axiom,
% 9.51/5.30      ! [Y: int,X: int] :
% 9.51/5.30        ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 9.51/5.30       => ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.51/5.30          = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7445_abs__sqrt__wlog,axiom,
% 9.51/5.30      ! [P: code_integer > code_integer > $o,X: code_integer] :
% 9.51/5.30        ( ! [X4: code_integer] :
% 9.51/5.30            ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
% 9.51/5.30           => ( P @ X4 @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.51/5.30       => ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7446_abs__sqrt__wlog,axiom,
% 9.51/5.30      ! [P: real > real > $o,X: real] :
% 9.51/5.30        ( ! [X4: real] :
% 9.51/5.30            ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 9.51/5.30           => ( P @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.51/5.30       => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7447_abs__sqrt__wlog,axiom,
% 9.51/5.30      ! [P: rat > rat > $o,X: rat] :
% 9.51/5.30        ( ! [X4: rat] :
% 9.51/5.30            ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 9.51/5.30           => ( P @ X4 @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.51/5.30       => ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7448_abs__sqrt__wlog,axiom,
% 9.51/5.30      ! [P: int > int > $o,X: int] :
% 9.51/5.30        ( ! [X4: int] :
% 9.51/5.30            ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 9.51/5.30           => ( P @ X4 @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.51/5.30       => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7449_abs__square__le__1,axiom,
% 9.51/5.30      ! [X: code_integer] :
% 9.51/5.30        ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 9.51/5.30        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7450_abs__square__le__1,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 9.51/5.30        = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7451_abs__square__le__1,axiom,
% 9.51/5.30      ! [X: rat] :
% 9.51/5.30        ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 9.51/5.30        = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7452_abs__square__le__1,axiom,
% 9.51/5.30      ! [X: int] :
% 9.51/5.30        ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 9.51/5.30        = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7453_abs__square__less__1,axiom,
% 9.51/5.30      ! [X: code_integer] :
% 9.51/5.30        ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 9.51/5.30        = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7454_abs__square__less__1,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 9.51/5.30        = ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7455_abs__square__less__1,axiom,
% 9.51/5.30      ! [X: rat] :
% 9.51/5.30        ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 9.51/5.30        = ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7456_abs__square__less__1,axiom,
% 9.51/5.30      ! [X: int] :
% 9.51/5.30        ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 9.51/5.30        = ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7457_power__mono__even,axiom,
% 9.51/5.30      ! [N: nat,A: code_integer,B: code_integer] :
% 9.51/5.30        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.51/5.30       => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
% 9.51/5.30         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7458_power__mono__even,axiom,
% 9.51/5.30      ! [N: nat,A: real,B: real] :
% 9.51/5.30        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.51/5.30       => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
% 9.51/5.30         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7459_power__mono__even,axiom,
% 9.51/5.30      ! [N: nat,A: rat,B: rat] :
% 9.51/5.30        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.51/5.30       => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
% 9.51/5.30         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7460_power__mono__even,axiom,
% 9.51/5.30      ! [N: nat,A: int,B: int] :
% 9.51/5.30        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.51/5.30       => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
% 9.51/5.30         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7461_add__log__eq__powr,axiom,
% 9.51/5.30      ! [B: real,X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ B )
% 9.51/5.30       => ( ( B != one_one_real )
% 9.51/5.30         => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30           => ( ( plus_plus_real @ Y @ ( log @ B @ X ) )
% 9.51/5.30              = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7462_log__add__eq__powr,axiom,
% 9.51/5.30      ! [B: real,X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ B )
% 9.51/5.30       => ( ( B != one_one_real )
% 9.51/5.30         => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30           => ( ( plus_plus_real @ ( log @ B @ X ) @ Y )
% 9.51/5.30              = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7463_minus__log__eq__powr,axiom,
% 9.51/5.30      ! [B: real,X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ B )
% 9.51/5.30       => ( ( B != one_one_real )
% 9.51/5.30         => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30           => ( ( minus_minus_real @ Y @ ( log @ B @ X ) )
% 9.51/5.30              = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7464_convex__sum__bound__le,axiom,
% 9.51/5.30      ! [I6: set_real,X: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
% 9.51/5.30        ( ! [I2: real] :
% 9.51/5.30            ( ( member_real @ I2 @ I6 )
% 9.51/5.30           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
% 9.51/5.30       => ( ( ( groups7713935264441627589nteger @ X @ I6 )
% 9.51/5.30            = one_one_Code_integer )
% 9.51/5.30         => ( ! [I2: real] :
% 9.51/5.30                ( ( member_real @ I2 @ I6 )
% 9.51/5.30               => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 9.51/5.30           => ( ord_le3102999989581377725nteger
% 9.51/5.30              @ ( abs_abs_Code_integer
% 9.51/5.30                @ ( minus_8373710615458151222nteger
% 9.51/5.30                  @ ( groups7713935264441627589nteger
% 9.51/5.30                    @ ^ [I4: real] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
% 9.51/5.30                    @ I6 )
% 9.51/5.30                  @ B ) )
% 9.51/5.30              @ Delta ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7465_convex__sum__bound__le,axiom,
% 9.51/5.30      ! [I6: set_nat,X: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
% 9.51/5.30        ( ! [I2: nat] :
% 9.51/5.30            ( ( member_nat @ I2 @ I6 )
% 9.51/5.30           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
% 9.51/5.30       => ( ( ( groups7501900531339628137nteger @ X @ I6 )
% 9.51/5.30            = one_one_Code_integer )
% 9.51/5.30         => ( ! [I2: nat] :
% 9.51/5.30                ( ( member_nat @ I2 @ I6 )
% 9.51/5.30               => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 9.51/5.30           => ( ord_le3102999989581377725nteger
% 9.51/5.30              @ ( abs_abs_Code_integer
% 9.51/5.30                @ ( minus_8373710615458151222nteger
% 9.51/5.30                  @ ( groups7501900531339628137nteger
% 9.51/5.30                    @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
% 9.51/5.30                    @ I6 )
% 9.51/5.30                  @ B ) )
% 9.51/5.30              @ Delta ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7466_convex__sum__bound__le,axiom,
% 9.51/5.30      ! [I6: set_int,X: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
% 9.51/5.30        ( ! [I2: int] :
% 9.51/5.30            ( ( member_int @ I2 @ I6 )
% 9.51/5.30           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
% 9.51/5.30       => ( ( ( groups7873554091576472773nteger @ X @ I6 )
% 9.51/5.30            = one_one_Code_integer )
% 9.51/5.30         => ( ! [I2: int] :
% 9.51/5.30                ( ( member_int @ I2 @ I6 )
% 9.51/5.30               => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 9.51/5.30           => ( ord_le3102999989581377725nteger
% 9.51/5.30              @ ( abs_abs_Code_integer
% 9.51/5.30                @ ( minus_8373710615458151222nteger
% 9.51/5.30                  @ ( groups7873554091576472773nteger
% 9.51/5.30                    @ ^ [I4: int] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
% 9.51/5.30                    @ I6 )
% 9.51/5.30                  @ B ) )
% 9.51/5.30              @ Delta ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7467_convex__sum__bound__le,axiom,
% 9.51/5.30      ! [I6: set_real,X: real > real,A: real > real,B: real,Delta: real] :
% 9.51/5.30        ( ! [I2: real] :
% 9.51/5.30            ( ( member_real @ I2 @ I6 )
% 9.51/5.30           => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
% 9.51/5.30       => ( ( ( groups8097168146408367636l_real @ X @ I6 )
% 9.51/5.30            = one_one_real )
% 9.51/5.30         => ( ! [I2: real] :
% 9.51/5.30                ( ( member_real @ I2 @ I6 )
% 9.51/5.30               => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 9.51/5.30           => ( ord_less_eq_real
% 9.51/5.30              @ ( abs_abs_real
% 9.51/5.30                @ ( minus_minus_real
% 9.51/5.30                  @ ( groups8097168146408367636l_real
% 9.51/5.30                    @ ^ [I4: real] : ( times_times_real @ ( A @ I4 ) @ ( X @ I4 ) )
% 9.51/5.30                    @ I6 )
% 9.51/5.30                  @ B ) )
% 9.51/5.30              @ Delta ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7468_convex__sum__bound__le,axiom,
% 9.51/5.30      ! [I6: set_int,X: int > real,A: int > real,B: real,Delta: real] :
% 9.51/5.30        ( ! [I2: int] :
% 9.51/5.30            ( ( member_int @ I2 @ I6 )
% 9.51/5.30           => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
% 9.51/5.30       => ( ( ( groups8778361861064173332t_real @ X @ I6 )
% 9.51/5.30            = one_one_real )
% 9.51/5.30         => ( ! [I2: int] :
% 9.51/5.30                ( ( member_int @ I2 @ I6 )
% 9.51/5.30               => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 9.51/5.30           => ( ord_less_eq_real
% 9.51/5.30              @ ( abs_abs_real
% 9.51/5.30                @ ( minus_minus_real
% 9.51/5.30                  @ ( groups8778361861064173332t_real
% 9.51/5.30                    @ ^ [I4: int] : ( times_times_real @ ( A @ I4 ) @ ( X @ I4 ) )
% 9.51/5.30                    @ I6 )
% 9.51/5.30                  @ B ) )
% 9.51/5.30              @ Delta ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7469_convex__sum__bound__le,axiom,
% 9.51/5.30      ! [I6: set_real,X: real > rat,A: real > rat,B: rat,Delta: rat] :
% 9.51/5.30        ( ! [I2: real] :
% 9.51/5.30            ( ( member_real @ I2 @ I6 )
% 9.51/5.30           => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
% 9.51/5.30       => ( ( ( groups1300246762558778688al_rat @ X @ I6 )
% 9.51/5.30            = one_one_rat )
% 9.51/5.30         => ( ! [I2: real] :
% 9.51/5.30                ( ( member_real @ I2 @ I6 )
% 9.51/5.30               => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 9.51/5.30           => ( ord_less_eq_rat
% 9.51/5.30              @ ( abs_abs_rat
% 9.51/5.30                @ ( minus_minus_rat
% 9.51/5.30                  @ ( groups1300246762558778688al_rat
% 9.51/5.30                    @ ^ [I4: real] : ( times_times_rat @ ( A @ I4 ) @ ( X @ I4 ) )
% 9.51/5.30                    @ I6 )
% 9.51/5.30                  @ B ) )
% 9.51/5.30              @ Delta ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7470_convex__sum__bound__le,axiom,
% 9.51/5.30      ! [I6: set_nat,X: nat > rat,A: nat > rat,B: rat,Delta: rat] :
% 9.51/5.30        ( ! [I2: nat] :
% 9.51/5.30            ( ( member_nat @ I2 @ I6 )
% 9.51/5.30           => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
% 9.51/5.30       => ( ( ( groups2906978787729119204at_rat @ X @ I6 )
% 9.51/5.30            = one_one_rat )
% 9.51/5.30         => ( ! [I2: nat] :
% 9.51/5.30                ( ( member_nat @ I2 @ I6 )
% 9.51/5.30               => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 9.51/5.30           => ( ord_less_eq_rat
% 9.51/5.30              @ ( abs_abs_rat
% 9.51/5.30                @ ( minus_minus_rat
% 9.51/5.30                  @ ( groups2906978787729119204at_rat
% 9.51/5.30                    @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( X @ I4 ) )
% 9.51/5.30                    @ I6 )
% 9.51/5.30                  @ B ) )
% 9.51/5.30              @ Delta ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7471_convex__sum__bound__le,axiom,
% 9.51/5.30      ! [I6: set_int,X: int > rat,A: int > rat,B: rat,Delta: rat] :
% 9.51/5.30        ( ! [I2: int] :
% 9.51/5.30            ( ( member_int @ I2 @ I6 )
% 9.51/5.30           => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
% 9.51/5.30       => ( ( ( groups3906332499630173760nt_rat @ X @ I6 )
% 9.51/5.30            = one_one_rat )
% 9.51/5.30         => ( ! [I2: int] :
% 9.51/5.30                ( ( member_int @ I2 @ I6 )
% 9.51/5.30               => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 9.51/5.30           => ( ord_less_eq_rat
% 9.51/5.30              @ ( abs_abs_rat
% 9.51/5.30                @ ( minus_minus_rat
% 9.51/5.30                  @ ( groups3906332499630173760nt_rat
% 9.51/5.30                    @ ^ [I4: int] : ( times_times_rat @ ( A @ I4 ) @ ( X @ I4 ) )
% 9.51/5.30                    @ I6 )
% 9.51/5.30                  @ B ) )
% 9.51/5.30              @ Delta ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7472_convex__sum__bound__le,axiom,
% 9.51/5.30      ! [I6: set_real,X: real > int,A: real > int,B: int,Delta: int] :
% 9.51/5.30        ( ! [I2: real] :
% 9.51/5.30            ( ( member_real @ I2 @ I6 )
% 9.51/5.30           => ( ord_less_eq_int @ zero_zero_int @ ( X @ I2 ) ) )
% 9.51/5.30       => ( ( ( groups1932886352136224148al_int @ X @ I6 )
% 9.51/5.30            = one_one_int )
% 9.51/5.30         => ( ! [I2: real] :
% 9.51/5.30                ( ( member_real @ I2 @ I6 )
% 9.51/5.30               => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 9.51/5.30           => ( ord_less_eq_int
% 9.51/5.30              @ ( abs_abs_int
% 9.51/5.30                @ ( minus_minus_int
% 9.51/5.30                  @ ( groups1932886352136224148al_int
% 9.51/5.30                    @ ^ [I4: real] : ( times_times_int @ ( A @ I4 ) @ ( X @ I4 ) )
% 9.51/5.30                    @ I6 )
% 9.51/5.30                  @ B ) )
% 9.51/5.30              @ Delta ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7473_convex__sum__bound__le,axiom,
% 9.51/5.30      ! [I6: set_nat,X: nat > int,A: nat > int,B: int,Delta: int] :
% 9.51/5.30        ( ! [I2: nat] :
% 9.51/5.30            ( ( member_nat @ I2 @ I6 )
% 9.51/5.30           => ( ord_less_eq_int @ zero_zero_int @ ( X @ I2 ) ) )
% 9.51/5.30       => ( ( ( groups3539618377306564664at_int @ X @ I6 )
% 9.51/5.30            = one_one_int )
% 9.51/5.30         => ( ! [I2: nat] :
% 9.51/5.30                ( ( member_nat @ I2 @ I6 )
% 9.51/5.30               => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 9.51/5.30           => ( ord_less_eq_int
% 9.51/5.30              @ ( abs_abs_int
% 9.51/5.30                @ ( minus_minus_int
% 9.51/5.30                  @ ( groups3539618377306564664at_int
% 9.51/5.30                    @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( X @ I4 ) )
% 9.51/5.30                    @ I6 )
% 9.51/5.30                  @ B ) )
% 9.51/5.30              @ Delta ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7474_floor__divide__upper,axiom,
% 9.51/5.30      ! [Q4: real,P4: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ Q4 )
% 9.51/5.30       => ( ord_less_real @ P4 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P4 @ Q4 ) ) ) @ one_one_real ) @ Q4 ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7475_floor__divide__upper,axiom,
% 9.51/5.30      ! [Q4: rat,P4: rat] :
% 9.51/5.30        ( ( ord_less_rat @ zero_zero_rat @ Q4 )
% 9.51/5.30       => ( ord_less_rat @ P4 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P4 @ Q4 ) ) ) @ one_one_rat ) @ Q4 ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7476_log__minus__eq__powr,axiom,
% 9.51/5.30      ! [B: real,X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ B )
% 9.51/5.30       => ( ( B != one_one_real )
% 9.51/5.30         => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30           => ( ( minus_minus_real @ ( log @ B @ X ) @ Y )
% 9.51/5.30              = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7477_powr__neg__numeral,axiom,
% 9.51/5.30      ! [X: real,N: num] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.51/5.30          = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7478_of__int__round__abs__le,axiom,
% 9.51/5.30      ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ X ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7479_of__int__round__abs__le,axiom,
% 9.51/5.30      ! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7480_round__unique_H,axiom,
% 9.51/5.30      ! [X: real,N: int] :
% 9.51/5.30        ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30       => ( ( archim8280529875227126926d_real @ X )
% 9.51/5.30          = N ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7481_round__unique_H,axiom,
% 9.51/5.30      ! [X: rat,N: int] :
% 9.51/5.30        ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 9.51/5.30       => ( ( archim7778729529865785530nd_rat @ X )
% 9.51/5.30          = N ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7482_lemma__interval,axiom,
% 9.51/5.30      ! [A: real,X: real,B: real] :
% 9.51/5.30        ( ( ord_less_real @ A @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ B )
% 9.51/5.30         => ? [D5: real] :
% 9.51/5.30              ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.51/5.30              & ! [Y4: real] :
% 9.51/5.30                  ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D5 )
% 9.51/5.30                 => ( ( ord_less_eq_real @ A @ Y4 )
% 9.51/5.30                    & ( ord_less_eq_real @ Y4 @ B ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7483_lemma__interval__lt,axiom,
% 9.51/5.30      ! [A: real,X: real,B: real] :
% 9.51/5.30        ( ( ord_less_real @ A @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ B )
% 9.51/5.30         => ? [D5: real] :
% 9.51/5.30              ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.51/5.30              & ! [Y4: real] :
% 9.51/5.30                  ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D5 )
% 9.51/5.30                 => ( ( ord_less_real @ A @ Y4 )
% 9.51/5.30                    & ( ord_less_real @ Y4 @ B ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7484_round__altdef,axiom,
% 9.51/5.30      ( archim8280529875227126926d_real
% 9.51/5.30      = ( ^ [X3: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X3 ) ) @ ( archim7802044766580827645g_real @ X3 ) @ ( archim6058952711729229775r_real @ X3 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7485_round__altdef,axiom,
% 9.51/5.30      ( archim7778729529865785530nd_rat
% 9.51/5.30      = ( ^ [X3: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X3 ) ) @ ( archim2889992004027027881ng_rat @ X3 ) @ ( archim3151403230148437115or_rat @ X3 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7486_arctan__double,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.51/5.30       => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
% 9.51/5.30          = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7487_powr__int,axiom,
% 9.51/5.30      ! [X: real,I: int] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
% 9.51/5.30           => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
% 9.51/5.30              = ( power_power_real @ X @ ( nat2 @ I ) ) ) )
% 9.51/5.30          & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
% 9.51/5.30           => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
% 9.51/5.30              = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7488_horner__sum__of__bool__2__less,axiom,
% 9.51/5.30      ! [Bs: list_o] : ( ord_less_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_size_list_o @ Bs ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7489_zabs__less__one__iff,axiom,
% 9.51/5.30      ! [Z2: int] :
% 9.51/5.30        ( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
% 9.51/5.30        = ( Z2 = zero_zero_int ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7490_arctan__less__zero__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( arctan @ X ) @ zero_zero_real )
% 9.51/5.30        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7491_zero__less__arctan__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ ( arctan @ X ) )
% 9.51/5.30        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7492_frac__of__int,axiom,
% 9.51/5.30      ! [Z2: int] :
% 9.51/5.30        ( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z2 ) )
% 9.51/5.30        = zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7493_frac__of__int,axiom,
% 9.51/5.30      ! [Z2: int] :
% 9.51/5.30        ( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 9.51/5.30        = zero_zero_rat ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7494_nat__1,axiom,
% 9.51/5.30      ( ( nat2 @ one_one_int )
% 9.51/5.30      = ( suc @ zero_zero_nat ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7495_nat__0__iff,axiom,
% 9.51/5.30      ! [I: int] :
% 9.51/5.30        ( ( ( nat2 @ I )
% 9.51/5.30          = zero_zero_nat )
% 9.51/5.30        = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7496_nat__le__0,axiom,
% 9.51/5.30      ! [Z2: int] :
% 9.51/5.30        ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
% 9.51/5.30       => ( ( nat2 @ Z2 )
% 9.51/5.30          = zero_zero_nat ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7497_zless__nat__conj,axiom,
% 9.51/5.30      ! [W2: int,Z2: int] :
% 9.51/5.30        ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
% 9.51/5.30        = ( ( ord_less_int @ zero_zero_int @ Z2 )
% 9.51/5.30          & ( ord_less_int @ W2 @ Z2 ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7498_nat__neg__numeral,axiom,
% 9.51/5.30      ! [K: num] :
% 9.51/5.30        ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.51/5.30        = zero_zero_nat ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7499_nat__zminus__int,axiom,
% 9.51/5.30      ! [N: nat] :
% 9.51/5.30        ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.51/5.30        = zero_zero_nat ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7500_zero__less__nat__eq,axiom,
% 9.51/5.30      ! [Z2: int] :
% 9.51/5.30        ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
% 9.51/5.30        = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7501_nat__ceiling__le__eq,axiom,
% 9.51/5.30      ! [X: real,A: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
% 9.51/5.30        = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7502_one__less__nat__eq,axiom,
% 9.51/5.30      ! [Z2: int] :
% 9.51/5.30        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
% 9.51/5.30        = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7503_numeral__power__less__nat__cancel__iff,axiom,
% 9.51/5.30      ! [X: num,N: nat,A: int] :
% 9.51/5.30        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 9.51/5.30        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7504_nat__less__numeral__power__cancel__iff,axiom,
% 9.51/5.30      ! [A: int,X: num,N: nat] :
% 9.51/5.30        ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 9.51/5.30        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7505_numeral__power__le__nat__cancel__iff,axiom,
% 9.51/5.30      ! [X: num,N: nat,A: int] :
% 9.51/5.30        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 9.51/5.30        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7506_nat__le__numeral__power__cancel__iff,axiom,
% 9.51/5.30      ! [A: int,X: num,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 9.51/5.30        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7507_arctan__less__iff,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) )
% 9.51/5.30        = ( ord_less_real @ X @ Y ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7508_arctan__monotone,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ X @ Y )
% 9.51/5.30       => ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7509_nat__abs__triangle__ineq,axiom,
% 9.51/5.30      ! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7510_nat__abs__int__diff,axiom,
% 9.51/5.30      ! [A: nat,B: nat] :
% 9.51/5.30        ( ( ( ord_less_eq_nat @ A @ B )
% 9.51/5.30         => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 9.51/5.30            = ( minus_minus_nat @ B @ A ) ) )
% 9.51/5.30        & ( ~ ( ord_less_eq_nat @ A @ B )
% 9.51/5.30         => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 9.51/5.30            = ( minus_minus_nat @ A @ B ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7511_nat__zero__as__int,axiom,
% 9.51/5.30      ( zero_zero_nat
% 9.51/5.30      = ( nat2 @ zero_zero_int ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7512_nat__mono,axiom,
% 9.51/5.30      ! [X: int,Y: int] :
% 9.51/5.30        ( ( ord_less_eq_int @ X @ Y )
% 9.51/5.30       => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7513_infinite__int__iff__unbounded__le,axiom,
% 9.51/5.30      ! [S2: set_int] :
% 9.51/5.30        ( ( ~ ( finite_finite_int @ S2 ) )
% 9.51/5.30        = ( ! [M2: int] :
% 9.51/5.30            ? [N2: int] :
% 9.51/5.30              ( ( ord_less_eq_int @ M2 @ ( abs_abs_int @ N2 ) )
% 9.51/5.30              & ( member_int @ N2 @ S2 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7514_infinite__int__iff__unbounded,axiom,
% 9.51/5.30      ! [S2: set_int] :
% 9.51/5.30        ( ( ~ ( finite_finite_int @ S2 ) )
% 9.51/5.30        = ( ! [M2: int] :
% 9.51/5.30            ? [N2: int] :
% 9.51/5.30              ( ( ord_less_int @ M2 @ ( abs_abs_int @ N2 ) )
% 9.51/5.30              & ( member_int @ N2 @ S2 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7515_frac__ge__0,axiom,
% 9.51/5.30      ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7516_frac__ge__0,axiom,
% 9.51/5.30      ! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7517_frac__lt__1,axiom,
% 9.51/5.30      ! [X: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X ) @ one_one_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7518_frac__lt__1,axiom,
% 9.51/5.30      ! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7519_nat__mono__iff,axiom,
% 9.51/5.30      ! [Z2: int,W2: int] :
% 9.51/5.30        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 9.51/5.30       => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
% 9.51/5.30          = ( ord_less_int @ W2 @ Z2 ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7520_of__nat__ceiling,axiom,
% 9.51/5.30      ! [R2: real] : ( ord_less_eq_real @ R2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R2 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7521_of__nat__ceiling,axiom,
% 9.51/5.30      ! [R2: rat] : ( ord_less_eq_rat @ R2 @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R2 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7522_zless__nat__eq__int__zless,axiom,
% 9.51/5.30      ! [M: nat,Z2: int] :
% 9.51/5.30        ( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
% 9.51/5.30        = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z2 ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7523_nat__le__iff,axiom,
% 9.51/5.30      ! [X: int,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
% 9.51/5.30        = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7524_zabs__def,axiom,
% 9.51/5.30      ( abs_abs_int
% 9.51/5.30      = ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7525_abs__mod__less,axiom,
% 9.51/5.30      ! [L: int,K: int] :
% 9.51/5.30        ( ( L != zero_zero_int )
% 9.51/5.30       => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7526_of__nat__floor,axiom,
% 9.51/5.30      ! [R2: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 9.51/5.30       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim6058952711729229775r_real @ R2 ) ) ) @ R2 ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7527_of__nat__floor,axiom,
% 9.51/5.30      ! [R2: rat] :
% 9.51/5.30        ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 9.51/5.30       => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R2 ) ) ) @ R2 ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7528_nat__less__eq__zless,axiom,
% 9.51/5.30      ! [W2: int,Z2: int] :
% 9.51/5.30        ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 9.51/5.30       => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
% 9.51/5.30          = ( ord_less_int @ W2 @ Z2 ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7529_nat__le__eq__zle,axiom,
% 9.51/5.30      ! [W2: int,Z2: int] :
% 9.51/5.30        ( ( ( ord_less_int @ zero_zero_int @ W2 )
% 9.51/5.30          | ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
% 9.51/5.30       => ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
% 9.51/5.30          = ( ord_less_eq_int @ W2 @ Z2 ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7530_nat__eq__iff2,axiom,
% 9.51/5.30      ! [M: nat,W2: int] :
% 9.51/5.30        ( ( M
% 9.51/5.30          = ( nat2 @ W2 ) )
% 9.51/5.30        = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 9.51/5.30           => ( W2
% 9.51/5.30              = ( semiri1314217659103216013at_int @ M ) ) )
% 9.51/5.30          & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
% 9.51/5.30           => ( M = zero_zero_nat ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7531_nat__eq__iff,axiom,
% 9.51/5.30      ! [W2: int,M: nat] :
% 9.51/5.30        ( ( ( nat2 @ W2 )
% 9.51/5.30          = M )
% 9.51/5.30        = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 9.51/5.30           => ( W2
% 9.51/5.30              = ( semiri1314217659103216013at_int @ M ) ) )
% 9.51/5.30          & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
% 9.51/5.30           => ( M = zero_zero_nat ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7532_le__mult__nat__floor,axiom,
% 9.51/5.30      ! [A: real,B: real] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ B ) ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7533_le__mult__nat__floor,axiom,
% 9.51/5.30      ! [A: rat,B: rat] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim3151403230148437115or_rat @ A ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7534_split__nat,axiom,
% 9.51/5.30      ! [P: nat > $o,I: int] :
% 9.51/5.30        ( ( P @ ( nat2 @ I ) )
% 9.51/5.30        = ( ! [N2: nat] :
% 9.51/5.30              ( ( I
% 9.51/5.30                = ( semiri1314217659103216013at_int @ N2 ) )
% 9.51/5.30             => ( P @ N2 ) )
% 9.51/5.30          & ( ( ord_less_int @ I @ zero_zero_int )
% 9.51/5.30           => ( P @ zero_zero_nat ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7535_le__nat__iff,axiom,
% 9.51/5.30      ! [K: int,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.51/5.30       => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
% 9.51/5.30          = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7536_nat__floor__neg,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.51/5.30       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 9.51/5.30          = zero_zero_nat ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7537_floor__eq3,axiom,
% 9.51/5.30      ! [N: nat,X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 9.51/5.30         => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 9.51/5.30            = N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7538_le__nat__floor,axiom,
% 9.51/5.30      ! [X: nat,A: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
% 9.51/5.30       => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7539_nat__2,axiom,
% 9.51/5.30      ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.51/5.30      = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7540_nat__less__iff,axiom,
% 9.51/5.30      ! [W2: int,M: nat] :
% 9.51/5.30        ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 9.51/5.30       => ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
% 9.51/5.30          = ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7541_frac__eq,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ( archim2898591450579166408c_real @ X )
% 9.51/5.30          = X )
% 9.51/5.30        = ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.51/5.30          & ( ord_less_real @ X @ one_one_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7542_frac__eq,axiom,
% 9.51/5.30      ! [X: rat] :
% 9.51/5.30        ( ( ( archimedean_frac_rat @ X )
% 9.51/5.30          = X )
% 9.51/5.30        = ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 9.51/5.30          & ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7543_floor__eq4,axiom,
% 9.51/5.30      ! [N: nat,X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 9.51/5.30         => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 9.51/5.30            = N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7544_frac__add,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 9.51/5.30         => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
% 9.51/5.30            = ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) ) )
% 9.51/5.30        & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 9.51/5.30         => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
% 9.51/5.30            = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7545_frac__add,axiom,
% 9.51/5.30      ! [X: rat,Y: rat] :
% 9.51/5.30        ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 9.51/5.30         => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
% 9.51/5.30            = ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) ) )
% 9.51/5.30        & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 9.51/5.30         => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
% 9.51/5.30            = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7546_diff__nat__eq__if,axiom,
% 9.51/5.30      ! [Z8: int,Z2: int] :
% 9.51/5.30        ( ( ( ord_less_int @ Z8 @ zero_zero_int )
% 9.51/5.30         => ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) )
% 9.51/5.30            = ( nat2 @ Z2 ) ) )
% 9.51/5.30        & ( ~ ( ord_less_int @ Z8 @ zero_zero_int )
% 9.51/5.30         => ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) )
% 9.51/5.30            = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z2 @ Z8 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z2 @ Z8 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7547_of__int__of__nat,axiom,
% 9.51/5.30      ( ring_1_of_int_rat
% 9.51/5.30      = ( ^ [K3: int] : ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri681578069525770553at_rat @ ( nat2 @ K3 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7548_of__int__of__nat,axiom,
% 9.51/5.30      ( ring_18347121197199848620nteger
% 9.51/5.30      = ( ^ [K3: int] : ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri4939895301339042750nteger @ ( nat2 @ K3 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7549_of__int__of__nat,axiom,
% 9.51/5.30      ( ring_1_of_int_int
% 9.51/5.30      = ( ^ [K3: int] : ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K3 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7550_of__int__of__nat,axiom,
% 9.51/5.30      ( ring_1_of_int_real
% 9.51/5.30      = ( ^ [K3: int] : ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K3 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7551_nat__dvd__iff,axiom,
% 9.51/5.30      ! [Z2: int,M: nat] :
% 9.51/5.30        ( ( dvd_dvd_nat @ ( nat2 @ Z2 ) @ M )
% 9.51/5.30        = ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 9.51/5.30           => ( dvd_dvd_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) )
% 9.51/5.30          & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 9.51/5.30           => ( M = zero_zero_nat ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7552_nat__intermed__int__val,axiom,
% 9.51/5.30      ! [M: nat,N: nat,F: nat > int,K: int] :
% 9.51/5.30        ( ! [I2: nat] :
% 9.51/5.30            ( ( ( ord_less_eq_nat @ M @ I2 )
% 9.51/5.30              & ( ord_less_nat @ I2 @ N ) )
% 9.51/5.30           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 9.51/5.30       => ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.30         => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 9.51/5.30           => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 9.51/5.30             => ? [I2: nat] :
% 9.51/5.30                  ( ( ord_less_eq_nat @ M @ I2 )
% 9.51/5.30                  & ( ord_less_eq_nat @ I2 @ N )
% 9.51/5.30                  & ( ( F @ I2 )
% 9.51/5.30                    = K ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7553_incr__lemma,axiom,
% 9.51/5.30      ! [D: int,Z2: int,X: int] :
% 9.51/5.30        ( ( ord_less_int @ zero_zero_int @ D )
% 9.51/5.30       => ( ord_less_int @ Z2 @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7554_decr__lemma,axiom,
% 9.51/5.30      ! [D: int,X: int,Z2: int] :
% 9.51/5.30        ( ( ord_less_int @ zero_zero_int @ D )
% 9.51/5.30       => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D ) ) @ Z2 ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7555_eucl__rel__int__remainderI,axiom,
% 9.51/5.30      ! [R2: int,L: int,K: int,Q4: int] :
% 9.51/5.30        ( ( ( sgn_sgn_int @ R2 )
% 9.51/5.30          = ( sgn_sgn_int @ L ) )
% 9.51/5.30       => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
% 9.51/5.30         => ( ( K
% 9.51/5.30              = ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R2 ) )
% 9.51/5.30           => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7556_nat__ivt__aux,axiom,
% 9.51/5.30      ! [N: nat,F: nat > int,K: int] :
% 9.51/5.30        ( ! [I2: nat] :
% 9.51/5.30            ( ( ord_less_nat @ I2 @ N )
% 9.51/5.30           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 9.51/5.30       => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 9.51/5.30         => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 9.51/5.30           => ? [I2: nat] :
% 9.51/5.30                ( ( ord_less_eq_nat @ I2 @ N )
% 9.51/5.30                & ( ( F @ I2 )
% 9.51/5.30                  = K ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7557_eucl__rel__int_Ocases,axiom,
% 9.51/5.30      ! [A12: int,A23: int,A32: product_prod_int_int] :
% 9.51/5.30        ( ( eucl_rel_int @ A12 @ A23 @ A32 )
% 9.51/5.30       => ( ( ( A23 = zero_zero_int )
% 9.51/5.30           => ( A32
% 9.51/5.30             != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
% 9.51/5.30         => ( ! [Q3: int] :
% 9.51/5.30                ( ( A32
% 9.51/5.30                  = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
% 9.51/5.30               => ( ( A23 != zero_zero_int )
% 9.51/5.30                 => ( A12
% 9.51/5.30                   != ( times_times_int @ Q3 @ A23 ) ) ) )
% 9.51/5.30           => ~ ! [R3: int,Q3: int] :
% 9.51/5.30                  ( ( A32
% 9.51/5.30                    = ( product_Pair_int_int @ Q3 @ R3 ) )
% 9.51/5.30                 => ( ( ( sgn_sgn_int @ R3 )
% 9.51/5.30                      = ( sgn_sgn_int @ A23 ) )
% 9.51/5.30                   => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A23 ) )
% 9.51/5.30                     => ( A12
% 9.51/5.30                       != ( plus_plus_int @ ( times_times_int @ Q3 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7558_eucl__rel__int_Osimps,axiom,
% 9.51/5.30      ( eucl_rel_int
% 9.51/5.30      = ( ^ [A1: int,A22: int,A33: product_prod_int_int] :
% 9.51/5.30            ( ? [K3: int] :
% 9.51/5.30                ( ( A1 = K3 )
% 9.51/5.30                & ( A22 = zero_zero_int )
% 9.51/5.30                & ( A33
% 9.51/5.30                  = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
% 9.51/5.30            | ? [L2: int,K3: int,Q5: int] :
% 9.51/5.30                ( ( A1 = K3 )
% 9.51/5.30                & ( A22 = L2 )
% 9.51/5.30                & ( A33
% 9.51/5.30                  = ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
% 9.51/5.30                & ( L2 != zero_zero_int )
% 9.51/5.30                & ( K3
% 9.51/5.30                  = ( times_times_int @ Q5 @ L2 ) ) )
% 9.51/5.30            | ? [R5: int,L2: int,K3: int,Q5: int] :
% 9.51/5.30                ( ( A1 = K3 )
% 9.51/5.30                & ( A22 = L2 )
% 9.51/5.30                & ( A33
% 9.51/5.30                  = ( product_Pair_int_int @ Q5 @ R5 ) )
% 9.51/5.30                & ( ( sgn_sgn_int @ R5 )
% 9.51/5.30                  = ( sgn_sgn_int @ L2 ) )
% 9.51/5.30                & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
% 9.51/5.30                & ( K3
% 9.51/5.30                  = ( plus_plus_int @ ( times_times_int @ Q5 @ L2 ) @ R5 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7559_floor__add,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 9.51/5.30         => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
% 9.51/5.30            = ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) )
% 9.51/5.30        & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
% 9.51/5.30         => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
% 9.51/5.30            = ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ one_one_int ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7560_floor__add,axiom,
% 9.51/5.30      ! [X: rat,Y: rat] :
% 9.51/5.30        ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 9.51/5.30         => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
% 9.51/5.30            = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) )
% 9.51/5.30        & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
% 9.51/5.30         => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
% 9.51/5.30            = ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ one_one_int ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7561_nat0__intermed__int__val,axiom,
% 9.51/5.30      ! [N: nat,F: nat > int,K: int] :
% 9.51/5.30        ( ! [I2: nat] :
% 9.51/5.30            ( ( ord_less_nat @ I2 @ N )
% 9.51/5.30           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 9.51/5.30       => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 9.51/5.30         => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 9.51/5.30           => ? [I2: nat] :
% 9.51/5.30                ( ( ord_less_eq_nat @ I2 @ N )
% 9.51/5.30                & ( ( F @ I2 )
% 9.51/5.30                  = K ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7562_arctan__add,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.51/5.30       => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 9.51/5.30         => ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y ) )
% 9.51/5.30            = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7563_powr__real__of__int,axiom,
% 9.51/5.30      ! [X: real,N: int] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
% 9.51/5.30           => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 9.51/5.30              = ( power_power_real @ X @ ( nat2 @ N ) ) ) )
% 9.51/5.30          & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
% 9.51/5.30           => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 9.51/5.30              = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7564_case__prod__Pair__iden,axiom,
% 9.51/5.30      ! [P4: produc8763457246119570046nteger] :
% 9.51/5.30        ( ( produc3906647086178084059nteger @ produc6137756002093451184nteger @ P4 )
% 9.51/5.30        = P4 ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7565_case__prod__Pair__iden,axiom,
% 9.51/5.30      ! [P4: produc1908205239877642774nteger] :
% 9.51/5.30        ( ( produc6512950862096126219nteger @ produc8603105652947943368nteger @ P4 )
% 9.51/5.30        = P4 ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7566_case__prod__Pair__iden,axiom,
% 9.51/5.30      ! [P4: produc2285326912895808259nt_int] :
% 9.51/5.30        ( ( produc8492565224438309093nt_int @ produc5700946648718959541nt_int @ P4 )
% 9.51/5.30        = P4 ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7567_case__prod__Pair__iden,axiom,
% 9.51/5.30      ! [P4: produc7773217078559923341nt_int] :
% 9.51/5.30        ( ( produc5122537100556696953nt_int @ produc4305682042979456191nt_int @ P4 )
% 9.51/5.30        = P4 ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7568_case__prod__Pair__iden,axiom,
% 9.51/5.30      ! [P4: product_prod_int_int] :
% 9.51/5.30        ( ( produc4245557441103728435nt_int @ product_Pair_int_int @ P4 )
% 9.51/5.30        = P4 ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7569_the__elem__eq,axiom,
% 9.51/5.30      ! [X: produc3843707927480180839at_nat] :
% 9.51/5.30        ( ( the_el221668144340439132at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
% 9.51/5.30        = X ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7570_the__elem__eq,axiom,
% 9.51/5.30      ! [X: product_prod_nat_nat] :
% 9.51/5.30        ( ( the_el2281957884133575798at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
% 9.51/5.30        = X ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7571_the__elem__eq,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( the_elem_real @ ( insert_real @ X @ bot_bot_set_real ) )
% 9.51/5.30        = X ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7572_the__elem__eq,axiom,
% 9.51/5.30      ! [X: nat] :
% 9.51/5.30        ( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 9.51/5.30        = X ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7573_the__elem__eq,axiom,
% 9.51/5.30      ! [X: int] :
% 9.51/5.30        ( ( the_elem_int @ ( insert_int @ X @ bot_bot_set_int ) )
% 9.51/5.30        = X ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7574_exp__ge__one__minus__x__over__n__power__n,axiom,
% 9.51/5.30      ! [X: real,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
% 9.51/5.30       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.30         => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7575_exp__ge__one__plus__x__over__n__power__n,axiom,
% 9.51/5.30      ! [N: nat,X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
% 9.51/5.30       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.30         => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7576_sum__diff1_H__aux,axiom,
% 9.51/5.30      ! [F3: set_complex,I6: set_complex,F: complex > complex,I: complex] :
% 9.51/5.30        ( ( finite3207457112153483333omplex @ F3 )
% 9.51/5.30       => ( ( ord_le211207098394363844omplex
% 9.51/5.30            @ ( collect_complex
% 9.51/5.30              @ ^ [I4: complex] :
% 9.51/5.30                  ( ( member_complex @ I4 @ I6 )
% 9.51/5.30                  & ( ( F @ I4 )
% 9.51/5.30                   != zero_zero_complex ) ) )
% 9.51/5.30            @ F3 )
% 9.51/5.30         => ( ( ( member_complex @ I @ I6 )
% 9.51/5.30             => ( ( groups808145749697022017omplex @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.51/5.30                = ( minus_minus_complex @ ( groups808145749697022017omplex @ F @ I6 ) @ ( F @ I ) ) ) )
% 9.51/5.30            & ( ~ ( member_complex @ I @ I6 )
% 9.51/5.30             => ( ( groups808145749697022017omplex @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.51/5.30                = ( groups808145749697022017omplex @ F @ I6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7577_sum__diff1_H__aux,axiom,
% 9.51/5.30      ! [F3: set_real,I6: set_real,F: real > complex,I: real] :
% 9.51/5.30        ( ( finite_finite_real @ F3 )
% 9.51/5.30       => ( ( ord_less_eq_set_real
% 9.51/5.30            @ ( collect_real
% 9.51/5.30              @ ^ [I4: real] :
% 9.51/5.30                  ( ( member_real @ I4 @ I6 )
% 9.51/5.30                  & ( ( F @ I4 )
% 9.51/5.30                   != zero_zero_complex ) ) )
% 9.51/5.30            @ F3 )
% 9.51/5.30         => ( ( ( member_real @ I @ I6 )
% 9.51/5.30             => ( ( groups5683813829254066239omplex @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.51/5.30                = ( minus_minus_complex @ ( groups5683813829254066239omplex @ F @ I6 ) @ ( F @ I ) ) ) )
% 9.51/5.30            & ( ~ ( member_real @ I @ I6 )
% 9.51/5.30             => ( ( groups5683813829254066239omplex @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.51/5.30                = ( groups5683813829254066239omplex @ F @ I6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7578_sum__diff1_H__aux,axiom,
% 9.51/5.30      ! [F3: set_complex,I6: set_complex,F: complex > real,I: complex] :
% 9.51/5.30        ( ( finite3207457112153483333omplex @ F3 )
% 9.51/5.30       => ( ( ord_le211207098394363844omplex
% 9.51/5.30            @ ( collect_complex
% 9.51/5.30              @ ^ [I4: complex] :
% 9.51/5.30                  ( ( member_complex @ I4 @ I6 )
% 9.51/5.30                  & ( ( F @ I4 )
% 9.51/5.30                   != zero_zero_real ) ) )
% 9.51/5.30            @ F3 )
% 9.51/5.30         => ( ( ( member_complex @ I @ I6 )
% 9.51/5.30             => ( ( groups5737402329758386879x_real @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.51/5.30                = ( minus_minus_real @ ( groups5737402329758386879x_real @ F @ I6 ) @ ( F @ I ) ) ) )
% 9.51/5.30            & ( ~ ( member_complex @ I @ I6 )
% 9.51/5.30             => ( ( groups5737402329758386879x_real @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.51/5.30                = ( groups5737402329758386879x_real @ F @ I6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7579_sum__diff1_H__aux,axiom,
% 9.51/5.30      ! [F3: set_real,I6: set_real,F: real > real,I: real] :
% 9.51/5.30        ( ( finite_finite_real @ F3 )
% 9.51/5.30       => ( ( ord_less_eq_set_real
% 9.51/5.30            @ ( collect_real
% 9.51/5.30              @ ^ [I4: real] :
% 9.51/5.30                  ( ( member_real @ I4 @ I6 )
% 9.51/5.30                  & ( ( F @ I4 )
% 9.51/5.30                   != zero_zero_real ) ) )
% 9.51/5.30            @ F3 )
% 9.51/5.30         => ( ( ( member_real @ I @ I6 )
% 9.51/5.30             => ( ( groups97945582718554045l_real @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.51/5.30                = ( minus_minus_real @ ( groups97945582718554045l_real @ F @ I6 ) @ ( F @ I ) ) ) )
% 9.51/5.30            & ( ~ ( member_real @ I @ I6 )
% 9.51/5.30             => ( ( groups97945582718554045l_real @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.51/5.30                = ( groups97945582718554045l_real @ F @ I6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7580_sum__diff1_H__aux,axiom,
% 9.51/5.30      ! [F3: set_complex,I6: set_complex,F: complex > rat,I: complex] :
% 9.51/5.30        ( ( finite3207457112153483333omplex @ F3 )
% 9.51/5.30       => ( ( ord_le211207098394363844omplex
% 9.51/5.30            @ ( collect_complex
% 9.51/5.30              @ ^ [I4: complex] :
% 9.51/5.30                  ( ( member_complex @ I4 @ I6 )
% 9.51/5.30                  & ( ( F @ I4 )
% 9.51/5.30                   != zero_zero_rat ) ) )
% 9.51/5.30            @ F3 )
% 9.51/5.30         => ( ( ( member_complex @ I @ I6 )
% 9.51/5.30             => ( ( groups2276542476275365739ex_rat @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.51/5.30                = ( minus_minus_rat @ ( groups2276542476275365739ex_rat @ F @ I6 ) @ ( F @ I ) ) ) )
% 9.51/5.30            & ( ~ ( member_complex @ I @ I6 )
% 9.51/5.30             => ( ( groups2276542476275365739ex_rat @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.51/5.30                = ( groups2276542476275365739ex_rat @ F @ I6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7581_sum__diff1_H__aux,axiom,
% 9.51/5.30      ! [F3: set_real,I6: set_real,F: real > rat,I: real] :
% 9.51/5.30        ( ( finite_finite_real @ F3 )
% 9.51/5.30       => ( ( ord_less_eq_set_real
% 9.51/5.30            @ ( collect_real
% 9.51/5.30              @ ^ [I4: real] :
% 9.51/5.30                  ( ( member_real @ I4 @ I6 )
% 9.51/5.30                  & ( ( F @ I4 )
% 9.51/5.30                   != zero_zero_rat ) ) )
% 9.51/5.30            @ F3 )
% 9.51/5.30         => ( ( ( member_real @ I @ I6 )
% 9.51/5.30             => ( ( groups3269169158384524137al_rat @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.51/5.30                = ( minus_minus_rat @ ( groups3269169158384524137al_rat @ F @ I6 ) @ ( F @ I ) ) ) )
% 9.51/5.30            & ( ~ ( member_real @ I @ I6 )
% 9.51/5.30             => ( ( groups3269169158384524137al_rat @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.51/5.30                = ( groups3269169158384524137al_rat @ F @ I6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7582_sum__diff1_H__aux,axiom,
% 9.51/5.30      ! [F3: set_complex,I6: set_complex,F: complex > int,I: complex] :
% 9.51/5.30        ( ( finite3207457112153483333omplex @ F3 )
% 9.51/5.30       => ( ( ord_le211207098394363844omplex
% 9.51/5.30            @ ( collect_complex
% 9.51/5.30              @ ^ [I4: complex] :
% 9.51/5.30                  ( ( member_complex @ I4 @ I6 )
% 9.51/5.30                  & ( ( F @ I4 )
% 9.51/5.30                   != zero_zero_int ) ) )
% 9.51/5.30            @ F3 )
% 9.51/5.30         => ( ( ( member_complex @ I @ I6 )
% 9.51/5.30             => ( ( groups2909182065852811199ex_int @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.51/5.30                = ( minus_minus_int @ ( groups2909182065852811199ex_int @ F @ I6 ) @ ( F @ I ) ) ) )
% 9.51/5.30            & ( ~ ( member_complex @ I @ I6 )
% 9.51/5.30             => ( ( groups2909182065852811199ex_int @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.51/5.30                = ( groups2909182065852811199ex_int @ F @ I6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7583_sum__diff1_H__aux,axiom,
% 9.51/5.30      ! [F3: set_real,I6: set_real,F: real > int,I: real] :
% 9.51/5.30        ( ( finite_finite_real @ F3 )
% 9.51/5.30       => ( ( ord_less_eq_set_real
% 9.51/5.30            @ ( collect_real
% 9.51/5.30              @ ^ [I4: real] :
% 9.51/5.30                  ( ( member_real @ I4 @ I6 )
% 9.51/5.30                  & ( ( F @ I4 )
% 9.51/5.30                   != zero_zero_int ) ) )
% 9.51/5.30            @ F3 )
% 9.51/5.30         => ( ( ( member_real @ I @ I6 )
% 9.51/5.30             => ( ( groups3901808747961969597al_int @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.51/5.30                = ( minus_minus_int @ ( groups3901808747961969597al_int @ F @ I6 ) @ ( F @ I ) ) ) )
% 9.51/5.30            & ( ~ ( member_real @ I @ I6 )
% 9.51/5.30             => ( ( groups3901808747961969597al_int @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.51/5.30                = ( groups3901808747961969597al_int @ F @ I6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7584_sum__diff1_H__aux,axiom,
% 9.51/5.30      ! [F3: set_nat,I6: set_nat,F: nat > complex,I: nat] :
% 9.51/5.30        ( ( finite_finite_nat @ F3 )
% 9.51/5.30       => ( ( ord_less_eq_set_nat
% 9.51/5.30            @ ( collect_nat
% 9.51/5.30              @ ^ [I4: nat] :
% 9.51/5.30                  ( ( member_nat @ I4 @ I6 )
% 9.51/5.30                  & ( ( F @ I4 )
% 9.51/5.30                   != zero_zero_complex ) ) )
% 9.51/5.30            @ F3 )
% 9.51/5.30         => ( ( ( member_nat @ I @ I6 )
% 9.51/5.30             => ( ( groups8515261248781899619omplex @ F @ ( minus_minus_set_nat @ I6 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
% 9.51/5.30                = ( minus_minus_complex @ ( groups8515261248781899619omplex @ F @ I6 ) @ ( F @ I ) ) ) )
% 9.51/5.30            & ( ~ ( member_nat @ I @ I6 )
% 9.51/5.30             => ( ( groups8515261248781899619omplex @ F @ ( minus_minus_set_nat @ I6 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
% 9.51/5.30                = ( groups8515261248781899619omplex @ F @ I6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7585_sum__diff1_H__aux,axiom,
% 9.51/5.30      ! [F3: set_nat,I6: set_nat,F: nat > real,I: nat] :
% 9.51/5.30        ( ( finite_finite_nat @ F3 )
% 9.51/5.30       => ( ( ord_less_eq_set_nat
% 9.51/5.30            @ ( collect_nat
% 9.51/5.30              @ ^ [I4: nat] :
% 9.51/5.30                  ( ( member_nat @ I4 @ I6 )
% 9.51/5.30                  & ( ( F @ I4 )
% 9.51/5.30                   != zero_zero_real ) ) )
% 9.51/5.30            @ F3 )
% 9.51/5.30         => ( ( ( member_nat @ I @ I6 )
% 9.51/5.30             => ( ( groups8560362682196896993t_real @ F @ ( minus_minus_set_nat @ I6 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
% 9.51/5.30                = ( minus_minus_real @ ( groups8560362682196896993t_real @ F @ I6 ) @ ( F @ I ) ) ) )
% 9.51/5.30            & ( ~ ( member_nat @ I @ I6 )
% 9.51/5.30             => ( ( groups8560362682196896993t_real @ F @ ( minus_minus_set_nat @ I6 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
% 9.51/5.30                = ( groups8560362682196896993t_real @ F @ I6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7586_inverse__zero,axiom,
% 9.51/5.30      ( ( invers8013647133539491842omplex @ zero_zero_complex )
% 9.51/5.30      = zero_zero_complex ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7587_inverse__zero,axiom,
% 9.51/5.30      ( ( inverse_inverse_real @ zero_zero_real )
% 9.51/5.30      = zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7588_inverse__zero,axiom,
% 9.51/5.30      ( ( inverse_inverse_rat @ zero_zero_rat )
% 9.51/5.30      = zero_zero_rat ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7589_inverse__nonzero__iff__nonzero,axiom,
% 9.51/5.30      ! [A: complex] :
% 9.51/5.30        ( ( ( invers8013647133539491842omplex @ A )
% 9.51/5.30          = zero_zero_complex )
% 9.51/5.30        = ( A = zero_zero_complex ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7590_inverse__nonzero__iff__nonzero,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( ( inverse_inverse_real @ A )
% 9.51/5.30          = zero_zero_real )
% 9.51/5.30        = ( A = zero_zero_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7591_inverse__nonzero__iff__nonzero,axiom,
% 9.51/5.30      ! [A: rat] :
% 9.51/5.30        ( ( ( inverse_inverse_rat @ A )
% 9.51/5.30          = zero_zero_rat )
% 9.51/5.30        = ( A = zero_zero_rat ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7592_exp__less__cancel__iff,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 9.51/5.30        = ( ord_less_real @ X @ Y ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7593_exp__less__mono,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ X @ Y )
% 9.51/5.30       => ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7594_inverse__nonpositive__iff__nonpositive,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 9.51/5.30        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7595_inverse__nonpositive__iff__nonpositive,axiom,
% 9.51/5.30      ! [A: rat] :
% 9.51/5.30        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 9.51/5.30        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7596_inverse__nonnegative__iff__nonnegative,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 9.51/5.30        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7597_inverse__nonnegative__iff__nonnegative,axiom,
% 9.51/5.30      ! [A: rat] :
% 9.51/5.30        ( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 9.51/5.30        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7598_inverse__less__iff__less,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ A )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.51/5.30         => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.51/5.30            = ( ord_less_real @ B @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7599_inverse__less__iff__less,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.51/5.30       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.51/5.30         => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.51/5.30            = ( ord_less_rat @ B @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7600_inverse__less__iff__less__neg,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( ord_less_real @ A @ zero_zero_real )
% 9.51/5.30       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.51/5.30         => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.51/5.30            = ( ord_less_real @ B @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7601_inverse__less__iff__less__neg,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.51/5.30       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.51/5.30         => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.51/5.30            = ( ord_less_rat @ B @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7602_inverse__negative__iff__negative,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 9.51/5.30        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7603_inverse__negative__iff__negative,axiom,
% 9.51/5.30      ! [A: rat] :
% 9.51/5.30        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 9.51/5.30        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7604_inverse__positive__iff__positive,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 9.51/5.30        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7605_inverse__positive__iff__positive,axiom,
% 9.51/5.30      ! [A: rat] :
% 9.51/5.30        ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 9.51/5.30        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7606_exp__zero,axiom,
% 9.51/5.30      ( ( exp_complex @ zero_zero_complex )
% 9.51/5.30      = one_one_complex ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7607_exp__zero,axiom,
% 9.51/5.30      ( ( exp_real @ zero_zero_real )
% 9.51/5.30      = one_one_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7608_sum_Oempty_H,axiom,
% 9.51/5.30      ! [P4: real > complex] :
% 9.51/5.30        ( ( groups5683813829254066239omplex @ P4 @ bot_bot_set_real )
% 9.51/5.30        = zero_zero_complex ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7609_sum_Oempty_H,axiom,
% 9.51/5.30      ! [P4: real > real] :
% 9.51/5.30        ( ( groups97945582718554045l_real @ P4 @ bot_bot_set_real )
% 9.51/5.30        = zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7610_sum_Oempty_H,axiom,
% 9.51/5.30      ! [P4: real > rat] :
% 9.51/5.30        ( ( groups3269169158384524137al_rat @ P4 @ bot_bot_set_real )
% 9.51/5.30        = zero_zero_rat ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7611_sum_Oempty_H,axiom,
% 9.51/5.30      ! [P4: real > nat] :
% 9.51/5.30        ( ( groups3904299218471019873al_nat @ P4 @ bot_bot_set_real )
% 9.51/5.30        = zero_zero_nat ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7612_sum_Oempty_H,axiom,
% 9.51/5.30      ! [P4: real > int] :
% 9.51/5.30        ( ( groups3901808747961969597al_int @ P4 @ bot_bot_set_real )
% 9.51/5.30        = zero_zero_int ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7613_sum_Oempty_H,axiom,
% 9.51/5.30      ! [P4: nat > complex] :
% 9.51/5.30        ( ( groups8515261248781899619omplex @ P4 @ bot_bot_set_nat )
% 9.51/5.30        = zero_zero_complex ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7614_sum_Oempty_H,axiom,
% 9.51/5.30      ! [P4: nat > real] :
% 9.51/5.30        ( ( groups8560362682196896993t_real @ P4 @ bot_bot_set_nat )
% 9.51/5.30        = zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7615_sum_Oempty_H,axiom,
% 9.51/5.30      ! [P4: nat > rat] :
% 9.51/5.30        ( ( groups1351286907653491341at_rat @ P4 @ bot_bot_set_nat )
% 9.51/5.30        = zero_zero_rat ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7616_sum_Oempty_H,axiom,
% 9.51/5.30      ! [P4: nat > nat] :
% 9.51/5.30        ( ( groups1986416967739987077at_nat @ P4 @ bot_bot_set_nat )
% 9.51/5.30        = zero_zero_nat ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7617_sum_Oempty_H,axiom,
% 9.51/5.30      ! [P4: nat > int] :
% 9.51/5.30        ( ( groups1983926497230936801at_int @ P4 @ bot_bot_set_nat )
% 9.51/5.30        = zero_zero_int ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7618_sum_Oeq__sum,axiom,
% 9.51/5.30      ! [I6: set_nat,P4: nat > nat] :
% 9.51/5.30        ( ( finite_finite_nat @ I6 )
% 9.51/5.30       => ( ( groups1986416967739987077at_nat @ P4 @ I6 )
% 9.51/5.30          = ( groups3542108847815614940at_nat @ P4 @ I6 ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7619_sum_Oeq__sum,axiom,
% 9.51/5.30      ! [I6: set_complex,P4: complex > complex] :
% 9.51/5.30        ( ( finite3207457112153483333omplex @ I6 )
% 9.51/5.30       => ( ( groups808145749697022017omplex @ P4 @ I6 )
% 9.51/5.30          = ( groups7754918857620584856omplex @ P4 @ I6 ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7620_sum_Oeq__sum,axiom,
% 9.51/5.30      ! [I6: set_nat,P4: nat > real] :
% 9.51/5.30        ( ( finite_finite_nat @ I6 )
% 9.51/5.30       => ( ( groups8560362682196896993t_real @ P4 @ I6 )
% 9.51/5.30          = ( groups6591440286371151544t_real @ P4 @ I6 ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7621_inverse__le__iff__le,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ A )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.51/5.30         => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.51/5.30            = ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7622_inverse__le__iff__le,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.51/5.30       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.51/5.30         => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.51/5.30            = ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7623_inverse__le__iff__le__neg,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( ord_less_real @ A @ zero_zero_real )
% 9.51/5.30       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.51/5.30         => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.51/5.30            = ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7624_inverse__le__iff__le__neg,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.51/5.30       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.51/5.30         => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.51/5.30            = ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7625_right__inverse,axiom,
% 9.51/5.30      ! [A: complex] :
% 9.51/5.30        ( ( A != zero_zero_complex )
% 9.51/5.30       => ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
% 9.51/5.30          = one_one_complex ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7626_right__inverse,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( A != zero_zero_real )
% 9.51/5.30       => ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
% 9.51/5.30          = one_one_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7627_right__inverse,axiom,
% 9.51/5.30      ! [A: rat] :
% 9.51/5.30        ( ( A != zero_zero_rat )
% 9.51/5.30       => ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
% 9.51/5.30          = one_one_rat ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7628_left__inverse,axiom,
% 9.51/5.30      ! [A: complex] :
% 9.51/5.30        ( ( A != zero_zero_complex )
% 9.51/5.30       => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
% 9.51/5.30          = one_one_complex ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7629_left__inverse,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( A != zero_zero_real )
% 9.51/5.30       => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
% 9.51/5.30          = one_one_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7630_left__inverse,axiom,
% 9.51/5.30      ! [A: rat] :
% 9.51/5.30        ( ( A != zero_zero_rat )
% 9.51/5.30       => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
% 9.51/5.30          = one_one_rat ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7631_one__less__exp__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
% 9.51/5.30        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7632_exp__less__one__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
% 9.51/5.30        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7633_sum_Oinsert_H,axiom,
% 9.51/5.30      ! [I6: set_real,P4: real > complex,I: real] :
% 9.51/5.30        ( ( finite_finite_real
% 9.51/5.30          @ ( collect_real
% 9.51/5.30            @ ^ [X3: real] :
% 9.51/5.30                ( ( member_real @ X3 @ I6 )
% 9.51/5.30                & ( ( P4 @ X3 )
% 9.51/5.30                 != zero_zero_complex ) ) ) )
% 9.51/5.30       => ( ( ( member_real @ I @ I6 )
% 9.51/5.30           => ( ( groups5683813829254066239omplex @ P4 @ ( insert_real @ I @ I6 ) )
% 9.51/5.30              = ( groups5683813829254066239omplex @ P4 @ I6 ) ) )
% 9.51/5.30          & ( ~ ( member_real @ I @ I6 )
% 9.51/5.30           => ( ( groups5683813829254066239omplex @ P4 @ ( insert_real @ I @ I6 ) )
% 9.51/5.30              = ( plus_plus_complex @ ( P4 @ I ) @ ( groups5683813829254066239omplex @ P4 @ I6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7634_sum_Oinsert_H,axiom,
% 9.51/5.30      ! [I6: set_nat,P4: nat > complex,I: nat] :
% 9.51/5.30        ( ( finite_finite_nat
% 9.51/5.30          @ ( collect_nat
% 9.51/5.30            @ ^ [X3: nat] :
% 9.51/5.30                ( ( member_nat @ X3 @ I6 )
% 9.51/5.30                & ( ( P4 @ X3 )
% 9.51/5.30                 != zero_zero_complex ) ) ) )
% 9.51/5.30       => ( ( ( member_nat @ I @ I6 )
% 9.51/5.30           => ( ( groups8515261248781899619omplex @ P4 @ ( insert_nat @ I @ I6 ) )
% 9.51/5.30              = ( groups8515261248781899619omplex @ P4 @ I6 ) ) )
% 9.51/5.30          & ( ~ ( member_nat @ I @ I6 )
% 9.51/5.30           => ( ( groups8515261248781899619omplex @ P4 @ ( insert_nat @ I @ I6 ) )
% 9.51/5.30              = ( plus_plus_complex @ ( P4 @ I ) @ ( groups8515261248781899619omplex @ P4 @ I6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7635_sum_Oinsert_H,axiom,
% 9.51/5.30      ! [I6: set_int,P4: int > complex,I: int] :
% 9.51/5.30        ( ( finite_finite_int
% 9.51/5.30          @ ( collect_int
% 9.51/5.30            @ ^ [X3: int] :
% 9.51/5.30                ( ( member_int @ X3 @ I6 )
% 9.51/5.30                & ( ( P4 @ X3 )
% 9.51/5.30                 != zero_zero_complex ) ) ) )
% 9.51/5.30       => ( ( ( member_int @ I @ I6 )
% 9.51/5.30           => ( ( groups267424677133301183omplex @ P4 @ ( insert_int @ I @ I6 ) )
% 9.51/5.30              = ( groups267424677133301183omplex @ P4 @ I6 ) ) )
% 9.51/5.30          & ( ~ ( member_int @ I @ I6 )
% 9.51/5.30           => ( ( groups267424677133301183omplex @ P4 @ ( insert_int @ I @ I6 ) )
% 9.51/5.30              = ( plus_plus_complex @ ( P4 @ I ) @ ( groups267424677133301183omplex @ P4 @ I6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7636_sum_Oinsert_H,axiom,
% 9.51/5.30      ! [I6: set_complex,P4: complex > complex,I: complex] :
% 9.51/5.30        ( ( finite3207457112153483333omplex
% 9.51/5.30          @ ( collect_complex
% 9.51/5.30            @ ^ [X3: complex] :
% 9.51/5.30                ( ( member_complex @ X3 @ I6 )
% 9.51/5.30                & ( ( P4 @ X3 )
% 9.51/5.30                 != zero_zero_complex ) ) ) )
% 9.51/5.30       => ( ( ( member_complex @ I @ I6 )
% 9.51/5.30           => ( ( groups808145749697022017omplex @ P4 @ ( insert_complex @ I @ I6 ) )
% 9.51/5.30              = ( groups808145749697022017omplex @ P4 @ I6 ) ) )
% 9.51/5.30          & ( ~ ( member_complex @ I @ I6 )
% 9.51/5.30           => ( ( groups808145749697022017omplex @ P4 @ ( insert_complex @ I @ I6 ) )
% 9.51/5.30              = ( plus_plus_complex @ ( P4 @ I ) @ ( groups808145749697022017omplex @ P4 @ I6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7637_sum_Oinsert_H,axiom,
% 9.51/5.30      ! [I6: set_real,P4: real > real,I: real] :
% 9.51/5.30        ( ( finite_finite_real
% 9.51/5.30          @ ( collect_real
% 9.51/5.30            @ ^ [X3: real] :
% 9.51/5.30                ( ( member_real @ X3 @ I6 )
% 9.51/5.30                & ( ( P4 @ X3 )
% 9.51/5.30                 != zero_zero_real ) ) ) )
% 9.51/5.30       => ( ( ( member_real @ I @ I6 )
% 9.51/5.30           => ( ( groups97945582718554045l_real @ P4 @ ( insert_real @ I @ I6 ) )
% 9.51/5.30              = ( groups97945582718554045l_real @ P4 @ I6 ) ) )
% 9.51/5.30          & ( ~ ( member_real @ I @ I6 )
% 9.51/5.30           => ( ( groups97945582718554045l_real @ P4 @ ( insert_real @ I @ I6 ) )
% 9.51/5.30              = ( plus_plus_real @ ( P4 @ I ) @ ( groups97945582718554045l_real @ P4 @ I6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7638_sum_Oinsert_H,axiom,
% 9.51/5.30      ! [I6: set_nat,P4: nat > real,I: nat] :
% 9.51/5.30        ( ( finite_finite_nat
% 9.51/5.30          @ ( collect_nat
% 9.51/5.30            @ ^ [X3: nat] :
% 9.51/5.30                ( ( member_nat @ X3 @ I6 )
% 9.51/5.30                & ( ( P4 @ X3 )
% 9.51/5.30                 != zero_zero_real ) ) ) )
% 9.51/5.30       => ( ( ( member_nat @ I @ I6 )
% 9.51/5.30           => ( ( groups8560362682196896993t_real @ P4 @ ( insert_nat @ I @ I6 ) )
% 9.51/5.30              = ( groups8560362682196896993t_real @ P4 @ I6 ) ) )
% 9.51/5.30          & ( ~ ( member_nat @ I @ I6 )
% 9.51/5.30           => ( ( groups8560362682196896993t_real @ P4 @ ( insert_nat @ I @ I6 ) )
% 9.51/5.30              = ( plus_plus_real @ ( P4 @ I ) @ ( groups8560362682196896993t_real @ P4 @ I6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7639_sum_Oinsert_H,axiom,
% 9.51/5.30      ! [I6: set_int,P4: int > real,I: int] :
% 9.51/5.30        ( ( finite_finite_int
% 9.51/5.30          @ ( collect_int
% 9.51/5.30            @ ^ [X3: int] :
% 9.51/5.30                ( ( member_int @ X3 @ I6 )
% 9.51/5.30                & ( ( P4 @ X3 )
% 9.51/5.30                 != zero_zero_real ) ) ) )
% 9.51/5.30       => ( ( ( member_int @ I @ I6 )
% 9.51/5.30           => ( ( groups1523912220035142973t_real @ P4 @ ( insert_int @ I @ I6 ) )
% 9.51/5.30              = ( groups1523912220035142973t_real @ P4 @ I6 ) ) )
% 9.51/5.30          & ( ~ ( member_int @ I @ I6 )
% 9.51/5.30           => ( ( groups1523912220035142973t_real @ P4 @ ( insert_int @ I @ I6 ) )
% 9.51/5.30              = ( plus_plus_real @ ( P4 @ I ) @ ( groups1523912220035142973t_real @ P4 @ I6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7640_sum_Oinsert_H,axiom,
% 9.51/5.30      ! [I6: set_complex,P4: complex > real,I: complex] :
% 9.51/5.30        ( ( finite3207457112153483333omplex
% 9.51/5.30          @ ( collect_complex
% 9.51/5.30            @ ^ [X3: complex] :
% 9.51/5.30                ( ( member_complex @ X3 @ I6 )
% 9.51/5.30                & ( ( P4 @ X3 )
% 9.51/5.30                 != zero_zero_real ) ) ) )
% 9.51/5.30       => ( ( ( member_complex @ I @ I6 )
% 9.51/5.30           => ( ( groups5737402329758386879x_real @ P4 @ ( insert_complex @ I @ I6 ) )
% 9.51/5.30              = ( groups5737402329758386879x_real @ P4 @ I6 ) ) )
% 9.51/5.30          & ( ~ ( member_complex @ I @ I6 )
% 9.51/5.30           => ( ( groups5737402329758386879x_real @ P4 @ ( insert_complex @ I @ I6 ) )
% 9.51/5.30              = ( plus_plus_real @ ( P4 @ I ) @ ( groups5737402329758386879x_real @ P4 @ I6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7641_sum_Oinsert_H,axiom,
% 9.51/5.30      ! [I6: set_real,P4: real > rat,I: real] :
% 9.51/5.30        ( ( finite_finite_real
% 9.51/5.30          @ ( collect_real
% 9.51/5.30            @ ^ [X3: real] :
% 9.51/5.30                ( ( member_real @ X3 @ I6 )
% 9.51/5.30                & ( ( P4 @ X3 )
% 9.51/5.30                 != zero_zero_rat ) ) ) )
% 9.51/5.30       => ( ( ( member_real @ I @ I6 )
% 9.51/5.30           => ( ( groups3269169158384524137al_rat @ P4 @ ( insert_real @ I @ I6 ) )
% 9.51/5.30              = ( groups3269169158384524137al_rat @ P4 @ I6 ) ) )
% 9.51/5.30          & ( ~ ( member_real @ I @ I6 )
% 9.51/5.30           => ( ( groups3269169158384524137al_rat @ P4 @ ( insert_real @ I @ I6 ) )
% 9.51/5.30              = ( plus_plus_rat @ ( P4 @ I ) @ ( groups3269169158384524137al_rat @ P4 @ I6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7642_sum_Oinsert_H,axiom,
% 9.51/5.30      ! [I6: set_nat,P4: nat > rat,I: nat] :
% 9.51/5.30        ( ( finite_finite_nat
% 9.51/5.30          @ ( collect_nat
% 9.51/5.30            @ ^ [X3: nat] :
% 9.51/5.30                ( ( member_nat @ X3 @ I6 )
% 9.51/5.30                & ( ( P4 @ X3 )
% 9.51/5.30                 != zero_zero_rat ) ) ) )
% 9.51/5.30       => ( ( ( member_nat @ I @ I6 )
% 9.51/5.30           => ( ( groups1351286907653491341at_rat @ P4 @ ( insert_nat @ I @ I6 ) )
% 9.51/5.30              = ( groups1351286907653491341at_rat @ P4 @ I6 ) ) )
% 9.51/5.30          & ( ~ ( member_nat @ I @ I6 )
% 9.51/5.30           => ( ( groups1351286907653491341at_rat @ P4 @ ( insert_nat @ I @ I6 ) )
% 9.51/5.30              = ( plus_plus_rat @ ( P4 @ I ) @ ( groups1351286907653491341at_rat @ P4 @ I6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7643_nonzero__norm__inverse,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( A != zero_zero_real )
% 9.51/5.30       => ( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A ) )
% 9.51/5.30          = ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7644_nonzero__norm__inverse,axiom,
% 9.51/5.30      ! [A: complex] :
% 9.51/5.30        ( ( A != zero_zero_complex )
% 9.51/5.30       => ( ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ A ) )
% 9.51/5.30          = ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7645_field__class_Ofield__inverse__zero,axiom,
% 9.51/5.30      ( ( invers8013647133539491842omplex @ zero_zero_complex )
% 9.51/5.30      = zero_zero_complex ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7646_field__class_Ofield__inverse__zero,axiom,
% 9.51/5.30      ( ( inverse_inverse_real @ zero_zero_real )
% 9.51/5.30      = zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7647_field__class_Ofield__inverse__zero,axiom,
% 9.51/5.30      ( ( inverse_inverse_rat @ zero_zero_rat )
% 9.51/5.30      = zero_zero_rat ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7648_inverse__zero__imp__zero,axiom,
% 9.51/5.30      ! [A: complex] :
% 9.51/5.30        ( ( ( invers8013647133539491842omplex @ A )
% 9.51/5.30          = zero_zero_complex )
% 9.51/5.30       => ( A = zero_zero_complex ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7649_inverse__zero__imp__zero,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( ( inverse_inverse_real @ A )
% 9.51/5.30          = zero_zero_real )
% 9.51/5.30       => ( A = zero_zero_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7650_inverse__zero__imp__zero,axiom,
% 9.51/5.30      ! [A: rat] :
% 9.51/5.30        ( ( ( inverse_inverse_rat @ A )
% 9.51/5.30          = zero_zero_rat )
% 9.51/5.30       => ( A = zero_zero_rat ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7651_nonzero__inverse__eq__imp__eq,axiom,
% 9.51/5.30      ! [A: complex,B: complex] :
% 9.51/5.30        ( ( ( invers8013647133539491842omplex @ A )
% 9.51/5.30          = ( invers8013647133539491842omplex @ B ) )
% 9.51/5.30       => ( ( A != zero_zero_complex )
% 9.51/5.30         => ( ( B != zero_zero_complex )
% 9.51/5.30           => ( A = B ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7652_nonzero__inverse__eq__imp__eq,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( ( inverse_inverse_real @ A )
% 9.51/5.30          = ( inverse_inverse_real @ B ) )
% 9.51/5.30       => ( ( A != zero_zero_real )
% 9.51/5.30         => ( ( B != zero_zero_real )
% 9.51/5.30           => ( A = B ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7653_nonzero__inverse__eq__imp__eq,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( ( inverse_inverse_rat @ A )
% 9.51/5.30          = ( inverse_inverse_rat @ B ) )
% 9.51/5.30       => ( ( A != zero_zero_rat )
% 9.51/5.30         => ( ( B != zero_zero_rat )
% 9.51/5.30           => ( A = B ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7654_nonzero__inverse__inverse__eq,axiom,
% 9.51/5.30      ! [A: complex] :
% 9.51/5.30        ( ( A != zero_zero_complex )
% 9.51/5.30       => ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
% 9.51/5.30          = A ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7655_nonzero__inverse__inverse__eq,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( A != zero_zero_real )
% 9.51/5.30       => ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
% 9.51/5.30          = A ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7656_nonzero__inverse__inverse__eq,axiom,
% 9.51/5.30      ! [A: rat] :
% 9.51/5.30        ( ( A != zero_zero_rat )
% 9.51/5.30       => ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
% 9.51/5.30          = A ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7657_nonzero__imp__inverse__nonzero,axiom,
% 9.51/5.30      ! [A: complex] :
% 9.51/5.30        ( ( A != zero_zero_complex )
% 9.51/5.30       => ( ( invers8013647133539491842omplex @ A )
% 9.51/5.30         != zero_zero_complex ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7658_nonzero__imp__inverse__nonzero,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( A != zero_zero_real )
% 9.51/5.30       => ( ( inverse_inverse_real @ A )
% 9.51/5.30         != zero_zero_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7659_nonzero__imp__inverse__nonzero,axiom,
% 9.51/5.30      ! [A: rat] :
% 9.51/5.30        ( ( A != zero_zero_rat )
% 9.51/5.30       => ( ( inverse_inverse_rat @ A )
% 9.51/5.30         != zero_zero_rat ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7660_exp__less__cancel,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 9.51/5.30       => ( ord_less_real @ X @ Y ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7661_exp__not__eq__zero,axiom,
% 9.51/5.30      ! [X: complex] :
% 9.51/5.30        ( ( exp_complex @ X )
% 9.51/5.30       != zero_zero_complex ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7662_exp__not__eq__zero,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( exp_real @ X )
% 9.51/5.30       != zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7663_sum_Onon__neutral_H,axiom,
% 9.51/5.30      ! [G: real > complex,I6: set_real] :
% 9.51/5.30        ( ( groups5683813829254066239omplex @ G
% 9.51/5.30          @ ( collect_real
% 9.51/5.30            @ ^ [X3: real] :
% 9.51/5.30                ( ( member_real @ X3 @ I6 )
% 9.51/5.30                & ( ( G @ X3 )
% 9.51/5.30                 != zero_zero_complex ) ) ) )
% 9.51/5.30        = ( groups5683813829254066239omplex @ G @ I6 ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7664_sum_Onon__neutral_H,axiom,
% 9.51/5.30      ! [G: nat > complex,I6: set_nat] :
% 9.51/5.30        ( ( groups8515261248781899619omplex @ G
% 9.51/5.30          @ ( collect_nat
% 9.51/5.30            @ ^ [X3: nat] :
% 9.51/5.30                ( ( member_nat @ X3 @ I6 )
% 9.51/5.30                & ( ( G @ X3 )
% 9.51/5.30                 != zero_zero_complex ) ) ) )
% 9.51/5.30        = ( groups8515261248781899619omplex @ G @ I6 ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7665_sum_Onon__neutral_H,axiom,
% 9.51/5.30      ! [G: int > complex,I6: set_int] :
% 9.51/5.30        ( ( groups267424677133301183omplex @ G
% 9.51/5.30          @ ( collect_int
% 9.51/5.30            @ ^ [X3: int] :
% 9.51/5.30                ( ( member_int @ X3 @ I6 )
% 9.51/5.30                & ( ( G @ X3 )
% 9.51/5.30                 != zero_zero_complex ) ) ) )
% 9.51/5.30        = ( groups267424677133301183omplex @ G @ I6 ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7666_sum_Onon__neutral_H,axiom,
% 9.51/5.30      ! [G: real > real,I6: set_real] :
% 9.51/5.30        ( ( groups97945582718554045l_real @ G
% 9.51/5.30          @ ( collect_real
% 9.51/5.30            @ ^ [X3: real] :
% 9.51/5.30                ( ( member_real @ X3 @ I6 )
% 9.51/5.30                & ( ( G @ X3 )
% 9.51/5.30                 != zero_zero_real ) ) ) )
% 9.51/5.30        = ( groups97945582718554045l_real @ G @ I6 ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7667_sum_Onon__neutral_H,axiom,
% 9.51/5.30      ! [G: nat > real,I6: set_nat] :
% 9.51/5.30        ( ( groups8560362682196896993t_real @ G
% 9.51/5.30          @ ( collect_nat
% 9.51/5.30            @ ^ [X3: nat] :
% 9.51/5.30                ( ( member_nat @ X3 @ I6 )
% 9.51/5.30                & ( ( G @ X3 )
% 9.51/5.30                 != zero_zero_real ) ) ) )
% 9.51/5.30        = ( groups8560362682196896993t_real @ G @ I6 ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7668_sum_Onon__neutral_H,axiom,
% 9.51/5.30      ! [G: int > real,I6: set_int] :
% 9.51/5.30        ( ( groups1523912220035142973t_real @ G
% 9.51/5.30          @ ( collect_int
% 9.51/5.30            @ ^ [X3: int] :
% 9.51/5.30                ( ( member_int @ X3 @ I6 )
% 9.51/5.30                & ( ( G @ X3 )
% 9.51/5.30                 != zero_zero_real ) ) ) )
% 9.51/5.30        = ( groups1523912220035142973t_real @ G @ I6 ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7669_sum_Onon__neutral_H,axiom,
% 9.51/5.30      ! [G: real > rat,I6: set_real] :
% 9.51/5.30        ( ( groups3269169158384524137al_rat @ G
% 9.51/5.30          @ ( collect_real
% 9.51/5.30            @ ^ [X3: real] :
% 9.51/5.30                ( ( member_real @ X3 @ I6 )
% 9.51/5.30                & ( ( G @ X3 )
% 9.51/5.30                 != zero_zero_rat ) ) ) )
% 9.51/5.30        = ( groups3269169158384524137al_rat @ G @ I6 ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7670_sum_Onon__neutral_H,axiom,
% 9.51/5.30      ! [G: nat > rat,I6: set_nat] :
% 9.51/5.30        ( ( groups1351286907653491341at_rat @ G
% 9.51/5.30          @ ( collect_nat
% 9.51/5.30            @ ^ [X3: nat] :
% 9.51/5.30                ( ( member_nat @ X3 @ I6 )
% 9.51/5.30                & ( ( G @ X3 )
% 9.51/5.30                 != zero_zero_rat ) ) ) )
% 9.51/5.30        = ( groups1351286907653491341at_rat @ G @ I6 ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7671_sum_Onon__neutral_H,axiom,
% 9.51/5.30      ! [G: int > rat,I6: set_int] :
% 9.51/5.30        ( ( groups2350640619554545897nt_rat @ G
% 9.51/5.30          @ ( collect_int
% 9.51/5.30            @ ^ [X3: int] :
% 9.51/5.30                ( ( member_int @ X3 @ I6 )
% 9.51/5.30                & ( ( G @ X3 )
% 9.51/5.30                 != zero_zero_rat ) ) ) )
% 9.51/5.30        = ( groups2350640619554545897nt_rat @ G @ I6 ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7672_sum_Onon__neutral_H,axiom,
% 9.51/5.30      ! [G: real > nat,I6: set_real] :
% 9.51/5.30        ( ( groups3904299218471019873al_nat @ G
% 9.51/5.30          @ ( collect_real
% 9.51/5.30            @ ^ [X3: real] :
% 9.51/5.30                ( ( member_real @ X3 @ I6 )
% 9.51/5.30                & ( ( G @ X3 )
% 9.51/5.30                 != zero_zero_nat ) ) ) )
% 9.51/5.30        = ( groups3904299218471019873al_nat @ G @ I6 ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7673_norm__inverse__le__norm,axiom,
% 9.51/5.30      ! [R2: real,X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ R2 @ ( real_V7735802525324610683m_real @ X ) )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ R2 )
% 9.51/5.30         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ X ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7674_norm__inverse__le__norm,axiom,
% 9.51/5.30      ! [R2: real,X: complex] :
% 9.51/5.30        ( ( ord_less_eq_real @ R2 @ ( real_V1022390504157884413omplex @ X ) )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ R2 )
% 9.51/5.30         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ X ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7675_inverse__less__imp__less,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.51/5.30         => ( ord_less_real @ B @ A ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7676_inverse__less__imp__less,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.51/5.30       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.51/5.30         => ( ord_less_rat @ B @ A ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7677_less__imp__inverse__less,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( ord_less_real @ A @ B )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.51/5.30         => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7678_less__imp__inverse__less,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( ord_less_rat @ A @ B )
% 9.51/5.30       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.51/5.30         => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7679_inverse__less__imp__less__neg,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.51/5.30       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.51/5.30         => ( ord_less_real @ B @ A ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7680_inverse__less__imp__less__neg,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.51/5.30       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.51/5.30         => ( ord_less_rat @ B @ A ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7681_less__imp__inverse__less__neg,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( ord_less_real @ A @ B )
% 9.51/5.30       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.51/5.30         => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7682_less__imp__inverse__less__neg,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( ord_less_rat @ A @ B )
% 9.51/5.30       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.51/5.30         => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7683_inverse__negative__imp__negative,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 9.51/5.30       => ( ( A != zero_zero_real )
% 9.51/5.30         => ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7684_inverse__negative__imp__negative,axiom,
% 9.51/5.30      ! [A: rat] :
% 9.51/5.30        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 9.51/5.30       => ( ( A != zero_zero_rat )
% 9.51/5.30         => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7685_inverse__positive__imp__positive,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 9.51/5.30       => ( ( A != zero_zero_real )
% 9.51/5.30         => ( ord_less_real @ zero_zero_real @ A ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7686_inverse__positive__imp__positive,axiom,
% 9.51/5.30      ! [A: rat] :
% 9.51/5.30        ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 9.51/5.30       => ( ( A != zero_zero_rat )
% 9.51/5.30         => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7687_negative__imp__inverse__negative,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( ord_less_real @ A @ zero_zero_real )
% 9.51/5.30       => ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7688_negative__imp__inverse__negative,axiom,
% 9.51/5.30      ! [A: rat] :
% 9.51/5.30        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.51/5.30       => ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7689_positive__imp__inverse__positive,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ A )
% 9.51/5.30       => ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7690_positive__imp__inverse__positive,axiom,
% 9.51/5.30      ! [A: rat] :
% 9.51/5.30        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.51/5.30       => ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7691_nonzero__inverse__mult__distrib,axiom,
% 9.51/5.30      ! [A: complex,B: complex] :
% 9.51/5.30        ( ( A != zero_zero_complex )
% 9.51/5.30       => ( ( B != zero_zero_complex )
% 9.51/5.30         => ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B ) )
% 9.51/5.30            = ( times_times_complex @ ( invers8013647133539491842omplex @ B ) @ ( invers8013647133539491842omplex @ A ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7692_nonzero__inverse__mult__distrib,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( A != zero_zero_real )
% 9.51/5.30       => ( ( B != zero_zero_real )
% 9.51/5.30         => ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
% 9.51/5.30            = ( times_times_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7693_nonzero__inverse__mult__distrib,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( A != zero_zero_rat )
% 9.51/5.30       => ( ( B != zero_zero_rat )
% 9.51/5.30         => ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
% 9.51/5.30            = ( times_times_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7694_nonzero__inverse__minus__eq,axiom,
% 9.51/5.30      ! [A: complex] :
% 9.51/5.30        ( ( A != zero_zero_complex )
% 9.51/5.30       => ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
% 9.51/5.30          = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7695_nonzero__inverse__minus__eq,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( A != zero_zero_real )
% 9.51/5.30       => ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
% 9.51/5.30          = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7696_nonzero__inverse__minus__eq,axiom,
% 9.51/5.30      ! [A: rat] :
% 9.51/5.30        ( ( A != zero_zero_rat )
% 9.51/5.30       => ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
% 9.51/5.30          = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7697_not__exp__less__zero,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7698_exp__gt__zero,axiom,
% 9.51/5.30      ! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7699_exp__total,axiom,
% 9.51/5.30      ! [Y: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ Y )
% 9.51/5.30       => ? [X4: real] :
% 9.51/5.30            ( ( exp_real @ X4 )
% 9.51/5.30            = Y ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7700_nonzero__abs__inverse,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( A != zero_zero_real )
% 9.51/5.30       => ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
% 9.51/5.30          = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7701_nonzero__abs__inverse,axiom,
% 9.51/5.30      ! [A: rat] :
% 9.51/5.30        ( ( A != zero_zero_rat )
% 9.51/5.30       => ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
% 9.51/5.30          = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7702_inverse__le__imp__le,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.51/5.30         => ( ord_less_eq_real @ B @ A ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7703_inverse__le__imp__le,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.51/5.30       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.51/5.30         => ( ord_less_eq_rat @ B @ A ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7704_le__imp__inverse__le,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ A @ B )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.51/5.30         => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7705_le__imp__inverse__le,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( ord_less_eq_rat @ A @ B )
% 9.51/5.30       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.51/5.30         => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7706_inverse__le__imp__le__neg,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.51/5.30       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.51/5.30         => ( ord_less_eq_real @ B @ A ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7707_inverse__le__imp__le__neg,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.51/5.30       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.51/5.30         => ( ord_less_eq_rat @ B @ A ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7708_le__imp__inverse__le__neg,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ A @ B )
% 9.51/5.30       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.51/5.30         => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7709_le__imp__inverse__le__neg,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( ord_less_eq_rat @ A @ B )
% 9.51/5.30       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.51/5.30         => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7710_inverse__le__1__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ ( inverse_inverse_real @ X ) @ one_one_real )
% 9.51/5.30        = ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.51/5.30          | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7711_inverse__le__1__iff,axiom,
% 9.51/5.30      ! [X: rat] :
% 9.51/5.30        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
% 9.51/5.30        = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 9.51/5.30          | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7712_one__less__inverse,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ A )
% 9.51/5.30       => ( ( ord_less_real @ A @ one_one_real )
% 9.51/5.30         => ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7713_one__less__inverse,axiom,
% 9.51/5.30      ! [A: rat] :
% 9.51/5.30        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.51/5.30       => ( ( ord_less_rat @ A @ one_one_rat )
% 9.51/5.30         => ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7714_one__less__inverse__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X ) )
% 9.51/5.30        = ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30          & ( ord_less_real @ X @ one_one_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7715_one__less__inverse__iff,axiom,
% 9.51/5.30      ! [X: rat] :
% 9.51/5.30        ( ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
% 9.51/5.30        = ( ( ord_less_rat @ zero_zero_rat @ X )
% 9.51/5.30          & ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7716_inverse__add,axiom,
% 9.51/5.30      ! [A: complex,B: complex] :
% 9.51/5.30        ( ( A != zero_zero_complex )
% 9.51/5.30       => ( ( B != zero_zero_complex )
% 9.51/5.30         => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 9.51/5.30            = ( times_times_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( invers8013647133539491842omplex @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7717_inverse__add,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( A != zero_zero_real )
% 9.51/5.30       => ( ( B != zero_zero_real )
% 9.51/5.30         => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.51/5.30            = ( times_times_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( inverse_inverse_real @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7718_inverse__add,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( A != zero_zero_rat )
% 9.51/5.30       => ( ( B != zero_zero_rat )
% 9.51/5.30         => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.51/5.30            = ( times_times_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( inverse_inverse_rat @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7719_division__ring__inverse__add,axiom,
% 9.51/5.30      ! [A: complex,B: complex] :
% 9.51/5.30        ( ( A != zero_zero_complex )
% 9.51/5.30       => ( ( B != zero_zero_complex )
% 9.51/5.30         => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 9.51/5.30            = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( plus_plus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7720_division__ring__inverse__add,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( A != zero_zero_real )
% 9.51/5.30       => ( ( B != zero_zero_real )
% 9.51/5.30         => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.51/5.30            = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( plus_plus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7721_division__ring__inverse__add,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( A != zero_zero_rat )
% 9.51/5.30       => ( ( B != zero_zero_rat )
% 9.51/5.30         => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.51/5.30            = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( plus_plus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7722_field__class_Ofield__inverse,axiom,
% 9.51/5.30      ! [A: complex] :
% 9.51/5.30        ( ( A != zero_zero_complex )
% 9.51/5.30       => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
% 9.51/5.30          = one_one_complex ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7723_field__class_Ofield__inverse,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( A != zero_zero_real )
% 9.51/5.30       => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
% 9.51/5.30          = one_one_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7724_field__class_Ofield__inverse,axiom,
% 9.51/5.30      ! [A: rat] :
% 9.51/5.30        ( ( A != zero_zero_rat )
% 9.51/5.30       => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
% 9.51/5.30          = one_one_rat ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7725_division__ring__inverse__diff,axiom,
% 9.51/5.30      ! [A: complex,B: complex] :
% 9.51/5.30        ( ( A != zero_zero_complex )
% 9.51/5.30       => ( ( B != zero_zero_complex )
% 9.51/5.30         => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 9.51/5.30            = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ B @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7726_division__ring__inverse__diff,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( A != zero_zero_real )
% 9.51/5.30       => ( ( B != zero_zero_real )
% 9.51/5.30         => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.51/5.30            = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ B @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7727_division__ring__inverse__diff,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( A != zero_zero_rat )
% 9.51/5.30       => ( ( B != zero_zero_rat )
% 9.51/5.30         => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.51/5.30            = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ B @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7728_nonzero__inverse__eq__divide,axiom,
% 9.51/5.30      ! [A: complex] :
% 9.51/5.30        ( ( A != zero_zero_complex )
% 9.51/5.30       => ( ( invers8013647133539491842omplex @ A )
% 9.51/5.30          = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7729_nonzero__inverse__eq__divide,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( A != zero_zero_real )
% 9.51/5.30       => ( ( inverse_inverse_real @ A )
% 9.51/5.30          = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7730_nonzero__inverse__eq__divide,axiom,
% 9.51/5.30      ! [A: rat] :
% 9.51/5.30        ( ( A != zero_zero_rat )
% 9.51/5.30       => ( ( inverse_inverse_rat @ A )
% 9.51/5.30          = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7731_sum_Odistrib__triv_H,axiom,
% 9.51/5.30      ! [I6: set_nat,G: nat > real,H2: nat > real] :
% 9.51/5.30        ( ( finite_finite_nat @ I6 )
% 9.51/5.30       => ( ( groups8560362682196896993t_real
% 9.51/5.30            @ ^ [I4: nat] : ( plus_plus_real @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 9.51/5.30            @ I6 )
% 9.51/5.30          = ( plus_plus_real @ ( groups8560362682196896993t_real @ G @ I6 ) @ ( groups8560362682196896993t_real @ H2 @ I6 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7732_sum_Odistrib__triv_H,axiom,
% 9.51/5.30      ! [I6: set_int,G: int > real,H2: int > real] :
% 9.51/5.30        ( ( finite_finite_int @ I6 )
% 9.51/5.30       => ( ( groups1523912220035142973t_real
% 9.51/5.30            @ ^ [I4: int] : ( plus_plus_real @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 9.51/5.30            @ I6 )
% 9.51/5.30          = ( plus_plus_real @ ( groups1523912220035142973t_real @ G @ I6 ) @ ( groups1523912220035142973t_real @ H2 @ I6 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7733_sum_Odistrib__triv_H,axiom,
% 9.51/5.30      ! [I6: set_complex,G: complex > real,H2: complex > real] :
% 9.51/5.30        ( ( finite3207457112153483333omplex @ I6 )
% 9.51/5.30       => ( ( groups5737402329758386879x_real
% 9.51/5.30            @ ^ [I4: complex] : ( plus_plus_real @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 9.51/5.30            @ I6 )
% 9.51/5.30          = ( plus_plus_real @ ( groups5737402329758386879x_real @ G @ I6 ) @ ( groups5737402329758386879x_real @ H2 @ I6 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7734_sum_Odistrib__triv_H,axiom,
% 9.51/5.30      ! [I6: set_nat,G: nat > rat,H2: nat > rat] :
% 9.51/5.30        ( ( finite_finite_nat @ I6 )
% 9.51/5.30       => ( ( groups1351286907653491341at_rat
% 9.51/5.30            @ ^ [I4: nat] : ( plus_plus_rat @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 9.51/5.30            @ I6 )
% 9.51/5.30          = ( plus_plus_rat @ ( groups1351286907653491341at_rat @ G @ I6 ) @ ( groups1351286907653491341at_rat @ H2 @ I6 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7735_sum_Odistrib__triv_H,axiom,
% 9.51/5.30      ! [I6: set_int,G: int > rat,H2: int > rat] :
% 9.51/5.30        ( ( finite_finite_int @ I6 )
% 9.51/5.30       => ( ( groups2350640619554545897nt_rat
% 9.51/5.30            @ ^ [I4: int] : ( plus_plus_rat @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 9.51/5.30            @ I6 )
% 9.51/5.30          = ( plus_plus_rat @ ( groups2350640619554545897nt_rat @ G @ I6 ) @ ( groups2350640619554545897nt_rat @ H2 @ I6 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7736_sum_Odistrib__triv_H,axiom,
% 9.51/5.30      ! [I6: set_complex,G: complex > rat,H2: complex > rat] :
% 9.51/5.30        ( ( finite3207457112153483333omplex @ I6 )
% 9.51/5.30       => ( ( groups2276542476275365739ex_rat
% 9.51/5.30            @ ^ [I4: complex] : ( plus_plus_rat @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 9.51/5.30            @ I6 )
% 9.51/5.30          = ( plus_plus_rat @ ( groups2276542476275365739ex_rat @ G @ I6 ) @ ( groups2276542476275365739ex_rat @ H2 @ I6 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7737_sum_Odistrib__triv_H,axiom,
% 9.51/5.30      ! [I6: set_nat,G: nat > nat,H2: nat > nat] :
% 9.51/5.30        ( ( finite_finite_nat @ I6 )
% 9.51/5.30       => ( ( groups1986416967739987077at_nat
% 9.51/5.30            @ ^ [I4: nat] : ( plus_plus_nat @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 9.51/5.30            @ I6 )
% 9.51/5.30          = ( plus_plus_nat @ ( groups1986416967739987077at_nat @ G @ I6 ) @ ( groups1986416967739987077at_nat @ H2 @ I6 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7738_sum_Odistrib__triv_H,axiom,
% 9.51/5.30      ! [I6: set_int,G: int > nat,H2: int > nat] :
% 9.51/5.30        ( ( finite_finite_int @ I6 )
% 9.51/5.30       => ( ( groups2985770679641041633nt_nat
% 9.51/5.30            @ ^ [I4: int] : ( plus_plus_nat @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 9.51/5.30            @ I6 )
% 9.51/5.30          = ( plus_plus_nat @ ( groups2985770679641041633nt_nat @ G @ I6 ) @ ( groups2985770679641041633nt_nat @ H2 @ I6 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7739_sum_Odistrib__triv_H,axiom,
% 9.51/5.30      ! [I6: set_complex,G: complex > nat,H2: complex > nat] :
% 9.51/5.30        ( ( finite3207457112153483333omplex @ I6 )
% 9.51/5.30       => ( ( groups2911672536361861475ex_nat
% 9.51/5.30            @ ^ [I4: complex] : ( plus_plus_nat @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 9.51/5.30            @ I6 )
% 9.51/5.30          = ( plus_plus_nat @ ( groups2911672536361861475ex_nat @ G @ I6 ) @ ( groups2911672536361861475ex_nat @ H2 @ I6 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7740_sum_Odistrib__triv_H,axiom,
% 9.51/5.30      ! [I6: set_nat,G: nat > int,H2: nat > int] :
% 9.51/5.30        ( ( finite_finite_nat @ I6 )
% 9.51/5.30       => ( ( groups1983926497230936801at_int
% 9.51/5.30            @ ^ [I4: nat] : ( plus_plus_int @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 9.51/5.30            @ I6 )
% 9.51/5.30          = ( plus_plus_int @ ( groups1983926497230936801at_int @ G @ I6 ) @ ( groups1983926497230936801at_int @ H2 @ I6 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7741_exp__gt__one,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7742_sum_Omono__neutral__cong__right_H,axiom,
% 9.51/5.30      ! [S2: set_real,T3: set_real,G: real > complex,H2: real > complex] :
% 9.51/5.30        ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: real] :
% 9.51/5.30              ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_complex ) )
% 9.51/5.30         => ( ! [X4: real] :
% 9.51/5.30                ( ( member_real @ X4 @ S2 )
% 9.51/5.30               => ( ( G @ X4 )
% 9.51/5.30                  = ( H2 @ X4 ) ) )
% 9.51/5.30           => ( ( groups5683813829254066239omplex @ G @ T3 )
% 9.51/5.30              = ( groups5683813829254066239omplex @ H2 @ S2 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7743_sum_Omono__neutral__cong__right_H,axiom,
% 9.51/5.30      ! [S2: set_real,T3: set_real,G: real > real,H2: real > real] :
% 9.51/5.30        ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: real] :
% 9.51/5.30              ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_real ) )
% 9.51/5.30         => ( ! [X4: real] :
% 9.51/5.30                ( ( member_real @ X4 @ S2 )
% 9.51/5.30               => ( ( G @ X4 )
% 9.51/5.30                  = ( H2 @ X4 ) ) )
% 9.51/5.30           => ( ( groups97945582718554045l_real @ G @ T3 )
% 9.51/5.30              = ( groups97945582718554045l_real @ H2 @ S2 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7744_sum_Omono__neutral__cong__right_H,axiom,
% 9.51/5.30      ! [S2: set_real,T3: set_real,G: real > rat,H2: real > rat] :
% 9.51/5.30        ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: real] :
% 9.51/5.30              ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_rat ) )
% 9.51/5.30         => ( ! [X4: real] :
% 9.51/5.30                ( ( member_real @ X4 @ S2 )
% 9.51/5.30               => ( ( G @ X4 )
% 9.51/5.30                  = ( H2 @ X4 ) ) )
% 9.51/5.30           => ( ( groups3269169158384524137al_rat @ G @ T3 )
% 9.51/5.30              = ( groups3269169158384524137al_rat @ H2 @ S2 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7745_sum_Omono__neutral__cong__right_H,axiom,
% 9.51/5.30      ! [S2: set_real,T3: set_real,G: real > nat,H2: real > nat] :
% 9.51/5.30        ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: real] :
% 9.51/5.30              ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_nat ) )
% 9.51/5.30         => ( ! [X4: real] :
% 9.51/5.30                ( ( member_real @ X4 @ S2 )
% 9.51/5.30               => ( ( G @ X4 )
% 9.51/5.30                  = ( H2 @ X4 ) ) )
% 9.51/5.30           => ( ( groups3904299218471019873al_nat @ G @ T3 )
% 9.51/5.30              = ( groups3904299218471019873al_nat @ H2 @ S2 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7746_sum_Omono__neutral__cong__right_H,axiom,
% 9.51/5.30      ! [S2: set_real,T3: set_real,G: real > int,H2: real > int] :
% 9.51/5.30        ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: real] :
% 9.51/5.30              ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_int ) )
% 9.51/5.30         => ( ! [X4: real] :
% 9.51/5.30                ( ( member_real @ X4 @ S2 )
% 9.51/5.30               => ( ( G @ X4 )
% 9.51/5.30                  = ( H2 @ X4 ) ) )
% 9.51/5.30           => ( ( groups3901808747961969597al_int @ G @ T3 )
% 9.51/5.30              = ( groups3901808747961969597al_int @ H2 @ S2 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7747_sum_Omono__neutral__cong__right_H,axiom,
% 9.51/5.30      ! [S2: set_nat,T3: set_nat,G: nat > complex,H2: nat > complex] :
% 9.51/5.30        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: nat] :
% 9.51/5.30              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_complex ) )
% 9.51/5.30         => ( ! [X4: nat] :
% 9.51/5.30                ( ( member_nat @ X4 @ S2 )
% 9.51/5.30               => ( ( G @ X4 )
% 9.51/5.30                  = ( H2 @ X4 ) ) )
% 9.51/5.30           => ( ( groups8515261248781899619omplex @ G @ T3 )
% 9.51/5.30              = ( groups8515261248781899619omplex @ H2 @ S2 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7748_sum_Omono__neutral__cong__right_H,axiom,
% 9.51/5.30      ! [S2: set_nat,T3: set_nat,G: nat > real,H2: nat > real] :
% 9.51/5.30        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: nat] :
% 9.51/5.30              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_real ) )
% 9.51/5.30         => ( ! [X4: nat] :
% 9.51/5.30                ( ( member_nat @ X4 @ S2 )
% 9.51/5.30               => ( ( G @ X4 )
% 9.51/5.30                  = ( H2 @ X4 ) ) )
% 9.51/5.30           => ( ( groups8560362682196896993t_real @ G @ T3 )
% 9.51/5.30              = ( groups8560362682196896993t_real @ H2 @ S2 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7749_sum_Omono__neutral__cong__right_H,axiom,
% 9.51/5.30      ! [S2: set_nat,T3: set_nat,G: nat > rat,H2: nat > rat] :
% 9.51/5.30        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: nat] :
% 9.51/5.30              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_rat ) )
% 9.51/5.30         => ( ! [X4: nat] :
% 9.51/5.30                ( ( member_nat @ X4 @ S2 )
% 9.51/5.30               => ( ( G @ X4 )
% 9.51/5.30                  = ( H2 @ X4 ) ) )
% 9.51/5.30           => ( ( groups1351286907653491341at_rat @ G @ T3 )
% 9.51/5.30              = ( groups1351286907653491341at_rat @ H2 @ S2 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7750_sum_Omono__neutral__cong__right_H,axiom,
% 9.51/5.30      ! [S2: set_nat,T3: set_nat,G: nat > nat,H2: nat > nat] :
% 9.51/5.30        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: nat] :
% 9.51/5.30              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_nat ) )
% 9.51/5.30         => ( ! [X4: nat] :
% 9.51/5.30                ( ( member_nat @ X4 @ S2 )
% 9.51/5.30               => ( ( G @ X4 )
% 9.51/5.30                  = ( H2 @ X4 ) ) )
% 9.51/5.30           => ( ( groups1986416967739987077at_nat @ G @ T3 )
% 9.51/5.30              = ( groups1986416967739987077at_nat @ H2 @ S2 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7751_sum_Omono__neutral__cong__right_H,axiom,
% 9.51/5.30      ! [S2: set_nat,T3: set_nat,G: nat > int,H2: nat > int] :
% 9.51/5.30        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: nat] :
% 9.51/5.30              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_int ) )
% 9.51/5.30         => ( ! [X4: nat] :
% 9.51/5.30                ( ( member_nat @ X4 @ S2 )
% 9.51/5.30               => ( ( G @ X4 )
% 9.51/5.30                  = ( H2 @ X4 ) ) )
% 9.51/5.30           => ( ( groups1983926497230936801at_int @ G @ T3 )
% 9.51/5.30              = ( groups1983926497230936801at_int @ H2 @ S2 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7752_sum_Omono__neutral__cong__left_H,axiom,
% 9.51/5.30      ! [S2: set_real,T3: set_real,H2: real > complex,G: real > complex] :
% 9.51/5.30        ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.51/5.30       => ( ! [I2: real] :
% 9.51/5.30              ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.51/5.30             => ( ( H2 @ I2 )
% 9.51/5.30                = zero_zero_complex ) )
% 9.51/5.30         => ( ! [X4: real] :
% 9.51/5.30                ( ( member_real @ X4 @ S2 )
% 9.51/5.30               => ( ( G @ X4 )
% 9.51/5.30                  = ( H2 @ X4 ) ) )
% 9.51/5.30           => ( ( groups5683813829254066239omplex @ G @ S2 )
% 9.51/5.30              = ( groups5683813829254066239omplex @ H2 @ T3 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7753_sum_Omono__neutral__cong__left_H,axiom,
% 9.51/5.30      ! [S2: set_real,T3: set_real,H2: real > real,G: real > real] :
% 9.51/5.30        ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.51/5.30       => ( ! [I2: real] :
% 9.51/5.30              ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.51/5.30             => ( ( H2 @ I2 )
% 9.51/5.30                = zero_zero_real ) )
% 9.51/5.30         => ( ! [X4: real] :
% 9.51/5.30                ( ( member_real @ X4 @ S2 )
% 9.51/5.30               => ( ( G @ X4 )
% 9.51/5.30                  = ( H2 @ X4 ) ) )
% 9.51/5.30           => ( ( groups97945582718554045l_real @ G @ S2 )
% 9.51/5.30              = ( groups97945582718554045l_real @ H2 @ T3 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7754_sum_Omono__neutral__cong__left_H,axiom,
% 9.51/5.30      ! [S2: set_real,T3: set_real,H2: real > rat,G: real > rat] :
% 9.51/5.30        ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.51/5.30       => ( ! [I2: real] :
% 9.51/5.30              ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.51/5.30             => ( ( H2 @ I2 )
% 9.51/5.30                = zero_zero_rat ) )
% 9.51/5.30         => ( ! [X4: real] :
% 9.51/5.30                ( ( member_real @ X4 @ S2 )
% 9.51/5.30               => ( ( G @ X4 )
% 9.51/5.30                  = ( H2 @ X4 ) ) )
% 9.51/5.30           => ( ( groups3269169158384524137al_rat @ G @ S2 )
% 9.51/5.30              = ( groups3269169158384524137al_rat @ H2 @ T3 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7755_sum_Omono__neutral__cong__left_H,axiom,
% 9.51/5.30      ! [S2: set_real,T3: set_real,H2: real > nat,G: real > nat] :
% 9.51/5.30        ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.51/5.30       => ( ! [I2: real] :
% 9.51/5.30              ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.51/5.30             => ( ( H2 @ I2 )
% 9.51/5.30                = zero_zero_nat ) )
% 9.51/5.30         => ( ! [X4: real] :
% 9.51/5.30                ( ( member_real @ X4 @ S2 )
% 9.51/5.30               => ( ( G @ X4 )
% 9.51/5.30                  = ( H2 @ X4 ) ) )
% 9.51/5.30           => ( ( groups3904299218471019873al_nat @ G @ S2 )
% 9.51/5.30              = ( groups3904299218471019873al_nat @ H2 @ T3 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7756_sum_Omono__neutral__cong__left_H,axiom,
% 9.51/5.30      ! [S2: set_real,T3: set_real,H2: real > int,G: real > int] :
% 9.51/5.30        ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.51/5.30       => ( ! [I2: real] :
% 9.51/5.30              ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.51/5.30             => ( ( H2 @ I2 )
% 9.51/5.30                = zero_zero_int ) )
% 9.51/5.30         => ( ! [X4: real] :
% 9.51/5.30                ( ( member_real @ X4 @ S2 )
% 9.51/5.30               => ( ( G @ X4 )
% 9.51/5.30                  = ( H2 @ X4 ) ) )
% 9.51/5.30           => ( ( groups3901808747961969597al_int @ G @ S2 )
% 9.51/5.30              = ( groups3901808747961969597al_int @ H2 @ T3 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7757_sum_Omono__neutral__cong__left_H,axiom,
% 9.51/5.30      ! [S2: set_nat,T3: set_nat,H2: nat > complex,G: nat > complex] :
% 9.51/5.30        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.51/5.30       => ( ! [I2: nat] :
% 9.51/5.30              ( ( member_nat @ I2 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.51/5.30             => ( ( H2 @ I2 )
% 9.51/5.30                = zero_zero_complex ) )
% 9.51/5.30         => ( ! [X4: nat] :
% 9.51/5.30                ( ( member_nat @ X4 @ S2 )
% 9.51/5.30               => ( ( G @ X4 )
% 9.51/5.30                  = ( H2 @ X4 ) ) )
% 9.51/5.30           => ( ( groups8515261248781899619omplex @ G @ S2 )
% 9.51/5.30              = ( groups8515261248781899619omplex @ H2 @ T3 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7758_sum_Omono__neutral__cong__left_H,axiom,
% 9.51/5.30      ! [S2: set_nat,T3: set_nat,H2: nat > real,G: nat > real] :
% 9.51/5.30        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.51/5.30       => ( ! [I2: nat] :
% 9.51/5.30              ( ( member_nat @ I2 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.51/5.30             => ( ( H2 @ I2 )
% 9.51/5.30                = zero_zero_real ) )
% 9.51/5.30         => ( ! [X4: nat] :
% 9.51/5.30                ( ( member_nat @ X4 @ S2 )
% 9.51/5.30               => ( ( G @ X4 )
% 9.51/5.30                  = ( H2 @ X4 ) ) )
% 9.51/5.30           => ( ( groups8560362682196896993t_real @ G @ S2 )
% 9.51/5.30              = ( groups8560362682196896993t_real @ H2 @ T3 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7759_sum_Omono__neutral__cong__left_H,axiom,
% 9.51/5.30      ! [S2: set_nat,T3: set_nat,H2: nat > rat,G: nat > rat] :
% 9.51/5.30        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.51/5.30       => ( ! [I2: nat] :
% 9.51/5.30              ( ( member_nat @ I2 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.51/5.30             => ( ( H2 @ I2 )
% 9.51/5.30                = zero_zero_rat ) )
% 9.51/5.30         => ( ! [X4: nat] :
% 9.51/5.30                ( ( member_nat @ X4 @ S2 )
% 9.51/5.30               => ( ( G @ X4 )
% 9.51/5.30                  = ( H2 @ X4 ) ) )
% 9.51/5.30           => ( ( groups1351286907653491341at_rat @ G @ S2 )
% 9.51/5.30              = ( groups1351286907653491341at_rat @ H2 @ T3 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7760_sum_Omono__neutral__cong__left_H,axiom,
% 9.51/5.30      ! [S2: set_nat,T3: set_nat,H2: nat > nat,G: nat > nat] :
% 9.51/5.30        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.51/5.30       => ( ! [I2: nat] :
% 9.51/5.30              ( ( member_nat @ I2 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.51/5.30             => ( ( H2 @ I2 )
% 9.51/5.30                = zero_zero_nat ) )
% 9.51/5.30         => ( ! [X4: nat] :
% 9.51/5.30                ( ( member_nat @ X4 @ S2 )
% 9.51/5.30               => ( ( G @ X4 )
% 9.51/5.30                  = ( H2 @ X4 ) ) )
% 9.51/5.30           => ( ( groups1986416967739987077at_nat @ G @ S2 )
% 9.51/5.30              = ( groups1986416967739987077at_nat @ H2 @ T3 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7761_sum_Omono__neutral__cong__left_H,axiom,
% 9.51/5.30      ! [S2: set_nat,T3: set_nat,H2: nat > int,G: nat > int] :
% 9.51/5.30        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.51/5.30       => ( ! [I2: nat] :
% 9.51/5.30              ( ( member_nat @ I2 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.51/5.30             => ( ( H2 @ I2 )
% 9.51/5.30                = zero_zero_int ) )
% 9.51/5.30         => ( ! [X4: nat] :
% 9.51/5.30                ( ( member_nat @ X4 @ S2 )
% 9.51/5.30               => ( ( G @ X4 )
% 9.51/5.30                  = ( H2 @ X4 ) ) )
% 9.51/5.30           => ( ( groups1983926497230936801at_int @ G @ S2 )
% 9.51/5.30              = ( groups1983926497230936801at_int @ H2 @ T3 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7762_sum_Omono__neutral__right_H,axiom,
% 9.51/5.30      ! [S2: set_nat,T3: set_nat,G: nat > complex] :
% 9.51/5.30        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: nat] :
% 9.51/5.30              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_complex ) )
% 9.51/5.30         => ( ( groups8515261248781899619omplex @ G @ T3 )
% 9.51/5.30            = ( groups8515261248781899619omplex @ G @ S2 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7763_sum_Omono__neutral__right_H,axiom,
% 9.51/5.30      ! [S2: set_nat,T3: set_nat,G: nat > real] :
% 9.51/5.30        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: nat] :
% 9.51/5.30              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_real ) )
% 9.51/5.30         => ( ( groups8560362682196896993t_real @ G @ T3 )
% 9.51/5.30            = ( groups8560362682196896993t_real @ G @ S2 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7764_sum_Omono__neutral__right_H,axiom,
% 9.51/5.30      ! [S2: set_nat,T3: set_nat,G: nat > rat] :
% 9.51/5.30        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: nat] :
% 9.51/5.30              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_rat ) )
% 9.51/5.30         => ( ( groups1351286907653491341at_rat @ G @ T3 )
% 9.51/5.30            = ( groups1351286907653491341at_rat @ G @ S2 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7765_sum_Omono__neutral__right_H,axiom,
% 9.51/5.30      ! [S2: set_nat,T3: set_nat,G: nat > nat] :
% 9.51/5.30        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: nat] :
% 9.51/5.30              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_nat ) )
% 9.51/5.30         => ( ( groups1986416967739987077at_nat @ G @ T3 )
% 9.51/5.30            = ( groups1986416967739987077at_nat @ G @ S2 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7766_sum_Omono__neutral__right_H,axiom,
% 9.51/5.30      ! [S2: set_nat,T3: set_nat,G: nat > int] :
% 9.51/5.30        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: nat] :
% 9.51/5.30              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_int ) )
% 9.51/5.30         => ( ( groups1983926497230936801at_int @ G @ T3 )
% 9.51/5.30            = ( groups1983926497230936801at_int @ G @ S2 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7767_sum_Omono__neutral__right_H,axiom,
% 9.51/5.30      ! [S2: set_int,T3: set_int,G: int > complex] :
% 9.51/5.30        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: int] :
% 9.51/5.30              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_complex ) )
% 9.51/5.30         => ( ( groups267424677133301183omplex @ G @ T3 )
% 9.51/5.30            = ( groups267424677133301183omplex @ G @ S2 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7768_sum_Omono__neutral__right_H,axiom,
% 9.51/5.30      ! [S2: set_int,T3: set_int,G: int > real] :
% 9.51/5.30        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: int] :
% 9.51/5.30              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_real ) )
% 9.51/5.30         => ( ( groups1523912220035142973t_real @ G @ T3 )
% 9.51/5.30            = ( groups1523912220035142973t_real @ G @ S2 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7769_sum_Omono__neutral__right_H,axiom,
% 9.51/5.30      ! [S2: set_int,T3: set_int,G: int > rat] :
% 9.51/5.30        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: int] :
% 9.51/5.30              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_rat ) )
% 9.51/5.30         => ( ( groups2350640619554545897nt_rat @ G @ T3 )
% 9.51/5.30            = ( groups2350640619554545897nt_rat @ G @ S2 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7770_sum_Omono__neutral__right_H,axiom,
% 9.51/5.30      ! [S2: set_int,T3: set_int,G: int > nat] :
% 9.51/5.30        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: int] :
% 9.51/5.30              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_nat ) )
% 9.51/5.30         => ( ( groups2985770679641041633nt_nat @ G @ T3 )
% 9.51/5.30            = ( groups2985770679641041633nt_nat @ G @ S2 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7771_sum_Omono__neutral__right_H,axiom,
% 9.51/5.30      ! [S2: set_int,T3: set_int,G: int > int] :
% 9.51/5.30        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: int] :
% 9.51/5.30              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_int ) )
% 9.51/5.30         => ( ( groups2983280209131991357nt_int @ G @ T3 )
% 9.51/5.30            = ( groups2983280209131991357nt_int @ G @ S2 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7772_sum_Omono__neutral__left_H,axiom,
% 9.51/5.30      ! [S2: set_nat,T3: set_nat,G: nat > complex] :
% 9.51/5.30        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: nat] :
% 9.51/5.30              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_complex ) )
% 9.51/5.30         => ( ( groups8515261248781899619omplex @ G @ S2 )
% 9.51/5.30            = ( groups8515261248781899619omplex @ G @ T3 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7773_sum_Omono__neutral__left_H,axiom,
% 9.51/5.30      ! [S2: set_nat,T3: set_nat,G: nat > real] :
% 9.51/5.30        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: nat] :
% 9.51/5.30              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_real ) )
% 9.51/5.30         => ( ( groups8560362682196896993t_real @ G @ S2 )
% 9.51/5.30            = ( groups8560362682196896993t_real @ G @ T3 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7774_sum_Omono__neutral__left_H,axiom,
% 9.51/5.30      ! [S2: set_nat,T3: set_nat,G: nat > rat] :
% 9.51/5.30        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: nat] :
% 9.51/5.30              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_rat ) )
% 9.51/5.30         => ( ( groups1351286907653491341at_rat @ G @ S2 )
% 9.51/5.30            = ( groups1351286907653491341at_rat @ G @ T3 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7775_sum_Omono__neutral__left_H,axiom,
% 9.51/5.30      ! [S2: set_nat,T3: set_nat,G: nat > nat] :
% 9.51/5.30        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: nat] :
% 9.51/5.30              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_nat ) )
% 9.51/5.30         => ( ( groups1986416967739987077at_nat @ G @ S2 )
% 9.51/5.30            = ( groups1986416967739987077at_nat @ G @ T3 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7776_sum_Omono__neutral__left_H,axiom,
% 9.51/5.30      ! [S2: set_nat,T3: set_nat,G: nat > int] :
% 9.51/5.30        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: nat] :
% 9.51/5.30              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_int ) )
% 9.51/5.30         => ( ( groups1983926497230936801at_int @ G @ S2 )
% 9.51/5.30            = ( groups1983926497230936801at_int @ G @ T3 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7777_sum_Omono__neutral__left_H,axiom,
% 9.51/5.30      ! [S2: set_int,T3: set_int,G: int > complex] :
% 9.51/5.30        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: int] :
% 9.51/5.30              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_complex ) )
% 9.51/5.30         => ( ( groups267424677133301183omplex @ G @ S2 )
% 9.51/5.30            = ( groups267424677133301183omplex @ G @ T3 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7778_sum_Omono__neutral__left_H,axiom,
% 9.51/5.30      ! [S2: set_int,T3: set_int,G: int > real] :
% 9.51/5.30        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: int] :
% 9.51/5.30              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_real ) )
% 9.51/5.30         => ( ( groups1523912220035142973t_real @ G @ S2 )
% 9.51/5.30            = ( groups1523912220035142973t_real @ G @ T3 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7779_sum_Omono__neutral__left_H,axiom,
% 9.51/5.30      ! [S2: set_int,T3: set_int,G: int > rat] :
% 9.51/5.30        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: int] :
% 9.51/5.30              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_rat ) )
% 9.51/5.30         => ( ( groups2350640619554545897nt_rat @ G @ S2 )
% 9.51/5.30            = ( groups2350640619554545897nt_rat @ G @ T3 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7780_sum_Omono__neutral__left_H,axiom,
% 9.51/5.30      ! [S2: set_int,T3: set_int,G: int > nat] :
% 9.51/5.30        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: int] :
% 9.51/5.30              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_nat ) )
% 9.51/5.30         => ( ( groups2985770679641041633nt_nat @ G @ S2 )
% 9.51/5.30            = ( groups2985770679641041633nt_nat @ G @ T3 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7781_sum_Omono__neutral__left_H,axiom,
% 9.51/5.30      ! [S2: set_int,T3: set_int,G: int > int] :
% 9.51/5.30        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.51/5.30       => ( ! [X4: int] :
% 9.51/5.30              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.51/5.30             => ( ( G @ X4 )
% 9.51/5.30                = zero_zero_int ) )
% 9.51/5.30         => ( ( groups2983280209131991357nt_int @ G @ S2 )
% 9.51/5.30            = ( groups2983280209131991357nt_int @ G @ T3 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7782_inverse__le__iff,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.51/5.30        = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.51/5.30           => ( ord_less_eq_real @ B @ A ) )
% 9.51/5.30          & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 9.51/5.30           => ( ord_less_eq_real @ A @ B ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7783_inverse__le__iff,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.51/5.30        = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.51/5.30           => ( ord_less_eq_rat @ B @ A ) )
% 9.51/5.30          & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 9.51/5.30           => ( ord_less_eq_rat @ A @ B ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7784_inverse__less__iff,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.51/5.30        = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.51/5.30           => ( ord_less_real @ B @ A ) )
% 9.51/5.30          & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 9.51/5.30           => ( ord_less_real @ A @ B ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7785_inverse__less__iff,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.51/5.30        = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.51/5.30           => ( ord_less_rat @ B @ A ) )
% 9.51/5.30          & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 9.51/5.30           => ( ord_less_rat @ A @ B ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7786_one__le__inverse,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ A )
% 9.51/5.30       => ( ( ord_less_eq_real @ A @ one_one_real )
% 9.51/5.30         => ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7787_one__le__inverse,axiom,
% 9.51/5.30      ! [A: rat] :
% 9.51/5.30        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.51/5.30       => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 9.51/5.30         => ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7788_inverse__less__1__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( inverse_inverse_real @ X ) @ one_one_real )
% 9.51/5.30        = ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.51/5.30          | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7789_inverse__less__1__iff,axiom,
% 9.51/5.30      ! [X: rat] :
% 9.51/5.30        ( ( ord_less_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
% 9.51/5.30        = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 9.51/5.30          | ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7790_one__le__inverse__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X ) )
% 9.51/5.30        = ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30          & ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7791_one__le__inverse__iff,axiom,
% 9.51/5.30      ! [X: rat] :
% 9.51/5.30        ( ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
% 9.51/5.30        = ( ( ord_less_rat @ zero_zero_rat @ X )
% 9.51/5.30          & ( ord_less_eq_rat @ X @ one_one_rat ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7792_inverse__diff__inverse,axiom,
% 9.51/5.30      ! [A: complex,B: complex] :
% 9.51/5.30        ( ( A != zero_zero_complex )
% 9.51/5.30       => ( ( B != zero_zero_complex )
% 9.51/5.30         => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
% 9.51/5.30            = ( uminus1482373934393186551omplex @ ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7793_inverse__diff__inverse,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( A != zero_zero_real )
% 9.51/5.30       => ( ( B != zero_zero_real )
% 9.51/5.30         => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
% 9.51/5.30            = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7794_inverse__diff__inverse,axiom,
% 9.51/5.30      ! [A: rat,B: rat] :
% 9.51/5.30        ( ( A != zero_zero_rat )
% 9.51/5.30       => ( ( B != zero_zero_rat )
% 9.51/5.30         => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
% 9.51/5.30            = ( uminus_uminus_rat @ ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7795_reals__Archimedean,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ? [N3: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7796_reals__Archimedean,axiom,
% 9.51/5.30      ! [X: rat] :
% 9.51/5.30        ( ( ord_less_rat @ zero_zero_rat @ X )
% 9.51/5.30       => ? [N3: nat] : ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7797_sum_Odistrib_H,axiom,
% 9.51/5.30      ! [I6: set_real,G: real > complex,H2: real > complex] :
% 9.51/5.30        ( ( finite_finite_real
% 9.51/5.30          @ ( collect_real
% 9.51/5.30            @ ^ [X3: real] :
% 9.51/5.30                ( ( member_real @ X3 @ I6 )
% 9.51/5.30                & ( ( G @ X3 )
% 9.51/5.30                 != zero_zero_complex ) ) ) )
% 9.51/5.30       => ( ( finite_finite_real
% 9.51/5.30            @ ( collect_real
% 9.51/5.30              @ ^ [X3: real] :
% 9.51/5.30                  ( ( member_real @ X3 @ I6 )
% 9.51/5.30                  & ( ( H2 @ X3 )
% 9.51/5.30                   != zero_zero_complex ) ) ) )
% 9.51/5.30         => ( ( groups5683813829254066239omplex
% 9.51/5.30              @ ^ [I4: real] : ( plus_plus_complex @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 9.51/5.30              @ I6 )
% 9.51/5.30            = ( plus_plus_complex @ ( groups5683813829254066239omplex @ G @ I6 ) @ ( groups5683813829254066239omplex @ H2 @ I6 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7798_sum_Odistrib_H,axiom,
% 9.51/5.30      ! [I6: set_nat,G: nat > complex,H2: nat > complex] :
% 9.51/5.30        ( ( finite_finite_nat
% 9.51/5.30          @ ( collect_nat
% 9.51/5.30            @ ^ [X3: nat] :
% 9.51/5.30                ( ( member_nat @ X3 @ I6 )
% 9.51/5.30                & ( ( G @ X3 )
% 9.51/5.30                 != zero_zero_complex ) ) ) )
% 9.51/5.30       => ( ( finite_finite_nat
% 9.51/5.30            @ ( collect_nat
% 9.51/5.30              @ ^ [X3: nat] :
% 9.51/5.30                  ( ( member_nat @ X3 @ I6 )
% 9.51/5.30                  & ( ( H2 @ X3 )
% 9.51/5.30                   != zero_zero_complex ) ) ) )
% 9.51/5.30         => ( ( groups8515261248781899619omplex
% 9.51/5.30              @ ^ [I4: nat] : ( plus_plus_complex @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 9.51/5.30              @ I6 )
% 9.51/5.30            = ( plus_plus_complex @ ( groups8515261248781899619omplex @ G @ I6 ) @ ( groups8515261248781899619omplex @ H2 @ I6 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7799_sum_Odistrib_H,axiom,
% 9.51/5.30      ! [I6: set_int,G: int > complex,H2: int > complex] :
% 9.51/5.30        ( ( finite_finite_int
% 9.51/5.30          @ ( collect_int
% 9.51/5.30            @ ^ [X3: int] :
% 9.51/5.30                ( ( member_int @ X3 @ I6 )
% 9.51/5.30                & ( ( G @ X3 )
% 9.51/5.30                 != zero_zero_complex ) ) ) )
% 9.51/5.30       => ( ( finite_finite_int
% 9.51/5.30            @ ( collect_int
% 9.51/5.30              @ ^ [X3: int] :
% 9.51/5.30                  ( ( member_int @ X3 @ I6 )
% 9.51/5.30                  & ( ( H2 @ X3 )
% 9.51/5.30                   != zero_zero_complex ) ) ) )
% 9.51/5.30         => ( ( groups267424677133301183omplex
% 9.51/5.30              @ ^ [I4: int] : ( plus_plus_complex @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 9.51/5.30              @ I6 )
% 9.51/5.30            = ( plus_plus_complex @ ( groups267424677133301183omplex @ G @ I6 ) @ ( groups267424677133301183omplex @ H2 @ I6 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7800_sum_Odistrib_H,axiom,
% 9.51/5.30      ! [I6: set_complex,G: complex > complex,H2: complex > complex] :
% 9.51/5.30        ( ( finite3207457112153483333omplex
% 9.51/5.30          @ ( collect_complex
% 9.51/5.30            @ ^ [X3: complex] :
% 9.51/5.30                ( ( member_complex @ X3 @ I6 )
% 9.51/5.30                & ( ( G @ X3 )
% 9.51/5.30                 != zero_zero_complex ) ) ) )
% 9.51/5.30       => ( ( finite3207457112153483333omplex
% 9.51/5.30            @ ( collect_complex
% 9.51/5.30              @ ^ [X3: complex] :
% 9.51/5.30                  ( ( member_complex @ X3 @ I6 )
% 9.51/5.30                  & ( ( H2 @ X3 )
% 9.51/5.30                   != zero_zero_complex ) ) ) )
% 9.51/5.30         => ( ( groups808145749697022017omplex
% 9.51/5.30              @ ^ [I4: complex] : ( plus_plus_complex @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 9.51/5.30              @ I6 )
% 9.51/5.30            = ( plus_plus_complex @ ( groups808145749697022017omplex @ G @ I6 ) @ ( groups808145749697022017omplex @ H2 @ I6 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7801_sum_Odistrib_H,axiom,
% 9.51/5.30      ! [I6: set_real,G: real > real,H2: real > real] :
% 9.51/5.30        ( ( finite_finite_real
% 9.51/5.30          @ ( collect_real
% 9.51/5.30            @ ^ [X3: real] :
% 9.51/5.30                ( ( member_real @ X3 @ I6 )
% 9.51/5.30                & ( ( G @ X3 )
% 9.51/5.30                 != zero_zero_real ) ) ) )
% 9.51/5.30       => ( ( finite_finite_real
% 9.51/5.30            @ ( collect_real
% 9.51/5.30              @ ^ [X3: real] :
% 9.51/5.30                  ( ( member_real @ X3 @ I6 )
% 9.51/5.30                  & ( ( H2 @ X3 )
% 9.51/5.30                   != zero_zero_real ) ) ) )
% 9.51/5.30         => ( ( groups97945582718554045l_real
% 9.51/5.30              @ ^ [I4: real] : ( plus_plus_real @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 9.51/5.30              @ I6 )
% 9.51/5.30            = ( plus_plus_real @ ( groups97945582718554045l_real @ G @ I6 ) @ ( groups97945582718554045l_real @ H2 @ I6 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7802_sum_Odistrib_H,axiom,
% 9.51/5.30      ! [I6: set_nat,G: nat > real,H2: nat > real] :
% 9.51/5.30        ( ( finite_finite_nat
% 9.51/5.30          @ ( collect_nat
% 9.51/5.30            @ ^ [X3: nat] :
% 9.51/5.30                ( ( member_nat @ X3 @ I6 )
% 9.51/5.30                & ( ( G @ X3 )
% 9.51/5.30                 != zero_zero_real ) ) ) )
% 9.51/5.30       => ( ( finite_finite_nat
% 9.51/5.30            @ ( collect_nat
% 9.51/5.30              @ ^ [X3: nat] :
% 9.51/5.30                  ( ( member_nat @ X3 @ I6 )
% 9.51/5.30                  & ( ( H2 @ X3 )
% 9.51/5.30                   != zero_zero_real ) ) ) )
% 9.51/5.30         => ( ( groups8560362682196896993t_real
% 9.51/5.30              @ ^ [I4: nat] : ( plus_plus_real @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 9.51/5.30              @ I6 )
% 9.51/5.30            = ( plus_plus_real @ ( groups8560362682196896993t_real @ G @ I6 ) @ ( groups8560362682196896993t_real @ H2 @ I6 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7803_sum_Odistrib_H,axiom,
% 9.51/5.30      ! [I6: set_int,G: int > real,H2: int > real] :
% 9.51/5.30        ( ( finite_finite_int
% 9.51/5.30          @ ( collect_int
% 9.51/5.30            @ ^ [X3: int] :
% 9.51/5.30                ( ( member_int @ X3 @ I6 )
% 9.51/5.30                & ( ( G @ X3 )
% 9.51/5.30                 != zero_zero_real ) ) ) )
% 9.51/5.30       => ( ( finite_finite_int
% 9.51/5.30            @ ( collect_int
% 9.51/5.30              @ ^ [X3: int] :
% 9.51/5.30                  ( ( member_int @ X3 @ I6 )
% 9.51/5.30                  & ( ( H2 @ X3 )
% 9.51/5.30                   != zero_zero_real ) ) ) )
% 9.51/5.30         => ( ( groups1523912220035142973t_real
% 9.51/5.30              @ ^ [I4: int] : ( plus_plus_real @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 9.51/5.30              @ I6 )
% 9.51/5.30            = ( plus_plus_real @ ( groups1523912220035142973t_real @ G @ I6 ) @ ( groups1523912220035142973t_real @ H2 @ I6 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7804_sum_Odistrib_H,axiom,
% 9.51/5.30      ! [I6: set_complex,G: complex > real,H2: complex > real] :
% 9.51/5.30        ( ( finite3207457112153483333omplex
% 9.51/5.30          @ ( collect_complex
% 9.51/5.30            @ ^ [X3: complex] :
% 9.51/5.30                ( ( member_complex @ X3 @ I6 )
% 9.51/5.30                & ( ( G @ X3 )
% 9.51/5.30                 != zero_zero_real ) ) ) )
% 9.51/5.30       => ( ( finite3207457112153483333omplex
% 9.51/5.30            @ ( collect_complex
% 9.51/5.30              @ ^ [X3: complex] :
% 9.51/5.30                  ( ( member_complex @ X3 @ I6 )
% 9.51/5.30                  & ( ( H2 @ X3 )
% 9.51/5.30                   != zero_zero_real ) ) ) )
% 9.51/5.30         => ( ( groups5737402329758386879x_real
% 9.51/5.30              @ ^ [I4: complex] : ( plus_plus_real @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 9.51/5.30              @ I6 )
% 9.51/5.30            = ( plus_plus_real @ ( groups5737402329758386879x_real @ G @ I6 ) @ ( groups5737402329758386879x_real @ H2 @ I6 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7805_sum_Odistrib_H,axiom,
% 9.51/5.30      ! [I6: set_real,G: real > rat,H2: real > rat] :
% 9.51/5.30        ( ( finite_finite_real
% 9.51/5.30          @ ( collect_real
% 9.51/5.30            @ ^ [X3: real] :
% 9.51/5.30                ( ( member_real @ X3 @ I6 )
% 9.51/5.30                & ( ( G @ X3 )
% 9.51/5.30                 != zero_zero_rat ) ) ) )
% 9.51/5.30       => ( ( finite_finite_real
% 9.51/5.30            @ ( collect_real
% 9.51/5.30              @ ^ [X3: real] :
% 9.51/5.30                  ( ( member_real @ X3 @ I6 )
% 9.51/5.30                  & ( ( H2 @ X3 )
% 9.51/5.30                   != zero_zero_rat ) ) ) )
% 9.51/5.30         => ( ( groups3269169158384524137al_rat
% 9.51/5.30              @ ^ [I4: real] : ( plus_plus_rat @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 9.51/5.30              @ I6 )
% 9.51/5.30            = ( plus_plus_rat @ ( groups3269169158384524137al_rat @ G @ I6 ) @ ( groups3269169158384524137al_rat @ H2 @ I6 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7806_sum_Odistrib_H,axiom,
% 9.51/5.30      ! [I6: set_nat,G: nat > rat,H2: nat > rat] :
% 9.51/5.30        ( ( finite_finite_nat
% 9.51/5.30          @ ( collect_nat
% 9.51/5.30            @ ^ [X3: nat] :
% 9.51/5.30                ( ( member_nat @ X3 @ I6 )
% 9.51/5.30                & ( ( G @ X3 )
% 9.51/5.30                 != zero_zero_rat ) ) ) )
% 9.51/5.30       => ( ( finite_finite_nat
% 9.51/5.30            @ ( collect_nat
% 9.51/5.30              @ ^ [X3: nat] :
% 9.51/5.30                  ( ( member_nat @ X3 @ I6 )
% 9.51/5.30                  & ( ( H2 @ X3 )
% 9.51/5.30                   != zero_zero_rat ) ) ) )
% 9.51/5.30         => ( ( groups1351286907653491341at_rat
% 9.51/5.30              @ ^ [I4: nat] : ( plus_plus_rat @ ( G @ I4 ) @ ( H2 @ I4 ) )
% 9.51/5.30              @ I6 )
% 9.51/5.30            = ( plus_plus_rat @ ( groups1351286907653491341at_rat @ G @ I6 ) @ ( groups1351286907653491341at_rat @ H2 @ I6 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7807_forall__pos__mono__1,axiom,
% 9.51/5.30      ! [P: real > $o,E2: real] :
% 9.51/5.30        ( ! [D5: real,E: real] :
% 9.51/5.30            ( ( ord_less_real @ D5 @ E )
% 9.51/5.30           => ( ( P @ D5 )
% 9.51/5.30             => ( P @ E ) ) )
% 9.51/5.30       => ( ! [N3: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
% 9.51/5.30         => ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.51/5.30           => ( P @ E2 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7808_forall__pos__mono,axiom,
% 9.51/5.30      ! [P: real > $o,E2: real] :
% 9.51/5.30        ( ! [D5: real,E: real] :
% 9.51/5.30            ( ( ord_less_real @ D5 @ E )
% 9.51/5.30           => ( ( P @ D5 )
% 9.51/5.30             => ( P @ E ) ) )
% 9.51/5.30       => ( ! [N3: nat] :
% 9.51/5.30              ( ( N3 != zero_zero_nat )
% 9.51/5.30             => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
% 9.51/5.30         => ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.51/5.30           => ( P @ E2 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7809_real__arch__inverse,axiom,
% 9.51/5.30      ! [E2: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.51/5.30        = ( ? [N2: nat] :
% 9.51/5.30              ( ( N2 != zero_zero_nat )
% 9.51/5.30              & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 9.51/5.30              & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E2 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7810_sum_OG__def,axiom,
% 9.51/5.30      ( groups5683813829254066239omplex
% 9.51/5.30      = ( ^ [P6: real > complex,I7: set_real] :
% 9.51/5.30            ( if_complex
% 9.51/5.30            @ ( finite_finite_real
% 9.51/5.30              @ ( collect_real
% 9.51/5.30                @ ^ [X3: real] :
% 9.51/5.30                    ( ( member_real @ X3 @ I7 )
% 9.51/5.30                    & ( ( P6 @ X3 )
% 9.51/5.30                     != zero_zero_complex ) ) ) )
% 9.51/5.30            @ ( groups5754745047067104278omplex @ P6
% 9.51/5.30              @ ( collect_real
% 9.51/5.30                @ ^ [X3: real] :
% 9.51/5.30                    ( ( member_real @ X3 @ I7 )
% 9.51/5.30                    & ( ( P6 @ X3 )
% 9.51/5.30                     != zero_zero_complex ) ) ) )
% 9.51/5.30            @ zero_zero_complex ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7811_sum_OG__def,axiom,
% 9.51/5.30      ( groups8515261248781899619omplex
% 9.51/5.30      = ( ^ [P6: nat > complex,I7: set_nat] :
% 9.51/5.30            ( if_complex
% 9.51/5.30            @ ( finite_finite_nat
% 9.51/5.30              @ ( collect_nat
% 9.51/5.30                @ ^ [X3: nat] :
% 9.51/5.30                    ( ( member_nat @ X3 @ I7 )
% 9.51/5.30                    & ( ( P6 @ X3 )
% 9.51/5.30                     != zero_zero_complex ) ) ) )
% 9.51/5.30            @ ( groups2073611262835488442omplex @ P6
% 9.51/5.30              @ ( collect_nat
% 9.51/5.30                @ ^ [X3: nat] :
% 9.51/5.30                    ( ( member_nat @ X3 @ I7 )
% 9.51/5.30                    & ( ( P6 @ X3 )
% 9.51/5.30                     != zero_zero_complex ) ) ) )
% 9.51/5.30            @ zero_zero_complex ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7812_sum_OG__def,axiom,
% 9.51/5.30      ( groups267424677133301183omplex
% 9.51/5.30      = ( ^ [P6: int > complex,I7: set_int] :
% 9.51/5.30            ( if_complex
% 9.51/5.30            @ ( finite_finite_int
% 9.51/5.30              @ ( collect_int
% 9.51/5.30                @ ^ [X3: int] :
% 9.51/5.30                    ( ( member_int @ X3 @ I7 )
% 9.51/5.30                    & ( ( P6 @ X3 )
% 9.51/5.30                     != zero_zero_complex ) ) ) )
% 9.51/5.30            @ ( groups3049146728041665814omplex @ P6
% 9.51/5.30              @ ( collect_int
% 9.51/5.30                @ ^ [X3: int] :
% 9.51/5.30                    ( ( member_int @ X3 @ I7 )
% 9.51/5.30                    & ( ( P6 @ X3 )
% 9.51/5.30                     != zero_zero_complex ) ) ) )
% 9.51/5.30            @ zero_zero_complex ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7813_sum_OG__def,axiom,
% 9.51/5.30      ( groups97945582718554045l_real
% 9.51/5.30      = ( ^ [P6: real > real,I7: set_real] :
% 9.51/5.30            ( if_real
% 9.51/5.30            @ ( finite_finite_real
% 9.51/5.30              @ ( collect_real
% 9.51/5.30                @ ^ [X3: real] :
% 9.51/5.30                    ( ( member_real @ X3 @ I7 )
% 9.51/5.30                    & ( ( P6 @ X3 )
% 9.51/5.30                     != zero_zero_real ) ) ) )
% 9.51/5.30            @ ( groups8097168146408367636l_real @ P6
% 9.51/5.30              @ ( collect_real
% 9.51/5.30                @ ^ [X3: real] :
% 9.51/5.30                    ( ( member_real @ X3 @ I7 )
% 9.51/5.30                    & ( ( P6 @ X3 )
% 9.51/5.30                     != zero_zero_real ) ) ) )
% 9.51/5.30            @ zero_zero_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7814_sum_OG__def,axiom,
% 9.51/5.30      ( groups1523912220035142973t_real
% 9.51/5.30      = ( ^ [P6: int > real,I7: set_int] :
% 9.51/5.30            ( if_real
% 9.51/5.30            @ ( finite_finite_int
% 9.51/5.30              @ ( collect_int
% 9.51/5.30                @ ^ [X3: int] :
% 9.51/5.30                    ( ( member_int @ X3 @ I7 )
% 9.51/5.30                    & ( ( P6 @ X3 )
% 9.51/5.30                     != zero_zero_real ) ) ) )
% 9.51/5.30            @ ( groups8778361861064173332t_real @ P6
% 9.51/5.30              @ ( collect_int
% 9.51/5.30                @ ^ [X3: int] :
% 9.51/5.30                    ( ( member_int @ X3 @ I7 )
% 9.51/5.30                    & ( ( P6 @ X3 )
% 9.51/5.30                     != zero_zero_real ) ) ) )
% 9.51/5.30            @ zero_zero_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7815_sum_OG__def,axiom,
% 9.51/5.30      ( groups5737402329758386879x_real
% 9.51/5.30      = ( ^ [P6: complex > real,I7: set_complex] :
% 9.51/5.30            ( if_real
% 9.51/5.30            @ ( finite3207457112153483333omplex
% 9.51/5.30              @ ( collect_complex
% 9.51/5.30                @ ^ [X3: complex] :
% 9.51/5.30                    ( ( member_complex @ X3 @ I7 )
% 9.51/5.30                    & ( ( P6 @ X3 )
% 9.51/5.30                     != zero_zero_real ) ) ) )
% 9.51/5.30            @ ( groups5808333547571424918x_real @ P6
% 9.51/5.30              @ ( collect_complex
% 9.51/5.30                @ ^ [X3: complex] :
% 9.51/5.30                    ( ( member_complex @ X3 @ I7 )
% 9.51/5.30                    & ( ( P6 @ X3 )
% 9.51/5.30                     != zero_zero_real ) ) ) )
% 9.51/5.30            @ zero_zero_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7816_sum_OG__def,axiom,
% 9.51/5.30      ( groups3269169158384524137al_rat
% 9.51/5.30      = ( ^ [P6: real > rat,I7: set_real] :
% 9.51/5.30            ( if_rat
% 9.51/5.30            @ ( finite_finite_real
% 9.51/5.30              @ ( collect_real
% 9.51/5.30                @ ^ [X3: real] :
% 9.51/5.30                    ( ( member_real @ X3 @ I7 )
% 9.51/5.30                    & ( ( P6 @ X3 )
% 9.51/5.30                     != zero_zero_rat ) ) ) )
% 9.51/5.30            @ ( groups1300246762558778688al_rat @ P6
% 9.51/5.30              @ ( collect_real
% 9.51/5.30                @ ^ [X3: real] :
% 9.51/5.30                    ( ( member_real @ X3 @ I7 )
% 9.51/5.30                    & ( ( P6 @ X3 )
% 9.51/5.30                     != zero_zero_rat ) ) ) )
% 9.51/5.30            @ zero_zero_rat ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7817_sum_OG__def,axiom,
% 9.51/5.30      ( groups1351286907653491341at_rat
% 9.51/5.30      = ( ^ [P6: nat > rat,I7: set_nat] :
% 9.51/5.30            ( if_rat
% 9.51/5.30            @ ( finite_finite_nat
% 9.51/5.30              @ ( collect_nat
% 9.51/5.30                @ ^ [X3: nat] :
% 9.51/5.30                    ( ( member_nat @ X3 @ I7 )
% 9.51/5.30                    & ( ( P6 @ X3 )
% 9.51/5.30                     != zero_zero_rat ) ) ) )
% 9.51/5.30            @ ( groups2906978787729119204at_rat @ P6
% 9.51/5.30              @ ( collect_nat
% 9.51/5.30                @ ^ [X3: nat] :
% 9.51/5.30                    ( ( member_nat @ X3 @ I7 )
% 9.51/5.30                    & ( ( P6 @ X3 )
% 9.51/5.30                     != zero_zero_rat ) ) ) )
% 9.51/5.30            @ zero_zero_rat ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7818_sum_OG__def,axiom,
% 9.51/5.30      ( groups2350640619554545897nt_rat
% 9.51/5.30      = ( ^ [P6: int > rat,I7: set_int] :
% 9.51/5.30            ( if_rat
% 9.51/5.30            @ ( finite_finite_int
% 9.51/5.30              @ ( collect_int
% 9.51/5.30                @ ^ [X3: int] :
% 9.51/5.30                    ( ( member_int @ X3 @ I7 )
% 9.51/5.30                    & ( ( P6 @ X3 )
% 9.51/5.30                     != zero_zero_rat ) ) ) )
% 9.51/5.30            @ ( groups3906332499630173760nt_rat @ P6
% 9.51/5.30              @ ( collect_int
% 9.51/5.30                @ ^ [X3: int] :
% 9.51/5.30                    ( ( member_int @ X3 @ I7 )
% 9.51/5.30                    & ( ( P6 @ X3 )
% 9.51/5.30                     != zero_zero_rat ) ) ) )
% 9.51/5.30            @ zero_zero_rat ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7819_sum_OG__def,axiom,
% 9.51/5.30      ( groups2276542476275365739ex_rat
% 9.51/5.30      = ( ^ [P6: complex > rat,I7: set_complex] :
% 9.51/5.30            ( if_rat
% 9.51/5.30            @ ( finite3207457112153483333omplex
% 9.51/5.30              @ ( collect_complex
% 9.51/5.30                @ ^ [X3: complex] :
% 9.51/5.30                    ( ( member_complex @ X3 @ I7 )
% 9.51/5.30                    & ( ( P6 @ X3 )
% 9.51/5.30                     != zero_zero_rat ) ) ) )
% 9.51/5.30            @ ( groups5058264527183730370ex_rat @ P6
% 9.51/5.30              @ ( collect_complex
% 9.51/5.30                @ ^ [X3: complex] :
% 9.51/5.30                    ( ( member_complex @ X3 @ I7 )
% 9.51/5.30                    & ( ( P6 @ X3 )
% 9.51/5.30                     != zero_zero_rat ) ) ) )
% 9.51/5.30            @ zero_zero_rat ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7820_ex__inverse__of__nat__less,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ? [N3: nat] :
% 9.51/5.30            ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.51/5.30            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7821_ex__inverse__of__nat__less,axiom,
% 9.51/5.30      ! [X: rat] :
% 9.51/5.30        ( ( ord_less_rat @ zero_zero_rat @ X )
% 9.51/5.30       => ? [N3: nat] :
% 9.51/5.30            ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.51/5.30            & ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ N3 ) ) @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7822_power__diff__conv__inverse,axiom,
% 9.51/5.30      ! [X: complex,M: nat,N: nat] :
% 9.51/5.30        ( ( X != zero_zero_complex )
% 9.51/5.30       => ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.30         => ( ( power_power_complex @ X @ ( minus_minus_nat @ N @ M ) )
% 9.51/5.30            = ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ M ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7823_power__diff__conv__inverse,axiom,
% 9.51/5.30      ! [X: real,M: nat,N: nat] :
% 9.51/5.30        ( ( X != zero_zero_real )
% 9.51/5.30       => ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.30         => ( ( power_power_real @ X @ ( minus_minus_nat @ N @ M ) )
% 9.51/5.30            = ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ M ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7824_power__diff__conv__inverse,axiom,
% 9.51/5.30      ! [X: rat,M: nat,N: nat] :
% 9.51/5.30        ( ( X != zero_zero_rat )
% 9.51/5.30       => ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.30         => ( ( power_power_rat @ X @ ( minus_minus_nat @ N @ M ) )
% 9.51/5.30            = ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ M ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7825_exp__divide__power__eq,axiom,
% 9.51/5.30      ! [N: nat,X: complex] :
% 9.51/5.30        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.30       => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
% 9.51/5.30          = ( exp_complex @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7826_exp__divide__power__eq,axiom,
% 9.51/5.30      ! [N: nat,X: real] :
% 9.51/5.30        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.30       => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
% 9.51/5.30          = ( exp_real @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7827_log__inverse,axiom,
% 9.51/5.30      ! [A: real,X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ A )
% 9.51/5.30       => ( ( A != one_one_real )
% 9.51/5.30         => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30           => ( ( log @ A @ ( inverse_inverse_real @ X ) )
% 9.51/5.30              = ( uminus_uminus_real @ ( log @ A @ X ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7828_plus__inverse__ge__2,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7829_real__inv__sqrt__pow2,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.51/5.30          = ( inverse_inverse_real @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7830_sum__diff1_H,axiom,
% 9.51/5.30      ! [I6: set_complex,F: complex > complex,I: complex] :
% 9.51/5.30        ( ( finite3207457112153483333omplex
% 9.51/5.30          @ ( collect_complex
% 9.51/5.30            @ ^ [I4: complex] :
% 9.51/5.30                ( ( member_complex @ I4 @ I6 )
% 9.51/5.30                & ( ( F @ I4 )
% 9.51/5.30                 != zero_zero_complex ) ) ) )
% 9.51/5.30       => ( ( ( member_complex @ I @ I6 )
% 9.51/5.30           => ( ( groups808145749697022017omplex @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.51/5.30              = ( minus_minus_complex @ ( groups808145749697022017omplex @ F @ I6 ) @ ( F @ I ) ) ) )
% 9.51/5.30          & ( ~ ( member_complex @ I @ I6 )
% 9.51/5.30           => ( ( groups808145749697022017omplex @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.51/5.30              = ( groups808145749697022017omplex @ F @ I6 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7831_sum__diff1_H,axiom,
% 9.51/5.30      ! [I6: set_real,F: real > complex,I: real] :
% 9.51/5.30        ( ( finite_finite_real
% 9.51/5.30          @ ( collect_real
% 9.51/5.30            @ ^ [I4: real] :
% 9.51/5.30                ( ( member_real @ I4 @ I6 )
% 9.51/5.30                & ( ( F @ I4 )
% 9.51/5.30                 != zero_zero_complex ) ) ) )
% 9.51/5.30       => ( ( ( member_real @ I @ I6 )
% 9.51/5.30           => ( ( groups5683813829254066239omplex @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.51/5.30              = ( minus_minus_complex @ ( groups5683813829254066239omplex @ F @ I6 ) @ ( F @ I ) ) ) )
% 9.51/5.30          & ( ~ ( member_real @ I @ I6 )
% 9.51/5.30           => ( ( groups5683813829254066239omplex @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.51/5.30              = ( groups5683813829254066239omplex @ F @ I6 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7832_sum__diff1_H,axiom,
% 9.51/5.30      ! [I6: set_int,F: int > complex,I: int] :
% 9.51/5.30        ( ( finite_finite_int
% 9.51/5.30          @ ( collect_int
% 9.51/5.30            @ ^ [I4: int] :
% 9.51/5.30                ( ( member_int @ I4 @ I6 )
% 9.51/5.30                & ( ( F @ I4 )
% 9.51/5.30                 != zero_zero_complex ) ) ) )
% 9.51/5.30       => ( ( ( member_int @ I @ I6 )
% 9.51/5.30           => ( ( groups267424677133301183omplex @ F @ ( minus_minus_set_int @ I6 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 9.51/5.30              = ( minus_minus_complex @ ( groups267424677133301183omplex @ F @ I6 ) @ ( F @ I ) ) ) )
% 9.51/5.30          & ( ~ ( member_int @ I @ I6 )
% 9.51/5.30           => ( ( groups267424677133301183omplex @ F @ ( minus_minus_set_int @ I6 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 9.51/5.30              = ( groups267424677133301183omplex @ F @ I6 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7833_sum__diff1_H,axiom,
% 9.51/5.30      ! [I6: set_complex,F: complex > real,I: complex] :
% 9.51/5.30        ( ( finite3207457112153483333omplex
% 9.51/5.30          @ ( collect_complex
% 9.51/5.30            @ ^ [I4: complex] :
% 9.51/5.30                ( ( member_complex @ I4 @ I6 )
% 9.51/5.30                & ( ( F @ I4 )
% 9.51/5.30                 != zero_zero_real ) ) ) )
% 9.51/5.30       => ( ( ( member_complex @ I @ I6 )
% 9.51/5.30           => ( ( groups5737402329758386879x_real @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.51/5.30              = ( minus_minus_real @ ( groups5737402329758386879x_real @ F @ I6 ) @ ( F @ I ) ) ) )
% 9.51/5.30          & ( ~ ( member_complex @ I @ I6 )
% 9.51/5.30           => ( ( groups5737402329758386879x_real @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.51/5.30              = ( groups5737402329758386879x_real @ F @ I6 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7834_sum__diff1_H,axiom,
% 9.51/5.30      ! [I6: set_real,F: real > real,I: real] :
% 9.51/5.30        ( ( finite_finite_real
% 9.51/5.30          @ ( collect_real
% 9.51/5.30            @ ^ [I4: real] :
% 9.51/5.30                ( ( member_real @ I4 @ I6 )
% 9.51/5.30                & ( ( F @ I4 )
% 9.51/5.30                 != zero_zero_real ) ) ) )
% 9.51/5.30       => ( ( ( member_real @ I @ I6 )
% 9.51/5.30           => ( ( groups97945582718554045l_real @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.51/5.30              = ( minus_minus_real @ ( groups97945582718554045l_real @ F @ I6 ) @ ( F @ I ) ) ) )
% 9.51/5.30          & ( ~ ( member_real @ I @ I6 )
% 9.51/5.30           => ( ( groups97945582718554045l_real @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.51/5.30              = ( groups97945582718554045l_real @ F @ I6 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7835_sum__diff1_H,axiom,
% 9.51/5.30      ! [I6: set_int,F: int > real,I: int] :
% 9.51/5.30        ( ( finite_finite_int
% 9.51/5.30          @ ( collect_int
% 9.51/5.30            @ ^ [I4: int] :
% 9.51/5.30                ( ( member_int @ I4 @ I6 )
% 9.51/5.30                & ( ( F @ I4 )
% 9.51/5.30                 != zero_zero_real ) ) ) )
% 9.51/5.30       => ( ( ( member_int @ I @ I6 )
% 9.51/5.30           => ( ( groups1523912220035142973t_real @ F @ ( minus_minus_set_int @ I6 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 9.51/5.30              = ( minus_minus_real @ ( groups1523912220035142973t_real @ F @ I6 ) @ ( F @ I ) ) ) )
% 9.51/5.30          & ( ~ ( member_int @ I @ I6 )
% 9.51/5.30           => ( ( groups1523912220035142973t_real @ F @ ( minus_minus_set_int @ I6 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 9.51/5.30              = ( groups1523912220035142973t_real @ F @ I6 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7836_sum__diff1_H,axiom,
% 9.51/5.30      ! [I6: set_complex,F: complex > rat,I: complex] :
% 9.51/5.30        ( ( finite3207457112153483333omplex
% 9.51/5.30          @ ( collect_complex
% 9.51/5.30            @ ^ [I4: complex] :
% 9.51/5.30                ( ( member_complex @ I4 @ I6 )
% 9.51/5.30                & ( ( F @ I4 )
% 9.51/5.30                 != zero_zero_rat ) ) ) )
% 9.51/5.30       => ( ( ( member_complex @ I @ I6 )
% 9.51/5.30           => ( ( groups2276542476275365739ex_rat @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.51/5.30              = ( minus_minus_rat @ ( groups2276542476275365739ex_rat @ F @ I6 ) @ ( F @ I ) ) ) )
% 9.51/5.30          & ( ~ ( member_complex @ I @ I6 )
% 9.51/5.30           => ( ( groups2276542476275365739ex_rat @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.51/5.30              = ( groups2276542476275365739ex_rat @ F @ I6 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7837_sum__diff1_H,axiom,
% 9.51/5.30      ! [I6: set_real,F: real > rat,I: real] :
% 9.51/5.30        ( ( finite_finite_real
% 9.51/5.30          @ ( collect_real
% 9.51/5.30            @ ^ [I4: real] :
% 9.51/5.30                ( ( member_real @ I4 @ I6 )
% 9.51/5.30                & ( ( F @ I4 )
% 9.51/5.30                 != zero_zero_rat ) ) ) )
% 9.51/5.30       => ( ( ( member_real @ I @ I6 )
% 9.51/5.30           => ( ( groups3269169158384524137al_rat @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.51/5.30              = ( minus_minus_rat @ ( groups3269169158384524137al_rat @ F @ I6 ) @ ( F @ I ) ) ) )
% 9.51/5.30          & ( ~ ( member_real @ I @ I6 )
% 9.51/5.30           => ( ( groups3269169158384524137al_rat @ F @ ( minus_minus_set_real @ I6 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.51/5.30              = ( groups3269169158384524137al_rat @ F @ I6 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7838_sum__diff1_H,axiom,
% 9.51/5.30      ! [I6: set_int,F: int > rat,I: int] :
% 9.51/5.30        ( ( finite_finite_int
% 9.51/5.30          @ ( collect_int
% 9.51/5.30            @ ^ [I4: int] :
% 9.51/5.30                ( ( member_int @ I4 @ I6 )
% 9.51/5.30                & ( ( F @ I4 )
% 9.51/5.30                 != zero_zero_rat ) ) ) )
% 9.51/5.30       => ( ( ( member_int @ I @ I6 )
% 9.51/5.30           => ( ( groups2350640619554545897nt_rat @ F @ ( minus_minus_set_int @ I6 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 9.51/5.30              = ( minus_minus_rat @ ( groups2350640619554545897nt_rat @ F @ I6 ) @ ( F @ I ) ) ) )
% 9.51/5.30          & ( ~ ( member_int @ I @ I6 )
% 9.51/5.30           => ( ( groups2350640619554545897nt_rat @ F @ ( minus_minus_set_int @ I6 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 9.51/5.30              = ( groups2350640619554545897nt_rat @ F @ I6 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7839_sum__diff1_H,axiom,
% 9.51/5.30      ! [I6: set_complex,F: complex > int,I: complex] :
% 9.51/5.30        ( ( finite3207457112153483333omplex
% 9.51/5.30          @ ( collect_complex
% 9.51/5.30            @ ^ [I4: complex] :
% 9.51/5.30                ( ( member_complex @ I4 @ I6 )
% 9.51/5.30                & ( ( F @ I4 )
% 9.51/5.30                 != zero_zero_int ) ) ) )
% 9.51/5.30       => ( ( ( member_complex @ I @ I6 )
% 9.51/5.30           => ( ( groups2909182065852811199ex_int @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.51/5.30              = ( minus_minus_int @ ( groups2909182065852811199ex_int @ F @ I6 ) @ ( F @ I ) ) ) )
% 9.51/5.30          & ( ~ ( member_complex @ I @ I6 )
% 9.51/5.30           => ( ( groups2909182065852811199ex_int @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.51/5.30              = ( groups2909182065852811199ex_int @ F @ I6 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7840_Maclaurin__exp__lt,axiom,
% 9.51/5.30      ! [X: real,N: nat] :
% 9.51/5.30        ( ( X != zero_zero_real )
% 9.51/5.30       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.30         => ? [T6: real] :
% 9.51/5.30              ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
% 9.51/5.30              & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
% 9.51/5.30              & ( ( exp_real @ X )
% 9.51/5.30                = ( plus_plus_real
% 9.51/5.30                  @ ( groups6591440286371151544t_real
% 9.51/5.30                    @ ^ [M2: nat] : ( divide_divide_real @ ( power_power_real @ X @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) )
% 9.51/5.30                    @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.30                  @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7841_log__base__10__eq1,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 9.51/5.30          = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7842_powr__def,axiom,
% 9.51/5.30      ( powr_real
% 9.51/5.30      = ( ^ [X3: real,A4: real] : ( if_real @ ( X3 = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A4 @ ( ln_ln_real @ X3 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7843_log__base__10__eq2,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 9.51/5.30          = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7844_cosh__zero__iff,axiom,
% 9.51/5.30      ! [X: complex] :
% 9.51/5.30        ( ( ( cosh_complex @ X )
% 9.51/5.30          = zero_zero_complex )
% 9.51/5.30        = ( ( power_power_complex @ ( exp_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.51/5.30          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7845_cosh__zero__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ( cosh_real @ X )
% 9.51/5.30          = zero_zero_real )
% 9.51/5.30        = ( ( power_power_real @ ( exp_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.51/5.30          = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7846_ln__one,axiom,
% 9.51/5.30      ( ( ln_ln_real @ one_one_real )
% 9.51/5.30      = zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7847_fact__0,axiom,
% 9.51/5.30      ( ( semiri5044797733671781792omplex @ zero_zero_nat )
% 9.51/5.30      = one_one_complex ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7848_fact__0,axiom,
% 9.51/5.30      ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
% 9.51/5.30      = one_one_rat ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7849_fact__0,axiom,
% 9.51/5.30      ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
% 9.51/5.30      = one_one_int ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7850_fact__0,axiom,
% 9.51/5.30      ( ( semiri2265585572941072030t_real @ zero_zero_nat )
% 9.51/5.30      = one_one_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7851_fact__0,axiom,
% 9.51/5.30      ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
% 9.51/5.30      = one_one_nat ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7852_cosh__0,axiom,
% 9.51/5.30      ( ( cosh_complex @ zero_zero_complex )
% 9.51/5.30      = one_one_complex ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7853_cosh__0,axiom,
% 9.51/5.30      ( ( cosh_real @ zero_zero_real )
% 9.51/5.30      = one_one_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7854_ln__inj__iff,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.51/5.30         => ( ( ( ln_ln_real @ X )
% 9.51/5.30              = ( ln_ln_real @ Y ) )
% 9.51/5.30            = ( X = Y ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7855_ln__less__cancel__iff,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.51/5.30         => ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
% 9.51/5.30            = ( ord_less_real @ X @ Y ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7856_fact__Suc__0,axiom,
% 9.51/5.30      ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
% 9.51/5.30      = one_one_complex ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7857_fact__Suc__0,axiom,
% 9.51/5.30      ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
% 9.51/5.30      = one_one_rat ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7858_fact__Suc__0,axiom,
% 9.51/5.30      ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
% 9.51/5.30      = one_one_int ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7859_fact__Suc__0,axiom,
% 9.51/5.30      ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
% 9.51/5.30      = one_one_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7860_fact__Suc__0,axiom,
% 9.51/5.30      ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
% 9.51/5.30      = one_one_nat ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7861_ln__le__cancel__iff,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.51/5.30         => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
% 9.51/5.30            = ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7862_ln__eq__zero__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ( ln_ln_real @ X )
% 9.51/5.30            = zero_zero_real )
% 9.51/5.30          = ( X = one_one_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7863_ln__gt__zero__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 9.51/5.30          = ( ord_less_real @ one_one_real @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7864_ln__less__zero__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 9.51/5.30          = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7865_exp__ln__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ( exp_real @ ( ln_ln_real @ X ) )
% 9.51/5.30          = X )
% 9.51/5.30        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7866_exp__ln,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( exp_real @ ( ln_ln_real @ X ) )
% 9.51/5.30          = X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7867_ln__le__zero__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 9.51/5.30          = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7868_ln__ge__zero__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 9.51/5.30          = ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7869_fact__nonzero,axiom,
% 9.51/5.30      ! [N: nat] :
% 9.51/5.30        ( ( semiri5044797733671781792omplex @ N )
% 9.51/5.30       != zero_zero_complex ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7870_fact__nonzero,axiom,
% 9.51/5.30      ! [N: nat] :
% 9.51/5.30        ( ( semiri773545260158071498ct_rat @ N )
% 9.51/5.30       != zero_zero_rat ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7871_fact__nonzero,axiom,
% 9.51/5.30      ! [N: nat] :
% 9.51/5.30        ( ( semiri1406184849735516958ct_int @ N )
% 9.51/5.30       != zero_zero_int ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7872_fact__nonzero,axiom,
% 9.51/5.30      ! [N: nat] :
% 9.51/5.30        ( ( semiri2265585572941072030t_real @ N )
% 9.51/5.30       != zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7873_fact__nonzero,axiom,
% 9.51/5.30      ! [N: nat] :
% 9.51/5.30        ( ( semiri1408675320244567234ct_nat @ N )
% 9.51/5.30       != zero_zero_nat ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7874_ln__less__self,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7875_fact__ge__zero,axiom,
% 9.51/5.30      ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7876_fact__ge__zero,axiom,
% 9.51/5.30      ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7877_fact__ge__zero,axiom,
% 9.51/5.30      ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7878_fact__ge__zero,axiom,
% 9.51/5.30      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7879_fact__not__neg,axiom,
% 9.51/5.30      ! [N: nat] :
% 9.51/5.30        ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7880_fact__not__neg,axiom,
% 9.51/5.30      ! [N: nat] :
% 9.51/5.30        ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7881_fact__not__neg,axiom,
% 9.51/5.30      ! [N: nat] :
% 9.51/5.30        ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7882_fact__not__neg,axiom,
% 9.51/5.30      ! [N: nat] :
% 9.51/5.30        ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7883_fact__gt__zero,axiom,
% 9.51/5.30      ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7884_fact__gt__zero,axiom,
% 9.51/5.30      ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7885_fact__gt__zero,axiom,
% 9.51/5.30      ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7886_fact__gt__zero,axiom,
% 9.51/5.30      ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7887_fact__ge__1,axiom,
% 9.51/5.30      ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7888_fact__ge__1,axiom,
% 9.51/5.30      ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7889_fact__ge__1,axiom,
% 9.51/5.30      ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7890_fact__ge__1,axiom,
% 9.51/5.30      ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7891_fact__mono,axiom,
% 9.51/5.30      ! [M: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.30       => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7892_fact__mono,axiom,
% 9.51/5.30      ! [M: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.30       => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7893_fact__mono,axiom,
% 9.51/5.30      ! [M: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.30       => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7894_fact__mono,axiom,
% 9.51/5.30      ! [M: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.30       => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7895_cosh__real__pos,axiom,
% 9.51/5.30      ! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7896_fact__dvd,axiom,
% 9.51/5.30      ! [N: nat,M: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ N @ M )
% 9.51/5.30       => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7897_fact__dvd,axiom,
% 9.51/5.30      ! [N: nat,M: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ N @ M )
% 9.51/5.30       => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7898_fact__dvd,axiom,
% 9.51/5.30      ! [N: nat,M: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ N @ M )
% 9.51/5.30       => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7899_fact__dvd,axiom,
% 9.51/5.30      ! [N: nat,M: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ N @ M )
% 9.51/5.30       => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7900_ln__bound,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7901_ln__gt__zero,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ one_one_real @ X )
% 9.51/5.30       => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7902_ln__less__zero,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ one_one_real )
% 9.51/5.30         => ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7903_ln__gt__zero__imp__gt__one,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30         => ( ord_less_real @ one_one_real @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7904_fact__less__mono,axiom,
% 9.51/5.30      ! [M: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.51/5.30       => ( ( ord_less_nat @ M @ N )
% 9.51/5.30         => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7905_fact__less__mono,axiom,
% 9.51/5.30      ! [M: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.51/5.30       => ( ( ord_less_nat @ M @ N )
% 9.51/5.30         => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7906_fact__less__mono,axiom,
% 9.51/5.30      ! [M: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.51/5.30       => ( ( ord_less_nat @ M @ N )
% 9.51/5.30         => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7907_fact__less__mono,axiom,
% 9.51/5.30      ! [M: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.51/5.30       => ( ( ord_less_nat @ M @ N )
% 9.51/5.30         => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7908_cosh__real__nonpos__less__iff,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.51/5.30       => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 9.51/5.30         => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
% 9.51/5.30            = ( ord_less_real @ Y @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7909_cosh__real__nonneg__less__iff,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.51/5.30         => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
% 9.51/5.30            = ( ord_less_real @ X @ Y ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7910_cosh__real__strict__mono,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ Y )
% 9.51/5.30         => ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7911_fact__mod,axiom,
% 9.51/5.30      ! [M: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.30       => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
% 9.51/5.30          = zero_zero_int ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7912_fact__mod,axiom,
% 9.51/5.30      ! [M: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.30       => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
% 9.51/5.30          = zero_z3403309356797280102nteger ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7913_fact__mod,axiom,
% 9.51/5.30      ! [M: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.30       => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
% 9.51/5.30          = zero_zero_nat ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7914_fact__le__power,axiom,
% 9.51/5.30      ! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7915_fact__le__power,axiom,
% 9.51/5.30      ! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7916_fact__le__power,axiom,
% 9.51/5.30      ! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7917_fact__le__power,axiom,
% 9.51/5.30      ! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7918_ln__ge__zero__imp__ge__one,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30         => ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7919_cosh__ln__real,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( cosh_real @ ( ln_ln_real @ X ) )
% 9.51/5.30          = ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7920_ln__mult,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.51/5.30         => ( ( ln_ln_real @ ( times_times_real @ X @ Y ) )
% 9.51/5.30            = ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7921_ln__eq__minus__one,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ( ln_ln_real @ X )
% 9.51/5.30            = ( minus_minus_real @ X @ one_one_real ) )
% 9.51/5.30         => ( X = one_one_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7922_ln__div,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.51/5.30         => ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
% 9.51/5.30            = ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7923_ln__ge__iff,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_eq_real @ Y @ ( ln_ln_real @ X ) )
% 9.51/5.30          = ( ord_less_eq_real @ ( exp_real @ Y ) @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7924_ln__inverse,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ln_ln_real @ ( inverse_inverse_real @ X ) )
% 9.51/5.30          = ( uminus_uminus_real @ ( ln_ln_real @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7925_choose__dvd,axiom,
% 9.51/5.30      ! [K: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.30       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7926_choose__dvd,axiom,
% 9.51/5.30      ! [K: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.30       => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7927_choose__dvd,axiom,
% 9.51/5.30      ! [K: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.30       => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7928_choose__dvd,axiom,
% 9.51/5.30      ! [K: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.30       => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7929_choose__dvd,axiom,
% 9.51/5.30      ! [K: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.30       => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7930_ln__2__less__1,axiom,
% 9.51/5.30      ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7931_ln__le__minus__one,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7932_ln__diff__le,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ Y )
% 9.51/5.30         => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y ) @ Y ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7933_ln__add__one__self__le__self2,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.51/5.30       => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7934_ln__realpow,axiom,
% 9.51/5.30      ! [X: real,N: nat] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
% 9.51/5.30          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7935_ln__one__minus__pos__upper__bound,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ one_one_real )
% 9.51/5.30         => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7936_ln__powr__bound,axiom,
% 9.51/5.30      ! [X: real,A: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ one_one_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.51/5.30         => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7937_ln__powr__bound2,axiom,
% 9.51/5.30      ! [X: real,A: real] :
% 9.51/5.30        ( ( ord_less_real @ one_one_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.51/5.30         => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7938_log__eq__div__ln__mult__log,axiom,
% 9.51/5.30      ! [A: real,B: real,X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ A )
% 9.51/5.30       => ( ( A != one_one_real )
% 9.51/5.30         => ( ( ord_less_real @ zero_zero_real @ B )
% 9.51/5.30           => ( ( B != one_one_real )
% 9.51/5.30             => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30               => ( ( log @ A @ X )
% 9.51/5.30                  = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7939_fact__num__eq__if,axiom,
% 9.51/5.30      ( semiri5044797733671781792omplex
% 9.51/5.30      = ( ^ [M2: nat] : ( if_complex @ ( M2 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7940_fact__num__eq__if,axiom,
% 9.51/5.30      ( semiri773545260158071498ct_rat
% 9.51/5.30      = ( ^ [M2: nat] : ( if_rat @ ( M2 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7941_fact__num__eq__if,axiom,
% 9.51/5.30      ( semiri1406184849735516958ct_int
% 9.51/5.30      = ( ^ [M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7942_fact__num__eq__if,axiom,
% 9.51/5.30      ( semiri2265585572941072030t_real
% 9.51/5.30      = ( ^ [M2: nat] : ( if_real @ ( M2 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7943_fact__num__eq__if,axiom,
% 9.51/5.30      ( semiri1408675320244567234ct_nat
% 9.51/5.30      = ( ^ [M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7944_fact__reduce,axiom,
% 9.51/5.30      ! [N: nat] :
% 9.51/5.30        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.30       => ( ( semiri5044797733671781792omplex @ N )
% 9.51/5.30          = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7945_fact__reduce,axiom,
% 9.51/5.30      ! [N: nat] :
% 9.51/5.30        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.30       => ( ( semiri773545260158071498ct_rat @ N )
% 9.51/5.30          = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7946_fact__reduce,axiom,
% 9.51/5.30      ! [N: nat] :
% 9.51/5.30        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.30       => ( ( semiri1406184849735516958ct_int @ N )
% 9.51/5.30          = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7947_fact__reduce,axiom,
% 9.51/5.30      ! [N: nat] :
% 9.51/5.30        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.30       => ( ( semiri2265585572941072030t_real @ N )
% 9.51/5.30          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7948_fact__reduce,axiom,
% 9.51/5.30      ! [N: nat] :
% 9.51/5.30        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.30       => ( ( semiri1408675320244567234ct_nat @ N )
% 9.51/5.30          = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7949_binomial__fact,axiom,
% 9.51/5.30      ! [K: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.30       => ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
% 9.51/5.30          = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7950_binomial__fact,axiom,
% 9.51/5.30      ! [K: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.30       => ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
% 9.51/5.30          = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7951_binomial__fact,axiom,
% 9.51/5.30      ! [K: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.30       => ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
% 9.51/5.30          = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7952_fact__binomial,axiom,
% 9.51/5.30      ! [K: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.30       => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) ) )
% 9.51/5.30          = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7953_fact__binomial,axiom,
% 9.51/5.30      ! [K: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.30       => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
% 9.51/5.30          = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7954_fact__binomial,axiom,
% 9.51/5.30      ! [K: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.30       => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) )
% 9.51/5.30          = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7955_ln__sqrt,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ln_ln_real @ ( sqrt @ X ) )
% 9.51/5.30          = ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7956_Maclaurin__zero,axiom,
% 9.51/5.30      ! [X: real,N: nat,Diff: nat > complex > real] :
% 9.51/5.30        ( ( X = zero_zero_real )
% 9.51/5.30       => ( ( N != zero_zero_nat )
% 9.51/5.30         => ( ( groups6591440286371151544t_real
% 9.51/5.30              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
% 9.51/5.30              @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.30            = ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7957_Maclaurin__zero,axiom,
% 9.51/5.30      ! [X: real,N: nat,Diff: nat > real > real] :
% 9.51/5.30        ( ( X = zero_zero_real )
% 9.51/5.30       => ( ( N != zero_zero_nat )
% 9.51/5.30         => ( ( groups6591440286371151544t_real
% 9.51/5.30              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
% 9.51/5.30              @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.30            = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7958_Maclaurin__zero,axiom,
% 9.51/5.30      ! [X: real,N: nat,Diff: nat > rat > real] :
% 9.51/5.30        ( ( X = zero_zero_real )
% 9.51/5.30       => ( ( N != zero_zero_nat )
% 9.51/5.30         => ( ( groups6591440286371151544t_real
% 9.51/5.30              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
% 9.51/5.30              @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.30            = ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7959_Maclaurin__zero,axiom,
% 9.51/5.30      ! [X: real,N: nat,Diff: nat > nat > real] :
% 9.51/5.30        ( ( X = zero_zero_real )
% 9.51/5.30       => ( ( N != zero_zero_nat )
% 9.51/5.30         => ( ( groups6591440286371151544t_real
% 9.51/5.30              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
% 9.51/5.30              @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.30            = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7960_Maclaurin__zero,axiom,
% 9.51/5.30      ! [X: real,N: nat,Diff: nat > int > real] :
% 9.51/5.30        ( ( X = zero_zero_real )
% 9.51/5.30       => ( ( N != zero_zero_nat )
% 9.51/5.30         => ( ( groups6591440286371151544t_real
% 9.51/5.30              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
% 9.51/5.30              @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.30            = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7961_Maclaurin__lemma,axiom,
% 9.51/5.30      ! [H2: real,F: real > real,J: nat > real,N: nat] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ H2 )
% 9.51/5.30       => ? [B8: real] :
% 9.51/5.30            ( ( F @ H2 )
% 9.51/5.30            = ( plus_plus_real
% 9.51/5.30              @ ( groups6591440286371151544t_real
% 9.51/5.30                @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 9.51/5.30                @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.30              @ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7962_sin__coeff__def,axiom,
% 9.51/5.30      ( sin_coeff
% 9.51/5.30      = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7963_gbinomial__code,axiom,
% 9.51/5.30      ( gbinomial_complex
% 9.51/5.30      = ( ^ [A4: complex,K3: nat] :
% 9.51/5.30            ( if_complex @ ( K3 = zero_zero_nat ) @ one_one_complex
% 9.51/5.30            @ ( divide1717551699836669952omplex
% 9.51/5.30              @ ( set_fo1517530859248394432omplex
% 9.51/5.30                @ ^ [L2: nat] : ( times_times_complex @ ( minus_minus_complex @ A4 @ ( semiri8010041392384452111omplex @ L2 ) ) )
% 9.51/5.30                @ zero_zero_nat
% 9.51/5.30                @ ( minus_minus_nat @ K3 @ one_one_nat )
% 9.51/5.30                @ one_one_complex )
% 9.51/5.30              @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7964_gbinomial__code,axiom,
% 9.51/5.30      ( gbinomial_rat
% 9.51/5.30      = ( ^ [A4: rat,K3: nat] :
% 9.51/5.30            ( if_rat @ ( K3 = zero_zero_nat ) @ one_one_rat
% 9.51/5.30            @ ( divide_divide_rat
% 9.51/5.30              @ ( set_fo1949268297981939178at_rat
% 9.51/5.30                @ ^ [L2: nat] : ( times_times_rat @ ( minus_minus_rat @ A4 @ ( semiri681578069525770553at_rat @ L2 ) ) )
% 9.51/5.30                @ zero_zero_nat
% 9.51/5.30                @ ( minus_minus_nat @ K3 @ one_one_nat )
% 9.51/5.30                @ one_one_rat )
% 9.51/5.30              @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7965_gbinomial__code,axiom,
% 9.51/5.30      ( gbinomial_real
% 9.51/5.30      = ( ^ [A4: real,K3: nat] :
% 9.51/5.30            ( if_real @ ( K3 = zero_zero_nat ) @ one_one_real
% 9.51/5.30            @ ( divide_divide_real
% 9.51/5.30              @ ( set_fo3111899725591712190t_real
% 9.51/5.30                @ ^ [L2: nat] : ( times_times_real @ ( minus_minus_real @ A4 @ ( semiri5074537144036343181t_real @ L2 ) ) )
% 9.51/5.30                @ zero_zero_nat
% 9.51/5.30                @ ( minus_minus_nat @ K3 @ one_one_nat )
% 9.51/5.30                @ one_one_real )
% 9.51/5.30              @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7966_ln__series,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.51/5.30         => ( ( ln_ln_real @ X )
% 9.51/5.30            = ( suminf_real
% 9.51/5.30              @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N2 ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7967_tanh__ln__real,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( tanh_real @ ( ln_ln_real @ X ) )
% 9.51/5.30          = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7968_sin__x__sin__y,axiom,
% 9.51/5.30      ! [X: complex,Y: complex] :
% 9.51/5.30        ( sums_complex
% 9.51/5.30        @ ^ [P6: nat] :
% 9.51/5.30            ( groups2073611262835488442omplex
% 9.51/5.30            @ ^ [N2: nat] :
% 9.51/5.30                ( if_complex
% 9.51/5.30                @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 )
% 9.51/5.30                  & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.51/5.30                @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) ) @ ( power_power_complex @ X @ N2 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P6 @ N2 ) ) )
% 9.51/5.30                @ zero_zero_complex )
% 9.51/5.30            @ ( set_ord_atMost_nat @ P6 ) )
% 9.51/5.30        @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ Y ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7969_sin__x__sin__y,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( sums_real
% 9.51/5.30        @ ^ [P6: nat] :
% 9.51/5.30            ( groups6591440286371151544t_real
% 9.51/5.30            @ ^ [N2: nat] :
% 9.51/5.30                ( if_real
% 9.51/5.30                @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 )
% 9.51/5.30                  & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.51/5.30                @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) ) @ ( power_power_real @ X @ N2 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P6 @ N2 ) ) )
% 9.51/5.30                @ zero_zero_real )
% 9.51/5.30            @ ( set_ord_atMost_nat @ P6 ) )
% 9.51/5.30        @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7970_scaleR__cancel__right,axiom,
% 9.51/5.30      ! [A: real,X: complex,B: real] :
% 9.51/5.30        ( ( ( real_V2046097035970521341omplex @ A @ X )
% 9.51/5.30          = ( real_V2046097035970521341omplex @ B @ X ) )
% 9.51/5.30        = ( ( A = B )
% 9.51/5.30          | ( X = zero_zero_complex ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7971_scaleR__cancel__right,axiom,
% 9.51/5.30      ! [A: real,X: real,B: real] :
% 9.51/5.30        ( ( ( real_V1485227260804924795R_real @ A @ X )
% 9.51/5.30          = ( real_V1485227260804924795R_real @ B @ X ) )
% 9.51/5.30        = ( ( A = B )
% 9.51/5.30          | ( X = zero_zero_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7972_scaleR__zero__right,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( real_V2046097035970521341omplex @ A @ zero_zero_complex )
% 9.51/5.30        = zero_zero_complex ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7973_scaleR__zero__right,axiom,
% 9.51/5.30      ! [A: real] :
% 9.51/5.30        ( ( real_V1485227260804924795R_real @ A @ zero_zero_real )
% 9.51/5.30        = zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7974_sin__zero,axiom,
% 9.51/5.30      ( ( sin_complex @ zero_zero_complex )
% 9.51/5.30      = zero_zero_complex ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7975_sin__zero,axiom,
% 9.51/5.30      ( ( sin_real @ zero_zero_real )
% 9.51/5.30      = zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7976_tanh__0,axiom,
% 9.51/5.30      ( ( tanh_complex @ zero_zero_complex )
% 9.51/5.30      = zero_zero_complex ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7977_tanh__0,axiom,
% 9.51/5.30      ( ( tanh_real @ zero_zero_real )
% 9.51/5.30      = zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7978_tanh__real__less__iff,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
% 9.51/5.30        = ( ord_less_real @ X @ Y ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7979_suminf__zero,axiom,
% 9.51/5.30      ( ( suminf_complex
% 9.51/5.30        @ ^ [N2: nat] : zero_zero_complex )
% 9.51/5.30      = zero_zero_complex ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7980_suminf__zero,axiom,
% 9.51/5.30      ( ( suminf_real
% 9.51/5.30        @ ^ [N2: nat] : zero_zero_real )
% 9.51/5.30      = zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7981_suminf__zero,axiom,
% 9.51/5.30      ( ( suminf_nat
% 9.51/5.30        @ ^ [N2: nat] : zero_zero_nat )
% 9.51/5.30      = zero_zero_nat ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7982_suminf__zero,axiom,
% 9.51/5.30      ( ( suminf_int
% 9.51/5.30        @ ^ [N2: nat] : zero_zero_int )
% 9.51/5.30      = zero_zero_int ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7983_scaleR__eq__0__iff,axiom,
% 9.51/5.30      ! [A: real,X: complex] :
% 9.51/5.30        ( ( ( real_V2046097035970521341omplex @ A @ X )
% 9.51/5.30          = zero_zero_complex )
% 9.51/5.30        = ( ( A = zero_zero_real )
% 9.51/5.30          | ( X = zero_zero_complex ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7984_scaleR__eq__0__iff,axiom,
% 9.51/5.30      ! [A: real,X: real] :
% 9.51/5.30        ( ( ( real_V1485227260804924795R_real @ A @ X )
% 9.51/5.30          = zero_zero_real )
% 9.51/5.30        = ( ( A = zero_zero_real )
% 9.51/5.30          | ( X = zero_zero_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7985_scaleR__zero__left,axiom,
% 9.51/5.30      ! [X: complex] :
% 9.51/5.30        ( ( real_V2046097035970521341omplex @ zero_zero_real @ X )
% 9.51/5.30        = zero_zero_complex ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7986_scaleR__zero__left,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( real_V1485227260804924795R_real @ zero_zero_real @ X )
% 9.51/5.30        = zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7987_tanh__real__pos__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X ) )
% 9.51/5.30        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7988_tanh__real__neg__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( tanh_real @ X ) @ zero_zero_real )
% 9.51/5.30        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7989_sin__coeff__0,axiom,
% 9.51/5.30      ( ( sin_coeff @ zero_zero_nat )
% 9.51/5.30      = zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7990_powser__zero,axiom,
% 9.51/5.30      ! [F: nat > complex] :
% 9.51/5.30        ( ( suminf_complex
% 9.51/5.30          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) )
% 9.51/5.30        = ( F @ zero_zero_nat ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7991_powser__zero,axiom,
% 9.51/5.30      ! [F: nat > real] :
% 9.51/5.30        ( ( suminf_real
% 9.51/5.30          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) )
% 9.51/5.30        = ( F @ zero_zero_nat ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7992_fact__mono__nat,axiom,
% 9.51/5.30      ! [M: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.30       => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7993_fact__ge__self,axiom,
% 9.51/5.30      ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7994_scaleR__right__imp__eq,axiom,
% 9.51/5.30      ! [X: complex,A: real,B: real] :
% 9.51/5.30        ( ( X != zero_zero_complex )
% 9.51/5.30       => ( ( ( real_V2046097035970521341omplex @ A @ X )
% 9.51/5.30            = ( real_V2046097035970521341omplex @ B @ X ) )
% 9.51/5.30         => ( A = B ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7995_scaleR__right__imp__eq,axiom,
% 9.51/5.30      ! [X: real,A: real,B: real] :
% 9.51/5.30        ( ( X != zero_zero_real )
% 9.51/5.30       => ( ( ( real_V1485227260804924795R_real @ A @ X )
% 9.51/5.30            = ( real_V1485227260804924795R_real @ B @ X ) )
% 9.51/5.30         => ( A = B ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7996_fact__less__mono__nat,axiom,
% 9.51/5.30      ! [M: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.51/5.30       => ( ( ord_less_nat @ M @ N )
% 9.51/5.30         => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7997_tanh__real__lt__1,axiom,
% 9.51/5.30      ! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7998_fact__ge__Suc__0__nat,axiom,
% 9.51/5.30      ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_7999_scaleR__right__mono__neg,axiom,
% 9.51/5.30      ! [B: real,A: real,C: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ B @ A )
% 9.51/5.30       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.51/5.30         => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ C ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8000_scaleR__right__mono,axiom,
% 9.51/5.30      ! [A: real,B: real,X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ A @ B )
% 9.51/5.30       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.51/5.30         => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8001_dvd__fact,axiom,
% 9.51/5.30      ! [M: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ one_one_nat @ M )
% 9.51/5.30       => ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.30         => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8002_vector__fraction__eq__iff,axiom,
% 9.51/5.30      ! [U: real,V: real,A: complex,X: complex] :
% 9.51/5.30        ( ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ U @ V ) @ A )
% 9.51/5.30          = X )
% 9.51/5.30        = ( ( ( V = zero_zero_real )
% 9.51/5.30           => ( X = zero_zero_complex ) )
% 9.51/5.30          & ( ( V != zero_zero_real )
% 9.51/5.30           => ( ( real_V2046097035970521341omplex @ U @ A )
% 9.51/5.30              = ( real_V2046097035970521341omplex @ V @ X ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8003_vector__fraction__eq__iff,axiom,
% 9.51/5.30      ! [U: real,V: real,A: real,X: real] :
% 9.51/5.30        ( ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A )
% 9.51/5.30          = X )
% 9.51/5.30        = ( ( ( V = zero_zero_real )
% 9.51/5.30           => ( X = zero_zero_real ) )
% 9.51/5.30          & ( ( V != zero_zero_real )
% 9.51/5.30           => ( ( real_V1485227260804924795R_real @ U @ A )
% 9.51/5.30              = ( real_V1485227260804924795R_real @ V @ X ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8004_eq__vector__fraction__iff,axiom,
% 9.51/5.30      ! [X: complex,U: real,V: real,A: complex] :
% 9.51/5.30        ( ( X
% 9.51/5.30          = ( real_V2046097035970521341omplex @ ( divide_divide_real @ U @ V ) @ A ) )
% 9.51/5.30        = ( ( ( V = zero_zero_real )
% 9.51/5.30           => ( X = zero_zero_complex ) )
% 9.51/5.30          & ( ( V != zero_zero_real )
% 9.51/5.30           => ( ( real_V2046097035970521341omplex @ V @ X )
% 9.51/5.30              = ( real_V2046097035970521341omplex @ U @ A ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8005_eq__vector__fraction__iff,axiom,
% 9.51/5.30      ! [X: real,U: real,V: real,A: real] :
% 9.51/5.30        ( ( X
% 9.51/5.30          = ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A ) )
% 9.51/5.30        = ( ( ( V = zero_zero_real )
% 9.51/5.30           => ( X = zero_zero_real ) )
% 9.51/5.30          & ( ( V != zero_zero_real )
% 9.51/5.30           => ( ( real_V1485227260804924795R_real @ V @ X )
% 9.51/5.30              = ( real_V1485227260804924795R_real @ U @ A ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8006_Real__Vector__Spaces_Ole__add__iff2,axiom,
% 9.51/5.30      ! [A: real,E2: real,C: real,B: real,D: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E2 ) @ D ) )
% 9.51/5.30        = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8007_Real__Vector__Spaces_Ole__add__iff1,axiom,
% 9.51/5.30      ! [A: real,E2: real,C: real,B: real,D: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E2 ) @ D ) )
% 9.51/5.30        = ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8008_suminf__finite,axiom,
% 9.51/5.30      ! [N6: set_nat,F: nat > complex] :
% 9.51/5.30        ( ( finite_finite_nat @ N6 )
% 9.51/5.30       => ( ! [N3: nat] :
% 9.51/5.30              ( ~ ( member_nat @ N3 @ N6 )
% 9.51/5.30             => ( ( F @ N3 )
% 9.51/5.30                = zero_zero_complex ) )
% 9.51/5.30         => ( ( suminf_complex @ F )
% 9.51/5.30            = ( groups2073611262835488442omplex @ F @ N6 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8009_suminf__finite,axiom,
% 9.51/5.30      ! [N6: set_nat,F: nat > int] :
% 9.51/5.30        ( ( finite_finite_nat @ N6 )
% 9.51/5.30       => ( ! [N3: nat] :
% 9.51/5.30              ( ~ ( member_nat @ N3 @ N6 )
% 9.51/5.30             => ( ( F @ N3 )
% 9.51/5.30                = zero_zero_int ) )
% 9.51/5.30         => ( ( suminf_int @ F )
% 9.51/5.30            = ( groups3539618377306564664at_int @ F @ N6 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8010_suminf__finite,axiom,
% 9.51/5.30      ! [N6: set_nat,F: nat > nat] :
% 9.51/5.30        ( ( finite_finite_nat @ N6 )
% 9.51/5.30       => ( ! [N3: nat] :
% 9.51/5.30              ( ~ ( member_nat @ N3 @ N6 )
% 9.51/5.30             => ( ( F @ N3 )
% 9.51/5.30                = zero_zero_nat ) )
% 9.51/5.30         => ( ( suminf_nat @ F )
% 9.51/5.30            = ( groups3542108847815614940at_nat @ F @ N6 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8011_suminf__finite,axiom,
% 9.51/5.30      ! [N6: set_nat,F: nat > real] :
% 9.51/5.30        ( ( finite_finite_nat @ N6 )
% 9.51/5.30       => ( ! [N3: nat] :
% 9.51/5.30              ( ~ ( member_nat @ N3 @ N6 )
% 9.51/5.30             => ( ( F @ N3 )
% 9.51/5.30                = zero_zero_real ) )
% 9.51/5.30         => ( ( suminf_real @ F )
% 9.51/5.30            = ( groups6591440286371151544t_real @ F @ N6 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8012_tanh__real__gt__neg1,axiom,
% 9.51/5.30      ! [X: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8013_fact__diff__Suc,axiom,
% 9.51/5.30      ! [N: nat,M: nat] :
% 9.51/5.30        ( ( ord_less_nat @ N @ ( suc @ M ) )
% 9.51/5.30       => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
% 9.51/5.30          = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8014_zero__le__scaleR__iff,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) )
% 9.51/5.30        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.51/5.30            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 9.51/5.30          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.51/5.30            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 9.51/5.30          | ( A = zero_zero_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8015_scaleR__le__0__iff,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ B ) @ zero_zero_real )
% 9.51/5.30        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.51/5.30            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 9.51/5.30          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.51/5.30            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 9.51/5.30          | ( A = zero_zero_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8016_scaleR__nonpos__nonpos,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.51/5.30       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 9.51/5.30         => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8017_scaleR__nonpos__nonneg,axiom,
% 9.51/5.30      ! [A: real,X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.51/5.30       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.51/5.30         => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8018_scaleR__nonneg__nonpos,axiom,
% 9.51/5.30      ! [A: real,X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.51/5.30       => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.51/5.30         => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8019_scaleR__nonneg__nonneg,axiom,
% 9.51/5.30      ! [A: real,X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.51/5.30       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.51/5.30         => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8020_split__scaleR__pos__le,axiom,
% 9.51/5.30      ! [A: real,B: real] :
% 9.51/5.30        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.51/5.30            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 9.51/5.30          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.51/5.30            & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 9.51/5.30       => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8021_split__scaleR__neg__le,axiom,
% 9.51/5.30      ! [A: real,X: real] :
% 9.51/5.30        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.51/5.30            & ( ord_less_eq_real @ X @ zero_zero_real ) )
% 9.51/5.30          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.51/5.30            & ( ord_less_eq_real @ zero_zero_real @ X ) ) )
% 9.51/5.30       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8022_scaleR__mono_H,axiom,
% 9.51/5.30      ! [A: real,B: real,C: real,D: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ A @ B )
% 9.51/5.30       => ( ( ord_less_eq_real @ C @ D )
% 9.51/5.30         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.51/5.30           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.51/5.30             => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ D ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8023_scaleR__mono,axiom,
% 9.51/5.30      ! [A: real,B: real,X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ A @ B )
% 9.51/5.30       => ( ( ord_less_eq_real @ X @ Y )
% 9.51/5.30         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.51/5.30           => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.51/5.30             => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ Y ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8024_fact__div__fact__le__pow,axiom,
% 9.51/5.30      ! [R2: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ R2 @ N )
% 9.51/5.30       => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8025_scaleR__left__le__one__le,axiom,
% 9.51/5.30      ! [X: real,A: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_eq_real @ A @ one_one_real )
% 9.51/5.30         => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8026_binomial__fact__lemma,axiom,
% 9.51/5.30      ! [K: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.30       => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
% 9.51/5.30          = ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8027_neg__less__divideR__eq,axiom,
% 9.51/5.30      ! [C: real,A: real,B: real] :
% 9.51/5.30        ( ( ord_less_real @ C @ zero_zero_real )
% 9.51/5.30       => ( ( ord_less_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
% 9.51/5.30          = ( ord_less_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8028_neg__divideR__less__eq,axiom,
% 9.51/5.30      ! [C: real,B: real,A: real] :
% 9.51/5.30        ( ( ord_less_real @ C @ zero_zero_real )
% 9.51/5.30       => ( ( ord_less_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
% 9.51/5.30          = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8029_pos__less__divideR__eq,axiom,
% 9.51/5.30      ! [C: real,A: real,B: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ C )
% 9.51/5.30       => ( ( ord_less_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
% 9.51/5.30          = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8030_pos__divideR__less__eq,axiom,
% 9.51/5.30      ! [C: real,B: real,A: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ C )
% 9.51/5.30       => ( ( ord_less_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
% 9.51/5.30          = ( ord_less_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8031_nonzero__inverse__scaleR__distrib,axiom,
% 9.51/5.30      ! [A: real,X: complex] :
% 9.51/5.30        ( ( A != zero_zero_real )
% 9.51/5.30       => ( ( X != zero_zero_complex )
% 9.51/5.30         => ( ( invers8013647133539491842omplex @ ( real_V2046097035970521341omplex @ A @ X ) )
% 9.51/5.30            = ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ A ) @ ( invers8013647133539491842omplex @ X ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8032_nonzero__inverse__scaleR__distrib,axiom,
% 9.51/5.30      ! [A: real,X: real] :
% 9.51/5.30        ( ( A != zero_zero_real )
% 9.51/5.30       => ( ( X != zero_zero_real )
% 9.51/5.30         => ( ( inverse_inverse_real @ ( real_V1485227260804924795R_real @ A @ X ) )
% 9.51/5.30            = ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ X ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8033_sin__gt__zero__02,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.51/5.30         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8034_binomial__altdef__nat,axiom,
% 9.51/5.30      ! [K: nat,N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.30       => ( ( binomial @ N @ K )
% 9.51/5.30          = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8035_fold__atLeastAtMost__nat_Oelims,axiom,
% 9.51/5.30      ! [X: nat > nat > nat,Xa2: nat,Xb2: nat,Xc: nat,Y: nat] :
% 9.51/5.30        ( ( ( set_fo2584398358068434914at_nat @ X @ Xa2 @ Xb2 @ Xc )
% 9.51/5.30          = Y )
% 9.51/5.30       => ( ( ( ord_less_nat @ Xb2 @ Xa2 )
% 9.51/5.30           => ( Y = Xc ) )
% 9.51/5.30          & ( ~ ( ord_less_nat @ Xb2 @ Xa2 )
% 9.51/5.30           => ( Y
% 9.51/5.30              = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb2 @ ( X @ Xa2 @ Xc ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8036_fold__atLeastAtMost__nat_Osimps,axiom,
% 9.51/5.30      ( set_fo2584398358068434914at_nat
% 9.51/5.30      = ( ^ [F5: nat > nat > nat,A4: nat,B4: nat,Acc: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A4 ) @ Acc @ ( set_fo2584398358068434914at_nat @ F5 @ ( plus_plus_nat @ A4 @ one_one_nat ) @ B4 @ ( F5 @ A4 @ Acc ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8037_pos__le__minus__divideR__eq,axiom,
% 9.51/5.30      ! [C: real,A: real,B: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ C )
% 9.51/5.30       => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
% 9.51/5.30          = ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8038_pos__minus__divideR__le__eq,axiom,
% 9.51/5.30      ! [C: real,B: real,A: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ C )
% 9.51/5.30       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
% 9.51/5.30          = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8039_neg__le__minus__divideR__eq,axiom,
% 9.51/5.30      ! [C: real,A: real,B: real] :
% 9.51/5.30        ( ( ord_less_real @ C @ zero_zero_real )
% 9.51/5.30       => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
% 9.51/5.30          = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8040_neg__minus__divideR__le__eq,axiom,
% 9.51/5.30      ! [C: real,B: real,A: real] :
% 9.51/5.30        ( ( ord_less_real @ C @ zero_zero_real )
% 9.51/5.30       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
% 9.51/5.30          = ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8041_neg__minus__divideR__less__eq,axiom,
% 9.51/5.30      ! [C: real,B: real,A: real] :
% 9.51/5.30        ( ( ord_less_real @ C @ zero_zero_real )
% 9.51/5.30       => ( ( ord_less_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
% 9.51/5.30          = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8042_neg__less__minus__divideR__eq,axiom,
% 9.51/5.30      ! [C: real,A: real,B: real] :
% 9.51/5.30        ( ( ord_less_real @ C @ zero_zero_real )
% 9.51/5.30       => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
% 9.51/5.30          = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8043_pos__minus__divideR__less__eq,axiom,
% 9.51/5.30      ! [C: real,B: real,A: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ C )
% 9.51/5.30       => ( ( ord_less_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
% 9.51/5.30          = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8044_pos__less__minus__divideR__eq,axiom,
% 9.51/5.30      ! [C: real,A: real,B: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ C )
% 9.51/5.30       => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
% 9.51/5.30          = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8045_binomial__code,axiom,
% 9.51/5.30      ( binomial
% 9.51/5.30      = ( ^ [N2: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K3 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8046_suminf__geometric,axiom,
% 9.51/5.30      ! [C: real] :
% 9.51/5.30        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 9.51/5.30       => ( ( suminf_real @ ( power_power_real @ C ) )
% 9.51/5.30          = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8047_suminf__geometric,axiom,
% 9.51/5.30      ! [C: complex] :
% 9.51/5.30        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 9.51/5.30       => ( ( suminf_complex @ ( power_power_complex @ C ) )
% 9.51/5.30          = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8048_sum__atLeastAtMost__code,axiom,
% 9.51/5.30      ! [F: nat > complex,A: nat,B: nat] :
% 9.51/5.30        ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 9.51/5.30        = ( set_fo1517530859248394432omplex
% 9.51/5.30          @ ^ [A4: nat] : ( plus_plus_complex @ ( F @ A4 ) )
% 9.51/5.30          @ A
% 9.51/5.30          @ B
% 9.51/5.30          @ zero_zero_complex ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8049_sum__atLeastAtMost__code,axiom,
% 9.51/5.30      ! [F: nat > rat,A: nat,B: nat] :
% 9.51/5.30        ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 9.51/5.30        = ( set_fo1949268297981939178at_rat
% 9.51/5.30          @ ^ [A4: nat] : ( plus_plus_rat @ ( F @ A4 ) )
% 9.51/5.30          @ A
% 9.51/5.30          @ B
% 9.51/5.30          @ zero_zero_rat ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8050_sum__atLeastAtMost__code,axiom,
% 9.51/5.30      ! [F: nat > int,A: nat,B: nat] :
% 9.51/5.30        ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 9.51/5.30        = ( set_fo2581907887559384638at_int
% 9.51/5.30          @ ^ [A4: nat] : ( plus_plus_int @ ( F @ A4 ) )
% 9.51/5.30          @ A
% 9.51/5.30          @ B
% 9.51/5.30          @ zero_zero_int ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8051_sum__atLeastAtMost__code,axiom,
% 9.51/5.30      ! [F: nat > nat,A: nat,B: nat] :
% 9.51/5.30        ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 9.51/5.30        = ( set_fo2584398358068434914at_nat
% 9.51/5.30          @ ^ [A4: nat] : ( plus_plus_nat @ ( F @ A4 ) )
% 9.51/5.30          @ A
% 9.51/5.30          @ B
% 9.51/5.30          @ zero_zero_nat ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8052_sum__atLeastAtMost__code,axiom,
% 9.51/5.30      ! [F: nat > real,A: nat,B: nat] :
% 9.51/5.30        ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 9.51/5.30        = ( set_fo3111899725591712190t_real
% 9.51/5.30          @ ^ [A4: nat] : ( plus_plus_real @ ( F @ A4 ) )
% 9.51/5.30          @ A
% 9.51/5.30          @ B
% 9.51/5.30          @ zero_zero_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8053_tanh__add,axiom,
% 9.51/5.30      ! [X: complex,Y: complex] :
% 9.51/5.30        ( ( ( cosh_complex @ X )
% 9.51/5.30         != zero_zero_complex )
% 9.51/5.30       => ( ( ( cosh_complex @ Y )
% 9.51/5.30           != zero_zero_complex )
% 9.51/5.30         => ( ( tanh_complex @ ( plus_plus_complex @ X @ Y ) )
% 9.51/5.30            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8054_tanh__add,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ( cosh_real @ X )
% 9.51/5.30         != zero_zero_real )
% 9.51/5.30       => ( ( ( cosh_real @ Y )
% 9.51/5.30           != zero_zero_real )
% 9.51/5.30         => ( ( tanh_real @ ( plus_plus_real @ X @ Y ) )
% 9.51/5.30            = ( divide_divide_real @ ( plus_plus_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8055_cosh__converges,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( sums_real
% 9.51/5.30        @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) @ zero_zero_real )
% 9.51/5.30        @ ( cosh_real @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8056_cosh__converges,axiom,
% 9.51/5.30      ! [X: complex] :
% 9.51/5.30        ( sums_complex
% 9.51/5.30        @ ^ [N2: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_complex @ X @ N2 ) ) @ zero_zero_complex )
% 9.51/5.30        @ ( cosh_complex @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8057_sums__cos__x__plus__y,axiom,
% 9.51/5.30      ! [X: complex,Y: complex] :
% 9.51/5.30        ( sums_complex
% 9.51/5.30        @ ^ [P6: nat] :
% 9.51/5.30            ( groups2073611262835488442omplex
% 9.51/5.30            @ ^ [N2: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 ) @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_complex @ X @ N2 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P6 @ N2 ) ) ) @ zero_zero_complex )
% 9.51/5.30            @ ( set_ord_atMost_nat @ P6 ) )
% 9.51/5.30        @ ( cos_complex @ ( plus_plus_complex @ X @ Y ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8058_sums__cos__x__plus__y,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( sums_real
% 9.51/5.30        @ ^ [P6: nat] :
% 9.51/5.30            ( groups6591440286371151544t_real
% 9.51/5.30            @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 ) @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ X @ N2 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P6 @ N2 ) ) ) @ zero_zero_real )
% 9.51/5.30            @ ( set_ord_atMost_nat @ P6 ) )
% 9.51/5.30        @ ( cos_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8059_cos__x__cos__y,axiom,
% 9.51/5.30      ! [X: complex,Y: complex] :
% 9.51/5.30        ( sums_complex
% 9.51/5.30        @ ^ [P6: nat] :
% 9.51/5.30            ( groups2073611262835488442omplex
% 9.51/5.30            @ ^ [N2: nat] :
% 9.51/5.30                ( if_complex
% 9.51/5.30                @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 )
% 9.51/5.30                  & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.51/5.30                @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_complex @ X @ N2 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P6 @ N2 ) ) )
% 9.51/5.30                @ zero_zero_complex )
% 9.51/5.30            @ ( set_ord_atMost_nat @ P6 ) )
% 9.51/5.30        @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8060_cos__x__cos__y,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( sums_real
% 9.51/5.30        @ ^ [P6: nat] :
% 9.51/5.30            ( groups6591440286371151544t_real
% 9.51/5.30            @ ^ [N2: nat] :
% 9.51/5.30                ( if_real
% 9.51/5.30                @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 )
% 9.51/5.30                  & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.51/5.30                @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ X @ N2 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P6 @ N2 ) ) )
% 9.51/5.30                @ zero_zero_real )
% 9.51/5.30            @ ( set_ord_atMost_nat @ P6 ) )
% 9.51/5.30        @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8061_Maclaurin__sin__expansion3,axiom,
% 9.51/5.30      ! [N: nat,X: real] :
% 9.51/5.30        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.30       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30         => ? [T6: real] :
% 9.51/5.30              ( ( ord_less_real @ zero_zero_real @ T6 )
% 9.51/5.30              & ( ord_less_real @ T6 @ X )
% 9.51/5.30              & ( ( sin_real @ X )
% 9.51/5.30                = ( plus_plus_real
% 9.51/5.30                  @ ( groups6591440286371151544t_real
% 9.51/5.30                    @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X @ M2 ) )
% 9.51/5.30                    @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.30                  @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8062_Maclaurin__sin__expansion4,axiom,
% 9.51/5.30      ! [X: real,N: nat] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ? [T6: real] :
% 9.51/5.30            ( ( ord_less_real @ zero_zero_real @ T6 )
% 9.51/5.30            & ( ord_less_eq_real @ T6 @ X )
% 9.51/5.30            & ( ( sin_real @ X )
% 9.51/5.30              = ( plus_plus_real
% 9.51/5.30                @ ( groups6591440286371151544t_real
% 9.51/5.30                  @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X @ M2 ) )
% 9.51/5.30                  @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.30                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8063_sinh__converges,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( sums_real
% 9.51/5.30        @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) )
% 9.51/5.30        @ ( sinh_real @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8064_sinh__converges,axiom,
% 9.51/5.30      ! [X: complex] :
% 9.51/5.30        ( sums_complex
% 9.51/5.30        @ ^ [N2: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_complex @ X @ N2 ) ) )
% 9.51/5.30        @ ( sinh_complex @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8065_sinh__real__less__iff,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
% 9.51/5.30        = ( ord_less_real @ X @ Y ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8066_sinh__real__neg__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( sinh_real @ X ) @ zero_zero_real )
% 9.51/5.30        = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8067_sinh__real__pos__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X ) )
% 9.51/5.30        = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8068_sinh__0,axiom,
% 9.51/5.30      ( ( sinh_complex @ zero_zero_complex )
% 9.51/5.30      = zero_zero_complex ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8069_sinh__0,axiom,
% 9.51/5.30      ( ( sinh_real @ zero_zero_real )
% 9.51/5.30      = zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8070_cos__zero,axiom,
% 9.51/5.30      ( ( cos_complex @ zero_zero_complex )
% 9.51/5.30      = one_one_complex ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8071_cos__zero,axiom,
% 9.51/5.30      ( ( cos_real @ zero_zero_real )
% 9.51/5.30      = one_one_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8072_cos__mono__less__eq,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_eq_real @ X @ pi )
% 9.51/5.30         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.51/5.30           => ( ( ord_less_eq_real @ Y @ pi )
% 9.51/5.30             => ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
% 9.51/5.30                = ( ord_less_real @ Y @ X ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8073_cos__monotone__0__pi,axiom,
% 9.51/5.30      ! [Y: real,X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.51/5.30       => ( ( ord_less_real @ Y @ X )
% 9.51/5.30         => ( ( ord_less_eq_real @ X @ pi )
% 9.51/5.30           => ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8074_sincos__principal__value,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30      ? [Y3: real] :
% 9.51/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 9.51/5.30        & ( ord_less_eq_real @ Y3 @ pi )
% 9.51/5.30        & ( ( sin_real @ Y3 )
% 9.51/5.30          = ( sin_real @ X ) )
% 9.51/5.30        & ( ( cos_real @ Y3 )
% 9.51/5.30          = ( cos_real @ X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8075_cos__monotone__minus__pi__0,axiom,
% 9.51/5.30      ! [Y: real,X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
% 9.51/5.30       => ( ( ord_less_real @ Y @ X )
% 9.51/5.30         => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 9.51/5.30           => ( ord_less_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8076_pi__gt__zero,axiom,
% 9.51/5.30      ord_less_real @ zero_zero_real @ pi ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8077_pi__not__less__zero,axiom,
% 9.51/5.30      ~ ( ord_less_real @ pi @ zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8078_sin__gt__zero,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ pi )
% 9.51/5.30         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8079_sinh__less__cosh__real,axiom,
% 9.51/5.30      ! [X: real] : ( ord_less_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8080_cos__one__sin__zero,axiom,
% 9.51/5.30      ! [X: complex] :
% 9.51/5.30        ( ( ( cos_complex @ X )
% 9.51/5.30          = one_one_complex )
% 9.51/5.30       => ( ( sin_complex @ X )
% 9.51/5.30          = zero_zero_complex ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8081_cos__one__sin__zero,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ( cos_real @ X )
% 9.51/5.30          = one_one_real )
% 9.51/5.30       => ( ( sin_real @ X )
% 9.51/5.30          = zero_zero_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8082_cos__gt__zero,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30         => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8083_sin__eq__0__pi,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ pi )
% 9.51/5.30         => ( ( ( sin_real @ X )
% 9.51/5.30              = zero_zero_real )
% 9.51/5.30           => ( X = zero_zero_real ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8084_sin__zero__pi__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( abs_abs_real @ X ) @ pi )
% 9.51/5.30       => ( ( ( sin_real @ X )
% 9.51/5.30            = zero_zero_real )
% 9.51/5.30          = ( X = zero_zero_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8085_cos__gt__zero__pi,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30         => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8086_sin__zero__norm__cos__one,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ( sin_real @ X )
% 9.51/5.30          = zero_zero_real )
% 9.51/5.30       => ( ( real_V7735802525324610683m_real @ ( cos_real @ X ) )
% 9.51/5.30          = one_one_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8087_sin__zero__norm__cos__one,axiom,
% 9.51/5.30      ! [X: complex] :
% 9.51/5.30        ( ( ( sin_complex @ X )
% 9.51/5.30          = zero_zero_complex )
% 9.51/5.30       => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X ) )
% 9.51/5.30          = one_one_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8088_pi__less__4,axiom,
% 9.51/5.30      ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8089_cos__two__less__zero,axiom,
% 9.51/5.30      ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8090_pi__half__less__two,axiom,
% 9.51/5.30      ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8091_sin__pi__divide__n__ge__0,axiom,
% 9.51/5.30      ! [N: nat] :
% 9.51/5.30        ( ( N != zero_zero_nat )
% 9.51/5.30       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8092_sincos__total__2pi,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.51/5.30          = one_one_real )
% 9.51/5.30       => ~ ! [T6: real] :
% 9.51/5.30              ( ( ord_less_eq_real @ zero_zero_real @ T6 )
% 9.51/5.30             => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.51/5.30               => ( ( X
% 9.51/5.30                    = ( cos_real @ T6 ) )
% 9.51/5.30                 => ( Y
% 9.51/5.30                   != ( sin_real @ T6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8093_pi__half__gt__zero,axiom,
% 9.51/5.30      ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8094_sin__gt__zero2,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8095_sin__lt__zero,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ pi @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.51/5.30         => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8096_m2pi__less__pi,axiom,
% 9.51/5.30      ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8097_arctan__ubound,axiom,
% 9.51/5.30      ! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8098_cos__double__less__one,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.51/5.30         => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8099_sinh__zero__iff,axiom,
% 9.51/5.30      ! [X: complex] :
% 9.51/5.30        ( ( ( sinh_complex @ X )
% 9.51/5.30          = zero_zero_complex )
% 9.51/5.30        = ( member_complex @ ( exp_complex @ X ) @ ( insert_complex @ one_one_complex @ ( insert_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ bot_bot_set_complex ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8100_sinh__zero__iff,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ( sinh_real @ X )
% 9.51/5.30          = zero_zero_real )
% 9.51/5.30        = ( member_real @ ( exp_real @ X ) @ ( insert_real @ one_one_real @ ( insert_real @ ( uminus_uminus_real @ one_one_real ) @ bot_bot_set_real ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8101_sin__le__zero,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ pi @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.51/5.30         => ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8102_minus__pi__half__less__zero,axiom,
% 9.51/5.30      ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8103_sin__less__zero,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ zero_zero_real )
% 9.51/5.30         => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8104_sin__mono__less__eq,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.51/5.30       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 9.51/5.30           => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30             => ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
% 9.51/5.30                = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8105_sin__monotone__2pi,axiom,
% 9.51/5.30      ! [Y: real,X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 9.51/5.30       => ( ( ord_less_real @ Y @ X )
% 9.51/5.30         => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30           => ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8106_arctan__bounded,axiom,
% 9.51/5.30      ! [Y: real] :
% 9.51/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
% 9.51/5.30        & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8107_arctan__lbound,axiom,
% 9.51/5.30      ! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8108_sin__pi__divide__n__gt__0,axiom,
% 9.51/5.30      ! [N: nat] :
% 9.51/5.30        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.51/5.30       => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8109_sinh__ln__real,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( sinh_real @ ( ln_ln_real @ X ) )
% 9.51/5.30          = ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8110_Maclaurin__minus__cos__expansion,axiom,
% 9.51/5.30      ! [N: nat,X: real] :
% 9.51/5.30        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.30       => ( ( ord_less_real @ X @ zero_zero_real )
% 9.51/5.30         => ? [T6: real] :
% 9.51/5.30              ( ( ord_less_real @ X @ T6 )
% 9.51/5.30              & ( ord_less_real @ T6 @ zero_zero_real )
% 9.51/5.30              & ( ( cos_real @ X )
% 9.51/5.30                = ( plus_plus_real
% 9.51/5.30                  @ ( groups6591440286371151544t_real
% 9.51/5.30                    @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X @ M2 ) )
% 9.51/5.30                    @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.30                  @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8111_Maclaurin__cos__expansion2,axiom,
% 9.51/5.30      ! [X: real,N: nat] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.30         => ? [T6: real] :
% 9.51/5.30              ( ( ord_less_real @ zero_zero_real @ T6 )
% 9.51/5.30              & ( ord_less_real @ T6 @ X )
% 9.51/5.30              & ( ( cos_real @ X )
% 9.51/5.30                = ( plus_plus_real
% 9.51/5.30                  @ ( groups6591440286371151544t_real
% 9.51/5.30                    @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X @ M2 ) )
% 9.51/5.30                    @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.30                  @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8112_tan__double,axiom,
% 9.51/5.30      ! [X: complex] :
% 9.51/5.30        ( ( ( cos_complex @ X )
% 9.51/5.30         != zero_zero_complex )
% 9.51/5.30       => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 9.51/5.30           != zero_zero_complex )
% 9.51/5.30         => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 9.51/5.30            = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8113_tan__double,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ( cos_real @ X )
% 9.51/5.30         != zero_zero_real )
% 9.51/5.30       => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 9.51/5.30           != zero_zero_real )
% 9.51/5.30         => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 9.51/5.30            = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8114_sin__tan,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30       => ( ( sin_real @ X )
% 9.51/5.30          = ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8115_cos__tan,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30       => ( ( cos_real @ X )
% 9.51/5.30          = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8116_tan__zero,axiom,
% 9.51/5.30      ( ( tan_complex @ zero_zero_complex )
% 9.51/5.30      = zero_zero_complex ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8117_tan__zero,axiom,
% 9.51/5.30      ( ( tan_real @ zero_zero_real )
% 9.51/5.30      = zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8118_cos__coeff__0,axiom,
% 9.51/5.30      ( ( cos_coeff @ zero_zero_nat )
% 9.51/5.30      = one_one_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8119_tan__gt__zero,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30         => ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8120_lemma__tan__total,axiom,
% 9.51/5.30      ! [Y: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ Y )
% 9.51/5.30       => ? [X4: real] :
% 9.51/5.30            ( ( ord_less_real @ zero_zero_real @ X4 )
% 9.51/5.30            & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30            & ( ord_less_real @ Y @ ( tan_real @ X4 ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8121_lemma__tan__total1,axiom,
% 9.51/5.30      ! [Y: real] :
% 9.51/5.30      ? [X4: real] :
% 9.51/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 9.51/5.30        & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30        & ( ( tan_real @ X4 )
% 9.51/5.30          = Y ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8122_tan__mono__lt__eq,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 9.51/5.30           => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30             => ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 9.51/5.30                = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8123_tan__monotone_H,axiom,
% 9.51/5.30      ! [Y: real,X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 9.51/5.30       => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.51/5.30           => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30             => ( ( ord_less_real @ Y @ X )
% 9.51/5.30                = ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8124_tan__monotone,axiom,
% 9.51/5.30      ! [Y: real,X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 9.51/5.30       => ( ( ord_less_real @ Y @ X )
% 9.51/5.30         => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30           => ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8125_tan__total,axiom,
% 9.51/5.30      ! [Y: real] :
% 9.51/5.30      ? [X4: real] :
% 9.51/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 9.51/5.30        & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30        & ( ( tan_real @ X4 )
% 9.51/5.30          = Y )
% 9.51/5.30        & ! [Y4: real] :
% 9.51/5.30            ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
% 9.51/5.30              & ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30              & ( ( tan_real @ Y4 )
% 9.51/5.30                = Y ) )
% 9.51/5.30           => ( Y4 = X4 ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8126_add__tan__eq,axiom,
% 9.51/5.30      ! [X: complex,Y: complex] :
% 9.51/5.30        ( ( ( cos_complex @ X )
% 9.51/5.30         != zero_zero_complex )
% 9.51/5.30       => ( ( ( cos_complex @ Y )
% 9.51/5.30           != zero_zero_complex )
% 9.51/5.30         => ( ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) )
% 9.51/5.30            = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8127_add__tan__eq,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ( cos_real @ X )
% 9.51/5.30         != zero_zero_real )
% 9.51/5.30       => ( ( ( cos_real @ Y )
% 9.51/5.30           != zero_zero_real )
% 9.51/5.30         => ( ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 9.51/5.30            = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8128_tan__pos__pi2__le,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30         => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8129_tan__total__pos,axiom,
% 9.51/5.30      ! [Y: real] :
% 9.51/5.30        ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.51/5.30       => ? [X4: real] :
% 9.51/5.30            ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 9.51/5.30            & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30            & ( ( tan_real @ X4 )
% 9.51/5.30              = Y ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8130_tan__less__zero,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ zero_zero_real )
% 9.51/5.30         => ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8131_tan__mono__le__eq,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 9.51/5.30           => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30             => ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 9.51/5.30                = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8132_tan__mono__le,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.51/5.30       => ( ( ord_less_eq_real @ X @ Y )
% 9.51/5.30         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30           => ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8133_tan__bound__pi2,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 9.51/5.30       => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8134_arctan,axiom,
% 9.51/5.30      ! [Y: real] :
% 9.51/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
% 9.51/5.30        & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30        & ( ( tan_real @ ( arctan @ Y ) )
% 9.51/5.30          = Y ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8135_arctan__tan,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30         => ( ( arctan @ ( tan_real @ X ) )
% 9.51/5.30            = X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8136_arctan__unique,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30         => ( ( ( tan_real @ X )
% 9.51/5.30              = Y )
% 9.51/5.30           => ( ( arctan @ Y )
% 9.51/5.30              = X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8137_tan__add,axiom,
% 9.51/5.30      ! [X: complex,Y: complex] :
% 9.51/5.30        ( ( ( cos_complex @ X )
% 9.51/5.30         != zero_zero_complex )
% 9.51/5.30       => ( ( ( cos_complex @ Y )
% 9.51/5.30           != zero_zero_complex )
% 9.51/5.30         => ( ( ( cos_complex @ ( plus_plus_complex @ X @ Y ) )
% 9.51/5.30             != zero_zero_complex )
% 9.51/5.30           => ( ( tan_complex @ ( plus_plus_complex @ X @ Y ) )
% 9.51/5.30              = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8138_tan__add,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ( cos_real @ X )
% 9.51/5.30         != zero_zero_real )
% 9.51/5.30       => ( ( ( cos_real @ Y )
% 9.51/5.30           != zero_zero_real )
% 9.51/5.30         => ( ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
% 9.51/5.30             != zero_zero_real )
% 9.51/5.30           => ( ( tan_real @ ( plus_plus_real @ X @ Y ) )
% 9.51/5.30              = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8139_tan__diff,axiom,
% 9.51/5.30      ! [X: complex,Y: complex] :
% 9.51/5.30        ( ( ( cos_complex @ X )
% 9.51/5.30         != zero_zero_complex )
% 9.51/5.30       => ( ( ( cos_complex @ Y )
% 9.51/5.30           != zero_zero_complex )
% 9.51/5.30         => ( ( ( cos_complex @ ( minus_minus_complex @ X @ Y ) )
% 9.51/5.30             != zero_zero_complex )
% 9.51/5.30           => ( ( tan_complex @ ( minus_minus_complex @ X @ Y ) )
% 9.51/5.30              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8140_tan__diff,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ( cos_real @ X )
% 9.51/5.30         != zero_zero_real )
% 9.51/5.30       => ( ( ( cos_real @ Y )
% 9.51/5.30           != zero_zero_real )
% 9.51/5.30         => ( ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
% 9.51/5.30             != zero_zero_real )
% 9.51/5.30           => ( ( tan_real @ ( minus_minus_real @ X @ Y ) )
% 9.51/5.30              = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8141_lemma__tan__add1,axiom,
% 9.51/5.30      ! [X: complex,Y: complex] :
% 9.51/5.30        ( ( ( cos_complex @ X )
% 9.51/5.30         != zero_zero_complex )
% 9.51/5.30       => ( ( ( cos_complex @ Y )
% 9.51/5.30           != zero_zero_complex )
% 9.51/5.30         => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) )
% 9.51/5.30            = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8142_lemma__tan__add1,axiom,
% 9.51/5.30      ! [X: real,Y: real] :
% 9.51/5.30        ( ( ( cos_real @ X )
% 9.51/5.30         != zero_zero_real )
% 9.51/5.30       => ( ( ( cos_real @ Y )
% 9.51/5.30           != zero_zero_real )
% 9.51/5.30         => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) )
% 9.51/5.30            = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8143_tan__total__pi4,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.51/5.30       => ? [Z4: real] :
% 9.51/5.30            ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z4 )
% 9.51/5.30            & ( ord_less_real @ Z4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 9.51/5.30            & ( ( tan_real @ Z4 )
% 9.51/5.30              = X ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8144_tan__sec,axiom,
% 9.51/5.30      ! [X: complex] :
% 9.51/5.30        ( ( ( cos_complex @ X )
% 9.51/5.30         != zero_zero_complex )
% 9.51/5.30       => ( ( plus_plus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.51/5.30          = ( power_power_complex @ ( invers8013647133539491842omplex @ ( cos_complex @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8145_tan__sec,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ( cos_real @ X )
% 9.51/5.30         != zero_zero_real )
% 9.51/5.30       => ( ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.51/5.30          = ( power_power_real @ ( inverse_inverse_real @ ( cos_real @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8146_complex__unimodular__polar,axiom,
% 9.51/5.30      ! [Z2: complex] :
% 9.51/5.30        ( ( ( real_V1022390504157884413omplex @ Z2 )
% 9.51/5.30          = one_one_real )
% 9.51/5.30       => ~ ! [T6: real] :
% 9.51/5.30              ( ( ord_less_eq_real @ zero_zero_real @ T6 )
% 9.51/5.30             => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.51/5.30               => ( Z2
% 9.51/5.30                 != ( complex2 @ ( cos_real @ T6 ) @ ( sin_real @ T6 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8147_cot__less__zero,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ zero_zero_real )
% 9.51/5.30         => ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8148_sum__pos__lt__pair,axiom,
% 9.51/5.30      ! [F: nat > real,K: nat] :
% 9.51/5.30        ( ( summable_real @ F )
% 9.51/5.30       => ( ! [D5: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D5 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D5 ) @ one_one_nat ) ) ) ) )
% 9.51/5.30         => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8149_arctan__def,axiom,
% 9.51/5.30      ( arctan
% 9.51/5.30      = ( ^ [Y2: real] :
% 9.51/5.30            ( the_real
% 9.51/5.30            @ ^ [X3: real] :
% 9.51/5.30                ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.51/5.30                & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30                & ( ( tan_real @ X3 )
% 9.51/5.30                  = Y2 ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8150_cot__gt__zero,axiom,
% 9.51/5.30      ! [X: real] :
% 9.51/5.30        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.30       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.30         => ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8151_cot__zero,axiom,
% 9.51/5.30      ( ( cot_complex @ zero_zero_complex )
% 9.51/5.30      = zero_zero_complex ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8152_cot__zero,axiom,
% 9.51/5.30      ( ( cot_real @ zero_zero_real )
% 9.51/5.30      = zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8153_summable__single,axiom,
% 9.51/5.30      ! [I: nat,F: nat > complex] :
% 9.51/5.30        ( summable_complex
% 9.51/5.30        @ ^ [R5: nat] : ( if_complex @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_complex ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8154_summable__single,axiom,
% 9.51/5.30      ! [I: nat,F: nat > real] :
% 9.51/5.30        ( summable_real
% 9.51/5.30        @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8155_summable__single,axiom,
% 9.51/5.30      ! [I: nat,F: nat > nat] :
% 9.51/5.30        ( summable_nat
% 9.51/5.30        @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8156_summable__single,axiom,
% 9.51/5.30      ! [I: nat,F: nat > int] :
% 9.51/5.30        ( summable_int
% 9.51/5.30        @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8157_summable__zero,axiom,
% 9.51/5.30      ( summable_complex
% 9.51/5.30      @ ^ [N2: nat] : zero_zero_complex ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8158_summable__zero,axiom,
% 9.51/5.30      ( summable_real
% 9.51/5.30      @ ^ [N2: nat] : zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8159_summable__zero,axiom,
% 9.51/5.30      ( summable_nat
% 9.51/5.30      @ ^ [N2: nat] : zero_zero_nat ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8160_summable__zero,axiom,
% 9.51/5.30      ( summable_int
% 9.51/5.30      @ ^ [N2: nat] : zero_zero_int ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8161_summable__cmult__iff,axiom,
% 9.51/5.30      ! [C: complex,F: nat > complex] :
% 9.51/5.30        ( ( summable_complex
% 9.51/5.30          @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
% 9.51/5.30        = ( ( C = zero_zero_complex )
% 9.51/5.30          | ( summable_complex @ F ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8162_summable__cmult__iff,axiom,
% 9.51/5.30      ! [C: real,F: nat > real] :
% 9.51/5.30        ( ( summable_real
% 9.51/5.30          @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 9.51/5.30        = ( ( C = zero_zero_real )
% 9.51/5.30          | ( summable_real @ F ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8163_summable__divide__iff,axiom,
% 9.51/5.30      ! [F: nat > complex,C: complex] :
% 9.51/5.30        ( ( summable_complex
% 9.51/5.30          @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
% 9.51/5.30        = ( ( C = zero_zero_complex )
% 9.51/5.30          | ( summable_complex @ F ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8164_summable__divide__iff,axiom,
% 9.51/5.30      ! [F: nat > real,C: real] :
% 9.51/5.30        ( ( summable_real
% 9.51/5.30          @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
% 9.51/5.30        = ( ( C = zero_zero_real )
% 9.51/5.30          | ( summable_real @ F ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8165_summable__If__finite__set,axiom,
% 9.51/5.30      ! [A2: set_nat,F: nat > complex] :
% 9.51/5.30        ( ( finite_finite_nat @ A2 )
% 9.51/5.30       => ( summable_complex
% 9.51/5.30          @ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8166_summable__If__finite__set,axiom,
% 9.51/5.30      ! [A2: set_nat,F: nat > real] :
% 9.51/5.30        ( ( finite_finite_nat @ A2 )
% 9.51/5.30       => ( summable_real
% 9.51/5.30          @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8167_summable__If__finite__set,axiom,
% 9.51/5.30      ! [A2: set_nat,F: nat > nat] :
% 9.51/5.30        ( ( finite_finite_nat @ A2 )
% 9.51/5.30       => ( summable_nat
% 9.51/5.30          @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8168_summable__If__finite__set,axiom,
% 9.51/5.30      ! [A2: set_nat,F: nat > int] :
% 9.51/5.30        ( ( finite_finite_nat @ A2 )
% 9.51/5.30       => ( summable_int
% 9.51/5.30          @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8169_summable__If__finite,axiom,
% 9.51/5.30      ! [P: nat > $o,F: nat > complex] :
% 9.51/5.30        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.51/5.30       => ( summable_complex
% 9.51/5.30          @ ^ [R5: nat] : ( if_complex @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8170_summable__If__finite,axiom,
% 9.51/5.30      ! [P: nat > $o,F: nat > real] :
% 9.51/5.30        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.51/5.30       => ( summable_real
% 9.51/5.30          @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8171_summable__If__finite,axiom,
% 9.51/5.30      ! [P: nat > $o,F: nat > nat] :
% 9.51/5.30        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.51/5.30       => ( summable_nat
% 9.51/5.30          @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8172_summable__If__finite,axiom,
% 9.51/5.30      ! [P: nat > $o,F: nat > int] :
% 9.51/5.30        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.51/5.30       => ( summable_int
% 9.51/5.30          @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8173_summable__geometric__iff,axiom,
% 9.51/5.30      ! [C: real] :
% 9.51/5.30        ( ( summable_real @ ( power_power_real @ C ) )
% 9.51/5.30        = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8174_summable__geometric__iff,axiom,
% 9.51/5.30      ! [C: complex] :
% 9.51/5.30        ( ( summable_complex @ ( power_power_complex @ C ) )
% 9.51/5.30        = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8175_summable__comparison__test_H,axiom,
% 9.51/5.30      ! [G: nat > real,N6: nat,F: nat > real] :
% 9.51/5.30        ( ( summable_real @ G )
% 9.51/5.30       => ( ! [N3: nat] :
% 9.51/5.30              ( ( ord_less_eq_nat @ N6 @ N3 )
% 9.51/5.30             => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 9.51/5.30         => ( summable_real @ F ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8176_summable__comparison__test_H,axiom,
% 9.51/5.30      ! [G: nat > real,N6: nat,F: nat > complex] :
% 9.51/5.30        ( ( summable_real @ G )
% 9.51/5.30       => ( ! [N3: nat] :
% 9.51/5.30              ( ( ord_less_eq_nat @ N6 @ N3 )
% 9.51/5.30             => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 9.51/5.30         => ( summable_complex @ F ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8177_summable__comparison__test,axiom,
% 9.51/5.30      ! [F: nat > real,G: nat > real] :
% 9.51/5.30        ( ? [N8: nat] :
% 9.51/5.30          ! [N3: nat] :
% 9.51/5.30            ( ( ord_less_eq_nat @ N8 @ N3 )
% 9.51/5.30           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 9.51/5.30       => ( ( summable_real @ G )
% 9.51/5.30         => ( summable_real @ F ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8178_summable__comparison__test,axiom,
% 9.51/5.30      ! [F: nat > complex,G: nat > real] :
% 9.51/5.30        ( ? [N8: nat] :
% 9.51/5.30          ! [N3: nat] :
% 9.51/5.30            ( ( ord_less_eq_nat @ N8 @ N3 )
% 9.51/5.30           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 9.51/5.30       => ( ( summable_real @ G )
% 9.51/5.30         => ( summable_complex @ F ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8179_summable__const__iff,axiom,
% 9.51/5.30      ! [C: complex] :
% 9.51/5.30        ( ( summable_complex
% 9.51/5.30          @ ^ [Uu3: nat] : C )
% 9.51/5.30        = ( C = zero_zero_complex ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8180_summable__const__iff,axiom,
% 9.51/5.30      ! [C: real] :
% 9.51/5.30        ( ( summable_real
% 9.51/5.30          @ ^ [Uu3: nat] : C )
% 9.51/5.30        = ( C = zero_zero_real ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8181_powser__insidea,axiom,
% 9.51/5.30      ! [F: nat > real,X: real,Z2: real] :
% 9.51/5.30        ( ( summable_real
% 9.51/5.30          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ N2 ) ) )
% 9.51/5.30       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( real_V7735802525324610683m_real @ X ) )
% 9.51/5.30         => ( summable_real
% 9.51/5.30            @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8182_powser__insidea,axiom,
% 9.51/5.30      ! [F: nat > complex,X: complex,Z2: complex] :
% 9.51/5.30        ( ( summable_complex
% 9.51/5.30          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X @ N2 ) ) )
% 9.51/5.30       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( real_V1022390504157884413omplex @ X ) )
% 9.51/5.30         => ( summable_real
% 9.51/5.30            @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8183_suminf__le,axiom,
% 9.51/5.30      ! [F: nat > real,G: nat > real] :
% 9.51/5.30        ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 9.51/5.30       => ( ( summable_real @ F )
% 9.51/5.30         => ( ( summable_real @ G )
% 9.51/5.30           => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8184_suminf__le,axiom,
% 9.51/5.30      ! [F: nat > nat,G: nat > nat] :
% 9.51/5.30        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
% 9.51/5.30       => ( ( summable_nat @ F )
% 9.51/5.30         => ( ( summable_nat @ G )
% 9.51/5.30           => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8185_suminf__le,axiom,
% 9.51/5.30      ! [F: nat > int,G: nat > int] :
% 9.51/5.30        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
% 9.51/5.30       => ( ( summable_int @ F )
% 9.51/5.30         => ( ( summable_int @ G )
% 9.51/5.30           => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8186_summable__finite,axiom,
% 9.51/5.30      ! [N6: set_nat,F: nat > complex] :
% 9.51/5.30        ( ( finite_finite_nat @ N6 )
% 9.51/5.30       => ( ! [N3: nat] :
% 9.51/5.30              ( ~ ( member_nat @ N3 @ N6 )
% 9.51/5.30             => ( ( F @ N3 )
% 9.51/5.30                = zero_zero_complex ) )
% 9.51/5.30         => ( summable_complex @ F ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8187_summable__finite,axiom,
% 9.51/5.30      ! [N6: set_nat,F: nat > real] :
% 9.51/5.30        ( ( finite_finite_nat @ N6 )
% 9.51/5.30       => ( ! [N3: nat] :
% 9.51/5.30              ( ~ ( member_nat @ N3 @ N6 )
% 9.51/5.30             => ( ( F @ N3 )
% 9.51/5.30                = zero_zero_real ) )
% 9.51/5.30         => ( summable_real @ F ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8188_summable__finite,axiom,
% 9.51/5.30      ! [N6: set_nat,F: nat > nat] :
% 9.51/5.30        ( ( finite_finite_nat @ N6 )
% 9.51/5.30       => ( ! [N3: nat] :
% 9.51/5.30              ( ~ ( member_nat @ N3 @ N6 )
% 9.51/5.30             => ( ( F @ N3 )
% 9.51/5.30                = zero_zero_nat ) )
% 9.51/5.30         => ( summable_nat @ F ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8189_summable__finite,axiom,
% 9.51/5.30      ! [N6: set_nat,F: nat > int] :
% 9.51/5.30        ( ( finite_finite_nat @ N6 )
% 9.51/5.30       => ( ! [N3: nat] :
% 9.51/5.30              ( ~ ( member_nat @ N3 @ N6 )
% 9.51/5.30             => ( ( F @ N3 )
% 9.51/5.30                = zero_zero_int ) )
% 9.51/5.30         => ( summable_int @ F ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8190_summable__mult__D,axiom,
% 9.51/5.30      ! [C: complex,F: nat > complex] :
% 9.51/5.30        ( ( summable_complex
% 9.51/5.30          @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
% 9.51/5.30       => ( ( C != zero_zero_complex )
% 9.51/5.30         => ( summable_complex @ F ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8191_summable__mult__D,axiom,
% 9.51/5.30      ! [C: real,F: nat > real] :
% 9.51/5.30        ( ( summable_real
% 9.51/5.30          @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 9.51/5.30       => ( ( C != zero_zero_real )
% 9.51/5.30         => ( summable_real @ F ) ) ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8192_summable__zero__power,axiom,
% 9.51/5.30      summable_real @ ( power_power_real @ zero_zero_real ) ).
% 9.51/5.30  
% 9.51/5.30  thf(fact_8193_summable__zero__power,axiom,
% 9.51/5.30      summable_int @ ( power_power_int @ zero_zero_int ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8194_summable__zero__power,axiom,
% 9.51/5.31      summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8195_suminf__eq__zero__iff,axiom,
% 9.51/5.31      ! [F: nat > real] :
% 9.51/5.31        ( ( summable_real @ F )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 9.51/5.31         => ( ( ( suminf_real @ F )
% 9.51/5.31              = zero_zero_real )
% 9.51/5.31            = ( ! [N2: nat] :
% 9.51/5.31                  ( ( F @ N2 )
% 9.51/5.31                  = zero_zero_real ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8196_suminf__eq__zero__iff,axiom,
% 9.51/5.31      ! [F: nat > nat] :
% 9.51/5.31        ( ( summable_nat @ F )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.51/5.31         => ( ( ( suminf_nat @ F )
% 9.51/5.31              = zero_zero_nat )
% 9.51/5.31            = ( ! [N2: nat] :
% 9.51/5.31                  ( ( F @ N2 )
% 9.51/5.31                  = zero_zero_nat ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8197_suminf__eq__zero__iff,axiom,
% 9.51/5.31      ! [F: nat > int] :
% 9.51/5.31        ( ( summable_int @ F )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 9.51/5.31         => ( ( ( suminf_int @ F )
% 9.51/5.31              = zero_zero_int )
% 9.51/5.31            = ( ! [N2: nat] :
% 9.51/5.31                  ( ( F @ N2 )
% 9.51/5.31                  = zero_zero_int ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8198_suminf__nonneg,axiom,
% 9.51/5.31      ! [F: nat > real] :
% 9.51/5.31        ( ( summable_real @ F )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 9.51/5.31         => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8199_suminf__nonneg,axiom,
% 9.51/5.31      ! [F: nat > nat] :
% 9.51/5.31        ( ( summable_nat @ F )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.51/5.31         => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8200_suminf__nonneg,axiom,
% 9.51/5.31      ! [F: nat > int] :
% 9.51/5.31        ( ( summable_int @ F )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 9.51/5.31         => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8201_suminf__pos,axiom,
% 9.51/5.31      ! [F: nat > real] :
% 9.51/5.31        ( ( summable_real @ F )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N3 ) )
% 9.51/5.31         => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8202_suminf__pos,axiom,
% 9.51/5.31      ! [F: nat > nat] :
% 9.51/5.31        ( ( summable_nat @ F )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.51/5.31         => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8203_suminf__pos,axiom,
% 9.51/5.31      ! [F: nat > int] :
% 9.51/5.31        ( ( summable_int @ F )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N3 ) )
% 9.51/5.31         => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8204_summable__0__powser,axiom,
% 9.51/5.31      ! [F: nat > complex] :
% 9.51/5.31        ( summable_complex
% 9.51/5.31        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8205_summable__0__powser,axiom,
% 9.51/5.31      ! [F: nat > real] :
% 9.51/5.31        ( summable_real
% 9.51/5.31        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8206_summable__zero__power_H,axiom,
% 9.51/5.31      ! [F: nat > complex] :
% 9.51/5.31        ( summable_complex
% 9.51/5.31        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8207_summable__zero__power_H,axiom,
% 9.51/5.31      ! [F: nat > real] :
% 9.51/5.31        ( summable_real
% 9.51/5.31        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8208_summable__zero__power_H,axiom,
% 9.51/5.31      ! [F: nat > int] :
% 9.51/5.31        ( summable_int
% 9.51/5.31        @ ^ [N2: nat] : ( times_times_int @ ( F @ N2 ) @ ( power_power_int @ zero_zero_int @ N2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8209_summable__norm__comparison__test,axiom,
% 9.51/5.31      ! [F: nat > complex,G: nat > real] :
% 9.51/5.31        ( ? [N8: nat] :
% 9.51/5.31          ! [N3: nat] :
% 9.51/5.31            ( ( ord_less_eq_nat @ N8 @ N3 )
% 9.51/5.31           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 9.51/5.31       => ( ( summable_real @ G )
% 9.51/5.31         => ( summable_real
% 9.51/5.31            @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8210_summable__rabs__comparison__test,axiom,
% 9.51/5.31      ! [F: nat > real,G: nat > real] :
% 9.51/5.31        ( ? [N8: nat] :
% 9.51/5.31          ! [N3: nat] :
% 9.51/5.31            ( ( ord_less_eq_nat @ N8 @ N3 )
% 9.51/5.31           => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 9.51/5.31       => ( ( summable_real @ G )
% 9.51/5.31         => ( summable_real
% 9.51/5.31            @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8211_suminf__pos__iff,axiom,
% 9.51/5.31      ! [F: nat > real] :
% 9.51/5.31        ( ( summable_real @ F )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 9.51/5.31         => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
% 9.51/5.31            = ( ? [I4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8212_suminf__pos__iff,axiom,
% 9.51/5.31      ! [F: nat > nat] :
% 9.51/5.31        ( ( summable_nat @ F )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.51/5.31         => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
% 9.51/5.31            = ( ? [I4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8213_suminf__pos__iff,axiom,
% 9.51/5.31      ! [F: nat > int] :
% 9.51/5.31        ( ( summable_int @ F )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 9.51/5.31         => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
% 9.51/5.31            = ( ? [I4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I4 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8214_suminf__pos2,axiom,
% 9.51/5.31      ! [F: nat > real,I: nat] :
% 9.51/5.31        ( ( summable_real @ F )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 9.51/5.31         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.51/5.31           => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8215_suminf__pos2,axiom,
% 9.51/5.31      ! [F: nat > nat,I: nat] :
% 9.51/5.31        ( ( summable_nat @ F )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.51/5.31         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 9.51/5.31           => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8216_suminf__pos2,axiom,
% 9.51/5.31      ! [F: nat > int,I: nat] :
% 9.51/5.31        ( ( summable_int @ F )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 9.51/5.31         => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
% 9.51/5.31           => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8217_suminf__le__const,axiom,
% 9.51/5.31      ! [F: nat > int,X: int] :
% 9.51/5.31        ( ( summable_int @ F )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
% 9.51/5.31         => ( ord_less_eq_int @ ( suminf_int @ F ) @ X ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8218_suminf__le__const,axiom,
% 9.51/5.31      ! [F: nat > nat,X: nat] :
% 9.51/5.31        ( ( summable_nat @ F )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
% 9.51/5.31         => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8219_suminf__le__const,axiom,
% 9.51/5.31      ! [F: nat > real,X: real] :
% 9.51/5.31        ( ( summable_real @ F )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
% 9.51/5.31         => ( ord_less_eq_real @ ( suminf_real @ F ) @ X ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8220_powser__inside,axiom,
% 9.51/5.31      ! [F: nat > real,X: real,Z2: real] :
% 9.51/5.31        ( ( summable_real
% 9.51/5.31          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ N2 ) ) )
% 9.51/5.31       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( real_V7735802525324610683m_real @ X ) )
% 9.51/5.31         => ( summable_real
% 9.51/5.31            @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8221_powser__inside,axiom,
% 9.51/5.31      ! [F: nat > complex,X: complex,Z2: complex] :
% 9.51/5.31        ( ( summable_complex
% 9.51/5.31          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X @ N2 ) ) )
% 9.51/5.31       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( real_V1022390504157884413omplex @ X ) )
% 9.51/5.31         => ( summable_complex
% 9.51/5.31            @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8222_summableI__nonneg__bounded,axiom,
% 9.51/5.31      ! [F: nat > int,X: int] :
% 9.51/5.31        ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
% 9.51/5.31         => ( summable_int @ F ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8223_summableI__nonneg__bounded,axiom,
% 9.51/5.31      ! [F: nat > nat,X: nat] :
% 9.51/5.31        ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
% 9.51/5.31         => ( summable_nat @ F ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8224_summableI__nonneg__bounded,axiom,
% 9.51/5.31      ! [F: nat > real,X: real] :
% 9.51/5.31        ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
% 9.51/5.31         => ( summable_real @ F ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8225_bounded__imp__summable,axiom,
% 9.51/5.31      ! [A: nat > int,B2: int] :
% 9.51/5.31        ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( A @ N3 ) )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B2 )
% 9.51/5.31         => ( summable_int @ A ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8226_bounded__imp__summable,axiom,
% 9.51/5.31      ! [A: nat > nat,B2: nat] :
% 9.51/5.31        ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( A @ N3 ) )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B2 )
% 9.51/5.31         => ( summable_nat @ A ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8227_bounded__imp__summable,axiom,
% 9.51/5.31      ! [A: nat > real,B2: real] :
% 9.51/5.31        ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B2 )
% 9.51/5.31         => ( summable_real @ A ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8228_summable__geometric,axiom,
% 9.51/5.31      ! [C: real] :
% 9.51/5.31        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 9.51/5.31       => ( summable_real @ ( power_power_real @ C ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8229_summable__geometric,axiom,
% 9.51/5.31      ! [C: complex] :
% 9.51/5.31        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 9.51/5.31       => ( summable_complex @ ( power_power_complex @ C ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8230_complete__algebra__summable__geometric,axiom,
% 9.51/5.31      ! [X: real] :
% 9.51/5.31        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ one_one_real )
% 9.51/5.31       => ( summable_real @ ( power_power_real @ X ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8231_complete__algebra__summable__geometric,axiom,
% 9.51/5.31      ! [X: complex] :
% 9.51/5.31        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ one_one_real )
% 9.51/5.31       => ( summable_complex @ ( power_power_complex @ X ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8232_suminf__split__head,axiom,
% 9.51/5.31      ! [F: nat > real] :
% 9.51/5.31        ( ( summable_real @ F )
% 9.51/5.31       => ( ( suminf_real
% 9.51/5.31            @ ^ [N2: nat] : ( F @ ( suc @ N2 ) ) )
% 9.51/5.31          = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8233_the__elem__def,axiom,
% 9.51/5.31      ( the_el8326832613380209454nt_int
% 9.51/5.31      = ( ^ [X6: set_Pr958786334691620121nt_int] :
% 9.51/5.31            ( the_Pr4378521158711661632nt_int
% 9.51/5.31            @ ^ [X3: product_prod_int_int] :
% 9.51/5.31                ( X6
% 9.51/5.31                = ( insert5033312907999012233nt_int @ X3 @ bot_bo1796632182523588997nt_int ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8234_the__elem__def,axiom,
% 9.51/5.31      ( the_el221668144340439132at_nat
% 9.51/5.31      = ( ^ [X6: set_Pr4329608150637261639at_nat] :
% 9.51/5.31            ( the_Pr4604535484834969198at_nat
% 9.51/5.31            @ ^ [X3: produc3843707927480180839at_nat] :
% 9.51/5.31                ( X6
% 9.51/5.31                = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8235_the__elem__def,axiom,
% 9.51/5.31      ( the_el2281957884133575798at_nat
% 9.51/5.31      = ( ^ [X6: set_Pr1261947904930325089at_nat] :
% 9.51/5.31            ( the_Pr7557018466319803784at_nat
% 9.51/5.31            @ ^ [X3: product_prod_nat_nat] :
% 9.51/5.31                ( X6
% 9.51/5.31                = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8236_the__elem__def,axiom,
% 9.51/5.31      ( the_elem_real
% 9.51/5.31      = ( ^ [X6: set_real] :
% 9.51/5.31            ( the_real
% 9.51/5.31            @ ^ [X3: real] :
% 9.51/5.31                ( X6
% 9.51/5.31                = ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8237_the__elem__def,axiom,
% 9.51/5.31      ( the_elem_nat
% 9.51/5.31      = ( ^ [X6: set_nat] :
% 9.51/5.31            ( the_nat
% 9.51/5.31            @ ^ [X3: nat] :
% 9.51/5.31                ( X6
% 9.51/5.31                = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8238_the__elem__def,axiom,
% 9.51/5.31      ( the_elem_int
% 9.51/5.31      = ( ^ [X6: set_int] :
% 9.51/5.31            ( the_int
% 9.51/5.31            @ ^ [X3: int] :
% 9.51/5.31                ( X6
% 9.51/5.31                = ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8239_sum__le__suminf,axiom,
% 9.51/5.31      ! [F: nat > int,I6: set_nat] :
% 9.51/5.31        ( ( summable_int @ F )
% 9.51/5.31       => ( ( finite_finite_nat @ I6 )
% 9.51/5.31         => ( ! [N3: nat] :
% 9.51/5.31                ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I6 ) )
% 9.51/5.31               => ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) ) )
% 9.51/5.31           => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I6 ) @ ( suminf_int @ F ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8240_sum__le__suminf,axiom,
% 9.51/5.31      ! [F: nat > nat,I6: set_nat] :
% 9.51/5.31        ( ( summable_nat @ F )
% 9.51/5.31       => ( ( finite_finite_nat @ I6 )
% 9.51/5.31         => ( ! [N3: nat] :
% 9.51/5.31                ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I6 ) )
% 9.51/5.31               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) ) )
% 9.51/5.31           => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I6 ) @ ( suminf_nat @ F ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8241_sum__le__suminf,axiom,
% 9.51/5.31      ! [F: nat > real,I6: set_nat] :
% 9.51/5.31        ( ( summable_real @ F )
% 9.51/5.31       => ( ( finite_finite_nat @ I6 )
% 9.51/5.31         => ( ! [N3: nat] :
% 9.51/5.31                ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I6 ) )
% 9.51/5.31               => ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) ) )
% 9.51/5.31           => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I6 ) @ ( suminf_real @ F ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8242_sum__less__suminf,axiom,
% 9.51/5.31      ! [F: nat > int,N: nat] :
% 9.51/5.31        ( ( summable_int @ F )
% 9.51/5.31       => ( ! [M5: nat] :
% 9.51/5.31              ( ( ord_less_eq_nat @ N @ M5 )
% 9.51/5.31             => ( ord_less_int @ zero_zero_int @ ( F @ M5 ) ) )
% 9.51/5.31         => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8243_sum__less__suminf,axiom,
% 9.51/5.31      ! [F: nat > nat,N: nat] :
% 9.51/5.31        ( ( summable_nat @ F )
% 9.51/5.31       => ( ! [M5: nat] :
% 9.51/5.31              ( ( ord_less_eq_nat @ N @ M5 )
% 9.51/5.31             => ( ord_less_nat @ zero_zero_nat @ ( F @ M5 ) ) )
% 9.51/5.31         => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8244_sum__less__suminf,axiom,
% 9.51/5.31      ! [F: nat > real,N: nat] :
% 9.51/5.31        ( ( summable_real @ F )
% 9.51/5.31       => ( ! [M5: nat] :
% 9.51/5.31              ( ( ord_less_eq_nat @ N @ M5 )
% 9.51/5.31             => ( ord_less_real @ zero_zero_real @ ( F @ M5 ) ) )
% 9.51/5.31         => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8245_powser__split__head_I1_J,axiom,
% 9.51/5.31      ! [F: nat > complex,Z2: complex] :
% 9.51/5.31        ( ( summable_complex
% 9.51/5.31          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) )
% 9.51/5.31       => ( ( suminf_complex
% 9.51/5.31            @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) )
% 9.51/5.31          = ( plus_plus_complex @ ( F @ zero_zero_nat )
% 9.51/5.31            @ ( times_times_complex
% 9.51/5.31              @ ( suminf_complex
% 9.51/5.31                @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z2 @ N2 ) ) )
% 9.51/5.31              @ Z2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8246_powser__split__head_I1_J,axiom,
% 9.51/5.31      ! [F: nat > real,Z2: real] :
% 9.51/5.31        ( ( summable_real
% 9.51/5.31          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) )
% 9.51/5.31       => ( ( suminf_real
% 9.51/5.31            @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) )
% 9.51/5.31          = ( plus_plus_real @ ( F @ zero_zero_nat )
% 9.51/5.31            @ ( times_times_real
% 9.51/5.31              @ ( suminf_real
% 9.51/5.31                @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z2 @ N2 ) ) )
% 9.51/5.31              @ Z2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8247_powser__split__head_I2_J,axiom,
% 9.51/5.31      ! [F: nat > complex,Z2: complex] :
% 9.51/5.31        ( ( summable_complex
% 9.51/5.31          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) )
% 9.51/5.31       => ( ( times_times_complex
% 9.51/5.31            @ ( suminf_complex
% 9.51/5.31              @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z2 @ N2 ) ) )
% 9.51/5.31            @ Z2 )
% 9.51/5.31          = ( minus_minus_complex
% 9.51/5.31            @ ( suminf_complex
% 9.51/5.31              @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) )
% 9.51/5.31            @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8248_powser__split__head_I2_J,axiom,
% 9.51/5.31      ! [F: nat > real,Z2: real] :
% 9.51/5.31        ( ( summable_real
% 9.51/5.31          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) )
% 9.51/5.31       => ( ( times_times_real
% 9.51/5.31            @ ( suminf_real
% 9.51/5.31              @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z2 @ N2 ) ) )
% 9.51/5.31            @ Z2 )
% 9.51/5.31          = ( minus_minus_real
% 9.51/5.31            @ ( suminf_real
% 9.51/5.31              @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) )
% 9.51/5.31            @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8249_summable__partial__sum__bound,axiom,
% 9.51/5.31      ! [F: nat > complex,E2: real] :
% 9.51/5.31        ( ( summable_complex @ F )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.51/5.31         => ~ ! [N9: nat] :
% 9.51/5.31                ~ ! [M3: nat] :
% 9.51/5.31                    ( ( ord_less_eq_nat @ N9 @ M3 )
% 9.51/5.31                   => ! [N5: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M3 @ N5 ) ) ) @ E2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8250_summable__partial__sum__bound,axiom,
% 9.51/5.31      ! [F: nat > real,E2: real] :
% 9.51/5.31        ( ( summable_real @ F )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.51/5.31         => ~ ! [N9: nat] :
% 9.51/5.31                ~ ! [M3: nat] :
% 9.51/5.31                    ( ( ord_less_eq_nat @ N9 @ M3 )
% 9.51/5.31                   => ! [N5: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M3 @ N5 ) ) ) @ E2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8251_suminf__exist__split,axiom,
% 9.51/5.31      ! [R2: real,F: nat > real] :
% 9.51/5.31        ( ( ord_less_real @ zero_zero_real @ R2 )
% 9.51/5.31       => ( ( summable_real @ F )
% 9.51/5.31         => ? [N9: nat] :
% 9.51/5.31            ! [N5: nat] :
% 9.51/5.31              ( ( ord_less_eq_nat @ N9 @ N5 )
% 9.51/5.31             => ( ord_less_real
% 9.51/5.31                @ ( real_V7735802525324610683m_real
% 9.51/5.31                  @ ( suminf_real
% 9.51/5.31                    @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N5 ) ) ) )
% 9.51/5.31                @ R2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8252_suminf__exist__split,axiom,
% 9.51/5.31      ! [R2: real,F: nat > complex] :
% 9.51/5.31        ( ( ord_less_real @ zero_zero_real @ R2 )
% 9.51/5.31       => ( ( summable_complex @ F )
% 9.51/5.31         => ? [N9: nat] :
% 9.51/5.31            ! [N5: nat] :
% 9.51/5.31              ( ( ord_less_eq_nat @ N9 @ N5 )
% 9.51/5.31             => ( ord_less_real
% 9.51/5.31                @ ( real_V1022390504157884413omplex
% 9.51/5.31                  @ ( suminf_complex
% 9.51/5.31                    @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N5 ) ) ) )
% 9.51/5.31                @ R2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8253_summable__power__series,axiom,
% 9.51/5.31      ! [F: nat > real,Z2: real] :
% 9.51/5.31        ( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
% 9.51/5.31       => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 9.51/5.31         => ( ( ord_less_eq_real @ zero_zero_real @ Z2 )
% 9.51/5.31           => ( ( ord_less_real @ Z2 @ one_one_real )
% 9.51/5.31             => ( summable_real
% 9.51/5.31                @ ^ [I4: nat] : ( times_times_real @ ( F @ I4 ) @ ( power_power_real @ Z2 @ I4 ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8254_Abel__lemma,axiom,
% 9.51/5.31      ! [R2: real,R0: real,A: nat > complex,M4: real] :
% 9.51/5.31        ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 9.51/5.31       => ( ( ord_less_real @ R2 @ R0 )
% 9.51/5.31         => ( ! [N3: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N3 ) ) @ ( power_power_real @ R0 @ N3 ) ) @ M4 )
% 9.51/5.31           => ( summable_real
% 9.51/5.31              @ ^ [N2: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N2 ) ) @ ( power_power_real @ R2 @ N2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8255_summable__ratio__test,axiom,
% 9.51/5.31      ! [C: real,N6: nat,F: nat > real] :
% 9.51/5.31        ( ( ord_less_real @ C @ one_one_real )
% 9.51/5.31       => ( ! [N3: nat] :
% 9.51/5.31              ( ( ord_less_eq_nat @ N6 @ N3 )
% 9.51/5.31             => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) ) ) )
% 9.51/5.31         => ( summable_real @ F ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8256_summable__ratio__test,axiom,
% 9.51/5.31      ! [C: real,N6: nat,F: nat > complex] :
% 9.51/5.31        ( ( ord_less_real @ C @ one_one_real )
% 9.51/5.31       => ( ! [N3: nat] :
% 9.51/5.31              ( ( ord_less_eq_nat @ N6 @ N3 )
% 9.51/5.31             => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) )
% 9.51/5.31         => ( summable_complex @ F ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8257_sum__less__suminf2,axiom,
% 9.51/5.31      ! [F: nat > int,N: nat,I: nat] :
% 9.51/5.31        ( ( summable_int @ F )
% 9.51/5.31       => ( ! [M5: nat] :
% 9.51/5.31              ( ( ord_less_eq_nat @ N @ M5 )
% 9.51/5.31             => ( ord_less_eq_int @ zero_zero_int @ ( F @ M5 ) ) )
% 9.51/5.31         => ( ( ord_less_eq_nat @ N @ I )
% 9.51/5.31           => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
% 9.51/5.31             => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8258_sum__less__suminf2,axiom,
% 9.51/5.31      ! [F: nat > nat,N: nat,I: nat] :
% 9.51/5.31        ( ( summable_nat @ F )
% 9.51/5.31       => ( ! [M5: nat] :
% 9.51/5.31              ( ( ord_less_eq_nat @ N @ M5 )
% 9.51/5.31             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M5 ) ) )
% 9.51/5.31         => ( ( ord_less_eq_nat @ N @ I )
% 9.51/5.31           => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 9.51/5.31             => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8259_sum__less__suminf2,axiom,
% 9.51/5.31      ! [F: nat > real,N: nat,I: nat] :
% 9.51/5.31        ( ( summable_real @ F )
% 9.51/5.31       => ( ! [M5: nat] :
% 9.51/5.31              ( ( ord_less_eq_nat @ N @ M5 )
% 9.51/5.31             => ( ord_less_eq_real @ zero_zero_real @ ( F @ M5 ) ) )
% 9.51/5.31         => ( ( ord_less_eq_nat @ N @ I )
% 9.51/5.31           => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.51/5.31             => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8260_diffs__equiv,axiom,
% 9.51/5.31      ! [C: nat > complex,X: complex] :
% 9.51/5.31        ( ( summable_complex
% 9.51/5.31          @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) )
% 9.51/5.31       => ( sums_complex
% 9.51/5.31          @ ^ [N2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( C @ N2 ) ) @ ( power_power_complex @ X @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
% 9.51/5.31          @ ( suminf_complex
% 9.51/5.31            @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8261_diffs__equiv,axiom,
% 9.51/5.31      ! [C: nat > real,X: real] :
% 9.51/5.31        ( ( summable_real
% 9.51/5.31          @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) )
% 9.51/5.31       => ( sums_real
% 9.51/5.31          @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( C @ N2 ) ) @ ( power_power_real @ X @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
% 9.51/5.31          @ ( suminf_real
% 9.51/5.31            @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8262_arcsin__lt__bounded,axiom,
% 9.51/5.31      ! [Y: real] :
% 9.51/5.31        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 9.51/5.31       => ( ( ord_less_real @ Y @ one_one_real )
% 9.51/5.31         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 9.51/5.31            & ( ord_less_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8263_pochhammer__code,axiom,
% 9.51/5.31      ( comm_s2602460028002588243omplex
% 9.51/5.31      = ( ^ [A4: complex,N2: nat] :
% 9.51/5.31            ( if_complex @ ( N2 = zero_zero_nat ) @ one_one_complex
% 9.51/5.31            @ ( set_fo1517530859248394432omplex
% 9.51/5.31              @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A4 @ ( semiri8010041392384452111omplex @ O ) ) )
% 9.51/5.31              @ zero_zero_nat
% 9.51/5.31              @ ( minus_minus_nat @ N2 @ one_one_nat )
% 9.51/5.31              @ one_one_complex ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8264_pochhammer__code,axiom,
% 9.51/5.31      ( comm_s4028243227959126397er_rat
% 9.51/5.31      = ( ^ [A4: rat,N2: nat] :
% 9.51/5.31            ( if_rat @ ( N2 = zero_zero_nat ) @ one_one_rat
% 9.51/5.31            @ ( set_fo1949268297981939178at_rat
% 9.51/5.31              @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A4 @ ( semiri681578069525770553at_rat @ O ) ) )
% 9.51/5.31              @ zero_zero_nat
% 9.51/5.31              @ ( minus_minus_nat @ N2 @ one_one_nat )
% 9.51/5.31              @ one_one_rat ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8265_pochhammer__code,axiom,
% 9.51/5.31      ( comm_s4660882817536571857er_int
% 9.51/5.31      = ( ^ [A4: int,N2: nat] :
% 9.51/5.31            ( if_int @ ( N2 = zero_zero_nat ) @ one_one_int
% 9.51/5.31            @ ( set_fo2581907887559384638at_int
% 9.51/5.31              @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A4 @ ( semiri1314217659103216013at_int @ O ) ) )
% 9.51/5.31              @ zero_zero_nat
% 9.51/5.31              @ ( minus_minus_nat @ N2 @ one_one_nat )
% 9.51/5.31              @ one_one_int ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8266_pochhammer__code,axiom,
% 9.51/5.31      ( comm_s7457072308508201937r_real
% 9.51/5.31      = ( ^ [A4: real,N2: nat] :
% 9.51/5.31            ( if_real @ ( N2 = zero_zero_nat ) @ one_one_real
% 9.51/5.31            @ ( set_fo3111899725591712190t_real
% 9.51/5.31              @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A4 @ ( semiri5074537144036343181t_real @ O ) ) )
% 9.51/5.31              @ zero_zero_nat
% 9.51/5.31              @ ( minus_minus_nat @ N2 @ one_one_nat )
% 9.51/5.31              @ one_one_real ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8267_pochhammer__code,axiom,
% 9.51/5.31      ( comm_s4663373288045622133er_nat
% 9.51/5.31      = ( ^ [A4: nat,N2: nat] :
% 9.51/5.31            ( if_nat @ ( N2 = zero_zero_nat ) @ one_one_nat
% 9.51/5.31            @ ( set_fo2584398358068434914at_nat
% 9.51/5.31              @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A4 @ ( semiri1316708129612266289at_nat @ O ) ) )
% 9.51/5.31              @ zero_zero_nat
% 9.51/5.31              @ ( minus_minus_nat @ N2 @ one_one_nat )
% 9.51/5.31              @ one_one_nat ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8268_or__nat__unfold,axiom,
% 9.51/5.31      ( bit_se1412395901928357646or_nat
% 9.51/5.31      = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M2 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8269_Cauchy__iff2,axiom,
% 9.51/5.31      ( topolo4055970368930404560y_real
% 9.51/5.31      = ( ^ [X6: nat > real] :
% 9.51/5.31          ! [J3: nat] :
% 9.51/5.31          ? [M9: nat] :
% 9.51/5.31          ! [M2: nat] :
% 9.51/5.31            ( ( ord_less_eq_nat @ M9 @ M2 )
% 9.51/5.31           => ! [N2: nat] :
% 9.51/5.31                ( ( ord_less_eq_nat @ M9 @ N2 )
% 9.51/5.31               => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8270_or_Oleft__neutral,axiom,
% 9.51/5.31      ! [A: nat] :
% 9.51/5.31        ( ( bit_se1412395901928357646or_nat @ zero_zero_nat @ A )
% 9.51/5.31        = A ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8271_or_Oleft__neutral,axiom,
% 9.51/5.31      ! [A: int] :
% 9.51/5.31        ( ( bit_se1409905431419307370or_int @ zero_zero_int @ A )
% 9.51/5.31        = A ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8272_or_Oright__neutral,axiom,
% 9.51/5.31      ! [A: nat] :
% 9.51/5.31        ( ( bit_se1412395901928357646or_nat @ A @ zero_zero_nat )
% 9.51/5.31        = A ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8273_or_Oright__neutral,axiom,
% 9.51/5.31      ! [A: int] :
% 9.51/5.31        ( ( bit_se1409905431419307370or_int @ A @ zero_zero_int )
% 9.51/5.31        = A ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8274_The__split__eq,axiom,
% 9.51/5.31      ! [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
% 9.51/5.31        ( ( the_Pr8210177043389155639nteger
% 9.51/5.31          @ ( produc127349428274296955eger_o
% 9.51/5.31            @ ^ [X9: code_integer > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
% 9.51/5.31                ( ( X = X9 )
% 9.51/5.31                & ( Y = Y6 ) ) ) )
% 9.51/5.31        = ( produc6137756002093451184nteger @ X @ Y ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8275_The__split__eq,axiom,
% 9.51/5.31      ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
% 9.51/5.31        ( ( the_Pr6653488032121699663nteger
% 9.51/5.31          @ ( produc6253627499356882019eger_o
% 9.51/5.31            @ ^ [X9: produc6241069584506657477e_term > option6357759511663192854e_term,Y6: produc8923325533196201883nteger] :
% 9.51/5.31                ( ( X = X9 )
% 9.51/5.31                & ( Y = Y6 ) ) ) )
% 9.51/5.31        = ( produc8603105652947943368nteger @ X @ Y ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8276_The__split__eq,axiom,
% 9.51/5.31      ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] :
% 9.51/5.31        ( ( the_Pr5445864913131713084nt_int
% 9.51/5.31          @ ( produc1573362020775583542_int_o
% 9.51/5.31            @ ^ [X9: produc8551481072490612790e_term > option6357759511663192854e_term,Y6: product_prod_int_int] :
% 9.51/5.31                ( ( X = X9 )
% 9.51/5.31                & ( Y = Y6 ) ) ) )
% 9.51/5.31        = ( produc5700946648718959541nt_int @ X @ Y ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8277_The__split__eq,axiom,
% 9.51/5.31      ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int] :
% 9.51/5.31        ( ( the_Pr6882841213465913158nt_int
% 9.51/5.31          @ ( produc2558449545302689196_int_o
% 9.51/5.31            @ ^ [X9: int > option6357759511663192854e_term,Y6: product_prod_int_int] :
% 9.51/5.31                ( ( X = X9 )
% 9.51/5.31                & ( Y = Y6 ) ) ) )
% 9.51/5.31        = ( produc4305682042979456191nt_int @ X @ Y ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8278_The__split__eq,axiom,
% 9.51/5.31      ! [X: nat,Y: nat] :
% 9.51/5.31        ( ( the_Pr7557018466319803784at_nat
% 9.51/5.31          @ ( produc6081775807080527818_nat_o
% 9.51/5.31            @ ^ [X9: nat,Y6: nat] :
% 9.51/5.31                ( ( X = X9 )
% 9.51/5.31                & ( Y = Y6 ) ) ) )
% 9.51/5.31        = ( product_Pair_nat_nat @ X @ Y ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8279_The__split__eq,axiom,
% 9.51/5.31      ! [X: int,Y: int] :
% 9.51/5.31        ( ( the_Pr4378521158711661632nt_int
% 9.51/5.31          @ ( produc4947309494688390418_int_o
% 9.51/5.31            @ ^ [X9: int,Y6: int] :
% 9.51/5.31                ( ( X = X9 )
% 9.51/5.31                & ( Y = Y6 ) ) ) )
% 9.51/5.31        = ( product_Pair_int_int @ X @ Y ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8280_pochhammer__0,axiom,
% 9.51/5.31      ! [A: complex] :
% 9.51/5.31        ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
% 9.51/5.31        = one_one_complex ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8281_pochhammer__0,axiom,
% 9.51/5.31      ! [A: real] :
% 9.51/5.31        ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
% 9.51/5.31        = one_one_real ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8282_pochhammer__0,axiom,
% 9.51/5.31      ! [A: rat] :
% 9.51/5.31        ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
% 9.51/5.31        = one_one_rat ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8283_pochhammer__0,axiom,
% 9.51/5.31      ! [A: nat] :
% 9.51/5.31        ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
% 9.51/5.31        = one_one_nat ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8284_pochhammer__0,axiom,
% 9.51/5.31      ! [A: int] :
% 9.51/5.31        ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
% 9.51/5.31        = one_one_int ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8285_or__nat__numerals_I4_J,axiom,
% 9.51/5.31      ! [X: num] :
% 9.51/5.31        ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 9.51/5.31        = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8286_or__nat__numerals_I2_J,axiom,
% 9.51/5.31      ! [Y: num] :
% 9.51/5.31        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 9.51/5.31        = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8287_or__nat__numerals_I3_J,axiom,
% 9.51/5.31      ! [X: num] :
% 9.51/5.31        ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 9.51/5.31        = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8288_or__nat__numerals_I1_J,axiom,
% 9.51/5.31      ! [Y: num] :
% 9.51/5.31        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 9.51/5.31        = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8289_or__eq__0__iff,axiom,
% 9.51/5.31      ! [A: nat,B: nat] :
% 9.51/5.31        ( ( ( bit_se1412395901928357646or_nat @ A @ B )
% 9.51/5.31          = zero_zero_nat )
% 9.51/5.31        = ( ( A = zero_zero_nat )
% 9.51/5.31          & ( B = zero_zero_nat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8290_or__eq__0__iff,axiom,
% 9.51/5.31      ! [A: int,B: int] :
% 9.51/5.31        ( ( ( bit_se1409905431419307370or_int @ A @ B )
% 9.51/5.31          = zero_zero_int )
% 9.51/5.31        = ( ( A = zero_zero_int )
% 9.51/5.31          & ( B = zero_zero_int ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8291_bit_Odisj__zero__right,axiom,
% 9.51/5.31      ! [X: int] :
% 9.51/5.31        ( ( bit_se1409905431419307370or_int @ X @ zero_zero_int )
% 9.51/5.31        = X ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8292_pochhammer__pos,axiom,
% 9.51/5.31      ! [X: real,N: nat] :
% 9.51/5.31        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.31       => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8293_pochhammer__pos,axiom,
% 9.51/5.31      ! [X: rat,N: nat] :
% 9.51/5.31        ( ( ord_less_rat @ zero_zero_rat @ X )
% 9.51/5.31       => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8294_pochhammer__pos,axiom,
% 9.51/5.31      ! [X: nat,N: nat] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ X )
% 9.51/5.31       => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8295_pochhammer__pos,axiom,
% 9.51/5.31      ! [X: int,N: nat] :
% 9.51/5.31        ( ( ord_less_int @ zero_zero_int @ X )
% 9.51/5.31       => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8296_pochhammer__neq__0__mono,axiom,
% 9.51/5.31      ! [A: complex,M: nat,N: nat] :
% 9.51/5.31        ( ( ( comm_s2602460028002588243omplex @ A @ M )
% 9.51/5.31         != zero_zero_complex )
% 9.51/5.31       => ( ( ord_less_eq_nat @ N @ M )
% 9.51/5.31         => ( ( comm_s2602460028002588243omplex @ A @ N )
% 9.51/5.31           != zero_zero_complex ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8297_pochhammer__neq__0__mono,axiom,
% 9.51/5.31      ! [A: real,M: nat,N: nat] :
% 9.51/5.31        ( ( ( comm_s7457072308508201937r_real @ A @ M )
% 9.51/5.31         != zero_zero_real )
% 9.51/5.31       => ( ( ord_less_eq_nat @ N @ M )
% 9.51/5.31         => ( ( comm_s7457072308508201937r_real @ A @ N )
% 9.51/5.31           != zero_zero_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8298_pochhammer__neq__0__mono,axiom,
% 9.51/5.31      ! [A: rat,M: nat,N: nat] :
% 9.51/5.31        ( ( ( comm_s4028243227959126397er_rat @ A @ M )
% 9.51/5.31         != zero_zero_rat )
% 9.51/5.31       => ( ( ord_less_eq_nat @ N @ M )
% 9.51/5.31         => ( ( comm_s4028243227959126397er_rat @ A @ N )
% 9.51/5.31           != zero_zero_rat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8299_pochhammer__eq__0__mono,axiom,
% 9.51/5.31      ! [A: complex,N: nat,M: nat] :
% 9.51/5.31        ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 9.51/5.31          = zero_zero_complex )
% 9.51/5.31       => ( ( ord_less_eq_nat @ N @ M )
% 9.51/5.31         => ( ( comm_s2602460028002588243omplex @ A @ M )
% 9.51/5.31            = zero_zero_complex ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8300_pochhammer__eq__0__mono,axiom,
% 9.51/5.31      ! [A: real,N: nat,M: nat] :
% 9.51/5.31        ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 9.51/5.31          = zero_zero_real )
% 9.51/5.31       => ( ( ord_less_eq_nat @ N @ M )
% 9.51/5.31         => ( ( comm_s7457072308508201937r_real @ A @ M )
% 9.51/5.31            = zero_zero_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8301_pochhammer__eq__0__mono,axiom,
% 9.51/5.31      ! [A: rat,N: nat,M: nat] :
% 9.51/5.31        ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 9.51/5.31          = zero_zero_rat )
% 9.51/5.31       => ( ( ord_less_eq_nat @ N @ M )
% 9.51/5.31         => ( ( comm_s4028243227959126397er_rat @ A @ M )
% 9.51/5.31            = zero_zero_rat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8302_pochhammer__nonneg,axiom,
% 9.51/5.31      ! [X: real,N: nat] :
% 9.51/5.31        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.31       => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8303_pochhammer__nonneg,axiom,
% 9.51/5.31      ! [X: rat,N: nat] :
% 9.51/5.31        ( ( ord_less_rat @ zero_zero_rat @ X )
% 9.51/5.31       => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8304_pochhammer__nonneg,axiom,
% 9.51/5.31      ! [X: nat,N: nat] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ X )
% 9.51/5.31       => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8305_pochhammer__nonneg,axiom,
% 9.51/5.31      ! [X: int,N: nat] :
% 9.51/5.31        ( ( ord_less_int @ zero_zero_int @ X )
% 9.51/5.31       => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8306_pochhammer__0__left,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( ( N = zero_zero_nat )
% 9.51/5.31         => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 9.51/5.31            = one_one_complex ) )
% 9.51/5.31        & ( ( N != zero_zero_nat )
% 9.51/5.31         => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 9.51/5.31            = zero_zero_complex ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8307_pochhammer__0__left,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( ( N = zero_zero_nat )
% 9.51/5.31         => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 9.51/5.31            = one_one_real ) )
% 9.51/5.31        & ( ( N != zero_zero_nat )
% 9.51/5.31         => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 9.51/5.31            = zero_zero_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8308_pochhammer__0__left,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( ( N = zero_zero_nat )
% 9.51/5.31         => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 9.51/5.31            = one_one_rat ) )
% 9.51/5.31        & ( ( N != zero_zero_nat )
% 9.51/5.31         => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 9.51/5.31            = zero_zero_rat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8309_pochhammer__0__left,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( ( N = zero_zero_nat )
% 9.51/5.31         => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 9.51/5.31            = one_one_nat ) )
% 9.51/5.31        & ( ( N != zero_zero_nat )
% 9.51/5.31         => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 9.51/5.31            = zero_zero_nat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8310_pochhammer__0__left,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( ( N = zero_zero_nat )
% 9.51/5.31         => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 9.51/5.31            = one_one_int ) )
% 9.51/5.31        & ( ( N != zero_zero_nat )
% 9.51/5.31         => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 9.51/5.31            = zero_zero_int ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8311_bit_Ocomplement__unique,axiom,
% 9.51/5.31      ! [A: code_integer,X: code_integer,Y: code_integer] :
% 9.51/5.31        ( ( ( bit_se3949692690581998587nteger @ A @ X )
% 9.51/5.31          = zero_z3403309356797280102nteger )
% 9.51/5.31       => ( ( ( bit_se1080825931792720795nteger @ A @ X )
% 9.51/5.31            = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.51/5.31         => ( ( ( bit_se3949692690581998587nteger @ A @ Y )
% 9.51/5.31              = zero_z3403309356797280102nteger )
% 9.51/5.31           => ( ( ( bit_se1080825931792720795nteger @ A @ Y )
% 9.51/5.31                = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.51/5.31             => ( X = Y ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8312_bit_Ocomplement__unique,axiom,
% 9.51/5.31      ! [A: int,X: int,Y: int] :
% 9.51/5.31        ( ( ( bit_se725231765392027082nd_int @ A @ X )
% 9.51/5.31          = zero_zero_int )
% 9.51/5.31       => ( ( ( bit_se1409905431419307370or_int @ A @ X )
% 9.51/5.31            = ( uminus_uminus_int @ one_one_int ) )
% 9.51/5.31         => ( ( ( bit_se725231765392027082nd_int @ A @ Y )
% 9.51/5.31              = zero_zero_int )
% 9.51/5.31           => ( ( ( bit_se1409905431419307370or_int @ A @ Y )
% 9.51/5.31                = ( uminus_uminus_int @ one_one_int ) )
% 9.51/5.31             => ( X = Y ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8313_pochhammer__eq__0__iff,axiom,
% 9.51/5.31      ! [A: complex,N: nat] :
% 9.51/5.31        ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 9.51/5.31          = zero_zero_complex )
% 9.51/5.31        = ( ? [K3: nat] :
% 9.51/5.31              ( ( ord_less_nat @ K3 @ N )
% 9.51/5.31              & ( A
% 9.51/5.31                = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8314_pochhammer__eq__0__iff,axiom,
% 9.51/5.31      ! [A: rat,N: nat] :
% 9.51/5.31        ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 9.51/5.31          = zero_zero_rat )
% 9.51/5.31        = ( ? [K3: nat] :
% 9.51/5.31              ( ( ord_less_nat @ K3 @ N )
% 9.51/5.31              & ( A
% 9.51/5.31                = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8315_pochhammer__eq__0__iff,axiom,
% 9.51/5.31      ! [A: real,N: nat] :
% 9.51/5.31        ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 9.51/5.31          = zero_zero_real )
% 9.51/5.31        = ( ? [K3: nat] :
% 9.51/5.31              ( ( ord_less_nat @ K3 @ N )
% 9.51/5.31              & ( A
% 9.51/5.31                = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8316_pochhammer__of__nat__eq__0__iff,axiom,
% 9.51/5.31      ! [N: nat,K: nat] :
% 9.51/5.31        ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 9.51/5.31          = zero_zero_complex )
% 9.51/5.31        = ( ord_less_nat @ N @ K ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8317_pochhammer__of__nat__eq__0__iff,axiom,
% 9.51/5.31      ! [N: nat,K: nat] :
% 9.51/5.31        ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 9.51/5.31          = zero_zero_rat )
% 9.51/5.31        = ( ord_less_nat @ N @ K ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8318_pochhammer__of__nat__eq__0__iff,axiom,
% 9.51/5.31      ! [N: nat,K: nat] :
% 9.51/5.31        ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 9.51/5.31          = zero_z3403309356797280102nteger )
% 9.51/5.31        = ( ord_less_nat @ N @ K ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8319_pochhammer__of__nat__eq__0__iff,axiom,
% 9.51/5.31      ! [N: nat,K: nat] :
% 9.51/5.31        ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 9.51/5.31          = zero_zero_int )
% 9.51/5.31        = ( ord_less_nat @ N @ K ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8320_pochhammer__of__nat__eq__0__iff,axiom,
% 9.51/5.31      ! [N: nat,K: nat] :
% 9.51/5.31        ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 9.51/5.31          = zero_zero_real )
% 9.51/5.31        = ( ord_less_nat @ N @ K ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8321_pochhammer__of__nat__eq__0__lemma,axiom,
% 9.51/5.31      ! [N: nat,K: nat] :
% 9.51/5.31        ( ( ord_less_nat @ N @ K )
% 9.51/5.31       => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 9.51/5.31          = zero_zero_complex ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8322_pochhammer__of__nat__eq__0__lemma,axiom,
% 9.51/5.31      ! [N: nat,K: nat] :
% 9.51/5.31        ( ( ord_less_nat @ N @ K )
% 9.51/5.31       => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 9.51/5.31          = zero_zero_rat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8323_pochhammer__of__nat__eq__0__lemma,axiom,
% 9.51/5.31      ! [N: nat,K: nat] :
% 9.51/5.31        ( ( ord_less_nat @ N @ K )
% 9.51/5.31       => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 9.51/5.31          = zero_z3403309356797280102nteger ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8324_pochhammer__of__nat__eq__0__lemma,axiom,
% 9.51/5.31      ! [N: nat,K: nat] :
% 9.51/5.31        ( ( ord_less_nat @ N @ K )
% 9.51/5.31       => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 9.51/5.31          = zero_zero_int ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8325_pochhammer__of__nat__eq__0__lemma,axiom,
% 9.51/5.31      ! [N: nat,K: nat] :
% 9.51/5.31        ( ( ord_less_nat @ N @ K )
% 9.51/5.31       => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 9.51/5.31          = zero_zero_real ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8326_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 9.51/5.31      ! [K: nat,N: nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.31       => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 9.51/5.31         != zero_zero_complex ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8327_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 9.51/5.31      ! [K: nat,N: nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.31       => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 9.51/5.31         != zero_zero_rat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8328_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 9.51/5.31      ! [K: nat,N: nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.31       => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 9.51/5.31         != zero_z3403309356797280102nteger ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8329_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 9.51/5.31      ! [K: nat,N: nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.31       => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 9.51/5.31         != zero_zero_int ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8330_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 9.51/5.31      ! [K: nat,N: nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.31       => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 9.51/5.31         != zero_zero_real ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8331_arcsin__less__arcsin,axiom,
% 9.51/5.31      ! [X: real,Y: real] :
% 9.51/5.31        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.51/5.31       => ( ( ord_less_real @ X @ Y )
% 9.51/5.31         => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.51/5.31           => ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8332_arcsin__less__mono,axiom,
% 9.51/5.31      ! [X: real,Y: real] :
% 9.51/5.31        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.51/5.31       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 9.51/5.31         => ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
% 9.51/5.31            = ( ord_less_real @ X @ Y ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8333_pochhammer__product,axiom,
% 9.51/5.31      ! [M: nat,N: nat,Z2: complex] :
% 9.51/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.31       => ( ( comm_s2602460028002588243omplex @ Z2 @ N )
% 9.51/5.31          = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8334_pochhammer__product,axiom,
% 9.51/5.31      ! [M: nat,N: nat,Z2: rat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.31       => ( ( comm_s4028243227959126397er_rat @ Z2 @ N )
% 9.51/5.31          = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8335_pochhammer__product,axiom,
% 9.51/5.31      ! [M: nat,N: nat,Z2: nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.31       => ( ( comm_s4663373288045622133er_nat @ Z2 @ N )
% 9.51/5.31          = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8336_pochhammer__product,axiom,
% 9.51/5.31      ! [M: nat,N: nat,Z2: int] :
% 9.51/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.31       => ( ( comm_s4660882817536571857er_int @ Z2 @ N )
% 9.51/5.31          = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8337_pochhammer__product,axiom,
% 9.51/5.31      ! [M: nat,N: nat,Z2: real] :
% 9.51/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.31       => ( ( comm_s7457072308508201937r_real @ Z2 @ N )
% 9.51/5.31          = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8338_cos__arcsin__nonzero,axiom,
% 9.51/5.31      ! [X: real] :
% 9.51/5.31        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.51/5.31       => ( ( ord_less_real @ X @ one_one_real )
% 9.51/5.31         => ( ( cos_real @ ( arcsin @ X ) )
% 9.51/5.31           != zero_zero_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8339_Cauchy__iff,axiom,
% 9.51/5.31      ( topolo6517432010174082258omplex
% 9.51/5.31      = ( ^ [X6: nat > complex] :
% 9.51/5.31          ! [E3: real] :
% 9.51/5.31            ( ( ord_less_real @ zero_zero_real @ E3 )
% 9.51/5.31           => ? [M9: nat] :
% 9.51/5.31              ! [M2: nat] :
% 9.51/5.31                ( ( ord_less_eq_nat @ M9 @ M2 )
% 9.51/5.31               => ! [N2: nat] :
% 9.51/5.31                    ( ( ord_less_eq_nat @ M9 @ N2 )
% 9.51/5.31                   => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) ) @ E3 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8340_Cauchy__iff,axiom,
% 9.51/5.31      ( topolo4055970368930404560y_real
% 9.51/5.31      = ( ^ [X6: nat > real] :
% 9.51/5.31          ! [E3: real] :
% 9.51/5.31            ( ( ord_less_real @ zero_zero_real @ E3 )
% 9.51/5.31           => ? [M9: nat] :
% 9.51/5.31              ! [M2: nat] :
% 9.51/5.31                ( ( ord_less_eq_nat @ M9 @ M2 )
% 9.51/5.31               => ! [N2: nat] :
% 9.51/5.31                    ( ( ord_less_eq_nat @ M9 @ N2 )
% 9.51/5.31                   => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) ) @ E3 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8341_CauchyI,axiom,
% 9.51/5.31      ! [X8: nat > complex] :
% 9.51/5.31        ( ! [E: real] :
% 9.51/5.31            ( ( ord_less_real @ zero_zero_real @ E )
% 9.51/5.31           => ? [M10: nat] :
% 9.51/5.31              ! [M5: nat] :
% 9.51/5.31                ( ( ord_less_eq_nat @ M10 @ M5 )
% 9.51/5.31               => ! [N3: nat] :
% 9.51/5.31                    ( ( ord_less_eq_nat @ M10 @ N3 )
% 9.51/5.31                   => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) ) @ E ) ) ) )
% 9.51/5.31       => ( topolo6517432010174082258omplex @ X8 ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8342_CauchyI,axiom,
% 9.51/5.31      ! [X8: nat > real] :
% 9.51/5.31        ( ! [E: real] :
% 9.51/5.31            ( ( ord_less_real @ zero_zero_real @ E )
% 9.51/5.31           => ? [M10: nat] :
% 9.51/5.31              ! [M5: nat] :
% 9.51/5.31                ( ( ord_less_eq_nat @ M10 @ M5 )
% 9.51/5.31               => ! [N3: nat] :
% 9.51/5.31                    ( ( ord_less_eq_nat @ M10 @ N3 )
% 9.51/5.31                   => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) ) @ E ) ) ) )
% 9.51/5.31       => ( topolo4055970368930404560y_real @ X8 ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8343_CauchyD,axiom,
% 9.51/5.31      ! [X8: nat > complex,E2: real] :
% 9.51/5.31        ( ( topolo6517432010174082258omplex @ X8 )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.51/5.31         => ? [M8: nat] :
% 9.51/5.31            ! [M3: nat] :
% 9.51/5.31              ( ( ord_less_eq_nat @ M8 @ M3 )
% 9.51/5.31             => ! [N5: nat] :
% 9.51/5.31                  ( ( ord_less_eq_nat @ M8 @ N5 )
% 9.51/5.31                 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X8 @ M3 ) @ ( X8 @ N5 ) ) ) @ E2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8344_CauchyD,axiom,
% 9.51/5.31      ! [X8: nat > real,E2: real] :
% 9.51/5.31        ( ( topolo4055970368930404560y_real @ X8 )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.51/5.31         => ? [M8: nat] :
% 9.51/5.31            ! [M3: nat] :
% 9.51/5.31              ( ( ord_less_eq_nat @ M8 @ M3 )
% 9.51/5.31             => ! [N5: nat] :
% 9.51/5.31                  ( ( ord_less_eq_nat @ M8 @ N5 )
% 9.51/5.31                 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X8 @ M3 ) @ ( X8 @ N5 ) ) ) @ E2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8345_floor__real__def,axiom,
% 9.51/5.31      ( archim6058952711729229775r_real
% 9.51/5.31      = ( ^ [X3: real] :
% 9.51/5.31            ( the_int
% 9.51/5.31            @ ^ [Z3: int] :
% 9.51/5.31                ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X3 )
% 9.51/5.31                & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8346_termdiff__converges,axiom,
% 9.51/5.31      ! [X: real,K4: real,C: nat > real] :
% 9.51/5.31        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ K4 )
% 9.51/5.31       => ( ! [X4: real] :
% 9.51/5.31              ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ K4 )
% 9.51/5.31             => ( summable_real
% 9.51/5.31                @ ^ [N2: nat] : ( times_times_real @ ( C @ N2 ) @ ( power_power_real @ X4 @ N2 ) ) ) )
% 9.51/5.31         => ( summable_real
% 9.51/5.31            @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8347_termdiff__converges,axiom,
% 9.51/5.31      ! [X: complex,K4: real,C: nat > complex] :
% 9.51/5.31        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ K4 )
% 9.51/5.31       => ( ! [X4: complex] :
% 9.51/5.31              ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ K4 )
% 9.51/5.31             => ( summable_complex
% 9.51/5.31                @ ^ [N2: nat] : ( times_times_complex @ ( C @ N2 ) @ ( power_power_complex @ X4 @ N2 ) ) ) )
% 9.51/5.31         => ( summable_complex
% 9.51/5.31            @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8348_Suc__0__or__eq,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.51/5.31        = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8349_or__Suc__0__eq,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.51/5.31        = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8350_pochhammer__times__pochhammer__half,axiom,
% 9.51/5.31      ! [Z2: complex,N: nat] :
% 9.51/5.31        ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ ( suc @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 9.51/5.31        = ( groups6464643781859351333omplex
% 9.51/5.31          @ ^ [K3: nat] : ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 9.51/5.31          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8351_pochhammer__times__pochhammer__half,axiom,
% 9.51/5.31      ! [Z2: rat,N: nat] :
% 9.51/5.31        ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 9.51/5.31        = ( groups73079841787564623at_rat
% 9.51/5.31          @ ^ [K3: nat] : ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 9.51/5.31          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8352_pochhammer__times__pochhammer__half,axiom,
% 9.51/5.31      ! [Z2: real,N: nat] :
% 9.51/5.31        ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ ( suc @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 9.51/5.31        = ( groups129246275422532515t_real
% 9.51/5.31          @ ^ [K3: nat] : ( plus_plus_real @ Z2 @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.51/5.31          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8353_floor__rat__def,axiom,
% 9.51/5.31      ( archim3151403230148437115or_rat
% 9.51/5.31      = ( ^ [X3: rat] :
% 9.51/5.31            ( the_int
% 9.51/5.31            @ ^ [Z3: int] :
% 9.51/5.31                ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X3 )
% 9.51/5.31                & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8354_sum__power2,axiom,
% 9.51/5.31      ! [K: nat] :
% 9.51/5.31        ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 9.51/5.31        = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8355_VEBT_Osize_I3_J,axiom,
% 9.51/5.31      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 9.51/5.31        ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 9.51/5.31        = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8356_is__singleton__the__elem,axiom,
% 9.51/5.31      ( is_sin2937591304547752795at_nat
% 9.51/5.31      = ( ^ [A6: set_Pr4329608150637261639at_nat] :
% 9.51/5.31            ( A6
% 9.51/5.31            = ( insert9069300056098147895at_nat @ ( the_el221668144340439132at_nat @ A6 ) @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8357_is__singleton__the__elem,axiom,
% 9.51/5.31      ( is_sin2850979758926227957at_nat
% 9.51/5.31      = ( ^ [A6: set_Pr1261947904930325089at_nat] :
% 9.51/5.31            ( A6
% 9.51/5.31            = ( insert8211810215607154385at_nat @ ( the_el2281957884133575798at_nat @ A6 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8358_is__singleton__the__elem,axiom,
% 9.51/5.31      ( is_singleton_real
% 9.51/5.31      = ( ^ [A6: set_real] :
% 9.51/5.31            ( A6
% 9.51/5.31            = ( insert_real @ ( the_elem_real @ A6 ) @ bot_bot_set_real ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8359_is__singleton__the__elem,axiom,
% 9.51/5.31      ( is_singleton_nat
% 9.51/5.31      = ( ^ [A6: set_nat] :
% 9.51/5.31            ( A6
% 9.51/5.31            = ( insert_nat @ ( the_elem_nat @ A6 ) @ bot_bot_set_nat ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8360_is__singleton__the__elem,axiom,
% 9.51/5.31      ( is_singleton_int
% 9.51/5.31      = ( ^ [A6: set_int] :
% 9.51/5.31            ( A6
% 9.51/5.31            = ( insert_int @ ( the_elem_int @ A6 ) @ bot_bot_set_int ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8361_or__negative__int__iff,axiom,
% 9.51/5.31      ! [K: int,L: int] :
% 9.51/5.31        ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
% 9.51/5.31        = ( ( ord_less_int @ K @ zero_zero_int )
% 9.51/5.31          | ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8362_finite__atLeastLessThan,axiom,
% 9.51/5.31      ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8363_atLeastLessThan__iff,axiom,
% 9.51/5.31      ! [I: set_nat,L: set_nat,U: set_nat] :
% 9.51/5.31        ( ( member_set_nat @ I @ ( set_or3540276404033026485et_nat @ L @ U ) )
% 9.51/5.31        = ( ( ord_less_eq_set_nat @ L @ I )
% 9.51/5.31          & ( ord_less_set_nat @ I @ U ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8364_atLeastLessThan__iff,axiom,
% 9.51/5.31      ! [I: real,L: real,U: real] :
% 9.51/5.31        ( ( member_real @ I @ ( set_or66887138388493659n_real @ L @ U ) )
% 9.51/5.31        = ( ( ord_less_eq_real @ L @ I )
% 9.51/5.31          & ( ord_less_real @ I @ U ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8365_atLeastLessThan__iff,axiom,
% 9.51/5.31      ! [I: set_int,L: set_int,U: set_int] :
% 9.51/5.31        ( ( member_set_int @ I @ ( set_or8585797421378605585et_int @ L @ U ) )
% 9.51/5.31        = ( ( ord_less_eq_set_int @ L @ I )
% 9.51/5.31          & ( ord_less_set_int @ I @ U ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8366_atLeastLessThan__iff,axiom,
% 9.51/5.31      ! [I: rat,L: rat,U: rat] :
% 9.51/5.31        ( ( member_rat @ I @ ( set_or4029947393144176647an_rat @ L @ U ) )
% 9.51/5.31        = ( ( ord_less_eq_rat @ L @ I )
% 9.51/5.31          & ( ord_less_rat @ I @ U ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8367_atLeastLessThan__iff,axiom,
% 9.51/5.31      ! [I: num,L: num,U: num] :
% 9.51/5.31        ( ( member_num @ I @ ( set_or1222409239386451017an_num @ L @ U ) )
% 9.51/5.31        = ( ( ord_less_eq_num @ L @ I )
% 9.51/5.31          & ( ord_less_num @ I @ U ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8368_atLeastLessThan__iff,axiom,
% 9.51/5.31      ! [I: nat,L: nat,U: nat] :
% 9.51/5.31        ( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
% 9.51/5.31        = ( ( ord_less_eq_nat @ L @ I )
% 9.51/5.31          & ( ord_less_nat @ I @ U ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8369_atLeastLessThan__iff,axiom,
% 9.51/5.31      ! [I: int,L: int,U: int] :
% 9.51/5.31        ( ( member_int @ I @ ( set_or4662586982721622107an_int @ L @ U ) )
% 9.51/5.31        = ( ( ord_less_eq_int @ L @ I )
% 9.51/5.31          & ( ord_less_int @ I @ U ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8370_prod__zero__iff,axiom,
% 9.51/5.31      ! [A2: set_nat,F: nat > complex] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( ( groups6464643781859351333omplex @ F @ A2 )
% 9.51/5.31            = zero_zero_complex )
% 9.51/5.31          = ( ? [X3: nat] :
% 9.51/5.31                ( ( member_nat @ X3 @ A2 )
% 9.51/5.31                & ( ( F @ X3 )
% 9.51/5.31                  = zero_zero_complex ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8371_prod__zero__iff,axiom,
% 9.51/5.31      ! [A2: set_int,F: int > complex] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ( ( groups7440179247065528705omplex @ F @ A2 )
% 9.51/5.31            = zero_zero_complex )
% 9.51/5.31          = ( ? [X3: int] :
% 9.51/5.31                ( ( member_int @ X3 @ A2 )
% 9.51/5.31                & ( ( F @ X3 )
% 9.51/5.31                  = zero_zero_complex ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8372_prod__zero__iff,axiom,
% 9.51/5.31      ! [A2: set_complex,F: complex > complex] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ A2 )
% 9.51/5.31       => ( ( ( groups3708469109370488835omplex @ F @ A2 )
% 9.51/5.31            = zero_zero_complex )
% 9.51/5.31          = ( ? [X3: complex] :
% 9.51/5.31                ( ( member_complex @ X3 @ A2 )
% 9.51/5.31                & ( ( F @ X3 )
% 9.51/5.31                  = zero_zero_complex ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8373_prod__zero__iff,axiom,
% 9.51/5.31      ! [A2: set_nat,F: nat > real] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( ( groups129246275422532515t_real @ F @ A2 )
% 9.51/5.31            = zero_zero_real )
% 9.51/5.31          = ( ? [X3: nat] :
% 9.51/5.31                ( ( member_nat @ X3 @ A2 )
% 9.51/5.31                & ( ( F @ X3 )
% 9.51/5.31                  = zero_zero_real ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8374_prod__zero__iff,axiom,
% 9.51/5.31      ! [A2: set_int,F: int > real] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ( ( groups2316167850115554303t_real @ F @ A2 )
% 9.51/5.31            = zero_zero_real )
% 9.51/5.31          = ( ? [X3: int] :
% 9.51/5.31                ( ( member_int @ X3 @ A2 )
% 9.51/5.31                & ( ( F @ X3 )
% 9.51/5.31                  = zero_zero_real ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8375_prod__zero__iff,axiom,
% 9.51/5.31      ! [A2: set_complex,F: complex > real] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ A2 )
% 9.51/5.31       => ( ( ( groups766887009212190081x_real @ F @ A2 )
% 9.51/5.31            = zero_zero_real )
% 9.51/5.31          = ( ? [X3: complex] :
% 9.51/5.31                ( ( member_complex @ X3 @ A2 )
% 9.51/5.31                & ( ( F @ X3 )
% 9.51/5.31                  = zero_zero_real ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8376_prod__zero__iff,axiom,
% 9.51/5.31      ! [A2: set_nat,F: nat > rat] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( ( groups73079841787564623at_rat @ F @ A2 )
% 9.51/5.31            = zero_zero_rat )
% 9.51/5.31          = ( ? [X3: nat] :
% 9.51/5.31                ( ( member_nat @ X3 @ A2 )
% 9.51/5.31                & ( ( F @ X3 )
% 9.51/5.31                  = zero_zero_rat ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8377_prod__zero__iff,axiom,
% 9.51/5.31      ! [A2: set_int,F: int > rat] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ( ( groups1072433553688619179nt_rat @ F @ A2 )
% 9.51/5.31            = zero_zero_rat )
% 9.51/5.31          = ( ? [X3: int] :
% 9.51/5.31                ( ( member_int @ X3 @ A2 )
% 9.51/5.31                & ( ( F @ X3 )
% 9.51/5.31                  = zero_zero_rat ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8378_prod__zero__iff,axiom,
% 9.51/5.31      ! [A2: set_complex,F: complex > rat] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ A2 )
% 9.51/5.31       => ( ( ( groups225925009352817453ex_rat @ F @ A2 )
% 9.51/5.31            = zero_zero_rat )
% 9.51/5.31          = ( ? [X3: complex] :
% 9.51/5.31                ( ( member_complex @ X3 @ A2 )
% 9.51/5.31                & ( ( F @ X3 )
% 9.51/5.31                  = zero_zero_rat ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8379_prod__zero__iff,axiom,
% 9.51/5.31      ! [A2: set_int,F: int > nat] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ( ( groups1707563613775114915nt_nat @ F @ A2 )
% 9.51/5.31            = zero_zero_nat )
% 9.51/5.31          = ( ? [X3: int] :
% 9.51/5.31                ( ( member_int @ X3 @ A2 )
% 9.51/5.31                & ( ( F @ X3 )
% 9.51/5.31                  = zero_zero_nat ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8380_atLeastLessThan__empty,axiom,
% 9.51/5.31      ! [B: real,A: real] :
% 9.51/5.31        ( ( ord_less_eq_real @ B @ A )
% 9.51/5.31       => ( ( set_or66887138388493659n_real @ A @ B )
% 9.51/5.31          = bot_bot_set_real ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8381_atLeastLessThan__empty,axiom,
% 9.51/5.31      ! [B: set_int,A: set_int] :
% 9.51/5.31        ( ( ord_less_eq_set_int @ B @ A )
% 9.51/5.31       => ( ( set_or8585797421378605585et_int @ A @ B )
% 9.51/5.31          = bot_bot_set_set_int ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8382_atLeastLessThan__empty,axiom,
% 9.51/5.31      ! [B: rat,A: rat] :
% 9.51/5.31        ( ( ord_less_eq_rat @ B @ A )
% 9.51/5.31       => ( ( set_or4029947393144176647an_rat @ A @ B )
% 9.51/5.31          = bot_bot_set_rat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8383_atLeastLessThan__empty,axiom,
% 9.51/5.31      ! [B: num,A: num] :
% 9.51/5.31        ( ( ord_less_eq_num @ B @ A )
% 9.51/5.31       => ( ( set_or1222409239386451017an_num @ A @ B )
% 9.51/5.31          = bot_bot_set_num ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8384_atLeastLessThan__empty,axiom,
% 9.51/5.31      ! [B: nat,A: nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ B @ A )
% 9.51/5.31       => ( ( set_or4665077453230672383an_nat @ A @ B )
% 9.51/5.31          = bot_bot_set_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8385_atLeastLessThan__empty,axiom,
% 9.51/5.31      ! [B: int,A: int] :
% 9.51/5.31        ( ( ord_less_eq_int @ B @ A )
% 9.51/5.31       => ( ( set_or4662586982721622107an_int @ A @ B )
% 9.51/5.31          = bot_bot_set_int ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8386_ivl__subset,axiom,
% 9.51/5.31      ! [I: rat,J: rat,M: rat,N: rat] :
% 9.51/5.31        ( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ I @ J ) @ ( set_or4029947393144176647an_rat @ M @ N ) )
% 9.51/5.31        = ( ( ord_less_eq_rat @ J @ I )
% 9.51/5.31          | ( ( ord_less_eq_rat @ M @ I )
% 9.51/5.31            & ( ord_less_eq_rat @ J @ N ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8387_ivl__subset,axiom,
% 9.51/5.31      ! [I: num,J: num,M: num,N: num] :
% 9.51/5.31        ( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ I @ J ) @ ( set_or1222409239386451017an_num @ M @ N ) )
% 9.51/5.31        = ( ( ord_less_eq_num @ J @ I )
% 9.51/5.31          | ( ( ord_less_eq_num @ M @ I )
% 9.51/5.31            & ( ord_less_eq_num @ J @ N ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8388_ivl__subset,axiom,
% 9.51/5.31      ! [I: nat,J: nat,M: nat,N: nat] :
% 9.51/5.31        ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 9.51/5.31        = ( ( ord_less_eq_nat @ J @ I )
% 9.51/5.31          | ( ( ord_less_eq_nat @ M @ I )
% 9.51/5.31            & ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8389_ivl__subset,axiom,
% 9.51/5.31      ! [I: int,J: int,M: int,N: int] :
% 9.51/5.31        ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I @ J ) @ ( set_or4662586982721622107an_int @ M @ N ) )
% 9.51/5.31        = ( ( ord_less_eq_int @ J @ I )
% 9.51/5.31          | ( ( ord_less_eq_int @ M @ I )
% 9.51/5.31            & ( ord_less_eq_int @ J @ N ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8390_atLeastLessThan__empty__iff2,axiom,
% 9.51/5.31      ! [A: real,B: real] :
% 9.51/5.31        ( ( bot_bot_set_real
% 9.51/5.31          = ( set_or66887138388493659n_real @ A @ B ) )
% 9.51/5.31        = ( ~ ( ord_less_real @ A @ B ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8391_atLeastLessThan__empty__iff2,axiom,
% 9.51/5.31      ! [A: rat,B: rat] :
% 9.51/5.31        ( ( bot_bot_set_rat
% 9.51/5.31          = ( set_or4029947393144176647an_rat @ A @ B ) )
% 9.51/5.31        = ( ~ ( ord_less_rat @ A @ B ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8392_atLeastLessThan__empty__iff2,axiom,
% 9.51/5.31      ! [A: num,B: num] :
% 9.51/5.31        ( ( bot_bot_set_num
% 9.51/5.31          = ( set_or1222409239386451017an_num @ A @ B ) )
% 9.51/5.31        = ( ~ ( ord_less_num @ A @ B ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8393_atLeastLessThan__empty__iff2,axiom,
% 9.51/5.31      ! [A: nat,B: nat] :
% 9.51/5.31        ( ( bot_bot_set_nat
% 9.51/5.31          = ( set_or4665077453230672383an_nat @ A @ B ) )
% 9.51/5.31        = ( ~ ( ord_less_nat @ A @ B ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8394_atLeastLessThan__empty__iff2,axiom,
% 9.51/5.31      ! [A: int,B: int] :
% 9.51/5.31        ( ( bot_bot_set_int
% 9.51/5.31          = ( set_or4662586982721622107an_int @ A @ B ) )
% 9.51/5.31        = ( ~ ( ord_less_int @ A @ B ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8395_atLeastLessThan__empty__iff,axiom,
% 9.51/5.31      ! [A: real,B: real] :
% 9.51/5.31        ( ( ( set_or66887138388493659n_real @ A @ B )
% 9.51/5.31          = bot_bot_set_real )
% 9.51/5.31        = ( ~ ( ord_less_real @ A @ B ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8396_atLeastLessThan__empty__iff,axiom,
% 9.51/5.31      ! [A: rat,B: rat] :
% 9.51/5.31        ( ( ( set_or4029947393144176647an_rat @ A @ B )
% 9.51/5.31          = bot_bot_set_rat )
% 9.51/5.31        = ( ~ ( ord_less_rat @ A @ B ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8397_atLeastLessThan__empty__iff,axiom,
% 9.51/5.31      ! [A: num,B: num] :
% 9.51/5.31        ( ( ( set_or1222409239386451017an_num @ A @ B )
% 9.51/5.31          = bot_bot_set_num )
% 9.51/5.31        = ( ~ ( ord_less_num @ A @ B ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8398_atLeastLessThan__empty__iff,axiom,
% 9.51/5.31      ! [A: nat,B: nat] :
% 9.51/5.31        ( ( ( set_or4665077453230672383an_nat @ A @ B )
% 9.51/5.31          = bot_bot_set_nat )
% 9.51/5.31        = ( ~ ( ord_less_nat @ A @ B ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8399_atLeastLessThan__empty__iff,axiom,
% 9.51/5.31      ! [A: int,B: int] :
% 9.51/5.31        ( ( ( set_or4662586982721622107an_int @ A @ B )
% 9.51/5.31          = bot_bot_set_int )
% 9.51/5.31        = ( ~ ( ord_less_int @ A @ B ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8400_infinite__Ico__iff,axiom,
% 9.51/5.31      ! [A: real,B: real] :
% 9.51/5.31        ( ( ~ ( finite_finite_real @ ( set_or66887138388493659n_real @ A @ B ) ) )
% 9.51/5.31        = ( ord_less_real @ A @ B ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8401_infinite__Ico__iff,axiom,
% 9.51/5.31      ! [A: rat,B: rat] :
% 9.51/5.31        ( ( ~ ( finite_finite_rat @ ( set_or4029947393144176647an_rat @ A @ B ) ) )
% 9.51/5.31        = ( ord_less_rat @ A @ B ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8402_prod_Oempty,axiom,
% 9.51/5.31      ! [G: real > complex] :
% 9.51/5.31        ( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
% 9.51/5.31        = one_one_complex ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8403_prod_Oempty,axiom,
% 9.51/5.31      ! [G: real > real] :
% 9.51/5.31        ( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
% 9.51/5.31        = one_one_real ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8404_prod_Oempty,axiom,
% 9.51/5.31      ! [G: real > rat] :
% 9.51/5.31        ( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
% 9.51/5.31        = one_one_rat ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8405_prod_Oempty,axiom,
% 9.51/5.31      ! [G: real > nat] :
% 9.51/5.31        ( ( groups4696554848551431203al_nat @ G @ bot_bot_set_real )
% 9.51/5.31        = one_one_nat ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8406_prod_Oempty,axiom,
% 9.51/5.31      ! [G: real > int] :
% 9.51/5.31        ( ( groups4694064378042380927al_int @ G @ bot_bot_set_real )
% 9.51/5.31        = one_one_int ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8407_prod_Oempty,axiom,
% 9.51/5.31      ! [G: nat > complex] :
% 9.51/5.31        ( ( groups6464643781859351333omplex @ G @ bot_bot_set_nat )
% 9.51/5.31        = one_one_complex ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8408_prod_Oempty,axiom,
% 9.51/5.31      ! [G: nat > real] :
% 9.51/5.31        ( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
% 9.51/5.31        = one_one_real ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8409_prod_Oempty,axiom,
% 9.51/5.31      ! [G: nat > rat] :
% 9.51/5.31        ( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
% 9.51/5.31        = one_one_rat ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8410_prod_Oempty,axiom,
% 9.51/5.31      ! [G: nat > int] :
% 9.51/5.31        ( ( groups705719431365010083at_int @ G @ bot_bot_set_nat )
% 9.51/5.31        = one_one_int ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8411_prod_Oempty,axiom,
% 9.51/5.31      ! [G: int > complex] :
% 9.51/5.31        ( ( groups7440179247065528705omplex @ G @ bot_bot_set_int )
% 9.51/5.31        = one_one_complex ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8412_prod_Oinfinite,axiom,
% 9.51/5.31      ! [A2: set_nat,G: nat > complex] :
% 9.51/5.31        ( ~ ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( groups6464643781859351333omplex @ G @ A2 )
% 9.51/5.31          = one_one_complex ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8413_prod_Oinfinite,axiom,
% 9.51/5.31      ! [A2: set_int,G: int > complex] :
% 9.51/5.31        ( ~ ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ( groups7440179247065528705omplex @ G @ A2 )
% 9.51/5.31          = one_one_complex ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8414_prod_Oinfinite,axiom,
% 9.51/5.31      ! [A2: set_complex,G: complex > complex] :
% 9.51/5.31        ( ~ ( finite3207457112153483333omplex @ A2 )
% 9.51/5.31       => ( ( groups3708469109370488835omplex @ G @ A2 )
% 9.51/5.31          = one_one_complex ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8415_prod_Oinfinite,axiom,
% 9.51/5.31      ! [A2: set_nat,G: nat > real] :
% 9.51/5.31        ( ~ ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( groups129246275422532515t_real @ G @ A2 )
% 9.51/5.31          = one_one_real ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8416_prod_Oinfinite,axiom,
% 9.51/5.31      ! [A2: set_int,G: int > real] :
% 9.51/5.31        ( ~ ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ( groups2316167850115554303t_real @ G @ A2 )
% 9.51/5.31          = one_one_real ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8417_prod_Oinfinite,axiom,
% 9.51/5.31      ! [A2: set_complex,G: complex > real] :
% 9.51/5.31        ( ~ ( finite3207457112153483333omplex @ A2 )
% 9.51/5.31       => ( ( groups766887009212190081x_real @ G @ A2 )
% 9.51/5.31          = one_one_real ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8418_prod_Oinfinite,axiom,
% 9.51/5.31      ! [A2: set_nat,G: nat > rat] :
% 9.51/5.31        ( ~ ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( groups73079841787564623at_rat @ G @ A2 )
% 9.51/5.31          = one_one_rat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8419_prod_Oinfinite,axiom,
% 9.51/5.31      ! [A2: set_int,G: int > rat] :
% 9.51/5.31        ( ~ ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ( groups1072433553688619179nt_rat @ G @ A2 )
% 9.51/5.31          = one_one_rat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8420_prod_Oinfinite,axiom,
% 9.51/5.31      ! [A2: set_complex,G: complex > rat] :
% 9.51/5.31        ( ~ ( finite3207457112153483333omplex @ A2 )
% 9.51/5.31       => ( ( groups225925009352817453ex_rat @ G @ A2 )
% 9.51/5.31          = one_one_rat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8421_prod_Oinfinite,axiom,
% 9.51/5.31      ! [A2: set_int,G: int > nat] :
% 9.51/5.31        ( ~ ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ( groups1707563613775114915nt_nat @ G @ A2 )
% 9.51/5.31          = one_one_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8422_dvd__prod__eqI,axiom,
% 9.51/5.31      ! [A2: set_real,A: real,B: nat,F: real > nat] :
% 9.51/5.31        ( ( finite_finite_real @ A2 )
% 9.51/5.31       => ( ( member_real @ A @ A2 )
% 9.51/5.31         => ( ( B
% 9.51/5.31              = ( F @ A ) )
% 9.51/5.31           => ( dvd_dvd_nat @ B @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8423_dvd__prod__eqI,axiom,
% 9.51/5.31      ! [A2: set_int,A: int,B: nat,F: int > nat] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ( member_int @ A @ A2 )
% 9.51/5.31         => ( ( B
% 9.51/5.31              = ( F @ A ) )
% 9.51/5.31           => ( dvd_dvd_nat @ B @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8424_dvd__prod__eqI,axiom,
% 9.51/5.31      ! [A2: set_complex,A: complex,B: nat,F: complex > nat] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ A2 )
% 9.51/5.31       => ( ( member_complex @ A @ A2 )
% 9.51/5.31         => ( ( B
% 9.51/5.31              = ( F @ A ) )
% 9.51/5.31           => ( dvd_dvd_nat @ B @ ( groups861055069439313189ex_nat @ F @ A2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8425_dvd__prod__eqI,axiom,
% 9.51/5.31      ! [A2: set_real,A: real,B: int,F: real > int] :
% 9.51/5.31        ( ( finite_finite_real @ A2 )
% 9.51/5.31       => ( ( member_real @ A @ A2 )
% 9.51/5.31         => ( ( B
% 9.51/5.31              = ( F @ A ) )
% 9.51/5.31           => ( dvd_dvd_int @ B @ ( groups4694064378042380927al_int @ F @ A2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8426_dvd__prod__eqI,axiom,
% 9.51/5.31      ! [A2: set_nat,A: nat,B: int,F: nat > int] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( member_nat @ A @ A2 )
% 9.51/5.31         => ( ( B
% 9.51/5.31              = ( F @ A ) )
% 9.51/5.31           => ( dvd_dvd_int @ B @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8427_dvd__prod__eqI,axiom,
% 9.51/5.31      ! [A2: set_int,A: int,B: int,F: int > int] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ( member_int @ A @ A2 )
% 9.51/5.31         => ( ( B
% 9.51/5.31              = ( F @ A ) )
% 9.51/5.31           => ( dvd_dvd_int @ B @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8428_dvd__prod__eqI,axiom,
% 9.51/5.31      ! [A2: set_complex,A: complex,B: int,F: complex > int] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ A2 )
% 9.51/5.31       => ( ( member_complex @ A @ A2 )
% 9.51/5.31         => ( ( B
% 9.51/5.31              = ( F @ A ) )
% 9.51/5.31           => ( dvd_dvd_int @ B @ ( groups858564598930262913ex_int @ F @ A2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8429_dvd__prod__eqI,axiom,
% 9.51/5.31      ! [A2: set_real,A: real,B: code_integer,F: real > code_integer] :
% 9.51/5.31        ( ( finite_finite_real @ A2 )
% 9.51/5.31       => ( ( member_real @ A @ A2 )
% 9.51/5.31         => ( ( B
% 9.51/5.31              = ( F @ A ) )
% 9.51/5.31           => ( dvd_dvd_Code_integer @ B @ ( groups6225526099057966256nteger @ F @ A2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8430_dvd__prod__eqI,axiom,
% 9.51/5.31      ! [A2: set_nat,A: nat,B: code_integer,F: nat > code_integer] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( member_nat @ A @ A2 )
% 9.51/5.31         => ( ( B
% 9.51/5.31              = ( F @ A ) )
% 9.51/5.31           => ( dvd_dvd_Code_integer @ B @ ( groups3455450783089532116nteger @ F @ A2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8431_dvd__prod__eqI,axiom,
% 9.51/5.31      ! [A2: set_int,A: int,B: code_integer,F: int > code_integer] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ( member_int @ A @ A2 )
% 9.51/5.31         => ( ( B
% 9.51/5.31              = ( F @ A ) )
% 9.51/5.31           => ( dvd_dvd_Code_integer @ B @ ( groups3827104343326376752nteger @ F @ A2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8432_dvd__prodI,axiom,
% 9.51/5.31      ! [A2: set_real,A: real,F: real > nat] :
% 9.51/5.31        ( ( finite_finite_real @ A2 )
% 9.51/5.31       => ( ( member_real @ A @ A2 )
% 9.51/5.31         => ( dvd_dvd_nat @ ( F @ A ) @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8433_dvd__prodI,axiom,
% 9.51/5.31      ! [A2: set_int,A: int,F: int > nat] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ( member_int @ A @ A2 )
% 9.51/5.31         => ( dvd_dvd_nat @ ( F @ A ) @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8434_dvd__prodI,axiom,
% 9.51/5.31      ! [A2: set_complex,A: complex,F: complex > nat] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ A2 )
% 9.51/5.31       => ( ( member_complex @ A @ A2 )
% 9.51/5.31         => ( dvd_dvd_nat @ ( F @ A ) @ ( groups861055069439313189ex_nat @ F @ A2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8435_dvd__prodI,axiom,
% 9.51/5.31      ! [A2: set_real,A: real,F: real > int] :
% 9.51/5.31        ( ( finite_finite_real @ A2 )
% 9.51/5.31       => ( ( member_real @ A @ A2 )
% 9.51/5.31         => ( dvd_dvd_int @ ( F @ A ) @ ( groups4694064378042380927al_int @ F @ A2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8436_dvd__prodI,axiom,
% 9.51/5.31      ! [A2: set_nat,A: nat,F: nat > int] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( member_nat @ A @ A2 )
% 9.51/5.31         => ( dvd_dvd_int @ ( F @ A ) @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8437_dvd__prodI,axiom,
% 9.51/5.31      ! [A2: set_int,A: int,F: int > int] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ( member_int @ A @ A2 )
% 9.51/5.31         => ( dvd_dvd_int @ ( F @ A ) @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8438_dvd__prodI,axiom,
% 9.51/5.31      ! [A2: set_complex,A: complex,F: complex > int] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ A2 )
% 9.51/5.31       => ( ( member_complex @ A @ A2 )
% 9.51/5.31         => ( dvd_dvd_int @ ( F @ A ) @ ( groups858564598930262913ex_int @ F @ A2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8439_dvd__prodI,axiom,
% 9.51/5.31      ! [A2: set_real,A: real,F: real > code_integer] :
% 9.51/5.31        ( ( finite_finite_real @ A2 )
% 9.51/5.31       => ( ( member_real @ A @ A2 )
% 9.51/5.31         => ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups6225526099057966256nteger @ F @ A2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8440_dvd__prodI,axiom,
% 9.51/5.31      ! [A2: set_nat,A: nat,F: nat > code_integer] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( member_nat @ A @ A2 )
% 9.51/5.31         => ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups3455450783089532116nteger @ F @ A2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8441_dvd__prodI,axiom,
% 9.51/5.31      ! [A2: set_int,A: int,F: int > code_integer] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ( member_int @ A @ A2 )
% 9.51/5.31         => ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups3827104343326376752nteger @ F @ A2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8442_ivl__diff,axiom,
% 9.51/5.31      ! [I: rat,N: rat,M: rat] :
% 9.51/5.31        ( ( ord_less_eq_rat @ I @ N )
% 9.51/5.31       => ( ( minus_minus_set_rat @ ( set_or4029947393144176647an_rat @ I @ M ) @ ( set_or4029947393144176647an_rat @ I @ N ) )
% 9.51/5.31          = ( set_or4029947393144176647an_rat @ N @ M ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8443_ivl__diff,axiom,
% 9.51/5.31      ! [I: num,N: num,M: num] :
% 9.51/5.31        ( ( ord_less_eq_num @ I @ N )
% 9.51/5.31       => ( ( minus_minus_set_num @ ( set_or1222409239386451017an_num @ I @ M ) @ ( set_or1222409239386451017an_num @ I @ N ) )
% 9.51/5.31          = ( set_or1222409239386451017an_num @ N @ M ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8444_ivl__diff,axiom,
% 9.51/5.31      ! [I: nat,N: nat,M: nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ I @ N )
% 9.51/5.31       => ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M ) @ ( set_or4665077453230672383an_nat @ I @ N ) )
% 9.51/5.31          = ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8445_ivl__diff,axiom,
% 9.51/5.31      ! [I: int,N: int,M: int] :
% 9.51/5.31        ( ( ord_less_eq_int @ I @ N )
% 9.51/5.31       => ( ( minus_minus_set_int @ ( set_or4662586982721622107an_int @ I @ M ) @ ( set_or4662586982721622107an_int @ I @ N ) )
% 9.51/5.31          = ( set_or4662586982721622107an_int @ N @ M ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8446_prod_Odelta_H,axiom,
% 9.51/5.31      ! [S2: set_real,A: real,B: real > complex] :
% 9.51/5.31        ( ( finite_finite_real @ S2 )
% 9.51/5.31       => ( ( ( member_real @ A @ S2 )
% 9.51/5.31           => ( ( groups713298508707869441omplex
% 9.51/5.31                @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = ( B @ A ) ) )
% 9.51/5.31          & ( ~ ( member_real @ A @ S2 )
% 9.51/5.31           => ( ( groups713298508707869441omplex
% 9.51/5.31                @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = one_one_complex ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8447_prod_Odelta_H,axiom,
% 9.51/5.31      ! [S2: set_nat,A: nat,B: nat > complex] :
% 9.51/5.31        ( ( finite_finite_nat @ S2 )
% 9.51/5.31       => ( ( ( member_nat @ A @ S2 )
% 9.51/5.31           => ( ( groups6464643781859351333omplex
% 9.51/5.31                @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = ( B @ A ) ) )
% 9.51/5.31          & ( ~ ( member_nat @ A @ S2 )
% 9.51/5.31           => ( ( groups6464643781859351333omplex
% 9.51/5.31                @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = one_one_complex ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8448_prod_Odelta_H,axiom,
% 9.51/5.31      ! [S2: set_int,A: int,B: int > complex] :
% 9.51/5.31        ( ( finite_finite_int @ S2 )
% 9.51/5.31       => ( ( ( member_int @ A @ S2 )
% 9.51/5.31           => ( ( groups7440179247065528705omplex
% 9.51/5.31                @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = ( B @ A ) ) )
% 9.51/5.31          & ( ~ ( member_int @ A @ S2 )
% 9.51/5.31           => ( ( groups7440179247065528705omplex
% 9.51/5.31                @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = one_one_complex ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8449_prod_Odelta_H,axiom,
% 9.51/5.31      ! [S2: set_complex,A: complex,B: complex > complex] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ S2 )
% 9.51/5.31       => ( ( ( member_complex @ A @ S2 )
% 9.51/5.31           => ( ( groups3708469109370488835omplex
% 9.51/5.31                @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = ( B @ A ) ) )
% 9.51/5.31          & ( ~ ( member_complex @ A @ S2 )
% 9.51/5.31           => ( ( groups3708469109370488835omplex
% 9.51/5.31                @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = one_one_complex ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8450_prod_Odelta_H,axiom,
% 9.51/5.31      ! [S2: set_real,A: real,B: real > real] :
% 9.51/5.31        ( ( finite_finite_real @ S2 )
% 9.51/5.31       => ( ( ( member_real @ A @ S2 )
% 9.51/5.31           => ( ( groups1681761925125756287l_real
% 9.51/5.31                @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = ( B @ A ) ) )
% 9.51/5.31          & ( ~ ( member_real @ A @ S2 )
% 9.51/5.31           => ( ( groups1681761925125756287l_real
% 9.51/5.31                @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = one_one_real ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8451_prod_Odelta_H,axiom,
% 9.51/5.31      ! [S2: set_nat,A: nat,B: nat > real] :
% 9.51/5.31        ( ( finite_finite_nat @ S2 )
% 9.51/5.31       => ( ( ( member_nat @ A @ S2 )
% 9.51/5.31           => ( ( groups129246275422532515t_real
% 9.51/5.31                @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = ( B @ A ) ) )
% 9.51/5.31          & ( ~ ( member_nat @ A @ S2 )
% 9.51/5.31           => ( ( groups129246275422532515t_real
% 9.51/5.31                @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = one_one_real ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8452_prod_Odelta_H,axiom,
% 9.51/5.31      ! [S2: set_int,A: int,B: int > real] :
% 9.51/5.31        ( ( finite_finite_int @ S2 )
% 9.51/5.31       => ( ( ( member_int @ A @ S2 )
% 9.51/5.31           => ( ( groups2316167850115554303t_real
% 9.51/5.31                @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = ( B @ A ) ) )
% 9.51/5.31          & ( ~ ( member_int @ A @ S2 )
% 9.51/5.31           => ( ( groups2316167850115554303t_real
% 9.51/5.31                @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = one_one_real ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8453_prod_Odelta_H,axiom,
% 9.51/5.31      ! [S2: set_complex,A: complex,B: complex > real] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ S2 )
% 9.51/5.31       => ( ( ( member_complex @ A @ S2 )
% 9.51/5.31           => ( ( groups766887009212190081x_real
% 9.51/5.31                @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = ( B @ A ) ) )
% 9.51/5.31          & ( ~ ( member_complex @ A @ S2 )
% 9.51/5.31           => ( ( groups766887009212190081x_real
% 9.51/5.31                @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = one_one_real ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8454_prod_Odelta_H,axiom,
% 9.51/5.31      ! [S2: set_real,A: real,B: real > rat] :
% 9.51/5.31        ( ( finite_finite_real @ S2 )
% 9.51/5.31       => ( ( ( member_real @ A @ S2 )
% 9.51/5.31           => ( ( groups4061424788464935467al_rat
% 9.51/5.31                @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = ( B @ A ) ) )
% 9.51/5.31          & ( ~ ( member_real @ A @ S2 )
% 9.51/5.31           => ( ( groups4061424788464935467al_rat
% 9.51/5.31                @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = one_one_rat ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8455_prod_Odelta_H,axiom,
% 9.51/5.31      ! [S2: set_nat,A: nat,B: nat > rat] :
% 9.51/5.31        ( ( finite_finite_nat @ S2 )
% 9.51/5.31       => ( ( ( member_nat @ A @ S2 )
% 9.51/5.31           => ( ( groups73079841787564623at_rat
% 9.51/5.31                @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = ( B @ A ) ) )
% 9.51/5.31          & ( ~ ( member_nat @ A @ S2 )
% 9.51/5.31           => ( ( groups73079841787564623at_rat
% 9.51/5.31                @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = one_one_rat ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8456_prod_Odelta,axiom,
% 9.51/5.31      ! [S2: set_real,A: real,B: real > complex] :
% 9.51/5.31        ( ( finite_finite_real @ S2 )
% 9.51/5.31       => ( ( ( member_real @ A @ S2 )
% 9.51/5.31           => ( ( groups713298508707869441omplex
% 9.51/5.31                @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = ( B @ A ) ) )
% 9.51/5.31          & ( ~ ( member_real @ A @ S2 )
% 9.51/5.31           => ( ( groups713298508707869441omplex
% 9.51/5.31                @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = one_one_complex ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8457_prod_Odelta,axiom,
% 9.51/5.31      ! [S2: set_nat,A: nat,B: nat > complex] :
% 9.51/5.31        ( ( finite_finite_nat @ S2 )
% 9.51/5.31       => ( ( ( member_nat @ A @ S2 )
% 9.51/5.31           => ( ( groups6464643781859351333omplex
% 9.51/5.31                @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = ( B @ A ) ) )
% 9.51/5.31          & ( ~ ( member_nat @ A @ S2 )
% 9.51/5.31           => ( ( groups6464643781859351333omplex
% 9.51/5.31                @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = one_one_complex ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8458_prod_Odelta,axiom,
% 9.51/5.31      ! [S2: set_int,A: int,B: int > complex] :
% 9.51/5.31        ( ( finite_finite_int @ S2 )
% 9.51/5.31       => ( ( ( member_int @ A @ S2 )
% 9.51/5.31           => ( ( groups7440179247065528705omplex
% 9.51/5.31                @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = ( B @ A ) ) )
% 9.51/5.31          & ( ~ ( member_int @ A @ S2 )
% 9.51/5.31           => ( ( groups7440179247065528705omplex
% 9.51/5.31                @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = one_one_complex ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8459_prod_Odelta,axiom,
% 9.51/5.31      ! [S2: set_complex,A: complex,B: complex > complex] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ S2 )
% 9.51/5.31       => ( ( ( member_complex @ A @ S2 )
% 9.51/5.31           => ( ( groups3708469109370488835omplex
% 9.51/5.31                @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = ( B @ A ) ) )
% 9.51/5.31          & ( ~ ( member_complex @ A @ S2 )
% 9.51/5.31           => ( ( groups3708469109370488835omplex
% 9.51/5.31                @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = one_one_complex ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8460_prod_Odelta,axiom,
% 9.51/5.31      ! [S2: set_real,A: real,B: real > real] :
% 9.51/5.31        ( ( finite_finite_real @ S2 )
% 9.51/5.31       => ( ( ( member_real @ A @ S2 )
% 9.51/5.31           => ( ( groups1681761925125756287l_real
% 9.51/5.31                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = ( B @ A ) ) )
% 9.51/5.31          & ( ~ ( member_real @ A @ S2 )
% 9.51/5.31           => ( ( groups1681761925125756287l_real
% 9.51/5.31                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = one_one_real ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8461_prod_Odelta,axiom,
% 9.51/5.31      ! [S2: set_nat,A: nat,B: nat > real] :
% 9.51/5.31        ( ( finite_finite_nat @ S2 )
% 9.51/5.31       => ( ( ( member_nat @ A @ S2 )
% 9.51/5.31           => ( ( groups129246275422532515t_real
% 9.51/5.31                @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = ( B @ A ) ) )
% 9.51/5.31          & ( ~ ( member_nat @ A @ S2 )
% 9.51/5.31           => ( ( groups129246275422532515t_real
% 9.51/5.31                @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = one_one_real ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8462_prod_Odelta,axiom,
% 9.51/5.31      ! [S2: set_int,A: int,B: int > real] :
% 9.51/5.31        ( ( finite_finite_int @ S2 )
% 9.51/5.31       => ( ( ( member_int @ A @ S2 )
% 9.51/5.31           => ( ( groups2316167850115554303t_real
% 9.51/5.31                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = ( B @ A ) ) )
% 9.51/5.31          & ( ~ ( member_int @ A @ S2 )
% 9.51/5.31           => ( ( groups2316167850115554303t_real
% 9.51/5.31                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = one_one_real ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8463_prod_Odelta,axiom,
% 9.51/5.31      ! [S2: set_complex,A: complex,B: complex > real] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ S2 )
% 9.51/5.31       => ( ( ( member_complex @ A @ S2 )
% 9.51/5.31           => ( ( groups766887009212190081x_real
% 9.51/5.31                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = ( B @ A ) ) )
% 9.51/5.31          & ( ~ ( member_complex @ A @ S2 )
% 9.51/5.31           => ( ( groups766887009212190081x_real
% 9.51/5.31                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = one_one_real ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8464_prod_Odelta,axiom,
% 9.51/5.31      ! [S2: set_real,A: real,B: real > rat] :
% 9.51/5.31        ( ( finite_finite_real @ S2 )
% 9.51/5.31       => ( ( ( member_real @ A @ S2 )
% 9.51/5.31           => ( ( groups4061424788464935467al_rat
% 9.51/5.31                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = ( B @ A ) ) )
% 9.51/5.31          & ( ~ ( member_real @ A @ S2 )
% 9.51/5.31           => ( ( groups4061424788464935467al_rat
% 9.51/5.31                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = one_one_rat ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8465_prod_Odelta,axiom,
% 9.51/5.31      ! [S2: set_nat,A: nat,B: nat > rat] :
% 9.51/5.31        ( ( finite_finite_nat @ S2 )
% 9.51/5.31       => ( ( ( member_nat @ A @ S2 )
% 9.51/5.31           => ( ( groups73079841787564623at_rat
% 9.51/5.31                @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = ( B @ A ) ) )
% 9.51/5.31          & ( ~ ( member_nat @ A @ S2 )
% 9.51/5.31           => ( ( groups73079841787564623at_rat
% 9.51/5.31                @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 9.51/5.31                @ S2 )
% 9.51/5.31              = one_one_rat ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8466_is__singletonI,axiom,
% 9.51/5.31      ! [X: produc3843707927480180839at_nat] : ( is_sin2937591304547752795at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8467_is__singletonI,axiom,
% 9.51/5.31      ! [X: product_prod_nat_nat] : ( is_sin2850979758926227957at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8468_is__singletonI,axiom,
% 9.51/5.31      ! [X: real] : ( is_singleton_real @ ( insert_real @ X @ bot_bot_set_real ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8469_is__singletonI,axiom,
% 9.51/5.31      ! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8470_is__singletonI,axiom,
% 9.51/5.31      ! [X: int] : ( is_singleton_int @ ( insert_int @ X @ bot_bot_set_int ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8471_prod_Oinsert,axiom,
% 9.51/5.31      ! [A2: set_real,X: real,G: real > complex] :
% 9.51/5.31        ( ( finite_finite_real @ A2 )
% 9.51/5.31       => ( ~ ( member_real @ X @ A2 )
% 9.51/5.31         => ( ( groups713298508707869441omplex @ G @ ( insert_real @ X @ A2 ) )
% 9.51/5.31            = ( times_times_complex @ ( G @ X ) @ ( groups713298508707869441omplex @ G @ A2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8472_prod_Oinsert,axiom,
% 9.51/5.31      ! [A2: set_nat,X: nat,G: nat > complex] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ~ ( member_nat @ X @ A2 )
% 9.51/5.31         => ( ( groups6464643781859351333omplex @ G @ ( insert_nat @ X @ A2 ) )
% 9.51/5.31            = ( times_times_complex @ ( G @ X ) @ ( groups6464643781859351333omplex @ G @ A2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8473_prod_Oinsert,axiom,
% 9.51/5.31      ! [A2: set_int,X: int,G: int > complex] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ~ ( member_int @ X @ A2 )
% 9.51/5.31         => ( ( groups7440179247065528705omplex @ G @ ( insert_int @ X @ A2 ) )
% 9.51/5.31            = ( times_times_complex @ ( G @ X ) @ ( groups7440179247065528705omplex @ G @ A2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8474_prod_Oinsert,axiom,
% 9.51/5.31      ! [A2: set_complex,X: complex,G: complex > complex] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ A2 )
% 9.51/5.31       => ( ~ ( member_complex @ X @ A2 )
% 9.51/5.31         => ( ( groups3708469109370488835omplex @ G @ ( insert_complex @ X @ A2 ) )
% 9.51/5.31            = ( times_times_complex @ ( G @ X ) @ ( groups3708469109370488835omplex @ G @ A2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8475_prod_Oinsert,axiom,
% 9.51/5.31      ! [A2: set_real,X: real,G: real > real] :
% 9.51/5.31        ( ( finite_finite_real @ A2 )
% 9.51/5.31       => ( ~ ( member_real @ X @ A2 )
% 9.51/5.31         => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X @ A2 ) )
% 9.51/5.31            = ( times_times_real @ ( G @ X ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8476_prod_Oinsert,axiom,
% 9.51/5.31      ! [A2: set_nat,X: nat,G: nat > real] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ~ ( member_nat @ X @ A2 )
% 9.51/5.31         => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X @ A2 ) )
% 9.51/5.31            = ( times_times_real @ ( G @ X ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8477_prod_Oinsert,axiom,
% 9.51/5.31      ! [A2: set_int,X: int,G: int > real] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ~ ( member_int @ X @ A2 )
% 9.51/5.31         => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X @ A2 ) )
% 9.51/5.31            = ( times_times_real @ ( G @ X ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8478_prod_Oinsert,axiom,
% 9.51/5.31      ! [A2: set_complex,X: complex,G: complex > real] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ A2 )
% 9.51/5.31       => ( ~ ( member_complex @ X @ A2 )
% 9.51/5.31         => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X @ A2 ) )
% 9.51/5.31            = ( times_times_real @ ( G @ X ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8479_prod_Oinsert,axiom,
% 9.51/5.31      ! [A2: set_real,X: real,G: real > rat] :
% 9.51/5.31        ( ( finite_finite_real @ A2 )
% 9.51/5.31       => ( ~ ( member_real @ X @ A2 )
% 9.51/5.31         => ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X @ A2 ) )
% 9.51/5.31            = ( times_times_rat @ ( G @ X ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8480_prod_Oinsert,axiom,
% 9.51/5.31      ! [A2: set_nat,X: nat,G: nat > rat] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ~ ( member_nat @ X @ A2 )
% 9.51/5.31         => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A2 ) )
% 9.51/5.31            = ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8481_atLeastLessThan__singleton,axiom,
% 9.51/5.31      ! [M: nat] :
% 9.51/5.31        ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
% 9.51/5.31        = ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8482_sum_Oop__ivl__Suc,axiom,
% 9.51/5.31      ! [N: nat,M: nat,G: nat > complex] :
% 9.51/5.31        ( ( ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = zero_zero_complex ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8483_sum_Oop__ivl__Suc,axiom,
% 9.51/5.31      ! [N: nat,M: nat,G: nat > rat] :
% 9.51/5.31        ( ( ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = zero_zero_rat ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8484_sum_Oop__ivl__Suc,axiom,
% 9.51/5.31      ! [N: nat,M: nat,G: nat > int] :
% 9.51/5.31        ( ( ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = zero_zero_int ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8485_sum_Oop__ivl__Suc,axiom,
% 9.51/5.31      ! [N: nat,M: nat,G: nat > nat] :
% 9.51/5.31        ( ( ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = zero_zero_nat ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8486_sum_Oop__ivl__Suc,axiom,
% 9.51/5.31      ! [N: nat,M: nat,G: nat > real] :
% 9.51/5.31        ( ( ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = zero_zero_real ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8487_prod_Ocl__ivl__Suc,axiom,
% 9.51/5.31      ! [N: nat,M: nat,G: nat > complex] :
% 9.51/5.31        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.51/5.31         => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = one_one_complex ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.51/5.31         => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8488_prod_Ocl__ivl__Suc,axiom,
% 9.51/5.31      ! [N: nat,M: nat,G: nat > real] :
% 9.51/5.31        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.51/5.31         => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = one_one_real ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.51/5.31         => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8489_prod_Ocl__ivl__Suc,axiom,
% 9.51/5.31      ! [N: nat,M: nat,G: nat > rat] :
% 9.51/5.31        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.51/5.31         => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = one_one_rat ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.51/5.31         => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8490_prod_Ocl__ivl__Suc,axiom,
% 9.51/5.31      ! [N: nat,M: nat,G: nat > int] :
% 9.51/5.31        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.51/5.31         => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = one_one_int ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.51/5.31         => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8491_prod_Ocl__ivl__Suc,axiom,
% 9.51/5.31      ! [N: nat,M: nat,G: nat > nat] :
% 9.51/5.31        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.51/5.31         => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = one_one_nat ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.51/5.31         => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8492_prod_Oop__ivl__Suc,axiom,
% 9.51/5.31      ! [N: nat,M: nat,G: nat > complex] :
% 9.51/5.31        ( ( ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = one_one_complex ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8493_prod_Oop__ivl__Suc,axiom,
% 9.51/5.31      ! [N: nat,M: nat,G: nat > real] :
% 9.51/5.31        ( ( ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = one_one_real ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8494_prod_Oop__ivl__Suc,axiom,
% 9.51/5.31      ! [N: nat,M: nat,G: nat > rat] :
% 9.51/5.31        ( ( ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = one_one_rat ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8495_prod_Oop__ivl__Suc,axiom,
% 9.51/5.31      ! [N: nat,M: nat,G: nat > int] :
% 9.51/5.31        ( ( ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = one_one_int ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8496_prod_Oop__ivl__Suc,axiom,
% 9.51/5.31      ! [N: nat,M: nat,G: nat > nat] :
% 9.51/5.31        ( ( ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = one_one_nat ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
% 9.51/5.31            = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8497_sgn__rat__def,axiom,
% 9.51/5.31      ( sgn_sgn_rat
% 9.51/5.31      = ( ^ [A4: rat] : ( if_rat @ ( A4 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A4 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8498_abs__rat__def,axiom,
% 9.51/5.31      ( abs_abs_rat
% 9.51/5.31      = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8499_prod_OatLeastLessThan__concat,axiom,
% 9.51/5.31      ! [M: nat,N: nat,P4: nat,G: nat > complex] :
% 9.51/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.31       => ( ( ord_less_eq_nat @ N @ P4 )
% 9.51/5.31         => ( ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) )
% 9.51/5.31            = ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8500_prod_OatLeastLessThan__concat,axiom,
% 9.51/5.31      ! [M: nat,N: nat,P4: nat,G: nat > real] :
% 9.51/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.31       => ( ( ord_less_eq_nat @ N @ P4 )
% 9.51/5.31         => ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) )
% 9.51/5.31            = ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8501_prod_OatLeastLessThan__concat,axiom,
% 9.51/5.31      ! [M: nat,N: nat,P4: nat,G: nat > rat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.31       => ( ( ord_less_eq_nat @ N @ P4 )
% 9.51/5.31         => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) )
% 9.51/5.31            = ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8502_prod_OatLeastLessThan__concat,axiom,
% 9.51/5.31      ! [M: nat,N: nat,P4: nat,G: nat > int] :
% 9.51/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.31       => ( ( ord_less_eq_nat @ N @ P4 )
% 9.51/5.31         => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) )
% 9.51/5.31            = ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8503_prod_OatLeastLessThan__concat,axiom,
% 9.51/5.31      ! [M: nat,N: nat,P4: nat,G: nat > nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.31       => ( ( ord_less_eq_nat @ N @ P4 )
% 9.51/5.31         => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) )
% 9.51/5.31            = ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8504_less__eq__rat__def,axiom,
% 9.51/5.31      ( ord_less_eq_rat
% 9.51/5.31      = ( ^ [X3: rat,Y2: rat] :
% 9.51/5.31            ( ( ord_less_rat @ X3 @ Y2 )
% 9.51/5.31            | ( X3 = Y2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8505_prod_Oivl__cong,axiom,
% 9.51/5.31      ! [A: nat,C: nat,B: nat,D: nat,G: nat > nat,H2: nat > nat] :
% 9.51/5.31        ( ( A = C )
% 9.51/5.31       => ( ( B = D )
% 9.51/5.31         => ( ! [X4: nat] :
% 9.51/5.31                ( ( ord_less_eq_nat @ C @ X4 )
% 9.51/5.31               => ( ( ord_less_nat @ X4 @ D )
% 9.51/5.31                 => ( ( G @ X4 )
% 9.51/5.31                    = ( H2 @ X4 ) ) ) )
% 9.51/5.31           => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
% 9.51/5.31              = ( groups708209901874060359at_nat @ H2 @ ( set_or4665077453230672383an_nat @ C @ D ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8506_obtain__pos__sum,axiom,
% 9.51/5.31      ! [R2: rat] :
% 9.51/5.31        ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 9.51/5.31       => ~ ! [S4: rat] :
% 9.51/5.31              ( ( ord_less_rat @ zero_zero_rat @ S4 )
% 9.51/5.31             => ! [T6: rat] :
% 9.51/5.31                  ( ( ord_less_rat @ zero_zero_rat @ T6 )
% 9.51/5.31                 => ( R2
% 9.51/5.31                   != ( plus_plus_rat @ S4 @ T6 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8507_atLeastLessThan__inj_I2_J,axiom,
% 9.51/5.31      ! [A: real,B: real,C: real,D: real] :
% 9.51/5.31        ( ( ( set_or66887138388493659n_real @ A @ B )
% 9.51/5.31          = ( set_or66887138388493659n_real @ C @ D ) )
% 9.51/5.31       => ( ( ord_less_real @ A @ B )
% 9.51/5.31         => ( ( ord_less_real @ C @ D )
% 9.51/5.31           => ( B = D ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8508_atLeastLessThan__inj_I2_J,axiom,
% 9.51/5.31      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.51/5.31        ( ( ( set_or4029947393144176647an_rat @ A @ B )
% 9.51/5.31          = ( set_or4029947393144176647an_rat @ C @ D ) )
% 9.51/5.31       => ( ( ord_less_rat @ A @ B )
% 9.51/5.31         => ( ( ord_less_rat @ C @ D )
% 9.51/5.31           => ( B = D ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8509_atLeastLessThan__inj_I2_J,axiom,
% 9.51/5.31      ! [A: num,B: num,C: num,D: num] :
% 9.51/5.31        ( ( ( set_or1222409239386451017an_num @ A @ B )
% 9.51/5.31          = ( set_or1222409239386451017an_num @ C @ D ) )
% 9.51/5.31       => ( ( ord_less_num @ A @ B )
% 9.51/5.31         => ( ( ord_less_num @ C @ D )
% 9.51/5.31           => ( B = D ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8510_atLeastLessThan__inj_I2_J,axiom,
% 9.51/5.31      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.51/5.31        ( ( ( set_or4665077453230672383an_nat @ A @ B )
% 9.51/5.31          = ( set_or4665077453230672383an_nat @ C @ D ) )
% 9.51/5.31       => ( ( ord_less_nat @ A @ B )
% 9.51/5.31         => ( ( ord_less_nat @ C @ D )
% 9.51/5.31           => ( B = D ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8511_atLeastLessThan__inj_I2_J,axiom,
% 9.51/5.31      ! [A: int,B: int,C: int,D: int] :
% 9.51/5.31        ( ( ( set_or4662586982721622107an_int @ A @ B )
% 9.51/5.31          = ( set_or4662586982721622107an_int @ C @ D ) )
% 9.51/5.31       => ( ( ord_less_int @ A @ B )
% 9.51/5.31         => ( ( ord_less_int @ C @ D )
% 9.51/5.31           => ( B = D ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8512_atLeastLessThan__inj_I1_J,axiom,
% 9.51/5.31      ! [A: real,B: real,C: real,D: real] :
% 9.51/5.31        ( ( ( set_or66887138388493659n_real @ A @ B )
% 9.51/5.31          = ( set_or66887138388493659n_real @ C @ D ) )
% 9.51/5.31       => ( ( ord_less_real @ A @ B )
% 9.51/5.31         => ( ( ord_less_real @ C @ D )
% 9.51/5.31           => ( A = C ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8513_atLeastLessThan__inj_I1_J,axiom,
% 9.51/5.31      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.51/5.31        ( ( ( set_or4029947393144176647an_rat @ A @ B )
% 9.51/5.31          = ( set_or4029947393144176647an_rat @ C @ D ) )
% 9.51/5.31       => ( ( ord_less_rat @ A @ B )
% 9.51/5.31         => ( ( ord_less_rat @ C @ D )
% 9.51/5.31           => ( A = C ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8514_atLeastLessThan__inj_I1_J,axiom,
% 9.51/5.31      ! [A: num,B: num,C: num,D: num] :
% 9.51/5.31        ( ( ( set_or1222409239386451017an_num @ A @ B )
% 9.51/5.31          = ( set_or1222409239386451017an_num @ C @ D ) )
% 9.51/5.31       => ( ( ord_less_num @ A @ B )
% 9.51/5.31         => ( ( ord_less_num @ C @ D )
% 9.51/5.31           => ( A = C ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8515_atLeastLessThan__inj_I1_J,axiom,
% 9.51/5.31      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.51/5.31        ( ( ( set_or4665077453230672383an_nat @ A @ B )
% 9.51/5.31          = ( set_or4665077453230672383an_nat @ C @ D ) )
% 9.51/5.31       => ( ( ord_less_nat @ A @ B )
% 9.51/5.31         => ( ( ord_less_nat @ C @ D )
% 9.51/5.31           => ( A = C ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8516_atLeastLessThan__inj_I1_J,axiom,
% 9.51/5.31      ! [A: int,B: int,C: int,D: int] :
% 9.51/5.31        ( ( ( set_or4662586982721622107an_int @ A @ B )
% 9.51/5.31          = ( set_or4662586982721622107an_int @ C @ D ) )
% 9.51/5.31       => ( ( ord_less_int @ A @ B )
% 9.51/5.31         => ( ( ord_less_int @ C @ D )
% 9.51/5.31           => ( A = C ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8517_atLeastLessThan__eq__iff,axiom,
% 9.51/5.31      ! [A: real,B: real,C: real,D: real] :
% 9.51/5.31        ( ( ord_less_real @ A @ B )
% 9.51/5.31       => ( ( ord_less_real @ C @ D )
% 9.51/5.31         => ( ( ( set_or66887138388493659n_real @ A @ B )
% 9.51/5.31              = ( set_or66887138388493659n_real @ C @ D ) )
% 9.51/5.31            = ( ( A = C )
% 9.51/5.31              & ( B = D ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8518_atLeastLessThan__eq__iff,axiom,
% 9.51/5.31      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.51/5.31        ( ( ord_less_rat @ A @ B )
% 9.51/5.31       => ( ( ord_less_rat @ C @ D )
% 9.51/5.31         => ( ( ( set_or4029947393144176647an_rat @ A @ B )
% 9.51/5.31              = ( set_or4029947393144176647an_rat @ C @ D ) )
% 9.51/5.31            = ( ( A = C )
% 9.51/5.31              & ( B = D ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8519_atLeastLessThan__eq__iff,axiom,
% 9.51/5.31      ! [A: num,B: num,C: num,D: num] :
% 9.51/5.31        ( ( ord_less_num @ A @ B )
% 9.51/5.31       => ( ( ord_less_num @ C @ D )
% 9.51/5.31         => ( ( ( set_or1222409239386451017an_num @ A @ B )
% 9.51/5.31              = ( set_or1222409239386451017an_num @ C @ D ) )
% 9.51/5.31            = ( ( A = C )
% 9.51/5.31              & ( B = D ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8520_atLeastLessThan__eq__iff,axiom,
% 9.51/5.31      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.51/5.31        ( ( ord_less_nat @ A @ B )
% 9.51/5.31       => ( ( ord_less_nat @ C @ D )
% 9.51/5.31         => ( ( ( set_or4665077453230672383an_nat @ A @ B )
% 9.51/5.31              = ( set_or4665077453230672383an_nat @ C @ D ) )
% 9.51/5.31            = ( ( A = C )
% 9.51/5.31              & ( B = D ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8521_atLeastLessThan__eq__iff,axiom,
% 9.51/5.31      ! [A: int,B: int,C: int,D: int] :
% 9.51/5.31        ( ( ord_less_int @ A @ B )
% 9.51/5.31       => ( ( ord_less_int @ C @ D )
% 9.51/5.31         => ( ( ( set_or4662586982721622107an_int @ A @ B )
% 9.51/5.31              = ( set_or4662586982721622107an_int @ C @ D ) )
% 9.51/5.31            = ( ( A = C )
% 9.51/5.31              & ( B = D ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8522_prod_OatLeast0__lessThan__Suc,axiom,
% 9.51/5.31      ! [G: nat > complex,N: nat] :
% 9.51/5.31        ( ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.51/5.31        = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8523_prod_OatLeast0__lessThan__Suc,axiom,
% 9.51/5.31      ! [G: nat > real,N: nat] :
% 9.51/5.31        ( ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.51/5.31        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8524_prod_OatLeast0__lessThan__Suc,axiom,
% 9.51/5.31      ! [G: nat > rat,N: nat] :
% 9.51/5.31        ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.51/5.31        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8525_prod_OatLeast0__lessThan__Suc,axiom,
% 9.51/5.31      ! [G: nat > int,N: nat] :
% 9.51/5.31        ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.51/5.31        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8526_prod_OatLeast0__lessThan__Suc,axiom,
% 9.51/5.31      ! [G: nat > nat,N: nat] :
% 9.51/5.31        ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.51/5.31        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8527_prod_OatLeast__Suc__lessThan,axiom,
% 9.51/5.31      ! [M: nat,N: nat,G: nat > complex] :
% 9.51/5.31        ( ( ord_less_nat @ M @ N )
% 9.51/5.31       => ( ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 9.51/5.31          = ( times_times_complex @ ( G @ M ) @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8528_prod_OatLeast__Suc__lessThan,axiom,
% 9.51/5.31      ! [M: nat,N: nat,G: nat > real] :
% 9.51/5.31        ( ( ord_less_nat @ M @ N )
% 9.51/5.31       => ( ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 9.51/5.31          = ( times_times_real @ ( G @ M ) @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8529_prod_OatLeast__Suc__lessThan,axiom,
% 9.51/5.31      ! [M: nat,N: nat,G: nat > rat] :
% 9.51/5.31        ( ( ord_less_nat @ M @ N )
% 9.51/5.31       => ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 9.51/5.31          = ( times_times_rat @ ( G @ M ) @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8530_prod_OatLeast__Suc__lessThan,axiom,
% 9.51/5.31      ! [M: nat,N: nat,G: nat > int] :
% 9.51/5.31        ( ( ord_less_nat @ M @ N )
% 9.51/5.31       => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 9.51/5.31          = ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8531_prod_OatLeast__Suc__lessThan,axiom,
% 9.51/5.31      ! [M: nat,N: nat,G: nat > nat] :
% 9.51/5.31        ( ( ord_less_nat @ M @ N )
% 9.51/5.31       => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 9.51/5.31          = ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8532_prod_OatLeastLessThan__Suc,axiom,
% 9.51/5.31      ! [A: nat,B: nat,G: nat > complex] :
% 9.51/5.31        ( ( ord_less_eq_nat @ A @ B )
% 9.51/5.31       => ( ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
% 9.51/5.31          = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8533_prod_OatLeastLessThan__Suc,axiom,
% 9.51/5.31      ! [A: nat,B: nat,G: nat > real] :
% 9.51/5.31        ( ( ord_less_eq_nat @ A @ B )
% 9.51/5.31       => ( ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
% 9.51/5.31          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8534_prod_OatLeastLessThan__Suc,axiom,
% 9.51/5.31      ! [A: nat,B: nat,G: nat > rat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ A @ B )
% 9.51/5.31       => ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
% 9.51/5.31          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8535_prod_OatLeastLessThan__Suc,axiom,
% 9.51/5.31      ! [A: nat,B: nat,G: nat > int] :
% 9.51/5.31        ( ( ord_less_eq_nat @ A @ B )
% 9.51/5.31       => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
% 9.51/5.31          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8536_prod_OatLeastLessThan__Suc,axiom,
% 9.51/5.31      ! [A: nat,B: nat,G: nat > nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ A @ B )
% 9.51/5.31       => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
% 9.51/5.31          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8537_prod_Oswap__restrict,axiom,
% 9.51/5.31      ! [A2: set_real,B2: set_nat,G: real > nat > nat,R: real > nat > $o] :
% 9.51/5.31        ( ( finite_finite_real @ A2 )
% 9.51/5.31       => ( ( finite_finite_nat @ B2 )
% 9.51/5.31         => ( ( groups4696554848551431203al_nat
% 9.51/5.31              @ ^ [X3: real] :
% 9.51/5.31                  ( groups708209901874060359at_nat @ ( G @ X3 )
% 9.51/5.31                  @ ( collect_nat
% 9.51/5.31                    @ ^ [Y2: nat] :
% 9.51/5.31                        ( ( member_nat @ Y2 @ B2 )
% 9.51/5.31                        & ( R @ X3 @ Y2 ) ) ) )
% 9.51/5.31              @ A2 )
% 9.51/5.31            = ( groups708209901874060359at_nat
% 9.51/5.31              @ ^ [Y2: nat] :
% 9.51/5.31                  ( groups4696554848551431203al_nat
% 9.51/5.31                  @ ^ [X3: real] : ( G @ X3 @ Y2 )
% 9.51/5.31                  @ ( collect_real
% 9.51/5.31                    @ ^ [X3: real] :
% 9.51/5.31                        ( ( member_real @ X3 @ A2 )
% 9.51/5.31                        & ( R @ X3 @ Y2 ) ) ) )
% 9.51/5.31              @ B2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8538_prod_Oswap__restrict,axiom,
% 9.51/5.31      ! [A2: set_int,B2: set_nat,G: int > nat > nat,R: int > nat > $o] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ( finite_finite_nat @ B2 )
% 9.51/5.31         => ( ( groups1707563613775114915nt_nat
% 9.51/5.31              @ ^ [X3: int] :
% 9.51/5.31                  ( groups708209901874060359at_nat @ ( G @ X3 )
% 9.51/5.31                  @ ( collect_nat
% 9.51/5.31                    @ ^ [Y2: nat] :
% 9.51/5.31                        ( ( member_nat @ Y2 @ B2 )
% 9.51/5.31                        & ( R @ X3 @ Y2 ) ) ) )
% 9.51/5.31              @ A2 )
% 9.51/5.31            = ( groups708209901874060359at_nat
% 9.51/5.31              @ ^ [Y2: nat] :
% 9.51/5.31                  ( groups1707563613775114915nt_nat
% 9.51/5.31                  @ ^ [X3: int] : ( G @ X3 @ Y2 )
% 9.51/5.31                  @ ( collect_int
% 9.51/5.31                    @ ^ [X3: int] :
% 9.51/5.31                        ( ( member_int @ X3 @ A2 )
% 9.51/5.31                        & ( R @ X3 @ Y2 ) ) ) )
% 9.51/5.31              @ B2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8539_prod_Oswap__restrict,axiom,
% 9.51/5.31      ! [A2: set_complex,B2: set_nat,G: complex > nat > nat,R: complex > nat > $o] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ A2 )
% 9.51/5.31       => ( ( finite_finite_nat @ B2 )
% 9.51/5.31         => ( ( groups861055069439313189ex_nat
% 9.51/5.31              @ ^ [X3: complex] :
% 9.51/5.31                  ( groups708209901874060359at_nat @ ( G @ X3 )
% 9.51/5.31                  @ ( collect_nat
% 9.51/5.31                    @ ^ [Y2: nat] :
% 9.51/5.31                        ( ( member_nat @ Y2 @ B2 )
% 9.51/5.31                        & ( R @ X3 @ Y2 ) ) ) )
% 9.51/5.31              @ A2 )
% 9.51/5.31            = ( groups708209901874060359at_nat
% 9.51/5.31              @ ^ [Y2: nat] :
% 9.51/5.31                  ( groups861055069439313189ex_nat
% 9.51/5.31                  @ ^ [X3: complex] : ( G @ X3 @ Y2 )
% 9.51/5.31                  @ ( collect_complex
% 9.51/5.31                    @ ^ [X3: complex] :
% 9.51/5.31                        ( ( member_complex @ X3 @ A2 )
% 9.51/5.31                        & ( R @ X3 @ Y2 ) ) ) )
% 9.51/5.31              @ B2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8540_prod_Oswap__restrict,axiom,
% 9.51/5.31      ! [A2: set_nat,B2: set_real,G: nat > real > nat,R: nat > real > $o] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( finite_finite_real @ B2 )
% 9.51/5.31         => ( ( groups708209901874060359at_nat
% 9.51/5.31              @ ^ [X3: nat] :
% 9.51/5.31                  ( groups4696554848551431203al_nat @ ( G @ X3 )
% 9.51/5.31                  @ ( collect_real
% 9.51/5.31                    @ ^ [Y2: real] :
% 9.51/5.31                        ( ( member_real @ Y2 @ B2 )
% 9.51/5.31                        & ( R @ X3 @ Y2 ) ) ) )
% 9.51/5.31              @ A2 )
% 9.51/5.31            = ( groups4696554848551431203al_nat
% 9.51/5.31              @ ^ [Y2: real] :
% 9.51/5.31                  ( groups708209901874060359at_nat
% 9.51/5.31                  @ ^ [X3: nat] : ( G @ X3 @ Y2 )
% 9.51/5.31                  @ ( collect_nat
% 9.51/5.31                    @ ^ [X3: nat] :
% 9.51/5.31                        ( ( member_nat @ X3 @ A2 )
% 9.51/5.31                        & ( R @ X3 @ Y2 ) ) ) )
% 9.51/5.31              @ B2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8541_prod_Oswap__restrict,axiom,
% 9.51/5.31      ! [A2: set_nat,B2: set_int,G: nat > int > nat,R: nat > int > $o] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( finite_finite_int @ B2 )
% 9.51/5.31         => ( ( groups708209901874060359at_nat
% 9.51/5.31              @ ^ [X3: nat] :
% 9.51/5.31                  ( groups1707563613775114915nt_nat @ ( G @ X3 )
% 9.51/5.31                  @ ( collect_int
% 9.51/5.31                    @ ^ [Y2: int] :
% 9.51/5.31                        ( ( member_int @ Y2 @ B2 )
% 9.51/5.31                        & ( R @ X3 @ Y2 ) ) ) )
% 9.51/5.31              @ A2 )
% 9.51/5.31            = ( groups1707563613775114915nt_nat
% 9.51/5.31              @ ^ [Y2: int] :
% 9.51/5.31                  ( groups708209901874060359at_nat
% 9.51/5.31                  @ ^ [X3: nat] : ( G @ X3 @ Y2 )
% 9.51/5.31                  @ ( collect_nat
% 9.51/5.31                    @ ^ [X3: nat] :
% 9.51/5.31                        ( ( member_nat @ X3 @ A2 )
% 9.51/5.31                        & ( R @ X3 @ Y2 ) ) ) )
% 9.51/5.31              @ B2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8542_prod_Oswap__restrict,axiom,
% 9.51/5.31      ! [A2: set_nat,B2: set_complex,G: nat > complex > nat,R: nat > complex > $o] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( finite3207457112153483333omplex @ B2 )
% 9.51/5.31         => ( ( groups708209901874060359at_nat
% 9.51/5.31              @ ^ [X3: nat] :
% 9.51/5.31                  ( groups861055069439313189ex_nat @ ( G @ X3 )
% 9.51/5.31                  @ ( collect_complex
% 9.51/5.31                    @ ^ [Y2: complex] :
% 9.51/5.31                        ( ( member_complex @ Y2 @ B2 )
% 9.51/5.31                        & ( R @ X3 @ Y2 ) ) ) )
% 9.51/5.31              @ A2 )
% 9.51/5.31            = ( groups861055069439313189ex_nat
% 9.51/5.31              @ ^ [Y2: complex] :
% 9.51/5.31                  ( groups708209901874060359at_nat
% 9.51/5.31                  @ ^ [X3: nat] : ( G @ X3 @ Y2 )
% 9.51/5.31                  @ ( collect_nat
% 9.51/5.31                    @ ^ [X3: nat] :
% 9.51/5.31                        ( ( member_nat @ X3 @ A2 )
% 9.51/5.31                        & ( R @ X3 @ Y2 ) ) ) )
% 9.51/5.31              @ B2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8543_prod_Oswap__restrict,axiom,
% 9.51/5.31      ! [A2: set_nat,B2: set_nat,G: nat > nat > nat,R: nat > nat > $o] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( finite_finite_nat @ B2 )
% 9.51/5.31         => ( ( groups708209901874060359at_nat
% 9.51/5.31              @ ^ [X3: nat] :
% 9.51/5.31                  ( groups708209901874060359at_nat @ ( G @ X3 )
% 9.51/5.31                  @ ( collect_nat
% 9.51/5.31                    @ ^ [Y2: nat] :
% 9.51/5.31                        ( ( member_nat @ Y2 @ B2 )
% 9.51/5.31                        & ( R @ X3 @ Y2 ) ) ) )
% 9.51/5.31              @ A2 )
% 9.51/5.31            = ( groups708209901874060359at_nat
% 9.51/5.31              @ ^ [Y2: nat] :
% 9.51/5.31                  ( groups708209901874060359at_nat
% 9.51/5.31                  @ ^ [X3: nat] : ( G @ X3 @ Y2 )
% 9.51/5.31                  @ ( collect_nat
% 9.51/5.31                    @ ^ [X3: nat] :
% 9.51/5.31                        ( ( member_nat @ X3 @ A2 )
% 9.51/5.31                        & ( R @ X3 @ Y2 ) ) ) )
% 9.51/5.31              @ B2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8544_prod_Oswap__restrict,axiom,
% 9.51/5.31      ! [A2: set_list_nat,B2: set_nat,G: list_nat > nat > nat,R: list_nat > nat > $o] :
% 9.51/5.31        ( ( finite8100373058378681591st_nat @ A2 )
% 9.51/5.31       => ( ( finite_finite_nat @ B2 )
% 9.51/5.31         => ( ( groups2907647131375434839at_nat
% 9.51/5.31              @ ^ [X3: list_nat] :
% 9.51/5.31                  ( groups708209901874060359at_nat @ ( G @ X3 )
% 9.51/5.31                  @ ( collect_nat
% 9.51/5.31                    @ ^ [Y2: nat] :
% 9.51/5.31                        ( ( member_nat @ Y2 @ B2 )
% 9.51/5.31                        & ( R @ X3 @ Y2 ) ) ) )
% 9.51/5.31              @ A2 )
% 9.51/5.31            = ( groups708209901874060359at_nat
% 9.51/5.31              @ ^ [Y2: nat] :
% 9.51/5.31                  ( groups2907647131375434839at_nat
% 9.51/5.31                  @ ^ [X3: list_nat] : ( G @ X3 @ Y2 )
% 9.51/5.31                  @ ( collect_list_nat
% 9.51/5.31                    @ ^ [X3: list_nat] :
% 9.51/5.31                        ( ( member_list_nat @ X3 @ A2 )
% 9.51/5.31                        & ( R @ X3 @ Y2 ) ) ) )
% 9.51/5.31              @ B2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8545_prod_Oswap__restrict,axiom,
% 9.51/5.31      ! [A2: set_set_nat,B2: set_nat,G: set_nat > nat > nat,R: set_nat > nat > $o] :
% 9.51/5.31        ( ( finite1152437895449049373et_nat @ A2 )
% 9.51/5.31       => ( ( finite_finite_nat @ B2 )
% 9.51/5.31         => ( ( groups4248547760180025341at_nat
% 9.51/5.31              @ ^ [X3: set_nat] :
% 9.51/5.31                  ( groups708209901874060359at_nat @ ( G @ X3 )
% 9.51/5.31                  @ ( collect_nat
% 9.51/5.31                    @ ^ [Y2: nat] :
% 9.51/5.31                        ( ( member_nat @ Y2 @ B2 )
% 9.51/5.31                        & ( R @ X3 @ Y2 ) ) ) )
% 9.51/5.31              @ A2 )
% 9.51/5.31            = ( groups708209901874060359at_nat
% 9.51/5.31              @ ^ [Y2: nat] :
% 9.51/5.31                  ( groups4248547760180025341at_nat
% 9.51/5.31                  @ ^ [X3: set_nat] : ( G @ X3 @ Y2 )
% 9.51/5.31                  @ ( collect_set_nat
% 9.51/5.31                    @ ^ [X3: set_nat] :
% 9.51/5.31                        ( ( member_set_nat @ X3 @ A2 )
% 9.51/5.31                        & ( R @ X3 @ Y2 ) ) ) )
% 9.51/5.31              @ B2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8546_prod_Oswap__restrict,axiom,
% 9.51/5.31      ! [A2: set_nat,B2: set_list_nat,G: nat > list_nat > nat,R: nat > list_nat > $o] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( finite8100373058378681591st_nat @ B2 )
% 9.51/5.31         => ( ( groups708209901874060359at_nat
% 9.51/5.31              @ ^ [X3: nat] :
% 9.51/5.31                  ( groups2907647131375434839at_nat @ ( G @ X3 )
% 9.51/5.31                  @ ( collect_list_nat
% 9.51/5.31                    @ ^ [Y2: list_nat] :
% 9.51/5.31                        ( ( member_list_nat @ Y2 @ B2 )
% 9.51/5.31                        & ( R @ X3 @ Y2 ) ) ) )
% 9.51/5.31              @ A2 )
% 9.51/5.31            = ( groups2907647131375434839at_nat
% 9.51/5.31              @ ^ [Y2: list_nat] :
% 9.51/5.31                  ( groups708209901874060359at_nat
% 9.51/5.31                  @ ^ [X3: nat] : ( G @ X3 @ Y2 )
% 9.51/5.31                  @ ( collect_nat
% 9.51/5.31                    @ ^ [X3: nat] :
% 9.51/5.31                        ( ( member_nat @ X3 @ A2 )
% 9.51/5.31                        & ( R @ X3 @ Y2 ) ) ) )
% 9.51/5.31              @ B2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8547_prod_Olast__plus,axiom,
% 9.51/5.31      ! [M: nat,N: nat,G: nat > complex] :
% 9.51/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.31       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.51/5.31          = ( times_times_complex @ ( G @ N ) @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8548_prod_Olast__plus,axiom,
% 9.51/5.31      ! [M: nat,N: nat,G: nat > real] :
% 9.51/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.31       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.51/5.31          = ( times_times_real @ ( G @ N ) @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8549_prod_Olast__plus,axiom,
% 9.51/5.31      ! [M: nat,N: nat,G: nat > rat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.31       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.51/5.31          = ( times_times_rat @ ( G @ N ) @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8550_prod_Olast__plus,axiom,
% 9.51/5.31      ! [M: nat,N: nat,G: nat > int] :
% 9.51/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.31       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.51/5.31          = ( times_times_int @ ( G @ N ) @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8551_prod_Olast__plus,axiom,
% 9.51/5.31      ! [M: nat,N: nat,G: nat > nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.31       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.51/5.31          = ( times_times_nat @ ( G @ N ) @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8552_prod__Suc__Suc__fact,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 9.51/5.31        = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8553_prod__Suc__fact,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 9.51/5.31        = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8554_prod_Onested__swap,axiom,
% 9.51/5.31      ! [A: nat > nat > nat,N: nat] :
% 9.51/5.31        ( ( groups708209901874060359at_nat
% 9.51/5.31          @ ^ [I4: nat] : ( groups708209901874060359at_nat @ ( A @ I4 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I4 ) )
% 9.51/5.31          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.51/5.31        = ( groups708209901874060359at_nat
% 9.51/5.31          @ ^ [J3: nat] :
% 9.51/5.31              ( groups708209901874060359at_nat
% 9.51/5.31              @ ^ [I4: nat] : ( A @ I4 @ J3 )
% 9.51/5.31              @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
% 9.51/5.31          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8555_prod_Ohead__if,axiom,
% 9.51/5.31      ! [N: nat,M: nat,G: nat > complex] :
% 9.51/5.31        ( ( ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.51/5.31            = one_one_complex ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.51/5.31            = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8556_prod_Ohead__if,axiom,
% 9.51/5.31      ! [N: nat,M: nat,G: nat > real] :
% 9.51/5.31        ( ( ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.51/5.31            = one_one_real ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.51/5.31            = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8557_prod_Ohead__if,axiom,
% 9.51/5.31      ! [N: nat,M: nat,G: nat > rat] :
% 9.51/5.31        ( ( ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.51/5.31            = one_one_rat ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.51/5.31            = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8558_prod_Ohead__if,axiom,
% 9.51/5.31      ! [N: nat,M: nat,G: nat > int] :
% 9.51/5.31        ( ( ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.51/5.31            = one_one_int ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.51/5.31            = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8559_prod_Ohead__if,axiom,
% 9.51/5.31      ! [N: nat,M: nat,G: nat > nat] :
% 9.51/5.31        ( ( ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.51/5.31            = one_one_nat ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ N @ M )
% 9.51/5.31         => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.51/5.31            = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8560_prod__mono,axiom,
% 9.51/5.31      ! [A2: set_real,F: real > real,G: real > real] :
% 9.51/5.31        ( ! [I2: real] :
% 9.51/5.31            ( ( member_real @ I2 @ A2 )
% 9.51/5.31           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 9.51/5.31              & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 9.51/5.31       => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8561_prod__mono,axiom,
% 9.51/5.31      ! [A2: set_nat,F: nat > real,G: nat > real] :
% 9.51/5.31        ( ! [I2: nat] :
% 9.51/5.31            ( ( member_nat @ I2 @ A2 )
% 9.51/5.31           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 9.51/5.31              & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 9.51/5.31       => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8562_prod__mono,axiom,
% 9.51/5.31      ! [A2: set_int,F: int > real,G: int > real] :
% 9.51/5.31        ( ! [I2: int] :
% 9.51/5.31            ( ( member_int @ I2 @ A2 )
% 9.51/5.31           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 9.51/5.31              & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 9.51/5.31       => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8563_prod__mono,axiom,
% 9.51/5.31      ! [A2: set_real,F: real > rat,G: real > rat] :
% 9.51/5.31        ( ! [I2: real] :
% 9.51/5.31            ( ( member_real @ I2 @ A2 )
% 9.51/5.31           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
% 9.51/5.31              & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 9.51/5.31       => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8564_prod__mono,axiom,
% 9.51/5.31      ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 9.51/5.31        ( ! [I2: nat] :
% 9.51/5.31            ( ( member_nat @ I2 @ A2 )
% 9.51/5.31           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
% 9.51/5.31              & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 9.51/5.31       => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8565_prod__mono,axiom,
% 9.51/5.31      ! [A2: set_int,F: int > rat,G: int > rat] :
% 9.51/5.31        ( ! [I2: int] :
% 9.51/5.31            ( ( member_int @ I2 @ A2 )
% 9.51/5.31           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
% 9.51/5.31              & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 9.51/5.31       => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8566_prod__mono,axiom,
% 9.51/5.31      ! [A2: set_real,F: real > nat,G: real > nat] :
% 9.51/5.31        ( ! [I2: real] :
% 9.51/5.31            ( ( member_real @ I2 @ A2 )
% 9.51/5.31           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
% 9.51/5.31              & ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 9.51/5.31       => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8567_prod__mono,axiom,
% 9.51/5.31      ! [A2: set_int,F: int > nat,G: int > nat] :
% 9.51/5.31        ( ! [I2: int] :
% 9.51/5.31            ( ( member_int @ I2 @ A2 )
% 9.51/5.31           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
% 9.51/5.31              & ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 9.51/5.31       => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8568_prod__mono,axiom,
% 9.51/5.31      ! [A2: set_real,F: real > int,G: real > int] :
% 9.51/5.31        ( ! [I2: real] :
% 9.51/5.31            ( ( member_real @ I2 @ A2 )
% 9.51/5.31           => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I2 ) )
% 9.51/5.31              & ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 9.51/5.31       => ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ G @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8569_prod__mono,axiom,
% 9.51/5.31      ! [A2: set_nat,F: nat > int,G: nat > int] :
% 9.51/5.31        ( ! [I2: nat] :
% 9.51/5.31            ( ( member_nat @ I2 @ A2 )
% 9.51/5.31           => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I2 ) )
% 9.51/5.31              & ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 9.51/5.31       => ( ord_less_eq_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ G @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8570_prod__nonneg,axiom,
% 9.51/5.31      ! [A2: set_nat,F: nat > nat] :
% 9.51/5.31        ( ! [X4: nat] :
% 9.51/5.31            ( ( member_nat @ X4 @ A2 )
% 9.51/5.31           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.51/5.31       => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8571_prod__pos,axiom,
% 9.51/5.31      ! [A2: set_nat,F: nat > nat] :
% 9.51/5.31        ( ! [X4: nat] :
% 9.51/5.31            ( ( member_nat @ X4 @ A2 )
% 9.51/5.31           => ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.51/5.31       => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8572_prod__ge__1,axiom,
% 9.51/5.31      ! [A2: set_real,F: real > real] :
% 9.51/5.31        ( ! [X4: real] :
% 9.51/5.31            ( ( member_real @ X4 @ A2 )
% 9.51/5.31           => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
% 9.51/5.31       => ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8573_prod__ge__1,axiom,
% 9.51/5.31      ! [A2: set_nat,F: nat > real] :
% 9.51/5.31        ( ! [X4: nat] :
% 9.51/5.31            ( ( member_nat @ X4 @ A2 )
% 9.51/5.31           => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
% 9.51/5.31       => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8574_prod__ge__1,axiom,
% 9.51/5.31      ! [A2: set_int,F: int > real] :
% 9.51/5.31        ( ! [X4: int] :
% 9.51/5.31            ( ( member_int @ X4 @ A2 )
% 9.51/5.31           => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
% 9.51/5.31       => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8575_prod__ge__1,axiom,
% 9.51/5.31      ! [A2: set_real,F: real > rat] :
% 9.51/5.31        ( ! [X4: real] :
% 9.51/5.31            ( ( member_real @ X4 @ A2 )
% 9.51/5.31           => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
% 9.51/5.31       => ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8576_prod__ge__1,axiom,
% 9.51/5.31      ! [A2: set_nat,F: nat > rat] :
% 9.51/5.31        ( ! [X4: nat] :
% 9.51/5.31            ( ( member_nat @ X4 @ A2 )
% 9.51/5.31           => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
% 9.51/5.31       => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8577_prod__ge__1,axiom,
% 9.51/5.31      ! [A2: set_int,F: int > rat] :
% 9.51/5.31        ( ! [X4: int] :
% 9.51/5.31            ( ( member_int @ X4 @ A2 )
% 9.51/5.31           => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
% 9.51/5.31       => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8578_prod__ge__1,axiom,
% 9.51/5.31      ! [A2: set_real,F: real > nat] :
% 9.51/5.31        ( ! [X4: real] :
% 9.51/5.31            ( ( member_real @ X4 @ A2 )
% 9.51/5.31           => ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
% 9.51/5.31       => ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8579_prod__ge__1,axiom,
% 9.51/5.31      ! [A2: set_int,F: int > nat] :
% 9.51/5.31        ( ! [X4: int] :
% 9.51/5.31            ( ( member_int @ X4 @ A2 )
% 9.51/5.31           => ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
% 9.51/5.31       => ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8580_prod__ge__1,axiom,
% 9.51/5.31      ! [A2: set_real,F: real > int] :
% 9.51/5.31        ( ! [X4: real] :
% 9.51/5.31            ( ( member_real @ X4 @ A2 )
% 9.51/5.31           => ( ord_less_eq_int @ one_one_int @ ( F @ X4 ) ) )
% 9.51/5.31       => ( ord_less_eq_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8581_prod__ge__1,axiom,
% 9.51/5.31      ! [A2: set_nat,F: nat > int] :
% 9.51/5.31        ( ! [X4: nat] :
% 9.51/5.31            ( ( member_nat @ X4 @ A2 )
% 9.51/5.31           => ( ord_less_eq_int @ one_one_int @ ( F @ X4 ) ) )
% 9.51/5.31       => ( ord_less_eq_int @ one_one_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8582_fact__prod__Suc,axiom,
% 9.51/5.31      ( semiri1406184849735516958ct_int
% 9.51/5.31      = ( ^ [N2: nat] : ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8583_fact__prod__Suc,axiom,
% 9.51/5.31      ( semiri2265585572941072030t_real
% 9.51/5.31      = ( ^ [N2: nat] : ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8584_fact__prod__Suc,axiom,
% 9.51/5.31      ( semiri1408675320244567234ct_nat
% 9.51/5.31      = ( ^ [N2: nat] : ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8585_prod__zero,axiom,
% 9.51/5.31      ! [A2: set_nat,F: nat > complex] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ? [X2: nat] :
% 9.51/5.31              ( ( member_nat @ X2 @ A2 )
% 9.51/5.31              & ( ( F @ X2 )
% 9.51/5.31                = zero_zero_complex ) )
% 9.51/5.31         => ( ( groups6464643781859351333omplex @ F @ A2 )
% 9.51/5.31            = zero_zero_complex ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8586_prod__zero,axiom,
% 9.51/5.31      ! [A2: set_int,F: int > complex] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ? [X2: int] :
% 9.51/5.31              ( ( member_int @ X2 @ A2 )
% 9.51/5.31              & ( ( F @ X2 )
% 9.51/5.31                = zero_zero_complex ) )
% 9.51/5.31         => ( ( groups7440179247065528705omplex @ F @ A2 )
% 9.51/5.31            = zero_zero_complex ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8587_prod__zero,axiom,
% 9.51/5.31      ! [A2: set_complex,F: complex > complex] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ A2 )
% 9.51/5.31       => ( ? [X2: complex] :
% 9.51/5.31              ( ( member_complex @ X2 @ A2 )
% 9.51/5.31              & ( ( F @ X2 )
% 9.51/5.31                = zero_zero_complex ) )
% 9.51/5.31         => ( ( groups3708469109370488835omplex @ F @ A2 )
% 9.51/5.31            = zero_zero_complex ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8588_prod__zero,axiom,
% 9.51/5.31      ! [A2: set_nat,F: nat > real] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ? [X2: nat] :
% 9.51/5.31              ( ( member_nat @ X2 @ A2 )
% 9.51/5.31              & ( ( F @ X2 )
% 9.51/5.31                = zero_zero_real ) )
% 9.51/5.31         => ( ( groups129246275422532515t_real @ F @ A2 )
% 9.51/5.31            = zero_zero_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8589_prod__zero,axiom,
% 9.51/5.31      ! [A2: set_int,F: int > real] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ? [X2: int] :
% 9.51/5.31              ( ( member_int @ X2 @ A2 )
% 9.51/5.31              & ( ( F @ X2 )
% 9.51/5.31                = zero_zero_real ) )
% 9.51/5.31         => ( ( groups2316167850115554303t_real @ F @ A2 )
% 9.51/5.31            = zero_zero_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8590_prod__zero,axiom,
% 9.51/5.31      ! [A2: set_complex,F: complex > real] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ A2 )
% 9.51/5.31       => ( ? [X2: complex] :
% 9.51/5.31              ( ( member_complex @ X2 @ A2 )
% 9.51/5.31              & ( ( F @ X2 )
% 9.51/5.31                = zero_zero_real ) )
% 9.51/5.31         => ( ( groups766887009212190081x_real @ F @ A2 )
% 9.51/5.31            = zero_zero_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8591_prod__zero,axiom,
% 9.51/5.31      ! [A2: set_nat,F: nat > rat] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ? [X2: nat] :
% 9.51/5.31              ( ( member_nat @ X2 @ A2 )
% 9.51/5.31              & ( ( F @ X2 )
% 9.51/5.31                = zero_zero_rat ) )
% 9.51/5.31         => ( ( groups73079841787564623at_rat @ F @ A2 )
% 9.51/5.31            = zero_zero_rat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8592_prod__zero,axiom,
% 9.51/5.31      ! [A2: set_int,F: int > rat] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ? [X2: int] :
% 9.51/5.31              ( ( member_int @ X2 @ A2 )
% 9.51/5.31              & ( ( F @ X2 )
% 9.51/5.31                = zero_zero_rat ) )
% 9.51/5.31         => ( ( groups1072433553688619179nt_rat @ F @ A2 )
% 9.51/5.31            = zero_zero_rat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8593_prod__zero,axiom,
% 9.51/5.31      ! [A2: set_complex,F: complex > rat] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ A2 )
% 9.51/5.31       => ( ? [X2: complex] :
% 9.51/5.31              ( ( member_complex @ X2 @ A2 )
% 9.51/5.31              & ( ( F @ X2 )
% 9.51/5.31                = zero_zero_rat ) )
% 9.51/5.31         => ( ( groups225925009352817453ex_rat @ F @ A2 )
% 9.51/5.31            = zero_zero_rat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8594_prod__zero,axiom,
% 9.51/5.31      ! [A2: set_int,F: int > nat] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ? [X2: int] :
% 9.51/5.31              ( ( member_int @ X2 @ A2 )
% 9.51/5.31              & ( ( F @ X2 )
% 9.51/5.31                = zero_zero_nat ) )
% 9.51/5.31         => ( ( groups1707563613775114915nt_nat @ F @ A2 )
% 9.51/5.31            = zero_zero_nat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8595_atLeastLessThan__subset__iff,axiom,
% 9.51/5.31      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.51/5.31        ( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ A @ B ) @ ( set_or4029947393144176647an_rat @ C @ D ) )
% 9.51/5.31       => ( ( ord_less_eq_rat @ B @ A )
% 9.51/5.31          | ( ( ord_less_eq_rat @ C @ A )
% 9.51/5.31            & ( ord_less_eq_rat @ B @ D ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8596_atLeastLessThan__subset__iff,axiom,
% 9.51/5.31      ! [A: num,B: num,C: num,D: num] :
% 9.51/5.31        ( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ A @ B ) @ ( set_or1222409239386451017an_num @ C @ D ) )
% 9.51/5.31       => ( ( ord_less_eq_num @ B @ A )
% 9.51/5.31          | ( ( ord_less_eq_num @ C @ A )
% 9.51/5.31            & ( ord_less_eq_num @ B @ D ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8597_atLeastLessThan__subset__iff,axiom,
% 9.51/5.31      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.51/5.31        ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
% 9.51/5.31       => ( ( ord_less_eq_nat @ B @ A )
% 9.51/5.31          | ( ( ord_less_eq_nat @ C @ A )
% 9.51/5.31            & ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8598_atLeastLessThan__subset__iff,axiom,
% 9.51/5.31      ! [A: int,B: int,C: int,D: int] :
% 9.51/5.31        ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D ) )
% 9.51/5.31       => ( ( ord_less_eq_int @ B @ A )
% 9.51/5.31          | ( ( ord_less_eq_int @ C @ A )
% 9.51/5.31            & ( ord_less_eq_int @ B @ D ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8599_infinite__Ico,axiom,
% 9.51/5.31      ! [A: real,B: real] :
% 9.51/5.31        ( ( ord_less_real @ A @ B )
% 9.51/5.31       => ~ ( finite_finite_real @ ( set_or66887138388493659n_real @ A @ B ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8600_infinite__Ico,axiom,
% 9.51/5.31      ! [A: rat,B: rat] :
% 9.51/5.31        ( ( ord_less_rat @ A @ B )
% 9.51/5.31       => ~ ( finite_finite_rat @ ( set_or4029947393144176647an_rat @ A @ B ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8601_all__nat__less__eq,axiom,
% 9.51/5.31      ! [N: nat,P: nat > $o] :
% 9.51/5.31        ( ( ! [M2: nat] :
% 9.51/5.31              ( ( ord_less_nat @ M2 @ N )
% 9.51/5.31             => ( P @ M2 ) ) )
% 9.51/5.31        = ( ! [X3: nat] :
% 9.51/5.31              ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 9.51/5.31             => ( P @ X3 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8602_ex__nat__less__eq,axiom,
% 9.51/5.31      ! [N: nat,P: nat > $o] :
% 9.51/5.31        ( ( ? [M2: nat] :
% 9.51/5.31              ( ( ord_less_nat @ M2 @ N )
% 9.51/5.31              & ( P @ M2 ) ) )
% 9.51/5.31        = ( ? [X3: nat] :
% 9.51/5.31              ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 9.51/5.31              & ( P @ X3 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8603_fact__prod__rev,axiom,
% 9.51/5.31      ( semiri1406184849735516958ct_int
% 9.51/5.31      = ( ^ [N2: nat] : ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ ( minus_minus_nat @ N2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8604_fact__prod__rev,axiom,
% 9.51/5.31      ( semiri2265585572941072030t_real
% 9.51/5.31      = ( ^ [N2: nat] : ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ ( minus_minus_nat @ N2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8605_fact__prod__rev,axiom,
% 9.51/5.31      ( semiri1408675320244567234ct_nat
% 9.51/5.31      = ( ^ [N2: nat] : ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ ( minus_minus_nat @ N2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8606_pochhammer__prod,axiom,
% 9.51/5.31      ( comm_s4028243227959126397er_rat
% 9.51/5.31      = ( ^ [A4: rat,N2: nat] :
% 9.51/5.31            ( groups73079841787564623at_rat
% 9.51/5.31            @ ^ [I4: nat] : ( plus_plus_rat @ A4 @ ( semiri681578069525770553at_rat @ I4 ) )
% 9.51/5.31            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8607_pochhammer__prod,axiom,
% 9.51/5.31      ( comm_s4660882817536571857er_int
% 9.51/5.31      = ( ^ [A4: int,N2: nat] :
% 9.51/5.31            ( groups705719431365010083at_int
% 9.51/5.31            @ ^ [I4: nat] : ( plus_plus_int @ A4 @ ( semiri1314217659103216013at_int @ I4 ) )
% 9.51/5.31            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8608_pochhammer__prod,axiom,
% 9.51/5.31      ( comm_s7457072308508201937r_real
% 9.51/5.31      = ( ^ [A4: real,N2: nat] :
% 9.51/5.31            ( groups129246275422532515t_real
% 9.51/5.31            @ ^ [I4: nat] : ( plus_plus_real @ A4 @ ( semiri5074537144036343181t_real @ I4 ) )
% 9.51/5.31            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8609_pochhammer__prod,axiom,
% 9.51/5.31      ( comm_s4663373288045622133er_nat
% 9.51/5.31      = ( ^ [A4: nat,N2: nat] :
% 9.51/5.31            ( groups708209901874060359at_nat
% 9.51/5.31            @ ^ [I4: nat] : ( plus_plus_nat @ A4 @ ( semiri1316708129612266289at_nat @ I4 ) )
% 9.51/5.31            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8610_lessThan__atLeast0,axiom,
% 9.51/5.31      ( set_ord_lessThan_nat
% 9.51/5.31      = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8611_prod_Ointer__filter,axiom,
% 9.51/5.31      ! [A2: set_real,G: real > complex,P: real > $o] :
% 9.51/5.31        ( ( finite_finite_real @ A2 )
% 9.51/5.31       => ( ( groups713298508707869441omplex @ G
% 9.51/5.31            @ ( collect_real
% 9.51/5.31              @ ^ [X3: real] :
% 9.51/5.31                  ( ( member_real @ X3 @ A2 )
% 9.51/5.31                  & ( P @ X3 ) ) ) )
% 9.51/5.31          = ( groups713298508707869441omplex
% 9.51/5.31            @ ^ [X3: real] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_complex )
% 9.51/5.31            @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8612_prod_Ointer__filter,axiom,
% 9.51/5.31      ! [A2: set_nat,G: nat > complex,P: nat > $o] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( groups6464643781859351333omplex @ G
% 9.51/5.31            @ ( collect_nat
% 9.51/5.31              @ ^ [X3: nat] :
% 9.51/5.31                  ( ( member_nat @ X3 @ A2 )
% 9.51/5.31                  & ( P @ X3 ) ) ) )
% 9.51/5.31          = ( groups6464643781859351333omplex
% 9.51/5.31            @ ^ [X3: nat] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_complex )
% 9.51/5.31            @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8613_prod_Ointer__filter,axiom,
% 9.51/5.31      ! [A2: set_int,G: int > complex,P: int > $o] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ( groups7440179247065528705omplex @ G
% 9.51/5.31            @ ( collect_int
% 9.51/5.31              @ ^ [X3: int] :
% 9.51/5.31                  ( ( member_int @ X3 @ A2 )
% 9.51/5.31                  & ( P @ X3 ) ) ) )
% 9.51/5.31          = ( groups7440179247065528705omplex
% 9.51/5.31            @ ^ [X3: int] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_complex )
% 9.51/5.31            @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8614_prod_Ointer__filter,axiom,
% 9.51/5.31      ! [A2: set_complex,G: complex > complex,P: complex > $o] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ A2 )
% 9.51/5.31       => ( ( groups3708469109370488835omplex @ G
% 9.51/5.31            @ ( collect_complex
% 9.51/5.31              @ ^ [X3: complex] :
% 9.51/5.31                  ( ( member_complex @ X3 @ A2 )
% 9.51/5.31                  & ( P @ X3 ) ) ) )
% 9.51/5.31          = ( groups3708469109370488835omplex
% 9.51/5.31            @ ^ [X3: complex] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_complex )
% 9.51/5.31            @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8615_prod_Ointer__filter,axiom,
% 9.51/5.31      ! [A2: set_real,G: real > real,P: real > $o] :
% 9.51/5.31        ( ( finite_finite_real @ A2 )
% 9.51/5.31       => ( ( groups1681761925125756287l_real @ G
% 9.51/5.31            @ ( collect_real
% 9.51/5.31              @ ^ [X3: real] :
% 9.51/5.31                  ( ( member_real @ X3 @ A2 )
% 9.51/5.31                  & ( P @ X3 ) ) ) )
% 9.51/5.31          = ( groups1681761925125756287l_real
% 9.51/5.31            @ ^ [X3: real] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_real )
% 9.51/5.31            @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8616_prod_Ointer__filter,axiom,
% 9.51/5.31      ! [A2: set_nat,G: nat > real,P: nat > $o] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( groups129246275422532515t_real @ G
% 9.51/5.31            @ ( collect_nat
% 9.51/5.31              @ ^ [X3: nat] :
% 9.51/5.31                  ( ( member_nat @ X3 @ A2 )
% 9.51/5.31                  & ( P @ X3 ) ) ) )
% 9.51/5.31          = ( groups129246275422532515t_real
% 9.51/5.31            @ ^ [X3: nat] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_real )
% 9.51/5.31            @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8617_prod_Ointer__filter,axiom,
% 9.51/5.31      ! [A2: set_int,G: int > real,P: int > $o] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ( groups2316167850115554303t_real @ G
% 9.51/5.31            @ ( collect_int
% 9.51/5.31              @ ^ [X3: int] :
% 9.51/5.31                  ( ( member_int @ X3 @ A2 )
% 9.51/5.31                  & ( P @ X3 ) ) ) )
% 9.51/5.31          = ( groups2316167850115554303t_real
% 9.51/5.31            @ ^ [X3: int] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_real )
% 9.51/5.31            @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8618_prod_Ointer__filter,axiom,
% 9.51/5.31      ! [A2: set_complex,G: complex > real,P: complex > $o] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ A2 )
% 9.51/5.31       => ( ( groups766887009212190081x_real @ G
% 9.51/5.31            @ ( collect_complex
% 9.51/5.31              @ ^ [X3: complex] :
% 9.51/5.31                  ( ( member_complex @ X3 @ A2 )
% 9.51/5.31                  & ( P @ X3 ) ) ) )
% 9.51/5.31          = ( groups766887009212190081x_real
% 9.51/5.31            @ ^ [X3: complex] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_real )
% 9.51/5.31            @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8619_prod_Ointer__filter,axiom,
% 9.51/5.31      ! [A2: set_real,G: real > rat,P: real > $o] :
% 9.51/5.31        ( ( finite_finite_real @ A2 )
% 9.51/5.31       => ( ( groups4061424788464935467al_rat @ G
% 9.51/5.31            @ ( collect_real
% 9.51/5.31              @ ^ [X3: real] :
% 9.51/5.31                  ( ( member_real @ X3 @ A2 )
% 9.51/5.31                  & ( P @ X3 ) ) ) )
% 9.51/5.31          = ( groups4061424788464935467al_rat
% 9.51/5.31            @ ^ [X3: real] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_rat )
% 9.51/5.31            @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8620_prod_Ointer__filter,axiom,
% 9.51/5.31      ! [A2: set_nat,G: nat > rat,P: nat > $o] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( groups73079841787564623at_rat @ G
% 9.51/5.31            @ ( collect_nat
% 9.51/5.31              @ ^ [X3: nat] :
% 9.51/5.31                  ( ( member_nat @ X3 @ A2 )
% 9.51/5.31                  & ( P @ X3 ) ) ) )
% 9.51/5.31          = ( groups73079841787564623at_rat
% 9.51/5.31            @ ^ [X3: nat] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_rat )
% 9.51/5.31            @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8621_atLeastLessThan0,axiom,
% 9.51/5.31      ! [M: nat] :
% 9.51/5.31        ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
% 9.51/5.31        = bot_bot_set_nat ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8622_prod__le__1,axiom,
% 9.51/5.31      ! [A2: set_real,F: real > real] :
% 9.51/5.31        ( ! [X4: real] :
% 9.51/5.31            ( ( member_real @ X4 @ A2 )
% 9.51/5.31           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 9.51/5.31              & ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
% 9.51/5.31       => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ one_one_real ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8623_prod__le__1,axiom,
% 9.51/5.31      ! [A2: set_nat,F: nat > real] :
% 9.51/5.31        ( ! [X4: nat] :
% 9.51/5.31            ( ( member_nat @ X4 @ A2 )
% 9.51/5.31           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 9.51/5.31              & ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
% 9.51/5.31       => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ one_one_real ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8624_prod__le__1,axiom,
% 9.51/5.31      ! [A2: set_int,F: int > real] :
% 9.51/5.31        ( ! [X4: int] :
% 9.51/5.31            ( ( member_int @ X4 @ A2 )
% 9.51/5.31           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 9.51/5.31              & ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
% 9.51/5.31       => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ one_one_real ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8625_prod__le__1,axiom,
% 9.51/5.31      ! [A2: set_real,F: real > rat] :
% 9.51/5.31        ( ! [X4: real] :
% 9.51/5.31            ( ( member_real @ X4 @ A2 )
% 9.51/5.31           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
% 9.51/5.31              & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
% 9.51/5.31       => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8626_prod__le__1,axiom,
% 9.51/5.31      ! [A2: set_nat,F: nat > rat] :
% 9.51/5.31        ( ! [X4: nat] :
% 9.51/5.31            ( ( member_nat @ X4 @ A2 )
% 9.51/5.31           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
% 9.51/5.31              & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
% 9.51/5.31       => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8627_prod__le__1,axiom,
% 9.51/5.31      ! [A2: set_int,F: int > rat] :
% 9.51/5.31        ( ! [X4: int] :
% 9.51/5.31            ( ( member_int @ X4 @ A2 )
% 9.51/5.31           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
% 9.51/5.31              & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
% 9.51/5.31       => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8628_prod__le__1,axiom,
% 9.51/5.31      ! [A2: set_real,F: real > nat] :
% 9.51/5.31        ( ! [X4: real] :
% 9.51/5.31            ( ( member_real @ X4 @ A2 )
% 9.51/5.31           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
% 9.51/5.31              & ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
% 9.51/5.31       => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ one_one_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8629_prod__le__1,axiom,
% 9.51/5.31      ! [A2: set_int,F: int > nat] :
% 9.51/5.31        ( ! [X4: int] :
% 9.51/5.31            ( ( member_int @ X4 @ A2 )
% 9.51/5.31           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
% 9.51/5.31              & ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
% 9.51/5.31       => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ one_one_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8630_prod__le__1,axiom,
% 9.51/5.31      ! [A2: set_real,F: real > int] :
% 9.51/5.31        ( ! [X4: real] :
% 9.51/5.31            ( ( member_real @ X4 @ A2 )
% 9.51/5.31           => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) )
% 9.51/5.31              & ( ord_less_eq_int @ ( F @ X4 ) @ one_one_int ) ) )
% 9.51/5.31       => ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ one_one_int ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8631_prod__le__1,axiom,
% 9.51/5.31      ! [A2: set_nat,F: nat > int] :
% 9.51/5.31        ( ! [X4: nat] :
% 9.51/5.31            ( ( member_nat @ X4 @ A2 )
% 9.51/5.31           => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) )
% 9.51/5.31              & ( ord_less_eq_int @ ( F @ X4 ) @ one_one_int ) ) )
% 9.51/5.31       => ( ord_less_eq_int @ ( groups705719431365010083at_int @ F @ A2 ) @ one_one_int ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8632_is__singletonI_H,axiom,
% 9.51/5.31      ! [A2: set_set_nat] :
% 9.51/5.31        ( ( A2 != bot_bot_set_set_nat )
% 9.51/5.31       => ( ! [X4: set_nat,Y3: set_nat] :
% 9.51/5.31              ( ( member_set_nat @ X4 @ A2 )
% 9.51/5.31             => ( ( member_set_nat @ Y3 @ A2 )
% 9.51/5.31               => ( X4 = Y3 ) ) )
% 9.51/5.31         => ( is_singleton_set_nat @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8633_is__singletonI_H,axiom,
% 9.51/5.31      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.51/5.31        ( ( A2 != bot_bo2099793752762293965at_nat )
% 9.51/5.31       => ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 9.51/5.31              ( ( member8440522571783428010at_nat @ X4 @ A2 )
% 9.51/5.31             => ( ( member8440522571783428010at_nat @ Y3 @ A2 )
% 9.51/5.31               => ( X4 = Y3 ) ) )
% 9.51/5.31         => ( is_sin2850979758926227957at_nat @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8634_is__singletonI_H,axiom,
% 9.51/5.31      ! [A2: set_real] :
% 9.51/5.31        ( ( A2 != bot_bot_set_real )
% 9.51/5.31       => ( ! [X4: real,Y3: real] :
% 9.51/5.31              ( ( member_real @ X4 @ A2 )
% 9.51/5.31             => ( ( member_real @ Y3 @ A2 )
% 9.51/5.31               => ( X4 = Y3 ) ) )
% 9.51/5.31         => ( is_singleton_real @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8635_is__singletonI_H,axiom,
% 9.51/5.31      ! [A2: set_nat] :
% 9.51/5.31        ( ( A2 != bot_bot_set_nat )
% 9.51/5.31       => ( ! [X4: nat,Y3: nat] :
% 9.51/5.31              ( ( member_nat @ X4 @ A2 )
% 9.51/5.31             => ( ( member_nat @ Y3 @ A2 )
% 9.51/5.31               => ( X4 = Y3 ) ) )
% 9.51/5.31         => ( is_singleton_nat @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8636_is__singletonI_H,axiom,
% 9.51/5.31      ! [A2: set_int] :
% 9.51/5.31        ( ( A2 != bot_bot_set_int )
% 9.51/5.31       => ( ! [X4: int,Y3: int] :
% 9.51/5.31              ( ( member_int @ X4 @ A2 )
% 9.51/5.31             => ( ( member_int @ Y3 @ A2 )
% 9.51/5.31               => ( X4 = Y3 ) ) )
% 9.51/5.31         => ( is_singleton_int @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8637_prod_Orelated,axiom,
% 9.51/5.31      ! [R: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
% 9.51/5.31        ( ( R @ one_one_complex @ one_one_complex )
% 9.51/5.31       => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
% 9.51/5.31              ( ( ( R @ X15 @ X23 )
% 9.51/5.31                & ( R @ Y15 @ Y23 ) )
% 9.51/5.31             => ( R @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
% 9.51/5.31         => ( ( finite_finite_nat @ S2 )
% 9.51/5.31           => ( ! [X4: nat] :
% 9.51/5.31                  ( ( member_nat @ X4 @ S2 )
% 9.51/5.31                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.51/5.31             => ( R @ ( groups6464643781859351333omplex @ H2 @ S2 ) @ ( groups6464643781859351333omplex @ G @ S2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8638_prod_Orelated,axiom,
% 9.51/5.31      ! [R: complex > complex > $o,S2: set_int,H2: int > complex,G: int > complex] :
% 9.51/5.31        ( ( R @ one_one_complex @ one_one_complex )
% 9.51/5.31       => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
% 9.51/5.31              ( ( ( R @ X15 @ X23 )
% 9.51/5.31                & ( R @ Y15 @ Y23 ) )
% 9.51/5.31             => ( R @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
% 9.51/5.31         => ( ( finite_finite_int @ S2 )
% 9.51/5.31           => ( ! [X4: int] :
% 9.51/5.31                  ( ( member_int @ X4 @ S2 )
% 9.51/5.31                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.51/5.31             => ( R @ ( groups7440179247065528705omplex @ H2 @ S2 ) @ ( groups7440179247065528705omplex @ G @ S2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8639_prod_Orelated,axiom,
% 9.51/5.31      ! [R: complex > complex > $o,S2: set_complex,H2: complex > complex,G: complex > complex] :
% 9.51/5.31        ( ( R @ one_one_complex @ one_one_complex )
% 9.51/5.31       => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
% 9.51/5.31              ( ( ( R @ X15 @ X23 )
% 9.51/5.31                & ( R @ Y15 @ Y23 ) )
% 9.51/5.31             => ( R @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
% 9.51/5.31         => ( ( finite3207457112153483333omplex @ S2 )
% 9.51/5.31           => ( ! [X4: complex] :
% 9.51/5.31                  ( ( member_complex @ X4 @ S2 )
% 9.51/5.31                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.51/5.31             => ( R @ ( groups3708469109370488835omplex @ H2 @ S2 ) @ ( groups3708469109370488835omplex @ G @ S2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8640_prod_Orelated,axiom,
% 9.51/5.31      ! [R: real > real > $o,S2: set_nat,H2: nat > real,G: nat > real] :
% 9.51/5.31        ( ( R @ one_one_real @ one_one_real )
% 9.51/5.31       => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
% 9.51/5.31              ( ( ( R @ X15 @ X23 )
% 9.51/5.31                & ( R @ Y15 @ Y23 ) )
% 9.51/5.31             => ( R @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
% 9.51/5.31         => ( ( finite_finite_nat @ S2 )
% 9.51/5.31           => ( ! [X4: nat] :
% 9.51/5.31                  ( ( member_nat @ X4 @ S2 )
% 9.51/5.31                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.51/5.31             => ( R @ ( groups129246275422532515t_real @ H2 @ S2 ) @ ( groups129246275422532515t_real @ G @ S2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8641_prod_Orelated,axiom,
% 9.51/5.31      ! [R: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
% 9.51/5.31        ( ( R @ one_one_real @ one_one_real )
% 9.51/5.31       => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
% 9.51/5.31              ( ( ( R @ X15 @ X23 )
% 9.51/5.31                & ( R @ Y15 @ Y23 ) )
% 9.51/5.31             => ( R @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
% 9.51/5.31         => ( ( finite_finite_int @ S2 )
% 9.51/5.31           => ( ! [X4: int] :
% 9.51/5.31                  ( ( member_int @ X4 @ S2 )
% 9.51/5.31                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.51/5.31             => ( R @ ( groups2316167850115554303t_real @ H2 @ S2 ) @ ( groups2316167850115554303t_real @ G @ S2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8642_prod_Orelated,axiom,
% 9.51/5.31      ! [R: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
% 9.51/5.31        ( ( R @ one_one_real @ one_one_real )
% 9.51/5.31       => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
% 9.51/5.31              ( ( ( R @ X15 @ X23 )
% 9.51/5.31                & ( R @ Y15 @ Y23 ) )
% 9.51/5.31             => ( R @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
% 9.51/5.31         => ( ( finite3207457112153483333omplex @ S2 )
% 9.51/5.31           => ( ! [X4: complex] :
% 9.51/5.31                  ( ( member_complex @ X4 @ S2 )
% 9.51/5.31                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.51/5.31             => ( R @ ( groups766887009212190081x_real @ H2 @ S2 ) @ ( groups766887009212190081x_real @ G @ S2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8643_prod_Orelated,axiom,
% 9.51/5.31      ! [R: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
% 9.51/5.31        ( ( R @ one_one_rat @ one_one_rat )
% 9.51/5.31       => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
% 9.51/5.31              ( ( ( R @ X15 @ X23 )
% 9.51/5.31                & ( R @ Y15 @ Y23 ) )
% 9.51/5.31             => ( R @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
% 9.51/5.31         => ( ( finite_finite_nat @ S2 )
% 9.51/5.31           => ( ! [X4: nat] :
% 9.51/5.31                  ( ( member_nat @ X4 @ S2 )
% 9.51/5.31                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.51/5.31             => ( R @ ( groups73079841787564623at_rat @ H2 @ S2 ) @ ( groups73079841787564623at_rat @ G @ S2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8644_prod_Orelated,axiom,
% 9.51/5.31      ! [R: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
% 9.51/5.31        ( ( R @ one_one_rat @ one_one_rat )
% 9.51/5.31       => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
% 9.51/5.31              ( ( ( R @ X15 @ X23 )
% 9.51/5.31                & ( R @ Y15 @ Y23 ) )
% 9.51/5.31             => ( R @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
% 9.51/5.31         => ( ( finite_finite_int @ S2 )
% 9.51/5.31           => ( ! [X4: int] :
% 9.51/5.31                  ( ( member_int @ X4 @ S2 )
% 9.51/5.31                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.51/5.31             => ( R @ ( groups1072433553688619179nt_rat @ H2 @ S2 ) @ ( groups1072433553688619179nt_rat @ G @ S2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8645_prod_Orelated,axiom,
% 9.51/5.31      ! [R: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
% 9.51/5.31        ( ( R @ one_one_rat @ one_one_rat )
% 9.51/5.31       => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
% 9.51/5.31              ( ( ( R @ X15 @ X23 )
% 9.51/5.31                & ( R @ Y15 @ Y23 ) )
% 9.51/5.31             => ( R @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
% 9.51/5.31         => ( ( finite3207457112153483333omplex @ S2 )
% 9.51/5.31           => ( ! [X4: complex] :
% 9.51/5.31                  ( ( member_complex @ X4 @ S2 )
% 9.51/5.31                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.51/5.31             => ( R @ ( groups225925009352817453ex_rat @ H2 @ S2 ) @ ( groups225925009352817453ex_rat @ G @ S2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8646_prod_Orelated,axiom,
% 9.51/5.31      ! [R: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
% 9.51/5.31        ( ( R @ one_one_nat @ one_one_nat )
% 9.51/5.31       => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
% 9.51/5.31              ( ( ( R @ X15 @ X23 )
% 9.51/5.31                & ( R @ Y15 @ Y23 ) )
% 9.51/5.31             => ( R @ ( times_times_nat @ X15 @ Y15 ) @ ( times_times_nat @ X23 @ Y23 ) ) )
% 9.51/5.31         => ( ( finite_finite_int @ S2 )
% 9.51/5.31           => ( ! [X4: int] :
% 9.51/5.31                  ( ( member_int @ X4 @ S2 )
% 9.51/5.31                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.51/5.31             => ( R @ ( groups1707563613775114915nt_nat @ H2 @ S2 ) @ ( groups1707563613775114915nt_nat @ G @ S2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8647_prod_Oinsert__if,axiom,
% 9.51/5.31      ! [A2: set_real,X: real,G: real > complex] :
% 9.51/5.31        ( ( finite_finite_real @ A2 )
% 9.51/5.31       => ( ( ( member_real @ X @ A2 )
% 9.51/5.31           => ( ( groups713298508707869441omplex @ G @ ( insert_real @ X @ A2 ) )
% 9.51/5.31              = ( groups713298508707869441omplex @ G @ A2 ) ) )
% 9.51/5.31          & ( ~ ( member_real @ X @ A2 )
% 9.51/5.31           => ( ( groups713298508707869441omplex @ G @ ( insert_real @ X @ A2 ) )
% 9.51/5.31              = ( times_times_complex @ ( G @ X ) @ ( groups713298508707869441omplex @ G @ A2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8648_prod_Oinsert__if,axiom,
% 9.51/5.31      ! [A2: set_nat,X: nat,G: nat > complex] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( ( member_nat @ X @ A2 )
% 9.51/5.31           => ( ( groups6464643781859351333omplex @ G @ ( insert_nat @ X @ A2 ) )
% 9.51/5.31              = ( groups6464643781859351333omplex @ G @ A2 ) ) )
% 9.51/5.31          & ( ~ ( member_nat @ X @ A2 )
% 9.51/5.31           => ( ( groups6464643781859351333omplex @ G @ ( insert_nat @ X @ A2 ) )
% 9.51/5.31              = ( times_times_complex @ ( G @ X ) @ ( groups6464643781859351333omplex @ G @ A2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8649_prod_Oinsert__if,axiom,
% 9.51/5.31      ! [A2: set_int,X: int,G: int > complex] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ( ( member_int @ X @ A2 )
% 9.51/5.31           => ( ( groups7440179247065528705omplex @ G @ ( insert_int @ X @ A2 ) )
% 9.51/5.31              = ( groups7440179247065528705omplex @ G @ A2 ) ) )
% 9.51/5.31          & ( ~ ( member_int @ X @ A2 )
% 9.51/5.31           => ( ( groups7440179247065528705omplex @ G @ ( insert_int @ X @ A2 ) )
% 9.51/5.31              = ( times_times_complex @ ( G @ X ) @ ( groups7440179247065528705omplex @ G @ A2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8650_prod_Oinsert__if,axiom,
% 9.51/5.31      ! [A2: set_complex,X: complex,G: complex > complex] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ A2 )
% 9.51/5.31       => ( ( ( member_complex @ X @ A2 )
% 9.51/5.31           => ( ( groups3708469109370488835omplex @ G @ ( insert_complex @ X @ A2 ) )
% 9.51/5.31              = ( groups3708469109370488835omplex @ G @ A2 ) ) )
% 9.51/5.31          & ( ~ ( member_complex @ X @ A2 )
% 9.51/5.31           => ( ( groups3708469109370488835omplex @ G @ ( insert_complex @ X @ A2 ) )
% 9.51/5.31              = ( times_times_complex @ ( G @ X ) @ ( groups3708469109370488835omplex @ G @ A2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8651_prod_Oinsert__if,axiom,
% 9.51/5.31      ! [A2: set_real,X: real,G: real > real] :
% 9.51/5.31        ( ( finite_finite_real @ A2 )
% 9.51/5.31       => ( ( ( member_real @ X @ A2 )
% 9.51/5.31           => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X @ A2 ) )
% 9.51/5.31              = ( groups1681761925125756287l_real @ G @ A2 ) ) )
% 9.51/5.31          & ( ~ ( member_real @ X @ A2 )
% 9.51/5.31           => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X @ A2 ) )
% 9.51/5.31              = ( times_times_real @ ( G @ X ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8652_prod_Oinsert__if,axiom,
% 9.51/5.31      ! [A2: set_nat,X: nat,G: nat > real] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( ( member_nat @ X @ A2 )
% 9.51/5.31           => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X @ A2 ) )
% 9.51/5.31              = ( groups129246275422532515t_real @ G @ A2 ) ) )
% 9.51/5.31          & ( ~ ( member_nat @ X @ A2 )
% 9.51/5.31           => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X @ A2 ) )
% 9.51/5.31              = ( times_times_real @ ( G @ X ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8653_prod_Oinsert__if,axiom,
% 9.51/5.31      ! [A2: set_int,X: int,G: int > real] :
% 9.51/5.31        ( ( finite_finite_int @ A2 )
% 9.51/5.31       => ( ( ( member_int @ X @ A2 )
% 9.51/5.31           => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X @ A2 ) )
% 9.51/5.31              = ( groups2316167850115554303t_real @ G @ A2 ) ) )
% 9.51/5.31          & ( ~ ( member_int @ X @ A2 )
% 9.51/5.31           => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X @ A2 ) )
% 9.51/5.31              = ( times_times_real @ ( G @ X ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8654_prod_Oinsert__if,axiom,
% 9.51/5.31      ! [A2: set_complex,X: complex,G: complex > real] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ A2 )
% 9.51/5.31       => ( ( ( member_complex @ X @ A2 )
% 9.51/5.31           => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X @ A2 ) )
% 9.51/5.31              = ( groups766887009212190081x_real @ G @ A2 ) ) )
% 9.51/5.31          & ( ~ ( member_complex @ X @ A2 )
% 9.51/5.31           => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X @ A2 ) )
% 9.51/5.31              = ( times_times_real @ ( G @ X ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8655_prod_Oinsert__if,axiom,
% 9.51/5.31      ! [A2: set_real,X: real,G: real > rat] :
% 9.51/5.31        ( ( finite_finite_real @ A2 )
% 9.51/5.31       => ( ( ( member_real @ X @ A2 )
% 9.51/5.31           => ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X @ A2 ) )
% 9.51/5.31              = ( groups4061424788464935467al_rat @ G @ A2 ) ) )
% 9.51/5.31          & ( ~ ( member_real @ X @ A2 )
% 9.51/5.31           => ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X @ A2 ) )
% 9.51/5.31              = ( times_times_rat @ ( G @ X ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8656_prod_Oinsert__if,axiom,
% 9.51/5.31      ! [A2: set_nat,X: nat,G: nat > rat] :
% 9.51/5.31        ( ( finite_finite_nat @ A2 )
% 9.51/5.31       => ( ( ( member_nat @ X @ A2 )
% 9.51/5.31           => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A2 ) )
% 9.51/5.31              = ( groups73079841787564623at_rat @ G @ A2 ) ) )
% 9.51/5.31          & ( ~ ( member_nat @ X @ A2 )
% 9.51/5.31           => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A2 ) )
% 9.51/5.31              = ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8657_fact__split,axiom,
% 9.51/5.31      ! [K: nat,N: nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.31       => ( ( semiri5044797733671781792omplex @ N )
% 9.51/5.31          = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N @ K ) @ N ) ) ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8658_fact__split,axiom,
% 9.51/5.31      ! [K: nat,N: nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.31       => ( ( semiri773545260158071498ct_rat @ N )
% 9.51/5.31          = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N @ K ) @ N ) ) ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8659_fact__split,axiom,
% 9.51/5.31      ! [K: nat,N: nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.31       => ( ( semiri1406184849735516958ct_int @ N )
% 9.51/5.31          = ( times_times_int @ ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N @ K ) @ N ) ) ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8660_fact__split,axiom,
% 9.51/5.31      ! [K: nat,N: nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.31       => ( ( semiri2265585572941072030t_real @ N )
% 9.51/5.31          = ( times_times_real @ ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N @ K ) @ N ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8661_fact__split,axiom,
% 9.51/5.31      ! [K: nat,N: nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ K @ N )
% 9.51/5.31       => ( ( semiri1408675320244567234ct_nat @ N )
% 9.51/5.31          = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N @ K ) @ N ) ) ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8662_prod__dvd__prod__subset,axiom,
% 9.51/5.31      ! [B2: set_complex,A2: set_complex,F: complex > nat] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ B2 )
% 9.51/5.31       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.51/5.31         => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ F @ B2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8663_prod__dvd__prod__subset,axiom,
% 9.51/5.31      ! [B2: set_nat,A2: set_nat,F: nat > int] :
% 9.51/5.31        ( ( finite_finite_nat @ B2 )
% 9.51/5.31       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.51/5.31         => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ F @ B2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8664_prod__dvd__prod__subset,axiom,
% 9.51/5.31      ! [B2: set_complex,A2: set_complex,F: complex > int] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ B2 )
% 9.51/5.31       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.51/5.31         => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ F @ B2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8665_prod__dvd__prod__subset,axiom,
% 9.51/5.31      ! [B2: set_nat,A2: set_nat,F: nat > code_integer] :
% 9.51/5.31        ( ( finite_finite_nat @ B2 )
% 9.51/5.31       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.51/5.31         => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A2 ) @ ( groups3455450783089532116nteger @ F @ B2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8666_prod__dvd__prod__subset,axiom,
% 9.51/5.31      ! [B2: set_complex,A2: set_complex,F: complex > code_integer] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ B2 )
% 9.51/5.31       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.51/5.31         => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A2 ) @ ( groups8682486955453173170nteger @ F @ B2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8667_prod__dvd__prod__subset,axiom,
% 9.51/5.31      ! [B2: set_int,A2: set_int,F: int > nat] :
% 9.51/5.31        ( ( finite_finite_int @ B2 )
% 9.51/5.31       => ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.51/5.31         => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ F @ B2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8668_prod__dvd__prod__subset,axiom,
% 9.51/5.31      ! [B2: set_int,A2: set_int,F: int > int] :
% 9.51/5.31        ( ( finite_finite_int @ B2 )
% 9.51/5.31       => ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.51/5.31         => ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ ( groups1705073143266064639nt_int @ F @ B2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8669_prod__dvd__prod__subset,axiom,
% 9.51/5.31      ! [B2: set_int,A2: set_int,F: int > code_integer] :
% 9.51/5.31        ( ( finite_finite_int @ B2 )
% 9.51/5.31       => ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.51/5.31         => ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A2 ) @ ( groups3827104343326376752nteger @ F @ B2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8670_prod__dvd__prod__subset,axiom,
% 9.51/5.31      ! [B2: set_nat,A2: set_nat,F: nat > nat] :
% 9.51/5.31        ( ( finite_finite_nat @ B2 )
% 9.51/5.31       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.51/5.31         => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ F @ B2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8671_prod__dvd__prod__subset,axiom,
% 9.51/5.31      ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 9.51/5.31        ( ( finite6177210948735845034at_nat @ B2 )
% 9.51/5.31       => ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 9.51/5.31         => ( dvd_dvd_nat @ ( groups4077766827762148844at_nat @ F @ A2 ) @ ( groups4077766827762148844at_nat @ F @ B2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8672_prod__dvd__prod__subset2,axiom,
% 9.51/5.31      ! [B2: set_real,A2: set_real,F: real > nat,G: real > nat] :
% 9.51/5.31        ( ( finite_finite_real @ B2 )
% 9.51/5.31       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.51/5.31         => ( ! [A3: real] :
% 9.51/5.31                ( ( member_real @ A3 @ A2 )
% 9.51/5.31               => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 9.51/5.31           => ( dvd_dvd_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ B2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8673_prod__dvd__prod__subset2,axiom,
% 9.51/5.31      ! [B2: set_complex,A2: set_complex,F: complex > nat,G: complex > nat] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ B2 )
% 9.51/5.31       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.51/5.31         => ( ! [A3: complex] :
% 9.51/5.31                ( ( member_complex @ A3 @ A2 )
% 9.51/5.31               => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 9.51/5.31           => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ G @ B2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8674_prod__dvd__prod__subset2,axiom,
% 9.51/5.31      ! [B2: set_real,A2: set_real,F: real > int,G: real > int] :
% 9.51/5.31        ( ( finite_finite_real @ B2 )
% 9.51/5.31       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.51/5.31         => ( ! [A3: real] :
% 9.51/5.31                ( ( member_real @ A3 @ A2 )
% 9.51/5.31               => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 9.51/5.31           => ( dvd_dvd_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ G @ B2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8675_prod__dvd__prod__subset2,axiom,
% 9.51/5.31      ! [B2: set_nat,A2: set_nat,F: nat > int,G: nat > int] :
% 9.51/5.31        ( ( finite_finite_nat @ B2 )
% 9.51/5.31       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.51/5.31         => ( ! [A3: nat] :
% 9.51/5.31                ( ( member_nat @ A3 @ A2 )
% 9.51/5.31               => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 9.51/5.31           => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ G @ B2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8676_prod__dvd__prod__subset2,axiom,
% 9.51/5.31      ! [B2: set_complex,A2: set_complex,F: complex > int,G: complex > int] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ B2 )
% 9.51/5.31       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.51/5.31         => ( ! [A3: complex] :
% 9.51/5.31                ( ( member_complex @ A3 @ A2 )
% 9.51/5.31               => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 9.51/5.31           => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ G @ B2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8677_prod__dvd__prod__subset2,axiom,
% 9.51/5.31      ! [B2: set_real,A2: set_real,F: real > code_integer,G: real > code_integer] :
% 9.51/5.31        ( ( finite_finite_real @ B2 )
% 9.51/5.31       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.51/5.31         => ( ! [A3: real] :
% 9.51/5.31                ( ( member_real @ A3 @ A2 )
% 9.51/5.31               => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 9.51/5.31           => ( dvd_dvd_Code_integer @ ( groups6225526099057966256nteger @ F @ A2 ) @ ( groups6225526099057966256nteger @ G @ B2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8678_prod__dvd__prod__subset2,axiom,
% 9.51/5.31      ! [B2: set_nat,A2: set_nat,F: nat > code_integer,G: nat > code_integer] :
% 9.51/5.31        ( ( finite_finite_nat @ B2 )
% 9.51/5.31       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.51/5.31         => ( ! [A3: nat] :
% 9.51/5.31                ( ( member_nat @ A3 @ A2 )
% 9.51/5.31               => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 9.51/5.31           => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A2 ) @ ( groups3455450783089532116nteger @ G @ B2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8679_prod__dvd__prod__subset2,axiom,
% 9.51/5.31      ! [B2: set_complex,A2: set_complex,F: complex > code_integer,G: complex > code_integer] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ B2 )
% 9.51/5.31       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.51/5.31         => ( ! [A3: complex] :
% 9.51/5.31                ( ( member_complex @ A3 @ A2 )
% 9.51/5.31               => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 9.51/5.31           => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A2 ) @ ( groups8682486955453173170nteger @ G @ B2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8680_prod__dvd__prod__subset2,axiom,
% 9.51/5.31      ! [B2: set_int,A2: set_int,F: int > nat,G: int > nat] :
% 9.51/5.31        ( ( finite_finite_int @ B2 )
% 9.51/5.31       => ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.51/5.31         => ( ! [A3: int] :
% 9.51/5.31                ( ( member_int @ A3 @ A2 )
% 9.51/5.31               => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 9.51/5.31           => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ B2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8681_prod__dvd__prod__subset2,axiom,
% 9.51/5.31      ! [B2: set_int,A2: set_int,F: int > int,G: int > int] :
% 9.51/5.31        ( ( finite_finite_int @ B2 )
% 9.51/5.31       => ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.51/5.31         => ( ! [A3: int] :
% 9.51/5.31                ( ( member_int @ A3 @ A2 )
% 9.51/5.31               => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 9.51/5.31           => ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ ( groups1705073143266064639nt_int @ G @ B2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8682_prod_Oreindex__bij__witness__not__neutral,axiom,
% 9.51/5.31      ! [S5: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T3: set_real,G: real > int,H2: real > int] :
% 9.51/5.31        ( ( finite_finite_real @ S5 )
% 9.51/5.31       => ( ( finite_finite_real @ T4 )
% 9.51/5.31         => ( ! [A3: real] :
% 9.51/5.31                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 9.51/5.31               => ( ( I @ ( J @ A3 ) )
% 9.51/5.31                  = A3 ) )
% 9.51/5.31           => ( ! [A3: real] :
% 9.51/5.31                  ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 9.51/5.31                 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 9.51/5.31             => ( ! [B3: real] :
% 9.51/5.31                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 9.51/5.31                   => ( ( J @ ( I @ B3 ) )
% 9.51/5.31                      = B3 ) )
% 9.51/5.31               => ( ! [B3: real] :
% 9.51/5.31                      ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 9.51/5.31                     => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
% 9.51/5.31                 => ( ! [A3: real] :
% 9.51/5.31                        ( ( member_real @ A3 @ S5 )
% 9.51/5.31                       => ( ( G @ A3 )
% 9.51/5.31                          = one_one_int ) )
% 9.51/5.31                   => ( ! [B3: real] :
% 9.51/5.31                          ( ( member_real @ B3 @ T4 )
% 9.51/5.31                         => ( ( H2 @ B3 )
% 9.51/5.31                            = one_one_int ) )
% 9.51/5.31                     => ( ! [A3: real] :
% 9.51/5.31                            ( ( member_real @ A3 @ S2 )
% 9.51/5.31                           => ( ( H2 @ ( J @ A3 ) )
% 9.51/5.31                              = ( G @ A3 ) ) )
% 9.51/5.31                       => ( ( groups4694064378042380927al_int @ G @ S2 )
% 9.51/5.31                          = ( groups4694064378042380927al_int @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8683_prod_Oreindex__bij__witness__not__neutral,axiom,
% 9.51/5.31      ! [S5: set_real,T4: set_int,S2: set_real,I: int > real,J: real > int,T3: set_int,G: real > int,H2: int > int] :
% 9.51/5.31        ( ( finite_finite_real @ S5 )
% 9.51/5.31       => ( ( finite_finite_int @ T4 )
% 9.51/5.31         => ( ! [A3: real] :
% 9.51/5.31                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 9.51/5.31               => ( ( I @ ( J @ A3 ) )
% 9.51/5.31                  = A3 ) )
% 9.51/5.31           => ( ! [A3: real] :
% 9.51/5.31                  ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 9.51/5.31                 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 9.51/5.31             => ( ! [B3: int] :
% 9.51/5.31                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 9.51/5.31                   => ( ( J @ ( I @ B3 ) )
% 9.51/5.31                      = B3 ) )
% 9.51/5.31               => ( ! [B3: int] :
% 9.51/5.31                      ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 9.51/5.31                     => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
% 9.51/5.31                 => ( ! [A3: real] :
% 9.51/5.31                        ( ( member_real @ A3 @ S5 )
% 9.51/5.31                       => ( ( G @ A3 )
% 9.51/5.31                          = one_one_int ) )
% 9.51/5.31                   => ( ! [B3: int] :
% 9.51/5.31                          ( ( member_int @ B3 @ T4 )
% 9.51/5.31                         => ( ( H2 @ B3 )
% 9.51/5.31                            = one_one_int ) )
% 9.51/5.31                     => ( ! [A3: real] :
% 9.51/5.31                            ( ( member_real @ A3 @ S2 )
% 9.51/5.31                           => ( ( H2 @ ( J @ A3 ) )
% 9.51/5.31                              = ( G @ A3 ) ) )
% 9.51/5.31                       => ( ( groups4694064378042380927al_int @ G @ S2 )
% 9.51/5.31                          = ( groups1705073143266064639nt_int @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8684_prod_Oreindex__bij__witness__not__neutral,axiom,
% 9.51/5.31      ! [S5: set_real,T4: set_complex,S2: set_real,I: complex > real,J: real > complex,T3: set_complex,G: real > int,H2: complex > int] :
% 9.51/5.31        ( ( finite_finite_real @ S5 )
% 9.51/5.31       => ( ( finite3207457112153483333omplex @ T4 )
% 9.51/5.31         => ( ! [A3: real] :
% 9.51/5.31                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 9.51/5.31               => ( ( I @ ( J @ A3 ) )
% 9.51/5.31                  = A3 ) )
% 9.51/5.31           => ( ! [A3: real] :
% 9.51/5.31                  ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 9.51/5.31                 => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 9.51/5.31             => ( ! [B3: complex] :
% 9.51/5.31                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 9.51/5.31                   => ( ( J @ ( I @ B3 ) )
% 9.51/5.31                      = B3 ) )
% 9.51/5.31               => ( ! [B3: complex] :
% 9.51/5.31                      ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 9.51/5.31                     => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
% 9.51/5.31                 => ( ! [A3: real] :
% 9.51/5.31                        ( ( member_real @ A3 @ S5 )
% 9.51/5.31                       => ( ( G @ A3 )
% 9.51/5.31                          = one_one_int ) )
% 9.51/5.31                   => ( ! [B3: complex] :
% 9.51/5.31                          ( ( member_complex @ B3 @ T4 )
% 9.51/5.31                         => ( ( H2 @ B3 )
% 9.51/5.31                            = one_one_int ) )
% 9.51/5.31                     => ( ! [A3: real] :
% 9.51/5.31                            ( ( member_real @ A3 @ S2 )
% 9.51/5.31                           => ( ( H2 @ ( J @ A3 ) )
% 9.51/5.31                              = ( G @ A3 ) ) )
% 9.51/5.31                       => ( ( groups4694064378042380927al_int @ G @ S2 )
% 9.51/5.31                          = ( groups858564598930262913ex_int @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8685_prod_Oreindex__bij__witness__not__neutral,axiom,
% 9.51/5.31      ! [S5: set_int,T4: set_real,S2: set_int,I: real > int,J: int > real,T3: set_real,G: int > int,H2: real > int] :
% 9.51/5.31        ( ( finite_finite_int @ S5 )
% 9.51/5.31       => ( ( finite_finite_real @ T4 )
% 9.51/5.31         => ( ! [A3: int] :
% 9.51/5.31                ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 9.51/5.31               => ( ( I @ ( J @ A3 ) )
% 9.51/5.31                  = A3 ) )
% 9.51/5.31           => ( ! [A3: int] :
% 9.51/5.31                  ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 9.51/5.31                 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 9.51/5.31             => ( ! [B3: real] :
% 9.51/5.31                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 9.51/5.31                   => ( ( J @ ( I @ B3 ) )
% 9.51/5.31                      = B3 ) )
% 9.51/5.31               => ( ! [B3: real] :
% 9.51/5.31                      ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 9.51/5.31                     => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
% 9.51/5.31                 => ( ! [A3: int] :
% 9.51/5.31                        ( ( member_int @ A3 @ S5 )
% 9.51/5.31                       => ( ( G @ A3 )
% 9.51/5.31                          = one_one_int ) )
% 9.51/5.31                   => ( ! [B3: real] :
% 9.51/5.31                          ( ( member_real @ B3 @ T4 )
% 9.51/5.31                         => ( ( H2 @ B3 )
% 9.51/5.31                            = one_one_int ) )
% 9.51/5.31                     => ( ! [A3: int] :
% 9.51/5.31                            ( ( member_int @ A3 @ S2 )
% 9.51/5.31                           => ( ( H2 @ ( J @ A3 ) )
% 9.51/5.31                              = ( G @ A3 ) ) )
% 9.51/5.31                       => ( ( groups1705073143266064639nt_int @ G @ S2 )
% 9.51/5.31                          = ( groups4694064378042380927al_int @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8686_prod_Oreindex__bij__witness__not__neutral,axiom,
% 9.51/5.31      ! [S5: set_int,T4: set_int,S2: set_int,I: int > int,J: int > int,T3: set_int,G: int > int,H2: int > int] :
% 9.51/5.31        ( ( finite_finite_int @ S5 )
% 9.51/5.31       => ( ( finite_finite_int @ T4 )
% 9.51/5.31         => ( ! [A3: int] :
% 9.51/5.31                ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 9.51/5.31               => ( ( I @ ( J @ A3 ) )
% 9.51/5.31                  = A3 ) )
% 9.51/5.31           => ( ! [A3: int] :
% 9.51/5.31                  ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 9.51/5.31                 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 9.51/5.31             => ( ! [B3: int] :
% 9.51/5.31                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 9.51/5.31                   => ( ( J @ ( I @ B3 ) )
% 9.51/5.31                      = B3 ) )
% 9.51/5.31               => ( ! [B3: int] :
% 9.51/5.31                      ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 9.51/5.31                     => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
% 9.51/5.31                 => ( ! [A3: int] :
% 9.51/5.31                        ( ( member_int @ A3 @ S5 )
% 9.51/5.31                       => ( ( G @ A3 )
% 9.51/5.31                          = one_one_int ) )
% 9.51/5.31                   => ( ! [B3: int] :
% 9.51/5.31                          ( ( member_int @ B3 @ T4 )
% 9.51/5.31                         => ( ( H2 @ B3 )
% 9.51/5.31                            = one_one_int ) )
% 9.51/5.31                     => ( ! [A3: int] :
% 9.51/5.31                            ( ( member_int @ A3 @ S2 )
% 9.51/5.31                           => ( ( H2 @ ( J @ A3 ) )
% 9.51/5.31                              = ( G @ A3 ) ) )
% 9.51/5.31                       => ( ( groups1705073143266064639nt_int @ G @ S2 )
% 9.51/5.31                          = ( groups1705073143266064639nt_int @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8687_prod_Oreindex__bij__witness__not__neutral,axiom,
% 9.51/5.31      ! [S5: set_int,T4: set_complex,S2: set_int,I: complex > int,J: int > complex,T3: set_complex,G: int > int,H2: complex > int] :
% 9.51/5.31        ( ( finite_finite_int @ S5 )
% 9.51/5.31       => ( ( finite3207457112153483333omplex @ T4 )
% 9.51/5.31         => ( ! [A3: int] :
% 9.51/5.31                ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 9.51/5.31               => ( ( I @ ( J @ A3 ) )
% 9.51/5.31                  = A3 ) )
% 9.51/5.31           => ( ! [A3: int] :
% 9.51/5.31                  ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
% 9.51/5.31                 => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 9.51/5.31             => ( ! [B3: complex] :
% 9.51/5.31                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 9.51/5.31                   => ( ( J @ ( I @ B3 ) )
% 9.51/5.31                      = B3 ) )
% 9.51/5.31               => ( ! [B3: complex] :
% 9.51/5.31                      ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 9.51/5.31                     => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
% 9.51/5.31                 => ( ! [A3: int] :
% 9.51/5.31                        ( ( member_int @ A3 @ S5 )
% 9.51/5.31                       => ( ( G @ A3 )
% 9.51/5.31                          = one_one_int ) )
% 9.51/5.31                   => ( ! [B3: complex] :
% 9.51/5.31                          ( ( member_complex @ B3 @ T4 )
% 9.51/5.31                         => ( ( H2 @ B3 )
% 9.51/5.31                            = one_one_int ) )
% 9.51/5.31                     => ( ! [A3: int] :
% 9.51/5.31                            ( ( member_int @ A3 @ S2 )
% 9.51/5.31                           => ( ( H2 @ ( J @ A3 ) )
% 9.51/5.31                              = ( G @ A3 ) ) )
% 9.51/5.31                       => ( ( groups1705073143266064639nt_int @ G @ S2 )
% 9.51/5.31                          = ( groups858564598930262913ex_int @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8688_prod_Oreindex__bij__witness__not__neutral,axiom,
% 9.51/5.31      ! [S5: set_complex,T4: set_real,S2: set_complex,I: real > complex,J: complex > real,T3: set_real,G: complex > int,H2: real > int] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ S5 )
% 9.51/5.31       => ( ( finite_finite_real @ T4 )
% 9.51/5.31         => ( ! [A3: complex] :
% 9.51/5.31                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
% 9.51/5.31               => ( ( I @ ( J @ A3 ) )
% 9.51/5.31                  = A3 ) )
% 9.51/5.31           => ( ! [A3: complex] :
% 9.51/5.31                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
% 9.51/5.31                 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 9.51/5.31             => ( ! [B3: real] :
% 9.51/5.31                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 9.51/5.31                   => ( ( J @ ( I @ B3 ) )
% 9.51/5.31                      = B3 ) )
% 9.51/5.31               => ( ! [B3: real] :
% 9.51/5.31                      ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 9.51/5.31                     => ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
% 9.51/5.31                 => ( ! [A3: complex] :
% 9.51/5.31                        ( ( member_complex @ A3 @ S5 )
% 9.51/5.31                       => ( ( G @ A3 )
% 9.51/5.31                          = one_one_int ) )
% 9.51/5.31                   => ( ! [B3: real] :
% 9.51/5.31                          ( ( member_real @ B3 @ T4 )
% 9.51/5.31                         => ( ( H2 @ B3 )
% 9.51/5.31                            = one_one_int ) )
% 9.51/5.31                     => ( ! [A3: complex] :
% 9.51/5.31                            ( ( member_complex @ A3 @ S2 )
% 9.51/5.31                           => ( ( H2 @ ( J @ A3 ) )
% 9.51/5.31                              = ( G @ A3 ) ) )
% 9.51/5.31                       => ( ( groups858564598930262913ex_int @ G @ S2 )
% 9.51/5.31                          = ( groups4694064378042380927al_int @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8689_prod_Oreindex__bij__witness__not__neutral,axiom,
% 9.51/5.31      ! [S5: set_complex,T4: set_int,S2: set_complex,I: int > complex,J: complex > int,T3: set_int,G: complex > int,H2: int > int] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ S5 )
% 9.51/5.31       => ( ( finite_finite_int @ T4 )
% 9.51/5.31         => ( ! [A3: complex] :
% 9.51/5.31                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
% 9.51/5.31               => ( ( I @ ( J @ A3 ) )
% 9.51/5.31                  = A3 ) )
% 9.51/5.31           => ( ! [A3: complex] :
% 9.51/5.31                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
% 9.51/5.31                 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 9.51/5.31             => ( ! [B3: int] :
% 9.51/5.31                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 9.51/5.31                   => ( ( J @ ( I @ B3 ) )
% 9.51/5.31                      = B3 ) )
% 9.51/5.31               => ( ! [B3: int] :
% 9.51/5.31                      ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 9.51/5.31                     => ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
% 9.51/5.31                 => ( ! [A3: complex] :
% 9.51/5.31                        ( ( member_complex @ A3 @ S5 )
% 9.51/5.31                       => ( ( G @ A3 )
% 9.51/5.31                          = one_one_int ) )
% 9.51/5.31                   => ( ! [B3: int] :
% 9.51/5.31                          ( ( member_int @ B3 @ T4 )
% 9.51/5.31                         => ( ( H2 @ B3 )
% 9.51/5.31                            = one_one_int ) )
% 9.51/5.31                     => ( ! [A3: complex] :
% 9.51/5.31                            ( ( member_complex @ A3 @ S2 )
% 9.51/5.31                           => ( ( H2 @ ( J @ A3 ) )
% 9.51/5.31                              = ( G @ A3 ) ) )
% 9.51/5.31                       => ( ( groups858564598930262913ex_int @ G @ S2 )
% 9.51/5.31                          = ( groups1705073143266064639nt_int @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8690_prod_Oreindex__bij__witness__not__neutral,axiom,
% 9.51/5.31      ! [S5: set_complex,T4: set_complex,S2: set_complex,I: complex > complex,J: complex > complex,T3: set_complex,G: complex > int,H2: complex > int] :
% 9.51/5.31        ( ( finite3207457112153483333omplex @ S5 )
% 9.51/5.31       => ( ( finite3207457112153483333omplex @ T4 )
% 9.51/5.31         => ( ! [A3: complex] :
% 9.51/5.31                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
% 9.51/5.31               => ( ( I @ ( J @ A3 ) )
% 9.51/5.31                  = A3 ) )
% 9.51/5.31           => ( ! [A3: complex] :
% 9.51/5.31                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
% 9.51/5.31                 => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 9.51/5.31             => ( ! [B3: complex] :
% 9.51/5.31                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 9.51/5.31                   => ( ( J @ ( I @ B3 ) )
% 9.51/5.31                      = B3 ) )
% 9.51/5.31               => ( ! [B3: complex] :
% 9.51/5.31                      ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 9.51/5.31                     => ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
% 9.51/5.31                 => ( ! [A3: complex] :
% 9.51/5.31                        ( ( member_complex @ A3 @ S5 )
% 9.51/5.31                       => ( ( G @ A3 )
% 9.51/5.31                          = one_one_int ) )
% 9.51/5.31                   => ( ! [B3: complex] :
% 9.51/5.31                          ( ( member_complex @ B3 @ T4 )
% 9.51/5.31                         => ( ( H2 @ B3 )
% 9.51/5.31                            = one_one_int ) )
% 9.51/5.31                     => ( ! [A3: complex] :
% 9.51/5.31                            ( ( member_complex @ A3 @ S2 )
% 9.51/5.31                           => ( ( H2 @ ( J @ A3 ) )
% 9.51/5.31                              = ( G @ A3 ) ) )
% 9.51/5.31                       => ( ( groups858564598930262913ex_int @ G @ S2 )
% 9.51/5.31                          = ( groups858564598930262913ex_int @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8691_prod_Oreindex__bij__witness__not__neutral,axiom,
% 9.51/5.31      ! [S5: set_real,T4: set_nat,S2: set_real,I: nat > real,J: real > nat,T3: set_nat,G: real > complex,H2: nat > complex] :
% 9.51/5.31        ( ( finite_finite_real @ S5 )
% 9.51/5.31       => ( ( finite_finite_nat @ T4 )
% 9.51/5.31         => ( ! [A3: real] :
% 9.51/5.31                ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 9.51/5.31               => ( ( I @ ( J @ A3 ) )
% 9.51/5.31                  = A3 ) )
% 9.51/5.31           => ( ! [A3: real] :
% 9.51/5.31                  ( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
% 9.51/5.31                 => ( member_nat @ ( J @ A3 ) @ ( minus_minus_set_nat @ T3 @ T4 ) ) )
% 9.51/5.31             => ( ! [B3: nat] :
% 9.51/5.31                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ T3 @ T4 ) )
% 9.51/5.31                   => ( ( J @ ( I @ B3 ) )
% 9.51/5.31                      = B3 ) )
% 9.51/5.31               => ( ! [B3: nat] :
% 9.51/5.31                      ( ( member_nat @ B3 @ ( minus_minus_set_nat @ T3 @ T4 ) )
% 9.51/5.31                     => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
% 9.51/5.31                 => ( ! [A3: real] :
% 9.51/5.31                        ( ( member_real @ A3 @ S5 )
% 9.51/5.31                       => ( ( G @ A3 )
% 9.51/5.31                          = one_one_complex ) )
% 9.51/5.31                   => ( ! [B3: nat] :
% 9.51/5.31                          ( ( member_nat @ B3 @ T4 )
% 9.51/5.31                         => ( ( H2 @ B3 )
% 9.51/5.31                            = one_one_complex ) )
% 9.51/5.31                     => ( ! [A3: real] :
% 9.51/5.31                            ( ( member_real @ A3 @ S2 )
% 9.51/5.31                           => ( ( H2 @ ( J @ A3 ) )
% 9.51/5.31                              = ( G @ A3 ) ) )
% 9.51/5.31                       => ( ( groups713298508707869441omplex @ G @ S2 )
% 9.51/5.31                          = ( groups6464643781859351333omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8692_atLeast0__lessThan__Suc,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 9.51/5.31        = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8693_subset__eq__atLeast0__lessThan__finite,axiom,
% 9.51/5.31      ! [N6: set_nat,N: nat] :
% 9.51/5.31        ( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 9.51/5.31       => ( finite_finite_nat @ N6 ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8694_atLeastLessThanSuc,axiom,
% 9.51/5.31      ! [M: nat,N: nat] :
% 9.51/5.31        ( ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.31         => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 9.51/5.31            = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
% 9.51/5.31        & ( ~ ( ord_less_eq_nat @ M @ N )
% 9.51/5.31         => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 9.51/5.31            = bot_bot_set_nat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8695_fact__eq__fact__times,axiom,
% 9.51/5.31      ! [N: nat,M: nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ N @ M )
% 9.51/5.31       => ( ( semiri1408675320244567234ct_nat @ M )
% 9.51/5.31          = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
% 9.51/5.31            @ ( groups708209901874060359at_nat
% 9.51/5.31              @ ^ [X3: nat] : X3
% 9.51/5.31              @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8696_fact__div__fact,axiom,
% 9.51/5.31      ! [N: nat,M: nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ N @ M )
% 9.51/5.31       => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
% 9.51/5.31          = ( groups708209901874060359at_nat
% 9.51/5.31            @ ^ [X3: nat] : X3
% 9.51/5.31            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8697_OR__upper,axiom,
% 9.51/5.31      ! [X: int,N: nat,Y: int] :
% 9.51/5.31        ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.51/5.31       => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.51/5.31         => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.51/5.31           => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8698_atLeast1__lessThan__eq__remove0,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.51/5.31        = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8699_Chebyshev__sum__upper__nat,axiom,
% 9.51/5.31      ! [N: nat,A: nat > nat,B: nat > nat] :
% 9.51/5.31        ( ! [I2: nat,J2: nat] :
% 9.51/5.31            ( ( ord_less_eq_nat @ I2 @ J2 )
% 9.51/5.31           => ( ( ord_less_nat @ J2 @ N )
% 9.51/5.31             => ( ord_less_eq_nat @ ( A @ I2 ) @ ( A @ J2 ) ) ) )
% 9.51/5.31       => ( ! [I2: nat,J2: nat] :
% 9.51/5.31              ( ( ord_less_eq_nat @ I2 @ J2 )
% 9.51/5.31             => ( ( ord_less_nat @ J2 @ N )
% 9.51/5.31               => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I2 ) ) ) )
% 9.51/5.31         => ( ord_less_eq_nat
% 9.51/5.31            @ ( times_times_nat @ N
% 9.51/5.31              @ ( groups3542108847815614940at_nat
% 9.51/5.31                @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( B @ I4 ) )
% 9.51/5.31                @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 9.51/5.31            @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8700_VEBT_Osize__gen_I1_J,axiom,
% 9.51/5.31      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 9.51/5.31        ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 9.51/5.31        = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8701_rat__inverse__code,axiom,
% 9.51/5.31      ! [P4: rat] :
% 9.51/5.31        ( ( quotient_of @ ( inverse_inverse_rat @ P4 ) )
% 9.51/5.31        = ( produc4245557441103728435nt_int
% 9.51/5.31          @ ^ [A4: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( A4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A4 ) @ B4 ) @ ( abs_abs_int @ A4 ) ) )
% 9.51/5.31          @ ( quotient_of @ P4 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8702_finite__atLeastLessThan__int,axiom,
% 9.51/5.31      ! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8703_quotient__of__number_I3_J,axiom,
% 9.51/5.31      ! [K: num] :
% 9.51/5.31        ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
% 9.51/5.31        = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8704_rat__one__code,axiom,
% 9.51/5.31      ( ( quotient_of @ one_one_rat )
% 9.51/5.31      = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8705_rat__zero__code,axiom,
% 9.51/5.31      ( ( quotient_of @ zero_zero_rat )
% 9.51/5.31      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8706_quotient__of__number_I5_J,axiom,
% 9.51/5.31      ! [K: num] :
% 9.51/5.31        ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 9.51/5.31        = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8707_quotient__of__number_I4_J,axiom,
% 9.51/5.31      ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.51/5.31      = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8708_finite__atLeastZeroLessThan__int,axiom,
% 9.51/5.31      ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8709_quotient__of__div,axiom,
% 9.51/5.31      ! [R2: rat,N: int,D: int] :
% 9.51/5.31        ( ( ( quotient_of @ R2 )
% 9.51/5.31          = ( product_Pair_int_int @ N @ D ) )
% 9.51/5.31       => ( R2
% 9.51/5.31          = ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8710_quotient__of__denom__pos,axiom,
% 9.51/5.31      ! [R2: rat,P4: int,Q4: int] :
% 9.51/5.31        ( ( ( quotient_of @ R2 )
% 9.51/5.31          = ( product_Pair_int_int @ P4 @ Q4 ) )
% 9.51/5.31       => ( ord_less_int @ zero_zero_int @ Q4 ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8711_rat__uminus__code,axiom,
% 9.51/5.31      ! [P4: rat] :
% 9.51/5.31        ( ( quotient_of @ ( uminus_uminus_rat @ P4 ) )
% 9.51/5.31        = ( produc4245557441103728435nt_int
% 9.51/5.31          @ ^ [A4: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A4 ) )
% 9.51/5.31          @ ( quotient_of @ P4 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8712_rat__abs__code,axiom,
% 9.51/5.31      ! [P4: rat] :
% 9.51/5.31        ( ( quotient_of @ ( abs_abs_rat @ P4 ) )
% 9.51/5.31        = ( produc4245557441103728435nt_int
% 9.51/5.31          @ ^ [A4: int] : ( product_Pair_int_int @ ( abs_abs_int @ A4 ) )
% 9.51/5.31          @ ( quotient_of @ P4 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8713_rat__less__code,axiom,
% 9.51/5.31      ( ord_less_rat
% 9.51/5.31      = ( ^ [P6: rat,Q5: rat] :
% 9.51/5.31            ( produc4947309494688390418_int_o
% 9.51/5.31            @ ^ [A4: int,C4: int] :
% 9.51/5.31                ( produc4947309494688390418_int_o
% 9.51/5.31                @ ^ [B4: int,D4: int] : ( ord_less_int @ ( times_times_int @ A4 @ D4 ) @ ( times_times_int @ C4 @ B4 ) )
% 9.51/5.31                @ ( quotient_of @ Q5 ) )
% 9.51/5.31            @ ( quotient_of @ P6 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8714_VEBT_Osize__gen_I2_J,axiom,
% 9.51/5.31      ! [X21: $o,X222: $o] :
% 9.51/5.31        ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 9.51/5.31        = zero_zero_nat ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8715_quotient__of__int,axiom,
% 9.51/5.31      ! [A: int] :
% 9.51/5.31        ( ( quotient_of @ ( of_int @ A ) )
% 9.51/5.31        = ( product_Pair_int_int @ A @ one_one_int ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8716_rat__plus__code,axiom,
% 9.51/5.31      ! [P4: rat,Q4: rat] :
% 9.51/5.31        ( ( quotient_of @ ( plus_plus_rat @ P4 @ Q4 ) )
% 9.51/5.31        = ( produc4245557441103728435nt_int
% 9.51/5.31          @ ^ [A4: int,C4: int] :
% 9.51/5.31              ( produc4245557441103728435nt_int
% 9.51/5.31              @ ^ [B4: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A4 @ D4 ) @ ( times_times_int @ B4 @ C4 ) ) @ ( times_times_int @ C4 @ D4 ) ) )
% 9.51/5.31              @ ( quotient_of @ Q4 ) )
% 9.51/5.31          @ ( quotient_of @ P4 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8717_rat__minus__code,axiom,
% 9.51/5.31      ! [P4: rat,Q4: rat] :
% 9.51/5.31        ( ( quotient_of @ ( minus_minus_rat @ P4 @ Q4 ) )
% 9.51/5.31        = ( produc4245557441103728435nt_int
% 9.51/5.31          @ ^ [A4: int,C4: int] :
% 9.51/5.31              ( produc4245557441103728435nt_int
% 9.51/5.31              @ ^ [B4: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A4 @ D4 ) @ ( times_times_int @ B4 @ C4 ) ) @ ( times_times_int @ C4 @ D4 ) ) )
% 9.51/5.31              @ ( quotient_of @ Q4 ) )
% 9.51/5.31          @ ( quotient_of @ P4 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8718_normalize__negative,axiom,
% 9.51/5.31      ! [Q4: int,P4: int] :
% 9.51/5.31        ( ( ord_less_int @ Q4 @ zero_zero_int )
% 9.51/5.31       => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q4 ) )
% 9.51/5.31          = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P4 ) @ ( uminus_uminus_int @ Q4 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8719_normalize__denom__zero,axiom,
% 9.51/5.31      ! [P4: int] :
% 9.51/5.31        ( ( normalize @ ( product_Pair_int_int @ P4 @ zero_zero_int ) )
% 9.51/5.31        = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8720_normalize__denom__pos,axiom,
% 9.51/5.31      ! [R2: product_prod_int_int,P4: int,Q4: int] :
% 9.51/5.31        ( ( ( normalize @ R2 )
% 9.51/5.31          = ( product_Pair_int_int @ P4 @ Q4 ) )
% 9.51/5.31       => ( ord_less_int @ zero_zero_int @ Q4 ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8721_normalize__crossproduct,axiom,
% 9.51/5.31      ! [Q4: int,S: int,P4: int,R2: int] :
% 9.51/5.31        ( ( Q4 != zero_zero_int )
% 9.51/5.31       => ( ( S != zero_zero_int )
% 9.51/5.31         => ( ( ( normalize @ ( product_Pair_int_int @ P4 @ Q4 ) )
% 9.51/5.31              = ( normalize @ ( product_Pair_int_int @ R2 @ S ) ) )
% 9.51/5.31           => ( ( times_times_int @ P4 @ S )
% 9.51/5.31              = ( times_times_int @ R2 @ Q4 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8722_rat__divide__code,axiom,
% 9.51/5.31      ! [P4: rat,Q4: rat] :
% 9.51/5.31        ( ( quotient_of @ ( divide_divide_rat @ P4 @ Q4 ) )
% 9.51/5.31        = ( produc4245557441103728435nt_int
% 9.51/5.31          @ ^ [A4: int,C4: int] :
% 9.51/5.31              ( produc4245557441103728435nt_int
% 9.51/5.31              @ ^ [B4: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A4 @ D4 ) @ ( times_times_int @ C4 @ B4 ) ) )
% 9.51/5.31              @ ( quotient_of @ Q4 ) )
% 9.51/5.31          @ ( quotient_of @ P4 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8723_rat__times__code,axiom,
% 9.51/5.31      ! [P4: rat,Q4: rat] :
% 9.51/5.31        ( ( quotient_of @ ( times_times_rat @ P4 @ Q4 ) )
% 9.51/5.31        = ( produc4245557441103728435nt_int
% 9.51/5.31          @ ^ [A4: int,C4: int] :
% 9.51/5.31              ( produc4245557441103728435nt_int
% 9.51/5.31              @ ^ [B4: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A4 @ B4 ) @ ( times_times_int @ C4 @ D4 ) ) )
% 9.51/5.31              @ ( quotient_of @ Q4 ) )
% 9.51/5.31          @ ( quotient_of @ P4 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8724_Frct__code__post_I5_J,axiom,
% 9.51/5.31      ! [K: num] :
% 9.51/5.31        ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
% 9.51/5.31        = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8725_divmod__step__integer__def,axiom,
% 9.51/5.31      ( unique4921790084139445826nteger
% 9.51/5.31      = ( ^ [L2: num] :
% 9.51/5.31            ( produc6916734918728496179nteger
% 9.51/5.31            @ ^ [Q5: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8726_Frct__code__post_I6_J,axiom,
% 9.51/5.31      ! [K: num,L: num] :
% 9.51/5.31        ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
% 9.51/5.31        = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8727_card__Collect__less__nat,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( finite_card_nat
% 9.51/5.31          @ ( collect_nat
% 9.51/5.31            @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
% 9.51/5.31        = N ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8728_card__Collect__le__nat,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( finite_card_nat
% 9.51/5.31          @ ( collect_nat
% 9.51/5.31            @ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
% 9.51/5.31        = ( suc @ N ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8729_divmod__integer_H__def,axiom,
% 9.51/5.31      ( unique3479559517661332726nteger
% 9.51/5.31      = ( ^ [M2: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8730_full__exhaustive__integer_H_Ocases,axiom,
% 9.51/5.31      ! [X: produc1908205239877642774nteger] :
% 9.51/5.31        ~ ! [F2: produc6241069584506657477e_term > option6357759511663192854e_term,D5: code_integer,I2: code_integer] :
% 9.51/5.31            ( X
% 9.51/5.31           != ( produc8603105652947943368nteger @ F2 @ ( produc1086072967326762835nteger @ D5 @ I2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8731_exhaustive__integer_H_Ocases,axiom,
% 9.51/5.31      ! [X: produc8763457246119570046nteger] :
% 9.51/5.31        ~ ! [F2: code_integer > option6357759511663192854e_term,D5: code_integer,I2: code_integer] :
% 9.51/5.31            ( X
% 9.51/5.31           != ( produc6137756002093451184nteger @ F2 @ ( produc1086072967326762835nteger @ D5 @ I2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8732_sgn__integer__code,axiom,
% 9.51/5.31      ( sgn_sgn_Code_integer
% 9.51/5.31      = ( ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8733_zero__natural_Orsp,axiom,
% 9.51/5.31      zero_zero_nat = zero_zero_nat ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8734_card__less,axiom,
% 9.51/5.31      ! [M4: set_nat,I: nat] :
% 9.51/5.31        ( ( member_nat @ zero_zero_nat @ M4 )
% 9.51/5.31       => ( ( finite_card_nat
% 9.51/5.31            @ ( collect_nat
% 9.51/5.31              @ ^ [K3: nat] :
% 9.51/5.31                  ( ( member_nat @ K3 @ M4 )
% 9.51/5.31                  & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
% 9.51/5.31         != zero_zero_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8735_card__less__Suc,axiom,
% 9.51/5.31      ! [M4: set_nat,I: nat] :
% 9.51/5.31        ( ( member_nat @ zero_zero_nat @ M4 )
% 9.51/5.31       => ( ( suc
% 9.51/5.31            @ ( finite_card_nat
% 9.51/5.31              @ ( collect_nat
% 9.51/5.31                @ ^ [K3: nat] :
% 9.51/5.31                    ( ( member_nat @ ( suc @ K3 ) @ M4 )
% 9.51/5.31                    & ( ord_less_nat @ K3 @ I ) ) ) ) )
% 9.51/5.31          = ( finite_card_nat
% 9.51/5.31            @ ( collect_nat
% 9.51/5.31              @ ^ [K3: nat] :
% 9.51/5.31                  ( ( member_nat @ K3 @ M4 )
% 9.51/5.31                  & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8736_card__less__Suc2,axiom,
% 9.51/5.31      ! [M4: set_nat,I: nat] :
% 9.51/5.31        ( ~ ( member_nat @ zero_zero_nat @ M4 )
% 9.51/5.31       => ( ( finite_card_nat
% 9.51/5.31            @ ( collect_nat
% 9.51/5.31              @ ^ [K3: nat] :
% 9.51/5.31                  ( ( member_nat @ ( suc @ K3 ) @ M4 )
% 9.51/5.31                  & ( ord_less_nat @ K3 @ I ) ) ) )
% 9.51/5.31          = ( finite_card_nat
% 9.51/5.31            @ ( collect_nat
% 9.51/5.31              @ ^ [K3: nat] :
% 9.51/5.31                  ( ( member_nat @ K3 @ M4 )
% 9.51/5.31                  & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8737_subset__eq__atLeast0__lessThan__card,axiom,
% 9.51/5.31      ! [N6: set_nat,N: nat] :
% 9.51/5.31        ( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 9.51/5.31       => ( ord_less_eq_nat @ ( finite_card_nat @ N6 ) @ N ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8738_card__sum__le__nat__sum,axiom,
% 9.51/5.31      ! [S2: set_nat] :
% 9.51/5.31        ( ord_less_eq_nat
% 9.51/5.31        @ ( groups3542108847815614940at_nat
% 9.51/5.31          @ ^ [X3: nat] : X3
% 9.51/5.31          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
% 9.51/5.31        @ ( groups3542108847815614940at_nat
% 9.51/5.31          @ ^ [X3: nat] : X3
% 9.51/5.31          @ S2 ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8739_card__nth__roots,axiom,
% 9.51/5.31      ! [C: complex,N: nat] :
% 9.51/5.31        ( ( C != zero_zero_complex )
% 9.51/5.31       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31         => ( ( finite_card_complex
% 9.51/5.31              @ ( collect_complex
% 9.51/5.31                @ ^ [Z3: complex] :
% 9.51/5.31                    ( ( power_power_complex @ Z3 @ N )
% 9.51/5.31                    = C ) ) )
% 9.51/5.31            = N ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8740_card__roots__unity__eq,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( finite_card_complex
% 9.51/5.31            @ ( collect_complex
% 9.51/5.31              @ ^ [Z3: complex] :
% 9.51/5.31                  ( ( power_power_complex @ Z3 @ N )
% 9.51/5.31                  = one_one_complex ) ) )
% 9.51/5.31          = N ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8741_Frct__code__post_I2_J,axiom,
% 9.51/5.31      ! [A: int] :
% 9.51/5.31        ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
% 9.51/5.31        = zero_zero_rat ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8742_Frct__code__post_I1_J,axiom,
% 9.51/5.31      ! [A: int] :
% 9.51/5.31        ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
% 9.51/5.31        = zero_zero_rat ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8743_Frct__code__post_I7_J,axiom,
% 9.51/5.31      ! [A: int,B: int] :
% 9.51/5.31        ( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B ) )
% 9.51/5.31        = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8744_Frct__code__post_I8_J,axiom,
% 9.51/5.31      ! [A: int,B: int] :
% 9.51/5.31        ( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B ) ) )
% 9.51/5.31        = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8745_Frct__code__post_I3_J,axiom,
% 9.51/5.31      ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
% 9.51/5.31      = one_one_rat ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8746_Frct__code__post_I4_J,axiom,
% 9.51/5.31      ! [K: num] :
% 9.51/5.31        ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
% 9.51/5.31        = ( numeral_numeral_rat @ K ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8747_integer__of__int__code,axiom,
% 9.51/5.31      ( code_integer_of_int
% 9.51/5.31      = ( ^ [K3: int] :
% 9.51/5.31            ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
% 9.51/5.31            @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 9.51/5.31              @ ( if_Code_integer
% 9.51/5.31                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.51/5.31                  = zero_zero_int )
% 9.51/5.31                @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.51/5.31                @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8748_signed__take__bit__negative__iff,axiom,
% 9.51/5.31      ! [N: nat,K: int] :
% 9.51/5.31        ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
% 9.51/5.31        = ( bit_se1146084159140164899it_int @ K @ N ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8749_less__integer_Oabs__eq,axiom,
% 9.51/5.31      ! [Xa2: int,X: int] :
% 9.51/5.31        ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
% 9.51/5.31        = ( ord_less_int @ Xa2 @ X ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8750_less__integer__code_I1_J,axiom,
% 9.51/5.31      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8751_abs__integer__code,axiom,
% 9.51/5.31      ( abs_abs_Code_integer
% 9.51/5.31      = ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8752_not__bit__Suc__0__Suc,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8753_bit__Suc__0__iff,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.51/5.31        = ( N = zero_zero_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8754_not__bit__Suc__0__numeral,axiom,
% 9.51/5.31      ! [N: num] :
% 9.51/5.31        ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8755_bit__imp__take__bit__positive,axiom,
% 9.51/5.31      ! [N: nat,M: nat,K: int] :
% 9.51/5.31        ( ( ord_less_nat @ N @ M )
% 9.51/5.31       => ( ( bit_se1146084159140164899it_int @ K @ N )
% 9.51/5.31         => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8756_int__bit__bound,axiom,
% 9.51/5.31      ! [K: int] :
% 9.51/5.31        ~ ! [N3: nat] :
% 9.51/5.31            ( ! [M3: nat] :
% 9.51/5.31                ( ( ord_less_eq_nat @ N3 @ M3 )
% 9.51/5.31               => ( ( bit_se1146084159140164899it_int @ K @ M3 )
% 9.51/5.31                  = ( bit_se1146084159140164899it_int @ K @ N3 ) ) )
% 9.51/5.31           => ~ ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.51/5.31               => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N3 @ one_one_nat ) )
% 9.51/5.31                  = ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8757_int__of__integer__code,axiom,
% 9.51/5.31      ( code_int_of_integer
% 9.51/5.31      = ( ^ [K3: code_integer] :
% 9.51/5.31            ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
% 9.51/5.31            @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 9.51/5.31              @ ( produc1553301316500091796er_int
% 9.51/5.31                @ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
% 9.51/5.31                @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8758_bit__cut__integer__def,axiom,
% 9.51/5.31      ( code_bit_cut_integer
% 9.51/5.31      = ( ^ [K3: code_integer] :
% 9.51/5.31            ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.51/5.31            @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8759_divmod__integer__def,axiom,
% 9.51/5.31      ( code_divmod_integer
% 9.51/5.31      = ( ^ [K3: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K3 @ L2 ) @ ( modulo364778990260209775nteger @ K3 @ L2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8760_less__integer_Orep__eq,axiom,
% 9.51/5.31      ( ord_le6747313008572928689nteger
% 9.51/5.31      = ( ^ [X3: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8761_integer__less__iff,axiom,
% 9.51/5.31      ( ord_le6747313008572928689nteger
% 9.51/5.31      = ( ^ [K3: code_integer,L2: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8762_binomial__def,axiom,
% 9.51/5.31      ( binomial
% 9.51/5.31      = ( ^ [N2: nat,K3: nat] :
% 9.51/5.31            ( finite_card_set_nat
% 9.51/5.31            @ ( collect_set_nat
% 9.51/5.31              @ ^ [K6: set_nat] :
% 9.51/5.31                  ( ( member_set_nat @ K6 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
% 9.51/5.31                  & ( ( finite_card_nat @ K6 )
% 9.51/5.31                    = K3 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8763_bit__cut__integer__code,axiom,
% 9.51/5.31      ( code_bit_cut_integer
% 9.51/5.31      = ( ^ [K3: code_integer] :
% 9.51/5.31            ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 9.51/5.31            @ ( produc9125791028180074456eger_o
% 9.51/5.31              @ ^ [R5: code_integer,S8: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S8 ) ) @ ( S8 = one_one_Code_integer ) )
% 9.51/5.31              @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8764_nat__of__integer__code,axiom,
% 9.51/5.31      ( code_nat_of_integer
% 9.51/5.31      = ( ^ [K3: code_integer] :
% 9.51/5.31            ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 9.51/5.31            @ ( produc1555791787009142072er_nat
% 9.51/5.31              @ ^ [L2: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ one_one_nat ) )
% 9.51/5.31              @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8765_divmod__abs__def,axiom,
% 9.51/5.31      ( code_divmod_abs
% 9.51/5.31      = ( ^ [K3: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L2 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8766_pair__leqI2,axiom,
% 9.51/5.31      ! [A: nat,B: nat,S: nat,T: nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ A @ B )
% 9.51/5.31       => ( ( ord_less_eq_nat @ S @ T )
% 9.51/5.31         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8767_pair__leqI1,axiom,
% 9.51/5.31      ! [A: nat,B: nat,S: nat,T: nat] :
% 9.51/5.31        ( ( ord_less_nat @ A @ B )
% 9.51/5.31       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8768_nat__of__integer__non__positive,axiom,
% 9.51/5.31      ! [K: code_integer] :
% 9.51/5.31        ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
% 9.51/5.31       => ( ( code_nat_of_integer @ K )
% 9.51/5.31          = zero_zero_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8769_nat__of__integer__code__post_I1_J,axiom,
% 9.51/5.31      ( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
% 9.51/5.31      = zero_zero_nat ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8770_divmod__abs__code_I6_J,axiom,
% 9.51/5.31      ! [J: code_integer] :
% 9.51/5.31        ( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
% 9.51/5.31        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8771_divmod__abs__code_I5_J,axiom,
% 9.51/5.31      ! [J: code_integer] :
% 9.51/5.31        ( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
% 9.51/5.31        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8772_divmod__integer__code,axiom,
% 9.51/5.31      ( code_divmod_integer
% 9.51/5.31      = ( ^ [K3: code_integer,L2: code_integer] :
% 9.51/5.31            ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 9.51/5.31            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
% 9.51/5.31              @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L2 )
% 9.51/5.31                @ ( produc6916734918728496179nteger
% 9.51/5.31                  @ ^ [R5: code_integer,S8: code_integer] : ( if_Pro6119634080678213985nteger @ ( S8 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S8 ) ) )
% 9.51/5.31                  @ ( code_divmod_abs @ K3 @ L2 ) ) )
% 9.51/5.31              @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 9.51/5.31                @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 9.51/5.31                  @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L2 )
% 9.51/5.31                    @ ( produc6916734918728496179nteger
% 9.51/5.31                      @ ^ [R5: code_integer,S8: code_integer] : ( if_Pro6119634080678213985nteger @ ( S8 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S8 ) ) )
% 9.51/5.31                      @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8773_wmin__insertI,axiom,
% 9.51/5.31      ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
% 9.51/5.31        ( ( member8440522571783428010at_nat @ X @ XS )
% 9.51/5.31       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
% 9.51/5.31         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
% 9.51/5.31           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8774_wmax__insertI,axiom,
% 9.51/5.31      ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
% 9.51/5.31        ( ( member8440522571783428010at_nat @ Y @ YS )
% 9.51/5.31       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
% 9.51/5.31         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
% 9.51/5.31           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8775_bezw__0,axiom,
% 9.51/5.31      ! [X: nat] :
% 9.51/5.31        ( ( bezw @ X @ zero_zero_nat )
% 9.51/5.31        = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8776_Suc__0__mod__numeral,axiom,
% 9.51/5.31      ! [K: num] :
% 9.51/5.31        ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 9.51/5.31        = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8777_wmax__emptyI,axiom,
% 9.51/5.31      ! [X8: set_Pr1261947904930325089at_nat] :
% 9.51/5.31        ( ( finite6177210948735845034at_nat @ X8 )
% 9.51/5.31       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X8 ) @ fun_max_weak ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8778_wmin__emptyI,axiom,
% 9.51/5.31      ! [X8: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X8 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8779_smin__insertI,axiom,
% 9.51/5.31      ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
% 9.51/5.31        ( ( member8440522571783428010at_nat @ X @ XS )
% 9.51/5.31       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
% 9.51/5.31         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
% 9.51/5.31           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8780_smax__insertI,axiom,
% 9.51/5.31      ! [Y: product_prod_nat_nat,Y7: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X8: set_Pr1261947904930325089at_nat] :
% 9.51/5.31        ( ( member8440522571783428010at_nat @ Y @ Y7 )
% 9.51/5.31       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
% 9.51/5.31         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X8 @ Y7 ) @ fun_max_strict )
% 9.51/5.31           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ X8 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8781_prod__decode__aux_Osimps,axiom,
% 9.51/5.31      ( nat_prod_decode_aux
% 9.51/5.31      = ( ^ [K3: nat,M2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M2 @ K3 ) @ ( product_Pair_nat_nat @ M2 @ ( minus_minus_nat @ K3 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M2 @ ( suc @ K3 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8782_prod__decode__aux_Oelims,axiom,
% 9.51/5.31      ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
% 9.51/5.31        ( ( ( nat_prod_decode_aux @ X @ Xa2 )
% 9.51/5.31          = Y )
% 9.51/5.31       => ( ( ( ord_less_eq_nat @ Xa2 @ X )
% 9.51/5.31           => ( Y
% 9.51/5.31              = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
% 9.51/5.31          & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
% 9.51/5.31           => ( Y
% 9.51/5.31              = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8783_Suc__0__div__numeral,axiom,
% 9.51/5.31      ! [K: num] :
% 9.51/5.31        ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 9.51/5.31        = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8784_quotient__of__denom__pos_H,axiom,
% 9.51/5.31      ! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8785_smin__emptyI,axiom,
% 9.51/5.31      ! [X8: set_Pr1261947904930325089at_nat] :
% 9.51/5.31        ( ( X8 != bot_bo2099793752762293965at_nat )
% 9.51/5.31       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X8 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8786_smax__emptyI,axiom,
% 9.51/5.31      ! [Y7: set_Pr1261947904930325089at_nat] :
% 9.51/5.31        ( ( finite6177210948735845034at_nat @ Y7 )
% 9.51/5.31       => ( ( Y7 != bot_bo2099793752762293965at_nat )
% 9.51/5.31         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y7 ) @ fun_max_strict ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8787_bezw_Osimps,axiom,
% 9.51/5.31      ( bezw
% 9.51/5.31      = ( ^ [X3: nat,Y2: nat] : ( if_Pro3027730157355071871nt_int @ ( Y2 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X3 @ Y2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X3 @ Y2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X3 @ Y2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Y2 ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8788_rat__sgn__code,axiom,
% 9.51/5.31      ! [P4: rat] :
% 9.51/5.31        ( ( quotient_of @ ( sgn_sgn_rat @ P4 ) )
% 9.51/5.31        = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P4 ) ) ) @ one_one_int ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8789_bezw__non__0,axiom,
% 9.51/5.31      ! [Y: nat,X: nat] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ Y )
% 9.51/5.31       => ( ( bezw @ X @ Y )
% 9.51/5.31          = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8790_bezw_Oelims,axiom,
% 9.51/5.31      ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
% 9.51/5.31        ( ( ( bezw @ X @ Xa2 )
% 9.51/5.31          = Y )
% 9.51/5.31       => ( ( ( Xa2 = zero_zero_nat )
% 9.51/5.31           => ( Y
% 9.51/5.31              = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 9.51/5.31          & ( ( Xa2 != zero_zero_nat )
% 9.51/5.31           => ( Y
% 9.51/5.31              = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8791_bezw_Opelims,axiom,
% 9.51/5.31      ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
% 9.51/5.31        ( ( ( bezw @ X @ Xa2 )
% 9.51/5.31          = Y )
% 9.51/5.31       => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
% 9.51/5.31         => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 9.51/5.31                 => ( Y
% 9.51/5.31                    = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 9.51/5.31                & ( ( Xa2 != zero_zero_nat )
% 9.51/5.31                 => ( Y
% 9.51/5.31                    = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) )
% 9.51/5.31             => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8792_normalize__def,axiom,
% 9.51/5.31      ( normalize
% 9.51/5.31      = ( ^ [P6: product_prod_int_int] :
% 9.51/5.31            ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P6 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P6 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P6 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) )
% 9.51/5.31            @ ( if_Pro3027730157355071871nt_int
% 9.51/5.31              @ ( ( product_snd_int_int @ P6 )
% 9.51/5.31                = zero_zero_int )
% 9.51/5.31              @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 9.51/5.31              @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P6 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P6 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8793_vebt__mint_Opelims,axiom,
% 9.51/5.31      ! [X: vEBT_VEBT,Y: option_nat] :
% 9.51/5.31        ( ( ( vEBT_vebt_mint @ X )
% 9.51/5.31          = Y )
% 9.51/5.31       => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X )
% 9.51/5.31         => ( ! [A3: $o,B3: $o] :
% 9.51/5.31                ( ( X
% 9.51/5.31                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.51/5.31               => ( ( ( A3
% 9.51/5.31                     => ( Y
% 9.51/5.31                        = ( some_nat @ zero_zero_nat ) ) )
% 9.51/5.31                    & ( ~ A3
% 9.51/5.31                     => ( ( B3
% 9.51/5.31                         => ( Y
% 9.51/5.31                            = ( some_nat @ one_one_nat ) ) )
% 9.51/5.31                        & ( ~ B3
% 9.51/5.31                         => ( Y = none_nat ) ) ) ) )
% 9.51/5.31                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
% 9.51/5.31           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.51/5.31                  ( ( X
% 9.51/5.31                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.51/5.31                 => ( ( Y = none_nat )
% 9.51/5.31                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 9.51/5.31             => ~ ! [Mi3: nat,Ma3: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.51/5.31                    ( ( X
% 9.51/5.31                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 9.51/5.31                   => ( ( Y
% 9.51/5.31                        = ( some_nat @ Mi3 ) )
% 9.51/5.31                     => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8794_vebt__maxt_Opelims,axiom,
% 9.51/5.31      ! [X: vEBT_VEBT,Y: option_nat] :
% 9.51/5.31        ( ( ( vEBT_vebt_maxt @ X )
% 9.51/5.31          = Y )
% 9.51/5.31       => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
% 9.51/5.31         => ( ! [A3: $o,B3: $o] :
% 9.51/5.31                ( ( X
% 9.51/5.31                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.51/5.31               => ( ( ( B3
% 9.51/5.31                     => ( Y
% 9.51/5.31                        = ( some_nat @ one_one_nat ) ) )
% 9.51/5.31                    & ( ~ B3
% 9.51/5.31                     => ( ( A3
% 9.51/5.31                         => ( Y
% 9.51/5.31                            = ( some_nat @ zero_zero_nat ) ) )
% 9.51/5.31                        & ( ~ A3
% 9.51/5.31                         => ( Y = none_nat ) ) ) ) )
% 9.51/5.31                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
% 9.51/5.31           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.51/5.31                  ( ( X
% 9.51/5.31                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.51/5.31                 => ( ( Y = none_nat )
% 9.51/5.31                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 9.51/5.31             => ~ ! [Mi3: nat,Ma3: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.51/5.31                    ( ( X
% 9.51/5.31                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 9.51/5.31                   => ( ( Y
% 9.51/5.31                        = ( some_nat @ Ma3 ) )
% 9.51/5.31                     => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8795_gcd__pos__int,axiom,
% 9.51/5.31      ! [M: int,N: int] :
% 9.51/5.31        ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
% 9.51/5.31        = ( ( M != zero_zero_int )
% 9.51/5.31          | ( N != zero_zero_int ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8796_gcd__le1__int,axiom,
% 9.51/5.31      ! [A: int,B: int] :
% 9.51/5.31        ( ( ord_less_int @ zero_zero_int @ A )
% 9.51/5.31       => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8797_gcd__le2__int,axiom,
% 9.51/5.31      ! [B: int,A: int] :
% 9.51/5.31        ( ( ord_less_int @ zero_zero_int @ B )
% 9.51/5.31       => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8798_gcd__non__0__int,axiom,
% 9.51/5.31      ! [Y: int,X: int] :
% 9.51/5.31        ( ( ord_less_int @ zero_zero_int @ Y )
% 9.51/5.31       => ( ( gcd_gcd_int @ X @ Y )
% 9.51/5.31          = ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8799_prod__decode__aux_Opelims,axiom,
% 9.51/5.31      ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
% 9.51/5.31        ( ( ( nat_prod_decode_aux @ X @ Xa2 )
% 9.51/5.31          = Y )
% 9.51/5.31       => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
% 9.51/5.31         => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X )
% 9.51/5.31                 => ( Y
% 9.51/5.31                    = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
% 9.51/5.31                & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
% 9.51/5.31                 => ( Y
% 9.51/5.31                    = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) )
% 9.51/5.31             => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8800_gcd__nat_Oeq__neutr__iff,axiom,
% 9.51/5.31      ! [A: nat,B: nat] :
% 9.51/5.31        ( ( ( gcd_gcd_nat @ A @ B )
% 9.51/5.31          = zero_zero_nat )
% 9.51/5.31        = ( ( A = zero_zero_nat )
% 9.51/5.31          & ( B = zero_zero_nat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8801_gcd__nat_Oleft__neutral,axiom,
% 9.51/5.31      ! [A: nat] :
% 9.51/5.31        ( ( gcd_gcd_nat @ zero_zero_nat @ A )
% 9.51/5.31        = A ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8802_gcd__nat_Oneutr__eq__iff,axiom,
% 9.51/5.31      ! [A: nat,B: nat] :
% 9.51/5.31        ( ( zero_zero_nat
% 9.51/5.31          = ( gcd_gcd_nat @ A @ B ) )
% 9.51/5.31        = ( ( A = zero_zero_nat )
% 9.51/5.31          & ( B = zero_zero_nat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8803_gcd__nat_Oright__neutral,axiom,
% 9.51/5.31      ! [A: nat] :
% 9.51/5.31        ( ( gcd_gcd_nat @ A @ zero_zero_nat )
% 9.51/5.31        = A ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8804_gcd__0__nat,axiom,
% 9.51/5.31      ! [X: nat] :
% 9.51/5.31        ( ( gcd_gcd_nat @ X @ zero_zero_nat )
% 9.51/5.31        = X ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8805_gcd__0__left__nat,axiom,
% 9.51/5.31      ! [X: nat] :
% 9.51/5.31        ( ( gcd_gcd_nat @ zero_zero_nat @ X )
% 9.51/5.31        = X ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8806_gcd__Suc__0,axiom,
% 9.51/5.31      ! [M: nat] :
% 9.51/5.31        ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
% 9.51/5.31        = ( suc @ zero_zero_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8807_gcd__pos__nat,axiom,
% 9.51/5.31      ! [M: nat,N: nat] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
% 9.51/5.31        = ( ( M != zero_zero_nat )
% 9.51/5.31          | ( N != zero_zero_nat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8808_Sup__nat__empty,axiom,
% 9.51/5.31      ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
% 9.51/5.31      = zero_zero_nat ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8809_Inf__nat__def1,axiom,
% 9.51/5.31      ! [K4: set_nat] :
% 9.51/5.31        ( ( K4 != bot_bot_set_nat )
% 9.51/5.31       => ( member_nat @ ( complete_Inf_Inf_nat @ K4 ) @ K4 ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8810_gcd__le2__nat,axiom,
% 9.51/5.31      ! [B: nat,A: nat] :
% 9.51/5.31        ( ( B != zero_zero_nat )
% 9.51/5.31       => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8811_gcd__le1__nat,axiom,
% 9.51/5.31      ! [A: nat,B: nat] :
% 9.51/5.31        ( ( A != zero_zero_nat )
% 9.51/5.31       => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8812_gcd__diff2__nat,axiom,
% 9.51/5.31      ! [M: nat,N: nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.31       => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
% 9.51/5.31          = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8813_gcd__diff1__nat,axiom,
% 9.51/5.31      ! [N: nat,M: nat] :
% 9.51/5.31        ( ( ord_less_eq_nat @ N @ M )
% 9.51/5.31       => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
% 9.51/5.31          = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8814_gcd__nat_Oelims,axiom,
% 9.51/5.31      ! [X: nat,Xa2: nat,Y: nat] :
% 9.51/5.31        ( ( ( gcd_gcd_nat @ X @ Xa2 )
% 9.51/5.31          = Y )
% 9.51/5.31       => ( ( ( Xa2 = zero_zero_nat )
% 9.51/5.31           => ( Y = X ) )
% 9.51/5.31          & ( ( Xa2 != zero_zero_nat )
% 9.51/5.31           => ( Y
% 9.51/5.31              = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8815_gcd__nat_Osimps,axiom,
% 9.51/5.31      ( gcd_gcd_nat
% 9.51/5.31      = ( ^ [X3: nat,Y2: nat] : ( if_nat @ ( Y2 = zero_zero_nat ) @ X3 @ ( gcd_gcd_nat @ Y2 @ ( modulo_modulo_nat @ X3 @ Y2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8816_gcd__non__0__nat,axiom,
% 9.51/5.31      ! [Y: nat,X: nat] :
% 9.51/5.31        ( ( Y != zero_zero_nat )
% 9.51/5.31       => ( ( gcd_gcd_nat @ X @ Y )
% 9.51/5.31          = ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8817_bezout__nat,axiom,
% 9.51/5.31      ! [A: nat,B: nat] :
% 9.51/5.31        ( ( A != zero_zero_nat )
% 9.51/5.31       => ? [X4: nat,Y3: nat] :
% 9.51/5.31            ( ( times_times_nat @ A @ X4 )
% 9.51/5.31            = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8818_bezout__gcd__nat_H,axiom,
% 9.51/5.31      ! [B: nat,A: nat] :
% 9.51/5.31      ? [X4: nat,Y3: nat] :
% 9.51/5.31        ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y3 ) @ ( times_times_nat @ A @ X4 ) )
% 9.51/5.31          & ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y3 ) )
% 9.51/5.31            = ( gcd_gcd_nat @ A @ B ) ) )
% 9.51/5.31        | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y3 ) @ ( times_times_nat @ B @ X4 ) )
% 9.51/5.31          & ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y3 ) )
% 9.51/5.31            = ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8819_gcd__nat_Opelims,axiom,
% 9.51/5.31      ! [X: nat,Xa2: nat,Y: nat] :
% 9.51/5.31        ( ( ( gcd_gcd_nat @ X @ Xa2 )
% 9.51/5.31          = Y )
% 9.51/5.31       => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
% 9.51/5.31         => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 9.51/5.31                 => ( Y = X ) )
% 9.51/5.31                & ( ( Xa2 != zero_zero_nat )
% 9.51/5.31                 => ( Y
% 9.51/5.31                    = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) )
% 9.51/5.31             => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8820_VEBT__internal_OminNull_Opelims_I1_J,axiom,
% 9.51/5.31      ! [X: vEBT_VEBT,Y: $o] :
% 9.51/5.31        ( ( ( vEBT_VEBT_minNull @ X )
% 9.51/5.31          = Y )
% 9.51/5.31       => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
% 9.51/5.31         => ( ( ( X
% 9.51/5.31                = ( vEBT_Leaf @ $false @ $false ) )
% 9.51/5.31             => ( Y
% 9.51/5.31               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
% 9.51/5.31           => ( ! [Uv2: $o] :
% 9.51/5.31                  ( ( X
% 9.51/5.31                    = ( vEBT_Leaf @ $true @ Uv2 ) )
% 9.51/5.31                 => ( ~ Y
% 9.51/5.31                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
% 9.51/5.31             => ( ! [Uu2: $o] :
% 9.51/5.31                    ( ( X
% 9.51/5.31                      = ( vEBT_Leaf @ Uu2 @ $true ) )
% 9.51/5.31                   => ( ~ Y
% 9.51/5.31                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
% 9.51/5.31               => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.51/5.31                      ( ( X
% 9.51/5.31                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 9.51/5.31                     => ( Y
% 9.51/5.31                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
% 9.51/5.31                 => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.51/5.31                        ( ( X
% 9.51/5.31                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 9.51/5.31                       => ( ~ Y
% 9.51/5.31                         => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8821_VEBT__internal_OminNull_Opelims_I2_J,axiom,
% 9.51/5.31      ! [X: vEBT_VEBT] :
% 9.51/5.31        ( ( vEBT_VEBT_minNull @ X )
% 9.51/5.31       => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
% 9.51/5.31         => ( ( ( X
% 9.51/5.31                = ( vEBT_Leaf @ $false @ $false ) )
% 9.51/5.31             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
% 9.51/5.31           => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.51/5.31                  ( ( X
% 9.51/5.31                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 9.51/5.31                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8822_VEBT__internal_OminNull_Opelims_I3_J,axiom,
% 9.51/5.31      ! [X: vEBT_VEBT] :
% 9.51/5.31        ( ~ ( vEBT_VEBT_minNull @ X )
% 9.51/5.31       => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
% 9.51/5.31         => ( ! [Uv2: $o] :
% 9.51/5.31                ( ( X
% 9.51/5.31                  = ( vEBT_Leaf @ $true @ Uv2 ) )
% 9.51/5.31               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
% 9.51/5.31           => ( ! [Uu2: $o] :
% 9.51/5.31                  ( ( X
% 9.51/5.31                    = ( vEBT_Leaf @ Uu2 @ $true ) )
% 9.51/5.31                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
% 9.51/5.31             => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.51/5.31                    ( ( X
% 9.51/5.31                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 9.51/5.31                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8823_finite__enumerate,axiom,
% 9.51/5.31      ! [S2: set_nat] :
% 9.51/5.31        ( ( finite_finite_nat @ S2 )
% 9.51/5.31       => ? [R3: nat > nat] :
% 9.51/5.31            ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S2 ) ) )
% 9.51/5.31            & ! [N5: nat] :
% 9.51/5.31                ( ( ord_less_nat @ N5 @ ( finite_card_nat @ S2 ) )
% 9.51/5.31               => ( member_nat @ ( R3 @ N5 ) @ S2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8824_divmod__integer__eq__cases,axiom,
% 9.51/5.31      ( code_divmod_integer
% 9.51/5.31      = ( ^ [K3: code_integer,L2: code_integer] :
% 9.51/5.31            ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 9.51/5.31            @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 9.51/5.31              @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
% 9.51/5.31                @ ( if_Pro6119634080678213985nteger
% 9.51/5.31                  @ ( ( sgn_sgn_Code_integer @ K3 )
% 9.51/5.31                    = ( sgn_sgn_Code_integer @ L2 ) )
% 9.51/5.31                  @ ( code_divmod_abs @ K3 @ L2 )
% 9.51/5.31                  @ ( produc6916734918728496179nteger
% 9.51/5.31                    @ ^ [R5: code_integer,S8: code_integer] : ( if_Pro6119634080678213985nteger @ ( S8 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S8 ) ) )
% 9.51/5.31                    @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8825_card_Ocomp__fun__commute__on,axiom,
% 9.51/5.31      ( ( comp_nat_nat_nat @ suc @ suc )
% 9.51/5.31      = ( comp_nat_nat_nat @ suc @ suc ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8826_root__powr__inverse,axiom,
% 9.51/5.31      ! [N: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.31         => ( ( root @ N @ X )
% 9.51/5.31            = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8827_drop__bit__negative__int__iff,axiom,
% 9.51/5.31      ! [N: nat,K: int] :
% 9.51/5.31        ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
% 9.51/5.31        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8828_real__root__Suc__0,axiom,
% 9.51/5.31      ! [X: real] :
% 9.51/5.31        ( ( root @ ( suc @ zero_zero_nat ) @ X )
% 9.51/5.31        = X ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8829_real__root__eq__iff,axiom,
% 9.51/5.31      ! [N: nat,X: real,Y: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ( root @ N @ X )
% 9.51/5.31            = ( root @ N @ Y ) )
% 9.51/5.31          = ( X = Y ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8830_root__0,axiom,
% 9.51/5.31      ! [X: real] :
% 9.51/5.31        ( ( root @ zero_zero_nat @ X )
% 9.51/5.31        = zero_zero_real ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8831_drop__bit__of__Suc__0,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.51/5.31        = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8832_real__root__eq__0__iff,axiom,
% 9.51/5.31      ! [N: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ( root @ N @ X )
% 9.51/5.31            = zero_zero_real )
% 9.51/5.31          = ( X = zero_zero_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8833_real__root__less__iff,axiom,
% 9.51/5.31      ! [N: nat,X: real,Y: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
% 9.51/5.31          = ( ord_less_real @ X @ Y ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8834_real__root__le__iff,axiom,
% 9.51/5.31      ! [N: nat,X: real,Y: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
% 9.51/5.31          = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8835_real__root__eq__1__iff,axiom,
% 9.51/5.31      ! [N: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ( root @ N @ X )
% 9.51/5.31            = one_one_real )
% 9.51/5.31          = ( X = one_one_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8836_real__root__one,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( root @ N @ one_one_real )
% 9.51/5.31          = one_one_real ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8837_real__root__lt__0__iff,axiom,
% 9.51/5.31      ! [N: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
% 9.51/5.31          = ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8838_real__root__gt__0__iff,axiom,
% 9.51/5.31      ! [N: nat,Y: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y ) )
% 9.51/5.31          = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8839_real__root__le__0__iff,axiom,
% 9.51/5.31      ! [N: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
% 9.51/5.31          = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8840_real__root__ge__0__iff,axiom,
% 9.51/5.31      ! [N: nat,Y: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y ) )
% 9.51/5.31          = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8841_real__root__lt__1__iff,axiom,
% 9.51/5.31      ! [N: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
% 9.51/5.31          = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8842_real__root__gt__1__iff,axiom,
% 9.51/5.31      ! [N: nat,Y: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y ) )
% 9.51/5.31          = ( ord_less_real @ one_one_real @ Y ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8843_real__root__le__1__iff,axiom,
% 9.51/5.31      ! [N: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
% 9.51/5.31          = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8844_real__root__ge__1__iff,axiom,
% 9.51/5.31      ! [N: nat,Y: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y ) )
% 9.51/5.31          = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8845_real__root__pow__pos2,axiom,
% 9.51/5.31      ! [N: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.51/5.31         => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 9.51/5.31            = X ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8846_real__root__less__mono,axiom,
% 9.51/5.31      ! [N: nat,X: real,Y: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_real @ X @ Y )
% 9.51/5.31         => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8847_real__root__le__mono,axiom,
% 9.51/5.31      ! [N: nat,X: real,Y: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_eq_real @ X @ Y )
% 9.51/5.31         => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8848_real__root__power,axiom,
% 9.51/5.31      ! [N: nat,X: real,K: nat] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( root @ N @ ( power_power_real @ X @ K ) )
% 9.51/5.31          = ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8849_real__root__abs,axiom,
% 9.51/5.31      ! [N: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( root @ N @ ( abs_abs_real @ X ) )
% 9.51/5.31          = ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8850_sgn__root,axiom,
% 9.51/5.31      ! [N: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( sgn_sgn_real @ ( root @ N @ X ) )
% 9.51/5.31          = ( sgn_sgn_real @ X ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8851_real__root__gt__zero,axiom,
% 9.51/5.31      ! [N: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.31         => ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8852_real__root__strict__decreasing,axiom,
% 9.51/5.31      ! [N: nat,N6: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_nat @ N @ N6 )
% 9.51/5.31         => ( ( ord_less_real @ one_one_real @ X )
% 9.51/5.31           => ( ord_less_real @ ( root @ N6 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8853_root__abs__power,axiom,
% 9.51/5.31      ! [N: nat,Y: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y @ N ) ) )
% 9.51/5.31          = ( abs_abs_real @ Y ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8854_real__root__pos__pos,axiom,
% 9.51/5.31      ! [N: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.31         => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8855_real__root__strict__increasing,axiom,
% 9.51/5.31      ! [N: nat,N6: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_nat @ N @ N6 )
% 9.51/5.31         => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.31           => ( ( ord_less_real @ X @ one_one_real )
% 9.51/5.31             => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8856_real__root__decreasing,axiom,
% 9.51/5.31      ! [N: nat,N6: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_eq_nat @ N @ N6 )
% 9.51/5.31         => ( ( ord_less_eq_real @ one_one_real @ X )
% 9.51/5.31           => ( ord_less_eq_real @ ( root @ N6 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8857_real__root__pow__pos,axiom,
% 9.51/5.31      ! [N: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.31         => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 9.51/5.31            = X ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8858_real__root__pos__unique,axiom,
% 9.51/5.31      ! [N: nat,Y: real,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 9.51/5.31         => ( ( ( power_power_real @ Y @ N )
% 9.51/5.31              = X )
% 9.51/5.31           => ( ( root @ N @ X )
% 9.51/5.31              = Y ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8859_real__root__power__cancel,axiom,
% 9.51/5.31      ! [N: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.51/5.31         => ( ( root @ N @ ( power_power_real @ X @ N ) )
% 9.51/5.31            = X ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8860_real__root__increasing,axiom,
% 9.51/5.31      ! [N: nat,N6: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_eq_nat @ N @ N6 )
% 9.51/5.31         => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.51/5.31           => ( ( ord_less_eq_real @ X @ one_one_real )
% 9.51/5.31             => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8861_sgn__power__root,axiom,
% 9.51/5.31      ! [N: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
% 9.51/5.31          = X ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8862_root__sgn__power,axiom,
% 9.51/5.31      ! [N: nat,Y: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) ) )
% 9.51/5.31          = Y ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8863_ln__root,axiom,
% 9.51/5.31      ! [N: nat,B: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.51/5.31         => ( ( ln_ln_real @ ( root @ N @ B ) )
% 9.51/5.31            = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8864_log__root,axiom,
% 9.51/5.31      ! [N: nat,A: real,B: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.51/5.31         => ( ( log @ B @ ( root @ N @ A ) )
% 9.51/5.31            = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8865_log__base__root,axiom,
% 9.51/5.31      ! [N: nat,B: real,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.51/5.31         => ( ( log @ ( root @ N @ B ) @ X )
% 9.51/5.31            = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8866_split__root,axiom,
% 9.51/5.31      ! [P: real > $o,N: nat,X: real] :
% 9.51/5.31        ( ( P @ ( root @ N @ X ) )
% 9.51/5.31        = ( ( ( N = zero_zero_nat )
% 9.51/5.31           => ( P @ zero_zero_real ) )
% 9.51/5.31          & ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31           => ! [Y2: real] :
% 9.51/5.31                ( ( ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N ) )
% 9.51/5.31                  = X )
% 9.51/5.31               => ( P @ Y2 ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8867_Suc__funpow,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( compow_nat_nat @ N @ suc )
% 9.51/5.31        = ( plus_plus_nat @ N ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8868_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 9.51/5.31      ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 9.51/5.31      @ ^ [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X3 )
% 9.51/5.31      @ ^ [X3: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X3 ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8869_push__bit__of__Suc__0,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.51/5.31        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8870_bit__push__bit__iff__nat,axiom,
% 9.51/5.31      ! [M: nat,Q4: nat,N: nat] :
% 9.51/5.31        ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q4 ) @ N )
% 9.51/5.31        = ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.31          & ( bit_se1148574629649215175it_nat @ Q4 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8871_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
% 9.51/5.31      ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
% 9.51/5.31      @ ^ [M2: nat,N2: nat] :
% 9.51/5.31          ( ( dvd_dvd_nat @ M2 @ N2 )
% 9.51/5.31          & ( M2 != N2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8872_min__rpair__set,axiom,
% 9.51/5.31      fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8873_max__rpair__set,axiom,
% 9.51/5.31      fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8874_push__bit__negative__int__iff,axiom,
% 9.51/5.31      ! [N: nat,K: int] :
% 9.51/5.31        ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
% 9.51/5.31        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8875_bit__push__bit__iff__int,axiom,
% 9.51/5.31      ! [M: nat,K: int,N: nat] :
% 9.51/5.31        ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
% 9.51/5.31        = ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.31          & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8876_Gcd__remove0__nat,axiom,
% 9.51/5.31      ! [M4: set_nat] :
% 9.51/5.31        ( ( finite_finite_nat @ M4 )
% 9.51/5.31       => ( ( gcd_Gcd_nat @ M4 )
% 9.51/5.31          = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M4 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8877_times__int_Oabs__eq,axiom,
% 9.51/5.31      ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
% 9.51/5.31        ( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
% 9.51/5.31        = ( abs_Integ
% 9.51/5.31          @ ( produc27273713700761075at_nat
% 9.51/5.31            @ ^ [X3: nat,Y2: nat] :
% 9.51/5.31                ( produc2626176000494625587at_nat
% 9.51/5.31                @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y2 @ V3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V3 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) )
% 9.51/5.31            @ Xa2
% 9.51/5.31            @ X ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8878_eq__Abs__Integ,axiom,
% 9.51/5.31      ! [Z2: int] :
% 9.51/5.31        ~ ! [X4: nat,Y3: nat] :
% 9.51/5.31            ( Z2
% 9.51/5.31           != ( abs_Integ @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8879_Gcd__in,axiom,
% 9.51/5.31      ! [A2: set_nat] :
% 9.51/5.31        ( ! [A3: nat,B3: nat] :
% 9.51/5.31            ( ( member_nat @ A3 @ A2 )
% 9.51/5.31           => ( ( member_nat @ B3 @ A2 )
% 9.51/5.31             => ( member_nat @ ( gcd_gcd_nat @ A3 @ B3 ) @ A2 ) ) )
% 9.51/5.31       => ( ( A2 != bot_bot_set_nat )
% 9.51/5.31         => ( member_nat @ ( gcd_Gcd_nat @ A2 ) @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8880_zero__int__def,axiom,
% 9.51/5.31      ( zero_zero_int
% 9.51/5.31      = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8881_int__def,axiom,
% 9.51/5.31      ( semiri1314217659103216013at_int
% 9.51/5.31      = ( ^ [N2: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N2 @ zero_zero_nat ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8882_uminus__int_Oabs__eq,axiom,
% 9.51/5.31      ! [X: product_prod_nat_nat] :
% 9.51/5.31        ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
% 9.51/5.31        = ( abs_Integ
% 9.51/5.31          @ ( produc2626176000494625587at_nat
% 9.51/5.31            @ ^ [X3: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X3 )
% 9.51/5.31            @ X ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8883_one__int__def,axiom,
% 9.51/5.31      ( one_one_int
% 9.51/5.31      = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8884_less__int_Oabs__eq,axiom,
% 9.51/5.31      ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
% 9.51/5.31        ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
% 9.51/5.31        = ( produc8739625826339149834_nat_o
% 9.51/5.31          @ ^ [X3: nat,Y2: nat] :
% 9.51/5.31              ( produc6081775807080527818_nat_o
% 9.51/5.31              @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) )
% 9.51/5.31          @ Xa2
% 9.51/5.31          @ X ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8885_less__eq__int_Oabs__eq,axiom,
% 9.51/5.31      ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
% 9.51/5.31        ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
% 9.51/5.31        = ( produc8739625826339149834_nat_o
% 9.51/5.31          @ ^ [X3: nat,Y2: nat] :
% 9.51/5.31              ( produc6081775807080527818_nat_o
% 9.51/5.31              @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) )
% 9.51/5.31          @ Xa2
% 9.51/5.31          @ X ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8886_plus__int_Oabs__eq,axiom,
% 9.51/5.31      ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
% 9.51/5.31        ( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
% 9.51/5.31        = ( abs_Integ
% 9.51/5.31          @ ( produc27273713700761075at_nat
% 9.51/5.31            @ ^ [X3: nat,Y2: nat] :
% 9.51/5.31                ( produc2626176000494625587at_nat
% 9.51/5.31                @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y2 @ V3 ) ) )
% 9.51/5.31            @ Xa2
% 9.51/5.31            @ X ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8887_minus__int_Oabs__eq,axiom,
% 9.51/5.31      ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
% 9.51/5.31        ( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
% 9.51/5.31        = ( abs_Integ
% 9.51/5.31          @ ( produc27273713700761075at_nat
% 9.51/5.31            @ ^ [X3: nat,Y2: nat] :
% 9.51/5.31                ( produc2626176000494625587at_nat
% 9.51/5.31                @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) )
% 9.51/5.31            @ Xa2
% 9.51/5.31            @ X ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8888_less__eq__int_Orep__eq,axiom,
% 9.51/5.31      ( ord_less_eq_int
% 9.51/5.31      = ( ^ [X3: int,Xa4: int] :
% 9.51/5.31            ( produc8739625826339149834_nat_o
% 9.51/5.31            @ ^ [Y2: nat,Z3: nat] :
% 9.51/5.31                ( produc6081775807080527818_nat_o
% 9.51/5.31                @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y2 @ V3 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
% 9.51/5.31            @ ( rep_Integ @ X3 )
% 9.51/5.31            @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8889_less__int_Orep__eq,axiom,
% 9.51/5.31      ( ord_less_int
% 9.51/5.31      = ( ^ [X3: int,Xa4: int] :
% 9.51/5.31            ( produc8739625826339149834_nat_o
% 9.51/5.31            @ ^ [Y2: nat,Z3: nat] :
% 9.51/5.31                ( produc6081775807080527818_nat_o
% 9.51/5.31                @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y2 @ V3 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
% 9.51/5.31            @ ( rep_Integ @ X3 )
% 9.51/5.31            @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8890_rat__floor__lemma,axiom,
% 9.51/5.31      ! [A: int,B: int] :
% 9.51/5.31        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
% 9.51/5.31        & ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8891_image__minus__const__atLeastLessThan__nat,axiom,
% 9.51/5.31      ! [C: nat,Y: nat,X: nat] :
% 9.51/5.31        ( ( ( ord_less_nat @ C @ Y )
% 9.51/5.31         => ( ( image_nat_nat
% 9.51/5.31              @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 9.51/5.31              @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 9.51/5.31            = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
% 9.51/5.31        & ( ~ ( ord_less_nat @ C @ Y )
% 9.51/5.31         => ( ( ( ord_less_nat @ X @ Y )
% 9.51/5.31             => ( ( image_nat_nat
% 9.51/5.31                  @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 9.51/5.31                  @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 9.51/5.31                = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 9.51/5.31            & ( ~ ( ord_less_nat @ X @ Y )
% 9.51/5.31             => ( ( image_nat_nat
% 9.51/5.31                  @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 9.51/5.31                  @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 9.51/5.31                = bot_bot_set_nat ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8892_less__rat,axiom,
% 9.51/5.31      ! [B: int,D: int,A: int,C: int] :
% 9.51/5.31        ( ( B != zero_zero_int )
% 9.51/5.31       => ( ( D != zero_zero_int )
% 9.51/5.31         => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 9.51/5.31            = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8893_zero__notin__Suc__image,axiom,
% 9.51/5.31      ! [A2: set_nat] :
% 9.51/5.31        ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8894_Rat__induct__pos,axiom,
% 9.51/5.31      ! [P: rat > $o,Q4: rat] :
% 9.51/5.31        ( ! [A3: int,B3: int] :
% 9.51/5.31            ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.51/5.31           => ( P @ ( fract @ A3 @ B3 ) ) )
% 9.51/5.31       => ( P @ Q4 ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8895_quotient__of__eq,axiom,
% 9.51/5.31      ! [A: int,B: int,P4: int,Q4: int] :
% 9.51/5.31        ( ( ( quotient_of @ ( fract @ A @ B ) )
% 9.51/5.31          = ( product_Pair_int_int @ P4 @ Q4 ) )
% 9.51/5.31       => ( ( fract @ P4 @ Q4 )
% 9.51/5.31          = ( fract @ A @ B ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8896_normalize__eq,axiom,
% 9.51/5.31      ! [A: int,B: int,P4: int,Q4: int] :
% 9.51/5.31        ( ( ( normalize @ ( product_Pair_int_int @ A @ B ) )
% 9.51/5.31          = ( product_Pair_int_int @ P4 @ Q4 ) )
% 9.51/5.31       => ( ( fract @ P4 @ Q4 )
% 9.51/5.31          = ( fract @ A @ B ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8897_quotient__of__Fract,axiom,
% 9.51/5.31      ! [A: int,B: int] :
% 9.51/5.31        ( ( quotient_of @ ( fract @ A @ B ) )
% 9.51/5.31        = ( normalize @ ( product_Pair_int_int @ A @ B ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8898_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 9.51/5.31        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8899_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 9.51/5.31        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8900_lessThan__Suc__eq__insert__0,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( set_ord_lessThan_nat @ ( suc @ N ) )
% 9.51/5.31        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8901_atMost__Suc__eq__insert__0,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( set_ord_atMost_nat @ ( suc @ N ) )
% 9.51/5.31        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8902_Fract__less__zero__iff,axiom,
% 9.51/5.31      ! [B: int,A: int] :
% 9.51/5.31        ( ( ord_less_int @ zero_zero_int @ B )
% 9.51/5.31       => ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 9.51/5.31          = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8903_zero__less__Fract__iff,axiom,
% 9.51/5.31      ! [B: int,A: int] :
% 9.51/5.31        ( ( ord_less_int @ zero_zero_int @ B )
% 9.51/5.31       => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 9.51/5.31          = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8904_Fract__less__one__iff,axiom,
% 9.51/5.31      ! [B: int,A: int] :
% 9.51/5.31        ( ( ord_less_int @ zero_zero_int @ B )
% 9.51/5.31       => ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
% 9.51/5.31          = ( ord_less_int @ A @ B ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8905_one__less__Fract__iff,axiom,
% 9.51/5.31      ! [B: int,A: int] :
% 9.51/5.31        ( ( ord_less_int @ zero_zero_int @ B )
% 9.51/5.31       => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
% 9.51/5.31          = ( ord_less_int @ B @ A ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8906_zero__le__Fract__iff,axiom,
% 9.51/5.31      ! [B: int,A: int] :
% 9.51/5.31        ( ( ord_less_int @ zero_zero_int @ B )
% 9.51/5.31       => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 9.51/5.31          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8907_Fract__le__zero__iff,axiom,
% 9.51/5.31      ! [B: int,A: int] :
% 9.51/5.31        ( ( ord_less_int @ zero_zero_int @ B )
% 9.51/5.31       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 9.51/5.31          = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8908_one__le__Fract__iff,axiom,
% 9.51/5.31      ! [B: int,A: int] :
% 9.51/5.31        ( ( ord_less_int @ zero_zero_int @ B )
% 9.51/5.31       => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
% 9.51/5.31          = ( ord_less_eq_int @ B @ A ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8909_Fract__le__one__iff,axiom,
% 9.51/5.31      ! [B: int,A: int] :
% 9.51/5.31        ( ( ord_less_int @ zero_zero_int @ B )
% 9.51/5.31       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
% 9.51/5.31          = ( ord_less_eq_int @ A @ B ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8910_uminus__int__def,axiom,
% 9.51/5.31      ( uminus_uminus_int
% 9.51/5.31      = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
% 9.51/5.31        @ ( produc2626176000494625587at_nat
% 9.51/5.31          @ ^ [X3: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X3 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8911_finite__int__iff__bounded,axiom,
% 9.51/5.31      ( finite_finite_int
% 9.51/5.31      = ( ^ [S7: set_int] :
% 9.51/5.31          ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S7 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8912_finite__int__iff__bounded__le,axiom,
% 9.51/5.31      ( finite_finite_int
% 9.51/5.31      = ( ^ [S7: set_int] :
% 9.51/5.31          ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S7 ) @ ( set_ord_atMost_int @ K3 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8913_infinite__int__iff__infinite__nat__abs,axiom,
% 9.51/5.31      ! [S2: set_int] :
% 9.51/5.31        ( ( ~ ( finite_finite_int @ S2 ) )
% 9.51/5.31        = ( ~ ( finite_finite_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ S2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8914_times__int__def,axiom,
% 9.51/5.31      ( times_times_int
% 9.51/5.31      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 9.51/5.31        @ ( produc27273713700761075at_nat
% 9.51/5.31          @ ^ [X3: nat,Y2: nat] :
% 9.51/5.31              ( produc2626176000494625587at_nat
% 9.51/5.31              @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y2 @ V3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V3 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8915_minus__int__def,axiom,
% 9.51/5.31      ( minus_minus_int
% 9.51/5.31      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 9.51/5.31        @ ( produc27273713700761075at_nat
% 9.51/5.31          @ ^ [X3: nat,Y2: nat] :
% 9.51/5.31              ( produc2626176000494625587at_nat
% 9.51/5.31              @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8916_plus__int__def,axiom,
% 9.51/5.31      ( plus_plus_int
% 9.51/5.31      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 9.51/5.31        @ ( produc27273713700761075at_nat
% 9.51/5.31          @ ^ [X3: nat,Y2: nat] :
% 9.51/5.31              ( produc2626176000494625587at_nat
% 9.51/5.31              @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y2 @ V3 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8917_infinite__UNIV__nat,axiom,
% 9.51/5.31      ~ ( finite_finite_nat @ top_top_set_nat ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8918_nat__not__finite,axiom,
% 9.51/5.31      ~ ( finite_finite_nat @ top_top_set_nat ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8919_UNIV__nat__eq,axiom,
% 9.51/5.31      ( top_top_set_nat
% 9.51/5.31      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8920_range__mod,axiom,
% 9.51/5.31      ! [N: nat] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( image_nat_nat
% 9.51/5.31            @ ^ [M2: nat] : ( modulo_modulo_nat @ M2 @ N )
% 9.51/5.31            @ top_top_set_nat )
% 9.51/5.31          = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8921_card__UNIV__unit,axiom,
% 9.51/5.31      ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
% 9.51/5.31      = one_one_nat ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8922_range__mult,axiom,
% 9.51/5.31      ! [A: real] :
% 9.51/5.31        ( ( ( A = zero_zero_real )
% 9.51/5.31         => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 9.51/5.31            = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 9.51/5.31        & ( ( A != zero_zero_real )
% 9.51/5.31         => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 9.51/5.31            = top_top_set_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8923_infinite__UNIV__int,axiom,
% 9.51/5.31      ~ ( finite_finite_int @ top_top_set_int ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8924_root__def,axiom,
% 9.51/5.31      ( root
% 9.51/5.31      = ( ^ [N2: nat,X3: real] :
% 9.51/5.31            ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
% 9.51/5.31            @ ( the_in5290026491893676941l_real @ top_top_set_real
% 9.51/5.31              @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N2 ) )
% 9.51/5.31              @ X3 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8925_DERIV__even__real__root,axiom,
% 9.51/5.31      ! [N: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.51/5.31         => ( ( ord_less_real @ X @ zero_zero_real )
% 9.51/5.31           => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8926_has__real__derivative__neg__dec__left,axiom,
% 9.51/5.31      ! [F: real > real,L: real,X: real,S2: set_real] :
% 9.51/5.31        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 9.51/5.31       => ( ( ord_less_real @ L @ zero_zero_real )
% 9.51/5.31         => ? [D5: real] :
% 9.51/5.31              ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.51/5.31              & ! [H4: real] :
% 9.51/5.31                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.51/5.31                 => ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S2 )
% 9.51/5.31                   => ( ( ord_less_real @ H4 @ D5 )
% 9.51/5.31                     => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8927_has__real__derivative__pos__inc__left,axiom,
% 9.51/5.31      ! [F: real > real,L: real,X: real,S2: set_real] :
% 9.51/5.31        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ L )
% 9.51/5.31         => ? [D5: real] :
% 9.51/5.31              ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.51/5.31              & ! [H4: real] :
% 9.51/5.31                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.51/5.31                 => ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S2 )
% 9.51/5.31                   => ( ( ord_less_real @ H4 @ D5 )
% 9.51/5.31                     => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8928_has__real__derivative__neg__dec__right,axiom,
% 9.51/5.31      ! [F: real > real,L: real,X: real,S2: set_real] :
% 9.51/5.31        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 9.51/5.31       => ( ( ord_less_real @ L @ zero_zero_real )
% 9.51/5.31         => ? [D5: real] :
% 9.51/5.31              ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.51/5.31              & ! [H4: real] :
% 9.51/5.31                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.51/5.31                 => ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S2 )
% 9.51/5.31                   => ( ( ord_less_real @ H4 @ D5 )
% 9.51/5.31                     => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8929_has__real__derivative__pos__inc__right,axiom,
% 9.51/5.31      ! [F: real > real,L: real,X: real,S2: set_real] :
% 9.51/5.31        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ L )
% 9.51/5.31         => ? [D5: real] :
% 9.51/5.31              ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.51/5.31              & ! [H4: real] :
% 9.51/5.31                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.51/5.31                 => ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S2 )
% 9.51/5.31                   => ( ( ord_less_real @ H4 @ D5 )
% 9.51/5.31                     => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8930_DERIV__neg__imp__decreasing,axiom,
% 9.51/5.31      ! [A: real,B: real,F: real > real] :
% 9.51/5.31        ( ( ord_less_real @ A @ B )
% 9.51/5.31       => ( ! [X4: real] :
% 9.51/5.31              ( ( ord_less_eq_real @ A @ X4 )
% 9.51/5.31             => ( ( ord_less_eq_real @ X4 @ B )
% 9.51/5.31               => ? [Y4: real] :
% 9.51/5.31                    ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.51/5.31                    & ( ord_less_real @ Y4 @ zero_zero_real ) ) ) )
% 9.51/5.31         => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8931_DERIV__pos__imp__increasing,axiom,
% 9.51/5.31      ! [A: real,B: real,F: real > real] :
% 9.51/5.31        ( ( ord_less_real @ A @ B )
% 9.51/5.31       => ( ! [X4: real] :
% 9.51/5.31              ( ( ord_less_eq_real @ A @ X4 )
% 9.51/5.31             => ( ( ord_less_eq_real @ X4 @ B )
% 9.51/5.31               => ? [Y4: real] :
% 9.51/5.31                    ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.51/5.31                    & ( ord_less_real @ zero_zero_real @ Y4 ) ) ) )
% 9.51/5.31         => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8932_DERIV__pos__inc__right,axiom,
% 9.51/5.31      ! [F: real > real,L: real,X: real] :
% 9.51/5.31        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ L )
% 9.51/5.31         => ? [D5: real] :
% 9.51/5.31              ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.51/5.31              & ! [H4: real] :
% 9.51/5.31                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.51/5.31                 => ( ( ord_less_real @ H4 @ D5 )
% 9.51/5.31                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8933_DERIV__neg__dec__right,axiom,
% 9.51/5.31      ! [F: real > real,L: real,X: real] :
% 9.51/5.31        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.51/5.31       => ( ( ord_less_real @ L @ zero_zero_real )
% 9.51/5.31         => ? [D5: real] :
% 9.51/5.31              ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.51/5.31              & ! [H4: real] :
% 9.51/5.31                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.51/5.31                 => ( ( ord_less_real @ H4 @ D5 )
% 9.51/5.31                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8934_DERIV__pos__inc__left,axiom,
% 9.51/5.31      ! [F: real > real,L: real,X: real] :
% 9.51/5.31        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ L )
% 9.51/5.31         => ? [D5: real] :
% 9.51/5.31              ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.51/5.31              & ! [H4: real] :
% 9.51/5.31                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.51/5.31                 => ( ( ord_less_real @ H4 @ D5 )
% 9.51/5.31                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8935_DERIV__neg__dec__left,axiom,
% 9.51/5.31      ! [F: real > real,L: real,X: real] :
% 9.51/5.31        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.51/5.31       => ( ( ord_less_real @ L @ zero_zero_real )
% 9.51/5.31         => ? [D5: real] :
% 9.51/5.31              ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.51/5.31              & ! [H4: real] :
% 9.51/5.31                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.51/5.31                 => ( ( ord_less_real @ H4 @ D5 )
% 9.51/5.31                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8936_MVT2,axiom,
% 9.51/5.31      ! [A: real,B: real,F: real > real,F6: real > real] :
% 9.51/5.31        ( ( ord_less_real @ A @ B )
% 9.51/5.31       => ( ! [X4: real] :
% 9.51/5.31              ( ( ord_less_eq_real @ A @ X4 )
% 9.51/5.31             => ( ( ord_less_eq_real @ X4 @ B )
% 9.51/5.31               => ( has_fi5821293074295781190e_real @ F @ ( F6 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 9.51/5.31         => ? [Z4: real] :
% 9.51/5.31              ( ( ord_less_real @ A @ Z4 )
% 9.51/5.31              & ( ord_less_real @ Z4 @ B )
% 9.51/5.31              & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 9.51/5.31                = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F6 @ Z4 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8937_DERIV__local__const,axiom,
% 9.51/5.31      ! [F: real > real,L: real,X: real,D: real] :
% 9.51/5.31        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ D )
% 9.51/5.31         => ( ! [Y3: real] :
% 9.51/5.31                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
% 9.51/5.31               => ( ( F @ X )
% 9.51/5.31                  = ( F @ Y3 ) ) )
% 9.51/5.31           => ( L = zero_zero_real ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8938_DERIV__ln,axiom,
% 9.51/5.31      ! [X: real] :
% 9.51/5.31        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.31       => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8939_DERIV__local__max,axiom,
% 9.51/5.31      ! [F: real > real,L: real,X: real,D: real] :
% 9.51/5.31        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ D )
% 9.51/5.31         => ( ! [Y3: real] :
% 9.51/5.31                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
% 9.51/5.31               => ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X ) ) )
% 9.51/5.31           => ( L = zero_zero_real ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8940_DERIV__local__min,axiom,
% 9.51/5.31      ! [F: real > real,L: real,X: real,D: real] :
% 9.51/5.31        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ D )
% 9.51/5.31         => ( ! [Y3: real] :
% 9.51/5.31                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
% 9.51/5.31               => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 9.51/5.31           => ( L = zero_zero_real ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8941_DERIV__ln__divide,axiom,
% 9.51/5.31      ! [X: real] :
% 9.51/5.31        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.31       => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8942_DERIV__pow,axiom,
% 9.51/5.31      ! [N: nat,X: real,S: set_real] :
% 9.51/5.31        ( has_fi5821293074295781190e_real
% 9.51/5.31        @ ^ [X3: real] : ( power_power_real @ X3 @ N )
% 9.51/5.31        @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 9.51/5.31        @ ( topolo2177554685111907308n_real @ X @ S ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8943_has__real__derivative__powr,axiom,
% 9.51/5.31      ! [Z2: real,R2: real] :
% 9.51/5.31        ( ( ord_less_real @ zero_zero_real @ Z2 )
% 9.51/5.31       => ( has_fi5821293074295781190e_real
% 9.51/5.31          @ ^ [Z3: real] : ( powr_real @ Z3 @ R2 )
% 9.51/5.31          @ ( times_times_real @ R2 @ ( powr_real @ Z2 @ ( minus_minus_real @ R2 @ one_one_real ) ) )
% 9.51/5.31          @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8944_DERIV__log,axiom,
% 9.51/5.31      ! [X: real,B: real] :
% 9.51/5.31        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.31       => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8945_DERIV__fun__powr,axiom,
% 9.51/5.31      ! [G: real > real,M: real,X: real,R2: real] :
% 9.51/5.31        ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 9.51/5.31         => ( has_fi5821293074295781190e_real
% 9.51/5.31            @ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ R2 )
% 9.51/5.31            @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 9.51/5.31            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8946_DERIV__powr,axiom,
% 9.51/5.31      ! [G: real > real,M: real,X: real,F: real > real,R2: real] :
% 9.51/5.31        ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 9.51/5.31         => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 9.51/5.31           => ( has_fi5821293074295781190e_real
% 9.51/5.31              @ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ ( F @ X3 ) )
% 9.51/5.31              @ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
% 9.51/5.31              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8947_arcosh__real__has__field__derivative,axiom,
% 9.51/5.31      ! [X: real,A2: set_real] :
% 9.51/5.31        ( ( ord_less_real @ one_one_real @ X )
% 9.51/5.31       => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8948_DERIV__real__sqrt,axiom,
% 9.51/5.31      ! [X: real] :
% 9.51/5.31        ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.31       => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8949_DERIV__real__sqrt__generic,axiom,
% 9.51/5.31      ! [X: real,D6: real] :
% 9.51/5.31        ( ( X != zero_zero_real )
% 9.51/5.31       => ( ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.31           => ( D6
% 9.51/5.31              = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.51/5.31         => ( ( ( ord_less_real @ X @ zero_zero_real )
% 9.51/5.31             => ( D6
% 9.51/5.31                = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.51/5.31           => ( has_fi5821293074295781190e_real @ sqrt @ D6 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8950_artanh__real__has__field__derivative,axiom,
% 9.51/5.31      ! [X: real,A2: set_real] :
% 9.51/5.31        ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.51/5.31       => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8951_DERIV__real__root,axiom,
% 9.51/5.31      ! [N: nat,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.31         => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8952_DERIV__arcsin,axiom,
% 9.51/5.31      ! [X: real] :
% 9.51/5.31        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.51/5.31       => ( ( ord_less_real @ X @ one_one_real )
% 9.51/5.31         => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8953_Maclaurin__all__le__objl,axiom,
% 9.51/5.31      ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
% 9.51/5.31        ( ( ( ( Diff @ zero_zero_nat )
% 9.51/5.31            = F )
% 9.51/5.31          & ! [M5: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 9.51/5.31       => ? [T6: real] :
% 9.51/5.31            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
% 9.51/5.31            & ( ( F @ X )
% 9.51/5.31              = ( plus_plus_real
% 9.51/5.31                @ ( groups6591440286371151544t_real
% 9.51/5.31                  @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
% 9.51/5.31                  @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.31                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8954_Maclaurin__all__le,axiom,
% 9.51/5.31      ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
% 9.51/5.31        ( ( ( Diff @ zero_zero_nat )
% 9.51/5.31          = F )
% 9.51/5.31       => ( ! [M5: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.51/5.31         => ? [T6: real] :
% 9.51/5.31              ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
% 9.51/5.31              & ( ( F @ X )
% 9.51/5.31                = ( plus_plus_real
% 9.51/5.31                  @ ( groups6591440286371151544t_real
% 9.51/5.31                    @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
% 9.51/5.31                    @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.31                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8955_DERIV__odd__real__root,axiom,
% 9.51/5.31      ! [N: nat,X: real] :
% 9.51/5.31        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.51/5.31       => ( ( X != zero_zero_real )
% 9.51/5.31         => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8956_Maclaurin__minus,axiom,
% 9.51/5.31      ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 9.51/5.31        ( ( ord_less_real @ H2 @ zero_zero_real )
% 9.51/5.31       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31         => ( ( ( Diff @ zero_zero_nat )
% 9.51/5.31              = F )
% 9.51/5.31           => ( ! [M5: nat,T6: real] :
% 9.51/5.31                  ( ( ( ord_less_nat @ M5 @ N )
% 9.51/5.31                    & ( ord_less_eq_real @ H2 @ T6 )
% 9.51/5.31                    & ( ord_less_eq_real @ T6 @ zero_zero_real ) )
% 9.51/5.31                 => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.51/5.31             => ? [T6: real] :
% 9.51/5.31                  ( ( ord_less_real @ H2 @ T6 )
% 9.51/5.31                  & ( ord_less_real @ T6 @ zero_zero_real )
% 9.51/5.31                  & ( ( F @ H2 )
% 9.51/5.31                    = ( plus_plus_real
% 9.51/5.31                      @ ( groups6591440286371151544t_real
% 9.51/5.31                        @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 9.51/5.31                        @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.31                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8957_Maclaurin2,axiom,
% 9.51/5.31      ! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
% 9.51/5.31        ( ( ord_less_real @ zero_zero_real @ H2 )
% 9.51/5.31       => ( ( ( Diff @ zero_zero_nat )
% 9.51/5.31            = F )
% 9.51/5.31         => ( ! [M5: nat,T6: real] :
% 9.51/5.31                ( ( ( ord_less_nat @ M5 @ N )
% 9.51/5.31                  & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 9.51/5.31                  & ( ord_less_eq_real @ T6 @ H2 ) )
% 9.51/5.31               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.51/5.31           => ? [T6: real] :
% 9.51/5.31                ( ( ord_less_real @ zero_zero_real @ T6 )
% 9.51/5.31                & ( ord_less_eq_real @ T6 @ H2 )
% 9.51/5.31                & ( ( F @ H2 )
% 9.51/5.31                  = ( plus_plus_real
% 9.51/5.31                    @ ( groups6591440286371151544t_real
% 9.51/5.31                      @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 9.51/5.31                      @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.31                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8958_Maclaurin,axiom,
% 9.51/5.31      ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 9.51/5.31        ( ( ord_less_real @ zero_zero_real @ H2 )
% 9.51/5.31       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31         => ( ( ( Diff @ zero_zero_nat )
% 9.51/5.31              = F )
% 9.51/5.31           => ( ! [M5: nat,T6: real] :
% 9.51/5.31                  ( ( ( ord_less_nat @ M5 @ N )
% 9.51/5.31                    & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 9.51/5.31                    & ( ord_less_eq_real @ T6 @ H2 ) )
% 9.51/5.31                 => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.51/5.31             => ? [T6: real] :
% 9.51/5.31                  ( ( ord_less_real @ zero_zero_real @ T6 )
% 9.51/5.31                  & ( ord_less_real @ T6 @ H2 )
% 9.51/5.31                  & ( ( F @ H2 )
% 9.51/5.31                    = ( plus_plus_real
% 9.51/5.31                      @ ( groups6591440286371151544t_real
% 9.51/5.31                        @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 9.51/5.31                        @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.31                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8959_Maclaurin__all__lt,axiom,
% 9.51/5.31      ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
% 9.51/5.31        ( ( ( Diff @ zero_zero_nat )
% 9.51/5.31          = F )
% 9.51/5.31       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31         => ( ( X != zero_zero_real )
% 9.51/5.31           => ( ! [M5: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.51/5.31             => ? [T6: real] :
% 9.51/5.31                  ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
% 9.51/5.31                  & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
% 9.51/5.31                  & ( ( F @ X )
% 9.51/5.31                    = ( plus_plus_real
% 9.51/5.31                      @ ( groups6591440286371151544t_real
% 9.51/5.31                        @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
% 9.51/5.31                        @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.31                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8960_Maclaurin__bi__le,axiom,
% 9.51/5.31      ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
% 9.51/5.31        ( ( ( Diff @ zero_zero_nat )
% 9.51/5.31          = F )
% 9.51/5.31       => ( ! [M5: nat,T6: real] :
% 9.51/5.31              ( ( ( ord_less_nat @ M5 @ N )
% 9.51/5.31                & ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) ) )
% 9.51/5.31             => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.51/5.31         => ? [T6: real] :
% 9.51/5.31              ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
% 9.51/5.31              & ( ( F @ X )
% 9.51/5.31                = ( plus_plus_real
% 9.51/5.31                  @ ( groups6591440286371151544t_real
% 9.51/5.31                    @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
% 9.51/5.31                    @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.31                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8961_Taylor__down,axiom,
% 9.51/5.31      ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ( Diff @ zero_zero_nat )
% 9.51/5.31            = F )
% 9.51/5.31         => ( ! [M5: nat,T6: real] :
% 9.51/5.31                ( ( ( ord_less_nat @ M5 @ N )
% 9.51/5.31                  & ( ord_less_eq_real @ A @ T6 )
% 9.51/5.31                  & ( ord_less_eq_real @ T6 @ B ) )
% 9.51/5.31               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.51/5.31           => ( ( ord_less_real @ A @ C )
% 9.51/5.31             => ( ( ord_less_eq_real @ C @ B )
% 9.51/5.31               => ? [T6: real] :
% 9.51/5.31                    ( ( ord_less_real @ A @ T6 )
% 9.51/5.31                    & ( ord_less_real @ T6 @ C )
% 9.51/5.31                    & ( ( F @ A )
% 9.51/5.31                      = ( plus_plus_real
% 9.51/5.31                        @ ( groups6591440286371151544t_real
% 9.51/5.31                          @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M2 ) )
% 9.51/5.31                          @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.31                        @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8962_Taylor__up,axiom,
% 9.51/5.31      ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ( Diff @ zero_zero_nat )
% 9.51/5.31            = F )
% 9.51/5.31         => ( ! [M5: nat,T6: real] :
% 9.51/5.31                ( ( ( ord_less_nat @ M5 @ N )
% 9.51/5.31                  & ( ord_less_eq_real @ A @ T6 )
% 9.51/5.31                  & ( ord_less_eq_real @ T6 @ B ) )
% 9.51/5.31               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.51/5.31           => ( ( ord_less_eq_real @ A @ C )
% 9.51/5.31             => ( ( ord_less_real @ C @ B )
% 9.51/5.31               => ? [T6: real] :
% 9.51/5.31                    ( ( ord_less_real @ C @ T6 )
% 9.51/5.31                    & ( ord_less_real @ T6 @ B )
% 9.51/5.31                    & ( ( F @ B )
% 9.51/5.31                      = ( plus_plus_real
% 9.51/5.31                        @ ( groups6591440286371151544t_real
% 9.51/5.31                          @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M2 ) )
% 9.51/5.31                          @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.31                        @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8963_Taylor,axiom,
% 9.51/5.31      ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( ( Diff @ zero_zero_nat )
% 9.51/5.31            = F )
% 9.51/5.31         => ( ! [M5: nat,T6: real] :
% 9.51/5.31                ( ( ( ord_less_nat @ M5 @ N )
% 9.51/5.31                  & ( ord_less_eq_real @ A @ T6 )
% 9.51/5.31                  & ( ord_less_eq_real @ T6 @ B ) )
% 9.51/5.31               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.51/5.31           => ( ( ord_less_eq_real @ A @ C )
% 9.51/5.31             => ( ( ord_less_eq_real @ C @ B )
% 9.51/5.31               => ( ( ord_less_eq_real @ A @ X )
% 9.51/5.31                 => ( ( ord_less_eq_real @ X @ B )
% 9.51/5.31                   => ( ( X != C )
% 9.51/5.31                     => ? [T6: real] :
% 9.51/5.31                          ( ( ( ord_less_real @ X @ C )
% 9.51/5.31                           => ( ( ord_less_real @ X @ T6 )
% 9.51/5.31                              & ( ord_less_real @ T6 @ C ) ) )
% 9.51/5.31                          & ( ~ ( ord_less_real @ X @ C )
% 9.51/5.31                           => ( ( ord_less_real @ C @ T6 )
% 9.51/5.31                              & ( ord_less_real @ T6 @ X ) ) )
% 9.51/5.31                          & ( ( F @ X )
% 9.51/5.31                            = ( plus_plus_real
% 9.51/5.31                              @ ( groups6591440286371151544t_real
% 9.51/5.31                                @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M2 ) )
% 9.51/5.31                                @ ( set_ord_lessThan_nat @ N ) )
% 9.51/5.31                              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8964_Maclaurin__lemma2,axiom,
% 9.51/5.31      ! [N: nat,H2: real,Diff: nat > real > real,K: nat,B2: real] :
% 9.51/5.31        ( ! [M5: nat,T6: real] :
% 9.51/5.31            ( ( ( ord_less_nat @ M5 @ N )
% 9.51/5.31              & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 9.51/5.31              & ( ord_less_eq_real @ T6 @ H2 ) )
% 9.51/5.31           => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.51/5.31       => ( ( N
% 9.51/5.31            = ( suc @ K ) )
% 9.51/5.31         => ! [M3: nat,T7: real] :
% 9.51/5.31              ( ( ( ord_less_nat @ M3 @ N )
% 9.51/5.31                & ( ord_less_eq_real @ zero_zero_real @ T7 )
% 9.51/5.31                & ( ord_less_eq_real @ T7 @ H2 ) )
% 9.51/5.31             => ( has_fi5821293074295781190e_real
% 9.51/5.31                @ ^ [U2: real] :
% 9.51/5.31                    ( minus_minus_real @ ( Diff @ M3 @ U2 )
% 9.51/5.31                    @ ( plus_plus_real
% 9.51/5.31                      @ ( groups6591440286371151544t_real
% 9.51/5.31                        @ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M3 @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ U2 @ P6 ) )
% 9.51/5.31                        @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M3 ) ) )
% 9.51/5.31                      @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M3 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M3 ) ) ) ) ) )
% 9.51/5.31                @ ( minus_minus_real @ ( Diff @ ( suc @ M3 ) @ T7 )
% 9.51/5.31                  @ ( plus_plus_real
% 9.51/5.31                    @ ( groups6591440286371151544t_real
% 9.51/5.31                      @ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M3 ) @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ T7 @ P6 ) )
% 9.51/5.31                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) )
% 9.51/5.31                    @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ T7 @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) ) ) ) )
% 9.51/5.31                @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8965_DERIV__arctan__series,axiom,
% 9.51/5.31      ! [X: real] :
% 9.51/5.31        ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.51/5.31       => ( has_fi5821293074295781190e_real
% 9.51/5.31          @ ^ [X9: real] :
% 9.51/5.31              ( suminf_real
% 9.51/5.31              @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
% 9.51/5.31          @ ( suminf_real
% 9.51/5.31            @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.51/5.31          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8966_DERIV__real__root__generic,axiom,
% 9.51/5.31      ! [N: nat,X: real,D6: real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( X != zero_zero_real )
% 9.51/5.31         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.51/5.31             => ( ( ord_less_real @ zero_zero_real @ X )
% 9.51/5.31               => ( D6
% 9.51/5.31                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 9.51/5.31           => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.51/5.31               => ( ( ord_less_real @ X @ zero_zero_real )
% 9.51/5.31                 => ( D6
% 9.51/5.31                    = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
% 9.51/5.31             => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.51/5.31                 => ( D6
% 9.51/5.31                    = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 9.51/5.31               => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D6 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8967_DERIV__arccos,axiom,
% 9.51/5.31      ! [X: real] :
% 9.51/5.31        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.51/5.31       => ( ( ord_less_real @ X @ one_one_real )
% 9.51/5.31         => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8968_DERIV__power__series_H,axiom,
% 9.51/5.31      ! [R: real,F: nat > real,X0: real] :
% 9.51/5.31        ( ! [X4: real] :
% 9.51/5.31            ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 9.51/5.31           => ( summable_real
% 9.51/5.31              @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X4 @ N2 ) ) ) )
% 9.51/5.31       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 9.51/5.31         => ( ( ord_less_real @ zero_zero_real @ R )
% 9.51/5.31           => ( has_fi5821293074295781190e_real
% 9.51/5.31              @ ^ [X3: real] :
% 9.51/5.31                  ( suminf_real
% 9.51/5.31                  @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X3 @ ( suc @ N2 ) ) ) )
% 9.51/5.31              @ ( suminf_real
% 9.51/5.31                @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X0 @ N2 ) ) )
% 9.51/5.31              @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8969_arccos__less__arccos,axiom,
% 9.51/5.31      ! [X: real,Y: real] :
% 9.51/5.31        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.51/5.31       => ( ( ord_less_real @ X @ Y )
% 9.51/5.31         => ( ( ord_less_eq_real @ Y @ one_one_real )
% 9.51/5.31           => ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8970_arccos__less__mono,axiom,
% 9.51/5.31      ! [X: real,Y: real] :
% 9.51/5.31        ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 9.51/5.31       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 9.51/5.31         => ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y ) )
% 9.51/5.31            = ( ord_less_real @ Y @ X ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8971_DERIV__isconst3,axiom,
% 9.51/5.31      ! [A: real,B: real,X: real,Y: real,F: real > real] :
% 9.51/5.31        ( ( ord_less_real @ A @ B )
% 9.51/5.31       => ( ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
% 9.51/5.31         => ( ( member_real @ Y @ ( set_or1633881224788618240n_real @ A @ B ) )
% 9.51/5.31           => ( ! [X4: real] :
% 9.51/5.31                  ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 9.51/5.31                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 9.51/5.31             => ( ( F @ X )
% 9.51/5.31                = ( F @ Y ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8972_arccos__lt__bounded,axiom,
% 9.51/5.31      ! [Y: real] :
% 9.51/5.31        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 9.51/5.31       => ( ( ord_less_real @ Y @ one_one_real )
% 9.51/5.31         => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y ) )
% 9.51/5.31            & ( ord_less_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8973_sin__arccos__nonzero,axiom,
% 9.51/5.31      ! [X: real] :
% 9.51/5.31        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.51/5.31       => ( ( ord_less_real @ X @ one_one_real )
% 9.51/5.31         => ( ( sin_real @ ( arccos @ X ) )
% 9.51/5.31           != zero_zero_real ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8974_finite__greaterThanLessThan,axiom,
% 9.51/5.31      ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8975_finite__greaterThanLessThan__int,axiom,
% 9.51/5.31      ! [L: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L @ U ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8976_LIM__fun__gt__zero,axiom,
% 9.51/5.31      ! [F: real > real,L: real,C: real] :
% 9.51/5.31        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 9.51/5.31       => ( ( ord_less_real @ zero_zero_real @ L )
% 9.51/5.31         => ? [R3: real] :
% 9.51/5.31              ( ( ord_less_real @ zero_zero_real @ R3 )
% 9.51/5.31              & ! [X2: real] :
% 9.51/5.31                  ( ( ( X2 != C )
% 9.51/5.31                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R3 ) )
% 9.51/5.31                 => ( ord_less_real @ zero_zero_real @ ( F @ X2 ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8977_LIM__fun__not__zero,axiom,
% 9.51/5.31      ! [F: real > real,L: real,C: real] :
% 9.51/5.31        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 9.51/5.31       => ( ( L != zero_zero_real )
% 9.51/5.31         => ? [R3: real] :
% 9.51/5.31              ( ( ord_less_real @ zero_zero_real @ R3 )
% 9.51/5.31              & ! [X2: real] :
% 9.51/5.31                  ( ( ( X2 != C )
% 9.51/5.31                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R3 ) )
% 9.51/5.31                 => ( ( F @ X2 )
% 9.51/5.31                   != zero_zero_real ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8978_LIM__fun__less__zero,axiom,
% 9.51/5.31      ! [F: real > real,L: real,C: real] :
% 9.51/5.31        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 9.51/5.31       => ( ( ord_less_real @ L @ zero_zero_real )
% 9.51/5.31         => ? [R3: real] :
% 9.51/5.31              ( ( ord_less_real @ zero_zero_real @ R3 )
% 9.51/5.31              & ! [X2: real] :
% 9.51/5.31                  ( ( ( X2 != C )
% 9.51/5.31                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R3 ) )
% 9.51/5.31                 => ( ord_less_real @ ( F @ X2 ) @ zero_zero_real ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8979_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 9.51/5.31      ! [N: nat,J: nat,I: nat] :
% 9.51/5.31        ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
% 9.51/5.31       => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
% 9.51/5.31          = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8980_summable__Leibniz_I2_J,axiom,
% 9.51/5.31      ! [A: nat > real] :
% 9.51/5.31        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.51/5.31       => ( ( topolo6980174941875973593q_real @ A )
% 9.51/5.31         => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
% 9.51/5.31           => ! [N5: nat] :
% 9.51/5.31                ( member_real
% 9.51/5.31                @ ( suminf_real
% 9.51/5.31                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 9.51/5.31                @ ( set_or1222579329274155063t_real
% 9.51/5.31                  @ ( groups6591440286371151544t_real
% 9.51/5.31                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 9.51/5.31                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) )
% 9.51/5.31                  @ ( groups6591440286371151544t_real
% 9.51/5.31                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 9.51/5.31                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8981_summable__Leibniz_I3_J,axiom,
% 9.51/5.31      ! [A: nat > real] :
% 9.51/5.31        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.51/5.31       => ( ( topolo6980174941875973593q_real @ A )
% 9.51/5.31         => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
% 9.51/5.31           => ! [N5: nat] :
% 9.51/5.31                ( member_real
% 9.51/5.31                @ ( suminf_real
% 9.51/5.31                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 9.51/5.31                @ ( set_or1222579329274155063t_real
% 9.51/5.31                  @ ( groups6591440286371151544t_real
% 9.51/5.31                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 9.51/5.31                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) @ one_one_nat ) ) )
% 9.51/5.31                  @ ( groups6591440286371151544t_real
% 9.51/5.31                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 9.51/5.31                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) ) ) ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8982_trivial__limit__sequentially,axiom,
% 9.51/5.31      at_top_nat != bot_bot_filter_nat ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8983_mult__nat__left__at__top,axiom,
% 9.51/5.31      ! [C: nat] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.51/5.31       => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8984_mult__nat__right__at__top,axiom,
% 9.51/5.31      ! [C: nat] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.51/5.31       => ( filterlim_nat_nat
% 9.51/5.31          @ ^ [X3: nat] : ( times_times_nat @ X3 @ C )
% 9.51/5.31          @ at_top_nat
% 9.51/5.31          @ at_top_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8985_LIMSEQ__inverse__zero,axiom,
% 9.51/5.31      ! [X8: nat > real] :
% 9.51/5.31        ( ! [R3: real] :
% 9.51/5.31          ? [N8: nat] :
% 9.51/5.31          ! [N3: nat] :
% 9.51/5.31            ( ( ord_less_eq_nat @ N8 @ N3 )
% 9.51/5.31           => ( ord_less_real @ R3 @ ( X8 @ N3 ) ) )
% 9.51/5.31       => ( filterlim_nat_real
% 9.51/5.31          @ ^ [N2: nat] : ( inverse_inverse_real @ ( X8 @ N2 ) )
% 9.51/5.31          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.51/5.31          @ at_top_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8986_LIMSEQ__root__const,axiom,
% 9.51/5.31      ! [C: real] :
% 9.51/5.31        ( ( ord_less_real @ zero_zero_real @ C )
% 9.51/5.31       => ( filterlim_nat_real
% 9.51/5.31          @ ^ [N2: nat] : ( root @ N2 @ C )
% 9.51/5.31          @ ( topolo2815343760600316023s_real @ one_one_real )
% 9.51/5.31          @ at_top_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8987_increasing__LIMSEQ,axiom,
% 9.51/5.31      ! [F: nat > real,L: real] :
% 9.51/5.31        ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.51/5.31       => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ L )
% 9.51/5.31         => ( ! [E: real] :
% 9.51/5.31                ( ( ord_less_real @ zero_zero_real @ E )
% 9.51/5.31               => ? [N5: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N5 ) @ E ) ) )
% 9.51/5.31           => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8988_LIMSEQ__realpow__zero,axiom,
% 9.51/5.31      ! [X: real] :
% 9.51/5.31        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.51/5.31       => ( ( ord_less_real @ X @ one_one_real )
% 9.51/5.31         => ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8989_LIMSEQ__divide__realpow__zero,axiom,
% 9.51/5.31      ! [X: real,A: real] :
% 9.51/5.31        ( ( ord_less_real @ one_one_real @ X )
% 9.51/5.31       => ( filterlim_nat_real
% 9.51/5.31          @ ^ [N2: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N2 ) )
% 9.51/5.31          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.51/5.31          @ at_top_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8990_LIMSEQ__abs__realpow__zero2,axiom,
% 9.51/5.31      ! [C: real] :
% 9.51/5.31        ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 9.51/5.31       => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8991_LIMSEQ__abs__realpow__zero,axiom,
% 9.51/5.31      ! [C: real] :
% 9.51/5.31        ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 9.51/5.31       => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8992_LIMSEQ__inverse__realpow__zero,axiom,
% 9.51/5.31      ! [X: real] :
% 9.51/5.31        ( ( ord_less_real @ one_one_real @ X )
% 9.51/5.31       => ( filterlim_nat_real
% 9.51/5.31          @ ^ [N2: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N2 ) )
% 9.51/5.31          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.51/5.31          @ at_top_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8993_DERIV__neg__imp__decreasing__at__top,axiom,
% 9.51/5.31      ! [B: real,F: real > real,Flim: real] :
% 9.51/5.31        ( ! [X4: real] :
% 9.51/5.31            ( ( ord_less_eq_real @ B @ X4 )
% 9.51/5.31           => ? [Y4: real] :
% 9.51/5.31                ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.51/5.31                & ( ord_less_real @ Y4 @ zero_zero_real ) ) )
% 9.51/5.31       => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
% 9.51/5.31         => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8994_filterlim__pow__at__bot__even,axiom,
% 9.51/5.31      ! [N: nat,F: real > real,F3: filter_real] :
% 9.51/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.31       => ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
% 9.51/5.31         => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.51/5.31           => ( filterlim_real_real
% 9.51/5.31              @ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N )
% 9.51/5.31              @ at_top_real
% 9.51/5.31              @ F3 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8995_eventually__sequentially,axiom,
% 9.51/5.31      ! [P: nat > $o] :
% 9.51/5.31        ( ( eventually_nat @ P @ at_top_nat )
% 9.51/5.31        = ( ? [N4: nat] :
% 9.51/5.31            ! [N2: nat] :
% 9.51/5.31              ( ( ord_less_eq_nat @ N4 @ N2 )
% 9.51/5.31             => ( P @ N2 ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8996_eventually__sequentiallyI,axiom,
% 9.51/5.31      ! [C: nat,P: nat > $o] :
% 9.51/5.31        ( ! [X4: nat] :
% 9.51/5.31            ( ( ord_less_eq_nat @ C @ X4 )
% 9.51/5.31           => ( P @ X4 ) )
% 9.51/5.31       => ( eventually_nat @ P @ at_top_nat ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8997_le__sequentially,axiom,
% 9.51/5.31      ! [F3: filter_nat] :
% 9.51/5.31        ( ( ord_le2510731241096832064er_nat @ F3 @ at_top_nat )
% 9.51/5.31        = ( ! [N4: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N4 ) @ F3 ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8998_eventually__at__left__real,axiom,
% 9.51/5.31      ! [B: real,A: real] :
% 9.51/5.31        ( ( ord_less_real @ B @ A )
% 9.51/5.31       => ( eventually_real
% 9.51/5.31          @ ^ [X3: real] : ( member_real @ X3 @ ( set_or1633881224788618240n_real @ B @ A ) )
% 9.51/5.31          @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).
% 9.51/5.31  
% 9.51/5.31  thf(fact_8999_DERIV__pos__imp__increasing__at__bot,axiom,
% 9.51/5.31      ! [B: real,F: real > real,Flim: real] :
% 9.51/5.31        ( ! [X4: real] :
% 9.51/5.32            ( ( ord_less_eq_real @ X4 @ B )
% 9.51/5.32           => ? [Y4: real] :
% 9.51/5.32                ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.51/5.32                & ( ord_less_real @ zero_zero_real @ Y4 ) ) )
% 9.51/5.32       => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
% 9.51/5.32         => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9000_filterlim__pow__at__bot__odd,axiom,
% 9.51/5.32      ! [N: nat,F: real > real,F3: filter_real] :
% 9.51/5.32        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.32       => ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
% 9.51/5.32         => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.51/5.32           => ( filterlim_real_real
% 9.51/5.32              @ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N )
% 9.51/5.32              @ at_bot_real
% 9.51/5.32              @ F3 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9001_finite__greaterThanAtMost,axiom,
% 9.51/5.32      ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9002_GreatestI__ex__nat,axiom,
% 9.51/5.32      ! [P: nat > $o,B: nat] :
% 9.51/5.32        ( ? [X_12: nat] : ( P @ X_12 )
% 9.51/5.32       => ( ! [Y3: nat] :
% 9.51/5.32              ( ( P @ Y3 )
% 9.51/5.32             => ( ord_less_eq_nat @ Y3 @ B ) )
% 9.51/5.32         => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9003_Greatest__le__nat,axiom,
% 9.51/5.32      ! [P: nat > $o,K: nat,B: nat] :
% 9.51/5.32        ( ( P @ K )
% 9.51/5.32       => ( ! [Y3: nat] :
% 9.51/5.32              ( ( P @ Y3 )
% 9.51/5.32             => ( ord_less_eq_nat @ Y3 @ B ) )
% 9.51/5.32         => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9004_GreatestI__nat,axiom,
% 9.51/5.32      ! [P: nat > $o,K: nat,B: nat] :
% 9.51/5.32        ( ( P @ K )
% 9.51/5.32       => ( ! [Y3: nat] :
% 9.51/5.32              ( ( P @ Y3 )
% 9.51/5.32             => ( ord_less_eq_nat @ Y3 @ B ) )
% 9.51/5.32         => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9005_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 9.51/5.32      ! [N: nat,J: nat,I: nat] :
% 9.51/5.32        ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
% 9.51/5.32       => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
% 9.51/5.32          = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9006_finite__greaterThanAtMost__int,axiom,
% 9.51/5.32      ! [L: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L @ U ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9007_isCont__inverse__function2,axiom,
% 9.51/5.32      ! [A: real,X: real,B: real,G: real > real,F: real > real] :
% 9.51/5.32        ( ( ord_less_real @ A @ X )
% 9.51/5.32       => ( ( ord_less_real @ X @ B )
% 9.51/5.32         => ( ! [Z4: real] :
% 9.51/5.32                ( ( ord_less_eq_real @ A @ Z4 )
% 9.51/5.32               => ( ( ord_less_eq_real @ Z4 @ B )
% 9.51/5.32                 => ( ( G @ ( F @ Z4 ) )
% 9.51/5.32                    = Z4 ) ) )
% 9.51/5.32           => ( ! [Z4: real] :
% 9.51/5.32                  ( ( ord_less_eq_real @ A @ Z4 )
% 9.51/5.32                 => ( ( ord_less_eq_real @ Z4 @ B )
% 9.51/5.32                   => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
% 9.51/5.32             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9008_isCont__arcosh,axiom,
% 9.51/5.32      ! [X: real] :
% 9.51/5.32        ( ( ord_less_real @ one_one_real @ X )
% 9.51/5.32       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcosh_real ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9009_DERIV__inverse__function,axiom,
% 9.51/5.32      ! [F: real > real,D6: real,G: real > real,X: real,A: real,B: real] :
% 9.51/5.32        ( ( has_fi5821293074295781190e_real @ F @ D6 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
% 9.51/5.32       => ( ( D6 != zero_zero_real )
% 9.51/5.32         => ( ( ord_less_real @ A @ X )
% 9.51/5.32           => ( ( ord_less_real @ X @ B )
% 9.51/5.32             => ( ! [Y3: real] :
% 9.51/5.32                    ( ( ord_less_real @ A @ Y3 )
% 9.51/5.32                   => ( ( ord_less_real @ Y3 @ B )
% 9.51/5.32                     => ( ( F @ ( G @ Y3 ) )
% 9.51/5.32                        = Y3 ) ) )
% 9.51/5.32               => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
% 9.51/5.32                 => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D6 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9010_isCont__arccos,axiom,
% 9.51/5.32      ! [X: real] :
% 9.51/5.32        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.51/5.32       => ( ( ord_less_real @ X @ one_one_real )
% 9.51/5.32         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arccos ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9011_isCont__arcsin,axiom,
% 9.51/5.32      ! [X: real] :
% 9.51/5.32        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.51/5.32       => ( ( ord_less_real @ X @ one_one_real )
% 9.51/5.32         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcsin ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9012_LIM__less__bound,axiom,
% 9.51/5.32      ! [B: real,X: real,F: real > real] :
% 9.51/5.32        ( ( ord_less_real @ B @ X )
% 9.51/5.32       => ( ! [X4: real] :
% 9.51/5.32              ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ B @ X ) )
% 9.51/5.32             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.51/5.32         => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
% 9.51/5.32           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9013_isCont__artanh,axiom,
% 9.51/5.32      ! [X: real] :
% 9.51/5.32        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 9.51/5.32       => ( ( ord_less_real @ X @ one_one_real )
% 9.51/5.32         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ artanh_real ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9014_isCont__inverse__function,axiom,
% 9.51/5.32      ! [D: real,X: real,G: real > real,F: real > real] :
% 9.51/5.32        ( ( ord_less_real @ zero_zero_real @ D )
% 9.51/5.32       => ( ! [Z4: real] :
% 9.51/5.32              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X ) ) @ D )
% 9.51/5.32             => ( ( G @ ( F @ Z4 ) )
% 9.51/5.32                = Z4 ) )
% 9.51/5.32         => ( ! [Z4: real] :
% 9.51/5.32                ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X ) ) @ D )
% 9.51/5.32               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) )
% 9.51/5.32           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9015_GMVT_H,axiom,
% 9.51/5.32      ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F6: real > real] :
% 9.51/5.32        ( ( ord_less_real @ A @ B )
% 9.51/5.32       => ( ! [Z4: real] :
% 9.51/5.32              ( ( ord_less_eq_real @ A @ Z4 )
% 9.51/5.32             => ( ( ord_less_eq_real @ Z4 @ B )
% 9.51/5.32               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
% 9.51/5.32         => ( ! [Z4: real] :
% 9.51/5.32                ( ( ord_less_eq_real @ A @ Z4 )
% 9.51/5.32               => ( ( ord_less_eq_real @ Z4 @ B )
% 9.51/5.32                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ G ) ) )
% 9.51/5.32           => ( ! [Z4: real] :
% 9.51/5.32                  ( ( ord_less_real @ A @ Z4 )
% 9.51/5.32                 => ( ( ord_less_real @ Z4 @ B )
% 9.51/5.32                   => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
% 9.51/5.32             => ( ! [Z4: real] :
% 9.51/5.32                    ( ( ord_less_real @ A @ Z4 )
% 9.51/5.32                   => ( ( ord_less_real @ Z4 @ B )
% 9.51/5.32                     => ( has_fi5821293074295781190e_real @ F @ ( F6 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
% 9.51/5.32               => ? [C3: real] :
% 9.51/5.32                    ( ( ord_less_real @ A @ C3 )
% 9.51/5.32                    & ( ord_less_real @ C3 @ B )
% 9.51/5.32                    & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C3 ) )
% 9.51/5.32                      = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F6 @ C3 ) ) ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9016_INT__greaterThan__UNIV,axiom,
% 9.51/5.32      ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
% 9.51/5.32      = bot_bot_set_nat ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9017_greaterThan__0,axiom,
% 9.51/5.32      ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
% 9.51/5.32      = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9018_eventually__at__right__real,axiom,
% 9.51/5.32      ! [A: real,B: real] :
% 9.51/5.32        ( ( ord_less_real @ A @ B )
% 9.51/5.32       => ( eventually_real
% 9.51/5.32          @ ^ [X3: real] : ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 9.51/5.32          @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9019_greaterThan__Suc,axiom,
% 9.51/5.32      ! [K: nat] :
% 9.51/5.32        ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
% 9.51/5.32        = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9020_GMVT,axiom,
% 9.51/5.32      ! [A: real,B: real,F: real > real,G: real > real] :
% 9.51/5.32        ( ( ord_less_real @ A @ B )
% 9.51/5.32       => ( ! [X4: real] :
% 9.51/5.32              ( ( ( ord_less_eq_real @ A @ X4 )
% 9.51/5.32                & ( ord_less_eq_real @ X4 @ B ) )
% 9.51/5.32             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
% 9.51/5.32         => ( ! [X4: real] :
% 9.51/5.32                ( ( ( ord_less_real @ A @ X4 )
% 9.51/5.32                  & ( ord_less_real @ X4 @ B ) )
% 9.51/5.32               => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 9.51/5.32           => ( ! [X4: real] :
% 9.51/5.32                  ( ( ( ord_less_eq_real @ A @ X4 )
% 9.51/5.32                    & ( ord_less_eq_real @ X4 @ B ) )
% 9.51/5.32                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ G ) )
% 9.51/5.32             => ( ! [X4: real] :
% 9.51/5.32                    ( ( ( ord_less_real @ A @ X4 )
% 9.51/5.32                      & ( ord_less_real @ X4 @ B ) )
% 9.51/5.32                   => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 9.51/5.32               => ? [G_c: real,F_c: real,C3: real] :
% 9.51/5.32                    ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 9.51/5.32                    & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 9.51/5.32                    & ( ord_less_real @ A @ C3 )
% 9.51/5.32                    & ( ord_less_real @ C3 @ B )
% 9.51/5.32                    & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
% 9.51/5.32                      = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9021_atLeast__0,axiom,
% 9.51/5.32      ( ( set_ord_atLeast_nat @ zero_zero_nat )
% 9.51/5.32      = top_top_set_nat ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9022_atLeast__Suc,axiom,
% 9.51/5.32      ! [K: nat] :
% 9.51/5.32        ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 9.51/5.32        = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9023_Gcd__eq__Max,axiom,
% 9.51/5.32      ! [M4: set_nat] :
% 9.51/5.32        ( ( finite_finite_nat @ M4 )
% 9.51/5.32       => ( ( M4 != bot_bot_set_nat )
% 9.51/5.32         => ( ~ ( member_nat @ zero_zero_nat @ M4 )
% 9.51/5.32           => ( ( gcd_Gcd_nat @ M4 )
% 9.51/5.32              = ( lattic8265883725875713057ax_nat
% 9.51/5.32                @ ( comple7806235888213564991et_nat
% 9.51/5.32                  @ ( image_nat_set_nat
% 9.51/5.32                    @ ^ [M2: nat] :
% 9.51/5.32                        ( collect_nat
% 9.51/5.32                        @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M2 ) )
% 9.51/5.32                    @ M4 ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9024_Max__divisors__self__nat,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( ( N != zero_zero_nat )
% 9.51/5.32       => ( ( lattic8265883725875713057ax_nat
% 9.51/5.32            @ ( collect_nat
% 9.51/5.32              @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ N ) ) )
% 9.51/5.32          = N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9025_bdd__above__nat,axiom,
% 9.51/5.32      condit2214826472909112428ve_nat = finite_finite_nat ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9026_card__le__Suc__Max,axiom,
% 9.51/5.32      ! [S2: set_nat] :
% 9.51/5.32        ( ( finite_finite_nat @ S2 )
% 9.51/5.32       => ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9027_Sup__nat__def,axiom,
% 9.51/5.32      ( complete_Sup_Sup_nat
% 9.51/5.32      = ( ^ [X6: set_nat] : ( if_nat @ ( X6 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X6 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9028_divide__nat__def,axiom,
% 9.51/5.32      ( divide_divide_nat
% 9.51/5.32      = ( ^ [M2: nat,N2: nat] :
% 9.51/5.32            ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
% 9.51/5.32            @ ( lattic8265883725875713057ax_nat
% 9.51/5.32              @ ( collect_nat
% 9.51/5.32                @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N2 ) @ M2 ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9029_gcd__is__Max__divisors__nat,axiom,
% 9.51/5.32      ! [N: nat,M: nat] :
% 9.51/5.32        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.32       => ( ( gcd_gcd_nat @ M @ N )
% 9.51/5.32          = ( lattic8265883725875713057ax_nat
% 9.51/5.32            @ ( collect_nat
% 9.51/5.32              @ ^ [D4: nat] :
% 9.51/5.32                  ( ( dvd_dvd_nat @ D4 @ M )
% 9.51/5.32                  & ( dvd_dvd_nat @ D4 @ N ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9030_MVT,axiom,
% 9.51/5.32      ! [A: real,B: real,F: real > real] :
% 9.51/5.32        ( ( ord_less_real @ A @ B )
% 9.51/5.32       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.51/5.32         => ( ! [X4: real] :
% 9.51/5.32                ( ( ord_less_real @ A @ X4 )
% 9.51/5.32               => ( ( ord_less_real @ X4 @ B )
% 9.51/5.32                 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 9.51/5.32           => ? [L4: real,Z4: real] :
% 9.51/5.32                ( ( ord_less_real @ A @ Z4 )
% 9.51/5.32                & ( ord_less_real @ Z4 @ B )
% 9.51/5.32                & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) )
% 9.51/5.32                & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 9.51/5.32                  = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9031_uniformity__complex__def,axiom,
% 9.51/5.32      ( topolo896644834953643431omplex
% 9.51/5.32      = ( comple8358262395181532106omplex
% 9.51/5.32        @ ( image_5971271580939081552omplex
% 9.51/5.32          @ ^ [E3: real] :
% 9.51/5.32              ( princi3496590319149328850omplex
% 9.51/5.32              @ ( collec8663557070575231912omplex
% 9.51/5.32                @ ( produc6771430404735790350plex_o
% 9.51/5.32                  @ ^ [X3: complex,Y2: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X3 @ Y2 ) @ E3 ) ) ) )
% 9.51/5.32          @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9032_uniformity__real__def,axiom,
% 9.51/5.32      ( topolo1511823702728130853y_real
% 9.51/5.32      = ( comple2936214249959783750l_real
% 9.51/5.32        @ ( image_2178119161166701260l_real
% 9.51/5.32          @ ^ [E3: real] :
% 9.51/5.32              ( princi6114159922880469582l_real
% 9.51/5.32              @ ( collec3799799289383736868l_real
% 9.51/5.32                @ ( produc5414030515140494994real_o
% 9.51/5.32                  @ ^ [X3: real,Y2: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X3 @ Y2 ) @ E3 ) ) ) )
% 9.51/5.32          @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9033_Rolle__deriv,axiom,
% 9.51/5.32      ! [A: real,B: real,F: real > real,F6: real > real > real] :
% 9.51/5.32        ( ( ord_less_real @ A @ B )
% 9.51/5.32       => ( ( ( F @ A )
% 9.51/5.32            = ( F @ B ) )
% 9.51/5.32         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.51/5.32           => ( ! [X4: real] :
% 9.51/5.32                  ( ( ord_less_real @ A @ X4 )
% 9.51/5.32                 => ( ( ord_less_real @ X4 @ B )
% 9.51/5.32                   => ( has_de1759254742604945161l_real @ F @ ( F6 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 9.51/5.32             => ? [Z4: real] :
% 9.51/5.32                  ( ( ord_less_real @ A @ Z4 )
% 9.51/5.32                  & ( ord_less_real @ Z4 @ B )
% 9.51/5.32                  & ( ( F6 @ Z4 )
% 9.51/5.32                    = ( ^ [V3: real] : zero_zero_real ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9034_mvt,axiom,
% 9.51/5.32      ! [A: real,B: real,F: real > real,F6: real > real > real] :
% 9.51/5.32        ( ( ord_less_real @ A @ B )
% 9.51/5.32       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.51/5.32         => ( ! [X4: real] :
% 9.51/5.32                ( ( ord_less_real @ A @ X4 )
% 9.51/5.32               => ( ( ord_less_real @ X4 @ B )
% 9.51/5.32                 => ( has_de1759254742604945161l_real @ F @ ( F6 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 9.51/5.32           => ~ ! [Xi: real] :
% 9.51/5.32                  ( ( ord_less_real @ A @ Xi )
% 9.51/5.32                 => ( ( ord_less_real @ Xi @ B )
% 9.51/5.32                   => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 9.51/5.32                     != ( F6 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9035_DERIV__pos__imp__increasing__open,axiom,
% 9.51/5.32      ! [A: real,B: real,F: real > real] :
% 9.51/5.32        ( ( ord_less_real @ A @ B )
% 9.51/5.32       => ( ! [X4: real] :
% 9.51/5.32              ( ( ord_less_real @ A @ X4 )
% 9.51/5.32             => ( ( ord_less_real @ X4 @ B )
% 9.51/5.32               => ? [Y4: real] :
% 9.51/5.32                    ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.51/5.32                    & ( ord_less_real @ zero_zero_real @ Y4 ) ) ) )
% 9.51/5.32         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.51/5.32           => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9036_DERIV__neg__imp__decreasing__open,axiom,
% 9.51/5.32      ! [A: real,B: real,F: real > real] :
% 9.51/5.32        ( ( ord_less_real @ A @ B )
% 9.51/5.32       => ( ! [X4: real] :
% 9.51/5.32              ( ( ord_less_real @ A @ X4 )
% 9.51/5.32             => ( ( ord_less_real @ X4 @ B )
% 9.51/5.32               => ? [Y4: real] :
% 9.51/5.32                    ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.51/5.32                    & ( ord_less_real @ Y4 @ zero_zero_real ) ) ) )
% 9.51/5.32         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.51/5.32           => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9037_DERIV__isconst__end,axiom,
% 9.51/5.32      ! [A: real,B: real,F: real > real] :
% 9.51/5.32        ( ( ord_less_real @ A @ B )
% 9.51/5.32       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.51/5.32         => ( ! [X4: real] :
% 9.51/5.32                ( ( ord_less_real @ A @ X4 )
% 9.51/5.32               => ( ( ord_less_real @ X4 @ B )
% 9.51/5.32                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 9.51/5.32           => ( ( F @ B )
% 9.51/5.32              = ( F @ A ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9038_DERIV__isconst2,axiom,
% 9.51/5.32      ! [A: real,B: real,F: real > real,X: real] :
% 9.51/5.32        ( ( ord_less_real @ A @ B )
% 9.51/5.32       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.51/5.32         => ( ! [X4: real] :
% 9.51/5.32                ( ( ord_less_real @ A @ X4 )
% 9.51/5.32               => ( ( ord_less_real @ X4 @ B )
% 9.51/5.32                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 9.51/5.32           => ( ( ord_less_eq_real @ A @ X )
% 9.51/5.32             => ( ( ord_less_eq_real @ X @ B )
% 9.51/5.32               => ( ( F @ X )
% 9.51/5.32                  = ( F @ A ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9039_Rolle,axiom,
% 9.51/5.32      ! [A: real,B: real,F: real > real] :
% 9.51/5.32        ( ( ord_less_real @ A @ B )
% 9.51/5.32       => ( ( ( F @ A )
% 9.51/5.32            = ( F @ B ) )
% 9.51/5.32         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.51/5.32           => ( ! [X4: real] :
% 9.51/5.32                  ( ( ord_less_real @ A @ X4 )
% 9.51/5.32                 => ( ( ord_less_real @ X4 @ B )
% 9.51/5.32                   => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 9.51/5.32             => ? [Z4: real] :
% 9.51/5.32                  ( ( ord_less_real @ A @ Z4 )
% 9.51/5.32                  & ( ord_less_real @ Z4 @ B )
% 9.51/5.32                  & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9040_eventually__prod__sequentially,axiom,
% 9.51/5.32      ! [P: product_prod_nat_nat > $o] :
% 9.51/5.32        ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
% 9.51/5.32        = ( ? [N4: nat] :
% 9.51/5.32            ! [M2: nat] :
% 9.51/5.32              ( ( ord_less_eq_nat @ N4 @ M2 )
% 9.51/5.32             => ! [N2: nat] :
% 9.51/5.32                  ( ( ord_less_eq_nat @ N4 @ N2 )
% 9.51/5.32                 => ( P @ ( product_Pair_nat_nat @ N2 @ M2 ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9041_mono__Suc,axiom,
% 9.51/5.32      order_mono_nat_nat @ suc ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9042_mono__times__nat,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.32       => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9043_mono__ge2__power__minus__self,axiom,
% 9.51/5.32      ! [K: nat] :
% 9.51/5.32        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.51/5.32       => ( order_mono_nat_nat
% 9.51/5.32          @ ^ [M2: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M2 ) @ M2 ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9044_inj__sgn__power,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.32       => ( inj_on_real_real
% 9.51/5.32          @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N ) )
% 9.51/5.32          @ top_top_set_real ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9045_log__inj,axiom,
% 9.51/5.32      ! [B: real] :
% 9.51/5.32        ( ( ord_less_real @ one_one_real @ B )
% 9.51/5.32       => ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9046_inj__Suc,axiom,
% 9.51/5.32      ! [N6: set_nat] : ( inj_on_nat_nat @ suc @ N6 ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9047_inj__on__diff__nat,axiom,
% 9.51/5.32      ! [N6: set_nat,K: nat] :
% 9.51/5.32        ( ! [N3: nat] :
% 9.51/5.32            ( ( member_nat @ N3 @ N6 )
% 9.51/5.32           => ( ord_less_eq_nat @ K @ N3 ) )
% 9.51/5.32       => ( inj_on_nat_nat
% 9.51/5.32          @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
% 9.51/5.32          @ N6 ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9048_inj__on__set__encode,axiom,
% 9.51/5.32      inj_on_set_nat_nat @ nat_set_encode @ ( collect_set_nat @ finite_finite_nat ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9049_powr__real__of__int_H,axiom,
% 9.51/5.32      ! [X: real,N: int] :
% 9.51/5.32        ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.51/5.32       => ( ( ( X != zero_zero_real )
% 9.51/5.32            | ( ord_less_int @ zero_zero_int @ N ) )
% 9.51/5.32         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 9.51/5.32            = ( power_int_real @ X @ N ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9050_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 9.51/5.32      ! [X: vEBT_VEBT,Xa2: nat] :
% 9.51/5.32        ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
% 9.51/5.32       => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.51/5.32                ( X
% 9.51/5.32                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.51/5.32           => ( Xa2 = one_one_nat ) )
% 9.51/5.32         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.51/5.32                ( ( X
% 9.51/5.32                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) )
% 9.51/5.32               => ( ( Deg2 = Xa2 )
% 9.51/5.32                  & ! [X4: vEBT_VEBT] :
% 9.51/5.32                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.51/5.32                     => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.51/5.32                  & ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.51/5.32                  & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.51/5.32                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.51/5.32                  & ( case_o184042715313410164at_nat
% 9.51/5.32                    @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 9.51/5.32                      & ! [X3: vEBT_VEBT] :
% 9.51/5.32                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.51/5.32                         => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 9.51/5.32                    @ ( produc6081775807080527818_nat_o
% 9.51/5.32                      @ ^ [Mi2: nat,Ma2: nat] :
% 9.51/5.32                          ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 9.51/5.32                          & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.51/5.32                          & ! [I4: nat] :
% 9.51/5.32                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.51/5.32                             => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 9.51/5.32                                = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 9.51/5.32                          & ( ( Mi2 = Ma2 )
% 9.51/5.32                           => ! [X3: vEBT_VEBT] :
% 9.51/5.32                                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.51/5.32                               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 9.51/5.32                          & ( ( Mi2 != Ma2 )
% 9.51/5.32                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma2 )
% 9.51/5.32                              & ! [X3: nat] :
% 9.51/5.32                                  ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.51/5.32                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 9.51/5.32                                   => ( ( ord_less_nat @ Mi2 @ X3 )
% 9.51/5.32                                      & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) )
% 9.51/5.32                    @ Mima ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9051_atLeastLessThan__add__Un,axiom,
% 9.51/5.32      ! [I: nat,J: nat,K: nat] :
% 9.51/5.32        ( ( ord_less_eq_nat @ I @ J )
% 9.51/5.32       => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
% 9.51/5.32          = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9052_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 9.51/5.32      ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
% 9.51/5.32        ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg4 )
% 9.51/5.32        = ( ( Deg = Deg4 )
% 9.51/5.32          & ! [X3: vEBT_VEBT] :
% 9.51/5.32              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.51/5.32             => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.51/5.32          & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.51/5.32          & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.51/5.32            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.51/5.32          & ( case_o184042715313410164at_nat
% 9.51/5.32            @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
% 9.51/5.32              & ! [X3: vEBT_VEBT] :
% 9.51/5.32                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.51/5.32                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 9.51/5.32            @ ( produc6081775807080527818_nat_o
% 9.51/5.32              @ ^ [Mi2: nat,Ma2: nat] :
% 9.51/5.32                  ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 9.51/5.32                  & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.51/5.32                  & ! [I4: nat] :
% 9.51/5.32                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.51/5.32                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X6 ) )
% 9.51/5.32                        = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 9.51/5.32                  & ( ( Mi2 = Ma2 )
% 9.51/5.32                   => ! [X3: vEBT_VEBT] :
% 9.51/5.32                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.51/5.32                       => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 9.51/5.32                  & ( ( Mi2 != Ma2 )
% 9.51/5.32                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma2 )
% 9.51/5.32                      & ! [X3: nat] :
% 9.51/5.32                          ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.51/5.32                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X3 )
% 9.51/5.32                           => ( ( ord_less_nat @ Mi2 @ X3 )
% 9.51/5.32                              & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) )
% 9.51/5.32            @ Mima2 ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9053_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 9.51/5.32      ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 9.51/5.32        ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
% 9.51/5.32          = Y )
% 9.51/5.32       => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.51/5.32                ( X
% 9.51/5.32                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.51/5.32           => ( Y
% 9.51/5.32              = ( Xa2 != one_one_nat ) ) )
% 9.51/5.32         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.51/5.32                ( ( X
% 9.51/5.32                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) )
% 9.51/5.32               => ( Y
% 9.51/5.32                  = ( ~ ( ( Deg2 = Xa2 )
% 9.51/5.32                        & ! [X3: vEBT_VEBT] :
% 9.51/5.32                            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.51/5.32                           => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.51/5.32                        & ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.51/5.32                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.51/5.32                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.51/5.32                        & ( case_o184042715313410164at_nat
% 9.51/5.32                          @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 9.51/5.32                            & ! [X3: vEBT_VEBT] :
% 9.51/5.32                                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.51/5.32                               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 9.51/5.32                          @ ( produc6081775807080527818_nat_o
% 9.51/5.32                            @ ^ [Mi2: nat,Ma2: nat] :
% 9.51/5.32                                ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 9.51/5.32                                & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.51/5.32                                & ! [I4: nat] :
% 9.51/5.32                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.51/5.32                                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 9.51/5.32                                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 9.51/5.32                                & ( ( Mi2 = Ma2 )
% 9.51/5.32                                 => ! [X3: vEBT_VEBT] :
% 9.51/5.32                                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.51/5.32                                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 9.51/5.32                                & ( ( Mi2 != Ma2 )
% 9.51/5.32                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma2 )
% 9.51/5.32                                    & ! [X3: nat] :
% 9.51/5.32                                        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.51/5.32                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 9.51/5.32                                         => ( ( ord_less_nat @ Mi2 @ X3 )
% 9.51/5.32                                            & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) )
% 9.51/5.32                          @ Mima ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9054_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 9.51/5.32      ! [X: vEBT_VEBT,Xa2: nat] :
% 9.51/5.32        ( ( vEBT_VEBT_valid @ X @ Xa2 )
% 9.51/5.32       => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.51/5.32                ( X
% 9.51/5.32                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.51/5.32           => ( Xa2 != one_one_nat ) )
% 9.51/5.32         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.51/5.32                ( ( X
% 9.51/5.32                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) )
% 9.51/5.32               => ~ ( ( Deg2 = Xa2 )
% 9.51/5.32                    & ! [X2: vEBT_VEBT] :
% 9.51/5.32                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.51/5.32                       => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.51/5.32                    & ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.51/5.32                    & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.51/5.32                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.51/5.32                    & ( case_o184042715313410164at_nat
% 9.51/5.32                      @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 9.51/5.32                        & ! [X3: vEBT_VEBT] :
% 9.51/5.32                            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.51/5.32                           => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 9.51/5.32                      @ ( produc6081775807080527818_nat_o
% 9.51/5.32                        @ ^ [Mi2: nat,Ma2: nat] :
% 9.51/5.32                            ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 9.51/5.32                            & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.51/5.32                            & ! [I4: nat] :
% 9.51/5.32                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.51/5.32                               => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 9.51/5.32                                  = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 9.51/5.32                            & ( ( Mi2 = Ma2 )
% 9.51/5.32                             => ! [X3: vEBT_VEBT] :
% 9.51/5.32                                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.51/5.32                                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 9.51/5.32                            & ( ( Mi2 != Ma2 )
% 9.51/5.32                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma2 )
% 9.51/5.32                                & ! [X3: nat] :
% 9.51/5.32                                    ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.51/5.32                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 9.51/5.32                                     => ( ( ord_less_nat @ Mi2 @ X3 )
% 9.51/5.32                                        & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) )
% 9.51/5.32                      @ Mima ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9055_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 9.51/5.32      ! [X: vEBT_VEBT,Xa2: nat] :
% 9.51/5.32        ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
% 9.51/5.32       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 9.51/5.32         => ( ! [Uu2: $o,Uv2: $o] :
% 9.51/5.32                ( ( X
% 9.51/5.32                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.51/5.32               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 9.51/5.32                 => ( Xa2 = one_one_nat ) ) )
% 9.51/5.32           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.51/5.32                  ( ( X
% 9.51/5.32                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) )
% 9.51/5.32                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) @ Xa2 ) )
% 9.51/5.32                   => ( ( Deg2 = Xa2 )
% 9.51/5.32                      & ! [X4: vEBT_VEBT] :
% 9.51/5.32                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.51/5.32                         => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.51/5.32                      & ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.51/5.32                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.51/5.32                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.51/5.32                      & ( case_o184042715313410164at_nat
% 9.51/5.32                        @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 9.51/5.32                          & ! [X3: vEBT_VEBT] :
% 9.51/5.32                              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.51/5.32                             => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 9.51/5.32                        @ ( produc6081775807080527818_nat_o
% 9.51/5.32                          @ ^ [Mi2: nat,Ma2: nat] :
% 9.51/5.32                              ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 9.51/5.32                              & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.51/5.32                              & ! [I4: nat] :
% 9.51/5.32                                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.51/5.32                                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 9.51/5.32                                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 9.51/5.32                              & ( ( Mi2 = Ma2 )
% 9.51/5.32                               => ! [X3: vEBT_VEBT] :
% 9.51/5.32                                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.51/5.32                                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 9.51/5.32                              & ( ( Mi2 != Ma2 )
% 9.51/5.32                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma2 )
% 9.51/5.32                                  & ! [X3: nat] :
% 9.51/5.32                                      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.51/5.32                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 9.51/5.32                                       => ( ( ord_less_nat @ Mi2 @ X3 )
% 9.51/5.32                                          & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) )
% 9.51/5.32                        @ Mima ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9056_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 9.51/5.32      ! [X: vEBT_VEBT,Xa2: nat] :
% 9.51/5.32        ( ( vEBT_VEBT_valid @ X @ Xa2 )
% 9.51/5.32       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 9.51/5.32         => ( ! [Uu2: $o,Uv2: $o] :
% 9.51/5.32                ( ( X
% 9.51/5.32                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.51/5.32               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 9.51/5.32                 => ( Xa2 != one_one_nat ) ) )
% 9.51/5.32           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.51/5.32                  ( ( X
% 9.51/5.32                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) )
% 9.51/5.32                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) @ Xa2 ) )
% 9.51/5.32                   => ~ ( ( Deg2 = Xa2 )
% 9.51/5.32                        & ! [X2: vEBT_VEBT] :
% 9.51/5.32                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.51/5.32                           => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.51/5.32                        & ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.51/5.32                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.51/5.32                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.51/5.32                        & ( case_o184042715313410164at_nat
% 9.51/5.32                          @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 9.51/5.32                            & ! [X3: vEBT_VEBT] :
% 9.51/5.32                                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.51/5.32                               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 9.51/5.32                          @ ( produc6081775807080527818_nat_o
% 9.51/5.32                            @ ^ [Mi2: nat,Ma2: nat] :
% 9.51/5.32                                ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 9.51/5.32                                & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.51/5.32                                & ! [I4: nat] :
% 9.51/5.32                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.51/5.32                                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 9.51/5.32                                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 9.51/5.32                                & ( ( Mi2 = Ma2 )
% 9.51/5.32                                 => ! [X3: vEBT_VEBT] :
% 9.51/5.32                                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.51/5.32                                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 9.51/5.32                                & ( ( Mi2 != Ma2 )
% 9.51/5.32                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma2 )
% 9.51/5.32                                    & ! [X3: nat] :
% 9.51/5.32                                        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.51/5.32                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 9.51/5.32                                         => ( ( ord_less_nat @ Mi2 @ X3 )
% 9.51/5.32                                            & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) )
% 9.51/5.32                          @ Mima ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9057_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 9.51/5.32      ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 9.51/5.32        ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
% 9.51/5.32          = Y )
% 9.51/5.32       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 9.51/5.32         => ( ! [Uu2: $o,Uv2: $o] :
% 9.51/5.32                ( ( X
% 9.51/5.32                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.51/5.32               => ( ( Y
% 9.51/5.32                    = ( Xa2 = one_one_nat ) )
% 9.51/5.32                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 9.51/5.32           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
% 9.51/5.32                  ( ( X
% 9.51/5.32                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) )
% 9.51/5.32                 => ( ( Y
% 9.51/5.32                      = ( ( Deg2 = Xa2 )
% 9.51/5.32                        & ! [X3: vEBT_VEBT] :
% 9.51/5.32                            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.51/5.32                           => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.51/5.32                        & ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.51/5.32                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.51/5.32                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.51/5.32                        & ( case_o184042715313410164at_nat
% 9.51/5.32                          @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 9.51/5.32                            & ! [X3: vEBT_VEBT] :
% 9.51/5.32                                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.51/5.32                               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 9.51/5.32                          @ ( produc6081775807080527818_nat_o
% 9.51/5.32                            @ ^ [Mi2: nat,Ma2: nat] :
% 9.51/5.32                                ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 9.51/5.32                                & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.51/5.32                                & ! [I4: nat] :
% 9.51/5.32                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.51/5.32                                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 9.51/5.32                                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 9.51/5.32                                & ( ( Mi2 = Ma2 )
% 9.51/5.32                                 => ! [X3: vEBT_VEBT] :
% 9.51/5.32                                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.51/5.32                                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
% 9.51/5.32                                & ( ( Mi2 != Ma2 )
% 9.51/5.32                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma2 )
% 9.51/5.32                                    & ! [X3: nat] :
% 9.51/5.32                                        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.51/5.32                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
% 9.51/5.32                                         => ( ( ord_less_nat @ Mi2 @ X3 )
% 9.51/5.32                                            & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) )
% 9.51/5.32                          @ Mima ) ) )
% 9.51/5.32                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9058_sup__nat__def,axiom,
% 9.51/5.32      sup_sup_nat = ord_max_nat ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9059_min__weak__def,axiom,
% 9.51/5.32      ( fun_min_weak
% 9.51/5.32      = ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9060_max__weak__def,axiom,
% 9.51/5.32      ( fun_max_weak
% 9.51/5.32      = ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9061_take__bit__num__simps_I1_J,axiom,
% 9.51/5.32      ! [M: num] :
% 9.51/5.32        ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 9.51/5.32        = none_num ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9062_take__bit__num__simps_I2_J,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( ( bit_take_bit_num @ ( suc @ N ) @ one )
% 9.51/5.32        = ( some_num @ one ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9063_take__bit__num__simps_I5_J,axiom,
% 9.51/5.32      ! [R2: num] :
% 9.51/5.32        ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ one )
% 9.51/5.32        = ( some_num @ one ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9064_take__bit__num__simps_I3_J,axiom,
% 9.51/5.32      ! [N: nat,M: num] :
% 9.51/5.32        ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
% 9.51/5.32        = ( case_o6005452278849405969um_num @ none_num
% 9.51/5.32          @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 9.51/5.32          @ ( bit_take_bit_num @ N @ M ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9065_take__bit__num__simps_I4_J,axiom,
% 9.51/5.32      ! [N: nat,M: num] :
% 9.51/5.32        ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
% 9.51/5.32        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9066_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
% 9.51/5.32      ! [N: nat,M: num] :
% 9.51/5.32        ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
% 9.51/5.32        = ( case_nat_option_num @ none_num
% 9.51/5.32          @ ^ [N2: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) )
% 9.51/5.32          @ N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9067_take__bit__num__simps_I6_J,axiom,
% 9.51/5.32      ! [R2: num,M: num] :
% 9.51/5.32        ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M ) )
% 9.51/5.32        = ( case_o6005452278849405969um_num @ none_num
% 9.51/5.32          @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 9.51/5.32          @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9068_take__bit__num__simps_I7_J,axiom,
% 9.51/5.32      ! [R2: num,M: num] :
% 9.51/5.32        ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit1 @ M ) )
% 9.51/5.32        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9069_pred__numeral__simps_I1_J,axiom,
% 9.51/5.32      ( ( pred_numeral @ one )
% 9.51/5.32      = zero_zero_nat ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9070_less__Suc__numeral,axiom,
% 9.51/5.32      ! [N: nat,K: num] :
% 9.51/5.32        ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 9.51/5.32        = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9071_less__numeral__Suc,axiom,
% 9.51/5.32      ! [K: num,N: nat] :
% 9.51/5.32        ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 9.51/5.32        = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9072_le__numeral__Suc,axiom,
% 9.51/5.32      ! [K: num,N: nat] :
% 9.51/5.32        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 9.51/5.32        = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9073_le__Suc__numeral,axiom,
% 9.51/5.32      ! [N: nat,K: num] :
% 9.51/5.32        ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 9.51/5.32        = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9074_atLeastLessThan__nat__numeral,axiom,
% 9.51/5.32      ! [M: nat,K: num] :
% 9.51/5.32        ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 9.51/5.32         => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 9.51/5.32            = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 9.51/5.32        & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 9.51/5.32         => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 9.51/5.32            = bot_bot_set_nat ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9075_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( ( bit_take_bit_num @ N @ one )
% 9.51/5.32        = ( case_nat_option_num @ none_num
% 9.51/5.32          @ ^ [N2: nat] : ( some_num @ one )
% 9.51/5.32          @ N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9076_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
% 9.51/5.32      ! [N: nat,M: num] :
% 9.51/5.32        ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
% 9.51/5.32        = ( case_nat_option_num @ none_num
% 9.51/5.32          @ ^ [N2: nat] :
% 9.51/5.32              ( case_o6005452278849405969um_num @ none_num
% 9.51/5.32              @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 9.51/5.32              @ ( bit_take_bit_num @ N2 @ M ) )
% 9.51/5.32          @ N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9077_Bit__Operations_Otake__bit__num__code,axiom,
% 9.51/5.32      ( bit_take_bit_num
% 9.51/5.32      = ( ^ [N2: nat,M2: num] :
% 9.51/5.32            ( produc478579273971653890on_num
% 9.51/5.32            @ ^ [A4: nat,X3: num] :
% 9.51/5.32                ( case_nat_option_num @ none_num
% 9.51/5.32                @ ^ [O: nat] :
% 9.51/5.32                    ( case_num_option_num @ ( some_num @ one )
% 9.51/5.32                    @ ^ [P6: num] :
% 9.51/5.32                        ( case_o6005452278849405969um_num @ none_num
% 9.51/5.32                        @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 9.51/5.32                        @ ( bit_take_bit_num @ O @ P6 ) )
% 9.51/5.32                    @ ^ [P6: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) )
% 9.51/5.32                    @ X3 )
% 9.51/5.32                @ A4 )
% 9.51/5.32            @ ( product_Pair_nat_num @ N2 @ M2 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9078_take__bit__num__def,axiom,
% 9.51/5.32      ( bit_take_bit_num
% 9.51/5.32      = ( ^ [N2: nat,M2: num] :
% 9.51/5.32            ( if_option_num
% 9.51/5.32            @ ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M2 ) )
% 9.51/5.32              = zero_zero_nat )
% 9.51/5.32            @ none_num
% 9.51/5.32            @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M2 ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9079_nat_Odisc__eq__case_I1_J,axiom,
% 9.51/5.32      ! [Nat: nat] :
% 9.51/5.32        ( ( Nat = zero_zero_nat )
% 9.51/5.32        = ( case_nat_o @ $true
% 9.51/5.32          @ ^ [Uu3: nat] : $false
% 9.51/5.32          @ Nat ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9080_nat_Odisc__eq__case_I2_J,axiom,
% 9.51/5.32      ! [Nat: nat] :
% 9.51/5.32        ( ( Nat != zero_zero_nat )
% 9.51/5.32        = ( case_nat_o @ $false
% 9.51/5.32          @ ^ [Uu3: nat] : $true
% 9.51/5.32          @ Nat ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9081_num__of__nat_Osimps_I1_J,axiom,
% 9.51/5.32      ( ( num_of_nat @ zero_zero_nat )
% 9.51/5.32      = one ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9082_less__eq__nat_Osimps_I2_J,axiom,
% 9.51/5.32      ! [M: nat,N: nat] :
% 9.51/5.32        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 9.51/5.32        = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9083_max__Suc1,axiom,
% 9.51/5.32      ! [N: nat,M: nat] :
% 9.51/5.32        ( ( ord_max_nat @ ( suc @ N ) @ M )
% 9.51/5.32        = ( case_nat_nat @ ( suc @ N )
% 9.51/5.32          @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ N @ M6 ) )
% 9.51/5.32          @ M ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9084_max__Suc2,axiom,
% 9.51/5.32      ! [M: nat,N: nat] :
% 9.51/5.32        ( ( ord_max_nat @ M @ ( suc @ N ) )
% 9.51/5.32        = ( case_nat_nat @ ( suc @ N )
% 9.51/5.32          @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ M6 @ N ) )
% 9.51/5.32          @ M ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9085_numeral__num__of__nat,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.32       => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
% 9.51/5.32          = N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9086_num__of__nat__plus__distrib,axiom,
% 9.51/5.32      ! [M: nat,N: nat] :
% 9.51/5.32        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.51/5.32       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.32         => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
% 9.51/5.32            = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9087_num__of__nat__One,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( ( ord_less_eq_nat @ N @ one_one_nat )
% 9.51/5.32       => ( ( num_of_nat @ N )
% 9.51/5.32          = one ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9088_diff__Suc,axiom,
% 9.51/5.32      ! [M: nat,N: nat] :
% 9.51/5.32        ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 9.51/5.32        = ( case_nat_nat @ zero_zero_nat
% 9.51/5.32          @ ^ [K3: nat] : K3
% 9.51/5.32          @ ( minus_minus_nat @ M @ N ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9089_num__of__nat__double,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.32       => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
% 9.51/5.32          = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9090_pred__def,axiom,
% 9.51/5.32      ( pred
% 9.51/5.32      = ( case_nat_nat @ zero_zero_nat
% 9.51/5.32        @ ^ [X24: nat] : X24 ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9091_min__Suc__Suc,axiom,
% 9.51/5.32      ! [M: nat,N: nat] :
% 9.51/5.32        ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
% 9.51/5.32        = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9092_min__0L,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( ( ord_min_nat @ zero_zero_nat @ N )
% 9.51/5.32        = zero_zero_nat ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9093_min__0R,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( ( ord_min_nat @ N @ zero_zero_nat )
% 9.51/5.32        = zero_zero_nat ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9094_min__diff,axiom,
% 9.51/5.32      ! [M: nat,I: nat,N: nat] :
% 9.51/5.32        ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
% 9.51/5.32        = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9095_nat__mult__min__left,axiom,
% 9.51/5.32      ! [M: nat,N: nat,Q4: nat] :
% 9.51/5.32        ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q4 )
% 9.51/5.32        = ( ord_min_nat @ ( times_times_nat @ M @ Q4 ) @ ( times_times_nat @ N @ Q4 ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9096_nat__mult__min__right,axiom,
% 9.51/5.32      ! [M: nat,N: nat,Q4: nat] :
% 9.51/5.32        ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q4 ) )
% 9.51/5.32        = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q4 ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9097_inf__nat__def,axiom,
% 9.51/5.32      inf_inf_nat = ord_min_nat ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9098_min__Suc2,axiom,
% 9.51/5.32      ! [M: nat,N: nat] :
% 9.51/5.32        ( ( ord_min_nat @ M @ ( suc @ N ) )
% 9.51/5.32        = ( case_nat_nat @ zero_zero_nat
% 9.51/5.32          @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ M6 @ N ) )
% 9.51/5.32          @ M ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9099_min__Suc1,axiom,
% 9.51/5.32      ! [N: nat,M: nat] :
% 9.51/5.32        ( ( ord_min_nat @ ( suc @ N ) @ M )
% 9.51/5.32        = ( case_nat_nat @ zero_zero_nat
% 9.51/5.32          @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ N @ M6 ) )
% 9.51/5.32          @ M ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9100_and__not__num_Osimps_I8_J,axiom,
% 9.51/5.32      ! [M: num,N: num] :
% 9.51/5.32        ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.51/5.32        = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 9.51/5.32          @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 9.51/5.32          @ ( bit_and_not_num @ M @ N ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9101_and__not__num_Osimps_I1_J,axiom,
% 9.51/5.32      ( ( bit_and_not_num @ one @ one )
% 9.51/5.32      = none_num ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9102_and__not__num_Osimps_I4_J,axiom,
% 9.51/5.32      ! [M: num] :
% 9.51/5.32        ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
% 9.51/5.32        = ( some_num @ ( bit0 @ M ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9103_and__not__num_Osimps_I2_J,axiom,
% 9.51/5.32      ! [N: num] :
% 9.51/5.32        ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
% 9.51/5.32        = ( some_num @ one ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9104_and__not__num_Osimps_I3_J,axiom,
% 9.51/5.32      ! [N: num] :
% 9.51/5.32        ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
% 9.51/5.32        = none_num ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9105_and__not__num_Osimps_I7_J,axiom,
% 9.51/5.32      ! [M: num] :
% 9.51/5.32        ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
% 9.51/5.32        = ( some_num @ ( bit0 @ M ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9106_and__not__num_Oelims,axiom,
% 9.51/5.32      ! [X: num,Xa2: num,Y: option_num] :
% 9.51/5.32        ( ( ( bit_and_not_num @ X @ Xa2 )
% 9.51/5.32          = Y )
% 9.51/5.32       => ( ( ( X = one )
% 9.51/5.32           => ( ( Xa2 = one )
% 9.51/5.32             => ( Y != none_num ) ) )
% 9.51/5.32         => ( ( ( X = one )
% 9.51/5.32             => ( ? [N3: num] :
% 9.51/5.32                    ( Xa2
% 9.51/5.32                    = ( bit0 @ N3 ) )
% 9.51/5.32               => ( Y
% 9.51/5.32                 != ( some_num @ one ) ) ) )
% 9.51/5.32           => ( ( ( X = one )
% 9.51/5.32               => ( ? [N3: num] :
% 9.51/5.32                      ( Xa2
% 9.51/5.32                      = ( bit1 @ N3 ) )
% 9.51/5.32                 => ( Y != none_num ) ) )
% 9.51/5.32             => ( ! [M5: num] :
% 9.51/5.32                    ( ( X
% 9.51/5.32                      = ( bit0 @ M5 ) )
% 9.51/5.32                   => ( ( Xa2 = one )
% 9.51/5.32                     => ( Y
% 9.51/5.32                       != ( some_num @ ( bit0 @ M5 ) ) ) ) )
% 9.51/5.32               => ( ! [M5: num] :
% 9.51/5.32                      ( ( X
% 9.51/5.32                        = ( bit0 @ M5 ) )
% 9.51/5.32                     => ! [N3: num] :
% 9.51/5.32                          ( ( Xa2
% 9.51/5.32                            = ( bit0 @ N3 ) )
% 9.51/5.32                         => ( Y
% 9.51/5.32                           != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
% 9.51/5.32                 => ( ! [M5: num] :
% 9.51/5.32                        ( ( X
% 9.51/5.32                          = ( bit0 @ M5 ) )
% 9.51/5.32                       => ! [N3: num] :
% 9.51/5.32                            ( ( Xa2
% 9.51/5.32                              = ( bit1 @ N3 ) )
% 9.51/5.32                           => ( Y
% 9.51/5.32                             != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
% 9.51/5.32                   => ( ! [M5: num] :
% 9.51/5.32                          ( ( X
% 9.51/5.32                            = ( bit1 @ M5 ) )
% 9.51/5.32                         => ( ( Xa2 = one )
% 9.51/5.32                           => ( Y
% 9.51/5.32                             != ( some_num @ ( bit0 @ M5 ) ) ) ) )
% 9.51/5.32                     => ( ! [M5: num] :
% 9.51/5.32                            ( ( X
% 9.51/5.32                              = ( bit1 @ M5 ) )
% 9.51/5.32                           => ! [N3: num] :
% 9.51/5.32                                ( ( Xa2
% 9.51/5.32                                  = ( bit0 @ N3 ) )
% 9.51/5.32                               => ( Y
% 9.51/5.32                                 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 9.51/5.32                                    @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 9.51/5.32                                    @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
% 9.51/5.32                       => ~ ! [M5: num] :
% 9.51/5.32                              ( ( X
% 9.51/5.32                                = ( bit1 @ M5 ) )
% 9.51/5.32                             => ! [N3: num] :
% 9.51/5.32                                  ( ( Xa2
% 9.51/5.32                                    = ( bit1 @ N3 ) )
% 9.51/5.32                                 => ( Y
% 9.51/5.32                                   != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9107_xor__num_Osimps_I8_J,axiom,
% 9.51/5.32      ! [M: num,N: num] :
% 9.51/5.32        ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.51/5.32        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9108_xor__num_Osimps_I1_J,axiom,
% 9.51/5.32      ( ( bit_un2480387367778600638or_num @ one @ one )
% 9.51/5.32      = none_num ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9109_xor__num_Oelims,axiom,
% 9.51/5.32      ! [X: num,Xa2: num,Y: option_num] :
% 9.51/5.32        ( ( ( bit_un2480387367778600638or_num @ X @ Xa2 )
% 9.51/5.32          = Y )
% 9.51/5.32       => ( ( ( X = one )
% 9.51/5.32           => ( ( Xa2 = one )
% 9.51/5.32             => ( Y != none_num ) ) )
% 9.51/5.32         => ( ( ( X = one )
% 9.51/5.32             => ! [N3: num] :
% 9.51/5.32                  ( ( Xa2
% 9.51/5.32                    = ( bit0 @ N3 ) )
% 9.51/5.32                 => ( Y
% 9.51/5.32                   != ( some_num @ ( bit1 @ N3 ) ) ) ) )
% 9.51/5.32           => ( ( ( X = one )
% 9.51/5.32               => ! [N3: num] :
% 9.51/5.32                    ( ( Xa2
% 9.51/5.32                      = ( bit1 @ N3 ) )
% 9.51/5.32                   => ( Y
% 9.51/5.32                     != ( some_num @ ( bit0 @ N3 ) ) ) ) )
% 9.51/5.32             => ( ! [M5: num] :
% 9.51/5.32                    ( ( X
% 9.51/5.32                      = ( bit0 @ M5 ) )
% 9.51/5.32                   => ( ( Xa2 = one )
% 9.51/5.32                     => ( Y
% 9.51/5.32                       != ( some_num @ ( bit1 @ M5 ) ) ) ) )
% 9.51/5.32               => ( ! [M5: num] :
% 9.51/5.32                      ( ( X
% 9.51/5.32                        = ( bit0 @ M5 ) )
% 9.51/5.32                     => ! [N3: num] :
% 9.51/5.32                          ( ( Xa2
% 9.51/5.32                            = ( bit0 @ N3 ) )
% 9.51/5.32                         => ( Y
% 9.51/5.32                           != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) )
% 9.51/5.32                 => ( ! [M5: num] :
% 9.51/5.32                        ( ( X
% 9.51/5.32                          = ( bit0 @ M5 ) )
% 9.51/5.32                       => ! [N3: num] :
% 9.51/5.32                            ( ( Xa2
% 9.51/5.32                              = ( bit1 @ N3 ) )
% 9.51/5.32                           => ( Y
% 9.51/5.32                             != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) )
% 9.51/5.32                   => ( ! [M5: num] :
% 9.51/5.32                          ( ( X
% 9.51/5.32                            = ( bit1 @ M5 ) )
% 9.51/5.32                         => ( ( Xa2 = one )
% 9.51/5.32                           => ( Y
% 9.51/5.32                             != ( some_num @ ( bit0 @ M5 ) ) ) ) )
% 9.51/5.32                     => ( ! [M5: num] :
% 9.51/5.32                            ( ( X
% 9.51/5.32                              = ( bit1 @ M5 ) )
% 9.51/5.32                           => ! [N3: num] :
% 9.51/5.32                                ( ( Xa2
% 9.51/5.32                                  = ( bit0 @ N3 ) )
% 9.51/5.32                               => ( Y
% 9.51/5.32                                 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) )
% 9.51/5.32                       => ~ ! [M5: num] :
% 9.51/5.32                              ( ( X
% 9.51/5.32                                = ( bit1 @ M5 ) )
% 9.51/5.32                             => ! [N3: num] :
% 9.51/5.32                                  ( ( Xa2
% 9.51/5.32                                    = ( bit1 @ N3 ) )
% 9.51/5.32                                 => ( Y
% 9.51/5.32                                   != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9110_xor__num_Osimps_I2_J,axiom,
% 9.51/5.32      ! [N: num] :
% 9.51/5.32        ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
% 9.51/5.32        = ( some_num @ ( bit1 @ N ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9111_xor__num_Osimps_I3_J,axiom,
% 9.51/5.32      ! [N: num] :
% 9.51/5.32        ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
% 9.51/5.32        = ( some_num @ ( bit0 @ N ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9112_xor__num_Osimps_I4_J,axiom,
% 9.51/5.32      ! [M: num] :
% 9.51/5.32        ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
% 9.51/5.32        = ( some_num @ ( bit1 @ M ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9113_xor__num_Osimps_I7_J,axiom,
% 9.51/5.32      ! [M: num] :
% 9.51/5.32        ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
% 9.51/5.32        = ( some_num @ ( bit0 @ M ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9114_xor__num_Osimps_I6_J,axiom,
% 9.51/5.32      ! [M: num,N: num] :
% 9.51/5.32        ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.51/5.32        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9115_and__num_Oelims,axiom,
% 9.51/5.32      ! [X: num,Xa2: num,Y: option_num] :
% 9.51/5.32        ( ( ( bit_un7362597486090784418nd_num @ X @ Xa2 )
% 9.51/5.32          = Y )
% 9.51/5.32       => ( ( ( X = one )
% 9.51/5.32           => ( ( Xa2 = one )
% 9.51/5.32             => ( Y
% 9.51/5.32               != ( some_num @ one ) ) ) )
% 9.51/5.32         => ( ( ( X = one )
% 9.51/5.32             => ( ? [N3: num] :
% 9.51/5.32                    ( Xa2
% 9.51/5.32                    = ( bit0 @ N3 ) )
% 9.51/5.32               => ( Y != none_num ) ) )
% 9.51/5.32           => ( ( ( X = one )
% 9.51/5.32               => ( ? [N3: num] :
% 9.51/5.32                      ( Xa2
% 9.51/5.32                      = ( bit1 @ N3 ) )
% 9.51/5.32                 => ( Y
% 9.51/5.32                   != ( some_num @ one ) ) ) )
% 9.51/5.32             => ( ( ? [M5: num] :
% 9.51/5.32                      ( X
% 9.51/5.32                      = ( bit0 @ M5 ) )
% 9.51/5.32                 => ( ( Xa2 = one )
% 9.51/5.32                   => ( Y != none_num ) ) )
% 9.51/5.32               => ( ! [M5: num] :
% 9.51/5.32                      ( ( X
% 9.51/5.32                        = ( bit0 @ M5 ) )
% 9.51/5.32                     => ! [N3: num] :
% 9.51/5.32                          ( ( Xa2
% 9.51/5.32                            = ( bit0 @ N3 ) )
% 9.51/5.32                         => ( Y
% 9.51/5.32                           != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
% 9.51/5.32                 => ( ! [M5: num] :
% 9.51/5.32                        ( ( X
% 9.51/5.32                          = ( bit0 @ M5 ) )
% 9.51/5.32                       => ! [N3: num] :
% 9.51/5.32                            ( ( Xa2
% 9.51/5.32                              = ( bit1 @ N3 ) )
% 9.51/5.32                           => ( Y
% 9.51/5.32                             != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
% 9.51/5.32                   => ( ( ? [M5: num] :
% 9.51/5.32                            ( X
% 9.51/5.32                            = ( bit1 @ M5 ) )
% 9.51/5.32                       => ( ( Xa2 = one )
% 9.51/5.32                         => ( Y
% 9.51/5.32                           != ( some_num @ one ) ) ) )
% 9.51/5.32                     => ( ! [M5: num] :
% 9.51/5.32                            ( ( X
% 9.51/5.32                              = ( bit1 @ M5 ) )
% 9.51/5.32                           => ! [N3: num] :
% 9.51/5.32                                ( ( Xa2
% 9.51/5.32                                  = ( bit0 @ N3 ) )
% 9.51/5.32                               => ( Y
% 9.51/5.32                                 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
% 9.51/5.32                       => ~ ! [M5: num] :
% 9.51/5.32                              ( ( X
% 9.51/5.32                                = ( bit1 @ M5 ) )
% 9.51/5.32                             => ! [N3: num] :
% 9.51/5.32                                  ( ( Xa2
% 9.51/5.32                                    = ( bit1 @ N3 ) )
% 9.51/5.32                                 => ( Y
% 9.51/5.32                                   != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 9.51/5.32                                      @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 9.51/5.32                                      @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9116_and__num_Osimps_I1_J,axiom,
% 9.51/5.32      ( ( bit_un7362597486090784418nd_num @ one @ one )
% 9.51/5.32      = ( some_num @ one ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9117_and__num_Osimps_I7_J,axiom,
% 9.51/5.32      ! [M: num] :
% 9.51/5.32        ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
% 9.51/5.32        = ( some_num @ one ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9118_and__num_Osimps_I3_J,axiom,
% 9.51/5.32      ! [N: num] :
% 9.51/5.32        ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
% 9.51/5.32        = ( some_num @ one ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9119_and__num_Osimps_I4_J,axiom,
% 9.51/5.32      ! [M: num] :
% 9.51/5.32        ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
% 9.51/5.32        = none_num ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9120_and__num_Osimps_I2_J,axiom,
% 9.51/5.32      ! [N: num] :
% 9.51/5.32        ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
% 9.51/5.32        = none_num ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9121_and__num_Osimps_I9_J,axiom,
% 9.51/5.32      ! [M: num,N: num] :
% 9.51/5.32        ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.51/5.32        = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 9.51/5.32          @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 9.51/5.32          @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9122_num__of__nat_Osimps_I2_J,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.32         => ( ( num_of_nat @ ( suc @ N ) )
% 9.51/5.32            = ( inc @ ( num_of_nat @ N ) ) ) )
% 9.51/5.32        & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.32         => ( ( num_of_nat @ ( suc @ N ) )
% 9.51/5.32            = one ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9123_Rats__eq__int__div__nat,axiom,
% 9.51/5.32      ( field_5140801741446780682s_real
% 9.51/5.32      = ( collect_real
% 9.51/5.32        @ ^ [Uu3: real] :
% 9.51/5.32          ? [I4: int,N2: nat] :
% 9.51/5.32            ( ( Uu3
% 9.51/5.32              = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 9.51/5.32            & ( N2 != zero_zero_nat ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9124_Rats__no__bot__less,axiom,
% 9.51/5.32      ! [X: real] :
% 9.51/5.32      ? [X4: real] :
% 9.51/5.32        ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 9.51/5.32        & ( ord_less_real @ X4 @ X ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9125_Rats__dense__in__real,axiom,
% 9.51/5.32      ! [X: real,Y: real] :
% 9.51/5.32        ( ( ord_less_real @ X @ Y )
% 9.51/5.32       => ? [X4: real] :
% 9.51/5.32            ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 9.51/5.32            & ( ord_less_real @ X @ X4 )
% 9.51/5.32            & ( ord_less_real @ X4 @ Y ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9126_pos__deriv__imp__strict__mono,axiom,
% 9.51/5.32      ! [F: real > real,F6: real > real] :
% 9.51/5.32        ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F6 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.51/5.32       => ( ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( F6 @ X4 ) )
% 9.51/5.32         => ( order_7092887310737990675l_real @ F ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9127_positive__rat,axiom,
% 9.51/5.32      ! [A: int,B: int] :
% 9.51/5.32        ( ( positive @ ( fract @ A @ B ) )
% 9.51/5.32        = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9128_strict__mono__imp__increasing,axiom,
% 9.51/5.32      ! [F: nat > nat,N: nat] :
% 9.51/5.32        ( ( order_5726023648592871131at_nat @ F )
% 9.51/5.32       => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9129_infinite__enumerate,axiom,
% 9.51/5.32      ! [S2: set_nat] :
% 9.51/5.32        ( ~ ( finite_finite_nat @ S2 )
% 9.51/5.32       => ? [R3: nat > nat] :
% 9.51/5.32            ( ( order_5726023648592871131at_nat @ R3 )
% 9.51/5.32            & ! [N5: nat] : ( member_nat @ ( R3 @ N5 ) @ S2 ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9130_less__rat__def,axiom,
% 9.51/5.32      ( ord_less_rat
% 9.51/5.32      = ( ^ [X3: rat,Y2: rat] : ( positive @ ( minus_minus_rat @ Y2 @ X3 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9131_Rat_Opositive_Orep__eq,axiom,
% 9.51/5.32      ( positive
% 9.51/5.32      = ( ^ [X3: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X3 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X3 ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9132_sorted__list__of__set__greaterThanAtMost,axiom,
% 9.51/5.32      ! [I: nat,J: nat] :
% 9.51/5.32        ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
% 9.51/5.32       => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
% 9.51/5.32          = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9133_sorted__list__of__set__greaterThanLessThan,axiom,
% 9.51/5.32      ! [I: nat,J: nat] :
% 9.51/5.32        ( ( ord_less_nat @ ( suc @ I ) @ J )
% 9.51/5.32       => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
% 9.51/5.32          = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9134_sorted__list__of__set__lessThan__Suc,axiom,
% 9.51/5.32      ! [K: nat] :
% 9.51/5.32        ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
% 9.51/5.32        = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9135_sorted__list__of__set__atMost__Suc,axiom,
% 9.51/5.32      ! [K: nat] :
% 9.51/5.32        ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
% 9.51/5.32        = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9136_upto__aux__rec,axiom,
% 9.51/5.32      ( upto_aux
% 9.51/5.32      = ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9137_upto_Opelims,axiom,
% 9.51/5.32      ! [X: int,Xa2: int,Y: list_int] :
% 9.51/5.32        ( ( ( upto @ X @ Xa2 )
% 9.51/5.32          = Y )
% 9.51/5.32       => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
% 9.51/5.32         => ~ ( ( ( ( ord_less_eq_int @ X @ Xa2 )
% 9.51/5.32                 => ( Y
% 9.51/5.32                    = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
% 9.51/5.32                & ( ~ ( ord_less_eq_int @ X @ Xa2 )
% 9.51/5.32                 => ( Y = nil_int ) ) )
% 9.51/5.32             => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9138_upto_Opsimps,axiom,
% 9.51/5.32      ! [I: int,J: int] :
% 9.51/5.32        ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
% 9.51/5.32       => ( ( ( ord_less_eq_int @ I @ J )
% 9.51/5.32           => ( ( upto @ I @ J )
% 9.51/5.32              = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
% 9.51/5.32          & ( ~ ( ord_less_eq_int @ I @ J )
% 9.51/5.32           => ( ( upto @ I @ J )
% 9.51/5.32              = nil_int ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9139_upto__empty,axiom,
% 9.51/5.32      ! [J: int,I: int] :
% 9.51/5.32        ( ( ord_less_int @ J @ I )
% 9.51/5.32       => ( ( upto @ I @ J )
% 9.51/5.32          = nil_int ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9140_upto__Nil2,axiom,
% 9.51/5.32      ! [I: int,J: int] :
% 9.51/5.32        ( ( nil_int
% 9.51/5.32          = ( upto @ I @ J ) )
% 9.51/5.32        = ( ord_less_int @ J @ I ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9141_upto__Nil,axiom,
% 9.51/5.32      ! [I: int,J: int] :
% 9.51/5.32        ( ( ( upto @ I @ J )
% 9.51/5.32          = nil_int )
% 9.51/5.32        = ( ord_less_int @ J @ I ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9142_upto__single,axiom,
% 9.51/5.32      ! [I: int] :
% 9.51/5.32        ( ( upto @ I @ I )
% 9.51/5.32        = ( cons_int @ I @ nil_int ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9143_nth__upto,axiom,
% 9.51/5.32      ! [I: int,K: nat,J: int] :
% 9.51/5.32        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 9.51/5.32       => ( ( nth_int @ ( upto @ I @ J ) @ K )
% 9.51/5.32          = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9144_length__upto,axiom,
% 9.51/5.32      ! [I: int,J: int] :
% 9.51/5.32        ( ( size_size_list_int @ ( upto @ I @ J ) )
% 9.51/5.32        = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9145_upto__rec__numeral_I1_J,axiom,
% 9.51/5.32      ! [M: num,N: num] :
% 9.51/5.32        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.51/5.32         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.51/5.32            = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 9.51/5.32        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.51/5.32         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.51/5.32            = nil_int ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9146_upto__rec__numeral_I2_J,axiom,
% 9.51/5.32      ! [M: num,N: num] :
% 9.51/5.32        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.51/5.32         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.51/5.32            = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 9.51/5.32        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.51/5.32         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.51/5.32            = nil_int ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9147_upto__rec__numeral_I3_J,axiom,
% 9.51/5.32      ! [M: num,N: num] :
% 9.51/5.32        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.51/5.32         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.51/5.32            = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 9.51/5.32        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.51/5.32         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.51/5.32            = nil_int ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9148_upto__rec__numeral_I4_J,axiom,
% 9.51/5.32      ! [M: num,N: num] :
% 9.51/5.32        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.51/5.32         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.51/5.32            = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 9.51/5.32        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.51/5.32         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.51/5.32            = nil_int ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9149_upto__code,axiom,
% 9.51/5.32      ( upto
% 9.51/5.32      = ( ^ [I4: int,J3: int] : ( upto_aux @ I4 @ J3 @ nil_int ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9150_upto__aux__def,axiom,
% 9.51/5.32      ( upto_aux
% 9.51/5.32      = ( ^ [I4: int,J3: int] : ( append_int @ ( upto @ I4 @ J3 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9151_distinct__upto,axiom,
% 9.51/5.32      ! [I: int,J: int] : ( distinct_int @ ( upto @ I @ J ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9152_atLeastAtMost__upto,axiom,
% 9.51/5.32      ( set_or1266510415728281911st_int
% 9.51/5.32      = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ J3 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9153_upto__split2,axiom,
% 9.51/5.32      ! [I: int,J: int,K: int] :
% 9.51/5.32        ( ( ord_less_eq_int @ I @ J )
% 9.51/5.32       => ( ( ord_less_eq_int @ J @ K )
% 9.51/5.32         => ( ( upto @ I @ K )
% 9.51/5.32            = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9154_upto__split1,axiom,
% 9.51/5.32      ! [I: int,J: int,K: int] :
% 9.51/5.32        ( ( ord_less_eq_int @ I @ J )
% 9.51/5.32       => ( ( ord_less_eq_int @ J @ K )
% 9.51/5.32         => ( ( upto @ I @ K )
% 9.51/5.32            = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9155_atLeastLessThan__upto,axiom,
% 9.51/5.32      ( set_or4662586982721622107an_int
% 9.51/5.32      = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9156_greaterThanAtMost__upto,axiom,
% 9.51/5.32      ( set_or6656581121297822940st_int
% 9.51/5.32      = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9157_upto_Oelims,axiom,
% 9.51/5.32      ! [X: int,Xa2: int,Y: list_int] :
% 9.51/5.32        ( ( ( upto @ X @ Xa2 )
% 9.51/5.32          = Y )
% 9.51/5.32       => ( ( ( ord_less_eq_int @ X @ Xa2 )
% 9.51/5.32           => ( Y
% 9.51/5.32              = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
% 9.51/5.32          & ( ~ ( ord_less_eq_int @ X @ Xa2 )
% 9.51/5.32           => ( Y = nil_int ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9158_upto_Osimps,axiom,
% 9.51/5.32      ( upto
% 9.51/5.32      = ( ^ [I4: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J3 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9159_upto__rec1,axiom,
% 9.51/5.32      ! [I: int,J: int] :
% 9.51/5.32        ( ( ord_less_eq_int @ I @ J )
% 9.51/5.32       => ( ( upto @ I @ J )
% 9.51/5.32          = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9160_upto__rec2,axiom,
% 9.51/5.32      ! [I: int,J: int] :
% 9.51/5.32        ( ( ord_less_eq_int @ I @ J )
% 9.51/5.32       => ( ( upto @ I @ J )
% 9.51/5.32          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9161_greaterThanLessThan__upto,axiom,
% 9.51/5.32      ( set_or5832277885323065728an_int
% 9.51/5.32      = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9162_upto__split3,axiom,
% 9.51/5.32      ! [I: int,J: int,K: int] :
% 9.51/5.32        ( ( ord_less_eq_int @ I @ J )
% 9.51/5.32       => ( ( ord_less_eq_int @ J @ K )
% 9.51/5.32         => ( ( upto @ I @ K )
% 9.51/5.32            = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9163_remdups__upt,axiom,
% 9.51/5.32      ! [M: nat,N: nat] :
% 9.51/5.32        ( ( remdups_nat @ ( upt @ M @ N ) )
% 9.51/5.32        = ( upt @ M @ N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9164_hd__upt,axiom,
% 9.51/5.32      ! [I: nat,J: nat] :
% 9.51/5.32        ( ( ord_less_nat @ I @ J )
% 9.51/5.32       => ( ( hd_nat @ ( upt @ I @ J ) )
% 9.51/5.32          = I ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9165_drop__upt,axiom,
% 9.51/5.32      ! [M: nat,I: nat,J: nat] :
% 9.51/5.32        ( ( drop_nat @ M @ ( upt @ I @ J ) )
% 9.51/5.32        = ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9166_length__upt,axiom,
% 9.51/5.32      ! [I: nat,J: nat] :
% 9.51/5.32        ( ( size_size_list_nat @ ( upt @ I @ J ) )
% 9.51/5.32        = ( minus_minus_nat @ J @ I ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9167_take__upt,axiom,
% 9.51/5.32      ! [I: nat,M: nat,N: nat] :
% 9.51/5.32        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
% 9.51/5.32       => ( ( take_nat @ M @ ( upt @ I @ N ) )
% 9.51/5.32          = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9168_upt__conv__Nil,axiom,
% 9.51/5.32      ! [J: nat,I: nat] :
% 9.51/5.32        ( ( ord_less_eq_nat @ J @ I )
% 9.51/5.32       => ( ( upt @ I @ J )
% 9.51/5.32          = nil_nat ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9169_sorted__list__of__set__range,axiom,
% 9.51/5.32      ! [M: nat,N: nat] :
% 9.51/5.32        ( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 9.51/5.32        = ( upt @ M @ N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9170_upt__eq__Nil__conv,axiom,
% 9.51/5.32      ! [I: nat,J: nat] :
% 9.51/5.32        ( ( ( upt @ I @ J )
% 9.51/5.32          = nil_nat )
% 9.51/5.32        = ( ( J = zero_zero_nat )
% 9.51/5.32          | ( ord_less_eq_nat @ J @ I ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9171_nth__upt,axiom,
% 9.51/5.32      ! [I: nat,K: nat,J: nat] :
% 9.51/5.32        ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
% 9.51/5.32       => ( ( nth_nat @ ( upt @ I @ J ) @ K )
% 9.51/5.32          = ( plus_plus_nat @ I @ K ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9172_upt__rec__numeral,axiom,
% 9.51/5.32      ! [M: num,N: num] :
% 9.51/5.32        ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.51/5.32         => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.51/5.32            = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
% 9.51/5.32        & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.51/5.32         => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.51/5.32            = nil_nat ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9173_upt__0,axiom,
% 9.51/5.32      ! [I: nat] :
% 9.51/5.32        ( ( upt @ I @ zero_zero_nat )
% 9.51/5.32        = nil_nat ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9174_greaterThanAtMost__upt,axiom,
% 9.51/5.32      ( set_or6659071591806873216st_nat
% 9.51/5.32      = ( ^ [N2: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ ( suc @ M2 ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9175_atLeast__upt,axiom,
% 9.51/5.32      ( set_ord_lessThan_nat
% 9.51/5.32      = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9176_atLeastLessThan__upt,axiom,
% 9.51/5.32      ( set_or4665077453230672383an_nat
% 9.51/5.32      = ( ^ [I4: nat,J3: nat] : ( set_nat2 @ ( upt @ I4 @ J3 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9177_atLeastAtMost__upt,axiom,
% 9.51/5.32      ( set_or1269000886237332187st_nat
% 9.51/5.32      = ( ^ [N2: nat,M2: nat] : ( set_nat2 @ ( upt @ N2 @ ( suc @ M2 ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9178_map__add__upt,axiom,
% 9.51/5.32      ! [N: nat,M: nat] :
% 9.51/5.32        ( ( map_nat_nat
% 9.51/5.32          @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
% 9.51/5.32          @ ( upt @ zero_zero_nat @ M ) )
% 9.51/5.32        = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9179_distinct__upt,axiom,
% 9.51/5.32      ! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9180_map__Suc__upt,axiom,
% 9.51/5.32      ! [M: nat,N: nat] :
% 9.51/5.32        ( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
% 9.51/5.32        = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9181_greaterThanLessThan__upt,axiom,
% 9.51/5.32      ( set_or5834768355832116004an_nat
% 9.51/5.32      = ( ^ [N2: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ M2 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9182_upt__conv__Cons__Cons,axiom,
% 9.51/5.32      ! [M: nat,N: nat,Ns: list_nat,Q4: nat] :
% 9.51/5.32        ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
% 9.51/5.32          = ( upt @ M @ Q4 ) )
% 9.51/5.32        = ( ( cons_nat @ N @ Ns )
% 9.51/5.32          = ( upt @ ( suc @ M ) @ Q4 ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9183_atMost__upto,axiom,
% 9.51/5.32      ( set_ord_atMost_nat
% 9.51/5.32      = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9184_upt__conv__Cons,axiom,
% 9.51/5.32      ! [I: nat,J: nat] :
% 9.51/5.32        ( ( ord_less_nat @ I @ J )
% 9.51/5.32       => ( ( upt @ I @ J )
% 9.51/5.32          = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9185_map__decr__upt,axiom,
% 9.51/5.32      ! [M: nat,N: nat] :
% 9.51/5.32        ( ( map_nat_nat
% 9.51/5.32          @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 9.51/5.32          @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
% 9.51/5.32        = ( upt @ M @ N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9186_upt__add__eq__append,axiom,
% 9.51/5.32      ! [I: nat,J: nat,K: nat] :
% 9.51/5.32        ( ( ord_less_eq_nat @ I @ J )
% 9.51/5.32       => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
% 9.51/5.32          = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9187_upt__eq__Cons__conv,axiom,
% 9.51/5.32      ! [I: nat,J: nat,X: nat,Xs: list_nat] :
% 9.51/5.32        ( ( ( upt @ I @ J )
% 9.51/5.32          = ( cons_nat @ X @ Xs ) )
% 9.51/5.32        = ( ( ord_less_nat @ I @ J )
% 9.51/5.32          & ( I = X )
% 9.51/5.32          & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
% 9.51/5.32            = Xs ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9188_upt__rec,axiom,
% 9.51/5.32      ( upt
% 9.51/5.32      = ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9189_upt__Suc,axiom,
% 9.51/5.32      ! [I: nat,J: nat] :
% 9.51/5.32        ( ( ( ord_less_eq_nat @ I @ J )
% 9.51/5.32         => ( ( upt @ I @ ( suc @ J ) )
% 9.51/5.32            = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
% 9.51/5.32        & ( ~ ( ord_less_eq_nat @ I @ J )
% 9.51/5.32         => ( ( upt @ I @ ( suc @ J ) )
% 9.51/5.32            = nil_nat ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9190_upt__Suc__append,axiom,
% 9.51/5.32      ! [I: nat,J: nat] :
% 9.51/5.32        ( ( ord_less_eq_nat @ I @ J )
% 9.51/5.32       => ( ( upt @ I @ ( suc @ J ) )
% 9.51/5.32          = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9191_sum__list__upt,axiom,
% 9.51/5.32      ! [M: nat,N: nat] :
% 9.51/5.32        ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.32       => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
% 9.51/5.32          = ( groups3542108847815614940at_nat
% 9.51/5.32            @ ^ [X3: nat] : X3
% 9.51/5.32            @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9192_card__length__sum__list__rec,axiom,
% 9.51/5.32      ! [M: nat,N6: nat] :
% 9.51/5.32        ( ( ord_less_eq_nat @ one_one_nat @ M )
% 9.51/5.32       => ( ( finite_card_list_nat
% 9.51/5.32            @ ( collect_list_nat
% 9.51/5.32              @ ^ [L2: list_nat] :
% 9.51/5.32                  ( ( ( size_size_list_nat @ L2 )
% 9.51/5.32                    = M )
% 9.51/5.32                  & ( ( groups4561878855575611511st_nat @ L2 )
% 9.51/5.32                    = N6 ) ) ) )
% 9.51/5.32          = ( plus_plus_nat
% 9.51/5.32            @ ( finite_card_list_nat
% 9.51/5.32              @ ( collect_list_nat
% 9.51/5.32                @ ^ [L2: list_nat] :
% 9.51/5.32                    ( ( ( size_size_list_nat @ L2 )
% 9.51/5.32                      = ( minus_minus_nat @ M @ one_one_nat ) )
% 9.51/5.32                    & ( ( groups4561878855575611511st_nat @ L2 )
% 9.51/5.32                      = N6 ) ) ) )
% 9.51/5.32            @ ( finite_card_list_nat
% 9.51/5.32              @ ( collect_list_nat
% 9.51/5.32                @ ^ [L2: list_nat] :
% 9.51/5.32                    ( ( ( size_size_list_nat @ L2 )
% 9.51/5.32                      = M )
% 9.51/5.32                    & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
% 9.51/5.32                      = N6 ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9193_card__length__sum__list,axiom,
% 9.51/5.32      ! [M: nat,N6: nat] :
% 9.51/5.32        ( ( finite_card_list_nat
% 9.51/5.32          @ ( collect_list_nat
% 9.51/5.32            @ ^ [L2: list_nat] :
% 9.51/5.32                ( ( ( size_size_list_nat @ L2 )
% 9.51/5.32                  = M )
% 9.51/5.32                & ( ( groups4561878855575611511st_nat @ L2 )
% 9.51/5.32                  = N6 ) ) ) )
% 9.51/5.32        = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N6 @ M ) @ one_one_nat ) @ N6 ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9194_tl__upt,axiom,
% 9.51/5.32      ! [M: nat,N: nat] :
% 9.51/5.32        ( ( tl_nat @ ( upt @ M @ N ) )
% 9.51/5.32        = ( upt @ ( suc @ M ) @ N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9195_sorted__wrt__upt,axiom,
% 9.51/5.32      ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9196_sorted__upt,axiom,
% 9.51/5.32      ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9197_sorted__wrt__less__idx,axiom,
% 9.51/5.32      ! [Ns: list_nat,I: nat] :
% 9.51/5.32        ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
% 9.51/5.32       => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
% 9.51/5.32         => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9198_sorted__wrt__upto,axiom,
% 9.51/5.32      ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9199_sorted__upto,axiom,
% 9.51/5.32      ! [M: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9200_pairs__le__eq__Sigma,axiom,
% 9.51/5.32      ! [M: nat] :
% 9.51/5.32        ( ( collec3392354462482085612at_nat
% 9.51/5.32          @ ( produc6081775807080527818_nat_o
% 9.51/5.32            @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ M ) ) )
% 9.51/5.32        = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M )
% 9.51/5.32          @ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ R5 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9201_natLess__def,axiom,
% 9.51/5.32      ( bNF_Ca8459412986667044542atLess
% 9.51/5.32      = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9202_finite__vimage__Suc__iff,axiom,
% 9.51/5.32      ! [F3: set_nat] :
% 9.51/5.32        ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F3 ) )
% 9.51/5.32        = ( finite_finite_nat @ F3 ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9203_vimage__Suc__insert__0,axiom,
% 9.51/5.32      ! [A2: set_nat] :
% 9.51/5.32        ( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A2 ) )
% 9.51/5.32        = ( vimage_nat_nat @ suc @ A2 ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9204_Restr__natLeq,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
% 9.51/5.32          @ ( produc457027306803732586at_nat
% 9.51/5.32            @ ( collect_nat
% 9.51/5.32              @ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) )
% 9.51/5.32            @ ^ [Uu3: nat] :
% 9.51/5.32                ( collect_nat
% 9.51/5.32                @ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) ) ) )
% 9.51/5.32        = ( collec3392354462482085612at_nat
% 9.51/5.32          @ ( produc6081775807080527818_nat_o
% 9.51/5.32            @ ^ [X3: nat,Y2: nat] :
% 9.51/5.32                ( ( ord_less_nat @ X3 @ N )
% 9.51/5.32                & ( ord_less_nat @ Y2 @ N )
% 9.51/5.32                & ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9205_Arg__bounded,axiom,
% 9.51/5.32      ! [Z2: complex] :
% 9.51/5.32        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z2 ) )
% 9.51/5.32        & ( ord_less_eq_real @ ( arg @ Z2 ) @ pi ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9206_natLeq__def,axiom,
% 9.51/5.32      ( bNF_Ca8665028551170535155natLeq
% 9.51/5.32      = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9207_Restr__natLeq2,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
% 9.51/5.32          @ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
% 9.51/5.32            @ ^ [Uu3: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
% 9.51/5.32        = ( collec3392354462482085612at_nat
% 9.51/5.32          @ ( produc6081775807080527818_nat_o
% 9.51/5.32            @ ^ [X3: nat,Y2: nat] :
% 9.51/5.32                ( ( ord_less_nat @ X3 @ N )
% 9.51/5.32                & ( ord_less_nat @ Y2 @ N )
% 9.51/5.32                & ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9208_Arg__correct,axiom,
% 9.51/5.32      ! [Z2: complex] :
% 9.51/5.32        ( ( Z2 != zero_zero_complex )
% 9.51/5.32       => ( ( ( sgn_sgn_complex @ Z2 )
% 9.51/5.32            = ( cis @ ( arg @ Z2 ) ) )
% 9.51/5.32          & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z2 ) )
% 9.51/5.32          & ( ord_less_eq_real @ ( arg @ Z2 ) @ pi ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9209_natLeq__underS__less,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
% 9.51/5.32        = ( collect_nat
% 9.51/5.32          @ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9210_cis__Arg__unique,axiom,
% 9.51/5.32      ! [Z2: complex,X: real] :
% 9.51/5.32        ( ( ( sgn_sgn_complex @ Z2 )
% 9.51/5.32          = ( cis @ X ) )
% 9.51/5.32       => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
% 9.51/5.32         => ( ( ord_less_eq_real @ X @ pi )
% 9.51/5.32           => ( ( arg @ Z2 )
% 9.51/5.32              = X ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9211_bij__betw__roots__unity,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.32       => ( bij_betw_nat_complex
% 9.51/5.32          @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.51/5.32          @ ( set_ord_lessThan_nat @ N )
% 9.51/5.32          @ ( collect_complex
% 9.51/5.32            @ ^ [Z3: complex] :
% 9.51/5.32                ( ( power_power_complex @ Z3 @ N )
% 9.51/5.32                = one_one_complex ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9212_bij__betw__nth__root__unity,axiom,
% 9.51/5.32      ! [C: complex,N: nat] :
% 9.51/5.32        ( ( C != zero_zero_complex )
% 9.51/5.32       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.32         => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 9.51/5.32            @ ( collect_complex
% 9.51/5.32              @ ^ [Z3: complex] :
% 9.51/5.32                  ( ( power_power_complex @ Z3 @ N )
% 9.51/5.32                  = one_one_complex ) )
% 9.51/5.32            @ ( collect_complex
% 9.51/5.32              @ ^ [Z3: complex] :
% 9.51/5.32                  ( ( power_power_complex @ Z3 @ N )
% 9.51/5.32                  = C ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9213_Arg__def,axiom,
% 9.51/5.32      ( arg
% 9.51/5.32      = ( ^ [Z3: complex] :
% 9.51/5.32            ( if_real @ ( Z3 = zero_zero_complex ) @ zero_zero_real
% 9.51/5.32            @ ( fChoice_real
% 9.51/5.32              @ ^ [A4: real] :
% 9.51/5.32                  ( ( ( sgn_sgn_complex @ Z3 )
% 9.51/5.32                    = ( cis @ A4 ) )
% 9.51/5.32                  & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A4 )
% 9.51/5.32                  & ( ord_less_eq_real @ A4 @ pi ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9214_bij__betw__Suc,axiom,
% 9.51/5.32      ! [M4: set_nat,N6: set_nat] :
% 9.51/5.32        ( ( bij_betw_nat_nat @ suc @ M4 @ N6 )
% 9.51/5.32        = ( ( image_nat_nat @ suc @ M4 )
% 9.51/5.32          = N6 ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9215_quotient__of__def,axiom,
% 9.51/5.32      ( quotient_of
% 9.51/5.32      = ( ^ [X3: rat] :
% 9.51/5.32            ( the_Pr4378521158711661632nt_int
% 9.51/5.32            @ ^ [Pair: product_prod_int_int] :
% 9.51/5.32                ( ( X3
% 9.51/5.32                  = ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
% 9.51/5.32                & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
% 9.51/5.32                & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9216_normalize__stable,axiom,
% 9.51/5.32      ! [Q4: int,P4: int] :
% 9.51/5.32        ( ( ord_less_int @ zero_zero_int @ Q4 )
% 9.51/5.32       => ( ( algebr932160517623751201me_int @ P4 @ Q4 )
% 9.51/5.32         => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q4 ) )
% 9.51/5.32            = ( product_Pair_int_int @ P4 @ Q4 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9217_normalize__coprime,axiom,
% 9.51/5.32      ! [R2: product_prod_int_int,P4: int,Q4: int] :
% 9.51/5.32        ( ( ( normalize @ R2 )
% 9.51/5.32          = ( product_Pair_int_int @ P4 @ Q4 ) )
% 9.51/5.32       => ( algebr932160517623751201me_int @ P4 @ Q4 ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9218_quotient__of__coprime,axiom,
% 9.51/5.32      ! [R2: rat,P4: int,Q4: int] :
% 9.51/5.32        ( ( ( quotient_of @ R2 )
% 9.51/5.32          = ( product_Pair_int_int @ P4 @ Q4 ) )
% 9.51/5.32       => ( algebr932160517623751201me_int @ P4 @ Q4 ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9219_Rat__induct,axiom,
% 9.51/5.32      ! [P: rat > $o,Q4: rat] :
% 9.51/5.32        ( ! [A3: int,B3: int] :
% 9.51/5.32            ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.51/5.32           => ( ( algebr932160517623751201me_int @ A3 @ B3 )
% 9.51/5.32             => ( P @ ( fract @ A3 @ B3 ) ) ) )
% 9.51/5.32       => ( P @ Q4 ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9220_Rat__cases,axiom,
% 9.51/5.32      ! [Q4: rat] :
% 9.51/5.32        ~ ! [A3: int,B3: int] :
% 9.51/5.32            ( ( Q4
% 9.51/5.32              = ( fract @ A3 @ B3 ) )
% 9.51/5.32           => ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.51/5.32             => ~ ( algebr932160517623751201me_int @ A3 @ B3 ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9221_Rat__cases__nonzero,axiom,
% 9.51/5.32      ! [Q4: rat] :
% 9.51/5.32        ( ! [A3: int,B3: int] :
% 9.51/5.32            ( ( Q4
% 9.51/5.32              = ( fract @ A3 @ B3 ) )
% 9.51/5.32           => ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.51/5.32             => ( ( A3 != zero_zero_int )
% 9.51/5.32               => ~ ( algebr932160517623751201me_int @ A3 @ B3 ) ) ) )
% 9.51/5.32       => ( Q4 = zero_zero_rat ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9222_quotient__of__unique,axiom,
% 9.51/5.32      ! [R2: rat] :
% 9.51/5.32      ? [X4: product_prod_int_int] :
% 9.51/5.32        ( ( R2
% 9.51/5.32          = ( fract @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) )
% 9.51/5.32        & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X4 ) )
% 9.51/5.32        & ( algebr932160517623751201me_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) )
% 9.51/5.32        & ! [Y4: product_prod_int_int] :
% 9.51/5.32            ( ( ( R2
% 9.51/5.32                = ( fract @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ Y4 ) ) )
% 9.51/5.32              & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y4 ) )
% 9.51/5.32              & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ Y4 ) ) )
% 9.51/5.32           => ( Y4 = X4 ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9223_coprime__Suc__0__right,axiom,
% 9.51/5.32      ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9224_coprime__Suc__0__left,axiom,
% 9.51/5.32      ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9225_coprime__diff__one__right__nat,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.32       => ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9226_coprime__diff__one__left__nat,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.51/5.32       => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9227_Gcd__nat__set__eq__fold,axiom,
% 9.51/5.32      ! [Xs: list_nat] :
% 9.51/5.32        ( ( gcd_Gcd_nat @ ( set_nat2 @ Xs ) )
% 9.51/5.32        = ( fold_nat_nat @ gcd_gcd_nat @ Xs @ zero_zero_nat ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9228_Rats__abs__nat__div__natE,axiom,
% 9.51/5.32      ! [X: real] :
% 9.51/5.32        ( ( member_real @ X @ field_5140801741446780682s_real )
% 9.51/5.32       => ~ ! [M5: nat,N3: nat] :
% 9.51/5.32              ( ( N3 != zero_zero_nat )
% 9.51/5.32             => ( ( ( abs_abs_real @ X )
% 9.51/5.32                  = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M5 ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
% 9.51/5.32               => ~ ( algebr934650988132801477me_nat @ M5 @ N3 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9229_sort__upt,axiom,
% 9.51/5.32      ! [M: nat,N: nat] :
% 9.51/5.32        ( ( linord738340561235409698at_nat
% 9.51/5.32          @ ^ [X3: nat] : X3
% 9.51/5.32          @ ( upt @ M @ N ) )
% 9.51/5.32        = ( upt @ M @ N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9230_sort__upto,axiom,
% 9.51/5.32      ! [I: int,J: int] :
% 9.51/5.32        ( ( linord1735203802627413978nt_int
% 9.51/5.32          @ ^ [X3: int] : X3
% 9.51/5.32          @ ( upto @ I @ J ) )
% 9.51/5.32        = ( upto @ I @ J ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9231_vanishes__mult__bounded,axiom,
% 9.51/5.32      ! [X8: nat > rat,Y7: nat > rat] :
% 9.51/5.32        ( ? [A8: rat] :
% 9.51/5.32            ( ( ord_less_rat @ zero_zero_rat @ A8 )
% 9.51/5.32            & ! [N3: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N3 ) ) @ A8 ) )
% 9.51/5.32       => ( ( vanishes @ Y7 )
% 9.51/5.32         => ( vanishes
% 9.51/5.32            @ ^ [N2: nat] : ( times_times_rat @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9232_vanishes__def,axiom,
% 9.51/5.32      ( vanishes
% 9.51/5.32      = ( ^ [X6: nat > rat] :
% 9.51/5.32          ! [R5: rat] :
% 9.51/5.32            ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.51/5.32           => ? [K3: nat] :
% 9.51/5.32              ! [N2: nat] :
% 9.51/5.32                ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.51/5.32               => ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N2 ) ) @ R5 ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9233_vanishesI,axiom,
% 9.51/5.32      ! [X8: nat > rat] :
% 9.51/5.32        ( ! [R3: rat] :
% 9.51/5.32            ( ( ord_less_rat @ zero_zero_rat @ R3 )
% 9.51/5.32           => ? [K7: nat] :
% 9.51/5.32              ! [N3: nat] :
% 9.51/5.32                ( ( ord_less_eq_nat @ K7 @ N3 )
% 9.51/5.32               => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N3 ) ) @ R3 ) ) )
% 9.51/5.32       => ( vanishes @ X8 ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9234_vanishesD,axiom,
% 9.51/5.32      ! [X8: nat > rat,R2: rat] :
% 9.51/5.32        ( ( vanishes @ X8 )
% 9.51/5.32       => ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 9.51/5.32         => ? [K2: nat] :
% 9.51/5.32            ! [N5: nat] :
% 9.51/5.32              ( ( ord_less_eq_nat @ K2 @ N5 )
% 9.51/5.32             => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N5 ) ) @ R2 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9235_and__not__num_Opelims,axiom,
% 9.51/5.32      ! [X: num,Xa2: num,Y: option_num] :
% 9.51/5.32        ( ( ( bit_and_not_num @ X @ Xa2 )
% 9.51/5.32          = Y )
% 9.51/5.32       => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X @ Xa2 ) )
% 9.51/5.32         => ( ( ( X = one )
% 9.51/5.32             => ( ( Xa2 = one )
% 9.51/5.32               => ( ( Y = none_num )
% 9.51/5.32                 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 9.51/5.32           => ( ( ( X = one )
% 9.51/5.32               => ! [N3: num] :
% 9.51/5.32                    ( ( Xa2
% 9.51/5.32                      = ( bit0 @ N3 ) )
% 9.51/5.32                   => ( ( Y
% 9.51/5.32                        = ( some_num @ one ) )
% 9.51/5.32                     => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
% 9.51/5.32             => ( ( ( X = one )
% 9.51/5.32                 => ! [N3: num] :
% 9.51/5.32                      ( ( Xa2
% 9.51/5.32                        = ( bit1 @ N3 ) )
% 9.51/5.32                     => ( ( Y = none_num )
% 9.51/5.32                       => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
% 9.51/5.32               => ( ! [M5: num] :
% 9.51/5.32                      ( ( X
% 9.51/5.32                        = ( bit0 @ M5 ) )
% 9.51/5.32                     => ( ( Xa2 = one )
% 9.51/5.32                       => ( ( Y
% 9.51/5.32                            = ( some_num @ ( bit0 @ M5 ) ) )
% 9.51/5.32                         => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
% 9.51/5.32                 => ( ! [M5: num] :
% 9.51/5.32                        ( ( X
% 9.51/5.32                          = ( bit0 @ M5 ) )
% 9.51/5.32                       => ! [N3: num] :
% 9.51/5.32                            ( ( Xa2
% 9.51/5.32                              = ( bit0 @ N3 ) )
% 9.51/5.32                           => ( ( Y
% 9.51/5.32                                = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
% 9.51/5.32                             => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 9.51/5.32                   => ( ! [M5: num] :
% 9.51/5.32                          ( ( X
% 9.51/5.32                            = ( bit0 @ M5 ) )
% 9.51/5.32                         => ! [N3: num] :
% 9.51/5.32                              ( ( Xa2
% 9.51/5.32                                = ( bit1 @ N3 ) )
% 9.51/5.32                             => ( ( Y
% 9.51/5.32                                  = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
% 9.51/5.32                               => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
% 9.51/5.32                     => ( ! [M5: num] :
% 9.51/5.32                            ( ( X
% 9.51/5.32                              = ( bit1 @ M5 ) )
% 9.51/5.32                           => ( ( Xa2 = one )
% 9.51/5.32                             => ( ( Y
% 9.51/5.32                                  = ( some_num @ ( bit0 @ M5 ) ) )
% 9.51/5.32                               => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
% 9.51/5.32                       => ( ! [M5: num] :
% 9.51/5.32                              ( ( X
% 9.51/5.32                                = ( bit1 @ M5 ) )
% 9.51/5.32                             => ! [N3: num] :
% 9.51/5.32                                  ( ( Xa2
% 9.51/5.32                                    = ( bit0 @ N3 ) )
% 9.51/5.32                                 => ( ( Y
% 9.51/5.32                                      = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 9.51/5.32                                        @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 9.51/5.32                                        @ ( bit_and_not_num @ M5 @ N3 ) ) )
% 9.51/5.32                                   => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 9.51/5.32                         => ~ ! [M5: num] :
% 9.51/5.32                                ( ( X
% 9.51/5.32                                  = ( bit1 @ M5 ) )
% 9.51/5.32                               => ! [N3: num] :
% 9.51/5.32                                    ( ( Xa2
% 9.51/5.32                                      = ( bit1 @ N3 ) )
% 9.51/5.32                                   => ( ( Y
% 9.51/5.32                                        = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
% 9.51/5.32                                     => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9236_and__num_Opelims,axiom,
% 9.51/5.32      ! [X: num,Xa2: num,Y: option_num] :
% 9.51/5.32        ( ( ( bit_un7362597486090784418nd_num @ X @ Xa2 )
% 9.51/5.32          = Y )
% 9.51/5.32       => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X @ Xa2 ) )
% 9.51/5.32         => ( ( ( X = one )
% 9.51/5.32             => ( ( Xa2 = one )
% 9.51/5.32               => ( ( Y
% 9.51/5.32                    = ( some_num @ one ) )
% 9.51/5.32                 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 9.51/5.32           => ( ( ( X = one )
% 9.51/5.32               => ! [N3: num] :
% 9.51/5.32                    ( ( Xa2
% 9.51/5.32                      = ( bit0 @ N3 ) )
% 9.51/5.32                   => ( ( Y = none_num )
% 9.51/5.32                     => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
% 9.51/5.32             => ( ( ( X = one )
% 9.51/5.32                 => ! [N3: num] :
% 9.51/5.32                      ( ( Xa2
% 9.51/5.32                        = ( bit1 @ N3 ) )
% 9.51/5.32                     => ( ( Y
% 9.51/5.32                          = ( some_num @ one ) )
% 9.51/5.32                       => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
% 9.51/5.32               => ( ! [M5: num] :
% 9.51/5.32                      ( ( X
% 9.51/5.32                        = ( bit0 @ M5 ) )
% 9.51/5.32                     => ( ( Xa2 = one )
% 9.51/5.32                       => ( ( Y = none_num )
% 9.51/5.32                         => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
% 9.51/5.32                 => ( ! [M5: num] :
% 9.51/5.32                        ( ( X
% 9.51/5.32                          = ( bit0 @ M5 ) )
% 9.51/5.32                       => ! [N3: num] :
% 9.51/5.32                            ( ( Xa2
% 9.51/5.32                              = ( bit0 @ N3 ) )
% 9.51/5.32                           => ( ( Y
% 9.51/5.32                                = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
% 9.51/5.32                             => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 9.51/5.32                   => ( ! [M5: num] :
% 9.51/5.32                          ( ( X
% 9.51/5.32                            = ( bit0 @ M5 ) )
% 9.51/5.32                         => ! [N3: num] :
% 9.51/5.32                              ( ( Xa2
% 9.51/5.32                                = ( bit1 @ N3 ) )
% 9.51/5.32                             => ( ( Y
% 9.51/5.32                                  = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
% 9.51/5.32                               => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
% 9.51/5.32                     => ( ! [M5: num] :
% 9.51/5.32                            ( ( X
% 9.51/5.32                              = ( bit1 @ M5 ) )
% 9.51/5.32                           => ( ( Xa2 = one )
% 9.51/5.32                             => ( ( Y
% 9.51/5.32                                  = ( some_num @ one ) )
% 9.51/5.32                               => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
% 9.51/5.32                       => ( ! [M5: num] :
% 9.51/5.32                              ( ( X
% 9.51/5.32                                = ( bit1 @ M5 ) )
% 9.51/5.32                             => ! [N3: num] :
% 9.51/5.32                                  ( ( Xa2
% 9.51/5.32                                    = ( bit0 @ N3 ) )
% 9.51/5.32                                 => ( ( Y
% 9.51/5.32                                      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
% 9.51/5.32                                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 9.51/5.32                         => ~ ! [M5: num] :
% 9.51/5.32                                ( ( X
% 9.51/5.32                                  = ( bit1 @ M5 ) )
% 9.51/5.32                               => ! [N3: num] :
% 9.51/5.32                                    ( ( Xa2
% 9.51/5.32                                      = ( bit1 @ N3 ) )
% 9.51/5.32                                   => ( ( Y
% 9.51/5.32                                        = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 9.51/5.32                                          @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 9.51/5.32                                          @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
% 9.51/5.32                                     => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9237_xor__num_Opelims,axiom,
% 9.51/5.32      ! [X: num,Xa2: num,Y: option_num] :
% 9.51/5.32        ( ( ( bit_un2480387367778600638or_num @ X @ Xa2 )
% 9.51/5.32          = Y )
% 9.51/5.32       => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X @ Xa2 ) )
% 9.51/5.32         => ( ( ( X = one )
% 9.51/5.32             => ( ( Xa2 = one )
% 9.51/5.32               => ( ( Y = none_num )
% 9.51/5.32                 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 9.51/5.32           => ( ( ( X = one )
% 9.51/5.32               => ! [N3: num] :
% 9.51/5.32                    ( ( Xa2
% 9.51/5.32                      = ( bit0 @ N3 ) )
% 9.51/5.32                   => ( ( Y
% 9.51/5.32                        = ( some_num @ ( bit1 @ N3 ) ) )
% 9.51/5.32                     => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
% 9.51/5.32             => ( ( ( X = one )
% 9.51/5.32                 => ! [N3: num] :
% 9.51/5.32                      ( ( Xa2
% 9.51/5.32                        = ( bit1 @ N3 ) )
% 9.51/5.32                     => ( ( Y
% 9.51/5.32                          = ( some_num @ ( bit0 @ N3 ) ) )
% 9.51/5.32                       => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
% 9.51/5.32               => ( ! [M5: num] :
% 9.51/5.32                      ( ( X
% 9.51/5.32                        = ( bit0 @ M5 ) )
% 9.51/5.32                     => ( ( Xa2 = one )
% 9.51/5.32                       => ( ( Y
% 9.51/5.32                            = ( some_num @ ( bit1 @ M5 ) ) )
% 9.51/5.32                         => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
% 9.51/5.32                 => ( ! [M5: num] :
% 9.51/5.32                        ( ( X
% 9.51/5.32                          = ( bit0 @ M5 ) )
% 9.51/5.32                       => ! [N3: num] :
% 9.51/5.32                            ( ( Xa2
% 9.51/5.32                              = ( bit0 @ N3 ) )
% 9.51/5.32                           => ( ( Y
% 9.51/5.32                                = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) )
% 9.51/5.32                             => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 9.51/5.32                   => ( ! [M5: num] :
% 9.51/5.32                          ( ( X
% 9.51/5.32                            = ( bit0 @ M5 ) )
% 9.51/5.32                         => ! [N3: num] :
% 9.51/5.32                              ( ( Xa2
% 9.51/5.32                                = ( bit1 @ N3 ) )
% 9.51/5.32                             => ( ( Y
% 9.51/5.32                                  = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) )
% 9.51/5.32                               => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
% 9.51/5.32                     => ( ! [M5: num] :
% 9.51/5.32                            ( ( X
% 9.51/5.32                              = ( bit1 @ M5 ) )
% 9.51/5.32                           => ( ( Xa2 = one )
% 9.51/5.32                             => ( ( Y
% 9.51/5.32                                  = ( some_num @ ( bit0 @ M5 ) ) )
% 9.51/5.32                               => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
% 9.51/5.32                       => ( ! [M5: num] :
% 9.51/5.32                              ( ( X
% 9.51/5.32                                = ( bit1 @ M5 ) )
% 9.51/5.32                             => ! [N3: num] :
% 9.51/5.32                                  ( ( Xa2
% 9.51/5.32                                    = ( bit0 @ N3 ) )
% 9.51/5.32                                 => ( ( Y
% 9.51/5.32                                      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) )
% 9.51/5.32                                   => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 9.51/5.32                         => ~ ! [M5: num] :
% 9.51/5.32                                ( ( X
% 9.51/5.32                                  = ( bit1 @ M5 ) )
% 9.51/5.32                               => ! [N3: num] :
% 9.51/5.32                                    ( ( Xa2
% 9.51/5.32                                      = ( bit1 @ N3 ) )
% 9.51/5.32                                   => ( ( Y
% 9.51/5.32                                        = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) )
% 9.51/5.32                                     => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9238_Field__natLeq__on,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( ( field_nat
% 9.51/5.32          @ ( collec3392354462482085612at_nat
% 9.51/5.32            @ ( produc6081775807080527818_nat_o
% 9.51/5.32              @ ^ [X3: nat,Y2: nat] :
% 9.51/5.32                  ( ( ord_less_nat @ X3 @ N )
% 9.51/5.32                  & ( ord_less_nat @ Y2 @ N )
% 9.51/5.32                  & ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) )
% 9.51/5.32        = ( collect_nat
% 9.51/5.32          @ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9239_wf__less,axiom,
% 9.51/5.32      wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9240_cauchyD,axiom,
% 9.51/5.32      ! [X8: nat > rat,R2: rat] :
% 9.51/5.32        ( ( cauchy @ X8 )
% 9.51/5.32       => ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 9.51/5.32         => ? [K2: nat] :
% 9.51/5.32            ! [M3: nat] :
% 9.51/5.32              ( ( ord_less_eq_nat @ K2 @ M3 )
% 9.51/5.32             => ! [N5: nat] :
% 9.51/5.32                  ( ( ord_less_eq_nat @ K2 @ N5 )
% 9.51/5.32                 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M3 ) @ ( X8 @ N5 ) ) ) @ R2 ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9241_cauchy__imp__bounded,axiom,
% 9.51/5.32      ! [X8: nat > rat] :
% 9.51/5.32        ( ( cauchy @ X8 )
% 9.51/5.32       => ? [B3: rat] :
% 9.51/5.32            ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 9.51/5.32            & ! [N5: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N5 ) ) @ B3 ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9242_cauchy__not__vanishes__cases,axiom,
% 9.51/5.32      ! [X8: nat > rat] :
% 9.51/5.32        ( ( cauchy @ X8 )
% 9.51/5.32       => ( ~ ( vanishes @ X8 )
% 9.51/5.32         => ? [B3: rat] :
% 9.51/5.32              ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 9.51/5.32              & ? [K2: nat] :
% 9.51/5.32                  ( ! [N5: nat] :
% 9.51/5.32                      ( ( ord_less_eq_nat @ K2 @ N5 )
% 9.51/5.32                     => ( ord_less_rat @ B3 @ ( uminus_uminus_rat @ ( X8 @ N5 ) ) ) )
% 9.51/5.32                  | ! [N5: nat] :
% 9.51/5.32                      ( ( ord_less_eq_nat @ K2 @ N5 )
% 9.51/5.32                     => ( ord_less_rat @ B3 @ ( X8 @ N5 ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9243_cauchy__not__vanishes,axiom,
% 9.51/5.32      ! [X8: nat > rat] :
% 9.51/5.32        ( ( cauchy @ X8 )
% 9.51/5.32       => ( ~ ( vanishes @ X8 )
% 9.51/5.32         => ? [B3: rat] :
% 9.51/5.32              ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 9.51/5.32              & ? [K2: nat] :
% 9.51/5.32                ! [N5: nat] :
% 9.51/5.32                  ( ( ord_less_eq_nat @ K2 @ N5 )
% 9.51/5.32                 => ( ord_less_rat @ B3 @ ( abs_abs_rat @ ( X8 @ N5 ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9244_cauchy__def,axiom,
% 9.51/5.32      ( cauchy
% 9.51/5.32      = ( ^ [X6: nat > rat] :
% 9.51/5.32          ! [R5: rat] :
% 9.51/5.32            ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.51/5.32           => ? [K3: nat] :
% 9.51/5.32              ! [M2: nat] :
% 9.51/5.32                ( ( ord_less_eq_nat @ K3 @ M2 )
% 9.51/5.32               => ! [N2: nat] :
% 9.51/5.32                    ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.51/5.32                   => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) ) @ R5 ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9245_cauchyI,axiom,
% 9.51/5.32      ! [X8: nat > rat] :
% 9.51/5.32        ( ! [R3: rat] :
% 9.51/5.32            ( ( ord_less_rat @ zero_zero_rat @ R3 )
% 9.51/5.32           => ? [K7: nat] :
% 9.51/5.32              ! [M5: nat] :
% 9.51/5.32                ( ( ord_less_eq_nat @ K7 @ M5 )
% 9.51/5.32               => ! [N3: nat] :
% 9.51/5.32                    ( ( ord_less_eq_nat @ K7 @ N3 )
% 9.51/5.32                   => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) ) @ R3 ) ) ) )
% 9.51/5.32       => ( cauchy @ X8 ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9246_le__Real,axiom,
% 9.51/5.32      ! [X8: nat > rat,Y7: nat > rat] :
% 9.51/5.32        ( ( cauchy @ X8 )
% 9.51/5.32       => ( ( cauchy @ Y7 )
% 9.51/5.32         => ( ( ord_less_eq_real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
% 9.51/5.32            = ( ! [R5: rat] :
% 9.51/5.32                  ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.51/5.32                 => ? [K3: nat] :
% 9.51/5.32                    ! [N2: nat] :
% 9.51/5.32                      ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.51/5.32                     => ( ord_less_eq_rat @ ( X8 @ N2 ) @ ( plus_plus_rat @ ( Y7 @ N2 ) @ R5 ) ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9247_not__positive__Real,axiom,
% 9.51/5.32      ! [X8: nat > rat] :
% 9.51/5.32        ( ( cauchy @ X8 )
% 9.51/5.32       => ( ( ~ ( positive2 @ ( real2 @ X8 ) ) )
% 9.51/5.32          = ( ! [R5: rat] :
% 9.51/5.32                ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.51/5.32               => ? [K3: nat] :
% 9.51/5.32                  ! [N2: nat] :
% 9.51/5.32                    ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.51/5.32                   => ( ord_less_eq_rat @ ( X8 @ N2 ) @ R5 ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9248_less__real__def,axiom,
% 9.51/5.32      ( ord_less_real
% 9.51/5.32      = ( ^ [X3: real,Y2: real] : ( positive2 @ ( minus_minus_real @ Y2 @ X3 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9249_positive__Real,axiom,
% 9.51/5.32      ! [X8: nat > rat] :
% 9.51/5.32        ( ( cauchy @ X8 )
% 9.51/5.32       => ( ( positive2 @ ( real2 @ X8 ) )
% 9.51/5.32          = ( ? [R5: rat] :
% 9.51/5.32                ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.51/5.32                & ? [K3: nat] :
% 9.51/5.32                  ! [N2: nat] :
% 9.51/5.32                    ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.51/5.32                   => ( ord_less_rat @ R5 @ ( X8 @ N2 ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9250_Real_Opositive_Orep__eq,axiom,
% 9.51/5.32      ( positive2
% 9.51/5.32      = ( ^ [X3: real] :
% 9.51/5.32          ? [R5: rat] :
% 9.51/5.32            ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.51/5.32            & ? [K3: nat] :
% 9.51/5.32              ! [N2: nat] :
% 9.51/5.32                ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.51/5.32               => ( ord_less_rat @ R5 @ ( rep_real @ X3 @ N2 ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9251_strict__mono__enumerate,axiom,
% 9.51/5.32      ! [S2: set_nat] :
% 9.51/5.32        ( ~ ( finite_finite_nat @ S2 )
% 9.51/5.32       => ( order_5726023648592871131at_nat @ ( infini8530281810654367211te_nat @ S2 ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9252_enumerate__Ex,axiom,
% 9.51/5.32      ! [S2: set_nat,S: nat] :
% 9.51/5.32        ( ~ ( finite_finite_nat @ S2 )
% 9.51/5.32       => ( ( member_nat @ S @ S2 )
% 9.51/5.32         => ? [N3: nat] :
% 9.51/5.32              ( ( infini8530281810654367211te_nat @ S2 @ N3 )
% 9.51/5.32              = S ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9253_le__enumerate,axiom,
% 9.51/5.32      ! [S2: set_nat,N: nat] :
% 9.51/5.32        ( ~ ( finite_finite_nat @ S2 )
% 9.51/5.32       => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9254_range__enumerate,axiom,
% 9.51/5.32      ! [S2: set_nat] :
% 9.51/5.32        ( ~ ( finite_finite_nat @ S2 )
% 9.51/5.32       => ( ( image_nat_nat @ ( infini8530281810654367211te_nat @ S2 ) @ top_top_set_nat )
% 9.51/5.32          = S2 ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9255_finite__le__enumerate,axiom,
% 9.51/5.32      ! [S2: set_nat,N: nat] :
% 9.51/5.32        ( ( finite_finite_nat @ S2 )
% 9.51/5.32       => ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
% 9.51/5.32         => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9256_bij__enumerate,axiom,
% 9.51/5.32      ! [S2: set_nat] :
% 9.51/5.32        ( ~ ( finite_finite_nat @ S2 )
% 9.51/5.32       => ( bij_betw_nat_nat @ ( infini8530281810654367211te_nat @ S2 ) @ top_top_set_nat @ S2 ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9257_Least__eq__0,axiom,
% 9.51/5.32      ! [P: nat > $o] :
% 9.51/5.32        ( ( P @ zero_zero_nat )
% 9.51/5.32       => ( ( ord_Least_nat @ P )
% 9.51/5.32          = zero_zero_nat ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9258_Least__Suc,axiom,
% 9.51/5.32      ! [P: nat > $o,N: nat] :
% 9.51/5.32        ( ( P @ N )
% 9.51/5.32       => ( ~ ( P @ zero_zero_nat )
% 9.51/5.32         => ( ( ord_Least_nat @ P )
% 9.51/5.32            = ( suc
% 9.51/5.32              @ ( ord_Least_nat
% 9.51/5.32                @ ^ [M2: nat] : ( P @ ( suc @ M2 ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9259_Least__Suc2,axiom,
% 9.51/5.32      ! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
% 9.51/5.32        ( ( P @ N )
% 9.51/5.32       => ( ( Q @ M )
% 9.51/5.32         => ( ~ ( P @ zero_zero_nat )
% 9.51/5.32           => ( ! [K2: nat] :
% 9.51/5.32                  ( ( P @ ( suc @ K2 ) )
% 9.51/5.32                  = ( Q @ K2 ) )
% 9.51/5.32             => ( ( ord_Least_nat @ P )
% 9.51/5.32                = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9260_mask__nat__positive__iff,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 9.51/5.32        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9261_less__eq__mask,axiom,
% 9.51/5.32      ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9262_not__mask__negative__int,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9263_less__mask,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.51/5.32       => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9264_mask__nat__less__exp,axiom,
% 9.51/5.32      ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9265_divmod__nat__code,axiom,
% 9.51/5.32      ( divmod_nat
% 9.51/5.32      = ( ^ [M2: nat,N2: nat] :
% 9.51/5.32            ( produc8678311845419106900er_nat @ code_nat_of_integer @ code_nat_of_integer
% 9.51/5.32            @ ( if_Pro6119634080678213985nteger
% 9.51/5.32              @ ( ( code_integer_of_nat @ M2 )
% 9.51/5.32                = zero_z3403309356797280102nteger )
% 9.51/5.32              @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 9.51/5.32              @ ( if_Pro6119634080678213985nteger
% 9.51/5.32                @ ( ( code_integer_of_nat @ N2 )
% 9.51/5.32                  = zero_z3403309356797280102nteger )
% 9.51/5.32                @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( code_integer_of_nat @ M2 ) )
% 9.51/5.32                @ ( code_divmod_abs @ ( code_integer_of_nat @ M2 ) @ ( code_integer_of_nat @ N2 ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9266_integer__of__nat__0,axiom,
% 9.51/5.32      ( ( code_integer_of_nat @ zero_zero_nat )
% 9.51/5.32      = zero_z3403309356797280102nteger ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9267_last__upt,axiom,
% 9.51/5.32      ! [I: nat,J: nat] :
% 9.51/5.32        ( ( ord_less_nat @ I @ J )
% 9.51/5.32       => ( ( last_nat @ ( upt @ I @ J ) )
% 9.51/5.32          = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9268_not__nonnegative__int__iff,axiom,
% 9.51/5.32      ! [K: int] :
% 9.51/5.32        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 9.51/5.32        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9269_not__negative__int__iff,axiom,
% 9.51/5.32      ! [K: int] :
% 9.51/5.32        ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 9.51/5.32        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9270_and__not__num__eq__Some__iff,axiom,
% 9.51/5.32      ! [M: num,N: num,Q4: num] :
% 9.51/5.32        ( ( ( bit_and_not_num @ M @ N )
% 9.51/5.32          = ( some_num @ Q4 ) )
% 9.51/5.32        = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 9.51/5.32          = ( numeral_numeral_int @ Q4 ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9271_and__not__num__eq__None__iff,axiom,
% 9.51/5.32      ! [M: num,N: num] :
% 9.51/5.32        ( ( ( bit_and_not_num @ M @ N )
% 9.51/5.32          = none_num )
% 9.51/5.32        = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 9.51/5.32          = zero_zero_int ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9272_Real_Opositive_Oabs__eq,axiom,
% 9.51/5.32      ! [X: nat > rat] :
% 9.51/5.32        ( ( realrel @ X @ X )
% 9.51/5.32       => ( ( positive2 @ ( real2 @ X ) )
% 9.51/5.32          = ( ? [R5: rat] :
% 9.51/5.32                ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.51/5.32                & ? [K3: nat] :
% 9.51/5.32                  ! [N2: nat] :
% 9.51/5.32                    ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.51/5.32                   => ( ord_less_rat @ R5 @ ( X @ N2 ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9273_Real_Opositive_Orsp,axiom,
% 9.51/5.32      ( bNF_re728719798268516973at_o_o @ realrel
% 9.51/5.32      @ ^ [Y5: $o,Z: $o] : ( Y5 = Z )
% 9.51/5.32      @ ^ [X6: nat > rat] :
% 9.51/5.32        ? [R5: rat] :
% 9.51/5.32          ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.51/5.32          & ? [K3: nat] :
% 9.51/5.32            ! [N2: nat] :
% 9.51/5.32              ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.51/5.32             => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) )
% 9.51/5.32      @ ^ [X6: nat > rat] :
% 9.51/5.32        ? [R5: rat] :
% 9.51/5.32          ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.51/5.32          & ? [K3: nat] :
% 9.51/5.32            ! [N2: nat] :
% 9.51/5.32              ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.51/5.32             => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9274_less__eq__natural_Orsp,axiom,
% 9.51/5.32      ( bNF_re578469030762574527_nat_o
% 9.51/5.32      @ ^ [Y5: nat,Z: nat] : ( Y5 = Z )
% 9.51/5.32      @ ( bNF_re4705727531993890431at_o_o
% 9.51/5.32        @ ^ [Y5: nat,Z: nat] : ( Y5 = Z )
% 9.51/5.32        @ ^ [Y5: $o,Z: $o] : ( Y5 = Z ) )
% 9.51/5.32      @ ord_less_eq_nat
% 9.51/5.32      @ ord_less_eq_nat ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9275_less__natural_Orsp,axiom,
% 9.51/5.32      ( bNF_re578469030762574527_nat_o
% 9.51/5.32      @ ^ [Y5: nat,Z: nat] : ( Y5 = Z )
% 9.51/5.32      @ ( bNF_re4705727531993890431at_o_o
% 9.51/5.32        @ ^ [Y5: nat,Z: nat] : ( Y5 = Z )
% 9.51/5.32        @ ^ [Y5: $o,Z: $o] : ( Y5 = Z ) )
% 9.51/5.32      @ ord_less_nat
% 9.51/5.32      @ ord_less_nat ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9276_less__integer_Orsp,axiom,
% 9.51/5.32      ( bNF_re3403563459893282935_int_o
% 9.51/5.32      @ ^ [Y5: int,Z: int] : ( Y5 = Z )
% 9.51/5.32      @ ( bNF_re5089333283451836215nt_o_o
% 9.51/5.32        @ ^ [Y5: int,Z: int] : ( Y5 = Z )
% 9.51/5.32        @ ^ [Y5: $o,Z: $o] : ( Y5 = Z ) )
% 9.51/5.32      @ ord_less_int
% 9.51/5.32      @ ord_less_int ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9277_Real_Opositive_Otransfer,axiom,
% 9.51/5.32      ( bNF_re4297313714947099218al_o_o @ pcr_real
% 9.51/5.32      @ ^ [Y5: $o,Z: $o] : ( Y5 = Z )
% 9.51/5.32      @ ^ [X6: nat > rat] :
% 9.51/5.32        ? [R5: rat] :
% 9.51/5.32          ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.51/5.32          & ? [K3: nat] :
% 9.51/5.32            ! [N2: nat] :
% 9.51/5.32              ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.51/5.32             => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) )
% 9.51/5.32      @ positive2 ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9278_plus__rat_Otransfer,axiom,
% 9.51/5.32      ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 9.51/5.32      @ ^ [X3: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y2 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) )
% 9.51/5.32      @ plus_plus_rat ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9279_one__rat_Otransfer,axiom,
% 9.51/5.32      pcr_rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ one_one_rat ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9280_zero__rat_Otransfer,axiom,
% 9.51/5.32      pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9281_Fract_Otransfer,axiom,
% 9.51/5.32      ( bNF_re3461391660133120880nt_rat
% 9.51/5.32      @ ^ [Y5: int,Z: int] : ( Y5 = Z )
% 9.51/5.32      @ ( bNF_re2214769303045360666nt_rat
% 9.51/5.32        @ ^ [Y5: int,Z: int] : ( Y5 = Z )
% 9.51/5.32        @ pcr_rat )
% 9.51/5.32      @ ^ [A4: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A4 @ B4 ) )
% 9.51/5.32      @ fract ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9282_uminus__rat_Otransfer,axiom,
% 9.51/5.32      ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
% 9.51/5.32      @ ^ [X3: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X3 ) ) @ ( product_snd_int_int @ X3 ) )
% 9.51/5.32      @ uminus_uminus_rat ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9283_times__rat_Otransfer,axiom,
% 9.51/5.32      ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 9.51/5.32      @ ^ [X3: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y2 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) )
% 9.51/5.32      @ times_times_rat ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9284_Rat_Opositive_Otransfer,axiom,
% 9.51/5.32      ( bNF_re1494630372529172596at_o_o @ pcr_rat
% 9.51/5.32      @ ^ [Y5: $o,Z: $o] : ( Y5 = Z )
% 9.51/5.32      @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
% 9.51/5.32      @ positive ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9285_inverse__rat_Otransfer,axiom,
% 9.51/5.32      ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
% 9.51/5.32      @ ^ [X3: product_prod_int_int] :
% 9.51/5.32          ( if_Pro3027730157355071871nt_int
% 9.51/5.32          @ ( ( product_fst_int_int @ X3 )
% 9.51/5.32            = zero_zero_int )
% 9.51/5.32          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 9.51/5.32          @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) )
% 9.51/5.32      @ inverse_inverse_rat ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9286_times__int_Otransfer,axiom,
% 9.51/5.32      ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 9.51/5.32      @ ( produc27273713700761075at_nat
% 9.51/5.32        @ ^ [X3: nat,Y2: nat] :
% 9.51/5.32            ( produc2626176000494625587at_nat
% 9.51/5.32            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y2 @ V3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V3 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) )
% 9.51/5.32      @ times_times_int ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9287_minus__int_Otransfer,axiom,
% 9.51/5.32      ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 9.51/5.32      @ ( produc27273713700761075at_nat
% 9.51/5.32        @ ^ [X3: nat,Y2: nat] :
% 9.51/5.32            ( produc2626176000494625587at_nat
% 9.51/5.32            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) )
% 9.51/5.32      @ minus_minus_int ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9288_zero__int_Otransfer,axiom,
% 9.51/5.32      pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9289_int__transfer,axiom,
% 9.51/5.32      ( bNF_re6830278522597306478at_int
% 9.51/5.32      @ ^ [Y5: nat,Z: nat] : ( Y5 = Z )
% 9.51/5.32      @ pcr_int
% 9.51/5.32      @ ^ [N2: nat] : ( product_Pair_nat_nat @ N2 @ zero_zero_nat )
% 9.51/5.32      @ semiri1314217659103216013at_int ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9290_uminus__int_Otransfer,axiom,
% 9.51/5.32      ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
% 9.51/5.32      @ ( produc2626176000494625587at_nat
% 9.51/5.32        @ ^ [X3: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X3 ) )
% 9.51/5.32      @ uminus_uminus_int ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9291_one__int_Otransfer,axiom,
% 9.51/5.32      pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9292_less__int_Otransfer,axiom,
% 9.51/5.32      ( bNF_re717283939379294677_int_o @ pcr_int
% 9.51/5.32      @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 9.51/5.32        @ ^ [Y5: $o,Z: $o] : ( Y5 = Z ) )
% 9.51/5.32      @ ( produc8739625826339149834_nat_o
% 9.51/5.32        @ ^ [X3: nat,Y2: nat] :
% 9.51/5.32            ( produc6081775807080527818_nat_o
% 9.51/5.32            @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) )
% 9.51/5.32      @ ord_less_int ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9293_less__eq__int_Otransfer,axiom,
% 9.51/5.32      ( bNF_re717283939379294677_int_o @ pcr_int
% 9.51/5.32      @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 9.51/5.32        @ ^ [Y5: $o,Z: $o] : ( Y5 = Z ) )
% 9.51/5.32      @ ( produc8739625826339149834_nat_o
% 9.51/5.32        @ ^ [X3: nat,Y2: nat] :
% 9.51/5.32            ( produc6081775807080527818_nat_o
% 9.51/5.32            @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) )
% 9.51/5.32      @ ord_less_eq_int ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9294_plus__int_Otransfer,axiom,
% 9.51/5.32      ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 9.51/5.32      @ ( produc27273713700761075at_nat
% 9.51/5.32        @ ^ [X3: nat,Y2: nat] :
% 9.51/5.32            ( produc2626176000494625587at_nat
% 9.51/5.32            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y2 @ V3 ) ) ) )
% 9.51/5.32      @ plus_plus_int ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9295_times__int_Orsp,axiom,
% 9.51/5.32      ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 9.51/5.32      @ ( produc27273713700761075at_nat
% 9.51/5.32        @ ^ [X3: nat,Y2: nat] :
% 9.51/5.32            ( produc2626176000494625587at_nat
% 9.51/5.32            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y2 @ V3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V3 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) )
% 9.51/5.32      @ ( produc27273713700761075at_nat
% 9.51/5.32        @ ^ [X3: nat,Y2: nat] :
% 9.51/5.32            ( produc2626176000494625587at_nat
% 9.51/5.32            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y2 @ V3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V3 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9296_intrel__iff,axiom,
% 9.51/5.32      ! [X: nat,Y: nat,U: nat,V: nat] :
% 9.51/5.32        ( ( intrel @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ U @ V ) )
% 9.51/5.32        = ( ( plus_plus_nat @ X @ V )
% 9.51/5.32          = ( plus_plus_nat @ U @ Y ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9297_uminus__int_Orsp,axiom,
% 9.51/5.32      ( bNF_re2241393799969408733at_nat @ intrel @ intrel
% 9.51/5.32      @ ( produc2626176000494625587at_nat
% 9.51/5.32        @ ^ [X3: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X3 ) )
% 9.51/5.32      @ ( produc2626176000494625587at_nat
% 9.51/5.32        @ ^ [X3: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X3 ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9298_zero__int_Orsp,axiom,
% 9.51/5.32      intrel @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9299_one__int_Orsp,axiom,
% 9.51/5.32      intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9300_less__int_Orsp,axiom,
% 9.51/5.32      ( bNF_re4202695980764964119_nat_o @ intrel
% 9.51/5.32      @ ( bNF_re3666534408544137501at_o_o @ intrel
% 9.51/5.32        @ ^ [Y5: $o,Z: $o] : ( Y5 = Z ) )
% 9.51/5.32      @ ( produc8739625826339149834_nat_o
% 9.51/5.32        @ ^ [X3: nat,Y2: nat] :
% 9.51/5.32            ( produc6081775807080527818_nat_o
% 9.51/5.32            @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) )
% 9.51/5.32      @ ( produc8739625826339149834_nat_o
% 9.51/5.32        @ ^ [X3: nat,Y2: nat] :
% 9.51/5.32            ( produc6081775807080527818_nat_o
% 9.51/5.32            @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9301_less__eq__int_Orsp,axiom,
% 9.51/5.32      ( bNF_re4202695980764964119_nat_o @ intrel
% 9.51/5.32      @ ( bNF_re3666534408544137501at_o_o @ intrel
% 9.51/5.32        @ ^ [Y5: $o,Z: $o] : ( Y5 = Z ) )
% 9.51/5.32      @ ( produc8739625826339149834_nat_o
% 9.51/5.32        @ ^ [X3: nat,Y2: nat] :
% 9.51/5.32            ( produc6081775807080527818_nat_o
% 9.51/5.32            @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) )
% 9.51/5.32      @ ( produc8739625826339149834_nat_o
% 9.51/5.32        @ ^ [X3: nat,Y2: nat] :
% 9.51/5.32            ( produc6081775807080527818_nat_o
% 9.51/5.32            @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9302_plus__int_Orsp,axiom,
% 9.51/5.32      ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 9.51/5.32      @ ( produc27273713700761075at_nat
% 9.51/5.32        @ ^ [X3: nat,Y2: nat] :
% 9.51/5.32            ( produc2626176000494625587at_nat
% 9.51/5.32            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y2 @ V3 ) ) ) )
% 9.51/5.32      @ ( produc27273713700761075at_nat
% 9.51/5.32        @ ^ [X3: nat,Y2: nat] :
% 9.51/5.32            ( produc2626176000494625587at_nat
% 9.51/5.32            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y2 @ V3 ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9303_minus__int_Orsp,axiom,
% 9.51/5.32      ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 9.51/5.32      @ ( produc27273713700761075at_nat
% 9.51/5.32        @ ^ [X3: nat,Y2: nat] :
% 9.51/5.32            ( produc2626176000494625587at_nat
% 9.51/5.32            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) )
% 9.51/5.32      @ ( produc27273713700761075at_nat
% 9.51/5.32        @ ^ [X3: nat,Y2: nat] :
% 9.51/5.32            ( produc2626176000494625587at_nat
% 9.51/5.32            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V3 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9304_plus__rat_Orsp,axiom,
% 9.51/5.32      ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 9.51/5.32      @ ^ [X3: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y2 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) )
% 9.51/5.32      @ ^ [X3: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y2 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9305_inverse__rat_Orsp,axiom,
% 9.51/5.32      ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
% 9.51/5.32      @ ^ [X3: product_prod_int_int] :
% 9.51/5.32          ( if_Pro3027730157355071871nt_int
% 9.51/5.32          @ ( ( product_fst_int_int @ X3 )
% 9.51/5.32            = zero_zero_int )
% 9.51/5.32          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 9.51/5.32          @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) )
% 9.51/5.32      @ ^ [X3: product_prod_int_int] :
% 9.51/5.32          ( if_Pro3027730157355071871nt_int
% 9.51/5.32          @ ( ( product_fst_int_int @ X3 )
% 9.51/5.32            = zero_zero_int )
% 9.51/5.32          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 9.51/5.32          @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9306_zero__rat_Orsp,axiom,
% 9.51/5.32      ratrel @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9307_one__rat_Orsp,axiom,
% 9.51/5.32      ratrel @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9308_Fract_Orsp,axiom,
% 9.51/5.32      ( bNF_re157797125943740599nt_int
% 9.51/5.32      @ ^ [Y5: int,Z: int] : ( Y5 = Z )
% 9.51/5.32      @ ( bNF_re6250860962936578807nt_int
% 9.51/5.32        @ ^ [Y5: int,Z: int] : ( Y5 = Z )
% 9.51/5.32        @ ratrel )
% 9.51/5.32      @ ^ [A4: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A4 @ B4 ) )
% 9.51/5.32      @ ^ [A4: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A4 @ B4 ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9309_uminus__rat_Orsp,axiom,
% 9.51/5.32      ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
% 9.51/5.32      @ ^ [X3: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X3 ) ) @ ( product_snd_int_int @ X3 ) )
% 9.51/5.32      @ ^ [X3: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X3 ) ) @ ( product_snd_int_int @ X3 ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9310_times__rat_Orsp,axiom,
% 9.51/5.32      ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 9.51/5.32      @ ^ [X3: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y2 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) )
% 9.51/5.32      @ ^ [X3: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y2 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9311_Rat_Opositive_Orsp,axiom,
% 9.51/5.32      ( bNF_re8699439704749558557nt_o_o @ ratrel
% 9.51/5.32      @ ^ [Y5: $o,Z: $o] : ( Y5 = Z )
% 9.51/5.32      @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
% 9.51/5.32      @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9312_plus__rat_Oabs__eq,axiom,
% 9.51/5.32      ! [Xa2: product_prod_int_int,X: product_prod_int_int] :
% 9.51/5.32        ( ( ratrel @ Xa2 @ Xa2 )
% 9.51/5.32       => ( ( ratrel @ X @ X )
% 9.51/5.32         => ( ( plus_plus_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X ) )
% 9.51/5.32            = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_snd_int_int @ X ) ) @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9313_inverse__rat_Oabs__eq,axiom,
% 9.51/5.32      ! [X: product_prod_int_int] :
% 9.51/5.32        ( ( ratrel @ X @ X )
% 9.51/5.32       => ( ( inverse_inverse_rat @ ( abs_Rat @ X ) )
% 9.51/5.32          = ( abs_Rat
% 9.51/5.32            @ ( if_Pro3027730157355071871nt_int
% 9.51/5.32              @ ( ( product_fst_int_int @ X )
% 9.51/5.32                = zero_zero_int )
% 9.51/5.32              @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 9.51/5.32              @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9314_one__rat__def,axiom,
% 9.51/5.32      ( one_one_rat
% 9.51/5.32      = ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9315_Fract_Oabs__eq,axiom,
% 9.51/5.32      ( fract
% 9.51/5.32      = ( ^ [Xa4: int,X3: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X3 ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9316_zero__rat__def,axiom,
% 9.51/5.32      ( zero_zero_rat
% 9.51/5.32      = ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9317_uminus__rat_Oabs__eq,axiom,
% 9.51/5.32      ! [X: product_prod_int_int] :
% 9.51/5.32        ( ( ratrel @ X @ X )
% 9.51/5.32       => ( ( uminus_uminus_rat @ ( abs_Rat @ X ) )
% 9.51/5.32          = ( abs_Rat @ ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9318_times__rat_Oabs__eq,axiom,
% 9.51/5.32      ! [Xa2: product_prod_int_int,X: product_prod_int_int] :
% 9.51/5.32        ( ( ratrel @ Xa2 @ Xa2 )
% 9.51/5.32       => ( ( ratrel @ X @ X )
% 9.51/5.32         => ( ( times_times_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X ) )
% 9.51/5.32            = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_fst_int_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9319_Rat_Opositive_Oabs__eq,axiom,
% 9.51/5.32      ! [X: product_prod_int_int] :
% 9.51/5.32        ( ( ratrel @ X @ X )
% 9.51/5.32       => ( ( positive @ ( abs_Rat @ X ) )
% 9.51/5.32          = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9320_inverse__rat__def,axiom,
% 9.51/5.32      ( inverse_inverse_rat
% 9.51/5.32      = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
% 9.51/5.32        @ ^ [X3: product_prod_int_int] :
% 9.51/5.32            ( if_Pro3027730157355071871nt_int
% 9.51/5.32            @ ( ( product_fst_int_int @ X3 )
% 9.51/5.32              = zero_zero_int )
% 9.51/5.32            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 9.51/5.32            @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9321_uminus__rat__def,axiom,
% 9.51/5.32      ( uminus_uminus_rat
% 9.51/5.32      = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
% 9.51/5.32        @ ^ [X3: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X3 ) ) @ ( product_snd_int_int @ X3 ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9322_plus__rat__def,axiom,
% 9.51/5.32      ( plus_plus_rat
% 9.51/5.32      = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 9.51/5.32        @ ^ [X3: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y2 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9323_times__rat__def,axiom,
% 9.51/5.32      ( times_times_rat
% 9.51/5.32      = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 9.51/5.32        @ ^ [X3: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y2 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9324_natLeq__on__wo__rel,axiom,
% 9.51/5.32      ! [N: nat] :
% 9.51/5.32        ( bNF_We3818239936649020644el_nat
% 9.51/5.32        @ ( collec3392354462482085612at_nat
% 9.51/5.32          @ ( produc6081775807080527818_nat_o
% 9.51/5.32            @ ^ [X3: nat,Y2: nat] :
% 9.51/5.32                ( ( ord_less_nat @ X3 @ N )
% 9.51/5.32                & ( ord_less_nat @ Y2 @ N )
% 9.51/5.32                & ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9325_pred__nat__trancl__eq__le,axiom,
% 9.51/5.32      ! [M: nat,N: nat] :
% 9.51/5.32        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
% 9.51/5.32        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9326_less__eq,axiom,
% 9.51/5.32      ! [M: nat,N: nat] :
% 9.51/5.32        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
% 9.51/5.32        = ( ord_less_nat @ M @ N ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9327_bit__concat__bit__iff,axiom,
% 9.51/5.32      ! [M: nat,K: int,L: int,N: nat] :
% 9.51/5.32        ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
% 9.51/5.32        = ( ( ( ord_less_nat @ N @ M )
% 9.51/5.32            & ( bit_se1146084159140164899it_int @ K @ N ) )
% 9.51/5.32          | ( ( ord_less_eq_nat @ M @ N )
% 9.51/5.32            & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9328_concat__bit__0,axiom,
% 9.51/5.32      ! [K: int,L: int] :
% 9.51/5.32        ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
% 9.51/5.32        = L ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9329_concat__bit__negative__iff,axiom,
% 9.51/5.32      ! [N: nat,K: int,L: int] :
% 9.51/5.32        ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
% 9.51/5.32        = ( ord_less_int @ L @ zero_zero_int ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9330_gcd__nat_Oordering__top__axioms,axiom,
% 9.51/5.32      ( ordering_top_nat @ dvd_dvd_nat
% 9.51/5.32      @ ^ [M2: nat,N2: nat] :
% 9.51/5.32          ( ( dvd_dvd_nat @ M2 @ N2 )
% 9.51/5.32          & ( M2 != N2 ) )
% 9.51/5.32      @ zero_zero_nat ) ).
% 9.51/5.32  
% 9.51/5.32  thf(fact_9331_bot__nat__0_Oordering__top__axioms,axiom,
% 9.51/5.32      ( ordering_top_nat
% 9.51/5.32      @ ^ [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X3 )
% 9.51/5.32      @ ^ [X3: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X3 )
% 9.51/5.32      @ zero_zero_nat ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 9.51/5.32      ! [X: int,Y: int] :
% 9.51/5.32        ( ( if_int @ $false @ X @ Y )
% 9.51/5.32        = Y ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 9.51/5.32      ! [X: int,Y: int] :
% 9.51/5.32        ( ( if_int @ $true @ X @ Y )
% 9.51/5.32        = X ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 9.51/5.32      ! [X: nat,Y: nat] :
% 9.51/5.32        ( ( if_nat @ $false @ X @ Y )
% 9.51/5.32        = Y ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 9.51/5.32      ! [X: nat,Y: nat] :
% 9.51/5.32        ( ( if_nat @ $true @ X @ Y )
% 9.51/5.32        = X ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
% 9.51/5.32      ! [X: num,Y: num] :
% 9.51/5.32        ( ( if_num @ $false @ X @ Y )
% 9.51/5.32        = Y ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
% 9.51/5.32      ! [X: num,Y: num] :
% 9.51/5.32        ( ( if_num @ $true @ X @ Y )
% 9.51/5.32        = X ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 9.51/5.32      ! [X: rat,Y: rat] :
% 9.51/5.32        ( ( if_rat @ $false @ X @ Y )
% 9.51/5.32        = Y ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 9.51/5.32      ! [X: rat,Y: rat] :
% 9.51/5.32        ( ( if_rat @ $true @ X @ Y )
% 9.51/5.32        = X ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 9.51/5.32      ! [X: real,Y: real] :
% 9.51/5.32        ( ( if_real @ $false @ X @ Y )
% 9.51/5.32        = Y ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 9.51/5.32      ! [X: real,Y: real] :
% 9.51/5.32        ( ( if_real @ $true @ X @ Y )
% 9.51/5.32        = X ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
% 9.51/5.32      ! [P: real > $o] :
% 9.51/5.32        ( ( P @ ( fChoice_real @ P ) )
% 9.51/5.32        = ( ? [X6: real] : ( P @ X6 ) ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 9.51/5.32      ! [X: complex,Y: complex] :
% 9.51/5.32        ( ( if_complex @ $false @ X @ Y )
% 9.51/5.32        = Y ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 9.51/5.32      ! [X: complex,Y: complex] :
% 9.51/5.32        ( ( if_complex @ $true @ X @ Y )
% 9.51/5.32        = X ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 9.51/5.32      ! [X: extended_enat,Y: extended_enat] :
% 9.51/5.32        ( ( if_Extended_enat @ $false @ X @ Y )
% 9.51/5.32        = Y ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 9.51/5.32      ! [X: extended_enat,Y: extended_enat] :
% 9.51/5.32        ( ( if_Extended_enat @ $true @ X @ Y )
% 9.51/5.32        = X ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 9.51/5.32      ! [X: code_integer,Y: code_integer] :
% 9.51/5.32        ( ( if_Code_integer @ $false @ X @ Y )
% 9.51/5.32        = Y ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 9.51/5.32      ! [X: code_integer,Y: code_integer] :
% 9.51/5.32        ( ( if_Code_integer @ $true @ X @ Y )
% 9.51/5.32        = X ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 9.51/5.32      ! [X: set_int,Y: set_int] :
% 9.51/5.32        ( ( if_set_int @ $false @ X @ Y )
% 9.51/5.32        = Y ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 9.51/5.32      ! [X: set_int,Y: set_int] :
% 9.51/5.32        ( ( if_set_int @ $true @ X @ Y )
% 9.51/5.32        = X ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 9.51/5.32      ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
% 9.51/5.32        ( ( if_VEBT_VEBT @ $false @ X @ Y )
% 9.51/5.32        = Y ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 9.51/5.32      ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
% 9.51/5.32        ( ( if_VEBT_VEBT @ $true @ X @ Y )
% 9.51/5.32        = X ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 9.51/5.32      ! [X: list_int,Y: list_int] :
% 9.51/5.32        ( ( if_list_int @ $false @ X @ Y )
% 9.51/5.32        = Y ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 9.51/5.32      ! [X: list_int,Y: list_int] :
% 9.51/5.32        ( ( if_list_int @ $true @ X @ Y )
% 9.51/5.32        = X ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 9.51/5.32      ! [X: list_nat,Y: list_nat] :
% 9.51/5.32        ( ( if_list_nat @ $false @ X @ Y )
% 9.51/5.32        = Y ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 9.51/5.32      ! [X: list_nat,Y: list_nat] :
% 9.51/5.32        ( ( if_list_nat @ $true @ X @ Y )
% 9.51/5.32        = X ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 9.51/5.32      ! [X: int > int,Y: int > int] :
% 9.51/5.32        ( ( if_int_int @ $false @ X @ Y )
% 9.51/5.32        = Y ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 9.51/5.32      ! [X: int > int,Y: int > int] :
% 9.51/5.32        ( ( if_int_int @ $true @ X @ Y )
% 9.51/5.32        = X ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 9.51/5.32      ! [X: option_nat,Y: option_nat] :
% 9.51/5.32        ( ( if_option_nat @ $false @ X @ Y )
% 9.51/5.32        = Y ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 9.51/5.32      ! [X: option_nat,Y: option_nat] :
% 9.51/5.32        ( ( if_option_nat @ $true @ X @ Y )
% 9.51/5.32        = X ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 9.51/5.32      ! [X: option_num,Y: option_num] :
% 9.51/5.32        ( ( if_option_num @ $false @ X @ Y )
% 9.51/5.32        = Y ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 9.51/5.32      ! [X: option_num,Y: option_num] :
% 9.51/5.32        ( ( if_option_num @ $true @ X @ Y )
% 9.51/5.32        = X ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 9.51/5.32      ! [X: product_prod_int_int,Y: product_prod_int_int] :
% 9.51/5.32        ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
% 9.51/5.32        = Y ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 9.51/5.32      ! [X: product_prod_int_int,Y: product_prod_int_int] :
% 9.51/5.32        ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
% 9.51/5.32        = X ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 9.51/5.32      ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 9.51/5.32        ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
% 9.51/5.32        = Y ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 9.51/5.32      ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 9.51/5.32        ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
% 9.51/5.32        = X ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 9.51/5.32      ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
% 9.51/5.32        ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
% 9.51/5.32        = Y ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 9.51/5.32      ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
% 9.51/5.32        ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
% 9.51/5.32        = X ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 9.51/5.32      ! [P: $o] :
% 9.51/5.32        ( ( P = $true )
% 9.51/5.32        | ( P = $false ) ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 9.51/5.32      ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
% 9.51/5.32        ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
% 9.51/5.32        = Y ) ).
% 9.51/5.32  
% 9.51/5.32  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 9.51/5.32      ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
% 9.51/5.32        ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
% 9.51/5.32        = X ) ).
% 9.51/5.32  
% 9.51/5.32  thf(conj_0,conjecture,
% 9.51/5.32      vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ na ).
% 9.51/5.32  ------------------------
% 9.52/5.84  % (14185)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/275Mi)
% 9.52/5.84  % (14180)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/183Mi)
% 9.52/5.84  % (14181)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/4Mi)
% 9.52/5.84  % (14182)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/27Mi)
% 9.52/5.84  % (14183)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/2Mi)
% 9.52/5.84  % (14184)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/2Mi)
% 9.52/5.84  % (14186)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/18Mi)
% 9.52/5.84  % (14183)Instruction limit reached!
% 9.52/5.84  % (14183)------------------------------
% 9.52/5.84  % (14183)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.84  % (14183)Termination reason: Unknown
% 9.52/5.84  % (14183)Termination phase: shuffling
% 9.52/5.84  
% 9.52/5.84  % (14183)Memory used [KB]: 11897
% 9.52/5.84  % (14183)Time elapsed: 0.003 s
% 9.52/5.84  % (14183)Instructions burned: 2 (million)
% 9.52/5.84  % (14183)------------------------------
% 9.52/5.84  % (14183)------------------------------
% 9.52/5.84  % (14184)Instruction limit reached!
% 9.52/5.84  % (14184)------------------------------
% 9.52/5.84  % (14184)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.84  % (14184)Termination reason: Unknown
% 9.52/5.84  % (14184)Termination phase: shuffling
% 9.52/5.84  
% 9.52/5.84  % (14184)Memory used [KB]: 11897
% 9.52/5.84  % (14184)Time elapsed: 0.003 s
% 9.52/5.84  % (14184)Instructions burned: 2 (million)
% 9.52/5.84  % (14184)------------------------------
% 9.52/5.84  % (14184)------------------------------
% 9.52/5.85  % (14181)Instruction limit reached!
% 9.52/5.85  % (14181)------------------------------
% 9.52/5.85  % (14181)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.85  % (14181)Termination reason: Unknown
% 9.52/5.85  % (14181)Termination phase: shuffling
% 9.52/5.85  
% 9.52/5.85  % (14181)Memory used [KB]: 12153
% 9.52/5.85  % (14181)Time elapsed: 0.007 s
% 9.52/5.85  % (14181)Instructions burned: 4 (million)
% 9.52/5.85  % (14181)------------------------------
% 9.52/5.85  % (14181)------------------------------
% 9.52/5.86  % (14186)Instruction limit reached!
% 9.52/5.86  % (14186)------------------------------
% 9.52/5.86  % (14186)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.86  % (14186)Termination reason: Unknown
% 9.52/5.86  % (14187)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/3Mi)
% 9.52/5.86  % (14186)Termination phase: shuffling
% 9.52/5.86  
% 9.52/5.86  % (14186)Memory used [KB]: 13816
% 9.52/5.86  % (14186)Time elapsed: 0.022 s
% 9.52/5.86  % (14186)Instructions burned: 19 (million)
% 9.52/5.86  % (14186)------------------------------
% 9.52/5.86  % (14186)------------------------------
% 9.52/5.86  % (14188)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/37Mi)
% 9.52/5.87  % (14187)Instruction limit reached!
% 9.52/5.87  % (14187)------------------------------
% 9.52/5.87  % (14187)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.87  % (14187)Termination reason: Unknown
% 9.52/5.87  % (14187)Termination phase: shuffling
% 9.52/5.87  
% 9.52/5.87  % (14187)Memory used [KB]: 12025
% 9.52/5.87  % (14187)Time elapsed: 0.005 s
% 9.52/5.87  % (14187)Instructions burned: 3 (million)
% 9.52/5.87  % (14187)------------------------------
% 9.52/5.87  % (14187)------------------------------
% 9.52/5.87  % (14189)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/15Mi)
% 9.52/5.87  % (14182)Instruction limit reached!
% 9.52/5.87  % (14182)------------------------------
% 9.52/5.87  % (14182)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.87  % (14182)Termination reason: Unknown
% 9.52/5.87  % (14182)Termination phase: shuffling
% 9.52/5.87  
% 9.52/5.87  % (14182)Memory used [KB]: 13944
% 9.52/5.87  % (14182)Time elapsed: 0.030 s
% 9.52/5.87  % (14182)Instructions burned: 28 (million)
% 9.52/5.87  % (14182)------------------------------
% 9.52/5.87  % (14182)------------------------------
% 9.52/5.88  % (14190)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/3Mi)
% 9.52/5.89  % (14191)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/1041Mi)
% 9.52/5.89  % (14190)Instruction limit reached!
% 9.52/5.89  % (14190)------------------------------
% 9.52/5.89  % (14190)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.89  % (14190)Termination reason: Unknown
% 9.52/5.89  % (14190)Termination phase: shuffling
% 9.52/5.89  
% 9.52/5.89  % (14190)Memory used [KB]: 12025
% 9.52/5.89  % (14190)Time elapsed: 0.005 s
% 9.52/5.89  % (14190)Instructions burned: 3 (million)
% 9.52/5.89  % (14189)Instruction limit reached!
% 9.52/5.89  % (14189)------------------------------
% 9.52/5.89  % (14189)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.89  % (14190)------------------------------
% 9.52/5.89  % (14190)------------------------------
% 9.52/5.89  % (14189)Termination reason: Unknown
% 9.52/5.89  % (14189)Termination phase: shuffling
% 9.52/5.89  
% 9.52/5.89  % (14189)Memory used [KB]: 12792
% 9.52/5.89  % (14189)Time elapsed: 0.022 s
% 9.52/5.89  % (14189)Instructions burned: 15 (million)
% 9.52/5.89  % (14189)------------------------------
% 9.52/5.89  % (14189)------------------------------
% 9.52/5.89  % (14192)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/7Mi)
% 9.52/5.90  % (14188)Instruction limit reached!
% 9.52/5.90  % (14188)------------------------------
% 9.52/5.90  % (14188)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.90  % (14188)Termination reason: Unknown
% 9.52/5.90  % (14188)Termination phase: shuffling
% 9.52/5.90  
% 9.52/5.90  % (14188)Memory used [KB]: 14072
% 9.52/5.90  % (14188)Time elapsed: 0.034 s
% 9.52/5.90  % (14188)Instructions burned: 37 (million)
% 9.52/5.90  % (14188)------------------------------
% 9.52/5.90  % (14188)------------------------------
% 9.52/5.90  % (14192)Instruction limit reached!
% 9.52/5.90  % (14192)------------------------------
% 9.52/5.90  % (14192)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.90  % (14192)Termination reason: Unknown
% 9.52/5.90  % (14192)Termination phase: shuffling
% 9.52/5.90  
% 9.52/5.90  % (14192)Memory used [KB]: 12281
% 9.52/5.90  % (14192)Time elapsed: 0.011 s
% 9.52/5.90  % (14192)Instructions burned: 7 (million)
% 9.52/5.90  % (14192)------------------------------
% 9.52/5.90  % (14192)------------------------------
% 9.52/5.91  % (14193)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/16Mi)
% 9.52/5.91  % (14194)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/3Mi)
% 9.52/5.91  % (14194)Instruction limit reached!
% 9.52/5.91  % (14194)------------------------------
% 9.52/5.91  % (14194)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.91  % (14194)Termination reason: Unknown
% 9.52/5.91  % (14194)Termination phase: shuffling
% 9.52/5.91  
% 9.52/5.91  % (14194)Memory used [KB]: 12025
% 9.52/5.91  % (14194)Time elapsed: 0.005 s
% 9.52/5.91  % (14194)Instructions burned: 3 (million)
% 9.52/5.91  % (14194)------------------------------
% 9.52/5.91  % (14194)------------------------------
% 9.52/5.92  % (14195)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/3Mi)
% 9.52/5.92  % (14195)Instruction limit reached!
% 9.52/5.92  % (14195)------------------------------
% 9.52/5.92  % (14195)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.92  % (14195)Termination reason: Unknown
% 9.52/5.92  % (14195)Termination phase: shuffling
% 9.52/5.92  
% 9.52/5.92  % (14195)Memory used [KB]: 12025
% 9.52/5.92  % (14195)Time elapsed: 0.005 s
% 9.52/5.92  % (14195)Instructions burned: 3 (million)
% 9.52/5.92  % (14195)------------------------------
% 9.52/5.92  % (14195)------------------------------
% 9.52/5.92  % (14196)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/7Mi)
% 9.52/5.93  % (14193)Instruction limit reached!
% 9.52/5.93  % (14193)------------------------------
% 9.52/5.93  % (14193)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.93  % (14193)Termination reason: Unknown
% 9.52/5.93  % (14193)Termination phase: shuffling
% 9.52/5.93  
% 9.52/5.93  % (14193)Memory used [KB]: 13816
% 9.52/5.93  % (14193)Time elapsed: 0.020 s
% 9.52/5.93  % (14193)Instructions burned: 16 (million)
% 9.52/5.93  % (14193)------------------------------
% 9.52/5.93  % (14193)------------------------------
% 9.52/5.93  % (14196)Instruction limit reached!
% 9.52/5.93  % (14196)------------------------------
% 9.52/5.93  % (14196)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.93  % (14196)Termination reason: Unknown
% 9.52/5.93  % (14196)Termination phase: shuffling
% 9.52/5.93  
% 9.52/5.93  % (14196)Memory used [KB]: 12281
% 9.52/5.93  % (14196)Time elapsed: 0.011 s
% 9.52/5.93  % (14196)Instructions burned: 7 (million)
% 9.52/5.93  % (14196)------------------------------
% 9.52/5.93  % (14196)------------------------------
% 9.52/5.93  % (14197)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/3Mi)
% 9.52/5.93  % (14197)Instruction limit reached!
% 9.52/5.93  % (14197)------------------------------
% 9.52/5.93  % (14197)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.93  % (14197)Termination reason: Unknown
% 9.52/5.93  % (14197)Termination phase: shuffling
% 9.52/5.93  
% 9.52/5.93  % (14197)Memory used [KB]: 12025
% 9.52/5.93  % (14197)Time elapsed: 0.005 s
% 9.52/5.93  % (14197)Instructions burned: 3 (million)
% 9.52/5.93  % (14197)------------------------------
% 9.52/5.93  % (14197)------------------------------
% 9.52/5.94  % (14185)Instruction limit reached!
% 9.52/5.94  % (14185)------------------------------
% 9.52/5.94  % (14185)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.94  % (14185)Termination reason: Unknown
% 9.52/5.94  % (14185)Termination phase: shuffling
% 9.52/5.94  
% 9.52/5.94  % (14185)Memory used [KB]: 17654
% 9.52/5.94  % (14185)Time elapsed: 0.103 s
% 9.52/5.94  % (14185)Instructions burned: 276 (million)
% 9.52/5.94  % (14185)------------------------------
% 9.52/5.94  % (14185)------------------------------
% 9.52/5.94  % (14198)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/4Mi)
% 9.52/5.94  % (14198)Instruction limit reached!
% 9.52/5.94  % (14198)------------------------------
% 9.52/5.94  % (14198)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.94  % (14198)Termination reason: Unknown
% 9.52/5.94  % (14198)Termination phase: shuffling
% 9.52/5.94  
% 9.52/5.94  % (14198)Memory used [KB]: 12153
% 9.52/5.94  % (14198)Time elapsed: 0.007 s
% 9.52/5.94  % (14198)Instructions burned: 4 (million)
% 9.52/5.94  % (14198)------------------------------
% 9.52/5.94  % (14198)------------------------------
% 9.52/5.95  % (14200)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/710Mi)
% 9.52/5.95  % (14199)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/18Mi)
% 9.52/5.95  % (14202)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_14050 for (2993ds/902Mi)
% 9.52/5.95  % (14201)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_14050 for (2994ds/6Mi)
% 9.52/5.96  % (14201)Instruction limit reached!
% 9.52/5.96  % (14201)------------------------------
% 9.52/5.96  % (14201)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.96  % (14201)Termination reason: Unknown
% 9.52/5.96  % (14201)Termination phase: shuffling
% 9.52/5.96  
% 9.52/5.96  % (14201)Memory used [KB]: 12281
% 9.52/5.96  % (14201)Time elapsed: 0.009 s
% 9.52/5.96  % (14201)Instructions burned: 6 (million)
% 9.52/5.96  % (14201)------------------------------
% 9.52/5.96  % (14201)------------------------------
% 9.52/5.96  % (14203)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_14050 for (2993ds/21Mi)
% 9.52/5.96  % (14180)Instruction limit reached!
% 9.52/5.96  % (14180)------------------------------
% 9.52/5.96  % (14180)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.96  % (14180)Termination reason: Unknown
% 9.52/5.96  % (14180)Termination phase: shuffling
% 9.52/5.96  
% 9.52/5.96  % (14180)Memory used [KB]: 16119
% 9.52/5.96  % (14180)Time elapsed: 0.122 s
% 9.52/5.96  % (14180)Instructions burned: 184 (million)
% 9.52/5.96  % (14180)------------------------------
% 9.52/5.96  % (14180)------------------------------
% 9.52/5.97  % (14199)Instruction limit reached!
% 9.52/5.97  % (14199)------------------------------
% 9.52/5.97  % (14199)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.97  % (14199)Termination reason: Unknown
% 9.52/5.97  % (14199)Termination phase: shuffling
% 9.52/5.97  
% 9.52/5.97  % (14199)Memory used [KB]: 13816
% 9.52/5.97  % (14199)Time elapsed: 0.021 s
% 9.52/5.97  % (14199)Instructions burned: 18 (million)
% 9.52/5.97  % (14199)------------------------------
% 9.52/5.97  % (14199)------------------------------
% 9.52/5.97  % (14204)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_14050 for (2993ds/5Mi)
% 9.52/5.98  % (14205)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_14050 for (2993ds/6Mi)
% 9.52/5.98  % (14204)Instruction limit reached!
% 9.52/5.98  % (14204)------------------------------
% 9.52/5.98  % (14204)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.98  % (14204)Termination reason: Unknown
% 9.52/5.98  % (14204)Termination phase: shuffling
% 9.52/5.98  
% 9.52/5.98  % (14204)Memory used [KB]: 12153
% 9.52/5.98  % (14204)Time elapsed: 0.006 s
% 9.52/5.98  % (14204)Instructions burned: 5 (million)
% 9.52/5.98  % (14204)------------------------------
% 9.52/5.98  % (14204)------------------------------
% 9.52/5.98  % (14203)Instruction limit reached!
% 9.52/5.98  % (14203)------------------------------
% 9.52/5.98  % (14203)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.98  % (14203)Termination reason: Unknown
% 9.52/5.98  % (14203)Termination phase: shuffling
% 9.52/5.98  
% 9.52/5.98  % (14203)Memory used [KB]: 13944
% 9.52/5.98  % (14203)Time elapsed: 0.019 s
% 9.52/5.98  % (14203)Instructions burned: 23 (million)
% 9.52/5.98  % (14203)------------------------------
% 9.52/5.98  % (14203)------------------------------
% 9.52/5.98  % (14205)Instruction limit reached!
% 9.52/5.98  % (14205)------------------------------
% 9.52/5.98  % (14205)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/5.98  % (14205)Termination reason: Unknown
% 9.52/5.98  % (14205)Termination phase: shuffling
% 9.52/5.98  
% 9.52/5.98  % (14205)Memory used [KB]: 12281
% 9.52/5.98  % (14205)Time elapsed: 0.007 s
% 9.52/5.98  % (14205)Instructions burned: 6 (million)
% 9.52/5.98  % (14205)------------------------------
% 9.52/5.98  % (14205)------------------------------
% 9.52/5.98  % (14206)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_14050 for (2993ds/377Mi)
% 9.52/5.99  % (14208)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_14050 for (2993ds/19Mi)
% 9.52/5.99  % (14209)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_14050 for (2993ds/879Mi)
% 9.52/6.00  % (14207)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_14050 for (2993ds/779Mi)
% 9.52/6.01  % (14208)Instruction limit reached!
% 9.52/6.01  % (14208)------------------------------
% 9.52/6.01  % (14208)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.52/6.01  % (14208)Termination reason: Unknown
% 9.52/6.01  % (14208)Termination phase: shuffling
% 9.52/6.01  
% 9.52/6.01  % (14208)Memory used [KB]: 13816
% 9.52/6.01  % (14208)Time elapsed: 0.017 s
% 9.52/6.01  % (14208)Instructions burned: 20 (million)
% 9.52/6.01  % (14208)------------------------------
% 9.52/6.01  % (14208)------------------------------
% 11.22/6.02  % (14210)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_14050 for (2993ds/17Mi)
% 11.22/6.04  % (14210)Instruction limit reached!
% 11.22/6.04  % (14210)------------------------------
% 11.22/6.04  % (14210)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.22/6.04  % (14210)Termination reason: Unknown
% 11.22/6.04  % (14210)Termination phase: shuffling
% 11.22/6.04  
% 11.22/6.04  % (14210)Memory used [KB]: 13816
% 11.22/6.04  % (14210)Time elapsed: 0.016 s
% 11.22/6.04  % (14210)Instructions burned: 17 (million)
% 11.22/6.04  % (14210)------------------------------
% 11.22/6.04  % (14210)------------------------------
% 11.22/6.05  % (14211)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_14050 for (2992ds/3Mi)
% 11.22/6.05  % (14211)Instruction limit reached!
% 11.22/6.05  % (14211)------------------------------
% 11.22/6.05  % (14211)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.22/6.05  % (14211)Termination reason: Unknown
% 11.22/6.05  % (14211)Termination phase: shuffling
% 11.22/6.05  
% 11.22/6.05  % (14211)Memory used [KB]: 12025
% 11.22/6.05  % (14211)Time elapsed: 0.004 s
% 11.22/6.05  % (14211)Instructions burned: 3 (million)
% 11.22/6.05  % (14211)------------------------------
% 11.22/6.05  % (14211)------------------------------
% 11.22/6.06  % (14212)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_14050 for (2992ds/30Mi)
% 11.22/6.08  % (14212)Instruction limit reached!
% 11.22/6.08  % (14212)------------------------------
% 11.22/6.08  % (14212)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.22/6.08  % (14212)Termination reason: Unknown
% 11.22/6.08  % (14212)Termination phase: shuffling
% 11.22/6.08  
% 11.22/6.08  % (14212)Memory used [KB]: 14072
% 11.22/6.08  % (14212)Time elapsed: 0.021 s
% 11.22/6.08  % (14212)Instructions burned: 32 (million)
% 11.22/6.08  % (14212)------------------------------
% 11.22/6.08  % (14212)------------------------------
% 11.22/6.10  % (14213)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_14050 for (2992ds/127Mi)
% 11.22/6.11  % (14206)Instruction limit reached!
% 11.22/6.11  % (14206)------------------------------
% 11.22/6.11  % (14206)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.22/6.11  % (14206)Termination reason: Unknown
% 11.22/6.11  % (14206)Termination phase: shuffling
% 11.22/6.11  
% 11.22/6.11  % (14206)Memory used [KB]: 18549
% 11.22/6.11  % (14206)Time elapsed: 0.132 s
% 11.22/6.11  % (14206)Instructions burned: 377 (million)
% 11.22/6.11  % (14206)------------------------------
% 11.22/6.11  % (14206)------------------------------
% 11.94/6.12  % (14214)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_14050 for (2992ds/100Mi)
% 11.94/6.15  % (14213)Instruction limit reached!
% 11.94/6.15  % (14213)------------------------------
% 11.94/6.15  % (14213)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.94/6.15  % (14213)Termination reason: Unknown
% 11.94/6.15  % (14213)Termination phase: shuffling
% 11.94/6.15  
% 11.94/6.15  % (14213)Memory used [KB]: 15351
% 11.94/6.15  % (14213)Time elapsed: 0.055 s
% 11.94/6.15  % (14213)Instructions burned: 128 (million)
% 11.94/6.15  % (14213)------------------------------
% 11.94/6.15  % (14213)------------------------------
% 11.94/6.16  % (14214)Instruction limit reached!
% 11.94/6.16  % (14214)------------------------------
% 11.94/6.16  % (14214)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.94/6.16  % (14214)Termination reason: Unknown
% 11.94/6.16  % (14214)Termination phase: shuffling
% 11.94/6.16  
% 11.94/6.16  % (14214)Memory used [KB]: 15095
% 11.94/6.16  % (14214)Time elapsed: 0.039 s
% 11.94/6.16  % (14214)Instructions burned: 102 (million)
% 11.94/6.16  % (14214)------------------------------
% 11.94/6.16  % (14214)------------------------------
% 11.94/6.16  % (14215)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_14050 for (2991ds/3Mi)
% 11.94/6.16  % (14215)Instruction limit reached!
% 11.94/6.16  % (14215)------------------------------
% 11.94/6.16  % (14215)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.94/6.16  % (14215)Termination reason: Unknown
% 11.94/6.16  % (14215)Termination phase: shuffling
% 11.94/6.16  
% 11.94/6.16  % (14215)Memory used [KB]: 12025
% 11.94/6.16  % (14215)Time elapsed: 0.004 s
% 11.94/6.16  % (14215)Instructions burned: 3 (million)
% 11.94/6.16  % (14215)------------------------------
% 11.94/6.16  % (14215)------------------------------
% 11.94/6.17  % (14216)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_14050 for (2991ds/20Mi)
% 11.94/6.18  % (14217)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_14050 for (2991ds/86Mi)
% 11.94/6.19  % (14216)Instruction limit reached!
% 11.94/6.19  % (14216)------------------------------
% 11.94/6.19  % (14216)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.94/6.19  % (14216)Termination reason: Unknown
% 11.94/6.19  % (14216)Termination phase: shuffling
% 11.94/6.19  
% 11.94/6.19  % (14216)Memory used [KB]: 13816
% 11.94/6.19  % (14216)Time elapsed: 0.016 s
% 11.94/6.19  % (14216)Instructions burned: 20 (million)
% 11.94/6.19  % (14216)------------------------------
% 11.94/6.19  % (14216)------------------------------
% 11.94/6.19  % (14200)Instruction limit reached!
% 11.94/6.19  % (14200)------------------------------
% 11.94/6.19  % (14200)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.94/6.19  % (14200)Termination reason: Unknown
% 11.94/6.19  % (14200)Termination phase: shuffling
% 11.94/6.19  
% 11.94/6.19  % (14200)Memory used [KB]: 23283
% 11.94/6.19  % (14200)Time elapsed: 0.249 s
% 11.94/6.19  % (14200)Instructions burned: 712 (million)
% 11.94/6.19  % (14200)------------------------------
% 11.94/6.19  % (14200)------------------------------
% 12.51/6.20  % (14218)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_14050 for (2991ds/107Mi)
% 12.51/6.21  % (14219)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_14050 for (2991ds/39Mi)
% 12.51/6.22  % (14217)Instruction limit reached!
% 12.51/6.22  % (14217)------------------------------
% 12.51/6.22  % (14217)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.51/6.22  % (14217)Termination reason: Unknown
% 12.51/6.22  % (14217)Termination phase: shuffling
% 12.51/6.22  
% 12.51/6.22  % (14217)Memory used [KB]: 14711
% 12.51/6.22  % (14217)Time elapsed: 0.041 s
% 12.51/6.22  % (14217)Instructions burned: 89 (million)
% 12.51/6.22  % (14217)------------------------------
% 12.51/6.22  % (14217)------------------------------
% 12.51/6.23  % (14219)Instruction limit reached!
% 12.51/6.23  % (14219)------------------------------
% 12.51/6.23  % (14219)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.51/6.23  % (14219)Termination reason: Unknown
% 12.51/6.23  % (14219)Termination phase: shuffling
% 12.51/6.23  
% 12.51/6.23  % (14219)Memory used [KB]: 14200
% 12.51/6.23  % (14219)Time elapsed: 0.022 s
% 12.51/6.23  % (14219)Instructions burned: 42 (million)
% 12.51/6.23  % (14219)------------------------------
% 12.51/6.23  % (14219)------------------------------
% 12.51/6.23  % (14220)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_14050 for (2991ds/448Mi)
% 12.51/6.24  % (14221)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_14050 for (2991ds/46Mi)
% 12.51/6.24  % (14218)Instruction limit reached!
% 12.51/6.24  % (14218)------------------------------
% 12.51/6.24  % (14218)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.51/6.24  % (14218)Termination reason: Unknown
% 12.51/6.24  % (14218)Termination phase: shuffling
% 12.51/6.24  
% 12.51/6.24  % (14218)Memory used [KB]: 14967
% 12.51/6.24  % (14218)Time elapsed: 0.044 s
% 12.51/6.24  % (14218)Instructions burned: 107 (million)
% 12.51/6.24  % (14218)------------------------------
% 12.51/6.24  % (14218)------------------------------
% 12.51/6.25  % (14222)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_14050 for (2990ds/98Mi)
% 12.51/6.26  % (14191)Instruction limit reached!
% 12.51/6.26  % (14191)------------------------------
% 12.51/6.26  % (14191)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.51/6.26  % (14191)Termination reason: Unknown
% 12.51/6.26  % (14191)Termination phase: shuffling
% 12.51/6.26  
% 12.51/6.26  % (14191)Memory used [KB]: 34029
% 12.51/6.26  % (14191)Time elapsed: 0.372 s
% 12.51/6.26  % (14191)Instructions burned: 1041 (million)
% 12.51/6.26  % (14191)------------------------------
% 12.51/6.26  % (14191)------------------------------
% 12.51/6.26  % (14221)Instruction limit reached!
% 12.51/6.26  % (14221)------------------------------
% 12.51/6.26  % (14221)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.51/6.26  % (14221)Termination reason: Unknown
% 12.51/6.26  % (14221)Termination phase: shuffling
% 12.51/6.26  
% 12.51/6.26  % (14221)Memory used [KB]: 14328
% 12.51/6.26  % (14221)Time elapsed: 0.025 s
% 12.51/6.26  % (14221)Instructions burned: 47 (million)
% 12.51/6.26  % (14221)------------------------------
% 12.51/6.26  % (14221)------------------------------
% 12.51/6.27  % (14224)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_14050 for (2990ds/507Mi)
% 12.51/6.27  % (14225)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_14050 for (2990ds/149Mi)
% 12.51/6.29  % (14222)Instruction limit reached!
% 12.51/6.29  % (14222)------------------------------
% 12.51/6.29  % (14222)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.51/6.29  % (14222)Termination reason: Unknown
% 12.51/6.29  % (14222)Termination phase: shuffling
% 12.51/6.29  
% 12.51/6.29  % (14222)Memory used [KB]: 15095
% 12.51/6.29  % (14222)Time elapsed: 0.038 s
% 12.51/6.29  % (14222)Instructions burned: 100 (million)
% 12.51/6.29  % (14222)------------------------------
% 12.51/6.29  % (14222)------------------------------
% 13.12/6.30  % (14226)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_14050 for (2990ds/39Mi)
% 13.12/6.32  % (14209)Instruction limit reached!
% 13.12/6.32  % (14209)------------------------------
% 13.12/6.32  % (14209)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.12/6.32  % (14209)Termination reason: Unknown
% 13.12/6.32  % (14209)Termination phase: shuffling
% 13.12/6.32  
% 13.12/6.32  % (14209)Memory used [KB]: 27249
% 13.12/6.32  % (14209)Time elapsed: 0.329 s
% 13.12/6.32  % (14209)Instructions burned: 880 (million)
% 13.12/6.32  % (14209)------------------------------
% 13.12/6.32  % (14209)------------------------------
% 13.12/6.32  % (14226)Instruction limit reached!
% 13.12/6.32  % (14226)------------------------------
% 13.12/6.32  % (14226)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.12/6.32  % (14226)Termination reason: Unknown
% 13.12/6.32  % (14226)Termination phase: shuffling
% 13.12/6.32  
% 13.12/6.32  % (14226)Memory used [KB]: 14200
% 13.12/6.32  % (14226)Time elapsed: 0.023 s
% 13.12/6.32  % (14226)Instructions burned: 41 (million)
% 13.12/6.32  % (14226)------------------------------
% 13.12/6.32  % (14226)------------------------------
% 13.12/6.32  % (14202)Instruction limit reached!
% 13.12/6.32  % (14202)------------------------------
% 13.12/6.32  % (14202)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.12/6.32  % (14202)Termination reason: Unknown
% 13.12/6.32  % (14202)Termination phase: shuffling
% 13.12/6.32  
% 13.12/6.32  % (14202)Memory used [KB]: 27760
% 13.12/6.32  % (14202)Time elapsed: 0.375 s
% 13.12/6.32  % (14202)Instructions burned: 904 (million)
% 13.12/6.32  % (14202)------------------------------
% 13.12/6.32  % (14202)------------------------------
% 13.12/6.33  % (14225)Instruction limit reached!
% 13.12/6.33  % (14225)------------------------------
% 13.12/6.33  % (14225)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.12/6.33  % (14225)Termination reason: Unknown
% 13.12/6.33  % (14225)Termination phase: shuffling
% 13.12/6.33  
% 13.12/6.33  % (14225)Memory used [KB]: 15479
% 13.12/6.33  % (14225)Time elapsed: 0.058 s
% 13.12/6.33  % (14225)Instructions burned: 151 (million)
% 13.12/6.33  % (14225)------------------------------
% 13.12/6.33  % (14225)------------------------------
% 13.12/6.33  % (14227)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_14050 for (2990ds/6Mi)
% 13.12/6.33  % (14228)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_14050 for (2990ds/1087Mi)
% 13.12/6.34  % (14227)Instruction limit reached!
% 13.12/6.34  % (14227)------------------------------
% 13.12/6.34  % (14227)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.12/6.34  % (14227)Termination reason: Unknown
% 13.12/6.34  % (14227)Termination phase: shuffling
% 13.12/6.34  
% 13.12/6.34  % (14227)Memory used [KB]: 12281
% 13.12/6.34  % (14227)Time elapsed: 0.007 s
% 13.12/6.34  % (14227)Instructions burned: 7 (million)
% 13.12/6.34  % (14227)------------------------------
% 13.12/6.34  % (14227)------------------------------
% 13.12/6.34  % (14230)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_14050 for (2990ds/44Mi)
% 13.12/6.34  % (14229)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_14050 for (2990ds/47Mi)
% 13.12/6.35  % (14231)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_14050 for (2989ds/432Mi)
% 13.12/6.36  % (14230)Instruction limit reached!
% 13.12/6.36  % (14230)------------------------------
% 13.12/6.36  % (14230)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.12/6.36  % (14230)Termination reason: Unknown
% 13.12/6.36  % (14230)Termination phase: shuffling
% 13.12/6.36  
% 13.12/6.36  % (14230)Memory used [KB]: 14328
% 13.12/6.36  % (14230)Time elapsed: 0.023 s
% 13.12/6.36  % (14230)Instructions burned: 45 (million)
% 13.12/6.36  % (14230)------------------------------
% 13.12/6.36  % (14230)------------------------------
% 13.12/6.37  % (14229)Instruction limit reached!
% 13.12/6.37  % (14229)------------------------------
% 13.12/6.37  % (14229)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.12/6.37  % (14229)Termination reason: Unknown
% 13.12/6.37  % (14229)Termination phase: shuffling
% 13.12/6.37  
% 13.12/6.37  % (14229)Memory used [KB]: 14328
% 13.12/6.37  % (14229)Time elapsed: 0.025 s
% 13.12/6.37  % (14229)Instructions burned: 47 (million)
% 13.12/6.37  % (14229)------------------------------
% 13.12/6.37  % (14229)------------------------------
% 13.12/6.37  % (14232)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_14050 for (2989ds/62Mi)
% 13.12/6.37  % (14220)Instruction limit reached!
% 13.12/6.37  % (14220)------------------------------
% 13.12/6.37  % (14220)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.12/6.37  % (14220)Termination reason: Unknown
% 13.12/6.37  % (14220)Termination phase: shuffling
% 13.12/6.37  
% 13.12/6.37  % (14220)Memory used [KB]: 22003
% 13.12/6.37  % (14220)Time elapsed: 0.147 s
% 13.12/6.37  % (14220)Instructions burned: 450 (million)
% 13.12/6.37  % (14220)------------------------------
% 13.12/6.37  % (14220)------------------------------
% 13.12/6.38  % (14233)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_14050 for (2989ds/31Mi)
% 13.12/6.39  % (14234)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_14050 for (2989ds/96Mi)
% 13.64/6.40  % (14233)Instruction limit reached!
% 13.64/6.40  % (14233)------------------------------
% 13.64/6.40  % (14233)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.64/6.40  % (14233)Termination reason: Unknown
% 13.64/6.40  % (14233)Termination phase: shuffling
% 13.64/6.40  
% 13.64/6.40  % (14233)Memory used [KB]: 14072
% 13.64/6.40  % (14233)Time elapsed: 0.022 s
% 13.64/6.40  % (14233)Instructions burned: 32 (million)
% 13.64/6.40  % (14233)------------------------------
% 13.64/6.40  % (14233)------------------------------
% 13.64/6.40  % (14232)Instruction limit reached!
% 13.64/6.40  % (14232)------------------------------
% 13.64/6.40  % (14232)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.64/6.40  % (14232)Termination reason: Unknown
% 13.64/6.40  % (14232)Termination phase: shuffling
% 13.64/6.40  
% 13.64/6.40  % (14232)Memory used [KB]: 14583
% 13.64/6.40  % (14232)Time elapsed: 0.028 s
% 13.64/6.40  % (14232)Instructions burned: 63 (million)
% 13.64/6.40  % (14232)------------------------------
% 13.64/6.40  % (14232)------------------------------
% 13.64/6.41  % (14235)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_14050 for (2989ds/453Mi)
% 13.64/6.41  % (14236)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_14050 for (2989ds/71Mi)
% 13.64/6.43  % (14234)Instruction limit reached!
% 13.64/6.43  % (14234)------------------------------
% 13.64/6.43  % (14234)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.64/6.43  % (14234)Termination reason: Unknown
% 13.64/6.43  % (14234)Termination phase: shuffling
% 13.64/6.43  
% 13.64/6.43  % (14234)Memory used [KB]: 14967
% 13.64/6.43  % (14234)Time elapsed: 0.042 s
% 13.64/6.43  % (14234)Instructions burned: 98 (million)
% 13.64/6.43  % (14234)------------------------------
% 13.64/6.43  % (14234)------------------------------
% 13.64/6.44  % (14207)Instruction limit reached!
% 13.64/6.44  % (14207)------------------------------
% 13.64/6.44  % (14207)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.64/6.44  % (14207)Termination reason: Unknown
% 13.64/6.44  % (14207)Termination phase: shuffling
% 13.64/6.44  
% 13.64/6.44  % (14207)Memory used [KB]: 26481
% 13.64/6.44  % (14207)Time elapsed: 0.436 s
% 13.64/6.44  % (14207)Instructions burned: 779 (million)
% 13.64/6.44  % (14207)------------------------------
% 13.64/6.44  % (14207)------------------------------
% 13.64/6.44  % (14237)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_14050 for (2988ds/83Mi)
% 13.64/6.44  % (14236)Instruction limit reached!
% 13.64/6.44  % (14236)------------------------------
% 13.64/6.44  % (14236)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.64/6.44  % (14236)Termination reason: Unknown
% 13.64/6.44  % (14236)Termination phase: shuffling
% 13.64/6.44  
% 13.64/6.44  % (14236)Memory used [KB]: 14583
% 13.64/6.44  % (14236)Time elapsed: 0.033 s
% 13.64/6.44  % (14236)Instructions burned: 73 (million)
% 13.64/6.44  % (14236)------------------------------
% 13.64/6.44  % (14236)------------------------------
% 13.64/6.45  % (14224)Instruction limit reached!
% 13.64/6.45  % (14224)------------------------------
% 13.64/6.45  % (14224)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.64/6.45  % (14224)Termination reason: Unknown
% 13.64/6.45  % (14224)Termination phase: shuffling
% 13.64/6.45  
% 13.64/6.45  % (14224)Memory used [KB]: 20340
% 13.64/6.45  % (14224)Time elapsed: 0.180 s
% 13.64/6.45  % (14224)Instructions burned: 509 (million)
% 13.64/6.45  % (14224)------------------------------
% 13.64/6.45  % (14224)------------------------------
% 13.64/6.45  % (14239)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_14050 for (2988ds/41Mi)
% 13.64/6.46  % (14238)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_14050 for (2988ds/11Mi)
% 13.64/6.46  % (14240)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_14050 for (2988ds/6Mi)
% 13.64/6.46  % (14240)Instruction limit reached!
% 13.64/6.46  % (14240)------------------------------
% 13.64/6.46  % (14240)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.64/6.46  % (14240)Termination reason: Unknown
% 13.64/6.46  % (14240)Termination phase: shuffling
% 13.64/6.46  
% 13.64/6.46  % (14240)Memory used [KB]: 12281
% 13.64/6.46  % (14240)Time elapsed: 0.006 s
% 13.64/6.46  % (14240)Instructions burned: 6 (million)
% 13.64/6.46  % (14240)------------------------------
% 13.64/6.46  % (14240)------------------------------
% 13.64/6.47  % (14238)Instruction limit reached!
% 13.64/6.47  % (14238)------------------------------
% 13.64/6.47  % (14238)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.64/6.47  % (14238)Termination reason: Unknown
% 13.64/6.47  % (14238)Termination phase: shuffling
% 13.64/6.47  
% 13.64/6.47  % (14238)Memory used [KB]: 12537
% 13.64/6.47  % (14238)Time elapsed: 0.015 s
% 13.64/6.47  % (14238)Instructions burned: 11 (million)
% 13.64/6.47  % (14238)------------------------------
% 13.64/6.47  % (14238)------------------------------
% 13.64/6.47  % (14237)Instruction limit reached!
% 13.64/6.47  % (14237)------------------------------
% 13.64/6.47  % (14237)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.64/6.47  % (14237)Termination reason: Unknown
% 13.64/6.47  % (14237)Termination phase: shuffling
% 13.64/6.47  
% 13.64/6.47  % (14237)Memory used [KB]: 14711
% 13.64/6.47  % (14237)Time elapsed: 0.037 s
% 13.64/6.47  % (14237)Instructions burned: 83 (million)
% 13.64/6.47  % (14237)------------------------------
% 13.64/6.47  % (14237)------------------------------
% 13.64/6.47  % (14239)Instruction limit reached!
% 13.64/6.47  % (14239)------------------------------
% 13.64/6.47  % (14239)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.64/6.47  % (14239)Termination reason: Unknown
% 13.64/6.47  % (14239)Termination phase: shuffling
% 13.64/6.47  
% 13.64/6.47  % (14239)Memory used [KB]: 14200
% 13.64/6.47  % (14239)Time elapsed: 0.022 s
% 13.64/6.47  % (14239)Instructions burned: 43 (million)
% 13.64/6.47  % (14239)------------------------------
% 13.64/6.47  % (14239)------------------------------
% 13.64/6.48  % (14241)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_14050 for (2988ds/150Mi)
% 13.64/6.48  % (14231)Instruction limit reached!
% 13.64/6.48  % (14231)------------------------------
% 13.64/6.48  % (14231)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.64/6.48  % (14231)Termination reason: Unknown
% 13.64/6.48  % (14231)Termination phase: shuffling
% 13.64/6.48  
% 13.64/6.48  % (14231)Memory used [KB]: 21748
% 13.64/6.48  % (14231)Time elapsed: 0.135 s
% 13.64/6.48  % (14231)Instructions burned: 434 (million)
% 13.64/6.48  % (14231)------------------------------
% 13.64/6.48  % (14231)------------------------------
% 13.64/6.48  % (14243)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_14050 for (2988ds/109Mi)
% 13.64/6.48  % (14242)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_14050 for (2988ds/343Mi)
% 13.64/6.49  % (14245)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_14050 for (2988ds/140Mi)
% 13.64/6.49  % (14244)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_14050 for (2988ds/253Mi)
% 14.31/6.53  % (14243)Instruction limit reached!
% 14.31/6.53  % (14243)------------------------------
% 14.31/6.53  % (14243)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.31/6.53  % (14243)Termination reason: Unknown
% 14.31/6.53  % (14243)Termination phase: shuffling
% 14.31/6.53  
% 14.31/6.53  % (14243)Memory used [KB]: 15223
% 14.31/6.53  % (14243)Time elapsed: 0.045 s
% 14.31/6.53  % (14243)Instructions burned: 109 (million)
% 14.31/6.53  % (14243)------------------------------
% 14.31/6.53  % (14243)------------------------------
% 14.31/6.53  % (14241)Instruction limit reached!
% 14.31/6.53  % (14241)------------------------------
% 14.31/6.53  % (14241)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.31/6.53  % (14241)Termination reason: Unknown
% 14.31/6.53  % (14241)Termination phase: shuffling
% 14.31/6.53  
% 14.31/6.53  % (14241)Memory used [KB]: 15735
% 14.31/6.53  % (14241)Time elapsed: 0.060 s
% 14.31/6.53  % (14241)Instructions burned: 152 (million)
% 14.31/6.53  % (14241)------------------------------
% 14.31/6.53  % (14241)------------------------------
% 14.31/6.54  % (14246)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_14050 for (2987ds/614Mi)
% 14.31/6.55  % (14247)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_14050 for (2987ds/202Mi)
% 14.31/6.55  % (14245)Instruction limit reached!
% 14.31/6.55  % (14245)------------------------------
% 14.31/6.55  % (14245)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.31/6.55  % (14245)Termination reason: Unknown
% 14.31/6.55  % (14245)Termination phase: shuffling
% 14.31/6.55  
% 14.31/6.55  % (14245)Memory used [KB]: 15607
% 14.31/6.55  % (14245)Time elapsed: 0.056 s
% 14.31/6.55  % (14245)Instructions burned: 141 (million)
% 14.31/6.55  % (14245)------------------------------
% 14.31/6.55  % (14245)------------------------------
% 14.31/6.55  % (14235)Instruction limit reached!
% 14.31/6.55  % (14235)------------------------------
% 14.31/6.55  % (14235)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.31/6.55  % (14235)Termination reason: Unknown
% 14.31/6.55  % (14235)Termination phase: shuffling
% 14.31/6.55  
% 14.31/6.55  % (14235)Memory used [KB]: 22003
% 14.31/6.55  % (14235)Time elapsed: 0.141 s
% 14.31/6.55  % (14235)Instructions burned: 453 (million)
% 14.31/6.55  % (14235)------------------------------
% 14.31/6.55  % (14235)------------------------------
% 14.31/6.56  % (14248)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_14050 for (2987ds/339Mi)
% 14.31/6.56  % (14249)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=283:si=on:rtra=on_0 on DTF2THF_14050 for (2987ds/283Mi)
% 14.66/6.61  % (14242)Instruction limit reached!
% 14.66/6.61  % (14242)------------------------------
% 14.66/6.61  % (14242)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.66/6.61  % (14242)Termination reason: Unknown
% 14.66/6.61  % (14242)Termination phase: shuffling
% 14.66/6.61  
% 14.66/6.61  % (14242)Memory used [KB]: 18421
% 14.66/6.61  % (14242)Time elapsed: 0.123 s
% 14.66/6.61  % (14242)Instructions burned: 345 (million)
% 14.66/6.61  % (14242)------------------------------
% 14.66/6.61  % (14242)------------------------------
% 14.66/6.62  % (14250)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_14050 for (2987ds/134Mi)
% 14.66/6.62  % (14247)Instruction limit reached!
% 14.66/6.62  % (14247)------------------------------
% 14.66/6.62  % (14247)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.66/6.62  % (14247)Termination reason: Unknown
% 14.66/6.62  % (14247)Termination phase: shuffling
% 14.66/6.62  
% 14.66/6.62  % (14247)Memory used [KB]: 16247
% 14.66/6.62  % (14247)Time elapsed: 0.075 s
% 14.66/6.62  % (14247)Instructions burned: 202 (million)
% 14.66/6.62  % (14247)------------------------------
% 14.66/6.62  % (14247)------------------------------
% 14.66/6.63  % (14251)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_14050 for (2987ds/298Mi)
% 14.66/6.65  % (14244)Instruction limit reached!
% 14.66/6.65  % (14244)------------------------------
% 14.66/6.65  % (14244)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.66/6.65  % (14244)Termination reason: Unknown
% 14.66/6.65  % (14244)Termination phase: shuffling
% 14.66/6.65  
% 14.66/6.65  % (14244)Memory used [KB]: 17270
% 14.66/6.65  % (14244)Time elapsed: 0.157 s
% 14.66/6.65  % (14244)Instructions burned: 254 (million)
% 14.66/6.65  % (14244)------------------------------
% 14.66/6.65  % (14244)------------------------------
% 14.66/6.65  % (14249)Instruction limit reached!
% 14.66/6.65  % (14249)------------------------------
% 14.66/6.65  % (14249)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.66/6.65  % (14249)Termination reason: Unknown
% 14.66/6.65  % (14249)Termination phase: shuffling
% 14.66/6.65  
% 14.66/6.65  % (14249)Memory used [KB]: 19701
% 14.66/6.65  % (14249)Time elapsed: 0.093 s
% 14.66/6.65  % (14249)Instructions burned: 285 (million)
% 14.66/6.65  % (14249)------------------------------
% 14.66/6.65  % (14249)------------------------------
% 14.66/6.66  % (14253)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_14050 for (2986ds/258Mi)
% 14.66/6.67  % (14250)Instruction limit reached!
% 14.66/6.67  % (14250)------------------------------
% 14.66/6.67  % (14250)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.66/6.67  % (14250)Termination reason: Unknown
% 14.66/6.67  % (14250)Termination phase: shuffling
% 14.66/6.67  
% 14.66/6.67  % (14250)Memory used [KB]: 15479
% 14.66/6.67  % (14250)Time elapsed: 0.053 s
% 14.66/6.67  % (14250)Instructions burned: 136 (million)
% 14.66/6.67  % (14250)------------------------------
% 14.66/6.67  % (14250)------------------------------
% 14.66/6.67  % (14252)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_14050 for (2986ds/192Mi)
% 14.66/6.68  % (14248)Instruction limit reached!
% 14.66/6.68  % (14248)------------------------------
% 14.66/6.68  % (14248)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.66/6.68  % (14248)Termination reason: Unknown
% 14.66/6.68  % (14248)Termination phase: shuffling
% 14.66/6.68  
% 14.66/6.68  % (14248)Memory used [KB]: 18038
% 14.66/6.68  % (14248)Time elapsed: 0.119 s
% 14.66/6.68  % (14248)Instructions burned: 339 (million)
% 14.66/6.68  % (14248)------------------------------
% 14.66/6.68  % (14248)------------------------------
% 14.66/6.68  % (14254)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_14050 for (2986ds/90Mi)
% 14.66/6.69  % (14255)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_14050 for (2986ds/231Mi)
% 14.66/6.70  % (14228)Instruction limit reached!
% 14.66/6.70  % (14228)------------------------------
% 14.66/6.70  % (14228)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.66/6.70  % (14228)Termination reason: Unknown
% 14.66/6.70  % (14228)Termination phase: Property scanning
% 14.66/6.70  
% 14.66/6.70  % (14228)Memory used [KB]: 28272
% 14.66/6.70  % (14228)Time elapsed: 0.365 s
% 14.66/6.70  % (14228)Instructions burned: 1087 (million)
% 14.66/6.70  % (14228)------------------------------
% 14.66/6.70  % (14228)------------------------------
% 15.34/6.71  % (14256)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_14050 for (2986ds/458Mi)
% 15.34/6.72  % (14254)Instruction limit reached!
% 15.34/6.72  % (14254)------------------------------
% 15.34/6.72  % (14254)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.34/6.72  % (14254)Termination reason: Unknown
% 15.34/6.72  % (14254)Termination phase: shuffling
% 15.34/6.72  
% 15.34/6.72  % (14254)Memory used [KB]: 14839
% 15.34/6.72  % (14254)Time elapsed: 0.039 s
% 15.34/6.72  % (14254)Instructions burned: 90 (million)
% 15.34/6.72  % (14254)------------------------------
% 15.34/6.72  % (14254)------------------------------
% 15.34/6.72  % (14251)Instruction limit reached!
% 15.34/6.72  % (14251)------------------------------
% 15.34/6.72  % (14251)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.34/6.72  % (14251)Termination reason: Unknown
% 15.34/6.72  % (14251)Termination phase: shuffling
% 15.34/6.72  
% 15.34/6.72  % (14251)Memory used [KB]: 19829
% 15.34/6.72  % (14251)Time elapsed: 0.095 s
% 15.34/6.72  % (14251)Instructions burned: 298 (million)
% 15.34/6.72  % (14251)------------------------------
% 15.34/6.72  % (14251)------------------------------
% 15.34/6.73  % (14257)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_14050 for (2986ds/31Mi)
% 15.34/6.74  % (14258)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_14050 for (2986ds/297Mi)
% 15.34/6.75  % (14257)Instruction limit reached!
% 15.34/6.75  % (14257)------------------------------
% 15.34/6.75  % (14257)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.34/6.75  % (14257)Termination reason: Unknown
% 15.34/6.75  % (14257)Termination phase: shuffling
% 15.34/6.75  
% 15.34/6.75  % (14257)Memory used [KB]: 14072
% 15.34/6.75  % (14257)Time elapsed: 0.020 s
% 15.34/6.75  % (14257)Instructions burned: 33 (million)
% 15.34/6.75  % (14257)------------------------------
% 15.34/6.75  % (14257)------------------------------
% 15.34/6.75  % (14246)Instruction limit reached!
% 15.34/6.75  % (14246)------------------------------
% 15.34/6.75  % (14246)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.34/6.75  % (14246)Termination reason: Unknown
% 15.34/6.75  % (14246)Termination phase: shuffling
% 15.34/6.75  
% 15.34/6.75  % (14246)Memory used [KB]: 23922
% 15.34/6.75  % (14246)Time elapsed: 0.214 s
% 15.34/6.75  % (14246)Instructions burned: 614 (million)
% 15.34/6.75  % (14246)------------------------------
% 15.34/6.75  % (14246)------------------------------
% 15.34/6.76  % (14253)Instruction limit reached!
% 15.34/6.76  % (14253)------------------------------
% 15.34/6.76  % (14253)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.34/6.76  % (14253)Termination reason: Unknown
% 15.34/6.76  % (14253)Termination phase: shuffling
% 15.34/6.76  
% 15.34/6.76  % (14253)Memory used [KB]: 17014
% 15.34/6.76  % (14253)Time elapsed: 0.094 s
% 15.34/6.76  % (14253)Instructions burned: 258 (million)
% 15.34/6.76  % (14253)------------------------------
% 15.34/6.76  % (14253)------------------------------
% 15.34/6.76  % (14259)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_14050 for (2985ds/987Mi)
% 15.34/6.76  % (14255)Instruction limit reached!
% 15.34/6.76  % (14255)------------------------------
% 15.34/6.76  % (14255)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.34/6.76  % (14255)Termination reason: Unknown
% 15.34/6.76  % (14255)Termination phase: shuffling
% 15.34/6.76  
% 15.34/6.76  % (14255)Memory used [KB]: 16886
% 15.34/6.76  % (14255)Time elapsed: 0.075 s
% 15.34/6.76  % (14255)Instructions burned: 231 (million)
% 15.34/6.76  % (14255)------------------------------
% 15.34/6.76  % (14255)------------------------------
% 15.34/6.76  % (14260)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_14050 for (2985ds/182Mi)
% 15.34/6.77  % (14261)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_14050 for (2985ds/182Mi)
% 15.34/6.77  % (14262)dis+10_1:8_cha=on:cnfonf=lazy_pi_sigma_gen:fe=axiom:hud=5:plsq=on:plsqr=32,1:prag=on:sos=all:i=154:si=on:rtra=on_0 on DTF2THF_14050 for (2985ds/154Mi)
% 15.34/6.79  % (14252)Instruction limit reached!
% 15.34/6.79  % (14252)------------------------------
% 15.34/6.79  % (14252)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.34/6.79  % (14252)Termination reason: Unknown
% 15.34/6.79  % (14252)Termination phase: shuffling
% 15.34/6.79  
% 15.34/6.79  % (14252)Memory used [KB]: 16119
% 15.34/6.79  % (14252)Time elapsed: 0.120 s
% 15.34/6.79  % (14252)Instructions burned: 192 (million)
% 15.34/6.79  % (14252)------------------------------
% 15.34/6.79  % (14252)------------------------------
% 16.46/6.81  % (14263)dis+1003_1:64_atotf=0.4:au=on:fsr=off:hflw=5:hfsq=on:hfsqc=20:hfsql=off:plsq=on:plsqc=1:plsqr=32,1:i=79:si=on:rtra=on_0 on DTF2THF_14050 for (2985ds/79Mi)
% 16.46/6.82  % (14262)Instruction limit reached!
% 16.46/6.82  % (14262)------------------------------
% 16.46/6.82  % (14262)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.46/6.82  % (14262)Termination reason: Unknown
% 16.46/6.82  % (14262)Termination phase: shuffling
% 16.46/6.82  
% 16.46/6.82  % (14262)Memory used [KB]: 15863
% 16.46/6.82  % (14262)Time elapsed: 0.053 s
% 16.46/6.82  % (14262)Instructions burned: 155 (million)
% 16.46/6.82  % (14262)------------------------------
% 16.46/6.82  % (14262)------------------------------
% 16.46/6.83  % (14258)Instruction limit reached!
% 16.46/6.83  % (14258)------------------------------
% 16.46/6.83  % (14258)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.46/6.83  % (14258)Termination reason: Unknown
% 16.46/6.83  % (14258)Termination phase: shuffling
% 16.46/6.83  
% 16.46/6.83  % (14258)Memory used [KB]: 19701
% 16.46/6.83  % (14258)Time elapsed: 0.094 s
% 16.46/6.83  % (14258)Instructions burned: 298 (million)
% 16.46/6.83  % (14258)------------------------------
% 16.46/6.83  % (14258)------------------------------
% 16.46/6.83  % (14260)Instruction limit reached!
% 16.46/6.83  % (14260)------------------------------
% 16.46/6.83  % (14260)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.46/6.83  % (14260)Termination reason: Unknown
% 16.46/6.83  % (14260)Termination phase: shuffling
% 16.46/6.83  
% 16.46/6.83  % (14260)Memory used [KB]: 15991
% 16.46/6.83  % (14260)Time elapsed: 0.067 s
% 16.46/6.83  % (14260)Instructions burned: 182 (million)
% 16.46/6.83  % (14260)------------------------------
% 16.46/6.83  % (14260)------------------------------
% 16.46/6.84  % (14264)dis+1010_5:2_anc=all_dependent:awrs=converge:bd=off:bet=on:c=on:cha=on:cnfonf=lazy_not_gen_be_off:fe=axiom:hud=5:kws=precedence:piset=and:prag=on:s2a=on:sac=on:slsq=on:slsqc=1:sp=reverse_arity:ss=axioms:st=1.5:i=216:si=on:rtra=on_0 on DTF2THF_14050 for (2985ds/216Mi)
% 16.46/6.84  % (14261)Instruction limit reached!
% 16.46/6.84  % (14261)------------------------------
% 16.46/6.84  % (14261)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.46/6.84  % (14261)Termination reason: Unknown
% 16.46/6.84  % (14261)Termination phase: shuffling
% 16.46/6.84  
% 16.46/6.84  % (14261)Memory used [KB]: 16375
% 16.46/6.84  % (14261)Time elapsed: 0.071 s
% 16.46/6.84  % (14261)Instructions burned: 184 (million)
% 16.46/6.84  % (14261)------------------------------
% 16.46/6.84  % (14261)------------------------------
% 16.46/6.84  % (14265)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_14050 for (2985ds/114Mi)
% 16.46/6.84  % (14266)dis+1002_1:1_acc=on:au=on:bd=off:e2e=on:sgt=8:ss=axioms:i=922:si=on:rtra=on_0 on DTF2THF_14050 for (2985ds/922Mi)
% 16.46/6.85  % (14267)lrs+10_1:128_au=on:bd=off:cnfonf=lazy_not_gen_be_off:nwc=10.0:piset=and:i=182:si=on:rtra=on_0 on DTF2THF_14050 for (2984ds/182Mi)
% 16.46/6.86  % (14263)Instruction limit reached!
% 16.46/6.86  % (14263)------------------------------
% 16.46/6.86  % (14263)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.46/6.86  % (14263)Termination reason: Unknown
% 16.46/6.86  % (14263)Termination phase: shuffling
% 16.46/6.86  
% 16.46/6.86  % (14263)Memory used [KB]: 14711
% 16.46/6.86  % (14263)Time elapsed: 0.054 s
% 16.46/6.86  % (14263)Instructions burned: 79 (million)
% 16.46/6.86  % (14263)------------------------------
% 16.46/6.86  % (14263)------------------------------
% 16.46/6.87  % (14256)Instruction limit reached!
% 16.46/6.87  % (14256)------------------------------
% 16.46/6.87  % (14256)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.46/6.87  % (14256)Termination reason: Unknown
% 16.46/6.87  % (14256)Termination phase: shuffling
% 16.46/6.87  
% 16.46/6.87  % (14256)Memory used [KB]: 19957
% 16.46/6.87  % (14256)Time elapsed: 0.161 s
% 16.46/6.87  % (14256)Instructions burned: 460 (million)
% 16.46/6.87  % (14256)------------------------------
% 16.46/6.87  % (14256)------------------------------
% 16.46/6.88  % (14269)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=50:si=on:rtra=on_0 on DTF2THF_14050 for (2984ds/50Mi)
% 16.46/6.88  % (14265)Instruction limit reached!
% 16.46/6.88  % (14265)------------------------------
% 16.46/6.88  % (14265)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.46/6.88  % (14265)Termination reason: Unknown
% 16.46/6.88  % (14265)Termination phase: shuffling
% 16.46/6.88  
% 16.46/6.88  % (14265)Memory used [KB]: 15223
% 16.46/6.88  % (14265)Time elapsed: 0.046 s
% 16.46/6.88  % (14265)Instructions burned: 114 (million)
% 16.46/6.88  % (14265)------------------------------
% 16.46/6.88  % (14265)------------------------------
% 16.46/6.88  % (14268)ott+1010_1:1024_au=on:av=off:c=on:cnfonf=conj_eager:piset=equals:s2a=on:sp=occurrence:spb=units:ss=axioms:i=164:si=on:rtra=on_0 on DTF2THF_14050 for (2984ds/164Mi)
% 16.46/6.90  % (14270)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_14050 for (2984ds/368Mi)
% 17.08/6.90  % (14269)Instruction limit reached!
% 17.08/6.90  % (14269)------------------------------
% 17.08/6.90  % (14269)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.08/6.90  % (14269)Termination reason: Unknown
% 17.08/6.90  % (14269)Termination phase: shuffling
% 17.08/6.90  
% 17.08/6.90  % (14269)Memory used [KB]: 14328
% 17.08/6.90  % (14269)Time elapsed: 0.024 s
% 17.08/6.90  % (14269)Instructions burned: 52 (million)
% 17.08/6.90  % (14269)------------------------------
% 17.08/6.90  % (14269)------------------------------
% 17.08/6.90  % (14264)Instruction limit reached!
% 17.08/6.90  % (14264)------------------------------
% 17.08/6.90  % (14264)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.08/6.90  % (14264)Termination reason: Unknown
% 17.08/6.91  % (14264)Termination phase: shuffling
% 17.08/6.91  
% 17.08/6.91  % (14264)Memory used [KB]: 16630
% 17.08/6.91  % (14264)Time elapsed: 0.071 s
% 17.08/6.91  % (14264)Instructions burned: 218 (million)
% 17.08/6.91  % (14264)------------------------------
% 17.08/6.91  % (14264)------------------------------
% 17.08/6.91  % (14271)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1021:si=on:rtra=on_0 on DTF2THF_14050 for (2984ds/1021Mi)
% 17.08/6.91  % (14267)Instruction limit reached!
% 17.08/6.91  % (14267)------------------------------
% 17.08/6.91  % (14267)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.08/6.91  % (14267)Termination reason: Unknown
% 17.08/6.91  % (14267)Termination phase: shuffling
% 17.08/6.91  
% 17.08/6.91  % (14267)Memory used [KB]: 16247
% 17.08/6.91  % (14267)Time elapsed: 0.066 s
% 17.08/6.91  % (14267)Instructions burned: 184 (million)
% 17.08/6.91  % (14267)------------------------------
% 17.08/6.91  % (14267)------------------------------
% 17.08/6.92  % (14272)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=267:si=on:rtra=on_0 on DTF2THF_14050 for (2984ds/267Mi)
% 17.08/6.93  % (14273)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=204:si=on:rtra=on_0 on DTF2THF_14050 for (2984ds/204Mi)
% 17.08/6.98  % (14268)Instruction limit reached!
% 17.08/6.98  % (14268)------------------------------
% 17.08/6.98  % (14268)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.08/6.98  % (14268)Termination reason: Unknown
% 17.08/6.98  % (14268)Termination phase: shuffling
% 17.08/6.98  
% 17.08/6.98  % (14268)Memory used [KB]: 15991
% 17.08/6.98  % (14268)Time elapsed: 0.099 s
% 17.08/6.98  % (14268)Instructions burned: 165 (million)
% 17.08/6.98  % (14268)------------------------------
% 17.08/6.98  % (14268)------------------------------
% 17.08/7.00  % (14274)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_14050 for (2983ds/26Mi)
% 17.08/7.00  % (14272)Instruction limit reached!
% 17.08/7.00  % (14272)------------------------------
% 17.08/7.00  % (14272)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.08/7.00  % (14272)Termination reason: Unknown
% 17.08/7.00  % (14272)Termination phase: shuffling
% 17.08/7.00  
% 17.08/7.00  % (14272)Memory used [KB]: 19445
% 17.08/7.00  % (14272)Time elapsed: 0.086 s
% 17.08/7.00  % (14272)Instructions burned: 268 (million)
% 17.08/7.00  % (14272)------------------------------
% 17.08/7.00  % (14272)------------------------------
% 17.08/7.01  % (14273)Instruction limit reached!
% 17.08/7.01  % (14273)------------------------------
% 17.08/7.01  % (14273)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.08/7.01  % (14273)Termination reason: Unknown
% 17.08/7.01  % (14273)Termination phase: shuffling
% 17.08/7.01  
% 17.08/7.01  % (14273)Memory used [KB]: 16375
% 17.08/7.01  % (14273)Time elapsed: 0.086 s
% 17.08/7.01  % (14273)Instructions burned: 204 (million)
% 17.08/7.01  % (14273)------------------------------
% 17.08/7.01  % (14273)------------------------------
% 17.08/7.01  % (14274)Instruction limit reached!
% 17.08/7.01  % (14274)------------------------------
% 17.08/7.01  % (14274)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.08/7.01  % (14274)Termination reason: Unknown
% 17.08/7.01  % (14274)Termination phase: shuffling
% 17.08/7.01  
% 17.08/7.01  % (14274)Memory used [KB]: 13944
% 17.08/7.01  % (14274)Time elapsed: 0.018 s
% 17.08/7.01  % (14274)Instructions burned: 27 (million)
% 17.08/7.01  % (14274)------------------------------
% 17.08/7.01  % (14274)------------------------------
% 17.08/7.01  % (14275)dis+1002_1:1_bd=preordered:c=on:cbe=off:cnfonf=lazy_not_gen_be_off:hud=10:ixr=off:prag=on:i=53:si=on:rtra=on_0 on DTF2THF_14050 for (2983ds/53Mi)
% 17.08/7.02  % (14270)Instruction limit reached!
% 17.08/7.02  % (14270)------------------------------
% 17.08/7.02  % (14270)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.08/7.02  % (14270)Termination reason: Unknown
% 17.08/7.02  % (14270)Termination phase: shuffling
% 17.08/7.02  
% 17.08/7.02  % (14270)Memory used [KB]: 18421
% 17.08/7.02  % (14270)Time elapsed: 0.129 s
% 17.08/7.02  % (14270)Instructions burned: 368 (million)
% 17.08/7.02  % (14270)------------------------------
% 17.08/7.02  % (14270)------------------------------
% 17.08/7.02  % (14277)dis+1010_1:1_acc=model:avsq=on:ixr=off:prag=on:s2a=on:sac=on:i=1539:si=on:rtra=on_0 on DTF2THF_14050 for (2983ds/1539Mi)
% 18.31/7.03  % (14276)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=4436:si=on:rtra=on_0 on DTF2THF_14050 for (2983ds/4436Mi)
% 18.31/7.03  % (14278)lrs+2_1:1_bd=off:e2e=on:fde=none:kws=inv_precedence:plsq=on:plsqr=32,1:sos=on:i=408:si=on:rtra=on_0 on DTF2THF_14050 for (2982ds/408Mi)
% 18.31/7.04  % (14275)Instruction limit reached!
% 18.31/7.04  % (14275)------------------------------
% 18.31/7.04  % (14275)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.31/7.04  % (14275)Termination reason: Unknown
% 18.31/7.04  % (14275)Termination phase: shuffling
% 18.31/7.04  
% 18.31/7.04  % (14275)Memory used [KB]: 14328
% 18.31/7.04  % (14275)Time elapsed: 0.025 s
% 18.31/7.04  % (14275)Instructions burned: 53 (million)
% 18.31/7.04  % (14275)------------------------------
% 18.31/7.04  % (14275)------------------------------
% 18.31/7.05  % (14279)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=5205:si=on:rtra=on_0 on DTF2THF_14050 for (2982ds/5205Mi)
% 18.31/7.05  % (14259)Instruction limit reached!
% 18.31/7.05  % (14259)------------------------------
% 18.31/7.05  % (14259)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.31/7.05  % (14259)Termination reason: Unknown
% 18.31/7.05  % (14259)Termination phase: shuffling
% 18.31/7.05  
% 18.31/7.05  % (14259)Memory used [KB]: 33389
% 18.31/7.05  % (14259)Time elapsed: 0.296 s
% 18.31/7.05  % (14259)Instructions burned: 987 (million)
% 18.31/7.05  % (14259)------------------------------
% 18.31/7.05  % (14259)------------------------------
% 18.31/7.07  % (14280)dis+1010_1:4_au=on:av=off:cbe=off:nwc=10.0:s2a=on:s2at=3.0:sd=2:ss=axioms:st=5.0:i=1913:si=on:rtra=on_0 on DTF2THF_14050 for (2982ds/1913Mi)
% 18.59/7.16  % (14266)Instruction limit reached!
% 18.59/7.16  % (14266)------------------------------
% 18.59/7.16  % (14266)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.59/7.16  % (14266)Termination reason: Unknown
% 18.59/7.16  % (14266)Termination phase: shuffling
% 18.59/7.16  
% 18.59/7.16  % (14266)Memory used [KB]: 28016
% 18.59/7.16  % (14266)Time elapsed: 0.318 s
% 18.59/7.16  % (14266)Instructions burned: 922 (million)
% 18.59/7.16  % (14266)------------------------------
% 18.59/7.16  % (14266)------------------------------
% 18.59/7.17  % (14281)lrs+2_1:1_bd=off:cbe=off:piset=equals:sd=1:sgt=20:sos=all:ss=axioms:i=4350:si=on:rtra=on_0 on DTF2THF_14050 for (2981ds/4350Mi)
% 18.59/7.18  % (14278)Instruction limit reached!
% 18.59/7.18  % (14278)------------------------------
% 18.59/7.18  % (14278)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.59/7.18  % (14278)Termination reason: Unknown
% 18.59/7.18  % (14278)Termination phase: shuffling
% 18.59/7.18  
% 18.59/7.18  % (14278)Memory used [KB]: 19317
% 18.59/7.18  % (14278)Time elapsed: 0.144 s
% 18.59/7.18  % (14278)Instructions burned: 408 (million)
% 18.59/7.18  % (14278)------------------------------
% 18.59/7.18  % (14278)------------------------------
% 18.59/7.19  % (14282)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=68:si=on:rtra=on_0 on DTF2THF_14050 for (2981ds/68Mi)
% 18.85/7.22  % (14282)Instruction limit reached!
% 18.85/7.22  % (14282)------------------------------
% 18.85/7.22  % (14282)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.85/7.22  % (14282)Termination reason: Unknown
% 18.85/7.22  % (14282)Termination phase: shuffling
% 18.85/7.22  
% 18.85/7.22  % (14282)Memory used [KB]: 14456
% 18.85/7.22  % (14282)Time elapsed: 0.032 s
% 18.85/7.22  % (14282)Instructions burned: 68 (million)
% 18.85/7.22  % (14282)------------------------------
% 18.85/7.22  % (14282)------------------------------
% 18.85/7.23  % (14283)dis+10_1:1_au=on:nwc=5.0:s2a=on:s2at=2.0:sac=on:sd=7:ss=axioms:st=2.0:i=358:si=on:rtra=on_0 on DTF2THF_14050 for (2981ds/358Mi)
% 18.85/7.26  % (14271)Instruction limit reached!
% 18.85/7.26  % (14271)------------------------------
% 18.85/7.26  % (14271)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.85/7.26  % (14271)Termination reason: Unknown
% 18.85/7.26  % (14271)Termination phase: Property scanning
% 18.85/7.26  
% 18.85/7.26  % (14271)Memory used [KB]: 28272
% 18.85/7.26  % (14271)Time elapsed: 0.347 s
% 18.85/7.26  % (14271)Instructions burned: 1024 (million)
% 18.85/7.26  % (14271)------------------------------
% 18.85/7.26  % (14271)------------------------------
% 18.85/7.27  % (14284)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_14050 for (2980ds/31Mi)
% 18.85/7.29  % (14284)Instruction limit reached!
% 18.85/7.29  % (14284)------------------------------
% 18.85/7.29  % (14284)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.85/7.29  % (14284)Termination reason: Unknown
% 18.85/7.29  % (14284)Termination phase: shuffling
% 18.85/7.29  
% 18.85/7.29  % (14284)Memory used [KB]: 13944
% 18.85/7.29  % (14284)Time elapsed: 0.020 s
% 18.85/7.29  % (14284)Instructions burned: 34 (million)
% 18.85/7.29  % (14284)------------------------------
% 18.85/7.29  % (14284)------------------------------
% 19.00/7.30  % (14285)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_14050 for (2980ds/279Mi)
% 19.00/7.36  % (14283)Instruction limit reached!
% 19.00/7.36  % (14283)------------------------------
% 19.00/7.36  % (14283)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.00/7.36  % (14283)Termination reason: Unknown
% 19.00/7.36  % (14283)Termination phase: shuffling
% 19.00/7.36  
% 19.00/7.36  % (14283)Memory used [KB]: 18421
% 19.00/7.36  % (14283)Time elapsed: 0.127 s
% 19.00/7.36  % (14283)Instructions burned: 360 (million)
% 19.00/7.36  % (14283)------------------------------
% 19.00/7.36  % (14283)------------------------------
% 19.00/7.37  % (14286)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_14050 for (2979ds/258Mi)
% 19.00/7.39  % (14285)Instruction limit reached!
% 19.00/7.39  % (14285)------------------------------
% 19.00/7.39  % (14285)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.00/7.39  % (14285)Termination reason: Unknown
% 19.00/7.39  % (14285)Termination phase: shuffling
% 19.00/7.39  
% 19.00/7.39  % (14285)Memory used [KB]: 19701
% 19.00/7.39  % (14285)Time elapsed: 0.091 s
% 19.00/7.39  % (14285)Instructions burned: 281 (million)
% 19.00/7.39  % (14285)------------------------------
% 19.00/7.39  % (14285)------------------------------
% 19.24/7.40  % (14287)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_14050 for (2979ds/26Mi)
% 19.24/7.42  % (14287)Instruction limit reached!
% 19.24/7.42  % (14287)------------------------------
% 19.24/7.42  % (14287)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.24/7.42  % (14287)Termination reason: Unknown
% 19.24/7.42  % (14287)Termination phase: shuffling
% 19.24/7.42  
% 19.24/7.42  % (14287)Memory used [KB]: 13944
% 19.24/7.42  % (14287)Time elapsed: 0.018 s
% 19.24/7.42  % (14287)Instructions burned: 28 (million)
% 19.24/7.42  % (14287)------------------------------
% 19.24/7.42  % (14287)------------------------------
% 19.24/7.43  % (14288)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_be_gen:ntd=on:pe=on:i=995:si=on:rtra=on_0 on DTF2THF_14050 for (2979ds/995Mi)
% 19.24/7.46  % (14286)Instruction limit reached!
% 19.24/7.46  % (14286)------------------------------
% 19.24/7.46  % (14286)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.24/7.46  % (14286)Termination reason: Unknown
% 19.24/7.46  % (14286)Termination phase: shuffling
% 19.24/7.46  
% 19.24/7.46  % (14286)Memory used [KB]: 17142
% 19.24/7.46  % (14286)Time elapsed: 0.094 s
% 19.24/7.46  % (14286)Instructions burned: 258 (million)
% 19.24/7.46  % (14286)------------------------------
% 19.24/7.46  % (14286)------------------------------
% 19.24/7.47  % (14289)dis+10_1:8_amm=sco:cbe=off:hflw=5:hfsq=on:hfsqc=14:hfsql=off:hfsqr=4,1:hud=5:ile=on:lma=on:nm=2:ntd=on:prag=on:sac=on:sp=frequency:i=373:si=on:rtra=on_0 on DTF2THF_14050 for (2978ds/373Mi)
% 19.38/7.60  % (14289)Instruction limit reached!
% 19.38/7.60  % (14289)------------------------------
% 19.38/7.60  % (14289)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.38/7.60  % (14289)Termination reason: Unknown
% 19.38/7.60  % (14289)Termination phase: shuffling
% 19.38/7.60  
% 19.38/7.60  % (14289)Memory used [KB]: 18805
% 19.38/7.60  % (14289)Time elapsed: 0.135 s
% 19.38/7.60  % (14289)Instructions burned: 374 (million)
% 19.38/7.60  % (14289)------------------------------
% 19.38/7.60  % (14289)------------------------------
% 19.38/7.62  % (14320)lrs+1010_1:128_au=on:cnfonf=lazy_pi_sigma_gen:hflw=4:hfsq=on:hfsqc=20:hfsql=off:i=253:si=on:rtra=on_0 on DTF2THF_14050 for (2977ds/253Mi)
% 19.38/7.65  % (14277)Instruction limit reached!
% 19.38/7.65  % (14277)------------------------------
% 19.38/7.65  % (14277)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.38/7.65  % (14277)Termination reason: Unknown
% 19.38/7.65  % (14277)Termination phase: Property scanning
% 19.38/7.65  
% 19.38/7.65  % (14277)Memory used [KB]: 28144
% 19.38/7.65  % (14277)Time elapsed: 0.629 s
% 19.38/7.65  % (14277)Instructions burned: 1540 (million)
% 19.38/7.65  % (14277)------------------------------
% 19.38/7.65  % (14277)------------------------------
% 19.38/7.66  % (14280)Instruction limit reached!
% 19.38/7.66  % (14280)------------------------------
% 19.38/7.66  % (14280)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.38/7.66  % (14280)Termination reason: Unknown
% 19.38/7.66  % (14280)Termination phase: SInE selection
% 19.38/7.66  
% 19.38/7.66  % (14280)Memory used [KB]: 28912
% 19.38/7.66  % (14280)Time elapsed: 0.599 s
% 19.38/7.66  % (14280)Instructions burned: 1913 (million)
% 19.38/7.66  % (14280)------------------------------
% 19.38/7.66  % (14280)------------------------------
% 19.38/7.67  % (14321)lrs+1002_1:14_au=on:bd=off:e2e=on:fde=unused:sac=on:sos=on:i=832:si=on:rtra=on_0 on DTF2THF_14050 for (2976ds/832Mi)
% 19.38/7.68  % (14322)lrs+1010_1:1_chr=on:cnfonf=lazy_not_be_gen:gs=on:hud=5:plsq=on:plsqc=1:plsqr=5,1:prag=on:rawr=on:i=668:si=on:rtra=on_0 on DTF2THF_14050 for (2976ds/668Mi)
% 19.38/7.70  % (14320)Instruction limit reached!
% 19.38/7.70  % (14320)------------------------------
% 19.38/7.70  % (14320)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.38/7.70  % (14320)Termination reason: Unknown
% 19.38/7.70  % (14320)Termination phase: shuffling
% 19.38/7.70  
% 19.38/7.70  % (14320)Memory used [KB]: 17270
% 19.38/7.70  % (14320)Time elapsed: 0.081 s
% 19.38/7.70  % (14320)Instructions burned: 255 (million)
% 19.38/7.70  % (14320)------------------------------
% 19.38/7.70  % (14320)------------------------------
% 20.77/7.71  % (14323)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=40:si=on:rtra=on_0 on DTF2THF_14050 for (2976ds/40Mi)
% 20.77/7.72  % (14288)Instruction limit reached!
% 20.77/7.72  % (14288)------------------------------
% 20.77/7.72  % (14288)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.77/7.72  % (14288)Termination reason: Unknown
% 20.77/7.72  % (14288)Termination phase: shuffling
% 20.77/7.72  
% 20.77/7.72  % (14288)Memory used [KB]: 33389
% 20.77/7.72  % (14288)Time elapsed: 0.296 s
% 20.77/7.72  % (14288)Instructions burned: 995 (million)
% 20.77/7.72  % (14288)------------------------------
% 20.77/7.72  % (14288)------------------------------
% 20.77/7.73  % (14323)Instruction limit reached!
% 20.77/7.73  % (14323)------------------------------
% 20.77/7.73  % (14323)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.77/7.73  % (14323)Termination reason: Unknown
% 20.77/7.73  % (14323)Termination phase: shuffling
% 20.77/7.73  
% 20.77/7.73  % (14323)Memory used [KB]: 14200
% 20.77/7.73  % (14323)Time elapsed: 0.022 s
% 20.77/7.73  % (14323)Instructions burned: 42 (million)
% 20.77/7.73  % (14323)------------------------------
% 20.77/7.73  % (14323)------------------------------
% 20.77/7.74  % (14324)lrs+1010_1:1_cnfonf=lazy_pi_sigma_gen:fd=off:ntd=on:sd=1:sgt=32:sos=on:ss=axioms:i=294:si=on:rtra=on_0 on DTF2THF_14050 for (2976ds/294Mi)
% 20.77/7.74  % (14325)dis+1010_1:1_avsq=on:fe=off:fsr=off:hud=10:plsq=on:plsqr=32,1:prag=on:i=187:si=on:rtra=on_0 on DTF2THF_14050 for (2976ds/187Mi)
% 21.20/7.81  % (14325)Instruction limit reached!
% 21.20/7.81  % (14325)------------------------------
% 21.20/7.81  % (14325)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.20/7.81  % (14325)Termination reason: Unknown
% 21.20/7.81  % (14325)Termination phase: shuffling
% 21.20/7.81  
% 21.20/7.81  % (14325)Memory used [KB]: 16119
% 21.20/7.81  % (14325)Time elapsed: 0.069 s
% 21.20/7.81  % (14325)Instructions burned: 188 (million)
% 21.20/7.81  % (14325)------------------------------
% 21.20/7.81  % (14325)------------------------------
% 21.20/7.82  % (14326)lrs+1010_2:3_au=on:fsr=off:sd=4:ss=axioms:st=1.5:i=371:si=on:rtra=on_0 on DTF2THF_14050 for (2975ds/371Mi)
% 21.20/7.83  % (14324)Instruction limit reached!
% 21.20/7.83  % (14324)------------------------------
% 21.20/7.83  % (14324)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.20/7.83  % (14324)Termination reason: Unknown
% 21.20/7.83  % (14324)Termination phase: shuffling
% 21.20/7.83  
% 21.20/7.83  % (14324)Memory used [KB]: 19829
% 21.20/7.83  % (14324)Time elapsed: 0.093 s
% 21.20/7.83  % (14324)Instructions burned: 297 (million)
% 21.20/7.83  % (14324)------------------------------
% 21.20/7.83  % (14324)------------------------------
% 21.20/7.84  % (14327)dis+1003_1:1_au=on:bd=preordered:fde=none:ixr=off:nwc=10.0:i=1827:si=on:rtra=on_0 on DTF2THF_14050 for (2975ds/1827Mi)
% 21.20/7.88  % (14322)Instruction limit reached!
% 21.20/7.88  % (14322)------------------------------
% 21.20/7.88  % (14322)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.20/7.88  % (14322)Termination reason: Unknown
% 21.20/7.88  % (14322)Termination phase: shuffling
% 21.20/7.88  
% 21.20/7.88  % (14322)Memory used [KB]: 24946
% 21.20/7.88  % (14322)Time elapsed: 0.201 s
% 21.20/7.88  % (14322)Instructions burned: 671 (million)
% 21.20/7.88  % (14322)------------------------------
% 21.20/7.88  % (14322)------------------------------
% 21.20/7.89  % (14328)lrs+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:tnu=1:i=70:si=on:rtra=on_0 on DTF2THF_14050 for (2974ds/70Mi)
% 21.59/7.91  % (14276)First to succeed.
% 21.59/7.92  % (14328)Instruction limit reached!
% 21.59/7.92  % (14328)------------------------------
% 21.59/7.92  % (14328)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.59/7.92  % (14328)Termination reason: Unknown
% 21.59/7.92  % (14328)Termination phase: shuffling
% 21.59/7.92  
% 21.59/7.92  % (14328)Memory used [KB]: 14583
% 21.59/7.92  % (14328)Time elapsed: 0.030 s
% 21.59/7.92  % (14328)Instructions burned: 70 (million)
% 21.59/7.92  % (14328)------------------------------
% 21.59/7.92  % (14328)------------------------------
% 21.59/7.92  % (14276)Refutation found. Thanks to Tanya!
% 21.59/7.92  % SZS status Theorem for DTF2THF_14050
% 21.59/7.92  % SZS output start Proof for DTF2THF_14050
% 21.59/7.92  thf(type_def_5, type, rat: $tType).
% 21.59/7.92  thf(type_def_6, type, int: $tType).
% 21.59/7.92  thf(type_def_8, type, real: $tType).
% 21.59/7.92  thf(type_def_9, type, set_Pr1261947904930325089at_nat: $tType).
% 21.59/7.92  thf(type_def_10, type, nat: $tType).
% 21.59/7.92  thf(type_def_11, type, product_prod_int_int: $tType).
% 21.59/7.92  thf(type_def_12, type, product_prod_nat_nat: $tType).
% 21.59/7.92  thf(type_def_13, type, complex: $tType).
% 21.59/7.92  thf(type_def_14, type, num: $tType).
% 21.59/7.92  thf(type_def_15, type, option_num: $tType).
% 21.59/7.92  thf(type_def_16, type, product_prod_num_num: $tType).
% 21.59/7.92  thf(type_def_17, type, code_integer: $tType).
% 21.59/7.92  thf(type_def_18, type, produc6271795597528267376eger_o: $tType).
% 21.59/7.92  thf(type_def_19, type, produc8923325533196201883nteger: $tType).
% 21.59/7.92  thf(type_def_20, type, set_fi4554929511873752355omplex: $tType).
% 21.59/7.92  thf(type_def_21, type, filter6041513312241820739omplex: $tType).
% 21.59/7.92  thf(type_def_22, type, set_fi7789364187291644575l_real: $tType).
% 21.59/7.92  thf(type_def_23, type, filter2146258269922977983l_real: $tType).
% 21.59/7.92  thf(type_def_24, type, set_nat: $tType).
% 21.59/7.92  thf(type_def_25, type, set_set_nat: $tType).
% 21.59/7.92  thf(type_def_26, type, filter_real: $tType).
% 21.59/7.92  thf(type_def_27, type, filter_nat: $tType).
% 21.59/7.92  thf(type_def_28, type, filter1242075044329608583at_nat: $tType).
% 21.59/7.92  thf(type_def_29, type, set_Pr5085853215250843933omplex: $tType).
% 21.59/7.92  thf(type_def_30, type, set_Pr6218003697084177305l_real: $tType).
% 21.59/7.92  thf(type_def_31, type, set_complex: $tType).
% 21.59/7.92  thf(type_def_32, type, set_list_nat: $tType).
% 21.59/7.92  thf(type_def_33, type, set_Product_unit: $tType).
% 21.59/7.92  thf(type_def_34, type, set_o: $tType).
% 21.59/7.92  thf(type_def_35, type, set_int: $tType).
% 21.59/7.92  thf(type_def_36, type, set_list_o: $tType).
% 21.59/7.92  thf(type_def_37, type, set_list_complex: $tType).
% 21.59/7.92  thf(type_def_38, type, set_list_int: $tType).
% 21.59/7.92  thf(type_def_39, type, set_li5450038453877631591at_nat: $tType).
% 21.59/7.92  thf(type_def_40, type, set_list_VEBT_VEBT: $tType).
% 21.59/7.92  thf(type_def_41, type, set_num: $tType).
% 21.59/7.92  thf(type_def_42, type, set_Pr4329608150637261639at_nat: $tType).
% 21.59/7.92  thf(type_def_43, type, set_rat: $tType).
% 21.59/7.92  thf(type_def_44, type, set_real: $tType).
% 21.59/7.92  thf(type_def_45, type, set_set_complex: $tType).
% 21.59/7.92  thf(type_def_46, type, set_set_int: $tType).
% 21.59/7.92  thf(type_def_47, type, set_se7855581050983116737at_nat: $tType).
% 21.59/7.92  thf(type_def_48, type, set_VEBT_VEBT: $tType).
% 21.59/7.92  thf(type_def_49, type, set_Pr8693737435421807431at_nat: $tType).
% 21.59/7.92  thf(type_def_50, type, produc1319942482725812455at_nat: $tType).
% 21.59/7.92  thf(type_def_51, type, list_nat: $tType).
% 21.59/7.92  thf(type_def_52, type, extended_enat: $tType).
% 21.59/7.92  thf(type_def_53, type, produc3843707927480180839at_nat: $tType).
% 21.59/7.92  thf(type_def_54, type, vEBT_VEBT: $tType).
% 21.59/7.92  thf(type_def_55, type, list_o: $tType).
% 21.59/7.92  thf(type_def_56, type, produc8763457246119570046nteger: $tType).
% 21.59/7.92  thf(type_def_57, type, produc7773217078559923341nt_int: $tType).
% 21.59/7.92  thf(type_def_58, type, produc1908205239877642774nteger: $tType).
% 21.59/7.92  thf(type_def_59, type, produc2285326912895808259nt_int: $tType).
% 21.59/7.92  thf(type_def_60, type, list_int: $tType).
% 21.59/7.92  thf(type_def_61, type, option_nat: $tType).
% 21.59/7.92  thf(type_def_62, type, set_Pr958786334691620121nt_int: $tType).
% 21.59/7.92  thf(type_def_63, type, list_complex: $tType).
% 21.59/7.92  thf(type_def_64, type, list_P6011104703257516679at_nat: $tType).
% 21.59/7.92  thf(type_def_65, type, list_real: $tType).
% 21.59/7.92  thf(type_def_66, type, list_set_nat: $tType).
% 21.59/7.92  thf(type_def_67, type, list_VEBT_VEBT: $tType).
% 21.59/7.92  thf(type_def_68, type, list_P7333126701944960589_nat_o: $tType).
% 21.59/7.92  thf(type_def_69, type, list_P3521021558325789923at_int: $tType).
% 21.59/7.92  thf(type_def_70, type, list_P5647936690300460905T_VEBT: $tType).
% 21.59/7.92  thf(type_def_71, type, option_o: $tType).
% 21.59/7.92  thf(type_def_72, type, option_int: $tType).
% 21.59/7.92  thf(type_def_73, type, list_num: $tType).
% 21.59/7.92  thf(type_def_74, type, option4927543243414619207at_nat: $tType).
% 21.59/7.92  thf(type_def_75, type, option_real: $tType).
% 21.59/7.92  thf(type_def_76, type, option_set_nat: $tType).
% 21.59/7.92  thf(type_def_77, type, option_VEBT_VEBT: $tType).
% 21.59/7.92  thf(type_def_78, type, list_C878401137130745250e_term: $tType).
% 21.59/7.92  thf(type_def_79, type, set_Co9149898834107579976e_term: $tType).
% 21.59/7.92  thf(type_def_80, type, list_i8448526496819171953e_term: $tType).
% 21.59/7.92  thf(type_def_81, type, set_in3461395444621081367e_term: $tType).
% 21.59/7.92  thf(type_def_82, type, list_P1316552470764441098e_term: $tType).
% 21.59/7.92  thf(type_def_83, type, set_Pr7604974323444597168e_term: $tType).
% 21.59/7.92  thf(type_def_84, type, list_P1743416141875011707e_term: $tType).
% 21.59/7.92  thf(type_def_85, type, set_Pr3642885161833720865e_term: $tType).
% 21.59/7.92  thf(type_def_86, type, list_P5311841565141990158nteger: $tType).
% 21.59/7.92  thf(type_def_87, type, set_Pr8056137968301705908nteger: $tType).
% 21.59/7.92  thf(type_def_88, type, list_P8915022641806594461nt_int: $tType).
% 21.59/7.92  thf(type_def_89, type, set_Pr1872883991513573699nt_int: $tType).
% 21.59/7.92  thf(type_def_90, type, list_P7828571989066258726nteger: $tType).
% 21.59/7.92  thf(type_def_91, type, set_Pr1281608226676607948nteger: $tType).
% 21.59/7.92  thf(type_def_92, type, list_P651320350408439699nt_int: $tType).
% 21.59/7.92  thf(type_def_93, type, set_Pr9222295170931077689nt_int: $tType).
% 21.59/7.92  thf(type_def_94, type, list_P4002435161011370285od_o_o: $tType).
% 21.59/7.92  thf(type_def_95, type, set_Product_prod_o_o: $tType).
% 21.59/7.92  thf(type_def_96, type, list_P3795440434834930179_o_int: $tType).
% 21.59/7.92  thf(type_def_97, type, set_Pr8834758594704517033_o_int: $tType).
% 21.59/7.92  thf(type_def_98, type, list_P6285523579766656935_o_nat: $tType).
% 21.59/7.92  thf(type_def_99, type, set_Pr2101469702781467981_o_nat: $tType).
% 21.59/7.92  thf(type_def_100, type, list_P5232166724548748803o_real: $tType).
% 21.59/7.92  thf(type_def_101, type, set_Pr6573716822653411497o_real: $tType).
% 21.59/7.92  thf(type_def_102, type, list_P7495141550334521929T_VEBT: $tType).
% 21.59/7.92  thf(type_def_103, type, set_Pr7543698050874017315T_VEBT: $tType).
% 21.59/7.92  thf(type_def_104, type, list_P5578671422887162913nteger: $tType).
% 21.59/7.92  thf(type_def_105, type, set_Pr4811707699266497531nteger: $tType).
% 21.59/7.92  thf(type_def_106, type, list_P5707943133018811711nt_int: $tType).
% 21.59/7.92  thf(type_def_107, type, list_P6863124054624500543t_real: $tType).
% 21.59/7.92  thf(type_def_108, type, set_Pr3538720872664544793t_real: $tType).
% 21.59/7.92  thf(type_def_109, type, list_P7524865323317820941T_VEBT: $tType).
% 21.59/7.92  thf(type_def_110, type, set_Pr8044002425091019955T_VEBT: $tType).
% 21.59/7.92  thf(type_def_111, type, list_P8469869581646625389at_nat: $tType).
% 21.59/7.92  thf(type_def_112, type, list_P3595434254542482545real_o: $tType).
% 21.59/7.92  thf(type_def_113, type, set_Pr4936984352647145239real_o: $tType).
% 21.59/7.92  thf(type_def_114, type, list_P4344331454722006975al_int: $tType).
% 21.59/7.92  thf(type_def_115, type, set_Pr1019928272762051225al_int: $tType).
% 21.59/7.92  thf(type_def_116, type, list_P6834414599653733731al_nat: $tType).
% 21.59/7.92  thf(type_def_117, type, set_Pr3510011417693777981al_nat: $tType).
% 21.59/7.92  thf(type_def_118, type, list_P8689742595348180415l_real: $tType).
% 21.59/7.92  thf(type_def_119, type, list_P877281246627933069T_VEBT: $tType).
% 21.59/7.92  thf(type_def_120, type, set_Pr6019664923565264691T_VEBT: $tType).
% 21.59/7.92  thf(type_def_121, type, list_P6254988961118846195et_nat: $tType).
% 21.59/7.92  thf(type_def_122, type, set_Pr5488025237498180813et_nat: $tType).
% 21.59/7.92  thf(type_def_123, type, list_P5464809261938338413at_nat: $tType).
% 21.59/7.92  thf(type_def_124, type, list_P3126845725202233233VEBT_o: $tType).
% 21.59/7.92  thf(type_def_125, type, set_Pr3175402225741728619VEBT_o: $tType).
% 21.59/7.92  thf(type_def_126, type, list_P4547456442757143711BT_int: $tType).
% 21.59/7.92  thf(type_def_127, type, set_Pr5066593544530342725BT_int: $tType).
% 21.59/7.92  thf(type_def_128, type, list_P7037539587688870467BT_nat: $tType).
% 21.59/7.92  thf(type_def_129, type, set_Pr7556676689462069481BT_nat: $tType).
% 21.59/7.92  thf(type_def_130, type, list_P2623026923184700063T_real: $tType).
% 21.59/7.92  thf(type_def_131, type, set_Pr7765410600122031685T_real: $tType).
% 21.59/7.92  thf(type_def_132, type, list_P7413028617227757229T_VEBT: $tType).
% 21.59/7.92  thf(type_def_133, type, set_Pr6192946355708809607T_VEBT: $tType).
% 21.59/7.92  thf(type_def_134, type, option6357759511663192854e_term: $tType).
% 21.59/7.92  thf(type_def_135, type, produc6241069584506657477e_term: $tType).
% 21.59/7.92  thf(type_def_136, type, produc8551481072490612790e_term: $tType).
% 21.59/7.92  thf(type_def_137, type, produc8243902056947475879T_VEBT: $tType).
% 21.59/7.92  thf(type_def_138, type, set_Pr6227168374412355847list_o: $tType).
% 21.59/7.92  thf(type_def_139, type, set_Pr765067013931698361st_int: $tType).
% 21.59/7.92  thf(type_def_140, type, set_Pr3451248702717554689st_nat: $tType).
% 21.59/7.92  thf(type_def_141, type, set_Pr1916528119006554503T_VEBT: $tType).
% 21.59/7.92  thf(type_def_142, type, product_prod_o_o: $tType).
% 21.59/7.92  thf(type_def_143, type, product_prod_o_int: $tType).
% 21.59/7.92  thf(type_def_144, type, product_prod_o_nat: $tType).
% 21.59/7.92  thf(type_def_145, type, produc2504756804600209347T_VEBT: $tType).
% 21.59/7.92  thf(type_def_146, type, product_prod_nat_o: $tType).
% 21.59/7.92  thf(type_def_147, type, product_prod_nat_int: $tType).
% 21.59/7.92  thf(type_def_148, type, produc8025551001238799321T_VEBT: $tType).
% 21.59/7.92  thf(type_def_149, type, produc334124729049499915VEBT_o: $tType).
% 21.59/7.92  thf(type_def_150, type, produc4894624898956917775BT_int: $tType).
% 21.59/7.92  thf(type_def_151, type, produc9072475918466114483BT_nat: $tType).
% 21.59/7.92  thf(type_def_152, type, set_Code_integer: $tType).
% 21.59/7.92  thf(type_def_153, type, produc4953844613479565601on_nat: $tType).
% 21.59/7.92  thf(type_def_154, type, produc2233624965454879586on_nat: $tType).
% 21.59/7.92  thf(type_def_155, type, produc8306885398267862888on_nat: $tType).
% 21.59/7.92  thf(type_def_156, type, produc3447558737645232053on_num: $tType).
% 21.59/7.92  thf(type_def_157, type, produc7036089656553540234on_num: $tType).
% 21.59/7.92  thf(type_def_158, type, produc1193250871479095198on_num: $tType).
% 21.59/7.92  thf(type_def_159, type, produc6121120109295599847at_nat: $tType).
% 21.59/7.92  thf(type_def_160, type, produc5491161045314408544at_nat: $tType).
% 21.59/7.92  thf(type_def_161, type, produc5542196010084753463at_nat: $tType).
% 21.59/7.92  thf(type_def_162, type, product_prod_o_real: $tType).
% 21.59/7.92  thf(type_def_163, type, produc679980390762269497t_real: $tType).
% 21.59/7.92  thf(type_def_164, type, produc1531783533982839933T_VEBT: $tType).
% 21.59/7.92  thf(type_def_165, type, produc7102631898165422375list_o: $tType).
% 21.59/7.92  thf(type_def_166, type, produc1186641810826059865st_int: $tType).
% 21.59/7.92  thf(type_def_167, type, produc1828647624359046049st_nat: $tType).
% 21.59/7.92  thf(type_def_168, type, produc9211091688327510695T_VEBT: $tType).
% 21.59/7.92  thf(type_def_169, type, product_prod_nat_num: $tType).
% 21.59/7.92  thf(type_def_170, type, produc859450856879609959at_nat: $tType).
% 21.59/7.92  thf(type_def_171, type, product_prod_real_o: $tType).
% 21.59/7.92  thf(type_def_172, type, produc8786904178792722361al_int: $tType).
% 21.59/7.92  thf(type_def_173, type, produc3741383161447143261al_nat: $tType).
% 21.59/7.92  thf(type_def_174, type, produc2422161461964618553l_real: $tType).
% 21.59/7.92  thf(type_def_175, type, produc3757001726724277373T_VEBT: $tType).
% 21.59/7.92  thf(type_def_176, type, produc7819656566062154093et_nat: $tType).
% 21.59/7.92  thf(type_def_177, type, produc5170161368751668367T_real: $tType).
% 21.59/7.92  thf(type_def_178, type, produc4411394909380815293omplex: $tType).
% 21.59/7.92  thf(func_def_0, type, produc1319942482725812455at_nat: $tType).
% 21.59/7.92  thf(func_def_1, type, produc5542196010084753463at_nat: $tType).
% 21.59/7.92  thf(func_def_2, type, list_P7828571989066258726nteger: $tType).
% 21.59/7.92  thf(func_def_3, type, set_Pr1281608226676607948nteger: $tType).
% 21.59/7.92  thf(func_def_4, type, produc1908205239877642774nteger: $tType).
% 21.59/7.92  thf(func_def_5, type, produc5491161045314408544at_nat: $tType).
% 21.59/7.92  thf(func_def_6, type, list_P651320350408439699nt_int: $tType).
% 21.59/7.92  thf(func_def_7, type, set_Pr9222295170931077689nt_int: $tType).
% 21.59/7.92  thf(func_def_8, type, produc2285326912895808259nt_int: $tType).
% 21.59/7.92  thf(func_def_9, type, list_P5311841565141990158nteger: $tType).
% 21.59/7.92  thf(func_def_10, type, set_Pr8056137968301705908nteger: $tType).
% 21.59/7.92  thf(func_def_11, type, produc8763457246119570046nteger: $tType).
% 21.59/7.92  thf(func_def_12, type, list_P1316552470764441098e_term: $tType).
% 21.59/7.92  thf(func_def_13, type, set_Pr7604974323444597168e_term: $tType).
% 21.59/7.92  thf(func_def_14, type, list_P1743416141875011707e_term: $tType).
% 21.59/7.92  thf(func_def_15, type, set_Pr3642885161833720865e_term: $tType).
% 21.59/7.92  thf(func_def_16, type, list_P8915022641806594461nt_int: $tType).
% 21.59/7.92  thf(func_def_17, type, set_Pr1872883991513573699nt_int: $tType).
% 21.59/7.92  thf(func_def_18, type, produc7773217078559923341nt_int: $tType).
% 21.59/7.92  thf(func_def_19, type, list_P5464809261938338413at_nat: $tType).
% 21.59/7.92  thf(func_def_20, type, set_Pr4329608150637261639at_nat: $tType).
% 21.59/7.92  thf(func_def_21, type, produc1193250871479095198on_num: $tType).
% 21.59/7.92  thf(func_def_22, type, produc8306885398267862888on_nat: $tType).
% 21.59/7.92  thf(func_def_23, type, produc6121120109295599847at_nat: $tType).
% 21.59/7.92  thf(func_def_24, type, produc7036089656553540234on_num: $tType).
% 21.59/7.92  thf(func_def_25, type, produc2233624965454879586on_nat: $tType).
% 21.59/7.92  thf(func_def_26, type, produc3843707927480180839at_nat: $tType).
% 21.59/7.92  thf(func_def_27, type, list_P8469869581646625389at_nat: $tType).
% 21.59/7.92  thf(func_def_28, type, list_C878401137130745250e_term: $tType).
% 21.59/7.92  thf(func_def_29, type, set_Pr8693737435421807431at_nat: $tType).
% 21.59/7.92  thf(func_def_30, type, set_Co9149898834107579976e_term: $tType).
% 21.59/7.92  thf(func_def_31, type, list_i8448526496819171953e_term: $tType).
% 21.59/7.92  thf(func_def_32, type, set_Pr1916528119006554503T_VEBT: $tType).
% 21.59/7.92  thf(func_def_33, type, produc859450856879609959at_nat: $tType).
% 21.59/7.92  thf(func_def_34, type, set_in3461395444621081367e_term: $tType).
% 21.59/7.92  thf(func_def_35, type, produc9211091688327510695T_VEBT: $tType).
% 21.59/7.92  thf(func_def_36, type, produc6241069584506657477e_term: $tType).
% 21.59/7.92  thf(func_def_37, type, set_fi4554929511873752355omplex: $tType).
% 21.59/7.92  thf(func_def_38, type, set_Pr3451248702717554689st_nat: $tType).
% 21.59/7.92  thf(func_def_39, type, set_Pr765067013931698361st_int: $tType).
% 21.59/7.92  thf(func_def_40, type, list_P7413028617227757229T_VEBT: $tType).
% 21.59/7.92  thf(func_def_41, type, produc8551481072490612790e_term: $tType).
% 21.59/7.92  thf(func_def_42, type, set_Pr6192946355708809607T_VEBT: $tType).
% 21.59/7.92  thf(func_def_43, type, list_P6254988961118846195et_nat: $tType).
% 21.59/7.92  thf(func_def_44, type, list_P5578671422887162913nteger: $tType).
% 21.59/7.92  thf(func_def_45, type, set_Pr5488025237498180813et_nat: $tType).
% 21.59/7.92  thf(func_def_46, type, set_Pr4811707699266497531nteger: $tType).
% 21.59/7.92  thf(func_def_47, type, option6357759511663192854e_term: $tType).
% 21.59/7.92  thf(func_def_48, type, produc3447558737645232053on_num: $tType).
% 21.59/7.92  thf(func_def_49, type, produc4953844613479565601on_nat: $tType).
% 21.59/7.92  thf(func_def_50, type, set_fi7789364187291644575l_real: $tType).
% 21.59/7.92  thf(func_def_51, type, filter6041513312241820739omplex: $tType).
% 21.59/7.92  thf(func_def_52, type, list_P2623026923184700063T_real: $tType).
% 21.59/7.92  thf(func_def_53, type, list_P877281246627933069T_VEBT: $tType).
% 21.59/7.92  thf(func_def_54, type, produc1828647624359046049st_nat: $tType).
% 21.59/7.92  thf(func_def_55, type, produc1186641810826059865st_int: $tType).
% 21.59/7.92  thf(func_def_56, type, set_Pr7765410600122031685T_real: $tType).
% 21.59/7.92  thf(func_def_57, type, set_Pr6019664923565264691T_VEBT: $tType).
% 21.59/7.92  thf(func_def_58, type, list_P7037539587688870467BT_nat: $tType).
% 21.59/7.92  thf(func_def_59, type, list_P4547456442757143711BT_int: $tType).
% 21.59/7.92  thf(func_def_60, type, list_P5647936690300460905T_VEBT: $tType).
% 21.59/7.92  thf(func_def_61, type, list_P7524865323317820941T_VEBT: $tType).
% 21.59/7.92  thf(func_def_62, type, set_Pr6227168374412355847list_o: $tType).
% 21.59/7.92  thf(func_def_63, type, set_li5450038453877631591at_nat: $tType).
% 21.59/7.92  thf(func_def_64, type, produc8243902056947475879T_VEBT: $tType).
% 21.59/7.92  thf(func_def_65, type, set_Pr7556676689462069481BT_nat: $tType).
% 21.59/7.92  thf(func_def_66, type, set_Pr5066593544530342725BT_int: $tType).
% 21.59/7.92  thf(func_def_67, type, set_Pr8044002425091019955T_VEBT: $tType).
% 21.59/7.92  thf(func_def_68, type, set_se7855581050983116737at_nat: $tType).
% 21.59/7.92  thf(func_def_69, type, set_Pr5085853215250843933omplex: $tType).
% 21.59/7.92  thf(func_def_70, type, produc7819656566062154093et_nat: $tType).
% 21.59/7.92  thf(func_def_71, type, produc8923325533196201883nteger: $tType).
% 21.59/7.92  thf(func_def_72, type, list_P3126845725202233233VEBT_o: $tType).
% 21.59/7.92  thf(func_def_73, type, list_P7495141550334521929T_VEBT: $tType).
% 21.59/7.92  thf(func_def_74, type, filter2146258269922977983l_real: $tType).
% 21.59/7.92  thf(func_def_75, type, set_Pr3175402225741728619VEBT_o: $tType).
% 21.59/7.92  thf(func_def_76, type, set_Pr7543698050874017315T_VEBT: $tType).
% 21.59/7.92  thf(func_def_77, type, option4927543243414619207at_nat: $tType).
% 21.59/7.92  thf(func_def_78, type, list_P8689742595348180415l_real: $tType).
% 21.59/7.92  thf(func_def_79, type, filter1242075044329608583at_nat: $tType).
% 21.59/7.92  thf(func_def_80, type, set_Pr6218003697084177305l_real: $tType).
% 21.59/7.92  thf(func_def_81, type, list_P6834414599653733731al_nat: $tType).
% 21.59/7.92  thf(func_def_82, type, list_P4344331454722006975al_int: $tType).
% 21.59/7.92  thf(func_def_83, type, list_P6863124054624500543t_real: $tType).
% 21.59/7.92  thf(func_def_84, type, produc5170161368751668367T_real: $tType).
% 21.59/7.92  thf(func_def_85, type, produc3757001726724277373T_VEBT: $tType).
% 21.59/7.92  thf(func_def_86, type, set_Pr3510011417693777981al_nat: $tType).
% 21.59/7.92  thf(func_def_87, type, set_Pr1019928272762051225al_int: $tType).
% 21.59/7.92  thf(func_def_88, type, set_Pr3538720872664544793t_real: $tType).
% 21.59/7.92  thf(func_def_89, type, produc7102631898165422375list_o: $tType).
% 21.59/7.92  thf(func_def_90, type, list_P6011104703257516679at_nat: $tType).
% 21.59/7.92  thf(func_def_91, type, list_P3521021558325789923at_int: $tType).
% 21.59/7.92  thf(func_def_92, type, list_P5707943133018811711nt_int: $tType).
% 21.59/7.92  thf(func_def_93, type, produc9072475918466114483BT_nat: $tType).
% 21.59/7.92  thf(func_def_94, type, produc4894624898956917775BT_int: $tType).
% 21.59/7.92  thf(func_def_95, type, produc8025551001238799321T_VEBT: $tType).
% 21.59/7.92  thf(func_def_96, type, produc1531783533982839933T_VEBT: $tType).
% 21.59/7.92  thf(func_def_97, type, set_Pr1261947904930325089at_nat: $tType).
% 21.59/7.92  thf(func_def_98, type, set_Pr958786334691620121nt_int: $tType).
% 21.59/7.92  thf(func_def_99, type, produc4411394909380815293omplex: $tType).
% 21.59/7.92  thf(func_def_100, type, list_P3595434254542482545real_o: $tType).
% 21.59/7.92  thf(func_def_101, type, list_P5232166724548748803o_real: $tType).
% 21.59/7.92  thf(func_def_102, type, set_Pr4936984352647145239real_o: $tType).
% 21.59/7.92  thf(func_def_103, type, set_Pr6573716822653411497o_real: $tType).
% 21.59/7.92  thf(func_def_104, type, list_P7333126701944960589_nat_o: $tType).
% 21.59/7.92  thf(func_def_105, type, list_P6285523579766656935_o_nat: $tType).
% 21.59/7.92  thf(func_def_106, type, list_P3795440434834930179_o_int: $tType).
% 21.59/7.92  thf(func_def_107, type, set_list_VEBT_VEBT: $tType).
% 21.59/7.92  thf(func_def_108, type, produc334124729049499915VEBT_o: $tType).
% 21.59/7.92  thf(func_def_109, type, produc2504756804600209347T_VEBT: $tType).
% 21.59/7.92  thf(func_def_110, type, set_Pr2101469702781467981_o_nat: $tType).
% 21.59/7.92  thf(func_def_111, type, set_Pr8834758594704517033_o_int: $tType).
% 21.59/7.92  thf(func_def_112, type, produc6271795597528267376eger_o: $tType).
% 21.59/7.92  thf(func_def_113, type, produc2422161461964618553l_real: $tType).
% 21.59/7.92  thf(func_def_114, type, produc3741383161447143261al_nat: $tType).
% 21.59/7.92  thf(func_def_115, type, produc8786904178792722361al_int: $tType).
% 21.59/7.92  thf(func_def_116, type, produc679980390762269497t_real: $tType).
% 21.59/7.92  thf(func_def_117, type, product_prod_num_num: $tType).
% 21.59/7.92  thf(func_def_118, type, product_prod_nat_num: $tType).
% 21.59/7.92  thf(func_def_119, type, product_prod_nat_nat: $tType).
% 21.59/7.92  thf(func_def_120, type, product_prod_nat_int: $tType).
% 21.59/7.92  thf(func_def_121, type, product_prod_int_int: $tType).
% 21.59/7.92  thf(func_def_122, type, list_P4002435161011370285od_o_o: $tType).
% 21.59/7.92  thf(func_def_123, type, set_list_complex: $tType).
% 21.59/7.92  thf(func_def_124, type, set_Product_prod_o_o: $tType).
% 21.59/7.92  thf(func_def_125, type, set_set_complex: $tType).
% 21.59/7.92  thf(func_def_126, type, option_VEBT_VEBT: $tType).
% 21.59/7.92  thf(func_def_127, type, option_set_nat: $tType).
% 21.59/7.92  thf(func_def_128, type, product_prod_real_o: $tType).
% 21.59/7.92  thf(func_def_129, type, product_prod_o_real: $tType).
% 21.59/7.92  thf(func_def_130, type, list_VEBT_VEBT: $tType).
% 21.59/7.92  thf(func_def_131, type, set_list_nat: $tType).
% 21.59/7.92  thf(func_def_132, type, set_list_int: $tType).
% 21.59/7.92  thf(func_def_133, type, product_prod_nat_o: $tType).
% 21.59/7.92  thf(func_def_134, type, product_prod_o_nat: $tType).
% 21.59/7.92  thf(func_def_135, type, product_prod_o_int: $tType).
% 21.59/7.92  thf(func_def_136, type, list_set_nat: $tType).
% 21.59/7.92  thf(func_def_137, type, set_VEBT_VEBT: $tType).
% 21.59/7.92  thf(func_def_138, type, set_set_nat: $tType).
% 21.59/7.92  thf(func_def_139, type, set_set_int: $tType).
% 21.59/7.92  thf(func_def_140, type, set_Code_integer: $tType).
% 21.59/7.92  thf(func_def_141, type, set_Product_unit: $tType).
% 21.59/7.92  thf(func_def_142, type, list_complex: $tType).
% 21.59/7.92  thf(func_def_143, type, set_list_o: $tType).
% 21.59/7.92  thf(func_def_144, type, product_prod_o_o: $tType).
% 21.59/7.92  thf(func_def_145, type, set_complex: $tType).
% 21.59/7.92  thf(func_def_146, type, option_real: $tType).
% 21.59/7.92  thf(func_def_147, type, filter_real: $tType).
% 21.59/7.92  thf(func_def_148, type, option_num: $tType).
% 21.59/7.92  thf(func_def_149, type, option_nat: $tType).
% 21.59/7.92  thf(func_def_150, type, option_int: $tType).
% 21.59/7.92  thf(func_def_151, type, filter_nat: $tType).
% 21.59/7.92  thf(func_def_152, type, list_real: $tType).
% 21.59/7.92  thf(func_def_153, type, set_real: $tType).
% 21.59/7.92  thf(func_def_154, type, list_num: $tType).
% 21.59/7.92  thf(func_def_155, type, list_nat: $tType).
% 21.59/7.92  thf(func_def_156, type, list_int: $tType).
% 21.59/7.92  thf(func_def_157, type, vEBT_VEBT: $tType).
% 21.59/7.92  thf(func_def_158, type, set_rat: $tType).
% 21.59/7.92  thf(func_def_159, type, set_num: $tType).
% 21.59/7.92  thf(func_def_160, type, set_nat: $tType).
% 21.59/7.92  thf(func_def_161, type, set_int: $tType).
% 21.59/7.92  thf(func_def_162, type, code_integer: $tType).
% 21.59/7.92  thf(func_def_163, type, option_o: $tType).
% 21.59/7.92  thf(func_def_164, type, extended_enat: $tType).
% 21.59/7.92  thf(func_def_165, type, list_o: $tType).
% 21.59/7.92  thf(func_def_166, type, complex: $tType).
% 21.59/7.92  thf(func_def_167, type, set_o: $tType).
% 21.59/7.92  thf(func_def_168, type, real: $tType).
% 21.59/7.92  thf(func_def_169, type, rat: $tType).
% 21.59/7.92  thf(func_def_170, type, num: $tType).
% 21.59/7.92  thf(func_def_171, type, nat: $tType).
% 21.59/7.92  thf(func_def_172, type, int: $tType).
% 21.59/7.92  thf(func_def_173, type, archim2889992004027027881ng_rat: rat > int).
% 21.59/7.92  thf(func_def_174, type, archim7802044766580827645g_real: real > int).
% 21.59/7.92  thf(func_def_175, type, archim3151403230148437115or_rat: rat > int).
% 21.59/7.92  thf(func_def_176, type, archim6058952711729229775r_real: real > int).
% 21.59/7.92  thf(func_def_177, type, archimedean_frac_rat: rat > rat).
% 21.59/7.92  thf(func_def_178, type, archim2898591450579166408c_real: real > real).
% 21.59/7.92  thf(func_def_179, type, archim7778729529865785530nd_rat: rat > int).
% 21.59/7.92  thf(func_def_180, type, archim8280529875227126926d_real: real > int).
% 21.59/7.92  thf(func_def_181, type, bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat).
% 21.59/7.92  thf(func_def_182, type, bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat).
% 21.59/7.92  thf(func_def_183, type, bNF_re728719798268516973at_o_o: ((nat > rat) > (nat > rat) > $o) > ($o > $o > $o) > ((nat > rat) > $o) > ((nat > rat) > $o) > $o).
% 21.59/7.92  thf(func_def_184, type, bNF_re4297313714947099218al_o_o: ((nat > rat) > real > $o) > ($o > $o > $o) > ((nat > rat) > $o) > (real > $o) > $o).
% 21.59/7.92  thf(func_def_185, type, bNF_re3403563459893282935_int_o: (int > int > $o) > ((int > $o) > (int > $o) > $o) > (int > int > $o) > (int > int > $o) > $o).
% 21.59/7.92  thf(func_def_186, type, bNF_re157797125943740599nt_int: (int > int > $o) > ((int > product_prod_int_int) > (int > product_prod_int_int) > $o) > (int > int > product_prod_int_int) > (int > int > product_prod_int_int) > $o).
% 21.59/7.92  thf(func_def_187, type, bNF_re3461391660133120880nt_rat: (int > int > $o) > ((int > product_prod_int_int) > (int > rat) > $o) > (int > int > product_prod_int_int) > (int > int > rat) > $o).
% 21.59/7.92  thf(func_def_188, type, bNF_re5089333283451836215nt_o_o: (int > int > $o) > ($o > $o > $o) > (int > $o) > (int > $o) > $o).
% 21.59/7.92  thf(func_def_189, type, bNF_re6250860962936578807nt_int: (int > int > $o) > (product_prod_int_int > product_prod_int_int > $o) > (int > product_prod_int_int) > (int > product_prod_int_int) > $o).
% 21.59/7.92  thf(func_def_190, type, bNF_re2214769303045360666nt_rat: (int > int > $o) > (product_prod_int_int > rat > $o) > (int > product_prod_int_int) > (int > rat) > $o).
% 21.59/7.92  thf(func_def_191, type, bNF_re578469030762574527_nat_o: (nat > nat > $o) > ((nat > $o) > (nat > $o) > $o) > (nat > nat > $o) > (nat > nat > $o) > $o).
% 21.59/7.92  thf(func_def_192, type, bNF_re4705727531993890431at_o_o: (nat > nat > $o) > ($o > $o > $o) > (nat > $o) > (nat > $o) > $o).
% 21.59/7.92  thf(func_def_193, type, bNF_re6830278522597306478at_int: (nat > nat > $o) > (product_prod_nat_nat > int > $o) > (nat > product_prod_nat_nat) > (nat > int) > $o).
% 21.59/7.92  thf(func_def_194, type, bNF_re5228765855967844073nt_int: (product_prod_int_int > product_prod_int_int > $o) > ((product_prod_int_int > product_prod_int_int) > (product_prod_int_int > product_prod_int_int) > $o) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > $o).
% 21.59/7.92  thf(func_def_195, type, bNF_re8699439704749558557nt_o_o: (product_prod_int_int > product_prod_int_int > $o) > ($o > $o > $o) > (product_prod_int_int > $o) > (product_prod_int_int > $o) > $o).
% 21.59/7.92  thf(func_def_196, type, bNF_re7145576690424134365nt_int: (product_prod_int_int > product_prod_int_int > $o) > (product_prod_int_int > product_prod_int_int > $o) > (product_prod_int_int > product_prod_int_int) > (product_prod_int_int > product_prod_int_int) > $o).
% 21.59/7.92  thf(func_def_197, type, bNF_re7627151682743391978at_rat: (product_prod_int_int > rat > $o) > ((product_prod_int_int > product_prod_int_int) > (rat > rat) > $o) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > (rat > rat > rat) > $o).
% 21.59/7.92  thf(func_def_198, type, bNF_re1494630372529172596at_o_o: (product_prod_int_int > rat > $o) > ($o > $o > $o) > (product_prod_int_int > $o) > (rat > $o) > $o).
% 21.59/7.92  thf(func_def_199, type, bNF_re8279943556446156061nt_rat: (product_prod_int_int > rat > $o) > (product_prod_int_int > rat > $o) > (product_prod_int_int > product_prod_int_int) > (rat > rat) > $o).
% 21.59/7.92  thf(func_def_200, type, bNF_re717283939379294677_int_o: (product_prod_nat_nat > int > $o) > ((product_prod_nat_nat > $o) > (int > $o) > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > (int > int > $o) > $o).
% 21.59/7.92  thf(func_def_201, type, bNF_re7408651293131936558nt_int: (product_prod_nat_nat > int > $o) > ((product_prod_nat_nat > product_prod_nat_nat) > (int > int) > $o) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > (int > int > int) > $o).
% 21.59/7.92  thf(func_def_202, type, bNF_re6644619430987730960nt_o_o: (product_prod_nat_nat > int > $o) > ($o > $o > $o) > (product_prod_nat_nat > $o) > (int > $o) > $o).
% 21.59/7.92  thf(func_def_203, type, bNF_re7400052026677387805at_int: (product_prod_nat_nat > int > $o) > (product_prod_nat_nat > int > $o) > (product_prod_nat_nat > product_prod_nat_nat) > (int > int) > $o).
% 21.59/7.92  thf(func_def_204, type, bNF_re4202695980764964119_nat_o: (product_prod_nat_nat > product_prod_nat_nat > $o) > ((product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > $o).
% 21.59/7.92  thf(func_def_205, type, bNF_re3099431351363272937at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > ((product_prod_nat_nat > product_prod_nat_nat) > (product_prod_nat_nat > product_prod_nat_nat) > $o) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > $o).
% 21.59/7.92  thf(func_def_206, type, bNF_re3666534408544137501at_o_o: (product_prod_nat_nat > product_prod_nat_nat > $o) > ($o > $o > $o) > (product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > $o).
% 21.59/7.92  thf(func_def_207, type, bNF_re2241393799969408733at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > product_prod_nat_nat) > (product_prod_nat_nat > product_prod_nat_nat) > $o).
% 21.59/7.92  thf(func_def_208, type, bNF_We3818239936649020644el_nat: set_Pr1261947904930325089at_nat > $o).
% 21.59/7.92  thf(func_def_209, type, binomial: nat > nat > nat).
% 21.59/7.92  thf(func_def_210, type, gbinomial_complex: complex > nat > complex).
% 21.59/7.92  thf(func_def_211, type, gbinomial_int: int > nat > int).
% 21.59/7.92  thf(func_def_212, type, gbinomial_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_213, type, gbinomial_rat: rat > nat > rat).
% 21.59/7.92  thf(func_def_214, type, gbinomial_real: real > nat > real).
% 21.59/7.92  thf(func_def_215, type, bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o).
% 21.59/7.92  thf(func_def_216, type, bit_and_not_num: num > num > option_num).
% 21.59/7.92  thf(func_def_217, type, bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o).
% 21.59/7.92  thf(func_def_218, type, bit_concat_bit: nat > int > int > int).
% 21.59/7.92  thf(func_def_219, type, bit_ri7919022796975470100ot_int: int > int).
% 21.59/7.92  thf(func_def_220, type, bit_ri6519982836138164636nteger: nat > code_integer > code_integer).
% 21.59/7.92  thf(func_def_221, type, bit_ri631733984087533419it_int: nat > int > int).
% 21.59/7.92  thf(func_def_222, type, bit_se3949692690581998587nteger: code_integer > code_integer > code_integer).
% 21.59/7.92  thf(func_def_223, type, bit_se725231765392027082nd_int: int > int > int).
% 21.59/7.92  thf(func_def_224, type, bit_se727722235901077358nd_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_225, type, bit_se8568078237143864401it_int: nat > int > int).
% 21.59/7.92  thf(func_def_226, type, bit_se8570568707652914677it_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_227, type, bit_se1345352211410354436nteger: nat > code_integer > code_integer).
% 21.59/7.92  thf(func_def_228, type, bit_se2159334234014336723it_int: nat > int > int).
% 21.59/7.92  thf(func_def_229, type, bit_se2161824704523386999it_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_230, type, bit_se2000444600071755411sk_int: nat > int).
% 21.59/7.92  thf(func_def_231, type, bit_se2002935070580805687sk_nat: nat > nat).
% 21.59/7.92  thf(func_def_232, type, bit_se1080825931792720795nteger: code_integer > code_integer > code_integer).
% 21.59/7.92  thf(func_def_233, type, bit_se1409905431419307370or_int: int > int > int).
% 21.59/7.92  thf(func_def_234, type, bit_se1412395901928357646or_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_235, type, bit_se545348938243370406it_int: nat > int > int).
% 21.59/7.92  thf(func_def_236, type, bit_se547839408752420682it_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_237, type, bit_se2793503036327961859nteger: nat > code_integer > code_integer).
% 21.59/7.92  thf(func_def_238, type, bit_se7879613467334960850it_int: nat > int > int).
% 21.59/7.92  thf(func_def_239, type, bit_se7882103937844011126it_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_240, type, bit_se1745604003318907178nteger: nat > code_integer > code_integer).
% 21.59/7.92  thf(func_def_241, type, bit_se2923211474154528505it_int: nat > int > int).
% 21.59/7.92  thf(func_def_242, type, bit_se2925701944663578781it_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_243, type, bit_se8260200283734997820nteger: nat > code_integer > code_integer).
% 21.59/7.92  thf(func_def_244, type, bit_se4203085406695923979it_int: nat > int > int).
% 21.59/7.92  thf(func_def_245, type, bit_se4205575877204974255it_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_246, type, bit_se6526347334894502574or_int: int > int > int).
% 21.59/7.92  thf(func_def_247, type, bit_se6528837805403552850or_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_248, type, bit_se1146084159140164899it_int: int > nat > $o).
% 21.59/7.92  thf(func_def_249, type, bit_se1148574629649215175it_nat: nat > nat > $o).
% 21.59/7.92  thf(func_def_250, type, bit_take_bit_num: nat > num > option_num).
% 21.59/7.92  thf(func_def_251, type, bit_un7362597486090784418nd_num: num > num > option_num).
% 21.59/7.92  thf(func_def_252, type, bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o).
% 21.59/7.92  thf(func_def_253, type, bit_un2480387367778600638or_num: num > num > option_num).
% 21.59/7.92  thf(func_def_254, type, bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o).
% 21.59/7.92  thf(func_def_255, type, code_bit_cut_integer: code_integer > produc6271795597528267376eger_o).
% 21.59/7.92  thf(func_def_256, type, code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger).
% 21.59/7.92  thf(func_def_257, type, code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger).
% 21.59/7.92  thf(func_def_258, type, code_int_of_integer: code_integer > int).
% 21.59/7.92  thf(func_def_259, type, code_integer_of_int: int > code_integer).
% 21.59/7.92  thf(func_def_260, type, code_integer_of_nat: nat > code_integer).
% 21.59/7.92  thf(func_def_261, type, code_nat_of_integer: code_integer > nat).
% 21.59/7.92  thf(func_def_262, type, comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex).
% 21.59/7.92  thf(func_def_263, type, comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real).
% 21.59/7.92  thf(func_def_264, type, complete_Inf_Inf_nat: set_nat > nat).
% 21.59/7.92  thf(func_def_265, type, comple7806235888213564991et_nat: set_set_nat > set_nat).
% 21.59/7.92  thf(func_def_266, type, complete_Sup_Sup_nat: set_nat > nat).
% 21.59/7.92  thf(func_def_267, type, arg: complex > real).
% 21.59/7.92  thf(func_def_268, type, cis: real > complex).
% 21.59/7.92  thf(func_def_269, type, complex2: real > real > complex).
% 21.59/7.92  thf(func_def_270, type, condit2214826472909112428ve_nat: set_nat > $o).
% 21.59/7.92  thf(func_def_271, type, differ6690327859849518006l_real: (real > real) > filter_real > $o).
% 21.59/7.92  thf(func_def_272, type, has_de1759254742604945161l_real: (real > real) > (real > real) > filter_real > $o).
% 21.59/7.92  thf(func_def_273, type, has_fi5821293074295781190e_real: (real > real) > real > filter_real > $o).
% 21.59/7.92  thf(func_def_274, type, adjust_div: product_prod_int_int > int).
% 21.59/7.92  thf(func_def_275, type, divmod_nat: nat > nat > product_prod_nat_nat).
% 21.59/7.92  thf(func_def_276, type, eucl_rel_int: int > int > product_prod_int_int > $o).
% 21.59/7.92  thf(func_def_277, type, unique6319869463603278526ux_int: product_prod_int_int > $o).
% 21.59/7.92  thf(func_def_278, type, unique6322359934112328802ux_nat: product_prod_nat_nat > $o).
% 21.59/7.92  thf(func_def_279, type, unique3479559517661332726nteger: num > num > produc8923325533196201883nteger).
% 21.59/7.92  thf(func_def_280, type, unique5052692396658037445od_int: num > num > product_prod_int_int).
% 21.59/7.92  thf(func_def_281, type, unique5055182867167087721od_nat: num > num > product_prod_nat_nat).
% 21.59/7.92  thf(func_def_282, type, unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 21.59/7.92  thf(func_def_283, type, unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int).
% 21.59/7.92  thf(func_def_284, type, unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat).
% 21.59/7.92  thf(func_def_285, type, comm_s8582702949713902594nteger: code_integer > nat > code_integer).
% 21.59/7.92  thf(func_def_286, type, comm_s2602460028002588243omplex: complex > nat > complex).
% 21.59/7.92  thf(func_def_287, type, comm_s4660882817536571857er_int: int > nat > int).
% 21.59/7.92  thf(func_def_288, type, comm_s4663373288045622133er_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_289, type, comm_s4028243227959126397er_rat: rat > nat > rat).
% 21.59/7.92  thf(func_def_290, type, comm_s7457072308508201937r_real: real > nat > real).
% 21.59/7.92  thf(func_def_291, type, semiri3624122377584611663nteger: nat > code_integer).
% 21.59/7.92  thf(func_def_292, type, semiri5044797733671781792omplex: nat > complex).
% 21.59/7.92  thf(func_def_293, type, semiri1406184849735516958ct_int: nat > int).
% 21.59/7.92  thf(func_def_294, type, semiri1408675320244567234ct_nat: nat > nat).
% 21.59/7.92  thf(func_def_295, type, semiri773545260158071498ct_rat: nat > rat).
% 21.59/7.92  thf(func_def_296, type, semiri2265585572941072030t_real: nat > real).
% 21.59/7.92  thf(func_def_297, type, invers8013647133539491842omplex: complex > complex).
% 21.59/7.92  thf(func_def_298, type, inverse_inverse_rat: rat > rat).
% 21.59/7.92  thf(func_def_299, type, inverse_inverse_real: real > real).
% 21.59/7.92  thf(func_def_300, type, at_bot_real: filter_real).
% 21.59/7.92  thf(func_def_301, type, at_top_nat: filter_nat).
% 21.59/7.92  thf(func_def_302, type, at_top_real: filter_real).
% 21.59/7.92  thf(func_def_303, type, eventually_nat: (nat > $o) > filter_nat > $o).
% 21.59/7.92  thf(func_def_304, type, eventu1038000079068216329at_nat: (product_prod_nat_nat > $o) > filter1242075044329608583at_nat > $o).
% 21.59/7.92  thf(func_def_305, type, eventually_real: (real > $o) > filter_real > $o).
% 21.59/7.92  thf(func_def_306, type, filterlim_nat_nat: (nat > nat) > filter_nat > filter_nat > $o).
% 21.59/7.92  thf(func_def_307, type, filterlim_nat_real: (nat > real) > filter_real > filter_nat > $o).
% 21.59/7.92  thf(func_def_308, type, filterlim_real_real: (real > real) > filter_real > filter_real > $o).
% 21.59/7.92  thf(func_def_309, type, princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex).
% 21.59/7.92  thf(func_def_310, type, princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real).
% 21.59/7.92  thf(func_def_311, type, prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat).
% 21.59/7.92  thf(func_def_312, type, finite_card_complex: set_complex > nat).
% 21.59/7.92  thf(func_def_313, type, finite_card_list_nat: set_list_nat > nat).
% 21.59/7.92  thf(func_def_314, type, finite_card_nat: set_nat > nat).
% 21.59/7.92  thf(func_def_315, type, finite410649719033368117t_unit: set_Product_unit > nat).
% 21.59/7.92  thf(func_def_316, type, finite_card_set_nat: set_set_nat > nat).
% 21.59/7.92  thf(func_def_317, type, finite_finite_o: set_o > $o).
% 21.59/7.92  thf(func_def_318, type, finite3207457112153483333omplex: set_complex > $o).
% 21.59/7.92  thf(func_def_319, type, finite_finite_int: set_int > $o).
% 21.59/7.92  thf(func_def_320, type, finite_finite_list_o: set_list_o > $o).
% 21.59/7.92  thf(func_def_321, type, finite8712137658972009173omplex: set_list_complex > $o).
% 21.59/7.92  thf(func_def_322, type, finite3922522038869484883st_int: set_list_int > $o).
% 21.59/7.92  thf(func_def_323, type, finite8100373058378681591st_nat: set_list_nat > $o).
% 21.59/7.92  thf(func_def_324, type, finite500796754983035824at_nat: set_li5450038453877631591at_nat > $o).
% 21.59/7.92  thf(func_def_325, type, finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_326, type, finite_finite_nat: set_nat > $o).
% 21.59/7.92  thf(func_def_327, type, finite_finite_num: set_num > $o).
% 21.59/7.92  thf(func_def_328, type, finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o).
% 21.59/7.92  thf(func_def_329, type, finite4343798906461161616at_nat: set_Pr4329608150637261639at_nat > $o).
% 21.59/7.92  thf(func_def_330, type, finite_finite_rat: set_rat > $o).
% 21.59/7.92  thf(func_def_331, type, finite_finite_real: set_real > $o).
% 21.59/7.92  thf(func_def_332, type, finite6551019134538273531omplex: set_set_complex > $o).
% 21.59/7.92  thf(func_def_333, type, finite6197958912794628473et_int: set_set_int > $o).
% 21.59/7.92  thf(func_def_334, type, finite1152437895449049373et_nat: set_set_nat > $o).
% 21.59/7.92  thf(func_def_335, type, finite9047747110432174090at_nat: set_se7855581050983116737at_nat > $o).
% 21.59/7.92  thf(func_def_336, type, finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_337, type, bij_be1856998921033663316omplex: (complex > complex) > set_complex > set_complex > $o).
% 21.59/7.92  thf(func_def_338, type, bij_betw_nat_complex: (nat > complex) > set_nat > set_complex > $o).
% 21.59/7.92  thf(func_def_339, type, bij_betw_nat_nat: (nat > nat) > set_nat > set_nat > $o).
% 21.59/7.92  thf(func_def_340, type, comp_C8797469213163452608nteger: ((code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 21.59/7.92  thf(func_def_341, type, comp_C1593894019821074884nteger: (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 21.59/7.92  thf(func_def_342, type, comp_int_nat_int: (int > nat) > (int > int) > int > nat).
% 21.59/7.92  thf(func_def_343, type, comp_nat_nat_nat: (nat > nat) > (nat > nat) > nat > nat).
% 21.59/7.92  thf(func_def_344, type, inj_on_nat_nat: (nat > nat) > set_nat > $o).
% 21.59/7.92  thf(func_def_345, type, inj_on_real_real: (real > real) > set_real > $o).
% 21.59/7.92  thf(func_def_346, type, inj_on_set_nat_nat: (set_nat > nat) > set_set_nat > $o).
% 21.59/7.92  thf(func_def_347, type, map_fu4960017516451851995nt_int: (int > product_prod_nat_nat) > ((product_prod_nat_nat > product_prod_nat_nat) > int > int) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > int > int > int).
% 21.59/7.92  thf(func_def_348, type, map_fu3667384564859982768at_int: (int > product_prod_nat_nat) > (product_prod_nat_nat > int) > (product_prod_nat_nat > product_prod_nat_nat) > int > int).
% 21.59/7.92  thf(func_def_349, type, map_fu4333342158222067775at_rat: (rat > product_prod_int_int) > ((product_prod_int_int > product_prod_int_int) > rat > rat) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > rat > rat > rat).
% 21.59/7.92  thf(func_def_350, type, map_fu5673905371560938248nt_rat: (rat > product_prod_int_int) > (product_prod_int_int > rat) > (product_prod_int_int > product_prod_int_int) > rat > rat).
% 21.59/7.92  thf(func_def_351, type, strict1292158309912662752at_nat: (nat > nat) > set_nat > $o).
% 21.59/7.92  thf(func_def_352, type, the_in5290026491893676941l_real: set_real > (real > real) > real > real).
% 21.59/7.92  thf(func_def_353, type, fun_max_strict: set_Pr4329608150637261639at_nat).
% 21.59/7.92  thf(func_def_354, type, fun_max_weak: set_Pr4329608150637261639at_nat).
% 21.59/7.92  thf(func_def_355, type, fun_min_strict: set_Pr4329608150637261639at_nat).
% 21.59/7.92  thf(func_def_356, type, fun_min_weak: set_Pr4329608150637261639at_nat).
% 21.59/7.92  thf(func_def_357, type, fun_pair_leq: set_Pr8693737435421807431at_nat).
% 21.59/7.92  thf(func_def_358, type, fun_pair_less: set_Pr8693737435421807431at_nat).
% 21.59/7.92  thf(func_def_359, type, fun_re2478310338295953701at_nat: produc1319942482725812455at_nat > $o).
% 21.59/7.92  thf(func_def_360, type, gcd_Gcd_nat: set_nat > nat).
% 21.59/7.92  thf(func_def_361, type, bezw: nat > nat > product_prod_int_int).
% 21.59/7.92  thf(func_def_362, type, bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 21.59/7.92  thf(func_def_363, type, gcd_gcd_int: int > int > int).
% 21.59/7.92  thf(func_def_364, type, gcd_gcd_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_365, type, gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 21.59/7.92  thf(func_def_366, type, abs_abs_Code_integer: code_integer > code_integer).
% 21.59/7.92  thf(func_def_367, type, abs_abs_complex: complex > complex).
% 21.59/7.92  thf(func_def_368, type, abs_abs_int: int > int).
% 21.59/7.92  thf(func_def_369, type, abs_abs_rat: rat > rat).
% 21.59/7.92  thf(func_def_370, type, abs_abs_real: real > real).
% 21.59/7.92  thf(func_def_371, type, minus_minus_int_o: (int > $o) > (int > $o) > int > $o).
% 21.59/7.92  thf(func_def_372, type, minus_1139252259498527702_nat_o: (list_nat > $o) > (list_nat > $o) > list_nat > $o).
% 21.59/7.92  thf(func_def_373, type, minus_minus_nat_o: (nat > $o) > (nat > $o) > nat > $o).
% 21.59/7.92  thf(func_def_374, type, minus_2270307095948843157_nat_o: (product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > product_prod_nat_nat > $o).
% 21.59/7.92  thf(func_def_375, type, minus_minus_real_o: (real > $o) > (real > $o) > real > $o).
% 21.59/7.92  thf(func_def_376, type, minus_6910147592129066416_nat_o: (set_nat > $o) > (set_nat > $o) > set_nat > $o).
% 21.59/7.92  thf(func_def_377, type, minus_8373710615458151222nteger: code_integer > code_integer > code_integer).
% 21.59/7.92  thf(func_def_378, type, minus_minus_complex: complex > complex > complex).
% 21.59/7.92  thf(func_def_379, type, minus_minus_int: int > int > int).
% 21.59/7.92  thf(func_def_380, type, minus_minus_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_381, type, minus_minus_rat: rat > rat > rat).
% 21.59/7.92  thf(func_def_382, type, minus_minus_real: real > real > real).
% 21.59/7.92  thf(func_def_383, type, minus_811609699411566653omplex: set_complex > set_complex > set_complex).
% 21.59/7.92  thf(func_def_384, type, minus_minus_set_int: set_int > set_int > set_int).
% 21.59/7.92  thf(func_def_385, type, minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat).
% 21.59/7.92  thf(func_def_386, type, minus_minus_set_nat: set_nat > set_nat > set_nat).
% 21.59/7.92  thf(func_def_387, type, minus_minus_set_num: set_num > set_num > set_num).
% 21.59/7.92  thf(func_def_388, type, minus_1356011639430497352at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 21.59/7.92  thf(func_def_389, type, minus_3314409938677909166at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat).
% 21.59/7.92  thf(func_def_390, type, minus_minus_set_rat: set_rat > set_rat > set_rat).
% 21.59/7.92  thf(func_def_391, type, minus_minus_set_real: set_real > set_real > set_real).
% 21.59/7.92  thf(func_def_392, type, minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat).
% 21.59/7.92  thf(func_def_393, type, one_one_Code_integer: code_integer).
% 21.59/7.92  thf(func_def_394, type, one_one_complex: complex).
% 21.59/7.92  thf(func_def_395, type, one_on7984719198319812577d_enat: extended_enat).
% 21.59/7.92  thf(func_def_396, type, one_one_int: int).
% 21.59/7.92  thf(func_def_397, type, one_one_nat: nat).
% 21.59/7.92  thf(func_def_398, type, one_one_rat: rat).
% 21.59/7.92  thf(func_def_399, type, one_one_real: real).
% 21.59/7.92  thf(func_def_400, type, plus_p5714425477246183910nteger: code_integer > code_integer > code_integer).
% 21.59/7.92  thf(func_def_401, type, plus_plus_complex: complex > complex > complex).
% 21.59/7.92  thf(func_def_402, type, plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat).
% 21.59/7.92  thf(func_def_403, type, plus_plus_int: int > int > int).
% 21.59/7.92  thf(func_def_404, type, plus_plus_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_405, type, plus_plus_num: num > num > num).
% 21.59/7.92  thf(func_def_406, type, plus_plus_rat: rat > rat > rat).
% 21.59/7.92  thf(func_def_407, type, plus_plus_real: real > real > real).
% 21.59/7.92  thf(func_def_408, type, sgn_sgn_Code_integer: code_integer > code_integer).
% 21.59/7.92  thf(func_def_409, type, sgn_sgn_complex: complex > complex).
% 21.59/7.92  thf(func_def_410, type, sgn_sgn_int: int > int).
% 21.59/7.92  thf(func_def_411, type, sgn_sgn_rat: rat > rat).
% 21.59/7.92  thf(func_def_412, type, sgn_sgn_real: real > real).
% 21.59/7.92  thf(func_def_413, type, times_3573771949741848930nteger: code_integer > code_integer > code_integer).
% 21.59/7.92  thf(func_def_414, type, times_times_complex: complex > complex > complex).
% 21.59/7.92  thf(func_def_415, type, times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat).
% 21.59/7.92  thf(func_def_416, type, times_times_int: int > int > int).
% 21.59/7.92  thf(func_def_417, type, times_times_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_418, type, times_times_rat: rat > rat > rat).
% 21.59/7.92  thf(func_def_419, type, times_times_real: real > real > real).
% 21.59/7.92  thf(func_def_420, type, uminus_uminus_int_o: (int > $o) > int > $o).
% 21.59/7.92  thf(func_def_421, type, uminus5770388063884162150_nat_o: (list_nat > $o) > list_nat > $o).
% 21.59/7.92  thf(func_def_422, type, uminus_uminus_nat_o: (nat > $o) > nat > $o).
% 21.59/7.92  thf(func_def_423, type, uminus8676089048583255045_nat_o: (product_prod_nat_nat > $o) > product_prod_nat_nat > $o).
% 21.59/7.92  thf(func_def_424, type, uminus_uminus_real_o: (real > $o) > real > $o).
% 21.59/7.92  thf(func_def_425, type, uminus6401447641752708672_nat_o: (set_nat > $o) > set_nat > $o).
% 21.59/7.92  thf(func_def_426, type, uminus1351360451143612070nteger: code_integer > code_integer).
% 21.59/7.92  thf(func_def_427, type, uminus1482373934393186551omplex: complex > complex).
% 21.59/7.92  thf(func_def_428, type, uminus_uminus_int: int > int).
% 21.59/7.92  thf(func_def_429, type, uminus_uminus_rat: rat > rat).
% 21.59/7.92  thf(func_def_430, type, uminus_uminus_real: real > real).
% 21.59/7.92  thf(func_def_431, type, uminus1532241313380277803et_int: set_int > set_int).
% 21.59/7.92  thf(func_def_432, type, uminus3195874150345416415st_nat: set_list_nat > set_list_nat).
% 21.59/7.92  thf(func_def_433, type, uminus5710092332889474511et_nat: set_nat > set_nat).
% 21.59/7.92  thf(func_def_434, type, uminus6524753893492686040at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 21.59/7.92  thf(func_def_435, type, uminus935396558254630718at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat).
% 21.59/7.92  thf(func_def_436, type, uminus612125837232591019t_real: set_real > set_real).
% 21.59/7.92  thf(func_def_437, type, uminus613421341184616069et_nat: set_set_nat > set_set_nat).
% 21.59/7.92  thf(func_def_438, type, zero_z3403309356797280102nteger: code_integer).
% 21.59/7.92  thf(func_def_439, type, zero_zero_complex: complex).
% 21.59/7.92  thf(func_def_440, type, zero_z5237406670263579293d_enat: extended_enat).
% 21.59/7.92  thf(func_def_441, type, zero_zero_int: int).
% 21.59/7.92  thf(func_def_442, type, zero_zero_nat: nat).
% 21.59/7.92  thf(func_def_443, type, zero_zero_rat: rat).
% 21.59/7.92  thf(func_def_444, type, zero_zero_real: real).
% 21.59/7.92  thf(func_def_445, type, groups8507830703676809646_o_nat: ($o > nat) > set_o > nat).
% 21.59/7.92  thf(func_def_446, type, groups7754918857620584856omplex: (complex > complex) > set_complex > complex).
% 21.59/7.92  thf(func_def_447, type, groups5690904116761175830ex_int: (complex > int) > set_complex > int).
% 21.59/7.92  thf(func_def_448, type, groups5693394587270226106ex_nat: (complex > nat) > set_complex > nat).
% 21.59/7.92  thf(func_def_449, type, groups5058264527183730370ex_rat: (complex > rat) > set_complex > rat).
% 21.59/7.92  thf(func_def_450, type, groups5808333547571424918x_real: (complex > real) > set_complex > real).
% 21.59/7.92  thf(func_def_451, type, groups7873554091576472773nteger: (int > code_integer) > set_int > code_integer).
% 21.59/7.92  thf(func_def_452, type, groups3049146728041665814omplex: (int > complex) > set_int > complex).
% 21.59/7.92  thf(func_def_453, type, groups4538972089207619220nt_int: (int > int) > set_int > int).
% 21.59/7.92  thf(func_def_454, type, groups4541462559716669496nt_nat: (int > nat) > set_int > nat).
% 21.59/7.92  thf(func_def_455, type, groups3906332499630173760nt_rat: (int > rat) > set_int > rat).
% 21.59/7.92  thf(func_def_456, type, groups8778361861064173332t_real: (int > real) > set_int > real).
% 21.59/7.92  thf(func_def_457, type, groups7501900531339628137nteger: (nat > code_integer) > set_nat > code_integer).
% 21.59/7.92  thf(func_def_458, type, groups2073611262835488442omplex: (nat > complex) > set_nat > complex).
% 21.59/7.92  thf(func_def_459, type, groups7108830773950497114d_enat: (nat > extended_enat) > set_nat > extended_enat).
% 21.59/7.92  thf(func_def_460, type, groups3539618377306564664at_int: (nat > int) > set_nat > int).
% 21.59/7.92  thf(func_def_461, type, groups3542108847815614940at_nat: (nat > nat) > set_nat > nat).
% 21.59/7.92  thf(func_def_462, type, groups2906978787729119204at_rat: (nat > rat) > set_nat > rat).
% 21.59/7.92  thf(func_def_463, type, groups6591440286371151544t_real: (nat > real) > set_nat > real).
% 21.59/7.92  thf(func_def_464, type, groups977919841031483927at_nat: (product_prod_nat_nat > nat) > set_Pr1261947904930325089at_nat > nat).
% 21.59/7.92  thf(func_def_465, type, groups4567486121110086003t_real: (product_prod_nat_nat > real) > set_Pr1261947904930325089at_nat > real).
% 21.59/7.92  thf(func_def_466, type, groups3860910324918113789at_nat: (produc3843707927480180839at_nat > nat) > set_Pr4329608150637261639at_nat > nat).
% 21.59/7.92  thf(func_def_467, type, groups7713935264441627589nteger: (real > code_integer) > set_real > code_integer).
% 21.59/7.92  thf(func_def_468, type, groups5754745047067104278omplex: (real > complex) > set_real > complex).
% 21.59/7.92  thf(func_def_469, type, groups1932886352136224148al_int: (real > int) > set_real > int).
% 21.59/7.92  thf(func_def_470, type, groups1935376822645274424al_nat: (real > nat) > set_real > nat).
% 21.59/7.92  thf(func_def_471, type, groups1300246762558778688al_rat: (real > rat) > set_real > rat).
% 21.59/7.92  thf(func_def_472, type, groups8097168146408367636l_real: (real > real) > set_real > real).
% 21.59/7.92  thf(func_def_473, type, groups8294997508430121362at_nat: (set_nat > nat) > set_set_nat > nat).
% 21.59/7.92  thf(func_def_474, type, groups771621172384141258BT_nat: (vEBT_VEBT > nat) > set_VEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_475, type, groups808145749697022017omplex: (complex > complex) > set_complex > complex).
% 21.59/7.92  thf(func_def_476, type, groups2909182065852811199ex_int: (complex > int) > set_complex > int).
% 21.59/7.92  thf(func_def_477, type, groups2911672536361861475ex_nat: (complex > nat) > set_complex > nat).
% 21.59/7.92  thf(func_def_478, type, groups2276542476275365739ex_rat: (complex > rat) > set_complex > rat).
% 21.59/7.92  thf(func_def_479, type, groups5737402329758386879x_real: (complex > real) > set_complex > real).
% 21.59/7.92  thf(func_def_480, type, groups267424677133301183omplex: (int > complex) > set_int > complex).
% 21.59/7.92  thf(func_def_481, type, groups2983280209131991357nt_int: (int > int) > set_int > int).
% 21.59/7.92  thf(func_def_482, type, groups2985770679641041633nt_nat: (int > nat) > set_int > nat).
% 21.59/7.92  thf(func_def_483, type, groups2350640619554545897nt_rat: (int > rat) > set_int > rat).
% 21.59/7.92  thf(func_def_484, type, groups1523912220035142973t_real: (int > real) > set_int > real).
% 21.59/7.92  thf(func_def_485, type, groups8515261248781899619omplex: (nat > complex) > set_nat > complex).
% 21.59/7.92  thf(func_def_486, type, groups1983926497230936801at_int: (nat > int) > set_nat > int).
% 21.59/7.92  thf(func_def_487, type, groups1986416967739987077at_nat: (nat > nat) > set_nat > nat).
% 21.59/7.92  thf(func_def_488, type, groups1351286907653491341at_rat: (nat > rat) > set_nat > rat).
% 21.59/7.92  thf(func_def_489, type, groups8560362682196896993t_real: (nat > real) > set_nat > real).
% 21.59/7.92  thf(func_def_490, type, groups5683813829254066239omplex: (real > complex) > set_real > complex).
% 21.59/7.92  thf(func_def_491, type, groups3901808747961969597al_int: (real > int) > set_real > int).
% 21.59/7.92  thf(func_def_492, type, groups3904299218471019873al_nat: (real > nat) > set_real > nat).
% 21.59/7.92  thf(func_def_493, type, groups3269169158384524137al_rat: (real > rat) > set_real > rat).
% 21.59/7.92  thf(func_def_494, type, groups97945582718554045l_real: (real > real) > set_real > real).
% 21.59/7.92  thf(func_def_495, type, groups8682486955453173170nteger: (complex > code_integer) > set_complex > code_integer).
% 21.59/7.92  thf(func_def_496, type, groups3708469109370488835omplex: (complex > complex) > set_complex > complex).
% 21.59/7.92  thf(func_def_497, type, groups858564598930262913ex_int: (complex > int) > set_complex > int).
% 21.59/7.92  thf(func_def_498, type, groups861055069439313189ex_nat: (complex > nat) > set_complex > nat).
% 21.59/7.92  thf(func_def_499, type, groups225925009352817453ex_rat: (complex > rat) > set_complex > rat).
% 21.59/7.92  thf(func_def_500, type, groups766887009212190081x_real: (complex > real) > set_complex > real).
% 21.59/7.92  thf(func_def_501, type, groups3827104343326376752nteger: (int > code_integer) > set_int > code_integer).
% 21.59/7.92  thf(func_def_502, type, groups7440179247065528705omplex: (int > complex) > set_int > complex).
% 21.59/7.92  thf(func_def_503, type, groups1705073143266064639nt_int: (int > int) > set_int > int).
% 21.59/7.92  thf(func_def_504, type, groups1707563613775114915nt_nat: (int > nat) > set_int > nat).
% 21.59/7.92  thf(func_def_505, type, groups1072433553688619179nt_rat: (int > rat) > set_int > rat).
% 21.59/7.92  thf(func_def_506, type, groups2316167850115554303t_real: (int > real) > set_int > real).
% 21.59/7.92  thf(func_def_507, type, groups2907647131375434839at_nat: (list_nat > nat) > set_list_nat > nat).
% 21.59/7.92  thf(func_def_508, type, groups3455450783089532116nteger: (nat > code_integer) > set_nat > code_integer).
% 21.59/7.92  thf(func_def_509, type, groups6464643781859351333omplex: (nat > complex) > set_nat > complex).
% 21.59/7.92  thf(func_def_510, type, groups705719431365010083at_int: (nat > int) > set_nat > int).
% 21.59/7.92  thf(func_def_511, type, groups708209901874060359at_nat: (nat > nat) > set_nat > nat).
% 21.59/7.92  thf(func_def_512, type, groups73079841787564623at_rat: (nat > rat) > set_nat > rat).
% 21.59/7.92  thf(func_def_513, type, groups129246275422532515t_real: (nat > real) > set_nat > real).
% 21.59/7.92  thf(func_def_514, type, groups4077766827762148844at_nat: (product_prod_nat_nat > nat) > set_Pr1261947904930325089at_nat > nat).
% 21.59/7.92  thf(func_def_515, type, groups6225526099057966256nteger: (real > code_integer) > set_real > code_integer).
% 21.59/7.92  thf(func_def_516, type, groups713298508707869441omplex: (real > complex) > set_real > complex).
% 21.59/7.92  thf(func_def_517, type, groups4694064378042380927al_int: (real > int) > set_real > int).
% 21.59/7.92  thf(func_def_518, type, groups4696554848551431203al_nat: (real > nat) > set_real > nat).
% 21.59/7.92  thf(func_def_519, type, groups4061424788464935467al_rat: (real > rat) > set_real > rat).
% 21.59/7.92  thf(func_def_520, type, groups1681761925125756287l_real: (real > real) > set_real > real).
% 21.59/7.92  thf(func_def_521, type, groups4248547760180025341at_nat: (set_nat > nat) > set_set_nat > nat).
% 21.59/7.92  thf(func_def_522, type, groups9116527308978886569_o_int: ($o > int) > int > list_o > int).
% 21.59/7.92  thf(func_def_523, type, groups4561878855575611511st_nat: list_nat > nat).
% 21.59/7.92  thf(func_def_524, type, the_int: (int > $o) > int).
% 21.59/7.92  thf(func_def_525, type, the_nat: (nat > $o) > nat).
% 21.59/7.92  thf(func_def_526, type, the_Pr8210177043389155639nteger: (produc8763457246119570046nteger > $o) > produc8763457246119570046nteger).
% 21.59/7.92  thf(func_def_527, type, the_Pr6882841213465913158nt_int: (produc7773217078559923341nt_int > $o) > produc7773217078559923341nt_int).
% 21.59/7.92  thf(func_def_528, type, the_Pr6653488032121699663nteger: (produc1908205239877642774nteger > $o) > produc1908205239877642774nteger).
% 21.59/7.92  thf(func_def_529, type, the_Pr5445864913131713084nt_int: (produc2285326912895808259nt_int > $o) > produc2285326912895808259nt_int).
% 21.59/7.92  thf(func_def_530, type, the_Pr4378521158711661632nt_int: (product_prod_int_int > $o) > product_prod_int_int).
% 21.59/7.92  thf(func_def_531, type, the_Pr7557018466319803784at_nat: (product_prod_nat_nat > $o) > product_prod_nat_nat).
% 21.59/7.92  thf(func_def_532, type, the_Pr4604535484834969198at_nat: (produc3843707927480180839at_nat > $o) > produc3843707927480180839at_nat).
% 21.59/7.92  thf(func_def_533, type, the_real: (real > $o) > real).
% 21.59/7.92  thf(func_def_534, type, if_int_int: $o > (int > int) > (int > int) > int > int).
% 21.59/7.92  thf(func_def_535, type, if_Code_integer: $o > code_integer > code_integer > code_integer).
% 21.59/7.92  thf(func_def_536, type, if_complex: $o > complex > complex > complex).
% 21.59/7.92  thf(func_def_537, type, if_Extended_enat: $o > extended_enat > extended_enat > extended_enat).
% 21.59/7.92  thf(func_def_538, type, if_int: $o > int > int > int).
% 21.59/7.92  thf(func_def_539, type, if_list_int: $o > list_int > list_int > list_int).
% 21.59/7.92  thf(func_def_540, type, if_list_nat: $o > list_nat > list_nat > list_nat).
% 21.59/7.92  thf(func_def_541, type, if_nat: $o > nat > nat > nat).
% 21.59/7.92  thf(func_def_542, type, if_num: $o > num > num > num).
% 21.59/7.92  thf(func_def_543, type, if_option_nat: $o > option_nat > option_nat > option_nat).
% 21.59/7.92  thf(func_def_544, type, if_option_num: $o > option_num > option_num > option_num).
% 21.59/7.92  thf(func_def_545, type, if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o).
% 21.59/7.92  thf(func_def_546, type, if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 21.59/7.92  thf(func_def_547, type, if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int).
% 21.59/7.92  thf(func_def_548, type, if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat).
% 21.59/7.92  thf(func_def_549, type, if_rat: $o > rat > rat > rat).
% 21.59/7.92  thf(func_def_550, type, if_real: $o > real > real > real).
% 21.59/7.92  thf(func_def_551, type, if_set_int: $o > set_int > set_int > set_int).
% 21.59/7.92  thf(func_def_552, type, if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_553, type, infini8530281810654367211te_nat: set_nat > nat > nat).
% 21.59/7.92  thf(func_def_554, type, abs_Integ: product_prod_nat_nat > int).
% 21.59/7.92  thf(func_def_555, type, rep_Integ: int > product_prod_nat_nat).
% 21.59/7.92  thf(func_def_556, type, int_ge_less_than: int > set_Pr958786334691620121nt_int).
% 21.59/7.92  thf(func_def_557, type, int_ge_less_than2: int > set_Pr958786334691620121nt_int).
% 21.59/7.92  thf(func_def_558, type, intrel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 21.59/7.92  thf(func_def_559, type, nat2: int > nat).
% 21.59/7.92  thf(func_def_560, type, pcr_int: product_prod_nat_nat > int > $o).
% 21.59/7.92  thf(func_def_561, type, power_int_real: real > int > real).
% 21.59/7.92  thf(func_def_562, type, ring_18347121197199848620nteger: int > code_integer).
% 21.59/7.92  thf(func_def_563, type, ring_17405671764205052669omplex: int > complex).
% 21.59/7.92  thf(func_def_564, type, ring_1_of_int_int: int > int).
% 21.59/7.92  thf(func_def_565, type, ring_1_of_int_rat: int > rat).
% 21.59/7.92  thf(func_def_566, type, ring_1_of_int_real: int > real).
% 21.59/7.92  thf(func_def_567, type, inf_inf_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_568, type, inf_in2572325071724192079at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 21.59/7.92  thf(func_def_569, type, semila1623282765462674594er_nat: (nat > nat > nat) > nat > (nat > nat > $o) > (nat > nat > $o) > $o).
% 21.59/7.92  thf(func_def_570, type, sup_sup_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_571, type, sup_sup_set_nat: set_nat > set_nat > set_nat).
% 21.59/7.92  thf(func_def_572, type, sup_su5525570899277871387at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat).
% 21.59/7.92  thf(func_def_573, type, lattic8265883725875713057ax_nat: set_nat > nat).
% 21.59/7.92  thf(func_def_574, type, lattic5364784637807008409ex_nat: (complex > nat) > set_complex > complex).
% 21.59/7.92  thf(func_def_575, type, lattic1922116423962787043ex_num: (complex > num) > set_complex > complex).
% 21.59/7.92  thf(func_def_576, type, lattic4729654577720512673ex_rat: (complex > rat) > set_complex > complex).
% 21.59/7.92  thf(func_def_577, type, lattic8794016678065449205x_real: (complex > real) > set_complex > complex).
% 21.59/7.92  thf(func_def_578, type, lattic5003618458639192673nt_num: (int > num) > set_int > int).
% 21.59/7.92  thf(func_def_579, type, lattic7811156612396918303nt_rat: (int > rat) > set_int > int).
% 21.59/7.92  thf(func_def_580, type, lattic2675449441010098035t_real: (int > real) > set_int > int).
% 21.59/7.92  thf(func_def_581, type, lattic4004264746738138117at_num: (nat > num) > set_nat > nat).
% 21.59/7.92  thf(func_def_582, type, lattic6811802900495863747at_rat: (nat > rat) > set_nat > nat).
% 21.59/7.92  thf(func_def_583, type, lattic488527866317076247t_real: (nat > real) > set_nat > nat).
% 21.59/7.92  thf(func_def_584, type, lattic5055836439445974935al_nat: (real > nat) > set_real > real).
% 21.59/7.92  thf(func_def_585, type, lattic1613168225601753569al_num: (real > num) > set_real > real).
% 21.59/7.92  thf(func_def_586, type, lattic4420706379359479199al_rat: (real > rat) > set_real > real).
% 21.59/7.92  thf(func_def_587, type, lattic8440615504127631091l_real: (real > real) > set_real > real).
% 21.59/7.92  thf(func_def_588, type, append_int: list_int > list_int > list_int).
% 21.59/7.92  thf(func_def_589, type, append_nat: list_nat > list_nat > list_nat).
% 21.59/7.92  thf(func_def_590, type, count_list_o: list_o > $o > nat).
% 21.59/7.92  thf(func_def_591, type, count_list_complex: list_complex > complex > nat).
% 21.59/7.92  thf(func_def_592, type, count_list_int: list_int > int > nat).
% 21.59/7.92  thf(func_def_593, type, count_list_nat: list_nat > nat > nat).
% 21.59/7.92  thf(func_def_594, type, count_4203492906077236349at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat > nat).
% 21.59/7.92  thf(func_def_595, type, count_list_real: list_real > real > nat).
% 21.59/7.92  thf(func_def_596, type, count_list_set_nat: list_set_nat > set_nat > nat).
% 21.59/7.92  thf(func_def_597, type, count_list_VEBT_VEBT: list_VEBT_VEBT > vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_598, type, distinct_int: list_int > $o).
% 21.59/7.92  thf(func_def_599, type, distinct_nat: list_nat > $o).
% 21.59/7.92  thf(func_def_600, type, drop_nat: nat > list_nat > list_nat).
% 21.59/7.92  thf(func_def_601, type, enumerate_o: nat > list_o > list_P7333126701944960589_nat_o).
% 21.59/7.92  thf(func_def_602, type, enumerate_int: nat > list_int > list_P3521021558325789923at_int).
% 21.59/7.92  thf(func_def_603, type, enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat).
% 21.59/7.92  thf(func_def_604, type, enumerate_VEBT_VEBT: nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT).
% 21.59/7.92  thf(func_def_605, type, find_o: ($o > $o) > list_o > option_o).
% 21.59/7.92  thf(func_def_606, type, find_int: (int > $o) > list_int > option_int).
% 21.59/7.92  thf(func_def_607, type, find_nat: (nat > $o) > list_nat > option_nat).
% 21.59/7.92  thf(func_def_608, type, find_num: (num > $o) > list_num > option_num).
% 21.59/7.92  thf(func_def_609, type, find_P8199882355184865565at_nat: (product_prod_nat_nat > $o) > list_P6011104703257516679at_nat > option4927543243414619207at_nat).
% 21.59/7.92  thf(func_def_610, type, find_real: (real > $o) > list_real > option_real).
% 21.59/7.92  thf(func_def_611, type, find_set_nat: (set_nat > $o) > list_set_nat > option_set_nat).
% 21.59/7.92  thf(func_def_612, type, find_VEBT_VEBT: (vEBT_VEBT > $o) > list_VEBT_VEBT > option_VEBT_VEBT).
% 21.59/7.92  thf(func_def_613, type, fold_nat_nat: (nat > nat > nat) > list_nat > nat > nat).
% 21.59/7.92  thf(func_def_614, type, last_nat: list_nat > nat).
% 21.59/7.92  thf(func_def_615, type, linord1735203802627413978nt_int: (int > int) > list_int > list_int).
% 21.59/7.92  thf(func_def_616, type, linord738340561235409698at_nat: (nat > nat) > list_nat > list_nat).
% 21.59/7.92  thf(func_def_617, type, linord2614967742042102400et_nat: set_nat > list_nat).
% 21.59/7.92  thf(func_def_618, type, cons_int: int > list_int > list_int).
% 21.59/7.92  thf(func_def_619, type, cons_nat: nat > list_nat > list_nat).
% 21.59/7.92  thf(func_def_620, type, nil_int: list_int).
% 21.59/7.92  thf(func_def_621, type, nil_nat: list_nat).
% 21.59/7.92  thf(func_def_622, type, hd_nat: list_nat > nat).
% 21.59/7.92  thf(func_def_623, type, map_nat_nat: (nat > nat) > list_nat > list_nat).
% 21.59/7.92  thf(func_def_624, type, set_Co8062243466402858685e_term: list_C878401137130745250e_term > set_Co9149898834107579976e_term).
% 21.59/7.92  thf(func_def_625, type, set_in5217446777445088012e_term: list_i8448526496819171953e_term > set_in3461395444621081367e_term).
% 21.59/7.92  thf(func_def_626, type, set_Pr8342322266483756581e_term: list_P1316552470764441098e_term > set_Pr7604974323444597168e_term).
% 21.59/7.92  thf(func_def_627, type, set_Pr16608062948090134e_term: list_P1743416141875011707e_term > set_Pr3642885161833720865e_term).
% 21.59/7.92  thf(func_def_628, type, set_o2: list_o > set_o).
% 21.59/7.92  thf(func_def_629, type, set_complex2: list_complex > set_complex).
% 21.59/7.92  thf(func_def_630, type, set_int2: list_int > set_int).
% 21.59/7.92  thf(func_def_631, type, set_nat2: list_nat > set_nat).
% 21.59/7.92  thf(func_def_632, type, set_num2: list_num > set_num).
% 21.59/7.92  thf(func_def_633, type, set_Pr2999063419360598313nteger: list_P5311841565141990158nteger > set_Pr8056137968301705908nteger).
% 21.59/7.92  thf(func_def_634, type, set_Pr1633835384712236856nt_int: list_P8915022641806594461nt_int > set_Pr1872883991513573699nt_int).
% 21.59/7.92  thf(func_def_635, type, set_Pr2135590979564877377nteger: list_P7828571989066258726nteger > set_Pr1281608226676607948nteger).
% 21.59/7.92  thf(func_def_636, type, set_Pr4943052134776177454nt_int: list_P651320350408439699nt_int > set_Pr9222295170931077689nt_int).
% 21.59/7.92  thf(func_def_637, type, set_Product_prod_o_o2: list_P4002435161011370285od_o_o > set_Product_prod_o_o).
% 21.59/7.92  thf(func_def_638, type, set_Pr2828948584524939422_o_int: list_P3795440434834930179_o_int > set_Pr8834758594704517033_o_int).
% 21.59/7.92  thf(func_def_639, type, set_Pr7006799604034136130_o_nat: list_P6285523579766656935_o_nat > set_Pr2101469702781467981_o_nat).
% 21.59/7.92  thf(func_def_640, type, set_Pr2600826154070092190o_real: list_P5232166724548748803o_real > set_Pr6573716822653411497o_real).
% 21.59/7.92  thf(func_def_641, type, set_Pr655345902815428824T_VEBT: list_P7495141550334521929T_VEBT > set_Pr7543698050874017315T_VEBT).
% 21.59/7.92  thf(func_def_642, type, set_Pr920681315882439344nteger: list_P5578671422887162913nteger > set_Pr4811707699266497531nteger).
% 21.59/7.92  thf(func_def_643, type, set_Pr2470121279949933262nt_int: list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int).
% 21.59/7.92  thf(func_def_644, type, set_Pr112895574167722958t_real: list_P6863124054624500543t_real > set_Pr3538720872664544793t_real).
% 21.59/7.92  thf(func_def_645, type, set_Pr8714266321650254504T_VEBT: list_P7524865323317820941T_VEBT > set_Pr8044002425091019955T_VEBT).
% 21.59/7.92  thf(func_def_646, type, set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat).
% 21.59/7.92  thf(func_def_647, type, set_Pr5518436109238095868at_nat: list_P8469869581646625389at_nat > set_Pr8693737435421807431at_nat).
% 21.59/7.92  thf(func_def_648, type, set_Pr5196769464307566348real_o: list_P3595434254542482545real_o > set_Pr4936984352647145239real_o).
% 21.59/7.92  thf(func_def_649, type, set_Pr8219819362198175822al_int: list_P4344331454722006975al_int > set_Pr1019928272762051225al_int).
% 21.59/7.92  thf(func_def_650, type, set_Pr3174298344852596722al_nat: list_P6834414599653733731al_nat > set_Pr3510011417693777981al_nat).
% 21.59/7.92  thf(func_def_651, type, set_Pr5999470521830281550l_real: list_P8689742595348180415l_real > set_Pr6218003697084177305l_real).
% 21.59/7.92  thf(func_def_652, type, set_Pr8897343066327330088T_VEBT: list_P877281246627933069T_VEBT > set_Pr6019664923565264691T_VEBT).
% 21.59/7.92  thf(func_def_653, type, set_Pr9040384385603167362et_nat: list_P6254988961118846195et_nat > set_Pr5488025237498180813et_nat).
% 21.59/7.92  thf(func_def_654, type, set_Pr3765526544606949372at_nat: list_P5464809261938338413at_nat > set_Pr4329608150637261639at_nat).
% 21.59/7.92  thf(func_def_655, type, set_Pr7708085864119495200VEBT_o: list_P3126845725202233233VEBT_o > set_Pr3175402225741728619VEBT_o).
% 21.59/7.92  thf(func_def_656, type, set_Pr2853735649769556538BT_int: list_P4547456442757143711BT_int > set_Pr5066593544530342725BT_int).
% 21.59/7.92  thf(func_def_657, type, set_Pr7031586669278753246BT_nat: list_P7037539587688870467BT_nat > set_Pr7556676689462069481BT_nat).
% 21.59/7.92  thf(func_def_658, type, set_Pr1087130671499945274T_real: list_P2623026923184700063T_real > set_Pr7765410600122031685T_real).
% 21.59/7.92  thf(func_def_659, type, set_Pr9182192707038809660T_VEBT: list_P7413028617227757229T_VEBT > set_Pr6192946355708809607T_VEBT).
% 21.59/7.92  thf(func_def_660, type, set_real2: list_real > set_real).
% 21.59/7.92  thf(func_def_661, type, set_set_nat2: list_set_nat > set_set_nat).
% 21.59/7.92  thf(func_def_662, type, set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT).
% 21.59/7.92  thf(func_def_663, type, size_list_VEBT_VEBT: (vEBT_VEBT > nat) > list_VEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_664, type, tl_nat: list_nat > list_nat).
% 21.59/7.92  thf(func_def_665, type, list_u4743598893156345252e_term: list_C878401137130745250e_term > nat > (code_integer > option6357759511663192854e_term) > list_C878401137130745250e_term).
% 21.59/7.92  thf(func_def_666, type, list_u8946639151299769843e_term: list_i8448526496819171953e_term > nat > (int > option6357759511663192854e_term) > list_i8448526496819171953e_term).
% 21.59/7.92  thf(func_def_667, type, list_u877304756163299468e_term: list_P1316552470764441098e_term > nat > (produc6241069584506657477e_term > option6357759511663192854e_term) > list_P1316552470764441098e_term).
% 21.59/7.92  thf(func_def_668, type, list_u3533491785856317309e_term: list_P1743416141875011707e_term > nat > (produc8551481072490612790e_term > option6357759511663192854e_term) > list_P1743416141875011707e_term).
% 21.59/7.92  thf(func_def_669, type, list_update_o: list_o > nat > $o > list_o).
% 21.59/7.92  thf(func_def_670, type, list_update_int: list_int > nat > int > list_int).
% 21.59/7.92  thf(func_def_671, type, list_update_nat: list_nat > nat > nat > list_nat).
% 21.59/7.92  thf(func_def_672, type, list_u1133519416628930960nteger: list_P5311841565141990158nteger > nat > produc8763457246119570046nteger > list_P5311841565141990158nteger).
% 21.59/7.92  thf(func_def_673, type, list_u4780935413889332127nt_int: list_P8915022641806594461nt_int > nat > produc7773217078559923341nt_int > list_P8915022641806594461nt_int).
% 21.59/7.92  thf(func_def_674, type, list_u234853988314817064nteger: list_P7828571989066258726nteger > nat > produc1908205239877642774nteger > list_P7828571989066258726nteger).
% 21.59/7.92  thf(func_def_675, type, list_u7736365598306452245nt_int: list_P651320350408439699nt_int > nat > produc2285326912895808259nt_int > list_P651320350408439699nt_int).
% 21.59/7.92  thf(func_def_676, type, list_u2254550707601501961nteger: list_P5578671422887162913nteger > nat > produc8923325533196201883nteger > list_P5578671422887162913nteger).
% 21.59/7.92  thf(func_def_677, type, list_u3002344382305578791nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int > list_P5707943133018811711nt_int).
% 21.59/7.92  thf(func_def_678, type, list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat).
% 21.59/7.92  thf(func_def_679, type, list_u4696772448584712917at_nat: list_P5464809261938338413at_nat > nat > produc3843707927480180839at_nat > list_P5464809261938338413at_nat).
% 21.59/7.92  thf(func_def_680, type, list_u6961636818849549845T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT > list_P7413028617227757229T_VEBT).
% 21.59/7.92  thf(func_def_681, type, list_update_real: list_real > nat > real > list_real).
% 21.59/7.92  thf(func_def_682, type, list_update_set_nat: list_set_nat > nat > set_nat > list_set_nat).
% 21.59/7.92  thf(func_def_683, type, list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_684, type, listrel1_o: set_Product_prod_o_o > set_Pr6227168374412355847list_o).
% 21.59/7.92  thf(func_def_685, type, listrel1_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int).
% 21.59/7.92  thf(func_def_686, type, listrel1_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat).
% 21.59/7.92  thf(func_def_687, type, listrel1_VEBT_VEBT: set_Pr6192946355708809607T_VEBT > set_Pr1916528119006554503T_VEBT).
% 21.59/7.92  thf(func_def_688, type, listrel1p_int: (int > int > $o) > list_int > list_int > $o).
% 21.59/7.92  thf(func_def_689, type, listrel1p_nat: (nat > nat > $o) > list_nat > list_nat > $o).
% 21.59/7.92  thf(func_def_690, type, nth_o: list_o > nat > $o).
% 21.59/7.92  thf(func_def_691, type, nth_int: list_int > nat > int).
% 21.59/7.92  thf(func_def_692, type, nth_nat: list_nat > nat > nat).
% 21.59/7.92  thf(func_def_693, type, nth_num: list_num > nat > num).
% 21.59/7.92  thf(func_def_694, type, nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o).
% 21.59/7.92  thf(func_def_695, type, nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int).
% 21.59/7.92  thf(func_def_696, type, nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat).
% 21.59/7.92  thf(func_def_697, type, nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT).
% 21.59/7.92  thf(func_def_698, type, nth_Pr112076138515278198_nat_o: list_P7333126701944960589_nat_o > nat > product_prod_nat_o).
% 21.59/7.92  thf(func_def_699, type, nth_Pr3440142176431000676at_int: list_P3521021558325789923at_int > nat > product_prod_nat_int).
% 21.59/7.92  thf(func_def_700, type, nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat).
% 21.59/7.92  thf(func_def_701, type, nth_Pr744662078594809490T_VEBT: list_P5647936690300460905T_VEBT > nat > produc8025551001238799321T_VEBT).
% 21.59/7.92  thf(func_def_702, type, nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o).
% 21.59/7.92  thf(func_def_703, type, nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int).
% 21.59/7.92  thf(func_def_704, type, nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat).
% 21.59/7.92  thf(func_def_705, type, nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT).
% 21.59/7.92  thf(func_def_706, type, nth_real: list_real > nat > real).
% 21.59/7.92  thf(func_def_707, type, nth_set_nat: list_set_nat > nat > set_nat).
% 21.59/7.92  thf(func_def_708, type, nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT).
% 21.59/7.92  thf(func_def_709, type, product_o_o: list_o > list_o > list_P4002435161011370285od_o_o).
% 21.59/7.92  thf(func_def_710, type, product_o_int: list_o > list_int > list_P3795440434834930179_o_int).
% 21.59/7.92  thf(func_def_711, type, product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat).
% 21.59/7.92  thf(func_def_712, type, product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT).
% 21.59/7.92  thf(func_def_713, type, product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o).
% 21.59/7.92  thf(func_def_714, type, produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT).
% 21.59/7.92  thf(func_def_715, type, product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o).
% 21.59/7.92  thf(func_def_716, type, produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int).
% 21.59/7.92  thf(func_def_717, type, produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat).
% 21.59/7.92  thf(func_def_718, type, produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT).
% 21.59/7.92  thf(func_def_719, type, remdups_nat: list_nat > list_nat).
% 21.59/7.92  thf(func_def_720, type, replicate_o: nat > $o > list_o).
% 21.59/7.92  thf(func_def_721, type, replicate_int: nat > int > list_int).
% 21.59/7.92  thf(func_def_722, type, replicate_nat: nat > nat > list_nat).
% 21.59/7.92  thf(func_def_723, type, replic4235873036481779905at_nat: nat > product_prod_nat_nat > list_P6011104703257516679at_nat).
% 21.59/7.92  thf(func_def_724, type, replic2264142908078655527at_nat: nat > produc3843707927480180839at_nat > list_P5464809261938338413at_nat).
% 21.59/7.92  thf(func_def_725, type, replicate_real: nat > real > list_real).
% 21.59/7.92  thf(func_def_726, type, replicate_set_nat: nat > set_nat > list_set_nat).
% 21.59/7.92  thf(func_def_727, type, replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_728, type, sorted_wrt_int: (int > int > $o) > list_int > $o).
% 21.59/7.92  thf(func_def_729, type, sorted_wrt_nat: (nat > nat > $o) > list_nat > $o).
% 21.59/7.92  thf(func_def_730, type, take_nat: nat > list_nat > list_nat).
% 21.59/7.92  thf(func_def_731, type, upt: nat > nat > list_nat).
% 21.59/7.92  thf(func_def_732, type, upto: int > int > list_int).
% 21.59/7.92  thf(func_def_733, type, upto_aux: int > int > list_int > list_int).
% 21.59/7.92  thf(func_def_734, type, upto_rel: product_prod_int_int > product_prod_int_int > $o).
% 21.59/7.92  thf(func_def_735, type, zip_Co8729459035503499408nteger: list_C878401137130745250e_term > list_P5578671422887162913nteger > list_P5311841565141990158nteger).
% 21.59/7.92  thf(func_def_736, type, zip_in8766932505889695135nt_int: list_i8448526496819171953e_term > list_P5707943133018811711nt_int > list_P8915022641806594461nt_int).
% 21.59/7.92  thf(func_def_737, type, zip_Pr8292346330294042792nteger: list_P1316552470764441098e_term > list_P5578671422887162913nteger > list_P7828571989066258726nteger).
% 21.59/7.92  thf(func_def_738, type, zip_Pr4168994715204986005nt_int: list_P1743416141875011707e_term > list_P5707943133018811711nt_int > list_P651320350408439699nt_int).
% 21.59/7.92  thf(func_def_739, type, zip_o_o: list_o > list_o > list_P4002435161011370285od_o_o).
% 21.59/7.92  thf(func_def_740, type, zip_o_int: list_o > list_int > list_P3795440434834930179_o_int).
% 21.59/7.92  thf(func_def_741, type, zip_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat).
% 21.59/7.92  thf(func_def_742, type, zip_o_real: list_o > list_real > list_P5232166724548748803o_real).
% 21.59/7.92  thf(func_def_743, type, zip_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT).
% 21.59/7.92  thf(func_def_744, type, zip_int_int: list_int > list_int > list_P5707943133018811711nt_int).
% 21.59/7.92  thf(func_def_745, type, zip_int_real: list_int > list_real > list_P6863124054624500543t_real).
% 21.59/7.92  thf(func_def_746, type, zip_int_VEBT_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT).
% 21.59/7.92  thf(func_def_747, type, zip_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o).
% 21.59/7.92  thf(func_def_748, type, zip_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat).
% 21.59/7.92  thf(func_def_749, type, zip_nat_VEBT_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT).
% 21.59/7.92  thf(func_def_750, type, zip_Pr4664179122662387191at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P8469869581646625389at_nat).
% 21.59/7.92  thf(func_def_751, type, zip_real_o: list_real > list_o > list_P3595434254542482545real_o).
% 21.59/7.92  thf(func_def_752, type, zip_real_int: list_real > list_int > list_P4344331454722006975al_int).
% 21.59/7.92  thf(func_def_753, type, zip_real_nat: list_real > list_nat > list_P6834414599653733731al_nat).
% 21.59/7.92  thf(func_def_754, type, zip_real_real: list_real > list_real > list_P8689742595348180415l_real).
% 21.59/7.92  thf(func_def_755, type, zip_real_VEBT_VEBT: list_real > list_VEBT_VEBT > list_P877281246627933069T_VEBT).
% 21.59/7.92  thf(func_def_756, type, zip_set_nat_set_nat: list_set_nat > list_set_nat > list_P6254988961118846195et_nat).
% 21.59/7.92  thf(func_def_757, type, zip_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o).
% 21.59/7.92  thf(func_def_758, type, zip_VEBT_VEBT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int).
% 21.59/7.92  thf(func_def_759, type, zip_VEBT_VEBT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat).
% 21.59/7.92  thf(func_def_760, type, zip_VEBT_VEBT_real: list_VEBT_VEBT > list_real > list_P2623026923184700063T_real).
% 21.59/7.92  thf(func_def_761, type, zip_VE537291747668921783T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT).
% 21.59/7.92  thf(func_def_762, type, suc: nat > nat).
% 21.59/7.92  thf(func_def_763, type, compow_nat_nat: nat > (nat > nat) > nat > nat).
% 21.59/7.92  thf(func_def_764, type, case_nat_o: $o > (nat > $o) > nat > $o).
% 21.59/7.92  thf(func_def_765, type, case_nat_nat: nat > (nat > nat) > nat > nat).
% 21.59/7.92  thf(func_def_766, type, case_nat_option_num: option_num > (nat > option_num) > nat > option_num).
% 21.59/7.92  thf(func_def_767, type, pred: nat > nat).
% 21.59/7.92  thf(func_def_768, type, semiri4939895301339042750nteger: nat > code_integer).
% 21.59/7.92  thf(func_def_769, type, semiri8010041392384452111omplex: nat > complex).
% 21.59/7.92  thf(func_def_770, type, semiri4216267220026989637d_enat: nat > extended_enat).
% 21.59/7.92  thf(func_def_771, type, semiri1314217659103216013at_int: nat > int).
% 21.59/7.92  thf(func_def_772, type, semiri1316708129612266289at_nat: nat > nat).
% 21.59/7.92  thf(func_def_773, type, semiri681578069525770553at_rat: nat > rat).
% 21.59/7.92  thf(func_def_774, type, semiri5074537144036343181t_real: nat > real).
% 21.59/7.92  thf(func_def_775, type, semiri2816024913162550771omplex: (complex > complex) > nat > complex > complex).
% 21.59/7.92  thf(func_def_776, type, semiri8420488043553186161ux_int: (int > int) > nat > int > int).
% 21.59/7.92  thf(func_def_777, type, semiri8422978514062236437ux_nat: (nat > nat) > nat > nat > nat).
% 21.59/7.92  thf(func_def_778, type, semiri7787848453975740701ux_rat: (rat > rat) > nat > rat > rat).
% 21.59/7.92  thf(func_def_779, type, semiri7260567687927622513x_real: (real > real) > nat > real > real).
% 21.59/7.92  thf(func_def_780, type, size_size_list_o: list_o > nat).
% 21.59/7.92  thf(func_def_781, type, size_s3451745648224563538omplex: list_complex > nat).
% 21.59/7.92  thf(func_def_782, type, size_size_list_int: list_int > nat).
% 21.59/7.92  thf(func_def_783, type, size_size_list_nat: list_nat > nat).
% 21.59/7.92  thf(func_def_784, type, size_size_list_num: list_num > nat).
% 21.59/7.92  thf(func_def_785, type, size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat).
% 21.59/7.92  thf(func_def_786, type, size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat).
% 21.59/7.92  thf(func_def_787, type, size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat).
% 21.59/7.92  thf(func_def_788, type, size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat).
% 21.59/7.92  thf(func_def_789, type, size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat).
% 21.59/7.92  thf(func_def_790, type, size_s2970893825323803983at_int: list_P3521021558325789923at_int > nat).
% 21.59/7.92  thf(func_def_791, type, size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat).
% 21.59/7.92  thf(func_def_792, type, size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat).
% 21.59/7.92  thf(func_def_793, type, size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat).
% 21.59/7.92  thf(func_def_794, type, size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat).
% 21.59/7.92  thf(func_def_795, type, size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat).
% 21.59/7.92  thf(func_def_796, type, size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat).
% 21.59/7.92  thf(func_def_797, type, size_size_list_real: list_real > nat).
% 21.59/7.92  thf(func_def_798, type, size_s3254054031482475050et_nat: list_set_nat > nat).
% 21.59/7.92  thf(func_def_799, type, size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_800, type, size_size_num: num > nat).
% 21.59/7.92  thf(func_def_801, type, size_size_option_nat: option_nat > nat).
% 21.59/7.92  thf(func_def_802, type, size_size_option_num: option_num > nat).
% 21.59/7.92  thf(func_def_803, type, size_s170228958280169651at_nat: option4927543243414619207at_nat > nat).
% 21.59/7.92  thf(func_def_804, type, size_size_VEBT_VEBT: vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_805, type, nat_prod_decode_aux: nat > nat > product_prod_nat_nat).
% 21.59/7.92  thf(func_def_806, type, nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 21.59/7.92  thf(func_def_807, type, nat_set_decode: nat > set_nat).
% 21.59/7.92  thf(func_def_808, type, nat_set_encode: set_nat > nat).
% 21.59/7.92  thf(func_def_809, type, root: nat > real > real).
% 21.59/7.92  thf(func_def_810, type, sqrt: real > real).
% 21.59/7.92  thf(func_def_811, type, bitM: num > num).
% 21.59/7.92  thf(func_def_812, type, inc: num > num).
% 21.59/7.92  thf(func_def_813, type, neg_nu7757733837767384882nteger: code_integer > code_integer).
% 21.59/7.92  thf(func_def_814, type, neg_nu6511756317524482435omplex: complex > complex).
% 21.59/7.92  thf(func_def_815, type, neg_nu3811975205180677377ec_int: int > int).
% 21.59/7.92  thf(func_def_816, type, neg_nu3179335615603231917ec_rat: rat > rat).
% 21.59/7.92  thf(func_def_817, type, neg_nu6075765906172075777c_real: real > real).
% 21.59/7.92  thf(func_def_818, type, bit0: num > num).
% 21.59/7.92  thf(func_def_819, type, bit1: num > num).
% 21.59/7.92  thf(func_def_820, type, one: num).
% 21.59/7.92  thf(func_def_821, type, case_num_option_num: option_num > (num > option_num) > (num > option_num) > num > option_num).
% 21.59/7.92  thf(func_def_822, type, size_num: num > nat).
% 21.59/7.92  thf(func_def_823, type, num_of_nat: nat > num).
% 21.59/7.92  thf(func_def_824, type, numera6620942414471956472nteger: num > code_integer).
% 21.59/7.92  thf(func_def_825, type, numera6690914467698888265omplex: num > complex).
% 21.59/7.92  thf(func_def_826, type, numera1916890842035813515d_enat: num > extended_enat).
% 21.59/7.92  thf(func_def_827, type, numeral_numeral_int: num > int).
% 21.59/7.92  thf(func_def_828, type, numeral_numeral_nat: num > nat).
% 21.59/7.92  thf(func_def_829, type, numeral_numeral_rat: num > rat).
% 21.59/7.92  thf(func_def_830, type, numeral_numeral_real: num > real).
% 21.59/7.92  thf(func_def_831, type, pred_numeral: num > nat).
% 21.59/7.92  thf(func_def_832, type, none_int: option_int).
% 21.59/7.92  thf(func_def_833, type, none_nat: option_nat).
% 21.59/7.92  thf(func_def_834, type, none_num: option_num).
% 21.59/7.92  thf(func_def_835, type, none_P5556105721700978146at_nat: option4927543243414619207at_nat).
% 21.59/7.92  thf(func_def_836, type, none_real: option_real).
% 21.59/7.92  thf(func_def_837, type, none_set_nat: option_set_nat).
% 21.59/7.92  thf(func_def_838, type, none_VEBT_VEBT: option_VEBT_VEBT).
% 21.59/7.92  thf(func_def_839, type, some_o: $o > option_o).
% 21.59/7.92  thf(func_def_840, type, some_int: int > option_int).
% 21.59/7.92  thf(func_def_841, type, some_nat: nat > option_nat).
% 21.59/7.92  thf(func_def_842, type, some_num: num > option_num).
% 21.59/7.92  thf(func_def_843, type, some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat).
% 21.59/7.92  thf(func_def_844, type, some_VEBT_VEBT: vEBT_VEBT > option_VEBT_VEBT).
% 21.59/7.92  thf(func_def_845, type, case_o184042715313410164at_nat: $o > (product_prod_nat_nat > $o) > option4927543243414619207at_nat > $o).
% 21.59/7.92  thf(func_def_846, type, case_option_num_num: num > (num > num) > option_num > num).
% 21.59/7.92  thf(func_def_847, type, case_o6005452278849405969um_num: option_num > (num > option_num) > option_num > option_num).
% 21.59/7.92  thf(func_def_848, type, map_option_num_num: (num > num) > option_num > option_num).
% 21.59/7.92  thf(func_def_849, type, size_option_nat: (nat > nat) > option_nat > nat).
% 21.59/7.92  thf(func_def_850, type, size_option_num: (num > nat) > option_num > nat).
% 21.59/7.92  thf(func_def_851, type, size_o8335143837870341156at_nat: (product_prod_nat_nat > nat) > option4927543243414619207at_nat > nat).
% 21.59/7.92  thf(func_def_852, type, the_nat2: option_nat > nat).
% 21.59/7.92  thf(func_def_853, type, the_num: option_num > num).
% 21.59/7.92  thf(func_def_854, type, the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat).
% 21.59/7.92  thf(func_def_855, type, order_underS_nat: set_Pr1261947904930325089at_nat > nat > set_nat).
% 21.59/7.92  thf(func_def_856, type, bot_bo5358457235160185703eger_o: (code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > $o).
% 21.59/7.92  thf(func_def_857, type, bot_bo1403522918969695512_int_o: (int > option6357759511663192854e_term) > product_prod_int_int > $o).
% 21.59/7.92  thf(func_def_858, type, bot_bo3000040243691356879eger_o: (produc6241069584506657477e_term > option6357759511663192854e_term) > produc8923325533196201883nteger > $o).
% 21.59/7.92  thf(func_def_859, type, bot_bo8662317086119403298_int_o: (produc8551481072490612790e_term > option6357759511663192854e_term) > product_prod_int_int > $o).
% 21.59/7.92  thf(func_def_860, type, bot_bot_int_int_o: int > int > $o).
% 21.59/7.92  thf(func_def_861, type, bot_bot_int_o: int > $o).
% 21.59/7.92  thf(func_def_862, type, bot_bot_list_nat_o: list_nat > $o).
% 21.59/7.92  thf(func_def_863, type, bot_bot_nat_nat_o: nat > nat > $o).
% 21.59/7.92  thf(func_def_864, type, bot_bot_nat_o: nat > $o).
% 21.59/7.92  thf(func_def_865, type, bot_bo482883023278783056_nat_o: product_prod_nat_nat > $o).
% 21.59/7.92  thf(func_def_866, type, bot_bot_real_o: real > $o).
% 21.59/7.92  thf(func_def_867, type, bot_bot_set_nat_o: set_nat > $o).
% 21.59/7.92  thf(func_def_868, type, bot_bo4199563552545308370d_enat: extended_enat).
% 21.59/7.92  thf(func_def_869, type, bot_bot_filter_nat: filter_nat).
% 21.59/7.92  thf(func_def_870, type, bot_bot_nat: nat).
% 21.59/7.92  thf(func_def_871, type, bot_bot_set_complex: set_complex).
% 21.59/7.92  thf(func_def_872, type, bot_bot_set_int: set_int).
% 21.59/7.92  thf(func_def_873, type, bot_bot_set_list_nat: set_list_nat).
% 21.59/7.92  thf(func_def_874, type, bot_bot_set_nat: set_nat).
% 21.59/7.92  thf(func_def_875, type, bot_bot_set_num: set_num).
% 21.59/7.92  thf(func_def_876, type, bot_bo3145834390647256904nteger: set_Pr8056137968301705908nteger).
% 21.59/7.92  thf(func_def_877, type, bot_bo4508923176915781079nt_int: set_Pr1872883991513573699nt_int).
% 21.59/7.92  thf(func_def_878, type, bot_bo5443222936135328352nteger: set_Pr1281608226676607948nteger).
% 21.59/7.92  thf(func_def_879, type, bot_bo572930865798478029nt_int: set_Pr9222295170931077689nt_int).
% 21.59/7.92  thf(func_def_880, type, bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int).
% 21.59/7.92  thf(func_def_881, type, bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat).
% 21.59/7.92  thf(func_def_882, type, bot_bo228742789529271731at_nat: set_Pr4329608150637261639at_nat).
% 21.59/7.92  thf(func_def_883, type, bot_bot_set_rat: set_rat).
% 21.59/7.92  thf(func_def_884, type, bot_bot_set_real: set_real).
% 21.59/7.92  thf(func_def_885, type, bot_bot_set_set_int: set_set_int).
% 21.59/7.92  thf(func_def_886, type, bot_bot_set_set_nat: set_set_nat).
% 21.59/7.92  thf(func_def_887, type, bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT).
% 21.59/7.92  thf(func_def_888, type, ord_Least_nat: (nat > $o) > nat).
% 21.59/7.92  thf(func_def_889, type, ord_less_int_o: (int > $o) > (int > $o) > $o).
% 21.59/7.92  thf(func_def_890, type, ord_less_nat_o: (nat > $o) > (nat > $o) > $o).
% 21.59/7.92  thf(func_def_891, type, ord_le549003669493604880_nat_o: (product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > $o).
% 21.59/7.92  thf(func_def_892, type, ord_less_real_o: (real > $o) > (real > $o) > $o).
% 21.59/7.92  thf(func_def_893, type, ord_less_set_nat_o: (set_nat > $o) > (set_nat > $o) > $o).
% 21.59/7.92  thf(func_def_894, type, ord_le6747313008572928689nteger: code_integer > code_integer > $o).
% 21.59/7.92  thf(func_def_895, type, ord_le72135733267957522d_enat: extended_enat > extended_enat > $o).
% 21.59/7.92  thf(func_def_896, type, ord_less_int: int > int > $o).
% 21.59/7.92  thf(func_def_897, type, ord_less_nat: nat > nat > $o).
% 21.59/7.92  thf(func_def_898, type, ord_less_num: num > num > $o).
% 21.59/7.92  thf(func_def_899, type, ord_less_rat: rat > rat > $o).
% 21.59/7.92  thf(func_def_900, type, ord_less_real: real > real > $o).
% 21.59/7.92  thf(func_def_901, type, ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o).
% 21.59/7.92  thf(func_def_902, type, ord_less_set_complex: set_complex > set_complex > $o).
% 21.59/7.92  thf(func_def_903, type, ord_less_set_int: set_int > set_int > $o).
% 21.59/7.92  thf(func_def_904, type, ord_less_set_nat: set_nat > set_nat > $o).
% 21.59/7.92  thf(func_def_905, type, ord_less_set_num: set_num > set_num > $o).
% 21.59/7.92  thf(func_def_906, type, ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o).
% 21.59/7.92  thf(func_def_907, type, ord_le2604355607129572851at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o).
% 21.59/7.92  thf(func_def_908, type, ord_less_set_rat: set_rat > set_rat > $o).
% 21.59/7.92  thf(func_def_909, type, ord_less_set_real: set_real > set_real > $o).
% 21.59/7.92  thf(func_def_910, type, ord_less_set_set_int: set_set_int > set_set_int > $o).
% 21.59/7.92  thf(func_def_911, type, ord_less_set_set_nat: set_set_nat > set_set_nat > $o).
% 21.59/7.92  thf(func_def_912, type, ord_le3636971675376928563eger_o: ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > $o) > ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > $o) > $o).
% 21.59/7.92  thf(func_def_913, type, ord_le2124322318746777828_int_o: ((int > option6357759511663192854e_term) > product_prod_int_int > $o) > ((int > option6357759511663192854e_term) > product_prod_int_int > $o) > $o).
% 21.59/7.92  thf(func_def_914, type, ord_le4340812435750786203eger_o: ((produc6241069584506657477e_term > option6357759511663192854e_term) > produc8923325533196201883nteger > $o) > ((produc6241069584506657477e_term > option6357759511663192854e_term) > produc8923325533196201883nteger > $o) > $o).
% 21.59/7.92  thf(func_def_915, type, ord_le5643404153117327598_int_o: ((produc8551481072490612790e_term > option6357759511663192854e_term) > product_prod_int_int > $o) > ((produc8551481072490612790e_term > option6357759511663192854e_term) > product_prod_int_int > $o) > $o).
% 21.59/7.92  thf(func_def_916, type, ord_le6741204236512500942_int_o: (int > int > $o) > (int > int > $o) > $o).
% 21.59/7.92  thf(func_def_917, type, ord_less_eq_int_o: (int > $o) > (int > $o) > $o).
% 21.59/7.92  thf(func_def_918, type, ord_le2646555220125990790_nat_o: (nat > nat > $o) > (nat > nat > $o) > $o).
% 21.59/7.92  thf(func_def_919, type, ord_less_eq_nat_o: (nat > $o) > (nat > $o) > $o).
% 21.59/7.92  thf(func_def_920, type, ord_le704812498762024988_nat_o: (product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > $o).
% 21.59/7.92  thf(func_def_921, type, ord_less_eq_real_o: (real > $o) > (real > $o) > $o).
% 21.59/7.92  thf(func_def_922, type, ord_le3964352015994296041_nat_o: (set_nat > $o) > (set_nat > $o) > $o).
% 21.59/7.92  thf(func_def_923, type, ord_le3102999989581377725nteger: code_integer > code_integer > $o).
% 21.59/7.92  thf(func_def_924, type, ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o).
% 21.59/7.92  thf(func_def_925, type, ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o).
% 21.59/7.92  thf(func_def_926, type, ord_less_eq_int: int > int > $o).
% 21.59/7.92  thf(func_def_927, type, ord_less_eq_nat: nat > nat > $o).
% 21.59/7.92  thf(func_def_928, type, ord_less_eq_num: num > num > $o).
% 21.59/7.92  thf(func_def_929, type, ord_less_eq_rat: rat > rat > $o).
% 21.59/7.92  thf(func_def_930, type, ord_less_eq_real: real > real > $o).
% 21.59/7.92  thf(func_def_931, type, ord_less_eq_set_o: set_o > set_o > $o).
% 21.59/7.92  thf(func_def_932, type, ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o).
% 21.59/7.92  thf(func_def_933, type, ord_le211207098394363844omplex: set_complex > set_complex > $o).
% 21.59/7.92  thf(func_def_934, type, ord_less_eq_set_int: set_int > set_int > $o).
% 21.59/7.92  thf(func_def_935, type, ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o).
% 21.59/7.92  thf(func_def_936, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 21.59/7.92  thf(func_def_937, type, ord_less_eq_set_num: set_num > set_num > $o).
% 21.59/7.92  thf(func_def_938, type, ord_le3216752416896350996nteger: set_Pr8056137968301705908nteger > set_Pr8056137968301705908nteger > $o).
% 21.59/7.92  thf(func_def_939, type, ord_le135402666524580259nt_int: set_Pr1872883991513573699nt_int > set_Pr1872883991513573699nt_int > $o).
% 21.59/7.92  thf(func_def_940, type, ord_le653643898420964396nteger: set_Pr1281608226676607948nteger > set_Pr1281608226676607948nteger > $o).
% 21.59/7.92  thf(func_def_941, type, ord_le8725513860283290265nt_int: set_Pr9222295170931077689nt_int > set_Pr9222295170931077689nt_int > $o).
% 21.59/7.92  thf(func_def_942, type, ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o).
% 21.59/7.92  thf(func_def_943, type, ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o).
% 21.59/7.92  thf(func_def_944, type, ord_le1268244103169919719at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o).
% 21.59/7.92  thf(func_def_945, type, ord_less_eq_set_rat: set_rat > set_rat > $o).
% 21.59/7.92  thf(func_def_946, type, ord_less_eq_set_real: set_real > set_real > $o).
% 21.59/7.92  thf(func_def_947, type, ord_le4403425263959731960et_int: set_set_int > set_set_int > $o).
% 21.59/7.92  thf(func_def_948, type, ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o).
% 21.59/7.92  thf(func_def_949, type, ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_950, type, ord_max_Code_integer: code_integer > code_integer > code_integer).
% 21.59/7.92  thf(func_def_951, type, ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat).
% 21.59/7.92  thf(func_def_952, type, ord_max_int: int > int > int).
% 21.59/7.92  thf(func_def_953, type, ord_max_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_954, type, ord_max_num: num > num > num).
% 21.59/7.92  thf(func_def_955, type, ord_max_rat: rat > rat > rat).
% 21.59/7.92  thf(func_def_956, type, ord_max_real: real > real > real).
% 21.59/7.92  thf(func_def_957, type, ord_max_set_int: set_int > set_int > set_int).
% 21.59/7.92  thf(func_def_958, type, ord_max_set_nat: set_nat > set_nat > set_nat).
% 21.59/7.92  thf(func_def_959, type, ord_ma7524802468073614006at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 21.59/7.92  thf(func_def_960, type, ord_max_set_real: set_real > set_real > set_real).
% 21.59/7.92  thf(func_def_961, type, ord_min_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_962, type, order_Greatest_nat: (nat > $o) > nat).
% 21.59/7.92  thf(func_def_963, type, order_mono_nat_nat: (nat > nat) > $o).
% 21.59/7.92  thf(func_def_964, type, order_5726023648592871131at_nat: (nat > nat) > $o).
% 21.59/7.92  thf(func_def_965, type, order_7092887310737990675l_real: (real > real) > $o).
% 21.59/7.92  thf(func_def_966, type, ordering_top_nat: (nat > nat > $o) > (nat > nat > $o) > nat > $o).
% 21.59/7.92  thf(func_def_967, type, top_top_set_int: set_int).
% 21.59/7.92  thf(func_def_968, type, top_top_set_nat: set_nat).
% 21.59/7.92  thf(func_def_969, type, top_to1996260823553986621t_unit: set_Product_unit).
% 21.59/7.92  thf(func_def_970, type, top_top_set_real: set_real).
% 21.59/7.92  thf(func_def_971, type, power_8256067586552552935nteger: code_integer > nat > code_integer).
% 21.59/7.92  thf(func_def_972, type, power_power_complex: complex > nat > complex).
% 21.59/7.92  thf(func_def_973, type, power_power_int: int > nat > int).
% 21.59/7.92  thf(func_def_974, type, power_power_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_975, type, power_power_rat: rat > nat > rat).
% 21.59/7.92  thf(func_def_976, type, power_power_real: real > nat > real).
% 21.59/7.92  thf(func_def_977, type, produc6137756002093451184nteger: (code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > produc8763457246119570046nteger).
% 21.59/7.92  thf(func_def_978, type, produc4305682042979456191nt_int: (int > option6357759511663192854e_term) > product_prod_int_int > produc7773217078559923341nt_int).
% 21.59/7.92  thf(func_def_979, type, produc4035269172776083154on_nat: (nat > nat > $o) > produc4953844613479565601on_nat > produc2233624965454879586on_nat).
% 21.59/7.92  thf(func_def_980, type, produc8929957630744042906on_nat: (nat > nat > nat) > produc4953844613479565601on_nat > produc8306885398267862888on_nat).
% 21.59/7.92  thf(func_def_981, type, produc3576312749637752826on_num: (num > num > $o) > produc3447558737645232053on_num > produc7036089656553540234on_num).
% 21.59/7.92  thf(func_def_982, type, produc5778274026573060048on_num: (num > num > num) > produc3447558737645232053on_num > produc1193250871479095198on_num).
% 21.59/7.92  thf(func_def_983, type, produc8603105652947943368nteger: (produc6241069584506657477e_term > option6357759511663192854e_term) > produc8923325533196201883nteger > produc1908205239877642774nteger).
% 21.59/7.92  thf(func_def_984, type, produc5700946648718959541nt_int: (produc8551481072490612790e_term > option6357759511663192854e_term) > product_prod_int_int > produc2285326912895808259nt_int).
% 21.59/7.92  thf(func_def_985, type, produc3994169339658061776at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > produc6121120109295599847at_nat > produc5491161045314408544at_nat).
% 21.59/7.92  thf(func_def_986, type, produc2899441246263362727at_nat: (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > produc6121120109295599847at_nat > produc5542196010084753463at_nat).
% 21.59/7.92  thf(func_def_987, type, product_Pair_o_o: $o > $o > product_prod_o_o).
% 21.59/7.92  thf(func_def_988, type, product_Pair_o_int: $o > int > product_prod_o_int).
% 21.59/7.92  thf(func_def_989, type, product_Pair_o_nat: $o > nat > product_prod_o_nat).
% 21.59/7.92  thf(func_def_990, type, product_Pair_o_real: $o > real > product_prod_o_real).
% 21.59/7.92  thf(func_def_991, type, produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT).
% 21.59/7.92  thf(func_def_992, type, produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o).
% 21.59/7.92  thf(func_def_993, type, produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger).
% 21.59/7.92  thf(func_def_994, type, product_Pair_int_int: int > int > product_prod_int_int).
% 21.59/7.92  thf(func_def_995, type, produc801115645435158769t_real: int > real > produc679980390762269497t_real).
% 21.59/7.92  thf(func_def_996, type, produc3329399203697025711T_VEBT: int > vEBT_VEBT > produc1531783533982839933T_VEBT).
% 21.59/7.92  thf(func_def_997, type, produc8435520187683070743list_o: list_o > list_o > produc7102631898165422375list_o).
% 21.59/7.92  thf(func_def_998, type, produc364263696895485585st_int: list_int > list_int > produc1186641810826059865st_int).
% 21.59/7.92  thf(func_def_999, type, produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat).
% 21.59/7.92  thf(func_def_1000, type, produc3897820843166775703T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > produc9211091688327510695T_VEBT).
% 21.59/7.92  thf(func_def_1001, type, product_Pair_nat_o: nat > $o > product_prod_nat_o).
% 21.59/7.92  thf(func_def_1002, type, product_Pair_nat_int: nat > int > product_prod_nat_int).
% 21.59/7.92  thf(func_def_1003, type, product_Pair_nat_nat: nat > nat > product_prod_nat_nat).
% 21.59/7.92  thf(func_def_1004, type, product_Pair_nat_num: nat > num > product_prod_nat_num).
% 21.59/7.92  thf(func_def_1005, type, produc599794634098209291T_VEBT: nat > vEBT_VEBT > produc8025551001238799321T_VEBT).
% 21.59/7.92  thf(func_def_1006, type, product_Pair_num_num: num > num > product_prod_num_num).
% 21.59/7.92  thf(func_def_1007, type, produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat).
% 21.59/7.92  thf(func_def_1008, type, produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num).
% 21.59/7.92  thf(func_def_1009, type, produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat).
% 21.59/7.92  thf(func_def_1010, type, produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat).
% 21.59/7.92  thf(func_def_1011, type, product_Pair_real_o: real > $o > product_prod_real_o).
% 21.59/7.92  thf(func_def_1012, type, produc3179012173361985393al_int: real > int > produc8786904178792722361al_int).
% 21.59/7.92  thf(func_def_1013, type, produc3181502643871035669al_nat: real > nat > produc3741383161447143261al_nat).
% 21.59/7.92  thf(func_def_1014, type, produc4511245868158468465l_real: real > real > produc2422161461964618553l_real).
% 21.59/7.92  thf(func_def_1015, type, produc6931449550656315951T_VEBT: real > vEBT_VEBT > produc3757001726724277373T_VEBT).
% 21.59/7.92  thf(func_def_1016, type, produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat).
% 21.59/7.92  thf(func_def_1017, type, produc2922128104949294807at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > produc3843707927480180839at_nat).
% 21.59/7.92  thf(func_def_1018, type, produc9060074326276436823at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > produc1319942482725812455at_nat).
% 21.59/7.92  thf(func_def_1019, type, produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o).
% 21.59/7.92  thf(func_def_1020, type, produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int).
% 21.59/7.92  thf(func_def_1021, type, produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat).
% 21.59/7.92  thf(func_def_1022, type, produc8117437818029410057T_real: vEBT_VEBT > real > produc5170161368751668367T_real).
% 21.59/7.92  thf(func_def_1023, type, produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT).
% 21.59/7.92  thf(func_def_1024, type, produc457027306803732586at_nat: set_nat > (nat > set_nat) > set_Pr1261947904930325089at_nat).
% 21.59/7.92  thf(func_def_1025, type, produc6499014454317279255nteger: (code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 21.59/7.92  thf(func_def_1026, type, produc8678311845419106900er_nat: (code_integer > nat) > (code_integer > nat) > produc8923325533196201883nteger > product_prod_nat_nat).
% 21.59/7.92  thf(func_def_1027, type, produc127349428274296955eger_o: ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > $o) > produc8763457246119570046nteger > $o).
% 21.59/7.92  thf(func_def_1028, type, produc3906647086178084059nteger: ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > produc8763457246119570046nteger) > produc8763457246119570046nteger > produc8763457246119570046nteger).
% 21.59/7.92  thf(func_def_1029, type, produc8604463032469472703et_int: ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > set_int) > produc8763457246119570046nteger > set_int).
% 21.59/7.92  thf(func_def_1030, type, produc3558942015123893603et_nat: ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > set_nat) > produc8763457246119570046nteger > set_nat).
% 21.59/7.92  thf(func_def_1031, type, produc815715089573277247t_real: ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > set_real) > produc8763457246119570046nteger > set_real).
% 21.59/7.92  thf(func_def_1032, type, produc2558449545302689196_int_o: ((int > option6357759511663192854e_term) > product_prod_int_int > $o) > produc7773217078559923341nt_int > $o).
% 21.59/7.92  thf(func_def_1033, type, produc5122537100556696953nt_int: ((int > option6357759511663192854e_term) > product_prod_int_int > produc7773217078559923341nt_int) > produc7773217078559923341nt_int > produc7773217078559923341nt_int).
% 21.59/7.92  thf(func_def_1034, type, produc8289552606927098482et_nat: ((int > option6357759511663192854e_term) > product_prod_int_int > set_nat) > produc7773217078559923341nt_int > set_nat).
% 21.59/7.92  thf(func_def_1035, type, produc8709739885379107790t_real: ((int > option6357759511663192854e_term) > product_prod_int_int > set_real) > produc7773217078559923341nt_int > set_real).
% 21.59/7.92  thf(func_def_1036, type, produc6253627499356882019eger_o: ((produc6241069584506657477e_term > option6357759511663192854e_term) > produc8923325533196201883nteger > $o) > produc1908205239877642774nteger > $o).
% 21.59/7.92  thf(func_def_1037, type, produc6512950862096126219nteger: ((produc6241069584506657477e_term > option6357759511663192854e_term) > produc8923325533196201883nteger > produc1908205239877642774nteger) > produc1908205239877642774nteger > produc1908205239877642774nteger).
% 21.59/7.92  thf(func_def_1038, type, produc1573362020775583542_int_o: ((produc8551481072490612790e_term > option6357759511663192854e_term) > product_prod_int_int > $o) > produc2285326912895808259nt_int > $o).
% 21.59/7.92  thf(func_def_1039, type, produc8492565224438309093nt_int: ((produc8551481072490612790e_term > option6357759511663192854e_term) > product_prod_int_int > produc2285326912895808259nt_int) > produc2285326912895808259nt_int > produc2285326912895808259nt_int).
% 21.59/7.92  thf(func_def_1040, type, produc1553301316500091796er_int: (code_integer > code_integer > int) > produc8923325533196201883nteger > int).
% 21.59/7.92  thf(func_def_1041, type, produc1555791787009142072er_nat: (code_integer > code_integer > nat) > produc8923325533196201883nteger > nat).
% 21.59/7.92  thf(func_def_1042, type, produc9125791028180074456eger_o: (code_integer > code_integer > produc6271795597528267376eger_o) > produc8923325533196201883nteger > produc6271795597528267376eger_o).
% 21.59/7.92  thf(func_def_1043, type, produc6916734918728496179nteger: (code_integer > code_integer > produc8923325533196201883nteger) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 21.59/7.92  thf(func_def_1044, type, produc6771430404735790350plex_o: (complex > complex > $o) > produc4411394909380815293omplex > $o).
% 21.59/7.92  thf(func_def_1045, type, produc4947309494688390418_int_o: (int > int > $o) > product_prod_int_int > $o).
% 21.59/7.92  thf(func_def_1046, type, produc4245557441103728435nt_int: (int > int > product_prod_int_int) > product_prod_int_int > product_prod_int_int).
% 21.59/7.92  thf(func_def_1047, type, produc73460835934605544et_int: (int > int > set_int) > product_prod_int_int > set_int).
% 21.59/7.92  thf(func_def_1048, type, produc4251311855443802252et_nat: (int > int > set_nat) > product_prod_int_int > set_nat).
% 21.59/7.92  thf(func_def_1049, type, produc1656060378719767003at_nat: (int > int > set_Pr1261947904930325089at_nat) > product_prod_int_int > set_Pr1261947904930325089at_nat).
% 21.59/7.92  thf(func_def_1050, type, produc6452406959799940328t_real: (int > int > set_real) > product_prod_int_int > set_real).
% 21.59/7.92  thf(func_def_1051, type, produc5233655623923918146et_nat: (int > int > set_set_nat) > product_prod_int_int > set_set_nat).
% 21.59/7.92  thf(func_def_1052, type, produc8739625826339149834_nat_o: (nat > nat > product_prod_nat_nat > $o) > product_prod_nat_nat > product_prod_nat_nat > $o).
% 21.59/7.92  thf(func_def_1053, type, produc27273713700761075at_nat: (nat > nat > product_prod_nat_nat > product_prod_nat_nat) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat).
% 21.59/7.92  thf(func_def_1054, type, produc6081775807080527818_nat_o: (nat > nat > $o) > product_prod_nat_nat > $o).
% 21.59/7.92  thf(func_def_1055, type, produc1917071388513777916omplex: (nat > nat > complex) > product_prod_nat_nat > complex).
% 21.59/7.92  thf(func_def_1056, type, produc2676513652042109336d_enat: (nat > nat > extended_enat) > product_prod_nat_nat > extended_enat).
% 21.59/7.92  thf(func_def_1057, type, produc6840382203811409530at_int: (nat > nat > int) > product_prod_nat_nat > int).
% 21.59/7.92  thf(func_def_1058, type, produc6842872674320459806at_nat: (nat > nat > nat) > product_prod_nat_nat > nat).
% 21.59/7.92  thf(func_def_1059, type, produc2626176000494625587at_nat: (nat > nat > product_prod_nat_nat) > product_prod_nat_nat > product_prod_nat_nat).
% 21.59/7.92  thf(func_def_1060, type, produc6207742614233964070at_rat: (nat > nat > rat) > product_prod_nat_nat > rat).
% 21.59/7.92  thf(func_def_1061, type, produc1703576794950452218t_real: (nat > nat > real) > product_prod_nat_nat > real).
% 21.59/7.92  thf(func_def_1062, type, produc478579273971653890on_num: (nat > num > option_num) > product_prod_nat_num > option_num).
% 21.59/7.92  thf(func_def_1063, type, produc5414030515140494994real_o: (real > real > $o) > produc2422161461964618553l_real > $o).
% 21.59/7.92  thf(func_def_1064, type, product_fst_int_int: product_prod_int_int > int).
% 21.59/7.92  thf(func_def_1065, type, product_fst_nat_nat: product_prod_nat_nat > nat).
% 21.59/7.92  thf(func_def_1066, type, product_snd_int_int: product_prod_int_int > int).
% 21.59/7.92  thf(func_def_1067, type, product_snd_nat_nat: product_prod_nat_nat > nat).
% 21.59/7.92  thf(func_def_1068, type, abs_Rat: product_prod_int_int > rat).
% 21.59/7.92  thf(func_def_1069, type, fract: int > int > rat).
% 21.59/7.92  thf(func_def_1070, type, frct: product_prod_int_int > rat).
% 21.59/7.92  thf(func_def_1071, type, rep_Rat: rat > product_prod_int_int).
% 21.59/7.92  thf(func_def_1072, type, field_5140801741446780682s_real: set_real).
% 21.59/7.92  thf(func_def_1073, type, normalize: product_prod_int_int > product_prod_int_int).
% 21.59/7.92  thf(func_def_1074, type, of_int: int > rat).
% 21.59/7.92  thf(func_def_1075, type, pcr_rat: product_prod_int_int > rat > $o).
% 21.59/7.92  thf(func_def_1076, type, positive: rat > $o).
% 21.59/7.92  thf(func_def_1077, type, quotient_of: rat > product_prod_int_int).
% 21.59/7.92  thf(func_def_1078, type, ratrel: product_prod_int_int > product_prod_int_int > $o).
% 21.59/7.92  thf(func_def_1079, type, real2: (nat > rat) > real).
% 21.59/7.92  thf(func_def_1080, type, cauchy: (nat > rat) > $o).
% 21.59/7.92  thf(func_def_1081, type, pcr_real: (nat > rat) > real > $o).
% 21.59/7.92  thf(func_def_1082, type, positive2: real > $o).
% 21.59/7.92  thf(func_def_1083, type, realrel: (nat > rat) > (nat > rat) > $o).
% 21.59/7.92  thf(func_def_1084, type, rep_real: real > nat > rat).
% 21.59/7.92  thf(func_def_1085, type, vanishes: (nat > rat) > $o).
% 21.59/7.92  thf(func_def_1086, type, real_V3694042436643373181omplex: complex > complex > real).
% 21.59/7.92  thf(func_def_1087, type, real_V975177566351809787t_real: real > real > real).
% 21.59/7.92  thf(func_def_1088, type, real_V1022390504157884413omplex: complex > real).
% 21.59/7.92  thf(func_def_1089, type, real_V7735802525324610683m_real: real > real).
% 21.59/7.92  thf(func_def_1090, type, real_V4546457046886955230omplex: real > complex).
% 21.59/7.92  thf(func_def_1091, type, real_V2046097035970521341omplex: real > complex > complex).
% 21.59/7.92  thf(func_def_1092, type, real_V1485227260804924795R_real: real > real > real).
% 21.59/7.92  thf(func_def_1093, type, field_nat: set_Pr1261947904930325089at_nat > set_nat).
% 21.59/7.92  thf(func_def_1094, type, algebr932160517623751201me_int: int > int > $o).
% 21.59/7.92  thf(func_def_1095, type, algebr934650988132801477me_nat: nat > nat > $o).
% 21.59/7.92  thf(func_def_1096, type, divide6298287555418463151nteger: code_integer > code_integer > code_integer).
% 21.59/7.92  thf(func_def_1097, type, divide1717551699836669952omplex: complex > complex > complex).
% 21.59/7.92  thf(func_def_1098, type, divide_divide_int: int > int > int).
% 21.59/7.92  thf(func_def_1099, type, divide_divide_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_1100, type, divide_divide_rat: rat > rat > rat).
% 21.59/7.92  thf(func_def_1101, type, divide_divide_real: real > real > real).
% 21.59/7.92  thf(func_def_1102, type, dvd_dvd_Code_integer: code_integer > code_integer > $o).
% 21.59/7.92  thf(func_def_1103, type, dvd_dvd_complex: complex > complex > $o).
% 21.59/7.92  thf(func_def_1104, type, dvd_dvd_int: int > int > $o).
% 21.59/7.92  thf(func_def_1105, type, dvd_dvd_nat: nat > nat > $o).
% 21.59/7.92  thf(func_def_1106, type, dvd_dvd_rat: rat > rat > $o).
% 21.59/7.92  thf(func_def_1107, type, dvd_dvd_real: real > real > $o).
% 21.59/7.92  thf(func_def_1108, type, modulo364778990260209775nteger: code_integer > code_integer > code_integer).
% 21.59/7.92  thf(func_def_1109, type, modulo_modulo_int: int > int > int).
% 21.59/7.92  thf(func_def_1110, type, modulo_modulo_nat: nat > nat > nat).
% 21.59/7.92  thf(func_def_1111, type, zero_n356916108424825756nteger: $o > code_integer).
% 21.59/7.92  thf(func_def_1112, type, zero_n1201886186963655149omplex: $o > complex).
% 21.59/7.92  thf(func_def_1113, type, zero_n2684676970156552555ol_int: $o > int).
% 21.59/7.92  thf(func_def_1114, type, zero_n2687167440665602831ol_nat: $o > nat).
% 21.59/7.92  thf(func_def_1115, type, zero_n2052037380579107095ol_rat: $o > rat).
% 21.59/7.92  thf(func_def_1116, type, zero_n3304061248610475627l_real: $o > real).
% 21.59/7.92  thf(func_def_1117, type, suminf_complex: (nat > complex) > complex).
% 21.59/7.92  thf(func_def_1118, type, suminf_int: (nat > int) > int).
% 21.59/7.92  thf(func_def_1119, type, suminf_nat: (nat > nat) > nat).
% 21.59/7.92  thf(func_def_1120, type, suminf_real: (nat > real) > real).
% 21.59/7.92  thf(func_def_1121, type, summable_complex: (nat > complex) > $o).
% 21.59/7.92  thf(func_def_1122, type, summable_int: (nat > int) > $o).
% 21.59/7.92  thf(func_def_1123, type, summable_nat: (nat > nat) > $o).
% 21.59/7.92  thf(func_def_1124, type, summable_real: (nat > real) > $o).
% 21.59/7.92  thf(func_def_1125, type, sums_complex: (nat > complex) > complex > $o).
% 21.59/7.92  thf(func_def_1126, type, sums_int: (nat > int) > int > $o).
% 21.59/7.92  thf(func_def_1127, type, sums_nat: (nat > nat) > nat > $o).
% 21.59/7.92  thf(func_def_1128, type, sums_real: (nat > real) > real > $o).
% 21.59/7.92  thf(func_def_1129, type, collect_Code_integer: (code_integer > $o) > set_Code_integer).
% 21.59/7.92  thf(func_def_1130, type, collect_complex: (complex > $o) > set_complex).
% 21.59/7.92  thf(func_def_1131, type, collect_int: (int > $o) > set_int).
% 21.59/7.92  thf(func_def_1132, type, collect_list_o: (list_o > $o) > set_list_o).
% 21.59/7.92  thf(func_def_1133, type, collect_list_complex: (list_complex > $o) > set_list_complex).
% 21.59/7.92  thf(func_def_1134, type, collect_list_int: (list_int > $o) > set_list_int).
% 21.59/7.92  thf(func_def_1135, type, collect_list_nat: (list_nat > $o) > set_list_nat).
% 21.59/7.92  thf(func_def_1136, type, collec3343600615725829874at_nat: (list_P6011104703257516679at_nat > $o) > set_li5450038453877631591at_nat).
% 21.59/7.92  thf(func_def_1137, type, collec5608196760682091941T_VEBT: (list_VEBT_VEBT > $o) > set_list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1138, type, collect_nat: (nat > $o) > set_nat).
% 21.59/7.92  thf(func_def_1139, type, collect_num: (num > $o) > set_num).
% 21.59/7.92  thf(func_def_1140, type, collec8663557070575231912omplex: (produc4411394909380815293omplex > $o) > set_Pr5085853215250843933omplex).
% 21.59/7.92  thf(func_def_1141, type, collec213857154873943460nt_int: (product_prod_int_int > $o) > set_Pr958786334691620121nt_int).
% 21.59/7.92  thf(func_def_1142, type, collec3392354462482085612at_nat: (product_prod_nat_nat > $o) > set_Pr1261947904930325089at_nat).
% 21.59/7.92  thf(func_def_1143, type, collec3799799289383736868l_real: (produc2422161461964618553l_real > $o) > set_Pr6218003697084177305l_real).
% 21.59/7.92  thf(func_def_1144, type, collec6321179662152712658at_nat: (produc3843707927480180839at_nat > $o) > set_Pr4329608150637261639at_nat).
% 21.59/7.92  thf(func_def_1145, type, collect_rat: (rat > $o) > set_rat).
% 21.59/7.92  thf(func_def_1146, type, collect_real: (real > $o) > set_real).
% 21.59/7.92  thf(func_def_1147, type, collect_set_complex: (set_complex > $o) > set_set_complex).
% 21.59/7.92  thf(func_def_1148, type, collect_set_int: (set_int > $o) > set_set_int).
% 21.59/7.92  thf(func_def_1149, type, collect_set_nat: (set_nat > $o) > set_set_nat).
% 21.59/7.92  thf(func_def_1150, type, collec5514110066124741708at_nat: (set_Pr1261947904930325089at_nat > $o) > set_se7855581050983116737at_nat).
% 21.59/7.92  thf(func_def_1151, type, pow_nat: set_nat > set_set_nat).
% 21.59/7.92  thf(func_def_1152, type, image_int_int: (int > int) > set_int > set_int).
% 21.59/7.92  thf(func_def_1153, type, image_int_nat: (int > nat) > set_int > set_nat).
% 21.59/7.92  thf(func_def_1154, type, image_nat_nat: (nat > nat) > set_nat > set_nat).
% 21.59/7.92  thf(func_def_1155, type, image_nat_set_nat: (nat > set_nat) > set_nat > set_set_nat).
% 21.59/7.92  thf(func_def_1156, type, image_5971271580939081552omplex: (real > filter6041513312241820739omplex) > set_real > set_fi4554929511873752355omplex).
% 21.59/7.92  thf(func_def_1157, type, image_2178119161166701260l_real: (real > filter2146258269922977983l_real) > set_real > set_fi7789364187291644575l_real).
% 21.59/7.92  thf(func_def_1158, type, image_real_real: (real > real) > set_real > set_real).
% 21.59/7.92  thf(func_def_1159, type, insert_complex: complex > set_complex > set_complex).
% 21.59/7.92  thf(func_def_1160, type, insert_int: int > set_int > set_int).
% 21.59/7.92  thf(func_def_1161, type, insert_list_nat: list_nat > set_list_nat > set_list_nat).
% 21.59/7.92  thf(func_def_1162, type, insert_nat: nat > set_nat > set_nat).
% 21.59/7.92  thf(func_def_1163, type, insert_num: num > set_num > set_num).
% 21.59/7.92  thf(func_def_1164, type, insert5033312907999012233nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int).
% 21.59/7.92  thf(func_def_1165, type, insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 21.59/7.92  thf(func_def_1166, type, insert9069300056098147895at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat).
% 21.59/7.92  thf(func_def_1167, type, insert_rat: rat > set_rat > set_rat).
% 21.59/7.92  thf(func_def_1168, type, insert_real: real > set_real > set_real).
% 21.59/7.92  thf(func_def_1169, type, insert_set_nat: set_nat > set_set_nat > set_set_nat).
% 21.59/7.92  thf(func_def_1170, type, insert_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1171, type, is_singleton_int: set_int > $o).
% 21.59/7.92  thf(func_def_1172, type, is_singleton_nat: set_nat > $o).
% 21.59/7.92  thf(func_def_1173, type, is_sin2850979758926227957at_nat: set_Pr1261947904930325089at_nat > $o).
% 21.59/7.92  thf(func_def_1174, type, is_sin2937591304547752795at_nat: set_Pr4329608150637261639at_nat > $o).
% 21.59/7.92  thf(func_def_1175, type, is_singleton_real: set_real > $o).
% 21.59/7.92  thf(func_def_1176, type, is_singleton_set_nat: set_set_nat > $o).
% 21.59/7.92  thf(func_def_1177, type, the_elem_int: set_int > int).
% 21.59/7.92  thf(func_def_1178, type, the_elem_nat: set_nat > nat).
% 21.59/7.92  thf(func_def_1179, type, the_el8326832613380209454nt_int: set_Pr958786334691620121nt_int > product_prod_int_int).
% 21.59/7.92  thf(func_def_1180, type, the_el2281957884133575798at_nat: set_Pr1261947904930325089at_nat > product_prod_nat_nat).
% 21.59/7.92  thf(func_def_1181, type, the_el221668144340439132at_nat: set_Pr4329608150637261639at_nat > produc3843707927480180839at_nat).
% 21.59/7.92  thf(func_def_1182, type, the_elem_real: set_real > real).
% 21.59/7.92  thf(func_def_1183, type, vimage_nat_nat: (nat > nat) > set_nat > set_nat).
% 21.59/7.92  thf(func_def_1184, type, set_fo1517530859248394432omplex: (nat > complex > complex) > nat > nat > complex > complex).
% 21.59/7.92  thf(func_def_1185, type, set_fo2581907887559384638at_int: (nat > int > int) > nat > nat > int > int).
% 21.59/7.92  thf(func_def_1186, type, set_fo2584398358068434914at_nat: (nat > nat > nat) > nat > nat > nat > nat).
% 21.59/7.92  thf(func_def_1187, type, set_fo1949268297981939178at_rat: (nat > rat > rat) > nat > nat > rat > rat).
% 21.59/7.92  thf(func_def_1188, type, set_fo3111899725591712190t_real: (nat > real > real) > nat > nat > real > real).
% 21.59/7.92  thf(func_def_1189, type, set_or1266510415728281911st_int: int > int > set_int).
% 21.59/7.92  thf(func_def_1190, type, set_or1269000886237332187st_nat: nat > nat > set_nat).
% 21.59/7.92  thf(func_def_1191, type, set_or7049704709247886629st_num: num > num > set_num).
% 21.59/7.92  thf(func_def_1192, type, set_or633870826150836451st_rat: rat > rat > set_rat).
% 21.59/7.92  thf(func_def_1193, type, set_or1222579329274155063t_real: real > real > set_real).
% 21.59/7.92  thf(func_def_1194, type, set_or370866239135849197et_int: set_int > set_int > set_set_int).
% 21.59/7.92  thf(func_def_1195, type, set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat).
% 21.59/7.92  thf(func_def_1196, type, set_or4662586982721622107an_int: int > int > set_int).
% 21.59/7.92  thf(func_def_1197, type, set_or4665077453230672383an_nat: nat > nat > set_nat).
% 21.59/7.92  thf(func_def_1198, type, set_or1222409239386451017an_num: num > num > set_num).
% 21.59/7.92  thf(func_def_1199, type, set_or4029947393144176647an_rat: rat > rat > set_rat).
% 21.59/7.92  thf(func_def_1200, type, set_or66887138388493659n_real: real > real > set_real).
% 21.59/7.92  thf(func_def_1201, type, set_or8585797421378605585et_int: set_int > set_int > set_set_int).
% 21.59/7.92  thf(func_def_1202, type, set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat).
% 21.59/7.92  thf(func_def_1203, type, set_ord_atLeast_nat: nat > set_nat).
% 21.59/7.92  thf(func_def_1204, type, set_ord_atMost_int: int > set_int).
% 21.59/7.92  thf(func_def_1205, type, set_ord_atMost_nat: nat > set_nat).
% 21.59/7.92  thf(func_def_1206, type, set_ord_atMost_num: num > set_num).
% 21.59/7.92  thf(func_def_1207, type, set_ord_atMost_rat: rat > set_rat).
% 21.59/7.92  thf(func_def_1208, type, set_ord_atMost_real: real > set_real).
% 21.59/7.92  thf(func_def_1209, type, set_or58775011639299419et_int: set_int > set_set_int).
% 21.59/7.92  thf(func_def_1210, type, set_or4236626031148496127et_nat: set_nat > set_set_nat).
% 21.59/7.92  thf(func_def_1211, type, set_or6656581121297822940st_int: int > int > set_int).
% 21.59/7.92  thf(func_def_1212, type, set_or6659071591806873216st_nat: nat > nat > set_nat).
% 21.59/7.92  thf(func_def_1213, type, set_or5832277885323065728an_int: int > int > set_int).
% 21.59/7.92  thf(func_def_1214, type, set_or5834768355832116004an_nat: nat > nat > set_nat).
% 21.59/7.92  thf(func_def_1215, type, set_or1633881224788618240n_real: real > real > set_real).
% 21.59/7.92  thf(func_def_1216, type, set_or1210151606488870762an_nat: nat > set_nat).
% 21.59/7.92  thf(func_def_1217, type, set_or5849166863359141190n_real: real > set_real).
% 21.59/7.92  thf(func_def_1218, type, set_ord_lessThan_int: int > set_int).
% 21.59/7.92  thf(func_def_1219, type, set_ord_lessThan_nat: nat > set_nat).
% 21.59/7.92  thf(func_def_1220, type, set_ord_lessThan_num: num > set_num).
% 21.59/7.92  thf(func_def_1221, type, set_ord_lessThan_rat: rat > set_rat).
% 21.59/7.92  thf(func_def_1222, type, set_or5984915006950818249n_real: real > set_real).
% 21.59/7.92  thf(func_def_1223, type, set_or890127255671739683et_nat: set_nat > set_set_nat).
% 21.59/7.92  thf(func_def_1224, type, topolo4422821103128117721l_real: filter_real > (real > real) > $o).
% 21.59/7.92  thf(func_def_1225, type, topolo5044208981011980120l_real: set_real > (real > real) > $o).
% 21.59/7.92  thf(func_def_1226, type, topolo6980174941875973593q_real: (nat > real) > $o).
% 21.59/7.92  thf(func_def_1227, type, topolo2177554685111907308n_real: real > set_real > filter_real).
% 21.59/7.92  thf(func_def_1228, type, topolo2815343760600316023s_real: real > filter_real).
% 21.59/7.92  thf(func_def_1229, type, topolo6517432010174082258omplex: (nat > complex) > $o).
% 21.59/7.92  thf(func_def_1230, type, topolo4055970368930404560y_real: (nat > real) > $o).
% 21.59/7.92  thf(func_def_1231, type, topolo896644834953643431omplex: filter6041513312241820739omplex).
% 21.59/7.92  thf(func_def_1232, type, topolo1511823702728130853y_real: filter2146258269922977983l_real).
% 21.59/7.92  thf(func_def_1233, type, arccos: real > real).
% 21.59/7.92  thf(func_def_1234, type, arcosh_real: real > real).
% 21.59/7.92  thf(func_def_1235, type, arcsin: real > real).
% 21.59/7.92  thf(func_def_1236, type, arctan: real > real).
% 21.59/7.92  thf(func_def_1237, type, arsinh_real: real > real).
% 21.59/7.92  thf(func_def_1238, type, artanh_real: real > real).
% 21.59/7.92  thf(func_def_1239, type, cos_complex: complex > complex).
% 21.59/7.92  thf(func_def_1240, type, cos_real: real > real).
% 21.59/7.92  thf(func_def_1241, type, cos_coeff: nat > real).
% 21.59/7.92  thf(func_def_1242, type, cosh_complex: complex > complex).
% 21.59/7.92  thf(func_def_1243, type, cosh_real: real > real).
% 21.59/7.92  thf(func_def_1244, type, cot_complex: complex > complex).
% 21.59/7.92  thf(func_def_1245, type, cot_real: real > real).
% 21.59/7.92  thf(func_def_1246, type, diffs_complex: (nat > complex) > nat > complex).
% 21.59/7.92  thf(func_def_1247, type, diffs_real: (nat > real) > nat > real).
% 21.59/7.92  thf(func_def_1248, type, exp_complex: complex > complex).
% 21.59/7.92  thf(func_def_1249, type, exp_real: real > real).
% 21.59/7.92  thf(func_def_1250, type, ln_ln_real: real > real).
% 21.59/7.92  thf(func_def_1251, type, log: real > real > real).
% 21.59/7.92  thf(func_def_1252, type, pi: real).
% 21.59/7.92  thf(func_def_1253, type, powr_real: real > real > real).
% 21.59/7.92  thf(func_def_1254, type, sin_complex: complex > complex).
% 21.59/7.92  thf(func_def_1255, type, sin_real: real > real).
% 21.59/7.92  thf(func_def_1256, type, sin_coeff: nat > real).
% 21.59/7.92  thf(func_def_1257, type, sinh_complex: complex > complex).
% 21.59/7.92  thf(func_def_1258, type, sinh_real: real > real).
% 21.59/7.92  thf(func_def_1259, type, tan_complex: complex > complex).
% 21.59/7.92  thf(func_def_1260, type, tan_real: real > real).
% 21.59/7.92  thf(func_def_1261, type, tanh_complex: complex > complex).
% 21.59/7.92  thf(func_def_1262, type, tanh_real: real > real).
% 21.59/7.92  thf(func_def_1263, type, transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 21.59/7.92  thf(func_def_1264, type, transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 21.59/7.92  thf(func_def_1265, type, vEBT_Leaf: $o > $o > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1266, type, vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1267, type, vEBT_size_VEBT: vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1268, type, vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o).
% 21.59/7.92  thf(func_def_1269, type, vEBT_VEBT_high: nat > nat > nat).
% 21.59/7.92  thf(func_def_1270, type, vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o).
% 21.59/7.92  thf(func_def_1271, type, vEBT_VEBT_low: nat > nat > nat).
% 21.59/7.92  thf(func_def_1272, type, vEBT_VEBT_membermima: vEBT_VEBT > nat > $o).
% 21.59/7.92  thf(func_def_1273, type, vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 21.59/7.92  thf(func_def_1274, type, vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o).
% 21.59/7.92  thf(func_def_1275, type, vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 21.59/7.92  thf(func_def_1276, type, vEBT_VEBT_valid: vEBT_VEBT > nat > $o).
% 21.59/7.92  thf(func_def_1277, type, vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 21.59/7.92  thf(func_def_1278, type, vEBT_invar_vebt: vEBT_VEBT > nat > $o).
% 21.59/7.92  thf(func_def_1279, type, vEBT_set_vebt: vEBT_VEBT > set_nat).
% 21.59/7.92  thf(func_def_1280, type, vEBT_vebt_buildup: nat > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1281, type, vEBT_v4011308405150292612up_rel: nat > nat > $o).
% 21.59/7.92  thf(func_def_1282, type, vEBT_vebt_delete: vEBT_VEBT > nat > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1283, type, vEBT_vebt_delete_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 21.59/7.92  thf(func_def_1284, type, vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1285, type, vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 21.59/7.92  thf(func_def_1286, type, vEBT_VEBT_bit_concat: nat > nat > nat > nat).
% 21.59/7.92  thf(func_def_1287, type, vEBT_VEBT_minNull: vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1288, type, vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1289, type, vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat).
% 21.59/7.92  thf(func_def_1290, type, vEBT_vebt_member: vEBT_VEBT > nat > $o).
% 21.59/7.92  thf(func_def_1291, type, vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 21.59/7.92  thf(func_def_1292, type, vEBT_VEBT_add: option_nat > option_nat > option_nat).
% 21.59/7.92  thf(func_def_1293, type, vEBT_VEBT_greater: option_nat > option_nat > $o).
% 21.59/7.92  thf(func_def_1294, type, vEBT_VEBT_less: option_nat > option_nat > $o).
% 21.59/7.92  thf(func_def_1295, type, vEBT_VEBT_lesseq: option_nat > option_nat > $o).
% 21.59/7.92  thf(func_def_1296, type, vEBT_VEBT_max_in_set: set_nat > nat > $o).
% 21.59/7.92  thf(func_def_1297, type, vEBT_VEBT_min_in_set: set_nat > nat > $o).
% 21.59/7.92  thf(func_def_1298, type, vEBT_VEBT_mul: option_nat > option_nat > option_nat).
% 21.59/7.92  thf(func_def_1299, type, vEBT_V4262088993061758097ft_nat: (nat > nat > nat) > option_nat > option_nat > option_nat).
% 21.59/7.92  thf(func_def_1300, type, vEBT_V819420779217536731ft_num: (num > num > num) > option_num > option_num > option_num).
% 21.59/7.92  thf(func_def_1301, type, vEBT_V1502963449132264192at_nat: (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat).
% 21.59/7.92  thf(func_def_1302, type, vEBT_VEBT_power: option_nat > option_nat > option_nat).
% 21.59/7.92  thf(func_def_1303, type, vEBT_vebt_maxt: vEBT_VEBT > option_nat).
% 21.59/7.92  thf(func_def_1304, type, vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1305, type, vEBT_vebt_mint: vEBT_VEBT > option_nat).
% 21.59/7.92  thf(func_def_1306, type, vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1307, type, vEBT_is_pred_in_set: set_nat > nat > nat > $o).
% 21.59/7.92  thf(func_def_1308, type, vEBT_vebt_pred: vEBT_VEBT > nat > option_nat).
% 21.59/7.92  thf(func_def_1309, type, vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 21.59/7.92  thf(func_def_1310, type, vEBT_is_succ_in_set: set_nat > nat > nat > $o).
% 21.59/7.92  thf(func_def_1311, type, vEBT_vebt_succ: vEBT_VEBT > nat > option_nat).
% 21.59/7.92  thf(func_def_1312, type, vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 21.59/7.92  thf(func_def_1313, type, accp_nat: (nat > nat > $o) > nat > $o).
% 21.59/7.92  thf(func_def_1314, type, accp_P1096762738010456898nt_int: (product_prod_int_int > product_prod_int_int > $o) > product_prod_int_int > $o).
% 21.59/7.92  thf(func_def_1315, type, accp_P4275260045618599050at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > product_prod_nat_nat > $o).
% 21.59/7.92  thf(func_def_1316, type, accp_P3113834385874906142um_num: (product_prod_num_num > product_prod_num_num > $o) > product_prod_num_num > $o).
% 21.59/7.92  thf(func_def_1317, type, accp_P2887432264394892906BT_nat: (produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o) > produc9072475918466114483BT_nat > $o).
% 21.59/7.92  thf(func_def_1318, type, accp_VEBT_VEBT: (vEBT_VEBT > vEBT_VEBT > $o) > vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1319, type, max_ex8135407076693332796at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat).
% 21.59/7.92  thf(func_def_1320, type, min_ex6901939911449802026at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat).
% 21.59/7.92  thf(func_def_1321, type, pred_nat: set_Pr1261947904930325089at_nat).
% 21.59/7.92  thf(func_def_1322, type, wf_nat: set_Pr1261947904930325089at_nat > $o).
% 21.59/7.92  thf(func_def_1323, type, fChoice_real: (real > $o) > real).
% 21.59/7.92  thf(func_def_1324, type, member1535805642427569193e_term: (code_integer > option6357759511663192854e_term) > set_Co9149898834107579976e_term > $o).
% 21.59/7.92  thf(func_def_1325, type, member8845023287901829240e_term: (int > option6357759511663192854e_term) > set_in3461395444621081367e_term > $o).
% 21.59/7.92  thf(func_def_1326, type, member4242434998011752849e_term: (produc6241069584506657477e_term > option6357759511663192854e_term) > set_Pr7604974323444597168e_term > $o).
% 21.59/7.92  thf(func_def_1327, type, member3222579708246209666e_term: (produc8551481072490612790e_term > option6357759511663192854e_term) > set_Pr3642885161833720865e_term > $o).
% 21.59/7.92  thf(func_def_1328, type, member_o: $o > set_o > $o).
% 21.59/7.92  thf(func_def_1329, type, member_complex: complex > set_complex > $o).
% 21.59/7.92  thf(func_def_1330, type, member_int: int > set_int > $o).
% 21.59/7.92  thf(func_def_1331, type, member_list_o: list_o > set_list_o > $o).
% 21.59/7.92  thf(func_def_1332, type, member_list_int: list_int > set_list_int > $o).
% 21.59/7.92  thf(func_def_1333, type, member_list_nat: list_nat > set_list_nat > $o).
% 21.59/7.92  thf(func_def_1334, type, member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1335, type, member_nat: nat > set_nat > $o).
% 21.59/7.92  thf(func_def_1336, type, member_num: num > set_num > $o).
% 21.59/7.92  thf(func_def_1337, type, member3068662437193594005nteger: produc8763457246119570046nteger > set_Pr8056137968301705908nteger > $o).
% 21.59/7.92  thf(func_def_1338, type, member7034335876925520548nt_int: produc7773217078559923341nt_int > set_Pr1872883991513573699nt_int > $o).
% 21.59/7.92  thf(func_def_1339, type, member4164122664394876845nteger: produc1908205239877642774nteger > set_Pr1281608226676607948nteger > $o).
% 21.59/7.92  thf(func_def_1340, type, member7618704894036264090nt_int: produc2285326912895808259nt_int > set_Pr9222295170931077689nt_int > $o).
% 21.59/7.92  thf(func_def_1341, type, member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o).
% 21.59/7.92  thf(func_def_1342, type, member7847949116333733898_o_int: product_prod_o_int > set_Pr8834758594704517033_o_int > $o).
% 21.59/7.92  thf(func_def_1343, type, member2802428098988154798_o_nat: product_prod_o_nat > set_Pr2101469702781467981_o_nat > $o).
% 21.59/7.92  thf(func_def_1344, type, member7400031367953476362o_real: product_prod_o_real > set_Pr6573716822653411497o_real > $o).
% 21.59/7.92  thf(func_def_1345, type, member5477980866518848620T_VEBT: produc2504756804600209347T_VEBT > set_Pr7543698050874017315T_VEBT > $o).
% 21.59/7.92  thf(func_def_1346, type, member157494554546826820nteger: produc8923325533196201883nteger > set_Pr4811707699266497531nteger > $o).
% 21.59/7.92  thf(func_def_1347, type, member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o).
% 21.59/7.92  thf(func_def_1348, type, member2744130022092475746t_real: produc679980390762269497t_real > set_Pr3538720872664544793t_real > $o).
% 21.59/7.92  thf(func_def_1349, type, member2056185340421749780T_VEBT: produc1531783533982839933T_VEBT > set_Pr8044002425091019955T_VEBT > $o).
% 21.59/7.92  thf(func_def_1350, type, member4159035015898711888list_o: produc7102631898165422375list_o > set_Pr6227168374412355847list_o > $o).
% 21.59/7.92  thf(func_def_1351, type, member6698963635872716290st_int: produc1186641810826059865st_int > set_Pr765067013931698361st_int > $o).
% 21.59/7.92  thf(func_def_1352, type, member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o).
% 21.59/7.92  thf(func_def_1353, type, member4439316823752958928T_VEBT: produc9211091688327510695T_VEBT > set_Pr1916528119006554503T_VEBT > $o).
% 21.59/7.92  thf(func_def_1354, type, member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o).
% 21.59/7.92  thf(func_def_1355, type, member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o).
% 21.59/7.92  thf(func_def_1356, type, member772602641336174712real_o: product_prod_real_o > set_Pr4936984352647145239real_o > $o).
% 21.59/7.92  thf(func_def_1357, type, member1627681773268152802al_int: produc8786904178792722361al_int > set_Pr1019928272762051225al_int > $o).
% 21.59/7.92  thf(func_def_1358, type, member5805532792777349510al_nat: produc3741383161447143261al_nat > set_Pr3510011417693777981al_nat > $o).
% 21.59/7.92  thf(func_def_1359, type, member7849222048561428706l_real: produc2422161461964618553l_real > set_Pr6218003697084177305l_real > $o).
% 21.59/7.92  thf(func_def_1360, type, member7262085504369356948T_VEBT: produc3757001726724277373T_VEBT > set_Pr6019664923565264691T_VEBT > $o).
% 21.59/7.92  thf(func_def_1361, type, member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o).
% 21.59/7.92  thf(func_def_1362, type, member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o).
% 21.59/7.92  thf(func_def_1363, type, member3307348790968139188VEBT_o: produc334124729049499915VEBT_o > set_Pr3175402225741728619VEBT_o > $o).
% 21.59/7.92  thf(func_def_1364, type, member5419026705395827622BT_int: produc4894624898956917775BT_int > set_Pr5066593544530342725BT_int > $o).
% 21.59/7.92  thf(func_def_1365, type, member373505688050248522BT_nat: produc9072475918466114483BT_nat > set_Pr7556676689462069481BT_nat > $o).
% 21.59/7.92  thf(func_def_1366, type, member8675245146396747942T_real: produc5170161368751668367T_real > set_Pr7765410600122031685T_real > $o).
% 21.59/7.92  thf(func_def_1367, type, member568628332442017744T_VEBT: produc8243902056947475879T_VEBT > set_Pr6192946355708809607T_VEBT > $o).
% 21.59/7.92  thf(func_def_1368, type, member_rat: rat > set_rat > $o).
% 21.59/7.92  thf(func_def_1369, type, member_real: real > set_real > $o).
% 21.59/7.92  thf(func_def_1370, type, member_set_int: set_int > set_set_int > $o).
% 21.59/7.92  thf(func_def_1371, type, member_set_nat: set_nat > set_set_nat > $o).
% 21.59/7.92  thf(func_def_1372, type, member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1373, type, deg: nat).
% 21.59/7.92  thf(func_def_1374, type, m: nat).
% 21.59/7.92  thf(func_def_1375, type, ma: nat).
% 21.59/7.92  thf(func_def_1376, type, maxi: nat).
% 21.59/7.92  thf(func_def_1377, type, mi: nat).
% 21.59/7.92  thf(func_def_1378, type, na: nat).
% 21.59/7.92  thf(func_def_1379, type, summary: vEBT_VEBT).
% 21.59/7.92  thf(func_def_1380, type, treeList: list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1381, type, xa: nat).
% 21.59/7.92  thf(func_def_1382, type, ya: nat).
% 21.59/7.92  thf(func_def_1401, type, sP0: list_VEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1402, type, sP1: vEBT_VEBT > list_VEBT_VEBT > nat > $o).
% 21.59/7.92  thf(func_def_1403, type, sP2: nat > nat > list_VEBT_VEBT > nat > nat > $o).
% 21.59/7.92  thf(func_def_1404, type, sP3: nat > nat > list_VEBT_VEBT > nat > nat > $o).
% 21.59/7.92  thf(func_def_1405, type, sP4: nat > nat > list_VEBT_VEBT > nat > nat > $o).
% 21.59/7.92  thf(func_def_1406, type, sP5: list_VEBT_VEBT > nat > $o).
% 21.59/7.92  thf(func_def_1407, type, sP6: nat > nat > list_VEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1408, type, sP7: nat > vEBT_VEBT > list_VEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1409, type, sP8: nat > list_VEBT_VEBT > nat > nat > $o).
% 21.59/7.92  thf(func_def_1410, type, sP9: nat > list_VEBT_VEBT > nat > nat > nat > $o).
% 21.59/7.92  thf(func_def_1411, type, sP10: nat > list_VEBT_VEBT > nat > nat > nat > $o).
% 21.59/7.92  thf(func_def_1412, type, sP11: $o > nat > $o).
% 21.59/7.92  thf(func_def_1413, type, sP12: vEBT_VEBT > nat > $o).
% 21.59/7.92  thf(func_def_1414, type, sP13: list_VEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1415, type, sP14: nat > vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1416, type, sP15: $o > nat > $o).
% 21.59/7.92  thf(func_def_1417, type, sP16: $o > nat > $o).
% 21.59/7.92  thf(func_def_1418, type, sP17: $o > nat > $o > $o).
% 21.59/7.92  thf(func_def_1419, type, sP18: $o > $o > nat > $o > $o).
% 21.59/7.92  thf(func_def_1420, type, sP19: list_VEBT_VEBT > nat > nat > $o).
% 21.59/7.92  thf(func_def_1421, type, sP20: list_VEBT_VEBT > nat > nat > nat > $o).
% 21.59/7.92  thf(func_def_1422, type, sP21: list_VEBT_VEBT > nat > nat > nat > nat > $o).
% 21.59/7.92  thf(func_def_1423, type, sP22: list_VEBT_VEBT > nat > nat > nat > nat > $o).
% 21.59/7.92  thf(func_def_1424, type, sP23: $o > list_VEBT_VEBT > nat > nat > nat > nat > $o).
% 21.59/7.92  thf(func_def_1425, type, sP24: $o > vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1426, type, sP25: vEBT_VEBT > $o > nat > $o).
% 21.59/7.92  thf(func_def_1427, type, sP26: $o > vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1428, type, sP27: $o > vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1429, type, sP28: nat > $o).
% 21.59/7.92  thf(func_def_1430, type, sP29: list_VEBT_VEBT > nat > nat > nat > nat > $o).
% 21.59/7.92  thf(func_def_1431, type, sP30: nat > list_VEBT_VEBT > nat > nat > nat > $o).
% 21.59/7.92  thf(func_def_1432, type, sP31: nat > nat > list_VEBT_VEBT > nat > nat > $o).
% 21.59/7.92  thf(func_def_1433, type, sP32: vEBT_VEBT > list_VEBT_VEBT > nat > $o).
% 21.59/7.92  thf(func_def_1434, type, sP33: list_VEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1435, type, sP34: nat > list_VEBT_VEBT > nat > nat > nat > $o).
% 21.59/7.92  thf(func_def_1436, type, sP35: list_VEBT_VEBT > nat > $o).
% 21.59/7.92  thf(func_def_1437, type, sP36: nat > vEBT_VEBT > list_VEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1438, type, sP37: list_VEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1439, type, sP38: (nat > vEBT_VEBT > $o) > nat > $o).
% 21.59/7.92  thf(func_def_1440, type, sP39: nat > (nat > vEBT_VEBT > $o) > $o).
% 21.59/7.92  thf(func_def_1441, type, sP40: nat > vEBT_VEBT > nat > $o).
% 21.59/7.92  thf(func_def_1442, type, sP41: vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1443, type, sP42: list_VEBT_VEBT > nat > nat > nat > nat > $o).
% 21.59/7.92  thf(func_def_1444, type, sP43: nat > list_VEBT_VEBT > nat > nat > nat > $o).
% 21.59/7.92  thf(func_def_1445, type, sP44: nat > vEBT_VEBT > list_VEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1446, type, sP45: list_VEBT_VEBT > nat > nat > nat > nat > $o).
% 21.59/7.92  thf(func_def_1447, type, sP46: list_VEBT_VEBT > vEBT_VEBT > nat > $o).
% 21.59/7.92  thf(func_def_1448, type, sP47: nat > list_VEBT_VEBT > nat > nat > nat > $o).
% 21.59/7.92  thf(func_def_1449, type, sP48: nat > vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1450, type, sP49: vEBT_VEBT > nat > $o).
% 21.59/7.92  thf(func_def_1451, type, sP50: nat > vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1452, type, sP51: vEBT_VEBT > nat > $o).
% 21.59/7.92  thf(func_def_1453, type, sP52: nat > nat > list_VEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1454, type, sP53: nat > list_VEBT_VEBT > nat > nat > $o).
% 21.59/7.92  thf(func_def_1455, type, sP54: nat > list_VEBT_VEBT > nat > nat > nat > $o).
% 21.59/7.92  thf(func_def_1456, type, sP55: $o > nat > $o).
% 21.59/7.92  thf(func_def_1457, type, sP56: $o > nat > $o).
% 21.59/7.92  thf(func_def_1458, type, sP57: $o > nat > $o > $o).
% 21.59/7.92  thf(func_def_1459, type, sP58: nat > vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1460, type, sP59: list_VEBT_VEBT > nat > nat > $o).
% 21.59/7.92  thf(func_def_1461, type, sP60: list_VEBT_VEBT > nat > nat > nat > $o).
% 21.59/7.92  thf(func_def_1462, type, sP61: list_VEBT_VEBT > nat > nat > nat > nat > $o).
% 21.59/7.92  thf(func_def_1463, type, sP62: nat > nat > list_VEBT_VEBT > nat > nat > $o).
% 21.59/7.92  thf(func_def_1464, type, sK63: nat > vEBT_VEBT > list_VEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1465, type, sK64: list_VEBT_VEBT > nat > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1466, type, sK65: nat > nat > list_VEBT_VEBT > nat > nat > nat).
% 21.59/7.92  thf(func_def_1467, type, sK66: nat > nat > list_VEBT_VEBT > nat > nat > nat).
% 21.59/7.92  thf(func_def_1468, type, sK67: list_VEBT_VEBT > nat > nat).
% 21.59/7.92  thf(func_def_1469, type, sK68: list_VEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1470, type, sK69: list_VEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1471, type, sK70: vEBT_VEBT > nat > $o).
% 21.59/7.92  thf(func_def_1472, type, sK71: vEBT_VEBT > nat > $o).
% 21.59/7.92  thf(func_def_1473, type, sK72: vEBT_VEBT > nat > nat).
% 21.59/7.92  thf(func_def_1474, type, sK73: vEBT_VEBT > nat > nat).
% 21.59/7.92  thf(func_def_1475, type, sK74: vEBT_VEBT > nat > list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1476, type, sK75: vEBT_VEBT > nat > nat).
% 21.59/7.92  thf(func_def_1477, type, sK76: vEBT_VEBT > nat > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1478, type, sK77: nat).
% 21.59/7.92  thf(func_def_1479, type, sK78: (list_VEBT_VEBT > $o) > list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1480, type, sK79: list_VEBT_VEBT > (vEBT_VEBT > $o) > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1481, type, sK80: nat > vEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1482, type, sK81: nat > vEBT_VEBT > list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1483, type, sK82: nat > vEBT_VEBT > option4927543243414619207at_nat).
% 21.59/7.92  thf(func_def_1484, type, sK83: list_VEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1485, type, sK84: list_VEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1486, type, sK85: vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1487, type, sK86: list_VEBT_VEBT > nat > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1488, type, sK87: nat > vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1489, type, sK88: nat > nat).
% 21.59/7.92  thf(func_def_1490, type, sK89: vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1491, type, sK90: vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1492, type, sK91: vEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1493, type, sK92: vEBT_VEBT > list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1494, type, sK93: vEBT_VEBT > product_prod_nat_nat).
% 21.59/7.92  thf(func_def_1495, type, sK94: vEBT_VEBT > product_prod_nat_nat).
% 21.59/7.92  thf(func_def_1496, type, sK95: vEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1497, type, sK96: vEBT_VEBT > list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1498, type, sK97: vEBT_VEBT > $o > nat > $o).
% 21.59/7.92  thf(func_def_1499, type, sK98: vEBT_VEBT > $o > nat > $o).
% 21.59/7.92  thf(func_def_1500, type, sK99: vEBT_VEBT > list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1501, type, sK100: vEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1502, type, sK101: vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1503, type, sK102: nat > $o > vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1504, type, sK103: nat > $o > vEBT_VEBT > list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1505, type, sK104: nat > $o > vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1506, type, sK105: nat > $o > vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1507, type, sK106: nat > $o > vEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1508, type, sK107: vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1509, type, sK108: vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1510, type, sK109: nat > list_VEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1511, type, sK110: nat > list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1512, type, sK111: list_VEBT_VEBT > (vEBT_VEBT > $o) > nat).
% 21.59/7.92  thf(func_def_1513, type, sK112: list_VEBT_VEBT > (vEBT_VEBT > $o) > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1514, type, sK113: list_VEBT_VEBT > list_VEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1515, type, sK114: nat > vEBT_VEBT > list_VEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1516, type, sK115: list_VEBT_VEBT > nat > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1517, type, sK116: list_VEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1518, type, sK117: list_VEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1519, type, sK118: list_VEBT_VEBT > nat > nat).
% 21.59/7.92  thf(func_def_1520, type, sK119: nat > nat > list_VEBT_VEBT > nat > nat > nat).
% 21.59/7.92  thf(func_def_1521, type, sK120: list_VEBT_VEBT > nat > nat > nat > nat > nat).
% 21.59/7.92  thf(func_def_1522, type, sK121: list_VEBT_VEBT > vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1523, type, sK122: list_VEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1524, type, sK123: list_VEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1525, type, sK124: nat > list_VEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1526, type, sK125: vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1527, type, sK126: nat > (nat > vEBT_VEBT > $o) > nat).
% 21.59/7.92  thf(func_def_1528, type, sK127: nat > (nat > vEBT_VEBT > $o) > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1529, type, sK128: list_VEBT_VEBT > (nat > vEBT_VEBT > $o) > nat > nat).
% 21.59/7.92  thf(func_def_1530, type, sK129: (nat > vEBT_VEBT > $o) > nat > list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1531, type, sK130: vEBT_VEBT > nat > nat).
% 21.59/7.92  thf(func_def_1532, type, sK131: vEBT_VEBT > nat > nat).
% 21.59/7.92  thf(func_def_1533, type, sK132: vEBT_VEBT > nat > nat).
% 21.59/7.92  thf(func_def_1534, type, sK133: vEBT_VEBT > nat > list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1535, type, sK134: vEBT_VEBT > nat > nat).
% 21.59/7.92  thf(func_def_1536, type, sK135: vEBT_VEBT > nat > nat).
% 21.59/7.92  thf(func_def_1537, type, sK136: vEBT_VEBT > nat > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1538, type, sK137: nat > vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1539, type, sK138: nat > vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1540, type, sK139: nat > vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1541, type, sK140: nat > vEBT_VEBT > list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1542, type, sK141: nat > vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1543, type, sK142: nat > vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1544, type, sK143: nat > vEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1545, type, sK144: vEBT_VEBT > nat > nat).
% 21.59/7.92  thf(func_def_1546, type, sK145: vEBT_VEBT > nat > nat).
% 21.59/7.92  thf(func_def_1547, type, sK146: vEBT_VEBT > nat > list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1548, type, sK147: vEBT_VEBT > nat > nat).
% 21.59/7.92  thf(func_def_1549, type, sK148: vEBT_VEBT > nat > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1550, type, sK149: nat > vEBT_VEBT > list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1551, type, sK150: nat > vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1552, type, sK151: nat > vEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1553, type, sK152: nat > vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1554, type, sK153: nat > vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1555, type, sK154: nat > list_VEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1556, type, sK155: nat > list_VEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1557, type, sK156: vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1558, type, sK157: vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1559, type, sK158: nat > vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1560, type, sK159: nat > vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1561, type, sK160: nat > vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1562, type, sK161: nat > vEBT_VEBT > list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1563, type, sK162: nat > vEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1564, type, sK163: vEBT_VEBT > list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1565, type, sK164: vEBT_VEBT > product_prod_nat_nat).
% 21.59/7.92  thf(func_def_1566, type, sK165: vEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1567, type, sK166: nat > vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1568, type, sK167: nat > vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1569, type, sK168: vEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1570, type, sK169: vEBT_VEBT > list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1571, type, sK170: vEBT_VEBT > product_prod_nat_nat).
% 21.59/7.92  thf(func_def_1572, type, sK171: vEBT_VEBT > list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1573, type, sK172: vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1574, type, sK173: vEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1575, type, sK174: vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1576, type, sK175: vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1577, type, sK176: (vEBT_VEBT > $o) > list_VEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1578, type, sK177: vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1579, type, sK178: vEBT_VEBT > $o).
% 21.59/7.92  thf(func_def_1580, type, sK179: vEBT_VEBT > nat).
% 21.59/7.92  thf(func_def_1581, type, sK180: vEBT_VEBT > list_VEBT_VEBT).
% 21.59/7.92  thf(func_def_1582, type, sK181: vEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1583, type, sK182: vEBT_VEBT > option4927543243414619207at_nat).
% 21.59/7.92  thf(func_def_1584, type, sK183: nat).
% 21.59/7.92  thf(func_def_1585, type, sK184: (vEBT_VEBT > $o) > list_VEBT_VEBT > vEBT_VEBT).
% 21.59/7.92  thf(func_def_1587, type, ph186: !>[X0: $tType]:(X0)).
% 21.59/7.92  thf(f25655,plain,(
% 21.59/7.92    $false),
% 21.59/7.92    inference(subsumption_resolution,[status(thm)],[f25654,f25279])).
% 21.59/7.92  thf(f25279,plain,(
% 21.59/7.92    ((ord_less_nat @ (divide_divide_nat @ ma @ (size_s6755466524823107622T_VEBT @ treeList)) @ (size_s6755466524823107622T_VEBT @ treeList)) != $true)),
% 21.59/7.92    inference(trivial_inequality_removal,[status(thm)],[f25278])).
% 21.59/7.92  thf(f25278,plain,(
% 21.59/7.92    ((ord_less_nat @ (divide_divide_nat @ ma @ (size_s6755466524823107622T_VEBT @ treeList)) @ (size_s6755466524823107622T_VEBT @ treeList)) != $true) | ($true != $true)),
% 21.59/7.92    inference(superposition,[status(thm)],[f25264,f24577])).
% 21.59/7.92  thf(f24577,plain,(
% 21.59/7.92    ( ! [X0 : nat,X1 : list_VEBT_VEBT] : (((member_VEBT_VEBT @ (nth_VEBT_VEBT @ X1 @ X0) @ (set_VEBT_VEBT2 @ X1)) = $true) | ($true != (ord_less_nat @ X0 @ (size_s6755466524823107622T_VEBT @ X1)))) )),
% 21.59/7.92    inference(cnf_transformation,[status(thm)],[f23916])).
% 21.59/7.92  thf(f23916,plain,(
% 21.59/7.92    ! [X0 : nat,X1 : list_VEBT_VEBT] : (($true != (ord_less_nat @ X0 @ (size_s6755466524823107622T_VEBT @ X1))) | ((member_VEBT_VEBT @ (nth_VEBT_VEBT @ X1 @ X0) @ (set_VEBT_VEBT2 @ X1)) = $true))),
% 21.59/7.92    inference(ennf_transformation,[status(thm)],[f23727])).
% 21.59/7.92  thf(f23727,plain,(
% 21.59/7.92    ! [X0 : nat,X1 : list_VEBT_VEBT] : (($true = (ord_less_nat @ X0 @ (size_s6755466524823107622T_VEBT @ X1))) => ((member_VEBT_VEBT @ (nth_VEBT_VEBT @ X1 @ X0) @ (set_VEBT_VEBT2 @ X1)) = $true))),
% 21.59/7.92    inference(fool_elimination,[status(thm)],[f23726])).
% 21.59/7.92  thf(f23726,plain,(
% 21.59/7.92    ! [X0 : nat,X1 : list_VEBT_VEBT] : ((ord_less_nat @ X0 @ (size_s6755466524823107622T_VEBT @ X1)) => (member_VEBT_VEBT @ (nth_VEBT_VEBT @ X1 @ X0) @ (set_VEBT_VEBT2 @ X1)))),
% 21.59/7.92    inference(rectify,[status(thm)],[f138])).
% 21.59/7.92  thf(f138,axiom,(
% 21.59/7.92    ! [X6 : nat,X18 : list_VEBT_VEBT] : ((ord_less_nat @ X6 @ (size_s6755466524823107622T_VEBT @ X18)) => (member_VEBT_VEBT @ (nth_VEBT_VEBT @ X18 @ X6) @ (set_VEBT_VEBT2 @ X18)))),
% 21.59/7.92    file('/export/starexec/sandbox/tmp/tmp.JxDoxSuNLY/DTF2THF_14050.p',fact_137_nth__mem)).
% 21.59/7.92  thf(f25264,plain,(
% 21.59/7.92    ((member_VEBT_VEBT @ (nth_VEBT_VEBT @ treeList @ (divide_divide_nat @ ma @ (size_s6755466524823107622T_VEBT @ treeList))) @ (set_VEBT_VEBT2 @ treeList)) != $true)),
% 21.59/7.92    inference(trivial_inequality_removal,[status(thm)],[f25258])).
% 21.59/7.92  thf(f25258,plain,(
% 21.59/7.92    ((member_VEBT_VEBT @ (nth_VEBT_VEBT @ treeList @ (divide_divide_nat @ ma @ (size_s6755466524823107622T_VEBT @ treeList))) @ (set_VEBT_VEBT2 @ treeList)) != $true) | ($true != $true)),
% 21.59/7.92    inference(superposition,[status(thm)],[f25150,f24696])).
% 21.59/7.92  thf(f24696,plain,(
% 21.59/7.92    ( ! [X0 : vEBT_VEBT] : (((vEBT_invar_vebt @ X0 @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) = $true) | ((member_VEBT_VEBT @ X0 @ (set_VEBT_VEBT2 @ treeList)) != $true)) )),
% 21.59/7.92    inference(definition_unfolding,[status(thm)],[f24517,f24659])).
% 21.59/7.92  thf(f24659,plain,(
% 21.59/7.92    (na = (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))),
% 21.59/7.92    inference(cnf_transformation,[status(thm)],[f2120])).
% 21.59/7.92  thf(f2120,axiom,(
% 21.59/7.92    (na = (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))),
% 21.59/7.92    file('/export/starexec/sandbox/tmp/tmp.JxDoxSuNLY/DTF2THF_14050.p',fact_2119__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062)).
% 21.59/7.92  thf(f24517,plain,(
% 21.59/7.92    ( ! [X0 : vEBT_VEBT] : (($true = (vEBT_invar_vebt @ X0 @ na)) | ((member_VEBT_VEBT @ X0 @ (set_VEBT_VEBT2 @ treeList)) != $true)) )),
% 21.59/7.92    inference(cnf_transformation,[status(thm)],[f24042])).
% 21.59/7.92  thf(f24042,plain,(
% 21.59/7.92    ! [X0 : vEBT_VEBT] : ((($true = (vEBT_invar_vebt @ X0 @ na)) & ($true = (sP41 @ X0))) | ((member_VEBT_VEBT @ X0 @ (set_VEBT_VEBT2 @ treeList)) != $true))),
% 21.59/7.92    inference(definition_folding,[status(thm)],[f23905,f24041,f24040])).
% 21.59/7.92  thf(f24040,plain,(
% 21.59/7.92    ! [X2 : nat,X0 : vEBT_VEBT,X1 : nat] : (($true = (sP40 @ X1 @ X0 @ X2)) <=> (($true = (vEBT_V8194947554948674370ptions @ X0 @ X2)) & (X1 != X2)))),
% 21.59/7.92    introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 21.59/7.92  thf(f24041,plain,(
% 21.59/7.92    ! [X0 : vEBT_VEBT] : (! [X1 : nat,X2 : nat] : (($true = (sP40 @ X1 @ X0 @ X2)) <=> ((vEBT_V8194947554948674370ptions @ (vEBT_vebt_delete @ X0 @ X1) @ X2) = $true)) | ~($true = (sP41 @ X0)))),
% 21.59/7.92    introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 21.59/7.92  thf(f23905,plain,(
% 21.59/7.92    ! [X0 : vEBT_VEBT] : ((($true = (vEBT_invar_vebt @ X0 @ na)) & ! [X1 : nat,X2 : nat] : ((($true = (vEBT_V8194947554948674370ptions @ X0 @ X2)) & (X1 != X2)) <=> ((vEBT_V8194947554948674370ptions @ (vEBT_vebt_delete @ X0 @ X1) @ X2) = $true))) | ((member_VEBT_VEBT @ X0 @ (set_VEBT_VEBT2 @ treeList)) != $true))),
% 21.59/7.92    inference(ennf_transformation,[status(thm)],[f18342])).
% 21.59/7.92  thf(f18342,plain,(
% 21.59/7.92    ! [X0 : vEBT_VEBT] : (((member_VEBT_VEBT @ X0 @ (set_VEBT_VEBT2 @ treeList)) = $true) => (($true = (vEBT_invar_vebt @ X0 @ na)) & ! [X1 : nat,X2 : nat] : ((($true = (vEBT_V8194947554948674370ptions @ X0 @ X2)) & (X1 != X2)) <=> ((vEBT_V8194947554948674370ptions @ (vEBT_vebt_delete @ X0 @ X1) @ X2) = $true))))),
% 21.59/7.92    inference(fool_elimination,[status(thm)],[f18341])).
% 21.59/7.92  thf(f18341,plain,(
% 21.59/7.92    ! [X0 : vEBT_VEBT] : ((member_VEBT_VEBT @ X0 @ (set_VEBT_VEBT2 @ treeList)) => (! [X1 : nat,X2 : nat] : ((vEBT_V8194947554948674370ptions @ (vEBT_vebt_delete @ X0 @ X1) @ X2) = (vEBT_V8194947554948674370ptions @ X0 @ X2) & (X1 != X2)) & (vEBT_invar_vebt @ X0 @ na)))),
% 21.59/7.92    inference(rectify,[status(thm)],[f24])).
% 21.59/7.92  thf(f24,axiom,(
% 21.59/7.92    ! [X10 : vEBT_VEBT] : ((member_VEBT_VEBT @ X10 @ (set_VEBT_VEBT2 @ treeList)) => (! [X12 : nat,X13 : nat] : ((vEBT_V8194947554948674370ptions @ (vEBT_vebt_delete @ X10 @ X12) @ X13) = (vEBT_V8194947554948674370ptions @ X10 @ X13) & (X12 != X13)) & (vEBT_invar_vebt @ X10 @ na)))),
% 21.59/7.92    file('/export/starexec/sandbox/tmp/tmp.JxDoxSuNLY/DTF2THF_14050.p',fact_23__C4_OIH_C_I1_J)).
% 21.59/7.92  thf(f25150,plain,(
% 21.59/7.92    ($true != (vEBT_invar_vebt @ (nth_VEBT_VEBT @ treeList @ (divide_divide_nat @ ma @ (size_s6755466524823107622T_VEBT @ treeList))) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))),
% 21.59/7.92    inference(backward_demodulation,[status(thm)],[f24874,f24672])).
% 21.59/7.92  thf(f24672,plain,(
% 21.59/7.92    ((size_s6755466524823107622T_VEBT @ treeList) = (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))),
% 21.59/7.92    inference(definition_unfolding,[status(thm)],[f24375,f24665])).
% 21.59/7.92  thf(f24665,plain,(
% 21.59/7.92    (m = (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))),
% 21.59/7.92    inference(definition_unfolding,[status(thm)],[f24510,f24659])).
% 21.59/7.92  thf(f24510,plain,(
% 21.59/7.92    (m = na)),
% 21.59/7.92    inference(cnf_transformation,[status(thm)],[f1])).
% 21.59/7.92  thf(f1,axiom,(
% 21.59/7.92    (m = na)),
% 21.59/7.92    file('/export/starexec/sandbox/tmp/tmp.JxDoxSuNLY/DTF2THF_14050.p',fact_0__C4_Ohyps_C_I3_J)).
% 21.59/7.92  thf(f24375,plain,(
% 21.59/7.92    ((size_s6755466524823107622T_VEBT @ treeList) = (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m))),
% 21.59/7.92    inference(cnf_transformation,[status(thm)],[f1705])).
% 21.59/7.92  thf(f1705,axiom,(
% 21.59/7.92    ((size_s6755466524823107622T_VEBT @ treeList) = (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m))),
% 21.59/7.92    file('/export/starexec/sandbox/tmp/tmp.JxDoxSuNLY/DTF2THF_14050.p',fact_1704__C4_Ohyps_C_I2_J)).
% 21.59/7.92  thf(f24874,plain,(
% 21.59/7.92    ((vEBT_invar_vebt @ (nth_VEBT_VEBT @ treeList @ (divide_divide_nat @ ma @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) != $true)),
% 21.59/7.92    inference(beta_eta_normalization,[status(thm)],[f24673])).
% 21.59/7.92  thf(f24673,plain,(
% 21.59/7.92    ((vEBT_invar_vebt @ (nth_VEBT_VEBT @ treeList @ ((^[Y0 : nat]: ((^[Y1 : nat]: (divide_divide_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))) @ ma @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) != $true)),
% 21.59/7.92    inference(definition_unfolding,[status(thm)],[f24376,f24619,f24491,f24659,f24659])).
% 21.59/7.92  thf(f24491,plain,(
% 21.59/7.92    (xa = ma)),
% 21.59/7.92    inference(cnf_transformation,[status(thm)],[f3])).
% 21.59/7.92  thf(f3,axiom,(
% 21.59/7.92    (xa = ma)),
% 21.59/7.92    file('/export/starexec/sandbox/tmp/tmp.JxDoxSuNLY/DTF2THF_14050.p',fact_2_True)).
% 21.59/7.92  thf(f24619,plain,(
% 21.59/7.92    (vEBT_VEBT_high = (^[Y0 : nat]: ((^[Y1 : nat]: (divide_divide_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))))),
% 21.59/7.92    inference(cnf_transformation,[status(thm)],[f21131])).
% 21.59/7.92  thf(f21131,plain,(
% 21.59/7.92    (vEBT_VEBT_high = (^[Y0 : nat]: ((^[Y1 : nat]: (divide_divide_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))))),
% 21.59/7.92    inference(fool_elimination,[status(thm)],[f21130])).
% 21.59/7.92  thf(f21130,plain,(
% 21.59/7.92    ((^[X0 : nat, X1 : nat] : (divide_divide_nat @ X0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ X1))) = vEBT_VEBT_high)),
% 21.59/7.92    inference(rectify,[status(thm)],[f2119])).
% 21.59/7.92  thf(f2119,axiom,(
% 21.59/7.92    ((^[X16 : nat, X14 : nat] : (divide_divide_nat @ X16 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ X14))) = vEBT_VEBT_high)),
% 21.59/7.92    file('/export/starexec/sandbox/tmp/tmp.JxDoxSuNLY/DTF2THF_14050.p',fact_2118_high__def)).
% 21.59/7.92  thf(f24376,plain,(
% 21.59/7.92    ((vEBT_invar_vebt @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ xa @ na)) @ na) != $true)),
% 21.59/7.92    inference(cnf_transformation,[status(thm)],[f23864])).
% 21.59/7.92  thf(f23864,plain,(
% 21.59/7.92    ((vEBT_invar_vebt @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ xa @ na)) @ na) != $true)),
% 21.59/7.92    inference(flattening,[status(thm)],[f14717])).
% 21.59/7.92  thf(f14717,plain,(
% 21.59/7.92    ~((vEBT_invar_vebt @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ xa @ na)) @ na) = $true)),
% 21.59/7.92    inference(fool_elimination,[status(thm)],[f14716])).
% 21.59/7.92  thf(f14716,plain,(
% 21.59/7.92    ~(vEBT_invar_vebt @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ xa @ na)) @ na)),
% 21.59/7.92    inference(rectify,[status(thm)],[f9374])).
% 21.59/7.92  thf(f9374,negated_conjecture,(
% 21.59/7.92    ~(vEBT_invar_vebt @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ xa @ na)) @ na)),
% 21.59/7.92    inference(negated_conjecture,[status(cth)],[f9373])).
% 21.59/7.92  thf(f9373,conjecture,(
% 21.59/7.92    (vEBT_invar_vebt @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ xa @ na)) @ na)),
% 21.59/7.92    file('/export/starexec/sandbox/tmp/tmp.JxDoxSuNLY/DTF2THF_14050.p',conj_0)).
% 21.59/7.92  thf(f25654,plain,(
% 21.59/7.92    ((ord_less_nat @ (divide_divide_nat @ ma @ (size_s6755466524823107622T_VEBT @ treeList)) @ (size_s6755466524823107622T_VEBT @ treeList)) = $true)),
% 21.59/7.92    inference(forward_demodulation,[status(thm)],[f24865,f24672])).
% 21.59/7.92  thf(f24865,plain,(
% 21.59/7.92    ($true = (ord_less_nat @ (divide_divide_nat @ ma @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ (size_s6755466524823107622T_VEBT @ treeList)))),
% 21.59/7.92    inference(beta_eta_normalization,[status(thm)],[f24721])).
% 21.59/7.92  thf(f24721,plain,(
% 21.59/7.92    ((ord_less_nat @ ((^[Y0 : nat]: ((^[Y1 : nat]: (divide_divide_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))) @ ma @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ (size_s6755466524823107622T_VEBT @ treeList)) = $true)),
% 21.59/7.92    inference(definition_unfolding,[status(thm)],[f24650,f24619,f24491,f24659])).
% 21.59/7.92  thf(f24650,plain,(
% 21.59/7.92    ((ord_less_nat @ (vEBT_VEBT_high @ xa @ na) @ (size_s6755466524823107622T_VEBT @ treeList)) = $true)),
% 21.59/7.92    inference(cnf_transformation,[status(thm)],[f15552])).
% 21.59/7.92  thf(f15552,plain,(
% 21.59/7.92    ((ord_less_nat @ (vEBT_VEBT_high @ xa @ na) @ (size_s6755466524823107622T_VEBT @ treeList)) = $true)),
% 21.59/7.92    inference(fool_elimination,[status(thm)],[f15551])).
% 21.59/7.92  thf(f15551,plain,(
% 21.59/7.92    (ord_less_nat @ (vEBT_VEBT_high @ xa @ na) @ (size_s6755466524823107622T_VEBT @ treeList))),
% 21.59/7.92    inference(rectify,[status(thm)],[f9])).
% 21.59/7.92  thf(f9,axiom,(
% 21.59/7.92    (ord_less_nat @ (vEBT_VEBT_high @ xa @ na) @ (size_s6755466524823107622T_VEBT @ treeList))),
% 21.59/7.92    file('/export/starexec/sandbox/tmp/tmp.JxDoxSuNLY/DTF2THF_14050.p',fact_8__092_060open_062high_Ax_An_A_060_Alength_AtreeList_092_060close_062)).
% 21.59/7.92  % SZS output end Proof for DTF2THF_14050
% 21.59/7.92  % (14276)------------------------------
% 21.59/7.92  % (14276)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.59/7.92  % (14276)Termination reason: Refutation
% 21.59/7.92  
% 21.59/7.92  % (14276)Memory used [KB]: 34668
% 21.59/7.92  % (14276)Time elapsed: 0.882 s
% 21.59/7.92  % (14276)Instructions burned: 2259 (million)
% 21.59/7.92  % (14276)------------------------------
% 21.59/7.92  % (14276)------------------------------
% 21.59/7.92  % (14163)Success in time 2.582 s
%------------------------------------------------------------------------------